Contents
PREFACE xiii
PART ONE CYCLE THE HYDROLOGIC CHAPTER1 lntroduction 1.1 t.2
r.3 t.4 1.5 1.6 I.7 1.8
3 Hydrology Defined 3 A Brief History 5 The Hydrologic Cycle 5 The Hydrologic Budget 11 HydrologicModels 11 HydrologicData 12 Common Units of Measurement Problems to Environmental Application of Hydrology
CHAPTER2 Precipitation 1 5 2.1 2.2 2.3 2,4 2.5 2.6 2.7
15 Water Vapor 17 Precipitation Distribution of the Precipitation Input 27 Point Precipitation 29 Areal Precipitation 34 Precipitation ProbableMaximum 36 Grossand Net PreciPitation
t2
vi
coNTENTS
CHAPTER3 Interception and Depression Storage 3.1 3.2 3.3
Interception 40 Throughfall 44 DepressionStorage
40
45
CHAPTER4 Infiltration 52 4.I 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10
MeasuringInfiltration 53 Calculation of Infiltration 53 Horton's Infiltration Model 57 Green-AMPT Model 64 Huggins-MonkeModel 67 Holtan Model 68 Recoveryof Infiltration Capacity 69 Temporal and Spatial Variability of Infiltration Capacity SCS Runoff Curve Number Procedure 73 76 @Index
70
CHAPTER5 Evaporation and Transportation 82 " 5.1 5.2 5.3 5.4 5.5 5.6 5.7
Evaporation 86 EstimatingEvaporation 86 EvaporationControl 95 Transpiration 95 TranspirationControl 100 Evapotranspiration 100 EstimatingEvapotranspiration
103
CHAPTER6 Streamflow 111 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8
DrainageBasinEffects 111 The Hydrograph 11,2 Units of Measurementfor Streamflow 113 Measuringand RecordingStreamflow 113 Measurements of Depth and Cross-Sectional Area II4 Measurementof Velocity lI4 RelatingPoint Velocity to Cross-SectionalFlow Velocity The Slope-AreaMethod for DeterminingDischarge II7
115
oONTENTS Vii
PART TWO AND MONITORING 121 fT'IEASUREMENTS HYDROLOGIG CHAPTER7 DataSources Hydrologic 7.1 7.2 7.3 7.4
123
I23 GeneralClimatologicalData 123 Precipitation Data 124 StreamflowData Evaporationand TranspirationData
I24
CHAPTER8 fnstrumentation126 8.1 8.2 8.3 8.4
126 Introduction 127 HYdrologicInstruments 135 TelemetrySYstems 135 RemoteSensing
CHAPTER9 Networks 144 Monitoring The Purposeof Monitoring 9.r 9.2 9.3 9.4
144
I45 SpecialConsiderations I47 Uie of ComPutersin Monitoring 147 Hydrological-MeteorlogicalNetworks
PART THREE SURFACEWATERHYDROLOGY151 CHAPTER1O Runoffandthe Catchnient 10.1 tO.2 10.3 IO.4 10.5
153
153 Catchments,Watersheds,and DrainageBasins 155 Basin CharacteristicsAffecting Runoff RudimentaryPrecipitation-RunoffRelationships 164 166 StreamflowFrequencyAnalysis 168 StreamflQwForecasting
CHAPTER11 Hydrographs 171 11.1 Il.2 11.3
StreamflowHYdrograPhs 171 FactorsAffecting HydrographShape HydrograPhComPonents 174
172
viii
ooNTENTS lI.4 11.5 11.6 Il.7
BaseFlow Separation I77 HydrographTime Relationships Time of Concentration I82 BasinLae Time I82
181
CHAPTER12 UnitHydrographs 188 l2.I 12.2 12.3 12.4 12.5 12.6
Unit HydrographDefinition 188 Derivation of Unit Hydrographsfrom StreamflowData 190 Unit HydrographApplications by Lagging Methods I94 S-HydrographMethod 198 The InstantaneousUnit Hydrograph 201 SyntheticUnit Hydrographs 205
CHAPTER13 Hydrograph Routing 13.1 13.2 13.3
HydrologicRiver Routing 235 HydrologicReservoirRouting 245 Hydraulic River Routing 248
CHAPTER14 SnowHydrology I4.l I4.2 I4.3 I4.4 14.5 14.6
234
265
Introduction 265 Snow Accumulation and Runoff 267 Snow Measurementsand Surveys 268 Point and Areal Snow Characteristics 269 The SnowmeltProcess 271. SnowmeltRunoff Determinations 284
CHAPTER15 Urbanand SmallWatershedHydrology 309 15.1 15.2 i5.3 15.4
Introduction 309 PeakFlow Formulasfor Urban Watersheds 311 PeakFlow Formulasfor Small Rural Watersheds 33I Runoff Effects of Urbanization 344
CHAPTER16 Hydrologic Design 16.l 16.2
359
Hydrologic DesignProcedures 360 Data for HydrologicDesign 363
CONTENTS
16.3 16.4 16.5 t6.6 16.7 16.8
HydrologicDesign-Frequency Criteria DesignStorms 373 Critical EventMethods 391 Airport DrainageDesign 400 Designof Urban Storm Drain Systems FloodplainAnalysis 409
365
402
PART FOUR GROUNDWATER HYDROLOGY 425 CHAPTER 17 Groundwater, Soils,and Geology l7.l I7.2 I7.3 I7.4 I7.5 I7.6
427
Introduction 427 GroundwaterFlow-General Properties 429 SubsurfaceDistribution of Water 429 GeologicConsiderations 430 Fluctuationsin GroundwaterLevel 433 Groundwater-Surface Water Relations 433
CHAPTER 18 Mechanics of Flow
435
18.1 t8.2 18.3 18.4 18.5 18.6
Hydrostatics 435 GroundwaterFlow 436 Darcy's Law 436 Permeability 438 Velocity Potential 440 HydrodynamicEquations 441 r8.7 ' Flowlines and EquipotentialLines 18.8 BoundaryConditions 447 18.9 Flow Nets 449 1 8 . 1 0 VariableHydraulic Conductivity 1 8 . 1 1 Anisotropy 452 18.t2 Dupuit's Theory 453
CHAPTER 19 Wellsand Collection Devices 19.1 19.2 19.3 19.4 19.5
444
451
460
Flow to Wells 460 SteadyUnconfinedRadial Flow Toward a Well 461 SteadyConfined Radial Flow Toward a Well 462 Well in A Uniform Flow Field 463 Well Fields 465
IX
X
CONTENTS
19.6 I9.7 19.8 I9.9 19.10 19.ll 19.12
The Method of Images 466 UnsteadyFlow 467 Leaky Aquifers 4'13 Partially PenetratingWells 473 Flow to an Infiltration Gallery 473 SaltwaterIntrusion 474 GroundwaterBasin Development 475
CHAPTER20 ModelingRegionalGroundwater Systems 20.I 20.2 20.3 20.4 20.5
RegionalGroundwaterModels 481 Finite-DifferenceMethods 484 Finite-ElementMethods 493 Model Applications 494 GroundwaterQuality Models 500
PART FIVE HYDROLOGIC MODELING
505
CHAPTER 21 Introduction to Hydrologic Modeling 2l.I 2t.2 21.3 21.4
481
5O7
HydrologicSimulation 508 GroundwaterSimulation 509 Hydrologic Simulation Protocol 524 Corps of EngineersSimulation Models
526
CHAPTER22 SyntheticStreamflows 535 22.I 22.2
SyntheticHydrology 536 Serially DependentTime SeriesAnalysis
CHAPTER23 Continuous Simulation Models 23.1 23.2
548
ContinuousStreamflowSimulationModels 549 ContinuousSimulation Model Studies 570
CHAPTER24 Single-Event Simulation Models 24.1 24.2 24.3
539
594
StormEventSimulation 594 FederalAgency Single-EventModels Storm SurgeModeling 625
597
CONTENTS Xi
CHAPTER25 UrbanRunotfSimulation Models 25.1 25.2 25.3
Urban StormwaterSystemModels 63I Urban Runoff Models Compared 659 Vendor-DevelopedUrbanStormwaterSoftware 663
PART SIX METHODS STATISTICAL
669
CHAPTER26 Probability and Statistics 26.1 26.2 26.3 26.4 26.5 26.6 26.7 26.8 26.9
2 6 .r0
671
RandomVariablesand StatisticalAnalysis 672 Conceptsof Probability 673 ProbabilityDistributions 676 Moments of Distributions 681 Distribution Characteristics 682 Types of Probability Distribution Functions 685 ContinuousProbabilityDistributionFunctions 685 Bivariate Linear Regressionand Correlation 690 Fitting RegressionEquations 692 Regressionand Correlation Applications 697
CHAPTER27 Frequency Analysis 27.1 27.2 27.3 27.4 27.5 27.6 27.7
630
708
FrequencyAnalysis 708 GraphicalFrequencyAnalysis 709 FrequencyAnalysis Using FrequencyFactors RegionalFrequencyAnalysis 7I9 Reliability of FrequencyStudies 730 FrequencyAnalysis of Partial Duration Series Flow Duration Analysis 737
APPENDICES 751 INDEX 757
7Il
734
Preface
Watermanagementis taking on new dimensions.New federalthrusts,the growprotecing list of global iisues,and strongpublic sentimentregardingenvironmental forces. driving tion havebeen the principal In the early yearsof the 20th century,waterresourcesdevelopmentand management were focuied almostexclusivd on water supplyand flood control' Today,these the environment,ensuring safe drinking issuesare still important, but protecting -and experiencescompete equally for recriatioinal water, and providing aesthetic consciouspublic is pressingfor environmentally an attentionand funds.Furthermore, with fewer structural compopractices, management greaterreliance on improved of continually striving to notion The problems. water ients, to solve this nuiion'. this preciousnatural husbanding of one by replaced provide more water has been resource. There is a growing constituencyfor allocatingwater for the-benefitof fish and wildlife, for protection-of marshesand estuary areas' and for other natural system uses.But estimatingthe quantitiesof water neededfor environmentalprotectionand for maintainingand/or restoringnatural systemsis difficult, and there are still many unknowns.Scilntific data are ,putt", and our understandingof the complexinteracof an scalesis rudimentary.Indeed,this is a critical issue' tions inherentin ecosystems sincethe quantitiesof water involvedin environmentalprotection can be substantial of and competitionfor thesewatersfrom traditional water usersis keen' The nations are that systems-decisions natural regarding decisions the world are facing major ladenwith significantectnomic and social impacts.Thus there is_anurgencyassociatedwith developinga betterunderstandingof ecologicsystemsand of their hydrologic components. Water policies of the future must thereforetake on broader dimensions'More emphasismustbe placedon regionalplanningand management,and regionalinstitutions to accommodatethis muJt be devised.Watermanagementmustbe practicedat, more andbetween,all levelsof government.Land useand wateruseplanningmustbe tightly coordinatedas well'
XIV
PREFACE
Water scientistsand engineersof tomorrow must be equipped to addressa diversity of issuessuch as: the design and operation of data retrieval and storage systems;forecasting;developingalternativewater use futures; estimatingwater requirementsfor natural systems;exploringthe impactsof climate change;developing more efficient systemsfor applyingwaterin all water-usingsectors;and analyzingand designingwater managementsystemsincorporating technical, economic, environmental, social, legal, and political elements.A knowledgeof hydrologicprinciplesis a requisitefor dealingwith suchibsues. This fourth edition hasbeendesignedto meet the contemporaryneedsof water scientistsand engineers.It is organizedto accommodatestudentsand practitioners who are concernedwith the development,management,and protection of water resources.The format of the book follows that of its predecessor, providing material for both an introductory and a more advancedcourse. Parts One through Four provide the basicsfor a beginninglevel course,while Parts Five and Six may be used for a more advancedcourseon hydrologicmodeling. This fourth edition has been updated throughout, and many solved examples havebeen added.In addition, new computer approacheshave been introduced and problem-solvingtechniquesincludethe use of spreadsheets as appropriate.New featuresof eachchapterincludean introductory statementof contentsand,at the conclusion of the chapter,a summaryof key points. Many sourceshave been drawn upon to provide subject matter for this book, and the authorshope that suitable acknowledgmenthas been given to them. Colleaguesand studentsare recognizedfor their helpful commentsand reviews,particularly the following reviewers. Gert Aron, ThePennsylvaniaStateUniversity JohnW. Bird, Universityof Nevada-Reno IstvanBogardi, Universityof Nebraska RonaldA. Chadderton,VillanovaUniversity RichardN. Downer,Universityof Vermont Bruce E. Larock, Universityof Califurnia-Davis Frank D. Masch,Universityof Texas-SanAntonio Philip L. Thompson,FederalHighwayAdministration A specialnote of thanks is due to Dr. John W. Knapp, Presidentof the Virginia Military Institute,coauthorof previouseditionsof this book, for his pastcontributions andvaluableguidance. WarrenViessman.Jr. Gary L. Lewis
PARTONE
CYCLE THE HYDROLOGIC
L.
Chapter1
lntroduction
I Prologue The purPoseof this chaPteris to: . , . .
Define hydrology. . earth -,.1Lscience' Give a brief niJiory of the evolution of this important Statethe fundamentalequationofhydrology' appliedto supplementdecision Demonstratetrow ffiofogic principle, "urib" management' support systemsfor water and environmental
DEFINED 1.1 HYDROLOGY Hydrologyisanearthscience'Itencompassestheoccuffence'distribution,moveA knowledgeof hydrologyis fundamenr, and propertiesof the watersof the earth. mentaltodecisionmutingp,o.",,e,*he,ewaterisu"ompon"nto.f.th.esystemof inextricably linked' and it is important concern.water and environmentalissuesare toclear$understandhowwaterisaffectedbyandhowwateraffectsecosystem maniPulations'
1.2 A BRIEFHISTORY Ancient philosophersfocusedtheir i flows Production of surfacewater oc"ur."n"e of water in variousstag from the seato the atmosPhereto t early speculationwas often faulty'l of large subterraneanreservoirsth is interestingto note, however'tha *suoeriornumbersindicatereferencesattheendofthechapter.
CHAPTER1
INTRODUCTION
Greek aqueductson both conveyancecrosssectionand velocity.This knowledgewas lost to the Romans,and the proper relation betweenarea,velocity, and rate of flow remainedunknownuntil Leonardoda Vinci rediscoveredit duringihe Italian Renaissance. During the first century s.c. Marcus Vitruvius, in Volume 8 of his treatiseDe Architectura Libri Decem (the engineer'schief handbookduring the Middle Ages), setforth a theory generallyconsideredto be the predecessorof modern notionsof the hydrologiccycle. He hypothesizedthat rain und ,no* falling in mountainous areas infiltrated the earth's surface and later appearedin the lowlands as streamsand springs. In spiteof the inaccuratetheoriesproposedin ancienttimes,it is only fair to state that practical applicationof varioustry-orotogic principleswas often carried out with considerablesuccess.For example,about4000 s.c. u du- was constructed acrossthe Nile to permit reclamation of previously barren lands for agricultural production. Severalthousandyears later a canal to convey fresh water from Cairo io Suezwas built. Mesopotamiantowns were protecteduguinrt floodsby high earthen walls. The Greek and Roman aqueductsand early Chineseirrigation and flood control works were also significantprojects. Nearthe endof the fifteenth century the trend towarda more scientific approach to hydrology based on the observationof hydrologic phenomena becameevident. Leonardo da Vinci and Bernard Palissyindepende-ntly reachedan accurateunderstandingof the watercycle.They apparentlybised theii theoriesmore on our"*iion than on purely philosophicalreasoning.Nevertheless, until the seventeenthcentury it seemsevident that little if any effort was directed toward obtaining quantitative measurementsof hydrologicvariables. The adventof what might be calledthe "modern" scienceof hydrology is usually consideredto beginwith the studiesof suchpioneersasPerrault,Mariotte, and Halley in the seventeenthcentury.r'aPerraultobtainedmeasurements of rainfall in the Seine River drainagebasin over a period of 3 years. Using these and measurements of runoff, and knowing,thedrainage areasize,he showeJthat rainfall was adequatein quantity to accountfor river flows. He also mademeasurements of evaporati,onand capillarity. Mariotte gaugedthe velocity of flow of the River Seine. Recordedvelocities were translatedinto termsof dischirgeby introducingmeasurements of the river crosssection'The English astronomerHalley measuredthe rate of evaporation of the MediterraneanSeaand concludedthat the amountof water evaporated was sufficient to accountfor the outflow of rivers tributary to the sea.Measurements suchas these, althoughcrude,permitted reliable conclusionsto be drawn reggrding the hydrologic phenomenabeing studied. brth numerousadvancesin hydraulic theory zometer,the Pitot tube, Bernoulli's theorem, ples.8 perimental hydrology flourished.Significant ydrology and in the measurementof surface water. Such significantcontributionsas Hagen-Poiseuille'scapillary flow equation, Darcy's law of flow in porous media, und th" Dupuit-Thiem well formula were evolved'e-lrThe beginningof systematicstream guoling can also be traced to this period' Although the basis for modern hydrology wui tirrr establishedin the nine,)
BUDGET 1.4 THE HYDROLOGIC
5
of teenth century, much of the effort was empirical in nature. The fundamentals early the In recognized. or widely established yet well been not physicalhydtotogyhad years of tle twJntieth ""niury the inadequaciesof many earlier empirical formulato tions becamewell known. As a result, interestedgovernmentalagenciesbegan rational 1950, to 1930 about From research' of hydrologic programs developtheir own analysis began to ieplace empiricism.3 Sherman's unit hydrograph, Horton's infiltration theory, und Th"it's nonequilibrium-approachto well hydraulicsare outstandingexamplesof the great progressmade'r2-'o Since 1930 a theoreiical approachto hydrologicproblemshas largely replaced permit a less sophisticatedmethods of ttre past. Advancesin scientific knowledge and ' better understandingofthe physicaibasisofhydrologic relations,and the advent continueddevelopnientof high-speeddigital computershavemadepossible,in both would a practical and an economiciense,extensivemathematicalmanipulationsthat past. in the havebeen overwhelming For a more compiehensivi historical treatment, the reader is referred to the works of Meinzer,Jonls, Biswas,and their co-workers'1'2'4'5'15
CYCLE 1 . 3THE HYDROLOGIC the The hydrologiccycle is a continuousprocessby which water is transportedfrom The exist' subcycles Many sea. the to back landind to the oceansto the atmosphere precipitation over land beforereturnevaporationof inlan-dwater and its subsequent for the global watertransportsystem force driving The example. ingio the oceanis one requiredfor evaporation'Note that energy the furnishes is providedby the sun,which the cycle; for example,sea passage through during changes the water quality also evaporation' through water fresh water is convertedto The completewater cycle is global in nature. world water problems require Practical studieson regional,national,internitional, continental,and global scales.16 earth is the to available water fresh of supply total the that significanceof the fact has limited and very small compared with ihe salt water content of the oceans at the receivedlittle attention.Thus watersflowing in one country cannotbe available u's' same time for use in other regions of the world. Raymond L' Nace of the thatoowaterresourcesare a global problem with GeologicalSurvey has aptly sta=ted are obligated to cope with problems requiring hydrologists local roots."tu Mtdern magnitudedifference.In addition, developing of oider of definition in varying scales careful attention, since climatological receive must weather techniquesto contiol hydrology and therefore the water the affect profoundly can changesin one area resourcesof other regions.
BUDGET 1 . 4 THE HYDROLOGIC Becausethe total quantity of water availableto the earth is finite and indestructible, subsysthe global hydrolojic ,yrt"* may be lookedupon as closed.Open hydrologic system' temsare abundantlhowever,and theseare usuallythe type analyzed'For any ' a water budgetcan be developedto accountfor the hydrologiccomponents'
CHAPTER1
,
INTRODUCTION
FiguresI'I,I.2, and 1.3 showa hydrologicbudgetfor the coterminousUnited States,a conceptualizedhydrologiccycle,andthe distributionof a precipitationinput, respectively.Thesefiguresillustrate the componentsof the water cycle with which a hydrologistis concerned.In a practicalsense,somehydrologicregionis dealtwith and a budgetfor that region is established.Suchregionsmay be topographicallydefined (watershedsand river basinsare examples),politically specified(e.g- couniy or city limits), or chosenon some other grounds. Watershedsor drainagi tasins are the easiestto deal with sincethey sharply define surfacewater boundaries.Thesetopographically determinedareasare drainedby a river/streamor systemof connecting rivers/streamssuch that all outflow is dischargedthrough a single outlet. Unfortunately,it is often necessaryto deal with regions that are not well suitedto tracking hydrologiccomponents.For theseareas,the hydrologistwill find hydrologicbudgeting somewhatof a challenge. The primary input in a hydrologicbudgetis precipitation.Figures 1.1-1.3 illustrate this. Someof the precipitation (e.g.,rain, snow,hail) may be interceptedby trees,grass,other vegetation,and structuralobjectsand will eventuallyreturn to the atmosphereby evaporation.Onceprecipitationreachesthe ground,someofit may fill (becomedepressionstorage),part may penetralethe ground (infiltraie) to depressions replenishsoil moisture and groundwaterreservoirs,and some may become surface runoff-that is, flow over the earth's surfaceto a definedchannelsuchas a stream. Figure 1'3 showsthe dispositionofinfiltration, depressionstorage,and surfacerunoff.
and vegetation
*r-d{i;
I
'1.'
Consumptive use 100bgd
bgd = billion gallons per day
Figure 1.1 Hydrologic budgetof cotermiriousunited States.(U.S. Geologicalsurvey.)
BUDGET 7 1.4 THEHYDROLOGIC Clouds and water vaPor
Clouds and water vaPor
ffi
(
)
" 1 ) t t l l ,
--1vtivv
p\--
P
P
P
P
P
P
E E
E
E
t t E' evaporation;P' Figure 1.2 The hydrologic 'surfac-e cycle: ?, transpiration; flow; and I' groundwater runoff; G, R, p.Sqipi*i"tt inflltration.
L
t Precipitation inPut (hyetogaph)
'l,r\ t SSeamflow (hyclrograPh)
Figure 1.3 Distributionof precipitationinput'
CHAPTER1
INTRODUCTION
water enteringthe ground may take severalpaths.Somemay be directly evaporatedif adequatetransferfrom the soil to the surfaceis maintained.This can easily occur where a high groundwater table (free water surface) is within the limits of capillary transportto the ground surface.Vegetationusingsoil moistureor groundwatei directly can also transmit infiltrated water to the atmosphereby a processknown as transpiration.Infiltrated water may likewise replenishsoil moisture deficiencies and enter storageprovided in groundwaterreservoirs,which in turn maintain dry weather streamflow.Important bodies of groundwaterare usually flowing so that inflltrated water reachingthe saturated,on" muy be transportedfor considerable' distancesbefore it is discharged.Groundwatermovement is subject, of course,to physicaland geologicalconstraints. Water storedin depressionswill eventually evaporateor infiltrate the ground I surface.Surfacerunoff otti-ut"ty reachesminor channels(gullies, rivulets, and the : like), flowsto major streamsand rivers,and finally reachesan ocean.Along the course of a stream,evaporationand infiltration can also occur. The foregoing discussion suggeststhat the hydrologic cycle, while simple in concept,is actually exceedinglycomplex.Pathstaken by particlesof water precipitated in any arcaare numerousand varied before the seais reached.The time scale may be on the order of seconds,minutes,days,or years. A generalhydrologicequationcan be developedbasedon theprocessesillustrated in Figs. 1.2 and 1.3. ConsiderFig. 1.4. In it, the hydrologicvariablesP, E, T, R, G, and l are as definedin Fig. 1.2. Subscriptss and g are introduced to denote vectorsoriginatingaboveand belowthe earth's surface,respectivd. For example,R,
[" Earth's surface Surface channels
R2
Level of plastic rock . (no water below this level)
Figure 1.4 Regionalhydrologiccycle.
BUDGET 1.4 THE HYDROLOGIC
9
signifies groundwaterflow that is effluent to a surface streamoand E, represents evaporationfrom surfacewaterbodiesor other surfacestorageareas.Letter S stands for storage.The regionunderconsiderationspecifiedasA hasa lower boundarybelow which water will not be found. The upper boundary is the earth's surface.Vertical boundsare arbitrarily set asprojectionsof the peripheryof the region.Remembering that the water budgetis a balancebetweeninflows, outflows,and changesin,storage, Fig. 1.4canbe translatedinto the following mathematicalstatements,whereall values are given in units of volume per unit time: 1. Hydrologicbudgetabovethe surface P+R1 -RrIRr-E"-7,-
1:AS"
(1.1)
AS,
(1.2)
2. Hydrologicbudgetbelow the surface I + Gt- G2- Rr- E, - 4:
3. Hydrologicbudgetfor the region (sum of Eqs' 1.2 and 1.3) p - (Rr- R,) - (E" + E) - (r" + Tr) - (Gr- G,) : a(S, + ss), (1.3) If the subscriptsare droppedfrom Eq. 1.3 sothat letterswithout subscriptsrefer to total precipitation and net valuesof surfaceflow, undergroundflow, evaporation, transpiration,and storage;the hydrologicbudgetfor a regioncan be written simply as p_R-G_E_T:LS
(1.4)
This is the basicequationof hydrology.For a simplifiedhydrologicsystemwhereterms G, E, and Z do not apply, Eq. 1.4 reducesto p-R:AS
(1.5)
Equation 1.4 is applicableto exercisesof any degreeof complexity and is therefore basic to the solution of all hydrologicproblems. The difficulty in solvingpractical problemslies mainly in the inability to measure or estimateproperly the various hydrologic equation terms. For local studies, reliableestimatesoften are made,but on a global scaleqqantificationis usuallycrude. Precipitationis measuredby rain or snowgaugeslocatedthroughoutan area.Surface flows can be measuredusing various devicessuchas weirs, flumes,velocity meters, and depthgaugeslocatedin the rivers and streamsof the area.Under goodconditions these measurementsare 95 percent or more accufate,but large floods cannot be measureddirectly by current methodsand dataon sucheventsare sorelyneeded.Soil moisturecan be measuredusingneutronprobesand gravimetricmethods;infiltration can be deterrnined locally by infiltrometers or estimated through the use of precipitation-runoff data. Areal estimatesof soil moistureand infiltration are generilly very crude,however.The extentandrate of movementof groundwaterareusually exceedinglydifficult to determine,and adequatedataon quantitiesof groundwaterare not alwaysavailable.Knowledgeof the geologyof aregion is essentialfor groundwater estimatesif they are to be more than just rough guides.The determinationof the
1O
CHAPTER1
INTRODUCTION
quantities of water evaporatedand transpired is also extremely difflcult under the presentstateof developmentiofthe science.Most estimatesof evapotranspirationare obtainedby usingevaporationpans,energybudgets,masstransfermethods,or empirical relations.A predicamentinherent in the analysisof large drainagebasinsis the fact that rates of evaporation,transpiration,and groundwatermovementare often assumedto be highly heterogeneous. The hydrologicequationis a usefultool; the readershouldunderstandthat it can be employed in various ways to estimate the magnitude and time distribution of hydrologicvariables.An introductory exampleis given here,and otherswill be found throushout the book. EXAMPLE 1.1 In a given year,a 10,000-mi2wabrshedreceived20 in. of precipitation.The average rate of flow measuredin the river drainingthe areawasfound to be 700 cfs (cubicfeet per second).Make a rough estimateof the combinedamountsof waterevaporatedand transpiredfrom the region during the year of record. Solution. Beginningwith the basic hydrologicequation P - R - G - E - Z : A S
(r.4)
and sinceevaporationand transpirationcan be combined, ET:P-ft-G-AS
{4 t:
.+ v' a'E-;.
(1.6)
The term EZ is the unknown to be evaluatedand P and R are specifled.The equation thus has flve variables and three unknowns and cannot tre solved without additional information. In order to get a solution, two assumptionsare made. First, since the drainageareais quite large (measuredin hundredsof squaremiles),a presumption that the groundwaterdivide (boundary)follows the surfacedivide is probably reasonable.In this casethe G componentmay be consideredzero. The vector R, existsbut is included in R. The foregoingassumptionis usually not valid forsmall areasand mustthereforebe usedcarefully.It is alsopresupposed that AS : 0, thus implying that the groundwaterreservoir volume has not changedduring the year. For such short periods this assumptioncan be very inaccurate,evenfor well-wateredregionswith balancedwithdrawalsand good rechargepotentials.In arid areaswheregroundwateris beingmined (AS consistently negative),it would be an unreasonablesuppositionin many cases.Nevertheless,the assumptionis made here for illustrative purposesand qualified by sayingthat pastrecordsof waterlevelsin the areahaverevealedan approximate constancyin groundwaterstorage.Hydrology is not an exact science,and reasonablewell-foundedassumptionsare required if practical problemsare to be solved. Using the simplificationsjust outlined, the working relation reducesto ET:P_R
'I1 1.6 HYDROLOGIC DATA which canbe solveddirectly.First, changeR into inchesper yearsothat the units are compatible: _ ft3 r
\
t
-
r
sec
R _
.
,
1
sec
area (m n-l
yt
,
6
^ :R,in.
II
7 0 0 x 8 6 ; 4 0 0 x 3 6 5 x 1 :2 104x (5280)'
0.95in.
Therefore,ET : 20 - 0.95 : 19.05in./yr. The amountof evapotranspirationfor the year in questionis estimatedto be 19.05in. This is admittedlya crudeapproximationbut could serveasa useful guide for water resourcesplanning. ll
1.5 HYDROLOGIC MODELS Hydrologic systemsare generally analyzedby using mathematicalmodels.'These modelsmay be empirical, statistical,or foundedon known physicallaws.They may be usedfor suchsimplepurposesas determiningthe rate of flow that a roadwaygrate mustbe designedto handle,or they may guidedecisionsaboutthe bestway to develop a river basinfor a rnultiplicity of objectives.The choiceof the modelshouldbe tailored to the purposefor which it is to be used.In general,'thesimplestmodel capableof producinginformation adequateto deal with the issueshouldbe chosen. Unfortunately,most waterresourcessystemsof practicalconcernhavephysical, social,political, environmental,andlegaldirnensions;andtheir interactionscannotbe exactly describedin mathematicalterms. Furthermore,the historical data necessary for meaningful hydrologic analysesare often lacking or unreliable. And when one considersthat hydrologic systemsare generallyprobabilistic in nature, it is easyto understandthat the modeler'stask is not a simpleone.In fact, it is often the casethat the best that can be hoped for from a model is an enhancedunderstandingof the systembeing analyzed.But this in itself can be of great value, leading,for example, to the implementationof datacollectionprogramsthat canultimately supportreliable modelingefforts. For the most part, mathematicalmodels are designedto describethe way a system'selementsrespondto sometype of stimulus(input). For example,a model of 'a groundwater systemmight be developedto demonstratethe effectson groundwater storageof various schemesfor pumping. Equations 1.1 and L2 are mathematical modelsof the hydrologicbudget,and Figure 1.3 can be considereda pictorial model of the rainfall-runoff process.In later chapters,a variety of hydrologicmodelswill be presentedand discussed.Thesemodelsprovidethe basisfor informed watermanagement decisions.
DATA 1.6 HYDROLOGIC Hydrologic dataarc neededto describeprecipitation; streamflows;evaporation;soil moisture; snow fields; sedimentation;transpiration;infiltration; water quality; air, s9i!, and water temperatures;and other variablesor componentsof hydrologicsys-
12
CHAPTER 1
INTRODUCTION
tems. Sources of data are numerous,with the U.S. Geological Survey being the primary one for streamflow and groundwaterfacts. The National Weather Service (NOAA or National Oceanicand AtmosphericAdministration)is the major collector of meterologicdata.Many other federal,state,and local agenciesand other organizations also compile hydrologicdata. For a completelisting of theseorganizationssee Refs.3 and 17.
1 . 7 COMMONUNITSOF MEASUREMENT
'
Streamandriver flowsare usuallyrecordedascubic metersper second(m3/sec),cubic feetper second(cfs),or second-feet(sec-ft);groundwaterflowsand watersupplyflo\i/s are commonly measuredin gallons per minute, hour, or day (gpm, gph, gpd), or millions of gallonsper day (mgd); flowsusedin agricultureor relatedto water storage are often expressedas acre-feet (acre-ft), acre-feet per unit time, inches (in.) or centimeters(cm) depth per unit time, or acre-inchesper hour (acre-in./hr). Volumesare often given as gallons,cubic feet, cubic meters,acre-feet,secondfoot-days,and inchesor centimeters.An acre-footis equivalentto a volume of water 1 ft deep over 1 acre of land (43,560 ft3). A second-foot-day(cfs-day,sfd) is the accuinulatedvolumeproducedby a flow of 1 cfs in a24-hr period.A second-foot-hour (cfs-hr) is the accumulatedvolume produced by a flow of 1 cfs in t hr. Inches or centimetersof depthrelate to a volume equivalentto that many inchesor centimeters of water over the areaof concern.In hydrologicmassbalances,it is sometimesuseful to note that 1 cfs-day : 2 acre-feetwith sufficient accuracyfor most calculations. Rainfall depthsare usually recordedin inchesor centimeterswhereasrainfall rates are given in inches or centimetersper hour. Evaporation,transpiration, and infiltration rates are usually given as inchesor centimetersdepthper unit time. Some usefulconstantsand tabulatedvaluesof severalof the physicalpropertiesof water are given in Appendix A at the end of the book.
1.8 APPLICATIONOF HYDROLOGYTO ENVIRONMENTALPROBLEMS It is true that humanscannot exist without water; it is also true that water, mismanaged,or during times of deficiency(droughts),or times of surplus(floods),can be life threatening.Furthermore,there is no aspectof environmentalconcernthat doesnot relate in someway to water. Land, air, and water are all interrelatedas are water and all life forms. Accordingly, the spectrum of issuesrequiring an understandingof hydrologicprocessesis almost unlimited. As waterbecomesmore scarceand as competition for its useexpands,the need for improved water managementwill grow. And to provide water for the world's expandingpopulation, new industrial developments,food production, recreational demands,and for the preservationand protection of natural systemsand other purposes,it will becomeincreasinglyimportant for us to achievea thoroughunderstanding of the underlyinghydrologicprocesseswith which we must contend.This is the challengeto hydrologists,waterresourcesengineers,planners,policymakers,lawyers, economists,and others who must strive to see that future allocationsof water are sufficient to meet the needsof human and natural svstems.
PROBLEMS
13
r summary distribution' moveHydrology is the scienceof water. It embracesthe occurrence, sense,an account-"nt, urii propertiesof the watersof the earth. In a mathematical sothat a history ing may be madeof the inputs,outputs,and waterStofagesof a region of water movementfor the region can be estimated' the hydrologic After reading this chapter you should be able to understand shouldalso You budgetand make a simpleu".ouniing of water transportin a region' to be used facilitate have gained an undersiandingof trow hydrologic analysescan designand managementprocessesfor water resourcessystems'
PROBLEMS of 50 mi2.Convertthis
1.1.. One-half inch of runoff resultsfrom a stormon a drainagearea
amount to acre-feetand cubic meters. surfacearea of t.2. Assume you afe dealing with a vertical walled reservoir having a will it take to hours many How occurs: m3/sec 1.0 of 500,000 m' and that anlnflow raise the reservoirlevel bY 30 cm? time is 15 acre-ft and 1.3. consider that the storageexistingin a river reachat a reference from the reachis outflow the and cfs 500 is reach tie at the sametime the inflow to 650cfs.onehourlater,theinflowis550cfsandtheoutflowis630cfs.Findthe meters' changein storageduring the hour in acre-feetand in cubic walled reservoir was t.4. During a24-hr time period, the inflow to a 500-acre vertical a rise or fall in there in. was 1 was evaporation interval, same the 100 cfs. During centimeters' and in inches surfacewaterelevation?How muchwasit? Give the answer areais 3000 acres' 1.5. The annualevaporationfrom a lake is 50 in. If the lake's surface centimeters? in and acre-feet in rate evaporation daity what would beiire time requiredto raise 1.6. A flow of 10 cfs entersa 1-mi2vertical walledreservoir.Find the the reservoirlevelbY 6 in. Iftheaverageannualrunoffis5l02cfsand areaof4511mi2. t.7. Adrainagebasinhasan lossesfor the areain the averalerainfall is 42.5 in.,estimatethe evaportranspiration is? estimate this you think 1 year. Iiow reliable do Determinethe storage 1.8. The storagein a reachof a river is 16.0acre-ft at a given time. during the hour are outflow and inflow of (u"r"-f""tj t hr later if the averagerates 700 and 650 cfs, resPectivelY. areafor 3 days' (a) L.9. Rain falls atataverage irrtensity of 0.4 in./hr over a 600-acfe the 3-day (b) determine per second; feet Determinethe averagerate ofrainfau in cubic volumeofrainfallinacre-feet;and(c)determinethe3-dayvolumeofrainfallininches of equivalentdepth over the 600-acrearea' 100 acre-ft/day'Deter1.10. The evaporationrate from the surfaceof a 3650-acrelake is year ifthe inflow to the lake a 365-day during lake (feet) in the mine the depthchange is25.2cfs.1s the changein lake depth an increaseor a decrease? acre-feetif the drainage 1.11. One and one-half inchesof runoff areequivalentto how many : ft"') 43,560 areais 25-mi2?lNote: I acte of how many cubic feet L.12. one-half inch of rain per day is equivalentto an averagerate per second? meters many p". ,".ona if the areais 500 acrei?How
14
cHAPTER1 INTRoDUCTIoN
REFERENCES 1. P. B. Jones,G. D. Walker,R. W. Harden,and L. L. McDaniels,"The Developmentof the Scienceof Hydrology," Circ. No. 60-03, TexasWater Commission,Apr. 1963. 2. W. D. Mead, Noteson Hydrology. Chicago:D. W. Mead, 1904. 3. Ven Te Chow (ed)., Handbook of Applied Hydrology. New York: McGraw-Hill,1964. 4. O. E. Meinzer,Hydrology,Vol. 9 of Physicsof the Earth.New York: McGraw-Hlll, 1942. Reprintedby Dover, New York, 1949. 5. P. D. Krynine, "On the Antiquity of Sedimentationand Hydrology," Bull. Geol. Soc. Am. 70. l7 2I - l7 26(1960\. 6. RaphaelG. Kazmann,Modern Hydrology.New York: Harper & Row, 1965. 7. H. Pazwoshand G. Mavrigian, "A Historical Jewelpiece-Discovery of the Millennium Hydrologic Works of Karaji," WaterResourcesBull. 16(6), 1094-1096(Dec. 1980), 8. Hunter Rouseand Simon Ince, History of Hydraulics, Iowa Institute of Hydraulic Re- :. search,State University of Iowa, 1957. 9. G. H. L. Hagen, "Ueber die Bewegungdes Wassersin engen cylindrischenRohren," Poggendorfs Ann. Phys. Chem.16, 423- 442(1839). 10. Henri Darcy, Les fontaines publiques de la ville de Dijon. Paris:.V. Dalmont, 1856. 11. J. Dupuit, Etudesthdoriqueset practiques sur le mouvementdes eaux dans les canauxs dtcouvertset d travers les terrainspermdables,2nded. Paris:Dunod, 1863. 12. L. K. Sherman,"Stream Flow from Rainfall by the Unit-Graph Method," Eng. NewsRec.108(1932). 13. R. E. Horton, "The Role of Infiltration in the Hydrologic Cycle," Trans.Am. Geophys. Union 14, 446- 460(1933). l 14. C. V. Theis,"The RelationBetweenthe Lowering of the PiezometricSurfaceand the Rate and Duration of a Well Using Ground WaterRecharge,"Trans.Am. Geophys.Union 16, 519-524(1935\. 15. Asit K. Biswas,"Hydrologic EngineeringPrior to 600 s.c.," Proc. ASCE J. Hyd. Div., Proc. Paper5431,Vol. 93, No. HY5 (Sept.1967). 16. RaymondL. Nace,"WaterResources:A GlobalProblemwith Local Roots,"Environ. Sci. Technol.1(7) (July i967). 17. D. K. Todd (ed.), The WaterEncylopedia.New York: Water Information Center, 1970.
Chapter2
Precipitation
r Prologue The purposeof this chapteris to: ' Define precipitation, discussits forms, and describeits spatial and temporal attributes. ' Illustrate techniquesfor estimatingareal precipitation amountsfor specific storm eventsand for maximum precipitation-generatingconditions. Precipitation replenishessurfacewater bodies,rbnewssoil moisture for plants, and rechargesaquifers.Its principal forms are rain and snow.The relative importanceof theseforms is determinedby ttre climate of the area under consideration.In many parts of the westernUnited States,the extentof the snowpackis a determiningfactor relative to the amountof waterthat will be availablefor the summergrowing season. In more humid localities, the timing and distribution of rainfall are of principal concern. Precipitatedwaterfollows the pathsshownin Figs. r.2 and,1.3.some of it may be intercepted,evaporated,infiltrated, and becomesurfaceflow. The actual disposition dependson the amountof rainfall, soil moistureconditions,topography,vegetal cover soil type, and other factors Hydrologic modeling and water resourcesassessments dependupon a knowledgeof the form and amountof precipitation occurringin a region of concernover a time period of interest.
2.1 WATERVAPOR The fraction of watervapor in the atmosphereis very small comparedto quantitiesof other gasespresent,but it is exceedinglyimportant to our way of life. Precipitationis derived from this atmosphericwater. The moisture centent of the air is also a significantfactor in local evaporationprocesses.Thus it is necessaryfor a hydrologist to be acquaintedwith waysfor evaluatingthe atmosphericwatervapor contentand to understandthe thermodynamiceffects of atmosphericmoisture.l
CHAPTER2
16
PRECIPITATION
Under most conditions of practical interest (modest ranges of pressureand temperature,provided that the condensationpoint is excluded),water vapor essentialiy obeys the gas laws. Atmospheric moisture is derived from evaporationand transpiration,the principal sourcebeing evaporationfrom the oceans.Precipitation overthe United Statescomeslargelyfrom oceanicevaporation,the watervaporbeing transporatedover the continentby the primary atmosphericcirculation system. Measuresof watervapor or atmospherichumidity are relatedbasicallyto conditions of evaporationand condensationoccurring over a level surfaceof pure water. Considera ilosed systemcontaining approximatelyequal volumesof water and air maintainedat the sametemperature.If the initial condition of the air is dry, evaporation takesplace and the quantity of water vapor in the air increases.A measurement of pressurein the airspacewill reveal that as evaporationproceeds,pressurein the airipaceincreasesbecauseof an increasein partial pressureof the watervapor (vapor preJsure).Evaporationcontinuesuntil vapor pressureof the overlying air equalsthe surfacevapor pressure[a measureof the excessof water moleculesleaving(evaporating from) the water surfaceover thosereturning]. At this point, evaporationceases, and if the temperaturesof the air spaceand water are equal,the airspaceis saidto be saturated.If the containerhad beenopeninsteadof closed,the equilibriumwould not havebeenreached,and all the water would eventuallyhaveevaporated.Somecommonly usedmeasuresof atmosphericmoistureor humidity are vapor pressure,absolute humidity, specifichumidity, mixing ratio, relative humidity, and dew point tem-
,
Perature.
Amount of PrecipitableWater .
Estimatesof the amount of precipitation that might occur over a given region with favorable conditions are often useful. These may be obtained by calculating the amountof water containedin a column of atmosphereextendingup from the earth's surface.This quantity is known as theprecipitable water 14{althoughit cannotall be removed from the atmosphereby natural processes.Precipitable water is usually expressedin centimetersor inches. An equationfor computingthe amountof precipitablewater in the atmosphere can be derived as follows. Considera column of air having a squarebase 1 cm on a side.The total water masscontainedin this column betweenelevationzero and some height z would be
W:
r p*dz
(2.1)
J^
wherep. : the absolutehumidity and IVis the depthof precipitablewaterin centimeters. The integral can be evaluatedgraphically or by dividing the atmosphereinto layersof approximatelyuniform specifichumidities,solvingfortheseindividually, and then summing.Figure 2.1 illustratesthe averageamountof precipitablewater for the continentalUnited Statesup to an elevationof 8 km.2
Geographicand TemporalVariations The quantity of atmosphericwater vapor varieswith location and time. Thesevariations may be attributed mainly to temperatureand sourceof supply considerations. The greatestconcentrationscan be found near the ocean surfacein the tropics, the
2.2 PRECIPITATION
17
Sault Ste.Mtrie Portled
0;7 VCT -NJ 0.8
0.9 1.0
0.8 b.z o.d1'o 1.1 r.2 t.J
Bromsville
Figure 2.L Mean precipitablewater, in inches,to an elevationof 8 km. (U.S. WeatherBureau.)2
concentrationsgenerallydecreasingwith latitude, altitude, and distanceinland from coastalareas. About half the atmosphericmoisturecanbe found within the first mile abovethe earth's surface.This is becausethe vertical transport of vapor is mainly through convectiveaction,which is slight at higher altitudes.It is also of interestthat there is not necessarilyany relation betweenthe amount of atmosphericwater vapor over a regionand the resultingprecipitation.The amountof water vapor containedover dry areasof the Southwest,for example,at times exceedsthat over considerablymore humid northern regions,eventhough the latter areasexperienceprecipitation while the former do not.
2.2 PRECIPITATION Precipitation is the primary input vector of the hydrologiccycle. Its forms are rain, snow,andhail andvariationsof thesesuchasdrizzle and sleet.Precipitationis derived from atmosphericwater, its form and quantity thus being influencedby the action of other climatic factors such as wind, temperature,and atmosphericpressure.Atmosphericmoistureis a necessarybut not sufficient condition for precipitation. Continental air massesare usuallyvery dry sothat mostprecipitationis derivedfrom moist maritime air that originatesoverthe oceans.In North America about50 percentof the evaporatedwater is taken up by continental air and movesback againto the sea.
18
CHAPTER2
PRECIPITATION
Formationof PreciPitation Two processesare consideredto be capableof supportingthe growth of dropletsof sufficient mass(dropletsfrom about 500 to 4000 p'min diameter)to overcomeair resistanceand consequentlyfall to the earth asprecipitation.Theseare known as the process' ice crystalprocessand the coalescence the small cloud dropletsincreasetheir which by process is one The c^oalescence collision. Water droplets may be through droplets other with size due to contact gravitationaland air resistance to both subjected are that bodies falling consideredas are proportional to the (terminal velocities) equilibrium at effects. Fall velocities descendmore quickly will droplets larger the thus the droplet; of squareof the radius overtakenby larger are often droplets smaller result, As a ones. than the smaller increasingly producing the drops, to unite tend collisions resulting droplets,and the into smallup break in diameter) mm (order 7 of drops large largir particles.Very effect. chain of a produce somewhat and process coalescence dropletsthat repeatitre significant generate to produced be may raindrops large In this *unn"r, sufficiently precipitation. This processis ionsidered to be particularly important in tropical regionsor in warm clouds. An important type of growth is known to occurif ice crystalsand waterdroplets -40'C- Under are found toexist togetherat subfreezingtemperaturesdown to about theseconditions,certain particles t saltsserveasfreezingnucleisothat theseconditionsis higher over the t condensationoccurson the surface unevenparticle sizedistributionsde' with otherparticles.This is considet mechanism. The artificial inducementof precipitation has been studied extensively,and thesestudiesare continuing.It has been demonstratedthat condensationnuclei supplied to cloudscan induceprecipitation.The ability of humansto ensurethe produciion of precipitation or to control its geographiclocation or timing has not yet been attained,however' Many legal as well as technologicalproblemsare associatedwith the prospects ..rain-makiig" processes. Of interesthereis the impacton hydrologicestimatesthat of partially controlled artificial precipitation might have. Many uncontrolled oi onty are consideredas statisticalvariatesthat are variables Lydrologic naturally occurring with a random component.If the distribudistributed or either randomty distrlUuted an inferenceasto the frequencyof modeled, can be variable the tion or time seiiesof given magnitude(suchas precipitaa of events hydrologic occurrenceof significant used and if the effectsof these are controls artificial however, If, tion) can be made. prove to be totally unreliable may analyses frequency predicted, cannot be reliably tools.
PrecipitationTyPes Dynamic or adiabaticcoolingis the primary causeof condensationand is responsible for most rainfall. Thus it can be seen that vertical transport of air massesis a requirementfor precipitation.Precipitationmay be classifiedaccordingto the condi-
'1 2.2 PRECIPITATIOI'|9 tions that generatevertical air motion. In this respect,the three major categoriesof precipitation type are convective,orographic, and cyclonic. Convective Precipitation Convectiveprecipitationis typical of the tropicsand is brought about by heatingof the air at the interfacewith the ground. This heatedair expandswith a resultantreductionin weight.During this period,increasingquantities of water vapor are taken up; the warm moisture-ladenair becomesunstable;and pronouncedvertical currents are developed.Dynamic cooling takes place, causing condensationand precipitation.Convectiveprecipitationmay be in the form of light showersor stormsof extremelyhigh intensity (thunderstormsare a typical example). Orographic Precipitation Orographicprecipitationresultsfrom the mechanical lifting of moist horizontal air currentsover natural barriers suchas mountainranges. This type of precipitation is very common on the West Coast of the United States where moistureladen air from the Pacific Oceanis interceptedby coastalhills and mountains.Factorsthat are important in this processinclude land elevation,local slope,orientation of land slope,and distancefrom the moisture source. In dealingwith orographicprecipitation,it is commonto divide the regionunder study into zonesfor which influencesasidefrom elevationare believedto be reasonably constant.For eachof thesezones,a relation betweenrainfall and elevationis developedfor usein producingisohyetalmaps(seeSection2,5). with the movementof Cyclonic Precipitation Cyclonicprecipitationis associated air massesfrom high-pressureregionsto low-pressureregions.Thesepressuredifferencesare createdby the unequalheatingof the earth's surface. Cyclonicprecipitationmay be classifiedasfrontal or nonfrontal. Any barometric low canproducenonfrontal precipitationasair is lifted throughhorizontalconvergenceof the inflow into a low-pressurearea. Frontal precipitation results from the lifting of warm air over cold air at the contact zone between air masseshaving different characteristics.If the air massesare moving so that warm air replacescolder warm air, the front is known asawarmfront; if , on the otherhand,cold air displaces air, the front is saidto be cold.If the front is not in motion,it is saidto be a stationary front. Figure 2.2 illustratesa vertical sectionthrough a frontal surface.
Figure 2.2 Vertical cross-sectionthrough a frontal surface.
20
CHAPTER 2 PRECIPITATION
Thunderstorms Many areasof the United Statesare subjectedto severeconvectivestorms,which are generallyidentifiedasthunderstormsbecauseof their electricalnature.Thesestorms, although usually very local in nature, are often productive of very intenserainfalls that are highly significantwhen local and urban drainageworks are considered. Thunderstormcells developfrom vertical air movementsassociatedwith intense surfaceheatingor orographiceffects.Thereare threeprimary stagesin the life history of a thunderstorm.Theseare the cumulusstage,the mature stage,andthe dissipating stage.Figure 2.3 illustrateseachof thesestages. All thunderstormsbeginascumulusclouds,althoughfew suchcloudseverreach the stage of developmentneededto produce such a storm. The cumulus stageis characterizedby strongupdrafts that often reachaltitudesof over 25,000ft. Vertical wind speedsat upperlevelsare often as greatas 35 mph. As indicatedinFig.2.3a, there is considerablehorizontal inflow of air (entrainment)during the cumulusstage. This is an important elementin the developmentof the storm, as additional moisture is provided.Air temperaturesinside the cell are greaterthart thoseoutside,as indicatedby the convexity of the isothermsviewed from above.The number and size of The duration ofthe cumulusstage the water dropletsincreaseasthe stageprogresses. is approximatelyi0-15 min. The strong updrafts and entrainmentsupport increasedcondensationand the developmentof waterdropletsand ice crystals.Firrally, whenthe particlesincreasein size and number so that surfaceprecipitation opcurs,the storm is said to be in the mature stage.In this stage strong downdrafts are created as falling rain and ice crystals cool the air below. Updraft velocities at the higher altitudes reach up to 70 mptrin the early periodsof the maturestage.Downdraft speedsof over20 mph are
El
o o F
(a)
(b)
(c)
Figure 2.3 Cumulus,mature,and dissipatingstagesof a thunderstormcell. (Department of the Army.)
2.2 PRECIPITATION 21
usual aboveabout 5000 ft in elevation.At lower levels,frictional resistancetendsto decreasethe downdraft velocity. Gusty surfacewinds move outwardfrom the region of rainfall. Heavyprecipitationis often derivedduring this preiod, which is usuallyon the order of 15-30 min. In the final or dissipatingstage,the downdraftbecomespredominantuntil all the air within the cell is descendingand being dynamically heated.Since the updraft ceases,the mechanismfor condensationends and precipitation tails off and ends.
PrecipitationData Considerabledata on precipitation are available in publications of the National WeatherService.a's Other sourcesincludevariousstateand federal agenciesengaged in water resourceswork. For critical regional studiesit is recommendedthat all possibledata be compiled; often the establishmentof a gauging network will be necessary(seealso Chapters7-9).
Precipitation VariabiIity Precipitationvariesgeographically,temporally, and seasonally.Figure 2.4 indrcates the mean annualprecipitation for the continentalUnited States,while Fig. 2.5 gives an exampleof seasonaldifferences.It should be understoodthat both regional and temporal variationsin precipitation are very important in water resourcesplanning and hydrologicstudies.For example,it may be very important to know that the cycle of minimum precipitationcoincideswith the peakgrowing seasonin a particular atea, or that the periodofheaviestrainfall shouldbe avoidedin schedulingcertainconstruction activities. Precipitation amounts sometimesvary considerablywithin short distances. Recordshaveshowndifferencesof 20 percentor more in the catchof rain gaugesless that2Oft apart.Precipitationis usuallymeasuredwith a rain gaugeplacedin the open so that no obstacleprojects within the inverted conical surfacehavingthe top of the gaugeas its apexand a slopeof45'. The catchofa gaugeis influencedby the wind, which usually causeslow readings.Variousdevicessuchas Nipher and Alter shields havebeendesignedto minimize this error in measurement.Precipitationgaugesmay be of the recording or nonrecordingtype. The former are requiredif the time distribution of precipitation is to be known. Information about the featuresof gaugesis readily available.3 Becauseprecipitationvariesspatially,it is usuallynecessaryto usethe datafrom severalgaugesto estimatethe averageprecipitation for an area and to evaluateits reliability (seeChapter27). This is especiallyimportant in forestedareaswhere the variation tendsto be large. Time variations in rainfall intensity are extremely important in the rainfallrunoff process,particularly in urban areas(seeFi g. 2.6a). The arealdistributionis also significantandhighly correlatedwith the time history of outflow (seeFig. 2.6b).These considerationsare discussedin greaterdetail in following chapters.
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CHAPTER2
PRECIPITATION
- 3
5
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Time(x 102sec) (a)
-/'?:;"
Isohyets-lines of equal rainfall depth
-"@l5 Y.s.
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Figure 2.6 (a) Rainfall distribution in a convective storm June 1960, Baltimore, Maryland. (b) Isohyetal pattern, storm of SeptemberL0, 195'1, Baltimore, Maryland. O, recording rain gauge.
INPUT 2.3 DISTRIBUTION OF THE PRECIPITATION
25
2.3 DISTRIBUTION OF THE PRECIPITATION INPUT Total precipitation is distributedin numerousways.That interceptedby vegetation and treesmay be equivalentto the total precipitationinput for relatively small storms. Once interception storageis fllled. raindrops begin falling from leavesand grass, where water storedon thesesurfaceseventuallybecomesdepletedthrough evaporation. Precipitationthat reachesthe ground may take severalpaths. Some water will fill depressionsand eventually evaporate;some will infiltrate the soil. Part of the infiltrated watermay strike relatively imperviousstratanearthe soil surfaceand flow approximatelyparallel to it as interflow until an outlet is reached.Other portions may replenishsoil moisturein the upper soil zone, and someinfiltrated water may reach the groundwaterreservoirthat sustainsdry weatherstreamflow.The componentof the precipitation input that exceedsthe local infiltration rate will developa film of water on the surface(surfacedetention)until overlandflow commences.Detention depths varying from I to 1j in. for various conditions of slope and surfacetype havebeen reported.3 Overland flow ultimately reaches defined channels and becomes streamflow. Figure 2.7 ilhxtrates in a generalway the dispositionof a uniform storm input to a natural drainagebasin.Although suchan input is not to be expectedin nature,the indicated relations are representativeof actual conditions. Modifications resulting from nonuniform stormswill be discussedas they arise. In Fig. 2.7anote that the storm input is distributeduniformly over time /o at a rateequalto i (dimensionally intocomponents equalto LT '). This inputis dissected I, through lo, the sum of which is equal to I at any time r. Figure 2.7b illustratesthe manner in which infiltrated water is further subdividedinto interflow, groundwater, and soil moisture. Figure 2.7c showsthe transition from overland flow supply into strearhflow.The mechanicsof theseprocesseswill be treatedin detail in later sections. The nature of the curvespresenteddepictsthe generalrunoff process.It should be realized,however,that actual graphsof infiltration and/or other factorsversustime might appear quite different in form and relative magnitudewhen comparedwith theseillustrations becatrseof the effects of nonuniform storm patterns,antecedent conditions,and other factors. The rate and areal distribution of runoff from a drainagebasin are determined by a combination of physiographicand climatiOfactors. Important climatic factors includethe form of precipitation (rain, snow,hail), the type of precipitation (convective, orographic,cyclonic),the quantity and time distributionof the precipitation,the characterof the regionalvegetativecover,prevailingevapotranspirationcharacteristics, and the statusofthe soil moisturereservoir.Physiographicfactorsof significance includegeometricpropertiesof the drainagebasin,land-usecharacteristics,soil type, geologicstructure,and characteristicsofdrainage channels(geometry,slope,roughness,and storagecapacity). Large drainagebasinsoften react differently from smalleroneswhen subjected to a precipitationinput. This can be explainedin part by suchfactorsas geologicage, relativeimpact ofland-use practices,sizedifferential,variationsin storagecharacteristics, and other causes.Chow definesa small watershedas a drainagebasin whose
CHAPTER2
PRECIPITATION
(a/
\ I I
Soil moisture \a
Int".flot
Mechanics of surface runoff
(c)
Figure 2.7 The runoff process:(a) dispositionof precipitation, (b) componentsof infiltration, and (c) dispositionof overland flow supply.
27 PRECIPITATION 2.4 POINT characteristicsdo not filter out (1) fluctuationscharacteristicofhigh-intensity, shortduration storms;or (2) the effectsof land managementpractices.6On this basis,small basinsmay vary from lessthan an acre up to 100 mi2. A large basin is one in which channelstorageeffectively filters out the high frequenciesof imposedprecipitation and effectsof land-usepractices.
2.4 POINTPRECIPITATION Precipitation eventsare recordedby gaugesat specificlocations.The resulting data p"rmit determinationof the frequency and charactei of precipitation eventsin the vicinity of the site. Point precipitation data are used collectively to estimateareal variability ofrain and snowand are alsousedindividually for developingdesignstorm characteristicsfor small urban er other watersheds.Design storms are discussedin detail in Chapter 16. Point rainfall data are usedto deriveintensity-duration-frequencycurvessuch as those shown in Fig. 2.8. Such curves are used in the rational method for urban storm drainagedesign(Chapter 25); thek constructionis discussedin Chapter27 'ln applyingthe rational method,a rainfall intensityis usedwhich representsthe average intensity of a storm of given frequencyfor a selectedduration.The frequencychosen should reflect the economics of flood damage reduction. Frequenciesof up to 100 yearsare commonlyusedwhereresidentialareasare to be protected.For highervaluedistricts and critical facilities,up to 500 yearsor higherreturn periodsare often selected.Local conditionsand practicenormally dictatethe selectionof thesedesign criteria. (ExecutiveOrder 11988,Floodplain Management,I97 7).
\ 00-yr frequency 50-yr frequency I -20-yr frequencyf, 10-yrfrequency
\ \
t\ B. a
\
7 ,
\
\1
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5-Yr,frequencl
tl=.t
Y
120 Duration (min)
Figure 2.8 Typical intensity-duration-frequencycurves for Baltimore, Maryland, and vicinity.
28
CHAPTER2
PRECIPITATION
Figure 2.9 Four quadrants surrounding precipitation stationA.
It is occasionallynecessaryto estimatepoint rainfall at a given location from recordedvaluesat surroundingsites.This can be doneto completemissingrecordsor to determinea representativeprecipitation to be used at the point of interest. The National WeatherServicehasdevelopeda procedurefor this which hasbeenverified on both theoreticaland empiricalbases.T Considerthat rainfall is to be calculatedfor point A in Fig. 2.9. Establisha set of axes running through A and determinethe absolutecoordinatesof the nearest surroundingpoints B, C, D, E, and F. These are recorded in columns 3 and 4 of Table 2.L The estimatedprecipitationat A is determinedas a weightedaverageof the other five points.The weightsarereciprocalsof the sumsof the squaresof AX and AY; that is, D2 : LX2 + LYz, and W : llDt. The estimatedrainfall at the point of interestis given by I (P x W)/> I{. In the specialcasewhere rainrall is known in only two adjacentquadrants(e.g.,I and II), the estimateis given asI (p x lV). This has the effect of reducing estimatesto zero as the points move from an area of TABLE 2.1 DETERMINATIONOF POINT RAINFALLFROM DATA AT NEARBY GAUGES
t1) Point
(2) Rainfall
(3) AX
(4) AY
(5) (D')
4 1 3 3
2 6 2 3 2
20 37 1,3 18 8
(6) wx103
(7) PxWx103
(in.)
B C D E F Sums
r.60 1.80 1.50 2.00 r .
7
0
?
*Note.'Estimatedprecipitation(P) at A = 567.7/334.5; P = 1.70 in.
50 27.O 76.9 55.6 125.0
,TT3
80.0 48.6 115.4 111,.2 2t2.5 567.7
_l
29 PRECIPITATION 2.5 AREAL precipitation to one with no records.This is consideredto be the most logical procedure for handlingthis unusualcase.7The estimatedresult will alwaysbe lessthan the greatestand greaterthan the smallestsurroundingprecipitation. For specialeffects suchas mountain influences,an adjustmentprocedurecan be applied.
2.5 AREALPRECIPITATION For most hydrologicanalyses,it is important to know the areal distributionof precipitation. Usually, averagedepths for representativeportions of the watershedare determinedand used for this pwpose. The most direct approachis to use the arithmetic averageof gaugedquantities.This procedureis satisfactoryif gaugesare uniformily distributedand the topographyis flat. Other commonly usedmethodsare the isohyetalmethodand the Thiessenmethod.The reliability of rainfall measuredat one gaugein representingthe averagedepth over a surroundingareais a function of (1) the distancefrom the gaugeto the centerof the representativearea,(2) the sizeof the area,(3) topography,(4) the natureofthe rainfall ofconcern (e.g.,stormeventversus For more information on meanmonthly), and (5) local stormpatterncharacteristics.8 errorsof estimation,the readershouldconsultRefs.7 and 8. Chapter27 alsocontains a discussionof areal variability of precipitation. Figures 2.10 and 2.11 illustrate how the measuredrainfall at a single gauge relatesto the averagerainfall over a watershedwith changein ( 1) the relativeposition of the gaugein the watershedand (2) the time period over which the averageis calculated.In the first caseit is clearthat the more centralthe gaugelocation,the more closely its observationswill match the averagefor a representativearea, providing that the regionis not too large.Figure 2.11 shows,not surprisingly,that areal averages
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(n 1
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Storm rainfall at one gauge in inches (a)
1
2
3
Stormrainfall at one gaugein inches (b)
Figure 2.L0 Errors resultingfrom useof a singlegaugeto estimatewatershedaverage (giuge location effect, Soil ConservationService),(a) Watershedarea is 0.75 mi2 and gaugeis near the center. (b) Watershedareais 0.75 mi2 and gaugeis 4 mi outsidethe watershedboundary.
30
PRECIPITATION
CHAPTER2
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Stormrainfall at one gaugein inches
(b)
estimatewatershedaverage Figure 2.L1 Errors resultingfrom use of a singlegaugeto areais 5'45 mi2 and the (aiWatershed Service). ConserVation (time period effect, Soil gaugeisontheboundary.(b)Watershedareais5.45In|zandthegaugeisonthe boundarv. to conform more closely over long time periods, in this case one year, may be expected event' This suggests storm to a single guog" uu"ruge than those for an individual space and time both with tempered that the Oerlgn-of guuging networks should be considerations.
lsohyetalMethod are the isohyetal The two principal methodsfor determiningareal averagesof rainfall on interpolation based method and the Thiessenmethod. The isohyetal method is surveyingand in between gauges.It closely resemblesthe calculation of contours gauge plot the^rain to -upptng."Tf" first step in developingan isohye?1..-up is an Next' (Fig' 2'I2)' locationson a suitablemap and to reJord the rainfall amounts increselected at interpolation betweengaugesis performed and rainfall amounts connectedto form mentsare plotted.tdenticaldepthsfrom eachinterpolationare then averageof weighted the isohyets(lines of equal rainfall depth)' The areafaverageis isohyetal The isohyets' depthsbetweenisohyets,that is, the meanvaluebetweenthe over an precipitation method is the most accurateapproachfor determiningaverage topographic to attention area,but its proper oserequir#a skilled analystand careful the represenand other tactori that impact on areal variability. Figure 2. 13 illustrates tationofamajorstormeventinNorthCarolinabyanisohyetalmap.
ThiessenMethod method'In this is theThiessen Anothermethodof calculatingarealrainfallaverages ascenters' usingraingauges intopolygonalsubareas theareais subdivided procedure depth'Thiessen average areusedasweightsin estimitingthewatershed Thesubareas as shownin Fig. i.t+. fnis procedureis not suitablefor diagramsare constructed
2.5 AREAL PRECIPITATION
31
Average
precipitation for area A4 is 4.25 in.
----x------*
-*!fl1
A2 --^ 3 in'
Average preciPitation = 2 A i P i \ i for entire basin
(c)
of an isohyetalmap:(a)locateraingaugesand Figure 2.12 Construction and(c) plot isohyets' betweengauges; pdt values;(b) interpolate network is fixed mountainousareasbecauseof orographicinfluences'The Thiessen any gaugesare if io, u glu"o gaugeconfiguation, and polygonsmustbe reconstructed relocated.
AccuracY locatior\of the Irrespectiveof the methodusedfor estimatingareal precipitation,the of the estimate application of point tothe guu* orra in derivingthe estimaterelative may be distances vertical lbcattieso mustbe takeninto consideration.In mountainous are spacings horizontal gentle landscapbs, For -ft" irnpottant than horizontal ones.
32
CHAPTER 2 PRECIPITATION a . !.1 a.l - o
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Averagedepthoverentirewatershed =
+t Figure 2.14 Constructionofa Thiessendiagram:(a) connectrain gaugelocations;(b) draw perpendicularbisectors;and (c) calculate Thiessenweights l,er, ,qr, A3). (d) A completednetwork. the most important. when a precipitation gauging network is to be developed, both spacing and arrangement of gaugei must be considered.
EXAMPLE 2.1 $yen 1rrcdrainagearea of Fig. 2.r5 and the rainfall data displayedin column 3 of Table2.2, calculatethe averagerainfall over the areausing tne arithmetic mean, 1aj and (b) the Thiessenpolygonweighting system.
Figure 2.15 Thiessendiagram for Example 2.1.
34
CHAPTER2
PRECIPITATION
TABLE 2.2 DATA AND THIESSEN POLYGON FOR EXAMPLE2,1. CALCULATION (1)
(2)
(3)
(4)
GaugeNo.
"/" Area
Precip.-in
(2) x (3)
1.56
0.08
2
4
2.95
0.12
3
3
3.44
0.10
t5
2.91
0.44
ll
4.17
0.46
6
19
4.21
0.80
7
4
2.'|
0.11
8
7
2.45
0.17
9
21
3.88
0.81
10
6
3.98
0.24
ll
5
2.51
0.13
Total
100
3.45
Solution. a. Identify thosegaugesfalling within the areaboundary.They includegauges 1, 4 through 6, 8, and 9. Averagingthe valuesfor thesesix gaugesyields an estimatedmean areal rainfall of 3.20 inches. b. Followine the Thiessen method as described in Section 2.5, construct polygonsusingtrianglesto connectgaugepoints. Thesepolygonsare shown with each on Fig. 2.15. Calculatethe percentof the total area associated gaugeandrecord asin column2 of Table2.2.The Thiessenweightedaverage is obtainedby multiplying the valuesin column 2by the yaluesin column 3. The Thiessenaverageis computedas 3.45 inchesof rainfall. The use of a spreadsheet(Table 2.2) facilitates computations and aids in organizing data. lr
2.6 PROBABLEMAXIMUMPRECIPITATION The probablemaximum precipitation (PMP) is the critical depth-duration-arearainfall relation for a given areaand seasonwhich would result from a storm containing the most critical meteorologicalconditionsconsideredprobable.eSuch storm events are used in flood flow estimatesby the U.S. Corps of Engineersand other water resourcesagencies.The critical meteorologicalconditions are basedon analysesof air-mass properties(effectiveprecipitablewater,depthof inffow layer,wind, temperature, and other factors),synoptic situationsduring recorded stormsin the region, topography,season,and locationofthe area.The rainfall derivedistermedprobable maximumprecipitation sinceit is subjectto limitations of meteorologicaltheory and data and is basedon the most e_ffectiwcqmbination of factors controlling rainfall
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36
CHAPTER2
PRECIPITATION 1000 800 600 400
$ zoo 9?
E
1oo
st r o9u 9 4 4 0 20 10 l0
20 30 40 50 6Q 70 80
90 100 110 120 130 140 150
Percentageof 200 miz, 24-hr values
Figure 2.17 Seasonal variation, depth-area-duration relations; percentage to be applied to 200 ni2-24 hr probable maximum precipitation values for August in Zone 6. (U.S. Department of Commerce,National WeatherService.)
intensity.eAn earlierdesignationof "maximumpossibleprecipitation" is synonymous. Additional information on PMP is given in Chapter 16. The seasonalvariation of PMP is important ip the designand operation of multipurposestructuresand in floodingconsiderationsthat may occurin combination with snowmelt. In both of these cases,annual probable maximums might be less important than seasonalmaximums.Figures2.16 and,2.r7 display24-hr pMp for the easternhalf of the United Statesfor 200-ffi2 watershedsduring the month of August (similar figuresare availablefrom the National WeatherService).
2.7 GROSSAND NETPRECIPITATION The net (excess)precipitation that contributesdirectly to surfacerunoff is equivalent to the gross precipitation minus lossesto interception, storm period evaporation, depressionstorage,and infiltration. The relation betweenexcessprecipitationP" and grossprecipitation P is thus P":P.-Ilosses
(2.2)
where the lossesinclude all deductionsfrom the grossstorm input. The paths that water precipitatedover an areamay take can be representedby flow diagramsof the type given in Fig. 1.3 and by equationsof the form ot&q. 2.i. Modelssuchastheseare the basisfor most hydrologicinvestigations,and muchbf the contentof this book is devotedto the conceptualizationof individual componentsof the various hydrologicprocessesand to synthesizingthesecomponentsinto holistic representationsof hydrologicevents.
PROBLEMS 37
r Summary precipitation is the sourceof fresh waterreplenishmentfor the planetEarth. Too much or too little canmeanthe differencebetweenprosperityand disaster.In betweenthese extremesare the normal precipitation eventsthat are experiencedwith a frequency and intensity relatedmainly to geographicposition and topographicfeatures' After reading this chaptei you should understandthat both the timing and considamountof precipitition occuiring over anareaareimportant and that thereis areal erable g"olrapttic variability in precipitation. You should be able to estimate process precipiLtiin amountsfrom gaugedata and conceptualizesimplehydrologic shed models.It shouldbe recognizedihataveragevaluesofprecipitation for a region uses, some light on the quantiiy of water that might be made availablefor various are while a i
average,estimatethe amount of rainfall for gaugeX' artdc compute the rainfall for gaugeX in Problem 2.1 if the storm readingsat A, B, 'werc respectively' 3.7,4.I, and4'8 in., 2.3. compute the meanannualprecipitationfor the watershedin the following figureusing The the arithmetic mean, the'itri"*"n polygon method, and the isohyetalmethod.
"r)
38
CHAPTER2 PRECIPITATION gaugereadingsfor gaugesA-K, respectively,are: 29.79,34.97,25.6,24.2i,24.6, 42.61,42.35,15.51,39.99,43.04,and28.41. 2.4. Computethe meanannualprecipitationfor the watershedin the figurefor Problem2.3 using the arithmetic mean and the Thiessenpolygonmethod.The gaugereadingsfor gaugesA-K, respectively,are: 28.1, 33.7, 25.6, 23.9, 24.6, 40.1, 41.3, 37.2, 38.7, 41.1,and29.3. 2.5. The chart from a rain gaugeshown in the sketchrepresentsa record that you must interpret.Find the averagerainfall intensity (rate)between6 e.r'1.and noon on August 10. Find also the total precipitation on August 10 and August 11.
d
0.30 o.25 0.20 0.15 0.10 0.05 0
Aus. l 0
/
Midnight 6 a.m.
6 p.m. Midnight 6 a.m.
Ais..I7
Noon
6 p.m. Midnight
Time
2.6. Refer to the chart of Problem 2.5. Calculate the rainfall intensity for the period between6.q..1u.and noon on August 11. Would you considerthis to be a period of intenserainfall? 2.7. Use the map of Fig. 2.6 andfrom it constructa setof Thiessenpolygons.Using these, estimatethe mean rainfall for the region. 2.8. A meandraft of 100mgd is producedfrom a drainageareaof 200 mi2.At the flow line the reservoiris estimatedto cover 4000 acres.The annualrainfall is 37 in., the mean annualrunoff is l0 in., and the mean annual lake evaporationis 30 in. Find the net gain or lossin storage.Computethe volume of water evaporated.How significantis this amount? 2.9. A meandraft of 380,000m3/dayis producedfrom a drainageareaof 330 km2.At the flow line, the reservoiris estimatedto cover about 1600hectares.The annualrainfall is 96.5 cm, the meanannualrunoff is 22.8 cm, and the meanannuallake evaporation is 77.1 cm. Find the net gain or loss in storageand compute the volume of water evaporated. Calculatevolumesin m3. 2.10. Drainage areas within each of the isohyetal lines for a storm are tabulated for a watershed.Use the isohyetal method to determinethe averageprecipitation depth within the basin for the storm. Make a conceptualsketch.
lsohyetalinterval(in.)
Area (acres)
0-2
2700 1900 1000 0
a
i
4-6 6-8
2.11. ReworkProblem2.10 if the valuesin the secondcolumn of the table are2,500, 2,100 1,200, and 300, respectively,
REFERENCES 39 2.12. Discusshow you would go about collectingdata for analysisof the water budgetof a region.What agencieswould you contact?What other sourcesof information would you seekout? 2.13. For an areaof your choice,plot the meanmonthly precipitation versustime. Explain how this fits the pattern of seasonalwater usesfor the area.Will the form of precipitation be an important consideration?
REFERENCES I
2. 3. A
5. 6. ,1
8. 9.
TennesseeValley Authority, "Heat and Mass TransferBetweena Water Surfaceand the Atmosphere,"Lab. Rep. No, 14, TVA EngineeringLab. Noiris, TN, Apr. 1972. A. L. Shands,"Mean PrecipitableWater in the United States," U.S. WeatherBureau, Tech.PaperNo. 10, 1949. R. K. Linsley, Jr., M. A. Kohler, and J. L. H. Paulhus,Applied Hydrology. New York, McGraw-Hill, 1949. D. W. Miller, J. J. Geraghty,and R. S. Collins, WaterAtlas of the United States.Port Washington,NY Water Information Center, 1963. U.S. WeatherBureau,Tech. Papers1-33. Washington,DC: U.S. GovernmentPrinting Office. Ven Te Chow (ed.), Handbook of Applied Hydrology. New York, McGraw-Hill,l9&. Staff, Hydrologic ResearchLaboratory, "National WeatherServiceRiver ForecastSystem ForecastProcedures,"NOAA Tech. Mem. NWS HYDRO 14, National Weather Service,SilverSpring,MD,Dec. 1972. V. Mockus, Sec.4, in SCSNational EngineeringHandbookon Hydrology,Washington, DC: Soil ConservationService, A:ug.1972. J. T. Riedel, J. F. Appleby, and R. W. Schloemer,"SeasonalVariation of the Probable Maximum Precipitation East of the 105th Meridian for Areas from 10 to 1000 Square Miles andDurationsof 6,12,24,and48 Hours,"Hydrometeorological Rept.No. 33, U.S. WeatherBureau,Washington,D.C., 1967.
Chapter3
Interception and Depression Storage
r Prologue The purposeof this chapteris to: . Define interception and depressionstorqge. ' Define the roles theseabstractingmechanismsplay in affectingthe amountof precipitatedwater ultimately availablefor other distribution. ' Providesomeapproachesto estimatingthe quantitiesof waterinterceptedand storedin depressionsduring precipitation events. Figure 1.3indicatesthe pathsthat precipitatedwatermay take asit reachesthe earth. The first encountersare with interceptingsurfacessuch as trees,plants, grass,and structures.Waterin excessof interceptioncapacitythen beginsto filI surfacedepressions.A film of water alsobuilds up overthe ground surface.This is known as surface detention.Once this film is of sufficientdepth, surfaceflow toward definedchannels commences,providing that the rate at which water seepsinto the ground is lessthan the rate of surfacesupply.This chapterdealswith the first two mechanismsby which the grossprecipitation input becomestransformedinto net precipitation.
3.1 INTERCEPTION Part ofthe stormprecipitationthat occursisinterceptedby vegetationandotherforms ofcover on the drainagearea.Interceptioncanbe definedasthat segmentofthe gross precipitationinput which wets and adheresto abovegroundobjectsuntil it is returned to the atmospherethrough evaporation.Precipitation striking vegetationmay be retained on leavesor blades of grass,flow down the stemsof plants and become stemflow,or fall off the leavesto becomepart of the throughfall. The modifying effect that a forest canopycan haveon rainfall intensity at the ground (the throughfall) can be put !o practical use in watershedmanagementschemes. The amountof waterinterceptedis a function of ( 1) the stormcharacter,(2) the species,age,and densityofprevailingplantsandtrees,and (3) the seasonofthe year. Usually about 10-20 percentof the precipitationthat falls during the growing season
41 3.1 INTERCEPTION is interceptedand returned to the hydrologiccycle by evaporation.Water lossesby interceptionare especiallypronouncedunderdenseclosedforest stands-as much as 25 percentof the total annualprecipitation.Schomakerhasreportedthat the average annualinterceptionlossby Douglasfir standsin westernOregonand Washingtonis about24 percent.l A lO-year-oldloblolly pine plantation in the South showedlosses on a yearlybasisof approximately14 percent,while Ponderosapine forestsin California were found to intercept abort 12 percent of the annual precipitation. Mean interception lossesof approximately 13 percent of gross summer rainfall were reported for hardwoodstandsin the White Mountainsof New Hampshire.Additional information given in Table 3.1 includes some data on interception measurements obtainedin Maine from a mature spruce-firstand, a moderatelywell-stockedwhite and gray birch stand,and an improved pasture.r Lull indicatesthat oak or aspenleavesmay retain as much as 100 drops of wdtter.zFor a well-developedtree, interception storageon the order of 0.06 in' of precipitationcould thereforebe expectedon the basisof an averageretentionof about CATCH OFSTANDARD PRECIPITATION TABLE 3,1 WEEKLYAVERAGE IN LOCATED RAINGAUGES BUREAU.WPE U,S.WEATHER A SPRUCE-FIR STAND, A HARDWOOD STAND, AND A PASTUREDURINGTHE WINTER OF 1965_1966
Measuring date'
tl/9t65 11tL6/65 ru23t65 12tr0t65" 12n7 t65 12/30t65 lt4t66 1tr2t66 U18t66 U25t66 2tr/66 2t8t66 2n1t66 2n5t66 2t21t66 3t2t66 3t7t66 3t15t66 3t29t66 Total
Weekly average precipitationcatch (in. ot equivalentrain)
Spruce-fir 0.24 1.01 1.01 1.41 0.55 0.66 0.20 0.36 Trace 0.25 1.38 0.05 0.29, 0.76 0.17 0.86 0.76 0 0.73 10.69
Birch
0.33 r.25 r.23 1.65 0.81 0.95 0.25 0.55 Trace 0.58 l.9l 0.07 0.02 0.81 0.22 l -za
0.84 0 1.13 13.83
Pasture 0.39 L45 r.36 t.79 0.87 1.08 0.26 0.61 Trace 0.59 1.96 0.06 Trace 0.98 0.22 t.45 0.97 0 1.27 15.31
Percent interception by forest cover
Spruce-fir
Birch
38 30 26 2l 37 39 23 4l
l5 l4 10 8 7 l2
58 30 l7
2
A
10
J
t6
4l 22
t7 0 I6 13
43 30.2
1l 9.5
22 ZJ
" The period betweenmeasuringdatesis 7 days,exceptwhen precipitation occurred on the seventhday. In this event, measurgmentwas postponeduntil precipitation ceased. bMeasurementswere delayeduntil a method was devisedto melt frozen precipitation on the site. 'This measurementin the spruce stand was the result of foliage drip during a thaw from previously intercepted snow. "The Effect of Forest and Pastureon the Disposition of Precipitation," Source: Aftet C, E. Schomaker, Maine Farm Res.(July 1966).
42
oHAPTER s rNrER;;-;;rr;-.r,o*
"ro*ou=
20 drops per leaf. For light showers(where gross precipitation P < 0.01 in.) 10t percentinterceptionmight occur, whereasfor showerswhereP > 0.04 in., lossesin the rangeof 10-40 percentare realistic.3 Figure 3.1 illustratesthe generaltime distribution pattern of interceptionloss intensity.Most interceptionlossdevelopsduring the initial stormperiod and the rate of interception rapidly approacheszero thereafter.l-6Potentialitorm interception lossescan be estimatedbv usins2'3,6 Zi:,S+KEt
(3.1)
where . L, : the volume of water intercepted(in.) S : the interceptionstoragethat will be retainedon the foliage againstthe forcesof wind and gravity (usuallyvariesbetween0.01 and 0.05 in.) K : the ratio of surfaceareaof interceptingleavesto horizontal projection:, of this area E : the amountof waterevaporatedper hour duringthe precipitationperiod ' (in.) : r time (hr)
i= it+i2+4+i4
Figure 3.1 Disposition of rainfall input in terms of interception, depressionstorage,infiltration, and overlandflow.
3.1 INTERCEPTION 43 Equation3.1 is basedon the assumptionthat rainfall is sufficientto fully satisfy the storageterm S. The following equationwas designedto accountfor the rainfall amountT-e L;:S(1 -e-P/s)+KEt
(3.2)
whereP : rainfall and e is the baseof natural logarithms.Note in Eqs. 3.1 and 3.2 that the storm time duration t is given in hours, while ,L,, S, and E are commonly measuredin in. or mm. It is important to recognizethat forms of vegetationother than trees can aiso interceptlarge quantitiesof water. Grasses,crops,and shrubsoften haveleaf-areato ground-arearatios that are similar to thosefor forests.Table3.2 summarizessome observationsthat havebeen madeon crops during growing seasonsand on a variety of grasses.Interceptedamountsare aboutthe sameasthosefor forests,but sincesome of thesetypes of vegetationexist only until harvest,their annualimpact on interception is generallylessthan that of forestedareas. Precipitationtype, rainfall intensityand duration,wind, and atmosphericconditions affecting evaporationare factors that serve to determineinterception losses. Snow interception,while highly visible,usually is not a major loss sincemuch of the interceptedsnowfall is eventuallytransmittedto the ground by wind action and melt. Interceptionduring rainfall eventsis commonly greaterthan for snowfall events.In both cases,wind velocity is an important factor. The importanceof interceptionin hydrologicmodelingis tied to the purposeof the model. Estimates of loss to gross precipitation through interception can be significantin annual or long-term models,but for heavy rainfalls during individual storm events,accountingfor interceptionmay be unnecessary.It is important for the modeler to assesscarefully both the time frame of the model and the volume of precipitation with which one must deal. TABLE3.2 OBSERVED PERCENTAGES OF INTERCEPTION BY VARIOUSCROPSAND GRASSES'
Vegetation type Crops Alfalfa Corn Soybeans Oats Grassesb Little bluestem Big bluestem Tall panic grass Bindweed Buffalo grass Blue grass Mixed species Natural grasses
Intercepted(%)
Comments
36 16 15 7 50-60-.1
s7 l s 7 f 17 l ) 31 17
Water applied at rate of ] in. in 30 min
Pdor to harvest
)A
t4-t9
"Valuesroundedto nearestpercent.Data for table were obtainedfrom Refs.2,4, and 5, 'Grass heightsvary up to about 36 in.
44
CHAPTER3
INTERCEPTIONAND DEPRESSIONSTORAGE
Equations3.1 and 3.2 canbe usedto estimatetotal interceptionlosses,but for detailedanalysesofindividual storins,it is necessaryto dealwith the areal variability of suchlosses.Generalequationsfor estimatingsuchlossesare not available,however. Most researchhas been related to particular speciesor experimentalplots strongly associatedwith a given locality. In addition, the lossfunction varieswith the storm's character.If adequateexperimentaldata are available,the nature of the varianceof interceptionversustime might be inferred. Otherwise,common priictice is to deduct the estimatedvolumeentirelyfrom the initial period of the storm(initial abstraction). EXAMPLE 3.1 Using the following equationsdevelopedby Horton6for interceptionby ash and oak trees, estimatethe interception loss beneaththesetrees for a storm having a total precipitationof 1.5 in. Solution 1. For ashtrees.
L ; : 0 . 0 1 5+ 0 . 2 3 P : 0.36in. : 0.015+ 0.23(1.5) 2. For oak trees,
L;:0.03+0.22P : 0.36in. rl : 0.03+ 0.22(1.5)
3.2 THROUGHFALL A numberof relationshipsfor estimatingthroughfall for a variety of foresttypeshave been developed.ntt Deiermining factors for throughfall quantities include canopy coverage,total leaf area,numberand type of layersof vegetation,wind velocity, and rainfall intensity.The arealvariability ofthesefactorsresultsin little or no throughfall in somelocationsand considerablethroughfall in others.In general,prediction equations for throughfall mustincludemeasuresof canopysurfaceareaand coverasprime variables.An example of a throughfall relationship for an easternUnited States hardwoodforest follows.l2 For the growing season T n : 0 . 9 0 1 P- 0 . O 3 l n
(3.3)
For the dormant season T n : 0 . 9 I 4 P- 0 . 0 1 5 n where ?1,: throughfall (in.) P : total precipitation (in.) n : number of storms
(3.4)
3.3
DEPRESSIONSTORAGE
45
3.3 DEPRESSION STORAGE Precipitationthat reachesthe ground may infiltrate, flow over the surface,or become trappedin numeroussmall depressionsfrom which the only escapeis evaporationor infiltration. The natureof depressions, aswell astheir size,is largely a funition of the original land form and local land-usepractices.Becauseof extremevariability in the nature of depressionsand the paucity of sufficient measurements,no generalized relation with enoughspecifiedparametersfor all casesis feasible.A rational model can, however,be suggested. Figure 3.1 illustratesthe dispositionof a precipitationinput. A studyof it shows that the tate at which depressionstorageis filled rapidly declinesafter the initiation of a precipitationevent.Ultimately, the amountof precipitation goinginto depression storagewill approachzero,given that thereis alargeenoughvolume of precipitation to exceedother lossesto surfacestoragesuch as inflltration and evaporation.Ultimately, all the water stored in depressionswill either evaporateor seep into the ground.Finally, it shouldbe understoodthat the geometryof a land surfaceis usually complex and thus depressionsvary. widely in size, degreeof interconnection,and contributingdrainagearea.In general,depressionsmay be looked upon as miniature reservoirsand as suchthey are subjectto similar analytical techniques. According to Linsley et a1.13 the volume of water storedby surfacedepressions at any given time can be approximatedusing Y:Sd(l
-e-kP")
r? 5)
where V : the volume actually in storageat sometime of interest S, : the maximum storagecapacityof the depressions P" : the rainfall excess(grossrainfall minus evaporation,interception,and infiltration) k : a constantequivalentto l/So The valueof the constantcan be determinedby consideringthat if P" : 0, essentially all the water will fill depressionsand dv/dp" will equal one. This requires that k : r/Sa. Estimatesof s, may be securedby making samplefleld measurementsof the areaunder study.Combiningsuchdata with estimatesof P" permits a determination of V. The mannerin which Vvarieswith time must still be estimatedif depression storagelossesare to be abstractedfrom the grossrainfall input. One assumptionregarding dVldt is that all depressionsmust be full before overlandflow supply begins.Actually, this would not agreewith reality unlessthe locationsof depressionswere gradedwith the largestonesoccurring downstream.If the depressionstoragewere abstractedin this manner, the total volume would be deductedfrom the initial storm period suchas shownby the shadedareain Fig.3.2. Such postulateshave been used with satisfactory results under special circumstances.ra Depressionstorageintensitycan alsobe estimatedusingEq. 3.5. If the overland flow supplyrale oplus depressionstorageintensityequali - /, wherei is the rainfall intensity reachingthe ground and/is the infiltration rate, then the ratio of overland flow supply to overlandflow plus depressionstoragesupply can be proved equal to
i - f
(3.6)
46
CHAPTER 3 INTERCEPTION ANDDEPRESSION STORAGE
o b0 o
!t o
E
d
4
8
1
2
16
20
Time (min)
Figure3.2 Simpledepression storageabstraction scheme. This expressioncan be derivedby adjudging c : i - f - o (3.7) i - f i - f and noting that o is equal to the derivativeof Eq. 3.5 with respectto time. Then )
o:fiso1t_e-kP")
(3.8)
u : (Soke-kg#
(3.e)
It was shown that k : 1/S, so that
u : ,-o'"d!" dt
(3.10)
The excessprecipitationP, equalsthe grossrainfall minus infiltrated water, and since the derivativewith respectto time canbe replacedby the equivalentintensity(i * f), the intensity of depressionstoragebecomes o:e-or.(i-f)
a
i - f
(i-f)*G-f)e-."" i - f
(3.11)
(3.r2)
o.125
o.25
0.50(turf)
0.315
Mass overlandflow and depressionstoragesupply ( P - F)
0.0625 All depressions filled before overland flow supply begins -= :\
----l
0.0938
0.125(pavements) 1.00
I
on
d
-
9)-
RO
0.80
E
l-
bl I
o "
7
n
Exponential relationship a
tl
i,t
-\
-lP-F)tS, " = -I - e I
F
I
,
i 6 0 6
o b!
OGEE sumrnationof the standardprobability curve
o
g
n5n
a
B o E
a
{ .:
-
o.7o I
-
40
6
B
9 d J U
o
o o0
s2 o
6zo
I o
!)
E
t
o
o
0
50
100
150
200
Mean depth as a percentage of overall depth of depression storage
Figure 3.3 Depth distributioncurve ofdepressionstorage.Enter graph from top, readdown to selectedcurve, and project right or left as desired.(After Tholin and Kiefer.r5) and
o
i * f
-
:(i
:
f)(I
- e-kP")
i - f |
-
g-kP"
(3.13) (3.r4)
Figure 3.3 illustratesa plot of this function versusthe massoverlandflow and depressionstoragesupply(P - F), whereF is the accumulatedmassinfiltrationl5 and
48
CHAPTER3
AND DEPRESSION STORAGE INTERCEPTION
P is the grossprecipitation.In the plot meandepthsof 0.25 in. for turf and 0.0625in. for pavementswere assumed.Maximum depthswere 0.50 and0.125in., respectively. The figure also depictsthe effect on estimatedoverlandflow supplyrate, which is derivedfrom the choiceof the depressionstoragemodel. Three modelsare shown in the figure: the flrst one assumesthat all depressionsare full before overlandflow begins.For a turf area having depressionswith a mean depth of 0.25 in., the figure showsthat for P - F valueslessthan 0.25 in., thereis no overlandflow supply,while for P - F valuesgreater than 0.25 in., the overlandflow supply is equal to i - f . For the exponentialmodel (Model 2), c alwayswill be greaterthanzero. Tholin and Kiefer haverecommendedthat a relation betweenthosepreviouslymentionedis A cumulative normal likely more representativeof fully developedurban areas.15 describedin Fig. 3.3 and is also probability curve was selectedfor this representation (Model 3). Depressionstoragedeductionsare usually madefrom the first part of the storm as illustrated in Fig. 3.2. The amount to be deductedis a function of topography, groundcover,and extentand type of land development.During major storms,this loss is often consideredto be negligible.Someguidelinesfor estimatingdepressionstorage losseshavebeen developedbasedon studiesof experimentaland other watersheds. Values for depressionstoragelossesfrom intense storms reported by Hicks are Tholin andKieferhaveused 0.20 in. for sand,0.15in. for loam,and0.10in. for c1ay.16 Studiesof in. forpavements.ls 0.0625 and valuesof 0.25 in. inpervious urban areas yielded information shownin the by Viessman four small imperviousdrainageareas This is slope. correlated with loss is highly Fig.3.4, where mean depressionstorage
0.15
I Ei
U.IU
0
1
2
3
4
Slope(70)
Figure 3.4 Depression storage loss versus slope for four impervious drainageareas.(From Viessman.ra)
PROBLEMS
49
h
oo
Antecedent rainfall during preceding 30 min o
Time (min)
Figure3.5 Depression storage intensityversustimefor animpervious area.(After Turner.l7)
easilyunderstood,sincea given depressionwill hold its maximum volumeif horizontally oriented. Using very limited data from a small, paved-streetsection, Turner devisedthe curvesshownin Fig. 3.5.17Other sourcesof datarelatedto surfacestorase are availablein the literature.2,18,1e
r Summary Accountingfor the dispositionof precipitation is an important part of the hydrologic modelingprocess.Two abstractionsfrom the precipitation input, intercepiion, and depressionstoragewere coveredin this chapter. Interception lossesduring the courseof a year may be substantial,but during intensestorms,they may be sufficiently small to neglect.Precipitationtype, rainfall intensity and duration, wind, and atmosphericconditions affeiting evaporationare factorsthat serveto determineinterceptionlossesfor a given foresi standor ground coverconfiguration.Interceptionduring rainfall eventsis commonly greaterihan for snowfall events. Depressionstoragedeductionsoccur early in a storm sequenceand they are a function of topography,ground cover, and extent and type of land development. During major storms,this loss is often consideredto be negligible.
PROBI.EMS 3.1. UsingFig. 3.2, estimatethe volume of depressionstoragefor a 3-acrepaveddrainage area' Statethe volume in cubic feet and cubic meters.Convert it to equivalentdep;h over the area in in. and cm.
50
CHAPTER3
INTERCEPTIONAND DEPRESSIONSTORAGE
3.2. Estimatethe percentageof the total volume of rainfall that is indicatedas depression storagein Fig. 3.2. 3.3. Using the averageannual precipitationfor your state,estimatethe annual amountof interceptionloss. 3.4. Refer to Fig. 2.4 and estimate the annual interception lossesin lllinois, Florida, California, and New Mexico. How good do you think theseestimatesare?In which estimatesdo you havethe most confldence?Why? In which of thesestateswould the water budgetbe most affectedby interception? 3.5. Using Fig. 3.4, estimatethe percentageof rainfall that would be lost to depression storagefor a l0-acre parking lot havinga mean slopeof 1 percent.Repeatfor a slope of 3 percent.Using the total rainfall volume determinedin Problem 3.2, estimatethe equivalentdepth over the area of the depressionstorageloss for both slopes.Stater depthsin mm and in. 3.6. Refer to Fig. 3.3 and estimatethe ratio of overlandflow supplyto overlandflow and depressionstoragesupplyif the areais turf, the OGEE summationcurve is the model, and the mean depth of depressionstorageis (a) 75 percent and (b) 125 percent. 3,7. Explain how a relation such as that given in Fig. 3.3 could be used in a simulation model of the rainfall-runoff process' 3.8. Using Eqs. 3.3 and 3.4, estimatethe throughfall in in. for 28 in. of rainfall during the growing season(21 events), and 17 in. of rainfall during the dormant season(13 events). 3.9. Using Horton's equationsgiven in Example 3.1, estimatethe interceptionlossesby ash and oak trees for a storm having a total precipitation of 1.33 in'
REFERENCES "The Effect of Forestand Pastureon the Dispositionof Precipitation," 1. C. E. Schomaker, Maine Farm Res.(July 1966). 2. ven Te chow (ed.), Handbook of Apptied Hydrology. New York: McGraw-Hill,1964. 3 . JosephKittredge, ForestInfluences.New York: McGraw-Hill' 1948. "Interception of Rainfall by HerbaceousVegetation,"Science86(2243), i + . O. n. Ctark,
59r-s92(r937).
"Resultsof the Mountain Home Rainfall Interceptionand Inflltration Project 5. J. S. Beard, on Black Wattle, 1953-1954," J. S. Afr. Foresty Assoc'27,72-85(1956)' "Rainfall Interception," Monthly WeatherRev.47,603-623(L9I9)' 6. R. E. Horton, "A Note on the InterceptionLoss Equation," J. Geoplrys.Res.65' 38507. R. A. Meriam, 1 ( 1 9 6 0 ) . 385 8. D. M. Gray (ed.),Hand.bookon the Principles of Hydrology.National ResearchCouncil, Canada,Port Washington:WaterInformation Center,Inc., 1973. 9. K. N. Brooks, P. F. Folliott, H. M. Gregersen,and J. L. Thames,Hydrology and the Managementof Watersheds.Ames, IA: Iowa StateUniversity Press/Ames,1991. "The InterceptionProcess."In Prediction in CatchmentHydrology,National 10. G. J. Blake, Symposiumon Hydrology,eds.T. G. Chapmanand F. X. Dunin, MelbourneAust. Acad. S c i . , 5 9 - 8 1 1, 9 7 5 . "Throughfall in PlantedStandsof Fourth SouthernPines 11. F. A. Roth, II, and M. Chang, Bulletin 17' 880-885(1981) Speciesin EastTexas,"WaterResources
REFERENCES 51 "canopy and Litter Interceptionby Hardwoodsof Eastern J. D. Helvey and J. H. Patric, Res.l, 193-206(1965)' Resour. United States,"Water New York: R. K. Linsley, Jr., M. A' Kohler, and J. L. H. Paulhus,Apptied Hydrology' McGraw-Hill, 1949. "A Linear Model for synthesizingHydrographsfor Small Drainage warren viessman,Jr., presented at the Forty-eighthAnnual Meetingof the AmericanGeophysical paper Areas," D.C., APr. 1967. Union, Washington, "The Hydrology of Urban Runoff," Trans. ASCE 125, A. L. Tholin una C. J. Kiefer, 1 3 0 8 -1 3 7 91( 9 6 0 ) . ..A Method of Computing Urban Runoff ,', Trans' ASCE |09, I2L,7W. I. Hicks,
Aue. 1966.
Chapter4
lnfiltration
Prologue The purposeof this chapteris to: . Defineinfiltration. ' Indicatethe role infiltration playsin affectingrunoffquantities and in replenishing soil moistureand groundwaterstorages. ' Presentmodelsfor estimatinginflltration and provide examplesof how they can be used. Infiltration is that processby which precipitation movesdownwardthrough the surface of the earth and replenishessoil moisture, rechargesaquifers,and ultimately supportsstreamflowsduring dry periods.Along with interception,depressionstorage, and stormperiodevaporation,it determinesthe availability,if any,of the precipitation input for generatingoverland flows (Fig. 1.3). Furthermore,infiltration rates influence the timing of overland flow inputs to channelizedsystems.Accordingly, infiltration is an important componentof any hydrologicmodel. The ratef at which infiltration occursis influencedby suchfactorsas the type and extent of vegetalcover, the condition of the surfacecrust, temperature,rainfall intensity,physicalpropertiesof the soil, and water quality. The rate at which wateris transmittedthrough the surfacelayeris highly dependent on the condition of the surface.For example,inwashof fine materialsmay seal the surfaceso that infiltration ratesare low evenwhen the underlyingsoils are highly permeable.After water crossesthe surfaceinterface,its rate of downwardmovement is controlled by the transmissioncharacteristicsof the underlying soil profile. The volume of storageavailablebelow ground is also a factor affecting infiltration rates. Considerableresearchon infiltration hastakenplace,but consideringthe infinite combinationsof soil and other factors existing in nature, no perfectly quantified generalrelation exists.
OF INFILTRATION 53 4.2 CALCULATION
4.1 MEASURING INFILTRATION
.
Commonly usedmethodsfor determininginfiltration capacityare hydrographanalysesandinfiltrometer studies.Infiltrometersareusuallyclassifiedasrainfall simulators or flooding devices.In the former, arlificalrainfall is simulatedover a small test plot and the inhltration calculatedfrom observationsofrainfall andrunoff, with consideration given to depressionstorageand surfacedetention.lFlooding infiltrometers are usually rings or iubes insertedin the ground. Water is applied and maintainedat a constantlevel and observationsmade of the rate of replenishmentrequired' Estimatesof infiltration basedon hydrographanalyseshavethe advantageover infiltrometers of relating more directly to prevailing conditions of precipitation and field. However,they areno betterthan the precisionwith which rainfall andrunoff are measured.Of partitular importancein suchstudiesis the areal variability of rainfall. Several meth;ds have been developedand are in use. Reference1 gives a good descriptionof thesemethods.
OF INFILTRATION 4.2 CALCULATION Infiltration calculationsvary in sophisticationfrom the applicationof reported averageratesfor specificsoil types and vegetalcoversto the useof differential equations g6verningthe flow of wateiin unsaturatedporousmedia.For small urban areasthat iespondiapidly to storm input, more precisemethodsare sometimeswarranted'On large waterlhedssubjectto peak flow production from prolongedstorms,averageor representativevaluesmay be adequate. The infiltration pto"".r is -omplicated at best. Even under ideal conditions (uniform soil propertiei andknown fluid properties),conditionsrarely encounteredin practice,the processis difflcult to characterize.Accordingly,therehasbeenconsiderabtestudyof the infiltration process.Most of theseeffortshaverelatedto the development of i1) empirical equationsbasedon field observationsand (2) the solution of equationsbasedon the mechanicsof saturatedflow in porous media.l'2 Later in this chapter,severalcommonly usedinfiltration modelsare discussed. As a prefaceto that discussion,a brief descriptionof the infiltration processfollows' It reviews the principal factors affecting infiltration and points out some of the problemsencounteredby hydrologicmodelers. Webeginour discussionwith an idealcase,onein which the soil is homogeneous throughout the profile and all the pores are directly interconnectedby capillary purru!"r. Furtheimore,it is assumedthat the rainfall is uniformly distributedovef the ur"u of "on"ern. Undertheseconditions,the infiltration processmay be chatactetized as one dimensional and the major influencing factors are therefore soil type and moisturecontent.3 through which The soil type characterizesthe sizeand numberof the passages and relative potential capillary the sets content moisture the the watermustflow while capillary to due head hydraulic the is potential Capillary conductivity of the soil. sign. opposite with potential but capillary as same is the forces. capillary suction
54
CHAPTER4
INFILTRATION
9 aoo
4
Ei o
. 300 -. d
d
t
2oo
0
0.1
0.2
0.3
0.4
0.5
Moisture content, 0 (vol/vol)
Figure 4.1 Typical capillary suction-relativeconductivity-moisture contentrelation.(AfterMein andLarson.e.) Capillary conductivity is the volume rate of flow of water through the soil under a gradient of unity (dependenton soil moisture content). Relative conductivity is the capillary conductivity for a specifiedmoisturecontent divided by the saturatedconductivity. Figure 4.1 illustratesthe relations amongthesevariables.Note that at low moisturecontents,capillary suctionis high while relative conductivityis low. At high moisture contentsthe reverseis true. With this background,an infiltration event can be examined.Consider that rainfall is occurringon an initially dry soil. As shownin Fig. 4.l,the relative conductivity is low at the outsetdue to the low soil moistureconditions.Thus, for the water to move downwardthrough the soil, a higher moisture level is needed.As moisture builds up, a wetting front forms with the moisturecontentbehindthe front beinghigh (essentiallysaturated)and that aheadof the front being low. At the wetting front, the capillary suctionis high due to the low moisture content aheadof the front. At the beginning of a rainfall event, the potential gradient that drives soil moisture movementis high becausethe wetting front is virtually at the soil surface. Initially, the infiltration capacity is higher than the rainfall rate and thus the infiltration rate cannot exceedthe rainfall rate. As time advancesand more water entersthe soil, the wetting zone dimensionincreasesand the potential gradient is reduced.Infiltration capacitydecreases until it equalsthe rainfall rate. This occursat the time the soil at the land surfacebecomessaturated.Figures4.2 and4.3 illustrate
l
4.2
CALCULATIONOF INFILTRATION
55
Moisturecontent.d
d
B
a
I with a constantrainfall Figure4.2 Typicalmoistureproflledevelopment rate. these conditions. Figure 4.2 showshow a moisture profile might developwhen a rainstormofconstant intensityoccurs.In the diagramthe soil moistureat the surface is shownto rangefrom its initial value at the top left to its saturatedvalue at the top right. Thus in moving downwardon the left-handside of the diagram,one can trace the downward progressionof the wetting front for varying levels of soil moisture
92
::
Time,r Figure 4.3 Infiltration rate versustime for a given rainfall intensity.(After Mein and Larson.e)
56
CHAPTER4
INFILTRATION
contentat the land surface.Figure 4.3 indicatesthat until saturationis reachedat the surface,the infiltration rate is constantand equalto the rainfall applicationrate at the surface.At Point 4, apoint that coriespondsto the time at which saturationoccursat the surface,the infiltration rate beginsto proceedat its capacity rate, the maximum rate at which the soil can transmit water acrossits surface.As time goes on, the infiltration capacity continues to decline until it becomesequal to the saturated conductivity of the soil, the capillary conductivity when the soil is saturated.This ultimate infiltration rate is shown by the dashedline to the right of K" in Fig. 4.3. Of particular interest is the determinationof Point 4 on the curve of Fig. 4.3. This is the point at which runoff would beginfor the conditions specifiedabove.It is also the point at which the actual infiltration rate/becomes equal to the infiltration capacityratefo ratherthan the rainfall intensityrate i. The time of occurrenceof this point depends,for a given soil type, on the initial moisture content and the rainfall rate. The shapeof the infiltration curve after this point in time is also influencedby thesefactors. Another factor that must be reckonedwith in the infiltration processis that of hysteresis. In Fig. 4.1 it can be seenthat the plot of capillary suctionversussoil moistureis a loop. The curve is not the samefor wetting and drying of the soil. The curves shown on the figure are the boundary wetting and boundary drying curves, curves applicableunder conditions of continuouswetting or drying. Betweenthese curves, an infinite number of possiblepaths exist that dependon the wetting and drying history of the soil. A numberof approachesto the hysteresisproblemhavebeen reportedin the literature.3 The illustration of the infiltration processpresentedwas basedon an ideal soil. Unfortunately,suchconditionsare not replicatedin natural systems.Natural soils are highly variable in composition within regions and soil cover conditions are also far-ranging. Becauseof this, no simpleinfiltration model can accuratelyportray all the conditionsencounteredin the fleld. The searchhas thus beenfor modelsthat can be called upon to give acceptableestimatesof the rates at which infiltration occurs durine rainfall events. Mein and Larson have describedthree generalcasesof infiltration associated with rainfall.3The first caseis one in which the rainfall rate is lessthan the saturated conductivity of the soil. Under this condition, shownas (4) in Fig. 4.4, runoff never occurs since all the rainfall infiltrates the soil surface.Nevertheless,this condition mustbe recognizedin continuoussimulationprocessessincethe level of soil moisture is affectedeven though runoff doesnot occur. The secondcaseis one in which the rainfall rate exceedsthe saturatedconductivity but is lessthan the infiltration capacity. Curves(I), (2), and (3) of Fig. 4.4 illustratethis condition.It shouldbe observed that the period from the beginningof rainfall to the time of surfacesaturationvaries with the rainfall intensity.The final caseis one in which the rainfall intensityexceeds the infiltration capacity.This condition is illustratedby the infiltration capacitycurve of Fig. 4.5 andthoseportionsof infiltration curves(l), (2), and (3) of Fig. 4.4 that are in their declining stages.Only under this condition can runoff occur. All three caseshaverelevanceto hydrologicmodeling,particularly when it is continuousover time.
4.3
tsal,
tsatt
HORTON'SINFILTRATIONMODEL
57
tsat,
Time, t
Figure 4.4 Inflltration curves for several rainfall intensities.(After Mein and Larson.e)
fo: f"+ Ao-f")"u' Infiltration capacitY curve
f, 0
o
,r*"
Figure 4.5 Horton's infiltration curve and hyetograph.
MODEL INFILTRATION .I3 HORTON'S The inflltration processwas thoroughly studiedby Horton in the early 1930s.oAn outgrowth of his work, shown graphically in Fig. 4.1, was the following relation for determininginfiltration capacity: (+.r; fo: f, + ("fr f")e'n'
58
CHAPTER4
INFILTRATION
where fo : k : f" : ,fo :
the infiltration capacity(depth/time) at sometime / a constantrepresentingthe rate of decreaseinf capacity d final or equilibrium capacity the initial infiltration capacity
It indicatesthat if the rainfall supply exceedsthe infiltration capacity,infiltration tendsto decreasein an exponentialmanner.Although simplein form, difficulties in determininguseful valuesfor/. and fr restrict the useofthis equation.The areaunder the curve for any time interval representsthe depth of water infiltrated during that interval. The infiltration rate is usually given in inches per hour and the time r in minutes,althoughother time incrementsare usedand the coefficientk is determined accordingly. By observingthe variation of inflltration with time and developingplots of / versus/ asshownin Fig. 4.5,we canestimatefsandft. Two setsof/and / are selected from the curve and enteredin Eq. 4.1. Two equationshavingtwo unknownsare thus obtained;they can be solvedby successiveapproximationsforfi and k. Typical infiltration ratesat the end of t hr ( f) areshownin Table4.1. A typical relation betweenf, and the infiltration rate throughout a rainfall period is shown graphically in Fig. 4.6a; Fig.4.6h showsan infiltration capacity curve for normal antecedentconditionson turf. The data given in Table4.1 are for aturf areaand must be multiplied by a suitablecover factor for other types of cover complexes.A range of cover factorsis listed inTable 4.2. Total volumes of inflltration and other abstractionsfrom a given recorded rainfall are obtainablefrom a dischargehydrograph(plot of the streamflowrate versus time) if one is available.Separationof the base flow (dry weather flow) from the dischargehydrographresultsin a direct runoff hydrograph(DRH), which accountsfor the direct surfacerunoff. that is. rainfall less abstractions.Direct surfacerunoff or precipitation excessin inchesuniformly distributedover a watershedcan readily be calculatedby picking valuesof DRH dischargeat equal time incrementsthrough the hydrographand applyingthe formulas P": where P" : 4r : A : r?7:
(0.0371e)() q')
( a) \
Ano
precipitation excess(in.) DRH ordinatesat equal time intervals (cfs) drainagearea(mi2) hurrber of time intervalsin a 24-hr period
For most casesthe differencebetweenthe original rainfall and the direct runoff can be consideredas infiltrated water. Exceptionsmay occur in areasof excessive subsurfacedrainageor tracts of intensiveinterceptionpotential.The calculatedvalue of infiltration can then be assumedasdistributedaccordingto an equationof the form of Eq. 4.1 or it may be uniformly spreadover the stormperiod. Choiceof the method employeddependson the accuracyrequirementsand size of the watershed. To circumvent some of the problems associatedwith the use of Horton's infiltration model,someadjustmentscanbe made.6ConsiderFig. 4.5. Notethat where the infiltration capacitycurve is abovethe hyetograph,the actual rate of infiltration
MODEL 4.3 HORTON'SINFILTRATION
h
o - 1
q
1
1.0 Time (hr) (a)
d
h
U
3.0 2.8
-TT
2.O
-
2.4 2.2 2.0 1.8 1.6 t.4 7.2 1.0 0.8 0.6 0.4 0.2
I
,4 f=0.53+2.4
l
l
l
0691ty
l, = U.UUUUJTIu . ) 9 ( l
l
e fle rOa"
9l
r\$4"
o\
rras!
-iro$dts"
l
l
l
l
l
l
Infiltration capacity curve (/.
0 10
70 80
100
t20
140
160
t80
Time (min) from beginning of infiltration capacity cuve' t/ (b)
Figure 4.6 (a) Typical infiltration curve. (b) Infiltration capacity and mass curvesfor normal antecedentconditionsof turf areas.fAfter A. L. Tholin and "The Hydrology of Urban Runoff," Proc. ASCE J. Sanitary Clint J. Kiefer, Ens. Div. S4(SA2),56 (Mar. 1959).1
59
60
CHAPTER4
INFILTRATION
TABLE 4.1 ryPICAL f, VALUES
Soilgroup
f, (in./hr)
0.50-1.00 0.10-0.50 0.01-0.10
High (sandysoils) Intermediate(loaps, clay, silt) Low (clays,clay loam)
f' (mm/h)
t2.50-25.00 2.so-r2.50 0.25-2.50
Source: After ASCE Manual of Engineering Practice,No.28.
TABLE4,2 COVERFACTORS Cover
Cover fac{or
Permanentforest and grass
Good(1 in. humus) Medium(f-1 in. humus; Poor(< j in. humus)
Close-growingcrops
Good Medium Poor Good Medium Poor
Row crops
3.0-7.5 2.0-3.0 1.2-1.4 2.5-3.0 t.6-2.0 1.11 - .3 1 . 3 -1 . 5 1.1-1 3 1 . 0 -1 . 1
Source: After ASCE Manual of Engineering Practice, No.2t.
is equal to that of the rainfall intensity, adjustedfor interception,evaporation,and other losses.Consequently,the actuafinfiltration is given by
f(t) : minlfof), i(t)l
(4.3)
where/(r) is the actual infiltration into the soil and i(l) is the rainfall intensity.Thus the infiltration rate at any time is equal to the lesserof the infiltration capacity,f,(t) or the rainfall intensity. Commonly,the typical valuesof foandf" are greaterthan the prevailingrainfall intensitiesduring a storm. Thus, when Eq.4.l is solvedforS as a function of time alone, it shpwsa decreasein infiltration capacity even when rainfall intensitiesare much lessthanfo. Accordingly,a reductionin infiltration capacityis maderegardless of the amount of water that entersthe soil. To adjust for this deficiency,the integratedform of Horton's equationmay be used,
F(tp):
lr'' ,
o, : f,tp +
(t - a' *,,1
(4.4)
whereF is the cumulativeinfiltration at time to, as shownin Fig. 4.7.rnthe figure, it is assumedthat the actual infiltration has been equal to$. As previouslynoted, this is not usually the case,and the tnie cumulativeinfiltration must be determined.This
MODEL 4.3 HORTON'SINFILTRATION
0 0
61
tptpt tl Equivalent time
Figure4.7 Cumulativeinfiltration. can be done using F\t) :
l,',u,o,
(4.s)
where/(r) is determinedusingF,q.4.3. Equations4.4 and4.5 may be us6djointly to calculatethe time t, that is, the equivalent time for the actual infiltrated volume to equal the volume under the infiltration capacity curve (Fig. 4.7). The actual accumulatedinfiltration given by Eq. 4.5 is equatedto the area under the Horton curve, F,q. 4.4, and the resulting expressionis solvedfor r' This equation, F:fJrt
(l-e-k'n)
(4.6)
cannotbe solvedexplicitly for to,but an iterativesolutioncan be obtained.It should be understoodthat the time to is lessthan or equal to the actual elapsedtime r. Thus the availableinfiltration capacityas shownin Fig. 4.7 is equalto or exceedsthat given described,fbecomesa functionof the actual by Eq. 4.1. By makingthe adjustments amount of water infiltrated and not just a variable with time as is assumedin the original Horton equation. In selectinga model for use in inflltration calculations,it is important to know its limitations. In somecasesa model can be adjustedto accommodateshortcomings; are not realisticfor the natureof the useproposed, in othercases,if its assumptions the model shouldbe discardedin favor of anotherthat better fits the situation. Part One of this book deals with the principal componentsof the hydrologic cycle. In later chapters,the emphasisis on putting these componentstogetherin
62
CHAPTER4
INFILTRATION
varioushydrolqgicmodelingprocesses.When thesemodelsare designedfor continuous simulation,the approachis to calculatethe appropriatecomponentsof the hydrologic equation,Eq. 1.4, continuouslyovertime. A discussionof how infiltration could be incorporatedinto a simulation model follows. It exemplifiesthe use of Horton's equationin a storm water managementmodel (SWMM)." First, an initial value of /o is determined.Then, consideringthat the value of$ dependson the actualamountof infiltration that hasoccurredup to that time, a value of the averageinfiltration capacity,fo, availableover the next time stepis calculated using
f o : * 1 " = ' *fo' o' ' : l W
(4.7)
-tp
Equation 4.3 is then usedto find the averagerate of infiltration, /.
'v - [f, tt
iti >1, iri
(4.8)
where i is the averagerainfall intensity over the time step. Following this, infiltration is incrementedusing the expression F(t + Lt) : F(t) * AF : F(t) + f Lt
(4.e)
whereAF : f Lt is the addedcumulativeinfiltration (Fig.4.7). The next stepis to find a new valueof ro.This is doneusingEq. 4.6.If LF : f o Lt, tp : to t Ar. But if the new /oris lessthanto + A/ (seeFig. 4.7),Eq. 4.6 must be solvedby iteration for the new value of lo. This can be accomplishedusing the Newton-Raphsonprocedure.6 When the valueof tp > I6f k, the Horton curve is approximatelyhorizontal and : f". Once this point hasbeenreached,there is no further needfor iteration since fo and equal to f" and no longerdependenton F. is constant f EXAMPLE 4.I
Givenan initial inflltrationcapacityfiof 3.0 in./hr anda time constantft of 0.29hr-', derive an infiltration capacity vs. time curve if the ultimate infiltration capacity is 0.55 in./hr. For the first ten hours, estimatethe total volume of water infiltrated in inchesover the watershed. Solution. Using Horton's equation(4.1), valuesof infiltration can be computed for various times. The equationis as follows: f : f" + Lfo + f")e-k, Substitutingthe appropriatevaluesinto the equationyields ,f = 0.55 + (3.0 0.55)s-o'zo' Table4.3, valuesof f atecomputedand Then for the times shownin spreadsheet enteredinto the table. Using the spreadsheetgraphics package,the curve of Fig. 4.8 is derived,
4,3 HORTON'SINFILTRATIONMODEL
63
TABLE 4.3 CALCULATIONSFOR EXAMPLE4.1
lnfiltration (in./h0
Time (hr)
Time (hr)
lnfiltration (in./hr)
0.00
3.00
5.00
t.l2
0.10
2.93
6.00
0.98
0.25
2.83
7.00
0.87
0.50
2.67
8.00
0.79
1.00
2.38
9.00
0.73
2.00
t.92
10.00
0.68
3.00
1.58
15.00
0.58
4.00
r.32
20.00
0.56
4.1 can To find the volumeof waterinfiltrated duringthe flrst 10hours,Eq' be integratedover the range of 0-10 v :
I J
[0.5s + (3.0 0.55)e-o2e' ldt
[0.55/ + (2.45I-0.29)e-o2s\o : 12.47in' V
Y:
The volume in inchesover the watershedis thus 12'47 in'
ll
3.0
\ ;
2.0
6 .:
15
l?
-\-\
1.0
"'-0 0.5
2
4
6
L 8
10
Time (hr) Figure 4.8
GraPh for ExamPle 4'1.
12
t4
18
64 4.4
CHAPTER4
INFILTRATION
MODEL GREEN-AMPT The Green-Ampt infiltration model,originallyproposedin 1911,has had a resurgenceof inte1gs1.3'6-1t This approachis basedon Darcy's law (seeChapter18).In its original form, it was intendedfor use where infiltration resultedfrom an excessof water at the ground surfaceat all times. In 1973, Mein and Larson presenteda methodologyfor applying the Green-Ampt model to a steadyrainfall input.eThey alsodevelopeda procedurefor determiningthe valueof the capillary suctionparameter usedin the model. In 1978,Chu demonstratedthe applicability of the model for useunder conditionsof unsteadyrainfall.lo As a result of theseand other efforts, the Green-Ampt model is now employedas an option in such widely used continuous simulation modelsas SWMM.6 The original formulation by Greenand Ampt assumedthat the soil surfacewas coveredby ponded water of negligible depth and that the water infiltrated a deep homogenoussoil with a uniform initial watercontent(seeFig. 4.9). Wateris assumed to enter the soil so as to define sharply a wetting front separatingthe wetted and unwettedregions as shown in the figure. If the conductivity in the wetted zone is definedas K", applicationof Darcy's law yields the equation
"
Io:
K"(r + s) r
(4.10)
where I is the distancefrom the ground surfaceto the wetting front and S is the capillary suction at the wetting front. Referringto Fig. 4.9, it can be seenthat the cumulativeinfiltration F is equivalentto the product of the depth to the wetting front L and the initial moisture deficit, 0, - 0, : IMD. Making these substitutionsin
Ponded depth considerednegligible
I Ho
I
I
I
II
I
Figure 4.9 Definition sketch for GreenAmpt model.
4.4 GREEN-AMPTMODEL
65
Eq. 4.10 and rearranging,we obtain - , \ -, f- : -K.( rp
*
t * jt' ) F
(4.11) /
Considering thatfp : dFldt. we can state
#: *,(t.'#)
(4.r2)
: 0 at t : 0' we obtain Integratingand substitutingthe conditionsthat F
+ I MD X {) : K.t
F - S x I M D X l o g " ({ IMD
X
s
(4.r3)
This form of the Green-Ampt equationis more convenientfor usein watershed than Eq. 4.l}beciuse it relatesthe cumulativeinfiltrationto the modelingprocesses ponded time at which infiltration began.The derivation of this equation assumesa at all capacity infiltration to the equal surfaceso that the actualrate of infiltration is a time, any at infiltration cumulative times.Using Eq. 4.13, we can determinethe equation the in parameters the All featuredesiiablefor continuoussystemsmodeling. are physicalpropertiesof the soil-water systemand are measurable.The determinaparticution of suitablevaluesfor the capillary suctions is often difficult, however, It can be 4.10. Fig. in soil larly for relationssuchas that ihown for a clay-type capillary of variation wide is a there observedfrom the figure that for this curve suctionwith soil moisturecontent.3 The Mein-Larson formulation using the Green-Ampt model incorporatestwo The first stagedealswith prediction of the volume of water that infiltrates stages.3,6 before the surfacebecomessaturated.The secondstageis one in which infiltration water capacityis calculatedusingthe Green-Ampt equation.In the widely usedstorm the of one is infiltration of model ,nunug"*"nt model, the irodif,ed Green-Ampt using made are Computations optiois that can be employedto estimateinfiltration.6
v)
U
Moisture content,0
Figure 4.10 Capillary suction versus moisture contentcurves.
66
CHAPTER4
INFILTRATION
the following equations:for F ( F,(f : i),
F. :7:
IM?
i/K" - |
rori > K"
\4.r4)
and thereis no calculationof F"for i < K"; for F > F,(f : fi):
*\ ' f,: x,(t where
t * j") F
l
(4.11)
f : fo: I : F : { :
acttal infiltration rate (ftlsec) infiltration capacity(ftlsec) rainfall intensity (ftlsec) cumulativeinfiltration volume in the event (ft) cumulative infiltration volume required to causesurfacesaturation (f0 S : averagecapillary suctionat the wetting front (ft of water) IMD : initial moisture deficit for the event (ftlft) K" : saturatedhydraulic conductivity of soil (ftlsec)
Equation 4.10 showsthat the volume of rainfall neededto saturatethe surface is a function of the rainfall intensity.In the modelingprocess,for eachtime stepfor which I ) K", the value of d is computedand comparedwith the volumeof rainfall infiltrated to that time. If F equalsor exceeds{, the surfacesaturatesand calculations for infiltrationthenproceedusingEq.4.I4. Notethat by substituting/fori in Eq.4.l4 and rearranging,the equationtakesthe sameform as Eq. 4.11. For rainfall intensitieslessthan or equal to K", all the rainfall infiltrates and its amount is used olly to update the initial moisture deficit, IMD.6 The cumulative infiltration volume {" is not altered. After saturationis achievedat the surface,Bq.4.l1 showsthat the infiltration capacityis a function of the infiltrated volume,and thus of the infiltration ratesduring previous time steps.To avoid making numerical errors over long time steps,the integratedform of the Green-Ampt equation(Eq. .B) is used.This equationtakes the following form as it is usedin SWMM: (4.15) &Gr- tr): Fz- Cln (F2+ C) - Fr* Cln(F1+ C) where C : IMD X .t (ft of water) / : times (sec) 1,2 : subscriptsindicating the starting and ending of the time steps.
-
Equation4.15 mustbe solvediteratively for F2,the cumulativeinfiltration at the end of the time step.A Newton-Raphsonroutine is used.6 In the SWMM model, infiltration during time step tz - tt is equal to (t, - tr)i if the surfaceis not saturatedand is equal to F, - F, if saturationhas previously occurred and there is a sufficient water supply at the surface.If saturationoccurs during an interval, the infiltrated volumesover each stageof the processwithin the time stepsare computedand summed.When the rainfall endsor becomeslessthan
MODEL 4.5 HUGGINS_MONK
67
theinfiltrationcapacity,anypondedwaterisallowedtoinfiltrateandisaddedtothe cumulative inflltration volume'
MODEL 4.5 HUGGINS-MONKE problemby introducing thetimedependency havecircumvented Severalinvestigators by the followingequationproposed soilmoistureas the depenO"niuutluUfe.2'10-13 HugginsandMonkeis an examPle:2
f : f,*A(
I
(4.16)
coefficients layer (Q ,,orug" potential of a soil overlying the impeding minus antecedentmoisture) F _ the total volume of water that infiltrates impeding stratum T o = the total porosity of soil lying over the sprinkling infiltrometer studies'The The coefficientsare determinedusing data from in the iteration process'At the variableF mustbe catculatedfor eachtime increment
where AandP:
s - itr"
exceedsthe inflltration capacity,the rate zone," which determinesthe soil moistu (1) wherethe moist evaluatedasfollows:10 the field capacity(amountof waterheld in drained),the drainagerate is consideredz to the infiltration rate when the soil is s constan| and (3) if the watercontentis be drainagerate is comPutedas
rate: ,"(t - 2)' drainage
(4.r7)
where P, : the unsaturatedpore volume G:maximumgravitationalwater,thatis,thetotalporosityminusthefield caPacitY Datafromsprinklinginfiltrometerstudiesofvariouswatershedsofinterestare usedto estimatethe coefficientsin Eq' 4'16''
5.6
EVAPOTRANSPIRATION
101
of a masstransferequationthat has often been employedfor this purpose.However, Linsley and co-workers indicate that there is some question as to the adequate The equation is verification of this model to estimate evapotranspirationlosses.27 expressed as 833x2(e1
^ E : @
E : K: €1,€2 : Vr, Vz : Z =
er)(V,
V,)
(s.2s)
evaporation(in./hr) von K6rm6n's constant(0.4) vaporpressures(in. Hg) wind speeds(mph) the mean temperature("F) of the layer betweenthe lower level zt and the upper level z2
It is assumedin Eq. 5.25 that the atmosphereis adiabatic and the wind speedand moistureare distributedlogarithmically in a vertical direction. In view of the sm4ll differencesbetweenwind and vapor pressureto be expectedat two levelsso closely spaced,and sincethesegradientsare directly relatedto the sought-afterevaporation, highly exactinginstrumentationis required to get reliable results.
PotentialEvapotranspiration "the waterlosswhich will occur Thornthwaitedeflnedpotential evapotranspirationas if at no time there is a deficiencyof water in the soil for the use of vegetation."In a practical sense,however,most investigatorshave assumedthat potential evapotranspiration is equal to lake evaporationas determinedfrom National WeatherService ClassA pan records.This is not theoreticallycorrect becausethe albedo(amountof incoming radiation reflectedback to the atmosphere)of vegetatedareas and soils rangesas high as 45 percent.28As a result, potential evapotranspirationshould be somewhatlessthan free water surfaceevaporation.Errors in estimatingfree water evapotranspirationfrom pan recordsare such,however,asto make an adjustmentfor potential evapotranspirationof questionablevalue. An equationfor estimatingpotential evapotranspirationdevelopedby the Agricultural ResearchService (ARS) illustratesefforts to include vegetalcharacteristics and soil moisturein sucha calculation.The evapotranspirationpotentialfor any given day is determinedas follows:2e
E T : G I x k ux , " ( # l where
(s.26)
ET : evapotranspirationpotential (in./day) GI : growth index of crop in percentageof maturity K : ratio of G1 to pan evaporation,usually 1.0-I.2 for short grasses' 1,.2-L6 for cropsup to shoulderheight,and 1'6-2.0 for forest Eo: pan evaporation(in./day) ,S: total porosity SA : availableporosity (unfilled by water) AWC : porosity drainableonly by evapotranspiration x : AWC/G (G : moisture freely drainedby gravity)
102
CHAPTER5
EVAPORATIONAND TRANSPIRATION
0.2 0.1 ,.} 0.0
ti
>
n ,
0.1
J
J
Months (b)
Figure 5.5 Averagedaily consumptionof water: (a) for year 1953 by corn, followed by winter wheat under irrigation; (b) for year 1955, with irrigated first-year freadow of alfalfa, red clover, and timothy. Both measurements taken on lysimeterY 102 C at the Soil and Water ConservationResearchStation. Coshocton.Ohio. (After Holtan et al.2e)
l x F rrllS N I
]l{
tl rH
0
8
1
6
2
r
'
.
3
2
4
0
4
8
Weeks
Figure 5.6 Growth index GI = ETfET^,, from lysimeterrecords,irrigatedcorn,andhayfor 1955,from Coshocton, Ohio.(AfterHoltanet al.2e)
5.7
ESTIMATINGEVAPOTRANSPIRATION 103
TABLE 5.5 HYDROLOGICCAPACITIESOF SOIL TEXTURECLASSES
sa
GD
Textureclass
(v")
(Y")
Coarsesand Coarsesandy loam Sand Loamy sand Loamy fine sand Sandyloam Fine sandy loam Very fine sandy loam Loam Silt loam Sandy clay loam Clay loam Silty clay loam Sandy clay Silty clay Clay
24.4 24.5 32.3 37.0 32.6 30.9 36.6 32.7 30.0
t'7.7 15.8 19.0 26.9 27.2 18.6 23.5 21.0 14.4 11.4 13.4 13.0 8.4 rl.6
JI,J
25.3 25.7 z5-J
19.4 z l.+
o 1
18.8
7.3
AWC" ('/")
x AWC/G
6.7 8.7 13.3 10.1 5.4 t2.3 13.1 11.7 15.6 19.9 tr.9 12.7 t4.9 7.8 12.3 11 . 5
0.38 0.55 0.70 0.38 0.20 0.66 0.56 0.56 1.08 1.74 0.89 0.98 r.77 0.6'l r.34 r.58
aS = total porosity - 15 bar moisture 7o' bG : total porosity - 0.3 bar moisture 7o. "AWC: S - G. "Land Capability: A Hydrologic ResponseUnit in Source: Adaptedfrom C. B. England, U.SlDepartmentof Agriculture,ARS 41-172' Sept' 1970' Asiicultural'Watersheds," Aiter H. N. Holtan et a1.2e
The GI curveshave been develoPed evapotranspirationfor severalcrops (Fig. : daily rate(Fig. 5.6).Equation5.26is used USDAHL-74 model of watershedhydrolq late daily evapotranspiration.Represental Table5.5.
EVAPOTRANSPIRATION 5.7 ESTIMATING Transpirationis an important componentin the hydrologicbudgetof vegetatedareas, but it is a difficult quantity to measurebecauseof its dependenceon phytological variables.It is a function of the number and types of plants, soil moisture and soil type, qeason,temperature,and averageannual precipitation. As noted previously, evaporationand tianspiration are commonly estimatedin their combinedevapotranform. spiration ' If the precipitation and net runoff for an atea ate known, and estimatesof of ET canbe had using groundwateiflow and storagecan be made,rough estimatesihe basic hydrologicequatiJn,Eq. 1.1. A more sophisticatedapproachdevelopedby peaman foilows.t3It iJ representativeof the methodsmost often used'
104
CHAPTER 5 EVAPORATION ANDTRANSPIRATION
The PenmanMethod Both the energybudgetand masstransportmethodsfor estimatingevapotranspiration (ET)have limitations dueto the difficulties encounteredin estimatingparametersand in making other required assumptions.To circumventsomeof theseproblems,penman developeda method to combinethe masstransport and energybudgettheories. This widely usedmethodis one of the more reliableapproachesto estimatingETrates usingclimaticdata.13'rs'23,30 The Penmanequationis of the form of Eq. 5.18; it is theoreticallybasedand showsthat EZ is directly related to the quantity of radiative energy gained by the exposedsurface.In its simplified form, the Penmanequationisls t s t : -
LH + 0.27E L + 0.21
(s.27)
where A : the slopeof the saturatedvapor pressurecurve of air at absolute temperature(mm Hg/'F) H : the daily heatbudgetat the surface(estimateof net radiation) (mm/day) E : daily evapoiation(mm) ET : the evapotranspirationor consumptiveusefor a given period (mm/day) The variablesE and.Fl are calculatedusing the following equations: E : 0.35(e"- e)(l'+
( s .28) 0.0098ar) : where eo the saturationvapor pressureat mean ak temperature(mm Hg) e6 : the saturationvapor pressureat meandew point (actualvapor pressure in the air) (mm Hg) u2 : the mean wind speedat 2 m abovethe ground (mi/day) The equationusedto determinethe daily heat budget at the surface,11,is 11 : R(1 - r)(0.18 + 0.55.t)- 8(0.56 * 0.092e2s)(0.10 (5.29) + 0.905) where R : the mean monthly extraterrestrialradiation (mm HrO evaporatedper dav) : the estimatedpercentageof reflecting surface B = a temperature-dependent coefficient s : the estimatedratio of actual duration of brisht sunshineto maximum possibleduration of bright sunshine. The empirical reflectivecoefficientr is a function of the time of year,the calmnessof the water surface,wind velocity, and water quality. Typical rangesfor r are 0.05 to 0.12.31 valuesof e" andA can be obtainedfrom Figs.5.7 and 5.8, thosefor R and B can be obtainedfrom Tables5.6 and 5.7. The use of Penman'sequationrequiresa knowledgeof vaporpressures,sunshineduration,net radiation,wind speed,and mean temperature.Unfortunately,regular measurements of theseparametersare often unavailableat sites of concern and they must be estimated.Another complication is making a reductionin the valueof EZwhen the calculationsare for vegetatedsurfaces. While results of experimentsto quantify reduction factors have not completelyresolved the problem, there is evidencethat the annual reduction factor is close to
ESTIMATINGEVAPOTRANSPIRATION
5.7 "C 60 50
't22 104
t r 1 n
86
20
68
10
111
40
r04
€ : o
86
!?
o B t ^
o
o F
.F
50
t40
I
€ + o
"C
.F
- 1 0 0
50 10 20
30 40
50 60
70
313 = ts
68
14
s0l. zzV | 0.2
105
|
0.4 0 6
| 0.8
|
I
I
I
303
ct
ro?
!d 6
z6J
6 F.
t273
1.0 1.2 1.4 1.6 1.8
80 90 100
satuated vapor pressure' ea (mm Hg)
Figure 5.7 Relation betweentemperatureand saturatedvapor Pressure.
Valueof A (mmHg/'F) Figure 5.8 TemperatureversusA relation for use *i?rt ,n" PenmanLquation. (After Criddle'23)
that using Thus,unlessthereis evidenceto supportanothervalue'it appears unity.32-34 surfaces for results satisfactory a value of 1 for the reduction coefficient may give evaporation water free of having varied vegetal covers.Accordingly, aty estimate by an appropriatereduction could be used ro estimaieEZ, providin[ it is modified coefficient.
EXAMPLE 5.4
to 5.29, estimateET, giventhe following data: ': using the PenmanMethod, Eqs.5.21 : 30 degreesc' 20 degreesc, temperatureof 1l temperatureat warer r"tru"" : (48 mi/day)' the month is relative humidity : +O p"r""nt, wind-velocity i mph S is found to be 0'75' Juneat latitude 30 degreesnorth, r is given ut 0'07' and Solution l.Giventhedatafortemperature,thevaluesofeoandeacanbedetermined. UsingFig.5.TorAppendixTableA.2,thesaturatedVaporpressuresare 31'83'andfor foundto be l7.53unO-:t'Sl mm Hg respectively' ThI'"-: : : 12'73' X 0'4 a relativehumidity of 40 percent,e,t 31'83 Then,usingEq. 5.28' .83 - 12.73)(1+ 0.0098x 48) E : 0.35(31 E:9.83 mm/daY 2.ThevalueofAisfoundusingFig.5.8;forthegivenlatitudeandmonth,R isobtainedfiomTable5.6;andBisgottenfromTable5.'Tforatemperature : 1'0' R : 16'5' andB : I7 '01' of 30"C.fne vatuesfound are A Then,usingEq.5.29, H: 16'5(I - 0'07)(0'18+ 0's5 x 0'75) - 17.01(0.5 6 - 0.092x 12j30)(0.10 + 0'90 x 0'75) H = 6.04 mm/daY
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SUMMARY
107
TABLE5.7 VALUESOF TEMPERATURE-DEPENDENT B FORUSEINTHEPENMAN COEFFICIENT EQUATION Tu (K)
B (mmHrO/day)
270 275 280 285 290 295 300 305 310 315 320 325
r0.73
T" fF) Jf
11.51 12.40 13.20 14.26 15.30 16.34 r7.46 18.60 19.85 21.15 22.50
40 A<
50 55 60 65 70 75 80 85 90 95 100
B (mm HrO/day)
11.48 I 1.96 12.45 12.94 13.45 13.96 14.52 15.10 15.65 16.25 16.85 t'7.46 18.10 18.80
Source: Afler Criddle.23Note that B = oT1 where o is the Boltzmann c o n s t a n t . 2 . 0xl l 0 - e m m / d a y .
3. UsingEq. 5.27, Er : 0.0 x 6.04 + 0.27 x 9.83)/(1+ 0.27) ET : 6.85 mm/day is 6.85 mm/day. ll Thus the estimatedevapotranspiration
Simulating Evapotranspiration The volume of water evaporatedor transpiredfrom a watershedover time can be substantial.Accordingly, continuoushydrologicmodelingprocessesshouldincorporate an EZcomponent.The modelsgiven in this chaptertypify suchan approach(see aboundon this subjes1.28'2e'rs':o':z alsothe flow chart of Fig. 1.3).References
Summary Figure 1.1 and Table5.1 showthe overall importanceof ET in the hydrologicbudget. In many regionsof the United States,annual ET exceedsannual precipitation by a significantamount.As a result, plans for water resourcesdevelopmentand use must incorporate estimatesof ET losses.Where irrigated agriculture is practiced, these estimatesare especiallyimportant. A numberof approachesto estimatingEThavebeendeveloped.They generally fall into the following classes:theoretical,basedon the physicsofthe process;analytical, basedon energyor water budgets;and empirical, basedon observations.Equations usedin making ET calculationsare usually of the type illustrated by Eqs. 5.1, 5.8,5.10,5.19,5.22,and5.26.
1oB
cHAp+R 5 EVApoRATtoN ANDTRANsptRAloN
PROBLEMS 5.1. An 8000-mi2watershedreceived20 in. of precipitationin a 1-yearperiod. The annual streamflowwas recorded as 5000 cfs. Roughly estimatethe combined amounts of water evaporatedand transpired.Qualify your answer. 5.2. Find the daily evaporationfrom a lake during which the following datawere obtained: air temperature 90oF,water temperature60oF,wind speed 20 mph, and relative humidity 30 percent. 5.3. Find the monthly consumptiveuse of an alfalfa crop when the mean temperatureis 70"F, the averagepercentageof daytime hours for the year is 10, and the monthly consumptiveuse coefficientis 0.87. 5.4. During a given month a lake having a surfacearea of 350 acreshas an inflow of 20 cfs, an outflow of 18 cfs, and a total seepagelossof 1 in. The total monthly precipitation is 1.5 in. and the evaporationloss is 4.0 in. Estimatethe changein storage. 5.5. What are two filethodsthat might be usedto reduceevaporationfrom a small pond? 5.6. Computethe daily evaporationfrom a ClassA pan if the amountsof water required to bring the level to the fixed point are as follows:
Day Rainfall (in.) Water added(in.) Evaporation
1 0 0.29
2 0.65 0.55
3 0.12 0.07
4 0 0.28
5 0.01 0.10
5.7. For Problem 5.6, the pan coefficientis 0.70. What is the lake evaporation(in inches) for the 5-day period for a lake with a 250-aqe surfacearea? 5.8. The pan coefficientfor a ClassA evaporationpan locatednear a lake is 0.7. A total of 0.50 in. of rain fell during a given day. Determinethe depth of evaporationfrom the lake during the sameday if 0.3 in. of water had to be addedto the pan at the end of the day in order to restorethe waterlevel to its original valueat the beginningof the day. 5.9. A 2500 mi2drainagebasinreceives25 in.lyr rainfall. The dischargeof the river at the basinoutlet is measuredat an averageof 650 cfs. Assumingthat the changein storage for the systemis essentiallyzero, estimatethe EZlossesfor the areain inchesand cm for the year. Stateyour assumptions. 5.10. Determine the daily evaporationfrom a lake for a day during which the following mean values were obtained: air temperature78'F; water temperature62oF; wind speed,8 mph; and relative humidity, 45 percent. 5.11. Usingthe Meyer and Dunne equations,find the daily evaporationrate for a lake given that the mean value for air temperaturewas 80T, for water temperature60'F, the ziveragewind speedwas 10 mph, and the relative humidity was 25 percent.Refer to Appendix Table A.2 for vapor pressrirevalues. 5.12. Determinethe seasonalconsumptiveuse of truck crops grown in Pennsylvaniaif the meanmonthly temperaturesfor May, June,July, and August are 62,71,16, and 75'F respectivelyandthepercentdaylighthoursforthegivenmonths arc10.02,10.1,10.3, and 9.6 as percent of the year respectively.
REFERENCES
109
5.13. Using the PenmanMethod, Eqs.5.27 to 5.29, estimateET, giventhe following data: temperatureat water surface: 20 degreesC, temperatureof air : 32 degreesC, relative humidity : 45 percent,wind velocity : 3 mph, the month is Juneat latitude 30 degreesnorth, r is given as 0.08, and S is found to be 0.73.
REFERENCES 1 . WaterResources Council,TheNation's WaterResources: 1975-2000,U.S. Govt. Print. Off., Washington, D.C., 1978. 2 . U.S. Department of Commerce, National Oceanic and Atmospheric Administration, National WeatherService,NOAA TechnicalReport 33,EvaporationAtlas of the Contiguous 48 UnitedStates,Washington, D.C., June1982. 3 . U.S. Department of Commerce, National Oceanic and Atmospheric Administration, National WeatherService,NOAA TechnicalReport NWS 34, "Mean, Monthly Seasonal, and Annual Pan Evaporationfor the United States,"Washington,D.C., June 1982. 4. "Water-LossInvestigations,"Vol. 1, Lake Hefner Studies,U.S. GeologicSurveyProfessionalPaperNo. 269 (1954). (Reprintof U.S. GeologicalSurveyCitc.229, 1952.) 5 . N. N. Gunaji, "EvaporationInvestigationsat ElephantButte Reservoir,New Mexico," ^lnl. Assoc.Sci.Hydrol. Pub. 18, 308-325(1968). 6. I. S. Bowen, "The Ratio of Heat Lossesby Conduction and by Evaporationfrom Any Water Surface,"Phys.Rev. 27, 779-7 81(1926). 7 . E. R. Anderson,L. J. Anderson,andJ. J. Marciano, "A Reviewof EvaporationTheory and Developmentof Instrumentation,'oLake Mead WaterLoss Investigation;Interim Report, Navy ElectronicsLab. Rept.No. 159 (Feb. 1950). 8 . O. G. Sutton, "The Application to Micrometeorologyof the Theory of Turbulent Flow over RoughSurfaces,"R. Meteor,Soc.Q. "I. 75(No. 236),335-350(Oct. 1949). 9 . A. F. Meyer, "Evaporationfrom Lakes and Reservoirs,"MinnesotaResourcesCommission,St. Paul,June1944. 1 0 . T. Dunne and L. B. Leopold, Waterin EnvironmentalPlanning, San Francisco:Freeman and Co., 1978. 1 1 . V. M. Ponce,EngineeringHydrology: Principles and Practices.EnglewoodCliffs. New Jersey:PrenticeHall, 1989. 1 2 . M. A. Kohler, T. J. Nordenson,and W E. Fox. "Evaporationfrom Pansand Lakes," U.S. Departmentof Commerce,WeatherBureau,Res.PaperNo. 38, Washington, D.C., 1955. 1 3 . H. T. Haan, H. P. Johnson,and D. L. Brakensiek(eds.),Hydrologic Modeling of Small Watersheds,ASAE MonographNo. 5. St. Joseph,MI: American Societyof Agricuitural Engineers,1982. 14. R. K. Linsley,M. A. Kohler, and J. L. H. Paulhus,Hydrologyfor Engineers,3rded. New York: McGraw-Hill, 1982. 1 5 . H. L. Penman,"Natural Evaporationfrom Open Water,Bare Soil, and Grass,"Proc. R. Soc.LondonSer.A 193(1032),120-145(Apr.1948). 16. F. G. Millar, "Evaporationfrom FreeWaterSurfaces,l'CanadaDepartmentof Transport, Division of MeteorologicalServices,Can, Meteor. Mem. vol. l, No. 2, 1937. 1 7 . J. B. Franzini, "Evaporation SuppressionResearch,"Water and SewageWorks (May 1961). 1 8 . Victor K. La Mer, "The Casefor EvaporationSuppression," Chem.Eng.(June10, 1963). 1 9 . D. R. Maidment(ed.),Handbookof Hydrology.New York: McGraw-Hill,1993. 20. H. F. Blaney,"Water and Our Crops," inWater, the Yearbookof Agriculture. Washington, D.C.: U.S. Departmentof Agriculture,1955.
110
ANDTRANSPIRATION 5 EVAPORATION CHAPTER "Monthly Consumptive Use Requirementsfor Irrigated Crops," Proc. ZI. H. F. Blaney, ASCE,J. Irrigation DrainageDiv. 85(IR1), 1-12(Mar' 1959)' 22. yen Te Chow (ed.), Handbook of Applied Hydrology. New York: McGraw-Hill ' 1964. "Methods of Computing ConsumptiveUse of Water." Proc. ASCE J. 23. W. D; Criddle, Irrigation DrainageDiv. 84(IR1), 1-27(Jan' 1958)' ,;Estimating Evaporation," Trans. Am. Geoplrys. union 37(l), 4324. H. L. penman,
5o(1es6).
"Reductionof Transpiration,"J. Geophy.Res.66(10), 3309-3312(Oct. 25. N. J. Roberts, 1961). "Researchon control of Phreatophytes,"Proc. Ninth Annual water con26. E. H. Hughes, ference,New Mexico StateUniversity,Las Cruces,Mat. 1964' 21. R. K. Linsley,Jr., M. A. Kohler, and J. L. H. Paulhus,Hydrologyfor Engineers.New York: McGraw-Hill, 1958. "National WeatherServiceRiver ForecastSys28. Staff, Hydrologic ResearchLaboratory, tem ForecastProcedures,"NWS HYDRO 14. U.S. Departmentof Commerce,Washington. D.C..Dec. 1972. "usDAHL-74 Revised 29. H. N. Holtan, G. J. Stiltner,w. H. Henson,and N. C. Lopez, ARS Tech.Bull. No. 1518.U.S.DepartmentofAgriculModelof WatershedHydrology," D.C., 1975. ture, Washington, "Evapotranspiration-Review of ReN. J. Rosenberg,H. E. Hart, and K. w. Brown, search,"MP 20, Agricultural ExperimentStation,University of Nebraska,Lincoln, 1968' R. 3 1 . H. McCuen, HydrologicAnalysis and Design.Englewoodcliffs, New Jersey:Prentice Hall, 1989. "Correlation of ClimatologicalData with WaterR'equire3 2 . W. O. Pruitt and F. J. Lourence, California Water ScienceEng. Paper9001. Davis, June of University ments of Crops." 1968. "Potential Evaporation:The cornbination concept and Its Experi33. C. H. M. Van Bavel, mental Verification," WaterResourcesRes.2' 455-467(1966)' "Discussionof Paperby H. L. Penman,'Estimating Evaporation,"' Trans' 34. H. F. Blaney, Am. Geoplrys.Union37,46-48(Feb. 1956). "Digital Simulationin Hydrology:StanfordWater35. N. H. Criwford and R. K. Linsley,Jr., shedModel IV," Tech. Rept. 39, Departmentof Civil Engineering,Stanford University' July 1966. 36. W. C. Huber, J. P. Heaney,S. J' Nix, R. E. Dickinson, and D"J' Polmann' Storm Water ManagementModel (lsei's Manual, Version /11, EPA-60012-84-109a(NTIS PB84198423).Cincinnati, oH: EnvironmentalProtection Agency, Nov. 1981. 37. L.A. Roesner,R. P. Shubinski,and J. A. Aldrich, StormWaterManagementModel User's Manual, versionIII: AddendumI, Extran, EPA-600/2-84-109b(NTIS PB84-198431). Cincinnati, OH: EnvironmentalProtection Agency,Nov' 1981'
Chapter6
Streamflow
r Prologue The purposeof this chapteris to: . Introduce the conceptof streamflow. . Describethe characteristicsof a hydrograph. ' Presentapproachesto measuringstreamflow. The amountof water flowing in surfacewater coursesat any instant of time is small in termsof the earth's total waterbudget,but it is of considerableimportanceto those concernedwith waterresourcesdevelopment,supply,and management.A knowledge of the quantity and quality of streamflowsis a requisite for: municipal, industrial, agriculiural, and other water supply endeavors;flood control; reservoir designand operation; hydroelectricpower generation;water-basedrecreation;navigation;fish und .ildlit" management;drainage; the managementof natural systemssuch as wetlands;and water and wastewatertreatment. Streamflowis generatedby precipitationduring stormeventsand by groundwater entering surface channels.During dry periods, streamflowsare sustainedby groundwaterdischarges.Where groundwaterreservoirsare below streamchannelsoften the casein arid regions-streams ceaseto flow during protractedprecipitationfree periods. Relations between precipitation and streamflow are complex, being influJnced by the factors discussedin the foregoing chapters.As a result, many approachesto relating theseimportant hydrologicvariableshavebeen developed.l-3 Severalof them are discussedin detail in Part Three of the text' Field measurementsof streamflow are based on the use of flow-measuring devicessuchas weirs and flumes and on the measurementof channelcross-sections along with streamflowvelocities(seeChapter 8).
DRAINAGEBASINEFFECTS The quality and quantity of streamflowgeneratedin a drainagebasin are affectedby Accordingly,it is important the bisin's physical,vegetative,and climatic features.a-e rocks,plants,topography, soils, ofthe good understanding a have hydrologist that the
CHAPTER 6 STREAMFLOW
J12
land-usepatterns,and otherbasincharacteristicsthat influencethe sequenceof events separatingprecipitation and runoff. It should be pointed out, however,that while natural basin featuresare very important elementsin the runoff process,land-use ieatures createdby humans (e.g., housingdevelopments,parking lots, agricultural patterns)may,in somecases,be the dominantones.Land managementpracticescan be beneficial,suchasin retardingerosion,and they can alsobe detrimentalwhenthey In Chapter10,the principal basin function to acceleratenatural hydrologicprocesses. are discussed. hydrologist concern to the of characteristics
6.2 THE HYDROGRAPH
,
Streamflow,at a givenlocationon a watercourse,is representedby a hydrograph.This continuousgraph displaysthe propertiesof streamflowwith respectto time, normally obtained by nteansof a continuousrecorder that shows stage(depth) versustime (stagehydrograph),and is then transformedinto a dischargehydrographby application of a rating curve. In general, the term hydrograph as used herein means a dischargehydrograph. As was shownin Fig. 1.3, the hydrographproducedin a streamis the result of various hydrologicprocessesthat occur-duringand after any precipitation event. A is given in Chapter11. A hydrographhas more completediscussionof theseprocesses (1) runoff, (2) interflow, (3) groundwateror surface elements: direct four component rising portion of a hydrographis known (4) precipitation. The channel baseflow, and vicinity of the peak is calledthe crest in the region curve; the as the concentration The shapeof a hydrographdepends portion the recession.lo falling is segrnent;andthe properties. Figure 6.1 illustratesthe basin pattem and precipitation characteristics on presented. definitions
Stormperiod hydrograph
'{1 oa-. . I'a
Endof direct runoff
\% .
\
Y
Continuous hydrograph Time Figure 6.1
Hydrograph definition.
6.4
MEASURINGAND RECORDINGSTREAMFLOW
113
FORSTREAMFLOW 6.3 UNITSOF MEASUREMENT Two types of units are usedin measuringwater flowing in streams.They are units of dischargeand units of volume.Discharge,or rate of flow, is the volume of water that passesa particularreferencepoint in a unit of time. The basicunits usedin connection of with streamgaugingin the United Statesare the foot and meter for measurements dimensionand the secondfor measurementsof time. Commonly usedunits of dischargemeasurementare cubic feet per second(cfs) and cubic meters per second (m3/sec).Other units of dischargein useare second-footper squaremile (sec-ft/mi2), for expressingthe averagerate of dischargefrom a drainagebasinor definedarea,and million gallonsper day (mgd),commonly usedin water supplycalculations.Units of volume used are the cubic foot, cubic meter, liter, gallon, and acre-foot (a volume equivalent to 1 ft of water over an acre, 43,560 ft2, of land). The latter unit is commonly used in irrigation practice in the westernUnited States.Irrespectiveof whetherEnglish or metric units are usedfor dimensions,the standardunit of time for streamflowobservationsis the second.
AND RECORDING STREAMFLOW 6.4 MEASURING Streamflowratesmay be determinedusinggaugingdevicessuchas flumes,weirs,and control sections,or they may be calculated from measurementsof head (depth), velocity, and cross-sectionalarea.t-t Usually, specific devices such as flumes are
' bI) 6 6 o bo
a
ri
)
3000 Discharge (sec-ft)
Figure 6.2 Station rating curve for RaquetteRiver at Piercefield,New York. (U.S. GeologicalSurvey.)
114
CHAPTER6
STREAMFLOW
limited to small streamsbecauseof problems of scale.For large stream systems, dischargeis normally estimatedby measuringvelocity and using the cross-sectional areato translatethis measurementinto discharge.Where flow-measuringdevicesare used,it is customaryto observethe head and use a rating curve to translatethis into discharge.When direct flow measurementscannot be made, dischargecalculations are often facilitated by use of velocity-area relations, chemical tracers, electrical methods,and empirical equationssuchas the Manning formula' In the United States,the primary responsibilityfor gaugingmajor streamslies of with the U.S. GeologicalSurvey,with a systematicrecord of streamflowin terms a at obtained is recording mean daily dischargebeing the norm. Usually, a stage several of more or one by gauging siie and this record is convertedinto discharge (seeChap-"tttoAr. Rating curves,tables, and formulas are used for this purpose and instruments The curve' ter 11).Figure 6.2 rho*r a typical stage-dischargerating the to adapted carefully methodsusedto convertstagerecordingsto dischargemustbe the to ensure as so gauging site natural or artiflcial conditions encounteredat the reliability of conversions.'
AREA OF DEPTHAND CROSS.SECTIONAL 6.5 MEASUREMENTS Unlessa direct flow-measuringdevicecanbe installedin a stream,aratity for streams of any scale,measurementsof depth of flow and cross-sectionalareawill be needed to permit dischargeto be calculated. Depth measurementsmay be taken using weighted soundin! [nes, calibrated rods, and ultrasonic soundingdevices.Crosssectionalareasat streamsectionscan be determinedusing ordinary surveyingtechniquescombinedwith soundingsor other depth measurementsthat ate taken below the water level at the time of the survey.Although the measurementof depth and simple,accuratedeterminationsrequirecarefulcalibration al areaseems cross-section of instrumentsand the ability to deal with submergedconditions.
OF VELOCITY 6.6 MEASUREMENT combinedwith thoseof cross-sectionalarea,permit calculaVelocity measurements, tion of dischargeat a given stream or river location. Point flow velocities can be determinedusing velocity-measuringdevicessuch as the Pitot tube, dynamometer' and current -"i"r. In the United States,the Price current meter has long been a standardin streamflowgauging.This deviceoperatesby exposingcuppedvanesto the direction of flow, *o"h lik" the anemometerused in measuringwind velocity. The cup-vaneassemblyrotatesin nearproportion to flow velocity and the rate of rotation is convertedto point velocity using a rating table or appropriateequation' , Various chemical and electrical methods are also employed in determining velocities.Commonly usedchemicalmethodsinclude salt velocity, salt dilution' and is the detectionof radioactivetracers. Of these methods,the salt velocity method into the introduced salt principle that perhapsthe most widely used.It is basedon the streamwill increaseits electricalconductivity.Electrodesplaceddownstreamof the
FLOWVELOCIry POINTVELOCIryTO CROSS-SECTIONAL 6.7 RELATING
115
Figure6.3 Verticalvelocityprofile
salt injection senseconductivity changesand thesecan be translatedinto velocity by knowing the spacingof electrodesand keeping track of time. Electrical methods includethe useof hot wire anemometers,electromagneticvoltagegeneration,oxygen polarography,and supersonicwaves.aStreamflowconditionsvary widely and thus no specificapproachto velocity determinationis universally suitable.The best method for use at a given site must be determinedon the basis of the characteristicsof streamflowat that site. Field observationshave shown that the mean velocity in a verticai stream sectionis closelyapproximatedby the averageof the velocitiesoccurringat depthsof 20percentand 80 percentof the total sectiondepthrespectively(seeFig.6.3).11'12 Where depthsarevery shallow,on the orderof 0.5 feet, singlemeterreadingsat about the 50 percentpoint havebeen shownto yield good results.t' Velocity measurements permit estimatingchannel and geometricdefinitionsof streamchannelcross-sections havebeen made. measurements and depth velocity flows at locationswhere
FLOWVELOCITY TO CROSS-SECTIONAL POINTVELOCITY 6.7 RELATING While point velocity measurementsare important, what is desiredis a method to translatethem into the averagecross-sectionalflow velocity. This averagevelocity, when multiplied by the cross-sectionalarea,yields the dischargeat a given stream section.One procedureis to take point velocity measurementsat numerousvertical and horizontal positions in a crosi section,plot them, and then determinevelocity contours.By calculatingthe areasbetweenthe contoursand assigningthe averageof the flow velocitiesof the two confining contoursto theseareas,a determinationof meanvelocity can be made.Once this is accomplished,dischargeis easilycalculated. Other approachesmake use of the geometric properties of stream channel One suchtechniqueis the mean-sectionmethod.To usethis approach, cross-sections. to it is necessary divide the streamchannelcross-sectionat a gauginglocation into a
116
CHAPTER6 STREAMFLOW Water surface
for Example6.1 Figure 6.4 Channelcross-section seriesof geometricshapes(seeFig. 6.4). AL eachvertical location along the crossThe averagevelocity of section,the meanvelocity is estimatedfrom measurements. to the averageof the be equal to is considered two verticals between area flow for the betweentwo vertidischarge The verticals. of the bordering for each meanvelocities of the section. area the by multiplied section for the velocity average the cals is thus flow for the total estimated provide an to summed are then discharges The individual to measurements enough to have is important it Note that location. at that channel 6.1. Example in illustrated is procedure The cross-section. the characterize EXAMPLE 6.1
Calculatethe dischargeat the sectiongiven in Fig. 6.4. Data from field observations are shownin Tables6.1 and6.2. 6.1 TABLE6.2 DATAFOREXAMPLE
TABLE 6.1 DATA FOR EXAMPLE6.1
Verticalsection#
Depth (ft.)
Avg. vel.
(rps)
0
0
0
I
4
2.1
2
5
2.3
'7.2
2.7
7.4
2.8
3
4
2.5
5 6
4.'1
2.2
7
0
0
DISCHARGE 6.8 THESLOPE-AREA METHODFORDETERMINING
,117
TABLE6.3 CALCULATIONS FOR EXAMPLE 6.1 Area (sq.ft.)
Vel. (fps)
A1
8.40
1.05
8.82
A2
14.85
2.20
32.67
A3
29.28
2.50
73.20
A4
3't.96
2,75
104.39
A5
26.83
2.65
7t.09
A6
30.09
2.35
70.71
A7
13.87
1.10
15.25
Area
t61,.27
Flow (cfs)
376.t3
Total estimateddischargeis 376.13 cfs
Solution. The first step is to calculatethe individual section areas(A1, A2, etc.). The calculatedareas(usingtriangular or trapezoidalformulas)are shown on spreadsheet Table6.3. Next, the estimatedmeanvelocitiesat the verticalsare multiplied by the section areasto obtain the individual area discharges(see Table 6.3). Thesedischargesare summedto yield an estimated376.13 cfs of flow being deliveredby the full channelwidth. r I
DISCHARGE METHODFORDETERMINING 6.8 THE SLOPE.AREA In somecasesit is difflcult to make velocity or other measurementsneededto deter: mine discharge.This is often the caseduring large flood events.Under suchcircumof high stances,it is sometimespossibleto estimatethe flow by taking measurements water lines (after the flood event),cross-sectionalareas,and channelslopesand then usingthesedatain an equationsuchasManning's to estimatethe flow. The applicable Manning equationis Q : \I.49ln)APzrzgrrz where Q : n : A : R: S:
(6.1)
discharge(cfs) Manning's roughnesscoefficient cross-sectionalarea(ft2) the hydraulic radius the headloss per unit length of channel
For streamflows,Manning's n valuesmay rangebetweenabout0.03 and 0.15. When reasonabledeterminationscan be made of n, A, R, and S, Eq. 6.1 can be used to estimate the streamflow that occurred during the high-water period. For a more completediscussionof this and other streamflowdeterminationmethods,the
118
CHAPTER6
STREAMFLOW
referencesat the end ofthe chaptershouldbe consulted.References4 and 10 give an excellent overview of techniques and include a valuable list of references on streamflow.
r Summary Streamflowis the result of storm-periodprecipitation, snowmelt,and groundwater discharge.l3It is a primary sourceof water for a host of instreamand offstreamuses. The graphicalrepresentationof streamflowis the hydrograph,a plot of flow versus time at a prescribedlocation alongthe water courseof interest.As illustratedby Fig. 1.3, the end product of many hydrologicmodelingprocessesis a hydrographwhich is derived from a precipitation input, modified appropriatelyby various abstractions suchas infiltration. Methodsfor measuringstreamflowin the field were presentedin this chapter.In later sectionsofthe book, a variety oftechniquesfor deriving hydrographs from precipitation and other hydrologic data are covered (seePart Three).
PROBLEMS 6.t. Considerthat you haveobtaineda gaugeheightreadingof4 ft at a gaugingsite on the RaquetteRiver (Fig. 6.2). What would you estimatethe dischargeto be in cfs and in m3/sec?If the gaugeheighthad been9 ft, what would the dischargebe?Which of the two estimatesdo you think would be the most reliable?Why? 6.2. SolveProblem 6.1 if the gaugeheight readingswere 5 ft and 7 ft. 6.3. Consulta USGSWaterSupplypaperand plot the streamflowdatafor a drainagebasin ofinterest.Discussthe factorsthat you believeinfluencedthe shapeofthe hydrograph. 6.4. For the major surfacewater course in your locality, discussthe value of making streamflowforecasts. 6.5. Calculatethe dischargeat the sectiongiven in Fig. 6.4 if the depth measurementsat Give resultsin the verticalswere:0, 3.8, 5.4, 7.1,8.1,7.0,4.5, and 0 ft respectively. cfs and m3/s. Calculatethe dischargeat the sectiongiven in Fig. 6.4 if the velocitieswere: 0, 2.3, 2.6,3.1,2.9,2.7,2.5, and0 fps respective$,and the depthsof Problem6.5 applied. Give resultsin cfs and m3/s.
REFERENCES 1 . N. C. Grover and A. W. Harrington, StreamF/ow. New York Wiley, 1943. 2. United StatesDepartmentof the Interior, Bureau of Reclamation,Water Measurement Manual. Washington,D.C.: U.S. GovernmentPrinting Office, 1967.
3 . I. E. Houk, "Calculation of Flow in Open Channels,"Stateof Ohio, The Miami Conservancy District, Tech. Rept. Part IV, Dayton, OH, 1918.
4. Ven Te Chow (ed.), Handbook of Applied Hydrology. New York: McGraw-Hill, 1964. 5 . American Society of Civil Engineers,"Hydrology Handbook," Manuals of Engineering Practice.No. 28. New York: ASCE. 1957.
REFERENCES 1 19 6 . O. E. Meinzer, Hydrology.New York: Dover, 1942. 7 . A. N. Strahler, "Geology-Part II," Handbook of Applied Hydrology. New York: McGraw-Hill. 1964. 8 . R. E. Horton, "Drainage Basin Characteristics,"Trans.Am. Geoplrys.Union l3r 35O36r(1932\. 9 . W. B. Langbeinet al., "TopographicCharacteristicsof DrainageBasins,"U.S. Geological Survey,Water Supply Paper,968-c, 1947. 1 0 . R. K. Linsley, Jr., M. A. Kohler, and J. L. H. Paulhus,Applied Hydrology. New York: McGrawHill, 1949. 1 1 . S. S. Butler, EngineeringHydrology.EnglewoodCliffs, New Jersey:Prentice-Hall, Inc., 1951. 12. C. H. Pierce,"Investigationof Methodsand EquipmentUsedin StreamGauging,"Water Supply Paper 868-A, U.S. GeologicalSurvey,Washington,D.C.: GovernmentPrinting Office,1941. 1 3 . D. R. Maidment (ed.),Handbookof Hydrology.New York: McGraw-Hill , 1993.
PART TWO
ENTS MEASUREM HYDROLOGIC AND MONITORING
Chapter7
HydrologicDataSources
r Prologue The purposeof this chaPteris to: . Describethe principal sourcesof data for hydrologicinvestigations. Data on hydrologicvariablesare fundamentalto analyses,forecasting,and modeling. Such data *uy 1" found in numerouspublications of state and federal agencies, researchinstitutes,universities,and otheroganizations.Severalofthe most significant sourcesof hydrologicdata are describedbriefly in this chapter'1-3
DATA 7 . 1 GENERALCLIMATOLOGICAL The most readily availablesourcesof data on temperature,solarradiation,wind, and humidity are ilimatological Data bulletins publishedby the EnvironrnentalData Service of the Nationai Oceanic and Atmospheric Administration (NOAA), and Monthly Summaryof Solar RadiationData publtshedby the National Climatic Data Center.The EnvironmentalData Service,in cooperationwith the World Meteorological Organization(WMO), also publishesMonthly Climatic Data for the World- A 1968 publication of the Environmental SciencesService Administration, entitled Climalic Atlasof the UnitedStates,summarizeswind, temperature,humidity, evaporation, precipitation, and solar radiation on a seriesof maps. In addition-to these federal sourcesof data, stateenvironmental,geologic,water resources,and agricultural agenciesshouldbe consulted.Most stateuniversitiesalso publish a variety of hydrologicdata through their researchcentersand extensionprograms.
DATA 7.2 PRECIPITATION There are probably more records of precipitation than of most otherhydrologic variables.The priniipal federalsourceof data on precipitationis NOAA. Climatological Data, publishedmonthly and annuallyfor eachstateor combinationof states,the pacific area,PuertoRico, and the Virgin Islandsby the EnvironmentalData Service,
E # = ,
124
CHAPTER7
HYDROLOGICDATA SOURCES
presentsa table of monthly averages,departuresfrom normal, and extremes of precipitation and temperatureas well as tables of daily precipitation, temperature, snowfall,snowon ground,evaporation,wind, and soil temperatarc.Hourly Precipitation Data is issuedmonthly and annuallyfor eachstateor combinationof statesand presentsalphabeticallyby station the hourly and daily precipitation amounts for stationsequippedwith recordin€gauges.A stationlocationmap is alsoincluded.This publication is availablefrom the EnvironmentalData Service.Another publication, World WeatherRecords,is issuedby geographicregionsfor 10-yearperiods.Data are listed by country or areaname, station name,latitude and longitude,and elevation. Monthly and annualmeanvaluesof stationpressure,sea-levelpressure,and temperaiure, and monthly and annualtotal precipitationare given in sequentialorder.Aside from NOAA, other federal and state agenciesand universitiespublish precipitation dataat varying intervals,often in a storm o1 site-specificcontext.In addition, many municipalitiesand water and wastewaterutilities also collect and maintain precipitation and other related data. Computeized precipitation data are availablefrom the National Climatic Data Centerin Asheville,North Carolina.
DATA 7.3 STREAMFLOW
.
The principal sourcesof streamflowdatafor the United Statesare the U.S. Geological Service(SCS),U.S.ForestService,andU.S. Survey(USGS),U.S. Soil Conservation Agricultural ResearchService(ARS). In addition,the U.S. Army Corpsof Engineers (COE), the Tennessee Valley Authority (TVA), and the U.S. Bureauof Reclamation (USBR) make somestreamflowmeasurementsand tabulatestreamflowdata relative to their missions.Stateagencies,universities,and variousresearchorganizationsalso compile and publish a variety of streamflowdata. The USGS Water Supply Papers(WSf; are the benchmark for referencing streamflowdata. Furthermore,computerizeddata are also availablefrom the USGS. Publications of the Geological Survey, publishedevery 5 years and supplemented annually, arean excellentsourceof information on that agency'sreports. The SCS historically published data on streamflow from small watershedsand plots in its Hydrologic Bulletin series,but much of the data have been republishedby ARS. "pilot watersheds"are publishedin cooperationwith the USGS. Recordsfrom SCS U.S. Forest Service streamflow data are publishedat irregular intervals in various technicalbulletins and professionalpapers.
DATA AND.TRANSPIRATION 7.4 EVAPORATION Monthly and annualissuesof ClimatologicalData, publishedby NOAA, includepan evaporationand related data. The ARS, agricultural colleges,and water utilities are other sourcesof information. In particular, data on evapotranspirationare often obtained by university researchersworking through their Agricultural Experiment Stations.
REFERENCES 125
r summary Climatic and other data are keystonesin hydrologicmodelingprocesses.Numerous sources of data exist and may be accessedto support model developmentand verification, statisticalanalyses,and specialstudies.
PROBLEM 7.1
Developa list ofdata sourcesin your stateorlocality by visiting the library or through other channels.
REFERENCES 1 . Soil ConservationService,U,S. Departmentof Agriculture, SCSNational Engineering Handbook, "Hydrology", Sec. 4. Washington,D.C.: U.S. GovernmentPrinting Offlce, Au,g.1972. J. F. Miller, "Annotated Bibliography of NOAA Publications of Hydrometeorological Interest," NOAA Tech. Mem. NWS HYDRO-22, National Oceanic and Atmospheric Administration, Washington,D.C., May 1975. , D. R. Maidment (ed.), Handbookof fudrology. New York: McGraw-Hill, 1993.
ChapterB
lnstrumentation
Prologue The purposeof this chapteris to: ' Describeinstrumentsusedin measuringhydrologicvariables. . Indicate waysin which data are recordedand transmitted. . Presentlimitations on measurements. The data neededto supporthydrologicanalysesmust be obtainedin sufficientquantity, with adequatefrequency,and in an appropriateform if they are to be of value. A variety of instrumentsareusedto obtain and transmitthe data.They are the subject of this chapter.
8.1 INTRODUCTION
'
.
Hydrologicinstrumentationsupportsarealinvestigations,problemanalyses,research, are planning,and environmentalpolicymakingand analysis.A hostof measurements neededto support efforts in water resourcesplanning, management,design, and constructionrelatedto suchsubjectsas aquifer systemsanalysis,solid wastemanagewater supplyavailability,water quality management, ment, flood hazardassessment, groundwaterrecharge,protection of fish and wildlife, and navigation. Historically,instrumentswere often usedto obtain cumulativerather than continuousinformation abouthydrologicvariablessuchasrainfall and evaporation.Fqrthermore,there was often no attemptto correlatewater quality constituentloadings, for example,with ratesof water flow. Consequently,many historic data havelimited utility, not somuchbecauseof lack of adequateinstrumentation,but ratherfrom using availableinstrumentsto measurethe wrong thing or in too limiting a fashion.Today it is widely recognizedthat it is important not only to selectappropriateinstruments but to selectthem in the contextof datanetworksthat meetthe needsof moderntimes. More will be said about this in Chapter9, In Section8.2, instrumentsfor measuring hydrologicvariablesand waysin which they can be usedjointly to createa complete representationof a functioning hydrologicsystemare discussed.
INSTRUMENTS 127 8.2 HYDROLOGIC
INSTRUMENTS 8.2 HYDROLOGIC Good sourcesof information abouthydrologicinstrumentsare the National Weather Service,U.S. GeologicalSurvey,U.S. Bureauof Reclamation,U.S. Army Corpsof Engineers,Soil ConservationService,and instrumentmanufacturers,Theseagencies and industrieshavelong been in the businessof measuringhydrologicvariablesand Someofthe of state-of-the-artmeasuringdevices. theycanprovidedetaileddescriptions is far from but the coverage here, are described major typesof measuringinstruments references.l-3 the appropriate consult should exhaustiveand the interestedreader
Precipitation
"
Gaugesfor measuringrainfall and snowfall may be recording or nonrecording.The mostcommonnonrecordinggaugeis the U.S. WeatherServicestandard8-in. gauge. The gaugemay be read at any desirableinterval but often this is daily. The gaugeis calibrated so that a measuringstick, when inserted, showsthe equivalentrainfall depth. Such gaugesare useful when only periodic volumes are required, but they cannotbe usedto indicate the time distribution of rainfall' Recordinggaugescontinuouslysensethe ratb of rainfall and its time of occurrence.Thesegaugesare usually either ofthe weighing-recordingtype or the tipping buckettype. Weighing-typegaugesusually run for a period of 1 week, at which time their chartsmustbe changed.The figure associatedwith Problem2.5 is typical of the recordedoutput. A masscurve of rainfall depth versustime is the product, and this curve can be translatedinto an intensity-time graph by calculating the ratios of accumulatedrainfall to time for whatevertime stepis desired.Tipping bucketgauges, on the other hand, senseeach consecutiverainfall accumulationwhen it reachesa prescribedamount, usually 0.01 in. or 1 mm of rain, A small calibratedbucket is located below the rainfall entry port of the gauge. When it fills to the 0.01-in. incrementit tips over,bringing a secondbucketinto position.Thesetwo small buckets are placed on a swivel and the bucketstip back and forth as they fiIl. Each time a bucketspills it producesan indication on a strip chart or other recordingform. In this way a record of rainfall depth versustime (intensity)is the outcome.For rain gauges to record snow accumulations,some modifications must be made. Usually these involveproviding a melting agentso that the snow can be convertedinto measurable water. Figure 8.la is the diagram of a self-reportingrain gaugingstation.The tipping bucketmechanismgeneratesa digital input signal whenever1 mm of rainfall drains through the funnel assembly.The signalfrom the gaugeis automaticallytransmitted to a receiving station where it records the station ID number and an accumulated amountof rainfall. The receiving stationrecordsthe time at which the messagewas received and rainfall rates for desiredperiods can be calculated accordingly' Figure 8.1b showsa similar gaugeequippedto measuresnow.In this case,a glycometh solution is usedto melt the snow.The melt water overflowsthrough a temperaturecompensatingmechanismand is measuredby the tipping bucket,which operatesthe station's transmitter.Gaugesof the type shownin Fig. 8.1 can easilybe incorporated into real-time monitoring systemsthat can be used in a variety of forecastingand operatingmodes.
128
CHAPTER 8 INSTRUMENTATION
Antenna
Antenna mast
Antenna
Temperatffe compensation overflow mechanism
Antenna mast
Funnel assembly
Glycometh collecting section
Tipping bucket
Tipping bucket Drain holes (4)
Signalcable
(a)
,
Signalcable
Lifting rope
Lifting rope
Antennacable
Antennacablb
(b)
Figure 8.1 Self-reporting (a) rain and (b) snow stations. (Courtesy of SierraMisco, Inc., EnvironmentalProducts,Berkeley,CA.)
Evaporationand Transpiration
..
Evaporationpans have been widely used for estimatingthe amount of evaporation from free water surfaces.Devicessuchasthat depictedin Fig. 8.2 areeasyto use,but relating measurements taken from them to actual field conditionsis difficult and the -data_lhey-produceare often of questionablevalue for making areal estimates.A
INSTRUMENTS 8.2 HYDROLOGIC
129
tr'igure 8.2 U.S. Weather Bureau Class A pan.
variety of pan types havebeen developedbut the U.S. WeatherBureauClassA pan is the standardin the United States.4Panevaporationobservationshavebeenusedto estimateboth free water(lake) evaporationand evapotranspirationfrom well-watered vegetation.Field experimentshave shown a high degreeof correlation of pan data with evapotranspirationfrom surroundingvegetationwhen there is full cover and goodwatersupply.a As in the caseof precipitationgauges,pan datacan be recorded and transmittedcontinuouslyto a central receiving station. Evapotranspirationmeasurementsare often made using lysimeters.Thesedevices are containersplaced in the field and filled with soil, on which sometype of vegetativegrowth is maintained.The object is to study soil-water-plant relationsin a natural surrounding.The main featureof a weighinglysimeteris a block of undisturbed soil, usually weighing about 50 tons, encasedin a steel shell that is 10 ft by 10 ft by 8 ft. The lysimeteris buried so that only a plasticborder marksthe top of the containedsoil. The entire block of soil and the steelcasingare placedon an underground scale sensitiveenough to record even the movement of a rabbit over its surface.The soil is weighed at intervals, often every 30 min around the clock, to measurechangesin soil water level. The scalesare set to counterbalancemost of the deadweight of the soil and measureonly the active changein weight of water in the soi1.5The scalescan weigh accuratelyabout 400 g (slightly under 1 1b), which is equivalent to 0.002 in. of water. The weight loss from the soil in the lysimeter representswaterusedby the vegetativecoverplus any soil evaporation.Added water is also weighedand thus an accountingof water contentcan be kept. Crops or cover are plantedon the areasurroundingthe lysimeterto provide uniformity of conditions surroundingthe instrument.Continuousrecordsat the set weighingintervalsprovide almostcontinuousmonitoring of conditions.The data obtainedcan be transmittedto any desirablelocation for analysisand/or other use.Weighinglysimeterscan produce accuratevaluesof evapotranspirationover short periodsof time. But they are expensive.Nonweighingtypes of lysirneters,which are lesscostly,havealsobeenused,but unless the soil moisture content can be measuredreliably by some independent method,the data obtainedfrom them cannotbe relied on exceptfor long-term measurementssuchas betweenprecipitationevents.s
and HumiditY Wind,Temperature, Measurementsof wind, temperature,and humidity are neededto supportmany types of hydrologicanalyses.Wind is commonly measuredusing an anemometer,a device that has a wind-propelledelementsuchas a cup (Fig. 8.2) or propellerwhosespeed is calibratedto reflect wind velocity. Wind direction is obtainedusing a vane,which orients itself with the direction of the wind.
130
CHAPTER 8 INSTRUMENTATIoN
Temperaturemeasurementsare madeusing standardthermometersof various types,while hurnidity is measuredusing a psychrometer.A psychrometerconsistsof two thermometers,one called a wet bulb, the other a dry bulb. Upon ventilation the thermometersmeasuredifferently, and this differenceis called the wet-bulb depression.By usingappropriatetables,dewpoint, vaporpressure,andrelativehumidity can , be determined.6 Figure 8.3 depictsa completeweatherstation incorporatingmeasurementsof precipitation, wind, temperature,barometricpressure,and humidity. Sucha station - can automatically report weatherdata from remote siteson either an event and/or
Funnel assembly Solar panel Tipping bucket
Lifting rope Antennacable
Main housing
Signal cable
Ground level
Transmitter
Figure 8.3 Self-reporting weather station. (Courtesy of Sierra-Misco, Inc., Environmental Products, Berkeley, CA.)
131 INSTRUMENTS 8.2 HYDROLOGIC timed basisto a central site. A station suchas this can be used for marine weather forecasting,quantitativedeterminationsof oncoming storms,determinationof wind effect on tidal areas,and establishinga data basefor irrigation.
OpenChannelFlow Measurementsof open channel(natural and created)flow are made using standard measuringdevicessuchas flumes and weirs, and they are also madeby calibrating special control sectionsalong rivers and streamssuchthat measurementsof depth (Jtage)of flow can be related to discharge.Flow-measuringdevicesare designedso that sensingsomeparametersuchas depth automaticallytranslatesthe observation into units of flow (discharge).When a control sectionis used,observationsof crosssectionalareafor variousdepthsmust be obtained,and averageflow velocitiesmust be ascertainedfor variousstagesso that a sectionrating curve can be established.In the United States,the U.S. GeologicalSurvey,the U.S. Bureau of Reclamation,the Soil ConservationService,andthe U.S. Army Corpsof Engineershavedoneextensive flow measuringand havebeen active in developinginstrumentsand proceduresfor ascertainingrates of flow.2'6 Weirs Weirs are common water-measuringdevices.When they are properly installed and maintainedthey can be a very simple and accuratemeansfor gauging discharge.The most often usedweir types are the rectangularweir and the V-notch weir (Flg. 8.4). To be effective,weirs usually require a fall of about0.5 ft or more in the channelin which they are placed.Basically,a weir is an overflow structureplaced acrossan open channel.For a weir of specificsize and shapewith free-flow steadystateconditionsand a proper weir-to-pool relation, only one depth of watercan exist in the upstreampool for a particular discharge.Flow rate is determinedby measuring the vertical distancefrom the crestof the overflowpart of the weir to the watersurface in the upstreampool. The weir's calibrationcurvethen translatesthis recordeddepth into rate of flow at the device. Parshall Flumes A Parshallflumeis a speciallyshapedopenchannelflow section that can be installedin a channelsection.Figure 8.5 depictsone of thesedevices.The flume has severalmajor advantages:(1) it can operatewith a relatively small head loss;(2) it is fairly insensitiveto the approachvelocity; (3) it can be usedevenundet submergedconditions;and (4) its flow velocity is usually sufficientto precludesediment depositsin the structure.2The Parshallflume was developedby the late Ralph L. Parshill and it is a particular form of venturi flume. The constrictedthroat of the flume producesa differential headthat canbe relatedto discharge.Thus,asin the case .ofthe weir, an observationofdepth (head)is all that is requiredto determinethe rate of flow at the control point. Weirs and flumes are generally best suitedto gauging small streamsand openchannels,althoughlargebroad-crestedweirs can be installed at dam sites as part of overflow structures.For major rivers, other measuringapproachessuchas developingfield ratings at a specifiedcontrol sectionmust be relied on.
132
CHAPTER8
INSTRUMENTATION
Figure 8.4 Field installation of weirs: (a) rectangular and (b) Vnotch. USDA CooperativeExtensionService,Mountain StatesArea.
Golttrol Sections where the installation of a weir, flume, or some other flowmeasuringdeviceis impractical, it is sometimespossibleto developa rating curve at somelocation alonga streamby taking measurements of depth,cross-sectionalarea, and velocity and calculatingthe rate of flow for a particular stageat the location.By doing this for a range of depths of flow, a station rating curve can be developed. Instrumentsrequired to developsuch a curve are depth-sensingdevices,surveying instruments,and velocity meters.The velocity meter is similar to an anemometer.It is placedat variouspositionsin the channeland a velocity is recorded.By doing this at a numberoflocations, a velocity profile for a given depth can be developed.From this an averageflow velocity can bJcomputed, ind uy uiing that determinationand
INSTRUMENTS 133 8.2 HYDROLOGIC Diverging sectlon Throat section a J
Altemate45" wing wall PLAN
t x t xf,tngle
SECTIONZ-f, Service.) Figure 8.5 Parshallflume.(U.S.Soil Conservation
the cross-sectionalarea,dischargecan be calculatedas the product of meanvelocity and cross-sectionalarea.If observationscan be madefor a rangeof depths,a rating of depthwill curvecanbe developedfor the control sectionsothat only measurements on this Additional information later time. flow at some to estimate rate of be needed proceduremay be found in Refs.2 and 6.
Valve shut-offkeys Connecting band Valves
Figure 8.6 Recorderhouseand stilling well for a streamgaugingstation. (U.S. Bureau of Reclamation.)
134
CHAPTER 8 INSTRUMENTATIoN
Depth (stage) Measurements Most depth measurementsare made using a float and cable arrangementin a stilling well or a bubbler gauge.In the first insiance,a stilling well connectedto the channel(Fig. 3.6) is used to housea float devicethat activatesa recorder as it movesup and down. Figure 8.7 illustratesa self-reporting stilling well liquid-level station. Data from this station can be transmitted to un! central location for analysisand/orother use.A bubbler-typeinstallation makesusl of dry air or nitrogenas a fluid for bubblingthrough an oriflce into a channelbed. As the depth of flow changes,the changein head above the bubbler orifice causesa
Antenna mast Cover with special accessscrews
Transmitter Hydraulic damping device Level sensor Counter weight Mounting brackets
lnlet tubes
Sidecleanout port Bottomcleanout port
Figure 8.7 Self-reportingstilling well liquid-levelstation. (Courtesy of Sierra-Misco,Inc., Environmental products. Berkeley,CA.)
8.4 REMOTE SENSING 135 correspondingpressurechange.This resultsin a fluid-levelchangein the manometer connectedto the gassupplyand this in turn is usedto reflect stagevariationover time. The foregoingdescriptionsare of a few of the instrumentsused in hydrologic work. Both the limitations associatedwith their use and their reliability must be understoodif they are to be used correctly and their outputs are to be considered credible.
8.3 TELEMETRY SYSTEMS Historically,many gaugeswere read periodically by an individual making the rounds of installations.This servedwell whenthe purposeof the data wasto establisha base record of somevariablesuchasrainfall. But today,undermany circumstances,it has becomenecessaryto make continuousrecordingsof rainfalls, streamflows,and evaporation rates and to have thesedata availablefor the real-time operation of water managementsystemsand for forecastinghydrologicevents.Someexamplesof activities requiring real-time hydrologicdataarc managingreservoirs,issuingflood warnings, allocating water for various usessuchas irrigation, monitoring streamflowsto ensurethat treatiesand pactsare honored,and monitoringthe quality and quantity of waterfor regulatoryand environmentalpurposes.Accordingly,gaugingstationscapable of electronicallytransmitting their data to a central location for immediateuse havenow becomecommon.The advantagesof suchstationsincludeproviding information to usersin a time frame that meetsmanagementneeds,reducingthe costsof collecting data, and providing a continuousand synchronousrecord of hydrologic events.Figure 8.8 showsa streamgaugereporting station using radio transmission. Figure 8.9 illustratesa satellitedata collection and transmitting operation.T-|2
8.4 REMOTE SENSING Sincethe 1960s,remote sensinghasbecomea commonhydrologictool. Examplesof aircraft and satellitedatacollectionand transmission abound.13-16 Figure8.10 illustratesthe useof aircraft and satellitesin a snowsurveysystem.Other typesof surveys suchasthoserelatedto determiningimperviousareas,classifyingland usesfor assessing basin'wide runoff indexes, determining lake evaporation, and groundwater prospectingcan be depictedin similar fashion.Table 8.1, which summarizesoperational uses of satellite data in hydrology circa 1981, showsthe great diversity of remote sensingand data transmissionoptions that can be exercised.l6 The principal value of remote sensingis its ability to provide regionalcoverage and at the sametime providepoint deflnition.Furthermore,satellitecommunications can be digitized and are thus compatiblewith the transferof computerizedinformation. Following the evolution of linkages between computer and communications technology,new softwaresystemsincorporatingpowerful data managementsystems havebeen developed.Thesesystemsfacilitate the storage,compaction,and random accessof large data banks of information. one data managementoption, geographic information systems(GIS), allows the overlayingof many setsof data (particularly satellite-deriveddata) for convenientanalysis.Versatilecolor pictorial and graphic display systemsare also becomingattractiveas their costshavedecreased.ra
136
CHAPTER 8 INSTRUMENTATIoN
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az*<3e -{/co '-Figure 8.8 Streamgaugereporting systemusing radio transmission.Water stage information is requestedfrom the gaugingstationsby VHF radio signal.In turn, this water stage information is obtained from the stream gaugesand automatically encodedand transmitted to the Boulder City receiving station. All downstream releasesfrom Hoover Dam are determinedand integrated with this streamflow information in controlling the flow of the lower ColoradoRiver. (U.S. Bureau of Reclamation.)
8.4
REMOTESENSING
Figure 8.9 Hydrologic data collection by satellite.(U.S. GeologicalSurvey.)
Low-altitude gamma-ray light.
Water 4gency forecastcenter
Figure 8.10. Satellite snow survey system.(AftenCalabreseand Thome'rs)
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CHAPTER 8 INSTRUMENTATIoN With the advancementof satellite technology,the use of satellitesas remote sensorplatformshasspread.Currently availablesensorscan operatein a multitude of electromagneticradiation wavelengthsand the information content of their signals can include Surfacetemperatures,radiation, atmosphericpollutants,and other types of meteorologicaldata. As remote sensorsare improved to permit the attainmentof greater radiometric and geographicresolution, and as computer image-enhancing techniquesbecomemore sophisticated,it is certain that this powerful watermanagement tool will seeeven sreaterand more diversifieduse.
r summary Hydrologic data are important componentsof model design and testing and of a variety of statistical analyses.The quality of data obtained relate to attributes of measuringinstrumentsand to the featuresof gaugingsites.It is important to understandthe pros and consof variousinstrumentsand to know how they canbestbe used.
REFERENCES 1. "Irrigation WaterMeasurement,"Mountain StatesRegionalPubl. 1, revisionof Extension Circ. 132,Irrigation Water Measurement,University of Wyoming, Laramie, June 1964. 2. U.S. Bureau of Reclamation,Water MeasurementManual, 2nd ed. Washington,D.C.: U.S. GovernmentPrinting Office, 1967. 3. Leupold & Stevens,Inc. StevensWater ResourcesData Book, 4th ed. Beaverton,OR: Leupoldand Stevens,Jan. 1987. 4. W. Brutsaert,Evaporationinto the Atmosphere.London: D. Reidel Publishing, 1982. 5. "TexasLab InstallsWeighingLysimeters,"Irrigation J. 37(3),8-12(May/June 1987). 6. R. K. Linsley, Jr., M. A. Kohler, and J. L. H. Paulhus,Applied Hydrology. NeW York: McGraw-Hill, 1982. 7. R. J. C. Burnashand T. M. Twedt, "Event-ReportingInstrumentationfor Real:TimeFlash Flood Warning," American Meteorological Society, Preprints, Conference on Flash Floods:Hydro-meteorological Aspects,May 1978. 8. D. E. Colton and R. J. C. Burnash,"A Flash-FloodWarningSystem,"American MeteorologicalSociety,Preprints,Conferenceon Flash Floods:Hydro-meteorologicalAspects, May 1978. Aug. 1980. 9. R. J. C. Burnash,"AutomatedPrecipitationMeasurements," 10. R. J. C. Burnashand R. L. Ferral, "A SystemsApproach to Real Time Runoff Analysis with a Deterministic Rainfall-Runoff Model," International Symposium on RainfallRunoff Modeling,Universityof Mississippi,May 18-21, 1981. 11. Hydrologic ServicesDivision, National WeatherService,WesternRegion, "Automated Local Evaluationin Real Time: A CooperativeFlood Warning Systemfor Your Community," Feb. 1981. 12. R. J. C. Burnashand R. L. Ferral, "Examplesof Benefitsand the TechnologyInvolvedin Optimizing HydrosystemOperation Through Real:Time Forecasting," Conferenceon Real:TimeOperation of Hydrosystems,Waterloo,Ontario, June24-26, 1981. 13. M. Deutsch,D. R. Wiesnet, and A. Rango (eds.),Satellite Hydrology. Bethesda,MD: AmericanWaterResources Association.1981.
.
REFERENCES 143
"Agricultural Meteorology: SystemsApproacli to Weather and Climate Thirty-Second Needsfor,tgricuttrire, Forestry,and Natiiral iesources,o'in Proceedings, ResearchInstitute, Agricultural MD: Bethesda, R'searcih'Institute. Agiicultural tuteeting, 1 9 8 3 p, p . 7 5 - 8 5 . *NASA Water Resources/HydrologyRemote Sensing 1 5 . M. A. Calabreseand p. G. Thome, Programinttrelg80's,"insatettiteHydrology(M'Deutsch'D'R'Wiesnet''andA'Rango' 1981' pp' 9-15' eds). Bethesda,MD: American Water ResourcesAssociation' 16. G.K.Moore,..AnlntroductiontosatelliteHydrology,''hsate.tlite-Hydrology(M. WaterResqurces Deutsch,D. R. Wiesnet,and A. Rango,eds.).Bethesda,MD: American t7-4L. Association,1981,PP.
t4. J. F. Bartholic,
ChapterI
MonitoringNetworks
Prologue The purposeof this chapteris to: . Describeelementsof systemsfor monitoring hydrologicvariables. . Indicate the importanceof real-time and continuousrecording of hydrologic events. Information (data)is the requisitefoundationfor designingschemesfor manipulating (managing)hydrologicsystems,for evaluatingthe efficacyof actionstaken to correct problem situations,and for identifying trouble spots deservingattention. But to be useful, the data must be of the right type, in the right form, and appropriatelyrepresentativeof critical spaceand time dimensions. Modeling hydrologic systemsrequires an understandingof how these syster-ns actually function; cleaningup a toxic wastedischargerequirestracking the effectsof remedial actions;enforcing environmentalregulationsrequires knowledgeof what has happenedsincethe rules were implemented;and regulatingreservoirreleasesto meet specifiedtargetsrequiresa continuousunderstandingof the stateof the system beingoperated.The key to meetingsuchrequirementslies in the productsof carefully designedand managedmonitoring networks.Developingsuchnetworksis no small task, however,as the numberof variablesthat must be observedmay be very large, the instrumentsto measurethem costlyto install and operate,andthe datastorageand managementrequirementsextensive.Accordingly, a monitoring network's design must beginwith a thoroughunderstandingof its purposeso that the degreeof resolution providedby its observationsis adequate,but not excessive,for the task at hand. A goodrule is to keepthe networkassimpleaspossible,within the constraintsof what must be accomplished.
9.1 THEPURPOSE OF MONITORING The purpose of monitoring is to gather information in a continuum such that the dynamics of the systemcan be ascertained.According to Dressing,objectives of monitoring for nonpoint source pollution control include developmentof baseline
9.2 SPECIAL CoNSIDERATIoNS 145 information, generatingdata for trend analysis,developingand/or verifying models, and investigatingsingleincidentsor events.zTheseobjectivesare alsovalid for hydrologic monitoring in general,but they shouldbe supplementedby the following obje"tives:planning,real-time systemoperating,enforcingregulatoryprograms,and environmental policymaking. In the flnal analysis,the ultimate purposeof monitoring is to enhancedecisionmaking,whetherit be for development,management,regulation, or researchaims.
9.2 SPECIALCONSIDERATIONS Beforean acceptablemonitoringplan can be devised,theremustbe a full understanding of the hydrologic systemto be monitored and of the objectivesto be met by monitoring. The costs of monitoring can be very high and thus it is essentialthat monitoring networksbe efflcient and cost effective.r-5
Timeand SpaceVariability In general, monitoring networks are designedto have both spatial and temporal dimensions.Although monitoringa specificpoint locationmay be all that is necessary undersomecircumstances, it is more commonthat what is happeningin a regional settingis of importance.The temporal aspectis similar. While a snapshotat some point in time may suffice for some purposes,the time varianceof conditions to be tracked is usually critical for effective analysesandlor decisionmaking.Both the short-termand long-term variabilities of many targefsof monitoring must be ascertained. For example,water quality in a streamcan changerapidly with time, while changesin lake levels,suchas thoseexperiencedin the GreatLakes in the 1980s,are the result of long-termhydrologicvariability. Spatial variability must also be representedin a monitoring network: for example, infiltration ratesmay vary considerablywithin a region, rainfall intensitiesmay be quite different within even short distances,and water quality in a river might be different in upstreamand downstreamlocations.Topography,soils, vegetalcovers, and many other factors affecting the performanceof a hydrologic systemare also distributed differently in space,and these differencesmust be recognizedin the monitoring plan. The trick is to developa monitoring systemthat can (1) provide the neededdata,(2) recognizeregionaland temporal variabilities,and (3) keep installation, operation,and maintenancecoststo a minimum. To do this requiresa comprehensiveknowledgeof the systemto be monitored,an understandingof what the data obtainedby the systemwill be used for, and a knowledgeof the level of detail in collecting the data that must be exercisedin spaceand time.
DataRequirements The amountand type of datato be generatedby a monitoringsystemmustbe carefully consideredin its design. Selectingappropriateinstruments,determining sampling frequency,and settingdata formats are elementsthat must be considered.Questions such as how much do we need to know and when do we need to know it must be answered.The form and extensiveness of data must be tightly relatedto monitoring
146
CHAPTER9
MONITORINGNETWORKS
Flood warnlng Water managemeff
---: Telephone
Lift station monitoring control
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lnfrared + + xgnr Water quality monitoring
(Courtesyof Sierra-Misco, Inc.,EnvironFigure9.1 A telemetrymonitoringsystem. mentalProducts. Berkeley. CA.
,
objectives.Furthermore,it might be necessaryto monitor surrogatesinsteadof the condition to be tracked.' For example,if lake eutrophicationis the issue,phosphorus If this approachis taken, andchlorophyllconcentrationsmight be surrogateflre&sureS; selectionof appropriatesurrogatesis very important and the foregoingcomments aboutdata formats and so on are also applicable.Hydrologic,waterquality, land use and treatment, topographic,soils, vegetativecover, meteorologic,and many other typesof datamaybe requiredin combinationor separatelyin a monitoringplan. It is easyto seethat the amount of data required for a monitoring program can be enormous.Consequently,great care must be taken to seethat the data collection effort is not in excessof the objectivesof the monitoring program. Figure 9.1, depicting a telemetrymonitoring system,gives an indication of the variety of data that might be collectedin a monitoring program.
QualityControland QualityAssurance The costsof monitoring are usually substantialand thus it is essentialthat the data generatedbe of consistentlyhigh quality. Accordingly, most monitoring systems include quality control and quality assurance(QA/QC) elements.Quality control is a plannedsystemof activitiesdesignedto producea quality product(datain this case) that meetsthe needsof the user. Quality assuranceis a plannedsystemof activities designedto guaranteethat the quality control programis being carried out properly. A quality managementplan should be part of the overall monitoring program and should be preparedwhen the monitoring program is being developedto ensure that the data collectedwill be of a satisfactorynature for the monitoring program's objectives.3
9,4
NETWORKS HYDROLOGICAL-METEOROLOGICAL
147
To useror
Figure 9.2 Microcomputeruse in streamgauging.
IN MONITORING 9.3 USEOF COMPUTERS With the rapid technologicaldevelopmentof computers,especiallyinexpensivemicrocomputers,the oppoitunities foiautomated collection of all types of hydrologic and water quality data have increased substantially.Microcomputers, used with analog-to-digital converters,pressureor liquid-level sensors,and the appropriate software can, for example, be used in hydrologic monitoring systemsas flow metering/dataacquisitionsystems(Fig.9.2).4Furthermore,suchsystemsare highly versatile and they are rblatively inexpensive.Computer systemscan be customdesignedfor almost any dataacquisitionapplicationand they are often lesscostly than other commerciallyavailablehardwaresystemsdesignedfor the samepurpose.Computerscan convert raw datainto other more usefulforms, storedata for later use,and communicatewith other computerterminalsif necessary.As such,they are a powerful and important componentof modern hydrologicmonitoring systems.Figure 9.3 illustratesthe use of computersin a real-time telemetry system.
NETWORKS ETEOROLOGICAL e.4 HYDROLOGICAL-M Most modern hydrologic-meteorologicnetworks are designedto provide real-time information for purposessuchas hydropowerscheduling,releasingflows for irrigation, developingand testinghydrologicsystemmodels,regulatingreservoirdischarges, allocating water from multiple sources,streamflow forecasting,tracking pollutant transport, and enforcing environmental regulations. Hydrological-meteorological
148
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MONITORINGNETWORKS
Water level
Weather station
Computer
:,/: Modem Data Collection
(Courtesy Figure9.3 Computer usein a real-timetelemetrysystem. of Sierra-Misco, Inc., Environmental Products, Berkeley,CA.)
networks may be designedto monitor physiographic,climatic, hydrologic,biologic, and chemicalfeatures,or combinationsof these,in a region or river basin.They must have gaugedensitiesand distributionsthat are sufficientto permit interpolationbetweengaugesitesin a mannerpermitting valid conclusionsto be drawn for the entire areacoveredby the network. Typically, measureinentsare madeof suchvariablesas precipitation, solar radiation, temperature,relative humidity, barometric pressure, snow depth, soil moisture,wind speed,streamflow,and water quality. In any event, special basin or regional climatic factors must be given due consideration.Each hydrological-meteorologicalnetwork is different in its purposeand setting and thus its designmust reflectboth the spatialand temporally varying featuresat the locality to be monitored along with the objectivesof the monitoring program.s
REFERENCES 149
r Summary Monitoring of hydrologicsystemsis essentialto better understandingof systeminteractions and to the designand testing of hydrologicmodels.It is also the meansby which a determinationcan be made of the effectivenessof measurestaken to alter watershedperformance.
REFERENCES t . S. J. Nix and P. E. Black (eds.),Proceedingsof the Symposiury. on Monitoring, Modeling, and Mediating Water Quality. Bethesda,MD: American Water ResourcesAssociation, 1987. 2. S. A. Dressing, "Nonpoint Source Monitoring and Evaluation Guide," in Ref. 1, pp. 69-78. J. J. Lawrenceand A. S. Y. Chau, "Quality Assurancefor EnvironmentalMonitoring," in Ref. I, pp. 165-176. 4. H. E. Postand T, J. Grizzard, "The Monitoring of StreamHydrology and Quality Using Microcomputers,"in Ref. 1, pp. 199-208. 5 . P. J. Gabrielsenand A. J. Carmeli, "Operation of a Hydrologic-MeteorologicMonitoring Networkin a SevereWinter Environment,"in Ref, l, pp. lI3-122.
PARTTHREE .
SURFACEWATER HYDROLOGY
C h a p t e r1 0
Runoffand the Catchment
r Prologue The purposeof this chaPteris to: . ' . . .
Expandon definitionsof termsfrequentlyusedin describingtherunoff process. Presentconceptsof surfacerunoff and drainagebasin discharge. Introduce elementsof drainagebasin geomorphology' Describequantitativemeasuresof watershedcharacteristics. Familiarize the readerwith elementsof frequencyanalysis'
Surfacewater hydrologydealswith the movementof water alongthe earth's surface as a result of precipitation and snow melt. Detailed analysisof surfacewater flow is highly important to suchfieldsasmunicipaland industrialwater supply,flood control, stieamflowforecasting,reservoirdesign,navigation,irrigation, drainage,water quality control, water-basedrecreation,and flsh and wildlife management. The relationbetweenprecipitationandrunoffis influencedby variousstormand basin characteristics.Becauseof thesecomplexitiesand the frequentpaucity of adequaterunoff data, many approximateformulashavebeendevelopedto relaterainfall andrunoff. The earliesfofihese were usuallycrudeempirical statements,whereasthe trend now is to developdescriptiveequationsbasedon physicalprocesses.
AND DRAINAGEBASINS WATERSHEDS 10.1 CATCHMENTS, Runoff occurswhen precipitationor snowmeltmovesacrossthe land surface-some of which eventuailyreachesnatural or artificial streamsand lakes'The land areaover which rain falls is called the catchmentand the land area that contributessurface runoff to any point of interestis called a watershed.This can be a few acresin size or thousandiof ,quut" miles.A large watershedcan contain many smaller subwatersheds. Streamsand rivers conveyboth surfacewater and groundwaterawayfrom high water areas,preventingsurfaceflooding and rising groundwaterproblems'The tract
154
CHAPTER 1O RUNOFF ANDTHECATCHMENT of land (both surfaceand subsurface)drainedby a river and its tributariesis called a drainage basin. A watershedsuppliessurfacerunoff to a river or stream,whereasa drainagebasinfor a given streamis the tract of land drainedof both surfacerunoff and groundwaterdischarge. Rain falling on a watershedin quantitiesexceedingthe soil or vegetationuptake becomessurface runoff. Waterinfiltrating the soil may eventuallyreturn to a stream and combine with surfacerunoff in forming the total drainage from the basin. The network of overlandflow coursesand defineddrainagechannelscomprisethe watershed.Surfacerunoff from tracts of land beginsits journey as overlandflory, often calledsheetflow,beforeit reachesa definedswaleor channel,usually beforeflowing more than a few hundredfeet. The lines separatingthe land surfaceinto watersheds are called divides.Thesenormally follow ridges and moundsand can be delineated using contourmaps,field surveys,or stereographpairs of aerial photographsto identify gradient directions.
ContributingArea In the majority of hydrologicanalyses,the magnitudeof total surfaceareacontributing direct runoff to somepoint of interestis needed.Becauseof variationsin topography, the true surfaceareacannotbe easilymeasured.The horizontalprojectionofland area is easilyobtainedand normally adoptedin hydrologiccalculations.This resultsin an error in actual watershedarea whereverthe projected area is less than the actual. Somesurfaceareain watershedsmay not contributeto surfacerunoff, so the error in using the projectedwatershedareais somewhatoffset.
PartialArea Hydrology For light storms,or for someflat areas,portions of the catchmentdo not contribute to runoff. Precipitationfalling on or flowing into depressedor blockedareascan exit only by seepageor evaporation,or by transpirationif vegetated.If sufficientrainfall occurs, such areasmay overflow and contribute to runoff. Thus the total area contributing to runoff varies with the intensity and duration of the storm. Methods for incorporating this phenomenonin hydrologic studies are calegorizedunder proceduresfor partial area lrydrology. In partial areahydrology,watershedareasare dividedby one of severalmethods into contributing (active) and noncontributing (passive)subareas.For infrequent (severe)storms,largerpercentagesof the watershedsurfacemay contributeto the peak flow and volume of runoff, which are the primary variables of interest to design engineers.For more frequent storms,significantly smaller portions of some waterpartial areahydrologyis seldomincorposhedsmay contribute.As a consequence, ratedin hydraulic structuredesign,and is of greaterinterestin watersupplyand water quality studies.As will be shownlater (Chapter12) unit hydrographtheory andrunoff curve numbermethodsare basedon linearity of rainfall and runoff, and assumethat the full watershedcontributesto runoff in all stormsand in proportional amountsat different times in the samestorm. Application of thesemethodsto watershedsthat havesignificantnoncontributingzonescould, and do, introduceerror ifthe zonesare nof first delineatedand the distributedeffectsproperly modeled.
10.2 BASINCHARACTERISTICS AFFECTINGRUNOFF
155
Subdivisionof contributing from noncontributingareashas traditionally been subjectivelyaccomplishedfrom siteinspection,topographicand soilsmaps,and aerial photos.Soils having good drainageclassifications,or dark tones or colors on aerial photos, can often be consideredas passiveareas.Other signs of noncontributing areaswould includepresehceof wetlands,grassedareas,rooftops (unlessconnected to the drainage),terraces,erosioncontrol structures,stockwateringponds,and flood control dams. Boughtontdevelopeda quantitativemethod of determiningthe proportion of a watershedthat contributessurfacerunoff in different storms,and at different times duringthe samestorm.by analyzingrainfall andrunoff records.His logic is asfollows: 1. Watershedscan be idealizedas a group of "surfacestoragecapacity" cells, eachrepresentinga fraction of the watershedarea and eachhaving some capacityto abstractrainfall into storage,infiltration, or evapotranspiration. 2. Runoff from eachcell occurs when rain fills the surfacestoragecapacity. 3. Runoff occursfrom the cell with the smallestcapacitybeforeflowing from the cell with the next largestcapacity(this is an assumptionby Boughton that has not been fully verified). 4. Using theseprinciples,storm data for the watershedare evaluatedfirst to find thosein which runoff occurs only from the area of smallestcapacity. This is done using a graphicalmethod outlined in the article that looks at slopechangesin the rainfall-runoff graph.Both the capacityof the cell and its area as a percentageof the watershedare estimated. 5. After subtractingthe contributionto runoff from the smallestcapacitycell, the capacityand contributingareafor the secondsmallestcapacitycell are determinedby the sameprocedure. 6. The process is repeateduntil all the runoff is accountedfor, or until 100 percent of the watershedis contributing,whicheveroccurs first. When Boughton applied the procedureto a test watershed,rit was found that runoff occurredfrom the entire watershedon only 3 of 30 eventsin the l5-year study period.In abouttwo-thirds of the runoff events,dischargeoccurredonly from the cell with the smallestsurfacestoragecapacity.
10.2 BASINCHARACTERISTICS AFFECTING RUNOFF The natureof streamflowin a regionis a function of the hydrologicinput to that region and the physical,vegetative,and climatic characteristics.As indicatedby the hydrologic equation,all the waterthat occursin an areaas a result ofprecipitation doesnot appearas streamflow.Fractionsof the grossprecipitationare divertedinto pathsthat do not terminatein the regional surfacetransport system.Precipitation striking the groundcango into storageon the surfaceor in the soil andinto groundwaterreservoirs beneaththe surface.The characterof the soil and rocks determinesto a large extent the storagesysteminto which precipitatedwaterwill enter.Opportunity for evaporation and transpirationwill also be affectedby the geologicand topographicnatureof the area.
156
CHAPTER1O RUNOFFAND THE CATCHMENT
StreamPatterns Wind, ice, and water act on land surfacesto createseveraltypes of drainagepatterns seen in nature. The particular design that results is a function of several factors including slope,underlying soil and rock properties,and the historiesof hydraulic action, freeze-thaw activity, and sedimenttransport.
(a) Dendritic system
(b) Trellis system
(c) Radial system
(e) Meandering stream
(f) Braided stream
(g) Anabranching stream
(h) Reticulate stream
Fi-gxre 10.1- -Streampatterns(combinedsystemsand individual streamshapes).
10.2 BASINCHARACTERISTICS AFFECTINGRUNOFF
157
Typical streampatternsare shownin Fig. 10.1.The most common,dendritic, is characterizedbynumeroussmall tributariesjoining at right anglesinto higher-order streams,eventually forming the major rivers in the region. The smaller tributaries often occur in sufficientquantitiesthat little land surfaceareais left uninterceptedby a definedchannelof someform, The maximum overlanddistancebetweenchannels in theseareasseldomexceedsa few hundredfeet. Trellis patternsare characterizedbylong main streamsinterceptedby numerous shorterright-angletributaries.They are commonin the Appalachians(EasternUnited States)and are also seenin the Rocky Mountains along the foothills, Multi-basin patterns,also called derangedsystems,occur in low gradient swampy areaswith numeroussurfacedepressionsand normally haveonly a few tributaries.Theseoccur in glaciated,windblown,and permafrostareas,and are common in plains and mountain valley regionsof the United States.Radial patternsare typically found in foothill areasor mountain areaswith more advancedsoil development. Individual streamsfavor one or more of the four patternsshown on the lower portion of Fig. 10.1.Streamsare rarely straightexcepton steepslopesin homogeneous materials. Braided streams are characterizedby numerous interconnected channelsflowing aroundand overislandsandbars,inundatingmostduringhigh flows. Braided streamsare generallytransportinglarge amountsof sediment,but often less than the amount supplied. They have been called incipient forms of meandering streamsdueto the fact that many revert to meanderingor other forms when sediment suppliesor other factorschange.Meanderingin an otherwisestraightchanneloccurs as a result of transversecurrents.Thesecurrents are consideredto be the result of forcesactingon the streamparticles,includingbed and bank shearforcesand coriolis effects. In evaluatingthe effects of changesin streamflows,a relationship known as Lane's Law is often applied.It statesthat the productof bed slopeand waterdischarge is proportional to the product of sedimentsize and transportrate. Changingany one of thesefour terms results in the likelihood of a shift in one or more of the others. Constructinga reservoit,for example,reducesthe sedimenttransportedinto the reach just downstream.By Lane's Law, either the slope or dischargemust decreaseor sedimentparticle sizemust increaseto offsetthe changein sedimenttransport.Most often, the slope decreaseswhen the sediment-hungryflows deepenthe bed in the reach.Degradation(downwardcutting) of the bed of the Missouri River, for example, has occurredbelow someof its upstreamreservoirs,isolatingboat marinasand water intake structuresin somelocations.
Geomorphology of DrainageBasins The principal geologicfactorsthat affect surfacewatersare classifiedaslithologic and structural. Lithologic effects are associatedwith the composition,texture, and sequenceof the rocks, whereasstructuraleffectsrelatemainly to discontinuitiessuchas faults and folds. A fault is a fracture that resultsin the relative displacementofrock that was previouslycontinuous.Folds are geologicstrata that are contortedor bent. Variationsin the erodibility of the different strata can easily lead to the creation of distinctive forms of drainagesystems. Both large-scaleandlocal effectson the storageand movementof surfacewaters existbecauseof geologicactivity and structure.For example,drainagepatternsare
1 58
CHAPTER10
RUNOFFAND THE CATCHMENT
determinedto a large extentby the nature of land forms. On the other hand, flowing surfacewatersalso affect the surfacegeometrythrough the processof erosion.Thus significant land forms resulting from volcanic activity, folding, and faulting affect drainage,whereasdrainagepatterns,havingbeengenerated,can alsomodify the land forms by creatingvalleys,deltas,and other geomorphicfeatures. Streamsare classifiedasbeingyoung,mature,or old on the basisof their ability to erodechannelmaterials.Youngstreamsare highly active and usually flow rapidly so that they are continually cutting their channels.The sedimentload imposed on thesestreamsby their tributariesis transportedwithout deposition.Mature streams are those in which the channel slope has been reduced to the point where flow velocitiesarejust able to transportincoming sedimentand where the channeldepth is no longerbeingmodifiedby erosion.A streamis classifiedasold whenthe channels in its systemhavebecomeaggraded.The flow velocitiesof old streamsare low dueto gentleslopesthat prevail.Wide meanderbelts,broad flood plains,anddeltaformation are alsocharacteristicof old streams.The lower reachesof the Mississippi,Rhine, and Nile are examples.Flows in young river basinsare often o'flashy,"*h"t"u, sluggish flows are common to older streams. The description of a drainagebasin in quantitative terms was an imponant forward stepin hydrologyand can be tracedback in largepart to the efforts of Robert E. Horton.2Strahler,Langbein,and othershaveexpandedHorton's original work.3-a To quantify the geometry of a basin, the fundamental dimensionsof length, time, and mass are used. Many drainagebasin featuresthat are important to the hydrologistcan be quantified in terms of length, length squared,orlength cubed. Examplesare elevation,streamlength, basin perimeter,drainagearea, and volume. The conceptof geometricsimilarity can be appliedto drainagebasinsjust as it is to many other systems.3 Most readerswill be awareof model-prototypestudiesof aircraft, dams,and turbomachinery.Suchstudiesinvolveconsiderationsof geometricas well as dynamic similarity. In the samemannerthat inferencesas to the operationof a prototype can sometimesbe drawn from a geometricallysimilar model, inferences canalsobe drawnaboutthe operationof one drainageareaon the basisof information obtained from a similar one. Perfect similarity will never be realized if natural drainagesystemsare compared,but striking similarities have been observedwhich can often be put to practical use.
Measuresof DrainageBasin Characteristics Important measuresof drainagebasin characteristicsinclude overlandflow lengths and streamlengths.The conceptof streamorder is often associatedwith the dimension of streamlength.
AFFECTINGRUNOFF 1O.2 BASIN CHARACTERISTICS
159
'/, "81 {,'v '\'rr
;)
Figure 10.2 Sketch indicating definition of streamorder.
to the streamorder, provided that a large enoughsampleis investigated.The order number permits comparisonsof drainage systemsthat are quite different in size quantity.Suchcomparisonsshouldbe madeat becausethe numberis a dimensionless locations in the two systemsthat have a similar geometry; that is, second-order streams,third-order streams,and so forth. Streamlengthsare determinedby the measurementof theh projectionsonto a If horizontal plane. Topographicmaps are useful for obtaining suchmeasurements. possible the meanlenglh of a streamsegmentZ, of order a is definedas L,, then it is to determine Z, using2
.
tr-
-2!i, L,,
(10.1)
N,
whereN, is the number of streamsegmentsof streamorder u. Another measurerelated to stream length is the distanceL"o from a point of intereston the main streamto a point on the primary channelthat is nearestthe center of gravity of the drainage arca (center of gravity of the plane atea of the drainage basin).Studiesof basinlag (time betweenthe centersof massof effectivestorminput and the resulting runoff ) havemadeuse of this dimension. Of particular significancein the physiographicdevelopmentof a drainagebasin is the overlandflow length Ls. This is the distancefrom the ridge line or drainage divide, measuredalongthe path of surfaceflow which is not confinedin apy defined channel,to the intersectionof this flow path with an establishedflow channel.If a drainagebasin of the first order is the basicelementof a larger drainagesystem,then a representativeoverlandflow length can be determinedfor thesefirst-orderbasins. One apprbachis to measurea numberof possibleflow paths from a map of the area and-toaveragethese.In somecases(for example,with the rational method,Chapter
160
CHAPTER1O RUNOFFAND THE CATCHMENT
15), the use of the longestoverland flow length is prescribed,measuredfrom the upstreamend of the first-order stream to the most remote point of flow that will terminateat this point.
Areal Measurements Just as linear measuresrelate to many factors of hydrologic interest, so do areal measures.For example,the quantity of dischargefrom any drainagebasinis obviously a function of the areal extentof that basin. Correlationshavebeenobservedbetweenthe averagearea,Au, of basinsof order u, andthe averagelength of streamsegments,2,. Thesevariablesare often relatedby an exponentialfunction. For example,studiesof sevenstreamsin the Maryland-Virginia areaby Hack haveproducedthe relationship6 L :
(10.2)
I.4Ao6
where L = the streamlength measuredin miles to the drainagedivide A : the drainagearea(miz) Hack's observationsindicatethat as the drainagebasinincreasesin size,it becomes longer and narrower; thus precisegeometricsimilarity is not preserved. Drainageareahas long beenusedas a parameterin precipitation*runoff equations or in simpleequationsindexingstreamflowto area or other parameters.Many early empirical equationsare of the form3 Q: cA: where Q a measureof flow suchas mean annual runoff A : the size of the contributine drainaeeatea
(10.3)
Valuesof c and m are determinedby regressionanalysis(seeChapter26); Fig. 10.3 illustratesa relation of this form.
€o r u
q
8
( ( ) 6 =
a
I
3
./
10
20
30 40 5060 80 100
200
Areatmi2) Figure 10.3 Runoff-drainage arca correlation for five Maryland streams (1933 storm
AFFECTINGRUNOFF 1O.2 BASINCHARACTERISTICS
161
Other arealmeasuresincludedefinitionsof the basinshapeandthe densityof the drainagenetwork or drainage density, definedas the ratio of total channelsegment lengthi cumulatedfor all streamorderswithin a basin to the basin area.The stream segmentsin a drainagebasin (total frequency is defined as the summation of all number of segmentsof all orders)divided by the drainagearea.
Channeland BasinGradients surface The slopesof a drainagebasinand its channelshavea very strongeffecton the characteristic the profiles exhibit channel stream runoff pro""r, of thairegion. Most this of decreasingslopeproceedingin a downstreamdirection.Figure 10.4illustrates particular.trait. Also illustrated in elevationdrop dividedby the channt suchthat the areasbetweenthe ave that is, A, : Azin the figure. The g asparametersto describedrainaget describe make this clear. Some mathemalicalfunctions that are used to more fully streamprofilesarelinear, exponenti ical value to representthe Primal Schwartz.TThis factor,known as th a uniform channelthat is equivalen sametraveltime. This factorhasber maximum from the centerof massof rainfall excessto the peak rate of runoff ) and discharge. In additionto the slopeofthe streamchannel,the overall land slopeofthe basin slopes is an important topographicfactor.A quantitativerelation betweenvalley wall method used commonly A Strahler.3 by and streamcnannetstopeshas been derived method of determiningthe slopesof a basin has been presentedby Horton'SThe area drainage of the map topographic a grid over involvessuperimposinga transparent drainage the with intersections its b"t*""n in question.nacfr grlJnne is measu."d needed' divide; the numberof intersectionsof eachgrid line with a contourline is also using made be then A determinationof the land slopecan nsec0, S:--t
!
Distance from head of stream
Figure 10.4 Typical streamprofile'
(10.4)
162
oHAPTER 10 RUNoFFAND THEoATCHMENT Drainage boundary Horizontal grid line
r
uContour'
Contour interval = 50 ft scaie
Figure 10.5 Determinationof meanland slope:numberof vertical intersections: 72; tumber of horizontal intersections= 120; total length of vertical grid segments: 103,900ft; total length of horizontalgrid segments: 101,200ft. 72x50 120 x 50 : 0.035ftlft : 0.059ftlft Ss " : 10 t , 2 0 0 103;900 .s + (.. 0.035+ 0.059 : 0.047ftlft Mean slope : 2
,s:
where n : the total number of contour intersectionsbv the horizontal and vertical grid lines t the total length of grid line segments(horizontal and vertical) h - the contour interval 0 * the anglemeasuredbetweena normal to the contoursand the grid line L
_
Because0 is very difficult to measureit is often neglected,and separatevaluesof averageslopein the horizontal and vertical are computedand then averagedto obtain an estimateof the meanland slope.This procedureis illustratedin Fig. 10.5.
Area-ElevationRelation How the areawithin a drainagebasinis distributedbetweencontours(Fig. 10.6)is of interestfor comparingdrainagebasinsand gaining insight into the storageand flow characteristicsof the basin.For suchstudies.an areadistribution curve suchas that shown in Fig. 10.7 is used.The curve can be obtained by planimeteringthe areas
RUNOFF AFFECTING 10.2 BASINCHARACTERISTICS
(
163
l l \ ' / t r 1 l l \ / / / / r r \ f t /
. ' , \: \ I . i/ z ' r ' \
,ffi_-_.w;i.i .Figure 10.6 Topographic map of WendyRun drainagearea showing20-,40-,and60-ftcontourlines. betweenadjacentcontoursor by using a grid as in Fig. 10.5 and forming the ratio of the number of squaresbetweencontoursto the total number of squarescontained within the drainageboundaries.The mean elevationis determinedas the weighted averageof elevationsbetweenadjacentcontours.The medianelevationcan be deterrnined from the area-elevationcurvesas the elevationat 50 percent,
-
e
300
E 2so o
€
2oo
Io
l)u
r! '
100
Median elevation
0
10
20
30
40
50
60
70
80
90
of area Percentage
Figure 10.7
An area-elevation distribution curve.
100
1 64
CHAPTER10
RUNOFFAND THE CATCHMENT
DrainageBasin Dynamics Geomorphology'likg hydrology,was largely qualitative in nature in its formative years'With the passingof time and the greatlr needfor reliablequantitativeinformation, the sciencehasprogressedto the point whererational relations betweenvariables are being developed.Theserelationsire usually intendedto quantify theinieractions betweenthe factorsthat modify the land forrnand the land iorm ifsef. In addition, equationsrelatingthegeomorphicpropertiesto hydrologic, climatologic,or vegetative factorsare beingsought.some of the iunctional ielations of particulir significanceto the hydrologistwill be discussedin the following chapters.
10.3 RUDIMENTARYPRECIPITATION-RUNOFF RELATIONSHIPS ritation and runoff has beento plot annual rend line, and estimatethe percentageof rtities determinedthis way, however, are :e ofreliability is higherfor drainageareas rnal or other types of variation, that is, an e procedure.The resulting equationtakes
the form
o:($)rr-Pr)
(10.5)
1902. I
1898r
^ 4 0
8 3 6
X 7 ,7 23
--l--
1 926
--i--I
. ) J +
32
a 903
a
1 9l 0 r - J 6 .E
l .
I t28
trg
a
Slope
s
0.57
29_
1920 1l 1924
: -1 9 1 8 i q)5-oL f--a 7921o
1932
1930
o1!
o
. 1897 90 0
1
2
3
4
5
6
7
8
9
1 0 1 1 t 2 t 3 ! 4
Runoff (in.)
Figure 10.8 Annual precipitation and annual runoff in the NeoshoRiver basin above Iola, Kansas.(U.S. GeologicalSurvey Data.)
165 10,3 RUDIMENTARYPRECIPITATION-RUNOFFRELATIONSHIPS
S : the slopeof the line (LPILQ) Pa : & baseprecipitation value below which Q is zero From Fig. 10.8 the relation for the examplewould be Q: 0'57(P 24) where Q and P are the annual runoff and precipitation, respectively,in inches' Considerablescatterof severaldata points from the assumedrelation indicatesthat this type of computation should be used with care. For rough approximationsin preliminary planning studies,suchmethodsare frequently helpful, however.EquationsresembiingEq. 10.5areimprovedif otherparameterssuchasantecedentprecipitation, soil moisture,season,and storm characteristicsare included. Suchrelations can be describedusing multiple regressiontechniquesor graphicalmethods'Linsley and co-workerspresenta very completetreatmentof methodsfor developingcorrelations involving severalvariables.e Soil moisturerelationsnormally havea soil moistureindex as the independent of actualantecedentsoil moistureare not genervariable,sincedirect measurements ally practical. Indexesthat havebeen insertedare groundwater,flowat the beginning of the storm, antecedentprecipitation, and basin evaporation.loGroundwatervalues shouldbe weightedto reflect the effectsof precipitation occurring within a few days of the storm becausesoil moisture changesfrom previousrains may affect results. Pan-evaporationmeasurementscan be employedto estimatesoil moistureamounts, since eviporation is related to soil moisture depletion.tl Antecedentprecipitation indexes(API) haveprobably receivedthe widestusebecauseprecipitationis readily measuredand relatesdirectly to moisture deficiencyof the basin. A typical antecedentprecipitation index is (10.6) Po: aP6'l bP, -f cP., where
P.:
po,p,." :
precipitationindex (in.) the antecedent
fi"HTlJ:;fJil:rr:Hifl3*ff"-e
for and presenr vear
This index links annualrainfall and runoff values.r2Coefficientsa, b, andc are found by trial and error or other fitting techniquesto producethe best correlationbetween the runoff and the antecedentprecipitation index. The sum of the coefficientsmust be 1. Kohler and Linsley13haveproposedthe following API for use with individual storms: P o : b r P l + b 2 P 2+ ' ' '
+ btPt
(10.7)
wherethe r subscripton P refersto precipitationwhich occurredthat many daysprior to the given storm,and the constantsb (lessthan unity) are assumedto be a function of t. Valuesfor the coefficientscan be determinedby correlationtechniques'In daily evaluationof the index, b, is consideredto be related to / by b,: K'
(10.8)
whereK is a recessionconstantnormallyrepoftedin the range0.85-0.98.The initial
166
cHAprER1o RUNoFFANDTHE cAToHMENT
value of the API (P"s)is coupled to the API / days later (P",) by Por:
PagK'
(10.e)
To evaluatethe index for a particular day basedon that of the precedingone, Eq. 10.9becomes Po, :
KPox
(10.10)
becauset : I. Various empirical relations for API have been proposed.Most are based on correlating two or three variablesand at best yield only rough approximations.In many casesthesewere developedwithout consideringphysical principlesor dimensional homogeneity.An addedshortcomingis that many formulas fit only a specific watershedand have little generalutility. Empirical equationsdemandgreat caution and an understandingof their origin. EXAMPLE 10.1 PrecipitationdepthsP, for a l4-day period are listed in Table 10.1. The API on April 1 is 0.00. Use K : 0.9 and determinethe,API for each successiveday. Solution. Equation 10.9 reducesto API,:
K(API,-1)+ P,
which was applied in developingthe successivevaluesof API, in Table 10.1. TABLE10.1 (A Precipitation April 1 2 J
4
)
'6 7 8 9 10 ll
t2 l3 t4
0.0 0.0 0.5 0.7 0.2 0.1 0.0 0.1 0.3 0.0 0.0 0.6 0.0 0.0
API,
0.00 0.00 0.50 1 .l 5 t,24 t.22 1.10 1.09 1.28 t.l5 1.04 1.54 1,39 |.25
10.4 STREAMFLOW FREOUENCY ANALYSIS Hydrologistsestimatestreamflows.Two approachesare employed.The first is a physical processesapproachin which runoff is computed on the basis of observedor expectedprecipitation.The secondis foundedon statisticalanalysesofrunoffrecords .- --vvi{}out resott-to precipitation data. Such investigationsusually include frequency
FREQUENCYANALYSIS 10.4 STREAMFLOW
167
C a o a ao
12 months
-
x.
1.0 Example 1.0 in drought with R.I. = 17 yr
0.1 10
100
RecurrenceInteryal (Yr)
for FiveRivFigure 10.9 Low-flowfrequencydataconsolidated ers.(AfterWhipple.r5; studies(Chapter 27) to indicatethe likelihood of certain runoff eventstaking place. A knowledgeof the frequencyof runoff eventsis helpful in determiningrisks associatedwith proposeddesignsor anticipatedoperatingschemes.Frequencyanalysesare usuallydiiectedtowardstudiesof miximum (flood) andminimum (drought)flows.ra'1s Figure 10.9 illustrates a typical drought frequency analysis.Unfortunately,many existingrunoff records are short-term; as a result they limit utility for reliable frequency analyses.Few adequaterecords are availableearlier than about 1900. In some cases,sequentialgeneratingtechniques(Chapter 22) can be used to develop syntheticrecords. Time-series analysesare particularly pertinent to the problem of estimating trends, cycles, and fluctuationsin hydrologic data. They also permit derivation of generatingprocessesby which syntheticrecordsof runoff can be developed.
RecurrenceIntervaland Frequency The recurrenceinterval (R.I.) is definedas the averageinterval over a long period of yearsduring which a correspondingmagnitudeof somehydrologicvariableis at least met. This parameteris also called the return period, and sometimes,though less appropriately,thefrequency of the event.For the examplein Fig. 10'9, droughtsless than 1 in. occurredin 8 ofthe 136yearsofrecords.The 1.0in. droughthasan average recurrenceinterval of about 17 years.Statedanotherway,on the average,one year of every l7-year sequenceis expectedto experiencea drought of at most 1.0 inch. Similarly,eachyearthe probabilityof a 1.O-in.droughtis 8/136 = 0'059, or about 6 percent. This is defined as the exceedenceprobability or frequency, and is the re-iprocal ofthe return period.It shouldbe obviousthat the 1.0-in.droughtcould
168
CHAPTER 10 RUNoFFAND THEoAToHMENT occur in any year,or in severalconsecutiveyears.This type ofanalysiscannottell the investigatorwhat will happenthis year or next, and allows only an estimateof the averagerecurrenceinterval and the probability of occurrencein any given year. This subjectis fully developedin Chapter27.
10.5 STREAMFLOW FORECASTING Surfacewaterhydrologyis basicto the designof many engineeringworks and important in waterquality managementschemes.In addition, the ability to providereliable forecastsof flowsfor shortperiodsinto the future is of greatvaluein operatingstorage and other works and in planning proper actions during times of flood.e'16A good exampleis the operationof a reservoirwith an uncontrolledinflow but with a means of regulatingthe outflow. If information on the natureof the inflow is determinablein advance,then the reservoircan be operatedby somedecisionrule to minimize downstreamflood damage.Suchoperationscan be computerizedto continually improve estimatesbasedon incoming dataandthus offer direction on the natureof the releases to be made.For river forecaststo be reliable, adequate,dependabledata on various watershedand meteorologicconditions are neededon a continuing basis.Modern monitoring stationscapableof telemeteringdata to computercontrol centersprovide an important supportfunction for forecasting.The methodsusedto forecastflows are basically the same ones empfoyedin design: precipitation-runoff equations,unit hydrographs,watershedmldels, and flow-routing techniques.
r Summary Runoff is probablythe most complexyet most important hydrologicprocessto understand.It has attractedthe attention and focus of engineersand scientistsand comprisesthe greatestpercentageby far of most hydrology textbooksand publications. The conceptsintroducedhere will be more fully developedin the next six chapters, as well as in significantportions of Parts Five and Six.
PROBLEMS
10'1'ffi,',T'#:fl :Jfr *:l;J:ff ;J3;:1,!;";ffi :?'fi#.n.}il:H:i* Hfi'#':iJf what purposesmight you use this? 10.2. Selecta rain gaugerecord of interest.Use the annual valuesas data to calculatethc coefficientsof an antecedentprecipitation index of the form of Eq. 10.8. 10.3. Determinethe drainagedensityof the basin shown.Area : 6400 acres.Lengths are in miles.
REFERENCES 169
Lengths of channel segments in mi between points
Figure for Problem 10.3 L0.4. Using any dictionary,plus indexesor glossariesfrom one or two other hydrologytexts, find and compare definitions of the following terms: runoff, direct runoff, direct surface runoff, surface ruryoff,surface water, overlandflow, streamflow,drainage, watershed,catchruen&inage basin, subbasin,dr ainage divide.
"systematic Procedure for Evaluating Partial Areas of Watershed 1. Boughton, W. C., Runoff," Proc. ASCE J. Irrigation and Drainage Engineering116, 1 (February 1990). "Drainage Basin Characteristics,"Trans.Am. Geophys.Union 13(1932). 2. R. E. Horton, "Geology-Part II," in Handhok of Applied Hydrology. New York: 3. A. N. Strahler, McGraw-Hill, 1964. "TopographicCharacteristicsof DrainageBasins,"U.S. Geological 4. W. B. L4ngbeinet al., Survey,Water Supply Paper,968-c(I941 ). "Erosional Developmentof Streamsand Their Drainage Basins:Hydro5. R. E. Horton, physicalApproach to Qualitative Morphology,"Bull. Geol. Soc.Am.56(1945). "studies of Longitudinal StreamProflles of Small Watersheds,"Tech. Rept. 6. J. T. Hack, 18, Columbia University,Departmentof Geology,New York, 1959. 7. A. B. Taylor and H. E. Schwartz,"IJnitHydrographT ag and PeakFlow Relatedto Basin Characteristics," Trans.Am. Geopltys. Union 33(1952). "Discussionof Paper,Flood Flow characteristicsby c. S. Jarvis," ?ans. R. E. Horton, 8. ASCE 89(1926)'. 9. R. K. Linsley, Jr., M. A. Kohler, and J. L. H. Paulhus,AppliedHydrology,2nd Ed' New York: McGraw-Hill, 1975. 10. ven Te chow (ed.), Handbook of Applied Hydrology. New York: McGraw-Hill,1964. 11. R.K.Linsley,Jr.,andWC.Ackerman,"MethodofPredictingtheRunofffromRainfall," Trans.ASCE 107(1942). 12. S. S. Butler, EngineeringHydrology. EnglewoodCliffs, NJ: Prentice-Hall, 1957.
170
oHAPTER10 RUNoFFANDTHEoAToHMEI 13. M. A. Kohler and R. K. Linslev. , / WeatherBureau,Res.Paper34, l95L T4, Leo R. Beard, "statistical Methodsin Hydrology," U.S. Army EngineerDistrict, Sacrf mento,CA,1962. 1 5 . William W. Whipple, Jr., "RegionalDrought FiequencyAnalysis," Proc. ASCEJ. Irrigd
16.
Chapter1'1
Hydrographs
r Prologue The purposeof this chapteris to: . Characterizea hydrographas a time plot of the dischargeof surfacerunoff arld groundwaterfrom drainagebasins. ' Introduce the componentsof hydrographsso that the reader can rel9le them of runoff presentedin subsequdnf-chapters. to the quantltative assessments . Describethe time relationshipsmost commonly usedin hydrographanalysis. Hydrographanalysisis the most widely usedmethod of analyzingsurfacerunoff. Its presentationin most textbooks is normally confined to one chapter. Becauseof numerousdevelopmentsin hydrographanalyses,three chaptersare dedicatedto the subjectin this text. Chapter 11 defineshydrographsand expandsthe conceptsintroduced in Chapter 6. The conceptsreferred to as unit hydrographtechniquesare packagedtogetherin Chapter 12. Individual streamflowhydrographshapesvary as flow travels downstream,and the conceptsfor analyzingthese changes,called hy' drographrouting methods,are presentedin Chapter 13.
HYDROGRAPHS 11.1 STREAMFLOW A streamflowhydrographprovidesthe rateof flow at all pointsin time duringand after a storm or snowmeltevent.l Hydrologistsdependon measuredor computed(synthesized) hydrographsto provide peak flow rates so that hydraulic structurescan be designedto accommodatethe flow safely.Becausea hydrographplots volumetric flow rates againsttime, integration of the area beneath a hydrographbetWeenany two points in time givesthe total volume of water passingthe point of interestduring the time interval. Thus, in addition to peak flows,hydrographsallow analysisof sizesof reservoirs,storagetanks, detentionponds,and other facilities that deal with volumes of runoff. A knowledge of the magnitude and time distribution of streamflow is essentialto many of theseaspectsof watermanagementand environmentalplanning.
172
CHAPTER11
HYDROGRAPHS Storm period hydrograph o u
q b0
Y41 oar.
Inflection point
\?
Hydrograph in period of no direct runoff and where no reservoir regulation exists rcflects discharge from groundwatet Storm period hydrograph End of Beginning of
direct runoff
direct runoff
,l
Continuous hydrograph
Figure 11.1 HydrographO"o;;.
SHAPE HYDROGRAPH 11.2 FACTORSAFFECTING A hydrographhas four componentelements:(1) direct surfacerunoff, (2) interflow, (3) groundwateror baseflow, and (4) channelpregipitation.2The rising pott(rn of a hydrographis known astheconcentrationcurye;the regionin the vicinity of the\eak is called the crest segment;and the falling portion is the recession.3The shapeof a hydrographdependson precipitation pattern characteristicsand basin properties. Figure 11.1illustratesthe definitionspresented.
Processes Precipitation-Streamflow Duriirg a given rainfall, water is continually being abstractedto saturatethe upper levelsof the soil surface;however,this saturationor infiltration is only one of many Rainfall is also interceptedby trees, plants, 4nd roof continuous abstractions.4-6 surfaces,and at the same time is evaporated.Once rain falls and fulfills initial requirementsof infiltration, natural depressionscollect falling rain to form small puddles,creatingdepressionstorage.In addition, numerouspools of water forming detentionstoragebuild up on permeableand impermeablesurfaceswithin the watershed.This storedwater gathersin small rivulets, which carry the wateroriginatingas overlandflow into small channels,then into larger channels,and flnally as channel flow to the watershedoutlet. Figure 11.2aillustratesthe distribution of a prolonged uniform rainfall. Although such an event is not the norm, the conceptis useful for showingthe mannerin which detentionand depressionstoragewould be distributed. a certain amountof baseflow In general,the channelof a watershedpossesses during most of the year. This flow comesfrom groundwateror spring contributions and may be consideredas the normal day-to-dayflow. Dischargefrom precipitation excess-that is, after abstractions are deducted from the original rainfallconstitutesthe direct runoff hydrograph(DRH). Arrival of direct ninoff at the outlet accountsfor an initial rise in the DRH. As precipitationexcesscontinues,enoughtime qlapqgsfor progressivelydistant areasto add to the outlet flow. Consequently,the
11.2 FACTORSAFFECTINGHYDROGRAPTISHNPC
179
Depressionstorage fi
o
a
d
Time (b)
Figure 11.2 (a) Distribution of a uniform storm rainfall for condition of is ultimately no interceptionloss.Note that all water storedin depressions evaporatedor infiltrated while somedetentionstorageis also subjectedto theselosses.(b) Equilibrium dischargehydrograph.
duration of rainfall dictatesthe proportionatearea of the watershedamplifying the peak, and the intensity of rainfall during this period of time determinesthe resulting greatestdischarge.
HydrographShapes If the rainfall maintainsa constantintensityfor a long enoughperiod of time, a state of equilibrium dischargeis reached, as depicted by curve A in Fig. 11.2b. The inflection point on curveA often indicatesthe time at which the entire drainagearea contributes to the flow. At this time maximum storase of the watershedis only
174
CHAPTER11
HYDROGRAPHS
and partially complete.As rainfall continues,maximum storagecapacityis attained of condition The is reached. (runoff)] iqrliUri"t" finflow (rainfall) equals outflow Extended nature' in attained maximum storageand equiiibrium is seldom if ever its duration negateany rainfall lnuy o"Iu., but viriations in intensity throughout intensity' rainfall constant possibility of u IRH of the theoretical shapefor Anormalsingle.peakDRHgenerallypossessestheshapeshownbycurveBin peak magnitudeof this Fig. 11.2b rather tian iy the "ur',oJin Fig. 11.2a.The time to and the size,slope, rainfall' ofthe hyirograph dependsonih" intensity and duration has beenreachedfor a flow peak shape,and storagecapacityof the watershed.once of supply coming its source given isolatedra'instorm,itt" ORg beginsto descend, and channel detention as such largely from water accumulatedwithin the watershed storage. processesinvolvedin forming the DRH can be better understoodby visualizing theprecipitationexcessaspartiallydisposedofimmediate$bysurfacerunoffwhile releasedlater from a portion remainsheld within the watershedboundariesand is of the duration effects integrated storage.Thus the shapeand timing of the DRH are asthe effect of well as factors and intensityofrainfall and other hydrometeorological capacity' storage the physiogiaphicfactorsof the watershedupon the
COMPONENTS 11.3 HYDROGRAPH
\
into its com\It is important to understandhow the hydrographcan be subdivided andwatershed ofprecipitation nenrpar15andto look at the effecton hydrographshape purpose' features.Figures11.3and Il'4 areusedfor this with respect A hydiographis a continuousgraphshowingthe rate of streamflow stage indicates that recorder strip to time, ,rtr*ityoutained by -"anr of a continuous hydrograph a discharge to vefsustime (stale hydrograph),which is then transformed generallytaken to by applicationof a rating curve.Hereafter,the term lrydrographis indicate a dischargehYdrograPh' a pgriod Figure t t.gi lttustrai"* ih" hydrographof a permanentstreamduring groundbecause lrydrograph between precipitation events,known i" i bot" flow the base of modification cause water sustainsthe flow. Four general conditions of sets following the using HortonT flow hydrographshape.They aL describedby inequalities:
i
Setl
i>f F(S, Set4 i>f
set3
F)Sr.
11.3 HYDROGRAPH COMPONENTS 175
I
\
(e)
(0
fo)
(h)
t
Figure 11.3 Effectsof stormandbasincharacteristics on hydrographshape. or addedgroundwatercomponentsdevelops.The entire effect of the storm would be to slightly reducethe soil moisturedeficiencysr. The field capacityis the amountof water held in the soil after excessgravitationalwater has drained. The conditions describedby Set 2 still do not produce direct surfacerunoff, although the comlponentsof interflow and groundwater flow are added to channel precipitation.The initial hydrographwould be modified,sincethe field capacityof the soil is exceeded.Figure 11.3b illustrates this condition. Note that deviation of the hydrographfrom the original baseflow curve is likely to be very small under these conditions. Figure 11.3c illustrates a casewhere surfacerunoff becomesa componentof flow becausei > f. In this example,interflow and groundwaterflow are zero,assoil
176
CHAPTER11 HYDROGRAPHS
End ofrainfall
9C
Groundwater flow
Time
of thehydrograph. Figure 11.4 Components
moisture deficiency still exists,although at a reduced level. Channel precipitation likewise constitutesa component. In the final set, Fig. 11.3d, all four componentsexist with rainfall intensity exceedinginflltration rate andthe field capacityofthe soil is reached.This casewould of a large storm event. be typical Figures 1l.je-h illustrate how hydrograph shape can be modified by areal variations in rainfall and rainfall intensity and by watershedconfiguration.8Minor fluctuationsshownin thesehydrographsare linked to variationsin stormintensity.In Fig. 11.3eonly the delayingeffectspertinent to a storm over the upstreamsectionof the areaareindicated.Figure 11.3fshowsthe reverseof this condition.Figures11'3g and h depict the comparativeeffects of basin geometry. In most hydrographanalyses,interflow and channelprecipitation are grouped with surfacerunoff rather than treated independently.Channelprecipitation begins with inceptionof rainfall and endswith the storm.Its distributionwith respectto time is highly iorrelated with the stormpattern.The relative volume contributiontendsto increasesomewhatas the storm proceeds,since stream levels rise and the water surfaceareatendsto increase.The fraction of watershedareaoccupiedby streamsand lakes is generallysmall, usually on the order of 5 percent or less,so the percentage of rrnoff relatedto channelprecipitation is usually minor during important storms. Distribution of interflow is commonly characterizedby a slowly increasingrate up to the end of the storm period, followed by a gradual recessionthat terminatesat the intersectionofthe surfaceflow hydrographand base flow hydrograph.Figure 11.4 illustrates the approximatenature of the componentsof channelprecipitation and interflow. The baseflow componentis composedof the water that percolatesdownward until it reachesthe groundwaterreservoirand then flowsto surfacestreamsasgroundwater discharge.The groundwaterhydrographmay or may not show an increase
11.4 BASE FLOW SEPARATION
'
177
during the actual storm period. Groundwateraccretion resulting from a particular storm is normally releasedover an extendedperiod, measuredin days for small watershedsand often in months or yearsfor large drainageareas. The surfacerunoff component consistsof water that flows overland until a streamchannelis reached.During large stormsit is the most significanthydrograph component.Figure 11.4illustratesthe surfacerunoff and groundwatercomponentsof a hydrograph.As pointed out in Fig. 11.3,the relative magnitudeof eachcomponent for a given storm is determinedby a combinationof many factors.Hydrographsare analyzedto provideknowledgeof the way precipitationand watershedcharacteristics interact to form them. The degreeof hydrographseparationrequireddependson the objective of the study. For most practical work, surfacerunoff and groundwater componentsonly are required.Researchprojectsor more sophisticatedanalysesmay dictate considerationof all components.When multiple storms occur within short periods,it is sometimesnecessaryto separatethe overlappingparts of consecutive surfacerunoff hydrographs.
11.4 BASEFLOWSEPARATION Severaltechniquesare usedto separatea hydrograph'ssurfaceand groundwaterflows. Most are basedon analysesof groundwaterrecessionor depletioncurves.If there is no addedinflow to the groundwaterreservoir,and if all groundwaterdischargefrom the upstreamareais interceptedat the stream-gaugingpoint of interest,then groundwater dischargecan be describedby either e'to er: eoK' or er: eo€-K' where eo : q, : K: e:
( 11 .1 )
a specifiedinitial discharge the dischargeat any time / after flow 46 arecessionconstant baseof natural logarithms
Time units frequently used are days for large watershedsand hours or minutes for small basins.A plot of either yields a straight line on semilogarithmicpaper by plotting / on the linear scale. For most watersheds,groundwaterdepletioncharacteristicsare approximately stable,sincethey closelyfit watershedgeology.Nevertheless, the recessionconstant varieswith seasonaleffectssuchas evaporationand freezingcyclesand other factors. Becauseq, dt is equivalentto -dS, where S is the quantity of water obtainedfrom storage,integrationofEq. 11.1produces t- 4o s - Qlog" K
(rr.2)
This equation determinesthe quantity of water releasedfrom groundwaterstorage betweenthe times of occurrenceof the two dischargesof interest,or it can be usedto calculatethe volume of water still in storageat a time some chosenvalue of flow occurs.To get the latter, 4, is setequalto zero and qsbecomesthe referencedischarge. Figure 11.5ais a plot of Eqs. 11.1and II.2 andprovidesadditionaldefinition.
178
CHAPTER11
HYDROGRAPHS
qo
.. ,?
q
K \r3
/
?r
t (a.)
I
l
I
I
qo discharge at rcginningof arr interval discharge at :nd of an inten,a\ Qt
I
II
Interval = unit period in / whicht is expreissed .
q d
/ qo Qt
Time (b) Figure 11.5 Baseflow model' to
.
t1
Groundwaterdepletioncurvescan be analyzedby variousmethodsto evaluate the recessionconstantK. One of thesewill be described.Data from a stream-gaugi.ng stationare a prerequisiteand shouldreflectrainlessperiodswith no upstreamregulation,'suchas a fesefvoir,to affect flow at the gaugingpoint. Otherwisean adjustment with its own enors is introduced. From the streamflow data, plot a portion of the recession hydrograph (Fig. 11.5b) to find values of dischargeat the beginning and end of selectedtime intervals.Flows at the beginningof eachinterval are analogousto 4e,whereasthose at the end are analogousto q1.Next, selectseveraltime intervalsand plot correspondvalues ing qe'sversusq,'s shownin Fig. 11.6.The time periodbetweenconsecutive of 4 shouldbe identical for eachdatum set. Figurestaken from recessioncurvesof times that still reflect surfacerunoff will usually fall below and to the right of a 45o line drawn on the plot. Thesevalueswill also be associatedwith larger numbersfor 4. Points taken from true groundwater recessionperiods should approximately
11,4 BASEFLOWSEPARATION 179 r.-,x. ^are points plotted from discharges for different groundwater discharge periods
,/ o
eo nd qt are taken for equal time
I
,(
I .5r"2'
9/ .,
/(^lI A
/#
- s l o p e o 0f A=
fi:
x
inEq11.1
/ Figure 11.6 Graphical rnethod for determiningrecessionconstant K. (U.S. Departmentof Agriculture, Soil ConservationService.)
describea straightline.Becauseh : eo : 0 when4o: 0, a straightlinecanbefitted graphicallyto the datapoints.The slopeof this line is qrfqo : K. Usingthis value,the depletioncurve plots as a straight line on semilogarithmicpaper (r is the linear scale variable)or as a curveon arithmeticpaper,Fig. 11.5a.
SeparationTechniques r
Severalmethodsfor baseflow separationare used when the actual amount of base flow is unknown. During large storms,the maximum rate of dischargeis only slightly affectedby baseflow, and inaccuraciesin separationmay not be important. The simplestbaseflow separationtechniqueis to draw a horizontalline from the point at which surfacerunoff begins,PointA in Fig. 11.7, to an intersectionwith the hydrographrecessionwherethe baseflow rate is the sameasat the beginningof direct runoff as indicatedby Point B. A secondmethodprojects the initial recessioncurve downwardfromA to C, which lies directly below the peak rate of flow. Then point D on the hydrograph,representingN daysafter the peak, is connectedto point C by a straightline definingthe groundwatercomponent.One estimateof N is basedon the formula3 N - Ao'2
(11.3)
where N : the time in days A = the drainageareain squaremiles A third procedureis to developa baseflow recessioncurve using Eq. 11.1 for data from the segmentFG, and then back-calculateall base flow to the left of Point fl
180
cHAPTER11 HYDRocRApHS
o
9p F
Time
Flgure 11.7 Illustrationof somehydrograph separation techniques. wherethe computedcurve beginsto deviatefrom the actualhydrograph,marking the end of direct runoff. The curve is projected backward arbitrarily to some Point E below the inflection point and its shapefrom C to E is arbitrarily assigned.A fourth widely usedmethod is to draw a line betweenA and F, and a fifth common method is to project the line AC alongthe slopeto the left of A , and then connectPointsC and
o d d
12N 6P
t21|'{{ 6A
12N 6P
tz]0|{ 6,{
I2N
6P
IzM
Figure 11.8 Illustration of base flow separation:hydrograph for the Uharie River near Trinity, North Carolina, February 25, 1939. (U.S. Departmentof Agriculture, Soil ConservationService).
TIME RELATIONSHIPS 181 11.5 HYDROGRAPH
B. All thesemethodsare approximatesincethe separationof hydrographsis partly a subjectiveprocedure. Figure 11.8illustratestwo graphicalseparationtechniquesto determinesurface runoff and groundwaterflow components. Line AD representsthe simpleprocedure of connectingthe point of the beginningof direct runoff with the flrst point on the groundwaterrecessioncurve (an advantageoverthe horizontal line techniquebecause the time baseof direct runoff is much shorter).Ctrve ABCD is constructedfrom the extensionof the baseflow recessioncurve.
TIMERELATIONSHIPS 11.5 HYDROGRAPH Wavetravel time is definedas the time required for direct runoff originating at the most remotepoint in the channelto reachthe outlet. The last drop of direct runoff to passthe outlet conceptuallytravelsoverthe watersurfaceand reachesthe outlet at the speedof a small surfacewave,rather than at a speedequal to the averagevelocity of flow. The wavetravel time is fasterthan the averagevelocity and varieswith channel shapeand other factors.For a rectangularchannel,the ratio is approximately5/3 (see Section 13.1 for other wavevelocities).The time base(Fig. 11.4) of a hydrographis consideredto be the time from which the concentrationcurve beginsuntil the directrunoff componentreacheszero. An equationfor time basemay take the form T6:t"*t,
(rr.4)
where To : the time baseof the direct runoff hydrograph /" : the duration of runoff-producingrain t- : the excessrainfall releasetime Watershedlag time, illustrated in Fig. 11.4, is definedas the time from the penterof tt massof effectiverainfall to the centerof massof direct runoff. Other definitionsand severalequationsrelating lag time to watershedcharacteristicsare providedin S_ecchapters. tion 11.7 and subsequent Becauseofits importancein unit hydrographtheory, the excess-rainfallrelease time is introduced.This is definedasthe time requiredfor the last, most remotedrop ofexcessrain that fell on the watershedto passthe outlet, signallingthe cessationof direct runoff. It is easilydeterminedas the time interval betweenthe end of rain and the end of direct runoff. Only that part of the outflow which classiflesas direct runoff (excessrain) is consideredin dqterminingthe releasetime. Watershedoutflow normally continuesafter cessationof direct runoff, in the form of interflow andbaseflow. Releasetime is very similar by definitionto wavetraveltime and time of concentration (Section11.6). A foundational assumptionof unit hydrographtheoryl2 is that the watershed excessreleasetime is a constant,regardlessof the storm duration, and is related to The excessreleasetime is also basinfactorsratherthan meteorologicalcharacteristics. conceptuallyidenticalwith the time baseof an instantaneousunit hydrograph(IUH). This is the runoff hydrographfrom 1.0 in. of excessrain applieduniformly over the watershedin an instant of time (seeChapter 12). Both wavetravel time and excessrainfall releasetime are often used synonymouslywith time of concentration.
182
CHAPTER11
HYDROGRAPHS
11.6TIMEOF CONCENTRATION The most common definition of time of concentrationoriginatesfrom consideration of overlandflow. If a uniform rain is appliedto atract, the portions nearestthe outlet contribute runoff at the outlet almost immediately.As rain continues,the depth of excesson the surfacegrowsand dischargeratesincreasethroughout.Runoff contributions from variouspoints upstreamarrive at later times, addingthemselvesto continuing runoff from nearerpoints, until flow eventually arrives from all points on the watershed,"concentrating" at the outlet. Thus, concentrationtime is the time required,with uniform rain, for 100percentof a tract of land to contributeto the direct runoff at the outlet.e As a secondpopular definition, the concentrationtime is often equatedwith either the excess-rainfallreleasetime or the wave travel time becausethe time for runoff to arrive at the outlet from the most remotepoint after rain ceasesis assumed to be indicative of the time required for 100 percent contribution from all points during any uniform storm having sufficient duration. The latter definition is often preferred becausefew storm durations exceedthe time of concentration,making determinationof /. possibleonly by examiningexcessrain recession. Becausetime of concentrationis conceptuallythe time requiredfor 100percent of the watershedto contribute, it is also often defined as the time from the end of excessrainfall to the inflection point on the hydrographrecessionlimb (e.g., see Fig. 12.2).The reasoningusedin this definitionis that direct runoff ceasesat the point of inflection. For a small tract of land experiencinguniform rain, the entire areacontributes at approximatelythe sametime that the runoff reachesan equilibrium.This givesrise to yet anotherdefinition of time of concentration.If rain abruptly ceased,the direct runoff would continueonly as long as the excess-rainfallreleasetime t,. On the basis of the seconddeflnition.excessreleasetime and time of concentrationcan be considered equivalent. Numerous equationsrelating time of concentrationto watershedparameters havebeen developed.Table 11.1 summarizesseveralpopular versions.Other variations are presentedin Chapters12, 15,16, and25.
11.7 BASIN LAG TIME The relative timing of rainfall and runoff must be known if drainageareashaving subbasinsare to be modeledor if continuoussimulation is desired.A basic measure of timing is basi'nlag, which locatesthe hydrograph'spositionrelativeto the causative stormpattern.It is most often definedas the differencein time betweenthe centerof massof effectiverainfall and the centerof massof runoff produced.Other definitions are also used.Two of theseare ( 1) the time interval from the maximum rainfall rate to the peakrate of runoff and(2) the time from the centerof massof effectiverainfall to the peak rate of flow. Time lag is characterizedby the ratio of a flow length to a mean velocity of flow and is thus a property that is influencedby the shapeof the drainagearea,the slopeof the main channel,channelroughnessand geometry,and thg'storm pattern.
11.7 BASINLAGTIME
183
TABLE11.1 SUMMARY OF TIMEOF CONCENTRATION FORMULAS Formula for t" (min)
Method and date Kirpich (1940)
tc : L : S :
USBR Design of Small Dams
t" : L :
(r973) Il :
lzzatd (7946)ts
0.00782077S-o38s length of channel/ditch from headwater to outlet, ft average watershed slope, ftlft
Developedfrom SCS data for sevenrural basins in Tennesseewith well-definedchanneland steepslopes(37o to 1O7o);for overlandflow on concreteor asphaltsurfacesmultiply t;by 0.4; for concretechannelsmultiply by 0.2; no adjustments for overland flow on bare soil or flow in roadsideditches.
60(lI.9L31H)o38s length of longest watercourse, ml elevation difference between divide and outlet, ft
Essentiallythe Kirpich formula; developedfrom small mountainousbasinsin California (U.S. Bureauof Reclamation,1973,pp. 67-7I).t4
41.025(0.0007, t
Developedin laboratory experimentsby Bureau of Public Roadsfor overland flow on roadwayand turf surfaces;valuesof the retardance coefficientrange from 0.0070 for very smooth pavementto 0.012 for concretepavementto 0.06 for denseturf; solution requiresiteration; product i times Z shouldbe = 500.
c)Lozz
-c S0.333i0.66?
I : c: Z : ,S: FederalAviation Administration ( 1970)r6
Kinematic WaveFormulas Morgali and Linsley (1965)'? Aron and Erborge (1973)18 SCS Lag Equation (1972)te
rainfall intensity,in/h retardancecoefficient length of flow path, ft slopeof flow path, ftlft
r" = 1.8(1.1- C)Losofso333 C : rational method runoff coefficient l, : length of overland flow, ft S : surface slope, Va O.94Lo6no6 (io 45o 3)
l, n I S
: : : :
length of overland flow, ft Manning roughnesscoefficient rainfall intensity in/h averageoverland slope ftlft t.67 Lo8[(tooo/cN)- 9]oi
,. ._- @
L : hydraulic length of watershed (Iongestflow path), ft CN : SCS runoffcurve number S : averagewatershedslope,Vo
SCS AverageVelocity Charts(1975, 1986),0
Sarrce.' After Ref. 13.
Remarks
',' : l v L 60- v Z : length of flow path, ft V : ureragevelocity in feet per secondfrom Fig. 3- 1 of TR 55 for various surfaces
Developedfrom air field drainagedata assembled by the Corps of Engineers;method is intended for use on airfield drainageproblems,but has been used frequently for overlandflow in urban basins. Overland flow equation developedfrom kinematic wave analysis of surface runoff from developed surfaces;method requiresiteration sinceboth i (rainfall intensity) arrdt, are unknown; superposition of intensity-duration-frequency curve gives direct graphical solution for t". Equation developedby SCS from agricultural watersheddata; it has been adaptedto small urban basinsunder 2000 acres;found generally good where areais completelypaved;for mixod areas it tendsto overestimate;adjustmentfactorsare applied to correct for channelimprovementand imperviousarea; the equationassumesthat t" : 1.67 X basinlag. Overland flow charts in Ref. 20 provide average velocity as function of watercourseslope and surfacecover.
184
CHAPTER11
HYDROGRAPHS
Various studieshave been conductedfor the purpose of developingrelations descriptiveof time lag. Most prominent of thesewas the work by Snyderon large natural watersheds.2lHis original equation has been widely used and modified in variouswaysby other investigators.Eaglesonhas proposedan equationfor lag time on sewereddrainageareashavinga minimum sizeof 147 acres.2z An early investigation (1936) on small drainageareas(2-4 acres)was conductedby Horner in his classicalwork on urbandrainagein St. Louis,Missouri.23 Horner'swork wasinconclusive in that it did not yield a deflned procedure,but he did conclude that the comparativelywide rangein the lag time at eachlocation led to the inferencethat the lag was a variable,its valuebeingdeterminedmore by rainfall characteristicsthan by characteristicsof the drainagearea. Snyder'sstudybasedon data from the AppalachianMountain regionproduced the following equationfor lag time:z1 t,1:
where
c,(L""L)o3
( 11 . s )
/1 : the lag time (hr) betweenthe centerof massof the rainfall excessfor a specifiedtype of storm and the peak rate of flow I'"o : the distancealongthe main streamfrom the baseto a point nearestthe centerof gravity of the basin (mi) I : length of the main stream channel (mi) from the base outlet to the upstreamend of the streamand including the additional distanceto the watersheddivide C, : & coefficientrepresentingvariationsof types and locationsof str'eams
For the areastudied,the constantC, was found to vary from I.8 to 2.2, with somewhatlower valuesfor basinswith steeperslopes.The constantis consideredto includethe effectsof slopeand storage.The value of 4 is assumedto be constantfor a given drainagearea,but allowanceis madefor the useof different valuesof lag for different types of storms. The relation is consideredapplicable to drainage areas rangingin sizefrom 10 to 10,000mi'. In a studyof seweredareasrangingin size from 0.22 to 7.51 mi2,Eagleson22 developedthe equation "t ', where
tr : L :
: -
L ( l . 5 / n ) R 2 / 3s t / 2
(11.6)
lag time, the center of mass of rainfall excess to the peak discharge (sec) the mean travel distance (ft), which is equal to the length of that portion
of the sewerwhich flows full n : the weightedManning's coefficientfor the main sewer : the weightedhydraulic radius of the main sewerflowing full { S : the weightedphysicalslopeof the main sewer Eagleson'sequationdirectly includesthe effects of channelgeometry and slope,as well as basin shape,and thus representsa refinementof the Snyderapproach.It also indirectly includesthe important effect of the storm pattern.
PROBLEMS185 Linsley and Ackerman give examplesof applicationof the following modified form of Snyder'sequation.24
t,:C,+!
(rr.1)
where s is a weighted slope of the channel and the other variablesare as defined previously. Other investigatorshaverepresentedtime lag by equationsof the form
t t =K +
( 11 . 8 )
Vs
Numerous other derivations of relations for watershedlag times can be found in standardhydrologictexts and periodical literature. Others are includedwith someof the syntheticunit hydrographdiscussionsin Chapter 12.
r Summary Understandingthe structureof hydrographSis important to many designand water supply applications.The hydrographrepresentsthe portion of the hydrologiccycle that engineersmostoften needin orderto determineratesof flow in streamsfor setting bridge lengthsand elevations,designingflood protection measures,and establishing areal extent of flooding. Similarly, the volume of drainageinto a reservoiror past a water supply diversionis determinedfrom the areaunder the hydrograph.Accurate estimatesof thesevolumesare important to designof dams,reservoirs,pipelines,and ' numerousother structures. After graspingthe fundamentalsof hydrographcomponents,including the time relationshipspresentedin this chapter,the reader should be well preparedfor the quantitativedevelopmentsof hydrographtheory and applicationspresentedthroughout Chapters12 through 16 and in Part Five.
PROBLEMS 11.1. Referto Fig. 11.1.Replotthis hydrographand usetwo different techniquesto separate the baseflow. 11.2. Obtain streamflowdata for a water courseof interest.Plot the hydrographfor a major runoff event and separatethe baseflow. 11.3. For the event of Problem I1.2, tabtlate the precipitation causingthe surfacerunoff and determinethe duration of runoff-producing.rain. Estimatethe time of concentration and useEq. I 1.4 to estimatethe time baseof the hydrograph.Comparethis with the time basecomputedfrom the hydrograph. 11.4. Tabulatedbelow are total hourly dischargerates at a cross sectionof a stream.The drainagearea abovethe sectionis 1.0 acre. a. Plot the hydrographon rectangular coordinate paper and label the rising limb (concentrationcurve), the crest segment,and the recessionlimb.
186
CHAPTERll HYDROGRAPHS b. Determinethe hour of cessationof the direct runoff usinga semilogplot of Q versus time. c. Use the base flow portion of your semilog plot to determine the groundwater recessionconstantK, d. Carefully constructand label baseflow separationcurveson the graph of Part a, using two different methods.
Time(hr)
Q (cfs)
Time (hr)
@ (cfs)
0 1 2 3
102 100 98 220 512 630 460 330
8 9 l0 11 t2
2lo 150 105 75 60 54 48.5 43.5
I
5 6 7
IJ
t4 15
11.5. On a neatsketchof a typical total runoff hydrograph,showor dimensionthe (a) storm
hyetograph,(b) beginningofdirectrunoff, (c) cessationtimeofdirectrunoff' (d) base fllw separationassumingthat additional contributions to base flow are negligible during ihe period of rise, and (e) crest segmentof the hydrograph' 11.6. For an urban watershedassignedby your instructor,obtain measuresof the watershed area,length, and slope,and compareestimatesof the time of concentrationusing the Kirpich, USBR, FAA, and SCSLag equationsin Table 11.1'
REFERENCES 1. American Society of Civil Engineers,Hydrology Handbook, Manuals of Engineering Practice,No. 28. New York: ASCE, 1957. "A 2. Donn G. DeCoursey, Runoff HydrographEquation," U.S. Departmentof Agricul.ture, AgriculturalResearchService,Feb. 1966,pp.4I-116' 3. R. K' Linsley, M. A. Kohler, and J' L. H' Paulhus,Applied Hydrology' New York: McGraw-Hill, 1949. "Erosional Developmentof Streamsand Their DrainageBasins:HydroA R. E. Horton, physicalApproach to QuantitativeMorphology,"Bull. Geol' Soc'Am' 56(1945)' "An Approach Toward a PhysicalInterpretationof Infiltration Capacity," 5 . if.-n. gorton, Sci. Soc.Am. 5,399-417(1940). Soil Proc. "An Inflltration Equation with Physical Significance,"Soil Sci.77(1954). 6 . J. R. Philip, 7 . R. E: Horion";Surface Runffi Phenomena.Ann Arbor, MI: EdwardsBros., 1935. 8 . R. J. M. DeWiest,Geohydrology.New York: Wiley' 1965' 9 . ,.Hydrology,,'in EngineeringHandbook, Sec. 4, U.s. Department of Agriculture, Soil ConservationService,1972. "Discussionof Analysis of Runoff characteristicsby o. H. Meyet," Trans. 1 0 . B. S. Barnes, ASCEl0s(1940). "Unit HydrographLag and PeakFlow Relatedto Basin 1 1 . A. B. Taylorand H. E. Schwartz, Characteristics," Trans,Am. Geophys. Union 33(1952).
REFERENCES 187 "Streamflow 12. L, K. Sherman, from Rainfall by the Unit-GraphMethod," Eng.News-Rec. 108(1932). 1 3 . D. F. Kilber, "Desk-top methods for urban stormwatercalculation," Ch. 4 in Urban Stormwater Hydrology, Water ResourcesMonograph No. 7, American Geophysical Union, Washington, D. C., 1982. 14. U.S. Bureauof Reclamation,Designof SmallDams,2nd ed., Washington,D.C.,1973. 1 5 . C.F.Izzafi,, "Hydraulicsof RunofffromDevelopedSurfaces,"Proceedings,26th Annual Meetingof the HighwayResearchBoad,26, pp. 129-146, December1946. 16. FederalAviation Administration,"Circular on Airport Drainage,"ReportA/C 050-532058, Washington, D.C., 1970. n. J. R. Morgali, andR. K. Linsley,"ComputerAnalysisof OverlandFlow,"./. Hyd.Div., Am. Soc.Civ.Eng.,9l, no. HY3, May 1965. 1 8 . G. Aron, and C. E. Egborge,"A PracticalFeasibility Study of Flood PeakAbatementin Urban Areas,"U.S. Army Corpsof Engineers,Sacramento, Calif., March 1973. 1 9 . Soil ConservationService,"National EngineeringHandbook, Sec. 4, Hydrology," U.S. Dept. of Agriculture,U.S. GPO, Washington, D.C., 1972. 20. Soil ConservationService,"Urban Hydrology for Small Watersheds,"TechnicalRelease 55, Washington, D.C., 1975(updated,1986). 2 t . F. F. Snyder,"Synthetic Unit Graphs," Trans.Am. Geophys.Union 19, 447-454(1938). 22. Peter S. Eagleson,"CharacteristicsofUnit Hydrographsfor SeweredAreas," paperpresentedbeforethe ASCE, Los Angeles,-CA,1959,unpublished. 23. W. W. Horner, and F. L. Flynt, "RelationBetweenRainfall and Runoff from Small Urban Areas," Trans.ASCE 62(101), 140-205(Oct 1956). aA R. K. Linsley,Jr.,andW. C. Ackerman,"Methodof PredictingtheRunofffromRainfall," Trans.ASCE 107(1942\.
C h a p t e r1 2
Unit Hydrographs
r Prologue The purposeof this chapteris to: . Define unit hydrographsand show their utility in hydrologicstudiesand design. . Developfully the current methodsof obtaining, analyzing,and synthesizing unit hydrographs. . Presentmethodsfor converting unit hydrographsfor one storm duration to other storm durations. Waysto predict flood peak dischargesand dischargehydrographsfrom rainfall events havebeenstudiedintensivelysincethe early 1930s.One approachreceivingconsiderable use is called the unit lrydrographmethod. 12.1 UNIT HYDROGRAPH DEFINITION The concept of a unit hydrographwas first introduced by Shermant'zin 1932. He defineda unit graph as follows:2 givendrainage area, Ifa givenone-day rainfallproduces a 1-in.depthofrunoffoverthe thehydrograph showingtheratesat whichtherunoffoccurredcanbe considered a unit graphfor that watershed. Thus, a unit hydrographis the hydrographof direct runoff (excludesbaseflow) for any stormthat producesexactly 1.0inch of net rain (the total runoff after abstractions).Sucha stormwould not be expectedto occur,but Sherman'sassumptionis that the ordinatesof a unit hydrographare t.O/P times the ordinatesof the direct runoff hydrographfor an equal-duration storm with P inchesof net rain. The term "unit" hasto do with the net rain amountof 1.0inch and doesnot mean to imply that the duration of rain that producedthe hydrographis one unit, whether an hour, day,or any other measureof time. The storm duration,X, that producedthe unithydrographmustbe specifiedbecausea watershedhasa differentunit hydrograph
12.1 UNIT HYDROGRAPHDEFINITION
189
for eachpossiblestorm duration. An X-hour unit lrydrograp,his defined as a direct runoff hydrographhavinga 1.0-in. volumeand resultingfrom anX-hour stormhaving a net rain rate of 1,/Xin.lhr. Az-hr unit hydrographwould havea 1.0-in. volume producedby a 2-hqstorm,and a 1-dayunit hydrographwould be producedby a storm having 1.0 in. of eicessrain uniformly producedduring a 24-hr period. The valueX is often a fraction. Figure 12.1illustratesa2-ltr,l2-hr, and24-hrwthydrograph for a given watershed.
t (b)
24hr
)z nl T (c)
Figure 12.1 Illustration of 2-br, I2-hr, and 24-ht unit hydrographsfor the same watershed(Note: a : b : c : 1' X A).
190
CHAPTER 12 UNITHYDROGRAPHS By Sherman'sassumption,applicationof an X-hourunit graphto designrainfall excessamountsother than 1 in. is accomplishedsimply by multiplying the rainfall excessamount by the unit graph ordinates,since the runoff ordinatesfor a given duration are assumedto be directly proportional to rainfall excess.A 3-hr storm producing2.0 in, of net rain would haverunoff rates2 times the valuesof the 3-hr unit hydrograph.One-half inch in 3 hr would produceflowshalf the magnitudeof the 3-hr unit hydrograph.This principle of proportional flows is expandedin Section 12.3 and appliesonly to equal duration storms. Implicit in deriving the unit hydrographis the assumptionthat rainfall is distributed in the sametemporal and spatialpattern for all storms.This is generallynot true; consequently,variationsin ordinatesfor different stormsof equal duration can be expected. This chapteris organizedto defineunit hydrographsfirst, then presentmethods of deriving unit hydrographsfrom actual rainfall and runoff records (Section 12.2). After familiarizing the readerwith the origin of unit hydrographs,Section I2.3 presentsmethodsof applyingunit hydrographsto generatedirect runoff hydrographsfor any storm with durationsthat are multiple integersof the U.H. duration. The constructionof unit hydrographsfor stormswith other than integermultiples of the derived duration is facilitated by a method known as the S-lrydrograph developedby Morgan and Hulinghorst.3The procedure,as explainedin Section 12.4, employs a unit hydrographto form an S-hydrographresulting from a continuous appliedrainfall. The need to alter duration of a unit hydrographled to studiesof the shortestpossiblestorm duration-the instantaneousunit rainfall. The concept of instantaneousunit hydrograph(IIJH) is tracedto Clark6and can also be used(Section 12.5) is constructingunit hydrographsfor other than the derived duration. The previousdiscussionassumesthat the analysthasrunoff and rainfall datafor deriving a unit hydrographfor the subject watershed.The application of unit hydrographtheory to ungaugedwatershedsreceivedearly attentionby Snyderaand also by Taylor and Schwartz,5who tried to relate aspectsof the unit hydrographto watershed characteristics.As a result, a full set of synthetic unit-hydrographmethods emerged.A numberof theseare presentedin Section12.6.
12.2 DERIVATION FROM OF UNITHYDROGRAPHS DATA STREAMFLOW Data collection preparatoryto deriving a unit hydrographfor a gaugedwatershedcan be extremelytime consuming.Fortunately,many watershedshaveavailablerecordsof streamflowand rainfall, and thesecan be supplementedwith office records of the Water ResourcesDivision of the U.S. GeologicalSurvey.TRainfall records pay be securedfrom ClimatologicalDatas publishedfor eachstatein the United Statesby the National Oceanicand AtmosphericAdministration (NOAA). Hourly rainfall records for recordingrainfall stationsare publishedas a Summaryof Hourly Observationsfor the location. Summariesare listed for approximately300 first-order situationsin the United States. To developa unit hydrograph,it is desirableto acquireas rnanyrainfall records aspossiblewithin the study areato ensurethat the amountand distributionof rainfall
12.2 DERIVATION,OFUNIT HYDROGRAPHSFROM STREAMFLOWDATA
191
over the watershedis accurate$ known. Preliminary selectionof storms to use in deriving a unit hydrographfor a watershedshouldbe restrictedto the following: 1. Stormsoccurring individually, that is, simple storm structure. 2. Storms having uniform distribution of rainfall throughout the period of rainfall excess. 3. Stormshavinguniform spatial distribution over the entire watershed. Theserestrictionsplace both upper and lower limits on size of the watershedto be employed.An upper limit of watershedsize of appro5imately1000 mi2 is overcautious, althoughgeneralstormsover suchareasare not unrealisticand somestudiesof areasup to 2000 mi2 have used the unit-hydrographtechnique.The lower limit of watershedextentdependson numerousother factorsand cannotbe preciselydefined. A generalrule of thumb is to assumeabout 1000acres.Fortunately,other hydrologic techniqueshelp resolveunit hydrographsfor watershedsoutsidethis range. The preliminary screeningof suitable storms for unit-hydrographformation shouldmeet more restrictive criteria before further analysis: 1. Duration of rainfall event should be approximately10-30 percent of the drainagearea lag time. 2. Direct runoff for the selectedstorm shouldrangefrom 0.5 to 1.75 in. 3. A suitablenumberof stormswith the sameduration shouldbe analyzedto obtain an average of the ordinates (approximately five events). Modificationsmay be madeto adjustdifferent unit hydrographsto a single duration by meansof S-hydrographsor IUH procedures. 4. Direct runoff ordinatesfor eachhydrographshouldbe reducedso that each eventrepresents1 in. of direct runoff. 5. The final unit hydrographof a specificdurationfor the watershedis obtained by averagingordinatesof selectedeventsand adjustingthe result to obtain 1 in. of direct runoff. Constructingthe unit hydrographin this way producesthe integratedeffect of runoff resultingfrom a representativeset of equal duration storms.Extremerainfall intensityis not reflectedin the determination.If intensestormsare needed,a study of recordsshouldbe madeto ascertaintheir influenceupon the dischargehydrographby comparingpeaks obtainedutilizing the derived unit hydrographand actual hydrographsfrom intensestorms. Essentialstepsin developinga unit hydrographfor an isolatedstorm are: l. Analyzethe streamflowhydrographto permit separationof surfacerunoff from groundwaterflow, accomplishedby the methodsdevelopedin Section l 1.4. 2. Measurethe total volume of surfacerunoff (direct runoff ) from the storm producingthe original hydrograph.This is the area under the hydrograph after groundwaterbaseflow has been removed. 3. Divide the ordinatesof the direct runoff hydrographby total direct runoff volume in inches,and plot theseresultsversustime as a unit graph for the basin. \-
192
12 UNITHYDROGRAPHS GHAPTER 4. Finally, the effective duration of the runoff-producing rain for this unit graphmustbe found from the hyetograph(time history of rainfall intensity) of the storm eventused. Proceduresother than thoselisted are requiredfor complex stormsor in developing synthetic unit graphs when data are limited. Unit hydrographscan also be transposedfrom one basinto anotherundercertain circumstances.An exampleillustratesthe derivationof a unit hydrograph.
EXAMPLE I2.1 Using the total direct runoff hydrographgiven in Fig. I2.2, derive a unit hydrograph for the l7I5 ac drainagearea. Solution 1. Separatethe base or groundwaterflow to get the total direct runoff hydrograph.A commonmethodis to draw a straightline AC that beginswhen
2-hr rainfall duration
I I
precipitaion = 4.2 in.
*zTotal
)
o
1l
l
t2
Time (hr)
Directrunoff. /T\ ordinate Yl
\
*zTotal directrunotf of l.4l5in.on1715ac
500 o FA
2-hr unit hydrograph of l.u rn. on I / I) ac
400 300 200 .100 0
Basef-tow
/
\,
/
t'-# -j
/^ 3
4
5
6
7
Baseflow separatlon
8
Time(hr) Dfuectrunoffduration
Figure 12.2 Illustration of the derivation of a unit hydrograph from an isolatedstorm.
12,2 DERIVATIONOF UNIT HYDROGRAPHSFROM STREAMFLOWDATA
193
the hydrographstartsan appreciablerise and endswherethe recessioncurve intersectsthe baseflow curve.The importantpoint hereis to be consistentin methodologyfrom storm to storm. 2. The depth of direct runoff over the watershedis calculatedusing > (DR x Ar) _ 2447 cfs-hr -: 1 4" ''* area l7l5 ac
(r2.r)
whereDR is the averageheightof the dfuectrunoff ordinateduring a chosen time period Ar (in this caseA/ : 1.0 hr) . The valuesof DR determinedfrom Fig. I2.2 are listedin Table 12.1. 3. Computeordinatesof the unit hydrographby using
8 "_ Q , 1 v,
(r2.2)
where Q, : the magnitudeof a hydrographordinate of direct runoff having a volume equal to % (in.) at someinstant of time after start of runoff Q, : the ordinateof the unit hydrographhavinga volume of 1 in. at someinstant of time In this examplethe valuesare obtainedby dividing the direct runoff ordinatesby 1.415.Table12.1outlinesthe computationof the unit-hydrograph ordinates. 4. Determinethe duration of effectiverainfall (rainfall that actuallyproduces surfacerunoff). As statedpreviously,the unit hydrographstorm duration
TABLE 12,1 DETERMINATION OFA 2.HRUNITHYDROGRAPH FROM AN ISOLATEDSTORM (1)
(2)
(3)
Time (hr)
Runoff (cfs)
Base flow (cfs)
I 2 3 4 4.7 5 6 7 8 9 10 10.5 lt t2
110 t22 230 578 666 645 434 293 202 160 1t7 105 90 80
t10 t10 t10 110 110 110 110 110 110 110 1r0 105 90 80
(4) Direct runoff, (2)-(3) (cfs)
0 t2 120 468 556 535 324 183 92 50 7 0 0 0
(5) 2-hr unit hydrograph ordinate,(4) + 1.415 (cfs) 0 8.5 84.8 i-l
I
393 379 229 129 65.0 35.3 4.9 0 0 0
194
CHAPTER12
UNIT HYDROGRAPHS
should not exceedabout 25 percent of the drainage atea lag time' but violatesthis rule for the example.From Fig. 12.2, the rain duration is 2 hr. 5. Using the values from Table I2.1, plot the unit hydrograph shown in Fig. 12.2. r I
BY LAGGINGMETHODS APPLICATIONS 12.3 UNITHYDROGRAPH Once an X-hr unit hydrographhasbeenderivedfrom streamflowdata (or synthesized from basin parameters,Section 12.6) it can be used to estimatethe direct runoff hydrographshapeand duration for virtually any rain event.Applications of the X-hr UH to other stormsbeginswithlagging procedures,usedfor stormshavingdurations that are integermultiples of the derived duration. Applications to stormswith fractional multiples of X, known as S-hydrographandIUH procedures,are discussedin Sections12.4 and 12.5. Becauseunit hydrographsare applicableto effective (net) rain, the processof applyrngUH theory to a storm beginsby first abstractingthe watershedlossesfrom the precipitation hyetograph,resulting in an effective rain hyetograph'Any of the proceduresdetailedin Chapter 4 can be applied. The remainder of this discussion assumesthat the analysthas already abstractedwatershedlossesfrom the storm. If the duration of anotherstorm is an integermultiple of X, the storm is'treated as a seriesof end-to-endX-hourstorms.First, the hydrographsfrom eachX increment ofrain are determinedfrom the X-hourunit hydrograph.The ordinatesare then added at correspondingtimes to determinethe total hydrograph. EXAMPLE 12.2 Dischargerates for the 2-hr unit hydrographshownin Fig' I2.3 are'.
Time (hr) O (cfs)
0 0
1 100
2 250
3 200
4 100
5 50
6 0
Develophourly ordinatesof the total hydrographresultingfrom a 4-hr designstorm havingthe following excessamounts:
Hour Excess(in.)
1 Q.5
L
0.5
J
1.0
4 1.0
Solution. The 4-hr duration of the designstorm is an integermultiple of the unit hydrographduration.Thus,the total hydrographcanbe foundby addingthe contributionsof two 2-hr incrementsof end-to-endrain, asshowninFig. l2'3c. The first 2-hr stormsegmenthas 1.0 in. of net rain and thus reproducesa unit 2.0in. of netrain (in 2 hr);thus The second2-hr stormsegmenthas hydrograph. its ordinatesare twice those of a 2-hr unit hydrograph.The total hydrograph,
12.3 F
r>
UNIT HYDROGRAPHAPPLICATIONSBY LAGGINGMETHODS
1.0
h
u.)
c
0 2
3
4 .
5
6
(a) 2-hourunit storn excess
^
300 200
i90 zoo € 1oo '6 0
1.0 ir
0.5 0 2
3
4
5
6
(c)Designstom excess 600 500 '6 400 :i
H 300 i5 200 100 0 r
2
3
4
5
6
7
(d) Contributionof each2-hourstorm 600 500 a
t +oo ; ff:oo '$ zoo -100 0 0
1
2
3
4
5
6
1
8
(e) Total design hydrograph
Figure 12.3 Example 12.2 deivation of total runoff hydrographusing a 2-hr unit hydrograph.
195
196
cHAPTERi2
uNtr HyDRocRApHS
Fig.12.3e, is found by summingthe fwo contributionsat correspondingtimes. Note in Fig, 12.3d that runoff from the secondstorm beginswhen the second rain begins,not at the beginningof the first storm. r I This methodof "lagging" is basedon the assumptionthat linear responseof the watershedis not influencedby previousstorms-that is, one can superimposehydrographs offset in time and the flows will be directly additive. The simplestway to developcompositedirect runoff hydrographsfor multiple-hourstormsis in a spreadsheet.Care must be taken, however,in visually confirming, as in ExampleI2.2, that the start and end points of runoff from eachcontributingX-hr incrementof rain are properly selected.A commonerror is to lag eachadditional contributinghydrograph by Ar, the time interval betweenreadings,rather than X, the associatedduration with the given unit hydrograph.Also, the multiplier for the UH ordinatesmust be the net rain occurring in X hours, not the rain occurring in the time increment A/. Example I2.3 illustratesthesepoints. EXAMPLE 12.3 Using the derived 2-hr lunit hydrographin Table 12.1, determinethe direct runoff hydrographfor a 4-hr. storm havingthe following excessrain amounts:
Hour
I
Excessrain. in.
o.7
2 0.7
t.2
1.2
Solution 1. Tabulatethe unit hydrographat intervalsof the selectedtime interval, A/, as shown inTable 12.2. TABLE 12.2 UNIT HYDROGRAPHAPPLICATIONOF EXAMPLE12.3
Time (h0
Effective rainfall (in.)
0 I
2 J
4 5 6 7 8 9 10 1l t2
0.7 0.7 1 . 2" 1.2
Unit hydrograph (cfs) 0 8.5 84.8 JJI
379 229 129 65 35.3 4.9 0
Contrib. of first 2-hrrain U HX 1 . 4 ' 0 I1.9 l19 463 531 321 181 91 49.4 6.9 0
Contrib. of second 2-hr rain
uH x 2.4'
0 20.4 203 794 910 550 310 156 84.7 11.8 0
Total outflow hydrograph (cfs)
0 11.9 119 483 734 11 1 5 1091 641 359 163 84.7 I 1.8 0
12.gUN|THYDRoGRAPHAPPL|CAT|oNSBYLAGG|NGMETHoDS197
2. Determine the correct UH multiplier for eachX-hr interval. BecauseX is 2 hrs for this example,the first two hours of the storm producea total net rain of 1.4 inches. Similar$, the last two hours of the storm produce 2.4 inchesof net rain. 3. Determinethe correct start and end times for eachof the two hydrographs and tabulatethe contributionof the l.4-inch and 2.8-inch rains at the : 3 hrs, appropriatelag times. Becausethe secondX-hr storm startedat / : I2'2. inTable shown hrs as 3 runoff for this-stormcannotbegin until / hydrographs runoff the total 4. Add the contributionsat eachtime to determine for the 4-hr storm. 5. Checkthe tabular solutionby plotting eachof the two hydrographsand sum rr the ordinatesat each/, as showninFig.I2'4' In addition to using a given X-hr UH for determiningthe runoff hydrographfor a given storm, tagging of ttt" X-hr UH can be used to developother duration unit hyirographs.The pioCedureis the sameas applyingthe X-hr UH to 1.0 in. of net rain in f nouri. As earlier, Y must be an integermultiple of X. For example,if a 1-hr unit hydrographis availablefor a given watershed,a unit hydrographresultingfrom a 2-hr siorm-is tbtained by plotting two L-hr unit hydrographs,with the secondunit hydrographlagged t hr, adding ordinates,and dividing by 2' This is demonstratedin nigl ti.S, riliere the dashedline rept"sentsthe resulting2-ht unit hydrograph'Thus the t in. of rainfall containedin the original 1-hr duration has been distributedover a Z-hr period.
t
2
3
4
5
6
7
8
9
1
0
1
1
1
Time units
Figure 12.4 Synthesizedhydrographfor'Example 12'3 derivedby the unit hydrographmethod'
\_
2
198
CHAPTER 12 UNITHYDRoGRAPHS
l-hr unit hydrogaph 2-hIunithy&ogaph
Time
Figure12.5 Unithydrographlaggingprocedurero developanotherunit hydrograph. Modificationsof the original unit-hydrographduration can be madeso that two 1-hr unit hydrographsare usedto form a 2-hr unit hydrograph;two 2-hr unit hydrographsresult in a 4-fu diagram,and so on. Care must be taken not to mix durations in the lagging procedure, since errors are introduced; a l-hr and a}-hr unit hydrograph do not representa 3-hr unit hydrograph.Lagging procedureis therefoie restrictedto multiples of the original duration accordingto the expression D1 : nD
(I2.3\
where Dl : the possibledurationsof the unit hydrographby lagging methods D : the original duration of any given unit hydrograph fl: I,2,3,..
12.4 S.HYDROGRAPHMETHOD The S-hydrographmethodovercomesrestrictionsimposedby the laggingmethodand allowsconstructionof any durationunit hydrograph.By observingthe lagging system just described,it is apparentthat for a l-hr unit hydrograph,the l-in. rainfall excess hasan intensityof 1 in./hr, whereasthe 2-hr unit hydrographis producedby a rainfall intensity of 0.5 in./hr. Continuouslagging of either one of theseunit hydrographsis comparableto a continuouslyappliedrainfall at either 0.5 in./hr or 1 in./hr intensity, dependingon wfuch unit hydrographis chosen. As an example,usingthe 1-hr unit hydrograph,continouslaggingrepresentsthe direct runoff from a constantrainfall of 1 in./hr as shownin Fig. r2.6a. The cumulative addition of the initial unit hydrographordinatesat time intervalsequalto the unit storm duration resultsin an S-hydrograph(seeFig. r2.7). &aphically, construction of an s-hydrographis readily accomplishedwith a pair of dividers. The maximum
12.4 S-HYDROGRAPHMETHOD
*l
199
Dhr
D-hr S-hydrographlaggedt hr
*l
Time
l-
(a)
Figure 12.6 S-hydrographmethod.
dischargeof the S-hydrographoccursat a time equalto D hourslessthan the time base of the initial unit hydrographas shown inFig. 12.6a. To constructa pictorial 2-hr unit hydrograph,simply lag the first S-hydrograph by a secondS-hydrographa time interval equalto the desiredduration.The difference in S-hydrographordinatesmust then be dividedby 2. Any duration r unit hydrograph may be obtained in the samemanner once another duration D unit hydrqgraphis known. Simply form a D-hr S-hydrograph;lag this S-hydrographf hr, andmultiply the difference in S-hydrographordinatesby D/t. Accuracy of the graphical procbdure dependson the scaleschosento plot the hydrographs.Tabular solution of the Shydrographmethodis also employed,but hydrographtabulationsmustbe at intervals of the original unit.hydrographduration. EXAMPLE 12.4 Given the following 2-hr unit hydrograph,use S-hydrographproceduresto construct a 3-hr unit hydrograph. Time (hr) 0 (cfs)
o 0
| 100
2 250
3 200
4 100
5 50
6 0
Solution. The 2-hr unit hydrographis the runoff from a 2-hr stormof 0.5 in./hr. The S-hydrograph is formedfrom a net rain rate of 0.5 in./hr lasting indefinitelyas shownin Fig. 12.6a.Its ordinatesarefoundby addingthe 2-hr (UH) runoff ratesfrom eachcontributing2-hr block of rain: unit-hydrograph
200
CHAPTER12
UNIT HYDROGRAPHS
I
i
/
i
/,/
I
a
90
-'{.3 u0 Figure 12.7
S-hydrograph.
Time (h0
1st2-hr
0 I z
3 4
5 6 7 8
0 100 250 200 100 5n
0
Time (min.)
2nd Z-hr
0 100 250 200 100 50 0
S-hydrograph
3rd2-hr
' 0 100 250 200 100
0 100 250
0 100 250 300 350 350 350 350 350
To find a 3.-hrhydrograph,the S-curveis laggedby 3 hr and subtractedasshown in Fig. 12.6b.This results in a hydrographfrom a 3-hr storm of 0.5 in./hr, or 1.5 in. total. Thus the ordinatesneed to be divided by 1.5 to producethe 3-hr unit hydrograph:
201
12.5 THE INSTANTANEOUSUNIT HYDROGRAPH
Time (hr)
S-hydrograph
0 1 2 3 4 5 6 7
0 100 250 300 350 350 350 350
Lagged S-hydrograph
0 100 250 300 350
Difference
0 100 250 300 250 r00 '50 0
3-hrUnit hydrograph 0 67 167 200 167 . 6 7 33 0
12.5 THE INSTANTANEOUS UNITHYDROGRAPH The unit-hydrographmethodof estimatinga runoff hydrographcanbe usedfor storms of extremelyshortduration.For example,if the durationof a stormis 1 min and a unit volume of surfacerunoff occurs, the resulting hydrograph is the 1-min unit hydrograph.The hydrographofrunofffor any 1-min storm ofconstant intensitycan be computedfrom the l-min unit hydrographby multiplying the ordinatesof the 1-min unit hydrographby the appropriaterain depth.A storm lasting for many minutescan be describedas a sequenceof 1-min storrns(Fig. 12.8).The runoffhydrographfrom each l-min storm in this sequencecan be obtainedas in the precedingexample.By superimposingthe runoff hydrograph from each of the l-r.nin storms, the runoff hydrographfor the completestorm can be obtained. From the unit hydrographfor any duration ofuniform rain, the unit hydrograph for any other durationcanbe obtained.As the durationbecomesshorter,the resulting unit hydrographapproachesan instantaneousunit hydrograph.The instant4neousunit hydrograph(IUH) is the hydrographof runoff thaf would result if 1 in. of waterwere spreaduniformly over an areain an instant and then allowed to run off.e To develop an IUH, any I in.lhr S-hydrographmust first be obtained. The resulting S-curve is laggedby the interval Ar to developa Ar-hour unit hydrograph. The resulting At-hour unit graph becomesan IUH when Ar is set to 0.0 in the limit. If a continuing 1 in./hr excessstorm produces the original and lagged Shydrographsof Fig. 12.6b,the Ar-hour unit hydrographis the differencebetweenthe two curves,divided by the amount of excessrain depth in A/ hours,or Qo- Q" Q,(Lt-hr UH) : ILt
(r2.4)
The Qo - Q" dtfferencesare dividedby I Lt to convertfrom a stormwith 1Al inches in Al hours to one with 1.0 in. in At hours,which is the definition of a Ar-hour unit graph. As Ar approacheszpro, Eq. 12.4 becomes
: !as 0,(ruH) Idt
(12.s)
202
CHAPTER12 UNIT HYDROGRAPHS
Figure 12.8 Unit-hy&ographdescriptionof the runoff process.(a) Unit hydrograph;(b) a sequenceof l-min storms;(c) superposition for eachof of runoffhydrographs the l-min storms.(After Schaake.e) which showsthat the flow at time I is proportional to the slopeof the S-hydrograph at time r. In applications,the slopeis approximatedbyLQ/A,I, and the IUH ordinates can be estimatedfrom pairs of closelyspacedpoints of the S-hydrograph. Ifan IUH is supplied,the aboveprocesscan be reversed,and any X-hour unit graph can be found by averagingIUH florvsat X-hr intervals, or
Q,(X-tuUH) : 1(IUH,+ IUH,_X)
{r2.6)
Use of this approximate equation is allowed for small X values and permits direct calculationof a unit graph from an IUH, bypassingthe normal S-hydrograph procedure.
UNITHYDROGRAPH 12.5 THEINSTANTANEOUS
203
EXAMPLE 12.5 Given the following 1.0 in./hr S-hydrograph,determinethe IUH, and then use it to estimatea 1-hr UH.
Time (hr) S-curve (cfs)
0 0
0.5 5 0
1.0 200
1.5 450
2,0 500
2.5 650
3.0 700
3.5 750
4.0 800
Solution. The IUH is foundfrom Eq. l2.5.The slopeat time r is approximated bY (Q,*o.,- Q,-o)lLt
IUH = AQlAf
0 0.5 1 1.5 2 2.5 3 3.5 +
5
0 200 400 300 200 200 100 100 50 0 0
0 JU
200 450 500 650 700 750 800 800 800
The 1-hr uH is obtainedfrom F,q.12.6,usingreadingsat 1-hr intervals:
0 1 2 3 A
5 6
luHf
luHr_1
1-hr UH
0 400 200 100 50 0 0
0 0 400 200 100 50 0
0 200 300 150 75 25 n "
l
l
The readershouldverify that the 1-hr UH obtainedthrough use of the IUH is approximatelyt[e sameas that obtainedby lagging the S-hydrographt hr, subtracting, and convertingthe differenceto a 1-hr UH. The ordinatesof the IUH representthe relative effect of antecedentrainfall intensitieson the runoff rate at any instant of time. By plotting the IUH with time increasingto the left rather than to the right (seeFig. I2.9), andthen superimposing this plot over the rainfall hyetograph(plotted with time increasingto the right as in pig.iZ.g),the'relative weight given to antecedentrainfall intensities(asa function of time into the past) is easily observed.In other words, the runoff rate at any time is
/
204
12 UNITHYDROGRAPHS CHAPTER
Time-reversed image of the instantaneous unit hydrograph
Time into the past Antecedent iainfall intensities
oo=[;f@xi(t-r)dr
unit hyFigure 12.9 Calculationof runoff rates with the instantaneous antecedent the of average is a weighted drograph.The runoff rateat arrytime unit hyimageof the instantaneous rainfall intensities.The time-reversed the weightingfurtction.(After Schaake'v) drographrepresents computedas a weighted averageof the previousrainfall intensities.Therefore,the computedrunoff hydrographis the weighted,moving averageof.the rainfall pattern and itre weighting irtt"iiott is the time-reversedimage of the unit hydrograph'e Statedmathematically,the runoff rate at any time is given by the convolution integral
Q(A:
l,',<,tu,r)dr
(r2.7) ll:
12.6 SYNTHETICUNIT HYDROGRAPHS
where
205
Q(t) : the surfacerunoff rate at time t f(r) : the ordinate of the IUH at time r i(t - r): the rainfall inten'sity (after abstraction of the appropriate infiltration losses)at time t - r
The variable 7 representstime into the past so that time r - r occursbefore time r. The limits on the integral allow r to vary betweena pastand presenttime (i.e., r : 0, t - r : 0)' The integral givesa continuousweightingof prbviousrainfall intensities by the ordinatesof the IUH.
12.6 SYNTHETIC UNITHYDROGRAPHS As previously noted, the linear characteristicsexhibited by unit hydrographsrf6i a watershedare a distinct advantagein constructingmore complex rto.- Oir.fruig. hydrographs.Generally,however,basicstreamflowand rainfall data arcnot available to allow construction of a unit hydrographexcept for relatively few watersheds; therefore, techniqueshave evolved that allow generationof syithetic unit lrydrographs.
GammaDistribution The shapesof hydrographsoften closely match a two-parametergamma function, given by
f(x) :
xoe-*/B
B"+lf(d + 1)
(12.8)
where0 ( r ( m. The parametera is a dimensionlessshapefactor (mustbe greater tltul - 1), and B is a positive scalefactor havingthe sameunits as x and contiolling the baselength. The product of a and B givesthe value-r correspondingto the apexl or maximum value ofl(x). For a ) 1, the distribution has a single upe* und'plot, similar to hydrograph shapes,as shown in Fig. 12.10. The dislribution mein is F@ + 1), and varianceis p2(a + I). . Many of the syntheticunit hydrographproceduresresult in only three to five points on the hydrograph,through which a smooth curve must be fitted. In addition to the requirementthat the curve passesthrough all the points, the area under the hydrographmust equal the runoff volume from one unit of rainfall excessover the watershed.This latter requirementis often left uncheckedand can result in considerablg errors in performing calculationsthrough the use of ordinatesof a hydrograph that do not representa "unit" of runoff. The mostusefulfeatureof the gammadistributionfunction (explainedin greater detail later) is thit it guaranteesa unit areaunder the curve. It can convenientlybe used to synthesizean entire hydrographif the calculatedpeak flow rate eo and its associatedtime to are known. This usesa proceduredevelopedby Aron und'White.ro If time r is substitutedfor x in Eq. 12.g, the time to peak tois aB. At this point, the function/(r) equalsthe peak flow rate e,, or
o,:ffi=ffrr*1
(r2.e)
206
CHAPTER12
UNIT HYDROGRAPHS
1.0
B--1
0 x
Figure 12.10 Gamma function shapesfor various shape and scale parameter values. whereC, A is the unit volumeof runoff from a basinwith arcaA. The conversionfactor C, = 1.008 is selectedto make @(a)dimensionless. The function f(a) is shownby Aron and white to be relatedto a by1l (12.10) a : 0.045+ 0.5d + 5.6Q2+ 0303 Collins showsthat this can be approximatedreasonablywell in the range 1
( a ( 8
byt'
q.:05Q+5.902
(r2.Lr)
Combiningthis with F,q. 12.9 *tu"t
o : o s, tf f i . t n ( H ) '
(r2.r2)
To fit a unit graph usingEqs. I2.9 and 12.12,the peak flow rate and time must be estimated.Severalof the methodsdescribedsubsequ-ntlyallow this. Next. @(a)is foundfrom Eq.12.9, and a from Eq.12.10 ot l2.Il. The unit hydrographcan now be constructedby calculatingQ at any convenientmultiple, a, of to. Substitutingalo for x in Eq. 12.8 gives the flow at t : atp as -
Qoto:
QraoeQ-o)"
(r2.r3)
which can be solvedfor all the flow rates of the hydrograph. EXAMPLE 12.6 The peak flow rate for the unit hydrographof a 36,000-acrewatershedis 1720cfs and o".oi, 12 hr following the initiation of runoff. Use Eq. 12.8 to synthesizethe rest of the hydrograph.
12.6 SYNTHETICUNIT HYDROGRAPHS
207
Solution. From Fq. 12.9,
r720(12) : 0.57 6@) : 1.008(36,000) From Eq. 12.10(and t2.It), q.:2.2 The hydrographis then found from Eq. L2.I3: a) e2'2(rQ*, : 1720a2'2 Solving for a few points, we obtain the followine values: 10.0
t = afo(hr)
O(ctu)
0 0.5 1.0 2.0 5.0 10.0
0 6 l2
0 tt25 1720 876 9 0
)4
60 120
sufficient intermediatepoints shouldbe generatedto define the entire shapeof the hydrograph. tl
Snyder's Method one- techniqueemployedby the corps of Engineersl3and many othersis basedon methods developedby Snyderaand expandedby Taylor and 3chwartz.sIt allows computation of lag time, time base,unit_hydrographduration, peak discharge,, and hydrographtime widths at 50 and75 perceniof peukflor. ny uring tttesesevenpoints, a sketchof the unit hydrographis obtained,rig.lz.rt,and cleckel to seeif it contains 1 in. of direct runoff.
Alternate recessions I to produce 1.0 in. ofrunoff
Time,r F'igure 12.11 Snyder'ssyntheticunit hydrograph.
208
CHAPTER12
UNIT HYDROGRAPHS
Time to Peak Snyder'smethodof synthesizinga unit hydrographassumesthat the peak flow rate occursat the watershedlag, estimatedfrom Eq. 11.5.Its locationis as shownon Fig. 12.11.The lag time and peak dischargerate are both established correlatedwith variousphysiographicwatershedcharacteristics.For the lag time, the variablesL and L"o for Eq. 11.5 are estimatedfrom map measurements,and C, is developedfor the locale,usingSnyder'sestimatesor other sources.TableL2.3summarizes a variety of C, valuesfor variousregions. It is assumedthat lag time is a constantfor a particular watershed-that'is, uninfluencedby variations in rainfall intensitiesor similar factors. The use of L"o accountsfor the watershedshape,andC, takescare of wide variationsin topography, from plains to mountainousregions. Steeperslopestend to generatelower valuesof C,, with extremesof 0.4 nqtec in SouthernCalifornia and 8.0 alongthe Gulf of Mexico and Rocky Mountains.W{ren snowpackaccumulationsinfluencepeak discharge,valuesof C' will be betweenone sixth to one third of Snyder'svalues. Time Base The time base of a syntheticunit hydrograph(seeFig. 12.11) by Snyder'smethod is t , : ?
(r2.r4)
, t l
T ;
6
where t6 : the basetime of the syntheticunit hydrograph(days) t1: the lag time (hr)
TABLE 12.3 TYPICALSNYDER'SCOEFF|CIENTSFOR U.S. LOCALITIES Location
Range of C,
Average Cr
Range of Co
Average Co
Appalachian Highlands West Iowa Southern California Ohio Eastern Gulf of Mexico Central Texas North and Mid-Atlantic states Sewered urban areas Mountainous watersheds Foothills areas Valley areas Easlern Nebraska Corps of Engineers training course
t.8-2.2 0.2-0.6
0.4-0.8 0.7-1.0
0.4-2.3
2.0 o.4 0.4 0.7 8.0 1.1
0.3-1.2
0.6 0.8 0.9 0.6 0.6 0.8
0.2-0.5
0.6/\4" 0.3
0.1-0.6
0.3
0.5-1.0
0.8
Great Plains Rocky Mountains SW desert NW coast and Cascades 21 urban basins Storm sewered areas "Channel slope S
0.6-0.8
0.4-1.0 0.4-8.0 0.8-2.0 1.5-8.8 0;t -1.9 2.0-4.4 0.3-0.9 0.2-0.3
1.2 o;7 0.4 0.8 0.3-0.9 1.3 1,4
3.1 0.6 0.2
0.6-0.7
12.6 SYNTHETIC UNITHYDROGRAPHS 209 Equation L2.14 gives reasonableeslimatesfor large watershedsbut will produce excessivelylarge valuesfor smaller areas.A generalrule of thumb for small areasis to use three to five times the time to peak as a base value when sketchinga unit hydrograph.In any event, the time base shouldbe adjustedas shown in rig. tz.lt until the areaunder the unit hydrographis 1.0". Duration The duration of rainfall excessfor Snyder's synthetic unit-hydrograph developmentis a function of lag time '
l -
-
-
tt
5.5
(r2.1s)
where /, : duration of the unit rainfall excess(hr) t1 : the lag time from the certtroidof unit rainfall excessto the peak of thb unit hydrograph This synthetictechniquealwaysresultsin an initial unit-hydrographduration equalto fi/5.5.However, sincechangesin lag time occur with changesin duration of tlie unit hydrograph,the following equation was developedto allow lag time and peak dischargeadjustmentsfor other unit-hydrographdurations. tm:h+0.25(t*-t) where t1p: tt : to: t, -
(r2.16)
the adjustedlag time (hr) the original lag time (hr) the desiredunit-hydrographduration (hr) the original unirhydrograph duration : t,/5.5 (hr)
Peak Discharge If one assumesthat a given duration rainfall produces1 in. of direct runoff, the outflow volume is some relatively constantpercentageof inflow volume.A simplified approximationof outflow volume is r, X er, andthe equation for peak dischargecan be written Vr:
-640CPA Ltn
(12.r7)
where Qp : peak discharge(cfs) C" : the coefficientaccountingfor flood waveand storageconditions;it is a function of lag time, durationof runoff ptoducingrain, effectivearea contributing to peak flow, and drainagearea A : watershedsize (mi2) tp : the lag time (hr) Thuspeakdischargetanbe calculatedgivenlag time andcoefficientofpeak discharge C.. Valuesfor Cp range from 0.4 to 0.8 and gengrally indicate retention or storage capacityof the watershed.Larger valuesof C" aregenerallyassociatedwith smaller valuesof C,. Typical valuesare tabulated inTable 12.3. Hydrograph Construction From Eqs: 1I.5, lZ,I4, I2.I5, and,12.1,7plot three points for the unit hydrographand sketcha syntheticunit hydrograph,remembering
210
CHAPTER12
UNIT HYDROGRAPI.IS
that total direct runoff amountsto 1 in. An analysisby the Corps of Engineers(see Fig. 12.12) gives additional assistancein plotting time widths for points on the hyAs a generalrule ofthumb, drographlocatedat 50 and 75 percentofpeak discharge.l3 eachsideof the pegk proportioned ttr" ti-l width at l/so and IV^ ordinatesshouldbe unit-hydrograph synthetic the left of in a ratio of I:2 with the short time sideon the gives valuesfor unrealistic Eq. I2.I4 peak. As noted earlier,for smallerwatersheds, to the time total multiplying by ihe bur" time. If this occurs,a value can be estimated and amount on the based modified peak by a value of from 3 to 5. This ratio can be boundaries. watershed ii." tut" of depletionof storagewater within the The envelopecurvesin Fig. I2.I2 ate definedby
/
Wso: 8301(Qo/A)"
(12.18)
W15: 470f(Qo/A)"
(t2"]a)
The sevenpoints formed through the use of theseequationscan be plotted and a smooth curve drawn. To assurea unit hydrograph,the curve shape and ordinates shouldbe adjusteduntil the areabeneaththe curve is equivalentto one unit of direct runoff depth over the watershedarea.This can be doneby hand-fittingand planimetering or by curve-fitting. Hudlow and Clarkla used least-squaresregressiontechniquesto fit a Pearson III type (gamma)probability densityfunction (referto Chapter26) throughthe seven Snyaeruoit-nyAtographpoints. This function has an areaof 1.0 and a shapesimilar 1000 800 400 N
,2
s
9 200 oo
) \
€ 1oo -? 8 0
\
4-l
'.,
E 6 0 9" 40
* -
n
F\
n
t *tS
d
o
08,
10 8 6 0.2
0 . 4 0 . 6 0 . 8I
2
4
6
810
\, \
\ $ \ 40 60 80100
Width of unit hYdrograPh (hr)
Figure L2.12 Unit hydrographwidth at 50 and 75 percent of peak flow' a, observedvalue of Wso.o, observedvalue of lfi5.
c uNrrHYDR.GRAPHS 211" to thator naturalhydrographs. rh" ,h;:l Q,:
nr(;
,lll::i e-G-tp')/b
(r2.20)
where a and b are shapeand scaleparameters.Hudlow and Clark presenta trial-anderror solution to the least-squaresnormal equations,using Newton's method, to developestirnatesof a and b. The application of Snyder's syntheticunit-hydrographmethod to areasother than the original study area should be precededby a reevaluationof coefficientsC, and C, in Eqs; 11.5 and I2.I7 . TIns analysiscan be accomplishedby the use of unit hydrographsin the region under study which havethe proper lag time-rainfall duration ratio; that is, t, : ttf 5.5. If anotherrainfall durationis selected, variationsof C and C, can be expected.
SCS Method A methoddevelopedby the Soil ConservationServicefor constructingsyntheticunit hydrographsis basedon a dimensionlesshydrograph(Fig. 12.13). This dimensionless graph is the result of an analysis of a large number of natural unit hydrographs from a wide range in size and geographii locations.The method requires only the
0.6
cSlo b
0.5
als 0.4 0.3 0.2 0.1
t tp
Figure 12.13 Dimensionless unit hydrographand masscurve. (After Mockus.1s)
212
CHAPTER 12 UNITHYDRoGRAPHS determination of the time to peak and the peak discharge as follovi,s:
tp:
D t+
tt
(r2.2r)
where to : the time from the beginningof rainfall to peak discharge(hr) D : the duration ofrainfall (hr) t1 : the lag time from the centroid of rainfall to peak discharge(hr) The ratios correspondingto Fig. 12.13 are listed in Table 12.4. The peak flow for the hydrographis developedby approximatingthe unit hydrographas a triangular shapewith basetime of ! t, and unit area.The readershouldverify that this produces ^
U r :
4844
-
t
(r2.22)
where Qp: peak discharge(cfs) A : drainagearea (mi2) tp : the time to peak (hr) The time base of ! ro is based on empirical values for averagerural experimental watershedsand shouldbe reduced(causingincreasedpeak flow) for steepconditions or increased (causing decreasedpeak flow) for flat conditions. The resulting coefficientinBq.12.22 rangesfrom nearly 600 for steepmountainousconditionsto 300 for flat swampyconditions. A relation of /, to size of watershedcan be used to estimatelag time. Typical relations from two geographicregionsare tt :
l.44A0'6 Texas
t,:
0.54A0'6 Ohio
TABLE 12.4 COORDINATESOF SCS DIMENSIONLESS UNIT HYDROGRAPHOF F I G U R E1 2 . 1 3 Q/Q,
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 t.2 1.3
0 0.015 0.075 0.16 0.28 0.43 0.60 0.77 0.89 o.9'7 1.00 0.98 0.92 0.84
Q/Qp
1.4 1.5 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.5 4.0 5.0
0.75 0.66 0.56 0.42 0.32 0.24 0.18 0.13 0.098 0.075 0.036 0.018 0.009 0.004
(r2.23a) (r2.23b)
12.6 SYNTHETIC UNITHYDROGRAPHS 213. The averagelag is 0.6/",where/" is the time of concentration,definedby SCS aseither the time for runoff to travel from the furthermostpoint in the watershed(calledthe upland method) or the time from the end of excessrain to the inflection of the unit hydrograph.For the first case, t":L1tp-D
(r2.24)
The dimensionlessunit hydrograph,Fig. 1.2.1.3,has a point of inflection at approximately 1..7t,.If the lag time of 0.6t" is assumed,Eqs. 12.2I and 12.24 give D - 0.2t0 D = 0.I33t"
0).zs) {r2,26)
A small variation in D is permissible,but it shouldnot exceed0.25t, or 0.I7t". pnce the 0.133r"-hourunit hydrographis developed,unit hydrographsfor other durdtions can be developedusing S-hydrographor IUH procedures. By finding a value of t,, a syntheticunit hydrographof chosenduration D is obtainedfrom Fig. 12.13. Atother equationusedby the SCS is . t - : "' where fi : I : Y : S:
/0.'(s * 1;o.z
lgooyo's
(r2.27)
the lag time (hr) length to divide in feet averagewatershedslopein percent the potentialmaximumretention(in.) : (1000/CN) - 10, where.CN is a curve number describedin Chapter4
The lag from Eq. 12.27is adjustedfor imperviousnessor improved watercourses,or both, if the watershedis in an urban area.The multiple to be applied to thq lag is M : 1.- P(-6.8 X 10-3 + 3.4 x 10-4CN- 4.3 x 10-?CN, -2.2 x 10-8CN3) (12.28) where CN is the curve number for urbanized conditions, and P can be either the percentageimperviousor the percentageof the main watercoursethat is hydraulically improved from natural conditions.If part of the area is imperviousand portions of the channelare improved,two valuesof M are determined,and both are multiplied by the lag. EXAMPLE 12.7
.\ For a drainagearea of 70 r4i2 having a lag time of 8 | hr, derive a unit hydrograph of duration 2 hr. Use the SCS dimensionlessunit hydrograph. Solution 1. Using Eq.l2.21we obtain
t,:?*8*=9*t'tr
214
CHAPTER12
UNIT HYDROGRAPHS
2. From Eq.12.22 _484x70 n YP 9.5
Qo:3J60 cfsoccurringatt :9|hr 3. Using Fig.12.13, we find the following: a: The peak flow occursar tfto: 1 or at t : 9+hr. b. The time baseof the hydrograph: 5toor 47.5 hr. c. The hydrographordinatesare: l . A t t / t o : 0 . 5 ,Q / Q r : 0 . 4 3 ; t h u sa t t : 4 . 7 5 h r ,Q : 1 5 3 1c f s . 2. At tlto : 2, Q/Q" - 0.32; thus at t : 19 ht, Q : 1139cfs. 3. At tft, : 3, Q/Q, : 0.07; thus at t : 28.5 hr, Q : 249 cfs. 4. CheckD/to:0.21; OK. rl
Gray'sMethod Another method of generatingsynthetic unit hydrographshas been developedby Gray.16An approximateupper limit of watershedsize for applicationof this method to the geographicareasof central Iowa, Missouri, Illinois, and Wisconsinis 94 mi2. The method is basedon dimensionalizingthe incomplete gamma distribution and resultsin a dimensionlessgraph of the form O,t" :
where
Q,lPo: q and y : f : e: Pp : I :
25.0(y')n
r(q)
t 1o_7,t/pR\( \s-1
(r2.2e)
\Pol
percentflow in 0.25 PRat any given r/P^ value shapeand scaleparameters,respectively the gammafunction of q, equalto (4 - 1)!* the baseof natural logarithms the period of rise (min) time (min)
The relationfor 7' is definedzs yt : yPpandq : l.+ y'. This form of the dimensionless unit hydrograph(Fig. 12.14)allowscomputation ofthe dischargeordinatesfor the unit hydrographat times equal to I intervals of the period of rise P*, that is, the time from the beginningof rainfall to the time of peak dischargeof the unit hydrograph. Correlationswith physiographiccharacteristicsof the watershedcan be developedto get the valuesofboth Poand y'. As 4qexample,the storagefactorP^fy' hasbeenlinked with watershedparameters Lf\/S", where L is the length of the main channel of the watershedin miles measuredfrom the outlet to the uppermostpart of the watershed(Fig. 12.15); S. is definedas an averageslopein percentobtainedby plotting the main channelprofile rrf4isnotaninreserf(q) : f(N + Z) : (N - I + z)(N- 2 + z)... (t + Z)/f(r + Z), where N equalsthe integer.""-rl
q, the function is approximatedby
: qe.-al';l. r(q) h. ;uF-&
- #fu.
)
12.6 SYNTHETICUNIT HYDROGRAPHS
215
Muckey Creek near Mapleton, lowa F< a-
K 1 6 F re o ' E o
p
-
6
-
x
1.0
1.5 time (min.)
2.0
l(Atloi -=:, penoo or nse (mrn.)
t PR
Figure 12.14 Dimensionlessgraph and fitted two-parameter gamma distribution for Watershed5. (After Gray.r6)
='(ft)' 957o confidence belts tlr
a-l>
Ttta
for Ppll'
b
IL, IA,MO,WI
": lH l0 e ttr x t9!
OH
11.4
NE andW. IA
o l H
al
9.27 0.562 A A
0.531 0.498
(After Gray.16)
(r = 0.92)
Rudo,tgilrgljggtq. lchannel
slope(Vo)
-L
r.ir '
VS"
Figure 12.15 Relationof storagefactor,Ppfy', and watershedparameter, (After LV 5",for in Nebraska, Iowa,Missouri,Illinois,andWisconsin. - watersheds Gray.r6) and drawing a straight line through the outlet elevationsuch that the positive and negativeareasbetweenthe streamprofile and the straightline are equal.The storage factorPofy' can also be correlatedwith the period of rise P^ as shownin Fig. 12.16. These two cerrelations allow solution of Eq. 12.29 and produce a synthetic unit hydrographof duration P^f4 for an ungaugedarea.
216
12 UNITHYDROGRAPHS CHAPTER
ql-
\to .
F
"!'
h
0.0139 2.6761Po+
x t!9
.Fl6 o t * A 'l
0
40
80
120
160
200
240
280
320
360
Period of rise, Pp (min.)
Figure12.16 Relationof storagefactorP*fy' andperiodof risePn.(After Gray.r6; The solutionproceedsas follows: 1. DetermineL, 5", and A for the ungaugedwatershed' 2. DetermineparametersP*, T', and q. a. With Lf\/5", useFig. I2.I5 to selectPpfy'. b. With P*/y', use Fig. 12.16 to obtain Pa. Compute y' as the ratio Pol@*ly'). : Substitute c. 7' obtained in Step 2b into the equation Q | + y', and solvefor 4. 3. Computethe ordinatesfor the dimensionlessgraph using Eq. 12'29' Compute the percent flow in 0.25PRfor values of tf P*: 0.125, 0.375, 0.625,. . . , and every succeedingincrementof tf P* : 0'250 until the sum of the percent flows approximates100 percent. Also compute the peak percentageby substitutingtfPR: 1. 4. Computethe unit hydrograph. a. Computethe necessaryfactorto convertthe volume of the direct runoff under the dimensionlessgraph to 1 in. of precipitation excessover the entire watershed. 1. The volume of the unit hydrograph: V
*
V : 1 i n . x A n i z x 6 4 0nuTf '
I
n'rnln
.- --^ ft2
x 43'560 ;
graph: Vo 2. Thevolumeof the dimensionless V o : 2 q , x 0 . 2 5 x P 'Rx 6 0 -
fiun 3. Solve fot 2 q, by equatingV and Vo, sincethey must be equal'
12.6 SYNTHETICUNIT HYDROGRAPHS
217
Rdinfall duration = 6 mln = Pnl 4
s 400 300
60 Time(min)
Figure 12.17 Derived hydrographof Example12.8usingGray'smethod.Notethat Gray's method results in a unit hydrograph for a P^/4-hr storm.
b. Convert the dimensionlessgraph ordinatesto the unit-hydrograph ordinates n _ perc€ntflow in 0.25PR\ u,-T.aqi c. Translate time base of dimensionlessgraph to absolutetime units by multiplying t/PR x P^ for each computedpoint. Rememberthat runoff doesnot commenceuntil the centroid of rainfall, or at a time P^/8. An exampleproblemdemonstrates the solutionof Gray'smethodfor a Missouri watershed. A plot representing the derivedhydrographis shownin Fig. l2.r7.Dkect runoff commeircesat the centroid of rainfall. Thus it is necessaryto add D/2 or Po/B to column 2,Table r2,.5,to obtain the proper times of the unit trydrographordinates, showsin Column 6. EXAMPLE 12.8
For the given data, use Gray's method to constructa unit hydrographfor the Green Acrewatershed, wheredrainagearea: 0.62mi2,length: 0.98mi, andS" : l.45%a. Procedure 1. a. b. c. 2. a.
Figure12.15;L/t/S: 0.813 ni: Pnfy' : 8.25min. Figure 12.161, P^/^y' : 8.25 min; P*: 24.9 min. q : I I Y' : 4.02; Y' : 3.02. Tabulatepercentflow in 0,25PRfortfP*: 0.250:
218
UNIT HYDROGRAPHS
CHAPTER12
TABLE 12.5 TABULATIONFOR EXAMPLE12.8
(1) r/P
0.000 0.125 0.375 0.625 0.875 1.000 r.t25 t.375 t.625 1.875 2.125 2.375 2.625 2.875 3.125 3.375
(2) Time, (min)
(3) Percent flowin 0.25Pn
(4) Cumulated flow
0 3.1 9.3 15.6 2r.8 24.9 28.0 34.2 40.4 46.7 52.9 59.3 65.5 71.7 78.0 84.2
0 0.45 5.80 t2.70 16.35 16.85 16.25 14.20 11 . 1 0 7.97 5.55 3.56 2.28
0 0.45 6.2 18.9 35.3
l.4l
0.86 0.50
5r . 5 65.7 76.8 84.8 90.4 93.9 96.2 97.6 98.5 99.0
(5) UH (cfs) 0 lt.J
490 631 651 628 548 428 308 214 138 88.0 54.4 JJ.J
19.3
(6) Actualtime (min) 0 6.1 12.3 18.6 24.8 27.9 31.0 37.2 43.4 49.7 " 55.9 62.3 68.5 '74.7 81.0 87.2
3. a. 1. V : I x 0.62 x 640 x 43,5601t2: 14.4X 10sft3' 2 . V o : 0 . 2 5 x 2 4 . 9x 6 0 X > q , : 3 7 3 . 5 2 q ' 3.2 q,: 3860. b. Column 5 is tabulated by multiplying 3860 times values in Column 3 divided by 100. c. Column 2 is obtainedby multiplying24.9 times valuesin Column 1. d. Column 3 comesfrom solutionof Eq. l2'2J' rr
Espey10-MinuteSyntheticUnit Hydrograph A regionalanalysisof 19 urbanwatershedswasconductedby EspeyandAltmanlTand resulted in a set of regressionequations.thatprovide sevenpoints of a 10-min hydrograph.The entire hydrographis developedby fitting a smooth curve through the points using eye-fitting or curve-fitting procedures.In either case, a unit area is necessary. The equationsfor time to peak (minutes),peak discharge(cfs),time base(minutes),and width at 50 and 75 percentof the peak flow rate are *
fo'1801 57 To : 3.ILo'23S-o'2s Qp : 3L62 X I03Ao'e6T-t'o7 Ta:
125.89 x I}3AQ;o'es
W5o: 16.22x I03Ao'e3Q-oe2 7eQ-o78 w 1 5: 3 . 2 4 x 1 0 3 4 0
where
(r2.30) (r2.3r) (r2.32) (12.33) (r2.34)
L : total distance (ft) along the main channel from the point being consideredto the upstreamwatershedboundary
12.6 SYNTHET|C UNITHYDROGRAPHS219 : S main channel slope (ftlft) defined by H/0.8L, where 11 is the differencein elevationbetweenthe point on the channelbottom at a distanceof 0.2L downstreamfrom the upstreamwatershedboundary and a point on the channelbottom at the downstreampoint being considered 1 : imperviousarea within the watershed(7o) Q : a dimensionlesswatershedconveyancefactor A : watersheddrainagearea (mi2) T, : time of rise of the unit hydrograph (min) Q, = peak flow of the unit hydrograph (cfs) I, :,time baseof the unit hydrograph(min) ITso: width of the hydrograph at 507o of Q, @in) I4zrr= width of the unit hydrographat75%oof p. (min) The coefficients of determination (explained in Chapter 26) for the five equationg ranged from 80 to 94 percent. The watershedconveyancefactor is found from Fig. 12.18.The WsoandWrt widths are normally drawnwith one-thirdof the calculated width placedto the left of the peak and two-thirds to the right.
Clark's IUH Time-AreaMethod A syntheticunit hydrographthat utilizes an instantaneousunit hydrograph(IUH) was developedin 1945 by clark.6It has been widely used,is often called the time-area method,and hasappearedin severalcomputerprogramsfor hydrographanalysis(see Chapters24 and25).
s
80
b 7 0 . 6 0
F s o ;o 4 0 ?
6 3 0 d
F
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 0..130.14 0.15 0.16 0.17 Main channel Manning z value
Figure 12.18 Watershedconveyancefactor Q as a function of percent watershed impervious cover l and weighted main channel Manning n value, for Espey method.
220
CHAPTER12
UNIT HYDROGRAPHS
The techniquerecognizesthat the dischargeat any point in time is a function of the translation and storagecharacteristicsof the watershed.The translation is obtained by estimatingthe overland and channeltravel time of runoff, which is then combinedwith an estimateof the delaycausedby the storageeffectsof a watershed. The translationof excessrainfall from its point of falling to the watershedmouth is accomplishedusing the time-area curve for the watershed.This is a histogramof incrementalrunoff versustime, constructedas shownin Fig. 12.19. The dashedlines in Fig. I2.I9a subdividethe basin into severalareas.Each line identifiesthe locus of "times" pointshavingequaltraveltimesto the outlpt. The isochronesare drawnequal apart, and sufficient zonesare selectedto fully definethe time-area relation. The time-area graph of Fig. l2.I9b is a form of unit hydrograph.The area beneaththe curve integratesto 1.0 unit of rain depth over the total areaA, and it has \ a translationhydrographshapeif sufflcient subareasare delineated. If one unit of net rain is placedon the watershedat t : 0, the runoff from At would passthe outlet during the first At period at an averagerate of At units of runoff pei unit of time. The volume dischargedwould be At units of areatimes one unit of rain. After all areascontribute,one unit of rainfall over the entire area would have passedthe outlet.
Lt
I I
J
.l
J ^3
A2
Lt A
As
^1
Time interval
ft) Figure L2.19 Developmentof time-area histogramfor use with Clark's method: (a) isochronesspacedLt apart (shown as dashedlines) and (b) time-area histogram.
12.6 SYNTHETICUNIT HYDROGRAPHS
221
The impact of watershedstorageon the translationhydrographis incorporated by routing the time-area histogramthrough a hypotheticallinear reservoirlocatedat the watershedoutlet, having a retardance coefficient K equivalentto that of the watershed.For the simplestform of reservoir,the storageS,at time f is linearly related to the outflow Q, at time /, or
(r2.3s)
S,': KQ,
whereKis a constantof proportionality calledthe storagecoefficient.It hastime units and is often approximatedby the lag time of the watershed. From continuity,the inflow, storage,and.oq,{flow for the reservoirare relatedby
I,-Q,:#:U#
(r2.36)
If the differentialis discretizedto LQ/LI, andif Q, andQ, are the flowsat t andt - l, thenEq. 12.36becomes Io,- Ao,: YQ'Lt
Q'
(12.37)
BecauseQ : (Q, + Qr)/2, the flow at the end of any Al is Qz: CoI * CtQ,
(12.38)
where
cr: #+T;
(r2.39)
and
- Lt -" ,t -- z2 K K+Lt
(r2.40)
The IUH is found from Eq. 12.38by solvingfor Q, at the end of eachsuccessive time interval. EXAMPLE 12.9 Given the following 15-min time-area curve, find the IUH for the 1000-acrewatershed.Then determinethe 15-min syntheticunit hydrograph.The storagecoefficient K is 30 min.
Timeinterval (min)
Areabetween isochrones (acres)
0 - 1 5" 15-30 30-45 45-60
100 300 500 100
Solution. From Eqs. 12.39and 12.40,Cs : 0.4 and Ct : 0.6.Routingis most easily accomplishedin a tableauas follows:
222
CHAPTER12
UNIT HYDROGRAPHS
Time (hr)
I (cfs)
I (acre-in./Af )
col + c1Q1
IUH (cfs)
0 100
400
160+0
0
300
1200
480 + 96
80
500
2000
800 + 346
368
100
400
160+ 688
861
0
0
0+509
99'7
0
0
0+305
6't9
0
0
0.25 0.50 0.75 1.00
l
1.25 1.50
The IUH has a characteristicallylong recessiondue to the magnitudeof K for this example.Note that after 1.25 hr, the flow becomes0.6 times the previous flow andcontinuesto decayat this rateindefinite$.As discussedin Section11.5, the time baseof the IUH shouldequal the excess-runoffreleasetime, which is one definition of time of concentration.Clark's method often producesprolongedrunoff becauseof this shortcoming. The 15-min unit hydrographis found using Eq. 12.6, ot
O,(l5-minUH) : lGUH, + IUH,-15) This resultsin: Time
IUH
(h0
(efs)
1S-minUH (cfs)
0 0.25 0.50 0.75 1.00 t.25 1.50
0 160 576 1146 848 509 :
0 80 368 861 997 679 :
:
tl
If the waterri"O ,uU time is not available,the K value can also be estimatedby recognizingthat Q, = KdQldt when the inflow is zero in Eq. 12.36.This occurs at approximatelythe inflection point on the recessionof Fig. t2.2, when inflow to the channelceases.If hydrographdataare available,the estimateof the Kvalue is the ratio of the flow rate to the slope of the hydrographat this particular point on the hydrograph.
12.6 SYNTHETIC UNITHYDROGRAPHS 223
Nash'sSyntheticIUH One of the earliest formulations of the IUH was developedby Nash.l8Instead of characteizingrunoff as translationfollowedby storagein a singlelinear reservoiras Clark did, Nash viewed the watershedas a seriesof n identical linear storagereservoirS,eachhavingthe samestoragecoefficientK. The first instantly (r : 0) receives a volume equal to a full inch of net rain from the entire watershed.This water then passesthrough reservoirsI,2,3, . . . , fl, with eachproviding an additional diffusion effecton the original I -in. rain. The number of reservoirs, n, is uniquely related to the reservoir storage coefficientK andthe watershedlag time. Once the IUH is developed;it can be used to synthesizeany other hydrograph by application of the convolution integral, Eq. I2.7, or from the approximatemethodsdiscussed in Section12.3. The derivation of Nash's equationfor IUH beginsfrom continuity at the first reservoir:
I,- Qu:#1,
(r2.4r)
wherc Qt, is the outflow from reservoir 1 at time t. SubstitutingS,: r ) 0 (for an IUH, I, is zero after t : 0), we obtain
_Qr,: u#1,,, #1,,,:
KQr, at time
(r2.42)
which can be written dQ"
eu:
- kI o' '
(r2.43)
Integrationfrom / : 0* to time t gives l n Q r , - l n Q r , l , - o :- -
*
(r2.44)
which reducesby taking antilogarithmsto Qt,
o,l;:
e ''-ttK
(r2.4s)
BecauseQr, : S,/K and S,=o: 1 in., then
Qu:f,"-'r*
(r2.46)
wherc Qy has units of depthper unit of tirne. Equation 12.46is an exponentialdecay function having an initial value of I/K at t : 0. This monotonically decreasing outflow from reservoir 1 becomesinflow to the secondreservoir. The secondreservoiris initially empty. The continuity relation Qu
*
Qz,:
K+ dt
(r2.47)
224
CHAPTER12 UNIT HYDROGRAPHS
is solved,giving | ,.. (r2.48) 9r, : V;te-"^ This equationhas a full hydrographshape,beginningwith zero flow at time zero, peaking at the maximum of the function, and eventuallyrecedingto zeto. Similarly derived,the hydrographflowing from ruthreservoirhas the form Qn,
:
|n-r n-t/K
, (12.49)
which is the two-parametergammafunction,
e,,:
(12.50)
#1n-1r-t/K
Becausethe outflow from the nth reservoirwas causedby 1 in. of excessrain falling Eq. 12.50describesan IUH. instantaneously,
EStimatiOn of K and n Valuesof K and n arc neededfor applicationof Nash's IUH. By integration,the centroid of the distribution (Eq. 12.50) occurs at t : nK. - 1). The From classicalcalculusmaximization,the peak flow occurs at t : K"(n : 0 is n(n + l)Kz. Trial combinationsof n and secondmoment of the IUH about / K can be usedto developthe IUH from Eq. 12.50, and the momentsof the plotted distribution can be estimatedto verify the productsnK and n(n * I)Kz ' If the IUH is discretizedinto m At increments,the momentsare approximatedby First moment = > ttQi Lt m
and
Secondmoment- 2 t?Q, Lt
(r2.sr) (r2.s2)
Another less tediousapproachis to use the definition of lag time as the time from centroid of rain to the centroid of the hydrograph.For an IUH, this is the sameas the centroidal distance.Thus, if the lag time can be determinedfrom equationssuchas those in Section 11.6, the product nK can be established,reducing the number of trials. Someinvestigatorshaveattemptedto relateNash'sK andn parametersto basin and storm characteristicsusingregressiontechniques.Rao et al.1edevelopedrelations for urban areasgreaterthan 5 mi2:
K : and
Lt -
po.zzz g.57 54o.zto (l + 1)o'azzPo:oa 0.83140.458D0.37r (I + 1yt'ezzPozat
(r2.s3) (r2.s4)
wherenK : tlandA is areain squaremiles,D is net rain duration in hours,Pn",is the net (effective)rain depth in inches,and / is the ratio of imperviousto total area.
12.6 SYNTHETICUNIT HYDROGRAPHS
225
ColoradoUrbanHydrographProcedure(CUHP) Synthetic unit hydrographscan be tailored for regional use. As an example,the Coloradourban hydrographprocedure20 provides 5-min syntheticunit hydrographs for use in the Denver metropolitan area. lt is based on Snyder's method and is consideredapplicableto watershedsin the size rangefrom 90 acresto 10 mi2, with "regular" shapes(length 4 times width). It was developedin the early 1980sand modifiedin 1984to reflectrefinementsfrom earlyapplications. In its pre-1984'form, the Snyder/CUHPC, and C, valuesfor use in Eqs. 11.5 and 12.17 are C, : 7.8I1P2378 and
Co:
0.89C?'46
(r2.ss) (r2.s6)
where P" is the percent impervious.Theseregressionequationswere developedfor P" > 30 percent, using data for 96 storms over 19 urban watersheds.The given equationsapply to normal watershedconditionsandneedto be adjustedfor steep,flat, or seweredbasins.If an urbanareais fully sewered,the calculatedC, from Eq. 12.55 is decreased10 percent. If sparselysewered,a 10 percent increaseis made. If the averageslope S of the lower 80 percent of the main water courseis flat (less than 0.01 ftlf| the C, value becomes C,:3.12/Po78so2
( r 2.57)
and for steepareas(,S> 0.025 ftlf$, C, : 3.75/Po78s'02
( 12.s 8)
The C, coefficient is determinedfrom Eq. 12.56 and adjustedto 10 percentup or down for fully or sparselyseweredconditions,respectively. For the CUHP applications,Eqs. 12.18 and 12.19become ryro: SO}A/Qo w75:260A/Qo
(r2.se) (12.60)
whereA andQohaveunits of squaremiles and cubic feet per second.For plotting ft., the smallerof 35 percentor 0.670is placedleft of the peak.For W15,45percentis placedto the left, or O.424To if 0.67pwas usedfor W'r6.I is the time from beginning of runoff of the unit rainfall to the peak time. Severalinvestigatorshave suggestedthat Snyder's C, and slope S are correlated.a'S The original CUHP procedurewas alteredto recognizethis relation, making the adjustmentsin C, unnecessary. For the modifiedversion,Eqs. 12.57and 12.58are bypassed, andthetime to peakratherthanlag time is usedin Snyder'sEq. 11.5,where tp: c,(LL,o/\8100'
(12.6r)
The revisedtime coefficientC, is obtainedfrom Fig. 12.20a,and the peak coefficient is found from Co : PC,Ao'1s
where
(r2.62)
S : weightedaverageslopeof basinalongthe streamto the upstreambasin boundary (ftlft)
CHAPTER12
UNIT HYDROGRAPHS 0.18 Basic equations
Equationsof curve: Ct= aI?+ blo+ c tro. I 00
0.16
0.14
I LL--\0.48 'r: "\fi)
\r
b c -0.00371 0.163 0.146 0.000023 -0M224 ] 3 x 1 o r -8.01 x l0- 0 12(
640c, --7-
4p :
tp = time to peak (br)
E q
L = watershed length (mi) Ic, = distance to centroid (mi)
o
So = waterway slope (ff/ft)
r i
U 0.10
0.08
Ec,.2l
3q. 0.10
Eq.3 80
40 60 Percentimpervious,l, (a)
20
100
12 Equations of curve: P = aI?+ blo+ c
10
Eq.
a
b -0 012
2 | -o.t)0091 +0.228 --T-| -T-l
:
\t<
+216
06
z-r ,ql.: v t Y
\
6
o 6
d
o
Eq. 1 0
20
84.2 40 60 Fercent impervious, l, (b)
80
100
Figure 12.20 Snyder's C, and C, coefficientsfor urban areas,for use with CUHP: (a) relation between C, and imperviousnessand (b) relationbetweenpeakingparameter and imperviousness.
PROBLEMS
227
P : coefflcient,dependingon imperviousness,from Fig. I2.20b A : drainagearea (mi2) L : length of the main stream channel (mi) from the outlet to the divide L"o : length along the main channel (mi) from the outlet to a channelpoint nearestthe watershedcentroid The peakrate andtime of the unit hydrographcanbe developedusingEqs. 12.17 After four additional points definedby Eqs. 12.59 and t2.60 areplotted, 12.6t. and the 5-min hydrographcan be fitted to provide a total area representing rest of the runoff. A hand fit is applied, or the mathematical curve-fitting in. of direct 1.0 early in Section 12.5 can be used if a gamma (or any other) described techniques appropriate. is considered distribution (lessthan 90 acres),the time to peak is watersheds For small Lp
* 0 . 0 7()1, ^. : o.zg(P=l:?3,6=P: '^" Pz - 0.49P"+ 0 .1 4
6P,)
02.63)
where I is the time of concentrationin minutes, and P^ is the percentimpervious.
r summary Unit hydrographmethodsallow the hydrologistto estimaterunoff volumesand rates for virtually any storm. By far, the greatestnumber of problems in practice are evaluatedusingunit hydrographprocedures.Most of the current computermodelsuse unit hydrographproceduresasdescribedin Chapters 23,24, and25. Thesemodelsare simply computerprogramsthat perform the unit hydrographsynthesesand convolution stepsdescribedin this chapter.Any softwareuser shouldunderstandthe origin, applicability,and parameterestimationproceduresfor eachunit hydrographmethod seiected.The most successfulusesof the computermodelswill resultfrom a thorough familiarity with the processesdescribedin this chapter.
PROBLEMS 12.1. Given the following storm pattern and assuminga triangular unit hydrographfor one time unit, determinethe compositehydrograph.
Stormpattern Time unit Rainfall
1 1
2 1
3 4
4 2
Unit hydrographbaselength : 6 time units; time of rise : 2 time units; and maximum ordinate : I rainfall unit height. 12.2. Given a rainfall duration of 1 time unit, an effectiveprecipitation of 1.5 in., and the following hydrograph,determine(a) the unit hydrographand (b) the compositehydrographfor the given storm sequence.
228
CHAPTERl2
UNIT HYDROGRAPHS Hydrographfor 1.S in. net rain in .t time unit Time units 1 2 3 Flow (cfs) r00 98 220
4
4.5
5
6
srz 620 585 460
7 8 330 2to
9 r 0 150 105
l 1 l2 13 75 60 54
Stormsequence Time units Precipitation (in.)
I
0.4
z l.l
J
2.0
1.5
12.3. Solve Problem lZ.2 if the storm sequenceis as follows:
Stormsequence Time units Precipitation (in.)
r 0.3
2 r.4
J
0.9
l2'4'
Using U.S. GeologicalSurveyrecords,or other data, selecta streamflow hydrograph for a large, preferably single-peakedrunoffevent. Separatethe base flow and determine a unit hydrograph for the area. 12.5. For the unit hydrographof problem 12.1, constructan S_hydrograph. 12.6. For the unit hydrqgraphcomputedin problem 12.2, construct an S_hydrograph. 12.7. use the S-hydrographof problem 12.6to find a 3 time-unit unit hydrograph. 12.8. Given a watershedof 100 mi2, assumethat C, = 1.g, the length of main stream channelis I 8 mi, and the length to a point nearestthe centroid is I O mi. Use Snyder,s methodto find (a) the time rag, (b) the duration of the syntheticunit hydrograpl, and (c) the peak dischargeof the unit hydrograph. l2'9' Apply Snyder's method to the determinationof a synthetic unit hydrographfor a drainage area of your choice. 12'10' Use Fig. 12.13to determine a2-hr pnrthydrographif the drainageareais 60 mi2 and Eq. 12.23ais applicable. 12.11. SolveProblem12.10using Eq.12.23b. 12.12. Assuminga Nebraskalocation, use Gray's method determine te a unit hydrograph: drainagearea = 1.0 mi2, length : 0.6 mi, S" : 1.3 percent. 12'13' A drainageareain Nebraskacontains30 mi2.The tenjtn ortne main channelis 10 mi andthe'representative watershedslopeis 2.5 percent.iJseGray's methodto determine a unit hydrograph. 12'14' Dischargerates for a flood hydrographpassingthe point of concentration for a 600acredrainagebasinare given in the tabl; belo;. The flood wasprotluced by a uniform rainfall rate of 2.15 in.rru, which started,at9 A.M., abruptry ended at il A.M: and resulted in 5.00 in. of direct surfacerunoff. The base flow (derived tiom influent seepage)prior to, during, and after the storm was 100 cfs.
229
PROBLEMS
8 a.u. Time Measured discharge 100
9
10
11
12
100 300 500 700
I P.M
800
2
3
5
4
6
600 400 300 200 100 100
cease? a. At what times did direct runoff begin and basin' the for (in'/hr) index the Determine b. @ (cfs)for eachtime.listed' c. Derive the 2-hr'unit-nydrograpirordinates releasetime) for the basin. (excess concJnffation d. Estimatethe tirne of e.Atwhattimewoulddirectsurfacerunoffceaseiftherainfallof2.T5in./hrhad and had lastedfor 8 hr rather than 2? begunat 9 .q.'l\a. (in.) for a uniform urg" .ur" (cfs) and the direct runoff f. Determined;;;i;t rainfall of 2j5 it'lhr and a duration of 8 hr' Lz.lS. Measuredtotal hourly dischargerates in the accompanyingtable' The hydrr uniform intensity of 2'60 in'/hr startir baseflow from 8 A.M' to 3 P'v' was a ' determined as the area under the dir
Time Measured discharge
8 a.v 100
100
10
11
12
I P.M
300
600
400
200
2 100
100
a. At what time did the direct runoff begin? to the volume of the direct surface b. betermine the net rain (in') corresponding runoff of 1000 cfs-hr' c. Determinethe { indexfor the basin' basin by tabulating time in hours and d. Derive " , nt ltii ttyJrog'api' for the dischargein cfs. basin? e. What is the excessreleasetime of the the direct unit hydrograptr-to-determine 2-hr tt" the derived f. For the ,"*"'t"'it, p.u. and 1 at began ;;; ;.*, on a day when excess-(netjralnfall runoff rate (.fr)'t"i P'M' 5 at abruptly ceasing iti"tt ity of 2 in'/hr for 4 hr' continued"t " |2.|6.A5-hrunithydrographfora425}-acre.basinisshownintheaccompanyingsketch. Thegivenhyd.og.-uphactuallyappearedasadirectrunoffhydrographfromthebasin' of 5 hr,beginning
by rainf;ffi;;;;ii"'"iriv caused
"i 0.30in./hrfor a duration
att = 0. the basin' a. Determinethe excessreleasetime of basin' the for index b. Determinethe @ contributingto direct runoff 4 hr after c. what n".""*"J" "'i,ti" JJr"g" t"r* was rain began(r = 4)? as shownin the sketch'Do not scale d. Use your responseto part c to determineQp' Qp from the drawing' t = 3 andr : 5' Why did the hydrograph e. Note that rain continuedto fall between = : rathet than continue to rise during those 5, t 3 and form a ptateaut"i*""n t 2 hours?
230
CHAPTER12
UNIT HYDROGRAPHS QO
o
,i: 700 = ouu I 500
e 40o
! :oo $ zoo 100
Time, t
f. Usethe given 5-hr unit hydrographto determinethe direct runoff rate (cfs)at 7 p.r'r. on a day when rain fell at an intensity of 0.60 in./hr from I p.rrr.to 11 p.vr. 12.17. The 2-hr unit hydrographfor a basin is given by the following tabie:
Time (hr) O@fs)
0 0
1 60
2 200
3 300
4 200
5 120
6 6 0
7 8 3 0 1
0
9 0
a. Determinethe hourly dischargevalues(cfs)from the basinfor a net rain of 5 in./hr and a rainfall durationof 2hr. b. Determinethe direct runoff (in.) for the storm of part a. What is the direct runoff for a net rain of 0.5 in./hr and a duration of Z Iv? c. Rain falls on the basinat arateof 4.5 in.lhr for a2-hr period and abruptly increases to a rate of 6.5 in./hr for a second2-hr period. convert theseactual intensitiesto netrain intensitiesusinga {index of 0.5 in./hr. Constructa tablethatproper$ lags and amplifiesthe 2-hr unit hydrograph,and determinethe hourly ordinates(cfs)of direct runoff for the storm.The deriveddirect runoff hydrographshouldbegin and end with zero dischargevalues. 12.18. Given the following 2-hr unit hydrograph for a drainage basin, determine hourly ordinatesof the 4-hr unit hydrograph:
Time (hr) 0@f9
0 0
1 50
2 300
3 400
4 200
5 s0
6 0
12.19. Usethe following 4-hr unit hydrographfor a basinto determinethe peak dischargerate (cfs)resulting from a net rain of 3.0 in./hr for a 4-hr duration foliowed immediately by 2.0 in.lhr for a 4-hr duration.
Time (hr) 0(cfO
0 0
2 200
4 300
6 100
8 50
10 0
PROBLEMS 231 12.20. Compare the time from the peak to the end of runoff for the SCS triangular unit hydrographwith the time of concentration,/". Discuss. 12.21. Prove that the areaunderthe rising limb of the SCSbasic dimensionlesshydrograph equalsthat of the triangular unit hydrograph,that is, 37.5 percefi of the total. 12.15occurswhenx:aB, 12.22. Bycalculus,showthatthemaximumvalueof/(x)inEq. for a > 1. Also solvefor the centroidaldistanceby taking the flrst moment aboutthe y axis. 12.23. Accordingto the rational method(seeChapter15) of estimatingpeakflow from small areasnthe peak rate for a storm with uniform continuingintensity is equal to the net rain rate and occurs at the time of concentration.For what conditions,if any, would ' Eqs. 12.63 and 12.17result in agreementof the peak magnitudeand time, estimated by CUHP, with those of the rational method?Discuss. L2.24. Describetwo methodsthat could be usedto constructa 2-hr unit hydrographusing a l-hr unit hydrographfor a basin. 12,25. Measuredtotal hourly dischargerates (cfs)from a 2.48-n12drainagebasin are tabulated below.The hydrographwas producedby a rainstormhavinga uniform intensity of 2.60 in./hr starting at 9 A.M.and abruptly ending at 11 l.rvl. The baseflow from 8 ,q,.lt.to 3 p.lrl.was a constant 100 cfs.
8 e.u. 100
Time Discharge(cfs)
11 450
10 300
9 100
1 p.vr. 2 150 100
12 300
J
100
a. At what time did direct runoff begin? b. Determinethe grossand net rain depths(inches). c. Derive a 2-hr unit hydrograph for the basin by tabulating time in hours and dischargein cubic feet per second. d. Derive a 4-hr unit hydrographfor the basin. e. Derive a l-hr unit hydrographfor the basin. 12,26. Given below is a 3-hr unit hydrographfor a watershed.The {-index is 1.5 in./hr. 3-hr rainfall ratesof 2.5, Desiredis the DRH for an 18-hr stormhavingsix successive 3 . 5 , 1 . 5 ,4 . 0 , 6 . 5 , 2 . 5i n . l h r .
Time (hr)
I
0
2
3
4
5
6
7
8
9
10 11 12 13 14
o(ruH) 0 10 40 60 80 100 90 70 60 50 40 30 2 0 1 0 0 12.27. Use the following 2-hr unit hydrographto determinethe peak direct-runoffdischarge rate (cfs)resulting from a net rain of 2.0 in./hr for 5 hr'
Time (hr) 0(cf9
0 0
1 50
2 200
3 300
4 200
5 1s0
6 100
7 0
12.28, The ordinatefor a 5-hr unit hydrographis 300 cfs at a time 4 hr after the beginning of net rainfall. A storm with a uniform intensity of 3 in,/hr and a duration of 5 hr occursover the basin.What is the runoff rate after 4 hr if the @index is 0.5 in./hr?
232
CHAPTER 12 UNITHYDROGRAPHS 12.29. Given below is an IUH for a watershed.Use the IUH to find hourly DRH rates for a net rain of 4 in. in a 2-hr oeriod.
Time (hr)
o(ruH)
0 0
1 10
2 40
3 50
4 60
5 80
6 100
7 80
8 20
r0
9 10 0
12.30. A 2-fu unit hydrographfor a basin is shownin the sketch. a. Determinethe peak discharge(cfs)for a net rain of 5.00 in./hr and a duration of 2 hr. b. What is the total direct surfacerunoff (in inches)for the storm describedin part a? c. A different storm with a net rain of 0.50 in./hr lastsfor 4 hr. What is the discharse at 8 p.vr.if the rainfall startedat 4 p.tvt.?
{, +oo po
.A 2oo
--t I I
--i
T I I I
I I I
I I I I I I
I I -T I I I
Time (hr)
12.31,. Recordedflow rates for a net rain of 1.92 inchesin 12 hours are shownin the table. ' If the baseflow is 375 cfs throughoutthe storm,determinethe 12-hrunit hydrograph, and convert it to a 6-hr unit hydrograph.Then apply the 6-hr unit hydrographto determinethe total hydrograph(including 400 cfs baseflow) for a24-hr storm having four 6-hr blocksof net rain at ratesof 0.7, 3.8, 10.8,and 1.8 in. per hour.
Time in hours
0 6 12 18 aA JU
36 A'
48 54 60 66 72 78
Observedflow (cfs)
375 825 2200 36s0 3900 3200 2375 1,725 1250 900 650 490 410 375
12.32. Starting with a triangular-shapedunit hydrographwith a baselength of 2.67toand a height of qo,deriveEq. 12.22,qo : 484A/tp. Statethe units of eachterm usedin the .- --- derivatign"
REFERENCES 233 12.33. The SCS syntheticunit hydrographis derivedby computingthe peak dischargerate (cubicfeet per second)from qo : 484A/tp.In the derivation,it was actually assumed thatqrin.lhr:0.7sv/tp,whereVisthevolumeofdirectrunoff(inches),roisthetime to peak flow (hours),and A is the basin area(squaremiles).Derive the first equation from the second. 12.34. Which of the techniquesfor synthesizinga unit hydrographrequiresthe leastcomputationai effort in developingthe entire unit hydrograph?Which probably requiresthe most?
REFERENCES 1. W. D. Mitchell, "Unit Hydrographsin Illinois," illinois WaterwaysDivision, 1948. 2. L.K. Sherman,"Stream-Flowfrom Rainfall by the Unit-GraphMethod," Eng.News-Rec. 108,501-505(Apr. 1932). "Unit Hydrographsfor Gaugedand Ungauged 3. Rand Morgan and D. W. Hulinghorst, Watersheds,"U.S. EngineersOffice, Binghamton,NY, July 1939. 4. F. F. Snyder,"synthetic Unit Graphs," Trans.Am. Geophys.Union 19,447 -454(1938). 5. A. B. Taylorand H. E. Schwartz,"lJnit HydrographLag and PeakFlow Relatedto Basin Characteristics," Trans. Am. Geoplrys. Union 33' 235-246(19 52). 6. C. O. Clark, "storageand the Unit Hydrograph,"ASCE Trans.ll0,1419-1446(1945). 7. Water SupplyPapers, U.S. GeologicalSurvey,Water ResourcesDivision. Washington, D.C.: U.S. GovernmentPrintingOffice, 1966-1970. 8. Hourly PrecipitationData, National Oceanicand AtmosphericAdministration.Washington, D.C.: U.S. GovernmentPrinting Office, L971. "Synthesisof the Inlet Hydrograph,"Tech.Rept.No. 3, Department 9. JohnC. Schaake,Jr., of Sanitary Engineeringand WaterResources,The JohnsHopkins University,Baltimore, MD, 1965. 10. G. Aron andE. White, "Fitting a GammaDistribution overa SyntheticUnit Hydrograph," Bull.18(l) (Feb.1982). WaterResources Bull. l9(2) (Apr. 1983). 11. G. Aron and E. White, "Replyto Discussion,"WaterResources 12. M. Collins, "Discussion-Fitting a Gamma Distribution over a Synthetic Unit HyBull. 19(2) (Apr. 1983). drograph,"WaterResources 13. "Flood-HydrographAnalysisand Computations,"U.S. Army Corpsof Engtneets,Engin' neering and Design Manuals,Emlll0-2-1405. Washington,D.C.: U.S. Government PrintingOffice,Aug. 1959. "Hydrograph Synthesisby Digital Computer," Proc. 14. M. D. Hudlow and R. A. Clark, ASCEJ. Hyd. Div. (May 1969). 15. V. Mockus, "Use of Storm and WatershedCharacteristicsin SyntheticHydrographAnalService,1957. ysisandApplication,"U.S.Departmentof Agriculture,Soil Conservation 16. D. M. Gray,"synthetic Unit Hydrographsfor Small DrainageAreas," Proc' ASCEJ. Hyd. Div. 87(HY4) (July 1961). 17. W. H. EspeyandD. G. Altman, "Nomographsfor Ten-minuteUnit Hydrographsfor Small Urban Witersheds,"EnvironmentalProtectionAgency,Rept.EPA-600/9-78-035,Washington,D.C., 1978. 18. J. E. Nash, "The Form of the InstantaneousUnit Hydrograph," IASH Publ. No. 45' Vol. ' 3. 1951. "ConceptualHydrologicModels for Urbanizing 19. R. A. Rao, J. W. Delleur, and B. Sarma, Basins,"Proc. ASCE J. Hyd. Div. (HY7) (July 1972). "First Short Courseon Urban Storm WaterModeling 20. University of Coloradoat Denver, Using ColoradoUrban HydrographProcedures,"Departmentof Civil Engineering,June 1985.
C h a p t e r1 3
HydrographRouting
r Prologue The purposeof this chapteris to: . Presenttechniquesfor determiningthe effect of streamsand reservoirson hydrographshapesasthe hydrographsmovedownstreamthrough the systems. . Distinguishbetweenthe two major classificationsof hydrographrouting techniques. . Familiarizethe readerwith proceduresfor determiningwhen to apply eachof the variousrouting methods. Flood forecasting,reservoirdesign,watershedsimulation,and comprehensivewater resourcesplanning generallyutilize someform of routing technique.Routingis used to predict the temporal and spatial variationsof a flood wave as it traversesa river reach or reservoir. Routing techniquesmay be classified into two categorieshydrologicrouting and hydraulic routing. Hydrologic routing employsthe equationof continuity with either a linear or curvilinear relation betweenstorageand dischargewithin a river or reservoir. Hydraulic routing, on the other hand, usesboth the equation of continuity and the equationof motion, customarilythe momentumequation.This particular form utilizes the partial differential equationsfor unsteadyflow in open channels.It more adequatelydescribesthe dynamicsofflow than doesthe hydrologicrouting technique. Applications of hydrologicrouting techniquesto problemsof flood prediction. of the effectsof urbanization evaluationsof flood control measures,and assessments are numerous.Most flood warning systemsinstituted by NOAA and the Corps of Engineersincorporatethis techniqueto predict flood stagesin advanceof a severe storm.It is the methodmost frequentlyusedto sizespillwaysfor small, intermediate. and large dams. Hydrologic river and reservoirrouting and hydraulic river routing techniquesare presentedin separatesectionsof this chapter.
13.1 HYDROLOGICRIVER ROUTING
235
RIVERROUTING 13.1 HYDROLOGIC The first referenceto routing a flood hydrographfrom one river station to another was by Graeff in 1883.1The techniquewasbasedon the useof wavevelocity and a rating curve of stageversus discharge.Hydrologic river routing techniquesare all founded upon the equationof continuity dS (1 3 . 1 ) I - O : dt where
1 : the inflow rate to the reach O : the outflow rate from the reach dS/dt : the rate of changeof storagewithin the reach
Three of the most popular hydrologicriver routing techniquesare describedin subsequent paragraphs,
MuskingumMethod Storagein a stableriver reachcan be expectedto dependprimarily on the discharge into and out of a reach and on hydraulic characteristicsof the channelsection.The storagewithin the reach at a given time can be expressedas2
s:2;y7^n+(1 - X)O-n1 a
(r3.2)
Constantsa andn reflectthe stagedischargecharacteristicsof control sectionsat each end of the reach,and b and m mirror the sta$e-volumecharacteristicsof the section. The factor X defines the relative weights given to inflow and outflow for the reach. The Muskingum method assumesthat mfn - L and lets b/a : K, resulting in
S: KIXI + (1 - X)o]
(13.3)
where K : the storagetime constantfor the reach x : a weighting factor that variesbetween0 and 0.5. Application of this equationhas shownthat K is usually reasonablycloseto the wave trarel time through the reach andX averagesabout 0.2. Behavior of the flood wave due to changesin the value of the weighting factor X is readily apparentfrom examinationof Fig. 13.1. The resulting downstreamflood waveis commonlydescribedby the amountof translation-that is, the time lag-and by the amount of attenuationor reduction in peak discharge.As can be noted from Fig. 13.1,'thevalueX : 0.5 resultsin a pure translationof the flood wave. Application of Eqs. 13.1 and 13.3 to a river reachis a straightforwardprocedure if Kand X are known. The routing procedurebeginsby dividing time into a number of equal increments,A/, and expressingEq. 13.1 in finite difference form, using subscriptsI and2 to denotethe beginningand ending times for Ar. This gives I t + 1 2_ O 1 + 0 2 _ S ' - S ' 2 2 A t
(13.4)
236
CHAPTER13
HYDROGRAPHROUTING
o
99 E
E
Time
Figure 13.1 Effect of weighting factor. The routing time interval A/ is normally assigned any convenient value between the limits of K/3 and K. The storagechangein the river reachduringthe routing interval from Eq. 1 3 . 3i s
s, - s' : Klx(I' - 1')+ (r - x)(o2- o')l
(13.s)
and substitutingthis into Eq. 13.4 resultsin the Muskingum routing equation O2:
CsI2+ CII. + CzOl
( 13.6)
in which
-KX + 0.5Ar K-KX+0.54/ KX + 0,5 Lt ct: K-KX+0.54t K - K X- 0 . 5 A r
co
cz:
K-KX+0.54t
(r3.7) (13.8)
(13.e)
Note that K and Ar must havethe sametime units and aiso that the three coefficients sum to 1.0. Theoretical stability of the nunrericalmethod is accomplishedif Al falls betweenthe limits 2KX and2K(I - X). The theoreticalvalue of K is the time required for an elemental(kinematic) waveto traversethe reach.It is approximatelythe titne interval betweeninflow and outflow peaks, if data are available.If not, the wave velocity can be estimatedfor variouschannelshapesas a function of averagevolocity Vfor any representativeflow rate Q. Velocityfor steadyuniform flow canbe estiinated by either thi Manning or Ch€zyequation: The approximatewavevelocities for different channel'shapesare given in Table 13.1. FORVARIOUS WAVEVELOCITIES 13.1 KINEMATIC TABLE ' CHANNELSHAPES
Channelshape Wide rectangular Triangular Wide parabolic
Manningequation t-V !v lJrl 9 '
Ch6zyequation 3-tl 2 l
tv Zrl 6 '
ROUTING 237 RIVER 13.1 HYDROLOGIC Since1, and Irareknown for every time increment,routing is accomplishedby time incrementsusing each02as Ol for the next time solvingEq. 13.6for successive incremeni. Example 13.1 illustratesthis row-by-row computation'
EXAMPLE 13.1 : 2 days.The Perform the flood routing for a reach of river given X : 0.2 and K : 1 day is shownin Table 13.2,column 1. Assumeequal inflow hydrographwith Ar inflow and outflow rates on the 16th. Solution. If Ar : l dayandX = 0.2 andK :2days, thenEqs.13.7toI3.9 give Co : 0.0477,C1 : 0.428, andC2 : 0'524' Row-by-rowcomputationis given in TabIe13.2. ll Determination of Muskingum K andX Valuesof K andX for Muskingumrouting are commonly estimatedusingK equal to the travel time in the reachand an average value of X : 0.2.If inflow and outflow hydrographrecords are availablefor one or more floods,the routing processis easilyreversedto provide better valuesof K and X for the reach. To illustrate the latter method, instantaneousvalues of S versus TABLE13.2 (1)
(2\
(3)
(4)
Date
lnflow
voI2
Crl't
C,Q,
3-t6 17 18 19 20 21, 22
4,260 7,646 11,167 16,730 21,590 20,950 26,570 46,000 59960 57,'t40 47,890 34,460 21,660 34,680 45,180 49,r40 41,290 33,830 20,5t0 t4,720 11,436
z3
24 25 26 27 28 29 30 J I
4-l 2 4 5 6 7 8 9
364 532 798 1,029 999 L,267 2,194 2,860
t,823 3,2'72
) )7)
7,160 9,240 8,966 tl,37l 19,688 25,662
2,315 3,206 4,602 6,702 8,877 10,013 12,355 18,289
)a1l)
)4 417
1,643 1,033 1,654 t r55
20,496 14,748 9,270 14,843
26,9'70
1q??7
r,969 t,613 9'78 702
2t,031, 17,672 14,479 8,778 6,300 4,894 3,977
l7,879 20,729 22,914
)'754 ) )24
o )04
443
7,831 6,228 6,083
3 t J
29'| 290
4 114
? ?{1 ) 66\
)\ 117 )'t 171 t7 1))
)t
11)
19,686 t5,283 11,595 8,872 6,928
(5) Computed outflow 4,260 4,419 6,rr9 8,783 12,791 16,941 1 9 ,1 0 23,578 34903 46,705 51,469 49,109 41,514 32,67' | 34,120 39,559 43,729 42,199 37,569 29,166 22,128 t6,932 13,222 10,576 8,497
238
CHAPTER13
HYDROGRAPHROUTING
XI + (l -- X)O are flrst graphedfor severalselectedvaluesof Xas shownin Example 13.2.BecauseS andXI + (1 - X)O are assumed to be linearlyrelatedvia Eq. 13.3, the acceptedvalue of X is that which gives the best linear plot (the narrowestloop). After plotting, the valuefor K is determinedas the reciprocalof the slopethrough the narrowestloop, sincefrom Eq. 13.3
K :
( 13 . 1 0 )
n+0-x)o
Instantaneousvaluesof S for the graphsin Example I3.2 were determinedby solvingfor S, in Eq. 13.4for successive time increinents.A valueof S, : 0 was [sed for the initial increment,but the value is arbitrary since only the slope and not the intercept of E_q.13.3 is desired.The 52 valuesare plotted against.average weighted discharges,XI + (l - X)O in Table 13.3. A preferablemethod would be to plot S, valuesagainstcorrespondingvaluesof instantaneous(rather than average)valuesof XIz + (l - X)Or, using recordedvaluesof inflow and outflow (not provided).
TABLE13,3 -
Date 3- 16 t7 18 t9 20 21 22 23 z+
25 3-26 27 28 29 30 3l 4-r 2 J
4 5 6 7 8
l.ll"
2 (cfs) 5,870 9,310 12,900 20,500 21,000 23,400 32,500 s5,400 62,700 52,600 43,200 25,200 22,800 41,200 s0,400 45,300 38,800 2?,000 16,200 12,400 10,200 8,080 6,010 5,050
u =
Oj+02
2 (cfs) 4,180 6,970 7,560 14,200 18,300 18,500 21,300 29,300 39,700 48,700 53,300 48,700 37,r00 35,800 35,800 35,800 42,700 44,lOO 35,400 25,200 t6,400 11,500 9,380 7,860
"Note: ,S2= ,Sr* f Ar - d Al fseeEq. 13.4]. ' E x a m p l e :4 3 5 0 : 0 . 1 ( 5 8 7 0 )+ ( 1 - 0 . 1 X 4 1 8 0 ) .
sz
Weighted discharge (cfs) X+(1 -X)O
(1O3cfs-days)
X: 0.1
X : 0.2
1.7 4.0 9.4 15.7 18.4
4,350b 7,200 8,090 14,800 18,600 19,000 22,400 31,900 42,000 49,100 52,300 46,400 35,700 36,300 37,300 36.800 42,300 42,400 33,500 23,900 15,800 1r,200| 9;040 7,300
4,520 't,440
z5-J
34.5 60.6 83.6 97.5 87.4 73.9 Jv.o
65.0 79.6 89.r 85.2 68.0 48.9 36.1 29.9 26.5 23.1 20.3
8,630 15,500 18,800 19,500 23,500 34,500 44,300 49,500 51,300 44,000 34,200 36,900 38,700 37,700 41,900 40,800 31,600 22,600 15,200 10,800 8,710 7,300
X: 0.3 4,690 7,670 9,160 16,100 19,100 20,000 24,700 37,100 46,600 50,000 50,300 41,700 32,800 37,400 40,200 38,600 41,500 39,000 29,600 2r,400 14,500 10,500 8,370 7,020
13.1 HYDROLOGIC RIVERROUTING
239.,
EXAMPLE 13.2 Given inflow and outflow hydrographson the Muckwamp River, determine K and X for the river reach.(SeeTable 13.3.)
o po
€
ro X
,a o x 5 A
I
E e .E b0 o
g t\ X
80.000 cfs-davs Storage,S
Solution. Selecting the narrowest loop gives X : 0.3; K : 80,000 cfsdays/40,000cfs : 2.0 days. These values could now be used to route other floods through the reachas in Example 13.1. Inherent in this procedureis the postulatethat the water surfacein the reachis a uniform unbrokensurfaceprofile betweenupstreamand downstream ends of the section.Additionally, it is presupposedthat K andX are constant throughoutthe range of flows. If significantdeparturesfrom theserestrictions are present,it may be necessaryto work with shorterreachesof the river or to employ a more sophisticatedapproach. I I Muskingum Crest Segment Routing Sometirnesit is desirableto solve for a single outflow rate or route only a portion of an inflow hydrographby the Muskingum method(e.g.,the crest segmentwhen only the peak outflow is desired).This is easily accomplishedby successivelynumbering the inflow rates as 11, 12, 13,. . . , In, In+t,. .. , and rewritingEq. 13.6as O,:
( 1 3 .11)
CsI, + Ctln-t + C2O^-l
where Onis the outflow rate at any time n. The outflow O,-1 is next eliminated from Eq. 13.11by makingthe substitution On-t :
Coln-t + ClIn-2 *
C2O,-2
(r3.r2)
By repeatedsubstitutionsfor the right-sideoutflow tetm On-2,On-2,. . . can eachbe eliminated and On can be expressedas a function only of the flrst n inflow rates or, finally, On : where Kr: Kz= K3: K,:
K l I n + K 2 I n - 1 + K 3 I n - 2+ '
Co CoC2+ C1 K2C2 K,-rC.rfori>.2
.' + K,I,
(13.13)
240
CHAPTER13
HYDROGRAPHROUTING
Using data from Example 13.1 to find the outflow rate on 3-26, we obtain Kr : C6: 0.0477 Kz: CoC2+ Cr : 0.0477(0.524) + 0.428:0.4530 : 0.2374 Kz : KzCz: 0.4530(0.524) Krt:
KrcG : 0.0013
Thus the outflow on 3-26 is calculatedas O11: 0.0477 X (47,g90)+ 0.4530 X (57,740)+ . . . + 0.0013(4260): 51,469cfs.
SCS ConvexMethod The U.S. Soil ConservationService (SCS) developeda coefficient channelrouting technique,similar to the Muskingum method, in their National EngineeringHanibook.3It has had widespreadapplication in planning and,designand can be used successfullyevenwhenlimited storagedatafor the reachare available.Until 1983,the procedurewas usedfor all streamflowhydrographrouting in TR-20, the SCS storm event simulation computer program describedin Chapter 24. Newer versions of TR-20 usethe att-kin methoddescribedin Section13.3. Analysis of Fig. 13.2 producesthe working equation for the convex routing method.Becausethe areasunderboth curvesare equal,and becausethe peak outflow
Figure 13.2 Geometricrelationsusedin the scs convexrouting method.(After U.S. Soil ConservationService.3)
RIVERROUTING 13.1 HYDROLOGIC
241
is lessthan (and occurslater than) the peak inflow, the curvescrossat somepoint A, resultingin the fact that the value Orwill alwaysfall betweenIl andOt. At any time, the vertical distanceof 02 aboveOr (or below 01 on the right of A) is a fraction C, of the differenceIt - 01 as shown in the inset of Fig. 13.2. By proportionatevertical distances Oz:
O t + C , ( 1 1- O )
(r3.r4)
This could be usedto route the entire inflow hydrographif C, could be established. FromEq. 13.14,
-'-
or- o, 1,.-o,
(13.1s)
BecauseAl is one limb of the triangle in the inset to Fig. 13.2, Lt _O"- O, Ir-O, K
(13.16)
where the constantK is the horizontal time from O, to the interseptionof the line passingthrough Ol and 02. Thus C, is a function of both A/ and K, or n r_- Lv t w
(r3.r7)
routing method. Determination of K and C, Proof that K from Fig. 13.2is a constantis left to the reader.It is a storageparameterwith time units and can be approximatedby the K from the Muskingummethod.Similarly, C, is approximatelytwice the MuskingumX. The reach length divided by wavevelocity (estimatedfrom Table 13.1) will provide another estimateof K, or actual measurementsof reach travel time can be used. Equation 13.l'l canthenbe solvedfor C,. The valuerecommendedby the SCS,in the absenceof other estimates,is C,:
(13.18)
where Vis the velocity for a representativesteadydischarge,andV * 1.7 approximatesthe celerity (speed)of a kinematic wavetravelingthrough the reach.The units of V in Eq. 13.18are feet per second(fps). Routingby Eq. 13.14is easilyaccomplishedafter the C, valueis estimated.The interval Ar shouldbe one-fifth or lessof the time to peak of the inflow hydrographto assurea sufficient number of calculatedoutflow rates to deflne the hydrograph.As with all routing methods,the time interval shouldbe selectedso that one point falls at or near the peak and other locationsof rapid change.
242
CHAPTER13
ROUTING HYDROGRAPH
Unlike otherrouting methods,the convexmethodequationfor O, is independent of 1r. Thus the proceduie can be used to forecast outflow from a reach without tnowing the concurrent inflow. This provides a method for early calculation and warninf for floods.Flow recorderscanbe linked through microprocessorsto warning systemsthat calculate downstreamflood potentials at least one full routing-time interval aheadof the flood. The procedurecan be reversedto find the inflow hydrographfor a given outflow hydrograph,or it can route a cumulativemasscurve of inflow to the reachinsteadof the hydrograPhitself.
Method Muskingum-Cunge been Severalattemptsto overcomethe limitations of the Muskingummethodhavelot physically in or difficulties complexity totally ,u"""riful becauseof computational interpretingthe routing parameteri.a'sThe Muskingum parametersare best derived fiom streamflowmeasurementsand are not easilyrelatedto channelcharacteristics. Cunge6blendedthe accuracyof the diffusion wave method (seeSection 13.3) with the simplicity of the Muskingum method, resulting in one of the most recomgives mendedtechniquesfor generaluse.It is classifiedas a hydrologicmethod,yet it methods' resultscomparablewith hydraulic Cungeshowedthat ihe finite-differenceform of the Muskingum equationbecomesthe diffusion waveequationif the parametersfor both methodsare appropriatelyrelated'From Eqs. 13.1 and I3'3, the Muskingumequationis s
-
K+ln+(l-x)ol:t-o
( 13. 1e)
dt-
obtain SubstitutingQ,for I andQi*rfot O, andrewritingin finite-differenceform' we
+ (1 - x)Q"+l- xQI- (1 - x)Qi*,] fir"a':' - Oi+i+ Qi- Q',*') : t(Q',*'
(13'20)
If K is set equalto Lxfc,Eq. 13.20is alsothe finite-differenceform of
u #* ' H : o
(t3.2r)
which is calledthe kinematicwave equation(seeEq. 13.59)and can be derivedby Ax combining the continuity and -oln"ntu- (or friction) equations.TThe variable denotesunincrerrr"ntofdistancealongthestreamaxisandcisthewavespeed' The equationto be usedfor routing is obtainedfrom Eq . 13.20by solvingfor the unknown flow rate,
O i I i : c o Q ' , * ' c+ t Q i * c r Q i * ,
(r3.22)
where
co
Lt/K - 2X 2(r-x)+Lt/K
(r3.23)
RIVERROUTING 13.1 HYDROLOGIC
LtlK + 2x
cr: 4 v2
-
2(r-x)+Lt/K 2(I-x)-cLt/Lx "
'
2(I-x)+Lt/K ./r
.-\
L
243
(r3.24) (r3.2s)
Aal?
BecauseK : A,xfc,it representsthe time for a waveto travelthe routing reachlength Ax, moving at velocity t. Congeshowsthat the velocity c is the celerity of a kinematic wavepreviouslydescribed(Table 13'1). When X : 0.5 and c Lt/A,x: 1.0, the routing equationproducestranslation without attenuation.When Ax : 0 (zeroreachlength),no translationor attenuation occurs. If previousflood data are available,the routing parameterc canbe extractedby reversingthe routing calculations.Estimatesof the parameterscan also be obtained from flow and channelmeasurements. The value of X for use in Cunge'sformulation is
x:t('-#t)
(r3.26)
where So : channelbottom slope (dimensionless) for the peak rate 4o : dischargeper unit width (cfs/fO,normally determined The value of celerity c can be estimatedas a function of the averagevelocity V by Q3.27)
mV
c:
where v is the averagevelocity QfA, andm is aboutI for wide natural channels.The coefficientm comesfrom the uniform flow equation Q:
(13.28)
bA-
which reduces,by taking partial derivatives,to
o Q: A
A
*g: A
*y
(r3.2e)
Substitutingthis into the continuity equation
Q*4:o 6x
(13.30)
At
givesEq. I3.2t if c : mv.If dischargedata ate available,m canbe estimatedfrom nq. tZig. Valuesfor commonshapechannelsare given in Table 13'1' The rout{rg can now be done using either constantm and c parameters(i.e., using a single ai'eragevelocity) or variableparameters(usingeachnew velocity v). equition I-3.2i rs solvedfor c, the valueX is derivedfrom Eq. 13.26,andE6' 13.23 to 13.25are solvedusing K : Lx/c. When using this rnethod,the valuesof Ax and Ar shouldbe selectedto assure that the flood wa-vedetailsare proper$ routed.Nominally' the time to peak of inflow is broken into 5 or 10 time incrementsAt. To give both temporal and spatialresolution, the total reachlength l, can be dividedinto severalincrementsof Ax length, and outflow from eachis treated as inflow to the next.
244
CHAPTER13
HYDROGRAPHROUTING
EXAMPLE 13.3 Usethe Muskingum-Cungemethodto route the hydrographfrom Example13.2.Use So: 0.0001,Lx : 545 mi, flow cross-sectionalareaat Q : 59,960is 5996 ft2, width at Q : 59,960is 60 ft, and Ar : 1.0 day (asin Example13.2). Solution. From the inflow, the peak rate of 59,960cfs gives
59.960
o^
So:T
59,960:,, V r-: TQo -
l0fps
c : Z V o : 1 6 . 7f p s From Eq. 13.26
":;1,
't
1000 :0.4 (16.7)s4s (s280) 0.0001 L
L* K :
7:
I I
-545(5280) ff: 1 7 2 , 8s 0e 0c C": -0.1765 Ct: 0'7647 Cz: 0'294I
and
The routing for a portion of the hydrographis as follows:
Date, t
3-16 17 18 t9 20 2l 22 L3
24 25 26 27 3-28
CoQ'*L*
- 1350 -r970 -2950 -3810 -3700 -4690 -8120 - 10,580 10,190 -8450 -6080 -3820 -6120
Ct Qti*ro*
3260 5850 8540 12,790 16,510 16,020 20,320 35,180 45,850 44,150 36,620, 26,3sO 16,560
Cz Qlutno*
0 560 1310 2030 3240 4720 4720 4980 8700 13,050 14,340 13,200 10,510
Qli*.* r9r0 (3-r7) 4440 6900 11,010 16,050 16,050 16p20 29,580 44,360 48,750 44,880 35,730 20,9s0(3-29)
Note that the peak outflow of 48,750 cfs on March 26 occurson the samedate as in Example I3.2 but has experiencedslightly greaterattenuationfrom the Muskingum-Cunge example. The value C, is alwayspositive,and negativevaluesof C, arenot particularly troublesome.Although C0is negativein this example,this condition should be avoidedin practice.As seenfrom Eq. 13.23, negativevaluesof Coare avoided
ROUTING 245 RESERVOIR 13.2 HYDROLOGIC
when
^|,
,* rr
(13.31)
Other Methods Other hydrolSgicriver routing procedureshavebeendeveloped,includingthe working method,Tatummethod,and multiple storagemethod. R&D method,straddle-stagger They all appear as options in HEC-I, the U.S. Army Corps of Engineer's event simulation and routing model describedin Chapter24.
ROUTING RESERVOIR 13.2 HYDROLOGIC The storageindication method of routing a hydrographthrough a reservoiris also calledthe modified Pulsmethod.8A flood wavepassingthrough a storagereservoiris both delayedand attenuatedas it enters and spreadsover the pool surface.Water storedin the reservoiris gradually releasedas pipe flow through turbines or outlet works, calledprincipal sprllways,or in extremefloods, over an emergencyspillway. Flow over an ungatedemergencyspillway weir sectioncan be describedfrom energy,momentum,and continuity considerationsby the form O:
CYH'
(r3.32)
where O : the outflow rate (cfs) Y : the length of the spillway crest (ft) H : deepestreservoirdepth abovethe spillway crest (ft) C : the dischargecoefficientfor the weir or section,theoretically 3'0 x : exponent,theoreticallyJ Flow through a free outlet dischargepipe is similarly describedby Eq. 13.32 where' I H C .r
: : : :
the cross-sectionalareaof the dischargepipe (ft') head above the free outlet elevation (ft) the pipe dischargecoefficient, theoretically\/29 exponent,theoreticallyj
Flow equations for other outlet conditions are availablein hydraulics textbooks. Storagevaluesfor variouspool elevationsin a reservoirare readily determinedfrom computationsof volumesconfinedbetweenvariouspool areasmeasuredfrom topographic maps. Since storageand outflow both dependonly on pool elevation,the resulting storage-elivationcurve and the outflow-elevationrelation (Eq. 13.32) can easily be combinedto form a storage-outflowgraph. Storagein a reservoirdepends only on the outflow, contrastedto the dependenceon the inflow and outflow in river routing(Eq. 13.3). "surchargestorage"or the storage For convenience,,Sis often defined as the abovethe emergencyspillwaycrest.Normally the overflowrate is zero when,Sis zero. Ifthe graphedstorage-outflowrelation is found to be linear, and ifthe slopeofthe line
246
CHAPTER13
HYDROGRAPHROUTING
is definedas K, then (13.33)
S : KO
is a and the reservoiris called a linear reservoir. Routing through a linear reservoir : 13.3' 0'0inEq. 13.l-usingr showninFig. Muskingumriverrouting specialcaseof exceeds Note alsothat the,outfl-owrate in Fig. tg.t is increasingonly while the inflow immeinflow the that assumptions the outflow.This observationis conJistentwith the only depends outflow the that and diatelygoesinto storageoverthe entirepool surface on this storage. time Routing through a linear reservoiris easily accomplishedby first dividing : and I3'4 Eq. Ko2into s2 into a numberof equal incrementsand then substituting increment' time each for solving for oz, wtrictris the only remainingunknown To route an emergencyRo'oathrough anonlinear reservoir,the storage-outflow the outflow relation and the continiity equation,Eq. L3'4,are combinedto determine as rewritten be can I3'4 and storageat the end of'eachtime inciement A/. Equation
r , + r n +.1( * - o , )
:'+
(r3.34)
t on+t
side'Pairs in which the only unknown for any time incrementis the tgt_*:n the right checked .34 and 13 Eq. satisfy that generated b" of trial valuesof S"*, andO,*1"ould procetrial this to resort than Rather confirmation. fot in the storage-outflow"uru" replottedas dure, a valrieof At is selectedand points on the storageoutflow curve are direct determithe ';storageindication" curve shownin Fig. 13.3' This graph allowsa I On+rhas been Zl,*rfLt ordinate the of value a o11ce nation of the outflow O,11 from the S-O calculatedfrom Eq. f :.:4' ftre secondunknown' S,*t, can be read 13.3)orfoundfromEq'13'34' curve(whichcouldilsobeplottedonthegraphinFig. t200 I
/ / {, 600
: .il<
0
1
0
2
0
3
0 4 0 Outflow (cfs)
5
Figure 13.3 Curveof zsl\t + O versusO'
0
19.2 HYDROLOGICRESERVOIRROUTING
247
This row-by-rownumericalintegrationof Eq. 13.34with Fig. 13.3is illustratedusing Ar : t hr in ExampleI3.4. EXAMPLE 13.4
Giventhe tria4gular-shapedinflow hydrographand the 2S/Lt + O curve ofFig. 13.3 find the outflow hydrograph for the reservoir assumingit to be completely full at the beginningof the storm.(SeeTable 13.4.)
o 90
o Time (hr)
In selectinga routing period Ar, generallyat leastfive points on the rising limb of the inflow hydrographare employedin the calculations.An increasednumber of points on the rising limb, that is, a small A/, improvesthe accuracy,sinceas A/ - 0 the numerical integrationapproachesthe true limit of the function being integrated, in this casedSfdt. Column 3 in Table 13.4 comesfrom the given inflow hydrograph,column 4 is simply the addition of I, + In*r, andColumns5 andT are initially zero, sincein this problem the reservoir is assumedfull at the commencementof inflow. Therefore, there is no availablestorage. TABLE 13.4 ROUTINGTABLE (1)
(2)
Time (hr) 0 I z 3 +
5 6 '7 8 9 t0 ll t2
(3) ln
(cfs)
1 2 3 4 5 6 7 8 9 10 ll 1 t
0 0 0 0 120, 150 180 135 9 0 45 0 2 0 3 0 3 6 9
(4) I, + In+l (cfs) -tt,
90 150 2lo 270 330 315 225 135 0 0 0
'n-o. (5)
(cfs) 0 20 74 160 284 450 664 853 948 953 870 746 630
(6) 2Sn+1 , a -;fT vn+1
(cfs) JU
(7)
(8)
an+t
Sn+t
(cfs)
(cfs-hr)
5 18
12.5 46 96 164 250 361 458 506 510 467 404 344 288
110 224 370
3Z
))4
52
780 979
)6
r078 1085 998 870 746 630
AA
63 65 65 64 62 58 54
248
ROUTING 13 HYDROGRAPH CHAPTER the sum of columns4 The starting value for n : 1 in column 6 is computedas and 5 from F,q.13.34
25" - + o2 Lt 25. 3 0 * 0 : - - + o2
(L + 12)- ( * - o ' )
:
Lt
column 6 gives a valuefor Enteringthe ordinateof Fig. 13.3 with the value 30 from end-of-time-interval o, of 5 cfs, which is recordedin column 7 , The corresponding 8. Moving to column in recorded and 6 and7 columns storage,s,, is calculatedfrom n be found fof : 2 usings' the secondfow, a value of the term in column 5 can now and O, from columns7 and 8' ThestepwiseprocedureusedtogetoutflowfiguresforallncanbeSumma. rized as l.Entriesincolumnsland3areknownfromthegiveninflow.hydrograph. column 3' - 2. Entries in column 4 arc the additionsof I' t I'*' in 3.Theinitialvalueofthetermincolumn5iszero,thoughitcouldalsobe 4 and 5 are based on any arbitrary starting storagevalue' and columns addedto producethe value in column 6' 4.The2SlA,t+oversusoplotisenteredwithknownvaluesotzSlLt+o to find valuesof O for column 7' 5.Columns6andlaresolvedforS,+r,whichisrecordedincolumn8. for column 5 usingthe 6. Advanceto the next row and calculatethe next value 7 and 8' valuesin the precedingrow for O and S {om columns 4 and enterthe 7. Add the value in cotuin 5 to the advancedsum in column result in column 6 for the new period under consideration' 8.ThenewoutflowforcolumnTisagainfoundfromtherelationof ZSlLt + O as in Fig. 13.3. g.Thecorrespondingnewstorageincolumn8isfoundbysolvingfrom columns6 andT' l.0.Steps6throughgarerepeateduntiltheentireoutflowhydrographisgenerated. rl
RIVERROUTING 13.3 HYDRAULIC ploys both the equationof continuity and utions to the complete hydrarllic routing
ili"'nTil::ll;lffi ffffii.;fiffi:::
tions. Hydraulicfoutingtechniquesarehelpfulinsolvingriverroutingproblems,overthe simultaneoussolution land flow, or sheetnow. uyaraulic routing proceedsfrom of the combinationfor forms general 1'ne of expression,of continoiti una-o*"ntuir, rivers are called the spatically varied unsteadyfl'ow equations'
13.3 HYDRAULIC RIVER ROUTING 249 Theseequationsalso apply to sheetflow or overland flow and include terms for laterally incoming rainfall. They can be simplified and used to resolveriver routing problems.eFor completeneisof presentation,a generalform of the spatially varied unsteadyflow equationswill be presentedfirst.
Equationof Continuity The equation of continuity statesthat inflow minus outflow equals the changein storage.To relatethis conceptto a river sectionundera condition ofrainfall or lateral inflow, consideran elementof length Ax and unit width into the page as shown in Fig. 13.4. The total inflow is
,,*.r)dtdx (1335) ,(, - X+)(, - * *) * * , [.' l,'*o' The total outflow is
+t*+)(,.#?) o' o(v
(13.36)
* ofrl,"
(r3.37)
The changein storageis
l(x, t)
,(,-#+)(,-
o ( v * o?t) ( r . * + ) l
f
A
x
L
x
2--*l-- 2(a)
i(x, t)
ft) Figure 13.4 Continuity and momentum elements(where p : the density of water, V : the averagevelocity. 1l : the depth. i : the lateral inflow per elemental Ax, and S = the slopeof the river bottom).
250
CHAPTER13
HYDROGRAPHROUTING
Consequently,continuity gives
-o(r{*tx + vfia') a, + pl A,xL, -
oX L,xL,t: o
(13.38)
where i is the averagelateral inflow resulting from rainfall over Ax and Ar. The continuity equation of unsteadyflow with lateral inflow is obtained by simplifyingEq. 13.38 6V .,0y , 0y "v6- x- | v - - 6r x At
:
( 13 .3e)
For otherthan a unit width, Eq. 13.39takesthe form
e d{x + v dYx * *d t-
l : 0
(13.40)
whereA is the width times the depth,y.
MomentumEquation In accordancewith Newton's secondlaw of motion, the changeof momentumper unit of time on a body is equal to the resultantof all externalforces acting on the body. The following derivationof the momentumequationof spatiallyvariedunsteadyflow is presentedsubjectto the following assumptions:(1) the flow is unidirectional and velocity uniform acrossthe flow section;(2) the pressureis hydrostatic;(3) the slope of the river bottom is relatively small; (4) the Manning formula may be used to evaluatethe friction lossdueto shearat the channelwall; (5) lateral inflow entersthe stream with no velocity componentin the direction of flow; and (6) the value of I representsthe spatial and time variationsof lateral inflow. Forcesactingon an elementof lengthAx andunit width are shownin Fig. i 3.4b. The forcesF1andF, representhydrostaticforceson the elementand are expressedas
atvA)axl - ylyA f-, - -;zl F,:
(13.41)
- - lL,!^ .f-^ - a(la)Axl rz: A- Z l
(r3.42)
wherey is the distancefrom the watersurfaceto the centroidofthe area.The resultant hydrostaticforce is F, - Fr, or \
1
r ,6:
-ra(!A) * x
(13.43)
By assuminga small slope for the river bottom, the gravitational force is given by Fr : yASA.x
(13.44)
The frictional force alongthe bottom is equalto the friction slope$multiplied by the weisht of water in an elementAx. \:
vASyLx
( 13 .4s )
13.3 HYDRAULICRIVER ROUTING
251
The rate of changeof momentumin the length Ax may be expressedas d(mv)
dV
:
,.dm
iii____ T
dt
(r3.46)
V____
dt
dt
in which m is the massof fluid. If it is assumedthat the incoming lateral inflow entersthe moving fluid with no velocity componentin the direction of flow and I representsthe spatial and time variationsof the lateral inflow, the rate of changeof momentumfor the elementcan be expressedas
dv dtuv) T:pALxi+pvi
(r3.47)
Lx
wherc dV/dt represents
dv av ,,av at
dt
T
(13.48)
v -
Ex
The rate of changeof momentumis therefore
(r3.4e)
oe*(Ya,. "#) + pviL,x
Equating the rate of changeof momentumto all externalforces acting on the elementresults in
{*r{+gaFA) A
t
O
x
A
O
+L:s(s-s) x
A
(13.50)
Now for a unit width element,the relation simplifiesto
{u,* r{u** tX+Yri : s{s- s) : o
( 1 3 .1s)
that can be expressions Equations13.50and 13.51form a set of simultaneouS solvedfor V and y subjectto the appropriateboundary conditions.
and DynamicWaves Kinematic,Diffusionn For the casewith zerclaterclinflow, l, Eq. 13.51can be solvedfor the friction slope c
_
Friction slope
c
_
Bed slope
Kinematic wave
Ev
:
Water surface slope
VAV
Convective acceleration
Diffusion wave Fu11dynamic wave
IAV
Temporai accelelation
(r3.s2)
252
ROUTING CHAPTER13 HYDROGRAPH
The three typesof analysisof unsteadyflow routing in open channelsare classifiedas kinematic,diffusion (also called noninertia), anddynamicwaveanalyses,depending on which terms in F;q. 13.52 are retained. The three techniquesdiffer not only by including different terms of Eq. 13.52 but also in the assumptionsregarding flow conditions for satisfyingthe momentum equation.Table 13.5 showstheseassumptions. HYDRAULIC USEDIN VARIOUS TABLE13.5 ASSUMPTIONS METHODS ROUTING Method Kinematic Diffusion Full dynamic
Watersurfaceprofile
Commonflow condition
Uniform Nonuniform Nonuniform
Steady Steady Unsteady
Steadyflow is definedas flow that doesnot changewith time, and uniform flow is flow with a water surfaceparalleling the bed slope.For steadyuniform flow, the curve)is a singlecurve without hysteresisloops. Steady rating curve (stage-discharge nonuniform flow has constantdischargebut varying water surfaceslopesuchas that found at the entranceto a reservoiror at the approachof a waterfall. One way of selectingthe applicablemethod is to examinethe rating curve and assesswhether it is the same for rising and falling stages.The choice of routing equation dependson whether the difference is small (kinematic), relatively large (dynamic),or somewherein-between(diffusion). The kinematic wave method assumesthat the inertia terms of Eq. 13.52 ate negligibleand that the friction slopeequalsthe bed slopeS. Momentumconservation is approxi(natedby assumingsteadyuniform flow, and routing is accomplishedby combiningthe continuity equationwith any form of friction lossequation.Typically, either the Manning equationor Ch6zyequationis used.The Chdzyequationis V :
C\/RS
( 13.s 3)
and the Manning equationis V :
l'486 n
O2/3St/2
( 13.s 4)
where C andn arefriction coefficients,S is the friction slope,and R is the hydraulic radius (areadividedby wettedperimeter).Both give velocity in feet per secondif area are input using squarefeet and feet units. and wetted qerimeter *of these equations can be substitutedinto the kinematic portion of Either Eq. 13.52,equatingslopeof energygradeline with bed slopeto accountfor momentum. The continuity equation,Eq. 13.39,for this casereducesto dQ *64:o At Ex
( 13.5s )
The Manning or Ch6zyequationhas the form Q:
b'4^
(13.56)
RIVERROUTING 253 13.3 HYDRAULIC
which, after taking derivatives,is 6Q = 6*4*-t AA
:
o
m--a : M V A
(r3.57)
Multiplying Eq. 13.55by dQ/dA gives
, ,a Q mv -dx
+ 9a t : o
( 13.s 8)
or. if c - mV,the kinematic routing equationis
.- dQx * 9 : o
(13.se)
"- : d Q : l d Q dA BdY
(13.60)
At
to the slope of the where B is the top width of the channel.Thus celerity is related c is constantand that assume rating curve,which varieswith stage.Most applications
9 * , 9 : dd{x9dx At
(13.61)
accounts for the The left-hand side is the kinematic wave equation and the right diffusion d is diffusion effect of nonuniform water surfaceproflles. The hydraulic given by .
(r3.62)
q
" 2 5 d
:
-
This term reveals where4 is the flow per unit width of channel,and s is the bed slope. in small d) (resulting why kinematiowaveanalysisis valid whenbed slopesare steep slopes,the bed flat or when the channells extremelywide (resultingin small 4)' For hydraulic diffusion coefflcientis particularly important' Numerical solutionsof Eq. 13.6I ate presentedby severalinvestigators'12-ra Thesenormally involve substitutionof the relation
ao A--
- B 3+v dt
(13.63)
254
CHAPTER13
HYDROGRAPHROUTING
into Eq. 13.61,whereB is the channeltop width. The Manning or Ch6zyequationis usedfor the friction slope,where
tr:#
(r3.64)
in which the conveyanceK is Q/\/Srfromeither equation.Diffusion wavesapply to a wider range of problems than kinematic formulations, but their use may not be warrantedbecauseit requiresabout the sameeffort as dynamic routing. The third type of waveanalysisaccountsfor all terms inBq.13.52, includingthe "dynamic" or nonuniform,unsteady,and inertia components.It is referredto as the "full dynamic" formulation. Dynamic wave solutionsare far more complicatedbut are often necessaryfor analysisof flow along very flat slopes,flow into large reservoirs, highly unsteadydam-breakflood waves,or reversing(e.g.,tidal) flows. These conditions are often encounteredon coastalplains. As a generalrule, full dynamic wave analysisbecomesnecessarywhen
s>
1 5 F 4\;
( 13 .6s )
S - bed slope T o : time to peak (sec) of the inflow hydrograph D : averageflow depth (ft) o : eravitationalacceleration
where
d
SCSAtt-Kin TR-20Method In 1983,the SCSreplacedtheconvexmethod(Section13.1)with themodifiedatt-kin (attematron-kinematic)methodas the agency'spreferredchannelrouting method.ls The 1.964SCS TR-20 (Chapter 24) single-eventsimulation model used the convex method but was subsequentlymodified to route by the att-kin method. The procedureis a blend of the storageindication and kinematicwavemethods. Figure 13.5 showsthe two-step processof simulating attenuationfirst by meansof
e
E
o PP
Flgure 13.5 Routing principlesusedin the SCS att-kin method,
RIVER ROUTING 255 r3.3 HYDRAULIC storagerouting and then translatingthe wavein time by the kinematic wavemethod to accountfor the fact that routed flow ratesnot only decreasein magnitudebut also require time to traversethe length of the routing reach.The storagerouting portion providesattenuationwith instantaneoustranslation,and the kinematic waverouting providestranslationand distortionbut doesnot attenuatethe peak. Both are needed to produce the desired effect. The previously rhentioned full dynamic equations simultaneouslyaccountfor both effectsbut are difficult to solve. Figure 13.5helpsto visualizethe process.The samevolume V1of waterflowing into the reachduring time /, would flow out of a hypotheticalstoragereservoirduring interval 12.This samevolume would translateand distort downstreamby kinematic actibn, flowing out of the reachduring interval /r. Through its theoretical developmentand selectionof routing coefficients,the att-kin methodequationssatisfythe physicalpropagationand timing of the peak flow rate first. Conservationof massis also assured(areasunder the three hydrographsof Fig. 13.5are equal). The actualprocessroutesthe inflow hydrographthrough storage,thentranslates the peak flow rate, without attenuation,to its final time location in the outflow hydrograph.The location in time of the peak outflow is assumedequal to that correspondingto the maximum storagein the reachduring passageof the flood. Becausecelerity changeswith storage,the other flows of the storage-routed hydrographare translated,pachby a different celerity,to their respectivefinal times and values. The storageindication routing is accomplishedby substitutionof the relation Q:
KS^
(13.66)
into the continuity equation, Eq. 13.30, where S is the storageand K and m are coefficients.Kinematic routing solvesthe unsteadyflow equation(13.59) with Q:
bA*
(r3.61)
whereb andm are input coefficients,and A is cross-sectionalarea.If Z is the length ofrouiing reach,andifthe cross-sectionalareathroughoutL is relativelyconstant,the storageis given by S: LA
(13.68)
Theseequatiohsare combined in an iterative fashion to assurethat the peak flow resultingfrom the kinematic routing equalsthe peak resulting from storagerouting, and simultaneouslyensuringthat the time of the peak outflow occurs at the time of maximum storagein the reach,or Qo: KS;
(13.69)
Input to the methodrequiresselectionof a reachlength and estimatesof b andm for for Table 13.1and Eq. 13.27,m canbe shownto be usein Eq. 13.59.As discussed a factor relating averagevelocity (under bankfull conditions)with wave celerity, or c
v
(13.70)
256
CHAPTER13
HYDROGRAPHROUTING
The larger ru becomes,the shorterthe travel time, A value of m I 1.0 would incorrectly makethe celerity slowerthan the averageflow velocity.Studiesby SCSresulted in a recommendationof ! for general use. Signif,canterrors resultedfor m values greater than2.0. Equation 13.67 is appropriatefor cross sectionshaving a single ihannel with regulaishape.Complexcrosssectionsare more diffcult to evaluate,but la valuescan be developid from a rating table for the stream'rs As the coefficient b decreases,attenuationof the peak flow increasesdue to reducedvelocity and increasedstoragein the reach.The value b canbe estimatedby plotting Q andA on log-logpaper and fitting the linear form of Eq. 13.67 (referlo faatei6.+1. The slope*oUa be m andtheinterceptut 4_: 1 would be b. The SCS has also developednomographsfor estimatingb and m'" As a generalguideline,the reach length L shouldbe increasedto a value that resultsin a kinematic wavetravel time c greaterthan the selectedtime increment,or Lo> cA,t Z*io ft
9,000 7,500 15,000
mV = wave celerity l*ln LR
mV ft/sec
0.5 0.7
LR ft 18,000
5,000
12,000 10,000
3,500
7,000
r
5,000 4,000
1,500
3,000
0.3
1,000
2,000
0.2'
750
1,500
= minimum acceptablereach length = minimum recommended reach length MainTime Increment Hours
2 I
6,000
0.5 n?5
3 /
0.1 0.05
5 7 10 T2
Example: mV = 4ftlsec Main time increment = 0.2 hr L*n= 1,450ft La = 2,900 ft
1.000 350 700 250
500
15
t00 for le ngth reach Figure 13.6 SCS nomographfor determining Service, Conservation (After Soil U.S. atl-kin method of routing. "ComputerProgramfor Project Formulation," TechnicalRelease 20, Revised,AppendixG, 1983.)
PROBLEMS 257 : mV' and Ar is the time increment' The where Lo is the recommendedlength' c minimumrecommendedLoisthatgivingawavetraveltimeequaltoabouthalfthe lengthy iray iesult in analytical difficulty when rime incremenr. This 1j[$;"1* inflowhydrographsorsteepStreams.areencountered.Italsoresultsinthepeak time incrementAr. If severalreacheswere outflow time being ,ourJ"J"rp io the full accumulate'Thus a reachlength between routed, this incrementattime^errorwould greater thal c. A/ is recommended' c Lt andc Lt/2 is acceptable'but a length Figure13.6providesth"rangeofminimumacc"eptableandminimumrecommended routing reachlengths.
ff ffi ff;;iicationsofhvdraulicroutingtec|1ioue11p.p"11'i^'"Tt*::?:Yi::
ff ;'il;;'n"'uu"'i11,, "u.n ;t'1":".1-':^:"::::T,it?ii: ffi ff Ji,ffiffi :3:[";ffi;i;;;ffi;"'"-*:i"o1j.u""]'mJ1't^:-:T:1T':1"::i"%i"$1 , i. wru,iilpresented iir"T-l"ii,'.i,"n" ;;lrt':? HX"J"H;il;il#il;;;;;,;;i;#
1"*Tl student ll :^11ti:: understand caniil::1,:: thataninteresred
modeling' the structuringprocessesof hydrologic
Summary ComputersoftwareforhydrographsynthesiS^androutingisavailablefromnumerous federal agelcy routines are detailed in public and private vendors. wid;ry used amflow processesincludesone or Chapter24. Virtually every ( m o r e h y d r o g r a p h r o a s H E C - I ( C h a hydroloeig p t e r 2 4a1d ) , hydraulic p r o v i d e lplied severalchoices.In I the models can be found in the routing procedures' g]]:i,:Ytditg suggestions .T" At sel 1iterarure.,6,17 ln rlver or engage will The readerwho when eachof the methodsshouldbe applied' reservoirroutingisencouragedtodwelopthecomparisonrequestedinProblem13.25 before leavingthis chaPter'
PROBLEMS hydrologic and hydraulic routing techniques' 13.1. Discussthe main diff-erencesbetween 13.2.
t+}i ;q":: Iit:t?;;i,Y :..ii+l ;; q :il# ffi?#t 1iT .'SX,ffff r" 191 :,-Yllli "'*ifi:'i I ilY?iBi'"".U'1-'"d;i"?';:fli!11iy":;T"':'.."t"jfl i:;#"J I"a'^#i'.h;r::;;'i;A--)-r",t'i'i""1'"'lT1,1l.o:'^"iui:.'* i:il3iil;
riverroutingequation' TheMuskingum ?:,:,':'?,*
12' v2' .xg 9z v\v 4-_ *u,'::9,":1i: i'||w p:.'i"l' oerro.; time the tlme orthe of Siif#ffi"#ru:=#;"i"e-i*ine outflow, and storage at the beginning lT"jll ;;:E;: n"t*#:J:'::3*4":::i3";ffi';
u,,t'l uutu", corresponding G "and ffi:Tt"'ff:fd;#?;##ir*"p"'i"a'lndAsistl"*"1c-",'istorage'Perrorm for Cs' C1' and C2' the describedA",inution
verify ttre equations
l"3.3.IftheMuslongumKvalueis12hrforareachofariver,andiftheXvalueis0.2'what purposes? would be a reasonablevalue of Ar for routing
258
ROUTING, 13 HYDROGRAPH CHAPTER
'
13.4. A river reachhas a storagerelation given by S; : ali + boi. Derive a routing equation for O2 analogousto the Muskingum equation (13.6). Give equationsfor the coefficientsof 11,01, and 12. 13.5. List the steps(starting with a measuredinflow and outflow hydrographfor a river reach)necessaryto determinethe Muskingum K and X values. If the inflow and outflow are recordedin cubic feet per second,statethe units that would result for K and X if your list of stepsis followed. 13.6. Given the following inflow hydrograph:
6 e.v. Noon 6 p.tu. Midnight 6 A.M. Noon 6 p.Iu. Midnight
lnflow (cfs)
Outflow (cfs)
100 300 680 500 400 310 230 100
100
Assumethat the outflow hydrographat a section 3-mi downstreamis desired. : a. compute the outflow hydrographby the Muskingum methodusing valuesof K : 0.13. 11 hr andX b. Plot the inflow and outflow hydrographs on a single graph' c. Repeatsteps(a) and (b) usingX : 0.00. I3,7. Giventhe following valuesof measureddischargesat both endsof.a30-mi river reach: a. Determinethe Muskingum K and X valuesfor this reach. b. Holding K constant (at your determined value), use the given inflow hydrograph to determineand plot three outflow hydrographsfor valuesofX equal to the computed value.0.5. and 0.0. Plot the actualoutflow and nurnericallycomparethe root mean squareof residualswhen each of the three calculatedhydrographsis compared with the measuredoutflow.
Time 6 e.Ira. Noon 6 p.v, Midftight 6 l.rra. Noon 6 p.tvt. Midnight 6 n.n. Noon 6 p.v. Midnight 6 A.M.
lnflow (cfs)
Outflow (cfs)
l0 30 68 50 40 3l
10 12.9 26.5 43.1 44.9 41.3 35.3 27.7 t9.4 1 5I. 12.7 11 . 5 10.8
ZJ
10 10 10 10 t0 10
PROBLEMS 259 13.8. Selecta streamin your geographicregionthat hasrunoffrecords' method of routing to find K and X'
Usethe Muskingum
l3.g.PrecipitationbeganatnoononJune14andcausedafloodhydrographinastredrn.As thehydrographpassed,thefollowingmeasuredstreamflowdataatcrosssectionsA and B were obtained: Outflow, SectionB (cfs)
Inflow, SectionA (cfs)
Time June14-17
10 10 t3 26 43
10 10 30 10 50 40 30 20 10 l0 l0 10 10 10
6 e.v. Noon 6 p.tvt. Midnight 6 e.u. Noon 6 p.u. Midnight 6 n.u. Noon 6 p.tvt. Midnight 6 e.u. Noon
4l 35 28 19 IJ I J
1l 10
river reach'
for the a. Determine the Muskingum K and X values b. DeterminethehydrographatSectionBifadifferentstormproducedthefollowing hydrographat SectionA:
Time 6 e.u. Noon 6 p.tt. Midnight 6 l.rra.
lnflow (cfs)
Time
lnflow (cfs)
100 100 200 500 600
Noon 6 p.tvt. Midnlght 6 e.u. Noon
400 300 200 100 100
emergencyspillway of a certain 13.10. The outflow rate (cts) and storage(cfs-hr) for an units of hours'Use
: Sl3,wherethe number3 has reservoirare linearly t"fui"a Uy d - or Ltl2to determine + 02 Ltlz = I tt + sr s2 equations continuity the this and event: inflow following the itr" p""f. outflow rad fr;m the reservoir for Time (h0
I (cfs)
0
0 400 600 200 0
L
4 6 8
D
(cfs-hr)
o (cfs)
7
260
CHAPTER13 HYDROGRAPHROUTING ' 13.11. A simple reservoir has a linear storage-indication curve defined by the equation
" : 2*. whereAr is equalto 10 hr. If s at 8 e.u. is 0 cfs-hr,usethe continuity equationto route the following hydrograph through the reservoir:
Time I (cf9
8 a.lra.
0
9 a.u. 200
10 e.Ira. 400
1l A.M.
200
Noon 0
I P.M.
0
13.12. For a vertical-walled reservoir with a surface area A show how the two routing equations(73.32 and 13.34) could be written to contain only o2, sz, and known, values(computedfrom or, s,, and so on). Eliminate .FIfrom all the equations.How could thesetwo equationsbe solvedfor the two unknowns? 13.13. Given: Vertical-walled reservoir, surface area: 1000 acres; emergency.spillway width : 97.l ft (ideal spillway);H : watersurfaceelevation(ft) abovethe spillway crest; and initial inflow and outflow are both 100 cfs. a. In acre-ft and cfs-days,determinethe valuesfor reservoirstorageS corresponding to the followingvaluesof I1..0, 0.5, l, 1.5,2,3, 4 ft. b. Determine the values of the emergencyspillway Q correspondingto the depths namedin part a. c. Carefully plot and label the discharge-stcirage curve (cfs versuscfs-days)and the storage-indicationcurve (cfsversuscfs,Fig. 13.3) on rectangularcoordinategraph paper. d. Determinethe outflow ratesoverthe spillwayat the endsof successive dayscorrespondingto the following inflow rates(instantaneousratesat the endsof successive days):100, 400, 1200,1500, 1100, 700, 400, 300, 200, 100, 100, 100. Use a routing table similar to the one used in Example 13.4 and continue the rotating procedureuntil the outflow drops below 10 cfs. e. Plot the inflow and outflow hydrographson a single graph. Where should these curvescross? 13.14. Routethe given inflow hydrograph through the reservoir by assumingfhe initial water level is at the emergencyspillway level (1160 ft) and that the principal spillway is plugged with debris. The reservoir has a 500-ft-wide ideal emergency spillway (C = 3.0) locatedat the 1160-ft elevation.Storage-area-elevation data are Elevation (ft)
Area of pool (ft2x 106)
Storage (ft3x 106)
1l{0 1120 1140 I 158 I 160 1162 1164 I 166 I 168 1180
0 0.85 3.75 9.8 10.8 I 1.8 12.8 13.8 14.85 25,0
0 4.25 50.25 172.15 r92.75 2r5.35 239.95 266.5s 295.20 528.55
PROBLEMS
261
The inflow hYdrograPhdataare
Time(h0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 J.)
4.0 A <
I (cfs)
0 3,630 10,920 to,'720 5 010
1,600 460 100 10 0
table. a.Findthel5-minunithydrographbySnyder'smethod.
063 30minand forthenrst l i"li:l! ff;ru#iHfllSiii.?om18in.ofrain .. il"i:i.,i[T;iitf;,ersus
of 15minandthe period o.cuweusinga routing
outflow and storagecurvesprovloeo' to the bottom the reservoirassumingit is full e. Route,n" uJ#f,iu.;;;n";;"dh of the sPillwaYelevation980' water^inthe reservor' f. Indicate maxi-mumheight of plu'"d to obtain 5 ft of freeboard? "f th;;;;;; g. At what "tJ;;;;iJ'n"'op
Elevation (ft)
960 970 980 990 1000
lncremental storage 1oo(ft")
Total storage 104(ft3)
0 40
zto 590 1240
40 250 840 2080
the reservoirinitially empty' 13.16. RepeatProblem 13'15 with l3,IT.AfloodhydrographistoberoutedtytheMuskingummethodthroughalO-mireach ro-ml fvl,r;ach be divided in with K = Zhr.tnto how *uny ,ubr"u"h"*-**rirt" Kl3 < Lt < Kl ;;J stitt satisry ttre ,tuuitity criterid order to or" lr ]'6.i;
262
CHAPTER 13 HYDROGRAPH ROUTING 13.18. RepeatProblem 13.6aby dividing the 3-mi reach into two subreaches with equal K valuesof 5.5 hr. Comparethe results. 13.19. Discussthe problemsassociatedwith the useof a reservoirrouting techniquesuchas the storage-indicationmethod in routing a flood through a river reach. 13.20. VerifyEq. 13.51. t3.21. Precipitationbeganat noon on June14 and causeda flood hydrographin a stream.As the storm passed,the following streamflow data at cross sectionsA and B were obtained:
Time June14-17
lnflow SectionA (cfs)
Outflow SectionB (cfs)
l0 10 30 70 50 40 30 20 10 10 l0 t0 l0 10
10 l0 13 26 43
6 .q,.N{. Noon 6 p.u. Midnight 6 .r.lr. Noon 6 p.lr. Midnight 6 a.Ira. Noon 6 p.rra. Midnight
6 e.u. Noon
+l
35 28 t9 l5 l3 ll
10
a. Determinethe Muskingum K and X valuesfor the river reach. b. Determine the hydrograph at Section B if a different storm produced the following hydrographat SectionA (continuecomputationsuntil outflow falls below 101 cfs):
Time 6 a.u. Noon 6 p.Ira. Midnight 6 a.Ira.
Inflow (cfs)
100 100 200 500 600
Time (cont.) Noon 6 p.u. Midnight 6 a.Ira. Noon
lnflow (cfs)
400 300 200 100 100
13.22. If the MuskingumK valueis 12 hr for a reachof a river, and if the X valueis 0.2, what would be a reasonablevalue of Ar for routing purposes?
PROBLEMS 263 30-mi river reach: 13.23. Giventhe following valuesof measureddischargesat both endsof a
Time
lnflow (cfs)
6 n.u. Noon 6 p.rra. Midnight 61.v. Noon 6 p.Ira. Midnight 6 n.rra. Noon 6 p.tu. Midnight 6 e.lr.
10 30 68 50 40 3l 23 10 10 10 10 10 10
Outflow (cfs)
10 12.9 26.5 43.r 44.9 4t.3 35.3 27.7 t9.4 15.1 12.7 I 1.5 10.8
a. Determinethe Muskingum K and X valuesfor this reach' b.HoldingKconstant(atyourdeterminedvalue),usethegiveninflowhydrographto computed determineandptot tt ree outflow hydrographsfor valuesofX equalto the value,0.5, and 0.0. 13.24. Given the following inflow hydrograph:
6 e.v. Noon 6 p.v. Midnight 6 e.l"l. Noon 6 p.v. Midnight
lnflow (cfs)
Outflow (cfs)
10 30 68 50 40 3l
10
ZJ
10
Assumethat the outflow hydrographat a section3-mi downstreamis desired' using values of a. Compute the outflow hydrograptrby the Muskingum method K - 1 1 h r a n dX : 0 . 1 3 ' b. Plot the inflow and outflow hydrographson a singlegraph' : 0'00' c. RepeatSteps(a) and (b) usingX textbookfor all refert3,25. Carefully review the chapterand consult one other hydrology presented' Compile the results encestoipplicability ofeach ofthe routing procedures frequently' consulted and retained into a list or table, ihis taUleshouldbe
264
ROUTING 13 HYDROGRAPH CHAPTER
REFERENCES 1. Graeff, "Trait6 d'hydraulique."Paris,1883,pp. 438-443. 2. Y. T. Chow, Open ChannelHydraulics. New York: McGraw-Hill , 1959. 3. U. S. Soil ConservationServiie, National EngineeringHandbook, Notice NEH 4-102. D.C.: U.S. GovernmentPrintingOffice,August 1972. Washington, 4. S. Hayami, On the Propagationof FloodWaves,Bulletin 1. Kyoto,Japan:DisasterPrevent i o n I n s t i t u t e1, 9 5 1 . "Multiple Linearization Flow Routing Model," Proc. 5. T. N. Keefer and R. S. McQuivey, ASCEL Hyd. Div.100(HY7) (Iuly 1974). "On the Subjectof a Flood PropagationMethod," J. Hyd. Res'IAHRT(2), 6. J. A. Cunge, 20s-230(1967). 7 . D. L. Brakensiek,"KinematicFlood Routing,"Trans.ASCE L0(3) (1967). "Water Studies,"Bureauof ReclamationManual, Yol. 8. U. S. Departmentof the Interior, U. S. GovernmentPrinting Offrce, 1941. D.C.: IV Sec. 6.10. Washington, "Numerical Techniquesfor Spatially VariedUnsteadyFlow," University 9. T. E. Harbaugh, of Missouri Water ResourcesCenter,Rept. No. 3, 1967. 10. R. K. Price, "Comparisonof Four Numerical Methods for Flood Routing," Proc. ASCE J. Hyd. Div. 100(HY7) (July 1974). "NonlinearKinematicWaveApproximation 11. R. M. Li, D. B. Simons,and M. A. Stevens, for WaterRouting,"WaterResourcesRes.AGU II(2) (Apr. 1975). "WavePropagationin Rivers," HEL Ser. 8' No. 1. 12. J. A. Harder and L. V. Armacost, Hydraulic Engineering Laboratory, College of Engineering,University of California, Berkeley,June 1966. "Numerical Solution of UnsteadyFlows in Open 13. D. J. Gunaratnamand F. E. Perkins, Channels"HydrodynamicsRept. 127,MIT, Cambridge,MA, July 1970. 14. G. DiSilvio, "Flood Wave Modiflcation Along Channels," Proc. ASCE J. Hyd. Div. 9s(HY7) (1e69). "Simplifled Dam-Breach 15. U.S. Department of Agriculture, Soil ConservationService, RoutingProcedure,"Tech.Release66,Mar. 1979. "ComparativeAnalysis of Flood Routing Methods," ResearchDoc. 16. Streldoff, T., et al., 24,Hydrologic EngineeringCenter,U. S. Army Corps of Engineers'Davis, CA, 1980. "Guidelinesfor Calculatingand Routinga Dam-Break l7 . D . L. Gunlachand W. A. Thomas, Flood," ResearchNote 5, HydrologicEngineeringCenter,U. S. Army Corpsof Engineers, Davis.CA. 1977.
C h a p t e r1 4
Snow Hydrology
Prologue The purposeof this chaPteris to: . Indicatethe importanceof snowmeltto water supplyand managementin cold regions. . Describemethodsfor measuringsnowfall and describingits water-producing caPabilities. ' Describethe physicsof snowmelt. . presentmodeisfor estimatingsnowmeltunder variousconditionsof temperature, relativehumidity, wind speed,topography,ground cover,and snowpack.
14.1 INTRODUCTION In many regions,snowis the dominantsourceof water supply.Mountainousareasin the Weit areprime examples.Goodellhasindicatedthat about90 percentof the year$ water suppliin the high Llevationsof the ColoradoRockiesis derivedfrom snowfall.' Equally high proportions are also likely in the Sierrasof California and numerous t"giotrr m tne Nortfrwest.A significantbut lessershareof the annualwateryield in the Northeastand Lake statesalso originatesas snow.It is important that the hydrologist understandthe natureand distributionof snowfallandthe mechanismsinvolvedin the snowmeltprocess. Snowmelt usually beginsin the spring. The runoff derived is normally out of phasewith the periodsof gieatestwaterneed;therefore,variouscontrol schemessuch as storagereseivoirshavJ been developedto minimize this problem. An additional is that $omeof the greatestfloods result from combinedlargepoint of significanc.e snowmelt.Streamflowforecastingis highly dependenton adeand rainstorms icale extentand characteristicsof snowfleldswithin the watershed. of the knowledge quate can be increasedby minimizing the vaporization of yield from snowfall water fhe yield can be managedwithin limits by controlling the Timing water. melt and snow can be be obtainedby speedingthe melt process' results Early snowmelt. rate of the whereasthe snowmeltperiod can be extendedor delayedby retarding it. The annual snowfall distribution in the United Statesis shownin Fig. 14'1'
&
f
N
-v /
S:.-
) \
I O
,4
00 co
,
I @
()
!1 a !? o
E q
{)^ > j .
o t
5E p P .=.Y
tu5
14,2 SNOW ACCUMULATIONAND RUNOFF
267
An adequateunderstandingof meteorologicalfactorsis as much a prerequisite in consideringthe snowmeltpro""rs as it is in dealingwith evapotranspiration.The atmospheresuppliesmoisturefor both snowfall and condensationof water vapor on the snowpack,regulatesthe exchangeof energy within a watershed,and is a controlling factor in snowmeltrates. As in the rainfall-runoffprocess, geographic,geologic,topographic,and vegetative factors also are operativein the snow accumulation-snowmeltrunoff process. For rainfall-runoff relations, point rainfall measuresare used in estimating areal and time distributions over the basin. A similar approachis taken in snow hydrologyalthoughthe point-arealrelationsare usuallymore cornplex.Mathematical equationscan be usedto determinethe various componentsof snowmeltat a given location. Adequatemeasuresof depth and other snowpackproperties can also be amount , obtainedat specificlocations.The use of thesemeasurementsin estimating less much is a areas and distribution in area and time of snow over large watershed subdiviparticular areal rigorous procedure.Usually, averageconditions related to sionsover time are usedas the foundationfor basin-widehydrologicestimates'Such proceduresare often in the categoryof index methods(Section 14.6). Snowmelt routines have been incorporated in numeroushydrologic models, some of which also include water quality dimensions.A good accountingof the fundamentalsof the snowmelt process and of contemporary snowmelt modeling approachesmay be found in Refs.2-15 listed at the end of this chapter.
AND RUNOFF 14.2 SNOWACCUMULATION Under the usual conditions encounteredin regionswith heavy snowfall, the runoff from the snowpackis the last occurrencein a seriesof eventsbeginningwhen the snowfallreachesthe ground.The time interval from the start to the end of the process might vary from aslittle asa day or lessto severalmonthsor more. Newly fallen snow hasa densityof about 10 percent(the percentageof snowvolumeits waterequivalent would occupy),but as the snow depth enlarges,settling and compactionincreasethe density.ls The temperaturein a deep layer of accumulatedsnow is often well below fueezingafter prolongedcold periods.When milder weather setsin, melting occurs flrst atlhe rno*pu"k surface.This initial meltwater moves only slightly below the surfaceand again freezesthrough contact with colder underlying snow.During the rcfreezingpro""rr, the heat of fusion releasedfrom meltwaterraisesthe snowpack t"*p"rutur". Heat is also transferredto the snowpackfrom overlying air and the ground. During persistent warm periods, the temperatureof th9 entire snowpack Iontinually rises'and finally reaches32'F. With continued melting, water begins flowing down through the pack. The initial melt component is retained on snow crystals in capillary films. Once the liquid water-holdingcapacity of the snow is reached,the snow is said tobe ripe. Throughoutthe foregoingprocess'pack density increasesdue to the refreezingof meltwaterand buildup of capillary films. After the water-holdingcapacityis reached,the densityremainsrelatively constantwith continued rnelt. Meltwater that exceedsthe water-holdingcapacity will continueto move
268
14 SNOWHYDRoLoGY CHAPTER down through the snowpackuntil the ground is finally reached.At this point runoff can occu.r.Three situationsthat may exist at the ground interfacewhen meltwater reachesit are describedby Horton.t6 First, considerthe casewherethe melt rate is lessthan the infiltration capacity of the soil. In addition, downward capillary pull of the soil coupled with gravity exceedsthe samepull of the snow lessgravity. The meltwaterdirectly entersthe soil and a slushlayer is not formed. The secondcaseoccurs when a soil's infiltration capacity is greaterthan the melt rate, but the net capillary pull of the snowpackexceedsthat of the soil aidedby gravity. Capillary water builds up in the overlying snow until equilibrium is reached at which upward and downwardforces balance.A slushlayer forms and providesa supply of water that infiltrates the soil as rapidly as it entersthe slushlayer. The final situationis one in which the melt rate exceedsthe infiltration capacity. A slush layer forms and water infiltrates the soil at the infiltration capacity rate. Excesswater acts in a manner analogousto surfacerunoff but at a much decreased overlandflow rate. As warm weathercontinues,the melt processis maintainedand accelerateduntil the snowcoveris dissipated.
14.3 SNOWMEASUREMENTS AND SURVEYS Snow measurementsare obtained through the use of standardand recording rain gauges,seasonalstorageprecipitation gauges,snow boards,and snow stakes.Rain gaugesare usually equippedwith shieldsto reducethe effect of wind.3Snow boards are about 16 in. square,laid on the snow so that new snowfall which accumulates betweenobservationperiodswill be found abovethem. Care must be taken to assure that adversewind effectsor other conditionsdo not producean erroneoussampleat the gauginglocation.Snow stakesare calibratedwoodenpostsdriven into the ground for periodic observationof the snowdepthor insertedinto the snowpackto determine its depth. Direct measurementsof snow depth at a single station are generally not very useful in making estimatesof the distributon over large areas,since the measured depth may be highly unrepresentativebecauseof drifting or blowing. To circumvent this problem, snow-surveyingprocedureshavebeendeveloped.Suchsurveysprovide information on the snowdepth,waterequivalent,density,and quality at variouspoints along a snow course.All thesemeasuresare of direct use to a hydrologist. The water equivalentis the depth of water that would weigh the sameamount as that of the sample.In this way snow can be describedin terms of inchesof water. Density is the percentageof snow volume that would be occupied by its water equivalent.The quality ofthe snow relatesto the ice contentofthe snowpackand is expressedas a decimal fraction. It is the ratio of the weight of the ice contentto the total weight. Snow quality is usually about0.95 exceptduring periodsof rapid melt, when it may drop to 0.70-0.80 or less.The thermal quality of snow,Q,, is the ratio of heatrequiredto producea particular amountof waterfrom the snow,to the quantity
269 CHABACTERISTICS SNOW ANDAREAL 14.4 POINT of heatneededto producethe sameamountof melt from pure ice at 32"F.Valuesof Q,may exceed100 percent at subfreezingtemperatures.The densityof dry snow is approximately10 percentbut thereis considerablevariability betweensamples.With grPater. aging, - - the densityof snow increasesto valueson the order of 50 percentor A snowcourseincludesa seriesof samplinglocations,normally not fewer than 10 in number.ttThe variousstationsare spacedabout50- 100ft apartin a geometric pattern designedin advance.Points are permanentlymarked so that the samelocations will be surveyedeachyear-very important if snow coursememorandaare to be correlatedwith areal snowcoverand depth, expectedrunoff potential, or other significantfactors.Survey dataareobtaineddirectly by forestersand others,by aerial photographsand observations,and by automatic recording stations that telemeter information to a central processinglocation. In the westernUnited Statesthe Soil ConservationServicecoordinatesmany snow surveys.Various states,federal agencies,and private enterprisesare also engagedin this type of activity. Sourcesof snowsurvey dataaresummarizedin Ref. 14.
14.4 POINTANDAREALSNOWCHARACTERISTICS The estimationof areal snow depth and water equivalentfrom point measurement data is highly important in hydrologicforecasting.
Estimatesof Areal Distributionof Snowfall Normally, taking arithmetic averagesor using Thiessenpolygonsdoesnot provide reliable results for estimating areal snow distribution from point gaugings'This is becauseorographicand topographiceffectsare often pronounced,and gaugingnetworks frequently are not denseenoughto permit the straightforwarduse of normal averagingtechniques.However, regional orographiceffects are relatively constant from yearto yearand stormto stormfor tractsthat are small whencomparedwith the areal extent of general storms occurring in the region.2This circumstancepermits many useful approachesin estimating the areal snow distribution once the basic patternhas beenfound for a region' One method used to estimatebasin precipitation from point observationsassumesthat the ratio of station precipitation to basin-precipitationis approximately constantfor a storm or storms.This can be statedas' P o _ Nb Po Pu:
or where
D t b
--
(14.1)
No PoNu
(r4.2)
No
the basin precipitation
P o : the observedprecipitation at a point or group of stations N t : the annual precipitation for the basin N o : the normal annualprecipitation for the control station or stations
27O
CHAPTER14 SNOW HYDROLOGY
The normal annualprecipitationis determinedfrom a map (carefullypreparedif it is to be representative)displayingthe meanannualisohyetsfor the region.The precipitation is determinedby planimeteringareasbetweenthe isohyets.If the number of stationsusedand their distribution adequatelydepict the basin,Eq. I4.2 canprovide a good approximation.For stationsnot uniformly distributed,weighting coefficients basedon the percentageof the basin areaportrayedby a gaugeare sometimesused in determiningN, for the group. Another systemusedin estimatingareal snowfallis the isopercentalmethod.In this approach,the storm or annual station precipitation is expressedas a percentage of the normal annualtotal. Isopercentallines are drawn and can be superimposedon a normal annualprecipitation map (NAP) to producenew isohyetsrepresentingthe storm of interest.A NAP map indicatesthe generalnature of the basin'stopographic effects,while the isopercentalmap showsthe deviationsfrom this pattern.The advantageof this methodover preparingan isohyetalmap directly is that relatively consistent storm pattern featuresof the NAP can be taken into considerationas well as observedindividual storm variations.
Estimatesof Basin-WideWater Equivalent A hydrologistmust be concernednot only with the amount and areal distribution of snowfall, but also with estimatingthe water equivalentof this snowpackover the basin, sincein the final analysisit is this factor that determinesrunoff. Basin water equivalentmay be given as an index or reported in a quantitative manner such as inchesdepth for the watershed. The customaryprocedurefor determiningthe basin water equivalentis to take observeddata from snowcoursestationsand to provide an index ofbasin conditions. Various proceduresemploy averages,weighted averages,and other approachesto accomplishthis.2The important point to rememberis that the usefulnessof any index is basedon how well it representsthe overall basinconditions,not on how favorably it describesa particular point value. Indexesdo not actually provide a quantitative evaluationofthe property they cover.Instead,they give relativechangesin the factor. By introducingadditional data, however,an index can be usedin a prediction equation. For example,if the basin water equivalentcan be estimatedby subtractingthe runoff and losscomponentsfrom the precipitationinput, the index can be correlated with actual basin water equivalentin a quantitativemanner.
Areal Snowcover Estimatesof the areal distribution of snowfall are very helpful in making hydrologic forecasts.A knowledgeof actualarealextentof snowcoveron the groundat any given time is also*applied in hydrograph synthesisand in making seasonalvolumetric forecastsofthe runoff. Observationsof snowcoverare generallyobtainedby ground and air reconnaissanceand photography.Between snowcoversurveys,approximations of the extentof the snowcoverare basedon availablehydrometeorologicaldata. Snowcoverdepletionpatternswithin a given basin are normally relatively uniform from yearto year; thus snowcoverindexescan often be developedfrom data gathered at a few representativestations.
14.5 THE SNOWMELTPROCESS
27'I
PROCESS 14.5 THE SNOWMELT The snowmeltprocessconvertsice content into water within the snowpack.Rates differ widely due to variationsin causativefactorsto be discussedlater. Thesedivergenciesare not as strikingly apparentwhen consideringdrainagefrom the snowpack, however,since the pack itself tends to filter out thesenon-uniformities so that the drainageexhibits a more consistentrate.
EnergySourcesfor Snowmelt The heatnecessaryto inducesnowmeltis derivedfrom short-andlong-waveradiation, condensationofvapor, convection,air and groundconduction,and rainfall. The most important ofthese sourcesare convection,vapor condensation,and radiation. Rainfall ranks aboutfourth in importancewhile conductionis usually a negligiblesource.
EnergyBudgetConsiderations If snowmeltis consideredas a heat transferprocess,an energybudgetequationcan be written to determinethe heatequivalentof the snowmelt.Suchan equationis of the form2 H^:
H,r + H," + H" + H" + Hs + He + Hq
(14.3)
where H*: the heat equivalentof snowmelt H4 : rrlt long-wave radiation exchange between the snowpack and surroundings H,, : the absorbedsolar radiation H" : the heat transferredfrom the air by convection H": the latent heat of vaporizationderived from condensation Hr : the heat conductionfrom the ground Ho : the rainfall heat content Hn: the internal energychangein the snowpack In this equationH,", Hs, andHo are all positive;I1,1is usually negativein the open; H" and Hnmay take on positive or negativevalues;andH" is normally positive.The actual amountof melt from a snowpackfor a given total heatenergyis a function of the snowpack'sthermal quality. The heatenergyrequiredto producea centimeterof waterfrom pure ice at32"F is 80 langleys(g-cal/cm2).Therefore,203.2langleysarc neededto get 1 in. of runoff from a snowpackof 100 percentthermal quality. If the term H. representsthe combinedtotal heat input in langleys,an equationfor snowmelt M in inches.is2 M-
H-
203.2Q,
(r4.4)
where Q, is the thermal quality of the snowpack.For sirbfreezingsnowpacks,Q,, exceeds1; for ripe snowpackshavingsomewatercontent,Q,is lessthan one.A typical value for these conditions is reported to be 0.97.6Figure 14.2 gives a graphical solution to this equationfor severalvaluesof Q,.
272
CHAPTER14
SNOW HYDROLOGY
w 0
100
200
300
400
500
600
Net heat flux to snow pack, f(langleys)
Figure 14.2 Snowmelt resulting from thermal energy' (After U.S. Army CoJpsof Engineers.2)
EXAMPLE 14.1 Given a snowpackwith thermal quality of 0.90, determinethe snowmeltin inchesif the total input is 137 langleys. Solution. UseEq. 14.4 M:
H^/203.2Q, : r37/203.2 x 0.90 M : in. rl 0.75 M
Turbulent Exchange The quantity of heat transferredto a snowpackby convectionand condensationis commonlydeterminedfrom turbulentexchangeequations.Suchan approachhasbeen of temperatureand vapor pressuremustbe madein widely used,sincemeasurements the turbulent zonewherevertical watervapor, temperature,and wind velocity gradients are controlled by the action of eddies.In the following two subsectionsseveral practical equationsfor estimatingcondensationand convectionmelt are given. Here a combinedtheoreticalequationis presentedto acquaintthe readerwith the theory of turbulent exchange. The basic turbulent exchangeequationcan be written'
o : Adz4 where
( 14.s )
Q : thetime rate of flow of a specifiedproperty of the air suchaswater vapor through a unit horizontal area = dqldT the vertical gradient of the property
PROCESS 273 14.5 THESNOWMELT
e : the property z : lhe elevation A: an exchangecoefficient Property 4 must be unaffectedby the vertical transport.Propertiespertinent to this discussionare the air temperature,water vapor, and wind velocity.Theoretically,the potential temperatureshouldbe used,but air temperaturesmeasuredat normal distancesabovethe snowpackdo not causeseriouserrors.The potential temperatureof dry air is that which the air would take if broughtadiabaticallyfrom its actualpressure to a standardpressure. Gradients of the various properties of importance here follow a power law distribution where conditions of atmosphericstability exist.rTThis qualification is characteristicof the atmosphere'sstate over snowfields.2Logarithmic profiles are more nearly representativeof neutral or unstableatmosphericconditions.The power law providesthat the ratio of valuesof a property determinedat two levelsabovethe snow is equivalentto the ratio of the levelsraised to somepower. Thus,
where
q
z
/ . \r/n
Qz_
(:31
Qt
\Zr,/
(r4.6)
the value of the property the elevation(with subscriptsdenotingthe level) the power law exponent
If 3, is made equal to I, q, assumedto be the property value at this height, and the subscriptdroppedfor the secondlevel,Eq. 14.6becomes q : qtz\/"
(r4.7)
The magnitude of q is takenasthe differencein valuesof 4 measuredat the level z and the snow surface.For example,if T : 38'F at height z, and temperatureis the property of interest,then q: (38 - 32) : 6"p. The gradient of the property dqldz can be obtainedby differentiatingEq. 14.7:
ff: (et),,,_",,"
(14.8)
If this expressionis substitutedin the basicturbulentexchangeequation(14.5),the followins relation is obtained:
o : AIL)zo-d/"
(r4.e)
Thus,eddyexchangeofthe property at a specifiedelevatione is determinedfrom observationsofthe ptoperty at unity level.The exchangecoefficientis alsorelatedto elevationz. For equilibrium conditions up to the usual levels of measurementof moistureand temperature,gradientsof thesevariablesare suchthat the eddy transfer of moistureand heatis constantwith heisht. Then the exchangecoefficientA mustbe
274
14 SNOWHYDROLOGY CHAPTER
inverselyrelatedto the prope"t r.:O*r,i - )n,Orr, dqld, h in Eq. 14.8for dq/dz givesthe followingresult: Substitution A - Ar4"-ttr"
(i4.10)
(14.11)
since(dq/dz)t : et/nfor I : 1. Now, if the valueof A from Eq. 14.11is insertedin Eq.14.9,
n: o,(T)
(r4.r2)
The exchangecoefficientat an observationlevel has been shownto be directly proportional to the wind velocity measuredat that elevation.lsTherefore,it may be written that A':
ko1
(14.13)
where o1 : the wind velocity at Level one k : a constantof proportionality Substitutingfor 41 in Eq. I4.I2 gives /k\
a: \;)a'"' Using the power law equation(14.7), we find that q, : q;1/" and
D, :
117-1/n
(r4'r4) (14.15) (14.16)
After making thesesubstitutionsinBq.14.14 and denotingthe observationlevel of o as Tu,andthat of q as za(sincethesemay be different),Eq. 14.14becomes
n: (!")rr,zu)-'/nq.ou
(r4.r7)
This is a generalizedturbulent exchangeequation. Considerationis now given to developingspecifictheoreticalequationsfor condensationand convectionmelt. First, considerthe caseof the condensationmelt. The property to be tlansported in this caseis water vapor, and sincethe exchangecoefficientexpressesthe transfer of an air maqs,it is necessaryto determinethe moisturecontentof the air mass.This can be accomplishedby using the specifichumidity, which gives the weight of the water vapor containedin a unit weight of moist air. Equation 14.18 can be usedto calculatespecifichumidity: q :
0.622e
where e : the vapor pressure . Po: the total pressureof the moist air
(14.18)
PROCESS 275 14.5 THESNOWMELT Inserting this expression in Eq. 14.17 for q"yields2
"u, )-'/^(9J4:), n" : (I)u"z
," : t s(l)u.,,)-'^(99),"u,
(r4.re)
(r4.20)
The proportionality constantk is a complexfunction relatedto the air density and other iactors.Sinceihe densityof air is a function of elevation,k also varieswith The constantmay be madeindependentof density,and thereforeof elevation, height. "introducing a factor to compensatedirectly for the density-elevationrelation' by Atmosphericpressureservesto accomplishthis, and the equationcan be adjustedby muttipiying Uy tne ratio pf po, wherepis the pressureat the snowfieldelevationand po is the se-atevetpr"rrur". Introducingthis ^ratioinBq. 14.20and a new constantk1, which is related to sealevel pressure,gives'
m" : a.s(\){r"r,r','(99),,u,
{r4.2r)
Forconvectionmelt,thepropertyofimportanceinEq. 14.17is airtemperature. b-e To convert air temperatureinto ttreimat units, the specific heat of air crmust H" introduced. Putting these values in Eq. I4.l7,heat transfer by eddy exchange convertsto2
I/nc o, : (I) uoz6)oror),
(14.22)
cgs Sincethe latent heatof fusion is 80 callg, convectivesnowmeltM" in gramsin the systemis givenby H"l8O, or / t \/lc\. (r4.23) ,. : (*J ())(,"20)-'h, or"uu Introducing the elevationdensitycorrectionp/po,we obtain
* : (#)(l)r,",,r',(o1),,r",
(14.24)
Equations I4.2I and 14.24 can be^ combined into a single convectioncondensationmelt M"" equationof the form' M"" =
Lk"ru)-,,"(#fir,r"
(r4.2s)
276
CHAPTER14
SNOW HYDROLOGY
This is a generalizedtheoreticalequationfor snowmeltthat resultsfrom the turbuleni tfansfer of water vapor and heat to the snowpack.It is assumedthat the exchange coefficientsfor heat and water vapor are equal. Their evaluationis accomplishedby experimentation.r A combined physical equation of the general nature of Eq. 14.25 has been developedby Light.le Widely used,its individual convectionand condensationmelt componentsare discussedin following sections.The combined form of the Light equationis
D : where
a
pk',
80 rn(alz) ln(b/zo)
ulc,r+ ( e - 6.IDryf P J
(r4.26)
D - the effective snowmelt(cm/sec) air density L _ von K6rm6n's coefficient= 0.38 ^ 0 the roughnessparameter: 0,25 r b : the levelsat which the wind velocity,temperature,and vaporpressure are measured,resPectivelY : the wind velocity U : co the sPecific heat of air 7 :;the air temperature e : the vapor pressureof the air p : the atmosPhericPressure
Convection Heat for snowmeltis transferredfrom the atmosphereto the snowpackby convection. The amountof snowmeltby this processis relatedto temperatureand wind velocity. The following equationcan be usedto estimatethe 6-hr depth of snowmeltin inches by convection:2O (r4.27) D: KV(T - 32) where V : the mean wind velocity (mph) Z - the air temperature('F) ,
On the basisof the theory of air turbulenceand heat transfer (turbulent exchange),a been theoreticalvaluefor the exchangecoefficientK of 0.00184 X 10-0'0000156'has given by Light.le In this relation h, the elevationin feet, is usedto reflect the change in barometric pressuredue to the difference in altitude. The expressionis said to representconditions for an open, level snowfieldwhere measurementsof wind and temperatureare madeat heightsof 50 and 10 ft, respectively,abovethe snow.Values vary from 1.0 at sea level to 0.70 at 10,000ft of of the exprtssion10-00000156' elevation.The actual valuesof K are normally lessthan the theoreticalflgure due to such factors as forest cover. Empirical 6-hr K values have been reported in the literature.20 due to Anderson and Crawfordls give an expressionfor the houily sno.wmelt convectionas M _
cv(T"_ 32) Q,
04.28)
14,5 THESNOWMELT PROCESS 277 where M : V : To : Q, : c:
the hourly melt (in.) the wind velocity (mi/hr) the surfaceair temperature('F) the snowquality a turbulent exchangecoefficientdeterminedempirically
Temperaturemeasurements are at 4 ft, with wind gaugedat 15 ft. The corresponding valueof c is reportedas 0.0002.
Condensation
snow.A total yield of around 8.5 in. of snowmeltincluding the condensateis thus derived. A water vapor supply at the snow surfaceis formed by the turbulent exchange process;cons€quently,a masstransferequationsimilar to thosepresentedfor evaporation studiesfits the melt process.An equationfor hourly snowmeltfrom condensation takesthe formrs
u : W n { r " 6- . 1 1 ) where
(14.2e)
b : an empirical constant eo : the vapor pressureof the air (mb), the numerical value 6.11 : the saturationvapor pressure(mb) over ice at 32'F (e, must exceed 6 . 1l )
Also, M, Q,, and v are as previouslydefined.The constantb hasa valueof 0.001 for temperatureand wind measurementsat 4 and 15 ft, respectively.ls A similar expressionbut for 6-hr snowmelt(D) is given as D:KrV(eo-6.11)
(14.30)
where the theoretical value of K, is said by Light to equal 0.00579 if wind and
p
-
- e,) kV(e" ' " Q,
where E : the hourly evaporationin inches es : the saturationvapor pressureover the snow k : an empirical constant
(14.3r)
278
14 SNOWHYDROLOGY CHAPTER Also, V, eo,andQ,areasdefinedbefore.lsIntheexpressionk : 0.0001,temperature and wind measurements are taken as for Eq. 14.30, and the temperature of the air is assumed equal to that of the snow surface for temperatures below 32oF.
RadiationMelt The net amount of short- and long-waveradiation receivedby a snowpackcan be a very important source of heat energy for snowmelt. Under clear skies, the most significantvariablesin radiationmelt are insolation,reflectivity or albedoof the snow, and air temperature.Humidity effects, while existent, are usually not important. When cloud cover exists,striking changesin the amount of radiation from an open snowfield are in evidence.The general nature of these effects is illustrated in Fig. 14.3.2Combinedshort- and long-waveradiation exchangeas a function of cloud height and coveris represented.Radiationmelt is shownto be more significantin the spring than in the winter. It should also be noted that winter radiation melt tendsto increasewith cloudcoveranddecreasingcloudheightasa resultof the more dominant role playedby long-waveradiation during that period. Forest canopiesalso exhibit important characteristicsin regulating radiative heatexchange.Theseeffectsdiffer somewhatfrom thoseexhibitedby the cloudcover, especiallywhereshort-waveradiation is concerned.Cloudsand treesboth limit insolation, but cloudsarevery reflective,while a largeamountof the interceptedinsolation is absorbedby the forest.Consequently,the forestis warmedand part of the incident energydirectly transferredto snow irl the form of long-waveradiation; an additional fraction is transferredindirectly by air also heatedby the forest. Figure 14.4illustratessomeeffectsof forestcanopyon radiation snowmelt.The figure typifies averageconditions for a coniferouscover in the middle latitudes.2In winter, the maximum radiation melt is associatedwith completeforest cover, and in spring the greatestradiation melt occursin the open. Generalizationsshouldnot be drawn from thesecurves,which indicaterelative seasonaleffects of forest cover on radiation melt for the conditionsdescribed.Another factor affectingradiation melt is the land slopeand its aspect(orientation).Radiationreceivedby north-facingslopes is lessthan that for south-exposureinclinesin the northern hemisphere,for example. Solar energy provides an important sourceof heat for snowmelt. Above the earth's atmosphere,the thermal equivalentof solarradiation normal to the radiation path is 1.97 lingleys/min (1 langleyis approximately3.97 x 10-3 Btu/cm2).The actual amount of radiation reachingthe snowpackis modified by many factorssuch asthe degreeofcloudiness,topography,and vegetalcover.The importanceofvegetal cover in influencingsnowmelt,long recognized,has promptedmany forest management schemesto regulatesnowmelt.t'to'"''o Two basic'lawsare applicableto radiation.Planck'slaw statesthat the temperature of a blackbodyis relatedto the spectraldistribution of energythat it radiates. Integrationof Planck's law for all wavelengthsproducesStefan'slaw, Ro: cT' where R, : the total radiation o : Stefan'sconstant[0.813 x 10-10langley/(min-K-')] T : the temperature(K)
(r4.32)
14.5 THE SNOWMELTPROCESS
crurar,.ier,iiTth
M,, = 2.0o[1_ (0.82- 0.0244N] Mt=4.4r[r-(1 -}.\UAM
Amount ofclouds, N (a)
M,, = 0.50U- (0.S2- O.O'U4I\tl Mt=4.84[1 * (1 _ 0.0242)t{l
J"*qv; o u.u
4.2
0.4
0.6
0.8
l.t
Amount ofclouds, N (b)
Daily radiation melr in the open with cloudy {igure.fa3 skies: (a) during spring, May 20; and (bf during wintei, February 15. (After U.S. Army Corps of Engineeis.r;
280
CHAPTER14 SNOW HYDROLOGY t t I
I
r.) ^ r _ -t "n"n t : t lr
I
> 10
\ .=
rt \
M,= M^
Mrt
.
-4 (r 'F))- 3.30 =3.821F+ .7s7
\M,
-o.5 0.2
0.0
0.4 .'
1.0
0.6
Forestcanopycover,F (a)
(l) ,*^=l,r*
0.5
H o.o Mf
-0.5
'--1
Mr, + Mr1
4
4
1-
Mrt
? 'qTF r
I
-n - 3.30 7s7Q.
-1.0 0.0
0.6
0.4
0.8
Forestcanopycover,F (b)
- Figure 14.4 Daily radiationmelt in the forestwith clear skies:(a) duringspring,May 20; and (b) duringwinter, February15.(AfterU.S.Army Corpsof Engineers.2) Becausesnow radiates as a blackbody, the amount of radiation is related to its temperature(Planck's law), and total energyradiated is accordingto Stefan'slaw. Long-waveradiation by a snowpackis determinedin a complexfashionthrough the interactionsof temperature,forest cover, and cloud conditions'
PROCESS 281' 14.5 THESNOWMELT Direct solar short-waveradiation receivedat the snow surfaceis not all transferredto sensibleheat.Part of the radiationis reflectedand thus lost for melt purposes. Short-wavereflection is known as albedo and ranges from about 40 percent for melting snow late in the seasonto approximately80 percent for newly fallen snow. This property Valuesas high as 90 percenthavealsobeenrepofted in severalcases.22 of the snowpackto reflect large fractions of the insolation explainswhy the covers persistand air temperaturesremain low during clear, sunny,winter periods. That portion of short-waveradiation not reflectedand availablefor snowmelt may becomelong-waveradiation or be conductedwithin the snowpack.Some heat may also be absorbedby the ground with no resultantmelt if the ground is frozen. An expressionfor hourly short-waveradiation snowmeltis given as2 M-
H^
203.2Q,
(r4.33)
H^ : the net absorbedradiation (langleys) where ' 203.2 : a conversionfactor for changinglangleysto inchesof water When the snowquality is 1, long-waveradiation is exchangedbetweenthe snowcover and its surroundings.Snowmelt from net positive long-wave radiation follows Eq. 14.33.If the net long-waveradiation is negative (back radiation), there is art equivalentheat loss from the snowpack. An approximatemethodof estimating12-hr snowmeltDn (periodsmidnight to noon, noon to midnight) from direct solarradiation has been given by Wilson.2oThe relation is ofthe form (14.34) Dp = Do(l - 0.75m) where Do : the snowmeltoccurring in a half-day in clear weather m : thedegreeof cloudiness(0 for clearweather,1.0 for completeovercast) Suggested valuesfor Do are 0.35 in. (March), 0.42 in. (April), 0.48 in. (May), and 0.53 in. (June)within latitudes 40-48".2o
Rainfall Heat derivedfrom rainfall is generallysmall, sinceduring thoseperiodswhenrainfall occurson a snowpack,the temperatureof the rain is probablyquite low. Nevertheless, at highertemperatures,rainfall may constitutea significantheat source;it affectsthe aging processof the snow and ffequently is very important in this respect. An equationfor hourly snowmeltfrom rainfall isrs
(14.3s) P : the rainfall (in.) T- : the web-bulb temperatureassumedto be that of the rain This equationis basedon the relation betweenheatrequiredto melt ice (I44 Btu per pound of ice) and the amountof heatgiven up by a pound of water when its temperature is decreasedby one degree. where
282
14 SNOWHYDROLOGY CHAPTER
Daily snowmeltby rainfall estimatesare given by Ma: 0.007Pd(T"- 32)
(r4.36)
where Md : the daily snowmelt(in.) , P d : the daily rainfall (in.) To : the meandaily air temperature('F) of saturatedair taken at the 10-ft
level23
Conduction Major sourcesof heatenergyto the snowpackare radiation, convection,and condensation. Under usual conditions, the reliable determinationof hourly or daily melt quantitiescanbe foundedon theseheatsourcesplus rainfall ifit occurs.An additional sourceof heat,negligiblein daily melt computationsbut perhapssignificantover an entire melt season,is ground conduction. Ground conductionmelt is the result of upward transferof heatfrom ground to snowpackdue to thermal energythat was storedin the ground during the preceding summerand earlyfall. This heatsourcecanproducemeltwaterduringwinter and eady springperiods when snowmeltat the surfacedoesnot normally occur. Heat transfer by ground conductioncan be expressedby the relation2 dT Hn: K-where
(14.37)
K : the thermal conductivity "r'J"n dTlda : the temperaturegradient perpendicularto soil surface
The snowmeltfrom ground conductionis generallyexceedinglysmall. Wilson notes that after about 30 days of continuoussnowcover,heat transferredfrom the The amountof snowmeltfrom ground conducground to the snow is insignificant.20 tlon during a snowmelt seasonhas been estimatedat approxirnately0.02 in.lday'23 Groundconductiondoesact to providemoistureto the soil; thus,whenotherfavorable conditionsfor snowmeltoccur, a more rapid developmentof runoff can be expected. This sectionhasemphasizedthephysicsof snowmelt.The mannerin which heat Equations14.27-I4.3I and canbe providedto initiatethemelt processwasdiscussed. L4.33-I4.37 inclusivecan be usedto estimatethe melt at a given point. The task of computingrunoff from snowmeltin a basin cannot be approachedin sucha simple fashion,sincethere are many complexfactorsoperative.The remainderof this chapter is devotedto the general subject of runoff from snowmelt investigations.Figure 14.5 illustrateshourly variation in the principal heat fluxes to a snowpackfor a cloudy day. EXAMPLE 14.2 During a completelycloudy April period of l2hr, the following averagesexistedfor a ripe snowpacklocated at 10,000 ft above sea lel'el at a latitude of 44" N: air temperature50' F; mean wind velocity, 10 mph; relative humidity, 65Vo;avenge rainfall intensity,0.03 in./hr for I2hr; wet bulb psychrometerreading,48oF. Estimate the snowmeltin in. of water for convection,condensation,radiation, and warm rain for the 12 hr period.
April 23 100
--- Short-waveradiation Long-wave radiation
80 9 6 0
ffi
Incident short-wave radiation
nm
Reflected shon-wave raoanon
G *
Absorbedshort-wave radiation
Eo
F
<2
Apnl24
Aprii 25
1600 2000 2400 0400 0800 1200 1600 2000 2400 0400 0800 1200
40
lt ii
Incident short-wave radiation (t)
l,-
long-wave radiation (R2)
Downwad
H . n E z v
t
o
s
i
l -
-20 _40
=
naiatipntn) J.bwaa]one-wa1e
o
Reflected shofr wave radiation (1,)
E E
o.o4 -
.,
On?
6
E 0.02 (Each bs reDresents mem value
E o.ot 0 1700-1800hr 2400*0100hr'
0.01 0.25 0.20 0.15
no condensation -
0900-1000Itr no condensation
Total Snow Melt (M) ud Runoff (O)
computed melt) I
E 0.10
no convgctiotr or -
meltllcondensationmeltmelt t t t t l E Radiation melt (Mr) Convrction condensation 6 mett (Mce) --- Radiationmelt total (Hourlyconvection-cotrde$ation meltis addedto of subtacbdftom totalhouly radiationmeltto arnveattotalhouly Totalcomputedmelt
Computedmelt (net for period0900- 1800hr = l22ir-022in = I 00 in.)
o n
_o* 0 Net nighttime loss = 0.22 in
-0.05
1600 2000 2400 0400 0800 1200 1600 2000 2400 0400 0800 1200 Apil23-----l--
Apfl24
April 25
Figure L4.5 Hourly variation in principal heat fluxesto a snowpackfor a cloudyday.(After U.S.Army Corpsof Enginers.2l
284
CHAPTER14
SNOW HYDROLOGY
Solution a. Convectionmelt, 6 hr
D:KV(r-32) D : 2 x 0 . 7x 0 . 0 0 1 8x4 1 0x ( 5 0 - 3 2 ) : 0 . 5 0 i n .
(14.27)
melt,6 hr b. Condensation D:KrV(e"-6.11) D : 2 x 0 . 0 0 5 7 8x 1 0 x ( 1 2 . 1 9x 0 . 6 5- 6 . 1 1 ) : 0.21in.
(14.30)
. Radiation melt, 12 hr D o ( 1- 0 ' 7 5m ) D r 2: 0 ' 4 2 x ( 1 - 0 . 7 5x 1 ) : 0 ' 1 1
Dn:
(r4.34)
d. Rainfall melt. hourly M : P(r-'- 32)lt44Q, 14 : 10.03x 12 x (48 - 32)l/(144 x 0.97) : s.64
( 14.3s )
Thus,total melt is 0.86 in. rr
RUNOFFDETERMINATIONS 14.6 SNOWMELT Variousapproachesto runoff determinationfrom snowmelthavebeenfollowed.They rangefrom relatively simplecorrelation analysesthat completelyignore the physical snowmeltprocessto relatively sophisticatedmethodsusing physicalequations.Most techniquescan be consideredas basedon degree-daycorrelations,analysesofrecession curves, correlation analyses,physical equations,or various indexes.Each is discussedin turn.
Purposesof SnowmeltRunoff Estimates Snowmeltrunoff estimatesare extremelyimportant for many regionsof the United Statesand other countriesin (1) forecastingseasonalwater yields for a diversity of water supply purposes,(2) regulating rivers and storageworks, (3) implementing flood control programs, and (4) selectingdesign floods for particular watersheds. Maximum floods in many areas are often due to a combination of rainfall and snowmeltrunoTf.In effect, the determinationof snowmeltrunoff hasthe sameutility as the calculationof runoff from rainfall. In someareasit will, in fact, be the more important of the two.
SnowpackCondition The mannerin which runoff from eitherrainfall or snowmeltis affectedby conditions prevalentwithin the snowpackis of primary interestto a hydrologist.Variousviewson storagecharacteristicsof a snowpackhave been advanced.These range from the
14.6 SNOWMELTRUNOFFDETERMINATIONS
285
conceptthat a snowpackcan retain large amountsof liquid water to the hypothesis that snowpackstoragi is negligible.Thereis no universallyapplicablerelation, and it becomesimportant to baseany runoff considerationson a knowledgeof the character of a snowpackat the time of study.Winter runoff is relatedto a snowpack'scondition, whereasin the spring,onceactivemelt begins,little or no delayin the transportof melt or rainfall through the snowpackoccurs. For drainagebasins in mountainousareas,snowpackstorageeffects may be approximatedby subdividingthe watershedinto relatively uniform areas.Normally, this will be accomplishedby usingelevationzones.Snowpackat the lowestlevelsmay be conditionedto transmit readily rain or meltwater,whereasin higher elevationsa liquid water deficit may prevail. At uppermostelevations,the snowpackmay be very dry and cold and thus in a condition for the optimxm storageof water. The storage of the potentialof the watershedzonesmustbe basedon representativemeasurements snow depth,density,temperature,water equivalent,moisturecontent,and snowpack character.The snowpackiharacterrelatesto the physicalstructureofthe pack' Unfortunately, adequate1;r"usrrr"rof all thesefactors are not always availableor easily obtained. Estimatesof changesbetween sampling periods are usually indexed to readily observedmeteorologicvariables. The formulation of snowpackstorageand time delay characteristicscan be pack.In this case,storageis relateddirectly to fashionedby assuminga homogeneous the liquid water defici1and cold content of the pack. Time delay is a function of the inflow rate. It is consideredthat the snowpackstoragepotential must be entfuely satisfiedbeforerunoff begins.In reality this is not the case,but the assumptionpermits an analysisto be made. As melt proceeds,the storagepotential of any snowpack diminishes. Storageof a snowpackbeforerunoff commencesis consideredto be the sum of the equivalentwaterrequilementto raisethe temperatureof the snowpackto 0'C (cold contentlV")and the HqJidwater-holdingcapacityof the snowpack.If the cold content is given in inchesof water neededto bring a snowpacktemperatureto 0oC,it may be representedby2
* r = fw"T" fi
(14.38)
7] : the mean snowpacktemperaturebelow OoC lyo : the initial watei equivalentof the snowpackin inchesfor an assumed specificheatof ice of 0'5 The time L in hoursneededto raisethe snowpacktemperatureto OoCis thus given by
where
WoT,
", -- r 6 o 1 i + * 7
(14.3e)
whereI is the rainfall intensity(in./hr) andm is the rate of melt (in'/hr)' Storage requiredto meet the liquid water deficit of the snowpackis given by
tr:#(%+w.) ( - : the amount of water stored(in.) where ".1 T _ the percent deficiencyin liquid water of the snowpack JP
(14.40)
286
CHAPTER14
SNOW HYDROLOGY
The time in hours r, neededto fill the storageSyis given by
+ t': fi\wg+ W") 100(i m)
(r4.4r)
It has been specifiedthat the total storagepotential Soto be met prior to the runoff is given as (r4.42) Sr:W'"*Sy This is also known as "permanent" storage,sinceit is not availableto the runoff until the snowpackhasfinally melted.An additional storagecomponenttransitory storage S,is that waterstoredin the snowpackwhile movingthroughit to becomerunoff. Until initiation of runoff, the transitory storagein inchescan be expressedas
(i+m) ^ D __T__ ",:
(r4.43)
where D : the depth of the snowpack (ft) V : the rate of transmissionthrough the snowpack(ftlhr) The delay time of water in passingthrough the snowpackt' is thus a
t
D v
(r4.44)
for /, in hours.Assumingthat I4z"is very small comparedwith Wo,the depth of the snowpackis given by
O:%
(r4.4s)
P'
Then with p" the densityof the snowpack. wo (r4.46) nv the total waterS storedin the snowpack,in Beforethe runoff commences, inches,is givenby t'-
S:W"+++S,
(r4.47)
which can also be written / r
t
t = * ( 1 f t + f i d i. +" mr ,\ r )
(14.48)
The total time in hours that passesbefore runoff is producedis thus2 t:t"+tr+tt
z : -, f ! , - r -+ { 1 t = w'1160(i f + r, Too1,-, p,v)
(r4,49) (14.50)
term theonlysignificant theactiverunofffromthesnowpack, After establishing in Eq. 14.49is /,, andthis is usuallysmallcomparedwith the overallbasinlag and
14.6 SNOWMELTRUNOFFDETERMINATIONS
287
33 End rain
31 -:
30
'*2 =
)
Q
6 - ' E .Z
te
d o
- 2 6
24 23
0
6
12
18 24
30
36 42
48
54
60
66
72
78
Time(br) duringrainfall' Figure 14.6 Waterbalancein a snowpack can be neglected.With increasedsnowmelt and runoff, additional increments of water prevlouslywithheld by snowblockageto drainageoutlets and other factorsare A deep released.Adequatequantifi;ation of this cannotbe accomplishedat present'24 -5oC, could storeabout 4 in. of of temperature a mean having ft, snowpack,say 15 fiquid waterbeforethe onsetof runoff. Figure 14.6illustratesthe nature of the water balancein a snowpackduring a rainstorm. EXAMPLE 14.3 A core sampleof a snowpackproducesthe following information: air temperature, 10 ft; 68"F; relative humidity, 2b percent; snowpackdensity,0'2; snowpackdepth' snowpacktemperature,22"F. a. b. c. d.
What is the vapor pressureof the air? Will condensationon the snowpackoccur, basedon the vapor pressure? what is the cold content of one sq ft of surfacearea of the snowpack? Is the snowpackripe?
Solution
-
a. From AppendixTableA.2, eo: 233 mb for a relative humidity of 20 percenr,€o = 4'66 mb b. Condensationwill not occur sincethe air is unsaturated" c. Cold content
160 W": WsT"f
288
CHAPTER14 SNOW HYDROLOGY
: 22oF: -5.6oC Temperature W": 0.20x l2Ox 5.6/160: 0.84in. is belowfreezing. I I is not ripe sinceits temperature d. The snowpack
Indexes Hydrologic indexesare made up of hydrologicor meteorologicvariablesto describe their functioning. The index variable is more easily measuredor handier than the elementit represents.When mean fixed relations are known to exist betweenpoint measurementsand watershedvalues,indexescan be used to record both areal and temporal aspectsof basin values. Indexes serve to permit (1) readily obtainable observationsto depict hydrologicvariablesor processeswhich themselvescannotbe easily measured,and (2) simplificationof computationalmethodsby allowing individual observationsor groupsof observationsto replacewatershedvaluesin time and space.The adequacyof an index is basedon (1) the ability ofthe index to describe , (2) the randomvariability of the obseradequatelythe physicalprocessit represents is typical of actual conditions, point observation (3) the to which the degree vation, and basin means.2 point measurement the beiween (4) variability nature of the and Indexesmay be equationsor simplecoefficients,and variable or constant. The types of data requiredto make comprehensivethermal budgetstudiesare or part for wut"tJh"dt otherthan experimentalones.As normally unavailablein wh^ole a result, a hydrologistmust make the best use of information at hand. The most commonly availabledata aredaily maximum and minimum temperatures,humidity, of thesedata, andfew and wind velocity.Lessprevalentare continuousmeasurements stationsrecord solarradiation or the durationof sunshine.Hourly cloudinessdatacan sometimesbe obtainedfrom local airport weatherstations. A completelygeneralindex for reliably describingsnowmelt-runoff relations for all basinshasnot beenestablished.Most indexesincludecoefficientsvalid only for specifictopographic,meteorologic,hydrologic,and seasonalconditionsand aretherefore limitedlnipplicability to other watersheds.Table 14.1 shows some types of indexesthat havebeen used successfullyin snowmeltinvestigations. VARIABLES BUDGET THERMAL USEDTODESCRIBE TABLE14,1 SOMEINDEXES Thermalbudgetcomponent
Index
Absorbed short-waveradiatron
Duration of sunshinedata Diurnal temperaturerange Air temperaturefor heavy forestedareas For open areaslong-waveradiation shouldbe estimated (7" - T)V, where f is air temperature, T6 the snow surface temperatureor basetemperature,and Vthe wind speed (e" - e")V,whereeoand e, are vapor pressuresof air and snow surfaceot a basevalue, and V is wind speed
Long-waveradiation' Convectiveheat exchange Heat of condensation
" Figure 14.7 iilustrates an approximatelinear relation betweenmelt and long-waveradiation usedby the U.S, Army Corps of Engineers for index purposes.
14,6 SNOWMELTRUNOFFDETERMINATIONS
289
The snowmeltrunoff equationstatedin terms of thermal budgetindexesis
( 14.s1)
Y:a+>b$t where
I : a : b, : 4 :
the snowmeltrunoff aregressionconstant the regressioncoefficients individual indexes
Variousindexesusableto representthe termsof Eq. 14.51are selectedand a standard regressionanalysisperformedto determinea andb,.It shouldbe notedthat everyterm in the heatbudgetequationis not alwayssignificantfor a particular analysis,and thus the numberof Xr will vary for different basinsand conditions.A final melt equation 1.6 Long-wave radiation melt in forest or with low overcast sky in open Ma=
1Z-L n:{-r'1 ,,
toTf,- oT!) {referto Kelvin scale)
= 604x lA-1274-3.355
'cE
Long-wave radiation melt in open
n / -v.+
o
1L O
Ma= zoittlpl rcrf,-orlt = 604x IO-12 Tx- 3.355 = o.o29(Ta_ 32) (refer to Fahrenheit scale)
r l t l M, = long-waveradiationmelt, in./day P = ftee water content of snow (taken as 0.03 in this case)
-r.o
rA = absolute air temperature, K ?s = absolute snow surface temp = 273K
-2.0
o = stefm's constant = 0.813 x 10. lolmgley/min-K-a Io = mean daily air temperatue, F
1 A
10
20
30 40 60 50 Meanair temperature, Zo("F)
260
265
270 2'75 280 285 Absolutetemperature,Z1(K)
290
300
Figure 14.7 Long-wave radiation melt, with linear approximation. (After U.S. Army Corps of Engineers.2)
290
CHAPTER14 SNOW HYDROLOGY
t.6 Equationderivedfrom combineddataof 1954and1955 G + 0.0245(T ^ -'77) RO = 0.00238 D=0.90 r=0.95 = 0.36in. S"- . = 0.1| in'
t.4
r
t
'
l
l
/
1 n z}
Z
"r_"n .C
ts E U.6
,/
a
.l
\C
5 0.6
Mean ru noff 0.59
ri
"y-x
,/,
0.4 aa,
'24.
/ a o
0.2 a .
0
0.2
0.8 0.4 0.6 runoff(in.) Estimated
1.0
1.2
Figure 14.8 Observedversusestimatedrunoff for (X) 1954 and (O) 1955. RO = the daily generatedsnowmelt runoff (in.) depth over a snow-coveredarea; G : the daily net all-waveradiation absorbedby snow in the open (langleys); 7-u* : the daily maximum temperature for Boise ("F); r = the coefficient of correlation; D : the coefficient of determination;S, = the standarddeviation of observedrunoff (in.); Sr-, : the standarderror of the stimated runoff (in.). (After U.S. Army Corps of
Engineers.2) devekipedby the Corps of Engineers2for the partly forestedBoiseRiver basin above Twin Springs,Idaho, was
* - 77) Q : 0.00238G+ 0.0245(T^
(r4.s2)
where Q = the daily snowmeltrunoff (in.) over the snow-coveredarea G = an estimatedvalue of the daily all-waveradiation exchangein the open (langleys) ZLu, : the daily maximum temperatureat Boise (T) The equationis said to predict the daily snowmeltrunoff valueswithin 0.11 in. of observedvalues about 67 percent of the time. Figure 14.8 illustrates this relation. In attemptingto developsuitableindexesfor snowmelt,a hydrologistshouldseek the approachmost closely resemblingthe thermal budget of the area, within the limitations of availabledata.
14.6 SNOWMELT RUNOFFDETERMINATIONS 291
TemperatureIndexes The atmospherictemperatureis an extremelyusefulparameterin snowmeltdetermination. It reflectsthe extentof radiation and the vapor pressureof the air; it is also sensitiveto air motion. Frequently,it is the only adequatemeteorologicvariable regulady on hand, so widespreaduse has been made of degree-dayrelations in snowmeltcomputations. I A degreeday is defrnedas a deviation of 1ofrom a given datum temperature consistentlyover a 24-hr period. In snowmeltcomputations,the referencetemperature is usually 32"F. rf the mean daily temperatureis 43oF, for example,this is equivalentto 11 degreedays above 32'F.If the temperaturedoes not drop below freezingduring the24-hr period, there will be24 degreehr for eachdegreedeparture above32'F. In this examplethere would be 264 degreehr for the day of observation. Variouswaysof estimatingthe meantemperaturehaveenabledinvestigatorsto take severalapproaches. one methodis simplyto averagethe maximumandminimum daily temperatures.Basesother than 32"F are also used.Regardlessof the particular attackemployed,a degreehour or degreeday is an index to the amountof heatpresent for snowmeltor other purposesand has proved useful in point-snowmeltand runoff from snowmeltdeterminations. The standardpracticein developingsnowmeltrelationson the basisof temperature is to correlatedegreedaysor degreehours with the snowmeltor basin runoff. In somecases,other factorsare introducedto defineforest covereffectsand/or other influences.Another approachoften usedis to calculatea degree-dayfactor-the ratio of runoff or snowmeltto accumulateddegreedaysthat producedthe runoff or melt. 2.8 1 A
; 1 A o
B 6
r-z
34
38
42
46 50 s4 58 Mean daily temperature,Z('F)
62
66
Figure 14.9 Mean temperature index. The equations are applicable only for the range of temperatures shown in the diagram. (After U.S. Army Corps of Engineers.2)
292
CHAPTER14 SNOW HYDROLOGY
Unfortunately,the degree-dayfactor has been found to vary seasonallyand between basins;therefore, single representativevalues should be used with caution. Pointbasinsrangefrom 0.015to 0.20 in. per degree degree-dayfactorsfor snow-covered per day when melting occurs.Gartskastatesthat an averagepoint value of 0.05 can be used to representspring snowmelt, provided that caution is used. Linsley and othersstatethat basinmean degree-dayfactorsare usuallybetween0.06 and 0.15 in./degreedayunderconditionsof continuoussnowcoverand at meltingtemperatures. Figure 14.9illustratestemperatureindex equationsfor springtimesnowmeltfor clear and forestedareas.'
GeneralizedBasinSnowmeltEquations Extensivestudiesby the U.S. Army Corps of Engineersat variouslaboratoriesin the West have produced several general equationsfor snowmelt during (1) rain-free periods and(2) periodsof rain.2aWhen rain is falling, heattransferby convectionand condensationis of prime importance.Solarradiationis slight, andlong-waveradiation can readily be determinedfrom theoretical considerations.When rain-free periods prevail,both solarand terrestrialradiationbecomesignificantand may require direct evaluation. Convection and condensationare usually less critical during rainless intervals.The equationsare summarizedas follows:2 1. Equationsfor periodswith rainfall. a. For open (coverbelow 10 percent)or partly forested(coverfrom 10 to 60 percent)watersheds,
M : (0.029+ 0.0084fto+ 0.007P)(7"- 32) + 0.09 b. For heavilyforestedareas(over80 percentcover), M : (0.074 + 0.007P,)(7.- 32) + 0.0s
(14.s3) (14.54)
where M : the daily snowmelt(in./daY) P , : the rainfall intensity (in./day) T o : the temperatureof saturatedair at 10-ft level ("F) the averagewind velocity at 50-ft level (mph) ^t . -- the basin constant,which includesforest and topographic effects,and representsaverageexposureof the areato wind. Valuesof k decreasefrom about 1.0 for clear plains areasto about 0.2 for denseforests ,,
Equationsfor rain-free periods. a. For heavyforestedareas, ,
M:0'074(0.537'"+ O.47fi)
(14.ss)
b. For forestedareas(coverof 60-80 percent),
+ 0.7870+ 0.0297" M : k(0.0084u)(0'22r'" c. For partly forestedareas, M : k'(I - rx0.00401,)(l- a) + 0.78r) + F(0'029r) + k(0.0084o)(0.227'"
(14'56)
Q4.57) )
14.6 SNOWMELT RUNOFF DETERMINATIONS 293 d. For open areas, M : k'(0.00s084)(1- a) + (1 -'N)(0.0zIzT: - 0.84) + N(0.02gr') + k(0.00Sao)Q.227'" + 0.78Ti)
(14.58)
where M, a, k : as previouslydescribed T'" : the difference betweenthe t0-ft air and the snow surface("F) temperatures T'o: the difference between the 10-ft dew-point and snow-surfacetemperatures('F) I, : the observedor estimatedinsolation (langleys) a : the observedor estimatedmeansnowsurfacealbedo k' : the basin short-waveradiation melt factor (varies , from 0.9 to 1.1),which is relatedto mean exposureof open areascomparedto an unshielded horizontal surface F : the mean basin forest-canopy cover (decimal fraction) Ti = the difference between the cloud-base and snow-surf,ace ternperatures(oF) N : the estimatedcloud cover (decimalfraction) Note that the use of equationsof the type given must be related to the areal extent of the snowcoverif realistic values are to be obtained. Presentmethods of determiningthis are not totally adequate. EXAMPLE 14.4 Use Eq. 14.53to estimatethe snowmeltat an elevationof 3000 ft in a partly forested arcaif the rainfall intensity is 0.3 in./day, the wind velocity is 20 mph, and the temperatureof the saturatedair is 42'F. b. Rework your solution for a denseforest cover and a saturatedair temperatureof 53"F. Solution A , M : (0.029+ 0.0084ftr-r + A.007P)(7,- 32) + 0.09 M _ (0.029+ 0.0084x 0.5 x 20 + 0.007x 0.3)(42- 32) + 0.09 I'tl -
I.24 in.lday
b . M = (0<074+ 0.007P)(7"- 32) + 0.05 M * (0.074+ 0.007x 0.3x53- 32) + 0.05 M _ 1.65in./day r:
The Water Budget The waterbudgetcanbe usedto estimatethe snowmeltrunoff from a watershed.2 Such an approachhasparticularmerit for areaswherehydrometeorologicrecordsare short. Difflculty with the methodis the usuallack of satisfactorydatato quantify the various
294
CHAPTER14
SNOW HYDROI-OGY
componentsproperly. A hydrologicbudgetequationfor the earth's surface(Eq. 1.1) can be written R : P - Z - A S where
P : R: L : AS :
(14.se)
the grossPreciPitation the runoff the losses the changein storage
For snowmeltcomputationsthis equationis modified somewhat' Grossprecipiiation for a given period P is now definedasthe sum of precipitations in the form of snow P" and rain P,, ot P:P,1-P"
(14.60)
This may also be written as P:P,*P":P^+Li
(14.61)
where P,: net precipitation L; : intercePtionloss A further refinementyields P: where P,n,P",: L,i, L"i:
P-+ L,i+ P"nt L,;
(r4'.62)
net rainfall and snowfall, respectively the rain and snowinterception,respectively
Figure 14.10indicatesthe nature of snow interceptionby forestedareas'Additional information on interceptioncan be found in Chapter 3' The total lossI is L : L,i + L,i + L" * Q"L" : the evapotranspirationloss Q,^: the changein availablesoil moisture The storageterm AS is then given as AS : (l7z - Wr) t Q,
$4.63)
where
04.64)
whereWr,Wr:thefinalandinitialwaterequivalentsofthesnowpack' resPectivelY Qr : the ground and channelstorage Inserting uilo"r for p, L, and AS from Eqs. 14.62-14.64 in Eq. 14.59 gives - L" - Q"^ - (W, - Wt) - Q, R : P,n + L,i + P", + L"i - L"i - L,i and cancelingpositive and negativevaluesof L,i and lr; produces - Q, - L" R = P,n+ P", - (W, - Wt) - Q,-
(14.65) I (14.66) 1 )
14.6 SNOWMELTRUNOFFDETERMINATIONS
295
4-')
a
Curv e A
/
.MC
-<
1\MC
(2)
o
,SP&
6 J U
o
625 F B ^ ^ o z v
PPM tr
o
,%?r
o
FQ 1<
c /rt
ro l 0 a h
6,
PPR
,/,
RF (1)
-Curv B
tr RF
WFM
:l
(4)
WFM 5
./
-10 0
l0
Figure 14.10 Engineers.2)
20
30
40 50 60 Canopydensity(%)
70
80
90
100
Snowfall interception loss. (After U.S. Army Corps of
The expressionP", - (W, - Wr1representsthe snowmeltM; thercforc, R:P,,+M-Q"^-Qr-L"
(14.67)
If reliable estimatesof the terms in Eq. 14.67 canbe secured,the basin dischargeR is computable.
Elevation-BandProcedure Runoff from snowmelton a watershedcan be estimatedfrom calculationsof excess watermadeavailtble on a seriesof contributingareas(bands)at variouselevationsin the watershed.The practiceis as follows: divide the watershedinto severalsubareas or bands;estimatethe quantity of snowmelt,rainfall, and lossesgeneratedon each band during a prescribedinterval of time; and usethe weightedsum of thesecontributionsto provide an estimateof the excesswateravailablefor runoff. For eachband, it is assumedthat snowmelt, rainfall, and lossesare uniform over the band. The subareasare consideredto be either snow-coveredor snow-freeand melting or not
296
CHAPTER14 SNOW HYDROLOGY
melting. For eachband, snowmeltis computedusing equationsof the type preSented earlier,rainfall is estimatedbasedon expectationsor historic information, and losses are estimatedas describedin Chapterb3 through 5. Ohce theseestimateshavebeen made for each band, the following equation servesto provide a weighted value of excesswater availablefor runoff from the basin.
)
tte + Mt - Lt)jAt (14.68)
M _
)a, whereM : snowmelt water availablefor runoff (cmlday), Pr is the rainfall on the 'band, M, is the snowmeltfrom the band, I, is the subarealoss,A, is the size of the subarea,and n is the total number of bands. EXAMPLE 14.5 Given the data in columns 1-5 of Table I4.2, estimatethe amount of excesswater availablefor runoff from the watershedusingthe elevation-bandmethod(Eq. 14.68). 14,5 TABLE14.2 DATAFOREXAMPLE (1)
(2)
(3)
(4)
(5)
(6)
(7)
Elevation band no.
Subarea slze sq km
Snowmelt
Rainfall
Losses
( 3 )+ ( 4 ) + ( 5 )
(6) x (2)
cmid
cm/d
cm/d
0.90 1 .l 0 1.80 1.90 2.20 "2.40
0.40 0.50 0.70 0.60 0.30 0.10
5 6
230 ))a 289 213 193 167
Totals
1316
I
z J A
0.02 0.40 0.60 0.70 0.3s 0.00
1,.32 2.00 3.10 3.20 2.85 2.50
303.60 448.00 895.90 681.60 5s0.05 411.50 3296.65
Solution. The solution for the numeratoris the sum bf the productsgiven in Table 14.2;the solutionfor the denominatoris the sum column 7 of spreadsheet of the subareassiven in column 2 of the table. The excess-water availabiefor runoff : 3269.6511316: 2.5I cmlday' II
HydrographRecessions Recessioncurveshavebeen discussedin Chapter 11 and take the generalform Q: '
where
, : -
Q : the dischargeat time t Qo : the initial rate of flow k:__afecessionconstant
Qoe-o'
(r4.6e)
14.6 SNOWMELTRUNOFFDETERMINATIONS
297
o u
€
Time
of a snowmelthydrograph' Figure 14.11 Separation runoff Studiesof daily streamflowby hydrographspermit evaluationof the amountof of the separation of one derived from snowmelt.The t""hniqutnred is essentially procethe illustrates daily hydrographs.Figure 14.11(not to scaleandoversimplified) dure. Assumeihat the first, second,and succeedingpeaks,respectively,fit snowmdlt point A the days.If the ultimate recessioncurve is extendedbackwardin time, at a from recessions between area recessioncurveft'om Hydrograph2 will intersectit. The Day to attributed melt is the Hydrograph1 and HydiogtipiZ (showncross-hatched) their determine to be studied 1. In like -unn"r, a seriesofsnowmelt hydrographscan to individual melt components.By observingsuchhydrographfeaturesas th€ height rate and volumetric recession, peak X, the height io trough i, andthe form of the of this forecastsof snowmeltrunoff can be made.A more comprehensivetreatment subjectcan be found in Ref. 25.
HydrographSYnthesis The synthesesofrunoffhydrographsassociatedwith snowhydrologyareoftwo-types' distribution flow of development the is kind second The first is a short-termforecist. for a comPletemelt seasonor a ' forecastingis very helpful in prel controls,while the synthesisof part lating designfloods. To forecasta snowfieldand streamflowneedbe k parameters ing, it is necessaryto havethe reliablepredictionof variousmeteorological canbe parameters historic Known conditions. initial in additionto a knowledgeof the satparameters generated or assumed whereas flows usedfor reconstructinghistoric hydrometeorologic common some displays 14.12 Figure isfy designflood syntlieses. data. In snowmelthydrographsyntheses,severalfactors(not of great conceln where snowonly rainfall exists)mustbe "utlfully considered.First, a drainagebasin with of the extent areal the since system, homogeneous a as cover cannot be accepted
298
SNOW HYDROLOGY
CHAPTER14
6 5 ^ ! 9 > >
v
B 5 (h
Short-waveradiationngtesj.^,-- .-
AbsorbedShort-wavefadiation
U
i hsolation obsened by USWB. of Boise Cily' lD 2. Basin albedo of snow surface ^ - = , | L Absorbed shon-wa\e ralialion compukd b) lomula 1ab. ll(
o
9.1 ' H 9 a t o
6
-8 Air Temperature Notes: 1. Veni;al bus show daily range of surface air temperature (max and min) at Idaho City' ID 2. aonnected points indicate daily (0700 hr) 700 mb temperature at Boise CitY, ID
90
t--
80 70 60
s
50 o 40
B
30 a 120
0.5
lrn
u 4 0.4 9 b
0.3
q
0.2
gB d
g > 7
E
0
1
0
2 0 MaY1955
31
20 10 June1955
Figurel4.l2Hydrometeorologicdataandcomputationofwatergenerated. (AfterU.S.Army Corpsof Engineers.') the contributblanket is highly important. where only snowmeltflows are developed, If rainfall ing areaneel not U"itr" entiredrainage-only that portion with-snowcover' while bare areas from o"-"o., during the snowcoverperiod, contributionscan come may cases in such losses other expansesmay producecombinedrunoff' The natureof differ greatly for nonsnowoverlayedand coveredlocations' subThe altitude is an exceedinglypertinent factor in the hydrology of tracts reducgeneral a to due jected to snowfall.Ratesof .no*tnitid"crease with elevation tion in temperaturewith height. orographic effects and the temperature-elevation snowcover relations tend to raise the amount of precipitation with altitude' Greater a result' As rates' melt depth occursbecauseof increasedprecipitation and reduced is apsnowline the as the basin-wide melt and cover-area increase with height
14.6 SNOWMELTRUNOFFDETERMINATIONS
299
proached,then diminish with elevationover the higher placesnormally completely snowcovereduntil late in the season.A snowpackexhibitsanotherimportant trait in relation to rainstorms.In the spring, relatively little runoff occurs from snow-free regionscomparedwith that from a snowfieldfor moderaterainfalls. During very cold weather,the situationduring heavyrains is often reversed,sincea dry snowpackcan retain significantamountsof water. Two basic approachesintroduce elevation effects into procedures for hydrographsynthesis.2 The first dividesthe basin into a seriesof elevationzoneswheie the snowdepth,precipitationlosses,and melt are assumeduniform. A secondmethod considersthe watershedas a unit, so adjustmentsare made to accountfor the areal extentof the snowcover,varying melt rates,precipitation, and other factors. To synthesizea snowmelthydrograph,information on the precipitation losses, snowmelt, and time distribution of the runoff are needed.Snowmelt is generally estimatedby index methodsfor forecasting,but in design flood synthesisthe heat budget approach,is the most used. Precipitation is determinedfrom gaugingsand historicor generateddata.Lossesare definid in two wayswheresnowmeltis involved. For rain-on-snowhydrographsall the water is considereda lossif delayedvery long in reaching a stream. This is basically the concept of direct runoff employed in rainstormhydrographanalysis.For hydrographsderivedprincipally from snowmelt, only that part of the waterwhich becomesevapotranspiration,or deeppercolation,or permanentlyretained in the snowpackis consideredto be lost. Assessingthe time distribution of runoff from snow-coveredareasis commonly done with unit hydrographsor storagerouting techniques.For rain-on-snow events,normal rainfall-type unit hydrographsare applied;for the distribution of strictly snowmeltexcess,special long-tailedunit graphsare employed.Storagerouting techniquesare widely exeicised to synthesizespring snowmelthydrographs,perhapsdividing them into severalcomponentsand different representativestoragetimes. The time distribution of snowmehrunoff differs from that of rainstormsdue mainly to large contrastsin the ratesof runoff generation.For flood flows associated with rainfall only, direct runoff is the prime concern,and time distribution of base flow is only approximated.Big errorsin estimatesof baseflow arenot generallyof any practical significance where major rainstorm floods occur. In rainstorm flows, infiltrated water is treated as part of the base flow component and little effort is directedtoward determiningits time distribution when it appearsas runoff. In using the unirhydrograph approachto estimatesnowmelthydrographs,it is customaryto separatethe surfaceand subsurfacecomponentsand route them independently. Storagerouting has beenusedextensivelyfor routing floodsthrough reservoirs or river reaches.It is also applicablein preparingrunoff hydrographs.In snowmelt runoff estimates,the rainfall and meltwaterare treatedasinputs to be routedthrough the basin, using storagetimes selectedfrom the hydrologic characteristicsof the watershed.Two basichydrologicrouting approachesare relatedto the assumptionof ( 1) reservoir-typestorageor (2) storagethat is a function of inflow and outflow.These methodswere treatedin depthin Chapter13. Storagerouting techniquesthat separaterunoff into surfaceand groundwater components,assigndifferent empirically derivedstoragetimesto each,and thenroute them separatelyhavebeen employed.26 An additional systemusesa multiple storage,
300
CHAPTER14
SNOW HYDROLOGY 120
,:
Rn
o 6 U il
Three 6-hr stages
One 18-hr stage
Time (hr)
storagerouting' Figure 14.13 Example of multiple-stagereservoir-type storageroutreservoir-type iti. ngur" illustratesih" ut" of multiple-storage to unit analogous ttrunn". u in runoff ing for"evaluatingtime distribution of Engineers'2) of Corps tiOrog.uptr. (Af1erU.S. Army
is routed throrrgh two or reservoir-type storagescheduling.2In this method inflow suchan approach'Any more stagesof storageru"""rtiu!fy. Figure 14.13 illustrates the storagetime and.the desiredtravel time can be obtainedby properly selecting varied to reflectvarious tf st4ges'Retentiontimes betweenstepsmay alsobe the. use of single-stage drologic characteristics.clark has suggestedthat to be^^simplified.27 rfter translatirrgirp", in time permits Jo=mputations hydrographshas runoff e most practicJd method for synthesizingsnowmelt primarily in differs graphs ,,nir hrrrrrnoranhThe characterof snowmeltunit 12' chapter in plots' As disclssed time base length fiom that of rainstorm unit In events' storm single isolated rainstorm unit hydrograph, ort"n are derived from snowmeltrunoff,ratesofwaterexcessaresmallandapproximate$continuous'Asa result, the use of S-hydrographsis indicated'2 since it allows (1) adiustThe S-hydrogrupfrri"titoA has considerableutility, generationrates,(2) adjusting ments to the derivedoni, hydrographfor nonuniform of the areaunder iirn" p"rtoo to; d;ired interval, (3) ready adjustments il;;;t; veragingseveralhydrographsto get a unit
;?ilt1,"1*T,::t1il1""t"1':*"ff rydiographmethodin adjustingfor nonuniform generationrates of water excess' Onceap"r""nrug"S-hyatogtupftisderived'aunithydrographofanydesired are of the S-hydrograph periodcanbe obtaineias indlcatedm nig. 14.15.ordinates
312
HYDROLOGY ANDSMALLWATERSHED 15 URBAN CHAPTER Both categoriesof peakflow determinationhavehad wide application;however, two relatively major difficulties are normally encounteredin applyingthe techniques. First, the rainfall-runoff formulas,suchas the rational formula, aie difficult to apply unless the return periods for rainfall and runoff are assumedto be equal. Also, estimatesof coefficientsrequiredby theseformulas are subjectiveand havereceived considerablecriticism. The empiric and correlativemethodsare limited in application becausethey are derivedfrom localizeddata and are not valid when extrapolatedto otherregions. The most fundamentalpeak flow formulas and empiric-correlativQmethods, ilue to their simplicity,persistin dominatingthe urban designscene,and severalof the most popular forms are briefly describedto acquaintthe reader with methodsand assu-ption*. Urban runoff simulationtechniquesare describedin Chapter25.
RationalFormula The rational formula for estimatingpeak runoff rates was introducedin the United Sincethen it has becomethe most widely used Statesby Emil Kuichlingin 1889.18 method for designingdrainagefacilities for small urban and rural watersheds.Peak flow is found from QO: CIA
(1s.1)
where Qo: the peak runoff rate (cfs) C _ the runoff coefficient(assumedto be dimensionless) I _ theaveragerainfall intensity(in./hr), for a stormwith a durationequal to a critical period of time /" t" : the time of concentration(seeChapter Ii) A : the size of the drainagearea (acres) cI : the averagenet rain intensity (in./hr) for a storm with duratiofl: t, The runoff coefficientcan be assumedto be dimensionlessbecause1.0 acre-in./hr is equivalentto 1.008 ft3lsec.Typical C valuesfor stormsof 5-10-year return periods are providedin Table 15.1. The rationale for the method lies in the conceptthat application of a steady, uniform rainfall intensity will causerunoff to reachits maximum rate when all parts of the watershedare contributingto the outflow at the point of design.That condition is met after the elapsedtime t", the time of concentration,which usually is taken as the time for a waveto flow from the most remotepart of the watershed.At this time, the runoff rate matchesthe net rain rate. Figure 15.1 graphically illustrates the relation. The IDF curve is the rainfall intensity-duration-frequencyrelation for the areaandthe peakintensityofthe runoff is Q/A: 4, which is proportional to the value of 1 defined at t". The constantof profottionatity is thus the runoff coefficient,C : (QIA)lL Note that QIA is a point value and that the relation, as it stands,yields nothing of the nature of the rest of the hydrograph. The definition chosenfor /" can adverselyaffect a designusing the rational formula. If the averagechannelvelocity is usedto estimatethe travel time from the most remote part of the watershed(a common assumption),the resulting design
FORURBANWATERSHEDS 313 15.2 PEAKFLOWFOHMULAS TABLE15,1 ryPICALC COEFFICIENTS FOR5TO 1O-YEAR FREQUENCY DESIGN Descriptionof area Business Downtown areas ' Neighborhoodareas Residential Single-family areas Multiunits, detached Multiunits, attached Residential(suburban) Apartment dwelling areas Industrial Light areas Heavy areas Parks,cemeteries Playgrounds Railroad yard areas Unimproved areas Streets Asphaltic Concrete Brick Drives and walks Roofs Lawns; sandy soil: Flat,2Vo Avenge,2-7Vo Steep,TVo Lawns; heavy soil: Flat,2Vo Average,2-7Vo Steep,TVo
Q
Flunoffcoefficients
0.70-0.95 0.50-0.70 0.30-0.50 0.40-0.60 0.60-0.75 0.25-0.40 0.50-0.70 0.50-0.80 0.60-0.90 0.10-0.25 0.20-0.35 0.20-0.40 0.10-0.30 0.70-0.95 0.80-0.95 0.70-0.85 0.75-0.85 0.75-0.9s 0.05-0.10 0.10-0.15 0.15-0.20 0.13-0.17 o.r8-0.22 0.25-0.35
H
o
Time (min)
Figure L5.1 Rainfall-runoff relation for the rational method.
314
CHAPTER15
URBANAND SMALLWATERSHEDHYDROLOGY
dischargecould be less than that which might actually occur during the life of the project. The reason is that wave travel time through the watershedis faster than averagedischargevelocity (seeSection 13.1).As a result of using the slowervelocity I{ the peak time (/.) is overestimated,the resultingintensityl from IDF curvesis too small, and the rational flow rate p is underestimated. Rational Method Applications Most applications of the rationalformulain determining peak flow rates utilize the following steps:, 1. Estimatethe time of concentrationof the drainagearea. 2. Estimatethe runoff coefficient,Table 15.1. 3. Selecta return period T, and find the intensity of rain that will be equaled or exceeded,on the average,once every I years.To produceequilibrium flows, this design storm must have a locally derived IDF curve such as Fig. 27.I3 or Fig. 15.2usinga rainfall durationequalto thetime of concentration. . 4. Determinethe desiredpeak flow Q,from Eq. 15.1. 5. Somedesignsituationsproducelargerpeak flowsif designstormintensities for durationslessthan /" are used.Substitutingintensitiesfor durationsless than t" is justified only if the contributingarea term in Eq. 15.1 is also reducedto accommodatethe shortenedstorm duration. One of the principal assumptionsof the rational method is that the predicted peak dischargehas the same'returnperiod as the rainfall IDF relation used in the IDF curves for storms in vicinitv of example site
-
a
4
E
"
0 5 10 15 20
30
40
50
60
70
80
90
100 110 r20
130
Time (min)
Figure 15.2 Intensity-duration-frequency curvesusedin Example 15.1.
15.2 PEAKFLOWFORMULASFORURBANWATERSHEDS 315
prediction.Another assumption,and onethat hasreceivedclosescrutinyby investigais the constancyof the runoff coefficient during the progressof individual tors,re'2o stormsand also from storm to storm. The"coefficientis usually selectedfrom a list basedon the degreeof imperviousnessand infiltration capacityof the drainagesurface. BecauseC : I,.rf I,the coefficientmust vary if it is to accountfor antecedent moisture,nonuniform rainfall, and the numerousconditions that causeabstractions and attenuationof flood-producingrainfalls. In practice,a composite,weightedaveragerunoff coefficientis computedfor the various surfaceconditions.Times of concentration are determinedfrom the hydraulic characteristicsof the principal flow path, which typically is divided into two parts, overland flow and flow in defined channels;the times of flow in eachsegmentare addedto obtain /". Another assumptionwith the rational method is that the equation is most applicableto antecedentmoisture conditions that exist for frequent storms,in the rangeof the 2- to 10-yrrecurrenceinterval,representativeof stormstraditionally used for design of residential storm drain systems.Becausemore severe,less frequent stormsoften have wetter antecedentmoisture conditions,the rational coefficient is increasedby multiplying it by a frequencyfactor.The commonly usedmultipliers for lessfrequent stormsare: Returnperiod(yrs)
2-to 25 )U
100
Multiplier
1.0 1.1 t.2 1.25
EXAMPLE 15.I Usethe rational.methodto find the 10-yearand 5O-yeardesignrunoff ratesfor the area shownin Fig. 15.3.The IDF rainfall curvesshownin Fig. 15.2arc applicable. Solution 1. Time of concentration: t,:tt*tz:15+5:20min
At = 3 acres
cr = o'3
tr = Az= Cz = tz =
15min 4acres o'7 5min
Figure 15.3 Hypotheticaldrainagesystem for Example15.1.
316
HYDROLOGY CHAPTER15 URBANAND SMALLWATERSHED
2. Runoff coefflcient: c : [(3 x 0.3) + (4 x 0.7)]lQ + 4) :0.53 for 10-yrevent C : 1.2(0.53): 0.64 for 50-yr event 3. Rainfall intensity-from Fig. 15.2: Irc : 4'2 in'/ht 1so: 5'3 in'/hr 4. Designpeak runoff: Q r c : C I A : 0 ' 5 3x 4 ' 2 x 7 Q s o : C I A : 0 ' 6 4x 5 ' 3 x 7
16 cfs 24 cfs rl
Rational Method Discussion The runoff coefficientin the rational formula is dependenton the soil type, antecedentmoisturecondition, recurrenceinterval, land use, slope, amount of urban development,rainfall intensity, surface and channel roughness,and durationof storm.Tablesand graphsgenerallyallow determinationof C from only two or three of thesefactors.Nomographsand regressionequationscan providerelationsamongmore factors.One suchrelation, applicableonly in the region for which it was derived,is2r 2(0.001CN1 48)0 ts-o't{(P+ I)/zfo j e1-s0 C : j .Z(t1-i)CN3To05[(0.01CN)o
n5.2) where CN : SCScurvenumber(Chapter4) T : recurrenceinterval ( years)
s:
averageland slope (7o) I : rain intensity(in./hr) P : percentimperviousness The rational formula is a simplemodelto expressa complexhydrologicsystem.Yet the methodcontinuesto be usedin practicewith resultsimplying acceptanceby designers, officials, and the public. The methodis easyto apply and givesconsistentresults. From the standpointof planning,for example,the methoddemonstratesin clearterms the effectsof development:runoff from developedsurfacesincreasesbecausetimes of concentrationdecreaseand runoff coefficientsincrease. For storm drainagesystems,the designeris normally askedto estimatethe peak flow rate that might be equalledor exceededat leastonce in a given numberof years (describedas the frequency- see Section 10.4). For designsusing the rational formula, the frequencyof the peak runoff eventis assumedequal to the frequencyof the rain event(an eventbeing deflnedas somerain depthin a given duration).Studies Figure 15.4 showscumulativelog-normal probabilhaveexploredthis assumption.z2 ity functions (Chapter 26) fitted to observationsof rainfall and runoff on a 47-acre area in Baltimore, Maryland, with an averageSurfaceimperviousnessof 0.44. The data are partial seriesfitted independentlyto the observedrainfall sequenceand the observedrunoff sequence.Thus the largestrunoff doesnot necessarilycorrespondto the largestranked rainfall, and a similar lack of correspondencebetweenany runoff
FORURBANWATERSHEDS 317 15,2 PEAKFLOWFORMULAS Percentage of sample values equal to or greater than indicated value
99. 5 9 9 9 8
95 90
80 7060504030
20
10
Rainfall frequency curve (rr = 7.5 min)
o
l"
!2 k
..t
y" o
tt
't'2'
5
2
10.s
./
o
" i/
o-' oo
2
o
-
q
a
1 0.9 0.8 0.7 0.6
a
a o
\
Peak runoff ftequency curve
0.5 cd
0.4 0.3
5 1020 Recurrenceinterval (Years)
Figure 15.4 Distributions of recorded rainfall and runoff' (After Schaake.22)
and the rainfall that producedit holds for the ranked position of the observationsin In Fig. 15.4, the 5-yearrainfall frequencyof the arraysof the two i"putut" sequences. io a runofffrequencyof4'0 cfs/acre;theratio indicatesa runoff 6.5in./hr corresponds coefficientof approximately0.6. Although the two sequencesare eachcloselylognormal, they tend to converge,which suggeststhat the runoff coefficient increases slightly with more intense,iess frequent storms.In the designrange,however,the t"*ttr tend to support the assumptionof the rational method that the recurrence interval of the runoff equalsthe reiurrence interval of the rainfall. It shouldbe noted that the rainfall distributionsin Figs. 15.1and 15.4havesimilarproperties.All IDF curvesare drawnthrough the averagerainfall intensitiesderivedfrom many different stormsof record; any single IDF curve dses not representthe progressof a single storm.For lack of historicalrunoff data, the designerturns to the rational methodto construct from the rainfall history what amounts to a runoff intensity-durationrelation. frequency The most critical (highestpeak) runoff eventis often assumedto be causedby a storm havinga duration -qual to the time of concentrationof the watershed'If the rainfall IDF curve is steepin the designrange,severaldurationsshouldbe testedfor the given frequency to assurethat no other storm of equal probability producesa higlier peak runoff iate. Most applicationsof the rational method do not includethis testbecausethe assumptionthai ihe peak occursat /" is commensuratewith the other inherent assumptions.
31 8
CHAPTER15
URBANAND SMALLWATERSHEDHYDROLOGY
The rational method is used in the designof urban storm drainagesystems servingareasup to six hundredacresin size.For areaslarger than I mir, liydrograph or other techniquesare generally warranted.Considerabl" 3udg-"nt is riquir-eAin selectingboth the runoff coefficientsand times of concentratloi. a common procedure is to selectcoefficientsand assumethat they remain constantthroughout the storm.As the designproceedsfrom point to point downstream,a compositeweighted C factoris computedfor the drainageareaaboveeachpoint. The time of concentration is composedof an inlet time (the overlandand any channelflow timesto the first inlet) plus the accumulatedtime of flow in the systemto the point of design. Figure 15.5 is an exampleof a designaid for prJdicting overland flow times. calculation of flow time in stormdrainscan readily be estimatid knowing the type of pipe, slope,size,and discharge.23 Generally,the pipe is assumedto flow full foi this calculation.(see Fig. 15.6.) Nomographsalso are availableto solvethe Manning equationfor flow in ditchesand gutters.The estimationof inlet time is frequentl| basedsolelyon judgment; reportedvaluesvary from 5 to 30 min. Denselyaevitopei areaswith impervioustractsimmediatelyadjacentto the inlet might be assignedinlet periods of 5 min, but a minimum value of 10-20 min is more uiual.
d o
10n "-"
4U
Fo F
Figure 15.5 Surfaceflow time curves.(After FederalAviation Agency.23)
15.2 PEAKFLOWFORMUI.AS FORURBANWATERSHEDS 3.I9 1,500
2,400 2,000
1,000 800
1,500
600 500 400
1,000 800
0.7
s00
0.9
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200
300 200
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100 80
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6
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: 8 o
0.00002 o h o
0 . 1 3 0.2
0.00001 0.000008 0.000006 0.000005 0.000004
Figure 15.6 Flow in pipes (Manning's formula); (After Ref. 24.)
Most designersapplyingthis methoddo not usethe time of concentrationin its strictestsense;rather, the largestsum of inlet time plus travel time in the storm drain systemis taken as the time of concentration.Caution is required in app$ing the method' Peak dischargeis not the summationof the individual dischapges, b"Jarr* peaks from subareasoccur at different times. The runoff from subareasshould be recheckedfor eacharea under consideration.The averageintensity / is that for the time of concentration of the total area drained. While I decreaiesas the design ploceeds downstream,the size of the contributing area increasesand normally e
320
CHAPTER15
URBANAND SMALLWATERSHEDHYDROLOGY
increasescontinuously.It shouldbe noted that the designat eachpoint downstream is a new solutionof the rational method.The only direct relation from point to point derivesfrom the meansfor determiningan incrementof time to be addedfor a new time of concentration.The effect is to provide an equal level of protection (i.e., an equal frequencyof surcharging)at all points in the system.Example 15.2 is reproduced from standarddesignreferencesto illustrate the application of the rational method to an urban sYstem.2a EXAMPLE 15.2 Basedon the storm sewerarrangementof Fig. 15.7a,determinethe outfall discharge. Assume that C : 0.3 for residentialareasand C = 0.6 for businesstracts. Use a 5-yearfrequencyrainfall from Fig. 15.7b andassumea minimum 20-min inlet time. Solution. The principal factors in the designare listed in Table 15.2' Additional columns can be provided to list elevationsof manhole inverts, sewer inverts, and ground elevations.This information is helpful in checkingdesigns use in drawingfinal designplans.(see Table 15.3.) lI and for subsequent 'orational"in that thepeak Modified Rational Method Therationalmethodis truly flow rate is simply set equal to the net rain rate after sufflcient time occurs for the entire watershedto contributerunoff. This resultsfor any storm equallingor exceed-
design,requiring volume of runoff as well as peak flow rates. OF COLUMNHEADINGSIN TABLE15.3 TABLE 15.2 DEFINITION Column
1 4 5 6 7 8 9 10 t1 T2 t3 14 15 t6 t7
Comment Line being investigated Inlet or manholebeing investigated Length of the line Subareaof the inlet Accumulatedsubareas Value of the concentrationtime for the area draining into the inlet Travel time in the pipe line WeightedC for the areabeing drained Rainfall intensitybasedon time of concentrationand a 5-yearfrequencyculve Uqitrunoff q: CI Accumulatedrunoff that must be carried by line Slopeof line Size of pipe Pipe capacity Velocity in full pipe Actual velocity in pipe
15.2 PEAK FLOW FORMULASFOR URBANWATERSHEDS
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I u\-q
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e
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l
Legend
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MH1-1 r ,\e / --
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contourline - areaoutline Drainage
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10-yr averagefrequencY
30
r
!
l
Ij-'ff:::i::::?
K:---t.-
40
50 60 70 80 Duration(min)
90
100 110 120
Figure 15.7 Sample storm drainage problem: (a) typical storm rainfall seier design plan anO (U) intensity-duration-frequency (After 24.) Ref' Iowa. for Davenport, curves
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WATERSHEDS323 FORURBAN 15,2 PEAKFLOWFORMULAS In the modified rational method.,a full hydrographis developqdrather than simply estimating the peak flow rate, using the following reasoning.If the storm duration exceedslhetime of concentration,the runoff rate would dse to the rational formula peakvalue,then stayconstantuntil net rain ceases.At that point, runoffrates would decreaseto zero as excessrain is releasedfrom the basin.Ifthe rainfall-excess releasetime (seeChapter 11) is equal to the time of concentration,the hydrograph : t",femaining flat would havean approximatetrapezoldshaperising to the peak al t D I t". until/: therainduration,D,andthenfallingalongastraightlineuntilt: rational modified the incorporate hydrology urban packages for Many software method for hydrographanalysis.the method is approximateand shouldnot be applied to watershedsover 50 acresin size.
SCSTR-55Method The U.S. Soil ConservationServicedevelopedproceduresfor estimatingrunoff volThey are known collectively as ume and peak ratei of dischargefrom urban areas.zs TR-55und indiuidually as thegraphical method,chart method, andtabular method. The threemethodsadjustrural proceduresin NEH-426to urbanconditionsby increasing the curve number CN foi impervious areas and reducing the lag time /1 for and channel improvements.Allowances are also made for various imlperviousness watershedshapes,slopes,and times of concentration.The SCS designedthe first two methodsto be usedfor estimatingpeak flows,and the third for synthesizingcomplete hydrographs.The tabular method and chart method (usedfor small watershedsup to aredescribedhereto help explainthe evolution 2000 acres)wererevisedin 1986,21but of the methods.All three were developedfor use with 24-hr storms'Use with other storm durationsis not advised. watersheds,up to 20 mr2 The graphicalmethodwasdevelopedfor homogeneous by the runoff curve represented in size, on which the land use and soil type may be a third variable is simply number. As shownin Chapter4, the runoff curve number in a graph of rainfall versusrunoff. Tie SCS peak dischargegraph shown in Fig. 15,8 is limited to applications (see where only the peak flow rate ii Oisired for 24 hr, Type-II storm distributions expethunderstorm the 24-hr of Chapter f 6l. A Type-II storm distributionis typical rienced in all staiei exceptthe Pacific Coaststates.Figure 15.8 was developedfrom numerousapplicationsofine SCSTR-20 eventsimulationmodeldescribedin Chapter 24. To apply Fig. 15.8, the watershedtime of concentrationin hours is enteredinto net ttre grapir'to prJdu." the peak dischargerate in cfs/mi2of watershedper inch of gross 24-hr the from rain during tk Z+-nr period. The 24-hr net rain is estimated amountusingthe scS curve number approachdescribedin chapter 4. the effectof urbanizationcanbe estimatedusingFig. 15.9.Oncethe composite curvenumber(CN) hasbeenestimatedfor the previousarea,a modifiedcurvenumber is determinedby enteringFig. 15.9 with the value of the percentimperviousareaon the modified watershed,r"ading vertically to the curve correspondingto the CN for the pervious watershed,and then reading horizontally to determinethe modified compositerunoff curve numberthat would be usedin determiningthe net rain depth for the urbanizedwatershed' Useof the 1975graphicalmethodis restrictedby the assumptionsof the tabular This method.The methodii a -ompositeof resultsfor one caseof the tabularmethod.
324
CHAPTER15
URBANNNO SUNU-WATERSHEDHYDROLOGY
'e 70n Ei 500 qF 400 E
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100 0.1
0.2
0.3 0.4 0.5 0.7 1.0 Time of concentration(hr)
2.5
3.0 4.0 5.0
(cfs/mi2lin.) Figure 15.8 Peakdischarge of runoff versustime of con(AfterU.S.SoilConsercentration /. for 24-hr,Type-IIstormdistribution. vationService.25) restricts its applicationsto runoff volumes greaterthan about 1.5 in. (if the curve numberis lessthan 60). Time of concentrationshouldrangebetween0.1 and 2.0hr, and the initial abstractionshould not exceedabout 25 percent of the precipitation. The chart method allows determinationof peak flows for 24-hr Type-II storms over watershedshaving a fixed length/width relation and no ponding areas.Three chartsareusedfor flat, moderate,or steepslopesof approximatelyI,4, or 16percent. Tablesof adjustmentsfor intermediateslopesare provided in the technicalrelease. Severalmicrocomputersoftwarepackagesfor urbanhydrologyhavebeendeveloped.28 Over two-thirdsarebasedon SCSprocedures, but cautionshouldbe applied
80
z
L) e .E
7n
u 6 0
0
l0
20
40 60 70 50 Connected imperviousarea(7o)
30
80
90
100
Figure 15.9 Percentage of impervious areas versus composite CNs for given pervious area CNs. (After U.S. Soil Conservation Service.2s)
WATERSHEDS325 FORURBAN 15,2 PEAKFLOWFORMULAS in assumingthat the commercialprogramsfully imitate TR-55 or other SCS handbook methods.An ideal TR-55 packagewould includeall three methods,would carry SCS endorsement,would stateall assumptionsand limitations, and would incorpopercentageof rate all SiS adjustmentsfor peak coefflcient,percentimperviousness, channelimproved,pondingor swampyareas,length/widthratio variations,and slope. Its use shoutAako be cautionedfor other than 24-hr stormshaving a Type-II SCS distribution. Packagesnot adheringto these limitations would not be qualified as TR-55 procedures. A significantproblem in someof the commercial softwarepackagesis the use of a trianlular-shaped unit hydrographfor convolutionto producehydrographsfor stormsof-various durations.The SCS used a triangular shapeto conceptualizethe peak flow rate of a curvilinear unit hydrograph,but has never endorseduse of other than either the curvilinear shapediscussedin Section I2.5 ot the tabulatedhydrg; graphsgiven in the TR-55 manual.For further reading,the SCS publisheda guide2e for the useof the 1975TR-55 intendedto clarify proceduresin the original technical release.
PrevailingSCSTR-55Method for use. ratherthan the 1975version,is recommended The 1986editionof TR-55,27 . It incorporatesseveralyearsof resultsof researchand experienceswith the original edition. The revisionsinclude the following:
1. Three additional rain distributions(seeFig. 16'17). ,, Expansionof the chapteron urban runoff curve numbers' 3. A procedurefor calculatingtravel times of sheetflow' 4. Deletion of the chart method.
Modifications to the graphical peak dischargemethod and tabular hydrographmethod. 6. TR-55 computerProgram.
Ratherthanrelyingtotally on Fig 15.9,the newTR-55usesTablet5.4 andFig.15'10 to provide urban runoff curve numbersfor certain instancesindicated in the table' For the new graphicalmethod,an urban curvenumberand the 24-hr designrain depth are estimated,ih"n un initial abstraction1, is determinedfrom the SCSrunoff equation(Chapter4) or from Table 15.5.The peak flow is found from linear interpolation of the curvesin Figs. 15.ll,15.I2, !5.13, or 15.I4, dependingon the rainfall distributiontype (Fig. rcn).If the computedI"f P ratio falls outsidethe curves,the nearestcurvi should be used. If the watershedcontains a pefcentageof ponds or swampyareas,the peak flow is multiplied by a reductioncoefficientfrom Table 15'6' EXAMPLE 15.3 concentration,CN: 75 A 1280-acreurbanTennesseewatershedhasa6.0-hrtimeof from Table15.4,and5 percentof the areais ponded.The25-year,24-httainis 6'0 in' Find the 25-yearpeak discharge.
TABLE 15.4 RUNOFFCURVE NUMBERSFOR URBANAREAS (see Sec. 4.9 foT other values)
Curvenumbers for hydrologic soilgroup'
Cover descriotion
Cover type and hydrologiccondition
Average percent imperviousareaD
A
Fully developed urban areas (vegetationestablished) Open space(lawns, parks, golf courses,cemeteries,etc.)" Poor condition (grasscover <50%) 68 Fair condition (grasscover 50-757o) 49 Good condition (grasscover > 757o) 39 Impervious areas Pavedparking lots, roofs, driveways,etc. (excluding right- of-way) 98 Streetsand roads Paved;curbs and storm sewers(excluding right-of-way) 98 Paved;open ditches(including right of-way) 83 Gravel (including righrof-way) 76 Dirt (including right-of-way) 72 Westerndeserturban areas Natural desertlandscaping(pervious areas only)' 63 Artificial desertlandscaping(impervious weed barrier, desertshrub with 1-2-in. sandor gravel mulch and basin borders) 96 Urban districts Commercial and business 89 85 Industrial 81 72 Residentialdistricts by averagelot size 77 65 f acre or less(town houses) j acre 61 38 57 30 I acre 25 I acre I acre 20 5l 2 aqes l2 46 Developingurban areas Newly gradedareas(pervious areasonly, no vegetation)" 77 Idle lands (CNs are determinedusing cover types similar to thosein Table 4.7).
B
c
79 69 6l
86 79 74
98
98
98 89 85 82
98 92 89 87
77
85
96
96
96
92 88
94 91
95 93
85 75 72 70 68 65
90 83 81 80 79 77
92 87 86 85 84 82
86
91
89 84 80
98 93 9r 89
'Average runoff condition, and 1" : 9.25. 'The averagepercent impervious area shown was used to developthe composite CNs. Other assumptionsare as follows: impervious areas are directly connected to the drainage system,impervious areas have a CN of 98, and pervious areas are consideredequivalent to open spacein good hydrologic condition. CNs for other combinations of conditionsmay-becomputedusingFig. 15.9 or 15.10 " CNs shown are equivalent to those of pasture. Composite CNs may be computed for other combinations of open spacecovol type. dComposite CNs for natural desertlandscapingshould be comppted using Fig. 15.9 or 15.10 basedon the impervious areapercentage(CN : 98) and the pervious area CN. The pervious area CNs are assumedequivalent to desert shrub in poor hydrologic condition. eComposite CNs to use for the designof temporary measuresduring grading and construction should be computed using Fig, 15.9 or 15.10basedon the degreeofdevelopment (impervious areapercentage)and the CNs for the newly graded pervious areas. Source: U.S. Soil ConservationService,2T
0.0
n5
> o
e
r.o I I
70
60
50
CompositeCN
Total impervious arca (Vo)
Figure 15.10 Graph of 1986 TR-55 composite CN with unconnected imperviousarea,or total imperviousarea,lessthan 30 percent.(After U.S. Soil . ConservationService.2T) TABLEls.s /aVALUESFORRUNOFFCURVENUMBERS Curvenumber
L fin.)
Curvenumber
L (in.)
40 4l 42 43 44 45 46 47 48 49 50
3.000 2.878 2.762 2.65r 2.545 2.444 2.348 2.255 2.167 2.082 2.000 1.922 r.846 1.774 1.704 1.636 1.571 r.509 1.448 1.390 1.333 1.279 1.226 I.t75 t.t25 1.077 1.030 0.985 0.941 0,899
70 71 72
0.857 0.817 0.'778 0.740 0.703 4.667 0.632 0.597 0.564 0.532 0.500 0.469 0.439 0.410 0.381 0.353 0.326 0.299 0.273 0.247 0.222 0.198 0.174 0.151 0.128 0.105 0.083 0.062 0.041
)l
52 53 54 55 56 57 58 59 60 6l 62 63 o+
65 66 67 68 69
Source:U.S. Soil ConservationService.
t5
74 IJ
76 77 78 79 80 8l 82 83 84 85 86 87 88 89 90 9l 92 93 94 95 96 o1 98
328
HYDROLOGY CHAPTER15 URBANAND SMALLWATERSHED
300
u
tnn
o P0 R I
3 S
100 R
o 60
40 01
0.6
0.8 l
Time of concentration, Z; (hr)
Figure 15.11 Unit peak discharge(q*) for SCS Type-I rainfall distribution. (After U.S. Soil ConservationService.)
9
100
Po
Ro
ci
60
30 0.1
0.2
0.4
0.6 0.8 1
2
4
6
810
Time of concentration,Zr (hr)
Figure 15.12 Unit peakdischarge(q,) for SCSType-IA rainfall distribution.(After U.S. Soil ConservationService.)
15.2 PEAK FLOW FORMULASFOR URBAN WATERSHEDS
329
1000 800
^
600 500
E
400
{, s 300 o 90 d
€ 2oo € 6
o
= P 1oo 80 60 50 0.1
0.2
0.4
0.6 0.8 1
2
4
6
810
Time of concentration,Z" (hr)
Figure 15.13 Unit peak discharge(q,) for SCSType-II rainfall distribution. (After U.S. Soil ConservationService.)
700 600 500 400 €< 300
-* d
,nn
E E
= 5 roo 80 60 40 0.1
0.2
0.4
0.6 0.8 1
2
4
6
810
Time of concentration, Z, (tn)
Figure 15.14 Unit peak discharge(q,) for SCSType-III rainfall distribution.(After U.S. Soil ConservationService.)
L
330
CHAPTER15
URBANAND SMALLWATERSHEDHYDROLOGY
TABLE 15.6 ADJUSTMENTFACTOR(Fp)FOR POND AND SWAMP AREAS THAT ARE SPREADTHROUGHOUTTHE WATERSHED Percentageof pond and swamp areas
0 0.2 1.0 3.0 5.0
1.00 0.97 0.87 0.75 0.72
Source:U.S. Soil ConservationService.
From Solution. From Fig. 16.17, the Type-II storm appliesto Tennessee. q,: csm/in. 96 Table15.5,I":0.667.Thus I,fP: 0.11.FromFig. 15.13, From Chapter4, tberunoff from 6.0 in. is 3.28 in. Since5 percentof the area is ponded,the peak flow is adjustedusingTable 15.6,giving 4 : 0.72. Thus g : (96 csm/in.)(3.28in.)(2.0 mr')(0.72): 453 cfs rr The graphicalmethodprovidespeak dischargesonly. If a hydrographis needed or watershedsubdivisionis required, the tabular method2Tshouldbe used.The event simulationmodel TR-20 shouldbe usedif the watershedis very'complexor a higher degreeof accuracyis required (seeChapter24). Assumptionsof the graphicalmethod include: The method shouldbe used only if the weighted CN is greaterthan 40. The ?i valueswith the method may range from 0.1 to 10 hr. The watershedmust be hydrologicallyhomogeneous,that is, describable by one CN. Land use, soils, and cover must be distributed uniformly throughoutthe watershed. The watershedmay haveonly one main streamor, if more than one, the branchesmust havenearly equal times of concentration' The method cannotperform channelor reservoirrouting. The Fofactorcan be appliedonly for pondsor swampsthat are not on the flow path. Accuracy of peak dischargeestimatedby this method will be reducedif I"fP va\uesare usedthat are outsidethe range given. When -this method is used to develop estimatesof peak dischargefor presentand developedconditionsof a watershed,usethe sameprogedure for estimating[. Both the graphicalandtabular methodsare derivedfrom TR-20 output.The use of I permits them to be usedfor any size watershedwithin the scopeof the curves or tables. The tabular method can be used for a heterogeneouswatershedthat is Hydrographsfor the subwatersubwatersheds. dividedinto a numberof homogeneous shedscan be routed and added.
15,3
PEAK FLOW FORMULASFOR SMALL RURAL WATERSHEDS
331
The tabularmethodis describedin the technicalreleaseand is not detailedhere. In using the method, the following stepsare employed:
'
1. Subdividedthe watershedinto areasthat are relatively homogeneousand haveconvenientrouting reaches' 2. Determinedrainageareaof eachsubareain squaremiles' 3. Estimate T"for eachsubareain hours. The procedurefor estimatingI is outlinedinTR-55. 4. Find the travel time for eachrouting reachin hours' 5. Developa weightedCN for eachsubarea. 6. Selectan appropriaterainfall distribution accordingto Fig. 16.17. 7. Determine the 24-hr rainfall for the selectedfrequency(Chapter 16). 8. Calculatetotal runoff in inchescomputedfrom CN andrainfall (Chapter4)' 9. Find I,fot eachsubareafrom Table 15.5. 10. Usingthe ratio of I,f P andT,for eachsubarea,selectone of the hydrographs tabulatedin TR55. 11. Multiply the hydrographordinates(csm/in.) by the area (mi2) and runoff (in.) of eachrespectivesubarea. 12. Route and combinethe hydrographs. The SCS recommendsthat TR-20, rather than the tabular method,be used if any of the following conditions apply: Travel time is greaterthan 3 hr. f is greaterthan2hr. Drainageareasof individual subareasdiffer by a factor of 5 or more. The TR-55procedureshavebeenincorporatedby SCSin a computerprogram.Copies are availablefrom the U.S. National TechnicalInformation Service.
FORSMALLRURALWATERSHEDS 15.3 PEAKFLOWFORMULAS SCSTP-149Method TR-55 is the SCS procedure for urban watersheds,TR-20 is the unit-hydrograph procedurefor larger agriculturalwatersheds(seeChapter24), andTP-149wasdevelbped to allow esiimation of peak flow rates from small (5-2000 acres)agricultural It consistsof a seriesof 42 charts from which the peak dischargeof a watersheds.3o 24-hr ruinfall can be determined. Input to the procedure is the drainage area, averagewatershedslope, storm distributiontype (I or II), watershedcompositecurve number, and depth of rainfall. Figures15.15ind 15.16illustratethe numerouschartsin the TP. Shownare type-I with CN : 70 for both. Similar and type-Il curvesfor moderatelyslopedwatersheds, 15.7.Applicationsof TP 149 Table given in chartsare availablefor the combinations incrementsof Table 15.7, 5-unit the than other to watershedshavingcurve numbers by arithmetic or be accomplished percent, can 16 4, or or for slopesother ihan I, values' chart adjacent logarithmic interpolationbetween
332
CHAPTER15
URBANAND SMALLWATERSHEDHYDROLOGY
h\or@e
g ?3eR3=
R
^^^ e H =HEF==
F
'/, 1000 800 7N 600 500 400
7
r!
'z ..\"\
o P!
rno
30
t0 8 ,7 6 5
z
,<\ 7 l
200
€ 8 0 & 7 0 3 a o 50 40
z
,4
300
.E
1000 800 700 600 500 400
l
2\
4
300 200
W
100 80 70 60 50 40
L
'//.2
L'//
& t/,, .M
30
7
a
ffi
20
7
v'
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H l/,v ,/:
/ttl h
9
r
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o o o o € r @
10 8 7 o
) O
o
R
O
O
O
O
O
O
O
g?fi€FAt
O
8 N
Drainage area (acres)
-tP-I49 peak Figure 15.15 ratesof dischargefor small watersheds,Type-I storms:24-hr rainfall, moderate slopes, and CN : 70. (After U.S. Soil ConservationService, "A Method of EstimatingVolumeandRateof Runoff in Small Watersheds,"U.S. Depaftment of Agriculture,Jan. 1968.)
TABLE 15.7 CHARTSAVAILABLEIN TP-149FOR PEAK FLOW RATESOF SMALL WATERSHEDS
Stormdistribution rype I,il
LU I,il
Slopetype
Slope range (%)
Curvenumber,CN
Flat, TVo Moderate,4To Steep,167a
0-3 3-8 8-30
60,65,70,75,80,85,90 60,65,70,75,80,85,90 60,65,10,75,80,85,90
FORSMALLRURALWATERSHEDS333 15,3 PEAKFLOWFORMULAS
n€r€e
R se8sRsc
R ==FeF=- F
1000 800 700 600 s00 400 300 200
;90 E
50 40
100 80 70 60 50 40
30
30
IUV
€ 8 0 r.1 70
fi o o
20
10
10 8 7 o 5
n€F-€=
8 7 6 5
R
3?38RAE
= = x = = x x X ; + E 6 F d 6
Drainage area(acres)
Figure 15.16 TP-149 peak rates of dischargefor small watersheds,Type-II storms: "A 24-hr rainfalT.moderate slopes,CN : 70. (After U.S. Soil ConservationService, Method of EstimatingVolumeand Rateof Runoff in Small Watersheds,"U.S.Department of Agriculture'Jan.1968.)
EXAMPLE 15.4 Comparethe peakflow ratesfrom Type-I and Type-II stormsusingFigs. 15.15and 15.16.Assumethat only stormtype changesand all other conditionsare equal. Solution. A 4-in. rain over200 acreson a watershedwith CN : T0lesultsin : II Qo : 52 cfs for a Type-I storm (Fig. 15.15) and Qp 9l cfs for Type in (Fig. 15.16).Thus the storm distributiontype makesa significantdifference results of peak flow estimationusing SCS techniques. I r
334
CHAPTER15
URBANAND SMALL WATERSHEDHYDROLOGY
FederalHighwayAdministrationscs PeakFlow DesignMethod The FederalHighway Administration (FHWA) lists in their HydrologicEngineering Circular No. tq, F{ydrology( 1995Ed.) a procedurefor estimatingpeak flow ratesfor homogeneour,,-ul1-to--"dium sizedwatershedshavingtimes of concentrationbetween 0.1 and 10 hours.It employsan SCSregressionequationthat has coefficients determinedfrom data on different rainfall distributiontypes and ratios of the initial abstraction1, (seeChapter4) and total precipitation,P. The peak dischargein metric units is calculatedfrom
( 1s .3)
Qp: 4,AQ,
where qois the peak dischargein m3/sec,A is the drainageareain sq. km., Q is the net rain depth in cm, andq, is the unit peak dischargefrom log qu:
Co + Cl log /" f
Crlogz t"
(1s.4)
in which /" is the time of concentrationin hours,and the regressioncoefficientsare obtainedfrom Table 15.8. TABLE 15.8 COEFFICIENTSFOR FHWA HEC-19SCS PEAK DISCHARGEMETHOD Rainfall
rype
IA
ilI
t,/P 0.10 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.10 0.20 0.25 0.30 0.50 0.10 0.30 0.35 0.40 *0.45 0.50 0.10 0.30 0.3s 0.40 0.45 0.50
co 2.30550 2.23537 2.18219 2.10624 2.00303 1.87733 t.76312 1.67889 2.03250 r.91978 1.83842 1.72657 1.63417 2.55323 2.46532 2.41896 2.36409 2.29238 2.20282 2.473t7 2.39628 2.35477 2.30726 2.24876 2.17772
c1 -0.51429 -0.50387 -0.48488 -0.45695 -0.40769 -032274 -0.15644 -0.06930 -0.31583 -0.282t5 -0.25543 -0.t9826 -0.09100 -0.61512 -0.62257 -0.61594 -0.59857 -0.57005 -0.51599 -0.51848 -0.51202 -0.49735 -0.4654r -0.41314 -0.36803
c2 -0.11750 -0.08929 -0.06589 -0.02835 0.01983 0.05754 0.00453 0.0 -0.13748 -0.07020 -0.02597 0.02633 0.0 -0.16403 -0.1t657 -0.08820 -0.05621 -0.0228r -0.01259 -0.17083 -0.13245 -0. l 1985 -0.1 1094 -0.11508 -0.0952s
Source: Afrer U.S. Federal Highway Administration' Hec-19, Hydrology' FHWAIP-95, 1995.
15.3 PEAK FLOW FORMULASFOR SMALL RURALWATERSHEDS
335
The procedurehas the following limitations: ' use with homogeneouswatersheds(CNsfrom zone to zone shouldnot differ . . . '
bYs)
CN shouldbe greaterthan 50 /. shouldbe between0. 1 and 10 hours I^/P shouldbe between0.1 and 0.5 /" shouldbe about samefor any of the main channels,if watershedhas more than one main channel . no channelor reservoirrouting is allowed . no storagefacility on main channel ' watershedareain storageponds and lakes shouldbe lessthan 5 percent
SyntheticUnit-Hydrograph PeakRateFormulas Peak flow rates from small watershedscan also be determinedusing the synthetic unit-hydrographtechniquesdescribedin Chapter 12. A storm having a duration definedby Eq. 12.22wirl produce,accordingto Snyder'smethodof synthesizingunit hydrographs, a peak dischargefor 1.0 in. of net rain given by Eq. 12.17,or
o^' -
64oct'A t
m
( 15.s )
Similarly, the peak flow rate resulting from a storm with duration D given by Eq. 12.22or 12.23is, accordingto the SCS methodfor constructingsyntheticunit hydrographs,equal to
484A ^ U^: -
(15.6)
where /o is the time from the beginningof the effectiverain to the time of the peak runoff rate, which by definitionis the watershedlag time plus half the stormduration. Both of Eqs. 15.5and 15.6 apply to 1.0 in. of netrain occurringin the durationD. Either can be multiplied by P"", for other storm depthswith equal durations.Peak flows for stormswith durationsother than D would need to be determinedby unithydrographmethods.
Discharge-Area and RegressionFormulas A multitude of peak flow formulas relating the dischargerate to drainageareahave beenproposedand applied.Gray3rlists 35 suchformulas,and Maidment32compares many others.Most of theseempiric equationsare derivedusingpairsof measurements of drainagearea and peak flow rates in a regressionequationhavingthe form where
Q: CA* : peak the dischargeassociatedwith a given return period Q A : the drainagearea C, ffi : regressionconstants
(15.7)
Popular discharge-areaformulas in the form of Eq. 15.7 include the Meyers equation33 ( 1s.8 ) O : 10,000405
336
CHAPTER15
URBANAND SMALLWATERSHEDHYDROLOGY
where A : the drainagearea,which must be 4 mi2 or more Q : the ultimate maximum flood flow (cfs) This examplegives only one flow rate of unknown frequencyand is chosenonly to illustrate the form of flood flow equations.A program of determiningflood magnitudesfor atange of frequencieson a state-by-statebasishas been completedby the USGSusing the multipl- regressiontechniquesdiscussedin Chapters26 and 27 and illustrated for Virginii in Problem 27.25. Similar formulas are availablefrom the USGSfor other states.Softwarecontainingall the USGSregressionequationsfor the United Statesis availablefrom the U.S. GeologicalSurvey and Federal Highway Administratioh as part of the HYDRAIN softwarepackage'
U.S.GeologicalSurveyIndex-FloodMethod The U.S. GeologicalSurveyindex-floodmethoddescribedin Section27.4 is a graphical regional c&relation of the recurrenceinterval with peak dischargerates. The stepsinvolvedin the derivationof a regionalflood index curve are outlined in Section 27.4.The first stepin applyingthe techniqueto a watershedis to determinethe mean annual flood, defineOas ttre flood magnitudehaving a return period of 2.33 years. Mean annual floods for ungaugedwatershedsare found from regressionequations similar in form to Eq. 15.7.For example,the USGSreport3aon flood magnitudesand frequenciesin Nebraskagives,in cfs, Qz'zt
= CA0T
(1s.e)
where A : the contributing drainage arcain mi2 C : aresionalcoefficientobtainedfrom Fig. 15'17
'
Once the mean annualflood magnitudeis obtained,other annual flood magnitudes can easily be determinedfrom the appropriate index-flood curve (see Fig. 26.4c).The usqof suchcurvesin urban hydrologyis limited becausethe USGS data network for the index-floodmethod seldomincludeswatershedssmallerthan 10 mi2. The USGSregressionequations,describedlater, areapplicablefor watershedsin the 1-10 mi2rangeand larger.
CyprusCreekFormula Extreme$ flat areasposeparticular difficulties to the hydrologist,includingestimates of infiltration, runoff volgme,and,peakrunoff rates.Flooding in theseareastends!o be shallow and widespread.Flow velocitiesare low, and water standson the surface for relative)ylong periodsof time. Theseareasare often distinguishedby networksof straight drainageihannels that have been constructedto store and eventually dischargethe excessrain. peak flow to calculatethe'instantaneous The Scs developeda procedure3s be would that of canals capacity the calculating first on based flatland areas prevent would that a duration to storm design for the to limit flat-xeaflooding crop damage,and then to apply a multiplier to this rate to obtain the instant peakfor the designof drainagestructures.The procedureis illustrateil in Fig. 15. 18.
OOIfr
N
jtoo
.oe
' -. 4- 6> *'-1
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6 . 1
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338
CHAPTER15
URBANAND SMALLWATERSHEDHYDROLOGY
.--Duratio4
of Overbank Flow-->l InstantaneousPeak
o o
Drainage Ditch CaPacitY "Removal" Rate Based on
Iime
Figure 15.18 Illustration of relation befweenCyplys Creek stJntaneousflow. (After Soil ConservationService'35)
"removal" rate and peak in-
time for The selectedduration was} hours,consideredto be the maximum allowable to developed was Creekformula, Cyprus the called inundationof crops.An equation, equation, The rate. removal 24-hr the called rate, flow determinethe canal design curve based on rainfall depth,-contributingdrainage area,and the SCS composite number is Q:
CA5/6
( 1s .10)
where Q : reqrriredchannelcapacity fot 24-ht removal (cfs) C : drainagecoefflcient A: drainageareaGq mi) bY The drainagecoefficient,C,fot Eq. 15.10is found from an equationdeveloPed Stephensand Mi1ls36 (15.1) C : 16.39+ (14.75Q,",) where designeventfrom Fig' 4'I4' Q"", : the scS direct runoff (in.) for the 24-hr peak flow rate is Once Eq. 15.10is solvedfor the given frequency,the instantaneous from 1 to about areas drainage obtainedfrom Fig. 15.19.The procedureis limited to to the 24-hr rate peak instantaneous that ratios of the 200 squaremiles.It is suggested areas flatland For 1.0. to equal canaliemoval rateUetirniLO b values greaterthan or peak flows the that that havepart of the areain storm ,"wJts, the SCS recornmends 15.20.TheSCSfurther fromFig. ts.tqu.increasedbytheamountsindicatedinFig. that are recomniendsrestrictingur" of thit procedureto watershedsthat haveslopes TR-55' TR-20, as such methods lessthan 0.002. For stJeperslopewatersheds,other TP l4g, or regressionequationsare recommended'
i)c<
- <
o11 I q
t * . 4 q
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€
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eleX ;;oung e8ere^V JnoH-tZ runrxrxehtr 01 luod snoeupluelsuJJo oqed
: r I
340
CHAPTER15
URBANAND SMALLWATERSHEDHYDROLOGY
d
s40 o !l
o 9 ^ ^ i 4rl
0 Percentageof Area Servedby Storm Sewers
Figure 15.20 Effect of urban storm sewerson peak dischargefor urban areas. (After U.S. GeologicalSurvey.aT)
EXAMPLE 15.5 Usethe Cypro, Creek methodto determinethe peak 50-yr flow rate fron a 1.0 sq mi drainageareathat hasa CN : 80,is 50 percentstormsewered,andhasa50-yr,24-hr rainfall depthof 12.0inches. Solution. From Fig. 4.I4, thedirectrunoff for 12 inchesof rain is 9.45in. The drainagecoefficient,C, is found from Eq. 15.11,
: 155.t C : 16.39+ (14.75X9.45)
15.3 PEAK FLOW FORMULASFOR SMALLRURALWATERSHEDS 341
The 24-hr removalrate is found from Eq. 15.10' : 0 : 155'8(1'0)5/6 155'8cfs rate to removalrate is 2.0, giving a From Fig. 15.19the ratio of instantaneous designflow rate of 311.6cfs if no stormsewersexisted.From Fig. 15.20,it is found that the unseweredareadischargeshouldbe increasedby 35 percentfor a watershedwith 50 percent storm sewers.The final designflow is 1.35 x 31'1.6: 420.7 cfs. rr
U.S.GeologicalSurveyRegressionEquations for UrbanAreas
.
The U.S. GeologicalSurvey,in cooperationwith the FederalHighway Administration, conducteda nationwidestudy of flood magnitudeand frequencyin urban waterat 56 citiesin 31 states,includinvolved26ggaugedbasins The investigation sheds.37 Basin sizesrangedfrom 0.2 to 15.21. Fig. in are shown locations The Hawaii. ing 100 mi2. Multiple linear regression(see,Chapter27) of a variety of independentparameters was conductedto developpeak flow equationsthat could be applied to small, ungaugedurban watershedsthroughoutthe United States.Similar USGS regression equationsfor large rural basinsare describedin Chaptet27' The simplestform of the developedregressionequationsinvolvesthe three most significantvariablesidentified.Thesewere contributing areaA (mi2),ba-sin^developandthe correspondingpeak flow RQ,(cfs)for the lth ment factorBDF (dimensionless), in the sameregionasthe urbanwatershed.The basin rural identical an from frequency and estimatescan be developedfrom variations, regional for accounts latter vaiiable (seeSection27.4). The threereports frequency flood USGS the applicable any of for the 2-,5-,IO-,25-,50-, 100-,and500-yearflowsaregiven parameter equations as37
ez: l3.2Ao.zt(13 BDFl-o.azpnotz es : 10.6Ao.rz(13 BDF)-o.3eReo18 t0(13- BDnl-otuRQ?dn Qto : 9.5rAo ts(13- BDF)-o'z+P9o'to Qt5 : 8.68Ao ts(13- BDFl-o'zzR03o" Qso: 8.o4Ao ts(13- BOrT-ot'RQ?r!& Qno: 7.70Ao - BDF)-'*RQ1i& : 7.47A0'16(13 Qsoo
(Ls.r2) ( 1s . 1 3) (15.14)
(1s.1s) (1s.16) (15.17) ( 1s . 18 )
Thesewere developedfrom data at 199of the 269 original sites.The other siteswere deletedbecauseof the presence.ofdetentionstorageor missingdata. All theseequations havecoefficientsof determinationabove0.90. of estimatedand observedvaluesused Figure 15.22showsthe correspondence fall within one standarddeviation values the percent of Forty 15.15. Eq. in devedping are similar to the 1O-year intervals recurrence for other line. Graphs regression of the graphshownin Fig. 15.22.
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PROBLEMS 353
15.5,
rainfall duration? The 4-hr unit hydrographfor a 5600-acrewatershedis n
Time (hr) 0 (cf9
0
1
400
8 4 6 1000 800 400
10 200
t2 0
: discussthe Rework Example 15.2 basedon a C : 0.2 and C 0'4' Compare and effect of C on the dischargeat the outfall' AwatershedhasareaA.Startingwithatriangular.shapedunithydrographwithabase : 484A/to' lengthof 2.67t, and a heightof [0, deriveEq. 15'9 (seealsoEq' t2'25)' Qo derivation' the State and carr units of eachterm usedin
UsingtheSCSdimensionlessunithydrographdescribedinChapter12,determinethe peak for a net storm of 101n.in 2 hr on a 400-acrebasinwith a time to peaidischarge ^of + nt and i lag time of 3 hr. Comparewith Eq' 12'17' rainfall at a A 10.00-mi2watershedwith a 100-min time of concentrationreceives min' 200 of period rate of 2.75 in.lhr for a (cfs)from the watershed1f C : 0'4' a. Determinethe peak d-ischarge rainfall. b. Estimatethe dischargerate lcfs) 150 min after the beginningof the beginning of after min 40 watershed c. Estimate the dischar"gerate from the rainfall. F
E * n
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0
100
354
CHAPTER15
URBANANDSMALLWATERSHED HYDROLOGY
1s.13. A storm gutter receivesdrainagefrom both sides.On the left it drains a rectangular 600-acreareaof t" : 60 min. On the right it drains a relatively steep300-acrearea of t" : 10 min. The f index on both sidesis 0.5 in.ihr. Use the intensity-durationfrequency curves in Fig. 15.7 to determinethe peak discharge(cf$ with a25-year recurrenceinterval for (a) the 600-acreareaalone, (b) the 300-acrearea alone,and (c) the combinedareaassumingthat the proportion of the 600-acreareacontributing to runoff at any time r after rain beginsis l/60. t5.14. A drainagebasin has a time of concentrationof 8 hr and producesa peak Q of 4032 cfs for a 10-hr storm with a net intensity of 2 in./hr. Determinethe peak flow rate and the time base(duration)of the direct surfacerunoff for a net rain of 4 in./hr lasting (a) 12 hr, (b) 8 hr, and (c) 4 hr. State any assumptionsused.
15.15. A 1.0-mi2parking lot has a runoff coefficientof 0.8 and a time of concentrationof 40 min. For the following three rainstorms,determinethe'peakdischarge(cf$ by the rationalmethod:(a) 4.0 in./hr for 10 min, (b) 1.0in./hr for 40 min, and (c) 0.5 in./hr for 60 min. State any assumptionregarding area contributing after various rainfall durations. 15.16. The concentration time varies with dischargebut is relatively constant for large discharges.From this statement,why do engineersfeel confidentin usingthe rational formula? 15.17. Determinethe 50-yearflood for a20-mi2 basin at the northwestcorner of Nebraska. Use the index-flood method and assumethat Fig. 26.4 appbes. 1s.18. Determinethe entire frequencycurve for the basin in Problem 15.17 and plot it on probability paper.
15.19. Use the index-flood method to determinethe 10- and 50-yearpeaksfor a 6400-acre drainagebasin near Lincoln, Nebraska.Assumethat Fig. 26.4 applies. 15.20. For the drainagebasin in Problem 15,19 determinethe probability that the 20-year peak will be equaledor exceededat leastonce (a) next year and (b) in a 4-yr. period. Referto Section26.1.
15.2L. For a 100-mi2drainagebasinnearLincoln, Nebraska,usethe index-flood methodto determinethe probability that next year's flood will equal or exceed3000 cfs. 15.22. UseFig. 26.4 to determinethe return period (years)of the meanannualflood for that region.How doesthis comparewith the theoreticalvalue for a Gumbel distribution? How doesit comparewith a normal distribution?Refer to Section26.6. ts.23. Usethe Cyprus Creekmethodto determinethe 25-yr peak dischargefor the watershed describedin Example 15.3. Assumethat the watershedis nearly flat.
15.24. You are asked to determinethe magnitudeof the S0-yearflood for a small, rural drainagebasin (nearyour town) that hasno streamflowrecords.Statethe namesof at leasttwo techniquesthat would provide estimatesof the desiredvalue. 't5.25. The drainageareas,channellengths,and relevantelevations(underlined)for several subbasinsof the Oak Creek Watershedat Lincoln, Nebraska,are shownin Fig. 24.8. The watershedhas a SCScurve numberof CN : 75 which may be usedto determine the direct runoff for any storm.Assumethat IDF curvesin Fig. 27.13 applyat Lincoln. Treat the entire watershedasa singlebasinand determinethe 50-yearflood magnitude at Point 8 using: a. The rational method. b. The SCSpeak flow graph,Fig. 15.8. c. Snyder'smethod of syntheticunit hydrographs,Eq. 15.5. d. The USGS index-flood method. Figure 26.4 applies.
REFERENCES 355
15.26. RepeatProblem 15.25 with SubareaI excluded.compare the results with Prob-
lem 15.25 and comment on the effectivenessat Point 8 for the 5O-yearevent of the BranchedOak Reservoirat Point 9. (This reservoirwill easilystorethe 100-yearflood from Area I.) 1s.27. RepeatProblem 15.25 for SubareaA' 15.28. RepeatProblem 15.25 for SubareaI' 15.29. Describecompletelyhow the magnitudeof the 30-yearflood for a watershedis determined by the USGS index-flood method. 15.30. A rural watershedwith a composite cN of 70 is being urbanized. Eventually' 36 percentof the areawill be impervious.Determinethe increasein runoff that can for a 6.2-in.rain. be expected peak flow for the SCS dimensionlessunit hydrographin Ch. l2,_determine 15.31 Using the the piak dischargefor a net storm of 10 in. in Zhr on a 400-actebasinwith a time to peak of 4 hr and a lag time of 3 hr. L5.32. A timber railroad bridgein Nebraskaat Milepost 27I.32 ontherailroad systemshown in the sketchis to be replacedwith a new concretestructure.The 50- and 100-year flood magnitudesare neededto establishthe low chord and embankmentelevations, respectively.Determine the designflow rates using the scs TP-149 method. The bridge drainsthe zone marked,about45 acres.The moderatelyslopedbasinlies in a rype-n stormregion,the curve numberis 70, and the 24-hr 50- and 100-yearrainfall depthsare 8.6" and g.4" respectively'
Bidge27l.32
The 15.33. Repeatproblem 15.32usingthe FHWA HEC-19 peakflow SCSdesignmethod. relationships the from be determined 1o can of Values hrs. is 0.2 time of concentration , in Fis. 414. Provide the answersin both metric and English units.
REFERENCES "A Critique of Current Methods in Hydrologic Systems 1 . J. Amorocho and W. E. Hart, Investigations ," Trans.Am. Geophys.Union 45(2),301-321(Jwe 1964)' ..NonlinearInstantaneousUnit-HydrographTheory," ASCE J. Hyd. Div. jingh, p. 2. f. 90(HY2), Par I, 313-347(Mar' 1964).
356
HYDROLOGY 15 URBANANDSMALLWATERSHED CHAPTER "ContinuousHydrographSynthesiswith an 3. W T. Sittner, C. E. Schauss,and J. C. Monro, API:Iype HydrologicModel," WaterResourcesRes.5(5), 1007- 1022(1969). 4. J.E.Nash,"TheFormoftheInstantaneousUnitHydrograph,"Int.Assoc.Sci.Hyd.3@5),
r14-L2r(r9s7).
"Mathematical Models of CatchmentBehavior," Proc. 5. D. R. Dawdy and T. O'Donnel, ASCEJ. Hyd. Div.91(HY4), 124-127(Iuly 1965). 6. S. L. S. Jacoby,"A MathematicalModel for Nonlinear Hydrologic Systems,"J. Geophy. Res. 7l(20), 48t | - 4824(0ct. 1966). 7. R. Prasad, "A Nonlinear Hydrologic System ResponseModel," Proc. ASCE J. Hyd. Div. 93(HY4)(1967). "Hydrology of Urban Runoff," J. ASCE 85, 418. A. L. Tholin and C. T. Keifer, 1959). 106(Mar. "Digital Simulationin Hydrology:StanfordWater9. N. H. Crawfordand R. K. Linsley,Jr., shedModel IV," Department of Civil Engineering,Stanford University, Stanford, CA, Tech.Rep.No. 39, July 1966. 10. JohnC. Schaake,Jr., "synthesisof the Inlet Hydrograph,"Tech.Rep. 3, Storm Drainage ResearchProject, JohnsHopkins University,Baltimore, MD, June 1965. "WaterPollution Aspectsof UrbanRunoff," Federal 11. AmericanPublic WorksAssociation, Water Pollution Control Administration, 1969. "Urban Water ResourcesRe12. Arnerican Society of Civil Engineers,First Year Report, search,"Sept. 1968. 13. W. Viessman,Jr., "Modeling of Water Quality Inputs from Urbanized Areas," Urban Water ResourcesResearch, Study by ASCE Urban Hydrology Research Council, Sept.1968,pp. A79-A103. "Characterization, 14. S. R. Weible,R. B. Weidner,A. G. Christianson,and R. J. Anderson, Treatment,and Disposal of Urban Storm Water," in Proceedingsof the Third International Conference,International AssociationonWater Pollution Researcft(S. H. Jenkins, ed.). Elmsford,NY PergamonPress,1969. "Pesticidesand Other 15. S. R. Weible,R. B. Weidner,J. M. Cohan,and A. G. Christianson, Contaminants in Rainfall and Runoff," '/. Am. Water Works Assoc. 58(8), 1675(Aug.1966). 16. Division of WaterResources,Departmentof Civil Engineering,University of Cincinnati, Cincinnati. OH. "Urban Runoff Characteristics,"Water Pollution Control ResearchSeries. EPA. 1970. 17. Metcalf and Eddy, Inc., University of Florida, Gainewille, Water ResourcesEngineers, Inc., "Storm Water ManagementModel," Environmental Protection Agency, Vol. 1,
r971.
18. E. Kuichling, "The Relation Betweenthe Rainfall and the Dischargeof Sewersin Populous Districts,"Tians.ASCE,20(1889). 19. W. W. Horner, "Modern Procedurein District SewerDesign," Eng. News 64,326(1910). "Relation BetweenRainfall and Runoff from Small Urban 20. W. W Horner and F. L. Flynt, Areas," Trans.ASCE 20(140),( 1936). 21. R. L. Rossnriller,"The Runoff Coefficient in the Rational Formula," EngineeringResearchInstitute, Iowa State University,Feb. 1981. "Experimental Examination of the 22. J. C. Schaake,Jr., J. C. Geye1,and J. W. Knapp, RationalMethod," Proc.ASCEJ. Hyd. Div.93(HY6) (Nov. 1967). "Airport Drainage," Advisory 23. FederalAviation Agency, Departmentof Transportation, D.C.: U.S. GovernmentPrintingOfflce, 1970. Circular, AIC 150-5320-58.Washington,
Chapter16
HydrologicDesign
Prologue The PurPoseof this chaPteris to:
r
. Introducethe hydrologistto proceduresusedin the United Statesfor designing structuresfor safe and effectivepassageof flood flows' , Give sufficient information for the designerto selectthe applicablecriteria for designinghYdrauhcstructures' , provide a discussionof designstorm hyetographsand provide methodsfor design. selectingthe duration, depth, and distribution of plecipitation for precipitation . DemonJtratehow designiloods can be developedwithout using data' ' Discussparticular designmethods including airport drainage'urban storm sewerdeiign, and flood control reservoirdesign' . Describethe U.S. Federal EmergencyManagementAgency (FEMA) flood of flood plain managementsystemand piesent the hydrologicfundamentals Plain analYsis. piotto studying Readersare encouragedto reviewthe materialin Chapters26 and27 designprocedurespresentedin this PredictingPeakdischargerater for use in designingminor and ma aspects of engineering hydrology' I small crodsroadculverts, levees,dt akPort drainage structuresto the lumped together'with major structr design information. GenerallY,a h dischargefor a designfrequencY,a dischargehYdrograPhfor a design rates,low-flow frequencYanalysis, are often conductedas part of a designproiect'
360
DESIGN 16 HYDROLOGIC CHAPTER Mostdesignsinvolvinghydrologicanalysesuseadesignfloodthatsimulates recordsare someseverefutirre eventorlmitates ime historicalevent.If streamflow records storm unavailable,designflood hydrographsare synthesizedfrom available are cases in rare Only usingthe rainfall--runoff proceduresof Chapters2, 12,and 15. watersheds' in small streamflowrecords adequatefor complex designs,particularly in Chapter 15 are Regionalanalysesand the empiric-coirelativemethodsdiscussed in chappresented Methods usefulfor determiningpeak flow ratesat ungaugedsites' for necessary hydrographs ter 12 andin this chapterare used for developingentire manYengineeringdesigns. are describedin uyirologic;ethJds for designingminor and major structures rnethodsfor levels, this chapter.included are discussions;f data needs,frequency dams' and for floodplains synthesizingdesignstorms,and hazard assessments
DESIGNPROCEDURES 16.1 HYDROLOGIC in either the peak flow Proceduresfor estimatingdesignflood flows (interestcan be historical or projected rate or the entire hydrogiaph)includemethodsthat examine and methodsthat flood flowsto arrive at i sultauteestimate(flow-basedmethods), to flood flow rates storms evaluatethe stormsthat producefloods,andthen convertthe on selectinga based be can In eachcase,the analysis (precipitation-basedmethods). meth' (callfrequency-based flood i"rign rr"qo"ncy and determiningthe ass,ociated final the narrowing and_ ods), developingdesignsfor a ringe of flood frequencies or methods)' (called risk-based choice on the basisof long-term c6sts and benefits designingonthebasisofanestimateoftheprobablemaximumstormormanmum RooAtnit could occur at the site (calledcritical-event methods). basedon frequencyMinor Structure Design Minor structuredesignis largely approachto hydrologic the in steps Several basedor sometimesrisk]basedmethods. techadopted and handbooks design most to minor structtJredesign are comrnon are: subsequently) (each is illustrated niques.The generalsteps to the time 1. Determinethe duration of the critical storm,usually equated concentrationof the watershed. t Choosethe designfrequencY. 3. Obtain the storri OeptltUaseaon the selectedfrequencyand duration' in 4. Qomputethe net direct runoff (severalmethodswere presented ter 4). 5. Selectthe time distribution of the rainfall excess' 6.Synthesizetheunithydrographforthewatershed(seeChapterl2). T.Applythederivedrainfall_excesspatterntothesyntheticunithydrograph get the runoff hYdrograPh. the frequen-cyiftn" calculatedflood (usuallyassumedequalto t Establish 8. designstorm frequencY).
DESIGNPROCEDURES 16.1 HYDROLOGIC
361
Major Structure Design Hydrologicdesignaspectsof maior structuresare considerably more complexthan thoseof a small dam, crossroadculvert, or urban drainage system.A designstorm hydrographfor a'large dam still is required but it is put to greater use. The designstorm hydrographis routed to determinethe adequacyof spillwaysand outlets operatedin conjunction with reservoir storage.The economic selectionofthe spillwaysizefrom the variouspossibilitiesdictatesthe final designand is a function of the degreeof protection providedfor downstreamlife and property, project economy,agencypolicy andconstructionstandards,andreservoiroperational requirements.Major structuredesignis largely basedon critical eventmethodspresentedin Section16.5. Water Resource System Design Most information and techniquespresentedin this chapterare directedtoward the flood protection aspectof small and large structures.Needlessto say,a major structureis designedfor more than just flood protection; it is multipurposeand may provide storagefor irrigation, power, water supply, navigation,and low-flow augmentation.The proper allocationof storageto theseuses requiresan understandingof the entire streamflowhistory in terms of the frequency of occurrenceof low flows and averagemonthly, seasonal,and yearly flows, as well as the historical and designfloods. Material is presentedin Part Five to provide a hydrologistwith the tools to developcomplete streamflowhistories for a complex multipurposesysteminvolving various combinationsof minor and major structures, water developmentprojects, and managementpractlces.
Flow-BasedMethods For designlocationswhererecordsof streamflows are available,or whereflows from anotherbasin can be transposedto the designlocation, a designflood magnitudecan be estimateddirectly from the streamflows by any of the following methods: '
1. Frequencyanalysisof flood flows at the designlocation or from a similar basin in the region. 2. Useof regionalflood frequencyequations,normally developedfrom regression analysis(seeChapter 26) of gaugedflood data. 3. Examination of the stream and floodplain for signs of highest historical floods and estimationof the flow rates using measurementsof the crosssectionand slopeof the stream.
Methods Precipitation-Based Where stream-gaugingrecordsare unavailableor inadequatefor streamflow.estimation, designfloods can be estimatedby evaluatingthe precipitation that would producethe flood, and then convertingthe frecipitation into runoffby any ofthe rainfallrunoff methodsdescribedin Chapters10-15 or 2l-27. Typical methodsinclude: 1. Design using the greateststorm of record at the site, by converting the precipitation to runoff. 2. Transpositionof a severehistoricalstormfrom anothersimilar watershedin the region.
362
CHAPTER16
HYDROLOGIC DESIGN
3. Frequency analysis of precipitation and conversion of design storm to runoff. 4. Useof a theoreticalprobablemaximumprecipitation (PMP), or fraction of PMP, basedon meteorologicalanalyses. Becausethe flood flow rate is desiredin all cases,the flow-basedmethodsare preferred over conversionof precipitation to runoff. Due to the relatively longer period of time and greaternumberof locationsat which precipitation amountshave been recorded, precipitation-basedmethods are used in the majority of designs, especiallywith small and very large basins.Flow-basedmethodsare typically used in the midrangeof basin sizes.
Frequency-BasedMethods Regardlessof whetherflow or precipitation dataareused,designsmost often proceed by selectinga minimum acceptablerecurrenceinterval and using proceduresfrom Chapter27 to determinethe correspondingworst condition storm or flood that could be equalledor exceededduring the selectedrecurrenceinterval. Criteria for selecting designrecurrenceintervals are summarizedin Section 16.3. Resultsfrom frequency analysisof flood flow data normally provide reliable estimatesof 2-, 5-, 10-, and 25-yearflows.Extrapolationbeyondthe rangeof the period of flow recordsis allowed, but is lessreliable.
Risk-BasedMethods Recenttrends in designof minor (and major) structuresare toward the use of economic risk analysesrather than frequgncy-based designs.The risk methodselectsthe structuresizeas that which minimizestotal expectedcosts.Tfreseare madeup of the structurecostsplus the potential flood lossesassociatedwith the particular structure. The procedureis illustrated in Fig. 16.1.The total expectedcost curve is the sum of
q
o b0 q
Optimal structure size, S* (least total expected cost) S^in
Structure size, S
Figure 16.1 Principlesof economicriskanalysisforstructure size selection. (U.S. Federal Highway Administration, Hydraulic EngineeringCircular No. 17).
16.2 DATAFORHYDROLOGIC DESIGN 363 the other two curves.Risk costs(flood damages,structuredamages,road and bridge losses,traffic interruptions)and structurecostsare estimatedfor eachof severalsizes. The optimal sizeis that with the smallestsum.Structuresselectedby risk analysisare normally constrainedto sizesequal to or larger than thoseresultingfrom traditional frequency-basedmethods.
CriticalEventMethods Becauseof the high risk to lives or property below major structures,their design generallyincludesprovisionsfor a flood causedby a combinationof the most severe meteorologicand hydrologic conditions that are possible.Instead of designingfor somefrequencyor leastexpectedtotal cost,flood handlingfacilities for the structures are sizedto safelystoreor passthe most critical storm or flood possible.Methodsfor designingby critical eventtechniquesinclude: Estimating the probable maximum precipitation (PMP) and determining the associatedflood flow rates and volumesby transformingthe precipitation to runoff. ) Determiningthe probablemaximum flood (PMF) by determiningthe PMP and convertingit to a flood by applicationof a rainfall-runoff model,including snowmeltrunoff if pertinent. 3. Examining the flood plain and stream to identify palaeo-floodevidences such as high-water marks, boulder marks on trees or banks, debris lines, historical accountsby local residents,or geologicor geomorphologic evidences. 4. In somecases,the critical eventmethodinvolvesestimatingthe magnitude of the 500-yr eventby various frequencyor approximatemethods.Often, suchas in mappingfloodplains,the 500-yr flood is estimatedas a multiple of the 100-yr event, ranging from 1.5 to 2.5. Due to lack of longer-term records, frequency-basedestimatesare seldom attemptedfor recurrence intervals exceeding500 years.
16.2 DATAFORHYDROLOGIC DESIGN The designof any structurerequires a certain amount of data, even if only a field estimateof the drainageareaand a descriptionof terraintype and cover.The following material identiflessomegeneraldata types and sources.
PhysiographicData The hydrologic study for any structurerequires a reliable topographicmap. United StatesGeologicalSurveytopographicmapsusually are available.The mappingof the United Statesis almost completewith 15-minute quadrangles,and many of these areasaremappedby 7.5-minutequadrangles.County mapsand aerialphotoscan also be usedto advantagein making preliminary studiesof the watershed.
364
DESIGN CHAPTER16 HYDROLOGIC
drainage Based on an area map, a careful investigation of the watershed's maps USGS from obtained be can information behaviormust be made. Additional erosive and inflltration the and types Soil that depict predominantrock formations. or univercharacteristicsof soilscanbe securedfrom U.S. Soil Conservationdistricts sity extensiondivisions. of an The drainageareascontributing to large dams require stricter analysis of a possibility The structures. minor designing in area,shydrotogyitranis necessary large for uniformly intenserainfall over the entire basin is an unrealisticassumption thus should the rainfall of variations spatial and The influenceof temporal watersheds. "worst possible" rainfall values are estimated the be considered.For major dams, in reservoir generallyconvertedto a designdischargehydrograph,which is then used and storage, surcharge size, spillway and reservoii routing calculationsto propoition downsustained or power requirements maintain any additional outlets neededto in hydrologic streamflow for navigation,irrigation, or watersupply.The basicconcern estimatefor realistic a using interests downstream designof a large Aamis to protect the designstorm hYdrograPh. purpose the Topographic'rnuf o"tuit necessarilyshifts with the type and .of of understanding the pi"ta increases always reconnaisiance structurebeing design"a. be. might structule the insignificant an area,shydrology*nomatter how
,
HydrologicData the regionunder one difficulty in hydrologicdesignis that of gettingadequatedatafor issuedby published-reports data canbe-acquiredfrom pr-eviously study.ConsiOerabie agencies federal of list is a governmentalagenciesand/or universities.The following that PublishhYdrologicdata: '
egricultural ResearchService Soil ConservationService Forest Service U.S. ArmY CorPsof Engineers National Oceanicand AtmosphericAdministration Bureauof Reclamation DePartmentof TransPortation U.S. GeologicalSurvey' TopographicDivision Division WaterResources U.S. Geological.Survey, governments,interAdditional dlta often canbe procuredfrom departmentsof state statecommissions,and regional and local agencies'
MeteorologicData and Atmospheric The National WeatherService, couchedin the National Oceanic in a variety published data meteorologic of source Administration, is the primary 16.2 Figure (HMR) series. Report of forms, including their Hyirometeorologic showstheapplicablereportsforvariousgeographicandtopographicregionsofthe
DESIGN-FREQUENCYCRITERIA 16.3 HYDROLOGIC
365
Figure16.2HydrometeorologicalreportseriescoverageofconterminousUnitedStates. (U.S.Bureauof Reclamation') state,and local agenciescollect and analyze United States.lNumerousother federal who design,inspect, or regulatelarge precipitation information-"tp""iuriv-tr'rore
"*"8l;.H[1;:1;ce
require:.5::S:1g:,:f stormhyetographs design forestimating or precrp-
in the region,maximum amount ttre meteoroioii" .huru"t"ristics of Jtot-t storm of precipitation,frequenciesof total itable moisturein the atmospheJo;;r;r the over storms for snowmelt and influence.of durationsbf U;t, depthsf*;;;i"", topography chains, mountain of major region.tn ,o1n" areassuch", f;;;iii'*gion, precipitation' on has a very distinct impact
CRITERIA DESIGN-FREQUENCY 16.3 HYDROLOGIC S e l e c t i o n o f f r e q u e i s m o s t o f t e n b a s e d o n p o t e n t i a l d a m a g e t o p r o p e r r : l o s s e s s u c h a s i n t e r r u p t i o n o f commefce'A stanc the worst conditio involved,a greal a an A11projects involve somerisks to property thror.t proceed can design human tife li absent'the tn quencylevel and designof the leastcost structure alternativetoleastStructurecost,economicriskana] rather than the final designfrequencyis optimized trequencret several for would accommodatestorms includenot only the actualconstfuction costs itr"r" u*"J. is leasttotal expectedcost "o** du9 to interruption of servicesand costsbut also the flood dama!;;irk una economicanalysescan be used' commerce.Either annual or p"resentworth
366
CHAPTER16
HYDROLOGICDESIGN
MinorStructures The designfrequenciesshown in Table 16.1 are typical of levels generallyencountered in minor structuredesign.An exampleof variationsthat do occur is the design backwatercould effectivelyhalt frequencyof a culvert,which undercasesof excessive trafflc. The Soil ConservationServicerecommendsthe use of a2l-year frequencyfor minor urban drainagedesignif there is no potential loss of life or risk of extensive damagesuch as first-floor elevationsof homes.A 100-yearfrequencyis commonly recommendedwhen extensiveproperty damagemay occur.t TABLE 16.1 MINORSTRUCTUREDESIGNFREQUENCIES
Typeof minorstructure Highwaycrossroad drainage" ADT' 0-400 400-1700 ADT 1700-5000 ADT ADT 5000Airfields Railroads Stormdrainage Levees Drainage ditches
Returnperiod,4 10yr 10-25yr 25 yr 50 yr 5yr 25-50 yr 2-10 yr 2-50 yr 5-50 yr
= 1/7, Frequency 0.10 0.10-0.04 0.04 o.o2 0.20 0.04-0.02 0.50-0.10 0.50-0.02 0.20-0.02
'ADT : averagedaily traffic. (After Ref. 3).
Large Dams Damsrequirehydrologicanalysisduringthe designof the original structureandduring periodic safetyevaluations.Significanteconomicand humanlossesarepossiblewhen large quantitiesof water are rapidly releasedfrom storage. Initial heightsof retardedwater behind the dam, disregardingthe total volume of stored water, can produce destructiveflood wavesfor a considerabledistance downstream.Basedon two criteria, the TaskForceon SpillwayDesignFloodsrecommeridedthe classificationof large damsas li,stedin Table 16.2.The type of construction has not been included in this grouping, althoughit affects the extentof failure resulting from overtopping. Many of the federalagencieshaveadopteddefinitionsfor hydraulic elementsof dams.The following list is usedby the Soil ConservationService: A spillwuy is an open or closedchannel,or both, used to convey excess water from a reservoir.It may contain gates,either manually or automatically controlled, to regulatethe dischargeof excesswater' Theprincipal spillwayis the ungatedspillwaydesignedto conveythe water from the retarding pool at releaserates establishedfor the structure. The emergencyspillway of a dam is the spillwaydesignedto conveywater in excessof that impoundedfor flood control or other beneficialpurposes.
16.3 HYDROLOGICDESIGN-FREQUENCYCRITERIA
367
FORI-ARGEDAMS TABLE16,2 DESIGNCRITERIA danger lmpoundment Potential Category (1) Major; failure cannot be tolerated
Storage (acre-ft)o (2)
Height (ft) (3)
Failure damage Potential' Loss of life (4) Considerable
>50,000
Damage (5) Excessiveor as matter of PolicY
lntermediate
1000-50.000 40-100
<1000
<50
Spillwaydesignflood (6) Probablemaximum; most severeflood considered reasonablypossible on the basin
Possiblebut small
Within financial capability of owner
Standardproject; based on most severe storm or meteorological conditions considered reasonablycharacteristic of the sPecific reglon
None
Of samemagnitude as cost of the dam
Frequencybasis; 50-1O0-year recurrence interval
and future potential oBased on consideration of height of dam above tailwater, stoarags volume, and length of damage reach, present floodplain' of population, and economic development tstorage at design spillway pool level. Sozrce: After SnYder.3
The retarding pool is the reservoir spaceallotted to the temporary rmpoundmentoi floodwater.Its upperlimit is the elevationof the crestof the emergencyspillway. Retardingstorageis the volume in the retarding pool' of The sedimentpool is the reservoir spaceallotted to the accumulation structure' the of incoming sedimentduring the life Sedimentstorageis the volume allocatedto total sedimentaccumulation' sedimentpool elevationis the elevationof the surfaceof the anticipated sedimentaccumulationat the dam. Anearthspitlwuyisanunvegetatedopenchannelspillwayinearthmaterials. Avegetatedspillwayisavegetatedopenchannelspillwayconstructedof earth materials. A ramp spillway is a vegetatedspillway constructedon the downstream faceof an earth dam. where The control section in an open channel spillway is that section acceleratedflow passesthrough critical depth' from The inlet channelof an emergencyspillwayis the channelupstream the control section.
368
CHAPTER16
HYDROLOGICDESIGN Top of dam
/ :Minimum spillwaycrestJ Emergency x r freeboard Surcharge Normalpoollevel\ v I storage Flood control retardingstorage
Emergency spillway
Reservoir Minimum Pool
Qo
reservoirpoollevelsandstoragezones' Figure 16.3 Multipurpose The exit channelof an emergencyspillwayis that portion of the channel downstreamfrom the control sectionwhich conductsthe flow safelyto a point where it may be releasedwithout jeopardizing the integrity of the structure. The emergencyspiltway lrydrographis that hydrographusedto establish the minimum designdimensionsof the emergencyspillway' Thefreeboard hydrographis the hydrographused to establishthe minimum elevationof the toP of the dam. Severalof thesefeaturesare illustrated in Fig' 16'3'
SmallDams provide Small damscustomarilyare designedusingtwo or more levelsof frequencyto 16.3 Figure an emergencyspillway and ensure an adequateallowable freeboard. (MF)showsa iypicit small dam with normal freeboard(NF) and minimal freeboard The freeboardvaluesfor earth dams with riprap protection on the upstreamslope' 100-mph shownin Tablq 16.3, atebasedon waverunup causedby storm winds with fetch The wind velocities.Minimal freeboardpertainsto wind velocitiesof 50 mph. If shore. is definedas the perpendiculardistancefrom the structureto the windward values smoothconcrerc;atirerthan riprap is usedon the upstreamface' the freeboard shown shouldbe increased50 percent'"
DESIGN-FREQUENCYCFIITERIA 16.3 HYDROLOGIC
369
TABLE 16.3 USBR RECOMMENDED NORMALAND MINIMUM FREEBOARDVALUES,FT Fetch (mi) 4
<1
3 +
I
6 8 10
2.5 5 10
5 o
7
Soarce:After Ref. 4.
of The U.S. Soil conservation Service designcriteria for principal spillways be should small dams are given in Table 16.4. The SCS TechnicalReleaseNo. 60 are frequency_requirements Design table.s this of consultedfor full interpretation SCS classifles selectedto fit the planned or foreseeableuse of the structures.The grouPs:u into three structures Class a. Structureslocated in rural or agricultural areas where failure mightdamagefarmbuildings,agriculturalland,ortownshiporcountry roads. ctass b. Structureslocated in predominantlyrural or agricultural areas where failure might damageiiolated homes, main highways.or minor railroads, o, "urrr]"interruption of use or serviceof relatively important
(
public utilities. Classc. Structureslocatedwhere failure might causeloss of life, serious public damageof homes,industrial and commercialbuildings,important utilities, main highways,or railroads, generally The physicalsizeof a small dam can rangeto over 100^ft in heightbut emerthe at of storage acre-ft is restrictedto structuresretarding lessthan 25,000 are if they attention gency spillway crest. Small dams generally receive.special Many life' of loss the cause could ion.i.o"i"d in populatedareaswhere dam failure possibilityexists,the flood deathshavebeencausedby dam or leveefailure. When this maximum precipiprobable the of designstorm for small damsis lstablishedby use of the maximization reasonable the tatio'n, PMP. The PMP is generally defined as definitions Other storm. maximum meteorologicalfactorsthaioperate to producea including: havebeen proPosed,T canbe 1. The p1itp is the rnaximumamountand duration of precipitationthat expectedto occur on a drainagebasin' 2. ThePMP is the flood that may be expectedfrom the most severecombination of critical meteorologicand hydrologicconditionsthat are reasonably possiblein the region. T'he pMp Las a low, but unknown, probability of occurrence.It is n-eitherthe maximumobserveddepthat the designlocation or region nor a value that is completelyimmune to exceedance'
370
CHAPTER16
HYDROLOGIC DESIGN
TABLE 16,4 SCS DESIGNCRITERIAFOR PRINCIPALSPILLWAYS OF SMALL DAMS
Class of dam (a)
Purpose of dam 5lngle-
irrigation only
SingIe or multiples
(b)
(c)
V"HI
Existing or planned upstream dams
Precioitationdata for maximum frequency2of use of emergencyspillwaytype:
Earth
Vegetated
Less than 30,000
None
0.sDrJ
O.5DL
Greaterthan 30,000
None
0.75DL
0.75DL
Lessthan 30,000 Greaterthan 30,000 All
None
!D5 0
None
0.5(Pso + Ploo)
Anyt
D r 100
Single or multiple
All
None or any
!
Single or multiple
All
None or any
Proo
D6 r25
0.5(Prs+ Pso) rD5 0
100
I Product of reservoir storagevolume V, (acre-feet) times effective height of dam 11, (feet). 2Precipitation depths for indicated return periods (years). 3Applies to irrigation dams on ephemeralstreamsin areaswhere mean annual rainfall is less than 25 in. aDL = designlife (years). 5Class (a) dams involving industrial or municipal water are to use minimum criteria equivalent to that of Class (b). 6In the case of a ramp spillway, the minimum criteria should be increasedfrom Prr to Pt66. ?Applies when the failure of the upstream dam may endangerthe lower dam. Soarce.'Soil Conservation Servtce.
Estimatesof PMP are basedon an investigationby the U.S. WeatherBureau conductedto establishthe maximumpossibleamountof precipitablewaterthat could Figure 16.4providesestimatesof precipbe achievedthroughoutthe United States.s'e itable water over watershedsbetweensealevel and 8,000 ft. Figure 16.5 extendsthe estimatesabove 8,000 ft.1 Point valuesof PMP for the samelocale may vary with duration of storm causingthe precipitation.Figure 16.6 providesPMP estimatesfor 6-hr storms.These and similar publishedcharts for other durations are helpful in selectingthe PMP for any region in the United States. the deiign frequenlies-for principal spillwaysfor small SCS Class aob, or c damsare providedin Table 16.5.Theseare basedon 6-hr rainfall depthsfor ( 1) the 1OO-year frequency(Fig. 16.7) and(2) the PMP (Fig. 16.6).Designstormdepthsfor all watershedshaving a time of concentration less than 6-hr are establishedin Table 16.5.For thosewatershedswith greatertime of concentration,adjustmentsare madeto the 6-hr storm depth to accountfor the gteateramountsof direct runoff in a longerperiod of time. Theseadjustmentsare discussedin Section 16.4.
16.3 HYDROLOGICDESIGN-FREQUENCYCRITERIA 371
DEPTHSOF PRECIPITABLE WATER IN A COLUMN OF MILLIBARS AIR OF GIVEN HEIGHT ABOVE 1OOO Assuming
saturation
with a pseudo-adiabatic
lapse
mte for the indicated surface temperatures Adapted ftom the U S W€ther Bueau Hyalrometeorological Repon No 23 TEMPEMrure
. 142230343842
46
505254
565860
62 64
66
68
'10
'72
74
'76
78
80 .F
800 a
-.: =
z p h
a H d
Extended to 200 mbar on Figure 16.5
1.5 1.0 ABLE WATER IN INCHES
Figure 16.4 Diagram for precipitable water determination from 1,000 to 700 millibars.(U.S. Bureauof Reclamation.)
L
372
16 HYDRoLoGIc CHAPTER DESIGN
DEPTHS OF PRECIPITABLE WATER IN A COLUMN OF AIR OF GIVEN HEIGHT ABOVE 1OOO MILLIBARS lapse Assuming saturation witha pseudo-adiabatic rate for the indicated surface temDeratwes Adapted from the U-S. Weather Bureau Hydrometeorolgical Report No 23 EMPEMTre
l8 26
66 70
74
200
2.0 2.5 3.0 3.5 1.5 WATERIN INCHES PRECIPITABLE
Figure 16.5 Diagram fbr precipitable water determination from 800 to 200 millibars.From 1281.103-D-1908.(U.S. Bureauof Reclamation.)
16.4 DESIGNSTORMS
373
MaiorStructures
16.4 DESIGNSTORMS Once the designfrequencyhasbeenestablished,the next stepin a structuredesignis the determinaiionof six ,io.- puru-eters: the stormduration,the durationof rainfall excess,the point depth, any ireal depth adjustment,the storm intensity and time distribution, and the areal distribution pattern'
Duration The length of storm usedby the SCS in designingemergencyand freeboardhydrographsior small damsis of 6-hr durationor /c, whicheveris greater' Often, the minor it*"tu.. being designedcannotbe justified economicallyon the basisof this length of storm. Foi many minor structures, particularly urban drainage structures, a designflood hydrogiaph is basedon a storm duration equal to the time of concentrati-onof the wateisneA.fnis procedureusesthe rational method of Chapter 15 or the synthetic unit hydrographs of Chapter 12 along with a critical storm pattern produced by arranging the rainfall excesspattern into the most critical sequence. fn" SCS uies 24-hr durations for all urban watershedstudies' but Durationsof approximately6hr or lessare satisfactoryfor small watersheds, 10 days' to up periods of for depths the lengthsof stormiln large areasrequire storm valuesare availablefor durationsof from 2 to I0 daysfor locations Freque-ncy-based within the United States.toSimilar data are also availablefor other selectedareas outsidethe United States.Generally,however,designcriteria for large damsrequire estimatesof storm depthsthat do not havefrequencylevelsassigned.
Durationof RainfallExcebs Initial rainfall during most stormsinfiltrates or is otherwiseabstracted,and the duration of excessrainTf,is lessthan the actual rain duration by an amount.equalto the time that initial abslractionsoccur. Excessrain duration 7s can be estimatedfor a 6-hr storm as a function of the curve number CN and precipitation P from
374
DESIGN CHAPTER16 HYDROLOGIC 125"
120"
€i 7t? "d, \ ,-o-
tv
Bureau, Figure 16.6 The 10-mi2or less PMP for 6-hr duration (in.). (U.S. Weather NOAA.)
storm Fig. 16.8. This family of curves was developedby the Sclrlt y-h:I" P is the zero represents 100 A CNof O""pttruna CNis a losi parameterdefined1lhaptgr {. lossessothatZg:6hrforCN:l00.Table16.6isusedtofindthedurationof is the excess rain for any storm duration greater than 6 hr. The rainfall ratio precipitation total the by divided (T'iUle 16.7) absrractionP* losi before runoff to amount P. The time ratio from Table 16.6 is multiplied by the rainfall duration obtain Ze.
0 s0 r00 95"
\ \
2?0 3?0 490 90.
Figure 16.6 Continued
Depth
available.
5P 85'
6u
386
CHAPTER16 HYDROLOGIC DESIGN 1.0 0.9 0.8 X \ o 0.7 0.6 .=a
0.5
0.3 v.z
0.1
I
2
3 4 Time(hr)
s
6
Figure16.18 A6-hrdesignstormdistribution for SCSdam design.(After Ref. 12.)
country. For more preciseinformation on boundariesin a statehavingmore than one storrntype, contact the respectiveSCS State ConservationEngineer. The greatestpeak flows from small basinsare usually causedby intense,brief rains. Thesecan occur as distinct eventsor as portions of a longer storm. The 24-hr storm duration is longerthan neededto determinepeaksfrom small watershedsbut is appropriatefor determiningrunoff volumes.In light of this, the SCS usesthem to studypeak flows, volumesof runoff, and direct iunoff hydrographsfrom watersheds normally studiedby the agency. Time distributionsfor PMP and other stormsusedin major structuredesigncan be constructedfrom Fig. 16.20.This family of curvesis usedby the U.S. Burlau of Reclamation6 in threegeographical zonesshownin Fig. 16.6.The corps of Engineers usesa distributioncurve similar to Fig. 16.18for 6-hr SpS analyses. Triangular Distribution The simplest design storm distribution is a triangular shape.Becausethe depth,P, andduration,D, of rain are alreadyestablished,the peak intensity,l-u,, is 2PfD, foundby solvingfor the height of the triangularhyetograih as shownin Fig. r6.2L.The only remainingdecisionis the time to the peak, The ratio 40. to/D has been investigatedfor a large number of storms at locationsin California, Illinois, Massachusetts, New Jersey,and North carolina. values range from about 0.3 to 0.5.17Once the triangle is constructed,the intensitiesat regulaiintervals may be graphicallyor analytically determinedfor input to the rainfall-runoff rnodelbeing usedfor design. Blocked IDF Distributions A frequentlyusedprocedurefor developinga design storm distribution for short duration storms (up to about 2 hr) is to successively construct blocks of a design storm.histogramby using the appropriate intensity-
16.4 DESIGNSTORMS
Rainfall distribution
f--l ryp"r ffi rvwIe i--l
rypeu
llffil rvnerrl rF-
V v
ooo
(b)
for zonesI' Figure 16.19 SCS 24-hr rainfall distributions:(a) 24-hr rainfall distributions (After Ref' 16') distributions' rainfall for scs IA, II, and III and (b) approximateboundaries
388
CHAPTER16
HYDROLOGICDESIGN 1.00 0.90 r
ORO
;
o.7o
E
o6n
tr
nsn
;
0.40
E o.3o o.20 0.10
Time (hr)
Figure 16.20 Distribution of 6-hr PMP for any area west of the 105' meridian. (After Ref. 6.)
o
Time, hr Figure 16.21 Triangulardesignhyetograph. duration-frequency curveto find therain intensitiesfor A/, 2 At,3 L,t,etc.,increments of time and then to organizethese"blocks" of rain intensitiesin somepattern,usually symmetrical,.around the center of the storm, making sure that the area under the hyetographis equal to the designstorm depth, P, spreadover the designstorm duration,D. To apply the procedure, successivedepths of equal-probability storms with durationsof A,t,2Lt,3A,t,4Lt,etc.,aredeterminedfrom the IDF curveandtabulated. Next, any of a variety of procedures,such as the alternating block method, the Chicagomethod,or the balancedmethod,are availablefor distributingtheseblocks and assuring that the total rain depth equals P. Most assumethat the highest
16.4 DESIGNSTORMS
389
intensityoccursin the middle of the storm,the secondhighestoccursnext, and so on, working out in both directions from the center block. The balanced method, for exampli, assumesthat a Ar-hr stormwith intensityia,from the IDF curvecould occur, with equal probability, during the middle of the D-hr designstorm. This intensity is plotted as the middle block of the designstorm hyetograph.Next, the rain depth for duration 2Lt is obtained from the IDF curve. Its distribution is assumedto be a two-bar histogramwith the first half matchingthe intensity of the Ar-hr storm; the secondhalf intensity is calculatedby spreadingthe rest of the rain depth for the 2 Al-hr durationunifoimly over the secondAr interval.The processis repeatedfor rain depthsfor stormswith durationsof 3A/, 4Lt, . .. , up to D. The goalis to developa storm hyetographsuchthat a storm of any duration,centeredat the middle of the blockedIDF hyetograph,will havea total rain depthmatchingthe rain depthfrom the IDF curve for the siven duration.
Areal Distribution precipitation depths can and do vary from point to point during a storm. Areal variation in designstorm depth is normally disregardedexceptin major structure designs.The usual approachin major structureanalysisis to selecta design(usually elliptical) or historic (transposed)isohyetalpattem for the PMP or SPS depth and assignprecipitation depthsio the isohyetsin a fashionthat givesthe desiredaverage depih over the basin. The averagedepth is determinedby the isohyetal method illustrated in ChaPter2. Four majorlypes of storm patterns are shown in Fig. 16.22 fot areasup to 400 mir. Thesewere identifiedby iluff in his analysisof midwesternstormpatterns.ll The letters H andL representareaswith high and low precipitation depths,respectively. The typical isohyetalpatternfor SPSstormshasbeenestablishedas generally Valley elliptical in itrapeas shownin nt. 16.23.This patternis usedby the Tennessee
?b
Figure 16.22 Major types of storm patterns: (a) closedelliptical, (b) open elliptical, (c) multicellular; and (d) banded.(After Huff.tl)
390
CHAPTER16 HYDROLOGIC DESIGN
t
t t Scale: miles
l
Figure 16.23 Generalizedpattern storm.
lsohyet
Area enclosed(mi2)
B C D E F G H
l1 45 114 279 546 903 1349 2508 4458
(After Ref.18.)
Authority (TVA)18for areasup to 3000 mi2.Variationsin the rainfall depth found in a standardproject storm will divergefrom a maximum at the storm centerto a value considerablylessthan the averagedepth at the edgesof the watershedboundaries. This variation can be determinedand incorporatedin the designstorm. A slightly modified isohyetalpattern for SPS storms is used by the Corps of Engineersre asshownin Fig. 16.24.Thepercentages shownfor isohyetsA, B, . . . , G are multiplied by the 96-hr SPS depth to give an elliptical pattern with the desired averagedepth.Similarmapsfor 24.,48-, or 72-hr stormscanbe obtainedsimplyby modifyingthe 96-hrpercentages of Fig. 16.24.Thisis accomplished usingthe deptharea-duration curvesin Fig. I6.24.For example, if a24-hr stormis used,first notethat theA isohyetof Fig. 16.24encloses an areaof 16nrr?.From Fig. 16,25thecorresponding SPSpercentagefor a24-hr stormis 116 percentrather than the 140percentvalue used with a 96-hr storm. Thereforethe pattern percentagesvary with the selected designstorm duration. An additional aid for constructingdesignstorm distributionsover smallermidwesternl8watersheds(up to 400 mi') is presentedin Table 16.10.The ratio of maximum point rainfall to mean rainfall over the basin is provided and can be used to estimatethe maximum depth occurring at a storm centerif the mean areal depth is
16.5 CRITICALEVENT METHODS
l
1
r
r
0
r
5
r
l
0
l
1
: miles
391
l
0
2
0
Figure 16.24 GeneralizedSPS isohyetal pattern for a 96-hr storm' The pattern may be orientedin any direction and may correspondto the deptharearelation representedby a 96-hr storm.
lsohyet A
B C D E F G
Area (mi2)
l6 100 320 800 1800 3700 7100
(After Ref. 19.)
.
known. Ratios for 50-, 100-, and 200-nr2 areasare equal to those in Table 16.10 multiplied by 0.91, 0.94, and 0.97, respectivd. For uniform rainfall the 95 percent ratios of the table are recornmended.With extreme Variability the 5 percent ratio applies.The 50 percentratios approximateaverageconditions'
EVENTULrnoos 16.5CRTilCAL
392
CHAPTER16
DESIGN HYDROLOGIC 20,000 10,000
\ +
5,000 4,000 3,000
\\\
s=
2,000 1,000 o
o
72-hperiod
,*Nr" 500 400 300
\\ \
200
I
\
100 50 40 30
\ \
20 10 40
60
120 100 80 value of SPS IsohyetPercentage
\
140
Figure 16.25 SPS depth-area-duration curves by 24-hr storm increments. (After Ref. 19.)
future event, and then design accordingly. These methods include the use of the probable rnaximum precipitation PMP, probable maximum flood PMF, record high storm depths, record high floods, multiples of frequency-basedfloods, and paleohydrology.
ProbableMaximumPrecipitation Probablemaximum precipitation depthsfor drainagebasinsin the United Statesare providedin the respectiveNational WeatherServiceHMRs2oidentified in Fig. 16.2. The probablemaximum stormis deflnedasthe most severestorm consideredreasonably possibleto occur.The resultingprobablemaximumflood is customarilyobtained by usingunit hydrographsand rainfall estimatesof the PMP preparedby the National rily'eather Service2l(seeFigs. 16.6 and 16.26).
EVENTMETHODS 393 16.5 CRTTTCAL ON 400 miz POINTTO MEANRAINFALL TABLE16.10 RATIOOF MAXIMUM Mean rainfall(in.) Rainfallperiod (hr)
1.0
1.5
2.O
4.0
5.0
1.30 1.35 1.41 1.46 1.52 1.60 1.65 r.69 1.77
t.26 1.30 r.33 r.36 1.43 1.50 1.54 1.57 1.63
1.22 1.25 r.28 1.30 1.35 1.40 1.43 1.45 1.50
1.16 1.20 1.24 t.28 1.33 r.39 1.43 1.47 1.53
|.14 1.18 Lzr 1.23 r.28 1.32 1.35 1.38 r.46
l.l2 1.16 1.19 t.22 1.26 1.30 1.32 1.33 1.38
1.13 l.l7 1.20 1.22 1.27 1 . 3I 1.33 1.35 1.40
1.11 1.15 1.18 r.20 1.24 1.28 1.30 t.32 1.36
1.10 1.r4 1.16 1.18 1.21 1.25 1.27 1.29 1.33
2.5
5% Probabilitylevel ratios (Storms with extreme variation in intensity)
0.5 1 2 3 6 l2 18
5.20 5.50 5.80 6.05
)4
48
0.5 I L
3 6 12 l8
2.66 3.03 3.46 3.77
JA
48
0.5 I
2 J
6 t2 l8 )4 48
2.38 2.75 3.15 3.46
r.4r 2.r8 1.70 1.48 1.80 2.29 1.55 1.90 2.44 1.99 1.61 2.53 r.72 2.r2 J.t I 2.69 1.83 2.25 2.86 4.01 1.90 2.33 2.96 4.14 1.96 2.40 3.05 4.27 2.08 2.55 3.25 levelratios 50% Probability (Stormswith averagetimedislributions) 1.22 1.32 1.5'1 2.02 r.27 1.65 L39 2.r5 1.32 1.46 r.75 2.29 1.38 1.85 2.42 1.52 r.43 r.63 1.98 2.59 1.50 r.75 2.12 2.78 1.57 1.81 2.20 2.89 1.60 2.28 1.87 3.00 2.44 3.r7 .1.68 ,1.99 levelratios 95%Probability (Stormswith uniformintensities) r.16 r.28 1.18 1.53 1.20 1.23 1,38 1.72 1.24 1.28 1.47 1.90 1.27 1.53 1.33 2.02 1.31 r.43 2.24 t.67 1.38 1.78 2.50 1.50 r.4l 1.53 1.89 2.67 r.43 1.58 1.92 2.77 r.47 r.64 2.04 3.07 3.00 3.21 3.38 3.54
Sorrce.'After Huff.lr
A proposedmethodto estimatePMP advocatedby Hershfield2lsuggeststhat the 24-hr PMP at a point be computedby the equation PMP24:P*KS,
where PMPZ : . F : K: , & :
(16.1)
the 24-hr probablemaximum precipitation the meanof the 24-hr annualmaximumsover the period of record a constantequalto 15 the standarddeviationof the 24-hr annual maximums
Adjustmentsto the value of F and S, for the record length are noted by Hershfield. However, for appraisalpurposesthese adjustmentsprobably will not significantly alter resultsmore than 5-10 percent.
394
CHAPTER16
HYDROLOGIC DESIGN
\
\
zo.t
,/
/
\\ - - - - -20.1(3) -"..-
I
ptvtp(in.). (1) Alexandria, Figure16.26 Twenty-four-hour 2000-mi2 LA, June13-17,1886.(2)Eautaw, AL, April 15-18,1900.(3)Elba,AL, March l1- 16,1929.(4) Yankeetown, NC, FL, September 3=7, 1950.(5) Altapass, July13-17,1916.(6)Jefferson, 10-13,1878.(AfterRef. 18.) OH,September The U.S.Bureauof Reclamationunderwentconsiderableevaluationof its design criteria for new damsand for safetyevaluationof existingdams,following the Teton, Idaho, dam failure in 1976.The policy adoptedfor modification of existingdamsis first to determinewhether they will accommodatethe peak dischargeof the PMF without overtopping.In addition, the dam and appurtenantfeaturesmust accommodateat leastthe first 80 percentof the PMF volumewithout failure. For embankment dams,failure is assumedto occur if overtoppinglevelsare reached.
Recordedbdremes-Creager, and Crippenand Bue EnvelopeCurves Where frequency-basedmethodsof PMP/PMF studiesare unwarranted,designfor critical eventscanbe basedon the greatestrecordedrain or flood flow for the location. Similarly, tablesor curvesof flood data can be developedto give the maximum floods of record in the region under study; seeCreagerflood envelopecurvesinFig. 16.27.
395
16.5 CRITICALEVENT METHODS 10.000 5,000 3,000 2,000 1,000
500 300
$ zoo q? 100
€ s 0 ii
JU
20 10
= {6gtrQs9ae
o 048-11
5 3 2 I
N-h
o R 9 -3 ' 8 8 8 8 8 3 8 3 I i;;". .N 6 -A 6 Drainage area (mi2)
o
o o
o
o
o *
o o
o
o
N o
h
o
Figure 16.27 Creagerenvelopecurves: O peak inflow for Harza Projects; 'recorded unusualflood discharges.(1) Congo at Inga, Congo.(2) Tigris at Samarra,Iraq. (3) Caroni at Guri, Venezuela.(4) Tigris at Eski Mosul, Iraq. (5) Jhelumat Mangla, Pakistan. (6) Diyala at DerbendiKhan, Iraq. (7) GreaterZab at Bekhme,Iraq. (8) Surinameat Brokopondo,Suriname.(9) LesserZab at DokenDam,Iraq. ( 10)PearlRiver,U.S"A.(11) Cowlitz at Mayfield, U.S.A. (12) Cowlitz at Mossyrock,U.S.A. (13) Karadj, Iran. (14) Agno at Ambuklao,Philippines.(15) Angat, Philippines.(16). Tachien,Formosa. (Nole.'Curves taken from Hydroelectric Handbook, by Creagerand Justin. New York: Wiley, 1950.)
In cases where estimates of PMP have not been made. volumes of rainfall to be expected can also be approximated from Creager rainfall envelope curves of the world record rainfalls as depicted in Fig. 16.28. Maximum flood flow data for 883 sites up to 25,900 sq km formed the basis for the Crippen and Bue envelope equation given by 4o
:
A)31 6rlosA+caGogA)2+ca(log lgfc r+
(16.2)
where qo is the maximum flow (m3/sec),A is the drainage area (sq km) and the coefficientsare from Table 16-11 usingFigure 16.29.
StandardProjectStorm The standardproject stormis anotherrainfall depththat is usedin the designof large dams. This value is usually obtainedfrom a survey of severestormsin the general vicinity of the drainagebasin. The storm selectedas the SPS may be oriented to produce the maximum amount of runoff for the SPF. Alternatively, severestorms experiencedin meteorologically"similar" areascan be transposedover the study atea.
396
HYDROLOGICDESIGN
CHAPTER16
1
2000 1000 600 400
A
/:fr>y%^"1", ndia
200
g
? 1oo ; 6 0 s 4 0 & 2 0 10 6 4 2
ffi"*"[,,"1,
anlalca
runkiKO , Formo t a aquio, Pldl .ppineIr landr
Fr-
-n"irpfii,W
Ihra]l, TX imethport, I A -
tri--ftTl- tIT ftL{
--J-f-1lHoit,MOreaDe Arses, Bglqeqla ?oint,Jamaica I | | I Plumb.
ssen, Bal
I Unionville \4d. 2 4 6810 20 4060
6
3
t2
5
24
10 2030
9t2
24
+.>+/>
j
Months
DaYs
Hours Dulatron
Figure16.28Creagercurvesofworld'sgreatestrainfalls.(AfterRef.18')
FOR CRIPPENAND BUE PEAK DISCHARGE TABLE 16,11 COEFFICIENTS CURVES ENVELOPE Fig.16.29 Region I
2 3 4 5
6 '7 8 9 10 11 12 IJ
I4 15 16 L I
Nationwide
Coefficients
Upperlimit(sqkm) 26000 7800 26000 26000 26000 26000 26000 26000 26000 2600 26000 18100 26000 26000 50 2600 26000 2600
Source: AfIer Crippen,J. R., and C. D. Bue, WaterSupPlYPaPer1887, 1977.
c1 3.203865 3.4'10923 3.330746 3.258400 3.126412 3.500489 3.326333 3.236183 3.503734 3.314692 3.231389 3.596209 3.461373 3.07349'l 3,451746 3.s65536 3.389030 3.743026
c2 .8049163 .74'72908 .8443r24 .8906783 .796472r .9123848 .8503960 .9193289 .8054884 1.0386350 .8867450 .8806263 .8519276 .64'727rO .9718339 .9699340 .9445212 .7918884
c3
c4
-.002975'7 -.0394382 - . 0 5 5 1 7 8 0 -.0000965 -.0642062. -.0021362 -.0870959 .0022803 -.0899000 .0022'744 -.1013380 .00496r4 -.0998'74'7 .0042129 -.0947436 .0029486 -.0890172 ,0018961 -.059'7463 -.0042542 -.102053s .0045531 -.0747598 .0000138 -.1094456 .0058948 -.0038285 -.0252243 -.00s7110 -.0617496 -.0034'776 -.0649503 - . 0 6 7 8 1 3 1 -.002'7647 .0244991 -.0192899
..Maximum Flood flows in The ConterminousUnited States,''U.S.G.S
o !
k c
3 F 6'* *
r
o F F : o o
ll
6 -
+
S B (hE
r N
u)
+ f
O
Q E ( o
e a e 6
h € d 6 h
6 o d d n
+ $ i r N n
i h n i o N d d r ; O N N n h n
oa
Fcr
3 .* @'a 9 -
F * ,'! o
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DESIGN CHAPTER16 HYDROLOGIC
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including storm Figure 16.36 Children's artist rendering of urban undergroundsysteT, with perReprinted Macaulay. David by 1976 A driins. From UNDERGROUND,Copyrighl missionof HoughtonMifflin Co' Al1 rights reserved'
and detention in gutters, house drains, catchbasins,and the storm sewer systems, interceptionin extensivelylandscapedlocations' Two items normally accountedfor in urban storm drain designare: many 1, Infil,tration. The ability of the soil to infiltrate water dependson given characteristicsof the soil as noted in chapter 3. The rangeof values continuous in the following table is typical of variousbare soils after t hr of rainfall' RATEs rvircnr- |NFILTRATIoN Soilgroup
(in./hr) Infiltration
High (sandy,oPen-structured) Intermediate(loam) Low (clay,close-structured)
0.50-1.00 0.10-0.50 0.01-0.10
The influenceof grasscover increasesthesevalues3 to 7'5 times'
16.8 FLOODPLAINANALYSIS
409
2. Retention This is usually assumedto be 0.10 in. for pervioussurfacessuch as lawns and normal urban pervious surfaces' Developmentof hydrologicparametersfor designof storm sewerpipes, street gurrers,or detentionbasinsis by the rational method(or modified rational rnethodiee Chapter25) when peak flow ratesand approximatehydrographsare adequate,or unit hydrographand kinematic wave hydrographsynthesismethodswhen greater detail is needed.The latter usually involveuseof public domainor vendor-developed stormwaterdesignsoftware.The hydrologicaspectsof computerizedhydrologicdetext' signtools are deiailedin Chapter25. In addition to the material presentedin this ILLUDAS' method, rational modified descriptionsof usesof the rational method, TR-sj, SWMM, DR3M, and other tools in designingurban storrrtdrainagefacilities are addressedin numerousurban drainagedesigntextsand handbooks.Additionally, many statedepartmentsof transportationor city and county engineer'soffices have developedtocattyapplicabledrainagedesignmanuals.As well, the American Society i'model" drainagedesignmanual for local adaptaof Civil Engineeishas developeda tion, availableby contacting ASCE in New York. Finally, the discussionof urhan "shopper'sguide" to urban drainageanalysis modelsin Chaptlr 25 includesa useful and designsoftware.
16.8 FLOODPLAIN ANALYSIS
.
Due to heavymonetary and other floodplainlossesover the years,the federalgovernment hasbeenconductingstudiesof floodplainsof the nation's waterwaysand methods of protecting life und prop"tty and pieventing overdevelopmentthat causesinin creasedwater levelsand more widespreadflooding. Hydrology is a key ingredient open dams, of effects studying rates, flow potential these studies for identifying channels,and.otherwater control structureson hydrographsand determiningvolumes of floodwatersthat will needto be safelystoredand conveyedby the waterwaysand floodplains.
U.S.NationalFlood lnsuranceProgram(NFIP) In 1968, the U.S. Department of Housing and Urban Development(HUD), later to called the FederalEmlrgency ManagementAgency (FEMA), initiated the NFIP of mapping with floodplains of identify flood hazard areasand to provide occupants the flood-proneareasand accesstolorv-costflood insurance'The NFIP requireslocal gou"rn-"nt5, to adoptand implementflood managementprogramsthat preventdevelopmentsin excessof national standards' Since the inception of the National Flood InsuranceProgram, flood hazatd The areas have been mapped in over 18,000 communities in the United States. maintenance a to converted since has and programcost over $t.O Uittion to complete effort of updating and expandingthe maps as developmentsoccur'-Each of these for studieshas required eithei approximateor detailedevaluationof peak flow rates has base the called discharge, flood, a range of recurrenceintervals. The 100-year beendeterminedin all cases.The portion ofthe floodplainoccupiedby the baseflood
4'10
oHAPTER 16 HYDRoLoGIc DESIGN
has been mapped,allowing communitiesto determinewhether a property is in the 100-yr floodplain,and in many cases,what water surfaceelevationwould be experiencedat the property during the baseflood. Figure 16.37illustratesthe typical NFIP mappingand floodplainmanagement procedure.Surveyedvalley and channelcross-sectionsare used in determiningthe 100-yr flow depth, allowing the hydrologistto delineatethe lateral extentof flooding duringthe 100-yrflood. Then afloodwaywidth is generallydeterminedasthat portion of the floodplain that is reservedin order to dischargethe 100-yearflood without cumulativelyincreasingthe water surfacemore than 1.0 ft. This procedureis illustrated in Fig, 16.38.The floodwayis most often centeredover the main stream channel,but can be offset or even split into severalzones. Developmentwithin the floodwayis allowed only if compensatedby relocating the floodway or mitigating the water surfaceincreasedue to the development.The flood fringe is that portion of the floodplain outsidethe floodwayin which development is allowed,up to a point of full encroachmentby buildings,roadbeds,berms,and so forth. As much as sevento ten percentof the total land area of the United States lies within the 100-yearfloodplain.The largestareasof floodplainare in the southern parts of the country, and the most populatedfloodplainsare alongthe north Atlantic coast,the GreatLakesregion,and in California. The floodplain mapping effort produced a large amount of data and analyses useful to designhydrologists.The productsof the program include: 1. The 10-, 50-, 100-, and 500-yearfrequencydischargefor streams. 2. The 10-, 50-, 100-,and 500-yearflood elevationsfor riverine,coastal,and lacustrinefloodplains.
Floodway Flood Fringe
i
_
I
Flood Fringe
',100_year"Floodplain
I
Channel
Figure 16.37 Definition sketchof floodplain delineations.
16,8 FLOODPLAINANALYSIS
411
100Year FloodPlain Floodway
Flood Fringe
Encroachment
the floodwaywidth' for determining Figure 16.38 Procedure 3.The100-and500-yearmappedfloodplaindelineationsatscalesranging from 1:4800to 1:24,000' 4. The 100-yearfloodwaydata and mapping' wave haz5. Coastalhigh hazard irea mapping 1*"u* subjectto significant ards). 6. FloodwaYflow velocities' 7. Insurancerisk zones' This information is provided in the form of three products: |.FloodlnsurancestudyReportsprovidegeneralprogramandcommunity floodway backgroundinformation,tauutateaflood dischatgedata,tabulated. tabuinformation' surcharge datalncluding velocity, floodwaywidth, and and 100-' 50-' 10-' the of lated flood insurance'zonedati, and profi'les flooding' riverine for 500-yearflood elevationversusstreamdistances of the 1002. Flood InsuranceRateMaps (FIRM maps)provide delineations and500-yearfloodplains,basefloodelevations'coastalhighhazardareas' andinsuranceriskzonesonaplanimetricbaseatascalebetweenl:4800 and 1:24,000. 3.FloodBoundaryHazardMapsprovidedelineationsofthel00-and500crosssections yearfloodplains,locationsof surveyedfloodplainand channel on a floodway 100-year the of usedin hydraulic analyses,and delineations 1:24'000' and 1:4800 planimetricor topographicbaseat a scalebetween
HydrologYfor FloodPlainStudies
NFIP studiesare basedon Flood flow frequency estimatesfor gaugedlocationsin records.Annual peak log-pearsonType III (seechapter 27; analysisof streamflow recommendedby FEMA' flows and historical data arcfitted accordingto procedures
412
CHAPTER16
HYDROLOGIGDESIGN
For ungaugedlocations,flood flow frequency'estimatesare developedthrough regionalfrequencyanalysisor throughrainfall-runoff modeling.Equationspublished by the U.S. GeologicalSurveyrelatepeak dfschargesof variousfrequenciesto various drainagebasincharacteristicssuchassize,slope,elevation,shape,andland use.These equationsare developedusing multiple regressiontechniques(see Chapter 26) at gaugedsitesthroughoutthe region. Rainfall-runoff modelingtechniques(Chapter24) use syntheticrainfall hyetographs.Storm-eventmodels, such as the Corps HEC-I and SCS TR-20 packages, employdesignstormsof particular frequenciesand then mathematicallysimulatethe physical runoff process.The resulting peak dischargeis assumedto have the same frequencyas the rainfall.
U.S.Flood Hazards Despiteconsiderableeffort and expenditurein identificationof floodplainsand flood hazardareas,dam failuresand other catastrophiescontinueto resultin severedamage to life, property, and the environment.Floods from hurricanes,intenserainstorms, and rapid snowmelt or structure failure have all contributed to the loss of life. A tabulationof eventscausingmore than 100 deathsin the United Statesis providedin Table 16.15. As indicated, the majority are hurricane related, principally concentrated in the east-coastand Gulf of Mexico regionsas sfown in Fig. 16.39. Monetary lossesfrom floodsare also large.Table 16.16showsa numberof past producingover $50 million in flood damageseach,given in 1966dollars. floods U.S. Collectively, these floods have produced flood dam4gesin billions of dollars, disI tributed through the yearsas shown in Fig. 16.40. The Federal InsuranceAdministration evaluatedthe floodplain areas in the communitiesmappedby FEMA. By using demographicand economicinformation, projectionsof future property at risk of flooding could be made.Results.suggest that property in have occurred of investments in flood damageable floodplains. billions Table 16.17lists the breakdown,by state,of estimated1990 developmentvalue that will be in harm's way.
Dam Break Hazards '
.-"-fabir- 16.18lists the outflow rates,peak depth, and storageat the time of failure for 18 significantdam failures in the United States.The death rate for dam failures is relatedto the polpulation at risk (PAR). This term describesthosepeoplewho would need to take someaction to avoid the rising water. Figures 16.41 and 16.42showthe lossesas functions of PAR for low (lessthan 1.5 hr) and high (greater than 1.5 hr) advancewarning times, respectively.The high-warning-timelossesare significantlyless.This strongly supportsthe incorporation of early warningand flood delayfeaturesin the designof any structure.Data used in plotting Figs. 16.41 and 16.42 are given in Table 16.19. Table 16.20providesa typical time line requiredfor alerting downstreamresidentsof a severestorm and potential dam failure. The valuesgiven are hypothetical, and apply to an assumed15-mi reach betweenthe storm center and the populated atea.
SUMMARY 413 TABLE 16.15 FLOODSCAUSING1OOOR MOREDEATHSIN THE UNITEDSTATES
Year
Streamor place
Lives lost
1831 1856 1874 1875 1886 1889
BaratariaIsle, LA Isle Derniere,LA ConnecticutRiver tributarY Indianola,TX
1893 1899 1900
Vic. GrandIsle,'LA PuertoRico Galveston,TX
150 320 t43 176 150 2t00 2000 3000 6000+ 100+ 247 151 700 46'7 177 550 284 t20 350 100+ t20 300 2400 350 ,))\
1903 1903 1906 1909 1913 1913 1915 l9l9 l92l 1926 1927 1927 1928 1928 1928 1932 1935 1935 1936 1937 1938 1955 195'l 1960 1972 ' 1972 t1976
Sabine,TX Johnstown,PA
Central States HePPner,OR Gulf coast Gulf coast-New Orleans Miami, Muskingham,and Ohio Rivers Brazos River, TX Louisianaand TexasGulf coast Louisianaand TexasGulf coast Upper ArkansasRiver Miami and Clewiston,FL Lower MississiPPiRiver Vermont Puerto Rico Lake Okeechobee,FL San Francisco,CA PuertoRico Florida KeYs RePublicanRiver, KS, NE NortheasternUnited States Ohio River New Englandcoast NortheasternUnited States Westcoast,LA PuertoRico Buffalo Creek, WV RaPid.Creek,SD Big Thompson,Co
400 110
ro7 137 200 115 556 r07 t25 245 r39
Cause Hurricane tidal flood Hurricane tidal flosd Dam failure Hurricane tidal flood Hurricane tidal flood Dam failure Hurricane tidal flood Hurricane tide and waves Hurricane tidal flood Rainfall-river floods Rainfall-river floods Hurricane tidal flood Hurricane tidal flood Rainfall-river floods Rainfall-river floods Hurricane tidal flood Hurricane tidal floo
414
DESIGN 16 HYDROLOGIC CHAPTER
Figure 16.39 Number by state of major hurricanesin the United States, I 899-1989.(Soarce.'National Hurricane Center, National WeatherService, NOAA.)
Figure 16.40 Average annual flood damagesin the U.S., 1916-85. (Source:National WeatherService,NOAA.)
Damages Damages (1985 $) Damages/200 million population (1985 $)
rg2o 25
30
35 40
45
50
55
60
65
Last Yearof five'YearPeriod
70
75
80 8s
TABLE 16.16 FLOODSRESULTINGIN DAMAGESEXCEEDING$50 MILLION IN THE UNITEDSTATES Damage($ millions)
Year
Streamor place
t844 Upper MississiPPiRiver 1889 Johnstown,PA 1900 Galveston,TX 1903 Passaicand DelawareRivers 1903 Missouri River basin 1 9 1 3 Ohio River basin t913 Brazos and ColoradoRivers, TX 1921 ArkansasRiver 1926 Miami and Clewiston,FL t926 Illinois River 1927 New England 1927 Lower MississiPPi 1928 Puerto Rico 1935 Susquehannaand DelawareRivers r936 NortheasternUnited States t936 Ohio River basin 1937 Ohio River basin 1 9 3 8 New England sffeams 1938 California streams t942 Mid-Atlantic. coastalstreams 1943 Central states 1944 South Florida 1944 Missouii River basin 1945 Hudson River basin 1945 South Florida 1945 Ohio River basin 1947 South Florida t947 Missouri River basin 1948 Columbia River basin 1950 San JoaquinRiver, CA KansasRiver basin 1 9 5I 1952 Missouri River basin 1952 Upper MississiPPiRiver 1954 New England streams 1955 NortheasternUnited States 1955 California and Oregonstreams 1957 Ohio River basin 1957 Texas rivers 1959 Ohio River basin 1960 South Florida 1 9 6 1 Texascoast 1964 Florida 1964 Ohio River basin 1964 California streams 1964 Columbia River -North Pacific t965 South Florida 1965 Upper MississiPPiRiver 1965 Platte River, CO, NE 1965 ArkansasRiver, CO, KS 1965 New Orleans and vicinitY ,N.A. : not available. Source:IJ.S.WaterResourcesCouncil, 1968.
dollars 1966 Dollars Contemporary N.A." .A
25 25 50 150 t28 t3 '7rl
N.A, 50 284 50 36 zzl
150 418 125 100 28 172 63 52 24 54 34 60 178 to2 883 180 198 180 684 271 65 144 lt4 78 300 32s 106 t'73 289 r39 158 19l 61 322
1,161 84 100 273 N.A, 516 349 64 130 ]I
178 N,A. 90 185 374 3tl
996 3'76 294 103 N.A. rt7 N.A. 75 98 6l 88 N.A. 226 57 N.A. N.A. N.A. 216 879 405 72 188 r20 86 336 3+Z
r12 183 3lL 144 r62 N.A, 65 338
Cause Rainfall-river flood Dam failure Hurricane tidal floods Rainfall and dam failure Rainfall-river flood Rainfall-river flood Hurricane rainfall-river floods Rainfall-river flood Hurricanetidal and river floods Rainfall-river floods Rainfall-river flood Rainfall-river flood Hurricane tide and waves Rainfall-river flood Rainfall-river flood Rainfall- snowmelt fl ood Rainfall-river flood Hurricanetidal and river floods Rainfall-river floods Rainfall-river floods Rainfall-river floods Huriicane tidal and river floodb Rainfall-river floods Rainfall-river floods Hurricane tidal and,rivet floods Rainfall-river floods Hurricanetidal and river floods Rainfall-river floods Rainfall-river floods Rainfall-river floods Rainfall-river floods Snowmeltfloods Rainfall-river floods Hurricane tidal floods Huiricane tidal and river floods Rainfall-river floods Rainfall-river floods Rainfall-river floods Rainfall-river floods Hurricanetidal and river floods Hurricane tidal floods Hurricanetidal and river floods Rainfall-river floods Rainfall-river floods Rainfall-river floods Hurricanetidal and river floods Rainfall-snowmelt river flood Rainfall-river floods Rainfall-river floods Hurricane tidal flood
PROPERryVALUEAT RISKFROMFLOODING, TABLE 16.17 ESTIMATED RANKEDIN DECREASING ORDER.BASEDON 1990COSTS Rank
State
1 2 3
California Florida Texas Louisiana Arizona New Iersey New York Illinois Massachusetts Pennsylvania Virginia Maryland Washington Ohio Michigan North Carolina Wisconsin Georgia Connecticut Missouri Indiana Minnesota Nebraska Oklahoma Alabama South Carolina Tennessee Colorado Oregon Mississippi New Mexico Kansas Iowa RhodeIsland Kentucky North Dakota Urah Nevada Arkansas Delaware Maine West Virginia New Hampshire South Dakota Idaho Hawaii Vermont Wyoming Montana Alaska
5 6 7 8 9 l0 tl 12 13 L4 15 16 l7 18 1,9 20 2l 22 z3 /)A
25 26 2'7 28 29 30 3l JZ
33 3+
35 36 3 t
38 39 40 4I 4) 43
44 4f
46 47 48 49 50
Propertyvalue,X $1000 163,323,1,92 131,548,814 72,376,950 45,402,322 45,094,183 38,945,265 32,005,900 26,880,755 23,8t3,115 18,888,390 17,441,420 16,330,448 t6,245,009 t5,273,r47 13,449,078 12,993,067 12,r8r,725 11,832,494 tt,1r7,290 11,654,861 r0,786,741 10,655,t64 10,360,574t 9,501,778 9,274,903 9,220,305 8,037,425 7,137,757 6,861,790 6,134,073 5,519,278 5,279,t94 5,26r,678 4,312,117 4,r70,637 3,924,872 3,812,936 3,437,813 3,005,rs0 2,954,467 2,416,322 2,098,262 t,991,453 1,430,610 1,39t,498 1,323,90s 1,091,099 1,081,460 881,661 647,81,8
"status of Floodplain Hazard Evaluation Under the National Flood Insurance Program," Source: B. R. Mrazik, Emergency ManagementAgency, Washington,DC, 1986. _ fed9r91
17.3 SUBSURFACE DISTRIBUTION OFWATER 429 locations it has become more important than overdrafts of groundwater supplies. Today,the hydrologistmust be concernedwith both the qualitli and quantity aspecrs of groundwater.Furthermore, there is emerging an increasing specialization in groundwaterquality modeling.This latter type of modelingis ge;erally beyondthe scopeof this text but information on this topic may be founOin'Refs. j-6.
17.2 GROUNDWATER FLOW-GENERALPROPEFTIES Understandingthe movementof groundwaterrequiresa knowledgeof the time and spacedependencies of the flow, the nature of thJporous medium and fluid, and the ,boundariesof the flow system. Groundwaterflowsareusuallythree-dimensional.Unfortunately,the solution of suchproblemsby analytic methodsis complexunlessthe systemis symmetric.7,8 In othercases,spacedependencyin oneofthe ioordinate direciions.uy 6" so slightthat assumptionof two-dimensionalflow is satisfactory.Many problemsof practical importancefall into this class.Sometimesone-dimensional flow can be assumed,thus further simplifying the solution. Fluid propertiessuchas velocity, pressure,temperature,density,and vis6osity often vary_intime and space.When timi dependencyoccurs,the issueis termed an unsteadyflow problem and solutionsare usually difficult. On the other hand, situations where-spacedependencyaloneexistsare iteady flow problems.Only nomogeneous (single-phase)fluids are consideredhere. For a discussionof muliiple phase flow, Refs.5 and g are recommended. Bou-ndaries to groundwaterflow systemsmay be fixed geologicstructuresor free water surfacesthat are dependentfor their position on the stateol the flow. A hydrologistmust be ableto definetheseboundariesmathematicallyif the groundwater flow problemsare to be solved. Porousmedia through which groundwatersflow may be classifiedas isotropic, anisotropic,heterogeneous, homogeneous, or severalpossiblecombinationsofthese. An isotropic medium has uniform properties in all directions from a given point. Anisotropic mediahaveone or more propertiesthat dependon a given diiection. For example,permeabilityof the medium might be greateialong a horizontal plane than alonga vertical.one.Heterogeneoas mediahavenonuniform-propertiesof umrotropy or isotropy, while homogeneous media are uniform in their ihaiacteristics.
17.3 SUBSURFACE DISTRIBUTION OF WATER
sotLs,ANDGEoLocy cHAprER17 GR.,NDWATER,
4;
l. Soilwater zone.A soil water zonebeginsat the ground surfaceand extends downward through the major root band. Its total depth is variable and
:ffi::J; ,:'##l":ff l;"lhr?ff fi',,nH#,:?'l*?,T#*.Ji'.i,'"1:f,
'
may be encounteredin this region: hygroscopicwater, which is adsorbed from the air; capillary water, held by surfacetension; and gravitational water, which is excesssoil water draining through the soil' 2. Intermediatezone.This belt extendsfrom the bottom of the soil-watetzone to the top of the capillary fringe and may ghangefrom nonexistenceto severalhundredfeet in thickness.The zoneis essentiallya connectinglink between a near-ground surface region and the near-water-tableregion through which infiltrating fluids must pass. 3. Capiltary zone. Acapillary zoneextendsfrom the watertable (Fig.I7 '2) to a height determinedby the capillary rise that can be generatedin the soil. The capillary band thicknessis a function of soil textureand may fluctuate not only from region to region but also within a local area. 4. Saturatedzone. In the saturatedzone, groundwaterfills the pore spaces completelyand porosity is thereforea direct measureof storagevolume. Part of this water (speciflcretention) cannot be removedby pumpirygor drainagebecauseofmolecular and surfacetensionforces.Specificretention is the ratio of volume of water retained againstgravity drainageto gloss volume of the soil.
Waterthat can be drainedfrom a soil by gravity is known as the specificyield. It is expressedasthe ratio of the volumeof waterthat can be drainedby gravity to the grossvolumeof the soil. Valuesof speciflcyield dependon the soil particle size,shape and distribution of pores, and degreeof compactionof the soil. Averagevaluesfor alluvial aquifersrange from 10 to 20 percent. Meinzer and others have developed proceduresfor determiningthe specificyield.12
CONSIDERATIONS 17.4 GEOLOGIC The determinationof groundwatervolumesand flow ratesrequiresa thoroughknowledgeof the geologyof a groundwaterbasin.In bedrock areas,hydrologiccharacteristics of the rocks,that is, their location,size,orientation,and ability to storeor transmit water,mustbe known. In unconsolidatedrock areas,basinsoften containhundredsto to unconsolidatedfill depositsthat originated thousandsof feet of semiconsolidated from the erosionof headwaterareas.Suchfllls often contain extensivequantitiesof storedwater. The characteristicsof thesebasin fills must be evaluated. A knowledge of the distribution and nature of geohydrologicunits such as aquifurs,aquifugis, andaquicludesis essentialto proper planningfor developmentor managementof groundwatersupplies.In addition,bedrockbasinboundariesmustbe locatedand an evaluationmadeof their leakagecharacteristics. An aquifer is a water-bearingstratumor formation that is capableof transmitting waterin quantitiessufficientto permit development.Aquifers may be considered
17,4 GEOLOGICCONSIDERATIONS P^^L ...u,ritlge
arc?
431
Discharge area
E] F
(/) a TT]
F
F zp
(b)
Figure 17.2 Deflnition sketches of groundwater systems and mechanisms for rechargeand withdrawal: (a) aquifernotationt0and (b) componentsofthe hydrologic II cycleaffectinggroundwater. as falling into two categories, confined and unconfined, depending on whether a water table or free surface exists under atmospheric pressure. Storage volume within an aquifer is changed whenever water is recharged to, or discharged from, an aquifer. In the case of an unconfined aquifer this may easily be determined as AS:SIAV
(17.1)
432
CHAPTER17
GROUNDWATER,SOILS,AND GEOLOGY
where AS : the changein storagevolume S, : the averagespecificyield of the aquifer AV : the volumeof the aquiferlying betweenthe original watertable andthe water table at somelater specifictime For saturated, confined aquifers, pressure changes produce only slight modificationsin the storagevolume. In this case,the weight of the overburdenis supportedpartly by hydiostatic pressureand somewhatby solid material in the uqoif"t. When hydrostaticpressurein a conflned aquifer is reducedby pumping or othermeans,theioad on the aquiferincreases,causingits compression,with the result that somewateris forced out. Decreasingthe hydrostaticpressurealsocausesa small expansion,which in turn produces an additional releaseof water. For confined aquifers,water yield is expiessedin terms of a storagecofficient S", definedas the volumeof wateian aquifei takesin or releasesper unit surfaceareaof aquiferper unit changein head normal to the surface.Figure 17.2 illtstrates the classificationsof aquifers. In additionto water-bearingstrataexhibitingsatisfactoryratesofyield, thereare also non-water-bearingand impermeablestratathat may contain large quantitiesof water but whosetransmissionrates are not high enoughto permit effectivedevelopment. An aquifugeis a formation impermeableand devoidof water; an aquicludeis an imPerviousstratum. In the following three chapters,the mechanicsof groundwaterflow and the elementsof groundwatermodelingwill be introduced.The techniquespresentedall dependon a=knowledgeof the physical systemto be modeled.Before a numerical model can be developed,a conciptual framework must be devised.This framework must take into accountthe region's topographyand geology;the types of aquifers, their thickness,lateral extent,boundaries,lithological variations,and characteristics; areas,their rates of dischargeand the nature and extent of rechargeand dischar_ge table'' recharge;and the elevationof the water
Topography To understandhow a groundwatersystemoperates,it is essentialto know something aboutthe region's surface.A topographicmap shouldbe compiledshowingall surface water bodies,including streami, iakis, and artificial channelsand/or ponds,as well as land surfacecontours.Furthermore,an inventory of pumping wells, observation wells, and explorationwells shouldbe madefor purposessuchas identifying types of soilsand rocks,pinpointingdischargelocationsandrates;and determiningwatertable elevations.
SubsurfaceGeologY governsthe occurrenceand movement The geologicstructureof a groundwater'basin of the grirndwater withinlt. Specifically,the number and types of water-bearing formations, their vertical dimensions,interconnections,hydraulic properties, and outcrop patternsmust be understoodbefore the systemcan be analyzed.uOnce the subsuriaceconditions have been identified, contour maps of the upper and lower
SUMMARY 433 boundariesof aquifers,watertable contourmaps,and mapsof aquifercharacteristics can be prepared.Well-drillers logs, experimentaltest wells, and other geophysical explorationmethodscan be usedto obtakr the neededgeologic data.s-e'13'14
17.5 FLUCTUATIONS IN GROUNDWATER LEVEL Any circumstancethat alters the pressureimposedon undergroundwater will also causea variation in the groundwaterlevel. Seasonalfactors,changesin stream and river stages,evapotranspiration,atmosphericpressurechanges,winds, tides,external loads,various forms of withdrawal and recharge,and earthquakesall may produce fluctuationsin the water table level or piezometricsurface,dependingon whetherthe aquifer is free or confined.eIt is important that an engineer concernedwith the developmentand utilization of groundwatersuppliesbe awareof thesefactors.The engineershould also be able to evaluatetheir importance relative to operation of a speciflcgroundwaterbasin.
17.6 GROUNDWATER-SURFACE WATERRELATIONS Notwithstandingthat water resourcedevelopmenthas often been basedon tL pr"dominantuseof either surfacewateror groundwater,it mustbe emphasizedthat these two componentsof the total water resource are interdependent.Changesin one componentcan havefar-reachingeffectson the other. Coordinateddevelopmentand managementof the combinedresourceare critical. Linkage betweensurfacewaters and groundwatersshouldbe investigatedin all regionalstudiesso that adverseeffects can be noted if they exist and opportunitiesfor joint managementunderstood. In Part Three it was shown how surface stream flows are sustainedby the groundwaterresource,and it was also pointed out that groundwatersare replenished by infiltration derived from precipitation on the earth's surface. Undergroundreservoirsare often extensiveand can serveto store water for a multitude of uses.If withdrawalsfrom thesereservoirsconsistentlyexceedrecharge, mining occursandultimate depletionof the resourceresults.By properly coordinating the use of surfacewater and groundwatersupplies,optimum regional water resource developmentseemsmost likely to be assured.Several studiesdirected toward this coordinatedusehavebeeninitiated.r5'16
r Summary The importance of groundwaterto the health and well-being of humans is well documented.Groundwateris a major sourceof freshwaterfor public consumption, industrial uses, and the irrigation of crops. For example, more than half of the freshwaterused in Florida for all purposescomes from groundwatersources,and about 90 percent of that state'spopulation dependson groundwaterfor its potable watersupply.The needto husbandthis resourceis clear.Quantity and quality dimensionsare both important.
4g4
CHNPTENTZ
GROUNDWATER,SOILS,AND GEOLOGY
Groundwaterprotectionandmanagementpracticesmustbebasedonanunder. groundwateris^distributed standing of groundwatersources,the rianner in which soil characteristicsof the region' below the earth's surface,geologic,topogiaphic,and andtheinterconnectionsbetweengroundwaterandsurfacewatersources.
REFERENCES "Ground Water,"Mech'Eng' (Jan' 1960)' 1 . J. G. Ferris, ..GrouniwaterProtection," Final Report of the National The Conservationrounoaiion, L. GroundwaterPolicy Forum, Washington,D'C'' 1987' p"rru.u, diUiu- G' Gray' and Gegpe-F Pirltder'.Groundwater E. F. Wood, Ra'ymonde. CoitaminationfromHazardousWastes'EnglewoodCliffs'NJ:Prentice-Hall'1984' in the United States. R'pu,ri"t, unJ f. euurt" s, Grouidwater Contarnination ;. ffi;;; ehitadelphia:University of PennsylvaniaPress' 1983' R.A.FreezeandJ.A.Cir"..y,Groundwater'EnglwoodCliffs'NJ:Prentice-Hall'19'19' of llydrology' New York: McGraw-Hill' 1993' D. R. Maidment (ed.), ii"aLook ,,Niurnerical Modeling of GroundwaterBasins,"InternaJ. Boonstraand N. A. O" nlJa"r, The Netherlands'1981'' tional Institute for Land Reclamationand Improvement' 1965' Wiley' R. J. M. DeWiest,Geolrydrology'New York: Wiley' 1960" D. K. Todd, GroundwaterHydrology' New York: ..croonowu# n"gi#, of the United States," GeologicalSurvey water R. c. Heath, U'S' GovernmentPrinting Office' 1984' Supply PaperNo. 22a2' iishiniton, D'C-: "The Report to Congress:^Waste DisposalPracproteciion ig"""y, 1 1 . U.S. Environmentat summary'U'S' EPA' PB 265-364' tices and Their Eft'ectson Groundwater,"Executive 1977. States,"U'S' Geological tz. o. E. Meinzer, "The occurrence of Groundwaterin the united 1923' Survey,Water-SupplyPaperNo' 489'"A GeneralizedGraphical Method -for Evaluating 1 3 . H. H. Cooper, Jr. and i. n. lacoU, Trans'Am' Geophys'Union Formation constants una so*-u.izing well-Field History," 27, 526-534(1946). "Outline of Methodsfor EstimatingCrounlw11e1fupplies"' U'S' Geolog1 4 . O. E. Meinzer, D C:' 19?2ical Survey,Water-SupplyPaper638-C, Washington' "Conjurrliu" Op"tution of Dams and Aquifers"' Proc' ASCEJ' Hyd' Div' 1 5 . Nathan Buras, S9(HY6) (Nov. 1963). 16. F'B.Clendenen,..AComprehensivePlanfortheConjunctiveU{ilizationofaSurface water SupplyDevelopment:Solano Reservoirwith undergroun'Js,orug" for Basin-wide 1959. project california,,, po.to. oi Eni thesis,university of california, Berkeley,
Chapter1B
Mechanicsof Flow
r. Prologue The purposeof this chaPteris to: i I . .
I Presentthe principlesof groundwaterflow. Describesoil propertiesthat affect groundwaterstorageand movement. Describethe relevanthydrodynamicequations. Relatethe mechanicsof groundwaterflow to modelingregional groundwater systemsand calculating flows to wells and other groundwater collection devices.
18.1 HYDROSTATICS Water locatedin pore spacesof a saturatedmedium is under pressure(calledpore pressure),which cin be determinedby insertinga piezometerin the mediumat a point it can be seenthat pore pressure of interest.If LocationA (Fig. 18.1)is considered, given by is P:h"l
(18.1)
where p : the pore pressure(gaugepressure) h o : the headmeasuredfrom the point to the water table v : the specificweight of water Pore pressureis consideredpositive or negative,dependingon whetherthe pressure headis measuredabove(poJitive)or below (negative)the point under consideration. If an arbitrary datum is established,the total head or piezometric head abovethe datum is (18.2) Po:Z'th wherePois known as the piezometricpotential. In Fig' 18.1 this is equal to ho 1- 7o for poiniA in the saturatedzone andza - h6forPointB in the unsaturatedzone.The
43r.
436
CHAPTER18
MECHANICSOF FLOW
Figure 18.1 Definitionsketchshowinghydrostaticpressuresin a porousmedium. Iermh"is the pore pressureof A while -hu denotestensionor vacuum(negative,pore pressure)at B.
FLOW 18.2 GROUNDWATER Analogiescan be drawn betweenflow in pipesunder pressureand in fully saturated confinid aquifers.The flow of groundwaterwith a free surfaceis also similar to that in an open channel.A major differenceis the geometryof a groundwatersystemflow channel as compared with common hydraulic pipe flow or channel systems'The problem "un eutily be recognizedby envisioninga dischargingcross section composedof a numbei of small openings,eachwith its own geometry,orientation, and iir" ,o that the flow velocity issuingfrom eachpore varies in both magnitudeand direction. Difflculties in analyzingsuch systemsare apparent. Computations are usually basedon macroscopicaveragesof fluid and medium propertiesover a given area. cross-sectional Unknown quantitiesto be determinedin groundwaterflow problemsare density, la pressure,and velocity if constanttemperatureconditionsare assumedto exist'r In general,water is consideredincompressible,so the number of working variablesis ieduced.An exceptionto this is discussedlater relative to the storagecoefficientfor a confined aquifei. Primary emphasishere will be placed on the flow of water in a porousmedium. saturated
18.3 DARCY'SLAW Darcy's law for fluid flow through a horizontal permeablebed is statedas' O:
- K A dh dx
(18.3)
437
LAW 18.3 DARCY'S where A : the total cross-sectionalareaincluding the spaceoccupiedby the porous material : the hydraulic conductivity of the material K Q: the flow acrossthe control areaA In Eq. 18.3 h : z *
P-+c v
(18.4)
where h - the piezometrichead the elevationabovea datum p : the hydrostaticpressure C _ an arbitrary constant If the specificdischargeS = QIA is substitutedin Eq. 18.3, q:
-K*(,.t)
( 18 .s )
, Note that 4 also equalsthe porosity n multiplied by the pore velocity Vo.Darcy's law is widely used in groundwaterflow problems.Severalapplicationsare illustrated in later sections.
No is definedherein as No:
PQd
(18.6)
lL
where q d P p
: : : :
the specificdischarge the mean grain diameter fluid densitY dynamic viscositY
For many conditions of practical importance (zones lying adjacent to collecting devicesare an exception),Darcy's law has been found to apply' Of specialinterestis the fact that the Darcy equationis analogousto Ohm's law
' : (*)u
(18.7)
where i:thecurrent R : the resistance E : the voltage current and velocity are analogous,asareK andI f R, andE anddhldx.The similarity of the two equationsis the basis for electric analog models of groundwaterflow svstems.2'3
I
438
CHAPTER18
MECHANICSOF FLOW
EXAMPLE 18.1
: Water temperaturein an aquifer is 60'F and the rate of water movement t .2 ft I day. The averageparticle diameterin the porous medium is 0.08 in. Find the Reynolds number and indicate whetherDarcy's law is applicable. .
Solution. Equation 18.6 gives the Reynoldsnumber as
X*:# This may also be written as qd N--'
t)
From TableA.2 in Appendix A, o is found to be 12l X 10-s ft2lsec.Converting the velocity4 into units of ftlsec gives4 : I.2/86,400 : 1.39 X 10-5' The meangrain diameterin ft : 0.08/12 : 0.0067.Substitutingthesevaluesin the equation,we obtain Nn:
r.39x10-sx0.0067 1 . 2 1X 1 0 - s 0.0077
SinceN,, < 1.0,Darcy's law doesapply' rl
18.4 PERMEABILITY The hydraulic conductivity K is an important paramet€rthat is often separatedinto two components,one related to the medium, the other to the fluid. The product k:Cdz
(18.8)
X:4
(18.e)
can be written lL
Dimensionsof intrinsic permeability areL2. Sincevaluesof k given asft2 or cm2 are extremelysmall, a unit of measureknown as the darcy has beenwidely adopted. 1 darcy : 0.987 x 10-s cm2 or
1.062x l0-1r f*
Severalwaysof expressinghydraulic conductivity are reportedin the literature. The U.S. GeologicalSurveyhasdefinedthe standardcoefficientof permeabilityK" as - the nunber of gallons per day of water passingthrough 1 ft2 of medium under a
446
CHAPTER18
MECHANICSOF FLOW
" If Eqs. 18.46 are substitutedinto Eq. 18.50,then
#o;+frat:o
(18.s2)
The total differential dry'is equal to zero,and ry'must be a constant.A seriesof curves {(r, y) equalto a successionof constantscan be drawn and will be tangentat all points to the velocity vectors.Thesecurvestrace the flow path of a fluid particle and are known as streamlinesor flowlines. An important property of the stream function is demonstratedwith the aid of Fig. 18.2. Consider the flow crossinga definedas f1 and tltr.lf the dischargeacross vertical sectionAB betweengfueamlines the sectionis designatedas Q, it is apparentthat
n:r,'
udy
( 1 8.s 3)
I d,t' J,t,^
(18.s4)
f*t
Q : and
(18.55)
Q:Qt-Q,
Equation 18.55illustratesthe important property that flow betweentwo streamlines is ionstant. Streamline spaclngreveals the relative magnitudesof flow velocities betweenthem. Higher valuesare associatedwith narrower spacings,and vice versa. The curvesin Fig. 18.2 designatedas @rand Q,, calledeqaipotentiallines, ate determinedby velocity potentials Q@, y): constant. These curves intersect the flowlines at right angles,illustrated in the following way. The total differential d@is given by
a 0 ,+ a + , d0: frdx fidt
(18.s6)
substituting for terms aslax and 0Q/sy their equivalentsu and o gives us
(18.s7)
udx-lody:O and
dy dx
u
(18.58)
t)
lineqarenormalto flowlines.Therystemsf flowlinesandequipoThusequipotential tentiallinesformsa flow net.
18.8 BOUNDARYCONDITIONS
447
'
One significantpoint of differencebetween{ and ry'functionsis that equipotential lines exist only when the flow is irrotational. For two-dimensional flow the condition of irrotationality is said to exist when the z componentof vorticity (. is ZA[O; Ot
t,: (x-,4): o
(18.se)
Proof of this is givenby Eskinazi.6substitutingforu and o in Eq. 18.59in termsof @,we obtain '
d-Q
6xdy
_ a26:0
(18.60)
EY0x
This indicatesthat when the velocity potential exists,the criterion for irrotationality is satisfied. Once either streamlinesor equipotentiallines in a flow domain are determined, the other is automaticallyknown becauseof the relations in Eq. 18.48.Thus r
and
r: [(X*- H,o.) - H*) r: [(Xo.
(18.61a) (18.6lb)
It is enough then to determineonly one of the functions, since the other can be obtainedusing relationsEqs. 18.61aand 18.61b.The complexpotentialgiven by
(r8.62) Of where l, the squareroot of -1, is widely used in analytic flow net analyses'a's specialimportanceis the fact that (18.63) y2w:V26+iYzt1r:g satisfiesthe conditions of continuity and irrotationality simultaneously. Equations presentedin this section have been limited to the case of twodimensional flow. Extension to three dimensionswould be obtained in a similar fashion.
CONDITIONS 18.8 BOUNDARY
surfaces). Impervious boundariesmay be artificial objects zuch as.concretedams, rock strata,oi soil stratathat arehighiy impervious.In Fig' 18.3the imperviousboundary
448
CHAPTER18
MECHANICSOF FLOW
AB representssucha limit. Sinceflow cannotcrossan imperviousboundary,velocity componentsnormal to it vanishand the imperviousboundaryis a streamline.In other words,at the boundary,V : constant. Next look at the upstreamface of the earth damBC. At any point of elevation y along BC the pressurecan be assumedhydrostatic,or p:y(h-y)
(18.64)
The definition of a velocity potential statesthat
o' : - * \( vz *' r/ )* .
(18.65)
Substitutingfor pressurein Eq. 18.65yields
o:-*l
*r]*t
(18.66)
and
6:-Kh+C Thusfor a constantreservoirlevelh andan isotropicmedium,
(18.67)
d : constant and surfaceBC, often termed a reservoir boundary, is an equipotentialline. The free surfaceor line of seepageCD in Fig. 18.3 is seento be a boundary betweenthe saturatedand unsaturatedzones.Since flow doesnot occur acrossthis boundary,it is obviouslyalso a streamline.Pressurealong this free surfacemust be constant,and thereforealong CD 6 + Ky: constant
(18.68)
This is a linear relation in $, and therefore equal vertical falls along CD must be associatedwith successiveequipotential drops. One important groundwater flow problem is to determinethe location of the line of seepage. The surfaceof seepageDE of Fig. 18.3 representsthe location at which water seepsthrough the downstreamface of the dam and trickles toward point E. The pressurealongDE is atmospheric.The surfaceof seepageis neither a flowline nor an equipotentialline.
ImPervious laYer
Figure 18.3 Some common boundary conditions.
18.9 FLOWNETS
449
18.9 FLOWNETS Flow nets, or graphicalrepresentationsof families of streamlinesand equipotential lines, are widely used in groundwaterstudies to determine quantities, rates, and directions of flow. The use of flow nets is limited to steadyincompressibleflow at constant viscosity and density for homogeneousmedia or for regions that can be compartmentalizedinto homogeneoussegments.Darcy's law must be applicableto the flow conditions. The mannerin which a flow net canbe usedin problem solvingis bestexplained with the aid of Fig. 18.4. This diagram showsa portion of a flow net constructedso that eachunit b.oundedby a pair of streamlinesand equipotentiallines is approximately square.The reasonfor this will be clear later. A flow net can be determinedexactly if functions Q and $ are known beforehand.This is often not the case,and as a result, graphicallyconstructedflow nets are widely used. The preparation of a flow net requires application of the concept of square elementsand adherenceto boundary conditions. Graphical flow nets ard usually difficult for a beginnerto create,but with reasonablepractice an acceptable net canbe drawn.Variousmechanicalmethodsfor graphicalflow net constructionare presentedin the literature and are not discussedhere.5'e After a flow net hasbeenconstructed,it can be analyzedusingthe geometryof the net and by applylngDarcy's law. Rememberingthat h : p/y + z, we find thatFig' 18'4 showsthat the hydraulic gradient G2betweentwo equipotentiallines is given by
o r:x
(18.6e)
Then by applyingDarcy's law, in the mannerof Todd,2the flow incrementbetween adjacentstreamlinesis (18.70)
Lq: '*(*)
whereLm representsthe cross-sectionalarea for a net of unit width normal to the plane of the diagram. If the flow net is constructedin an orthogonalmanner and h- Ll Equalpotential lines (0 = constant) Lm
r d " Qz
Figure 18.4 Segmentof an orthogonalflow net.
450
CHAPTER18
MECHANICSOF FLOW
composedof approximatelysquareelements, (18'71)
Lm: L,s and Lq = K Lh
Now if there are n equipotentialdrops betweenthe equipotentiallines, it is evident that h Lh: IL
sections wherc his the total headloss over the n spaces.If the flow is divided into m is medium of the per width unit by the flowlines,then the discharge
o:iu:Y
(r8.72)
when the medium's hydraulic conductivityis known, the dischargecan be computed using Eq. 18.72 and a knowledgeof flow net geometry' Where the flow net has a iree surfaceor line of seepage,the entranceand exil of these conditions given in Fig. 18.3 are useful. A more comprehensivediscussion conditionsis given in Ref. 10. Some trouble arisesin flow net construction at locations where the velocity according becomesinfinite or vanishes.Suchpoints are known assingularpoints and classification In the first to De'Wiestmay be placed in three separatecategories.a Sucha situationoften angles' right at intersect not do lines equipotential and flowlines occurswhen a boundarycoincideswith a flowline; PointA in Fig. 18.5is-anelamnl-e. The secondclassificationhas a discontinuityalongthe boundary that abruptly such changesthe slope of the streamline.In Fig. 18.6 PointsA, B, and c represent zero' is B it discontinuities.At PointsA and c the velocity is infinite, while at Point flow If the angleof discontinuitymeasuredin a counterclockwisedirection insidethe angle The field is le-ssthan 180o,the velocity is zero; if larger than 180o'it is inflnite' atA is 270",for examPle. net' The third categoiyincludesthe casewherea sourceor sink existsin the flow net flow of the Under these circumstancesthe velocity is infinite, since squares
Line of seepage Tangent Surface of seepage tt
a<90o,0<90" (a)
ft)
(After Figure 18.5 Some entranceand exit conditions for the line of seepage' Casagrande.lo)
18..!O VARIABLEHYDRAULICCONDUCTMry
451
Figure 18.6 Flowlineslopediscontinuities' approach zero sizeas the source or sink is approached.Wells and rechargewells ' representsinks and sourcesin a practical senseand are discussedlater.
CONDUCTIVITY 18.10VARIABLEHYDRAULIC It is commonfor flow within a porousmedium of one hydraulic conductivityto enter another region with a different hydraulic conductivity. When such a boundary is crossed.flowlines are refracted.The changein direction that occurs can be determined as a function of the two permeabilitiesinvolved in the manner of Todd and Figure 18.7illustratesthis. DeWiest.2'a consider two soils of permeabilitiesK, and K, which are separatedby the boundary lR shown in Fig. 18.7. The directions of the flowlines before and after crossingthe boundary are definedby angles0, and 0r'
Figure 18.7 Flowline refraction.
452
CHAPTER18
"
MECHANICSOF FLOW
For continuity to be preserved,the velocity componentsin media K, and Kt, which are normal to the boundary,mustbe equal,sincethe cross-sectionalareaat the boundaryis AB for a unit depth.UsingDarcy's law and noting the equipotentialdrops h" and hr, we flnd
o#cos 92 K,L*cos o' : u, -
(18.73)
From the geometry of the flgure it is apparentthat AC : AB sin 0t BD : AB sin 0z The headlossbetweenA andB is shownon the figure to be equalto both Lh" andA,hu, and sincethere can be only a singlevalue, | Lh": Ah6 in Eq. 18.73produces Introducingtheseexpressions K t _ K , tan 0,
tan 02
(18.74)
For refractedflow in a saturatedporous medium, the ratio of the tangentsof anglesformed by the intersectionof flowlineswith normals to the boundary is given by the ratio of hydraulic conductivities.As a result of refraction, the flow net on the K2 sideof the boundarywill no longerbe squaresif the equipotentialline spacingDB is maintained.To adjust the net on the K2 side,the relation Lhu
Kl
Lt%- It
( 18 .7s )
can be usedwhere Lhb + Lh". Equipotentiallines are also refractedin crossingpermeability boundaries.The relation for this is K, K2
_
tan at tan a,
(18.76)
where a is the anglebetweenthe equipotentialline and a normal to the boundary of permeability.a
18.11ANISOTROPY In many caseshydraulic conductivity is dependenton the direction of flow within a givenlayerof soil. This condition is saidto be anisotropic.Sedimentarydepositsoften fit this aspect,with flow occurring more readily along the plane of depositionthan acrossit. Where the permeability within a plane is uniform but very small acrossit as comparedto that along the plane, a flow net can still be usedafter proper adjustNonhomogeneous ments are made. A discussionof this is given elsewhere.a's'11'12
THEORY 453 18.12 DUPUIT'S 'sometimesbe analyzed by Yslng aquifers require special considerationbut may outsidethe scopeof this representativeor averageparameters'A detailid study is book.3-5'r2
THEORY 18.12DUPUIT'S free surfacecan be analyzed Groundwaterflow problemsin which one boundaryis a theory is foundedon two This flow. on the basis of Dupuit's theory of unconfined seepageis only slightly of line ptptl the if in tgO:.t' First, assumptionsmadeby equlp:tencorrespondingly, and, inclined, streamlinesr""y b" consideredhorizontal tl" seepage of line the of slopes -utfd tial lines will be essentiallyvertical. Second, satisfactorily be to known at" hydraulic gradient *r "fit. Whel fleld conditioos to Dupuit's theory representedby theseusru-ption*, the resultsobtainedaccording "o'-pur"veryfavorablywiththosearrivedatbymorerigoroustechniques' into a mathemdtFigure 18.8 is u,"fol in translatingthe foregoingassumptions base'ateadx dy a has which figut" th" ical statement.Consideran elementgiien in the x direction and considand a vertical height h,Writing the cJntinuity equaiionin ering steadYflow to be the case, (18.77) infloqo : velocitSo X area,o The velocity at x : 0 is given by Darcy's law as ,ro:
(18.78)
_K*
: 0 is Thus the dischargeacrossthe elementat x - K- - a h . d Y Qo: *h
(18.7e)
expansionas The outflow at x = dx is obtainedby a Taylor's series -K*.h _ _ *(-**n .. -dv " + o,_ Qor: dr 6x\ Ax,.
'
' ' aY\+ _ /
(18.80)
Figure 1,8.8 Definition sketch for development of DuPuit's equation'
454
CHAPTER18
MECHANICSOF FLOW
Subtractingthe outflow from the inflow if K is consideredconstant,we obtain
I* - o,: K dxa, " !(n*)
tt*.
r,_ o.: rydr,A+fg) 2 Ax\6x/
(r8.
dx\
or
6x/
'x
wheredx anddy are consideredfixed lengths.A similar considerationin the y tion yields
r,-o,:ryy*W) i
; )
, (18
Assuminsthat thereis no movementin the vertical direction.theseare the componentsof the inflow and outflow.Furthermore,still dealingwith steadyflow, I changein storagemust be zero. As a result,
Kd:dy Kd:dy : o * +fg) *(#) 2 lx\Ex / 2 ay\6y /
o8
and since (K dx dy)12 is constant,this reducesto
or
A2h2 -:0 - r . Azhz 0y' 6x' Y 2 h z: 0
(18.
Consequently,. accordingto Dupuit' s assumptions,Laplace's equation for functionh2 must be satisfied.la In the particular casewhererechargeis occurring as a resultofinfiltrated reachingthe water table, a simple adjustmentmay be made to Eq. 18.85. If rechargeintensity(dimensionallyLT-t) is specifiedasR, thenthe total rechargeto elementof Fig. 18.8 is R dx dy and the continuity equationfor steadyflow beco
*4!*(#.#).Rdxdy:s
(18
or more simply,
v2h2+?o:o Now, applyingDupuit's theory to the flow problem illustratedin Fig. 18.9, assumingone-dimensionalflow in the x direction only, we obtain the discharge unit width of the aquifer given by Darcy's law:
O: -Kh# In this instanceh is the height of the line of seepageat any position x along imperviousboundary.For the one-dimensionalexampleconsideredhere, Eq. 18
18.12 DUPUIT'STHEORY
455
fto=50ft
free surface
(b)
Figure 18.9 Steady flow in a porous medium between two witer bodies: (a) free surface with infiltration and (b) free surfacewithout infi ltration. becomes d 2h2 --;-;:
u
(18.e0)
h2:ax*b
( 18.e 1)
clx'
Upon integration,
wherea andb are constants. Then for boundarv conditions at x : O,h : hs,
b=ht
(18.e2)
Differentiationof Eq. 18.91yields ^. dh Ztl';-
ax
= A
( 18.e 3)
-QIK. Making this substitution,we obtain Also from Darcy's equation,h dhldx :
-2Q
o:-T-
(18.e4)
and inserting the valuesof the constantsin Eq' 18.91,we obtain
h2=-2f*+nt
(18,e5)
456
.CHAPTER18
MECHANICSOF FLOW
(often calledDupuit' s parabola)' This is the equationof a free surface.It is a parabola andonotingthat at * : L, If the existenceof a surfaceof seepage.afBi. ignor.d, h : hr, we f,nd that Eq. 18'95becomes
or
2QL hL: _ x n ra
(18.e6)
o l-n?) - = L2 o L'"
(18.e7)
which is known as the Dupuit equation' EXAMPLE 18.3 RefertoFig.ls.ga.GiventhedimensionsshownandarechargeintensityRof : 1000 ft using Dupuit's equation'Assumerthat 0.01 ftlday, find the Oi."ft"tg" ut x K: 8. Solution. Note that
Q=n dx
or Q=Rxi.C
Atx:0, .
Qo
Q:
therefore, Rx -l Qo
Q:
Also, Q
- oh IntegratingYields
dh -Kh=
-
nh
Rx ' r Qo
i:
- Kh,l,, : o*,il' * o^*1, 2lo"
2lo
-"lo
and inserting the limits, we obtain
-K(h'-Lh7):ry o L
r
-
)
* QoL
K(h?- h'r) z L
1)^:----:----
RL 2
PROBLEMS
457
Then since Q : Rx + Qo,
K(h'^- h?\ . ,) 2L R 0 . 0 1 7 . 5 : 0.075gpd/ft2 _ soo)+ 8(50, 40r) o : 0.075(1000 2000 0.075x 500 , 8 x 9 0 0 2000 3 7 . 5+ 3 . 6 4Ll gpdlftz T I
o
R(.
+
r summary Understandingthe movementof groundwaterrequiresa knowledgeof the time and space dependencyof the flow, nature of the porous medium and fluid, and the boundariesof the flow system.In particular, groundwaterdevelopmentand management dependon understandingthe storagepropertiesofthe associatedsoils and rocks ' and the ability of thesesubsurfacematerialsto transmit water. Fundamentalto the mechanicsof groundwaterflow is Darcy's law (Eq. 18.3).Usingthis equationalong with a knowledgeof the hydraulic conductivity K, estimatesof flow can be had. The hydrodynamicequationspresentedin this chapter serve as models for a variety of groundwatgrflow calculations.Applications are given in Chapters19 and 20.
PROBLEMS' 18.1. What is the Reynoldsnumber for flow in a soil when the water temperatureis 55oF, the velocity is 0.5 ftlday, and the mean grain diameteris 0.08 in.? 18.2. The water temperaturein an aquifer is 60'F, the velocity is 1.0 ftlday. The average particle diameterof the soil is 0.06 in. Find the Reynoldsnumberand indicatewhether Darcy's law applies. 18.3. ReworkProblem 18.2assumingthe temperatureis 65"F and the velocity is 0.8 ftlday. 18.4. A laboratory test of a soil gives a standardcoeff,cientofpermeability of 3.8 x 102 gpdlft2.If the prevailingfield temperatureis 60"F,find the field coefficientof permeability. " 18.5. ReworkProblem 18.4 assumingK" is 3.8 x r02 gpd/ft2and the temperatureis 65'F. 18.6. Given the well and flow net datain the following flgure, find the dischargeusinga flow net solution.The well is fully penetrating;K : 2.87 X 10-4 ftlsec, a: 180 ft, b = 43 ft, and c : 50 ft. 18.7. ReworkProblem 18.6assumingK : 8.2 x 10-5 mlsec,a: 85 m, b : 2l m, and c:26m.
458
CHAPTER18
MECHANICSOF FLOW Stagnation flowline
Axis of
5 4 3 2 I
symmetry -1 -2 -3 /
18.8. ReworkProble1 m8 . 6 a s s u m i nKg: 8 ' 4 X 1 0 - 5m l s e c , a : 1 0 0 m ,b : 2 2 m , a n d c:35m. 18.9. A stratum of clean sand and gravel 15 ft deep has a coefficient of permeability of K : 3.25 X 10-3 ftlsec, and is suppliedwith waterfrom achannelthat penetratesto the bottom of the stratum.If the water surfacein an infiltration gallery is 2 ft above the bottom of the stratum.and its distanceto the channelis 50 ft, what is the flow into a foot of gallery?UseEq. 18.97.
REFERENCES 1. 2. 3. 4. 5. 6. 7.
Henri Darcy, Lesfontaines publiquesde la ville de Dijon. Paris: V. Dalmont, 1856. D. K. Todd, GroundwaterHydrology. New Yorkl V/iley, 1960. William C. Walton, GroundwaterResourceEvaluation.New York: McGraw-Hill, 1970. R. J. M. DeWiest,Geohydrology.New York: Wiley, 1965. M. E. Harr, Groundwaterand Seepage.New York: McGraw-Hill, 1962. SalamonEskinazi,Principles of Fluid Mechanics.Boston:Allyn and Bacon,7962"Flow of Groundwater,"in EngineeringHydraulics (Hunter Rouse,ed.) New C.E. Jacob, York Wiley, 1950.
Chapter '1I
Wellsand Collection Devices
I
r Prologue Thepurposeof this chapteris to:
. Presentmethodsfor calculatingconfined and unconfinedsteadyradial flow toward a well. . Describeproceduresfor dealingwith unsteadygroundwaterflow conditions. . Describea method for estimatingflow to an infiltration gallery. Groundwateris collectedprimarily by wells, althoughinfiltration galleriesare sometimes usedwherethe circumstancesare appropriate.tOutflows from natural springs are also amenableto collection,but once thesewatersexit the ground, they become surfaceflows and are handled as such. Wells are holes or shafts, usually vertical, excavatedin the earth for the purpose of bringing groundwater to the surface. Infiltration galleriesarehorizontalconduitsfor interceptingand collectinggroundwater by gravity flow. Problemsof groundwaterflow to wells and infiltration galleries can be solvedby applyingDarcy's law. ,
19.1 FLOW TO WELLS A well systemcan be consideredas composedof three elements-the well structure, pump, and dischargepiping.2The well itself containsan open sectionthrough which waterentersand a casingto transportthe flow to the ground surface.The open section is usually a perforatedcasingor slottedmetal screenpermitting waterto enter and at the sametime preventingcollapseof the hole. Occasionally,gravel is placed at the bottom of the well casingaround the screen. When a well is pumped,wateris removedfrom the aquiferimmediatelyadjacent to the screen.Flow then becomesestablishedat locationssomedistancefrom the well in order to replenishthis withdrawal. Becauseof flow resistanceoffered by the soil. a head loss results and the piezometric surfaceadjacent to the well is depressed. producinga coneof depression(Fig. 19. 1), which spreadsuntil equilibriumis reached and steady-stateconditions are established.
19.2 STEADYUNCONFINEDRADIALFLOW TOWARDA WELL
461
Impervious
Figure 19.1 Well in an unconfinedaquifer.
The hydraulic characteristicsof an aquifer (which are describedby the storage coefficient and aquifer permeability) can be determinedby laboratory or field tests' "o111*onlyusedfield methodsare thp applicationof tracers,the use The three 111ort of field permeameterr,uni aquiferperformancetests.3A discussionof aquiferper!o.r. mancetestsis given here alongwith the developmentof flow equationsfor wells.2'a's Aquifer performancetestsmay be either equilibrium or nonequilibriumtests.In an equilibrium testthe coneof depressionmustbe stabilizedfor a flow equationto be derived.For a nonequilibriumtesithe derivationincludesa condition that steady-state conditionshavenot beenreached.Adolph Thiem publishedthe first performancetests basedon equilibriumconditionsin 1906.6
RADIALFLOWTOWARDA WELL 19.2 STEADYUNCONFINED
Q: hrxYKrfi where 2rrxy : . Kr: dyfdx : Q:
(le.1)
the areathrough any cylindrical shell, in ft2 with the well as its axis the hydraulic conductivity (ftlsec) the water table gradient at any distancex the well discharge(ft'lsec)
Integratingover the limits specified,we find that f,2
r..
fo, Il -gYr : 2rrK1ly ay
Jt
n
Jhl
(re.2)
462
CHAPTER19 WELLS AND COLLECTIONDEVICES
- h?) 2rrKr(h2z 2 : nKr(hi hl)
Q n f12 _
^
o
and
-
( 1e.3) g
ln(rr/r1)
(te.4)
converting K, to the field units of gpdlftz, Q to gpm, and ln to 1og,we can rewrite Eq. 19.4as ,ar: "
1055Qlog(r2lrr) n;- ni
------;-;----;;-
(1e.s)
If the drawdownin the well doesnot exceedone half of the original aquiferthickness estimatesof Q or Krcanbe obtainedby usingEq.19.4 or tr9.5,even ho,reasonable if the heighth, is measuredat the well peripherywhere1L: r*, the radius of the well boring. EXAMPLE 19.1 An 18-in. well fully penetratesan unconfinedaquifer of 100-ft depth.Two observa: tion wells located100 and 235 ft from the pumpedwell areknown to havedrawdowns of 22.2 and 21.ft, respectively.If the flow is steady and K1 : 1320 gpdlft2, what would be the discharge? Solution. Equation 19.4 is applicable,and for the given units this is Q :
K(h?- h?)
1055Iog(r2/r1) :0.37rlt = log(r2/r') rwQ35l1'00) hz: 100-21=79ft
100- 22.2: 77.8ft - 77.82) t32O(792 Q : 1055x 0.37107 634.44gPm lr
ht=
RADIALFLOWTOWARDA WELL 19.3 STEADYCONFINED The basic equilibrium equationfor a confined aquifer can be obtainedin a similar manner, using the notation of Fig. I9.2. The sameassumptionsapply. Mathematically, the flow in ft3lsecis found from
o : 2nxmXr! "dx
(1e.6)
h"-h, O:2rrK'm " :# ln\r2/h)
(re.7)
Integrating,we obtain
.I9.4 WELL IN A UNIFORMFLOWFIELD
463
--12+l
Figure 19.2 Radialflow to a well in a confinedaquifer. The coefficient of permeability may be determinedby rearrangingEq. 19.7 to the form - -_ 528Q log(rz/r'\ nr *&; h) where
(1e.8)
Q : gpm K f : the permeability (gpd/ft') r , h : ft
EXAMPLE 19.2 Determinethe permeabilityof an artesianaquiferbeingpumpedby a fully penetrating well. The aquifer is 90 ft thick and composedof medium sand. The steady-state pumpingrate is 850 gpm. The drawdownof an observationwell 50 ft awayis 10 ft; in a secondobservationwell 500 ft awayit is 1 ft. Solution Kf:
52SQroe?Jr') m(h2 - h')
528 x 850 x log 10 90x(10-1) : 554gpdlfr2 rl FIELD 19.4 WELL IN A UNIFORM"FLOW For a steady-statewell in a uniform flow field wherethe original piezometricsurface is not horizontal, a somewhatdifferent situation from that previously assumedprevails. Consider the artesian aquifer shown in Fig. 19.3. The heretofore assumed circular area of influencebecomesdistortedin this case.A solution is possibleby applyingpotentialtheory,by usinggraphicalmeans,or, if the slopeof the piezometric surfaceis very slight, Eq. 19.7 may be employedwithout seriouserror.
464
CHAPTER19 WELLS AND COLLECTIONDEVICES
Original piezometric surface
Figure 19.3 Wellin a uniformflow fieldandflow net definition' . Figure 19.3providesa graphicalsolutionto a uniform flow field problem.First, an ortholgonalflow net consistingof flowlines and equipotentiallines must be constructed.This shouldbe done so that the completedflow net will be composedof a number of elementsthat approachlittle squaresin shape.Once the net is complete, if can be analyzedUy consideringthe net geom€try and using Darcy's law in the manner of Todd.3 EXAMPLE 19.3 Find the dischargeto the well of Fig. 19.3by using an applicableflow net. Consider the aquiferto be 35 ft thick, 4 : i.65 x 1d-4 fps, and other dimensionsas shown' Solution. UsingEq. 18'72,we find that
n : * *n' whereh:35*25:60ft m : 2 X 5 : 1 0 ,
n:I4 3.65X10-4x60x10
t4 : 0.0156cfsper unit thlcknessof the aquifer The total dischargeQ is thus r r Q : 0.0156 X 35 : 0.55 cfs ot 245 gPm
19.5 WELLFIELDS 465
19.5 WELL FIELDS When more than one unit in a well fleld is pumped,there is a compositeeffect on the is illustratedby Fig. 19.4in which the conesof free water surface.This consequence depressionare seento overlap.The drawdownat a given location is equalto the sum of the individual drawdowns. If, within a particular well field, pumpingratesof the pumpedwells are known, the compositedrawdownat a point can be determined.In like manner,if the drawdown at one point is known, the well flows can be calculated' 'n If the drawdownat a given point is designatedas m, andsubscripts1,2, ' the drawdown to mt fefers ., well(e.g particular are usedto relate this drawdownto a for I7,), for the total drawdownmr at somelocation' n
lTlr:
sr ^Z
:
lTli
(1e.e)
The numberof wells, their rate of pumping,and well-fieldgeometryand charac' teristicsdeterminethe total drawdownat a specifiedlocation. Again consideringEq. 19.4,we obtain
h3- h'
: #*t"1
(1e.10)
lt can be seenthat the drawdownfor a well pumped atrate Q canbe computedif ho, ro, and r are known. It follows then from Eq. 19.9 that for n pumped wells in an unconfinedaquifer
h 3h- ' : 2 * . " ?
Figure 19.4 Combinedeffect of pumping severalwells at equal rates'
( 1 e .11)
466
CHAPTER19 WELLS AND COLLECTIONDEVICES
where ho : the original height of the water table h : the combined effect height of the water table after pumping n wells r Q, : the flow rate of the ith well foi : distance of the lth well to a location at which the drawdown is considerednegligible r, : the distancefrom well i to the point at which the drawdownis being investigated Todd indicatesthat valuesof rousedin practiceoften rangefrom 500 to 1000ft.3 The impact of this assumptionis softenedbecauseQ inEq. 19.10is not very sensitiveto 16.Equation19.11shouldbe usedonly wheredrawdownsare relativd small. For flow in a confinedaquifer the expressionfor combineddrawdownbecomes n
h o - h : 2 =Q:, ,n'o' z1rKm
f1
(r9.r2)
Equationsfor well flow covering a variety of particular well-field patternsare reported in the literature.3'7Those given here are applicablefor steadyflow in a homogeneousisotropic medium.
19.6THEMETHODOF IMAGES Some groundwaterflow problems subjectedto boundary conditions negating the direct useof radial flow equationscanbe transformedinto infinite systemsfitting these equationsby applyingthe methodof images.2'8'e When a streamis locatednear a pumped well and the streamand aquifer are interconnected,the drawdowncurve of a pumpedwell may be affectedas shownin Fig. 19.5. Another boundary condition often affecting the drawdownof a well is an imperviousformation that limits the extentof the aquifer.The cone of depressionof a pumpedwell is not affecteduntil the boundary is intersected.After that, the shape of the drawdown curve will be changedby the boundary. Boundary effects can frequently be evaluatedby meansof "image wells." The boundary condition is replacedby either a rechargingor a dischargingwell that is pumped or rechargedat a rate equivalentto that of the pumpedwell. That is, in an infinite aquifer,drawdowns of the real and imagewells would be identical.The imagewell is locatedat a distance from the boundary equal to that of the real well but on the oppositeside(Fig. 19.5). Streamsare replacedby rechargewells while impermeableboundariesare supplanted by pumpedimagewells.Computationsfor the caseof a well and imperviousboundary directly follow the proceduresoutlined under the sectionon well fields.For the well and streamsystem,the rechargeimagewell is consideredto havea negativedischarge. The headsare then addedaccordingto this sign convention. The procedurefor combiningdrawdowncurvesof real and imagewells to obtain an actual drawdown curve is illustratetl graphically for the example shown in Fis. 19.5.More detailedinformationon othercasescan be found elsewhere.e'10
19.7 UNSTEADY FLOW
467
Cone of depressionof real well without sffeam
connectedto Figure L9.5 Drawdown in a pumping well whose aquifer is a stream'
FLOW 19.7 UNSTEADY directly from when a new well is first pumped, alargeportion of the dischargecomes circumthese u.nder the storagevolume releasedas the cone of depression_d91elons, yield of the stancesthe equilibrium equationsoverestimatepermeabilityandtherefore situathe is usually the well. When steady-stateconditionsare not encountered-as can,be approaches tion in practice-a nonequilibrium gOgltlgl must be used' Two suchasthat procedure a simplified or Theis V. of C. method taken,t^heratherrigorous proposedby Dcob.l1'12 consideration In tgjS treis publisheda nonequilibriumapproachthat takesinto an analogybetween time and storagecharacteristicsof the aquifer.ll His methoduses flow to a well' The heattransferdescribedby the Biot-Fouiier law and groundwater boundary conditions' method providesa solution to Eq. 18.41 for given initial and constantthickness' Application of the methodis appiopriatefor confinedaquifersof of flow must be For'use under conditions of unconhneOflow, vertical components
468
CHAPTER19 WELLS AND COLLECTIONDEVICES
negligible,and changesin aquifer storagethrough water expansionand aquifer compressionmust also be negligiblerelative to the gravity drainageof poresas the water table drops as a result of pumping.l3 Theis statesthat the drawdown(s) in an observationwell locatedat a distance r from the pumpedwell is given by
O f * " - o" ', s : -4rrT J, ;
(1e.13)
in which Q : (constant) pumping rate (L3T-r units), Z : aquifer transmissivity (LtT 'units), and a is a dimensionless variabledefinedby
, ,s" u : r--4tT
(re.r4)
where r is the radial distancefrom the pumpingwell to an observationwell, S. is the aquiferstorativity(dimensionless), and r is time. The integralin Eq. 19.13is usually known asthewellfunctionof u andiscommonlywritten asW(u).It maybe evaluated from the infinite series a
1
- ln u * u - =+ - + -]-w(u): _'0.577216 2x21 3x3l
(1e.1s)
Usingthis notation,Eq. 19.13can be written as
Qw(u) "( : 4irT
(1e.i6)
The basic assumptiopsemployedin the Theis equation are essentiallythe sameas thosein Eq.I9.7 exceptfor the nonsteady-statecondition. Some valuesof the well function are given in Table 19.1. In American practice,Eqs. 19.13 and 19.I4 commonly appearin the following form,
t:
,:
rI4.6Qf* "-' , , J,;o' I . 8 71 2S "
n
(re.r7) (1e.18)
where 7 is given in units of gpd/ft, Q hasunits of gpm, and / is the time in dayssince the start of pumping. Equations19.13and 19.14canbe solvedby comparinga log-logplot of a versus I4z(z)known as i type curve, with a log-1ogplot of the observeddatar2ft versuss. In plotting type curves,W(u) ands are ordinates,z and rt ft areabscissas. The two curves are superimposedand movedaboutuntil segmentscoincide.In this operationthe axes must remain parallel. A coincidentpoint is then selectedon the matchedcurvesand both plots marked.The type curve then yieldsvaluesof u and W(u) for the desired point. Correspondingvaluesofs and r'ft aredeterminedfrom a plot ofthe observed
FLOW 1g7 UNSTEADY VALUESOF U TABLE19.1 VALUESOF W(U}FORVARIOUS 'L0 5.0 4.0 3,0 2.0 XI x 10-r x 10-2 x 10-3 x 10-4 x 10-5 x 10-6 x 10-7 x 10-8 x 10-e x 10-10 x 10-11 x 10-t2 x 10-13 x 10-14 x 10-15
0.219 t.82 4.04 6,33 8.63 10.94 t3.24 15.54 17.84 20.15 aa A<
24.75 27.0s 29.36 31.66 33,96
0.049 1.22 3.35 5.64 7.94 t0.24 12.55 14.85 t7,15 19,45 21.76 24.06 26.36 28,66 30.97
0 . 0 1 3 0.0038 0.70 0.91 2.68 2.96 4.95 5.23 7.25 1.53 9.55 9.84 12.14 11 . 8 5 14.15 14.44 16.46 16.74 19.05 t8.76 2 r . 3 5 2r.06 23.65 2 3 . 3 6 25.96 25.67 27.97 28.26 30.56 30.27 32.86 32.58
0.0011 0.56 2.47 4.' 13 7,02 9.33 11 . 6 3 1.3.93 16.23 18.54 20.84 23.14 25.44 27.75 30.05 32.35
0.00036 0.45 z,Jv
4.54 6.84 9.r4 r1.45 13.75 16.05 18.35 20.66 22,96 25.26 27.56 29.8'7 32.1,7
9.0
8.0
6.0 -' 0,00012 0.37 2.1,5 4.39 6,69 8.99 rt.29 13.60 15.90 18,20 20.50 22.8t
25.rr 2'1.41 29.71 32,02
0,000038 0.31 2.03 4,26 6.55 8.86 I 1.16 t3.46 t5.'16 18.07 20.37 22.67 24.9'1 27.28 29.58 31.88
469'
0.000012 0.26 1.92 4.14 6.44 8.74 1l.04 13.34 15.65 17.95 20.25 22,5s 24.86 27.16 29.46 3t.76
..Methodsfor Determinlng Permeability of water Bearing M.aterialswith Special Referenceto Dischargt.fig Source: AfterL,K. Wenzel, Washington'DC' 1942' Well Methods," U. S. GeologiJSt.""V, W"t*-Supp-ly Paper 887'
values for data. Inserting these values in Eqs. 19.13 and 1,9.1'4and,rearranging, found' coefficientS" can be transmissibilityI and storage "can be shortenedand simplified' when r is small and t often this procedure in the seriesof large, Jacobfound that valuesof u arc generallysmall'12Thus terms for Zbecomes expression the and fql te.tS bpyondthe secondonebecomenegligible 264Q(loet2 - log tt) (1e.1e) ' h"- h which can be further reducedto
264Q T _
(re.20)
Lh
where
Ah Q ho,h T
: : : :
drawdownper log cycle of time l(ho well discharge(gPm) as definedinFig' I9.2 the transmissibilitY(gPd/ft)
- n)lloe t' - log t')l
paper' The Field data on drawdown(ho h) versust arc dtaftedon semilogarthmic a straightline drawdownis plotted on an arithmetic soale,Fig. 19.6. Thisplotforms and *to*" slopepermits computingformation constantsusing Eq' 1'9'20 0.3Tto \^- : -r '
(1,9.2t)
obtainedthrough with rothe time correspondingto zerodrawdown.Equation 19.21is manipulationof Eq. 19'13.
470
CHAPTER19 WELLS AND COLLECTIONDEVICES
, = t4
r\too'
= 4e,8oo sd/ft
tl
II I
I
Time since PumPing began (min)
Figure 1.9.6 Pumpingtest data, Jacobmethod. EXAMPLE 19.4
Usingthe following data, find the formation constantsfor an aquiferusinga graphical solution to the Theis equation.Dischargeequals540 gpm. Distancefrom pumpedwell, r (ft) 50 100 150 200 300 400 500 600 700 800
1 2/ t
-
1,250 5,000 il,250 20,000 45,000 80,000 125,000 180,000 245,000 320.000
Average drawdown, s (ft)
3.04 2.16 1,.63 r.28 0.80 0.51 0.33 0.22 0.15 0 .l 0
FLOW 19.7 UNSTEADY
1.0 0.9 0.8 0.7
't I | , I W(u)vs. u
€ o.o ? 0.s €E o.+ B
E 0.3
5 6 7 891x105
I rrzldav) ' Figure L9.7 Graphicalsolutionto Theis'sequation' Solution.Plotsversusr2ftand}V(a)versusaasshowninFig.lg.T.Determine Eqs' 19'7 and 19'8:the match point as noted and computeS" and Z using T :
lr4t6Q w(u) s 1 1 4 . 6X 5 4 0 -
: --18---c u"
^ 1.9 :
91,860gpdift
uT 1.87r'1t
_ 0.09x 91,860: 0,22 tr 1.87X 20,000 EXAMPLE 19.5
7"andstorage of transmissibility usingthedatagiv.enin Fig. 19.6,findthe_coefficient : ft' 300 S, for an aquiflr, givenQ: 1000gprn andr coeffrcient Solution.FindthevalueofAhfromthegraph,5.3ft.ThenbyEq.|9.20 2640 264 - - Tx31000
T: -ff:
: 49,800gpd/ft Using Eq. 19.21.,we find that
s":
O.3Tto r'
472
CHAPTER19 WELLS AND COLLECTIONDEVICES
Note from Fig. 19.6 that ts : 2.6 min. Converting to days, we find q6il)ti\--l becomes , 0 -
s":
and
' 1 . 8 1x l b - ' d a y s 0 . 3 x 4 9 , 8 0 0 X 1 .X 81 0 - 3
(3oo)'
: 0.0003 I T EXAMPLE 19.6 Find the drawdownat an observationpoint 200 ft awayfrom a pumping well. Given that T : 3.0 x 10a gpd/ft, the pumping time is 12 days,S" : 3 X 10-4, and
0 = 3oogpm,
Solution. From Eq. L9.L8,u can be computed,
x 104x l2l : 6.23x I}-s u : u.87 x (200)'x 3 x L0-41113.0 Referringto Table 19.1 and interpolating,we estimateW(u) to be 9.1. Then, using Eq. 19.17, the drawdownis found to be s : [114.6x 9.1 x 300]/[3.0x 104]: 10.41ft rr EXAMPLE 19.7 A well is beingpumpedat a constantrate of 0.0038 m3/s.Giventhat I : 0.0028m2ls, r : 90 meters, and the storagecoefficient : 0.00098, find the drawdown in the observationwell for a time period of (a) 1,000 sec.and (b) 20 hours. Solution a. Using Eq. 19.14,u canbe computedas follows, u :190 x 90 x 0.000981/[4x 1000 x 0.0028] u : 0.71 Then from Table 19.1,W(u) is found to be 0.36. Applying Eq. 19'16,the drawdowncan be determined,
x n x 0.0028] x 0.367114 r : [0.0038 s : 0.039m b. Follow the procedureusedin (a) ' u : 1 9 0 x 9 0 x 0 . 0 0 0 9 8 1 / [x47 2 , 0 0 0 x 0 . 0 0 2 8 ] a : 0.0098 Then from Table 19.1, W(u) is found to be 4.06 Applying Eq. 19'16,the drawdowncan be determined,
x n x 0.00281 x 4.06]114 s = [0.0038 : 0 . 4 4 m r r s
19.10 FLOW TO AN INFILTRATIONGALLERY
1e.8LEAKYAQUIFERS
473
a_)
imperThe foregoinganalyseshavedealtwith free aquifersor thosecopfinedbe#een comnot strata are vious str-ata.1nt"ulity, many casesexist wherein the confining PletelYimPerviousand water is at aquifer. The flow regime is altere< about 1930,leakYaquifershaveber De Glee, Jacob,Hantush, DeWies ers.t'to-'uA thorough treatmentc interestedreadersshouldconsulttheindicatedreferences.
WELLS 19.9 PARTIALLYPENETRATING of the well' The ln many actual situations there is only partial penetration previouslyfor full questionthen arisesasto the applicability of proceduresdeveloped Penetration. ln 1957Hantush Numerousstudiesof this problemhavebeenconducted'7'2't'28 aquiferbecomes reportedthat steadyflow to a well just penetratingan infinite leaky aquiferthickness'28 very nearlyradial at a distancefrom the well of about1.5timesthe increasingly As depth of penetrationincreases,the approachto,radial flow becomes apparent.Therefore,computationofdrawdownsforpartiallypenetrating'wellsare provided that the made using equationsfoi total penetration with relative safety, aquifer thickness'At distancefr"o-ifr" pumped we11ls greaterthan 1.5 times the other relations points closerto the weli, it is frequJntly possibleto use a flow net or develoPedfor this region.
GALLERY 19.10FLOWTO AN INFILTRATION conduit constructed An infiltration gallery may be defined as a partially pervious part of this flow will be acrossthe path of t6" to"uigroundwater flow iuch that all or parallel to a streamso that intercepted.Thesegalleries-areoften built in a valley area under gravity-flow they can convey the collected flow to some designatedlocation .ondition*.Figurelg.Sshowsatypicalcrosssectionthroughagallerywithone 'perviousface. pervious wall corrrputation of dischargeto an inflltration gallery with one Severalassumptions inihe manneroutlinedby Dupuit-2e (Fig. f g.Sjis accomplished tangentof the angleof mustbe madelo effectthe solution.They are that the sineand that the velocity vectorsare evefyinclination ofthe water table are interchangeable; is incompressibleand where horizontal and uniformly distribuied; that the soil effects are negligible' isotropic; and that the gallery is of sufficient length that end do limit the utility of While permitting u ,oluion oi the problem, theseassumptions the results. Basedontheseassumptions,andfollowingtheproceduregiveninSecper unit width' using the tion 18.12,Eq. 18.97canbe usedto calculatethe discharge
L
474
CHAPTER 19 WELLS AND COLLECTIONDEVICES Ground surface
Intersection of assumed and actual water tables
X+
Figure 19.8 Cross-section throughan infiltrationgallery. nomenclature of Fig. 19.8,Eq. 18.97becomes K . - "-
s :;(hi
he)
This equationindicatesthat the computedwatertableis parabolic.This is often called Dupuit's parabola.Figure 19.8 showsthat the computedwater table differs from the actual water table in an increasingmanner as the gallery face is approached.It is thereforeapparentthat the computedparaboladoesnot accuratelydescribethe real water table. The differences,however,are small except near the point of outflow, providing the initial assumptionsare satisfled.The calculateddischargeapproximates the true dischargemore closelyas the ratio of Z/h, increases. EXAMPLE 19.8
A stratum of clean sand and gravel20 ft deep has a coefficient of permeability of K : 3.25 X 10-3ftlsec,andis suppliedwith waterfrom a channelthat penetrates to the bottom of the stratum.If the water surfacein an infiltration gallery is 3 ft above the bottom of the stratum,and its distanceto the channelis 50 ft, what is the flow into a foot of gallery?UseEq. 18.97. Solution q : 0.5(3.25x 10-3)(20x 20 - 3 x 3)/50 \ : 0.072 cfs, the flow into one foot of gallery I r
19.11SALTWATER INTRUSION The contaminationof fresh groundwaterby the intrusion of salt water often presents a seriousquality problem. Islands and coastalregions are particularly vulnerable. Aquifers locatedinland sometimescontaiq highly salinewaters as well. Freshwater
BASINDEVELOPMENT 475 15.12 GROUNDWATER
BASINDEVELOPMENT 19.12GROUNDWATER To use groundwaterresourcesefficiently while simultaneouslypermitting the maximum dJvelopmentof the resource,equilibrium must be establishedbetweenwilhdrawals and replenishments.Economic, legal, political, social, and water quality aspectsrequire full consideration. Lasting suppliesof groundwaterwill be assuredonly whenlong-termwithdrawls arebalanceJby rechargeduringthe correspondingperiod. The potentialof a groundby employingthe water budgetequation, water basin can be assessed
)r-)o:As where the inflow ) l includesall forms of recharge,the total outflow ) O includes everykind ofdischarge,and AS representsthe changein storageduringthe accounting perioO. The most significant forms of recharge and dischargeare those listed in Table t9.2. A groundwaterhydrologistmust be able to estimatethe quantity of water that canbe economicallyand safeiyproducedfrom a groundwaterbasinin a specifiedtime of imposing period. He or sheshouldalsobe competentto evaluatethe consequences supply. underground an variousrates of withdrawal on Developmentof groundwaterbasins should be based on careful study, since groundwaterresourcesare finite and exhaustible.If the various types of recharge balance the withdrawals from a basin over a period of time, no difficulty will be encountered.Excessivedrafts, however,can depleteundergroundwater suppliesto a point whereeconomicdevelopmentis not feasible.The mining of waterwill ultimately depletethe entire supply. ANDDISCHARGE OFRECHARGE TABLE19,2 SdUETONT"TS
i-_
Recharge
Discharge
Seepagefrom streams,Ponds,lakes Subsurfaceinflows Infiltrated precipitation Water rechargedartificallY
Seepageto lakes, streams,spnngs Subsurfaceoutflows Evapotranspiration Pumping or other artificial meansof collection
476
CHAPTER19 WELLS AND COLLECTIONDEVICES
r Summary The collection of groundwateris accomplishedprimarily through the constructionof wells, and many Tactorsinfluence the numerical estimation"of their performance. Some situation, ur" amenableto solutionthrough the utilization of relatively simple mathematicalexpressions.Others dependupon sophisticatedapplicationof the hydrodynamicequationsundervariousConditionsof nonuniformity of aquifermaterials and a variety oiboundary conditions.The readeris cautionednot to be misledby the simplicity oi ,o-" of thqsolutionspresentedand to observethat many of theserl\ to speciaiconditions and are not applicableto all groundwater-flowsituations.(. , ) The rate of movementof water through the ground is of a different magnr'tudd than that through natural or artificial channelsor conduits.Typical flow ratesrange of from 5 ft/day ti afew feet per year. Theselow ratesof flow exacerbatethe impact natural since cleanup complicate and sources groundwater contaminani spills on flushing from the site may take many yearsto occur' The methodsdescribedin this chapterfor estimatingflows to collection devices arebasedmainly on the principlesof fluid flow embodiedin Darcy's law. Applicafions arelimited to flowsin the laminar range,but undermostconditionsencounteredin the field, Darcy's law applies.Examplesof the ure of equationsdescribingthe mechanics of flow to wells andinfiltration gilleries were given in this chapter.Both steady-state and unsteadvflow conditions were addressedas well'
PROBLEMS coefflcient of 19.1. A 12-in. well fully penetfatesa confined aquifer 100 ft thick. The permeabilityis 60d gpd/ft'. Two test wells located40 and 120 ft awayshow a difference in drawdown U-"i*een them of 9 ft. Find the rate of flow delivered by the well. of 1g.2. A l2-in. well fully penetratesa confined aqpifer 100 ft thick. The coefficient permeability is 600 gpd/ft2. Two test wells located 45 and 120 ft away show a the difference in drawdoin between them of 8 ft. Find the rate of flow delivered by well. well' The 19.3. Determine the permeability of an artesianaquifer for a fully penetrating pumping steady-state The ft thick. aquifer is composedof medium sand and is 100 ft, and the 14 is away 75 ft well observation in an rate is 1200 gpm. The drawdown per day gallons Kin Find is l.2ft. away ft 500 well observation drawdownin a second per squarefoot. ftj/day/ft' Lg.4. Considera confined aquifer with a coefficientof transmissibilityT of 680 Att:5min,thedrawdowns:5.6ft;at50min,s:23'Ift;andat100min's: 28.2 ftrThe observationwell is 75 ft awayfrom the pumpingwell. Find the discharge of the well. 19.5. Giventhefollowingdata:0:59,000ft3lday'T:630ft3/day't,,3}days'r:1ft' the drawand s" : 6.4 x L6,4. Considerthis to be a nonequilibriumproblem.Find down s. Note that for a:8.0x10-e u:8.2 X 10-e
W(a): 13'sa W(u)= 19.94
'
PROBLEMS 477
19.6. ' Determinethe permeabilityof an artesianaquiferbeingpumpedby a fully penetrating pumping well, The aquifercomposedof medium sandis 130ft thick. The steady-state ft, and in a is 12 away well 6{ft in an observbtion gpm. The drawdown is 1300 rate secondwell 500 ft awayis 1.2 ft. Find Ky in gpdlft2. 1g.7. Considera confined aquifer with a coefficientof transmissibilityT : 700 ft3lday-ft. Atr : 5 minthe drawdown: 5.1ft; at50min,s = 20.0ft; at 100min,s = 26'2ft. The observationwell is 60 ft from the pumping well. Find the dischargeof the well. 19.8. Assumethat an aquiferbeing pumpedat atate of 300 gpm is confined and pumping testdata are given as follows.Find the cbefficientof transmissibilityTand the sttragq coefficientS. Assumer = 55 ft. | ) Time sincepumping started (min) Drawdowns (ft)
1,3 4.6
2.5 8.1
4.2 9.3
8.0 12.0
11.0 15.1
100.0 29.0
19.9. We are given the following dat4: Q = 60,000 ft3ldaY
r : 650fcl(dar(fo
t:30days r:lft s"=6.4 x 1o-4
Assumethis to be a nonequilibriumproblem. Find the drawdowns. Note for a : 8.0 x 10-e u : 8.2 X 10-e u : 8.6 x 10-e
lV(u) : 18.06 W(u) = 19.94 W(u) : 17.99
19.10. An 18-in. well fully penetratesan unconfinedaquifer 100 ft deep.Ttvo observation wells located90 and 235 ft from the pumped well are known to havedrawdownsof 225 and20.6ft, respectivd. If the flow is steadyandKy: 1300gpd/ft2, what would be the discharge? lg.1l. A confinedaquifer 80 ft deepis being pumpedunderequilibrium conditionsat a rate ' of 700 gpm. The well fully penetratesthe aquifer. Watpr levelsin observationwells 150 and 230 ft from the pumped well are 95 and 97 ft, respectively.Find the field coefflcient of permeability. lg.112. A well is pumpedat the rate of 500 gpm undernonequilibriumconditions.For the data listed, find the formation constantsS and Z. Use the Theis method.
r"/t 1,250. 5,000 tt,250 20,000 45,000 80,000 125,000 180,000 245,000 320,000
Averagedrawdown, h (ft) 3.2+
2.t8 .1.93 r.28 0.80 0.56 0.38 0.22 0.15 0.10
478
DEVICES 19 WELLSANDCOLLECTION CHAPTER "We are given a well pumping at arate of 590 gpm. An observationwell is locatedat 19.13. r : 180 ft. Find S and Z using the Jacobmethod for the following test data.
Drawdown (ft)
0.43 0.94 l 08 1.20 r.34 |.46 1.56 1.63 1.68 1.71 l 85 1.93
Time (min)
Drawdown (ft)
26 78 99 t3r t'73 2t8 266 303 331 364 481 5'13
2.00 2.06 2.12 2.15 2.20 2.23 2.28 2.30 2.32 2.36 2.38
Time (min)
66r 732 843 926 IO34 r134 1272 1351 I4t9 r520 1611
o
A24-1n.diameterwell penetratesthe full depthof an unconfinedaquifer.The original watertable and a bedrockaquifugewere located50 and 150ft, respectively,belowthe land surface.After pumping al arate of 1700 gpm continuously for 1920 days. equilibrium drawdownconditions were established,and the original water levels in observationwells located 1000 and 100 ft from the center of the pumpedwell were lowered 10 and 20 ft, respectively. a. Determinethe field permeability (gpdlftz) of the aquifer. b. For the samewell, zero drawdownoccurredoutsidea circle with a 10,000-ftradius measuredfrom the centerof the pumpedwell. Insidethe circle, the averagedrawdown in the water table was observedto be 10 ft. Determine the coefficient of storageof the aquifer. 19.15. A well fully penetratesthe 100-ft depth of a saturatedunconflned aquifer. The drawdownat the well casingis 40 ft whenequilibriumconditionsare establishedusing a constantdischargeof 50 gpm. What is the drawdownwhenequilibriumis established using a constantdischargeof 66 gpm? 19.16. After a long rainless period, the flow in Wahoo Creek decreasesby 8 cfs from Memphisdownstream8 mi to Ashland.The streampenetratesan unconfinedaquifer, wherethe water table contoursnear the creek parallel the westbank and slopeto the streamby 0.00020, while on the east side the contours slope awayfrom the stream toward-theLincoln wellfield at 0.00095. Computethe transmissivityof the aquifer knowing Q : TIt, where1 is the slopeand I is the length. 19.17. The time-drawdown data for an observationwell located 300 ft from a pumped artesianwell (500 gpm) are given in the following table.Find the coefficientof storage (ft3 of water/ft3of aquifer)and the transmissivity(gpd/ft) of the aquiferby the Theis method. Use 3 x 3 cycle log PaPer.
tg:t4.
REFERENCES
Time (hr)
Drawdown (ft)
1.8 2.\
0.27 0.30 0.3'l 0.42 0.50 0.61 0.84
. A
3.0 3.7 4.9 7.5
Time
(h0 9.8 12.2 t4.7 16.3 18.4 21.0 24.4
479
Drawdown (ft)
r.09 t.25 1.40 1.50 l 60 t.70 1.80
19.18. Over a 100-mi2surfacearea, the averagelevel of the water table for an unconfined
aquiferhas dropped I 0 ft becauseof the removalof 128,000 area-ftof waterfrom the aquifer.Determinethe storagecoefficientfor the aquifer.The specificyield is 0.2 and the porosityis 0.22. 19.19. Over a 100-mi2 surface area, the averagelevel of the piezometric surface for a confined aquifer in the Denver area has declined 400 ft as a result of long-te1m pumping.Determinethe amountof the water (acre-ft) pumpedfrom the aquifer' The porosity is 0.3 and the coefficient of storageis 0.0002. L9.20. Find the drawdownat an observationpoint 250 ft awayfrom a pumping well, given -: thatT = 3.1 x 104 gpdlft,the pumpingtime is 10 days,S" 3 x 10-4, andQ 280 gpm. 19.21. Find the permeabilityof an artesianaquiferbeingpumpedby a fully penetratingwell. The aquiier is 130 ft thick and is composedof medium sand.The steady-statepumping raie is 1300 gpm. The drawdownin an observationwell 65 ft awayis 12 ft' and in a secondwell 500 ft awayit is 1.2 ft. Find Kyin gpdlft2. 19.22, An 18 in. well fully penetratesan unconflnedaquifer 100 ft deep.Two observation wells located90 and 235 ft from the pumpedwell are known to havedrawdownsof 22.5 ft and20.6 ft respectively.If the flow is steady arld Ky: 1300 gpd/ft2, what would be the discharge? : 0.0028m2ls, 19.23. A well is being pumpedat a constantrate of 0.004 m3/s.Given thatT : in the obserdrawdown find the : 0.001, 100meters,and the storagecoefficient r (b) hours' 24 (a) hr, and t period of time vation well for a = 0.0028mzls, 19.24. A well is being pumpedat a constantrate of 0.003 m3/s.Given thatT is 12 hours,find pumplng began = since time 0.001, and the the storage"oiin"i"nt (b) 500 m' (a) m, and 150 of distance radial for a well observation the drawdownin an
REFERENCES 1 . J. W Clark, w. viessman, Jr., and M. J. Hammer,water supply and Pollution conffol, 2nd ed. New York: Thomas Y. Crowell' 1965. "Field TestsDeterminePotential Quantity, Quality of Ground Water 2. John F. Hoffman, Supply," Heating, Piping, and Air Conditioning(Aug' 1961)' 3 . D. K. Todd, GroundwaterHydrology, New York: Wiley, 1960'
Chapter20
ModelingRegional GroundwaterSYstems
r Prologue The PurPoseof this chaPteris to: ' Describethe featureSof large-scalegroundwatersystems. . Introduce the principlesof finite differenceapproachesto modelingregional groundwatersystems. . Iilustrate the application of groundwatermodeling techniquesto the Upper Big Blue basin in Nebraska.
-
flow of The analytical methodsdescribedso far havebeen applicablemainly to the groundregional warcr rc individual wells. In this chapter,the conceptsof analyzing water systemsare introduced.Suchanalysesare requisitesfor the wise development, management,andoperationofexpansivegroundwaterresources' jointly in most Given that waier quantity an-dquality aspectsmustbe dealt with mrrstoften models waterresourcesdecision-makingprol"tt"t, t"gional groundwater of groundaspects The fluid flow be designedto includeboth of tfuse dimensions.l-2s and complex are water modelsare presentedin this chapter.Solutetransportmodels. end of the given at is beyondthe scope-ofthis book, but a biief introduction to them of role and the chapter.It is important for the readerto understandthe importance thesewater quality-orientedmodels'
MODELS GROUNDWATER 20.1 REGIONAL (mathematical) Groundwatersystemsmodels may be of the analog or the digital type most variety. The focus of this chapter is on the digital type of lodel, the equations of a set by commonly employedtoday. Such models are characterized be models.may These ,"pr"r"nting the ihysical i.o""*"* occurring in an aquifer. discussed are deierministt or pio-babilisticin nature, but only deterministic-models featuresof the here. They describethe cause-effectrelations stemmingfrom known physicalsystemunder study'
482
CHAPTER20
MODELINGREGIONALGROUNDWATERSYSTEMS
Approximate equations numerically resulting in a matrix equation that may be solved using a computer
Simplify equation so that solutions may be obtained by analytical methods l _
t_
model' a mathematical Figure 20.1 Logic diagramfor develqping Worthington'OH') (Courtesyof theNationalWaterWellAssociation,
the procedurefor developinga deterministicmatheFigure 20.1 characterizes matical model. A conceptual model is formulated based on a knowledge of the characteristicsof the region and an understandingof the mechanicsof groundwater flow. The next stepis to translatethe conceptualmodel into a mathematicalmodel, usuallyrepresentedby a partial differential equationor setof equationsaccompanied Uy approplia@boundaryand initial conditions.Conditionsof continuity and conseruution of momentum, usually describedby Darcy's law, are incorporated in the model,Other modelfeaturesincludeartesianor watertable condition designationand dimensionality(one-,two-, or three-dimensional).If water quality and/or heattransfer considerationsareto be incorporatedin the model,additionalequationsdescribing conservationof massfor the chemicalspeciesinvolvedand conservationof energyare required.Typically usedrelationsare Fick's law for chemicaldiffusion and Fourier's law for heat transport. Once the mithematical model has been formulated, it can be applied to the situation at hand. This requiresconvertingthe governingequationsinto forms that facilitate solution.Ordinarily this is achievedthrough the use of numericalmethods suchasfinite differencesor dnite elementsto representthe applicablepartial differential equations.In using a finite differenceapproach,for example,the regionis divided into grid elementsund th" continuousvariablesare representedas discretevariables at the nodal points. In this manner,the governingdifferential equationis replacedby that canbe solvedin an iterativeway.Models a finite nu-b"r of algebraicexpressions of this type find wide applicationin the estimationof site-specificaquifer behavior. They havepro\iento be effectiveunderirregularboundaryconditions,wherethereare and where highly variablepumping or rechargerates are expected'l heterogeneities, Severil types of groundwater models and their applications are summarizedin Figure 20.2. A numberof stepsmustbe followedin modelinga targetedgroundwaterregion' Figure 20.3 is illustritive. The f,rst step is to define the boundaries.They may be ptiysical,suchas an imperviouslayer, or arbitrary, suchas the choiceof a politically,
Applications 'Geothermal
Watersupply
Seawaterintrusion
Regionai aquifer analysis
Land fills
Thermal storage
Near-well performance
Waste injection
Heat PumP
Groundwatelsurface water interactions
Radioactive waste storage
Thermal poliution
Dewatering operauons
Holding pontls
Land subsidence
Groundwater pollution
Figure20.2Typesofgroundwatermodelsandtypicalapplications.(Courtesyof OH') Worthington, theNationalWaterWellAssociation, elementsby or otherwise,definedsubregion.Next, the region is divided into discrete 20'4)' (see Figute grid a rectangularor polygonal superimposing ' (s. and z) and once th-egrid is determined,the controlling aquiferparameters is includedin the transport solute If grid element. the initial conditionsare setfor each must also properties dispersion model, additional parameterssuchai hydrodynamic operated be can model the met, been hive be specifigd.Atter atl of thesespecifications of comparisons matching). (history and its output comparedwith recordedhistory permit predictions model counterpart with recordedvaluesof headand other features considered parameteradjustmentsto be madeuntil observedand computeddata are by the modelerto be in closeagreement. to analyzea Upon completion of thelodel's calibration, it can be applied of the prediction model's The options. variety of managementand/or development decision-making to aid valuable a be can outcomesof these alternative strategies include: the processes.Examplesof the types oI problems that can be addressed aquifer of an on impact the use; of utilty of an aquifer to suppoit various levels storageof underground on effects the varying naturaiand artificial rechargerates; contamsubsurface of movement of rate weli lo-cation,spacing,andpumpingrate; the inants;and saltwaterintrusion. caution while numerical groundwatermodelshavemuch to recommendthem, Prickett appropriately. interpreted and must be exercisedto enJurethat they are used three waysin notesthat overkill, inappropriatepiediction, and misinterpretationare both the modeler pitfalls, these avoid To which groundwater-#"t, can Uemisused.s
484
CHAPTER20
MODELINGREGIONALGROUNDWATERSYSTEMS
illiiT3:':,",*itlffi,ffi:ff tlSl
(courtesv arwater ortheNation use'
and user must understandthe underlying assumptionsupon which the model was founded,its limitations, andits sourcesof errors.Usedwisely,modelscanbe powerful decision-makingaids. Used inappropriately,they can lead to erroneousand sometimes damagingproposals.
20.2 FINITE-DIFFERENCE.METHODS Digital simulation requires an adequatemathematical description of the physical processesto be modeled.For groundwaterflow this descriptionconsistsof a partial differential equationand accompanyingboundaryand initial conditions.The governing equationis integratedto produce a solution that gives the water levels or heads associatedwith the aquifer being studied at selectedpoints in space and time. The model can simulateyears of physical activity in a span of seconds,so that the
METHODS 485 2A.2 FINITE-DIFFERENCE
(a)
A.r Ly ..,--V ^t , /_-/
| 1"
', / 1 . 1 , / 4Finitedifference /
/
,
,
.
,
a
srld DlocK
(b) boundaries.(b) Finite-difFigure 20.4 (a) Map view of aquifershowingwell field and At' is the spacing t i"r""n"" grid{oi aquifir study,wGre Ax is the spacTqii lhu *ection' nodes;open block-center dots: Solid thicknesg. in tt" y-at"rtion, and f is itre aquifer worthingAssociation, well Water National the of circles:source-sink nooo tcoo*sy ton. OH.)
486
CHAPTER20
MODELINGREGIONALGROUNDWATERSYSTEMS
of proposedactionscan be evaluatedbefore decisionsinvolving conconsequences struction or social changeare implemented.The expectationis that the model runs * will lead to wiser and more cost-effective'decisions. The finite-differencemethodis basedon the subdivisionof an aquiferinto a grid and the analysisof flows associatedwith zonesof the aquifer.The equationthat must be solvedis derivedfrom continuity considerationqand Darcy's law for groundwater motion. This yieldsthe followirigpartial differential equation(a versionof Eq. 18.41), describingflow through an areally extensiveaquifer.Note that the equationpresented here describesthe two-dimensionalcase:
s#P.N#ot:s!+w where h : x : y : S: Z : V[ :
(20.r)
total hydraulic head (L), x direction in a cartesiancoordinatesystem(L), y direction in a cartesiancoordinatesystem(L), specificyield of the aquifer (dimensionless), transmissivityof the aquifer(I]lT) sourceand sink term (L/T)
In the aboveequation,vertical flow velocitiesare consideredto be negligible everywherein the aquifer.The following assumptionsareimplict in the derivation:the flow is two-dimensional;fluid densityis constantin time and space;hydraulicconductivity is uniform within the aquifer; flow obeysDarcy's law; and the specificyield of the aquifer is constantin spaceand time. Equation 20.1 is nonlinear for unconfined aquifersbecausetransmissivityis a function of headand thus the dependentvariable. In order to integrateEq.20.I, initial valuesof head, transmissivity,saturated thicknessof the aquifer,andthe amountsof waterproducedby sourcesand sinksmust be identified for every point in the region of the integration.The specificyield and location of geometric boundaries must also be defined. Unfortunately, analytic solutionsto Eq. 2O.l are impossibleto obtain exceptfor the most trivial cases.It is thus necessaryto resort to numerical integration techniquesto obtain the desired answers.t'to-to
Application of finite-difference techniquesto groundwaterflow problems requiresthat the region of concernbe divided into many small subregionsor elements (Fig. 20.5). For each of theseelements,characteristicvaluesof all the variablesin Eq. 20.I are specified.Thesevaluesare assignedto the centersof the elements,which are called nodes.The headsin adjacentnodesare relatedthrough a finite-difference equation,which is derivedfrom Eq. 20.1. Thesedifferenceequationscan be derived The by an appropriateTaylor's seriesexpansionor by massbalanceconsiderations.8 yield heads at to the simultaneously then be solved equations can resulting algebraic for each time step considered. eachnode It should be understoodthat the simulation methodspresentedin this chapter pointed toward the analysisof regional rather than localized groundwaterprobare as lems such the prediction of the drawdownat a particular well. In suchcases,the methodsdiscussedin Chapter 19 are usually the most appropriate.Here we,are mainly concernedwith waterlevel or headchangesthat might occur over a large area due to prescribedwater-usepractices.
2O.2 FINITE-DIFFERENCEMETHODS 487
t, I
o m,1 J
{ Node
eAx'---
Subregion
I Ay
l,,i
/
I -
Boundary ofregion of integration
\ \
a m,n
l, nd
for a finiteFigure 20.5 Subdivisionof a region of integrationinto computationalelements differenceProblemformulation.
BoundaryConditions In order to integrateEq.20.l,the governingboundary conditionsmust be-specified' ChapTwo typesof bJundary condition aie discussedhere.Otherswere presentedin ters 18 and 19. chosen Where the region of integrationis limited by a political or arbitrarily condition'10 boundary constant-gradient boundary,it is oftei the policy to employ a change In this "ur", unar.umption is madethat the gradientof the watertable will not with streams Where fall. or rise may llvel alongthe boundaryeventhoughthe water condiboundary stream encountered, are interconnectionsto the groundwatersystem as tions are employed.Constant-gradientboundariesare expressedmathematically a
h
l
\
(20.2)
*:81x'!) at the location x, y throughoutthe period of specified y): a constant g\x, where simulation (dimensionless) /z : hydraulic head (L) s : direction perpendicularto the boundary (L) Streamboundariesare expressedas h : f(x, Y, t) wheref(x,y,/)=anunknownfunctionoftimeatthelocationx'y (dimensionless) = hydraulic head (L) h
(20.3)
488
CHAPTER20
MODELINGREGIONALGROUNDWATERSYSTEMS
The volumetric rate of flow acrossthe constant-headboundariesdescribedby Bq.20.2 can be modeledat eachtime stepusing the Darcy equ-ation:ro
o: rfrtr
(20.4)
where h:head(L) Al = dummy variable denoting the length of the side of the subregion perpendicularto s (L) s : dummy variable denoting the direction of flow perpendicularto the boundary (L) p : volumetricdischarge(LilT) Z : transmissivityat the boundary (IllT) Use of this equationat a boundary is illustrated by the notation of Fig. 20.6. Considerthe flow from left to right in the x direction acrossthe left-handsideof the elementalregiondepicted.The node i - I, j lies outsidethe regionof integrationand thus it may be assumedthat no information aboutit is available.An assumptionmay be madeto circumventthis problem.It is that the transmissivityacrossthe boundary is uniform and equal to Tt,t. In finite-differenceform the head changeterm in Eq. 18.26 can be statedas ah _hi.j-hi-l,j
6x
(20.s)
A,x
But the headh,-r,, doesnot exist, and anotherapproximationis required, h,,i -
h,-r,i :
hi*r,j -
hi,j
(20.6)
Thesetwo expressionsare then substitutedin Eq. 20.4 to yield
f,,,u#'LY Q;-r/",i:
Boundary where 6 h= r . dx
.
_ 1^ ' i' 'i a
Figure 20t6 Subregions adjacent to a constantgradientboundary.
(20.7)
2O.2 FINITE.DIFFERENCEMETHODS 489
At the beginningof each time step, a new volumetric flux is calculatedalong each boundary.This is accomplishedby usingthe headsand transmissivconstant-gradient . ities computedin the previoustime interval. in groundboundaries as constant-head treated are sometimes Surfacestreams body in the surface level water the where problems. is adequate The assumption water process. In modeling period of the the time during remain unchanged is expectedto by affected are significantly heads, hence flows, and surface however, many instances, withdrawalsor rechargesto the interconnectedgroundwatersystem.They may then be a limited sourceof water supplyfor the groundwater.system.To accommodatethe surfacewater-groundwaterlinkage, a leakageterm may be applied.toThis expression may take the form
= -fi{n,.,.r leakage,,r,1 where
h,i,o)
(20.8)
b, , : thicknessof the streambed(L) hi,i.t : head in the aquifer at node i, j, at time k; k : 0 indicatesinitial conditions(L) t k;,; : hydraulic conductivity at node i,.i (LlT).
When Eq. 20.8 is used,the streamis consideredto coverthe entire arearepresentedby the related node. After eachtime step the leakagefrom the streamto the aquiferis calculatedand streamflowsare depletedaccordingly.If the streamflowat a particular node becomeszero, the model can be madeto note that the streamis dry and break the hydraulic connection atthat point.l0
Time Stepsand ElementDimensions The successof any finite-difference schemedependson the incremental valuesassignedthe elementdimensionsand the time steps.In general,the smallerthe dimensionsof elementsand time increments,the closerthe finite-differenceapproximation to the differential equation.However,as thesepartitions are madesmaller,a price in computationalcostsand data needsmustbe paid. Furthermore,oversubdivisionmay evenbring about computationalintractability. Thus the object is to selectthe degree of definition that results in an adequaterepresehtationof the systemwhile keeping data and computationalcosts at a minimum. There are proceduresfor making such selections,but, exceptfor a brief discussionin the following section, they are not presentedhere.lo-la
Flow Model One.Dimensional To illustrate the finite-difference approachto groundwaterproblem solving, a oneAlthough mostpractical-scalemodelsare dimensionalconceptualizationis discussed. their developmentis only an extensionof the in character, two- or three-dimensional the more complex modelsthe reader of some of For details case. one-dimensional The book by McWhorter and references.6-8'i0-1s the appropriate should consult problems.8The treatment of example excellent includes to read and Sunadais easy of that reference. the approach here follows taken flow one-dimensional
490
CHAPTER20
MODELINGREGIONALGROUNDWATERSYSTEMS Downstream head I{,
Upstream headHn
H6*Huatt=0 Flow
hi*t
-----w
IlD-constantatt>0
Confined aquifer
l+Ay+l groundwater flowcase.(After Figure20.7 Gridnotationfor a one-dimensional McWhorter andSunada.8) Let us considera one-dimensionalflow in a confinedaquifersystemsuchasthat illustratedby Fig. 20.7.lt is assumedthat the flow is unsteadyand that the flowlines are parallel and not time dependent.On this basis,a unit width of the aquifercan be studiedand observationsmadeaboutit can easilybe translatedto the total system.As shownin the figure, the unit width of the aquiferis A.r. The flow regionis overlaidby a grid, and for eachgrid element,valuesof hydraulic conductivity Kr, elementlength y,, aquiferthicknessb,, storagecoefficientSi, and the initial valuesof headh, must be specified.The massbalancefor grid elementI requiresthat the inflow (Qr-t-,) from elementI - 1 to elementI minusthe outflow{Qt-,*r)fromelementlto elementI * I must be balancedby the rate of changein storagewhich occursin elementi, LV,/LI. To simplify the problem, let us further considerthat the aquifer is of uniform and isotropic. Thus the valuesof K, ,S,b, and Ay thicknessand that it is homogeneous are constant,and we shall considerthat from studiesof the aquifer properties,they are also known. Thereforeit mav be statedthat K t : K z : ! . . : K ^ : K
Sr:Sz:..':S.:S bt:bz:',,:b*:b N r : L Y ,= ' ' ' : L Y - = N
(20.e)
wherethe subscriptln representsthe total numberof grid elements.Usingthis notation and Fig. 20.7, we can seethat the flow from elementi - 1 to I is .
Oi-r-i:
'
-UOO'|
(20.10)
;i
where i : the elementnumber n : the selectedtime Equation20.10is recognizedas Darcy's equation.It is assumedin this representationthat the head generatingthe flow at time n is the differencebetweenthe averageheadsat the two adjacent elementsdivided by the distancebetweentheir centers(nodes).This approximationapproachesexactnessas Ay diminishesto zero.
2O,2 FINITE.DIFFERENCEMETHODS 491
area of flow and is The ireaA appearingin Eq. 20.10 is the cross-sectional obtainedasthe productof Al and b. Sincewe are dealingwith a unit width of aquifer, A.r : 1 and sinceb is a constantby definition here, Eq. 20.10 mtry be written o,- -':
hl - "h,! i-t -7"i '
Ay
(20.11)
where 7 : Kb. A similar expressionfor the flow from element ito i * 1 may be obtained:
(20.r2) Equations20.11 and 20.12 representthe inflow and outflow from elementi. Considering that continuity conditionsmust be met, this changein flow acrossthe element *urt b" balancedby the changein storagewhich occursduring the time step.This is siven as
fr:'.(ry)
' (zo.tz)
in the continuity equation(inflow - outflow : Now insertingthesethreeexpressions changein storage),we get
t''i) (-rhi -,ni ) - f -rhi-,-- hi) : , 6,(h'lo'^. '\ ^t AY / / Av / \ \
1zo.r+i
the equationbecomes By rearrangement,
(20.1s) which is known as the explicit or forward difference form of the finite-difference equationif n is designatedas the current value of time. If, on the other hand, n is definedas / + Ar, thin the equationis the implicit or backwarddifferenceequation. The explicit solutionto Eq. 20.15 Eachof theseforms hasits own solutiontechniques.8 here. will be discussed By letting n : t inF;q.2O.l and rearranging,one obtains
+ tur+^, ffirr;, +hi-) nllr ##]
(20.16)
In this casethe spabederivativesare centeredat the beginningof the time stepand the singleunknownls h!*^'. Equation 20.16 canbe solvedexplicitly at eachelementfor the headat the next period of time. The solutiondependsonly on a knowledgeof the
492
CHAPTER20
MODELINGREGIONAI-GROUNDWATERSYSTEMS
an unstable condition. Irt the one-dimensionalhomogeneouscase discussedhere, stability is assuredif
<1 s/,vY t TLt
(20.r7)
The equationshowsthat the choiceof time and spaceincrementsis not independent. Satisfactionof Eq. 20.17 doesnot guaranteean accurateapproximation,however;it only providesfor a stablesolution.8 EXAMPLE 2O.T Refer to the one-dimensionalflow problem of Fig: 20.8. Let us assumethat the elementlength is 4 m and that the thicknessof the confinedaquiferis 2 m. It is further : 0 and that assumedthit the headat the left and right sidesof the region is 8 m at / zero. K : greatet than the head on the right side takes on the value 2 mfot all t : 0.5 m/day and S 0.02. As shown in the figure, there are five elements.Using the notation of eq. 20.16,the initial condition is hf; : 8.0 m. Use the explicit methodto determinefuture heads. Solution 1. First a determinationmust be made of the time step to use. This may be usingEq. 20.17. accomplished Lt <-
1 s(Ay)': ; W :
0 . 1d6a v s
The valueof Zused in the aboveexpressionwas obtainedusingthe relation T:Kb: Z:0.5X2.0:1.0 To be on the safeside,we shall choosea time stepof 0.1 days,althoughany value lessthan 0.16 would have assuredstability.
8m
A)=4m Figtrre 20.8
Sketch for Example 20.1.
493 20.3 FINITE-ELEMENTMETHODS ^' 2."For the first time step,t : 0.1, we can calculateh'a* and corresponding headsfor the other elementsusing Eq. 20.16.Thus
h,;",:
*, ",,411 0 f , --s(Ay),1 rful + hg) f# @?
and substitutingnumerical values,we get
^f. - 2(1.0)(0.1)l ,n, : 1.0(ol) + 8.olt /,1' (ffi(2.0 + 8.0) drn+Fl : 3 . 1 + 3 . 0 : 6 . 1 3m Sinceft? andho2: 8.0 m and sinceh, : 8,0 by definition,it can easilybe shownusingEq. 20.16 thatthe valuesof hlr andhl't ate not changedfrom their original level of 8.0 during the first time step. 3. Now considerthe secondtime step,t + Lt = 0.2 days.For element4,
: h2,
#(h2,
+ hg\+ nZ'lt-
#]
: 0.31Q.0+ 8.0) + 6.13(0.37 ) : 5.4 m
I
For element3
hg':ffi(h:'+h9\+l,l'[r ffi] : 0 . 3 1 ( 6 . 1+3 8 . 0 )+ 8 . 0 ( 0 . 3 7: 7 ) '4m
Element 2 doesnot have a head changeuntil the third time step. 4. The processdemonstratedis repeateduntil the headshavebeencalculated for the total time period of interest.For this example,they will ultimately reachequilibriumconditions. rl This exampleproblem illustratesthe mechanicsof the finite-differenceprocedure. Problemsof practical scalewould require the use of a computer,but the approach would still be the same.
METHODS 20.3 FINITE.ELEMENT The mostwidely usednumericaltechniquesfor solvinggroundwaterflow problemsare the finite-differenceand finite-elementmethods.The finite-elementmethodis similar to the finite-differencemethodin that both approacheslead to a set of N equationsin Nunknowns that canbe solvedby relaxation.6Nodesin the finite-elementmethodare usually the corner points of an irregular triangular or quadrilateral mesh for twodimensionalapplications,while for three-dimensionalapplications,bricks or tetrahedrons are commonly used.The sizeand shapeof the elementsselectedare arbitrary. They are chosento fit the applicationat hand.They differ from the regularrectangular grid elementsusedin finite-differencemodeling.Elementsthat are closestto points of flow concentratiensuchas wells are usually smaller than those further removed
494
CHAPTER20
MODELINGREGIONALGROUNDWATERSYSTEMS
from suchinfluences.Aquifer parameterssuchashydraulic conductivity may be kept constantfor a given elementbut may vary from one to another. To minimize the variational function, its partial derivativewith respectto head,,isevaluatedfor each node and equatedto zero.The procedureresultsin a set of algebraicequations that can be solved by iteration, matrix solution, or a combination of thesemethods.ra Finite-elementmodelersmust understandpartial differential equationsand the calculus of variations.6 The finite-elementapproachoffers someadvantagesover the finite-difference technique.Often, a smaller nodal grid is required, unA tnir offers economies in computer effort. The finite-elementapproachcan also accommodateone condition that the finite-difference approach is unable to handle.6When using the finitedifferencemethod,the principal directionsof anisotropyin an anisotroplcformation are parallel to the coordinatedirections.In caseswhele two anisotropicformations having different principal directions occur in a flow field, the finite-difference approach cannot produce a solution, whereasthe finite-element approach can. TLe finite-elementtechniquecan be used to simulate transient aquifei-performance. A detaileddiscussionof the finite-elementtechniqueis beyondthe scopeof this boor<, but there are many good referencesfor the interestedreader.6'rsts.zi ,
20.4 MODEL APPLICATIONS To illustrate how simulation modelscan be usedto provide insightsinto water managementschemes,a model analysisof the Upper Big Blue basin aquiferin Nebraska is presented'The studywasconductedby the Conservationand SurveyDivision of the University of Nebraskaunder the direction of Huntoon.l0 The useof groundwaterfor irrigation in the Upper Big Blue basinwas observed to be rapidly increasingand by 1972 about3.: wetlslmit iere in operation.At that time farmerswere becomingconcernedaboutthe progressivedeclineof water levels and were seekingguidanceaboutthe efficiencyof implementingsomeform of basinwide water managementpro€ram. The Universiry of Nebraski designeda model to evaluatethe situationand to explorevariousproposalsfor recharging:theaquifer and for estimatingthe long-term consequences of siveral scenariosof water use in the basin. The study area is shown in Fig. 20.9. Generallythe water table is free in the regionof interest'Figure 20.10 showsthe configurationof the watertable asobserved in 1953.For modelingpurposes,the water-levelcontoursshown were considered to be representativeof predevelopmentconditions.This assumptionwas basedon the fact that groundw-aterwithdrawals before this time were not extensive.It was also surmisedthat the Lonburs representeda water table in which an equilibrium existed betweennatural rechargeand dischargein the region.Transmissivitieswereestimated from drill-hole sample logs recorded in the area. These values are needed for modelingand are also important indicesof the potential yield of wells that might be constructed. As might be suspected,the information of most concernto the local landowners and water plannerswas the rate of decline of the water table. In particular, it was
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desiredto knbw how rapidly the groundwaterresourcewould Le depleted,where and when waterlevel declineswould posean economicconstrainton water use,and what impactsfuture developmentsand/or managementwould haveon thdrate of decline. The model developedto explorethesefeatureswas a two-dimensionalrepresentation of flow through an areally extensiveaquifer.loEquation 20.1, alongwith the appropriate boundary conditions, constituted the model. The region shown in n1g.Z0.Swas divided into a finite-differencegrid and, after substitutionof the nodal ualu"t of Z and S, the model was operatedto predict water-levelchangesto the year 2020 for variouspolicies of rechargeand for severallevelsof development.Calibration of the model was accomplishedusinghistoricdata. The model was operatedover the period 1953-1972 using the known distribution of wells and the averagenet pu-pug" per well to establisha match betweenobservedand estimatedwater-level "ttung"i. Once this was accomplished,the simulation of future trends proceeded. achievedin the matchingprocess. Figures20.II and20.12showthe correspondence On the basisof the model studies,it wasdeterminedthat waterlevelsin the study areawould continueto declineevenif developmentwas limited to the l912level. It wasfurther predictedthat someparts of the areawould experienceseveregroundwater shortageiby the year 2000. It was found, however,that by employingartificial rechargefrethods, permanentgroundwatersuppliescould be assured.To assessthe effectsof artificial recharge,two water delivery systemswere modeled.Both of these delivered water from Platte River Valley sourcesto rechargewells located in the project area. Using thesetwo water delivery systems,three rechargeschemeswere ii-ulut"d. The grosseffect of introducingthe rechargewells was the cancellationof the effects of the proportionatenumber of pumping wells. Figure 2O-I3 showsthe computedwater-levelchangesat one location under a graduateddevelopmentplan (projected on the basisof the 1972rate of development)with no rechargeand.then wlth graauateddevelopmentfor eachof the three rechargeschemes.The continual downwardtrend in waier level with no recharge(curve 1) clear$ showsthe nature of the problemin the UpperBig Bluebasin.The othercUrvesdepictingthe threeartificial ,""hurg" options show that stability can be achievedif such an approachis taken. While the costs of implementing artificial rechargemight be excessive,it is apparentthat any long-terrn solution to the declining water table problem, short of reducinguse, would require a supplementalsourceof water. Operationof the modelprovidedusefulinsightsinto the natureof the watertable problem and suggestedthat irrigators shouldbe making someimportant water managementdecisionsabout their future mode of operation..^ ^The modeling of groundwatersystemsis complex.lo-25In structuring models suchasthat just discussed,simplifying assumptionsmustusuallybe made.Thesehave to do with aquifer pbrameterssuch as transmissivity,specificyield (for unconfined aquifers),and storagecoefficient (for confined systems)'Furthermore,the boundary conditions are normally approximationsof what occurs in the physical system.and assumptionsaboutthe uniformity of materialsin varioussubsurfacestrataare sometimes crude.This doesnot meanthat groundwatermodelscannotbe expectedto yield useful results.It doesimply that the usersof the modelsmust be cautiousabouthow they interpretthe output.For example,an areally extensiveaquifermodel suchasthat developedby Huntoon for analyzingthe Blue River problem can be expectedto give
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trends'(AfterHuntoon'10)' water-level andcomputed Figure20.1.2 Measured reliable information about water-leveltrends for various conflgurationsof development. It shouldnot, on the other hand, be consideredan accuratepredictivetool for monitoringthe water-levelchangeat somespecificpoint in the regionof concern.This type of information could be derived only from a more detailed modeling of the lolafif surroundingthe point. The informationprovidedby the Blue River modelwas targetedto showlocal landownerswhat the future might hold for severaldevelopment levelsand for severalmanagementoptions.The actual water levelspredictedby the model were not of central concern;what was of interestwas the determinationthat unlessfuture developmentwas restrictedand supplementalwater provided,or unless current usescould be significantlyreduced,the outlook in the next 50 yearswas not good for irrigated farming. The model thus providedthe basisfor making somequantitativeobservations aboutthe future. It also providedinsightsinto the relief that might be expectedfrom artificial recharge.Beyondthat, it could be usedto model other possiblemanagement
2010
2020
20.13 Computed water-level Figure changesunder a plan of graduateddevelopment for conditions of (1) no techatge,(2) rechargeunder Scheme1, (3) rechargeunder Scheme 2, and (4) recharge under Scheme3. (After Huntoon.lo)
500
CHAPTER20
MODELINGREGIONALGROUNDWATERSYSTEMS
options.A,model suchas this, carefully usedand properly interpreted,can thus add a powerful dimensionto decision-makingprocesses.
20.5 GROUNDWATER QUALIW MODELS Groundwaterquality hasbecomea major sourceof concernin recentyears.This has comeaboutfrom the realizationthat many groundwatersourcesthat wereat onetime consideredalmost pristine have now been degradedin quality by seepagesfrom dumps, leakagefrom industrial waste holding ponds, and by other waste disposal and/or industrial and agricultural practices.To deal with suchproblems,'therehas been an expandingmovementto developquantitative techniquesto understandthe mechanicsof groundwaterquality. Thesemodels,althoughnot as advancedas their surfacewater counterparts,are now beginningto play an important role in water quality management. The subjectof groundwaterquality modelingis complexand underrapid development.Accordingly,a thoroughtreatmentof the subjectis beyondthe scopeof this book. The importance of this topic cannot be overemphasized,however, and the readeris encouragedto consultthe referencesat the end of the chapter,specifically Refs.6 and 26-30. In 1974, Gelhar and Wilson developeda lumped parametermodel for dealing with water quality in a stream-aquifersystem.The nomenclatureand conceptualization of their model are shown in Fig. 20.14.2eThe rationale for using a lumped parameterapproachwasthat when dealingwith changesin groundwaterquality over long periods of time, temporal rather than spatial variationsare most important. Changesin water table in the Gelhar-Wilson (GW) model are representedby the following equation: dh 'nd- t=
where
-q+e*q,-ep
(20.18)
h - averagethicknessof the saturatedzone p : averageeffectiveporosity e : natural rechargerate natural outflow from the aquifer 4 , : artificial recharge/unitarea Q p : pumping ratelunit area T _ time
This is just anotherform of the continuity equationrelating inflow, outflow, and the changein storage(lefrhand term in Eq. 20.18).The changein concentrationof a constituentis given by ,dc Pndt-
(e -l q, -t aph)c : ec. * q,c,
(2o.re)
20.5 GROUNDWATERQUALITYMODELS €. Ct
(b)
of the Gelhar-Wilsonmodel'(After Figure 20.L4 Schematic NovotnyandChesters.2a) where
c: c; : c, : c:
concentration coflcefltrationof the natural recharge concentrationof the artificial recharge a first-orderrate constantfor degradationof the contaminant
The GW model assumesthat dispersionis negligible.This assumptionmay be made on the basis that the objective of the model iS to estimateregional-average The model also provides for the determinationof hydraulic and concentratiolrs.2e soluteresponsetimes for the system.Theseare measuresof the lag that occursin the moue-"ni of both water and constituentinputs to the system.Gelhar and Wilson assumethat the responseof an aquiferto a specificinput can be likened to that of a well-mixed linear reservoir.Their studiesshowedthat the model's determinationof the concentrationof constituentsleavingan aquifer is representativeof the average concentrationof the constituentin the aquifer.On this basis,,itappearsthat the model
502
CHAPTER20
MODELINGREGIONALGROUNDWATERSYSTEMS
is well suitedto estimatingthe quality of groundwaterdischargingto a surfacestream, providing-ihe aquifer is narrow relative to the length along which dischargeoccurs.
r summary Groundwaterin a regionalaquifer systemis constantlyin motion. The amountstored at any time is affectedby artificial and natural-recharge,evapotranspiration,flow to springs and surface water courses,and by collection devices such as wells and infi ltration galleries. Natural hydrologic statesmay be significantly affected by human activities. Aquifer depletionshaving regional and national economicimplications are not uncorlmon. Depletionof the Ogallala aquiferin the central United Statesby long-term and extensivewater withdrawalsfor irrigation is a good example.On the other hand, waterlevelshavebeenmadeto rise, sometimesinadvertently,by humanintervention. Leaky irrigation canalsin central Nebraskawere at one time responsiblefor groundwaterlevel risesin somefarming locationsof a magnitudesufficientto jeopardizeuse of the land. Once major problemsof depletionor over-replenishmentoccur, they are not easilydealt with. In general,a safe-yieldpolicy for groundwatermanagementhas merit and shouldbe considered.6'30. Regional groundwater flow problems are usually modeled by an equation combiningDarcy's law and the equationof continuity.The resultingpartial differential equationoor setof equations,describesthe hydraulicrelationswithin the aquifer. To effect a solution to the governingequation(s),the aquifer's hydraulic features, geometry,and initial and boundary conditions must be determined.Unfortunately, many groundwaterproblems exist for which exact analytic solutionscannot be obtained.In suchcases,it is necessaryto rely on numericalmethodsfor modeling.Under suchcircumstances,an approximatesolutionis obtainedby replacingthe basicdifferential equationswith another set of equationsthat can be solved iteratively on a computer.Both finite differenceand finite elementmethodsare applicable. The finite differenceapproachdescribedin this chapterreplacesthe governing partial differential equationswith a setof algebraicequations.Thesecanbe solvedon the computerto producea set of water table elevationsat a finite numberof locations in the aquifer. Once the groundwatermodel has been calibrated,it can be usedto predict the outcomes(impacts) of alternativedevelopmentand/or managementstrategiesproposed for an aquifer. Such analysesare valuable adjuncts to decision-makingprocesses.Models can, for example,simulate the effects of opening new well fields, analyzechangedbperating practices for existing well fields, explore schemesfor artificial recharge,and predict the impactsof proposedirrigation developmentplans. Groundwatermodelscan be applied to unconfinedaquifers,semiconfinedaquifers, confined aquifers,or any combinationthereof. They can accommodatelarge variations in aquiferparameterssuchashydraulicconductivityand storagecoefficient,and they can be usedto analyzeunsteadyas well as steadyflow problems.
REFERENCES 503
PROBLEMS 20.t. Refer to Fig. 20.8. Assumethat the elementlength is 5 m and the thicknessof the
confined aquifer is 2.5 m. The head at the left and right sidesis 8.1 m at r : 0, and the head on the right is 2.5 m for all r > 0. K : 0.5 m/day and S : 0.02. Use the explicit method to determineheadsat future times. 20.2. Referto Fig. 20.8.Assumethe elementlengthis 10 ft andthe thicknessof the confined aquiferis 8 ft. The headat the left and right is 21 ft at t : 0, and it drops on the right sideto 8 ft for all t > 0. K: 1.5 ftlday-ands : 0.02. use the explicitmethodto calculatefuture heads. 20.3. Referto Fig. 20.12.Asidefrom the trend, what elsecanyou deducefrom studyingthis figure? 20.4. Discusshow you would go about designinga grid for a regional groundwaterstudy. What types of boun{ary conditions might you specify?Why?
REFERENCES 1. J. W. Mercer and C. R. Faust,Ground-WaterModeling.Worthington,OH: National Water Well Association,1981. 2. C.A.AppelandJ.D.Bredehoeft,"statusofGroundwaterModelingintheU.S.Geological Survey," U.S, Geol. Survey Circular 737(1976). "Utilization of Numerical Groundwa3. Y. Bachmat,B. Andres,D. Holta, and S. Sebastian, ter Models for Water ResourceManagement,"U.S. Environmental Protection Agency ReportEPA-600/8-78-012. "Contribution of Ground-waterModeling to Planning," J. Hydrol' 43(Oct. 4. J.E. Moore,
r979). 5. T. A. Prickett,
"Ground-water Computer Models-State of the Att,'l Ground Water
r7(2),t2r-r28(r979). 6. R. A. Freezeand J. A. Cherry, Groundwater.EnglewoodCliffs, NJ: Prentice-Hall,1979. 7. G. D. Bennett, Introduction to Ground Water Hydraulics, book 3, Applications of Hydraulics. Washington,D.C.: U.S. GeologicalSurvey,U.S. GovernmentPrinting Offlce, 1976. 8. D. B. McWhorter and D. K. Sunada, Ground Water Hydrology and Hydraulics. Fott Publications,1977' Collins,CO: WaterResources 9. D. K. Todd,GroundwaterHydrology,2d ed.New York: Wiley' 1980' "PredictedWater-LevelDeclinesfor Alternative GroundwaterDevelop10. P. W. Huntoon, Big Blue River Basin,Nebraska,"ResourceRep.No. 6, Conservation in the Upper ments and Survey Div., University of Nebraska,Lincoln, 1974. "The Numerical Solution of Parabolic and 11. D. W Peacemenand H. H. Rachford, Jr., Afpl. Math. J.3,28-4I(I955)' Indust. Elliptic DiffetentialEquations,"Soc. "Application of th€ Digital Computer for Aquifer D. Bredehoeft, and J. F. Pinder 12. G. Evaluation,"WaterResourcesRes.4(4), 1069- 1093(1968). 13. I. Remson,G. M. Hornberger,and F. J. Molz, NumericalMethodsin SubsurfaceHydrology. New York: Witey, 1971. for Groundwa14. T. A. Prickett and C. G. Lonnquist,SelectedDigital ComputerTechniques ter ResourceEvaluation,Illinois StateWater Survey Bull. No. 55, 1971.
PARTFIVE
MODELING HYDROLOGIC
Chapter21
lntroductionto Hydrologic Modeling
r Prologue The purposeof this chapteris to: . Introduce the types and classesof hydrologicmodels. . Illustrate the limitations, alternatives,steps,general components,and data needsof hydrologicsimulation models. . Presenta philosophicalprotocol for performing successfulmodelingstudies. . Give an overview of groundwatermodel types. . Distinguishthe need for separate,specificproceduresdetailedi4 subsequent Chapters22, 23, 24, and25. Information regardingratesandvolumesof flow at any point of interestalonga stream is necessaryin the analysisand designof many types of water projects. Although many streamshavebeen gaugedto provide continuousrecords of streamflow,plannersand engineersare sometimesfacedwith little or no availablestreamflowinformation and must rely on synthesisand simulatlon as tools to generateartificial flow sequencesfor use in rationalizing decisionsregardingstructuresizes,the effects of land use, flood control measures,water supplies,water quality, and the effects of natural or inducedwatershedor climatic changes. The problemsof decisionmaking in both the designand operationof large-scale systemsof flood control reservoirs,canals,aqueducts,and water supplysystemshave resultedin a need for mathematicalapproachessuchas simulation and synthesisto investigatethe total project. Simulationis definedas the mathematicaldescriptionof the responseof a'hydrologic water resourcesystemto a seriesof eventsduring a selectedtime period.For example,simulationcanmeancalculatingdaily, monthly, or seasonalstreamflowbasedon rainfall; or computingthe dischargehydrographresulting from a known or hypotheticalstorm; or simply fllling in the missingvaluesin a streamflowrecord. Simulation is commonly used in generating streamflow hydrographsfrom rainfall and drainagebasin data. The philosophiesand overall concepts used in simulation are introduced in this chapter. Chapter 22 summarizesconcepts of
508
To HyDRoLocrcMoDELING cHAprER21 rNTRoDUcroN streamdowsynthesisby stochasticmethods.Chapters23-25provide detailsregarding determinislic continuousmodels, single-eventmodels,urban runoff and storm sewerdesignmodels,and water quality models. Stochastictechniquesusedto extendrecords,either rainfall or streamflow,are classifiedas synthesismethods.This procedurerelies on the statisticalpropertiesof an existingrecord or regionalestimatesof theseparameters.An overviewof synthesis techniquesis presentedin Chapter22.
21.1 HYDROLOGIC SIMULATION In this chapter, simulation of all or parts of a surface,groundwater,or combined systemimplies the use of computersto imitate historical eventsor predict the future responseof the physicalsystemto a specificplan or action. Physical,analog,hybrid, or other modelsfor simulatingthe behaviorof hydraulic and hydrologicsystemsand systemcomponentshave had, and will continue to have, application in imitating prototype behSviorbut are not discussedhere. A few of the numerousevent,continuous,andurbanruneff computermodelsfor simulatingthe hydrologiccycle are comparedin Table 21.1. As shown in the tabfe, mostof the modelsweredevelopedfor, or by, universitiesor federalagencies.All have moderate-to-extensive input data requirements,and all have from 1 to 10 percentof TABLE 21.1 DIGITALSIMULATIONMODELSOF HYDROLOGICPROCESSES
Code name
Model name
Percentage of inputsby Date of original Agencyor organization judgmenf development
Continuous simulationmodels-Chapter23 streamflow API USDAHL SWM-IV HSPF NWSRFS SSARR PRMS SWRRB
Antecedent Frecipitation Index Model 1970,1973,1974 RevisedWatershedHydrology Stanford WatershedModel IV Hydrocomp Simulation Program-FORTRAN National Weather Service Runoff Forecast System Streamflow Synthesisand Reservoir Regulation Precipitation-Runoff Modeling System Simulator for Water Resourcesin Rural Basins
HEC-1 TR-20 USGS HYMO SWMM
Rainfall-runoff event-simulation models-Chapter HEC-I Flood HydrographPackage Corps Computer Program for Project Hydrology scs USGS Rainfall-Runoff Model USGS Hydrologic Model Computer Language ARS Storm Water ManasementModel EPA
UCUR STORM MITCAT SWMM ILLUDAS DR3M PSURM
University of Cincinnati Urban Runoff Model Quantity and Quality ofUrban Runoff MIT Catchment Model Storm Water ManagementModel Illinois Urban Drainage Area Simulator Distributed Routing Rainfall-Runoff Model PennsylvaniaState Urban RunoffModel
I
Private ARS Stanford University EPA
I 10 10 10 3 5 10
Corps USGS USDA
1969 t970 1959 196l t972 1958 1982
r990
24
10 1 5
1973 I 965 t972 r972 t971.
z 3 5 5 I 5 5
t972 r974 t970 I9'71 1972 1978 1979
I 5
models-Chapter 25 Urbanrunoffsimulation University of Cinci4natr Corps MIT EPA Illinois State Survey USGS PennsylvaniaState University
"Judgment percentagesare from U.S. Army WaterwaysExperiment Station.r
SIMULATION509 21.1 HYDROLOGIC by repeatedtrials inputs that arejudginent pafameters.Theseare normally validated modelsbut simulation event primarily are with the models.The urban runoff models deferred are models urban of descriptions havebeenisolatedin Table2 1. 1 becausethe to ChaPter25. modelsshown Severalof the major eventand continuousstreamflowsimulation models HEC-1 and Stanford and24.The 23 in Table2!.1aredescribedin chapters briefly are 2I't Table in listed models most are emphasized.For further referince, "Models and Methpublication in the models, described,alongwith about 100 other Fleming's text plesentscomplete ods Applicable to Corps of EngineersStudi-es."1 and other models'2 USDAHL' descriptionsof the SSARR, S'[iM, HSP,
Classificationof SimulationModels In recent decadesthe scienceof ct water resourcesystemshas Passed engineeringprocedure.The variedni has causeda proliferation of catego classiflcationsare Presented. physical vs. Mathematical Models Physicalmodelsinclude analogtechnologies In contrast' mathematical and principlesof similitude appliedto smali-scalemodels' A laboratory flume system' the represent to models,"iy on mathematicafJtut"-.nt, theory of hydrograph unit the while rhay be a 1:10 physical model of a stream, effective various to watershed a of response Chapter12 is a mathematicalmodelof the rain hYetograPhs.
'
by consideris achieved Continuous vs. Discrete Models A secondclassification processes the because continuous as models ing physical,analog,and some digital occur and are modeledcontinuous necessityand advantagesof slicing qualifY as discrete models' A we indication method for routing a flo instantaneousreservoirdischargerz time. over time and timeDynamic vs. static Models Processesthat involve changes modelsthat contrast' In models' dynamic varying interactionscan be simrrlatedby hydrologic Few static' called irequently ui" examine time-independentprocer*"* simulation modelsfall into the latter category' havehad the greatest Descriptive vs. conceptual Models Descriptivemodels becausethey are appticaiionand are of particular interestto practicinghydrologists and the useof basic designedto accountfor observedphenomenathrough empiricisrn concepassumptions' conservation momentum fundamentalssuchas continuity or rather interpletphenomena to theory on heavily tual models,on the other hand, rely on based models include latter the of Examples than torepresentthe physicalpto""tt.
510
CHAPTER21
INTRODUCTIONTO HYDROLOGICMODELING
probabiliiy theory.Recenttrendsin the useof artificial intelligenceand expertsystems in water systemmodelingwould classifyas conceptualmethods. Lumped vs. Distributed Parameter Models Modelsthat ignore spatialvariations in parametersthroughoutan entire systemare classifiedas lumpedparameter models.An exampleis the use of a unit hydrographfor predicting time di;tributions of surfacerunoff for different stormsover a homogeneous drainagearea.The "lumped parameter" is the X-hour unit hydrographusedfor convolutionwith rain to givelhe storm hydrograph.The time from end of rain to end of runoff is also a lumped parameteras it is held constantfor all storms.Distributedparametermodelsaccount for behavior variations from point to point throughout the *yst"*. Most modern groundwatersimulationmodelsare distributedin that they allow variationsin storage andtransmissivityparametersovera grid or lattice systemsuperimposedoverthe plan of an aquifer.More recently,surfacewater systemsare being analyzedthroughuseof distributedparameterGeographicalInformation System(GIS) technologies. Black-Box vs. Structure-lmitating Models Both of thesemodelsacceptinput and transform it into output. In the former case,the transformationis accomplished by techniquesthat havelittle or no physicalbasis.The alchemist'spurported ability to transform lead into gold or plants into medicinewas accomplishedin a black-box fashion- In hydrology, black-box models may sometimestransform "plants" into "medicine" even though the reasonsfor successare not clearly understood.For example,a modelthat acceptsa sequenceofnumbers,reduceseachby 20 percent,and outputsthe resultsmight be entirely adequatefor predictingthe attenuationof a flood waveas it travelsthrough a reachof a given stream.In contrast,a structure-imitating modelwould be designedto useacceptedprinciplesof fluid mechanicsand hydraulics to facilitate the transformation.
'
Stochastic vs. Deterministic Models Many stochasticprocessesare approximated by deterministic approachesif they exclude all considerationof random parametersor inputs. For example,the simulation of a reservoir systemoperating policy for water supply would properly include considerationsof unceitainties in natural inflows, yet many water supply systemsare designedon a deterministic basisby masscurve analyses,which assumethat sequencesof historical inflows are repetitive. Deterministic methods of modeling hydrologic behavior of a watershedhave becomepopular. Deterministic simulation describesthe behaviorof the hydrologic cycle in terms of ma[hematicalrelations outlining the interactionsof variousphases of the hydrologicclcle. Frequently,the modelsare structuredto simulatea streamflow value,hourly or daily, from given rainfall amountswithin the watershedboundaries. The model is "verified" or "calibrated" by comparingresults of the simulation with existingrecords.Oncethe modelis adjustedto fit the known period of data,additional periods of streamflowcan be generated. Event'Based vs. Contlnuous Models Hydrologicsystemscan be investigatedin greaterdetail if the time frame of simulationis shortened.Many short-termhydrologic modelscould be classifiedas event-simulationmodelsas contrastedwith seauential
' 21.1 HYDROLOGIC SIMULATION
.
511
or coniinuousmodels.An exampleof the former is the Corps of Engineers-singlemodel eventmodel, HEC-1,3 and an exampleof the latter is the Stanfordwatershed three' simulate to operated normally is which developedby Crawford and Linsley,a use might model simulation event A typical four, five, oi *or" yearsof streamflow. min' I perhaps even a time incrementof t hr or have arisen water Budget vs. Predictive Models sevelal model classiflcations comparison important One that distingtish betweenthe purposesof the model types. precipiis whetherthe model proposesto predict future conditionsusing synthesized tation and watershedconditions or model is definedas a model or set of of inflows, outflows,and changesin advisedthat simulationmodel studier use the Parametersthat affirm the b shed.For example,meteorologicdat watershed'A water application amounts might be known for a given agricultural (ET) formula budgetmodelwould be uied to determinethe correctevapotranspiration equation continuity in the parametersby testing a range of valuesuntil a balance month-byor day-by-day occursfor all time increments.This is often performedon a budget model' month basis. once the ET parametersare derived from the water or farming conditions' predictive simulationsof diffirent crop patterns,meteorologic the model in relationships practicescould be performedwith the satisfactionthat the corroboratehistorical water budget outputsare measured(precipitation studiesrequire the simultaneousdt ondarY Processessuch as ET, infil spatial distribution of water applications'
Limitationsof Simulation
t-
systems'some Becausesimulation entails a mathematicalabstractionof real-world to which the extent The can^occur. behavior system degreeof *i,,"p,",entation of "depends a developed of test The factors' many on model and systemoutputs vary is consisbehavior the that demonstrating by simulationmodelconsistsof u"iifi"uiion system' physical the tent with the known behaviorof resources Even verified simulation models have limitations in usesfor water of assessments performance allow will models planhing and analysis.Simulation particularly optispecificschemesbut cannotbe usedefficiently to generateoptions, some formulated^by plan is near-optimal a once mal plans, for stated objectives. testing for effective normally are runs simulation of othertechnique,a limited number variablesusing ranand improving the plan by modifying combinationsof decision optimal plans are generating for Techniques dom or systeriaticsamplingtechniques. describedin Section21'3' proceAnother limitation of simulationmodelsinvolveschangingthe operating Programming modeled' being system the of duresfor potentialor existingcomponents for example,requires u "o*pot", to handle reservoir storageand releaseprocesses' is rereprogramming considerable and rules, large portions to define the operatin! qoir"a if other operatingproceduresare to be investigated.
512
oHAPTER 21 INTRoDUoTIoN To HYDRoLoGIc MoDELING A'fourth limitation of simulationmodelsis the potential overrelianceon sophisticated output when hydrologicand economicinputs are inadequate.The techniques of operational hydrology can be used to obviate data inadequacies,but these also require input. Controversyover the use of syntheticdata centerson the questionof whetheroperationalhydrologyprovidesbetter information than that containedin the input.
Utilityof Simulation Computersimulation of hydrologicprocesseshas severalimportant advantagesthat shouldbe recognizedwheneverconsideringthe merits of a simulation approachto a problem that has other possiblesolutions.One alternativeto digital simulationis to build and operateeither the prototype systemor a physically scaledversion.Simulation by physical modeling has been applied successfullyto the analysisof many componentsof systemssuchas the designof hydraulic structuresor the investigation of streambank stability.However,for the analysisof complexwaterresourcesystems comprisedof many interactingcomponents,computersimulation often provesto be the only feasibletool. Another alternative to digital simulation is a hand solution of the governing equations.Simulation models,once formulated, can accomplishidentical results in lesstime. Also, solutionsthat would be impossibleto achieveby hand are frequently achievedby simulation.In addition, the systemcan be nondestructivelytested;prdposedmodificationsof the designsof systemelementscan be testedfor feasibility or comparedwith alternatives;andmanyproposalscanbe studiedin a shorttime period. An often overlookedadvantageof simulation includes the insight gained by gathering,organizing,and processingthe data, and by mentally and mathematically formulating the model algorithmsthat reproducebehaviorpatternsin the prototype.
Stepsin DigitalSimulation A simulationmodel is a set of equationsand algorithms(e.g.,operatingpolicies for reservoirs)that describethe real systemand imitate the behaviorof the system.A fundamentalfirst stepin organizinga simulationmodelinvolvesa detailedanalysisof all existingand proposedcomponentsof the systemand the collection of pertinent data. This stepis called the systemidentification or inventory phase.Includeditems of interestare site locations,reservoircharacteristics.rainfall and streamflowhistories, water and power demands,and so forth. Typical inventory items requiredfor a simulatiol study and data needsthat are specificto someof the modelsare detailed in subsequentparagraphs. The second-phase is model conceptualiTation,which often providesfeedbackto the first phasebf defining actual data requirementsfor the planner and identifying systemcomponentsthat areimportant to the behaviorof the system.This stepinvolves (1) selectinga techniqueor techniquesthat are to be used to representthe system elements,(2) formulating the comprehensivemathematicsof the techniques,and (3) translatingthe proposedformulation into a working computerprogramthat interconnects all the subsystems and algorithms. Following the systemidentification and conceptualizationphasesare several stepsof the implementationphase. Theseinclude(1) validatingthe model,preferably
SIMULATION 513 21.1 HYDROLOGIC for the by demonstrating that the model reproduces any available observed behavior the improve to necessary as algorithms (2) the modifying actual or a similir system; simulathe out carrying by work to model (3) putting the accuracy of the model; and tion exPeriments.
Model Protocoi Five axiomsfor performingsuccessfulmodelstudies,adaptedfrom recommendations by Friedrich,5are: 1. Evaluatethe data beforebeginning. t Document assumPtions. 3. Plan and control the sequenceof computerruns' of output. 4. Insist on reasonableness 5 . Document,document,document. Examining and evaluatingthe basic data are essential.An annotated,bibliographic program \ record of the data ,orrr."* shouldbe maintained.It is alwaysgood adviceto numerical of the modelsthat output (echo)datavaluesasthey arereadin. Verification valuesand proper entry of the data can be establishedfrom the echo. Statisticssuchasih" 1n"un,mode,median,range,standarddeviation,skewness, for kurtosis, and rank order are often helpful in locating entry errors' Checking Are the rainfall? of ahead inconsistenciescan identify errors. Didthe runoff occur characters waterlevelsgraduallyvaried,or are there discontinuities?Do alphabetical or missing as be interpreted appearin th! data?Will blank valuesin the data sets hyFor zeto)? (division by zeros?Will zeros in the data sets result in overflows drographrouting, doesthe time interval selectedfall betweenthe limits recommended for staUltityand convergenceof the numerical method usedto solvethe differential equations? ^ Assumptionsare also important to the successof a simulation' Assumptions assumpwere madeby the p.og.u--"iwhen developingthe mod91,and additional deviation standard the tions aremadeby userJ.Fo, "*u-ple, a programthat calculates must be from an unbiased estimating equation assumesthat the sample size = considered often is 30 N sufficiently lafge to validate ihe estimate.A value of used' as minimal. For a TP-149 (Chapter 15) application, is a 24-hr storm being and reading prior to used be assumedby the method?No computerprogram should of the aware becoming and undershndingthe assumptionsmadeUy ttre programmer programmed' assumptionsimplicit in the hydrologicprocessthat was 'ih" lor cpst of simulationcan reiult in unnecessaryruns and may enticeusers bstantively to the information originally purposeofruns) and working the plan can
asmarr isonrY dtli|l,""i:#THfftime monetary limits to
r alproximate time and in a single useasa guideduringa simulationproject. combining severalinvestigations available models of the Some run is anotherway to conductarrefficient simulation. allowthis.Forexample,TR-20(seeChapter24)allowsthegenerationofflood
CHAPTER21
514
'
MODELING TO HYDROLOGIC INTRODUCTION
hydrographsfrom severalstormsat once.It is often desirableto generatethe2-,5-, 10-, 25-, 50-,100-, and 500-yearflood dischargeat a singlewatershedlocation. The computer is able to generatefar more output than the hydrologistcan analyze.Most modelsincorporateoptionsallowingthe userto specifyoutputquantity. In additionto controlling outpul, a predeterminationshouldbe madeof which specific analyseswill be performed. A tabulation of key output data can be developedto compile and evaluatetrends (and make coursgcorrections)after eachrun. Because deterministichydrologyis about 80 percent acbounting,many opportunitiesexist in water budgelbalances.If the total rechargeto an aquifer is simulation for assessing less than the total outflow and withdrawls, but simulatedwater tables are rising, a checkof input and model parametersshouldbe made.Writing important conclusions on the printed output of simulationruns helpsdocumentthe study and guiderevisions in future runs. Documentation of simulation studies is generally deficient in practice. The record should communicatethe findings in a way that provides a later reviewer generalunderstandingof the work plan followed,decisionsmade,andreasonsfor each run. The documentationshouldstateassumptionsmade,provide samplesof the input and output,explaininput preparationrequirements,statehow sensitivethe resultsare to parameterchangesand assumptions,and documentreasonsout-of-rangeparameters were accepted. Documentationis an ongoingand continual task. It is especiallycrucial if the model will be employedin regulatory proceduresor litigation. A comprehensive processwould6: documentation Include an outline descriptionofthe problem being studied. Identify the equations,techniques,and methodsused. Demonstratethe model's validity to this problem. Discussthe code. Include all assumptionsusedin the code and in preparingthe input. List publishedor known limitations or rangesof the applicability of the model. 7. Characterizethe uncertaintiesin the model; describesensitivitytests. 8. Describeparametersand data setsused. 9. Statethe regulatory or legal criteria incorporatedin the model. 10. Describe the verification, whether with test data or analytical solutions. 11. Include a narrative descriptionof the results,indicating any unexpectedor unusualoutcomes. 12.Presentany other details deemedrelevant. 13.Discusithe modelused. 14.Documentchansesmade in the model code. 1. 2. 3. 4. 5. 6.
Componentsof HydrologicSimulationModels Numerousmathematicalmodelshavebeen developedfor the purposeof simulating various hydrologicphenomenaand systems.A generalconceptualmodel including mostof the importantcomponentsis shownin Fig. 21.1;severalothersare described subsequeht$.Irnportedwater in the lorygl leJtcould be input to reservoiror ground-
)
SIMULATION 21,1 HYDROLOGIC
515
System outflow
Snow accumulation and melt Overiand flow direct runoff
Depression storage
System outflow
SYsteminflow
System rntlow
Figure2l..l.Componentsofasurfaceandsubsurfacewaterresourcesystem. allocationson water storageor channelflow, or it might be guideddirectly to water The routing of the far rlgtriif either storageor distribuiion were deemedunnecessary. parameter channelflo* o, overlandflow could be accompdbhedby simple lumped of segments for discrete flow eq-uations techniques,or solutionsof the unsteady-state algorithms and the channelcould be used.In other words,the selectionof techniques as output desired refinement of d9ere9 the on depends to fepresenteachcomponent justifled is and also on knowledgl of the system.A distributedparameterapproach in described only when availableinformation is adequate.Componentsof modelsare ChaPters22-25.
DataNeedsfor HydrologicSimulation
,
aspart of the The simulation6f all or part of a water systemrequiresa data inventory or mofe) are (90 percent initial planningproc"ts. Most modelinput datarequirements engineering from obtained or map or tield ariaitable,or canbe empiricallydetermined encompasses that topics inventory handbooksand equations.A generallist of data most hydrologic- economicmodelingneedsfollows' A. Basin and SubbasinCharacteristics l, Lagtimes, travel times in reaches,times of concentration' 2.Contributingareas,depressions,meanoverlandflowdistancesandslopes'
\--
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3. Designstorm abstractions:evapotranspiration, infiltration, depression,and interception losses.composite curve numbers,infiltr_ationcapacitiesand parameters,@indexes. 4. Land-usepractices,soil types, surfaceand subsurfacedivides. 5. Water-usesites for recreation,irrigation, flood damagereduction, diversions,flow augmentation,and pumping. 6. Numbering systemfor junctions, subareas,gaugingand precipitation statrons. 7. Imprevious areas, forested areas, areas between isochrones, irrigable acreages. B. ChannelCharacteristics 1". Channelbed and valley floor profiles and slopes. 2. Manning or Ch6zycoefficientsfor variousreaches,or hydrhulicor field data from which thesecoefficientscould be estimated. 3. Channeland valley cross-sectionaldata for eachriver reach. 4. Seepageinformation; channellossesand baseflows. 5. channel and overbank storagecharacteristics,existingor proposedchannelization and leveedata. 6. Sedimentloads,bank stability, and vegetativegrowth. c. MeteorologicData 1. Hourly and daily precipitation for gaugesin or near the watershed. 2. Temperature,relative humidity, and solar radiation data. 3. Data on wind speedand direction. 4. Evaporationpan data. D. WaterUse Data 1. Flows returned to streamsfrom treatmentplants or industries. 2. Diversionsfrom streamsand reservoirs. 3. Transbasindiversionsfrom and to the basin. 4. Stream and ditch geometricpropertiesand seepagecharacteristics. 5. Irrigated acreagesand irrigation practices,including water useefficiencies. 6. Crop types and water consumptionrequirements. 7. Pastconservationpracticessuchas terracing,insfallation of irrigation return pits, and conservationtillage. 8. Stock wateringpractices. 9. Presenceand types of phreatophytesin stream valleys and along ditch banks. E. StreamflowData 1. Hourly, daily, monthly, annual streamflow data at all gauging stations, includin$ statisticalanalyses. 2. Flood frequencydata ani curvesat gaugingstations,or regionalcurvesfor ungaugedsites,preferablyon an annual and seasonalbasis. _3. Flow duration data and curves at gauging stations (also any synthesized data for ungaugedareas). 4. Rating curves; stage-discharge, velocity-discharge, depth-discharge curvesfor certain reaches.
SIMULATION517 21.1 HYDROLOGIC $, Flooded area curves. 6. Stageversusarea flooded' 7. StageversusfrequencYcurves' 8. Stageversusflooi damagecurves,preferablyon a seasonalbasis. 9. Hydraulic radius versusdischargecurves' 10. Sireamflowsat ungaugedsitesas fractions of gaugedvalues. ll. Returnflows as fractionsof water-useallocationsdivertedfor consumptive use. 12. Seasonaldistributionsof allocationsto users' 13. Minimal streamflowto be maintainedat eachsite' 14.Masscurvesandstorage_yieldanalysesatgaugedsites' F. Design Floods and Flood Routing l. Designstormand flood determination;temporaland spatialdistributionand mtenslty. 2. Maximum regional stormsand floods' 3. Selection.andverificationof flood routing techniquesto be usedand necessary routrng parameters. 4. Baseflow estimatesduring designfloods' 5. Availablerecords of historic floods' Information Reservoir G. L. List of potential sitesand location data' 2. Elevation-storagecurves. 3. Elevation-area curves. 4. Normal, minimal, other pool levels' 5. Evaporationand seepageloss data or estimates' 6. Sediment,dead storagerequirements' 7. Reservoireconomiclife. 8. Flood control operatingpolicies and rule curves' 9. Outflow characteristics,weir and outlet equations,controls. 10. Reservoir-basedrecreationbenefit functions' L1. Costsversusreservoirstoragecapacities' 12. Purposesof eachreservoirand beneficiariesand benefits.
NonmodelingAssessments can be After researchingthe availabledata and information, the needfor simulation can be model If a decisionis made to proceed,the appropriatesimulation assessed. selected,a sequenceplanned,and data prepared' Transformationof raw data into usabieform doesnot alwaysrequire a simulacan tion model.Much of the usualinformationneededfor waterresourcesassessments microcomputer in be preparedby hand or by using analytical proceduresavailable format.Typicalnonmodelinganalysesincludethefollowing: 1. Identify water-user groups and all basin sites for hydropower producirrigation, flood damagereduction from tion, reservoir-based-recreation,
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INTRODUCTION TO HYDRoLOGICMODELING
reservoircapacity,industrial and municipal water supply,diversions,and flow augmentation. t Compile annualand seasonalstreamflowsand flood recordsat eachgauged site for the period of record at eachsite. 3. Determine the fraction of the allocation to each consumptiveuse that is assumedto return to the streamat eachuser site in the basin. 4. Perform frequencyanalysesof annual streamflowand flood valuesat each gaugesite in the basin. Determinefor eachreservoirsite the eVaporationand seepagelosses. 6. Selectmeanprobabilitiesto be usedin the firm and secondaryyield analySES.
7. Developflood peak probability distributionsat eachpotential flood damage center or reachin the basin. 8. Determinethe fraction of water to be allocatedduring eachperiod to each water-usesite in the basin. 9. Determine existing and proposed hydropower plant capacitiesand load factors. 10. Identify any minimal allowablestreamflowsto be maintainedfor flow augmentation at eachflow augmentationreachin the basin. 11. Specify any maximal or minimal constraintson any of the annual or seasonal water allocations,storagecapacities,or target yields. 12. Specify any constraintson maximal or minimal dead storage,active storage, flood control storage,or total storagecapacitiesat any or all of the reservoirsitesin the basin. 13. Determine annual capital, operation, maintenance, and replacement (OMR) costsat eachreservoirsite as functions of a rangeof total reservoir 'capacities or scalesof development. 14. Determinebenefitsas functions of energyproduced. L5. Determineannual capital and OMR costsat eachhydropowerproduction site as functions of variousplant capacities. 16. Determine benefit-loss functions for a variety of allocationsto domestic, commercial,industrial, and diversionuses. 17. Determine short-run lossesas functions of deviations (both deficit and surplus)in plannedor target allocationsto user sites. 18. Developbenefit functions at eachirrigation site in the basin.This analysis requiresinformation on the area of land that can be irrigated per unit of water allocated,the quantitiesof eachcrop that can be producedper unit areaof land, the total fixed and variablecostsof producingeachcrop, and the unit"pricesthat will clear the market of any quantity of eachcrop. 19. Developflood-damage-reductionbeneflt functions at eachpotential flood damagesite. This analysisrequiresrecords of historical and/or simulated floods, channel storagecapacities,and flood control reservoir operating policies. 20. Developreservoir-basedrecreationbenefitfunctions at eachrecreationsite in the basin.
SIMULATION 21.2 GROUNDWATER
519
21.2 GROUNDWATER SIMULATION Digital simulation models are used in a different manner to study the storageand movementof water in a porous medium. Distributedrather than lumped parameter models are used to imitate observedevents and to evaluatefuture trends in the developmentand managementof groundwatersystems.The equationsdescribingthe flow of waterin a poroui mediumwerederivedin Chapter18 andmodelingof regional systemswas discirssedin Chapter 20. This section deals primarily with techniques used in solving the hydrodynamicequationsof motion and continuity, followed by brief discussionsof (i) typical input requirements,(2) techniquesof calibrating and verifying the models, and (3) the sensitivity of groundwatermodels to parameter changeslAnexampleof the calibrationand applicationof a groundwatermodelis also provided.
ModelTypes Groundwaterstudiesinvolve the adaptationof a particular code to the problem at hand. Severalpopular public domain computer codesfor solving various types of groundwaterflow problemsare listedin Table2L2.The codesbecomemodelswhen the systembeingsiudiedis describedto the codeby inputting the systemgeometryand known internai operandi (aquifer and flow field parameters,initial and boundary conditions, and water use and flow stressesapplied in time to all or parts of the system).Codes have emerged in four general categories:groundwaterflow codes,solutetransport codes,particle tracking codes,andaquifer testdata analysis programs.to Groundwaterflow codesprovide the user with the distribution of headsin an aquifer that would result from a simulated set of distributed recharge-discharge stressesat cells or line segments.From Darcy's law, the flow passingany two points can be calculated from the head differential. The codes are used to model both confined and unconfinedaquifers.Eachcan be structuredto model regional flow, or flow in proximity of a singG well or wellfield. Steady-stateand transientconditions canbe evaluated.Boundaiiescanbe barriers,full or partially penetratingstreamsand lakes,leaky zones,or constantheador constantgradientperimeters.By application of Darcy's 1aw,the seepagevelocitiesof groundwatercan be determinedafter solving for the head differentials. When groundwaterseepagevelocitiesareknown, the advection,dispersion,and changesin concentrationof iolutes can be modeled.Solutetransportmodelsbuild on groundwaterflow modelsby the addition of advection,dispersion,and/or chemical reactionequations.If the chemical,dispersion,or dilution concentrationchangesdue m groundwaterflow are not important, particle tracking codesmodel transport by advectionand providean easiermethodthan solutetransportmodelsto track the path and traveltimes of solutesthat moveunderthe influenceof headdifferentials.Aquifer test data programsprovide userswith computersolutionsto many of the hand calculations (Ctrapter li) neededto graph and interpret aquifer test data for determining aquifer and well Parameters' I
\.
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TABLE 21.2 GROUNDWATER MODELINGCODES
Acronymfor code PLASM MODFLOW AQUIFEM-1 GWFLOW GWSIM-II GWFL3D MODRET SUTRA RANDOMWALK MT3D AT123D MOC HST3D FLOWPATH PATH3D MODPATH WHPA TECTYPE PUMPTEST THCVFIT TGUESS
Description Groundwaterflow models Two-dimensionalfi nite difference Three-dimensionalflnite difference T!wo-and three-dimensionalfinite element Packageof1 analytical solutions Storageand movementmodel' Three-dimensionalfinite difference Seepagefrom retention ponds Solutetransooftmodels Dissolvedsubstancetransport model Two-dimensionaltransientmodel Three-dimensionalsolutetransport Analytical solution package Two-dimensionalsolutetransport 3-D heat and solutetransport model Particletrackingmodels Two-dimensionalsteadystate Three-dimensionaltransient solutions Three-dimensionaltransient solutions Analyticai solution package Aquifertest analyses Pump and slug test by curve matching Pumping and slug test Pumping and slug test Specific capacitydetermination
Source
ru.sws USGS MIT IGWMC TDWR TDWR USGS
1971 1988 r979 1975 1 9 81 r991 1992
USGS ill. SWS EPA DOE USGS USGS
1980 1981 1990 1981 t978 r992
SSG Wisc GS USGS EPA
1990 1989 t991 1990
ssG
1988 1980 1989 1990
IGWMC IGWMC IGWMC
Note: IGWMC : International Groundwater Modeling Center; Ili. SWS : Illinois State Water Survey; SSG : Scientific Software Group; EPA : Environmental Protection Agency; USGS : U.S. Geological Survey; Wisc. GS = Wisconsin Geological Survey; MIT : MassachusettsInstitute of Technology; TDWR : Texas Department of Water Resources;DOE : Department of Energy.
SolutionTechniques With few exceptions,the hydrodynamic equations for groundwaterflow have no analyticalsolutions,and groundwatermodelingreliesonfinite-dffirence and,finiteelementmethodsto provide approximatesolutionsto a wide variety of groundwater problems.The choice of method is normally driven by the systemto be modeled. Other numericalmethodsincludeboundary integral methods,integratedfinitedffirence methods,and analytic elementmethods. Thesesolutions,as with streamflowsimulation models,are facilitated by first subdividingthe region to be modeledinto subareas.Groundwatersystemsubdivision dependsmore on geometriccriteria and lesson topographiccriteria in the sensethat the region is overlaidby a regular or semiregularpattem of node points at which (or betweenwhich) specificmeasuresof aquifer and water systemparametersare input and other parametersare calculated.Approximate solutionsof simultaneouslinear and nonlinearequationsare found by making initial estimatesof the solution values, testingthe estimatesin the equationsof motion and continuity, adjustingthe values, andfinally acceptingminor violationsin the basicprinciplesor making further adjustments of-the parametersin an orderly and convergingfashion.
SIMULATION 21.2 GROUNDWATEFT
r '
521
Theorderly solutionof finite differenceanalogsof the steady-stateor unsteadypartial difierential equation of motion for flow of groundwaterin a confined stare aquifer or an unconfinedaquifer is obtainedby rela_xationmethods'An eady relaxation solutionof the equationis discussedby Jacob.TFor two-dimensionalproblems, the iterativealternating-direction-implicit(ADI) methoddevelopedby Peacemanand Rachfordsis often adoPted. prickett and Lonnquisteused the ADI techniqueto calculate fluctuations in watertable elevationsat all nodesin an aquifermodelby proceedingthrough time in small incrementsfrom a known initial state.Their modelis computationallyefficient and readily appliedand is particularly attractivefor usewith problemsinvolvingtime variablesind nu-"tous nodes.The primary aquiferparametersare the permeability and storagecoefficient,which, if assumedconstantover the aquifer plan, result in a homogenJousandisotropigcondition.For thosefamiliar with relaxationmethods'the over-relaxation(SOR)methodshavehad application Gauss-seidelandthe successive equations. in solvingdifference
DataRequirements Input to groundwatersystemmodelsmay be classifiedas spatialand temporal' Spatial input inciudesinitial oi projectedwater table maps,saturatedthicknessdata over the ."gion, land surfacecontourmaps,transmissivitymaps,regionalvariationsin storage coefficients,locationsand typei of wells and canals,locationsand types of aquifer boundariesboth lateral and vertical, a nodecoordinatesystem,actualor net pumpage rates,percolationand rechargerates for precipitation and other appliedwaters,logs of drilled wells, geologicstratigraphy,and soil types and cropping patterns. Time-dependenidatarequirementsfor aquifer models involve principally the formulation of ti-" schedules,using a rangeof time incrementsfor suchvariablesas pumpingrates,precipitationhyetographs,canaland streamflowhydrographs,groundrates,and developmentvariablessuchasthe timing of added waterevapotranspiration Becauseeachtemporal schedulecan apply only system components. wells or other the time-dependentrequirementsare also positions, node of particular iubset to a spatial. In addition to the listed input parameters,aquifer modelsrequire reliable estimatesof the percentagesof waters in the land phasethat actually percolateto the aquifer being modeled.Theseestimatescan be basedon knowledgeof the physical processesinvolved in unsaturatedflow through porous medium but are most often obtainedasjudgmentparametersthat are modifiedduringthe calibrationphaseof the simulation. Simply stated,the lateral movementand the changesin piezometeror watertable leneli are easilymodeledif the node-by-nodestresses(withdrawalratesor rechargerates)aie known. The latter parametersare governedby the complexmovement of water in the unsaturatedsoil zone and by the random precipitation and consumptiveusepatternsof the region.The art of modelinggroundwatersystemslies in the ability to evaluatetheseparameters'
Calibration
I L,
Groundwatermodelcalibrationremovessomeof the guessworkinvolvedin parameter determination.Severalcombinationsof parameters,basedon availableknowledgeof the physical syqtem,are testedin the model during a period for which records are
522
21 INTRODUCTIoN CHAPTER To HYDRoLoGIc MoDELING availabi-e. Simulatedresultsare then comparedwith historical events.After structuring the model,calibrationis achievedby operatingthe model during the studyperiod by imposinghistoricalprecipitationamounts,canaldiversions,evaporationand evapotranspirationrates, streamflowsand stream levels, pumping rates during known periods, and other stresseson the aquifer. Calibration is achievedafter the flow, storage,and other parametershavebeenadjustedwithin reasonablelirnits to produce the best imitation of recordedevents.
CaseExample A typical finite-differencestudyinvolvingsurfacewaterand groundwatermodelingin central Nebraskawas performed by Marlette and Lewis.tt The region involved is shownin Fig. 2I.2.In additionto the surfaceirrigation systemrepresented by the severalcanalsand laterals,over 1200wells withdraw waterfrom the aquiferbetween the PlatteRiver and the Gothenburgand DawsonCounty canals.The aquiferrecharge and withdrawal amountsas percentagesof precipitation, snowfall, pumped water, deliveredcanal water, evaporation,and evapotranspirationwere estimatedusing a mix of judgment andphysicalprocessevaluations.The resultingsetthat producedthe bestcomparisonwith recordedeventsat the six observationwells showninFig.2l.2 is summarizedin Table 21.3. Samplesof the comparisonbetweenrecordedand simulatedwaterlevelsin the DawsonCounty study during aZ-yearcalibrationperiod are shownin Figs. 21.3 and2l.4. The Prickettand Lonnquistmodelwasappliedin the DawsonCounty study.The storagecoefflcientfor this unconfinedaquiferwas establishedby calibrationtrials as 0.25 and the adopted permeability was 61 mlday. Other trials were made using variouscombinations of S andK, with S rangingfrom 0.10to 0.30andwith Kranging between4I and I02 mlday. As with most unconfined aquifer models,water table elevationswere most sensitiveto fluctuationsin the storasecoefflcient.Fisure 21.5 is
tt'iorX \F
'q;
Figure 21.2. Grid coordinatesfor Dawson County, Nebraska,aquifer model. o : observationwell.
SIMULATION 21.2 GROUNDWATER
523
FORWATERALLOCATIONS CRITERIA RECHARGE TABLE..21.3 ADOPTED AQUIFER COUNTY OVERTHEDAWSON
Systemcomponent Rainfall Snowfall Pumpedwater Delivered canal water
Evaporation Evapotranspiration
Aquiferrecharge/withdrawal ot as a percentage appliedamount
Allocationand appliedamounts Recordeddepth if daily amount exceeded 0.25 cm at all nodes 25Voofrecorded depthsat all nodes Averagerate of 50 l/sec at-all well nodes during irrigation seasons Recordeddaily rates,applied to land surfaceone node laterally uphill and two nodesdownhill from canal Observeddaily lake evaporationdepth at all marsh and water surfacenodes I25Voof daily lake evaporation,applied at all alfalfa nodes
30 30 50 30 100 15
d
o o 6
B
0.999
d
0.998 A M J J A S O N D J F M A M J J A S O N
1970
1971
Figure2L.3. Simulatedandrecordedwaterlevelsat observation w e l l Di n F i g . 2 1 . 2 .:I8 2 ; i : 3 7 . a typical summary of the calibration results at a single observationwell located at PositionF in Fig. 21.2. After vefification, the Dawson County model was applied to investigatethe short-terminfluenceof severalmanagementschemes.Included among the schemes were investigationsinvolving the completeremoval or shutdownof the surfacewater canals,and other testsin which isolatedcanal contributionsto rechargewere determined by operatingthe model with singlecanals and comparingresults with water table fluctuationsfor identicalconditionswith all canalsremoved.Many other applicationsof the model are possible.This particular study revealedthat rechargefrom
524
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INTRODUCTIONTO HYDROLOGICMODELING
o a d
F
0.999
0.998 A M J J A S O N D J F M A M J J A S O N l97l
1970
Figure 21.4. Simulated and recorded water levels at observation well F in Fig. 21.2. I : 97; i : 42.
a
lu3
P (J
o q)
(.1 'F1
H 0)
i3 -
702
A
M
J
J
A
S
O
N
D
t9'70
J
F
M
A
M
J
J
A
S
O
r97|
with constantpermeabilityandvarying changes Figure 21.5. Water-level Well4 Fig. 21.2;K : 6Im/day;I : 97;J : 42. storagecoefficients. the existingcanal systemcontributesto the waterbalancqof the aquiferbut is not the dominantfactor\n the shortrun. The naturalrechargefrom precipitationandfrom the Platte River accountfor the long-term water table stability in the region.
PROTOCOL SIMULATION 21.3 HYDROLOGIC The useof hydrologicsimulationas a tool in the decision-makingprocessis not new but is of a different, more sophisticatedand more encompassingform. A model is a rgplesentationof an actualor proposedsystemthat permitsthe evaluationand-rmanip-
21.g HYDROLOGICSIMULATIONPROTOCOL
525
useof ulation of-manyyearsof prototypebehavior.This is the featurethat makesthe largest, the of even thesetooii so attractiveand hoids suchpotential for the analysis so well mostcomplexsystems.It is alsothe prin-ipal featurethat makesthis approach suitedto water resourcessystemplanning and analysis' Apart from the useof cot1entionalhandmethodsand someelementarymodels, planninghastraditionally beena practiceof judgment.This is changing,however,as quantititive tools are developedthat permit the analysisof large numbersof alternaout but is tives and plans.Judgment,a-nessentialelementof the process,is not ruled planning the in to those strengthenedthrough new insights that were not available professiona few years ago. "What if ?" Plannersare conti;ually required to anticipatethe future and ask and "What's best?" questions.Quantitativeplanning techniques,suchas simulation cost than can provide detailedinformation aboutmore planning alternativesfor less at principally occurred has uny oth", approachavailable.Developmentof thesetools universitiesand federal agencies.
CombinedUse of Simutationand OptimizationModels at this An important secondtype of quantitativeplanningtool shouldbe mentioned a select to information point.Screeningmodelsare designedto utilize llmiPd system Hence objectives. of set or bestplan u*oni many alternativesfor a specifiedobjective toward screeningmodJs, or optimizationmodelsasthey are often called,are oriented plan formulation rn contrast to th€ Simulation models are suited to de reliable information on which to bi "If modelsaddressthe question, out tion models,on the otherhand,ask, will the systemlook like after we arr special merits of each, these two Completeder planningtechnologies. ing, oPtimization,and simulation r Final designvaluesshouldbe and operatinga detailedsimulation model over time using a to the systemel-ements and/or streamflows,while at sequenceof known or synthesizedprecipitation-amounts reservoirand streamcontrol, the sametime accumulatingbenefiisovertime for flood siderecreation,wateryields,strean and anY other factorsnot consider model. Severalsimulationruns wi resultin a plan that bestmeetsthe c generatedby coilventionalmethods' inforResultsof optimization modelswill provide readily obtainedand useful promismost the test to in order mation for initiating more refinedsimulationanalyses of ing measuresand arrive at final plans for the design,construction,and operation a water resourcesystem.Even thor for decisionsregardingboth the de postoptimizationsimulation is rec assumptionsoften requiredin prel
L
526
CHAPTER21
INTRODUCTIONTO HYDROLOGICMODELING
for preliminary screeningof developmentalternativesowing to time and cost limitations.Unlessa new generationof computersevolves,currenttime and sizelimitations do not allow screeningby simulating all iilternativesunlesssubstantialsacrificesin realismare made.For the present,preliminary screeningfollowedby detailedsimulation appearsto be the most effectivemeansfor arriving at optimal waterdevelopment and managementPlans.
MODELS SIMULATION 21.4 CORPSOF ENGINEERS In 1964,the U.S. Army Corps of Engineersdevelopeda specialtybranch locatedat the HydrologicEngineeringCenter(HEC) in Davis,California. The facility provides a centerforipplying academicresearchresultsto practical needsof the Corps fleld offices.In addition, the centerprovidestraining and technical assistanceto governmenr agenciesin advancedhydrology,hydraulics,and reservoiroperations. Over the years, a large number of analytical tools were developedat HEC. Table 2I.4 summarizesthe computerprogramsin categoriesof hydrology,river/reservoir hydraulics, reservoir operations, stochastichydrology, river/reservoir water TABLE 21.4 HEC WATER RESOURCECOMPUTERPROGRAMS Name
HEC-I, Flood HydrograPhPackage
Basin Rainfall and SnowmeltComputation
Date of latest version HydrologyModels September1980
Purpose Simulatesthe precipitation runoff processin any comPlexriver
July 1966 many subbasilsof a river basin using gaugedata and weightings (includedin HEC-1).
Unit Graph and HydrographComputation
Iuly 1966
Unit Graph Loss Rate OPtimization
August 1966
HydrographCombining and Routing
August 1966
Computessubbasin interception/infiltration, unit hydrographsbaseflow, and runoff hydrograph(included in HEC-1). Estimatesbest-fitvaluesfor unit graph and lossrate Parameters from given precipitation and subbasinrunoff (included in HEC-1). Combinesrunoff from subbasinsat confluencesand routes througha river hydrographs network using hydrologicrouting methods(includedin HEC-1).
21.4 CORPSOF ENGINEERSSIMULATIONMODELS TABLE21.4
527
(Continued)*
Name
Dateof latestversion
Purpose
Streamflow Routing OptiffIization
July 1966
Estimatesbest-fit valuesfor hydrologic streamflow routing parameterswith given upstream' downstream,and local inflow hydrographs(includedin HEC-l).
Interior Drainage Flood Routing
November1978
Computes seepage,gravitY and pressue flow, pumping and overtopping dischargesfor Pond areasbehind leveesor other flow obstructions.Main river elevation and ponding area elevation-area-capacitydata are usedin computingdischarges.
Storage,Treatment, Overflow, Runoff Model (..STORM")
JuJy1976
Simulatesthe precipitation runoff processfor a single, usuallY urban, basin for manY Yearsof hourly precipitation data. Simulates qualitY of urban runoff and dry weather sewageflow. EvaluatesquantitY and qualilY of overflow for combinations of sewagetreatmenl plant storage and treatment rate.
hYdraulics River/reservoir HEC-2, Water SurfaceProfiles
August 1979
super-critical. AnalYzes allowableencroachmentfor a given rise in water surface. Gradually Varied UnsteadyFlow Profiles
Iarnary L976
Simulatesone-dimensional, unsteady,free surfaceflows in a branching river network. Natural and artificial cross sectionsmaY be used. Uses an exPlicit centered difference computational scheme.
Geometric Elements from Cross Section Coordinates("GEDA')
June 1976
Computestables of hYdraulic elementsfor use bY the GraduallY Varied UnsteadYFlow Profiles or other programs.InterPolates values for area, top width, n value, and hydraulic radius at evenly spacedlocations along a reach.
534
CHAPTER21
INTRODUCTIONTO HYDROLOGICMODELING
PROBLEMS 21.1. Simulation and synthesisare treatedseparatelyin Chapters'22and 23. List the most distinguishingcharacteristicsof eachmethod and give an exampleof each. 21.2. Listatleastthreereasonsmanyofthedevelopedmodelsoftherainfall-runoffprocess might not be usedby hydrologists. 21.3. You are askedto determinea designinflow hydrographto a reservoirat a site where no recordsof streamfloware'available.I,!st generalstepsyou would take as a hydrol' ogist in developingthe entire designinflow hydrograph.
REFERENCES "Models and MethodsApplicableto Corps of 1. U.S. Army WaterwaysExperimentStation, EngineersUrban Studies,"MiscellaneousPaperH-74-8, National TechnicalInformation Service,Aug.1974. 2. GeorgeFleming, ComputerSimulation Techniquesin Hydrology. New York: American Elsevier,1975. "HEC-I Flood HydrographPackage,"Users and ProU.S. Army Corps of Engineers, grammersManuals,HEC Program723-X6-L20I0, Jan.1973. "Digital Simulationin Hydrology:StanfordWater+ . N. H. Crawford and R. K. Linsley,Jr., shedModel IV," Department of Civil Engineering,Stanford University, Stanford, CA, Tech.Rep.No. 39, July 1966. 5 . A. J. Friedrich, "Managementof ComputerUse in SolvingEngineeringProblems,"U.S. Army Corps of Engineers,Hydrologic EngineeringCenter,Davis,CA, 1979. 6 . National ResearchCouncil, Ground WaterModels-Scientific and RegulatoryApplications. Water Scienceand TechnologyBoard, Commissionon PhysicalSciences,MatheD.C., 1990. National AcademyPress,Washington, matics,and Resources, 7 . C. E. Jacob,"Flow of Groundwater,"in EngineeringHydraulics (HunterRouse,ed.)' New York: Wiley, 1950. "The Numerical Solution of Parabolicand Elliptic 8 . D. W. Peacemanand N. H. Rachford, Indust. Appl. Math.3' (1955). DifferentialEquations,"J. Soc. "selectedDigital ComputerTechniquesfor Ground9 . T. A. Prickett and C. G. Lonnquist, waterResourceEvaluation,"Ilinois StateWaterSurveyBull. No. 55,197I. 1 0 . D. R. Maidment, (ed.), Handbookof Hydrology.New York: McGraw-Hill, 1993. "Digital Simulationof Conjunctive-Useof Groundwater 1 1 . R. R. Marlette and G. L. Lewis, in DawsonCounty, Nebraska,"Civil EngineeringReport, University of Nebraska,Lincoln, 1973. 12. W. K. Johnson,"Use of SystemsAnalysis in Water ResourcePlanning," Proc- ASCE J. Hyd. Div. (1974). 1 3 .R. deNeufvi[e and D. H. Marks, SystemsPlanning and Design CaseStudiesin Modeling . EnglewoodCliffs, NJ: Prentice-Hall,I914. Optimizationand EvaluatioiT 14. D. P. Loucks, "stochasticMethodsfor Analyzing River Basin Systems."Cornell Univer', sity Water Resourcesand Marine SciencesCenter,Ithaca, NY, Aug. 1969. 1 5 . A. Maass,(ed.),Designof WaterResourcesSystem*Cambridge,MA: HarvardUniversity Press,1962. t6. A. F. Pabst, "Next Generation HEC Catchment Modeling," Proceedings,ASCE HydraulicsDivision Symposiumon EngineeringHydrology,SanFrancisco,CA, July 25-30' 1993.
Chapter22
Time":?J5: Hydrologic
r Prologue The purposeof this chaPteris to: . Show how time seriesanalysisis used for generatingsynthetic hydrologic records. . Give definitionsof termsusedto describethe stochasticaspectsof hydrologic series. . Introduce fundamentalsof streamflowsynthesisincluding masscurve analysis, random generationof sequences,serial-dependentsequences,and sequenceshavingprescribedfrequencydistributions' Time-seriesanalysisof hydrologicvariableshas becomea practical methodology for generatingsyntheticsequencesof precipitation or steamflowvaluesthat can ui-usedlor u tung" of applicationsfrom filling in missingdata in a gaugedrecord to, extendingmonthiy streamflowrecordsl, and from analyzinglong-term reliability of or reservoirst'oto forecastingfloodsor snowmeltrunoff quantiyields of-watersheds2 Thesesynthetic hydrologytechniques iies from syntheticprecipitation sequences.5 wideaugmentttre simulationtools describedin Chapter21. Both have,experienced generation of a spiead use by hydrologistsand engineers.6Synthesisinvolves the or annual). ,"qu"n"" of valuesfor somehydrologicvariable(daily, monthly, seasonal, The techniquesare most often applied to produce streamflowsequencesfor use in reservoir designor operation studiesbut can also be used to generaterainfall sequencesthat can subsequentlybe input to simulation models' If historioalflows iould be consideredto be representativeof all possiblefuture variationsthat someproject will experienceduring its lifetime, there would be little needfor synthetichyOroiogy.The hiitorical record is seldomadequatefor predicting future eventswith certainty,however.The exacthistoricalpatternis unlikely to recur, sequencesof dry years (or wet years) may not have been as severeas they may beiome, and the singlehistorical record gives the planner limited knowledgeof the magnitudeof risks involved.
536
CHAPTER22
HYDROLOGICTIME SERIESANALYSIS
particularly if Syhthesisenableshydrologiststo deal with data inadequaci.es, of hydrologic records historical Short record lengths are not sufficiently extensive. synhydrologic sequenceslrsing longer variables*"fu u, streamfloware extendedto hydrology'1 as operational known science thesisand other techniquesof the broad either preservethe statisticalcharacterof the historThesenew, syntheticsequences ical records or follow a prescribedprobability distribution, or both. When coupled with computer simulation techniques,the techniquesprovide hydrologistswith improved designand analysiscapabilities. The rnethodsdescribedin this chapterare basedon probability and statistics. The material presentedin Chapter 26 should be reviewed prior to studying this chapter.
HYDROLOGY 22.1 SYNTHETIC
,
Hydrologic synthesistechniquesare classifiedas (I) historical repetition methods, suchas masscurve analyses,which assumethat historical recordswill repeatthemsevlesin as many end-to-endrepetitionsas required to bracketthe planningperiod; (2) random generationtechniques,such as Monte Carlo techniques,which assume that the historicalrecordsare a numberof random,independentevents,any of which could occur within a definedprobability distribution; and (3) persistencemethods, such as Markov generationtechniques,which assumethat flows in sequenceare dependentand thit the next flow in.sequenceis influencedby some subsetof the previousflows. Historical repetition or random generationtechniquesare normally applied only to annual or seasonalflows. Successiveflows for shortertime intervals are usually correlated,necessitatinganalysisby the Markov generationmethod. As with most subfieldsof hydrology,a number of computerprogramsfor timeseriesanalysisandhydrologicdatasynthesishavebeendeveloped.One ofthe first, and one of the most widely app1i"d,wasthe U.S. Army Corps of Engineersmodel HEC-4 (seeSection21.a) pubfishedin I97IJ Its use is limited, though, to synthesizing of seriaitydependentmonthly streamflowsin a river reach.Other codess'e sequences are avarlableto thi hy-drologist,however. Additional models and descriptionsof theory and applicationsof time-seriesanalysisof precipitation and streamfloware detailedin a number of availabletexts and publications'10-l3
MassCurveAnalysis One of the earliestand simplestsynthesistechniqueswas devisedby Ripplla to investigate reservoirptoragecapacityrequirements.His analysisassumesthat the future inflowsto a reseivoirwill be a duplicateof the historicalrecord repeatedin its entirety as many times end fo end as is necessaryto span the useful life of the reservoir. Sufficient storageis then selectedto hold surpluswaters for releaseduring critical periods when inflows fall short of demands.Reservoirsize selectionis easily accompmfr"a from an analysisof peaks and troughs in the mass curve of accumulated Future flows can be similar, but are unlikely to be syntheticinflow versustime.15-17 identicalto pastflows.Randomgenerationand Markov modelingtechniquesproduce t-hatare difJerentfrom, although still representativeof, historical flows. seqUences
HYDROLOGY 537 22.1 SYNTHETIC
-
EXAMPLE 22.1 Streamflowspast a proposedreservoir site during a 5-year period of record were, in eachyear14,000,10,000,6000,8000,and 12,000acre-ft.use Rippl's respectively, to determinethe size of reservoir neededto provide a yield of method ,nur. Crrru" 9000 acre-ft in eachof the next 10 years. solution. A lo-year sequenceof syntheticflows,usingRippl's assumptions,is shown in Table Z2.l.Inflows are set equal to the historical record repeated twice. rl FOREXAMPLE22'1 TABLE22.1 STREAMFLOWS Flows (thousandsof acre-ft) Year Inflow Cumulative inflow
r 1 14
2 4
3 4 0 6 8 38 30
1 24
5 6 1 2 1 64 50
7 8 9 1 0 4 1 0 6 8 1 2 100 88 80 74
draft of 9000 acre-ft per year for 10 years.
90,000 80,000
^ I
70,000
B
60,000
Storagerequfued for 9000-acre-ft/Yr draft is maximum
Cumulative draft, slopeof9000 acre-fl/Yr
,' ,
50,000 4000 acre ft storagerequired
40,000 U
30,000 . 20,000 10.000 1
2
3
4
5
6
Year
Eigarc 22.1 Mass curve for ExamPle 22.1: inflow; --- cumulativedtaft.
cumulative
538
CHAPTER 22 HYDRoLoGIc TIMESERIES ANALYSIS
RandomGeneratiOn one method of generatingsequencesof future flows is a simplerandom rearrangement of past records.If the streamis ungaugedand recordsare not available,a probability distributioncan be selectedand a sequenceof future flows that follow the distribution and haveprescribedstatisticalmomentsis generated. Wheneverhistorical flows are available,a reasonablesequenceof future flows can be synthesizedby first consultinga table of randomnumbers,selectinga number, matchingthis with the rank-in-file numberof a pastflow, and listing thecorresponding flow as the first value in the new sequence.The next random numberwould be used in a similar fasion to generatethe next flow, and so on. Randomnumbershavingno correspondingflows are neglectedand the next randomnumberis selected.Table-B.3 in Appendix B is a table of uniformly distributedrandom numbers(eachsuccessive numberhas an equalprobability of taking on any of the possiblevalues).To illustrate the use of Table B.3 in the random generationprocess,the first three yearsof a syntheticflow sequencecould be generatedby selectingthe 53rd, 74th, and23rdfrom the list of past flows. Alternatively, the flows in 1953, 1974, and 1923 could also be selectedas the new randomsequence. Most computershave random number generationcapabilitiesin their system libraries. Rather than storing large tables of numberssuchas Table B.3, successive random integersare usually generatedby the computer. EXA]I/IPLE22.2 Annualflowsin CrookedCreekwere 19,000,14,000,21,000,8000,11,000,23,000, 1 0 , 0 0 0 , a n d 9 0 0 0 a c r e - f t , r e s p e c t i v e l y , f ol ,r2y,e3a, 4 r s, 5 , 6 , 7 , a n d 8 . G e n e r a t e a 5-year sequenceof annual flows, O,, by matching five random numberswith year numbers. Solution. Randomintegersbetween0 and 9 are generatedfrom the computer. The Q, valuesin Table22.2 areselectedfrom the eight given flowsby matching the respectiveyear numberwith the random number.The digit t has no correspondingflow in the 8-year sequence,so the next random number, 2, places the 14,000-cfs flow in Year 2 in the first position of the synthetic 5-year sequence.rI TABLE 22.2 DEVELOPMENTOF s.YEAR SYNTHETIC SEQUENCE Randomdigit I
o
2 3
2 5 8 I
I
5 6
I
Q(acre-ft)
Skip 14,000 11,000 9,000 19,000 8.000
544
CHAPTER22
HYDROLOGICTIME SERIESANALYSIS
3.- a regressioncoefficient of the standarddeviation for the daily flow logarithms within each month of record to the logarithm of the monthly total flow.,
'
Given the calculatedstatistics,the simulationof daily flows could be structured in the following manner: variatesfrom Eq. 22'1,0. 1. Generatestandardized 2. Use the logarithm of the monthly mean flow as an initial estimateof the mean of the logarithmic daily flows for the month' 3. Calculatethe standarddeviationof flow logarithmsby the previouslydetermined regressionequation. 4. Apply the inverseof Eq. 22.1I, k--
?lz(' -:)
.
'l-; 13
1
(22.r2)
to transformthe standardizedvariatesto flows,multiplying by the appropriate standarddeviationand addingthe mean. 5. Add the differencebetweenthe total monthly flow generatedand the given monthly flow to the given monthly flow, and repeatthe simulation. 6. Multiply daily results of the secondsimulation by the ratio of the given monthly total to the generatedmonthly total. Each simulation techniquecommonly requiresmodificiationswhen applied to individual problems.Methodsoutlined thus far can be utilized as guidesin establishing a procedureto follow in synthesizingflows.Simulationof flows for a given station hasbeenpresentedwith serial correlation,skewness,means,and standarddeviations for an entire system,the preserrnaintained.When generatingstreamflowsequences vation of cross-correlationbetweenstationsbecomesa significantfactor. are employedto determinethe capacSyntheticallygeneratedrunoff sequences ities of reservoirsto satisfy specifieddemands.Individually generatedflow magnitudesare uncertain, as are the syntheticallygeneratedsequencesof flows. Hydrologistscanestimateprobabilitiesof flowsby generatingseveralequally likd sequences of flowsand then evaluatingrecurrencesof certain values.Herein lies one of the most useful applicationsof Markov generatingtechniques.
i Summary Hydrologicmodelingis often presentedas comprisingonly the deterministicmodels of the rainfall-runoff processdescribedin Chapters2I,23,24, and 25. The fully equippedhydrologistincorporatesthe synthetichydrology models describedin this chapterin the analysisand designof water resotrces systems.A growing numberof projects are constructedor operatedon the basis of synthetichydrology and timesedesanalysiseachyear.
PROBLEMS
545
PROBLEMS 22'2 and detet22.1. plot cumulativeinflows versustime for the S-yearrecord in Example yield of
neededto provide a mine by mass curve analysisthe size of the r_eservoir maximumyield possible? is,the What 12,000acre-ft in eachoithe next 24years. sequenceof synthetic 'r, t use the annual rainfall trom Table 26.2to generatea lO-year assumption' annual rain depthsfor Richmond using Rippl's masscurve curve methods'Use 22.3. RepeatProblem 22.2 wingrandom generationrather than mass with the last two digits of these match and taUl'n.: numberJfrom i*o-Olglt.unaom the yearnumbersin Problem26.32.
22.4. RepeatProblem22.2wingrandomgenerationtogenefateal0-yearsyntheticsestandard tr<
and qu"n"" of annual rain defths that has.a normal CDF with a mean 1 26'32' Problem from data rain annual of ihe to that equal deviation follow a log-Pearson Repeat Problem 22.4 assumingthat the annual rain depths and skewofthe deviation, standard mean, the use statisiics Type III distribution.For logarithmsof annualrain at Richmond'
22.6. SelectagaugedStreaminyourgeographiclocationandprepareaquarterlymodel Type III (c) Pearson using (a) normal distribution, (Uitog-nbrmal distribution' and distribution.
)a1
CanyouconvertthesimulationprobleminExamp|e22.4toalog_normaldistribution given in the example? simulation?What difficulties aie encounteredwith the data
22.8. Selectamonthofthunderstormactivityinyourregion.FrompublishedNoAAhourly storms'for duration of rainfall data, flt a distributionto the time betweenstorms,and a computerpfogram Prepare month. selected the covering data 20 yearsof recorded of storms' to randomly generatethe times betweenstormsand the durations
22.9. Flowsduring6yearsofrecordwereusedinsynthesizingthemasscurveshownonthe following page. a.UseRippl'sassumptionandthegraphtodeterminethemissingmagnitudeofthe flow for the 12th Year. yield of 2000 acreb Determinethe reservoircapacityrequired to allow an annual acte-ftlYr' ftlyr. RePeatfor 500 does this value relate c. Determine the maximum yieid possibleat the site. How statisticallYto the flows? a table of randomprecip22.10. Describewith words and equationshow you would develop of 4 in' and a standard mean a have itation depthsthat follow a normal distributionand deviationof3in.Useyourmethodtocalculatethefirstthreedepths. is given below' use random 22.LI. A sequenceof uniformly distributedrandom numbers depths that will follow a rain annual generationto generatei 5_y"u, sequenceof in'' a standarddeviation 25'8 of mean a have F"u..on fype iU distribution and will of4'0in.,andaskewcoefficientof_2.2o.Randomnumberstobeusedare20,0I' 9 0 . 0 3 .a n d 8 0 . to a PearsonType III 22.L2. Total July runoff from a basinis randomly distributedaccording is 1000 acre-ft' the deviation standard the acre-ft', 10,000 is distribution. The mean Start with Q1 : is 0.50. skewis -0.6, and the lag-oneserialcorrelationcoefficient selected randomly of sequence if a flows 10,000and find nu" -oiJuu.kov-generated years' 50 and return periodsgives2, 100, 10,2,
546
CHAPTER22
HYDROLOGICTIME SERIESANALYSIS 30,000
25,000
20,000 9l B
15,000
U
10.000
0
1
2
3
4
5
6
7
8
9
101112
Year Figtre 22.9
Mass curve.
REFERENCES 1 . U.S. Army Corps of Engineers,*HEC-4 Monthly Streamflow Sirnulation," Hydrologic EngineeringCenter, 197l. 2. R. M. Hirsch, "synthetic Hydrology and Water Supply Reliability," Water Resources Research, v. 15, no. 6, L979. J. R. M. Vogel,andJ. R. Stedinger,"The Valueof StochasticStreamflowModelsin Overyear ReservoirDesign Applications," WaterResourcesResearch,v. 25, no. 9, 1988. A D. K. Frevert,et al., "Use of StochasticHydrologyin ReservoirOperation," J. Irrigation and DrainageEngineering,ASCE,v. 115,no. 3, 1989. "An Evaluationof the Practicality and Complexityof SomeRainfall 5 . J. W. Delleur, et al., and Runoff Time SeriesModels," WaterResourcesResearch,v. 12, no. 5, 1976. 6. J. D. Salas, et al., "Applied Modeling of Hydrologic Time Series," Water Resources Publications,Littleton, CO, 1980. "Operational HydrologyUsing Residuals,"J. Hydraulics 1 . G. K. Young,and W. C. Pisano, Division,ASCE,v.94, no. HY4, 1968. "Applied StochasticTechniques,PersonalComputer Ver8 . U.S. Bureau of Reclamation, sion 5.2. User'sManual," Earth SciencesDivision,Denver,CO, 1990. "SPIGOT, A Synthetic StreamflowGenerationSoft9 . J. C. Grygier, and J. R. Stedinger, ware Package," School of Civil and Environmental Engineering, Cornell University Ithaca.NY. 1990.
Chapter23
ContinuousSimulationModels
r Prologue The purposeof this chaPteris to: , Introduce and describecomputercodesavailablefor performing contrnuous simulation of surfacerunoff and streamflow' . Presentin detail how one of the programs-the Stanfordwatershedmodelcycle' simulatesthe miscellan"ou, "otnponentsof the hydrologic .Showthemajorsimilaritiesanddifferencesoftheleadingmodels. . Providea detailedcasestudyof how the modelparametersare developedfrom availableinformation. . Illustrate, by using the casestudy,the stepsinvolvedin calibratinga continuous model and verifying the results' . show how well the moleh are able to replicate gaugedstreamflows. with tools for estiSimulation modelsdescribedin this chapterprovide hydrologists direct runoff ' precipitation' for tittt" in accounting mating streamflowby continuously and streamflow' base percolation' deep infiltration, evaportranspiration,interflow, track models simulation continuous stormi, flow. During rain-free intervalsbetween flow, base and percolation, deep evaporation, to the storageof water.andits depletion until the next rain or snow eventoccurs' ThemodelsarebasedonthephysicalprocessesdescribedinChaptersl_14.As
yasdeterministiciools't1;:H':1;ffi ff;ff t: such,theycrassif :T:TLi""ll':'ff
rain and snow into runoff and streamflow' or other similar procedures'can be usedto r are then input to continuoussimulation in Table2l'l ate Severalof the continuoussimulationmodelsidentifiedearlier is presented (swM-IV)' IV version model, describedhere.The Stanford watershed indetailastypicaloftheothermodels.Manyoftheothersare,infact,basedon
MODELS 549 SIMUIATION STREAMFLOW 2g.1 CONTINUOUS in the SWM{V, and several simulate various componentsof the hydrologic cycle of studies case independent samemanner. Section 23.2 ptesentsand comparestwo the how and Stanford model studies,showinghow the parameterswere {etermined modelswere calibratedand applied to the problemsbeing assessed'
MODELS SIMUI-ATION STREAMFLOW 23.1 CONTINUOUS API Model of a This model was one of the earlieststructuredto give a deterr.ninisticsimulation 68 and of watersheds on tested originally was It continuousstreamflowhydrograph. g:i miz and must be calibrated^to each watershedto obtain a reliable method of r givenin Fig. 23' 1' simulatint the streamflow.lA flow diagramshowingthe structureis of to pertaining interrelations ttie Four basic components describe .this .model index, precipitation (antecedent API an hydrograph, streamflowin a river: a unit groundwater introduced in chapter 2 and iilustrated in Sec. 10'3), a relation for asa function hydrograph flow groundwater the recession,and a relation for computing flow and groundwater both generates model This of the diiect runoff hydrograph. to enjoy continues model API The values. precipitation direct runoff dischargefrom modeling' simulation in widespreadpopularity and use
Direct runoff hydrograph method) lunit-hydrograPh
Groundwater outflow hydrograph
Figure 23.1 Schematic diagram of API-type hYdrologic model. (After Sittner et al'r)
550
CHAPTER23
CONTINUOUSSIMULATIONMODELS
StanfordWatershedModel lV (SWM-|V) Crawford and Linsley designedthis digital computerprogram to simulateportions (the land phase)of the hydrologic cycle for an entire waftished.2The model has undergonemuchdevelopmentsinceits conceptionand is currently availablefrom the U.S. Environmental Protection Agency under the name HSPF, which is a public domain FORTRAN version (discussedsubsequently)of the original program. The SWM-IV hasbeenwidely acceptedas a tool to synthesizea continuoushydrographof hourly or daily streamflowsat a watershed,outlet.A lumped parameterapproachis used and data requirementsare much less than for alternative distributedmodels. Hour$ and daily precipitationdata,daily evaporationdata,and a variety of watershed parametersare input. The relationsand linkage of the variouscomponentsof SWM-IV are shownin Fig.23.2. Hydrologicfundamentalsare usedat eachpoint to transformthe input data into a hydrographof streamflowat the basinoutlet. Rainfall and evaporationdata are first enteredinto the program.Incoming rainfall is distributed,as showninFig.23.2, amonginterception,imperviousareassuchas lakes and streams,and water destined to be infiltrated or to appearin the upper zone as surfacerunoff or interflow, both of which contributeto the channelinflow. The infiltration and upper zone storageeventually percolateto lower zone storageand to activeand inactive groundwaterstorage. User-assignedparametersgovern the rate of water movementbetweenthe storage zonesshownin Fig. 23.2. Three zonesof moistureregulatesoil moistureprofiles and groundwaterconditions. The rapid runoff responseencounteredin smaller watershedsis accountedfor in the upper zone,while both upper and lower zonbscontrol suchfactorsas overland flow, infiltration, and groundwaterstorage.The lower zone is responsiblefor longerterm inflltration and groundwaterstoragethat is later releasedas base flow to the stream.The total streamflowis a combinationof overlandflow, groundwaterflow, and interflow. Model Structure The SWM-IV is madeup of a sequenceof computationroutines for eachprocessin the hydrologiccycle (interception,infiltration, routing, and so on). Separatediscussionsof each componentare provided in the following paragraphs. Actual calculationsproceedfrom processto processas ilfustrated by the arrows in Fig.23.2. All the moisturethat was originally storedin the watershedor wasinput as precipitation during any time period is balancedin the continuity equation P : E + R + A S
(23.r)
P : precipitation E :.evapotranspiration R : runoff AS : the total change in storagein the upper, lower, and groundwater storagezones The changein storagefor each zone is calculated as the difference between the volumesof inflow and outflow. Furthermore,all hydrologicactivity in a time interval is simulatedand balancedbeforethe programproceedsto the next time interval. The simulationterminateswhen no additional data are input. where
2g.1 CONTINUOUSSTREAMFLOWSIMULATIONMODELS
551
maximumsare providedin Table23.1. TABLE23.1 ryPICALMAXIMUM INTERCEPTIONRATES Watershed cover
Interceptionrate (in./hr)
Glassland Moderateforest cover Heavy forest cover
0.10 0.15 0.20
Source: After Crawford and Linsley.2
Evapotranspiration In SWM-IV evapotranspiration(ET) is assumedto occur at "upper" storagezone' The upper the potential rate from interceptionstorageand the andhighly permeablesurfacesoils.The lower soil zone zonesimulatesthe depressions simulatesthe linkage to the groundwaterstoragezone' , Evapotranspirationfrom the lower zone is set equal to the ET opportunity, defined in fig. ZZ.Z.W opportunity is defined as the maximum amount of water availablefor ET at a partiiular location durrng a prescribedtime interval. In the modelinglogic, ET occursfrom severallocations(seeFig. 23'2) includingthe interceptionstorage,upperzonestofage,lower zonestorage,streamand lake surfaces,and groundwaterttotug". Evapotranspirationfrominterception arrdlpper zone storageis Jet equal to the potential rate, Eo, which is assumedto be the lake evaporationrate calcuiatedasthe productof a pan coefficienttimes the input valuesof the evaporation pan data.The evaporationof any interceptedwateris assumedto occur at a rate equal io the potential evapotranspirationrate and ceaseswhen the interceptionstoragehas been depleted. Evaporationfrom streamand lake surfacesalsooccursat the potentialrate. The total volgmeis governedby the total surfaceareaof streamsand lakes(ETL) defined as the ratio of the total str-eamand lake areain the watershedto the total watershed area.Evapotranspirationfrom groundwaterstoragealso occurs at the potential rate and is caiculatedin a similar fashion using a surfacearea equal to a factor K24EL multiplied by the watershedarea.Thus the parameterK24EL representsthe fraction of tfre total watershedarea over which evapotranspirationfrom the groundwater storagewill occur.Most investigatorssetthis parameterat avalue equalto the fraction of the watershedarea coveredby phreatophytes.Its value is normally small but can be large, for eiample, in an agri-ultural area that has many acresof subirrigated alfalfa. If interceptionstorageis depleted,the modelwill attemptto satisfythe potential for ET by drawingfrom the upper zone storageat the potential rate. Once the upper zone sto;ageis dJpleted,ET ociurs from the lower zonebut not at the potential rate; the ET ratb from the lower zone is alwayslessthan Eo. When interception and the uppef zone storagedo not satisfythe potential,any excessentersas EoinFig-23.3,
MODEI.S 23 CONTINUOUS SIMUIATION CHAPTER
/-------'-------\ /r \. \
Actual evaootransoiration -'-r---
t., / /
\--------1-------J I
l* - - - - - - - - - - - - - -----\
Figure 23.2 Stanford watershedmodel IV flowchart. (After Crawford and Linsley.2)
29.1 CONTINUOUSSTREAMFLOWSIMULATIONMODELS
F".""I (: "iPrf) @ Channel inflow
@ Channel inflow
Channel inflow
zi \
,
Simulated streamflow
-k^-*ff^." \ - - - - ; - - J
Eigure 23.2 Continued
tt t
,'
553
"7 554
CHAPTER23
CONTINUOUSSIMULATIONMODELS
''E. h a'.: va' x x n l -
25
50
75
100
Percentase oraretxTi* SfttaffiTffi,:'JTl?J."pportunitv Figure 23.3 Evapotranspiration relation used in the Stanfordwatershedmodel. (After Crawford and Linsley'1 and the rate of evapotranspiration from the lower zone is determined from the shaded afea, or
E:Eo_% Q3.2) 2r The variable r is the evapotranspirationopportunity, deflnedas the maximum water amountavailablefor ET at a particular location during a prescribedtime period. This factorvariesfrom point to point over any watershedfrom zero to a maximumvalue of -.^ LZS (23.3) r : ,''J LZSN where LZS : the current soil moisture storagein the lower zone (in.) LZSN : a nominal storagelevel, normally set equal to the medianvalue of the lower zone storage(in.) K3 : an input parameterthat is a function of watershedcover as shown inTable 23.2 The ratio LZSILZSN is known as the lower zone soil moisture ratio and is usedto comparethe actuallower zone storagewith the nominal value at any time. Valuesof ET opportunity are assumedto vary over a watershedfrom zero to r along.thestraight line shownin Fig. 23.3. This assumedlinear cumulativedistributionof the parameter over an area is also usedin evaluatingareal disftibutionsof infiltration rates. Infiltration Like the erapotranspirationopportunity, the infiltration capacity of a watershedis highly variable from point to point and is assumedto be distributed accordingto a linear cumulativedistributionfunction shownas a line from the origin to Point b inFig.23.4. TABLE23.2 ryHCALLOWER ZONE EVAPOTMNSPIRATION PARAMETERS
Watershedcover Open land Grassland Light forest Heavy forest
0.20 0.23 0.28 0.30
555
23.1 CONTINUOUSSTREAMFLOWSIMULATIONMODELS
a
-
Percentage of area with an infiltration capacity equal to or less than the indicated value
Figure 23.4 Assumedlinear areal variation of inflltration capacity over a watershed. (After Crawford and Linsley.2)
Infiltration into the lower and groundwaterstoragezones is determinedas a function of the moisture supply 7 available for infiltration. Steps to determine infiltration for a given moisture supply7 are: "infiltration" in 1. The net infiltration is determinedfrom the area labeled Fig. 23.4.This wateris assumedto infiltrate into the lower and groundwater storagezones.The areaenclosedby the trapezoidis given by the equations in the first row of Table23.3.If themoisturesupply7 exceedsthe maximum infiltration capacityb, the maximum allowed net infiltration is b/2, which is the median infiltration capacity. 2. Some of the moisture supply contributesto an increasein the interflow detentionduring any time increment and is calculatedas the region indicatedby an arrow in Fig. 23.4.Eqtationsfor this areausingvariousranges in x are providedin the secondrow of Table 23.3. The volume of water in a stateofbeing transportedasinterflow at any instantis calledthe interflow detentionor detainedinterflow. 3. Any remainingmoisturesupplied,AD in Fig. 23.4, contributesto increasing the surface detention during the time increment. Equations for this triangular-shapedarea are included in Table 23.3 for various valuesof 7. TABLE 23,3 EQUATIONSFOR THE SHADEDAREAS IN FIG. 23.3
x
Component Net infiltration Increasein interflow detention
_
x-
x>cb
b
v
2
b 2
b
zb
t
i2/ r\ - l l - - f c/ 2r\
-
b 2
u-t,ts
i2 2cb
2'
- c b Increasein surfacedetention
/
Percentageof increaseddetention assignedto interflow Searce; After Crawford and Linsley.2
x-t
2rb
2rb r \
r oo(-r:)
/
r oo(-r "*
, t
\
_ r tr )
c - l
too2- _ ilbj
556
CHAPTER23 CONTINUOUSSIMULATIONMODELS
ThEquantity of net infiltration is contrblledlargelyby the maximuminfiltration capacityb, while the pbrameterc significaltly affectshydrographshapesbecausethe parametercoritrols the amount of water'detainedduring thelime increment. The valuesof b andc foi ahy time intetval dependon the soil moistureratio, LZS/LZSN, and on the input parametersCB and CC; CB is an index that controls the rate of infilffation and dependson the soil permeabilityand the volume of moisturethat can be storedin the soil. Valuesin the rangefrom 0.3 to 1.2 arecommon.The parameter CC is an input value that fixes the level of interflow relative to the overland flow. Valuesof CC rangefrom 1.0 to 5.0. If the soil moistureratio is lessthan 1.0, the variable b is found from "u
cB
:
(23.4)
2ALzs/LzsN)
and whenLZS/LZSN is greaterthan 1.0,the equationfor b is
. 0 :
C
B
,n-rr"tvttt*,
rr?5)
*n
Theseequationswere developedby Crawfordand Linsley from num0rousttials using SWM-IV in many different watersheds.When the soil moistureratio reachesa value of 2.0, the variable b reachesits minimum value of *r of CB. The parameterc is determinedfrctm c :
(23.6)
.(CC)ZQzslLzsN)
Variations in parametetsb and c with changesinLZS/LZSN aie shown in Figs. 23.5 and23.6.Midrangevaluesof CB : 1.0and CC : 1.0wereusedin developing thesecurves. Figure 23.7 is a graph of distributionof water amonginfiltration, intefflow,'and overland flow for various valuesof the moisture supply 7. Different valuesof b and c would producea different set of curves. Water stoied as overland flow surface detention will either contribute to streamflowof enterthe upper zone storageas'depictedin Fig. 23.2.The portion that
0.0
0.2
0.4 0.6 0.8
1.0 1.2
r.4
1.6 1.8 2.O
ratt" (-!ZL'J Lowerzonesoilmoisture Figure 23.5 Variationin patarneterb for variousvaluesofthe soil moistureratio. (After Crawfordand Linsley.2)
0.2
0.4 0.6
0.8
1.0
1.2
1.6
1.4
/ Lzs)
Lowerzone soll molstufe rauo \-I_ZSN' Figure 23.6 Variation in parameterc for various values of the soil moisture ratio. (After Crawford and Linsley.2)
o o I
lncrease in overland flow surface detention
E
o
b=L.0 c=1.5
O
0.2 0.4 0.6 0.8 1.0 1.2 1,.4 1.6 Moisturesupply7
0
Figure 23.7 Typical SWM-IV responseto moisture supply variations.(After Crawford and Linsley.2)
100 ; Y
8F o
80
9 ^
b€R
60 , Inflection
0 0
0.5
1.0
1.5
\
1
2.0
2.5
3.0
rati" (ffi) Upperzonesoilmoisture Figure 23.8 Delayedinfiltration as a function ofupper zone soilmoisture ratio. (After Crawford and Linsley.z)
558
CHAPTER23
CONTINUOUSSIMULATIONMODELS
entersthe upper zone storageis called delayedinfiltration and is a function of the upper zone soil moistureratio,\JZS/UZSN, as shownin Fig. 23.8. The inflection point occursat a soil moistureratio of 2.0. If the ratio is lessthhn 2.A, thepercentage retainedby the upper zone is given by
- (ffi)(--ri-)"'' e: roolr' ]
(23.7)
where UZI| is determinedfrom
r J Z r r : r r l # k - r . o+] r . o
(23.8)
The curve is definedto the right of the inflection point by
f/
t.0_)r,',-l
P': lool(tt . uzLz/ r
(23.e)
whereUZI2 is determinedfrom
rrzr2:rrlffi-z.+ o ]r . o
(23.r0)
Upper Zone Storage The upper storagezone,as shownin Fig. 23.2; receivesa large portion of the rain during the flrst few hours of the storm, while the lower and groundwaterstoragezonesmay or may not receiveany moisture.The portion of the upper zone storagethat is not evaporatedor transpiredis proportionedto the surface runoff, interflow, and percolation.Percolation(upperzone depletion)from the upper LZS/LZSN. zoneto thelowerzonein Fig.23.2ocursonly whenUZS/UZSN exceeds When this occurs,the percolationrate in in./hr is determinedfrom
: o.oo3(cBxuzs$(ffi PERC ffi)'
(23.rr)
where CB is an index that controls the rate of infiltration. It rangesfrom 0.3 to 1.2 dependingon the soil permeability and on the volume of moisturethat can be stored in the soil. The variablesUZS and UZSN are definedas the actual and nominal soil moisturestorageamountsin the upper zone.The nominal value of UZSN is approximately a function of watershedtopographyand cover and is alwaysconsideredto be much smaller than the nominal LZSN value.The initial estimatesof UZSN relative to LZSN are found ftomTable 23.4. TABLE 23.4 VALOESOF UZSN AS A FUNCTIONOF LZSN FOR INITIAL ESTIMATESIN SIMULATIONWITH SWM-IV Watershed Steepslopes,limited vegetation,low depressionstorage Moderateslopes,moderatevegetaion,moderatedepressionstorage Heavy vegetalor forest cover, soils subjectto cracking,high depression storage,very mild slopes Source: After Crawford and Linesley.2
O.O6LZSN O.OSLZSN O.14LZSN
29.1 CONTINUOUSSTREAMFLOWSIMULATIONMODELS
559
The parametersLZSN and CB must also be estimatedat the beginningof a simulation study.The combinationthat will most satisfactorilyreproduceboth longand short-termhistorical responsesto hydrologic inputs can be determinedby the following procedure:2 1. Assumean initial value for LZSN equal to one quarterof the mean annual rainfall plus 4 in. (usedin arid and semiaridregions),or one eighth of the annual mean rainfall plus 4 in. (usedin coastal,humid, or subhumidclimarcsJ. ) Determinethe initial value of UZSN fromTable 23.4. 3. Assumea valuefor CB in the normal rangefrom 0.3 to I.2. 4. Simulatea period of record usingthe streamflow,rainfall, and evaporation data and systematicallyadjust LZSN, UZSN, CB, and other parameters until agreementbetweensynthesizedand recordedstreamflowsis satisfactory. If the annualwaterbudgetsdo not balance,LZSN is adjusted;CB is adjustedon the basis of comparisonsbetween synthesizedand recorded flow rates for individual storms. Lower Zone Storage and Groundwater The lower groundwaterstoragezonein Fig.23.2 receiveswaterfrom the net infiltration and from percolation.The percolaof net infiltrationthat reaches tion rateis determinedfrom F;q.23.I1.Thepercentage groundwaterstoragedependson the soil moisture ratio LZS ILZSN as shownin Fig. 23.9.If this ratio is lessthan 1.0,the percentageP, is found from
P,:1oo[#(-#t,"',
(23.r2)
andif LZS/LZSN is greaterthan 1.0,the percentageis
+:roofr'-(--rg l"]
equations, tn",*:r* rnboth :, _ i'E;il,
]
.,.,
(23.r3)
(23.r4)
Notefrom Fig.23.9 thatthe nominal storageLZSN equalsthe lower zonestorageLZS when half or 50 percent of all the incoming moisture enters groundwaterstorage. The outflow from the groundwaterstorage,GWE at any time is basedon the commonly usedlinear semilogarithmicplot of baseflow dischargeversustime. This techniquewasdelcribedin Section11.4andillustratedin Fig. 11.8.In modifledform the baseflow equationis GwF : (LKK4)[1.0 + KV(Gws)](sGw)
(23.rs)
where LKK4 is definedby LKK4:1.0-(KK24)t/e6
(23.16)
in which KK24 is the minimum of all the observeddaily recessionconstants(see Secti,on11.4), whereeachconstantis the ratio of the groundwaterdischargerateto the
560
CHAPTER23
CONTINUOUSSIMULATIONMODELS
Inflection 9oo oF \
ltt
i
1.0
0.5
1.5
2.5
*u" (;R) Soilmoisture of infiltratedwaterthat reaches Figure 23.9 Percentage (After CrawfordandLinsley'2) storage. groundwater (K n groundwaterdischargerute 24 hr earlier. Thus the recessionconstantKK24 values has Eq. 11.1)is determinedusingt : I day.The variableGWS in Eq.23.15 given that dependon the long-terminflows to groundwaterstorage.Its value on any adjusted day (e.i., the lth day)is calculatedas97 percenLofthepreviousday'svalue, ground: foi any inflow to groundwaterstorage,or GWS; 0'97 (GWS'-1 * inflow to water storageduring daYl)' In Eq. 23.J{, SGW is a groundwater storage parametel that reflects the term fluctuationJin the volume of water storedand rangesfrom 0.10 to 3.90 in' The groundwater KV in Eq. 23.15 allows for changesthat are known to exist in the when KV is zero, E,q.23.15reducesto Eq. 11.1and recessionratesas time passes. follows the linear semilogrelation- If the usual dry season the groundwaterrecessilon being recessionrate KKz4is too largefor wet periods(whengroundwaterslofaggsare the so that from the streams)ihe parameterKV is hand-adjusted rechargedby seepage during value term i0 +-KVaGWS) will reduce the effective rate to some desired recesrechargeperiods.Table 23.5illustratesthis computationby showingeffective to 1'0' equal set sion rates for variouscombinationsof KK24 and GWS when KV is or to deep lost The fraction of activeor deepgroundwaterstoragethat is either drainage the or is diverted as flow across inactive groundwaterstorage Gi;.b.D This fraction is the total inflow to K24L. pu.u*tt"r ur input is basin boundary FOR RATES RECESSION TABLE23.5 EFFECTIVE VARIOUSCOMBINATIONSOF KK24 AND GWS WHEN l(/ : 1'0 GWS
0.99 0.98 0.97 0.96
0.99 0.98 0.97 0.96
0,5
1.0
0.985 0.970 0.955 0.940
0.98 0.96 0.94 0.92
Source: After Crawford and Linsley.2
n07
0.94 0.91 0.88
MODELS 561 SIMULATION S1REAMFLOW 23.1 CONTINUOUS groundwaterand representsall the active groundwaterstoragethat does not contribute to streamflow. Overland Flow
The overlandflo* pro".r. hasbeenstudiedLy -uny investigators.
hydraulictechniques. Averagevalues of lengths, slopes,and roughnessesof overland flow in the Manning and continuity equationsare usedin SWM-IV to continuouslycalculatethe surfacedetentionstorageD".The overlandflow dischargerate q is then relatedto D,. As the rain supplyrate continuesin time, the amountof water detainedon the surfaceincreasesuntil an equilibrium depth is established.The amount of surface detentionat equilibrium estimatedby SWM-IV is 6 0.000818i0 no'6L1'6
D":
(23.r7)
s0'3
where D" : the surfacedetention at equilibrium (ft3lft of overland flow width) j : the rain rate (in./hr) S : the slope(ftlft) L : the length of overlandffow (ft) ru : Manning's roughnesscoefficient The overland flow dischargerate is next determinedas a function of detention storage from
, : # y,,(?)"'[,0* o,o(r2)']"'
(23.18)
where e : the overland flow dischargerate (cfs per ft of width) D : the averagedetention storagedurrng the time interval The equationalso appliesduring the recessionthat occurs after rain ceases,but the ratio blD" is assumedto be 1.0. Typical overland flow roughnesscoefficientsafe providedin Table23.6. The time at which detention storagereachesan equilibrium is determirtedfrom t
.e
: -
0.94L3/5n3/s
(23.te)
i2/s S3/ro
where /, is the time to equilibrium (min). Crawford and Linsley show that these equationsvery accuratelyieproducemeasuredoverlandflow hydrographs.2 For eachtime interval Lt, an end-of-intervalsurfacedetentionD, is calculated from the initial value D, plus any water addedAD (Fig. n.q to surfacedetention from storageduring the time interval, lessany ovedandflow dischargeQthatescapes .4.
562
CHAPTER23
SIMULATIONMODELS CONTINUOUS
TABLE;3.6
ryPICAL MANNINGEQUATIONOVERLANDFLOW NN ROUGHNESSPARAMETERS,
Watershed cover Smooth asphalt Asphalt or concretePaving Packedclay Light turf Denseturf DenseshrubberYand forest litter
Manning'sn for overlandflow
0.012 0.014 0.03 0.20 0.35 0.40
Source: After Crawfordand Linsley.2
continudetentionstorageduring the time interval. This is simply an expressionof itv. or Dz:Dt+LD-4Lt
(23.20)
(D: + D)/2' EquaThe discharge@is found from Eq' 23.18usinga valueo! D . flow using easily of overland tions 23.17- 2i.20 allow the completedetermination overlandflow' roughness and found basin-widevaluesof the averagelength, slope, same Interflow The watertemporarilydetainedasinterflow storageis treatedin the was detention interflow to inflow fashion as overland flow detention storage.The similar constant recession a daily definedin Fig. 23.4.Theoutflowis simulatedusing IRC is the to that definedfor groundwaterdischarge.The interflow recessionconstant 24 hr discharge interflow to the averageratio of the interflow dischargeat any time is storage detention from outflow earlier.For each15-min time intervalmodeled,the
rNTF : LIFC4(SRGX) where
LIFC4:1.0-(tRc;'inu
(23.2r) ()1n\
Its The variableSRGXis the water storedin the interflow detentionat any time. time each to applied value continuouslychangeswhen the continuity equation is on the interval. The end-of-inteival value of SRGX depends,accordingto continuity, interflow the from valueat the beginningofthe interval and any inflow to or discharge detentionduring the interval. a hychannel Translation and Routing The Stanfordwatershedmodelutilizes watershed to the drologicwatershedrouting techniqueto translatethe channelinflow as outlei. Clark's IUH time-ar"a to"ihod describedin Section 12.6is adoptedalmost presentedinchapterl2.Inplaceofthenetrainhyetograph,theStanfordmodelviews "inflow" hyetograph.This inflow is the sum of all channelinflow componentsas an routed then translatedin time through the channelto the basinoutlet, whereit is next storage by caused through an equivalentstoragesystemto accountfor the attenuation sensethat in the-channelsystem.Roulingthrough the linear reservoir (linear in the is accom12.35) Eq. storageis assumedto be directly proportional to the outflow' plishedfrom (23.23) I or:7-KS1(1 -01) )
MODELS 563 SIMUI-ATION STREAMFLOW 29.1 CONTINUOUS where Oz:
the outflow rate aI the end of the time interval
o t : the outflow rate at the beginning I _ the average inflow rate during the time interval Also,
KSI
: vK *- 6Ltlz *
(23.24)
Examplesof the determinationof K and othernecessaryparametersfrom watershed data areincludedin Section23.2. Applications of the SWM-IV Applicationsof the model typically beginwith data for a three- to six-yearcalibration period for which rainfall and runoff data arc available.Thesedata are usedto allow successiveadjustmentsof severalparameters until the simulatedand recordedhydrographsof the streamflowagree.If sufficient data areavailable,a secondperiod of record may be reservedfor use as a control to checkthe accuracyof the paiametersderivedfrom a calibration with the first half of the data. The Stanfordwatershedmodelwasoriginally developedin 1959and hasundergoneseveralmodificationssincethat time. James3translatedthe Crawfordand Linsley model from ALGOL to FORTRAN. Several modifications of the FORTRAN versionhave evolvedfrom a variety of investigations.Included amongtheseare the Kentucky watershed model (KWM),4'5 the Kentucky self-calibrating_version (OpSETi,4the Ohio State University version, the Texas version,6the Hydrocomp SimulationProgram (HSP) written in PL/1, and EPA-produced,nonproprietary FORTRAN u"rrion of HSP called HSPF, and the National WeatherServicerunoff forecastingmodel. Brief descriptionsof severalof theseare includedbelow.
ARS RevisedModelof WatershedHydrology(USDAHL) Growing interest in the effects of soil types, vegetation,pavements,and farming practicei on infiltration and overlandflow hasresultedin the growth of the USDAHL continuoussimulation model. The 1974 versionTof this model was developedby investigatorsat the Agricultural ResearchServiceHydrographLaboratory. Input data to the model are relatively extensive.Continuousrecords of the precipitation,the weekly averagetemperatures,the weekly averagepan-evaporation amounts,and detaileddata onioils, vegetation,land use, and cultural practicesare required' The studywatershedis initially dividedinto asmany asfour distinctland-useor soil-type ,on".. Fourteen subroutinesand a main calling routine computefor each zone the snowmelt, inflltration, overland flow, channel flow, evapotranspiration, groundwaterevaporationand movement,groundwaterrecharge,and return flow' Evapotranspirationpotentialsare estirnatedby app$ing assignedcoefficientsto pun-"uuplrution data. Infiltration for eachsoil or land-usezone is computedusing a modifiedHoltan equation.Waterstoredin cracksin dry soilsis simulatedasa function of soil moisture.Manning's equationand the continuity equationare used to route overlandflow. The streamflowis routedby a simultaneoussolution of the continuity
564
CHAPTER23
MODELS CONTINUOUS SIMULATION
equdiion and a storagefunction. Groundwatermovementsare calculatedby Darcy's equation.The daily snowmelton each-zoneis calculatedas a function of the temperature at which snowmeltstarts,the weightedaveragevegetativedensityfor the zone, the weekly averageair temperature,and the potential snowmeltper day in the zone snowpack. Precipitation falling during a snowmelt day also contributes to the snowmeltequation. Among otheruses,the modelhasbeenappliedby the Soil ConservationService in preparing environmentalimpact statements.Figure 23.10 showsthe results of applying the 1974 version to annual runoff'from four widely separatedand widely diversifiedARS experimentalwatersheds.In addition to the runoff, the model computes the evapotranspirationamounts, soil moisture changes,return flows, and groundwaterrechargedepthsfor eachof the zones. Although other modificationsare possible,the USDAHL model is specifically designedfor relatively small rural watersheds,generallyunder 20 squaremiles.
$-i
o
E U
Cumulative computed runoff, O (in.)
Figure 23.10 Chart showing the accuracy of USDAHL-74 model for estimatingthe cumulativecomputedrunoff as comparedwith the cumulative measuredrunoff at four watersheds.o W-97, Coshocton,OH: A W- 11, Hastings,NE; I W-3, Ft. Lauderadale,FL; x W-G, Riesel, TX (After Holtan andLopez.l)
2g.1 CONTINUOUSSTREAMFLOWSIMULATIONMODELS
565
NationalWeatherServiceRiver Forecast System(NWSRFS) yet anotherversionof the Stanfordwatershedmodelwasdevelopedby the Hydrologic The ResearchLaboratory staff at the National WeatherServiceOffice of Hydrology.8 by the stages and flows river forcecasting in use NWSRFS model wis developedfor river several to successfully applied been has National WeatherService.The model River basinsrangingin sizefrom 70 mi2in North Carolinato 1000mi2in Oklahoma' for SWM in incorporated detail the require not does forecastin! in"largeriver basins changes major two includes model NWS the smaller watersheds.For this reason' fewer process involvingthe useof a longertime increment,simplifiedprogramming, parameters watershed optimal determining for comput;ions, and a rapid procedure accurately. flows historical that allow the model to reproduce inputs and A 6-hr time incrementis usedby the model, allowing fewer rainfall flow that overland as such processes of ,nor" i.po.tant, fewer detailedcalcuiations with than rapidly mo_re completed thus are occur in'shorter time periods.Iterations optimization Service Weather National the the SWM. As with the OPSETmodel, available procedurefor determiningparametervaluesgivesthe model a strengthnot with the SWM-IV. Other modificationsincludeh uPPerand lower zone retentionand the uPPersoil zone to groundwate groundwaterevapotranspirational jointlY comPutedin the NWS versi and groundnatedand is replacedby three types ofrunoff: surfacerunoff, interflow' response' slow and water flow-representing fast, medium, instantaInput data for modll calibration consistof meandaily dischargesand continuous a as is input Rainfall events. runoff neoushydrographsfor a few selected techniques.Berecord of 6-hr basin-widemeansdeterminedfrom areal averaging process simulation'the of detail in the and causeof the changesin routing increment outputfromtheNwsiersionissimilarinmakeuptotheSWM-IVoutput.
COEStreamflowSynthesisand ReservoirRegulation Model (SSARR) for large Another widely used continuous streamflow simulation model designed developed was model SSARR The basinswas devllopedby the corps of Engineers.e operation primarily for streamflowand flood fo.ecastingand for reservoirdesignand Weather National at the hydrologists studies.i,rior to the developmentof NWSRFS, rain and both to applied been has model Service used.the SSARR model. The snowmeltevents. basin into Applications of the model begin with a subdivisionof the drainage subdivides, with consistent character and homogeneoushydrologic units of i size otherdistinguishchannelconfluences,rJservoirsites,diversionpoints,soil types,and throughoutthe points significant ail fbr ing features.The streamflowsare computed river sYstem.
566
MODELS 23 CONTINUOUS SIMUI-ATION CHAPTER duintutt data can be input at any numberof stationsin the basin.The part that will run off is divided into the baseflow, subsurfaceor interflow, and surfacerunoff. The division is based on indices and on the intensity of the direct runoff. Each componentis simply delayedaccordingto different processes,and all are then combined to producethe final subbasinoutflow hydrograph.This subarearunoff is then routed through stream channelsand reservoirsto be combined with other subarea hydrographs,all of which becomepart of the output. Routingsthrough channelsand reservoirsare accomplishedby the sametechnique.This requiresan assumptionof shortstreamreaches,and occasionalallowances for backwatereffectsare necessaryin the channelrouting process.Streamflowsare synthesizedon the basisof rainfall and snowmeltrunoff. Snowmeltcanbe determined on the basis of the precipitation depth, elevation,air and dew point temperatures, albedo,radiation, and wind speed.Snowmeltoptions include the temperatureindex method or the energybudgetmethod. Input includesthe precipitation depths,the watershed-runoffindicesfor subdividing flow among the three processes,initial reservoir elevationsand outflows, drainageareas,bounds on usablestorageand allowabledischargefrom reservoirs, total computationperiods,routing intervals,and other specialinstructionsto control plots, prints, and other input-outputalternatives. This model was one of the earliestcontinuousstreamflowsimulation models using a lumped parameterrepresentationand has its primary strengthin its verified accuracyindicatedby testsconductedin severallarge drainagebasinsincluding the ColumbiaRiver basin and the Mekong River basin.
HydrocompSimulationProgram(HSP) A commercial version of the Stanford water model was developedat Hydrocomp, incorInc., namedthe HydrocompSimulationProgram.toAmong severaladvantages poratedin HSP are hydraulic reservoirrouting techniquesand kinematic-wavechannel routing techniques.Other major changesinclude the addition of water quality simulation capabilities.Due to theseadditions,the model is often referredto as the Hydrocomp water quality model. The HSP model has been usedroutinely for severaltypes of hydrologicstudy including floodplain mapping,water quality studies,storm water and urban flooding studies,urban drainagefacility design,and water quality aspectsof urban runoff. The model consistsof three computerroutines: l, Library allowsthe useof direct accessdisk storageto handleinput datawith efficient data managementroutines. 2. Lands handlesthe usual SWM lands phasealong with addedprocessesin calculating soil moisture budgets,groundwaterrecharge and discharge, inflow to streamchannels,and eutrophication. 3. Channel is responsiblefor assemblingand routing all channel inflow through channelnetworks,lakes, and reservoirs. The HSP model incorporatesa continuouswater balanceby trackingprecipitation through all possibleavenuesof the hydrologicand water resourcesystem.The
29j
CONTINUOUSSTREAMFLOWSIMULATIONMODELS
567
groundwaterinflow, interflow, and surfacerunoff are individually simulated,lagged, ind combinedat appropriatetimes as the channelinflow. The routing of computed inflow through the ihannel networkutilizes a modified kinematic-wavemodel.Water quality constituentsare relatedto variabledischargeratesand other waterparameters so that the coupling of quantity and quality of the runoff is accomplished. Inputs foisimulating waterquality includethe temperature,radiation,wind, and humidity and observedvalues of the factors under study which form the basis for calibration.At lest two yearsof data are preferablefor calibration; however,calibration hasbeenachievedwith lessthan one year of data. Other requiredinput includes hourly precipitationin 5-, 15-, or 30-min or greater time incrementsand potential evapoffanspiration;if snow or water quality simulation is desired,the temperature, radiation, wind, and humidity factorsare needed' Outputsfrom HSP can be obtainedfor any desiredpoint within the watershed. Included in the output options are valuesof quality dataat outfalls or other points, river stages,,"r"ruoi, levels,hourly and meandaily dischargerates,streamand lake remperarures,dissolvedoxygenand total dissolvedsolids,algaecounts,phosphorus' nitr;te, nitrite, ammonia,iotal nitrogen, phosphate,pH, carbonaceousBOD, coliforms, conservativemetals,and the usual daily, monthly, and annualwater budgets, snow depths,and end-of-periodmoistureequivalents' Typical simulation periods in HSP applicationsrange from 20 to 50 years. Hour-by-hour data are not viewed as an exact sequenceof future flows. Rather, the data arcusedfor analysisof the probability of occurrencesof rangesin the factorsof interest.When usedin this -unir"r, the model is functioning with a purposesimilar to that of someof the operationalhydrologytechniquesdescribedin Chapter22.
EPA HydrocompSimulationProgram-Fortran(HSPR Following developmentof the HSP versionof the Stanfordmodel,the U.S. Environmental Piotection Agency contractedin 1980 to have public-domain version made availablefor continuousstreamflowsimulationand water quality modeling.The original program,written in ALGOL, wasconvertedto FORTRAN 77'11The HSPF code is availa|le for PC applications.Substantialportions of water quality modeling algorithms were addedio HSP in developingHSPF. The hydrologiccycle processes, however, are essentiallythe same as in SWM-IV and HSP' One exceptionis the addition of severalrouting proceduresnot previouslyavailable. parametersthat drive ihe routinesin HSPFmustbe estimatedfor all the hydrologic processes,making verification of the model difficult becauseof the numerous combinationsof paramiter values.Methodsof estimatingtheseparametersand calibratingthe modeiareillustratedin the casestudiesin Section23.2 (Table23.8 defines over 35 parametersusedin the Stanfordmodel)'
ModelingSystem(PRMS) USGSPrecipitation'Runoff After developingtheir urban storm-eventmodel, DR3M (seeChapter 25), the U.S' GeologicalSurv"y developedseveralother computercodesto model continuoushydrologicpro""rr"r. fne pRUS performssimulationof daily streamflowsfor a variety of precipitation,climate,and land usecombinations.It is availablefrom the USGSin I
L.
568
CHAPTER23
CONTINUOUSSIMULATIONMODELS
PC, riinicomputeroor mainframeformat.12During storms,the model givesoutput on any prescribedtime interval. Betweenstorm periods,the model tracks soil moisture and other storage/depletionzones on a daily basis until the next storm interval. Streamflowis output as mean daily flow rates. A lumped-parameterapproachis utilized in PRMS.The smallestsubdivisionof to behaveas a the study watershedis a hydrologic responseunit that is assumed. hydrologicelement.The USGS has delineatedHRUs in most areasof homogeneous the United States.Like the D.R3M model,the streams,storm sewers,reservoirs,and detentionponds in the watershedare modeledas nodes and interconnectinglinks' Hydrographrouting in channelsand reservoirsis accomplished.bykinematic-wave and storage-indicationmethods,respectively.
ARS Simulatorfor WaterResourcesin Rural Basins(SWRRB) Through a cooperativeprogram with TexasA&M University, the Agricultural Research Service, U.S. Department of Agriculture, developeda continuous daily streamflow simulation ptog.a- for use in modeling ungaugedagricultural areas.t3 This FORTRAN 77 water budgermodel is availablefrom the ARS in PC format. Its developmentfocusedon a model that would allow the user to predict impacts of variouswatershedmanagementpracticessuchas crop rotation, fall plowing, urbanization, conservation tillage, terracing, fallowing, and floodwater detention on monthly and annual water and sedimentyields from rural basins. and for the total basin, are Sedimentyields for each of the subwatersheds, calculatedusingthe universalsoil lossequation.laThe methoduseswatershedhydrology outputsfrom the rainfall-runoff portion to estimatesedimentyield, using inputs of runoff volume,peak flows,soil type anderodibility, crop types,erosionmanagment factors,and watershedslopeand length. The subbasinsare modeledin a lumped-parameterstyle.Oncerainfall or snowalgorithmsare linked as shownin Fig. 23.11 fall recordsare input, physical-process for solar radiation, snowmelt,surfacerunoff, ET, conveyancelosses,sedimentproductionfrom individual storms,evqporationfrom watersurfaces,percolation,and soil moistureaccounting.Crop productionis alsocalculatedbasedon crop types,temperature, consumptiveuse of water, and irrigation practices. are estimatedby the SCS curve Hydrologicabstractionsfor eachsubwatershed number(CN) method(Chapter4, Section4.9). Soil waterbudgetingis performedby adding the net moistureinput to the soil profile from precipitation after subtracting direct runoff, ET (consumptiveuseby the cropsand evaporationfrom soils),percolation, and retu{n flow (groundwaterflow back to the streams).Direct surfacerunoff is set equal to the net rain from the CN method. For input to the sedimentyield component, a peak flow rate for individual storms is estimatedusing a modified rational method (Chapter 15). Snowmelt is calculatedby the degree-daymethod ' describedin Chapter14.
ExpertSystems Recenttrends in systemsanalysisare leadingto developmenlof expert system(ES) techniquesthat rely on artificial intelligencefor use in planning and designof water -,
.oo^,,r^^o
-r^io;id--7[-
mrfian-.m
h-c-ciw?i'- ii-FiYmqfinn
nhfqined
frnm
extencive
29.1 CONTINUOUSSTREAMFLOWSIMULATIONMODELS
C*.') '
/READTNPUTDATA/ INITIALIZE PARAMETERS
READ OR GENERATE PRECIPITATION AND MAX/MIN TEMPERATURE GENERATE SOLAR RADIATION
CoMPI.ITE SOIL TEMPERATTTRE COMPUTE SNOWFALL AND SNOWMELT
ACE
COMPUTEPEAK RATE TRANSMISSIONLOSSES. OUTI-ET SBDT,M}{T YIEIO ENO ROUTE SEDIMENT TO BASIN
COMPUTEANDPRINT FINALBASIN STATISTICS
Figure 23.11 Flowchart of the SWRRB hydrologicpr6""*, algorithms.(After Arnold't3)
570
CHAPTER23
CONTINUOUSSIMULATIONMODELS
interviJws of one or more experts in some field. The computer can then make "decisions"in muchthe sameway asthe,experts,applyingtheir judgment and experience and making theseavailableto othersthrough the expert systemmodel. Streamflowmodels,especiallythosethat perform continuoussimulation,incorporateinput parametersthat require considerablejudgment. Developersand usersof ahe watershed models have accumulated decades of experience in assigning coefflcientsand parameters.Their experienceand judgment can be extractedby an interview processinvolving hundredsof questionsto build an expert systemmode'. uncertaintiesabout The modelnot only incorporatesdirect answersbut alsoaddresses promise.ls considerable show technique modeling with this each.Early applicdLtions potential to be the have systems expert In addition to streamflowsimulation, reserdams, of systems basin river of complex useful in the designand management voirs,powerplants,diversioncanals,and flood control structures.Operationsfor such systemsinvolve independentand collective decisionsby dozens of professionals' Theseexpertsare normally in radio or telephonecontact with numerousother controllers and decision-makers.If ES data could be developedfrom theseteams,the potential for improvedmanagementexists.A prime incentiveof implementingexpert iystemsin waterresourcessystemsinvolvescapturinginsightsof experiencedprofessionalsbefore they retire or move into other positions.
MODELSTUDIES SIMULATION 23.2 CONTINUOUS This sectiondescribesin detail two independentapplicationsof the Kentuckyversion of the Stanfordwatershedmodel to small basinsin Kentucky and Nebraska.Results obtainedby Clarkel6in modelingthe CaveCreek (CC) watershedin Kentuckyand by the authorslTusing KWM for the Big BordeauxCreek (BBC) watershedin Nebraska are compared. Both are small, homogeneouswatershedshaving relative$ good recordsof precipitationandrunoff. The two casestudiesare describedsimultaneously to showhow different analystsdealt with the decisionsrequiredto developinput data and parameters.
Selectionof WatershedSize and Study Period
.
Severalguidelinesexist for selectinga watershedsubareasize and time period to be watersheds modeledin a simulation study.The use of relatively small, homogeneous difficulty any minimize to recommended is or subdivisionsof larger watersheds also This areas. larger over precipitation in causedby ignoring spatial variations soil types, as soil such characteristics watershed minimizes the'effects of lumping entire the parameters representing into single uses profiles,imperviousareas,andland catchment.Ross suggestsan upper limit of 25 fii12for study-watersheddrainage areas.18 One difficulty in restrictingwatershedsizearisesfrom the fact that few strearns for small drainageareasare continuouslygauged,and refiable streamflow data are difficult to obtain. For example,the BBC watershedwas selectedbecauseit was one of few small watershedsin Nebraskawith sufficientrecords.The CaveCreek watershedwas selegtedon the basisof the following criteria'1e
MODELSTUDIES SIMULATION 2g.2 CONTINUOUS
571
'A
minimum of 10 yearsof continuousrunoff recordsin order to establish the existingrainfall-runoff relation. 2. A drainas" ur"u of less than 5 mi2 so as to be reptesentativeof small drainage6asinsfor which better runoff coefficientsare needed' 3. A locaiion in closeproximity to a rain guagefor which hour$ precipitation dataarc available. 4. The availability of soil surveysfor the watershedunder study. 1.
DataSources Hourly precipitation data were availableat sites L20 fiil from the CC watershedand 8.0 mi fromthe BBC gaugingstation.Soil surveyrecordsand runoff data were availablefor both watersheds.Daily pan evaporationdata were availableapproximately 30 mi south of the BBC watershedand 25 mi south of the CC watershed. d and9 .22 mi2for the BBC watershed Drainageareasof 2.53 mi2for the CC watershe were found from U.S. GeologicalSurveyquadranglemaps.Input parametersfor the Stanfordand Kentuckyversionsare comparedand definedin Table23.7. Numerical values,usingthe Stanfordversionparameternames,aretabulatedfor both watersheds in Table 23.8. Each parameteris describedin detail in the following sections.
Time-Area HistogramData The time-area histograms for the BBC and CC watershedsare developedin Traveltimes and times of concentrationfor the Figs.23.12 and23.Ii, respectively. the Kirpich equation(for watershedslarger from found Stanfordwatershedmodel are than 15 acres),ies1
a : ooo78(#)"
(23.2s)
where T" : the time of concentration(min) L : the horizontal projection of the channellength from the most distant point to the basin outlet (ft) : S the slopebetweenthe two points In developingthe time-area histogramfor Big BordeauxCreek, 30 points and correspondingi.auettimes (min) were plottedas shownin Fig. 23.12.The dashed isochrones(lines of equal travel time to the basin outlet) were constructedby linear interpolation between the plotted points. Areas betweenparis of isochroneswere determinedfrom planimeteiing.To contrast,the time-area histogramfor CaveCreek was construcrcdfas shown in Fig. 23.13, by assumingthat the flow velocity was constant"u"ryrli"r", equalto the averagestreamflowvelocity obtainedby dividing f into the channel length Z. This procedure simply places all points along each isochroneat equal distancesto the basin outlet.
WatershedParameters The watersheddrainage area (AREA), the impervious fraction of the watershed surfacedraining directly into the stream (A), the fractional streamand lake surface area(ETL), the averageground slopeof overlandflow perpendicularto the contours
572
CHAPTER23
CONTINUOUSSIMULATIONMODELS
TABLE 2i,7
PARAMETERNAME COMPARISONSFOR KENTUCKYAND STANFORDWATERSHEDMODELS
Parametername
Kentucky verston
Stanford verston
NCTRI CTRI RMPF RGPMB AREA FIMP FWTR VINTMR BUZC SUZC LZC ETLF SUBWF GWETF SIAC BMIR BIVF
Z C MINH K1 AREA
oFss OFSL OFMN OFMNIS IFRC CSRX FSRX CHCAP EXQPV BFNLR BFRC GWS UZS LZS BENX IFS CONOPT
aoa DIV
ETL EPXM CX EDF LZSN K3 K24L K24EL EF CB CY SS L NN NNU IRC KSC KSF CHCAP RFC
t
DKN
Qaa DIV
ParameterdescriPtion Integernumber of elementsin the time-area histograq Time-area histogramordinate (cfs) DischargevalueLelow which no synthesizeddata are to be printed statton a distant for data precipitation for factor Multiplication Watersheddrainagearea (mi2) Fraction of watershedareathat is impervious Fraction of watershedarea in lakes or swamps Maximum rate of vegetativeinterceptionfor a dry watershed(in'/hr) lndex for estimatingsurfacestoragecapacity Ind,exfor estimatingsoil surfacestoragecapacityduring summers Index of moisture storagein soil profile abovewater table (in') Evapotranspirationparameterfor lower zone soil moisture flow from the basin Subsurface Groundwaterevapotranspirationby phreatophytes Factorvarying infiltration by season Index of infiltration rate Index of rate and quantity of water enteringinterflow Averagebasin ground slope (ftlft) Averageoverland flow distance(ft) Manning roughnesscoeff,cientfor overland flow Manning roughnesscoefficientfor flow over impervious areas Daily intelflow recessionconstant Streamflowrouting parameterfor 1owflows Streamflowrouting parameterfor flood flows Index capacity of existingchannel,bank-full (cfs) Exponentof flow for nonlinearrouting Daily baseflow nonlinearrecessionadjustmentfactor Daily baseflow recessionconstant Indexof groundwaterstorage(in.) (in') Depth of interception and depressionstorageat beginningof year Cuirent equivalentdepth of moisturein the soil profile (in') Cunent value of BFNLR Interflow storage Control options for input, output, and internal branching Title of computer simulation output, alphanumericinput Mean daily diversion into or out of the basin (cfs)
(SS),and the nban length of overlandflow (r) for the Big Bordeauxcreek watershed series were determinedfrom-areas,elevations,and lengths measuredfrom 7.5-min parameter BBC Other values). BBC for usGS topographicmaps (see Table 23.8 : half the product of the stream values utl gri : 0.0d5 (determinedfrom ETL : ftlft (determinedfrom Fig '23'14 0.088 length and channelwidth at the outlet), SS at eachgridline intersegcontours 20-ft u, ih- -"un of'140 measuredvaluesbetween averageof l40lengths the as Fig.23.14 183.2ft (determinedfrom tion), and L: points to the intersection gridline from lines measuredperpendicularto contour
TABLE 23.8 SUMMARYOF INPUT PARAMETERSFOR BIG BORDEAUXCREEK AND CAVE CREEK WATERSHEDS
Parameter name
Description
a^ BBC value(s) Units value(s)
min 105 min 15 7 0.129 0.158 0.221 , 0.151 0.126 0.145 0.070 mi2 9.22 0.0
TCONC TINC Z C
Time of concentration Routing interval Number of elementsin the time-area histogram Time-area histogramordinates
AREA
Watersheddrainagearea Impervious fraction of the watershedsurface Watershedstream and lake surfacearea as fraction of 0.005 watershedarea 0.088 ft/ft Averageoverland flow ground sloPe ft 183.2 flow of overland Averagelength cfs 39 Bank-full flow in channelat gaugingstation 0.485 constant recession Daily interflow 0.977 \ Daily baseflow recessionconstant 0.989 low flows for parameter Streamflowrouting 0.989 Streamflowrouting parameterfor flood flows cfs 1.0 printed be to rate Minimum hourly flow 1.0 Precipitation adjustmentfactor for distant gauge 0.3'7 Manning roughnesscoefficientfor overlandflow 0.0i3 Manning roughnesscoefficientfor impervious areas 0.5 seasons by infiltration Factorfor varying 0.15 in./hr Maximum interceptionrate for dry watershed 0.80 index capacitY Surfacestorage 1.10 Soil surfacemoisturesioragecapacity ln. Il./d index storage profile moisture Soil 0.28 Soil evaporationparameter 0.0 Index of inflow to deepinactive groundwater 0.0 Fraction of watershedareain phreatophytes 1.0 rates infiltration for seasonal Factor allowing 0.'75 Factorcontrolling infiltration rates 3.0 Index controlling water enteringinterflow 1.0 Parameterfor allowing nonlinear recesslon in. 0.1 parameter volume storage Groundwater in. 0.0 Equivalentdepth of upper zone storage in. 7.0 storage zone of lower Equivalentdepth in. 0.2 Index of antecedentmoisture conditions acre-ft 0.0 storage in swamP water Volume of 0.911 Jan. Monthly evaporationpan coefficients 0.911 Feb. 0.911 Mar. 0.911 Apr. 0.552 May 0.677 June 0.654 July 0.651 Aug. 0.642 Sep. 0.911 Oct. 0.911 Nov. 0.911 Dec.
^ ETL
ss L
CHCAP IRC KK24 KSC KSF MINH K1 NN NNU EMiN EPXM CX EDF LZSN K3 K24L K24EL EF CB CY KV24 SGW UZS LZS GWS VOLUME EVCR
60 i5 4
0.18 0.29 0.31 0.22
2.53 0.0 0.0 0.075 300.0 40 0.75 0.94 0.90 0.90 0.2 1.0 0.10 0.015 0.10 0.90 1.25 4:85 0.25 0.0 0.0 0.15 0.65 3.50 0.99
574
MODELS SIMULATION 23 CONTINUOUS CHAPTER 0.3 I
15-min isochrone
0.2
"15 rs 30 4s 60 90 105 Time(min)
3li
a5-min {
s
-l
Ll--:Ja
$.(-t 68o
Time.area
15 30 45 60 75 90 105 Time (min)
-l \t
30-minisochrone 45-min isochrone
60-min isochrone N
--.1-^
t-\-75-minisochrone 90-min isochrone n . , u miles r IJJJJ-JJJJJ
1 l
Figure 23.12 Time-areahistogramdevelopmentfor Big BordeauxCreek' The valuesrepresenttravel times (min.) to the outlet.
nearestchannel).The averageoverlandflow distancescan also be estimatedas the
resulting in an estimateof the bank-full BBC flood of 39 cfs. Another techniquefor rating curveif th€ determiningCHCAP involvesthe useof the stream-gauging-station gaugeheightfor bank-full flow is known or can be estimatedfrom topographicmaps. thJCave Creekvalueof 40 cfs wasdeterminedfrom a hydraulic analysisof the cross spctigl and profile of the main channel.As a rule of thumb, CHCAP is selectedto )
i
.../
3l.ovo _
Time-arca histogtam Contributing area (Vo)
Time (min)
18.0 29.0 31.0 22.O
0 -1 5 15-30 30-45 45-60
Figure 23.13 Derivation of the time-area histogramfor thecave creek Watershed.Dashedlines representisochrones.(After Clarke'16)
l
1
miles
0 r
r
r
r
0 1000
r
r
t
r
l
N
h -w" -q
f""t
:h &"{
, TGrid interval:fm=tlZOft \I
Figure 23.14 Grid overlay for determinationof the mean overland - slopeand distancefor Big BordeauxCreek.
576
CHAPTER23
CONTINUOUSSIMULATIONMODELS
producewell-definedoverbank and floodplain flow if Q exceedstwice the value of bffCAp. Similarly, low, shallow flows are assumedwheneverQ is lessthan half of CHCAP.
StreamflowRecessionand Routing Parameters The rate at which the model allows water to passthrough the upper soil zonesto the channelsis controlled by the daily interflow recessionconstant(IRC)' A graphical semilogarithmictechniqueof hydrographanalysisdevelopedby Barnes is used to The determinationof the interflow recessionconstantfor estimatethis parameter:21 Big BordeauxCreek is illustrated in Fig. 23.1.5,which is a semilogarithmicplot of
I o.s E o.+ o
K 0.3 E o
V'L
d
p9
€ o.ro
lnterflow Llog Q = -0;725 Lt = 3.2 days K=0.485=IRC
q
o.os S o
& 0.04 0.03
1 4 1 6 1 8 2 0 2 2 2 4 2 6 2 8 3 0 1 3
5
1969) Date(Ju1Y-August' Figure 23.15 Determination of the interflow recessionconstant (fnC) for th9 Big BordeauxCreekflood eventof July 20, 1969.The K: IRC, C' : QoK'andr: 1day. determinationof
2g.2 CONTINUOUSSIMULATIONMODEL STUDIES
!2 F
o b0
5 logQ,=1ogQ6+tlogK logK= LlogQlLt
d o o
f f i z o
|
' 3 " ' s1 9 1 1 3
1969) Date(July-August, Figure 23.16 Determination of the base flow recession constant (K"K24) for the Big Bordeaux Creek flood event of July 20, 1969' The determinationof K : KK24.
577
578
.
MODELS SIMULATION 23 CONTINUOUS CHAPTER
'
alternative method for determining Kl<24 using data collected several days after severalToodpeaksis illustratedfor CaveCreek in Fig. 23.I7. The line representsan envelopedrawn to the right of the data points. Two streamflowrouting parameteisare usedby the sinlulation model to route inflow hydrographsto the point of interest for the basin (in this caseto the stream garglng;tatiin ror compa;isonwith measuredflows). The first parameter1_519t_lt thechannelaapacity(CHCAP), usedinroutingonlyif channelflowsarelessthanhalf storageduring flood flows floodplain and (KSF) accountsfor channel and the second data for Big _B-ordeaux Respective capacity). (flows greaterthan twice the channel in Figs. 23.18 and shown hydrographs runoff Creek were determinedusing the two inflection point on the Because respectively. Z3.lg,representinglow and flood flows, a reversalof which at the time represents the recessionportion of eachhydrograph KSC and KSF parameters the then flow directionihrough the channelbanks occurs, The channel. in the storage in water representdaily recessionconstantsfor the routing hydrologic a from is derived parameters equationused in determiningboth techniqueequatingl/K timei the differencein the averageinflow and outflow during the routing period-withthe time rate of changeof the outflow in the channelreach.As shownin the figures,the low flow and flood flow routing parametersfor Big_Bordeaux Creek are identical. Table 23.8 showsthat a similar result was observedfor Cave Creek.
HydrologicParametersand Data In addition to the describedwatershedcharacteristics,severalhydrologicparameters and an impressiveamount of hydrologicdata are required as input to the simulation program. bue to the excessive'bulkof hydrologicdata for daily evaporation,hourly pr"-lpitution, and daily streamflowfot a 4-yearperiod in BBC and a 10-yearperiod
4.0 o d
3.0
1.0
.
2.0 3.0 4,0 5.0 24hr later(cfs) Discharge
Figure 23.17 Determinationof KK24 for CaveCreek.(After Clarke.l6)
MODELSTUDIES 579 SIMULATION 23.2 CONTINUOUS 0.36 t = routinginterval= 15min = 0.0104days 0.32
,, ',/ --#
,
o?o
6 0.26
K = - f foi , = - : 60.264 6=0e43dals x S C =K _ o . s t = 0 . 9 8 9 K + 0.5t
E o.rt o P0
= - 0'28at inflectionPoint --
,r'
-
In{lection
pomr
E
! b0
8 0.22 0.20 0.18 0.16 0.14 '
Date (October 1968)
Figure23.1sDeterminationoflowflowstreamflowroutingparametei(rsc) for the Big Bordeauxcreek low flow eventof october3, 1968. in CC, the actual hydrologicdata valueshavebeen compiled but'are not includedin this text. , Oneusefulhydrologicparameterusedasinput only in the BBC applicationis the hourly flow at the gaugingsiation below which no printing of simulateddischargesis desired.For the initial simulation of runoff from Big Bordeaux Creek, a MINH of 1.0 cfs was selectedto minimize the printed output. This can easilybe decreasedin successivesirnulationtrials" after the uncertaintiesin someof the other parameters havebeen reduced. A precipitation-weightingadjustmentparameter(K1), representingthe longterm ratio of av"rage precipitation over the basin to averageprecipitation at the precipitationgaugp,might ue greateror lessthan 1.0 if the gaugewere locatedat any di*tun." frornthJstudy basin.The recordinggaugesfor both watershedswere within closeproxirtrity, and both K1 parameterswere set to 1.0. ivlonthly-evaporationpan coefficients(EVCR) are usedby the simulationprogram asmultlpters in convJrtinginput pan-evaporationdatato lake evaporationdata The nearestevaporation iand alsofor determiningpotentiale.'upottuntpirationrates)' Box Butte Reservoir, the at mi south 20 is approximately Cieet Bordeaux pan for Big
580
CHAPTER23
CONTINUOUSSIMULATIONMODELS t = routing interval = 15 min = 0.0104 days
do =-t'' at lnrlectlon Polnt al
o a9 =0.873days *n - ).) du/dt KSF= K-0.5t = 0.989 K + O.st
K=-
'?
5.0 ,r'
!)
-
Inflection pomt
o E
4.0 d o
oo 6 J.U
t6
I7
18
t9
20 21 22 23 24 25 1969) Date(March Figure 23.19 Determination of flood flow streamflowrouting parameter (KSF)for theBig BordeauxCreekfloodflow eventof March17,1969. providing records of daily and monthly pan-evaporationdepthsfor the months between and including May and September.Estimatesof correspondingmonthly lake evaporationamountsat Box Butte Reservoirwere obtained from maps and charts developedby Shaffer.22 Valueslisted in Table23.8 are assumedto apply to eachyear of the simulation even though the simulation program allows changesfrom year to year. Manning's roughnessparametersfor flow over soil and impervioussurfacesare both required as input to the program. For the Big BordeauxCreek area,the initial estimatesfor qverlandflow (NN) and impervioussurfaceflow (NNU) were 0.37 and 0.013,representingcoefficientsfor denseshrubberyand forestlitter for overlandflow and smoothconcletefor impervioussurfaceflow. The later n is significantonly if the fraction of impervious area (A) is nonzero. The Cave Creek analysisincorporated NN : 0.1Qfor light turf and NNU : 0.015for concretepavement.
ModelCalibration Severalof the following parametersare determinedby trial and adjustmentuntil the comparisonbetweenthe simulatedand recordedstreamflowsis satisfactory.Guidelines-forestablishinginitial valuesexistfor only a few of the parameters,whereasmost
STUDIES 581 MODEL SIMULATION 2g,2 CONTINUOUS ranges.The BBC datain Table23'8 represent are initially determinedfrom suggested initial, unmodified estimates;those shown for CC are the optimal result of many repetitive runs. One factor for varying infiltration by seasons(EMIN) rangesbetween0'1 and 1.0 and has been shownby Briggs to be signiflcantin matchingmeasuredand simuBecauseno guidelinesfor estimatingthis parameterare lated winter peak flow rates.23 presently u*ilubl", the suggestedmidvalue of 0.5 was selectedfor Big Bordeaux Creek. Severalsoil moistureandrouting parainetersrequireinitial input valuesthat are improve difficult to estimatefrom availabledata.Calibrationproceduressuccessively parameters the initial estimatesby trial and adjustment.For Big BordeauxCreek,the and initial estimatesare as follows. EPXM-the ma*imum interception rate (in./hr) for a dry watershed. crawford and Linsley2suggesttrial valuesof 0.10, 0.15, and 0.20 fot grasslands,moderateforest covers,and heavyforest covers,respectively. ifr" O.ts-in./hr valuewas selectedfor the moderateforestcoveralongBig BordeauxCreek. Clarke used0.10 in the CaveCreek study' CX-an index of the surfacecapacityto store water as interceptionand storage.This parameternormallyrangesfrom 0.10to 1.65,and l depression the midvalu" of o.go was selectedfor Big Bordeaux creek although a greaternumber might be indicative of the forest cover. Clarke independently.arrivedat a final, similar value (0.90) for CaveCreek' EDF-an index of soil-surfacemoisturestoragecapacity,representingthe additional moisturestoragecapacityavailableduring warmer months due to vegetation.Dependingon the soil type, the index ranggsfrom 0.45 to 2.00. Sandysoilsiimilar to thosein the BBC areareadily give up moisture to vegetation,resultingin increasedstoragecapacity.lnitia,l valuesof 1.10 and 1.25 were independentlyselectedfor the Big Bordeaux and cave Creek areas. LZSN-a soil-profile moisturestorageindex (in.) approximatelyequalto the volume of water storedabovethe water table and below the ground surface.This parameteris a major runoff-volume parameter,inversely related to the basin yields, interflow, and groundwaterflow. The LZSN index,dependingon porosity andthe specificyield of the soil, rangesfrom 2.0 to 20.0,and 4 in. plus half the meanannualrainfall can be usedas an initial estimatein areasexperiencingseasonalrainfall' By the use of a lg3I-Ig52 averageof 15.55 in. of annual precipitation, the Chadron, : 11.78in. for Big Nebraskl, precipitation station givesan initial LZSN BordeauxCreek. A similar analysiswas usedfor CaveCreek' K3-a soil evaporationparameterthat controlsthe rateof evapotranspiration lossesfrom the lower soil zone.The parameterrangesfrom 0.2 to 0.9 dependingon the type and extentof the vegetativecover.As an initial estimatefor the light forestcoYerin BBC, 0.28 was selected,which agreed with the estimatis suggestedby Crawford and Linsley.2 Also, K3 is and _ approximatelyequal to ihe fraction of the basin covered by forest
582
CHAPTER23
CONTINUOUSSIMULATIONMODELS
grassd€ep-rooted vegetation.Recommendations2for barren ground, lands,andheavyforestsare'respectively,0'20'0'23'and0'30;0'25was oPtimal for the CaveCreek studY' K24L - a Parametercontrollinl from active groundwater stori drainagebasinboundarY'It also groundwaterthat Percolatesto K}4LPatameter canbe estimatr small waterlevels,or it is often assumedto be zerobecausethesel0ssesare runoff' and comparedto the magnitudesof rainfall K24FjL-'the friLctionof the total watershedover which evapotranspirarate' tion from groundwaterstorageis assumedto occul at the potential of vegetation quantity significant a unless zero d This pararneteris assume water, drawswater directly from the watertable.Plantsthat seekphreatic phreatophytes' called are alfalfa, or suchas cedar or cottonwood EF-afactorrangingfrom0.lto4.0thatrelatesinfiltrationratesto evaporationrates.Thisparametersimplyallowsamolerapidinfiltration during raim"t seasons.A normal starting value of 1'0 was rate recovery "nig Boideaux Creek, whereasthe CaveCreek value was set selectedfor much lower. the soil CB-an index that controls the rate of infiltration, dependingon permeabilityandthevolumeofmoisturethatcanbestoredinthesoil.A Big midvalue oi O.ZS(0.3-1,.2) was selectedfor the sandy soils around BordeauxCreek;asmallervalueof0.65wasoptimalfortheCaveCreek study. moisture cY-an index controlling the time distributionand quantitiesof moderately a enteringinterflow' This index rangesfrom 0'55 \" 4':' and waterhigh valueof 3.0 was selectedfor Big BordeauxCreekbecausethe moderately a shedcontainsmany pine needfemats. clarke also selected high CY for CaveCreek. KY24-adailybaseflowrecessionadjustmentfaclorusedtoproducea for Big simulated"oruilin"u, baseflow recession.An initial value of 1'0 for the recession flow base the Bordeaux creek was selectedbecause hydrographinFig.23.16islinear.Lateradjustmentsmightberequiredin matchingsimulatedand recordedbaseflow recessions' SGW-agroundwaterStorageincrement(in.),reflectingthefluctuations for Big in storaqelolume. Usually, an initial estimate(0.10 was selected trials; simulation several after adjusted and Bordeaui creek) is made parameters four following the and Thib in. 3.90 SGW rangesfrom 0.10 to were usedonly in the BBC studY' UZS-thecurrentvolume(in.)ofsoilsurfacemoistureasinterception anddepressionstorage'Becausethesimulationbeginsonoctoberlofthe as zero first calibration yeai the parametermay initially be designated unlessprecipitation occutt during the last few daysof September'
23.2 CONTINUOUSSIMULATIONMODEL STUDIES
.
583
LZS-the current volume (in.) of soil moisturestoragebetweenthe land surfaceand the watertable.Sixty percentof LZSN, or 7.0 in., wasselected to initiate the Big BordeauxCreek iimulation. GWS-the current groundwaterslopeindex (in.). This index providesan indication of antecedentmoistureconditions.Suggestedinitial valuesfall between0.15 and 0.25, or the value assignedto SGW can be used.A midrange0.20 in. was selectedfor Big BordeauxCreek. VOLUME-the volume (acre-ft) assignedto swamp storageand dry groundrecharge,accountingfor the runoffrequired to rechargeswampsin late summer.Since no swampswere visible on the USGS 7.5-min topographicmaps,an initial valueof 0.0 acre-ft was selectedfor Big Bordeaux Creek.
HydrologicData In addition to the parametervaluesof Table23.8, the Stanfordmodelrequiresa large volume of hydrologic data for each water year of the simulation. The following description of requried input data for each water year illustrates the data that were compiledand reducedto necessaryinput form for 4 wateryearsbeginningon October 1, 1968andendingon September30,1972.Only the input for BBC is described. The input includesthe following data for eachwater year: 1. The new year identificationentry containsthe water year and the recorded annual streamflow.The valuesof the annual streamflowfor Big Bordeaux Creek are listed in Table 23.9. 2. A descriptionofthe streamgaugingsite. 3. The title to be appliedto the ordinatefor graphicalplotsof the simulatedand recordedrunoff hydrograph;namely,the daily averageflow rate (cfs). 4. Pan-evaporationdata, read as 365 or 366 single entries containing daily evaporationamounts(in.). 5. The monthly evaporationpan coefficients,comprising the data listed in Table 23.8, beginningwith October. 6. Recordeddaily streamflows(averageflow for the day,cfs)readas 365or 366 entriesfor October I through September30. 7. Hour$ rainfall data, read for each water year. Two entries per day, each containingan identificationofthe gaugeanddate,areusedto providehourly TABLE23,9 ANNUAL BIGBORDEAUX CREEK STREAMFLOWS Water years
1968*1969 1969-1970 r970*1971 r97t-1972
Recorded annual streamflow (acre-ft)
434.0 465.4 296.4
584
23 CONTINUOUS SIMUI.ATIoN MoDELS cHAPTER depthsin inchesbefore noon on the first and after noon on the second.Valuesare requiredonly for half-daysexperiencingprecipitation.Becauseof the variablenumber of possiblerainfall values,a sentinelentry with the year setequalto 2001 is placed at the end of the data, indicatingthat all the precipitation data for the wateryear has beenread.
Outputfrom the KentuckyVersion Dependingon which optimal input, output, and branchingparametersare selected,a variety of output data are availablefrom the Kentucky version, including plotted graphs of measuredand synthesizeddaily streamflow rates. Options include the following: .
1. A table of synthesizedaveragedaily streamflowrates (cfs). 2. A table of monthly and annual totals of synthesizeddaily flow rates. 3. Synthesizedmonthly and annualtotals ofrunoffin equivalentinchesover the watershed. 4. Svnthesizedmonthlv and annual interflow amounts (in.) over the watershed. 5. Svnthesizedmonthlv and annual baseflow amounts(in.) over the water-
s[eo. 6. The volume of synthesizedstreamflowrunoff from the watershedfor the entire water year (acre-ft). 7. A summationof all the recordeddaily streamflowrates(cfs)for eachmonth and year. 8. The recordedannual total of runoff (in.) over the watershed. 9. The recordedvolume of runoff from Novemberthrough March (in.) over the watershed. 10. The amountof synthesizedsnowfrom Novemberthrough March (in.) over the watershed. 11. The volume of the recordedannual streamflow(acre-ft). 12. The sum of the recorded precipitation for each month and for the year. 13. The synthesizedmonthly and annual totals of evapotranspiration(in.). 14. The monthly and annualrecordedlake evaporationamounts(in.). 15. End-of-the-monthlevelsof UZS, the current surfacemoisturestorage(in.). 16. End-of-the-month levels of LZS, the current soil moisture storage(in.). 17. End-of-the-month values of SGW the current groundwater storage fluctuation(in.). 18. An annual moisture balance (in.), which representsthe moisture not accountedfor within the program.This is illustratedin the CaveCreekoutput.
GaveCreekModelCalibration Synthetic and actual flow rates at the Cave Creek gauging station are shown for a singleday in Fig. 23.20. Other typical output for portions of one water year of the simulationis presentedin Tables23.10and23.ll. The former providesan hour-byhour listing of all flow ratesin excessof the specifiedvalueof MINH, Table23.8. Note
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for and recordedhydrographs Figurp 23.20 Comparisonof synthesized CaveCreek.(After C1arke.t6.1 that the streamflowwas zero from Octoberthrough Decemberand exceededMINH only during two daysin January. Table23.ll containsmostof the informationdescribedin the 18 items of outpuf for the Kentuckyversion.The daily flows are followedby the syntheticand recorded monthly totals, monthly interflow and base flow amounts, monthly precipitation totals, monthly actual and potential ET amounts,and end-of-month storagesin'the soil profile and groundwaterzones in inches. Of particular interest is the annual summaryin the lower right. Of the 37.5 in. of precipitation,23.8 in. went to ET, 11.6in. ran off or was discharged from storage,and the remaining2.1 in. recharged the soil profile. Moisture not accountedfor during the year was 0.0844 in.
Sensitivityof ModelResponseto ParameterChanges One interestingand useful aspectof simulation is the easewith which changesin watershedparameterscan be evaluated.Clarke testedthe sensitivity of KWM by varying severalof the parametersin Table23.8 over reasonablerangeswhile holding all other parametersconstant.The resultsof his analysisfor CaveCreek on a typical day in March are summarizedin Table 23.12. Theseobservationswere taken from graphssuchasFigs.23.2l and23.22,which illustratethe sensitivityof flood magnitude and timing to changesin Z and KSC. These results and the summary in Table 23.12 are applicableonly to the CC watershedand should not be viewed as generally applicable.
r Summary If actual or synthesizedprecipitation recordsare available,one of the most effective meansof analyzinghistoricalflowsor evaluatingfuture possibleflowsunderchanging land use patterns is through any of the continuousstreamflow simulation models describedin this chapter.Hydrologicproblemsand applicationsthat can be analyzed
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23.2 CONTINUOUSSIMUI-ATIONMODEL STUDTES
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Figwe 23.22 Sensitivityof modelresponse to thechannelroutingparameter. (After Clarke.16) usingcontinuousmodelingincludewatershedyield studies,reservoirdesignand operation studies,sedimentyield estimatingfor reservoirdesignor analysisof impactsof erosion controls on water quality, water supply studiesfor municipal, industrial or agricultural demands,litigation over impactsof wellfieldsor direct diversions,determination of hydropower production potential, evaluationsof flows that will pass through critical in*stream or riparian habitat reachesof a stream,identification of flows that will be availablefor recreationalusesof a stream,and, amongnumerous other applications,analysisof water quantity and quality impacts of removingdams or rnaking other major upstreamchanges. The modelsdescribedin this chapter,and other similar continuoussimulation codes,are availablefrom the federal or state agenciesthat originatedthe code, or from numerouspublic outlets or vendorswho havebeen authorizedto distributethe software.
592
MODELS SIMULATION CHAPTER23 CONTINUOUS
Problems 23.r. Assumethat a 30-mi2rural watershedin your localereceivesh 3-in. rain in a 10-day
period. Reconstructthe block diagram ofFig. 23.2 and plot approximatepercentages io show,for averageconditions,how the rain would be distributed(a) initially and (b) after 30 days. 23.2. A sloping,concreteparking lot experiencesrain at afate of 3.0 in.i hr for 60 min' The tot is SOOft deepand has a slopeof 0.000-1ftlft. If the water detentionon the lot is zero at the start of the storm,calculatethe completeoverlandflow hydrographfor 1 ft of width using the SWM-IV equations.Use a 5-min routing interval and continue computationsuntil all the detainedwater is discharged. 23.3. Calculatethe SWM-IV overlandflow time-to-equilibrium for the lot of Problem23'2 and compareit with the Kirpich time of concentrationfor the lot. Should thesebe equal? 23.4. Cornpare,by listing traits and capabilitesof each,the SWM-IV with its more sophisticated offspring HSP and HSPF. 23.5. Discussthe primary differencesamong the four versionsof the Stanfordwatershed model describedin this chaPter. 23.6. Verify Eqs. 23.23 and 23.24by starting from Eqs. 13.4 and 13 '33' "typical" 23.7. Discuss the watershedbehavior that is depicted in Fig. 23.7. Is this a watershed? 23.8. Comparethe differencesbetweenthe two U.S. Departmentof Agriculture continuous simuiationmodels,USDAHL and SWRRB,and discussthe applicationsthat would be best suitedto each. 23.9. Review the differencesbetweenwater budget and simulation models discussedin Chapter2l anddeterminewhich of the continuoussimulationmodelsdescribedhere could be usedto perform water budgetcalculations. 23.10. For what applicationsmight the following be best suited? API model USDAHL HSPF PRMS SWRRB 23.1r. For the continuoussimulationmodel selectedby your instructor,describefour different types of problemsthat could be analyzedif you were given the ful1, calibrated model.
REFERENCES "ContinuousHydrographSynthesiswith an 1 . W. T. Sittner, C. E. Schauss,and J. C. Monro, API-Type Hydrologic Model," WaterResourcesRes'5(5), 1007'1022(1969)' "Digital Simulationin Hydrology:StanfordWaterL. N. H. diawford and R. K. Linsley,Jr., shedModel IV," Departmentof Civil Engineering,Stanford University, Tech. Rep. No. J.
3 9 .J u l y 1 9 6 6 . "An Evaluationof RelationshipBetweenStreamflowPatternsand Watershed L. D. James, CharacteristicsThrough the Use of OPSET," ResearchRep. No' 36, Water Resources Institute, University of Kentucky,Lexington, 1970'
Chapter24
Simulation Single-Event Models
r Prologue The purposeof this chaPteris to: . Describehow storm event models are structuredand how they are used to simulatedirect runoff hydrographsfor singlestorms. . Describethe five most widely usedfederal agencysingle-eventmodels(note that popular single-eventurban runoff simulation models are describedin Chapter25). ' Provide a detailedcasestudy using one of the models,HEC- 1' . Introducethe emergingtechnologyof storm surgemodeling-the simulation of hydraulic surgesresulting from wind energy acting on the ocean surface. Many severefloodsare causedby short-duration,high-intensityrainfall events. A single-eventwatershedmodel simulatesrunoff during and shortly following these discreterain events.Usersof single-eventmodelsare normally interestedin the peak flow rate, or the entire direct runoff hydrographif timing or volume of runoff is needed.Single-eventmodelssimulatethe rainfall-runoff processand makeno special effort to account for the rest of the hydrologic cycle. Few, if any, simulate soil moisture,evapotranspiration,interflow, baseflow, or other processesoccurring between discreterainfall events. Modelsdescribedin this chapterare applicableto studiesof watershedsthat are primarily rural in makeup.Urbanizedsubareasare allowed,but for watershedsthat ire principally urbanized, the single-event and continuous models described in Chapter2i aremoreapplicable.Coastalfloodingthat is inducedby surgescreatedby wind action on the ocean surfaceis modeledby a different class of single-event models,describedin Section24'3.
24.1 STORMEVENTSIMULATION Event simulation model structurescloselyimitate the rainfall and runoff processes suchasunit-hydrograph developedin earlierchapters.Lumpedparameterapproaches, methods,are generally incorporated even though some use distributed parameter
SIMULATION595 EVENT 24,1 STORM Preparationfor implementingmost single-eventsimulationstudiesbegins techniques.. with a watershedsubdivisioninto homogeneoussubbasinsas illustrated inFig' 24'1' Computationsproceed from the most remote upstream subbaslnin a downstream direction, In any single-eventmodelfor a typical basin(Fig. 24.I),the runoff hydrographs for eachof subbasinsA, B, . . . , E are computedindependently,and then routed and cornbinedat appropriatepoints (callednodes)to obtain designhydrographsthroughout the basin.The modelieads input parametersfor the storm; then appliesthe storm to the first upstreamsubbasin,Bf computesthe hydrographresultingfrom the storm event; repeatsthe hydrographcomputationfor subbasinA; combinesthe two computed'hydrographsinto a singlehydrograph;routesthe hydrographby conventional iechniquest[rough reachC to the upstreamend of reservoirR, whereit is combined with the compuied hydrographfor subbasinC; and so on through the procedure detailedinFig.24.l. Hydrographcomputationsfor subbasinsare most often determinedusing unitis hydrogiaph proceduresas illustrated in Fig. 24.2. The precipitation hyetograph leaving abstracted, are losses input iniiormly overthe subbasinarea,andptecipitation un "*..r, prelipitation hyetographthat is convoluted (see Chapter 12) with the
Outlet (point of concentration)
1. Subdividebasinto accommodatereservoirsites,damagecenters' diversionpoints, surfaceand subsurfacedivides,gaugingstations' precipitationstations,land uses,soil types,geomorphologicfeatures' 2. Lomputation sequencein eventsimulationmodels: a. Computehydrographfor subbasinB' b. Computehydrographfor subbasinA. c. Add hydrograPhsforA and B. d. Routecomtined hydrographto upstreamend of reservoirR' b. Compute hydrograph for subbasinC. f. Computehydrographfor subbasinD. at R. g. CombinethreehYdrograPhs h. Route combinedhydrographthroughreservoirR' i. Route reservoir outflow hydrograph to outlet' j. Computehydrographfor subbasinE' k. Combinetwo hydrographsat outlet. Figure 24.1 Typical watershed subdivision and computation sequence for event-simulation models.
Precioitation
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MODELS 597 AGENCY SINGLE.EVENT 24.2 FEDERAL
'
prescribedunit hydrographto producea surfacerunoff hydrographfor the subbasin. The abstractedlossesare divided among the loss componentson the basis of prescribedparameters.Subsurfaceflows and waters derived from groundwaterstorage are transformedinto a subsurfacerunoff hydrograph,which when combinedwith the surfacerunoff hydrographforms the total streamflow hydrographat the subbasin outlet. This hydrographcan then be routed downstream,combined with another contributing hydrograph,or simply output if this subbasinis the only, or the final, subbasinbeine considered.
MODELS 24.2 FEDERALAGENCYSINGLE-EVENT depictedin Figs.24.1and24.2 arerecognizedby most The rainfall-runoff processes of the eventsimulationmodelsnamedin Table21.1. Specificcomputationtechniques for losses,unit hydrographs,river routing, reservoirrouting, and baseflow are comparedin Table24.1for five of the major federalagencyrainfall-runoff eventsimulation models.All the modelsallow selectionamongavailabletechniques.Brief descriptions of eachof thesemodelsare followedby an illustrative exampleof an application of the HEC-I model to a single storm occurring over a 250-mf watershednear Lincoln, Nebraska.
U.S.GeologicalSurveyRainfall-RunoffModel The USGSmodel can be usedin evaluatingshort streamflowrecordsand calculating peak flow ratesfor natural drainagebasins.lThe programmonitorsthe daily moisture content of the subbasinsoil and can be used as a continuousstreamflowsimulation model.The model is classifiedas an eventsimulationmodelbecauseits calibrationis based on short-termrecords of rainfall, evaporation,and dischargesduring a few documentedfloods.It hasbeenmodified severaltimes and hasevolvedinto the USGS urban continuoussimulation model, DR3M, describedin detail in Chapter25. Input to the model consistsof initial estimatesof 10 parameters,which are modified by the model through an optimization fitting procedurethat matchessimulated and recordedflow rates. Other input includesdaily rainfall and evaporation, close-interval(5-60 min) rainfall and dischargedata, drainageareas,impervious areas,and baseflow rates for eachflood. Phillip's2infiltration equationis usedto determinea rainfall excesshyetograph, which is translatedto the subbasinoutlet and then routed through a linear reservoir, using the time-.areawatershedrouting techniquedescribedin Chapter 13. The USGSrainfall-runoff model can be usedto simulatestreamflowsfor relatively short periodsfor small basinswith approximatelylinear storage-outflowcharacteristicsin regionswheresnowmeltor frozen groundis not significant.Output from the model includes a table showingpeak discharges,storm runoff volumes, storm rainfall amounts,and an iteration-by-iterationprintout of magnitudesof parameters and residualsin fitting volumesand peak flow rates.
598
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SINGLE-EVENTSIMULATIONMODELS
TABLE 2II"1 HYDROLOGYPROCESSESAND OPTIONSUSED BY SEVERAL AGENCY RAINFALL-RUNOFFEVENTSIMULATIONSMODELS Model code names (sleeTable 21.1)
Modeledcomoonents Infiltration and losses Holtan's equation Horton's equation Green-Ampt Phillip's equation SCS curve number method Exponentiallossrate Standardcapacitycurves Unit hydrograph Input Clark's Snyder's Two-parametergamma response SCS dimensionlessunit hydrograph River routing Kinematic wave Full dynamic wave Muskingum Muskingum-Cunge Modified Puls Normal depth Variable storagecoefficient Att-kin method Translation only Reservoirrouting Storage-indication(Puls) Baseflow Input Constantvalue , Recessionequation Snowmeltroutine
HEC-1 (Corps)
TR-20
(scs)
USGS (USGS)
HYMO (ARS)
SWMM (EPA)
x X
x
X X X X X
X X
X X X Yes
No
No
Yes
ComputerProgramfor ProjectFormulation Hydrology(TR-20) A particular$ powerful hydrologicprocessand water surfaceprofile computerprogram was developedby CEIR, Inc.3and is known by the codename TR-20, which is an acronymfor the U.S. Soil ConservationServiceTechnicalReleaseNuntber20. The model is a computerprogram of methodsusedby the Soil ConservationServiceas presentedin lhe National Engineering Handbook.a The program 'is recognized as an engineer-orientedrather than computeroriented package,having been developedwith easeof use as a purpose.Input data sheetsand output data are designedfor easein use and interpretationby field engi-
24.2 FEDERALAGENCY SINGLE-EVENTMODELS
599
neers,and the programcontainsa liberal numberof operationsthat are user-accommodating,even at the expenseof machinetime. The TR-20 was designedto use soil and land-use information to determine runoff hydrographsfor known stormsand to perform reservoirand channelrouting of It is a single-eventmodel,with no provisionfor additional hyOrograpns. the generateO lossesor inflltration beiweendiscretestorm events.The programhasbeenusedin all 50 statesby engineersfor flood insuranceand flood hazardstudies,for the designof reservoirand channelprojects, and for urban and rural watershedplanning. Surfacerunoff is computedfrom an historicalor syntheticstormusingthe SCS curvenumberapproachdescribedin Chapter4 to abstractlosses'The standarddimensionlesshydrogiaphshownin Fig. 12.13is usedto developunit hydrographsfor each subareain the watershed.The ixcess rainfall hyetographis constructedusing the effectiverain and a given rainfall distributionand is then appliedincrementallyto the unit hydrographto obtain the subarearunoff hydrographfor th9 storm' As shown in Table 24.I, TR-20 usesthe storage-indicationmethod to route hydrographsthroughreservoirs(seeSection13.2).The baseflow is addedto the direct .onoffhyatographs at any time to producethe total flow rates.The programusesthe logic depictel in fig. Z+.iby computingthe total flow hydrographs,routing the flows thiough streamchannelsandreservoirs,combiningthe routedhydrographswith those from6ther tributaries,and routing the combinedhydrographsto the watershedoutlet' Prior to 1983, the model routed stream inflow hydrographsby the convex method (Sec(Section13.1),which has sincebeenreplacedby the modifiedatt-kinmethod tion 13.3).Asmanyas200channelreachesandggreservoirsorfloodwater-retarding structurescanbe accommodatedin any singleapplicationof the model.To add to this the capability,the programallowsthe concurrgntinput of up to 9 different stormsover watershedarea. Subdivisionof the watershedis facilitatedby determiningthe locationsof control points. Control points are defined as stream locations correspondingto crossor sectionaldata, reservoirsites,damagecenters,diversionpoints, gaugingstations, requiredata Subarea desired. be may tributary confluenceswhere hydrographdata ments include the drainagearea,the time of concentration,the reachlengths,strucreach ture data as describedin-Section 21.1, andeither routing coefflcientsfor each proare dala cross-sectional Whenever or cfoss-sectionaldata along the channels. peak flow the to addition in elevations vided, the model calculatesihe water surface by the dictated are sizes Subarea rates and time of occurrence at each section. it is information, hazatd flood and routing locations of control points. To provide of the characteristics hydraulic the so that necessaryto defineenoughcontiol points instreamaie definedbetwJencontrof sections.Applications with TR-20 normally The apart' more or 2 mi to feet hundred few corporatecont*rolpoints spacedbetween-a mi2' ."rolting subarLasthat contributerunoff to a control point are usually lessthan 5 is no there though even mi2 25 than less Common subareasizesfor structuresare program' the within limitation on reachlength or subareasize Minimal input daia requirementsto TR-20 includethe watershedcharacteristics, at leastone actual or syntheiicstorm including the depth, duration, and distribution; routing the discharge,capaciiy, and elevation data for each structure; and the
600
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SINGLE-EVENTSIMULATIONMODELS
coefficientsor cross-sectionaldata for eachreach.Input can be describedaccording to the following outline: 1. Watershedcharacteristics. a. The area(in mi2)contributingrunoff to eachreservoirand crosssection. b. Runoff curve number CN for eachsubarea.(SeeChapter4.) c. The antecedentmoisturecondition associatedwith eachsubarea,coded as dry, normal, or wet. d. The time of concentrationfor eacfi subarea(hr). e. The length of eachchannelrouting reach and subareamainstream. 2. Velocity-routingcoefficienttable. a. A table containingrouting coefficientsfor a rangeof velocities(ftlsec). This table is containedwithin the program and need only be enteredif the user desiresdifferent velocities. 3. Dimensionlesshydrograph. a. This table is containedwithin the program and need only be enteredif the user desiresa different hydrograph. 4. Actual hydrograph. a. Actual hydrographscan be introduced at any point in the watershed. Hydrographordinates are read as dischargerates (cfs) spacedat equal time incrementsapart, up to a maximum of 300 entries. b. Baseflow rates (cfs)can also be specifiefl' 5. Baseflow. a. In additionto the option of specifyingthe baseflow ratesassociatedwith a hydrographthat was input, the baseflow can be specifiedor modified at any other control Point. 6. Stormdata. a. Stormsare numberedfrom 1 to 9 and are input as cumulativedepthsat equally spacedtime increments. b. As an alternativeto specifyingcumulativedepthsat varioustime increments, a dimensionlessstorm can be input, and up to 9 stormscan be synthesizedby specifyingeachstorm depth and duration' data. 7. Streamcross-sectional a. Up to 200 crosssectionsmay be input for a singlerun. Cross-sectional data consistof up to 20 pairs of valuesof the dischargeversusflow area. b. If cross-sectionaldata are provided,the routing coefficientsare determined from them. In the absenceof suchdata, the user must specify a routing coefficientfor eachreach. 8. Structure.data. a. The rbservoirdataconsistof up to 20 pairsof outflow dischargerates(cfs) versusstorage(acre-ft). b. A maximum of 99 structuresare allowed in a run. The desiredoutput from TR-20 must be specifiedby a set of input file control variables.Hydrographsat eachcontrol point for eachstormcanbe printedby specifying the control point identification in the control cards. Any combination gf the following items can be producedat eachcontrol point:
MODELS 601' AGENCY SINGLE-EVENT 24,2 FEDERAL The peak dischargerate, time of peak, and peak watet'surfaceelevations. The dischargerates in tabular form for the entire hydrograph' Water surfaceelevationsfor t'heentire duration ofJunoff' The volume of direct runoff, determinedfrom the area under the hydrograph. 5. Hydrographordinatesin any specifiedformat. 6. Summary tablescontainingwater balanceinformation'
* . 2. 3. 4.
Basicdata neededby the computerprogramare determinedfrom field surveys' Rainfall frequency data are input from daia in the U.S. Weather Bureau TP-40 report.s Peak-dischargeand area-floodedinformation for presentand future conditions for severalreturn periodsare output by TR-20 in a form suitablefor direct use in an economicevaluationmodel.
ComputerLanguagefor Hydrologic Problem-Oriented Modeling(HYMO) A unique computerlanguagedesignedfor use by hydrologistswho haveno conventional computir programming experiencewas developedby Williams and Hann.6 Once the progru- has been compiled,the user forms a sequenceoJ commandsthat synthesiz{ route, stoie, plot, or add hydrographsfor subareasof any watershed. Seventeencommandsare availableto usein any sequenceto transformrainfall data into runoff hydrographsand to route thesehydrographsthrough streamsand.reservoirs. The HYMO model also computesthe sedimentyield irt tons for individual stormson the watershed. Watershedrunoff hydrographsare cornputed by HYMO using unit-hydrograph techniques.Unit hydrographscan either be input or synthesizedaccordingto the dimensionlessunit hydrographshown in Fig. 24.3. Tetms in the equationsare 1.0 .
-
i l - I
q = q' " ( - \ \'p
ttr-'trtrtr-tt I
t6 inflection point
s ls'o
q = qos(to-t)tx
tp
Figure 24.3 Dimensionlessunit hydrographused in HYMO. (After Williams and Hann.6)
602
CHAPTER24
SINGLE-EVENTSIMULATIONMODELS
4 : flow rate (ft3lsec)at time t q, : peak flow rate (ft3lsec) te = time to peak (hr) n : dimensionlessshapeparameter q6 : flow rate at the inflection point (cfs) /o : time at the inflection point (hr) K : recessionconstant(hr) Once K and to and 4oare known, the entire hydrographcan be computedfrom the three segmentequationsshownin Fig. 24.3. The peak flow rate is computedby the equation u ^ - -
BAO
(24.r)
where B : a watershedparameter,related to n as shown inFig.24.4 A : watershedarea (mi2) O: volume of runoff (in.), determined by HYMO from the SCS rainfall-runoff equationdescribedin Chapter4 t The duration of the unit hydrographis equatedwith the selectedtime increment.The runoff Q for the unit hydrographwould of course be 1.0 in. The parametern in Fig. 24.4 is obtainedfrom Fig. 24.5. ParametersK and to for ungaugedwatersheds
500
50
10 0
2
4
6
8
1
0
1
2
n
Figure 24.4 Relation between dimensionless shape parametern and watershedParameterB. (After Williams and Hann.6)
MODELS 24.2 FEDEMLAGENCYSINGLE-EVENT
603
L2
10
8
s
6
4
2
0
0
.
5
1
1
.
5
2
2
.
5
3
K tp
shape' Figure 24.5 Relation betweendimensionless parametern and ratio of recessionconstantto time to peak.(AfterWilliamsandHann.6.1 are determinedfrom regional regressionequationsbasedon 34 watershedslocated in Texas,Oklahoma, Arkansas,Louisiana, Mississippi,and Tennessee,ranging in size from 0.5 to 25 mi2, or
and
sLp-oi.'t(#,)"K : 27.oAo 231
(24.2)
! tp : 4.63Ao+"5rr-o''u(
(24.3)
)t'"
where SLP : the differencein elevation(ft), divided by floodplain distance(mi), betweenthe basin outlet and the most distant point on the divide LfW : the,basinlength/width ratio River routing is accomplishedin HYMO by a revised variable storqge cofficient (VfC) method.TThe continuity equation,I - O : dS/dt, and the storage equation,S : KO, are combinedand discretizedaccordingto the methodsoutlined in Chapter 13. The VSC methodrecognizesthe variability in K as the flow leavesthe confinesofthe streamchannelandinundatesthe floodplainand valley area.Relations betweenK and O are determinedby HYMO from the input cross-sectionaldata, or HYMO will calculatethe relation using Manning's equation if the floodplain and channelroughnesscoefficientsare specified.The bed slopeand reachlength are also part of the required input.
604
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SINGLE-EVENTSIMULATIONMODELS
The Widelyadoptedstorage-indicationmethod(seeChapter 13) is usedto route inflow hydrographsthroughreservoirs.The storage-outflowcurvemustbe determined externallyby the userand is input to the prolram asa table containingpairedstorages and outflows,using storagedefinedas zero wheneverthe outflow is zero. The user-orientedcomrnandsand the datarequirementsfor eachcommandare as follows: 1. Sta;t: the time rainfall beginson the watershed. 2. Storehydrograph:the time incremenfto,be used,the lowestflow rate to be stored, the watershed atea, and the successiveflow rates spacedat the specifledtime increment. 3. Develophydrograph:the desiredtime increment, the watershedatea, the SCSrunoff curve numberCN, the watershedchannellength and maximum differencein elevation,and the cumulativerainfall beginningwith zero and accumulatedat the end of eachtime incrementuntil the end of the storm. 4. Computethe rating curve:cross-sectionalidentificationnumber,numberof points in the crosssection,the maximum elevationof the crosssection,the main channel and left and right floodplain slopes,Manning's n fot each segment,and finally pairs of horizontal and vertical cpordinatesof the points describingthe crosssection. 5. Reachcomputations:the numberof crosssectionsin the routing reach,the time increment to be used in routing, the reach length, and the discharge rates for which the variable storagecoefficientis to be computed' 6. Print hydrograph:the idpntification nurnberof the cross sectionat which hydrographsare to be printed. 7. Plot hydrograph:the identificationnumberof the crosssectionsat which the hydrographsare to be plotted. 8. Add hydrographs:the identification numbers of the hydrographsto be added. 9. Route reservoir:the identificationriumbersof the locationsof .the outflow relation for the reservoir. andinflow hydrographs,andthe discharge-storage L0. Compute travel time: the reach identification number, reach length, and reach slope. 11. Sedimentyield: severalfactors describingsoil erodibility, cropping management,erosioncontrol practices,slopelength, and slopegradient. Output from HYMO includes the synthesizedor user-providedunit hydrographs,the storm runoff hydrographs,the river- bnd reservoir-routedhydrographs, and the sedimentyield for individual storms on each subwatershed.Hydrographs computedby HYMO comparedcloselywith measuredhydrographsfrom the 34 test watersheds.
Storm Water ManagementModel (SWMM) The EnvironmentalProtectionAgency model,SWMM,s is Hstedin Table21.1 in two locationscorrespondingto rainfall-runoff eventsimulationand urban runoff simulation. The model is primarily an urban runoff simulation model and is describedin detail-in Chapter25.
MODELS 605 SINGLE-EVENT AGENCY 24.2 FEDERAL and Lilie mostothers,the SWMM modelhasundergonenumefousmodiflcations a single-event improvementssinceits first releasein 1972.The initial version8was urban storm of modeling in continuous ri'se its allow model, and newer versionse,lo a new snowmelt waterflows and waterquality parameters.The latestreleaseincludes scour and routine, a new storm watei tto.ugt and treatment package'a sediment depositionroutine, and a revisedinfiltration simulation' hydrologic' swMM's hydrographandrouting routinesarehydraulicratherthan of single A distributed parameier approachis used for subcatchmentsconsisiting as routed parking lots, city lots, and so on. Accumulatedrainfall on theseplots is first closed or overlandflow to gutter or storm drain inlets,whereit is then routed asopen Of the five channelflow to the receivingwatefsor to sometype of treatmentfacility' detail greatest event-simulationmodelscoripared in Table 24.l,the SWMM givesthe in simulation,but cannotbe usedin large rural watershedsimulations. step using overland flow depths and flow rates are computedfor each time depth over a Manning's equation along with the continuity equation. The water depth reachesa subcatchmentwill increale without inducing an outflow until the over the subspecifieddetentionrequirement.If and wheneverthe resulting depth rate D., is largei than the specifieddetentionrequirement,Da, an outflow c^atchment, is computedusing a modified Manning's equation
r .49.-
V:-J1-(D,-
and
Q*:
D,)2/3St/2
vw(D, - D)
(24.4)
(24.s)
where V:thevelocity n : Manning's coefflcient S : the ground slope W : the width of the overlandflow Q. : the outflow dischargerate they are After flow depthsand rates from all subcatchmentshave been computed, the total to form guttef combinedalong with the flow from the immediateupstream flow in eachsuccessivegutter. The gutter and PiPeflows are routed to any points of interestin the network wh ordinatesfor each time step in the routir incrementsuntil the runoff from the storm ters of the gutter shape,slope,and length and are availroughnessJoef{cienti ior ttre pipes or channelsmust also be supplied able in most hydraulicstextbooks' includethe Other iniut requiredfor a typical simulationwith the SWMM model following: percent im1. Watershedcharacteristicssuchas the infiltration parameters, pervious area, slope, area, detention storage depth' and Manning's coefficientsfor overlandflow' 2. The rainfall hyetographfor the storm to be simulated'
606
SINGLE-EVENTSIMULATIONMODELS
CHAPTER24
3. The land-use data, averagemarket values of dwellings in subareas,and populationsof subareas. 4. Characteristicsof gutters such as the gutter geornetry,slope, roughness coefficients,maximum allowabledepths,and linkageswith other connecting inlets or gutters. 5 . Street cleaningfrequency. 6. Treatmentdevicesselectedand their sizes. 7. Indexesfor costsof facilities. 8. Boundary conditionsin the receiving waters. 9. Storagefacilities, location, and volume. 1.0. Inlet characteristicssuchas surfaceelevationsand invert elevations. 11. Characteristicsof pipes such as type, geometry,slope,Manning's n, and downstreamand upstreamjunction data.
HEC-I Flood HydrographPackage(HEC-1) The U.S. Army Corpsof EngineersHydrologicEngineeringCenterdevelopeda series of comprehensive computerprogramsascomputationalaidsfor consultants,universities, and federal, state, and local agencies(see Section 21.4). Programsfor flood hydrographcomputations,watersurfaceprofile computations,reservoirsystemanalyses,monthly streamflowsynthesis,and reservoir systemoperation for flood control comprisethe series.The single-eventflood hydrographpackage,HEC-1, is described here.tt
The HEC-1 model consistsof a calling program and six subroutines.Two of these subroutinesdeterminethe optimal unit hydrograph,loss rare, or streamflow routing parametersby matchingrecordedand simulatedhydrographvalues.The other subroutinesperform snowmelt computations, unit-hydrographcomputations,hydrographrouting and combiningcomputations,and hydrographbalancingcomputations. In addition to being capable of simulating the usual rainfall-runoff event processes,HEC-1 will also simulatemultiple floods for multiple basin development plansandperform the economicanalysisof flood damagesby numericallyintegrating areasunder damage-frequencycurvesfor existingand postdevelopmentconditions. HEC- 1 underwentrevisionsin the early 1970sand againin the 1980s.Several features were added (e.g., SCS curve number method, hydraulic routing), and a microcomputerversion was developedin 1984. The 1985 releaseexpandedearlier versionsto include kinematic hydrographrouting, simulation of urban runoff, hydrographanalysisfor flow over a dam or spillway,analysisof downstreamimpactsof dam failures,multistagepumpingplants for interior drainage,and flood control sysfem economics.The 1990versionof HEC- 1, availablefor PCs or Harris minicomputers, incorporatBsyet other improvements.It adds report-quality graphic and table capability, storageand retrieval of data from other programs,and new hydrologic proceduresincluding the popular Green and Ampt infiltration equation(Chapter4) and the Muskingum-Cungeflood routing method (Chapter 13). In addition to the unit-hydrographtechniquesof the earlier versions, the modified HEC-I allows hydrographsynthesesby kinematic-waveoverland runoff techniques,similar to those developedfor use in SWMM. The runoff can either be concentratedat the outlet of the subareaor uniformly addedalong the watercourse leogth through the subarea,distributingthe inflow to the channelor gutter in linearly
24.2 FEDERALAGENCY SINGLE-EVENTMODELS
607
increasingamountsin the downstreamdirection. The 1990 versionallows the use of the Muskingum-Cungerouting method in a land surfacerunoff calculation mode. Precipitation can be directly input, or one of three synthetic storms (refer to Chapter 16) can be selected.A standardproject storm (SPS) is availablefor large basins (over 10 mi2) located east of 105' longitude, using proceduresdescribedin Corpsof Engineersmanuals.A 96-hr durationis synthesized,but the stormhasa 6-hr peak during eachday. A secondtype of storm is the probablemaximum precipitation (PMP), using estimatesfrom National Weather Service hydrometeorologicreports availablefor different locations(Chapter 16 describesthese).A minimum duration is 24ht, and stormsup to 96 hr long may be analyzed.The third method allows synthesisof any duration from 5 min to 10 days.The userneedonly specify the desiredduration and depth,and the programbalancesthe depth aroundthe central portion of the duration usingthe blockedIDF methodof Section16.4. The later versionsof HEC-I include all the precipitation loss, syntheticunit hydrograph,and routing functions developedfor earlier versions.Additional loss methodsincludeboth the SCS curve numbermethodand Holtan's lossrate equation (an exponentialdecayfunction). Becauseof the popularity of SCS techniques,the HEC-1 now includesTR-20 proceduresfor lossesand hydrographsynthesis.The duration ofthe SCS dimensionlessunit hydrographis interpretedby HEC-1 as approximately0.2 times the time to g 0.25 times the time to peak (this convertsto 0.29 times the peak, but not exceedin lag time). For routing through streamsand reservoirs,the newestversion of HEC-1 includesall previousmethods,and additionally performskinematic-wavechannelrouting for severalstandardgeometriccross-sectionshapes. In comparisonto other event-simulationmodels,HEC-1 is relatively compact and still able to executea variety of computationalproceduresin a singlecomputer run. The model is applicableonly to single-stormanalysisbecausethere is no provision for precipitation loss rate recovery during periods of little or no precipitation. After dividing the watershedinto subareasand routing reachesas shown in Fig. 24.6,the precipitation for a subareacan be determinedby one of four methods: (1) nonrecordingand/or recordingprecipitation station data,(2) basinmeanprecipitation, (3) standardproject or probablemaximum hypotheticalprecipitation distributions,or (4) syntheticbalancedstormmethodusingIDF data (Section16.4).Either actual depthsor net rain amountsmay be input, dependingon the user's choice of techniquesfor abstractinglosses.The HEC-1 lossrate function is easily bypassedif the net rain is availablefor direct input. The programlogic for HEC-1 is shownin Fig. 24.6.Hydrologic processessuch as the subarearunoff computation, routing computation, hydrographcombining, subtractingdivertedflow, balancing,comparing,or summarizingare specifiedin the illustratedinFig.24.L input usingthe sequence One lossrate in the HEC-1 modelis an exponentialdecayfunction that depends on the rainfall intensity and the antecedentlosses.The instantaneousloss rate, in in./hr, is
L, = K'Pf
(24.6)
608
CHAPTER24
SINGLE-EVENTSIMULATIONMODELS
READINPUTDATA; REFORMATDATAAND WRITE TO WORKINGFILE
REA,DANDPRINT JOB SPECIFICATION;
READ ANDPRINT DATA
COMPUTE RL]NOFF HYDROGRAPH
1 Figure 24.6 HEC-1 Program OperationsOverview.l
where
L, : P, : E : K' :
the instantaneousloss rate (in./hr) intensity of the rain (in./hr) the exponentof recession(rangeof 0'5-0'9) acoefficient, decreasingwith time as lossesaccumulate K, : KoC-cuMlllo+ AK
where
Q4.7)
Ko : the loss coefficient at the beginning of the storm (when CUML = 0), an averagevalue of 0'6 CUML : the accumulatedloss(in.) from the beginningof the stormto time t C : a coefficient,an averageof 3.0
If AK is zero, the loss rate coefficientK' becomesa parabolicfunction of the accumulatedloss,CUML, and would thus plot asa straight-linefunction of CUML on semilogarithmicgraphpaperif K were plotted on the logarithmic scale.The straightln Fig.24.7, showingthe decreasein the lossrate coefficient fine reLtion t aepiCteO asthe lossesaccumulateduring any storm.Becauselossratestypically decreasemuch more rapidly during the initial minutesof a storm,the loss tate K' is increasedabove the straightJineamountby an amountequal to AK, which in turn is madea function of the amountof lossesth;t will accumulatebefore theK' value is againequalto the
24.2 FEDERALAGENCY SINGLE-EVENTMODELS
0.2 cuMLl
609
A,K=Oif CUML> CUMLy
Ko
0
(in.) loss Accumulated ' CUML Figure 24.7 Variationof the lossrate coefficientK' with the lossamountCUML. accumulated
straight-linevalue,K. This initial accumulatedloss,CUML,, is user-specifieda-nd-rs rela6d to AK in sucha fashionthat the initial lossrateK' is 20 percenttimes CUMLI greaterthan Ko (seeFig. 24.7).Initialloss coefficientsKo are difficult to estimate,and itandard purvesin Chapters4 and 23 are availableto determineinitial infiltration rateS,26. For gaugedbasins,HEC- 1 allows the userto input rainfall and runoff data from which the lossrate parametersare optimizedto give a bestfit to the information provided. Estimatesof parametersfor ungaugedbasinsfall in the judgment realm noted in Table 21.1. An alternativeto the describedlossrate function is availablein HEC-1, which is an initial abstractionfollowed by a constantloss rate, similar to a @index. The HEC-1 model provides separatecomputations of snowmelt in up to 10 elevationzones.The precipitationin any zoneis consideredto be snowif the zone temperatureis lessthan abase temperature,usually 32'F, plus 2".Thp snowmeltis computedby the degree-dayor energybudgetmethodswheneverthe temperatureis "qou1to or greaterthan the basetemperature.The elevationzonesareusually considered in incrementsof 1000 ft althoughany equal incrementscan be used. unit hydrographsfor eaph subareacan be provided by the user, or clark's methodl3 o{ synthesizingan instantaneousunit hydrograph (IUH) can be used. Clark's method is more commonly recognizedas the time-area curve method of hydrographsynthesisdescribedin Section12.6.The time-area histogram,determined from an isochronal map of the watershed,is convolutedwith a unit design-storm hyetographusingEq. 12.38,as illustratedin Fig. t2.l9.The methodsdescribedin Section 1.3.2arc then usedto route the resultinghydrographthrough linear reservoir storageusingEq. 12.35with a watershqdstoragecoefficientK. Input data for Clark's
610
24 SINGLE-EVENT MoDELS cHAPTER SIMULATIoN
method eonsistsof the time-area curve ordinates,the time of concentrationfor the Clark unit graph,and the watershedstoragecoefficientK. If the time-area curve for the watershedunder considerationis not'available,the model-providesa synthetic time-arca curve at,the user's request. Becausethe Corps of EngineerscommonlyusesSnyder'smethod(Chapter 12) in unit-hydrographsynthesisfor large basins,the Snydertime lag from Eq. 11.5 and Snyder'speakingcoefficientCofrom Eq.12.17 can be input, and Clark's parameters will be determinedby HEC-1 from the Snydercoefficients.The actual or synthetic time-area curve is still required. Baseflow is treatedby HEC-1 as an exponentialrecessionusingan exponentof 0.1 in the following equation:
O r :#
(24.8)
where Q, : the flow rate at the beginningof the time increment Q2 : the flow rate at the end of the time increment R : the ratio of the base flow to the base flow 10 time incrementslater The base flow determinedfrom this equation is added to the direct runoff hydrographordinatesdeterminedfrom unit-hydrographtechniques.The startingpoint for the entire computationis the user-prescribedbaseflow rateat the beginningof the simulation, which is normally the flow severaltime incrementsprior to any direct runoff. Ifthe initial baseflow rate is specifiedas zero, the computerprogram output containsonly direct runoff rates. "storageThe HEC-I packageallows the user a choiceof severalhydrologicor routing" techniquesfor routing floodsthroughriver reachesand reservoirs.All usethe continuity equation and some form of the storage-outflow relation; some are described in more detail in Chapter 13. The five routing proceduresincluded in the programare the following: 1. Modified Puls-this methodis also calledthe storageor storage-indication methodand is a level-pool-routingtechniquenormally reservedfor usewith reservoirsor flat streams.The techniquewas describedin detail in Section I3.2. . 2. Muskingum-describedin detail in Section13.1. 3. Muskingum-Cunge-a blendedhydrologicand hydraulic routing method detailedin Section13.1. 4. Kinematicwave-described in Section13.3. 5. Straddle-stagger-also known as the progressiveaveragelag method.The techniqpesimply averagesa subsetof consecutiveinflow rates, and the averagedinflow value is lagged a specifiednumber of time incrementsto form the outflow rate. Input to HEC- 1 is facilitatedby arrangingthree categoriesof datain a sequence compatiblewith the desiredcomputationsequence,summarizedin Table 24.2. The individual input records are precededby a two-charactercode. The first character
MODELS 611 AGENCYSINGLE.EVENT 24.2 FEDERAL OF INPUTDATAFORHEC-111 TABLE24.2 SUBDIVISIONS Jobcontrol I -, Job initialization V-, Variableoutput summary O , Optimization J -" Job type
andhydraulics Hydrology K-, Job step control H-, Hydrographtransformation Q-, Hydrographdata B_, Basin data P _, Precipitation data L-, Loss (infiltration) data U_, Unit graph da-ta M-, MeIt data R_, Routing data S-, Storagedata D_, Diversion data W-, Pump withdrawal data
andendof job Economics E_, etc., Economics, data ZZ,End of Job
in Example24.1.
EXAMPLE 24.1 In June 1963 the Oak Creek watershedshown in Fig. 24.8 experienceda severe
over Subarea1. For the remainingelongatedwatershedareaA-H within the boldface border,usethe Junestormto simulatehydrographsat eachof Points1 8 usinga single 3 and 8' Points at run of HEC-I and comparepeak flows with recordedvalues
SCS runoff equationaand a basin-widecompositecurve number of 73 (see Chapter4).
612
CHAPTER 24 SINGLE.EVENT SIMULATION MODELS
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Figure 24.8 Oak Creek watershedsubareamap.anddata sheet,June 1963.
5
MODELS 613 SINGLE-EVENT AGENCY 24.2 FEDERAL *
The computation logic to simulate runoff for this storm consistsof the following22 steps: L. Computethe hydrographfor AteaA at Point 1. 2. Routethe A hydrographfrom Point 1 to Point 2. 3. Computethe hydrographfor Area B atPoint 2. 4. Combine the two hydrographsatPoint 2. 5. Route the combinedhydrographto Point 3. 6. Computethe hydrographfor Area C-at Point 3. 7. Combine the two hydrographsat Point 3. 8. Route the combinedhydrographto Point 4. 9. Computethe hydrographfor Atea D at Point 4. 10. Combinethe two hydrographsat Point 4. 1L. Routethe combinedhydrographto Point 5. 12. Computethe hydrographfor AreaE at Point 5. 13. Combinethe two hydrographsat Point 5. 14. Routethe combinedhydrographto Point 6. 1.5. Computethe hydrographfor Atea H at Point 6. 16. Combine the two hydrographsat Point 6' 17. Route the combinedhydrographto Point 7. 18. Computethe hydrographfor Area F at Point 7' 19. Combine the two hydrographsat Point 7. 20. Route the combineAhyJrographto Point 8. 21. Computethe hydrographfor Area G at Point 8' 22. Combinethe two hydrographsat Point 8. Runoff hydrographsfor subareasare simulatedby convoluting the net storm hyetographwith unit hydrographssynthesizedby Clark's method using Snyder'i coiffiiients (seeChapter I2). A Co vabteof 0.8 is applied for Oak Crlek becauseof the moderately high retention capacity of the watershed. Subareatime lag valuesfor eachsubareaare found from Eq. (11.5) usinga C' valueof 2.0. Hydrographstreamrouting is performedusing the Muskingumtechnique (chapter 13) with x : 0.15 andK: the approximatereachtraveltime, using length dividedby the avefagevelocity. A Ch6zy averagevelocity determinedas 100 times the squareroot of the averagereachslopeis used.If K exceedsthree routing increments, the reach is further subdividedby HEC-1 into shorter lengthsto ensurecomputationalresolution. A sampleof the input and output for this job is shownasTable24.3.Each of the 22 computationalstepsare separatedin sequence.Only Steps 1-5 are includedin the sample.Note that the HEC-1 lossrate function was not usedso that the end-of-perlod excessand rain depths are equal. Note also that hydrographrouting of the A hydrographfrom Point 1 to Point 2 was facilitated (AMSKK) of l.2hr. using - three equal reachlengthseachwith a K-value ratesfor eachof steps flow peak time-averaged and 1 HECof summary A peak at Point 3 is 27,539 simulated the that Note given Table 24.4. in is l-22 The correspond27 of val:ue recorded the ,500' well with very agrees cfs, which -ing simulatedand observedpeak flows at Point 8 arc 22,453and 21,600cfs, -' - - - -respectively. Il
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6-hr
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24.3 STORMSURGEMODELING Coastalareasnot only experiencefloodsfrom single-eventstormsbut alsofrom storm surges(short-termchangesin sealevel) normally causedby hurricanes.Severalcomputer models are availableto analyZehurriCane-producedstorm surges.l2Most are of momentumtransferfrom the atmosdeterministic,modelingthe physicalprocesses phereto the ocean.Parametersin thesemodelscan be adjustedto allow analysisof actual or hypothetical storms such as the probable maximum hurricane (PMH) or standardprojecthurricane(SPH). Federal agencies,private consultants,and universitieshave developedsurge models.Severalnonproprietary,open-coastsurgemodelsare listed inTable 24.5. User's manualsare availablefor all the modelslisted. Storm surgemodels simulatethe effects of wind momentum on ocean water and wavehydrodymasses.This involvesprinciplesfrom meteorology,oceanography, geometry, and water-level namics,which-all operatefrom assumptionsabout storm, from surges damage conditions.Tide levelsare includedin mostof the modelsbecause peak level. surge often dependsentirely on concurrencewith the The equationssolvedby the first three models in Table 24.5 are two-dimensionalversionsof Eq. 13.51and the equatiotrof continuity,Eq. 13.40.The fourth model disregardsthe continuity equation and solvesEq. 13.51 through a seriesof
626
MODELS SIMULATION 24 SINGLE.EVENT CHAPTER
TABLE24.5 STORMSURGEMODELS User/agencf
Programname
Data input
Differentialequation solutionmethod
SPLASH
NWS
Atmosphericpressures,radius to maximum wind, storm speed
Finite-difference
SSURGE
coE
Finite-difference
FIA model
FIA
BATHYSTROPHIC
CERC
Atmosphericpressures,radius to maximum wind, storm speed Atmosphericpressures,radius'to maximum wind, storm speed, maximum wind speed,depth of shelf Wind field, pressuredifferences, radius to maximum wind, forward speed
Applicablecoasts Mildly curved, Gulf and East coasts All coasts
Finite-difference
Gulf and Atlantic coasts
Finite-difference
All coasts
,NWS, National WeatherService;COE, U.S. Army Corps of Engineers;FIA, FederalInsuranceAdministration;CERC, Coastal Engineering ResearchCenter, Corps of Engrneers.
The modelsarenot truly dynamicbecausethey treat time asa succession assumptions. of steadystates.Output is a file of water depthsat the end of eachtime stepusedin the simulation. Storms that can be simulatedinclude the SPH, PMH, or any prescribedwind field.
r summary By far the largestnumberof hydrologicmodel applicationsinvolvesthe useof singleeventsimulationmodels,whetherthe generalversionsdescribedin this chapteror the urban runoff models about to be presentedin Chapter 25. Data requirementsfor single-eventmodelsare nominal-far lessthan for continuousmodelingstudies.In the majority of cases,the data required for any subareaare easily obtained from readily availabletopographicand soils maps.Given the basin area,slope,soil types, land use,and location,estimatesof peak flow ratesand shapesof runoff hydrographs ar severallocationsin the watershedcan be obtainedfor given stormswithin a few hours' time usingthesemodels.They continueto be the primary tool usedby practicing engineersin analysisand designof stormwaterhandling facilities.
PROBLEMS 24.1. Six numberedsubareasfor a river basin are as shownin the sketch.Preparea sche-
matic diagramfor a modelstudyusingboxesas subbasinrunoff components'connecting lines as channel routing links, circles as hydrograph combination nodes, and trianglesas reservoirrouting nodes.Then describethe computationsequencefor this basinin the samemannershownin Fig.24.1. (Seesketchon nextpage). 24.2. Synthesizea unit hydrographfor a watershedin your locale using the HYMO model equations.Comparewith correspondingunit hydrographsfrom Snyder'smethodand the SCSmethodin Chapter12.
)
PROBLEMS
627
t-r'\t-;
Sketch for Problem 24.1 24.3. Usethe HYMO modelequationsto synthesizea unit hydrographfor the entire 258 sq. mi Oak Creekwatershedin Fig. 24.8. 24.4. A watershedexperiencesa l2-hr rainstormhaving a uniform intensity of 0.1 in./hr. Using E : 0.7, Ko : 0.6, C : 3.0, and AK : 0.0, calculatethe hourly lossratesl, asdeterminedby the HEC- 1 event-simulationmodel.Determinethe total andpercent lossesfor the storm. 24.5. RepeatProblem 24.4 wing CUML1 : 0.5 in. 24,6. Routethe inflow hydrographin Pr-oblem13.7to the outlet of the 30-mi reachusingthe HEC- 1 straddle-staggermethod by lagging averagedpairs of flows two time increments (12 hr). Comparethe routed and measuredoutflow rates. 24.7, Routethe inflow hydrographin Problem 13.7through the reachby dividing the 30-mi and treat the outflow from eachas inflow to the next in reach into three subreaches line. Lag flows one time increment in eachsubreachand comparethe f,nal outflows with the measuredvalues. 24,8. Study Table 24.3 andFig.24.8, and then definethe following terms from Table24.3: AMSKK,X, TAREA, NP, STORM, TP, CP, TC, R, RAIN, ANdEXCESS' COMP Q. 24.9, Search the HEC-I printout in Table 24.3 to determinevalues (give units) of the following: a. The time incrementusedin the model run. b. Snyder'sCe,Eq. 12.17input for SubareaB. c. The pbak flow rate for the synthesizedSubarea-Aunit hydrograph. d. The total runoff (in.) from SubareaA. e. The peak outflow rate from SubareaA. f. The peak-to-peaktime lag in routing the outflow hydrographfrom Point 1 to Point 2,Fig.24.8. g. The percent attenuationcausedby the reachbetweenPoints,l and 2. h. The Subarea-Bpeak outflow rate if SubareaA is neglected. i. The simulatedSubarea-Bpeak outflow rate.
628
MODELS SIMULATION SINGLE-EVENT *Refer questions: to the HEC-1 output in Tables24.3 and24.4to answerthe following 24.10. "PRECIPITATION PATTERN" actual rainfall depths' or a. Are the valueslabeled
CHAPTER24
Points 1 and 2 of Oak Creek. of Branched 24.11. Describehow, for a given storm,you wciulddeterminethe effectiveness enumerAnswer^by 8. Point at flooding "as24.8 to reduce oak reservoirat Point 9 in Fig. "runs" two of a maximum allowed are You ating your computationlogic illustrated. as you describeyour wlttr ilBc- t, und uny nui-rb", of subareasmay be usedas long subareas. 24.12.AhydrologistwishestomodelawatershedusingthesCScurvenumbermethodfo through a reservoir determinenet rain, and to route the watershedrunoff hydrograph usingthestorage-indicationmethod.BaseflowistobeincorporatedaSaconstant model or modelswould valuethroughoutthe storm duration.Which event-simulation accomPlishthe task? a hydrographusing the 24.13. The following data were preparedfor a Yl card for routing methodof UgC-t' The time incrementis 1'0 hr' straddle-stagger
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4.
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5
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Chapter25
UrbanRunoffSirnulation Models
r Prologue The purposeof this chaPteris to: . Describenine of the most commonly usedcomputerpackagesfor simulation of urban rainfall-runoff processes' . Show that the modelsdo more than simulaterunoff; they allow the user to systems' analyzeand designcompleteurban stormwatermanagement resultsobcomparing by precision . Demonstratethe'models;capabilitiesand tainedwhensimulatingtheSamewatershedwithdifferentmodels. . provide a ,,shopper'sgiid"" to commercialandpublic domainurban stormwater software. of runoff and also urbanization generallyhasthe effectof increasingthe volume to apply the attempts Early runoff. of tendsto result in earlier ani greaterpeak rates successwere design and analysis single-eventmodelsof Chapter24 inurban system total decreasing analogs' rainfall-runoff iutiy u""o*plished by seleciingmore intense area' pervious total the from zones rainfaUabstiactions6y deductingthe impervious channels,andby increasingthe speedoi traveloverthe land surfacesand in improved of approximations wave kinematic adding componentsto the codes to allow for stormwater urban and sewers in storm flow and hydrologicrouting of flow overla=nd retention or detentionPonds. urban watershed Discrepanciesbetweenthesemodels'predictionsand observed and continof single-event class whole responsehaveresulteJin the dwelopmentof a systems' in operating processes uous streamflow models of the oniq.t" -urbanized userto the allow also but process, Thesemodelsnot only simulatethe raintitt-runoff or to facilities management stormwater analyzeanexistingnetwork of interconnected ponds, detention sewers, storm (underground designnew componentsof the system ditches.streetinlet sizesand locations,etc')'
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Island'N9wYork' Figure 25.3 RRL resultsfor WoodoakDrive basin'Long (After (c) hydrographs' the and stormof October19,1966r(a)peaks,(b)volumes, Stall andTerstrieP.e) ofthese surfaces,gutters,pipes,and openchannels'Physicalunderstanding many the of understanding present ii muctrgreaier than the flow phen-omena complexphenomenagov"erningrunofffromruralareassuchasantecedent "infiltratilon, soil moisture movement, transpiration, moisture conditions, evaporation,and so forth. provide a function for 5. A modification of the RRL method that would grasSedareacontributionstorunoffcouldbedevelopedintoavaluable possiblein spiteof the iesign tdol for urban drainage.This is believedto be by the be could flexibility *unly "o*plexities involvei. Further ^offered storage' surface through runoff addiiional irovision for routing surface on an urban basin 6. The input data r"qoir"m"nt' foi o'" of the RRL methods for storm drainage a basin of are reasonablefor the engineeringevaluation than the data or_elaborate complex design.The necessarydaia are no more design' drainage ;;dlly compiled foi a traditional storm
25.1 URBANSTORMWATERSYSTEMMODELS
639
l.)
E zoo o
E
/
$ roo
U a
a
a
a
a
a
a
t
a
a
a
ot<
a
3 1
a
3!
, +
,,,' ,9riir.t,'ol '
100
1a "
1.0
i.'
'J - ;'i^':ls
o
r-'f;3:'r ,"ni
Q
8
200
h
1 z oo" a t 2 a 1l
300
0.5 1.0 Observedrunoff (in.) (b)
Observed peaks (cfs) (a)
1.5
150
o bo
k 100
0
1
2 Time (hr)
3
4
5
(c)
Figure25.4 RRLresultsfor EchoParkAvenue basin,LosAngeles, California, stormofApril 18,1965:(a)peaks,(b) volumes, (c)thehydrogriphs. (AfterStall andTerstriep.e) 7. It appearsthat rainfall occursin greateramountsin the United Statesthan in Great Britain. This may accountfor the fact that the RRL method is successfuland widely used in Great Britain and yet suffers the abovedescribedbreakdownsfor someof the basinsstudiedin the united States. 8. Better urban rainfall and runoff data are required for the proper testingof all mathematicalmodels.Researchbasinsthat do not havehydraulic pioblems,suchasundersizeddrainsor inadequateinlets, shouldbe selectedand instrumented.
lllinois UrbanDrainageArea Simulator,ILLUDAS As mentioned, the RRL method only simulatesrunoff from paved areas of the basin that are directly connectedto the storm drainagesystem.Grassedareasand nonconnectedpavedareasare excludedfrom consideration.The ILLUDAS10model
640
MODELS SIMULATION RUNOFF 25 URBAN CHAPTER but also incorporaiesthe direct$ connectedpavedareatechniqueofthe RRL method paved areas' nolconnected grassed and , ,""oggrir", and incorporatesrunoff from very similar to the Computationof grassedareahydrogriphsfor the subbasins.is usedto approachfbr pavedar-eahydrographs.Figuie 25.5 showsthe samesubbasin grassed contributing the is area shaded illustrate pavedarearunofi inFig. ZS.Z.ihe area,which is largely the front yards ofresidences.Rain falling on any not-directlygrassed connectedpavedarea is assumedto run off instantly onto the surrounding often yards side and back from area,and grassedareahydrologytakes over. Runoff The street' nearest the to laterally then drainsgraluaily to a commonbick lot line and consideration grassed areas such from traveltime requiredfor this virtually eliminates duringre1ative$shortintenseStormSnorma1lyusedfordrainagedesign. can After the contributinggrassedareahasbeenidentified,the curvein Fig' 25'5 of time the to equivalent are grass strips be constructed.Travel times acrossthe equilibrium from Izzatd's equation,2 t":
0.033KLq"o'67
(2s.r)
of4,, that is, which is the time when the overlandflow dischragereaches97 percent ()\ )\ 4" : 0.00002311L (cfs/ft of width) at equilibrium where 4, -i = dischargeof overlandflow : raii supplyrate (in./hr), assumedto be 1'0 in ILLUDAS L : length of overlandflow (ft) and
6:(0.00071 *c)S-o33
(2s.3)
where S : surfaceslope (ftlft) c : coefficienthaving a value of 0.046 for bluegrassturf' illustrated in The time-area curve is assumedto be a straightline. The endpoint,as the traveltime from the farthestpoint on the contributinggrassed Fig. 25.5represents atea,
varied' In ILLUDAS, depressionstorageis normally set at 0.20 in. but can be Infiltration is modeledusing Holtan's equation,lr
f:a(s-F)'o+f"
(2s.4)
,f : infiltration rate at time / (in'/hr) a : avegetativefactor : 1'0 for bluegrassturf S : storige availablein the soil mantle(in.) (storageat the soil porosity minus storageat the wilting point) F ; water alreaJy storedin the soil at time /, in excessof the wilting point(in.)(amountaccumulatedfrominfiltrationpriortotimet) s_F:Storagespaceremaininginthesoilmantleattime'(in') (in./hr), generallyequivalentto the f" : finallonstant infiltration rate saturatedhydraulic conductivity (in./hr) of the tightesthorizon Presentin the soil Profile to computean If physicalpropertiesof the soil are known, the equationcan be_used the various between interrelation general innttiation curve. Figure 25.6 showsthe infiftratisn rates-andstoragefactorsinvolved'
where
25j
URBAN STORMWATERSYSTEMMODELS (c) Rainfall
(a) Subbasln map (contributing grassedarea shaded)
o
(d) Runoff from supplemental paved area
(e) Losses
o
(b) Time versus grassed area curve
92
n
cAoce't
a
(f) Grassed area supply mte
GA:
al
GAz GA 0
1
2
3
4
5
6
7
Travel time to inlet (min)
3 4 s 6 7 8 9 1 0 Time (min) (g) Hydrograph Qr= GAt (GASRT) + GAI(GASRI) Qz= GAz(GASR1) + GA2(GASR)+ GA1(GASR3) Qz= GAz(GASRI) + "'+ GA1(GASR,) Qn=GAn(GASnr)
,? o
u F I
0
2
4
6
8 10121416
Time from start of rainfall (min)
hydrographs.(After Terstriepand Figure 25.5 Elementsin the developmentof grassed-area Stall.lo)
641
642
CHAPTER25
URBAN RUNOFFSIMULATIONMODELS
Infiltrat lon curve ial foinilial infiltration rate
S = Area cross-hatchedbelow curve equals total storage in soil (in.)
!.1 E
H
\%
s-r'
Available smragem soil (in.) Final, constant -; " infiltration rate
o
Time(hr)
Figure 25.6 Diagram of inflltration^relations used in ILLUDAS, Eq. 25.4. (After Terstriepand Stall.'u) _ TabIe 25.4 provides an example computation of an infiltration curve for bluegrass on a silt loam soil in which soil moisture S of 6.95 in. is available. The equation is
r r 55 )
f:r(6.9s-F)t++0.50
Standardinfiltration curveshavebeen devisedfor use in ILLUDAS for soils having SCS hydrologicgroupsA, B, C, and D (Chapter4). Thesecurveswere synthesized FORSILTLOAM CURVE OFINFILTRATION TABLE25,4 COMPUTATION Available srorage,
s-F (in.) 6.95 6.00 5.0 4.0 3.0 2.0 1.0 0
Water stored, A F F (in.) (in.) 0.9s "1.0 1.0 1.0 1.0 1.0 1.0
0 0.95 1.95 2.95 3.95 4.95 5.95 6.95
"Incremental rime, Lt : A'F/f*". Source: Tersftiep and Stall.ro
Time
lnfiltrationrate
(S- D'+
t (in./hr)
15.0 12.3 9.5 7.0 4.65 2.64 1.0 0
15.5 12.8 10.0 7.5 5 .l 5 3.r4 1.50 0.50
'avg
(in./hr) t+.1 I 1.4
8.7 6.3 A 1
2.3 o.'l
Lt' (hr)
t (h0
0.07 0.09 0.11 0.16 0.24 0.43 r.43
0 0.07 0.16 0.27 0.43 0.67 1.10 2.53
SYSTEMMODELS 643 25.1 URBANSTORMWATER
from the Horton equation where /o : ,f" : e: k : / :
f: f"+ (fo- f,)e-k, initial infiltration rate (in'lhr) ultimate infiltration rate baseof natural logarithms a shapefactor selectedas k : 2 time from start of rainfall
(2s.6)
This equationis solvedin ILLUDAS by the Newton-Raphsontechnique.The curves are showninFig.25.7. To accountfor wet versus dry conditions, II-LUDAS divides the antecedent moisturecondition (AMC) into four user-selectedranges'shownin Table 25'5'Each is based on the total 5-day precipitation prior to the storm day. Infiltration from F;q.25.6 is varied, dependingon the AMC value specified' ILLUDAS allows the user to operatein two modes,analysisand design.For designmode,the modelgenerateshydrographsandprovidesnominal stormsewerpipe diameters thatare adequate,without surcharge,to passthe peak flows. In analysis mode,the modelg"n".ul"t hydrographsthroughoutthe basinnodesandlinks andthen alerts the user if iny input pipe diametersare too small' It also sumsthe volume of runoff water backed up ai inlets becauseit could not be accommodatedby the undersizedstorm sewers.
Time(hr) Figure 25.7 Standardinfiltration curvesfor bluegrassturf on four SCS soil types used in ILLUDAS. (After Terstriep and Stall.lo)
644
MODELS 25 URBANRUNOFFSIMULATION CHAPTER CONDITIONS MOISTURE TABLE25.F ANTECEDENT LAWNS FORBLUEGRASS AMC number I
2 J A
Description Bone dry Rather dry Rather wet Saturated
Total rainfallduring 5 days precedingstorm (in.)
0 0-0.s 0 . 5 -I Over1
Source: Terstriep and Stall. lo
The ILLUDAS model requires estimationof severalinput parameters.Other evaluatedthe sensitivityof ILLUDAS to variationsin parameters.The studies12'13,14 study in Ref. 13 concludes: L. The sensitivityof the peak flows to changesin AMC increasesas the soil group changesfrom D to A. 2. The rangesof sensitivityto the soil groupsand AMC are approximatelythe 3. XT;;rr" in the AMC from2to 3 and a changein the soil group from B to C are critical for large designreturn periods, and from C to D for small designreturn periods. 4. The ti'me to peak for various combinationsof soil group and AMC, for different return periods,remainsthe same. 5. The peak flow and runoff volume increaseas the AMC changesfrom 1 to 4. This increaseis particularly important betweenAMC 2 and 3 in general and betweenAMC 3 and 4 for soil group A. The peak flow and the runoff volumeincreaseas the soil group changesfrom A to D for constantAMC. This increaseis particularly important between soil groups B and C in generaland soil groupsC and D for AMC 1. 6. The peak flows decreasemarkedly for time inerementslarger than 5 min, and the pipe diametersdecreasesignificantly for time incrementslarger than 10 min. 7. The time increment should not substantiallyexceedthe paved area inlet time.
Storage,Treatment,Ovefflow Runoff Model (STORM) STORM is the Corps of Engineerscontinuoussimulationmodel of the quantity and quality of urban Stormwater resulting from single eventsor continuousdaily rainfall.ls It also simulatesdry weatherflow from domestic,commercial, or industrial discharges.Wet weather hydrographs,simulated from intermittent or continuous hourly rainfall, can be used for a variety of hydrologicstudy purposes. Wet weatherpollutographs(hydrographsthat alsoprovidewaterquality characteristics) can be predicted for individual historical or synthetic eventsand used in of impactsof runoff on receiving streams.The pollutographsconsistof assessments rates,amountsof pollutants,and pollutant concentrations. runoff hourly
SYSTEMMODELS 645 25.1 URBANSTORMWATER
RainfalVsnowmelt
Treatment
Figure25.8ConceptualviewofurbansystemasusedinSTORM.(AfterU'S.Army CorPsof Engineers'") by the degreeThe model is conceptualizedin Fig. 25.8. Snowmeltis simulated selectionof in the aid to output is inlormation day method (Chapter 14). Statistical desiredcontrol of storm sbrage "upu"iti., und treatment rates required to achieve annualerosion' aver*ut". rurr&f. Statistics,suchas averageannualrunoff, avefage overflowfrom pollutant annual average and ageannualoverflowvolumefrom storige, storage,are all Provided' (to signal The model simulatesthe interactionof precipitation,air temperature erosion, dry weather flow' snowfall), runoff, pollutant accumulation,land Jurface or treatment system'The storage,treatmentiates, and overflowsfrom the storage prograncomputescontinuousorsingle-.event.runofffromrainfall' rainfall and depresnunotf is computedas a fractio-nof the differencebetween in excessof the Runoff use. land on depends ,ion ,torug". The fiaction selected treatment'Runoff speciteOiTeatmentcapacityis divertedinto storagefor subsequent becomesoverflow and is in excessof both the treatment rate and stofagecapacity diverted directly into the receiving waters'
SCSTechnicalReleaseNo. 55 GR'55) The SCS TR-55 Proceduresfor an urbanizedareaswere describedin theseProceduresrecommendsmat Procedures,severalvendorshave1
646
CHAPTER25
URBAN RUNOFFSIMULATIONMODELS
them avai{ablethrough a numberof outlets.A public domainversionis availablefrom the U.S. National TechnicalInformation Service.l6 renditionsof TR-55 should-performinitial studUsersof the vendor-developed ies usinghand-checksto verify the code.An ideal TR-55 program would be one that usesSCS sourcecode or has SCS endorsement,statesall assumptionsused,notifies the user of range violations, and incorporatesoptions for making adjustmentsto accountfor all or most of the following: 1. Changesin the 484 coefficient of Eq. 12.22 for steep, avefage,or flat watersheds. 2. Per centimperviousness. 3. Percentof channelthat is improved. 4. Pondingarea. 5. Subarealength over width ratios that fall outsidethe assumedrange. 6. Slope. 7. Antecedentmoisture conditions' 8. Different storm distributions' 9. Proper lag time equation. 10. Recognitionof the SCS recommendationthat the duration for the derived unit hydrographbe about 13 percentof the subareatime of concentration' 11. Allowance for watershedsthat have initial abstractions,1,, greater than 20 percentof the potential maximum retention, S. Whether by manual or computer operations,the SCS cautions that TR-55 hydrograph methods should not be used to perform final design if an error of ZS peicettt,inpredictedvolumecannotbe tolerated.Their adviceis to useTR-20, after -aking appropriateparameteradjustments,if the urban wathershedis very complex or if a higher degreeof accuracyis required.lTOther precautionsregardingthe useof the graphical and tabular methodsare identified in Section 15.2' SCS Urban Time Relationships The relationshipsamong time parametersin SCShydrologicmethodshavenot beencompletelyreconciledwith observedphenomena or time relationshipsin other modelsof urban rainfall-runoff processes.Several formulations for lag time, with miscellaneousadjustmentsfor urban effects, are mentioned in Section ll.7 and elsewherein SCS literature, and have substantial impact on the shape of the hydrographs.The rational formula (Chapter 15), to illustrate, assumesthat the time of concentration,deflnedas the time for rain falling at the most remote location to reachthe outlet, equalsthe time to peak of the urban hydrograph.lzzwd found this to hold approximatelytrue in observingrunoff hydrographsfrom pavedareas.2 The greatestdiscrepancyfound when comparing SCS and known urban time relationshipsis the prolongedtime basethat resultswhen SCSunit hydrographmethodsin Chapter 12 areapplied.The hydrographshapeis basedon observedrunoff from undeveloped,rural watersheds,As shownin Fig. 12.13,the time basefor the dimensionlessunit hydrographis about 5.0 times the time to peak. It was shownin Chapter I}thatlinear superpositionof unit hydrographsrequiresthat the releasetime must equal the time baseof the IUH, which in turn is the time of concentration,/"' Recall
25j
URBAN STORMWATERSYSTEMMODELS
647
TABLE 25.6 * COMPARISONOF TIME RELATIONSHIPSFOR A D-HR UNIT HYDROGRAPHBY SCS AND URBAN RUNOFFMETHODS Rational/lzzardllUHModels of Urban Runoff Given: D: t" Time to peak: t, Releasetime : tc Solving: D:t" Time to peak = t Releasetime : tc Timebase:D+t,:2t,
UnitHydrograph SCSDimensionless D : O.2 X time to peak Time to peak : Iag time + D/2 Lag time : 0.6 t" (Mockus Equation) Time base = 5.0 X time to peak D : 0.133 t" Time of peak : 0.666 r. Time base: 3.33 t, Releasetime : Time base - D : 3.20 t,
that the excess-rainfallreleasetime, t,, was definedin Chapter 11 as the time from end of excessrain to end of direct runoff. As shownin the Table 25.6, time relationshipsfor the SCS dimensionlessunit hydrographof Fig. 12.13 giveprolongedrunoff durationscomparedto other urban runoff models.Only the time to peak is approximately equivalentin this comparison.Urban runoff modelsbasedon SCSdimdnsionless unit Mrograph proceduresmay result in longer time basesand hydrograph recessionsthan other methods.
USGSDistributedRoutingRainfall-RunoffModel(DR3M) The U.S. GeologicalSurvey simulationmodel for urban rainfall-runoff applications originatedin tgZg as a lumped parametersingle-eventtnodel for small watersheds (deicribed in Chapter 24) and subsequentlywas expandedto distributedparameter status,intendedprimarily for urban applicability.lsAlso, a soil moistureroutine was addedallowing quasicontinuoussimulation. The modelian be appliedto watershedsfrom a few acresto severalsquaremiles in size (an upper limit ol10 mi2 is recommended).It doesnot simulatesubsurfaceor interflow coniributionsto streamflow,and thesemust be externally addedif considered imPortant to the simulation. Routing of rainfall to channelsis by unsteadyoverland flow hydraulics,and routing hydrographsthrough channel reachesis accomplishedby kinematic-wave methods(."t"ito Ctrupter13). The differential routing equationsare solvedby one of three optional numericalmethods.The usermay specify an explicit or implicit finitedifferencealgorithm, or the method of characteristics' Time may be discretizedby the user in as small as l-min increments.The smallesttime increment is used by the program during any dayshaving short-time interval rainfall, calledunit days. Otherdaysare simulatedas24-hr intervals.Movement of surfacewater is simulatedonly during unit days.For the rain-free intervals, daily rainfall is input and usedto modify the soil moisturebalanceleadinginto the nextunit day(s).The formatforrain datais compatiblewith that of the U.S. Geological Survey systlm, WATSTORE(Water Data Storageand Retrieval System).Input data can also be obtainedfrom any local National WeatherServiceoffice. practically any basincanbe studiedby breakingit into severalsetsof four types of model ,"g*"ntr. Theseinclude overland flow segments(must be approximately
648
CHAPTER25
URBANRUNOFFSIMULATIONMODELS
Legend: Overland flow boundary Stream channel
oF1
Overland flow segment I
R-ESIReservoir1 CH1 Channel1
Segment
oF1 oF2 cHl RES oF3 oF4 CHz
Lateral inflow
Upstream inflow
Rahfall.excess Rainfall excess oF1, OFz cH1 Rainfall excess Rainfall excess oF3, OF4
RESl
Watemhed outlet
for DR3M' of watershed Figure25.9 Segmentation rectangular),detentionstoragefacilities, channels,and nodes.This is illustrated in fig.2i.9. Eachsegmentin the flgure may haveinflow from either lateral or upstream sources(or both, as occurs for segmentCHz). Rainfall is uniformly distributedover the overlandflow rectangles'Each has a Laminar flow is assumed given length, slope,roughness,and percentimperviousness. 13'56arefound 13.28and bandmforEqs. of Ioo".or&.rtheiesegrnents.Thevalues fromm: 3 and '
n - : K-
88So u
(t\ 1\
where56is the slope,r: is the kinematic viscosityof water, equalto 0.0000141ftlsec (for 50"i water), and K is a coefficientrelating the Reynold's numberN. to Darcy's frictionfactorfbyK: fN,.FlowoverroughsurfacesislaminarifK)24.Thevalue K is related to rainfall intensity by (25.8) K: Ko+ 101 where K6 is fouhd from Table 25.7, andl is the rainfall intensity (in./hr). Channelsin Fig. 25.9 representeither natural or artificial gutters or storm sewers(either op"n .hunntls oi nonpressurepipes are allowed). Inflow to channels comesfrom otherchannels,overlandflow (aslateral inflow), or nodes.Nodesare used contributeto a channelor reservoi-r,or whenthe user whenmore than three Segments wishesto specify an input or baseflow hydrograph. Channel routing is by kinematic-wavetechniques,describedin Section 13'3' Input is the channei length, slope, and routing parametersb and m. These arc )
25.1 URBAN STORMWATERSYSTEMMODELS
649
TABLE 25.7 ROUGHNESSCOEFFICIENTS FOR OVERLANDAND CHANNELSEGMENTS Sudacetype Concreteasphalt Bare sand Graveledsurface Bare clay-loam soil (eroded) Spatsevegetation Short grassprairie Bluegrasssod
Laminarflow Ko
Turbulentflow Manning'sn
24-r08 30-120 90-400 100-500 1,000-4,000 3,000-10,000 7,000-40,000
0.01-0.013 0.01-0.016 0.012-0.03 0.012-0.033 0.053-0.13 0.10-0.20 0.17-0.48
Source: Alley and Smith.r8
developedfrom the Manning equation and slope of the channel,respectively.The equationsare m : 1.67 and l.4gsto/z
,
D : -
n
(25.e)
where ruis obtainedfrom Table 25.7 or similar information. The model adjustsboth b and m for various shapes,including circular and triangular (seeTable 13.1 for severalru values).If overbankflow is possible,a secondsetof b andmparameterscan be input for all flows in excessof the channelcapacity. Reservoirinflow hydrographsare routed by either of two storagerouting methods. If a linear reservoir model is appropriate,the storagecoefflcient K, relating outflow to storageby O : KS,is input. If the modifiedPuls methodis desired,a table of outflows and correspondingstoragelevels must be input. The model assumesan initial reservoirlevel equal to that correspondingto an outflow of 0.0 cfs. Pondingbehind culverts can be modeledas a reservoir if an S-O relation is input. This should include data points correspondingto roadway overflow to allow simulation of this common phenomenon. Excessrainfall (runoff) from perviousareasis developedfrom the precipitation input, minus severalabstractions.Infiltration is simulatedby
r:r(' .{of)
(2s.10)
where K is the hydraulic conductivity, P is the averagesuction head acrossthe capillary zone, andmo andm are the soil moisturecontentsbeforeand after wetting. The term SMS i$the soil moisture storase.The rate of excessrain is found from
r : and
*
ir1
, : r - ti f 1 > , s
where1 is the rate of rain suppliedto infiltration.
Qs.rr) (2s.1,2)
650
CHAPTER25
URBAN RUNOFFSIMULATIONMODELS
Runoff from impervious areasdependson whetherthe areasare directly connected to the drainagesystem.Those not directly connectedare assumedto flow immediatelyonto perviousareas,wherethey are addedaslateral inflow. One third of the rain on direcdy connectedareasis abstracted,and the rest is lateral inflow to the gutter or channel. Soil moisture is accountedfor in a two-layer hypothetical storagezone. The amount in storageaffects the infiltration rate and allows continuoussoil moisture accountingbetweenrain events.During unit days(dayswith short-durationrain input) all infiltraied moisturefrom Eq. 25.10 is addbdto the upper storagezone. Between unit days, a user-specifiedproportion of the daily rain is added to SMS. During rainlessdays,evapotranspirationoccursfrom SMS, usinginput pan evaporationrates multiplied by a coefficient.ttris processcontinuesuntil the next rhin event,at which time infiltration is governedby the amount of soil moisture; Applications of On:U havebeen documentedacrossthe continentalU.S. and in Alaska and Hawaii.leCalibration of computedand measuredrunoff for almost 400 storms over 37 watershedsreveal a median error in peak flow estimatesand volume of 21 and 24 percent,respectively,with the best results obtainedfor highly imperviouswatershedi.Indicationsfrom verificationstudiesto datearethat the model may overestimatethe peak flow ratesfor simulatedfloodsfrom stormshavingmagnitudesin the designrangeof flood control facilities, and give better resultsfor smaller stormstypically used in runoff quality studies.
FHWAStorm SewerDesignModel (HYDRA) As part of a packageof integrateddesigncomputerprogramscalledHYDRAIN,2othe U.S. Federal Highway Administration developedthe HYDRA storm drain design model for useby federaland other engineers.The model is distributedundercontract with the FHWA through Milrans SoftwareCenterat the Civil EngineeringDepartment of the Universityof Florida at Gainesville.HYDRA hasbeenlinked by commercial vendorsto an integratedCAD/GIS system.The program'sprimary useis analyzing adequacyof existing storm drains or designingnew storm drains and inlets by the rational merhod dJscribedin Chapter 15 or by a modified rational method which representsthe hydrographas a Eapezoidhaving a volume equal to the calculated net rain. Cornmercialversionsof HYDRA allow designby the modifiedrational method, SCS methods,or revised Santa Barbara hydrographmethods.2lHYDRA has one advantageover other storm sewerdesignmodelsin that hydraulic gradelines through the systerncan be checkedby hydraulic backwatercomputationsto determinetotal systemlossesandwhetherinlets,manholes,or junction boxesare surcharged.Another usefulfeatureis'that streetand gutter flowsthat exceedthe inlet capacityofthe storm sewersare routed by HYDRA to the next downstreamlocation and added to the hYdrograPhat that Point'
Storm Water ManagementModel (SWMM) A very widely acceptedand appliedstormrunoff simulationmodel wasjointly developed by Meicalf und Eddy, Inc., the University of Florida, and Water Resources for useby the U.S. EnvironmentalProtectionAgency (EPA). This model Engineers22
25.1 URBAN STORMWATERSYSTEM MODELS
651
is designei to simulatethe runoff of a drainagebasin for any predescribedrainfall pattern. The total watershedis broken into.a finite number of smaller units or subcatchmentsthat can readily be describedby their hydraulic or geometricproperties. A flowchart for the processis shownin Fig. 25.10. The SWMM model hasthe capability of determining,for short-durationstorms of given intensity,the locationsand magnitudesof local floodsaswell asthe quantity and quality of storm water runoff at severallocationsboth in the systemand in the receiving waters. The original SWMM was an event-simulationmodel, and later versionsitkeep track of long-term water budgets. The fine detail in the designon the model allows the simulation of both water quantity and quality aspectsassociatedwith urban runoff and combinedsewersystems.Only the water quantity aspectsare describedhere.Information obtainedfrom SWMM would be usedto designstorm sewersystemsfor storm waterrunoff control, Use of the model is limited to relatively small urban watershedsin regions where seasonaldifferencesin the quality aspectsof water are adequatelydocumented.
Subcatchments
F
Gutterflow 1. Overlandinput 2. Gutterinput 3. Flow (Manning's) 4. Depth(continuity)
Figure 25.10 Flowchart for SWMM Runoff Block hydrographiccompu-tation.({fter Metcalf and Eddy, Inc.22)
652
CHAPTER25
URBAN RUNiOFFSIMULATIONMODELS
fhe simulation is facilitated by five main subroutineblocks. Each block has a specificfunction, and the resultsof eachblock are enteredonlworkingstoragedevices to be usedas part of the input to other blocks. The main calling programof the modelis calledthe ExecutiveBlock. This block is the first and last to be usedand performs all the necessaryinterfacingamongthe other blocks. The Runoff Block usesManning's equationto route the uniform rainfall intensity over the overlandflow surfaces,through the small guttersand pipesof the sewer systeminto the main sewerpipes,and out of the sewerpipesinto the receivingstreams. This block also providestime-dependentpollutional graphs(pollutographs). A third packageof subroutines,the TransportBlock, determinesthe quality and quantity of dry weather flow, calculatesthe systeminfiltration, and calculatesthe water quality of the flows in the system. A usefulpackageof subroutinesfor water quality determinationis containedin the StorageBlock. The StorageBlock allows the user to specify or have the model selectsizesof severaltreatmentprocessesin an optional wastewatertreatmentfacility percentageof the peak flow. If used,this block simulates that receivesa user-selected the changesin the hydrographsand pollutographsof the sewageas the sewagepasses through the selectedsequenceof unit processes. The earlier version22allowed simulationof any reservoirfor which the outflow could be approximatedaseither a weir or orifice, or if the waterwaspumpedfrom the reservoir. The newer,version23allows input of 11 points of any storage-outflow relation and routes hydrographsthrough natural or artiflcial reservoirs,including backwaterareas behind culverts. Routing is by the modified Puls method, which assumesthat the reservoiris small enoughthat the water surfaceis alwayslevel. Evaporationfrom reservoirsis simulatedby a monthly coefficient(suppliedby the user) multiplied by the surfacearea. The Extran Block2acompletesthe hydraulic calculationsfor overlandflows, in channels,and in pipes and culverts.It solvesthe completehydrodynamicequations, surcharging,performsdynarnicrouting, and providesall the depth,velocity, assesses and energygradeline information requested. Subcatchmentareas,slopes,widths, and linkagesmustbe specifiedby the user. Manning's roughnesscoefficientscan be suppliedfor perviousand imperviousparts of eachsubcatchment. As indicated in Table 24.1, SWMM is the only event-simulationmodel listed that usesHorton's equationfor calculatingwatershedinfiltration losses.If parameters for Horton's equationareunavailable,the usercan specifyASCE standardinfiltration capacitycurves.Infiltration amountsthusdeterminedfor eachtime stepare compared with instantanEousamountsof water existingon the subcatchmentsurfaceplus any rainfall that occurredduring the time step,and if the infiltration lossis larger,it is set equalto the amountavailable.Input for Horton's equationconsistsof the maximal and minimal infiltration ratesand the recessionconstantk in Eq. 4.1. The Green-Ampt equationis also usedin SWMM. Urban storm drainagecomponentsare modeledusing,Manning'sequationand the continuity equation.The hydraulic radius of the trapezoidalguttersand circular pipesis calculatedfrom componentdimensionsand flow depths.A pipe surchargesif
MODELS 653 SYSTEM 25j URBAN STORMWATER it is full;provided that the inflow is greaterthan the outflow capacity.In this case,the surchargedamountwill be computedand storedin the Runoff and TransportBlocks at the head end of the pipe. The pipe will remain full until the stored water is completelydrained.Alternatively,the Extran Block canbe usedto conducta dynamic simulation of the systemunder pressure-flowconditions. Necessaryinputs in the model are the surfacearea, width of subcatchment, ground slope,Manning's roughnesscoefficient,infiltration rate, and detentiondepth. Channel descriptionsare the length, Manning's roughnesscoefficient,invert slope, diameter for pipes, or cross-sectionaldimensions.General data requirementsare summarizedin Table25.8. A step-by-stepprocessaccountsfor all inflow, infiltration losses,and flow from upstreamsubcatchmentareas,providing a calculateddischarge hydrographat the drainagebasin outlet. The following descriptionof the simulation processincorporatedin early versionsof SWMM will aid in understandingthe logic of the model.2s to the specifiedhyetograph: 1. Rainfall is addedto the subcatchment'according Dt: D,+ &Lt
(25.1,3)
where D, : the water depth after rainfall D, = the water depth of the subcatchmentat time / R, : the intensity of rainfall in time interval At 2. Infiltration 1, is computed by Horton's exponential function, I, = * f" + (fo f")e-o', and subtractedfrom the water depth existing on the subcatchment (2s.r4) Dz: D, - I, A,t where f",fr, k : coefficientsin Horton's equation(Eq. 4.1) Dt: the intermediate water depth after accounting for infiltration TABLE 25.8 GENERALDATA REQUIREMENTSFOR STORMWATER MANAGEMENTMODEL (SWMM) Item 7. Define the StudyArea. Land use, topography,population distribution, censustract data, aerial photos,and area boundaries. Item 2. Define the System.Plans of the collection systemto definebranching,sizes,and slopes; types and generallocationsof inlet structures. Item3. Define the SystemSpecialties.Flow diversions,regulators,and storagebasins. Item 4. Define the SystemMaintenance.Street sweeping(descriptionand frequency),catchbasin cleaning;ffouble spots (flooding). Item 5. Define the BaseFIow (DWF). Measureddirectly or through seweragefacility operatingdata; hourly variation and weekdayversusweekend;the DWF characteristics(compositedBOD and SS results);industrial flows (locations,averagequantities,and quality). Item 6. Define the Storm Flow. Duly rainfall totals over an extendedperiod (6 months or longer) encompassingthe study events;continuousrainfall hyetographs,continuousrunoff hydrographs,and combinedflow quality measurements(BOD and SS) for the study events;discreteor compositedsamplesas available(describefully when and how taken).
654
CHAPTER25
URBAN RUNOFFSIMULATIONMODELS
3. If the resultingwater depth of subcatchmentD, is larger than the specified detentiondepthDr, an butflow rateis computedusinga modifiedManning's equation. tf Y
and
_
Q-:
- Do),/,s,r,
!Ep, n
vw(D2 - Dd)
(2s.rs) (2s.16)
where V:thevelocity n : Manning's coefficibnt S : the ground slope IV : the width Q. : the outflow rate 4. The continuity equationis solvedto determinewater depthof the subcatchments resulting from rainfall, infiltration, and outflow' Thus
o..
D,.L,:Dz-ftLt
(2s.r7)
whereA is the surfaceareaof the subcatchment. are com5. Steps I-4 arerepeateduntil computationsfor all subcatchments pleted. 6. inflow (OJ to a gutter is computedas a summationof outflow from tributary subcatchments(Q..,) and flow rate of immediate upstream gutters (Qr,)
Q^:2Q-.,t2Qr.,
( 2s .18)
7. Theinflow is addedto raisethe existingwater depth of the gutter according to its geometry.Thus O',
Y.:Y,+7M
Q5.19)
where Yr, Y, : the water depth of the gutter A, : the mean water surfaceareabetweenY, and Y, 8. The outflow is calculatedfor the gutter using Manning's equation: y :
1' ' A l - O'
Pz/t51/z
(2s.20)
VA"
(2s.2r)
fL
and* where
Qr:
R : the hydraulic radius S; : the invert sloPe A" : lhe cross-sectionalarcaaI Y1
9. The continuity equationis solvedto determinethe waterdepth of the gutter resultine from the inflow and outflow. Thus
Yt+^t: Y, t (Qr^- oJf,
(25.22) )
25,1 URBAN STORMWATERSYSTEMMODELS
655
t.0. Steps6-9 are repeateduntil all the gutters are finished. 11.The flows reachingthe point concernedare addedto producea hydrograph coordinatealongthe time axis.
12. Theprocesses from 1 to 11 arerepeatedin succeeding time periodsuntil the completehydrographis computed. Three general types of output are provided by SWMM. If waste treatment processesare simulatedor proposed,the capital,land, and operationand maintenance costs are printed. Plots of water quality constituentsversustime form the secondtype of output. Thesepollutographsare producedfor severallocationsin the systemand in the receiving waters.Quality constituentshandledby SWMM include suspendedsolids,settleablesolids,BOD, nitrogen,phosphorus,and grease.The third type of outputis hydrologic.Hydrographsat any point, for example,the end of a gutter or inlet, are printed for designatedtime periods. The StatisticsBlock will provide frequencyanalysisof storm eventsfrom a continuoussimulation.
Universityof CincinnatiUrbanRunoffModel(UCURM) The University of cincinnati urban runoff model (ucuRM) was developedby the Division of WaterResources,the Departmentof Civil Engineering,of the University of Cincinnati.26 A flowchartis reproducedin Fig. 25.11.The programconsistsof three sections:(l) MAlN-infiltration and depressionstorage,and two subroutines, (2) GUTFL-gutter flow, and (3) PIROU-pipe rouring.It is similar to the EpA model and divides the drainagebasin into subcatchmentswhose flows are routed overland into gutters and sewerpipes. The rainfall is read in as a hyetograph.The infiltration and depressionstorageare summedand subtractedfrom the rainfall to give overlandflow. This is routed through the gutter systemto storm water inlets and the pipe network. Starting at the upstreaminlet, the flows are calculatedin successive segmentsof the sewersystem,including dischargesfrom inlets, to producethe total outflow. The drainage areais divided into small subcatchmentswith closely matched characteristics.The rainfall data are introduced and the infiltration is computedfor eachsubcatchment.Principal elementsof the modelingprocessfollow: 1. It is assumedthat runoff begins wheneverthe rainfall rate equals the infiltration rate and the mass of precipitation balancesinfiltration. The equationsrepresentingtheseconditionsare
t. / : - k r n 1I
I |
Qs.zt1
and
t*.'#0
- e k): mi(r) +
+ + r) - ,rrlr]-1 {;trr ffiliv
(2s.24)
7
MODELS 25 URBANRUNOFFSIMUI-ATION CHAPTER Read Data RainfallintensitY 2. Infiltration constant storage caPacitidr 5. rj"p.o.ion
,/ '1.
Infiltration (Horton'sequation) Depression, storagesupply
t /
/-----ffi
Data 1 Slop 2. Roughness J. Lenl ;th 4. Widrh
|
Overlanil flow 1. Detention 2. Dis:harge
ReadData 1. Layout 2. Length Gutter flow 1. Overlandinput 2. Gutterin..put 3. Dischargo (continuitY) Gutters '@
1. 2. 3. 4.
1. SloPe 2. Roughness 3. Diarieterorheight 4. Length
Flow routlng Gutter input Average velocitY (Manning's) Time offset AddoffsethYdrograPh and new qutter input
Pipes
Print hydrograPh
--16A \ -,/
Figure 25.11 UCURM model flowchart' (ATterPapadakisand Preul'26)
25j
URBANSTORMWATERSYSTEMMODELS
657
* where mi(I) : i(1) : k : "fo: f": DT : t :
the massprecipitateduntil time r (in.) the ordinatesof rainfall intensity curve the decayrate of infiltration (units/min) the initial infiltration capacity(in./hr) the ultimateinfiltration capacity(in./hr) the time incrementof rainfall intensity curve the time to intersectionof rainfall curve and infiltration curve x : arrincreinent of DT
The infiltration curve is computedfrom the equationsand t, I, andx arc stored. 2. Surfaceretentionis relatedto depressionstorageby an equationderivedby Linsleyet al..2s (25.25) s : (t _ f)e-@-F)/sd where 57 : P : F : i : f : s:
the total depressionstorage(in.) the accumulatedrainfall in storage(in.) the accumulatedinfiltratioh (in.) the rainfall intensity (in./hr) the infiltration (in./hr) the surfaceretention (in./hr)
The infiltration and surfaceretentionare subtractedfrom the rainfall intensity to yield the runoff. 3. The hydrographof the overlandflow is derivedby solving
- T+?: where
,/,Dinlt. *ry ou(#)'f''' (2s.26)
(25.27) D" : (0.009'79n'o'6ro6lo'6)fso'3 Dr.z : the detention storageat the beginning and end of time interval I (in./unit area) : rr, rz the overlandflow supplyat the beginningand end of time interval t (cfs/min) n : Manning's coefficient l: the lengthofoverland flow s : the slope (ftlft) 4 : discharge(in./hr per unit area)
4. For the initial time increment, q1 : 0 and Dr : 0 are substituted,D, is calculated,and q, is found from
q: yfnl ,,,,p,,,1r * o.o(S)'l',' L
\D"/ )
es.z8)
where the symbolsare as previously defined.The determinedD" and q, becomeDt and qt. The overlandflow hydrographis derivedby repeating this cycle. I
L
MODELS SIMULATION 25 UEBANRUNOFF CHAPTER
658
5.Thegutterflowiscomputedusingthecontinuityequation
60 *6!7: At 6x
(2s.2e)
q,
where Z is the width of the water surface' Thetelm(ayla|risneglectedbecausethechangeinthedepthofthe gutterflow is very small with respectto time. After integration,the equation becomes (25.30) Q: q"L * Qo where Qs : L : q": Q:
upstreamgutter contributions the length of the gutter (ft) the overlandflow from the hydrograph the flow from the gutter system
METHODS TABLE 25.9 COMPARISONOF URBAN RUNOFFMODELSAND
Model
Surfaoe routing
Pipe flow routing
Quality routing
Degree of sophistication of surface flow routing
Degree of sophistication of pipe flow routing
Accurate modelingof surcharging
Flexibilityof modeling of storm draln components
Peak flows only
No
Low
Low
No
Low
Chicago
Peak flows only Yes
No
No
Moderate
NA
NA
NA
Unit hydrograph
Yes
No
Low
No
Low
STORM
Yes
RRL
Yes
In combination with surface In combination with surface Yes
MIT
Yes
EPA-SWMM
Ratiorlal method
:"* Low
Yes
Low
Low
No
Moderate
No
Low
No
No
High
Lowmoderate NA
NA
NA
Yes
Ye$
Yes
High
Moderate
Yes
High
Cincinnati
Yes
Yes
Yes
High
Low
No
Low
HSPF
Yes
Yes
Yes
Moderate
Moderate
No
Low
Yes
No
Modetate
High
No
Low
(ucuRM)
ILLUDAS Sour ce;,}/ialifred.sftef
I,age13nd Smit!'-8
25.2 URBAN RUNOFFMODELSCOMPARED
659
25.2 URBANRUNOFFMODELSCOMPARED Severalquantitative comparisonsof the RRL, SWMM, UCURM, ILLUDAS, and STORM models have been reported in the literature. A qualitative comparisonof severalwas preparedby Lager and Smiths and is shownas Table 25.9. Table 25.10 provides a bullit matrix showing componentsof most of the same models. Other comparisonsinvolve quantitative analysisof results of the models when applied to actual guagedstorm events.One of the first was an applicationby Heepsand Mein6 of three modelsto two urban catchmentsin Australia for a total of 20 storm events. A similar statisticalcomparisonof the samethree modelsapplied to 12 stormsover eachof three urban watershedswas performedby Marsaleket al.7Significantresults from theseindependentevaluationsare summarizedhere.Another quantitativecomparisonis providedby Huber.27 The Heepsand Mein conclusionsof model performanceare:6
1. The degreeof subdivisionof the catchmenthasa significantinfluenceon the
t
Explicit modeling of insystem Treatmenl modeling storage
peak dischargepredicted by each of the models.The RRL and SWMM methodsgive lower peaks and the UCURM gives higher peaks for finer subdivision. The UCURM containsseveraldeficiencies.The major ones,the effectsof which canbe seenin the predictedhydrographs,arelhat depressionstorages
Receiving model available
Degree of calibration/ verification required
Simulation period
Availability
Documentation
Data requirements
No
NA
No
Usually not verified
Individual storms
Nonproprietary
Good
Low
NA
NA
No
Moderate
Nonproprietary
Fair
Moderate
No
NA
No
High
Individual storms Individual storms
Nonproprietary
Fair
Moderate
No
Yes
No
Low
Long term
Nonproprietary
Good
Moderate
No
NA
No
Moderate
Nonproprietary
Good
Moderate
NA
NA
No
Moderate
Proprietary
Fair
Moderate
Yes
Yes
Yes
Moderate
Nonproprietary
Good
Extensive
No
No
No
Moderate
Nonproprietary
Fair
Extensive
No
No
Yes
High
Individual storms Individual storms Individual or conttnuous storms Individual storms Individual storms or long-term Individual storms
Nonproprietary
Fair
Extensive
Nonproprietary
Good
Extensive
Moderate
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COMPARED 661 MODELS RUNOFF 25.2 URBAN are assignedfull when the rainfall intensity falls below the infiltration capacity,and that depressionstoragesare not depletedby infiltration. The useof instantaneousvaluesof the rainfall intensity(difficult to obtain from recordercharts)can causevolume errors. ' 3. The SWMM was the model with the best overall performancebut at the ' expenseof large computerstorageand time requirements. 4. The RRL model predictedpoorly for stormsin which perviousrunoff was significantbut performedreasonaf;lywell for many other types of storms. The results,in general,supportthoseof Stall and Terstriep'e 5 . A major problem with using noncontinuousmodels is the prediction of antecedeniconditions. This problem is further aggravatedby use of the Horton infiltration equationfor which prediction of the parametersis virtually impossible. The Marsalek study results,Tusingthe samethree modelsfor three watersheds in Illinois, Ontario, and Maryland, indicated that the SWMM model performed slightly better than the RRL model and both thesemodelswere more accuratethan the UCURM model for the small watershedsstudied. TabIe25.11providesdescriptionsof the urban watershedsusedby Marsalekin the runoff modelevaluation.Typical comparisonsof observedand calculatedtimesto peak,peak flows,and runoff volumesfor the three modelsare providedinFig.25.12 for the Calvin Park watershedin Kingston, Ontario. Marsaleket al. usedthe information describedto developthe qualitative comparisonin Table 25.12 and,arrived at the following conclusions:7 1. Uncalibrateddeterministicmodelsfor urbanrunoff, suchasRRL, SWMM, and UCURM, yielded a fairly good agreementbetweenthe simulatedand measuredrunoff eventson typical urban catchmentsof small size. 2. Onthe average,about 60 percentof the simulatedpeak flows were within 20 percentof the measuredvalues.About the samescatterwasfound for the simulatedtimes to peak and runoff volumes.The agreementbetweenthe measuredand simulatedvaluescould be further improvedby modelcalibration. 3. Wherr comparing the entire simulated and measuredhydrographsusing statisticalmeasuies,the agreementwasfound goodfor SWMM, goodto fair in the caseof RRL, and fat in the caseof UCURM' TABLE25.11DESCR|PT|oNoFTESTURBANWATERSHEDSUSEDFoRTHE EVALUATIONOF URBAN RUNOFFMODELS
Testurban drainage basin
Location
Area (acres)
lmoerviousness
Land use
Oakdale
Chicago, IL
13.0
4f .6
Residential
Calvin Park
Kingston, Ontario
89.4
27
Residentialand institutional
Gray Haven
Baltimore, MD
zt,J
52
Residential
PART SIX
STATIST]CAL METHODS
Chapter26
Probabilityand Statistics
r Prologue The purposeof this chapteris to: . Introducethe basictenetsof probability theory as appliedto random,hydrologic variables,with particular emphasison the relativefrequency definition of probability-a conceptthat is foundationalto the frequencyanalysisprocedurespresentedin Chapter27 andthroughoutmany other chaptersof this text. . Describecommonprobability distributionsand showhow they are appliedto hydrologicphenomena. . Relate the fundamentalsof probabitity theory to hydrologic designcriteria describedin Chapter16, Section16.3. . Acquaint the readerwith the theory behind linear regressionand show how this powerful techniqueis usedin hydrologyto predict how a study watershed will respondto somechangeby examiningresponsesof the watershedto past inputs or by statisticallyscrutinizingresponsesof other similar watershedsin order to developa predictive equationfor the subjectwatershed. ' Show how to transform many hydrologicvariablesthat havenonlinear relationships into new variatesthat can then be analyzedby performing linear regressionon the transformedvariates. . Providethe theoreticaland practical foundationnecessaryto fully capitalize on the hydrologicdesignprinciplesdiscussedin Chapter16 andthe time-series analysis*andmodelingproceduresdescribedin Chapter22. commonly, the study of hydrologyis undertakenby readerswho lack the prerequisite backgroundin principles of statistics,probability theory, and frequency most hydrologycoursesreview thesesubjectsearly in the analysis.As a consequence, schedule.Practically all hydrology texts include chapterson ;tatistical methodsto summarizethe basic principlesof statistics,probability theory, probability distributions, bivariate and multiple linear correlation and regression,time-seriesanalysis, and frequencyanalysis.Thus,despitethe placementof this material at the end of this text, the authorsassumethat the readerhasthis backgroundor will studythe material
672
ANDSTATISTICS 26 PROBABILIry CHAPTER in Part Sif, prior to beginninga study of Part Three. Readerswith an understanding of statisticalmethods,regressionanalysis,and the basicsof probability distribution functions may wish to turn directly to Chaptet27.
ANALYSIS 26.1 RANDOMVARIABLESAND STATISTICAL A random variable is one that demonstrates.variabilitythat isn't sufflciently explained by physicalprocesses.Many hydrologicphenomenahavethis tendency,appearing at times to be fully subjectto chancethemselves,or driven by some other closelyrelated factor. In practice,hydrologistsoften analyzeproblemsas systemsof connectedrandom and deterministicprocesses.For example,precipitation is often evaluatedstatisticallyas a randomvariablebecauseof the complexityof understanding and modelingthe atmosphericprocessesthat are known to drive the precipitation system.Runoff that results from the precipitation, on the other hand, is viewed deterministically,usingthe rainfall-runoff analogsthat arethe nucleusof the majority of this textbook. Hydrology relies heavily on principles from probability theory, statistics,and information analysis.Whole texts on frequencyanalysismethods,stochasticgeneration of data, regressionand analysisof variance,and regional analysesare available containingthorough descriptionsof the principles.l'2Many hydrologicprocessesare so complexthat they can be interpretedand explainedonly in a probabilislic sense. Hydrologic eventsappear as uncertaintiesof nature and are the result, it must be assumed,of an underlyingprocesswith randomor stochasticcomponents.The information to investigatetheseprocessesis containedin recordsof hydrologicobservations.Methods of statistical analysisprovidewaysto reduceand summarizeobserved data, to presentinformation in preciseand meaningfulform, to determinethe underlying characteristicsof the observedphenomena,and to makepredictionsconcerning future behavior.Statisticalanalysisdealswith methodsfor drawinginferencesabout the population basedon examinationof samplevaluesfrom the population. These inferencesincludeinformation aboutthe central tendency,range,distributionwithin the range, variability around the central tendency,degreeof uncertainty, and frequencyof occurrenceof values. Statistical analysisinvolves two basic sets of problems, one descriptive, the otherinferentiat. The former is a straightforwardapplicationof statisticalmethods, requiringfew decisionsand representinglittle risk. The inferential problem,however, entails decisionsbearing some risk, and requires an understandingof the methods employedand the dangersinvolvedin predicting and estimating.The most common inferential problem is to describethe whole classof possibleoccurrenceswhen only a portion of themhasbeenobserved.The wholeclassis thepopulation andthe portion observedis the sample. The randomvariablesin the processunderstudyare continuousif they may take on all values in the range of occurrence,including figures differing only by an infinitesimal amount; they are discreteif they are restrictedto specific,incremental values.Distribution of the variablesover the rangeof occurrenceis definedin terms of the frequencyor probabitity with which different valueshave occurredor might occur'"
26.2 coNcEPTSOF PROBABILITY 673
26.2 CONCEPTS OF PROBABILITY The laws ofprobability underlieany study ofthe statisticalnature ofrepeatedobservations or trials. The probability of a singleevent, say Et, is definedas the relative number of occurrencesof the event after a long seriesof trials. Thus P(E ), the probability of eventEr, is nrfN for n, occurrencesof the sameeventin N trials if N is sufficiently large. The number of occurrencesn, is thefrequency. and nr/N the relatiw freQuenffobabilities and the rules governingtheir manipurationare known intuitively or from experience.In the familiar coin-tossingexperiment,P(heads) : P(tails) : |. Eachoutcomeof a singletoss(a trial) has a finite probability, and the sumof the probabilitiesof all possibleoutcomesis 1. Also, the outcomesaremutually exclusive;that is, if one occurs,saya head,then a tail cannotoccur.In two successive tests,there are four possibleoutcomes-HH, TT, HT, TH-each with a probability of|. In this case,becauseeachtrial is independentofthe other one, probabilitiesfor eachoutcomeare foundby P(first trial) X P(secondtrial) : L,x L : j. Again, the sum of the probabilities of the possibleoutcomesis 1. Note that the probability of gettingexactlyoneheadand one tail duringthe experiment(without any regardto the order)is P(HT) + P(TH) : *. Summarizingthe rules of probability indicated by coin tossing,we find the following:
1. The probability of an event is nonnegative and never exceeds 1. 0
( 26.r )
2. The sum of the probabilitiesof all possibleoutcomesin a singletrial is 1.
s
.1J
P(8,) : 1
(26.2)
I
3. The probability of a numberof independentandmutually exclusiaeevents is the sum of the probabilitiesof the separateevents. P ( 4 U E 2 ) : P ( E y )+ P ( E 2 )
(26.3)
The probability statement,P(Et U Et), signifies the probability of the "the probability of Et ot Er." union of two eventsand is read 4. The probability of two independenteventsoccurring simultaneouslyor in successionis the product of the individual probabilities. P(q ) E,) : P(E) x P(E)
(26'4)
P(& a E ) is called probability of the intersectionof two eventsor ioint probability and is read "the probability of El and Er." Considerthe following exampleof eventsthat are not independentor mutually exclusive:An urban drainagecanal reachesflood stageeach summer with relative freqlrencyof 0.10; power failuresin industriesalongthe canal occur with probability
674
CHAPTER26
PROBABILITYAND STATISTICS
of 0.20; experienceshowsthat whenthereis a flood the ch'ancesof a powerfailure for whateverreasonare raisedto 0.40. The probability statementsare P(power failure) : P(P) : 0'20 P ( f l o o d ) :P ( F ) : 0 ' 1 0 P(no powerfailure) : P(P) : 0'80 P(no flood) : P(F) : 0'90 P(power failure given that a flood occurs) : 0'40 The last statementis called a conditionalprobability. It signifiesthejoint occurrence of events and is usually written P(P I F).. Rules 3 and 4 no longer are strictly : 0.3. If the eventsreapplicable.If Rule 3 applied,P(F u P) : P(F) + P(P) maineOindependent,theionditional probability P(P I F) would gqualthe marginal probability f (p). T'nor the eventsare independentif the probability of either is not i'conditionedby" or changedby knowledgethat the other has occurred'For independent events,the joint probabilitieswould be P(F)P):0.1 x0.2:0'02 P(FaF):o.tXo.8:0.08 P(FnP):0.9X0.2:0.18
P(FaF):o.gxo.8:0.72
The probability of a flood or a power failure during the summerwould be the sum of the flrst three joint probabilitiesabove. P(FUP):P(F nP)+ P(F.P)+p(F lP):e'23 The eventsare dependent,however,from the statementof conditionalprobability: When a flood o""u,. with P(F) : 0'1, a powerfailure will occurwith probability : 0'1 x 0'4 : 0'04 : 0".4, and true joint probability is P(F') x P(P I F) : P(F) + P(P) P(F a P). The proUaUitityof the union is then P(F U P) P(F ) P) : 0.1 + 0.2 - 0.04 : 0.26' Note the contrast: P(F U P) : 0'30 for mutually exclusiveevents P(r U P) : 0'28 for joint but independentevents P(F u P) : 0.26 otherwise The new, more generalrule for the union of probabilitiesis s. P(& u E,) : P(81) + P(E,) P(h . E2)
(26.s)
and a sixth rule shouldbe addedfor conditional probabilities: 6_ " ' P ( E , , l E P(Er r ) : -nwE2)
{26.6)
)
26.2 CONCEPTSOF PROBABILITY
675
be exceeded.Becausethe probability of any single, exact value of a continuous "occur" can also mean the level will be reachedor exceeded.In the variableis 0.0, long run, the levelwould be reachedor exceededon the averageoncein 10 years'lhus the averagereturn period* Z in yearsis definedas T :
1 P(Ft:
(26.7)
and the following generalprobability relation.hold: 1. The probability that F will be equalledor exceededin any year
P@):+
(26.8)
2. The probability that F will not be exceededin any year
(26.e)
P(F):l-P(F):l-+
3. The probability that F will not be equalledor exceededin any of n successlveyears / l\n
p,(F)x &(Fl x . . . x P,(F): P(F)': ( t - ;l \ T l
(26.10)
4. The probability R, called risk, that F will be equalledor exceededat least once in n successiveyears
R : 1 - ( r - + l : 1- I P ( F ) F
(26.rr)
Table 26.1showsreturn periods associatedwith variouslevelsof risk. TABLE 26,1 RETURNPERIODSASSOCIATEDWITH VARIOUSDEGREES OF RISKAND EXPECTEDDESIGNLIFE Expecteddesignlife (Years) Risk
e/.) 75 50 40 30 25 20 15 10 5 2 I
10 4.O2 2.00 7.74 3.43 10.3 4.44 14.5 6.12 7.46 " 17.9 22.9 9.47 31.3 12.8 4 8 I. 19.5 98.0 39.5 248 99.5 498 198.4
6.69 t4.9 20.1 28.5 35.3 45.3 62.0 95.4 195.s 496 996
100
15
11.0 22,t 29.9 42.6 52.6 6'7.7 90.8 r42.9 292.9 743 1492
14.9 29.4 39.7 56.5 70.0 90.1 t23.6 190.3 390 990 1992
18.0 36.6 49.5 70.6 87.4 1r2.5 154.3 238 488 t238 2488
35.6 72.6 98.4 140.7 174.3 224.6 308 475 976 2475 : 4975
72.7 144.8 196.3 28r 348 449 616 950 1949 4950 9953
* The terms return period arldrecurrenceinterval are usedinterchangeablyto denotethe reciprocal of-the.annualprobability of exceedence.
676
CHAPTER 26 PRoBABILITANDSTATISTICS
EXAMPLE 26.1 If Zis the recurence interval for a flood with magnitudeQ*findtbe probability (risk) that the peak flow rate will equalor exceedQ" atleastoncein two consecutiveyears. Assumethe eventsare i4dependent. Solution. The solutionis easilyobtainedby substitutioninto Eq. 26.11.To assistin understapdingthe equations,an alternativederivation follows. The four possibleoutcomesfor the tyo years are: a.' nonexceedance in both years b.' exceedancein the first year only c.' exceedancein the secondyear only d: exceedance in both years Becausethesefour representall possibleoutcomes,the probability of the union o f a , b , c , a n d d i s1 . 0 , o r f r o m E q . 2 6 . 2 , P ( a U b U c U d): l.0.Exceedanceinat least one year is satisfiedby b, c, or d, but not a. Thus the risk of at least one exceedanceis P(b U c U d), which is the total less the probability of a. From Eqs.26.2and26.3,we find that ' z - y e a r r i s :k P ( b U c U d ) : 1 P(a) From Eq. 26.3, we find that P(a) : P(Q < Q"inYear I) x P(Q I Q"inYear 2)
:(t-l-)ft-1-) r/ \ r/\ and
rr
R i s k :r - P ( a ) - 1 - ( t - 1 - ) ' \
.r./
EXAMPLE 26.2 What return period must a highway engineeruse in designinga critical underpass drain to accept only a 10 percent risk that flooding will occur in the next 5 years? Solution R : 1 - (' -;)" .10: 1-
('-i)'
Z : 48.1years
IT
26.3 PROBABILITY DISTRIBUTIONS Randomvariables,either discreteor continuous,are characteizedby the distribution of probabilitiesattachedto the specificvaluesthat the variablemay assume.A random variablethroughoutits rangeof occurrenceis generallydesignatedby a capital letter, and a specificvalueor outcomeof the randomprocessis designatedby a small letter.
26.3 PROBABILIry DISTRIBUTIONS
^ b 0.2 5 ^
P(0)= 0.0s P(1= ) s.15 P(2)= 0.2s P(3)= g.2s
677
P(4)= s.15 P(s)= s.1s P(6) = 9.63 P(7) = s.s2
o.l
Number of cloudY daYsPer week, x
Figure26.L Probabilitydistributionof cloudydaysper week' For example, P(X : x,) is the probability that random variableX takeson the value x,. A shoiter version is p(x,). Figur" 26.1 showsthe probability distribution of the number of cloudy daysin a weet. ft is a discretedistributionbecausethe number of daysis exact;in ihe rlcord from which the relative frequencieswere taken, a day had to-be describedas cloudy or not. Observethat eachof the seveneventshas a finite probability and the sum is 1; that is,
) r(-t,) = t Another important property of random variablesis the cumulativedistribution in X is lessthan or equal function, CDF, definedur ttr" ptotubility that any outcome to a stated,limiting value x. The cumulative diitribution function is denotedF(;r). Thus F(x):P(X=x)
(26.12)
andthe function increasesmonotonicallyfrom a lower limit of zeroto an upperbound of unity. Figure 26.2 isthe CDF of the numberof cloudy daysin a week derivedfrom fig. Z6JUy taking cumulativeprobabilities.The function showsthat the probability is gOqothai the numberof cloudy daysin the week will be 5 or less.Conversely,there is a 10 percentprobability that it will be cloudy for 6 or ! days.This complementary cumula-tiveprolability is sometimescalled G(x), where3 Q6'13) r(x) = P(X> x) G(x):1continuous variables present a slightly different picture. Figure 26.3 is the histogramof an 85-yearr""oid of annualstreamflows.The observationswere grouped into nine intervalsranging from 0 to 900 cfs and the number falling in eachinterval plot the was plotted as frequeniy on the left ordinate.A convenientalternativeis to record streamflow the for cDF The ordinate. right by the relative frequencyas shown districontinuous the increase, observations of number is shownin Fig. 26.4.As the broken the limit, In the intervals. the of size the reducing bution will be developedby curvesof Figs. 26.3 and'26.4 wilt appearas thosein Fig' 26'5' There ls a difference between the ordinates of Figs. 26'3 and 26.5a. Since relative frequencyis synonymouswith probability, it is convenientto reconstitutethe
.7' CHAPTER 26
PROBABILITYAND STATISTICS
vl
3 o.a P(x)
II
^ -' .F o.o
o
.l
0 1 2 3 4 5 6 7
0.4
U
0.05 0.15 0.25 0.20 0.15 0.10 0.08 0.02
0.05 0.20 0.45 0.65 0.80 0.90 0.98 1.00
o.2
0 Figure 26.2
1 2 3 4 5 6 Numberof cloudy daysper week,.r
7
Cumulative distribution of cloudy days per week.
RIR
b il
lt
a q
I o '5 d
o.top
0
1
Figure 26.3
2
3 4 5 6 7 Mean annualflow, -r (100 cfs)
Frequency distribution
8
9
of mean annual flows.
26.9 PROBABILITYDISTRIBUTIONS 679
'
Q . o- ' "r Y o q d
e
0.6
o I o >
04
(J
0.0 0
1
2
3 4 5 6 7 Meanannualflow, x (100cfs)
8
9
Figure 26.4 Cumulative frequency distribution of mean annual flows.
r
Figure 26.5, Csntinuousprobabilitydistributions:(a) probabilitydensityfunction and(b) cumulativedistributionfunction.
680
CHAPTER26
PROBABILITYAND STATISTICS
histograrfrso that the area in each interval representsprobability; the total area containedis thus unity. To do this, the ordinatein eachinterval, sayn/N for relative frequencyor probability, is divided by the interval width, Ax.-The ratio nfN Lx rs literally the probability per unit length in the interval and thereforerepresentsthe densityof probability.The probability n/N inthe interval is fepresentedon the average as AF(x), or F(x + Lxlz) - F(x - L'xl2).We CDF (beforethelimitingprocess) then can define l\x)
:
AF(x)
dF(x)
.. --;liln Ax-o Ax
(26.r4)
dx
which is called the probability densityfunction, PDF.3This function is the density (or intensity) of probability at any point; f(x) dx is describedas the differential probability. For continuousvariables,f(r) > 0, sincenegativeprobabilitieshaveno meaning. Also, the function has the property that ()61\\
1
f(x) dx:
which again is the requirement that the probabilities of all outcomessum to 1. Furthermore,the probability that x will fall betweenthe limits a and b is written
p(a-x-b):lu,r*,o*
(26.16)
Note that the probability that x takeson a particular value,saya, is zero;that is, fo
J
ft-l dx : o
(26.17)
which emphasizesthat nnite proUalilities are definedonly as areasunder the PDF betweenflnite limits. The CDF can now be deflnedin terms of the PDF as P(-*
< X < x) : P(X= x) : F(x) :
t'
I f(u) du
(26.18)
J_*
whereu is usedas a dummy variableto avoidconfusionwith the limit of integration. The area under the CDF has no meaning,only the ordinates,or the difference in ordinates.For example,P(*r3 X = x2),which is equivalentto Eq. 26.16, can be evaluated as.F(x) - F(xt). that cannotbe summarizedin integral form, there are For discrete.distributions analogousarithmetic statementscorrespondingto the propertiesgiven in Eqs. 26.15, 26.16, and26.18.In particular, the distributionof sampleddatetaken from a continuousdistributionis a specialcaseof discretedistributionsand canbe givenin the form Thus of arithmeticsummations.s
),f(',):
t
(26.re)
x1= b
P(a-X
).f(r) r 7 -
(26.20)
26.4 MOMENTSOF DISTRIBUTIONS
681
k
P(x
(26.2r)
For a finite numberof observationsin the sample,/(x) is the probability of xr for each outcomein the samplespaceand thereforeP(x,) : P(xr) : P(xt) : . . . : I /N. Hence/(x) can be replacedwith P(x,) in Eqs.26.19,26.20,and26.21. DXAMPLE26.3 Table B.1 containsthe areabeneatha "standardnormal" bell-shapedPDF.Because the distributionis symmetrical,areasare providedonly on one sideof the center.Use the distribution to determinethe valuesof 1 . P ( 0< z ' 2 ) . 2.P(-2=z=2). 3. P(z > 2\. 4. P(z< -1) Solution 1. P ( 0 = z < 2 ) : . 4 7 7 2 . ) F r o m s y m m e t r y ,P ( - 2 - z < 0 ) : Since P ( 03 2 3 2 ) : . 4 7 7 2 . P(-Z = z < 2) : P(-2 3 z 30) + P(0 3 z 3 2), thenP(=2 = 7 < 2) : .4772+ .4772: .9544. 3. This is the areaunderthe curvein the right tail beyondz : 2.0. Becausethe area right of center (z : 0) is .5000, P(z = 2) : P(z > 0) - P(0 = z = 2 ) ,o r P ( z > 2 ) : . 5 0 0 0 - . 4 7 7 2 : . 0 2 2 8 . 4 . F r o m t h es o l u t i o n t(o3 ) ,P ( z = - 1 ) : P ( z = 0 ) - P ( - l = z < 0 ) ' B y s y m m e t r y , P ( - 1= z = 0 ) - P ( 0 3 z s ! ) : . 3 4 1 3 , a n d P (=z - 1 ) : .5000- .3413: .1587. rr
16.4MOMENTS OF DISTRIBUTIONS The propertiesof many random variablescan be definedin terms of the momentsof the distribution. The moments representparametersthat usually have physical or geometripalsignificance.Readersshould recognizethe analogybetweenstatistical momentsand the momentsof'areasstudiedin solid mechanics. The rth moment of a distribution about the oriein is definedaso
tri : p::
x'f(x) dx
2*,rr,r:12,,,
(26.22) (26.23)
The first moment aboutthe origin is the mean,or as it is commonly known, the average.It determinesthe distancefrom the origin to the centroid ofthe distribution frequencyfunction. The prime is normally usedto signify momentstaken about the origin, but the mean is often written as p insteadof y'.
682
CHAPTER26
PROBABILITYAND STATISTICS
Moments can be definedabout axesother than the origin; the axis usedextensively in deflninghigher momentsis lhe mean or, as given above,the first moment about the origin. Thus I
p,: or
|
I
(x - rd'f\xl dx
1<
tr,:
;)-G,
- p)'
(26.24\ (26.25)
Wheneverp', or p', are defined for r =l 1, . , the distribution/(x) is completd defined.It seldomis necessaryto computemore than the first three rnoments; ieveral important distributionsrequire only two. The momentsare usedto specifythe parametersand descriptivecharacteristicsof distributions that follow in the next section.Becausevarious characteristicsof distributionsare describedby combinations of the momentsaboutthe meanand origin, the following relationsare occasionally helpfull'a: Ft: 0 -p' Pz: FL ttt
:
- 3plr, * 2p'
tJ"'z
(26.26) (26.27) (26.28)
CHARACTERISTICS 26.5 DISTRIBUTION ,'
Characteristicsof statisticaldistributionsare describedby the parametersof probability functions, which in turn are expressedin terms of the moments.The principal are centraltendency,the groupingofobservationsorprobability about characteristics a central value;variability, the dispersionof the variate or observations;and skewness,thedegreeof asymmetryof the distribution.The theoreticalfunctions shownin Fig.26.6 exhibit approximatelythe samegrouping about a central value,but/2 has a pronouncedright-skewwhilefi is much greatervariability thanfl, andf possesses symmetrical.
Symbol Gonvention In introducingthe parametersof distributions,the usual sequenceof statisticalproblemswill be followed-that is, parametersare derivedfrom the distributionof sample data and usedas estimatesof the parametersof the population distribution' Summa-
Figure 26.6 Symmetrical and skewed probability distribution for continuousvariables.
26.5 DISTRIBUTIONCHARACTERISTICS
683
tion forms of integralsare usedto computemomentsfor samples.For example,the meanof sampledatais designatedTandit is usedasthe bestestimateof the population mean.By convention,Greek letters are usedto denotepopulation parameters.
CentralTendency The familiar arithmetic average,the mean, is the most used measureof central tendency.It is the first moment about the origin and is designated
r:lf*,
(26.2e)
n i=t
of the population mean p. The statistic 'Meansx is the best estimate mean-for example, the geometric mean the arithmetic other than : ,l> (Ilx,)-are alsoused.Two addi. . . : (x624 meani or harmonic x,)1/" T median,which is the middle value of the the are tional measuresof central tendency into equal areas,and the mode, which in the distribution observeddata and divides and in continuousvariables frequently most discretevariablesis the value occurring in Fig. 26.6. illustrated three are probability A11 density. is the peak value of
Variability Dispersioncan be representedby the total rangeof valuesor by the averagedeviation aboutthe mean;however,the parameterof statisticalimportanceis the meansquared deviation as measuredby the secondmoment about the mean. The parameteris termed the varianceand is designatedby
c r : rn f
G , -p ) '
i=l
(26.30)
But the population mean /-{,is not known precisely and therefore it is necessaryto computeinstead n .
s-:
\
r
Z\xt-xf
(26.3r)
- 1 in place of n in As the best estimateof o2, the quantity s2 is found using n loss of a degreeof freedom the Eq. 26.30.The reasoningfor this substitutioninvolves text. of this scope by using 7 insteadof p, but a proof is beyondthe The squareroot of the varianceis a statisticknown as the standard deviation (o or s), in which form variability is measuredin the sameunits asthe variateand the mean,and henceis easierto interpretand manipulate.The coefficientof variation C", definedas cf p, or sfi, is an expressionuseful in comparingrelative variability.
Skewness A fully symmetricaldistribution would exhibit the property that all odd moments weightto either side equalLero.A skeweddistribution,however,would haveexcessive of the centerand the odd momentswould exist. The third moment a is
o : ! n2 @ , - p ) ' i:l
(26.32)
684
CHAPTER26
PROBABILITYAND STATISTICS
The best-estimateof the third moment is computedby n
u -
(n-r)(n-2)
s
Z-J
\xi-
x)'
(26.33)
The cofficient of skewnessis the ratio afc3 and is estimatedby n -- Q---; (--
(26.34)
J
For syfnmetricaldistributions,the third moment is zero and C" : 0; for right skewness(i.e.,thelongtailtotherightside)C">0,andforleftskewnessC"<0.ThePDF forfi shownin Fig. 26.6has a right or positive skew.The property of skewnessis of questionablestatisticalvaluewhenit mustbe estimatddfrom lessthan 50 sampledata points. EXAMPLE 26.4 Determinethe distributionparametersand comparethe distributionsof annualrainfall for the records shown inTable 26.2.
TABLE 26,2 ANNUAL RAINFALLFOR SELECTEDCITIES Annual rainfall(in.)
Year
t928 r927 t926 t925 r924 1923 1922 t92l 1920 1919 1918 t9t7 1916 1915 t914 t913 t9t2 1 9 1I 1910 1909 1908 1907 1906
Anniston, AL 48 49 )) 98 43 53 56 47 69 57 61 64 99 54 40. 47 58 44 44 64 44 51
LosAngeles, CA
Richinond, VA.
9
43 44 38 31 47 49 52 31 51 40 4l 43
lo
t9 9 8 6 15 20 11 o 18 8 23 t7 23 17 10 18 5
3 t
36 J+
)4
38 36 37 43 34
l9 l5 21
49 47
JJ
26.7 CONTINUOUSPROBABILITYDISTRIBUTIONFUNCTIONS
685
Solution Parameter Mean, ;r Standarddeviation,s Coefficientof variation,C, : s/V Coefficient of skewness,C": a/s3
Anniston 57.2 in. 15.5in. 0.27 1.69
LosAngeles
Richmond
14.9 in. 5.9 in. 0.40 -0.16
41.5in. 6.7in. 0.16 0.16
Comments.(1) Anniston's record shows a high annual averageand a fair$ large variability. In particular,Anniston'sdistributionhas a pronouncedright skew,caused principally by two very large observedvaluesin this short period of record. (2) Los Angeleshasa small annualaveragebut a very largevariability and a slightly negative skewness.(3) Richmondhasthe mostuniform distribution:a relativelysmall variability and only a slight positive skewness. I I
FUNCTIONS DISTRIBUTION 26.6 TYPESOF PROBABILITY
r
Many standardtheoreticalprobability distributionshavebeenusedto describehydrologic processes.It should be emphasizedthat any theoretical distribution is not an exactrepresentationof the natural processbut only a descriptionthat approximates the under$ing phenomenonand has proved usefui in describingthe observeddata. Table26.3summarizes{he commondistributions,giving the PDF,mean,andvariance of the functions. The distributions presentedin the table have experiencedwide applicationand are derivedand discussedin many standardtextbookson statistics.In the material to follow, only aspectsof the most usbd distributionsare given. The usesof binomial and Poissondiscreteprobability distributionsinTabIe26.3 arerestrictedgenerallyto thoserandomeventsin which the outcomecanbe described trials are independentand either as a successor failure. Furthermore,the successive to trial.3'a In a sense,the trial from constant probability remains of success the techniques' or enumerating counting are distributions discrete common The binomial distributionis frequentlyusedto approximateother distributions, and vice versa.For example,with discretevalues,when n is large andp small (such thal np ( 5 preferably),the binomial approachesthe Poissondistribution.This is a single-parameterdistribution (i : np) and is very useful in describringarrivals in queueingtheory. When p approachesI and n grows large, the binomial becomes indistinguishablefrom the normal distribution describedin the next section.
FUNCTIONS PROBABILIryDISTRIBUTION 26.7 CONTINUOUS Most hydrologicvariablesare assumedto be continuousrandom processes,and the as in frequency commoncontinuousdistributionsare usedto fit historical sequences, for continalso important are (Chapter applications 27). Other for example analysis, for computing is the basis distribution uniform The elementary distributions. uous
7 686
CHAPTER 26
PROBABILITYAND STATISTICS
^ ,-", i l {
+
N H
n d
, C)N
g b
.gb
G
\
B l
r
q ^
o d
*
J
co.
l
l
, L t ,
i
I
B -'i.l
I
(UN
> o
a
l
,
'\-..7
d
+
.i l \ F t.x
x -:
Nl'\a
t v
bl tl H
T
b
, tl h
x el
r-
a f f tx o o
s
1
+
d
\,
Ir o
o (!
tr
s \/" x rir
: "
" vr x
'
v
I
o \/t :'
x
\/r "'
8 vr '' R \,r vt
-r
IL
tr l dl
o
fr
o
z
r
;
Vl
.,
/\r
o o
a o z o
E U' 0)
o o lt
o tu J o
s
o E E
(!
I
IL
F
a? (o 6|
UJ @
E€E E ;F = .E
d
E E6 a. f
'.
-E
r
E
E "
= i ' 5E
E 3 E !
E6iE
l
s
F[ E S EZ B i ' oO eF E
l
l
-{.
a
-\? .
8
+
+ -
^r
8 vl ^
z o ut a U)
B l
+ x J CO-
N I N t i - l xrnl -il
f
sr
{ o J o
dlJa , +'
o ,? ,9, k
vl x
"'
'
'
\/l
M
u ;
d
6
I
l\t
^
26.7 CONTINUOUSPROBABILIW DISTRIBUTIONFUNCTIONS
687
randomnumberssoimportant in simulationstudies.The wholebody of materialin the area of reliability and estimatingdependson derived distributionslike Student'sl, chi-squared,and the F distribution. The explanationsthat follow concernthe more The readeris referred common distributionsappliedin fitting hydrologicsequences. to standardtexts for more detailedtreatment.3-6
NormalDistribution frequencyfunction, alsoknown The normal distributionis a symmetrical,be,ll-shaped as the Gaussiandistributionor the natural law of errors. It describesmany processes that are subjectto random and independentvariations.The whole basisfor a large body of statisticsinvolving testing and quality control is the normal distribution. Although it often does not perfectly fit sequencesof hydrologic data, it has wide application, for example, in dealing with transformed data that do follow the normal distributionand in estimatingsamplereliability by virture of the central limit theorem. The normal distribution has two parameters,the mean p, and the standard deviationa, for which 7 and s, derivedfrom sampledata, aresubstituted.By a simple transformation,the distribution can be written as a single-parameterfunction only. Definingz : \x - t-i/o, dx : o dz,the PDF becomes
(26.3s) and the CDF becomes F\z) :
-r*l-
e-"2/2du
(26.36)
The variablez is called the standardnormal variate; it is normally distributed with zero mean and unit standard deviation. Tables of areas under the standard normal curve, as given in Appendix B, TableB.1, serveall normal distributionsafter standardizationof the variables.Given a cumulativeprobability,the deviatez is found in the table of areasand x is found from the inversetransform: x:p+za
or x:7lzs
(26.37)
EXAMPLE 26.5 Assumethat the Richmond,Virginia, annualrainfall in Table 26.2 follows a noimal distribution.Usethe standardnormal transformationto find the rain depththat would havea recurrqnceinterval of 100 years. Solution. From example26.4,the meanis 41.5 in. and the standarddeviation is 6.7 in. This gives x:
4L5 + z(6.7)
Equation 26.18 showsthat the areaunder the PDF to the right of z is the exceedence probabilprobability of the event.For the 100-yrevent,F;q.26.7 givesthe exceedence Table 8.1 in From the figure accompanying tlI00:0.01. ity Pk): llT,:
688
CHAPTER26
PROBABILIry AND STATISTICS
Appendix B, F(z) : 0.5 - p(z) : 0.49, and z : 2.326 by interpolating the table' The expected100-yr rain depth is therefore x : 4t.5 + O.32O X 6.7 :57.1 in: The 100-yr event for a normal distribution is 2.326 standarddeviationsabove the mean. rt
Log-NormalDistribution partly dueto the influence Many hydrologicvariablesexhibit a markedright skewness, other lower limit, and or some zero, greater than values phenomena having of natural frequencieswill cases, In such range. the upper in theoreiically, beingunconstrained, distribunormal a follow may logarithms but their distribution, normal not follow the y : substituting from comes log-normal for the 26.3 in Table PDF shown tion.7The ln x in the normal. With p, andcy as the mean and standarddeviation,respectively, the following relations have been found to hold betweenthe characteristicsof the untransformid variatex and the transformedvariatey:r'7
p:exp(p'y+412) oz:p,2lexp(d)_11 a : lexp(3fi) - 3 exp(fi) + 2lC3 C,:lexp(dr) - r1t'' C,:3C" * Cl
(26.38) (26.3e) (26.40) (26.4r) (26.42)
Also p, : lfl M, whereM is the median value and the geometricmean of the x's. The log-normal is especiallyusefulbecausethe,transformationopensthe extensive body oI theoreticaland applied usesof the normal distribution. Since both the normal and log-normal are two-parameterdistributions,it is necessaryonly to compute the mean-andvarianceof the untransformedvariatex and solveEqs. 26.38 and 26.39 simultaneously.Information on three-parameteror truncatedlog-normal distributionscan be found in the literature.r'7
Gamma(and PearsonTYPelll)
-
The gammadistributionhaswide applicationin mathematicalstatisticsandhas been usedincreasinglyin hydrologicstudiesnow that computingfacilities make_iteasyto evaluate the gimma functioi insteadof relying on the painstakingmethod of using tablesof the incompletegammafunction that lead to the CDF, P(X < x). In greater useis a specialcaseof gamma: tbePearsonType/1L This distributionhasbeenwidely adoptedas the standardmethodfor flood frequencyanalysisin a form known as the log-pearson /11 in which the transform y :1og x is used to reduce skewness.8-r0 Aithough all three momentsare requiredto fit the distribution,it is extremelyflexible in that a zeroskewwill reducethe log-PearsonIII distribution to a log-normal and the pearsonType III to a normal. Tablesof the cumulativefunction are availableand A very important property of gamm-avariates will be explainedin a later section.lo'11 aswell asnormal variates(includingtransformednormals)is that the sumof two such variablesretains the samedistribution. This feature is important in generatingsyn-thefie hy-drologic sequences.l''''
26J
CONTINUOUSPROBABILITYDISTRIBUTIONFUNCTIONS
689
Gumbel'sExtremalDistribution The theory of extreme valuesconsidersthe distribution of -the largest(qr smallest) observationsoccurringin eachgroup of repeatedsamples.The distribution of the nt extreme values taken from n, samples,with each samplehaving n2 observations, dependson the distribution of the nrn, total observations.Gumbel was the first to employ extremevalue theory for analysisof flood frequencies.laChow has demonstratedthat the Gumbel distribution is essentiallya log-normal with constantskewness.tsThe CDF of the densityfunction given in Table 26.3 is
P(X - x) : F(x) : exp{-expl-o(, - u)l}
(26.43)
a convenientform to evaluatethe function. Parametersa andu are given asfunctions of the meanand standarddeviationin Table26.3.Tablesof the doubleexponentialare usually in terms of the reducedvariate,y - a(x - u).tuGumbel also has proposed anotherextremevaluedistributionthat appearsto fit instantaneous(minimumannual) droughtflows.17'18
CDFsin Hydrology Normal and Pearsondistributionscan often be usedto describehydrologicvariables if the variableis the sum or mean of severalother random variables.The sum of a numberof independentrandom variablesis approximatelynormally distributed.For example,the annualrainfall is the sumof the daily rain totals,eachof which is viewed as a random variable. Other examplesinclude annual lake evaporation, annual pumpagefrom a well, annual flow in a stream,and mean monthly temperature. The log-normal CDF hasbeen successfullyusedin approximatingthe distribution of variablesthat are the product of powersof many other randomvariables.The logarithm of the variableis approximatelynormally distributedbecausethe logarithm of productsis a sum of transformedvariables. Examplesof variablesthat havebeenknown to follow a log-normal distribution include:
1. Annual seriesof peak flow rates. 2. Daily precipitationdepthsand stremflowvolumes(alsomonthly, seasonal, and annual).
3. Daily peak dischargerates. 4. Annual precipitation and runoff (primarily in the westernUnited States). Earthquakemagnitudgs, 6. Intervalsbetweenearthquakes. 7. Yield stressin steel. 8. Sediment sizes in streamswhere fracturing and breakageof larger into smaller sizesis involved. 5.
The PearsonType III (a form of gamma) has been applied to a number of variablessuch as precipitation depths in the easternUnited Statesand cumulative watershedrunoff at any point in time during a given storm event.The transformed log-PearsonType III is most usedto approximatethe CDF for annualflood peaks.If the skew coefficient C" of the variable is zero, the CDF reverts to a log-normal.
690
ANDSTATISTICS 26 PROBABILIry CHAPTER It hasalso.beenusedwith monthly precipitationdepthandyield strengthsof concrete members. A useful CDF for values of annual extremeis the Gurnbel or extreme value probability of distribution.The meanof the distributionhas a theoreticalexceedance streamshave peaks in natural years. Flood 2.33 T of interval 0.43 and a recurrence 2.33-year with means including disffibution, to this conformance exhibited strong variptraight-line fit for Gumbel a produces paper that Graph recurrenceintervals. is A sample extremes. annual of graphical tests for useful and ables is a available peak rates, discharge annual peak to applie! has been 27 The CDF shownin Fig. .2. wind velocities,drought magnitudesand intervals, maximum rainfall intensitiesof given durations,and other hydrologicextremesthat are independentevents.
AND CORRELATION LINEARREGRESSION 26.8 BIVARIATE Correlation and regressionproceduresare widely used in hydrology and other sciences.teThe premiseof the methodsis that one variableis often conditionedby the value of another,or of severalothers,or the distribution of one may be conditioned by the value of another. Just as there are probability density functions (PDFs) for evaluatingthemarginal probability of a variable(seeSection26.2), so also are there PDFsforlhe conditional probabilities (also describedin Section 26.2) of variables. The conceptis illustrated in Fig. 26.7. For two variables,the bivariatedensityfunction,/(y li,), ptottedin the vertical on the frgure,changesfor eachvalueof x' The one shownappliesonly to variationsin y whenx : xr. Different distributionsmight occur for other valuesof x. A measureof the degreeof linear correlationbetweentwo variablesx and y is thelinear correlation cofficient, P*,y. Avalue of p',, : 0.0 indicatesa lack of linear
Pylr regressionline
Figure 26.7 Bivariate regressionwith conditional probability function.
ANDCORRELATION691 REGRESSION LINEAR 26.8 BIVARIATE correlation andp,,, : + 1.0 meansperfect correlation.The correlationcoefficientis found from cr.v
cov(x, y)
u - , , :
(26.44)
-
(l*ay
aroy
where o, ando, are the variancesof eachvariable,respectively,(seeEq. 26.30), and cov(x, y) is the covariancesharedby the two variables,definedas
y) : c,.,: cov(.r,
f _f _Q'-
p)(y - p,)f(x,v)dvdx
(26'4s)
The samplecorrelationcoefficient,r : s,.rfs*s* is usedto estimatep',r. The sample covarianceis found from the squareroot of
s?,,:
(26.46)
The regressionline shown on Fig. 26.7 is derived to passthrough the mean valuesof the distributions,so that for any given value of x, the mean value of y I x (read"y givenx ") can be estimatedby the regressionline. The standarderror of the estimateof y I x is depictedby the line drawnthrough the conditional distributionsat a distanceof one standarddeviationfrom the mean.If the conditionaldistributionsat all x-valuesare normal,it canbe shownthat the meanvalue,&y1,,of the conditional distribution is related to the meansof .r and H or a & , 1 , : l t ', * ( p 2,( * and the variance is
4t.
: #['" -
-
tr,)
, @- p)'1
d
l
t26.47)
(26.48)
where
a?: 40 - p')
(26.4e)
which is the variance of the residualsof the regression.Just as the mean of the distribution requiressubstitutionof the given value of x into Eq. 26.47, so also does thevariance,Eq.26.48. Whenthe valueof x in Fig. 26.7is setiequalto A the standard error of the meanis O-t=:
ae -----7
(26.s0)
VN
Equation 25.47 is linear and expressesthe linear dependencebetweeny and .r as slrownin Fig. 26J.The meanvalue of y can be computedfor fixed valuesof x' Also, if the correlationbetweenthem is significant,one can predict the valuesof y with less error than the marginal distribution of y alone.In fact, from Eq. 26.49, the fraction of the original varianceexplainedor accountedby the regressionis
o":t-*
(26.5r)
692
CHAPTER26
PROBABILIry AND STATISTICS
It can be seenalso from Eq. (26.47) that the slopeof the regressionline is cy
PA:
tl,yl,
-
lLy
(26.s2)
r - rr^
or, ifx andy are standardized,then p itself is the slope,where - p,r)/a, }rnt. (26.s3) p = (x - t*)/o, The bivariatecasecan be expandedto coverhigher-order,multivariatedistributions.
EQUATIONS 26.9 FITTINGREGRESSION
ing value of x. The line to be fitted is
(26.s4)
-l Bx; Y,: a
N 6 I
Year
, ? ; xbo
A1
J . F
/1 !
Q
43 44
3[i e
o
46 47 48 49 50 51 52
F r ;
h
d
B U r40 120 100 80 Lowest annualflow for 1 day (cfs) JacksonRiver at FallingSprings,Virginia, 1941-1952 60
Mean = Standard deviation =
Jackson River
Cowpasture River
61 92 65 72 82 67 74 t 18 t24 r08 65 88-
58 81 70 63 68 58 74 105 134 108 93 85
84.7
83.1
21.7
23.2
: 0.86. : Figure 26.8 Cross-correlationof low floWs.Regressionline: Iz 4.94 + 0.923X; r
EQUATIONS 693 26.9 FITTINGREGRESSION The best estimates of o and B are sought. Thus to minimize
) 0, - f), : ) ly, (o + px,)1,-
(26.ss)
wherey, are the observedvalues and!,ate the estimatedvaluesfrom Eq. 26.54,take partial derivativesas follows:
- (q+ n')r} *{> b, - (o* B';l'} tv, #{>
(26.s6) (26.s7)
After carrying out the differentiationsand summations,two equationsresultin a and B, callednormal equations.
o
(26.s8)
2*,y,-")xt-F2*?:o
(26.se)
)y,
- n d- B ) " , :
SolvingEqs. 26,58 and 26.59 simultaneouslyyields
2 v , F 2 ^' i : y - B T
e:--
n
(26.60)
n
P: =7 - (>;y[
(26.6r)
Recall the slopeis p(arf o), or as estimatedfrom sampledata B:
rt;
(26.62)
Also, the unexplainedvariancein the regressionequationis
4:4Q-p')
(26.63)
and is the squareroot of which is the standarddeviationof residuals(seeFig' 26'8) cailed the standard error of estimate.Thesecan be estimatedfrom
*-: 4 ns -l (z r - r , )
(26.64)
s2":
(26.6s)
2(y,-il'
wherey, and i, are as definedpreviously(seeEq' 26'55)' ivtanynyarotogicvariablesare linearly related,and after estimatingthe regresrangeof sion coeffici"ntr, p."di"tion of y can be madefor any value of x within the be should but performed often is observedx values.Extrapolationoutsidethe range
7
694
CHAPTER26
PROBABILIry AND STATISTICS
done wfth caution.Equation 26.48 showsthat the variancein the estimateof y for a givenx valuebecomeslargewhenx is severalstandarddeviationsaboveor belowthe mean. EXAMPLE 26.6 The lowestannualflows for a l2-yr period on the Jacksonand CowpastureRivers are tabulatedin Fig. 26.8. The stationsare upstr€amof the confluenceof the two rivers that form the JamesRiver. Find the regressionequationand the correlationbetween low flows. Solution
t
: The basicstatisticsare2 x : 1016;) y : 997;2 x2 : 9 1 , 2 1 6 ; 2 l " 88,777;and2 xy : 89,209. For the two-variableregressiona and B are found from Eqs.26;60and 26.61,. _ [ ( 8 9 , 2 0 e ) ( l 0 t 6 x e e 7 ) / ( 1 2 ) ] :o Q o ? (9r,216)- (tor6)' 102)
o:
ee7 (0.e23x1016) : 4'9r i
The regressionis y : 4.91 + 0.923x. 3. The correlation coefficientfrom Eq.26.62 is
(0.e23)(2r.7) : 23.2
0:86
4. From F;q.26.64the standarderror of estimate,s,, is 11.7,which is plotted line in Fig. 26.8. rl as limits aroundthe regression
Coefficientof Determinationfor the Regression A regressionequationreplaces(and extends)the data used in its development.Becauseit cannot reproduceall the basedata, the processresults in the loss of some information. This not only includeslossof information aboutparticular pairs of data, but alsoaboutthe variability of the data.The variancesf is a statisticalmeasureof the variability of the measuredvaluesof y. The greaterthe valueof sl, the wider the spread of points aroundthe mean.The percentageof information aboutthe variancein y that is retained,or explainedby, the regrdssionequationis called the cofficient of determination, Cr. To determineits value,the residualsor departures(differencesbetween actual and estimatedy values)haveknown variance4, which representsthe unaccountedvariance in the regressionequation.The explainedvariance would be the difference, 4 - o2, and the percentageretained (coefficient of determination)is
(26.66)
695
26.9 FITTINGREGRESSIONEQUATIONS
Comparisonwith Eq. 26.49 revealsthat Co= Pz
,
(26.67)
Thus the squareofthe correlationcoefficientp is the percentageof d, explained by the regression.For any sample of data the coefficient of determinationr2 is estimatedas sl,rlslsl . A large r2 indicatesa goodfit of the regressionequationto the databecausethe equationaccountsfor or is able to explaina large percentageof the variation in the data. EXAMPLE 26.7
Determinethe coefficientof determinationfor the regressionin Example26.6. Solution, From Eq. 26.67,the coefficientof determination,r2, is 0.7396. "accountsfor" about 74 Thus, the regressionequationadequatelyexplainsor pefcent of the original information abouty containedin the raw data. Twentysix percentof the information is lost. I r For examThe bivariateexamplecan be extendedto multiple linear r'bgressions. ple, the linear model in three variables,with y the dependentvariable andx1 andx2 the independentvariables,has the form y:a*F$t*Fzxz
(26.68)
y : an-r Br) t' * FrZ *,
(26.6e)
The normal equationsare )
xtt FrZ*?+ Fr2*,*,
(26.70)
2 yr r : * ) xzI Fr 2 *r *, + F"2 *7
(26.7r)
)y"':
")
The squareof the standarderror of estimateis
s7:
2(y,-r,)'
(26.72)
where y, are the observedvaluesand y-,are predictedby Eq. (26.68). The multiple correlation coefficient is
+) ^2\ | /2
n : ( t -
s;/
(26.73)
in Hydrology LinearTransformations Strongnonlinearbivariateand multivariatecorrelationsare also commonin hydrology, ind various mathematical models have been used to describethe relations. Piiabolic, exponential, hyperbolic, power, and other forms have provided better
696
CHAPTER26
PROBABILITYAND STATISTICS
graphicalfits than straight lines. Becauseof difficulties in the derivation of normal equationsusing least squaresfor thesemodels,many can be transformedto linear forms. The most familiar transformationis a linearizationof mtiltiplicative nonlinear relationsby using logarithms.For example,the equation Y :
(26.74)
g,xft1$z
becomes linear when logarithms are taken, or
log y : log * + Bllo$ x1 -f B"Iog x2
(26.7s)
The log transformationprocedureresultsin a linear form when the logarithms are substitutedin Eqs. 26.60 and 26.61.For of one or both setsof measurements example,if a bivariateparabolicform I : qXb is suggestedby the data, logarithms allow use of the linear form log Y : log a -t b log X. The normal equationscan be usedby redefiningy : log Y,x : logX, e : log a, andF : b,thereby transforming the equationto y : a * Bx. The regressioncannow be performedon the logarithms, valuesof a and B determined,and the estimateof a is found by taking the antilog of a. This transformationis possiblefor severalother nonlinearmodels,someof which are shown in Table 26.4. The variablesx and y must be nonnegative,with values preferably greaterthan 1.0 to avoid problemswith the log transformation. OF NONLINEARFORMS TABLE 26,4 LINEARTRANSFORMATIONS
Equation Y=A+BX Y = BeAx Y:AXB Y:ABx
Abscissa X log X X
Ordinate I
log Y log Y log Y
Eouationin linearform
lY): A + B[x] tbc rl : locB + A(1oge)[x] tbc rl : bc A + B[tocX] lloCrl : loCA + (loCB)[X]
Note.'Variables in brackets are the regressionvariates.
EXAMPLE 26.8 In the following exhibit (Table 26.5) preparedby Beard,20the regionalcorrelationis soughtof the standarddeviationof flow logarithmswith the logarithmsof the drainage areasize'andthe numberof rainy daysper year;X, is setequalto ( 1 t log s) to avoid negativevalues.Find the regressionequationand the multiple correlationcoefflcient. Solution 1. From Eqs.26.69,26.70, and26.7l, the parametersare -0.49 a : 1.34; Fr : -0.013; Fz: and the regressionequationis Xt : 1.34- 0.0I3X2- 0.49X3 log s : 0.34 - 0.013log(DA) - 0.49 log(days) or 2. The multiple correlation coefficient from Eqs' 26.72 and 26.73 is R : 0.56. ll
697
APPLICATIONS AND CORRELATION 26.10 REGRESSION TABLE 26.5 LOGARITHMICDATA FOR 50 GAUGINGSTATIONS Xr:1 + logs Station number (1) I
2 3 4 5 o
7 8 q
t0 11 t2 I J
t4 15 16 17 18 19 20 21 22 LJ
24 25 26 27 28 29 30 3l 32
Xz: logDA
X2 (2)
x3 (3)
x1 (4)
1.61 2.89 4.38 3.20 3.92 1.61 3.2r 3.65
2.11 2.12 2.ll 2.04 2.07 2.04 2.09 1.99
0.29 0.18 o.l7 0.44 0.38 0.3'7 0.30 0.35 0.16 0.11 0.32 0.34 0.25 0.43 0.2'l 0.25 0.52 0.18 0.39 0.40 0.25 0.23 0.54 0.51 0.45 0.63 0.45 0.59 0.46 0.32 0.96 0.12
3.23 z.rs
2.08 4.33 1.60 2.09 2.00 2.82 2.00 2.40 2.09 3.69 2.18 2.19 2.09 2.17 |.91 4.48 1.9s 4.95 2.21. r.97 2.08 3.4r 4.82 l 88 r.93 r.78 L.74 4.39 3.23 2.01 3.58 2.04 1.64 1.78 1 . 76 4.58 3.26 1.93 1.81 4.29 1.23 1.89 1.48 3.44 |.97 2.lt
Per Year & : log numberof rainY-daYs
Station numDer (5) 33 34 35 36 37 38 39 40 4I 42 43 44 46 4'7 48 49 50
>X x 2 XX, 2 X2 X2/n
x2 (6)
xs
1.94
1.87 t.36 1.81 1.58 1.48 1.89
z.tJ
3.63 1.91 2.26 2.97 0.70 0.30 3.38 2.87 2.42 4.53 3.04 4.13 |.49 5.37 1.36 2.31
r.32 1.54 1.62 2.03 2.26 1.93 1.78 2.00 2.Or 1.95 2.tl 2.23
x1 (8) 0.20 0.58 0.64 0.37 o.27 0.54 0.63 0.78 0.46 0.44 0.24 -0.03 0.30 0.17 0.14 0.10 0.27 0.18
r4'7.55 2.951 503.7779 435.4200 68.3579
96.24 1.925 285.5627 284.0042 1.5585
17.89 0.358 51.1527 52.7934 -r.640' 7
1.5585
r8'7.59r2 185.2428 2.3484
33.2598 34.4347 -r.r749
2 XX, 2 X2 X,ln
2X2X'ln 2 xx1
(7\
-1.6407
Note: x = X - X. (AfterBeard.2o)
2 6 ' I o R E GR E S S |o N A N D C o RRELAT|oNAPPL|CAT|oNS
8.1635 6.4010 L7625
698
26 PROBABILIry ANDSTATISTICS CHAPTER desiredJtatisticalparameteras dependentvariable,and the appropriatephysicaland The proceclimatic variableswithin the basinor regionas the independenlvariables. dures are signiflcantly better than using relatively short historical sequencesand point-frequencyanalysis.Not only doesthe methodreducethe inherently large sampling errors but it furnishesa meansto estimateparametersat ungaugedlocations. There are limitations to the techniquesof Section 26.9. First, the analyst assumesthe form of the model that can expressonly linear, or logarithmically linear, dependence.Second,the independentvariablesto be includedin the regressionanalysis are selected.And, third, the theory assumesthat the independentvariablesare indeedindependentand are observedor determinedwithout error. Advancedstatistical methodsthat are beyondthe scopeof this text offer meansto overcomesomeof theselimitations but in practiceit may be impossibleto satisfy them. Therefore,care must be exercisedin selectingthe model and in interpretingresults. Accidental or casualcorrelationmay existbetweenvariablesthat are not functionally correlated.For this reason,correlationshouldbe determinedbetweenhydrologic variablesonly when a physicalrelation can be presumed.Becauseof the natural dependencebetween many factors treated as independentvariables in hydrologic studies,the correlationbetweenthe dependentvariableand eachof the independent variablesis different from the relative effect of the sameindependentvariableswhen analyzedtogetherin a multivariatemodel. One way to guard againstthis effect is by screeningthe variablesinitially by graphical methods.Another is to examine the results of the final regressionequationto determinephysicalrelevance. Alternatively,regressiontechniquesthemselvesaid in screeningsignificantvariables.When electroniccomputationis available,a procedurecanbe followedin which successive independentvariableSare addedto the multiple regressionmodel, and the relative effect of eachis judged by the increasein the multiple correlationcoefficient. Although statisticaltestscan be employedto judge significance,it is useful otherwise to specify that any variable remain in the regressionequation if it contributesor explains,say,1 or 5 percentofthe total variance,or ofR2. A frequentlyusedrqethod is to computethe partial correlation cofficients for each variable.This statistic representsthe relative decreasein the varianceremaining( 1 - R') by the addition of the variablein question.If the varianceremainingwith the variableincludedin the regression is (I - Rz) : pz and the variance remaining after removal is (l - R'') : D'', then the partial regression correlation coefficient is
\D'' - D')lD''.
Most PC spreadsheetsoftware packageshave statistical routines for all the analysesdescribedhere and many more. Most are extremelyflexible,requiringminimal instructions-andinput data other than raw data. Specialmanipulationscan effect an interchangeol dependentand independentvariables,bring one variableat a time into the regression equation, rearrange the independent variables in order of significance,and perform various statisticaltests.
ExtendingHydrologicRecords Regressiontechniquesfrequently can be used to extend short records if significant correlation existsbetweenthe station of short record and a nearby station with a - lorigerreeord.Iq Example26.6,if the JacksonRiver recordswerecompletefromI94l
PROBLEMS 699 to datebut the Cowpasturerecordswere incompleteafter 1952,the cross-correlation could be usedto estimatethe missingyearsby solvingthe regressionequationfor I from 1953 on usingthe X flows as observed.The reliability of suchmethodsdepends on the correlationcoefficientand the length ofthe concurrentrecords.Ifthe concurrent record is too short or the correlation weak, the standarderror of the parameter to be estimatedcan be increasedand nothing is gained.The limiting value of crosscorrelation for estimatingmeansis approximatelyp : l/\/ n, where n is the length of the concurrentrecord.21Thus any correlation above0.3 would improve the Cowpasturerecords.Estirnatesof other parameterswith larger standarderrors require highercross-correlationfor significantimprovement.Extendingor filling in deficient recordsoften is necessaryfor regional studiesin which every record usedshould be adjustedto the samelength.
HydrologicVariables Regionalized Predicting.
'
Cruff and Rantzz2studiedvariousmethodsof regional flood analysisand found the multiple regressiontechniquea better predictor than either the index-floodmethod (Chapter27) or the fitting of theoreticalfrequencydistributionsto individual historical records.They flrst usedregressiontechniquesto extendall recordsto a common base length. Next they extrapolatedby various methods to estimate the 50- and 100-yearflood events and with multiple correlation examined several dependent variables including the drainage areaA, the basin-shapefactor (the ratio of the diameterof a circle of sizeA to the length of the basin measuredparallel to the main channel)Sa,channelslopeS, the annualprecipitationP, and others.They found only A and S, to be significant, which resulted in prediction equations of the form Q, : cAS!,. These equationswere superior to those of the other techniques.The multiple correlation coefficientwas as high as 0.954.It is interestingthat regression techniqueswereemployedin still a third way,that is, to estimateregionalvaluesof the mean and standarddeviation after adjustingthe record length. Example26.8 illustrated the applicationof regressionanalysisto regionalizethe standarddeviationof annual maximum flow logarithms as a function of the drainage area size and the number of rainy dayseachyear.
r summary Statisticsis a diversesubject,and the treatmentin this chapterhasbeennothing more than an introduction. Seriousstudentsand practitionersmust return againand again to the theory in standardworks.23They will find that evaluatingnew developments of statisand techniquesmust claim a large shareof their time. Only certain aspects, tical hydrologyhave been presented,principally the common distributionsand the methodsfor analyzingfrequency of eventsobservedat a single point. In the next chapterthis information is extendedto common applicationsin hydrology.
PROBLEMS 26.1. The probabilitiesof eventsE1 andE2 arc each.3. What is the probability that E1 or E2 will occur when (a) the eventsare independentbut not mutually exclusive,and - (b) whenthe probabilityof Et, given E2is .l?
700
CHAPTER26
ANDSTATISTICS PROBABILITY
26.2. EventsA and B are independenteventshaving marginal probabilities of.4 and .5, respectively.Determine for a single trial (a) the probability that both A and B will occur simultaneously,and (b) the probability that neither occurs. 26.3. The conditional probability, P(E, I E,r),of a power failure (given that a flood occurs) is .9, and the conditionalprobability,P(Ez I E), of a flood (given that a powerfailure occurs)is .2. If the joint probability, P (\ andE), of a power failure and a flood is .1, determinethe marginal probabilities,P(E) and P(E). 26.4. Describetwo random eventsthat are (a) mutually exclusive,(b) dependent,(c) both mutually exclusiveand dependent,and (d)"neithermutually exclusivenor dependent. 26.5. A temporar;1cofferdamis to be built to protect the 5-yearconstructionactivity for a major crossvalley dam. If the cofferdam is designedto withstand the 20-yearflood, what is the probability that the structurewill be overtopped(a) in the flrst year, (b) in the third year exactly,(c) at leastonce in the 5-yearconstrucfionperiod, and (d) not at all during the 5-yearperiod? 26.6. A 33-yearrecord of peak annualflow rateswas subjectedto a frequencyanalysis.The median value is defined as the midvalue in the table of rank-ordered magnitudes. Estimatethe following probabilities. , a. The probability that the annualpeak will equalor exceedthe medianvaluein any singleyear. b. The averageretlrrn period of the median value. c. The probability that the annual peak in 1993 will equal or exceedthe median value. d. The probability that the peak flow rate next year will be less than the median value. yearswill e. The probability that the peak flow rate in all of the next 10 successive value. be lessthan the median f. The probability that the peak flow rate will equal or exceedthe median value at years. leastoncein l0 successive g. The probability that the peak flow rates in both of two consecutiveyears will equal or exceedthe median value. h. The probability that, for a2-yearperiod, the peak flow rate will equal or exceed the median value in the secondyear but not in the first' 26.7. What return period must an engineeruse in his or her designof a bridge openingif there is to be only a 50 percent risk that flooding will occur at least once in two successiveyears?Repeatfor a risk of 100 percent. 26.8. A temporary flood wall has been constructedto protect several homes in the floodplain. The wall was built to withstand any dischargeup to the 20-year flood magnitude.The,wall will be removed at the end of the 3-year period after all the homeshavebeen relocated.Determinethe probabilities of the following events: a. The wall will be overtoppedin any year. b. The wall will not be overtoppedduring the relocation operation. c. The wall will be overtoppedat leastonce before all the homesare relocated. d. The wall will be overtoppedexactly once before all the homesare relocated. e. The wall will be adequatefor the flrst 2 years and then overtoppedin the third year. 26.9. Waveheightsand their respectivereturn periods(shownon the next page)are known for a 40-mi long reservoir.Ownersof a downstreamcampsitewill accepta 25 percent risk that a proiective wall will be overtoppedby wavesat least once in a 2}-yeat period. Determinethe minimum height of the protective wall.
PROBLEMS 701 Waveheight (ft)
Returnperiod (years)
10.0 8.5 7.4 5.0 3.5
100 50 30 10 5
26.10. Assumethat the channel capacityof 12,000cfs near a private home was equaledor exceededin 3 of the past 60 years.Find the following: a. The frequencyof the 12,000-cfsvalue. b. The probability that the home will be floodednext year. c. The return period of the 12,000-cfsvalue. d. The probability that the home will not be floodednext year. e. The probability of two consecutive,safeyears. f. The probability of a flood at leastonce in the next 20 years. g. The probability of a flood in the second,but not the first, of two consecutiveyears. h. The 20-yearflood risk. 26,11, The distribution of mean annual rainfall at 35 stations in the JamesRiver Basin, Virginia, is given in the following summary:
Interval(2-in. groupings) Numberof observations
36 or 37 in.
38 or 39 in. 4
40 or 41 in. j
42 or 43 in.
Z
Interval (2-in. groupings) Numberofobservations
44 or 45 in. 5
46 or 47 in. 4
48 or 49 in. 2
50 or 51 in. 2
Computethe relative frequencies(seeChapter27) andplot the frequencydistribution andthe cumulativedistribution.Estimatethe probability that the meanannualrainfall (a) will exceed40 in., (b) will exceed50 in., and (c) will be betweenthesevalues. 26.12. Write a simpleprogram to READin N data points and compute the mean, standard deviation,and skewnesscoefficient. 26.13. A normally distributedrandom variablehas a mean of 4.0 and a standarddeviation of 2.0. Determinethe value of f@
I
-
dx I f(x)
"8
26.14. For a standardnormal densitv ' function. use Table B.1 to determinethe value of fr+'o
I
fG) dx
J*-ro 26.15. A normal variableX has a meanof 5.0 and a standarddeviationof 1.0.Determinethe value of X that has a cumulativeprobability of 0.330. 26.16. If the mode of a PDF is considerablylarger than the median, would the skew most likely be positive or negative?
l
702
ANDSTATISTICS 26 PROBABILIW CHAPTER 26.17. tomplete the following mathematicalstatementsabout the properties of a PDF by insertingin the boxeson the left the correct item numberfrom the right. Assumethat X is a seriesof annual occuffencesfrom a normal distributibn. I r.Zerc a. I f(x) dx: J t " b.
d.
: f'_ro,dx r
2.Unity
dx: '34 l-.o 'o'
3. Valuewith 5 percentchanceofexceedanceeachyear
f,o"
dx:r
dx: .5 f-to, f(x) dx : Z
I rt, dx: .02
4. 0.68 5. Valueexpectedevery 50 r"urc on the average 6. P(X < mr) 'X'*r) 7.P(m1 8 . P ( m 1- X = m z )
l.
9. Median
10. Standarddeviation 26.18, The mean monthly temperaturefor Septemberat a weather station is found to be normally distributed.The mean is 65.5" F, the varianceis 39.3'F2, and the record is completefor 63 years.With the aid of TableB.1, find (a) the midrangewithin which two thirds of all future mean monthly valuesare expectedto fall, (b) the midrange within which 95 percentof all future valuesare expected,(c) the limit below which 80 percentof all future valuesare expected,and (d) the valuesthat are expectedto be exceededwith a frequencyof oncein l0 yearsand oncein 100years.Verify the results by plotting the cumulativedistribution on normal probability paper. 26.L9. The total annualrunoff from a small drainagebasinis determinedto be approximately normal with a mean of 14.0 in. and a varianceof 9.0 in.2.Determinethe probability that the annualrunoff from the basinwill be lessthan I 1.0 in. in all three of the next three consecutiveyears. 26.20. In the past60 years,a dischargeof 30,000cfs at a streamgaugingstationwasequaled or exceededonly three times. Determine the averagereturn period (years) of this value. 26.21. EventsA and B are independentand havemarginal probabilitiesof .4 and .5, respectively. Determine the following for a single trial: a. The probability that both A and B occur. b. The probability that neither occurs. c. The probability that B, but notA, occurs. Existingrecordsrevealthe following information aboutEventsA and 4 whereA = a ?6il,
IongMerch'warmspellandB 1q
:!94!$99!.
PROBLEMS
703 1n
Year A : warm March? B : April flood?
No Yes
No No
Yes No
No Yes
Yes Yes
No Yes
Yes No
No Yes
Yes Yes
No No
On the basisof the 10-yearrecord, answerthe following: a. Are variablesA and B independent?Prove. b. Are variablesA andB mutually exolusive?Prove. c. Determinethe marginal probability of an April flood. d. Determinethe probability of having a cold March next year. e. Determinethe probability (onevalue)of havingboth a cold March and a floodfree April next year. f. If a long March warm spell hasjust endedtoday,what is the best estimateof the probability of a flood in April? 26.23. Two dependenteventsarc A : a flood will occur in Omaha next year and B : an ice-jam will form near Omahain the Missouri River next year. Useyour judgment to rank from largestto smallestthe following probabilities:P(A), P(A andB), P (A or B), P(A I B). 26.24. The probability of having a specifiedreturn period, [, is definedas: I P(annualvaluewill be equaledor exceeded : /, \'-' exactlyoncein a periodof r : I years) T,f \' Also, p (annualvalue will be equaledor exceeded _ exactly r times in a period of n years)
pn_r(l _ p)r
the secondprobability shouldequal a. According to the descriptionsin parentheses, the first when n and r are equal to what values? b. Showthat both equationsresultin the sameprobability for an annualvaluewhose years.Discusss. frequencyis 33{ percentand the return period is Z.: /:3 26.25. For the function describedbelow, find (a) the number b that will make the function a probability densityfunction, and (b) the probability that a singlemeasurementof x will be lessthanl.
.l \^)
-
forx ( 0 for0
{i",,'
26.26. The random variabler representsdepth of precipitation in July. Betweenvaluesof -r : 0 and x : 30, the probability densityfunction has the equation/(-r) : x/40p',. In the past, the averageJuly precipitation p,, was 30 in. a. Determine the probability that next July's precipitation will not exceed20 in. b. Determinethe singleprobability that the July precipitatjon will equal or exceed 30 in. in all of five consecutiveyears. 26.27. The random variable-r representsdepth of precipitation in July. Betweenvaluesof x : 0 and .r : 30, the probability densityfunction has the equation/(.x) : x/1200. a. Determine the probability that next July's precipitation will not exceed20 in. b. Determinethe probability that next July's precipitationwill equalor exceed30 in.
l---
704
ANDSTATISTICS 26 PROBABILITY CHAPTER 26.28.
26.29.
26.30. 26.31.. 26.32.
26.33. 26.34.
discharge Measured (1000acre-ft)
recharge Estimated (1000acre-ft)
12.2 10.4 10.6 1.2.6 14.2 13.0 14.0 t2.0 10.4 tl.4
t2.o 9.8 I 1.0 t3.z
14.6 14.0 14.0 I z,+
10.4 11.6
26.35. F i t a r e g r e s s i o n e q u a t i o n t o t h e d a t a i n P r o b l e m 2 6 ' 3 4 , t r e a t i l g d i s c h a r g e a s t h e dependentvariable.computethestandarderrorofestimate.Estimatetheexpeoted be the estimateof dischargeif no dischargewhen recharg"ir 13 Kti what would relative improvementprovided the is information were availatie on recharge?what by the regressionestimate? linear regression'The program 26.36. prepare a computerprogramfor simple,two-variable, (b) computethe means'variances' X' should(a) read in N pain oioUt"tuuiions, Y and the regressionconstants'the (c) find and andX, Y and standarddeviationsoiUott' coefficient. verify with the data in standarderror of estimal-, u"J,ir" correlation Problem26.34. of the mean annual rainfall with the 26.37. From the following observations of variation How iinea, predi.tion equationfor the catchment. altitude of the gauge,d";;;;; well correlatedare rainfall and altitude?
)
i
PROBLEMS Gauge number
Mean annual rainfall(in.)
1 2 3
22 28 25 31
4
5 6 7 8 9 10 1l 12
705
Altitudeof gauge (1000ft) A A
4.4 1 < < A
5.6 5.6 5.8 6.0 6.6 6.6 6.8 7.0
JZ J I
36 35 36 46 4l 4I
26.38. Estimatethe expectedrainfall in Problem 26.31 for a gaugeto be installed at an altitude of 5500 ft. estimatesof A and B in the bivariateregressionequationY : A + 26.39. The least-squares definedaslogro)andXis BXarcA: 2.0 andB : 3.0,whereYis atransformation : the valuesof a and b. determine y axb, by related Ify and r are as 1o916;r. defined for 26.40. The time of rise of flood hydrographs(Z), deflnedasthe time a streamto rise from low waterto maximum depthfollowing a storm,is relatedto the streamlength (L) and the averageslope(S). From the information given below for 11 watershedsin Texas, New Mexico, and Oklahoma, derive a functional relation of the form T,: aLbS'.
Watershed number I
2 3 A
5 6 7 8 9 l0 11
Ir
L
D
(min)
(1000ft)
(fv10oo ft)
150 90 60 60 100 75 90 30 30 45 50
18.5 14.2 25.3 tt.7 9.7 8.1 21.'7 3.9 1.2
7.93 19.0 t2.a 13.3 11.0 15.0 16.7 146.0 20.0 64.0 33.0
J.J J.)
26.4r, Repeat the exercise in Problem 26'40 by fitting the relation T,: dF", whete
with t in mi and S in ftimi. Plot the results on log-log paper. 26.42. The squareof the linear correlationcoefficientis calledthe proportion of the variance that is "explainedby the regression."Describethe meaningof this phraseby evaluating the equationsgivenin the text.What varianceis explained,and what doesthe term "explained" mean? 26.43. Twenty measuredpairs of valuesof normally distributedvariablesX and Y are analyzed, yielding valuesof X : 3O,V : 20, s' = 20, ands, : 0. Determinethe values F : L/{S
706
CHAPTER26
PROBABILITY ANDSTATISTICS
d a andb and the standarddeviationof residualsfor a least-squares fit usingthe linear equationY:a+bX. 26.44. The least-squares estimatesof A and B in the bivariateregres*sion equationy : A + BXareA:2.OandB : I.0, whereyis atransformation definedaslog1eLIf Iand X are relatedby Y : a(.b)',determinethe valuesof a and b.
26.45. Given a table of ten valuesof mean annual floods and correspondingdrainageareas for a numberofdrainagebasins,statehow linear regressiontechniqueswould be used to determinethe coefficientand exponent(p and 4) in the equation Qzzz : pAq.
26.46. What choiceof transformedvariablesI and X would provide a linear transformation for y : a/(x3 + b)? Also, if a regressionon these transformed variables yields I : 100 + 10X determinethe correspondingvaluesof a and b. Would the linear transformationbe applicableto all possiblepairs and valuesofx and y? 26.47. Which measureof variation in a regressionY : a * bX is generallylargerin magnitude, the standarddeviation of I or the standarddeviation of residuals?For what condition would the two valuesbe equal?
REFERENCES 1. Ven T. Chow, "Statisticaland ProbabilityAnalysisof HydrologicData," Sec. 8-1, in Handbookof Applied Hydrology (V. T. Chow, ed.). New York: McGraw-Hill, 1964. 2. M. B. Fiering, "Information Analysis," in Water Supply and Waste Water Removal (G. M. Fair, J. C. Geyer,and D. A. Okun, eds.).New York: Wiley, 1966,Chap.4. 3. J. R. Benjamin and C. Cornell, Probability, Statisticsand Decisionfor Civil Engineers. New York: McGraw-Hill. 1969. 4. A. M. Mood and F. A. Graybill, Introduction to the Theory of Statistics, 2nd ed. New York: McGraw-Hill. 1963. 5. A. J. Duncan,Quality Control and Statistics.Homewood,IL: RichardD. Irwin, Inc., 1959. 6. P. G. HoeI, Introduction to MathematicalStatistics,3rd ed. New York: Wiley, 1962. 7. J. Aitchison and J. A. C. Brown, The Log-Normal Distributlon. New York: Cambridge UniversityPress,1957. 8. H. A. Foster,"TheoreticalFrequencyCurves,"Trans.ASCE 87,142-203(1924). 9. L. R. Beatd, Statistical Methods in Hydrology, Civil Works Investigations,U.S. Army Corps of Engineers,SacramentoDistrict, 1962. 10. "A Uniform Techniquefor DeterminingFlood Flow Frequencies,"Bull. No. 178, U.S. GeologicalSurvey,1989. 11. "New Tablesof PercentagePoints of the PearsonType III Distribution," Tech. Release No. 38, CentralTechnicalUnit, U.S. Departmentof Agriculture,1968. 12. M. B. Fiering, StreamflowSynthesis.Cambridge,MA: Harvard University Press,1967. 13. F. E. Perkins,SimulationLecture Notes,SummerInstitute, "Applied MathematicalProgramming in WaterResources,"University of Nebraska,1970. 14. E. J. Gumbel, "The Return Period of Flood Flows," Ann. Math. Statist. l2(2), L63190(June1941). 15. Ven T. Chow, "The Log-Probability and Its EngineeringApplication," Proc. ASCE 80, 1-25(Nov. 1954). 16. "Probability Tables and Other Analysis of Extreme Value Data," Series22, National Bureauof StandardsApplied Mathematics,1953. 17. E. J. Gumbel, Statisticsof Extremes.New York: ColumbiaUniversity Press,1958.
Chapter27
FrequencyAnalysis
r Prologue The purposeof this chapteris to: . Presentmethodsused in hydrology to evaluatethe recurrenceof particular magnitudesand durationsof random hydrologicvariables. . Elaborateon the definitionsof freqUency,reiurrence interval, return period, and risk analysisintroducedin Chapter10, Section10.4. . Illustrate the diverseapplicationsof frequencyanalysisin hydrology. . Teachseveralmethodsfor conductingfrequencyanalyses,includingthe useof frequencyfactors that allow estimationof recurrenceintervals for variables that follow conventionalprobability distribution functions. . Introduce methods of point and regional frequency analysis and describe regional USGSregressionequationsthat have been 4doptedthroughoutthe U.S. for estimating flood flows for use in structure design and floodplain analysis. . Establishhow to estimatethe reliability of estimatesderived from point or regional frequencyanalyses. Explainthe widelyusedBulletin No. 17BLog-PearsonType III proceduresfor performing uniform flood flow frequencyanalyses. Describehow variousfederal agenciesapply frequencymethodsin designor analysisof water resourcessystems. In Chapter 26, probability and statistical characteristicsof random variables were introduced,alongwith common distributionfunctions and principlesof regression and correlatiqn.The presentchapterprovidesapplicationsof theseprinciplesto common hydrologicvariables.
27.1 FREQUENCY ANALYSIS The statisticalmethodspresentedin Chapter26 areusedmostfrequentlyin describing hydrologicdata suchas rainfall depthsand intensities,peak annual discharge,flood flows, low-flow durations,and the like. Frequencyanalysiswas introduced in Sec-
ANALYSIS 709 FREQUENCY 27.2 GRAPHICAL tion 10.4 and is defined as the inve recurrencAor probabilitiesof magnit wise, the frequencYof a hYdrologic discretevariablewill occur or some' exceededin anY given Year'The lat freq-uenc probability or exceedance ir"qo"n"Y is a ProbabilitYand has ceedancsfrequencY,as shownbYEc Two methods of frequencYa plotting techniqueto obtain the cum iu"to.t. The cumulativedistribution the probability of an eventequal to is used to obtain recurrencerntervz tioned whenworking with recordssl ofexpectedhydrologiceventsgreaterthantwicetherecordlength.
ANALYSIS FREQUENCY 27.2 GRAPHICAL The frequencYof an event can be When annualmaximum valuesare imated as the meantime in Years'\ exceededonceon the average'The t be shownto be m
tuf,lj
x:
(21.r)
the mean number of exceedances the number of future trials -oi of values the number to I descendingvalues,with largestequal therunt t : 1' N : T' and If the mean number of exceedances
where
7 = N = n : m:
4 I
-
_
n ) l
(21.2)
m
indicatingthattherecurfenceintervalisequaltothenumberofyearsofrecordplus 1, divided bY the rank of the event' They give different results as Severalpictting p*liion formulas are available'l for 10 yearsof record .1,.The rangein recurrencerntervalsoutained notedin Table2'7' plotting position formulasdo not account is illustratedin the right-hani column.Most formula ihat doesaccountfor samplesize for the samplesizeor length of record. One generalform wa, giuen by Gringorten' and has the .r1 -
r -
n * l - 2 a m - a
(27.3)
710
CHAPTER2TFREQUENCYANALYSIS TABLE 27.1 PLOTTINGPOSITIONFORMULAS
Form:1 andn=10 Method
Solve for P (X > x\
P
m
.10
l0
.05
20
.067
14.9
.091
11
m-0.3 n+0.4
.06'l
14.9
Blom
,m-i n+i
.061
t6.4
Tukey
3m-l 3n*l
.065
15.5
California
n 2m-l
Hazen
1 - (0.5)'/'
Beard
fn
Weibull
n * l
Chegadayev
where n : the number of yearsof record the rank a parameterdependingon n as follows:
n a
10 0.448
20 0.443
30 0.442
40 0.441
50 0.440
n a
60 0.440
70 0.440
80 0.440
90 0.439
100 0.439
In general,a : 0.4 is recommendedin the Gringortenequation.If the distribution is approximatelynormal, , : fi is used.A value of a : 0.44 is usedif the data follows a Gumbel distribution. The techniquein all casesis to arrangethe datain increasingor decreasingorder of magnitudeand to assignorder number m to the ranked values.The most efficient formula for computingplotting positionsfor unspecifieddistributions,l and the one now commonly usedfor most sampledata, is the Weibull equation P _
m n-fI
(27.4)
Whenm is rankedfrom lowestto highest,P is an estimateof the probability of values being equalto or lessthan the rankedvalue,that is, P(X < x); whenthe rank is from the value highestto lowest,P is P(X > x). For probabilitiesexpressedin percentages, is IA\ml@ + 1). The probability that X: .x is zero for any continuousvariable.
27.4 REGIONALFREQUENCYANALYSIS
721
regiond'lstudies.Methodsof "smoothing" and averagingregionalvaluesof skewness havealsobeenproposed.ro'15 Many techniquesused in the past for generalizingregi6nal characteristicsdid not rely on statisticalconsiderations.The so-calledstation-yearmethodof extending rainfall recordshasprovedhelpful but has questionablestatisticalvalidity, especially areas.The method if applied to dependentseriesor to stationsin nonhomogeneous has been used to combine, say,two 25-yearrecords to obtain a single 50-year sequence. In practice, the analyst may have to use imagination and ingenuity to summarizeregional characteristics,while remaining awareof actual and theoretical considerations, lndex Flood Method The index-floodmethodusedin the pastby the U.S. GeologicalSurveyis an example of summarizingregional characteristicssuccessfully.t''tuThe method usesstatistical andgenerally databut combinesthem in graphicalsummaries.It canbe supplemented improved by using statisticalmethods,employing,for example,the regressiontechniquesexplainedin Chapter26.The index method,as illustrated in Fig. 27.4, canbe outlined as follows. 1. Preparesingle-stationflood-frequencycurves for each station within the homogeneous region (Fig. 27.4a). 2. Compute the ratio of flood dischargestaken from the curves at various frequenciesto the mean annual flood for the samestation. 3. Compileratios for all stationsand find the medianratio for eachfrequency (Fi5.27.4b). 4. Plot the median ratios againstrecurrenceinterval to produce a regional frequencycurve (Fig. 27.4c). Two statistical considerationsinvolved are (1) a homogeneitytest to justify definitionof a region, and(2) a methodfor extendingshortrecordsto placeall stations on the samebaseperiod. A somewhatsimilar techniquewas developedby Potterfor the Bureau of Public Roads.rTIt relies on the graphical correlation of floods with physical and climatic variables and is thus a techniquethat refers in part to the in Chapter26 (seealso Chapter16). discussion
U.S.G.S.RegionalRegressionEquations Early in the 1950s,the U.S. Geologicalsurvey institutedaprocessof correlatingflood flow magnitudqsand frequencieswith drainagebasin characteristics.Setsof regressionequationsfor the 2-,5-, l0-,25-,50-, and 100-yearfloodshavebeendeveloped regionin every state.The work was for practically everyhydrologicallyhomogeneous largely inauguratedto developmethodsfor estimatingpeak flow rates for designof highway structuresat ungaugedbasins.Data from gaugedsites was evaluatedby regional analysisto provide the best fit of regressionmodelsto the data. Continuouswater stagerecordersand crest-stagegaugedata were consultedto developfrequencycurvesfor all gaugedwatersheds.Given the frequencycurves,a
6000 5000 o
4000 a
o po
3000 2000 1000 0
I
I ts c") c..l
I
I
i' 2 5 1 0 2 interval(Yr) Recurrence
1.01 1.1
0
50
100
(al Recurrenceintervals (Yr)
station 1
5
1.5
1.1 ^ ALt
n ?<
1 AA
10 1 q?
20 ).55
50 3.03
(b)
3.0 2.8
I
2.2 6
2.0
o
1.8 l.o
o
po
' :
7.4 t.2 1.0 0.8 0.6 0.4 1.01 1.1
I
10 interval(Yr) Recurrence
t.52
5
50
100
(cl
Figure27.4 Index-floodmethod of regional flood frequency an-alysis:(a) single-stationflood frequencycurve, (b) ratios Q' to Qr.rrfor 15 stations,and (c) regional flood frequencycurve'
ANALYSIS 723 FREQUENCY 27.4 REGIONAL numberof correlationtestswere madeusingmultiple linear regressionto predict the peak flows from various easily obtained independentparameterssuch as drainage area,basinslope,watershedaspect,elevation,meantemperatureduringthe snowmelt season,and hundredsof other variables' Each study was reportedby state.The open file or water resourceinvestigation reports are availablefrom the USGS and include discussionsof the equations,com1n"ntron rangeof applicability,information on the reliability of the equations,copies of all the gaugedbasin frequencycurves,and.setsof equationsfor estimatingfloods Equationsfor all stateshavebeen compiledby the U.S. in ungaugid watersheds. Ceotogicit Survey into a PC software packagecalled NFF (National Flood Frequ"n"y;, availablefrom tJreusGS (or FHWA as part of their package,HYDRAIN)' Figure 27.5 showsthe six regionsfor the stateof Texas.Regionalregressionwas conductedindependentlyby regionusingavailablegaugedstationdata.As in many of the reports, the Texasmanual revealsthat different independentvariableswere selectedfor eachregion.18The equationsdevelopedfgr Region2 are: Q, Qs
: : :
389 Ao6a6So'2ta 236 = 485 Ao 668510 Qr, : 555 Ao 6825'0'250 Qro : 628 Ao 6e4s0261 Qrco Qro
(27.r2) (27.r3) (27.r4) (27.rs)
216 Aos74So'12s 184 322 Ao62oso
'
(27.16)
(27.r7)
where g : peak dischargefor given frequency,cfs A : drainagearea,squaremiles S : averageslopeof the streambedbetweenpoints 10 and 85 percentof the distancealons the main streamchannelfrom the mouth to the basin divide, feet per mile
EXAMPLE 27.6 Develop estimates of flood peaks for a 200-square-mile rural watershed near Dallas. The mean slope between the 10 and 85 percent points is 3.4 ft per mile. Solution.
Dallas is in Region 2. Equations 2712-27.17 2 1 6 A o s 1 a s o l 2:s 5 , 2 7 0 c f s : O s : 3 2 2 4 0 6 2 0 5 0 1 8 o 1 0 , 7 7 0c f s : Q r o : 3 8 9 A o ' 6 a 6 s o ' 2 1 a1 5 ' 4 9 0c f s :22'300 cfs Q r r : 4 8 5A 0 6 6 8 5 0 2 3 6 : Q r o : 5 5 5 A o 6 8 2 s o 2 5 o 2 7 , 8 0 0c f s : ll Qrco : 628 Ao6e4so261 34,170 cfs Qr:
-
give:
724
CHAPTER2T FREQUENCYANALYSTS
UNDEFINED
Pacoa
,'7I
N
N D R E
Figure 27.5 Hydrologicregionsin Texasfor 1976USGSregional regressionequations.(From Ref. 18.).
27.4 REGIONALFREQUENCYANALYSIS
725
726
CHAPTER2TFREQUENCYANALYSIS
NationalFlood Frequency(NFD Program equationslike Eqs.27.I2 through27:t7 for estimatingfloodSince1973,regression peak dischargesfor rural, unregulatedwatershedshavebeenpublished,at leastonce, for every stateand the Commonwealthof PuertoRico. In 1993the USGS,in cooperation with the FederalHighwayAdministration and the FederalEmergencyManagement Agency,compiled all of the current statewideand metropolitanarearegression equationsinto a microcomputerprogramtitled the National Flood Frequency(NFF) Program.reThis program summarizesregressionequationsfor estimatingflood-peak techniquesfor estimatinga typical flood dischargesfor all52 states.It also addresses probability peak discharge hydrographfor a given recurrenceinterval or exceedence prograr4 lists statewideregression for unregulatedrural and urban watersheds.The information and reference equationsfor rural watershedsand providesmuch of the for estimating equations input data neededto run the computerprogram.Regression least 13 statesare in at urban flood-peak dischargesfor severalmetropolitan areas also available. Information on computerspecificationsand the computerprogramare given.rT Instructionsfor installing NFF on a personalcomputerand a descriptionof the NFF program and the associateddata base of regressionstatistics are also given. The program is available as part of the Federal Highway Administration package, HYDRAIN, or by itself. Thoughthe USGSand FHWA do not distributeor servicethe software,information about vendorswho provide softwaresalesand servicecan be obtainedby contactingthe agencies.
Flood Frequencyfrom ChannelGeometry
.
Stream channelsin alluvial systemsdeveloptheir width, depth, slope, and other hydraulicgeometrycharacteristicsfrom the compositehydrographsthat flow through their valleys. It has been demonstratedthat the shapeof some streamchannels,if properly evaluatedby trained hydrologists,can be correlatedwith the mean annual flow, peak annual flow, bank-full flow, and the dominant, or channel=forming,discharge.Regressionequations,similar to Eqs. 27.12-27.I7, havebeen successfully derivedfor many perennialstreamswith very reasonablestandarderrors of estimate. Measurementsfor these studiesare normally obtained during low flow. The channelof interestis that channelbeing maintainedby the current flow regime.It is by the activechannel,limited laterally by the point barsand mostrecent characterized geologicfloodplaindeposits.It is felt that theserepresentthe most recentdepositions, and are thereforeindicative of the width neededby the current flow and sediment transport regipe. Figure 27.6 rll:ustratesthe principle in determining the active, floodplain-buildingchannel,established'forthe exampleas the width A-A . Suchstudieshavebeenconductedin Nevadd,California, Kansas,Colorado,and elsewhere.A USGSinvestigationof 53 gaugedstreamsin mountainregionsof Colorado resultedin the following equations.2o Qz = 0.666 Wr'eoaD*o Q, : 1.53 W1.682D-o2stAo'017 Qr6
:
2.38 W1
t43 2se 53o A.o D-o
(27.r8) (27.r9) (27.20)
27.4 REGIONALFREQUENCYANALYSIS
727
Referenceline Low-flow water level
Distance, in feet
Figure27.6 Ref. 20.)
Typical streamcross-section,illustrating active channeldimensions'(From
Q25 Qro
where Q W D A
: = : :
:
3.70 Wr'372D-o263Ao2rs
:
4.93 W127aD-02s6Ao'257
(27.2r) (27.22)
peakflow for the given frequency,cfs iop width of streamat bank-full condition, ft mean depth for bank-full flow, ft cross-sectionareaat bank-full flow, sq ft
The multiple correlation coefficientsfor theseequationsranged from 0-80 for the 42' 1 ranged_from 50-yrflow to 0.89 for the2-yr event.Standarderrors,respectively, use in for tool yet another offer investigations of types These percentto 32.2percent. site-specific by floods evaluate to hydrologist the and allow p"uk flo*r, estimating conditionsversusmore uncertain regionalparameters'
RegionalRainfallCharacteristics The variation of rainfall frequencieswith duration was introduced in Chapter 2' Regressionanalysiscan be usedto I thoseshownin ChaPter15, and the Many formulashavebeenusedin t a form with intensity(i) inverselypt of the form i : AIQ + B) to fit constantsA andB thereforeserveascharacteristicfeaturesof both the rainfall region and the frequencyof occurrencein eacharea'
728
CHAPTER27
FREQUENCYANALYSIS
EXAMPLE 27.7
Fit the following rainfall datato determinethe 10-yearintensity-duration-frequency curve.
r : duration (min) i : intensity (in./hr) Ui
r0
5
7.r o.t4
5.9 o.l7
15 5.1 0.20
30 3.8 0.26
t20
60 z-)
1
0.43
0.71,
A
Solution 1.. A model of the form i : AIG * B) can be expressedin linear form as ,,
lli:tlA+BlA.
The regressionof l.li versust yields l/i : 0.005t + 0'12, from which A : 2 0 0 a n d B: 2 4 . 3. Thus the rainfall formula is i : 200/(t + 24). The correlation coefficient is -0.997. ll
Maximum averagerainfall depths have been publishedby the U.S. Weather for durationsbetween30 min and24hr and for recurrenceintervalsbetween Bureau22
depthrelation showninFig.27.7. c o F
100
o
'a
9u
N
t 8 0
24ft
6hr 3hr
\\
F
e 'ro
lhr mln
q
* 6 0 b - "
50
\
150
350
400
Area(mi2) for use with duration frequency curves Area-depth Figare 27,7 values.(U.S. WeatherBureau.)
FREQUENCY ANALYSIS 729 27.4 REGIONAL The accuracyof arearainfall data dependshedvily on the densityand location of gaugesthroughoutthe areaconsidered.The simpleaveragingof the accumulation in all gaugesgivesno considerationof the effectiveareaaround eachgaugeor to the stormpattern.Two methodsare availablein calculatingthe weightedaverageof gauge records, the Thiessen polygon method and the isohyetal method. The Thiessen methodassumesa linear variation of rainfall betweeneachpair of gauges.Perpendicular bisectorsof the connectinglines form polygonsaround each gauge(or partial polygonswithin the areaboundary).If a sufficientnumberof gaugesare availableto constructcontoursof rainfall depth (isohy.ets),the weightingprocesscan be carried out by using the averagedepth betweenisohyetsand the area includedbetweenthe isohyetsand the areaboundaries.Figure 27.8 showsboth schemes. An exampleof the effect of gaugelocation and density is shown inFig.27.9. stormrainFigure 27.9ashowsthe increasein variability betweenThiessen-weighted falls and rainfall at a singlegaugeas the distanceof singlegaugesfrom the watershed centerincreases.Figure 27.9b showsthe effect of gaugedensityand total areaon the standarderror of the mean.Completestudiesof precipitationpatternsoverlargeareas requiredetailedanalysisof depth-area-duration datathat dependon the masscurves of accumulationfrom a network of gauges.The methodis describedin detail in other Figure 27.I0 depictsthe depth-area relation for the 24-hr storm references.23-25. shownin Fig. 27.8.It also required observationstaken at variousdurationsand the successivedeterminationof averagedepthsby the isohyetalmethod.
(a)
(b)
Figure 27.8 Methods of determiningrainfall averages:(a) Thiessennetwork (24-hr total; averagebasin precipitation = 2.54 in.) and (b) isohyetal map (24-hr total; averagebasin precipitation = 2.50 in.), The arithmetic averageover the basin = 39.10/15 * 2.61 in.
730
OHAPTER27
FREQUENCYANALYSIS H
U.+
>.= O F
(n n
Distance from rain gauge to watershed center (mi) (a)
H 6 6 6 o
b g
9 R 6
500 (mi2) Areapergauge (b) Figure27.9
(a) Relationbetweenthe standarddeviationof the watershed
a4tlt,ililliT,i',l:##-:"::: :1111i:ifli'ffi ill"lH"rffi cipitation as a function of the network density and drainageare for the Muskingumbasin.(U.S. WeatherBureau')
STUDIES 27.5 RELIABILIWOF FREQUENCY A significantdevelopmentof theoretical statisticsis the central limit theorem.As a .on*-rqu"n"" of the law of large numbers,the central limit theorem statesthat for a populationwith finite varianceoz anda meanp, the distributionof samplemeansitrut ir, a numberof equally good meansfrom repeatedsamples-will be distributed themselvesas a normal disiribution with meanp, anda varianceequalto a2fn, where or is the population standard deviation. This theorem does not limit the type of under$ing population distributionbut saysthat the distribution of the sampl" *9-uts will approacha normal distribution as the samplesizeincreases.The statisticalY n is the siandarddeviationof the distributionof meansand is called lhe standarderror
27.5 RELIABILIWOF FREQUENCYSTUDIES
731
r
9
z
o 00
<
1
500
0
1500
1000 Area (mi2)
2000 Figure27.10 Depth-area-duration curves for24-hr stormofFig.27'8. (FromRef' 23')
their of the mean. Listed in Table 27.6 are severalparametersof distributionsand are reliability, therefore and standard errors. It is apparent that standard errors, almost completelya function of the samplesize'
ConfidenceLimits based It is possibleto placeconfidencelimits on the measurementof a samplemean population' underlying the of on the normal distribution of all meansand regardless variate As mentionedearlier,approximatelytwo thirds of the observationsof a normal thirds two Therefore, deviation' shouldfall betweentheiimits of + i and 1 standard +olfi. percent 95 The limits of all sample means should occur between the confidencelimits for the mean are ment requires knowledgeof the u only s2insteadof o2 is known and confidencelimits for a samPlemeat For more the use of samplingdistributionsthat are beyond the scope of this text. and testing hypothesis information in ine neta of inferential statistics-in particular, statisticaldecisiontheory-the readermust turn to other sources. Approximateerror limits or control cufvescan be placedon freqlency curves. curve metdd proposedby Beardsinvolvesplacinglines aboveand below the fitted TABLE27.6 error Standard
Measure Mean Standarddeviation Coefficientof variation Coefficient of skewness
"/{!_ a/\/2n
c,\/r + zcl/vzn x6t"
- D/tn + l)(n - 2\(n + 3)
732
CHAPTER2T FREQUENCYANALYSIS TABLE27.7
Yearsof record (n) 5 10 I.)
20 30 40 50 '70 100
ERROR LIMITSFOR FLOOD FREQUENCYCURVES Exceedancefrequency (7o,at 5ololevel) 99.9
99
r.22 o.94 0.80 0.71 0.60 0.53 0.49 0.42 0.37
1.00 0.76 0.65 0.58 0.49 0.43 0.39 0.34 0.29
0.1
50
10
0.76 0.57 0.48 0.42 0.35 0.31 0.28 0.24 o.2l
0.95 0.58 0.46 0.39 0.31 0.27 0.24 0.20 0,17
2.12 1.07 0.79
10
50
90
Q.64
0.50 0.42 0.36 0.30 0.25
0.1
3.4r 1.65 1.19 0.97 0.74 0.61 0.54 0.44 0.36 99
4.41 2.11 1.52 t.23 0.93 0.77 0.67 0.s5 0.45 99.9
Exceedancefrequency (%, at 95% level) Nate: Tabular valuesare multiples ofthe standarddeviation of the variate. Five percent error limits are addedto the flood value from the fitted curve at the sameexceedancefrequency and the sum plotted. Ninety-five percent limits are subtractedfrom the flood value at the sameexceedancefrequency. Log values are added or subtractedbefore antilogging and plotting.
to form a reliability band. Table27.7 showsthe factorsby which the standarddeviations of the variatemust be multiplied to mark off a 90 percentreliability band above and below the frequencycurve. The 5 percentlevel, for example,meansthat only 5 percentof future valuesshouldfall higherthan the limit, and,similarly,only 5 percent shouldfall under the 95 percentlimit. Nine of ten should fall within the band. EXAMPLE 27.8
The maximum annual instantaneousflows from the Maury River near Lexington, Virginia, for a26-yearperiod are listed in Table 27.8. Plot the log-PearsonIII curve of best fit and determinethe magnitudeof the flood to be equaledor exceededonce in 5, 10, 50, and 100 years.Using Table 27.7, also plot the upper and lower confidencelimits.
Water (year)
Discharge (cfs)
1926 1927 1928 1,929 1930 t93l 1932 1933 1934
6,730. 9,150 6,310 10,000 15,000 2,950 8,650 I 1,100 6,360
1935 t936 1937 1938 1939 1940 t941 1942 t943
13,800 40,000 10,200 13,400 8,950 11,900 5,840 20:t00 12,300
Water (year)
Discharge (cfs)
1944 1945 1946 t947 1948 1949 1950 1951
6,680 6,540 5,560 7,700 8,630 14,500 23,700 15,100
STUDIES 733 OFFREQUENCY 27.5 RELIABILITY Solution 1. The statisticalcalculationsare summarizedas follows:
Log
Arithmetic Mean 7 Variances2 Skew coefficient C"
11,606
4.001
53.87 x 106 " 2.4
0.051b 0.38
2. After forming an array and computingplotting positions,the data are plotted in Fig. 27.11. 3. Plottingdatafor log-PearsonIII (Table27.9) aredevelopedfrom TableB.2' Confidencelimits are plotted in Fig. 27.II usingTable 27.7. r I 0.01 0.05 0.1 0.2 0.5 1 2
Log-Pearson IIIFit
V
5 F t 0
',;/
g r o d
-
:l
"
I bo 30
t.,
? + o d
5r)
'"
360 9" 70
t
d
6 a o
a
o
/-
o . 9 0
a,
95 98 99 99.8 99.9 99.99
r
2
4
6
1 0 2 0 4 0 6 (1000cfs) discharge Annualmaximum
0
1
0
Figure 27.11 Maximum instantaneous annual flows, Maury River, Lexington,Virginia.
0
;
734
CHAPTER2T FREQUENCYANALYSIS TABLE 27,9
Chance
('/") 99 95 90 80 50 20 l0 4 2 I
0.5
/ (vr)
(c" = 0.38) K
1.01 1.05 1.11 1.25 2 5 l0 25 50 100 200
-2.044 - 1.530 -
\.zJ+
-0.855 -0,062 0.818 1.315 t.874 2.251, 2.60r 2.930
(t: 4.001) (sy= 0'227) y + Ksy: logQ 3.537 3.653 3.721 3.760 3.987 4.ft1 4.300 4.426 4.512 4.591, 4.666
1 44?
4,498 < )6n
9,705 15,380 19,950 26,690 32,5t0 39,030 46,360
SERIES ANALYSISOF PARTIALDURATION 27.6 FREQUENCY In earlier examplesof frequency analysis,only the seriesof annual maximum or minimum occurrencesin the hydrologicrecord havebeen described.Theseextremes constitutean annual series thal is consistentwith frequencyanalysisand the manipulation of annualprobabilitiesof occurrence.All the observeddata-say, all floods or all the daily streamflows-would constitutea completeseries.Any subsetof the completeseriesis a partial series.In selectingthe maximumannualeventsfrom a record, it often happensthat the secondgreatesteventin one year exceedsthe annual maximum in some other year. Analysis of the annual seriesneglectssuch events. Although they generally contain the same number of events,the extreme values analyzedwithout regardfor the period (i.e., year)of occurrence,is usuallytermedthe partial duration series. In Table27.10 themaximumrainfall depthsthat occurredfor any 30-min period rainfallsat Baltimore,Maryland,1.945-1954,areshownin the order duringexcessive representa completeseries,The 11 maximum The 65 observations of occurrence. the annual series.the greatest11 events represent and annual eventsare underlined and representthe partial duration asterisk by an are identified throughoutthe record series. The larger numbersoccur in both series,and hencerecurrenceintervalsfor the less-frequenteventsare the same.The theoreticaldifferencesin recurrenceintervals based on annual and partial duration series of the same length are shown in TabIe27.Il. The differencefor intervalsgreaterthan 10 years is negligible.The following exampleis illustrative. EXAMPLE 27.9 Performa frequencyanalysisof the 30-min Baltimore rainfall data in Table27.L0 as an annual and apafiial duration seriesand plot the results. Solution. SeeTable27.I2. The data are plottedin Fig' 27'1'2. r r
DURATION SERIES ANALYSIS OF PARTIAL 27.6 FREQUENCY
735
MD,1945_1954 BALTIMORE, RAINFALL DEPTHS, TABLEi7.1O MMIMUM3O-MIN
Year 1945
Storm num0er 1 z J I
5 6 7 8 9
RF depth (in.)
Year 1.33* 0.65 0.47 0.84 0.68 0.63 0.47
0.38 0.47 0.39 0.76 0.56 0.35 0.43 0.40 0.36
Storm number
1953
RF depth (in.)
5 6 7 8
0.40 0.45 0.53 2.50* r.03 0.75 0.70 1.00*
1 2 3 4
0.42 0.70 0.85 0-30
1 2 3 4 5 6 'l
0.70 0.95 t.o2 0.50 0.65 0.55 0.52 0.45 0.54 0.60 0.80 0.95
1 z J A
0,52 0.49 1 2 J
4
5 6 7 8 9
1947
I z J A
5 6 7 8
0.62 0.55 0.88 0.47 0.36 1.15* 0.75 1.53* 0.51 0.88 2.04* 0.76 0,97 0.71 1.07* 0.94 r.20*
0.55 0.63 0.69 t.27* 1.10*
1955
0.88 0.97 0.59 0.46 0.50 0.55 t952
I 2 3 4 5 6 7 8
, 0.47 1.20* 0.93 0.70 0.57 0.46 0.48 1.30*
Nole: Underlineditems are the annual series.Asterisks identify the partial duration series.
BETWEEN THE TABLE27.1'I RELATION PARTIAL DURATION SERIES SERIES ANDTHEANNUAL Recurrenceinterval(yr) Partialdurationseries
Annualseries
0.5 1.0 1.5 2.0 5.0 10.0
t.2 1.6 2.0 2.5 5.5 10.5
8 9 0 I 2
736
CHAPTER2T FREQUENCYANALYSIS TABLE27.12
Recurrence interval (n + 1)/m
Depth(in.) Annual series
2.50 2.04 1.53 1.33 1.30 1.27 1.02 0.97 0.85 0.76 0.52
1 z J i
5 6 7 8 9 10 l1
Partial series
12 6
2.50 2.04 1.53 1.33 1.30 1.27 r.20 t.20 1.15 1.10 t.o7
A J A A
z
1.7 1.5 1.3 1.2 1.1
The preceding example leads to considerationof the frequency analysis of rainfall depth or iniensity for various durations of rainfall. Design problems often require the estimationof expectedintensitiesfor a critical time period. Frequency analysisof the rainfall record for.periodsother than the 30-min duration-for example, ihe maximum 5-, 10-, 20-, and 60-min occurrences-would yield a family of .oru"r similar to thoseof Fig. 27.10. The usual methodof presentingthesedata is to conveftdepthin inchesto an intensityin in. /hr andto summaize thedatain intensityduration-frequency curves as shown in Fig. 27.I3. Thesecurves are typical of the point analysisof rainfall data.It shouldbe emphasizedthat the frequencycurves.join -o""u.t"tt""t that are not necessarilyfrom the samestorm; that is, they do not repre-
/^
2.5
.i E
t <
0) E
t
_x--
1.0
d
I
zAnnta
1.01
1.r1.21.1 3.5
>-P
series
2,
3
4
5 678910
20
Recunenceinterval (Yr) Figune 27.12 Difference in annual and partial duration series 1 1-year record of maximum 30-min durations' Baltimore, Maryland'
PROBLEMS 741 Expand the computerprogram of Problem 26.12 to include the computationof the mean,standarddeviation;and skewnesscoefflcientofthe logarithmsofthe input data. Also, include a routine to sort the data by placing them in descendingorder and computethe correspondingplotting positions.verify, usiiig the datain Ptoblem27.4. 27.g, perform a completefrequencyanalysison one of the three 33-yearrecordsgiven in the tablebelow.Fit a Pearsontype III or log-PearsonIII and comparewith the normal or log-normal ofbest fit. Plot the dataand placecontrol curvesaroundthe theoretical curve of best flt usins.Table27.7.
27.t
Year t928 t929 1930 r93l 1932 1933 1934 r935 1936 193'7 r938 1939 1940 t94l 1942 1943 1944 1945 t946 1947 1948 t949 1950 1951 1952 1953 1954 1955 1956 1957' 1958 1959 1960
River Trempeuleau Dodge,Wl (DA : 643 mi') Qp..r.(cfs)
Blow River Banff, Alberta, Canada (DA = 858 mi' ) Opua,(cfs)
3,700 1,700 3,360 1,650 3,600 11,000 2,5'70 4,490 7,180 1,780 3,170 6,400 3,120 2,890 5,680 5,060 2,040 8,120 4,570 5,4r0 4,840 1920 3,600 4,840 6,950 4,040 5,710 10,400 r7,400 713
10,200 7,590 9,280 6,610 9,850 11,000 9,490 6,940 7,'720 5,210 7;770 6,270 7,220 4,450 5,850 7,380 5,590 4,450 7,2r0 5,880 r0,320
r,r40
6,730 7,480 6,440
8,000 1,480
4 )qo
10,080 8,570 5,460 9,180 10,120 8,680 9,060 5 160
James River Scottsville,VA (DA : 4570 mi2) Qp""r (cls)
75,600 44,700 45,800
2t,roo 31,400 5q 500
38,800 93,400 126,000 62,200 87,400 68,400 130,000 27,100 80,600 95,200 133,000 57,000 41,200 33,200 59,600 94,200 64900 54 snn 67,000 62,900 70,000 20,400 64,200 44,500 ,o ?no 64.200
742
CHAPTER 27 FREQUENCY ANALYSIS
27.10. Compare results of Problem 27.9 wfih estimatesby Gumbel's extreme-valuedistribu' tion for the 50- and 100-yearevents.
27.11. The pan-evaporationdata (in.) for the'month of July at a site4nMissouri are 9.7 11.2 9.3 9.8
lt.7 8.8 9.2 8.7
It.2 tt.4 9.3 I1.5
11.3 I 1.8 9.3 10.9
l 1.5 8.9 10.4 10.2
Determinethe mean, standarddeviation,and coefficient of variation. What are the standarderrors of thesestatistics?Establishthe approximate95 percentconfldence limits of the mean. 27.12. On which type of plotting paper(probability,log-probability, rectangularcoordinate, log-log, semilog,extreme-value,none)would eachof the following plot as a straight line? a. Normal frequencydistribution. b. Gumbel frequencydistribution. c.Y=3X+4 d. Log-normal frequencydistribution. e. PearsonType III with a skew of zero. t' Q: 43Ao1' g. Log-PearsonType III with a skew of logarithmsequal to zero. h. PearsonType III with a skew of 3.0. 27.13. Determine the 50-yearpeak (cfs) for a log-PearsonType III distribution of annual peaksfor a major river if the skew coefficient of logarithms (base 10) is -0.1, the mean'oflogarithms(base10) is 3.0, and the standarddeviationof basel0logarithms is 1.0. 27.14. A 4Oaear record of rainfall indicatesthat the mean monthly precipitation during April is 3.85 in. with a standarddeviationof 0.92. The distribution is normal. With 95 percentconfidence,estimatethe limits within which (a) next April's precipitation is expectedto fall, and (b) the mean April precipitation for the next 40 years is expectedto fall. 27.15. Given a table of valuesof mean annual floods and correspondingdrainageareasfor a number of basinsin a region, describehow regressionanalysiscould be used to determinethe coefficientsp and q in the relation Qz.tz: pAq. 27.16. The 80-yearrecord of annualprecipitationat a midwesterngaugelocationhasa range between14 in. in 1936and 42 in. in 1965.The recordha5a meanof 27.6 in. anda standarddeviationof 6.06 in. Assuminga normal distribution, (a) plot the frequency curve on probability paper,(b) determinethe probability of a droughtworsethan the 1936value,and (c) determinethe recurrenceinterval.ofthe 1965maximumdepthand comparei&ith the apparentrecurrenceinterval. 27.17. A reservoirin the localeof Problem27.16 wlll overfill when the annualprecipitation exceeds30 in. Determinethe probability that the reservoirwill overfill (a) next year, (b) at leastonce in three successiveyears,and (c) in each of three successiveybars.
PROBLEMS
743
27.18. *Using Eqs. 26.38 and 26.39 and log-probability paper, solve Problems27.16 and 27.17 assumingthat the annual precipitation is log-normal. 27.19. Given the following valuesof peak flow ratesfor a small stre.am,determinethe return period (years)for a flood of 100 cfs by flrst using annual peaksfor an annual series and then using all the data for a partial duration series.
Year
1963 t964 1965 1966 1967
June I Aug.3 June7 July 2 May 18 June 3 July 4
Peak (cfs)
Year
Date
Peak (cfs)
90 300 60 80 r00 90 40
1968
May 11 June 8 Sept.4 Aug. 8 May 9 Sept.8 May 4
800 700 90 400 30 700 80
1969 r970 t97r r972
27,20. Recordedmaximum depths (in.) of precipitation for a 30-min duration at a single station are:
Year
Date
Depth(in.)
Year
Date
Depth (in.)
1963
May 3 June 3 June7 June 2 June 1 Aug. 3 July 4
2.0 1.0 1.0 1.0 1.0 3.0 1.0
1968 1969
Aug.8 May 6 June 8 Sept.4 May 4 Sept.8 May 9
4.O 6.0 5.0 1.0 1.0 5.0 1.0
1964 1965 1966 1967
a. Determine the return period (years) for a depth of 2.0 in. using the california method with an annual series. b. RepeatPart (a) using a partial duration serles. c. Determinefrom the partial duration seriesthe depth of 30-min rain expectbdto be equaledor exceeded(on the average)once every 8 years. 27.21. For a 60-yearrecord of precipitation intensitiesand durations,a 30-min intensity of 2.50 in.lhr was equaledor exceededa total of 85 times. Al1 but 5 of the 60 years experiencedone or more 30-min intensitiesequaling or exceedingthe 2.50-in./hr value.Use the Weibull formula to determinethe return period of this intensity using (a) a paltial seriesand (b) an annual series.
744
ANALYSIS 27 FREQUENCY CHAPTER 27,22. from the data given in the accompanyingtable of low flows, prepare a set of low-flow frequency curves for the daily, weekly, and monthly durations. (cfs)FORTHE LOWESTMEANDISCHARGE NUMBER OFCONSECUTIVE FOLLOWING DAYS,MAURYRIVERNEARBUENAVISTA, VIRGINIA Year
1-day
7-day
1939 1940 194l 1942 1943 t944 1945 1946 1947 t948 1949 1950 195I 1952 1953 1954 1955 t956 1957 1958 1959 1960 1961 1962 1,963 1964 1965
100.0 167.0 22.0 101.0 86.0 62.0 78.0 76.0 97.0 r54.0 136.0 113.0 95.0 115.0 85.0 68.0 83.0 64.0 62.0 88.0 76.0 83.0 99.0 90.0 60.0 51.0 64.0
103.0 171.0 59.4 127.0 93.9 65.9 80.7 78.6 102.0 176.0 138.0 125.0 95.3 116.0 86.1 70.0 96.1 66.3 64.1 92.6 80.9 91.7 103.0 95.0 60.6 54.1 68.7
3O-day
r25.0 t94.0 69.1 1'73.0 103.0 77.4 90.3 87.1 123.0 215.0 163.0 139.0 101.0 r20.0 90.8 81.7 99.9 7r.7 75.8 107.0 117.0 103.0 152.0 105.0 70.8 62.0 76.2
27.23. For the 7-day low flows at Buena Vista given in Problem 27.22, attempt to fit a straight-linefrequencycurve on log-normalor extreme-valueprobability paper' proceedingas follows:From the original plot of the data, estimatethe lowestflow (say;4) - q: at the high recurrenceintervals;subtractthis flow from all observedflows (Q Qr); and rcplot Qr versusthe original recurrenceintervals.Repeatif necessary.The best fittiBg curve will be a three-parameterfrequencydistribution.
PROBLEMS 745 27.24. The following table surirmarizesthe number of occurrencesof intensitiesof various durationsfor a34-yearrecordof rainfall. Maximum intensitiesfor the given durations were determinedfor all excessivestorms and a count made of the exceedances' expectedon a 5-yearfrequencyand Interpolatefor the averagenumberof exceedances plot the 5-yearintensity-duration-frequency curve' Numbei of times stated ihtensitieswere equaledor exceeded Intensity(in./hr) Duration (min)
5 10 15 30 60 120
2.0 68
3s l7
72 29 1 8
5
6 l
3.0
4.0
5.0
6.0
7.0
'73
48 26 11 3 1
2I l1
o
2
3
I
51 LJ
7 2
I I
27.25. The resultsof a multiple regressionanalysisof over 200 flood recordsin Virginia led to the following regional flood frequencyequations: . Qt'z'v': 9'l3AeoeS2e3 Qzzt-v':20'84861S30e 3oo Qs'v,: 38'1A83oS 283 Qn', : 63'0A8o2S : 104A71e5266 Qzs-v, : l18A7esS21e Qso'v'
27.26.
27.27.
21.28.
27,29.
whefe the flood dischargefor the given frequencyis in cfs, A is the drainageareain mi2, and S is the channeislopein ftlmi (measuredbetweenthe points that are 10 and g5 percent of the total rivei miles upstreamof the gauging station to the drainage diviAe;. Devise a method for graphically portraying theseregional flood frequency relations.(Note that there are four factors,Q, T, A' and S') Using the regressionequationsin Problem 2'1.25, flnd the predicted floods for the River, at cootes Store.Drainage atea : 215 mi2and chanNorth Fork, shenandoa-h : nel slope 44.3 ftlmi. Comparethe predictions from the regressionequationsin Problem 27 '25 with the : valuesestimatedby the frequencyanafsis inqxample27.8. Drainageatea 487 fii2 : and channelsloPe 2I.l ftltrl'l. Referringto Fig. 2.6b, comparethe averagestorm rainfall over the city of Baltimore on SeptemberI0, Ig57, computedby the isohyetalmethod, with the simpleaverage of total accumulationat the riin gaugeswithin the city. Neglectthe areato the south of the 1.0-in.isohYet' Fit the forrnula i : AIQ + B) to the data derived in Problem 21.24 for the 5-year intensity-duration-frequency curve.
746
27 FREQUENCYANALYSIS CHAPTER 27.30. Deielop a regional flood index curve for the RappahannockRiver basin from the flood frequency data given in the following table: PEAK FLOOD FREQUENCYDISCHARGES(ft3/sec)FOR STATIONSlN THE RAPPAHANNOCKRIVER BASIN
Station
Drainage Typeof area(mi2) series 192
RappahannockRiver near Warrenton, VA Rush River at Washington,VA Thornton River near Laurel Mills, VA Hazel River at fuxeyville, VA RappahannockRiver at Remington,VA RappahannockRiver at Kellys Ford, VA Mountain Run near Culpeper,VA Rapidan River near Ruckerwille, VA RobinsonRiver near Locust Dale, VA Rapidan Rivet near Culpeper,VA RappahannockRiver near Fredericksburg, VA
15.2 142 286 616 641 14.'7 111 180 456 | sqg
Returnperiodin years
2.33 (mean).
10
25
Annual Partial
4,150 4,600
8,350 8,650
9,000 9,20A
14,000 14,000
19,250 19,250
Annual Partial Annual Partial Annual Partial Annual Partial Annual Partial Annual Partial Annual Partial Annual Partial Annual Partial Annual Partial
530 610 5,900 7,200 7300 8,300 11,000 12,000 12,300 14,000 '750
860 900 11,500 12,500 11,800 t2,400 14,500 15,200 19,000 20,000 t,750 1,900 7,100 7,700 7,000 7,300 16,400 17,600 39,900 42,000
r,290 1,310 19,900 20,500 17,200 18,000 18,100 18,900 26,800 27,500 3,350 3,550 11,600 12,000 9,800 10,100 26,900 27,600 55,000 57,500
2,100 2,100 34,000 34,000 25,000 25,500 24,500 25,000 42,000 42,000 6,000 6,000 21,000 21,000 !5,400 15,800 50,000 50,000 85,000 85,000
3,000 3,000 48,000 48,000 41,000 41,000 31,000 31,000 57,500 57,500 10,000 10,000 34,000 34,000 2r,5oo 21,500 78,000 78,000 117,000 117,000
950 3,950 4,700 4,600 5,150 9,100 10,800 26,000 29.300
27.31. From the information given in Problem 27.30, find the relation betweenthe mean annual flow and the drainagearea.(Note that the functional expressionshouldbe of the form Qz.zz: rA".) 27.32. Using the resultsof Problems27.30 and27.3l, estima{ethe 30-yearflood for an ungaugedwatershedwith a drainagearcaof 540 mi2. 27.33. Annual flood recordsfor a lO-yearperiod are given by:
Year Flood
1 300
2 700
3 200
4 400
5 1000
6 900
7 800
6
500
o
100
10 600
Mean : 550 cfs, median : 550 cfs, standarddeviation : 300 cfs. Use an annual seriesand the definition offrequency in a frequencyanalysisto determinethe magnitude of the 4-yearflood. Comparethis historicalvaiuewith the analytical4-yearflood obtained assuminsfloods follow a normal distribution.
PROBLEMS 747 27.34. For a 60-yearrecordof precipitationintensitiesanddurations,a 30-min intensityof 2.50 in.i hr wasequaledor exceededa total of 85 times.All but 5 of the 60 yearsexperienced one or more 30-min intensitiesequalingor exceedingthe 2.50-in./hr value. Use the Kimball formula to determine the return period of this intensity using (a) a partial duration seriesand (b) an annualseries. 27.35. The total annualrunoff from a small drainagebasinis determinedto be approximately normal with a mean of 14.0 in. and a standarddeviation of 3 in. Determine the probabilitythat the annualrunofffrom the basinwill be lessthan 8.0 in. in the second year only of the next three consecutiveyeafs. 27.36. Six yearsof peakrunoff ratesare given below.Assumethat the floodsfollow exactlya normal distributionand determinethe magnitudeof the 5O-yearpeak.
Year Runoff (cfs)
1 200
2 800
3 500
4 600
5 400
6 500
21.37. Annual floodsfor a streamare normally distributedwith a mean of 30,000 cfs and a returnperiodT,of a32,000-cfsflood varianceof I x 106cfs2.Determinethe average in the stream. 27.38. Annual floodsfor a streamhavea normal frequencydistribution.The 2-yearflood is 40,000cfs and the l0-year flood is 52,820cfs.Determinethe magnitudeof the 25-year flood. 27.39. The 80-yearrecord of annual precipitation at Linclon, Nebraska,yields a range of valuesbetween10 and 50 in. with a meanannualvalueof 25.00 in. and a standard deviationof 5.30 in. Becauseannualprecipitationrepresentsa sum of many random variables(i.e., depth of precipitation for eachday of the year), assumethat annual precipitation is normally distributed. a. In 1936 the precipitation at Lincoln was a mere 14 in. Determinethe probability that the annual precipitation will be 14 in. or lessnext year. b. In 1965Lincoln received42 in. On the average,this amountwould be equaledor exceededonce in how many years? c. Comparethe theoreticaland apparentreturn periods of the record-highvalue of 50.00in. 27.40. Annual floods (cfs) at a particular site on a river follow a zero-skew log-Pearson Type III distributions.If the mean of logarithms(base 10) of annual floods is 2.946 and the standarddeviationof base-10 logarithmsis 1.000,determinethe magnitude of the 50-yearflood. 27,41,. Annual floods (cfs) at a particular site on a river follow a zero-skew log-Pearson 1.733and annualfloodsis Type Illdistribution.Ifthemeanoflogarithms(base10)of the standarddeviationof base-l0logarithms is L.420,determinetfe magnitudeof the 10O-year flood. 27.42 The 100-yearrecord for a drainagebasin gives 10- and 5O-yearflood magnitudesof 12,500 and22,000 cfs. Determinethe magnitudeof the mean annual flood if (a) the flood peaksfollow the index-flood curve of Fig. 27.4c atd (b) the flood peaksfollow a Gumbel extreme-valuedistribution.
748
CHAPTER27
FREQUENCYANALYSIS *The following parameters were computed for a stream: 27.43.
Period ofrecord : 1960-1984, inelusive. Mean annual flood : 7000 cfs Standarddeviationof annual floods : 1000 cfs Skew coefficientof annual floods : 2.0 Mean qf logarithms(base l0) of annual floods : 3.52 Standarddeviationof logarithms : 0.50 Coefficientof skew of logarithms : -2I Determine the magnitudeof the 25-yearflood by assumingthat the peaks follow a (d) log-PearsonType III distribution, (b) Gumbel distribution, and (c) log-normal distribution. 27.44. Peak annual dischargerates in the Elkhorn river at Waterloo,Nebraska,yield the following statistics: Period of record : 1930-1969, inclusive Mean flood : 16,900cfs Standarddeviation : 17,600cfs Skew of annual floods : 0.8 Mean of logarithms(base 10) : 4.0923 Standarddeviationof logarithms : 0.3045 Skew of loearithms : 2.5 Determinethe 100-yearflood magnitudeusing the uniform techniqueadoptedby the U.S. Water ResourcesCouncil for all federal evaluations. b. Determinethe'100-yearflood magnitudeassumingthat the floods follow a twoparametergamma distribution. 27.45. A PearsonType III variableX has a mean of 4.0, a standarddeviationof 2.0, and a of I:f.(X)dX. coefficientofskewof 0.0.Determinethevalue(foursignificantfigures) .
27.46, A timber railroad bridge in Hydrologic Region2 of Texasat Milepost 738.04 on the railroad systemshownin the sketchis to be replacedwith a new concretestructure. The 50- and 100-yearflood magnitudesare neededto establishthe low chord and embankmentelevations,respectively.Determinethe designflow ratesusingthe USGS RegressionEquations.The drainagearea is 0.43 sq. mi, and the streambedslopeis 62 ftoer mi.
REFERENCES t . M. A. Benson,"Plotting Positionsand Economicsof EngineeringPlanning,"Proc. ASCE J. Hyd. Div. 88057-71(Nov. 1962). L I. Gringorten,"A Plotting Rule for ExtremeProbability Paper,"J. Geoplrys.Res'68(3), 8 1 3 - 8 1 4 ( F e b1. 9 6 3 ) .
ucENoftp.*
l l * *l* --) .J$ ^ " : 1- ' .
!
,
I
i
I
t \
\l Sketch for Problem 27.46
I I
Slrip
]
750
CHAPTER2TFREQUENCYANALYSIS 3. VerfT. Chow, "A General Formula for Hydrologic FrequencyAnalysis," Trans. Am. Geophys. Union 32, 231-237 (1951). 4. Water ResourcesCouncil, Hydrology Committee, "Guidelines:for Determining Flood Frequency,"Bulletin 17B, (Revised)U.S. Water ResourcesCouncil, Washington,D.C., Sept.,1981. 5. L. R. Beard, Statistical Methods in Hydrology, Civil Works Investigations,U.S. Army Corps of Engineers,SacramentoDistrict, 1962. 6. A. Hazen, Flood Flows. New York: Wiley, 1930. 7. V. T. Chow, "Statistical and Probability Analyqisof Hydrologic Data," in Handbookof Applied Hydrology.New York: McGraw-Hill, 1964. 8. P. Victorov, "Effect of Period of Record on Flood Prediction," Proc. ASCE J. Hyd. Div. 97(Nov.1971). 9. L. R. Beard, Statistical Methodsin Hydrology, Civil Works Investigations;Sacramento District, U.S. Army Corps of Engineers,1962. 10. M. A. Benson and N.C. Matalas, "Synthetic Hydrology Based on Regional Statistical " Water Re Parameters, sources Res. 3(4)(1967). 11. N. C. Matalas,"MathematicalAssessment of SyntheticHydrology," WaterResourcesRes.
3(4)(re67).
12. Yet T. Chow, "Statistical and Probability Analysis of Hydrologic Data," Sec. 8-I, in Handbookof Applied Hydrology (V. T. Chow, ed.). New York: McGraw-Hill, 1964. 13. T. Dalrymple, "Flood-FrequencyAnalysis,"Manual of Hydrology,Part 3, U.S. Geological $urveyWater-SupplyPaper1543=A.Washington,D.C.: U.S. GovernmentPrinting Offlce, 1960. 14. G. M. Fair,J. C. Geyer,andD. A. Okun,WaterandWasteWaterEngineerlng. New York: Wiley, 1966. 15. "Monthly Stream Simulation," Hydrologic EngineeringCenter, ComputerProgram23C-L267, SacramentoDistrict, U.S. Army Corps of Engineers,July 1967. 16. R.W.CruffandS.E.Rantz,"AComparisonofMethodsUsedinFloodFrequencyStudies for CoastalBasinsin California," Flood Hydrology,U.S.G.S.Water Supply Paper 1580: Washington,D.C.: U.S. GovernmentPrinting Office, 1965. 17. W. D. Potter, "Peak Ratesof Runoff from Small Watersheds,"Hydraulic Design Series No. 2, Bureau of Public Roads,Washington,D.C.: U.S. GovernmentPrinting Offlce, Apr. 1961. 18. U.S. GeologicalSurvey, "Technique for Estimating the Magnitude and Frequencyof Floods in Texas," WaterResourcesInvestigationsReport 77 -110, 7977. t9. M. E. Jennings,W. O. Thomas, Jr., and H. C. Riggs, "Nationwide Summary of U.S. GeologicalSurvey's RegionalRegressionEquationsfor Estimating Magnitude and Frequencyof Floodsat UngaugedSites,"U.S.G.S.WRI 93-1, Reston,VA, 1993. 20. U.S. Geological Survey, "Selected Streamflow Characteristicsas Related to Channel Geometry of Perennial Streamsin Colorado," Open-File Report 12-160, Water ResourcesDivision, Lakewood,Colorado,May 1972. 2 r . E. W. Steel,l{ater Supplyand Sewerage,4th ed. New York: McGraw-Hill, 1960. 22. D. M. Hershfield,"Rainfall FrequencyAtlas of the United States,"Tech.PaperNo. 40, U.S. WeatherBureau.1961. 23. HydrologyHandbook, ASCE Manual of Practice,No. 28, 1949. 24. W. G. Knisel, Jr.,and R. W Baird, in Al?SPrecipitation Facilities and RelatedStudies. Washington,D.C.: U.S. Departmentof Agriculture, Agricultural ResearchService,1971, Chap.14, , \ R. K. Linsley, Jr., M. A, Kohler, and J. L. H. Paulhus,Applied llydrology. New York: McGraw-Hill, 1949.
Appendixes A APPENDIX: FACTORS ANDCONVERSION CONSTANTS. TABLEA.1 WATERPROPERTIES. Heat of vaporizationof water at 1-.0atm Gasconstants(R) 540 callg: 970 Btu/Ib R : 0.0821(atm)(liter)/(g-mol)(K) R : 1.987g-ca1/(g-mol)(K) R : 1.987Btu/(lb-mo1)('R) Specificheat of air Accelerationof gravity (standard) g : 32.17ftlsecz:980.6 cm/sec2 Cp : 0.238call(g)("C) Density of dry air at OoCand 760 mm Hg 0.001293g/cm3 Heat of fusion of water 19.7 callg: l44BtulIb Conversionfactors I second-foot-dayper squaremile : 0.03719 inch
*t" 1inchofrunofrpersquare
: 33:3:::::r1;?t-0"t, : 2,323,200cubic feet
I cubic rootpersecond t3jL":'"i'ff1":* n'"' : ? I horsepower : 3Jr-f,f*:llld,p".,..ond
e = 2.71828 log e = 0.43429 ln 10 : 2.30259 Metricequivalents foot : 0.3048 meter mile : 1.609 kilometers acre : 0.4047 hectare 4047 squaremeters I squaremile (mi') : 259 hectares 2.59 squarekilometers (km2) 1 acre foot (acre-ft) : 1233 cubic meters I million cubic feet (mcf ) : 28,320 cubic meters I cubic foot per second(cfs) : 6.6rta, cubic metersper second 1.699 cubic metersPer minute I acre-in. per hour : 1.008cubic feet per second(cfs) " 1 second-foolday (cfsd) = 2447 cubic meters I million gallons (mg) = 3785 cubic meters 3.785 million liters 1 million gallons per day (mgd) = 694.4 gallons per minute (gpm) 2,629 cubic meters per :ninute 3785 cubic metersper day
752
APPENDIXA TABLE A,2* PROPERTIESOF WATER
Traditional U.S.Units
iz
40 50 60 70 80 90 100
Heat of vaporization (Btu/lb)
Unit weight 0b/ft3)
Temperature Specific gravity ('F) 0.99987 0.99999 0.99975 099907 0.99802 0.99669 0.99510 0.99318
62.416 62.423 62.408 62.366 62.300 62.217 62.11,8 61.998
t073 1066 1059 r054 r049 l044 1039 1033
Vaporpressure
Kinematic viscositY (ft'lsec)
1,.93x L67 x - 1.41 X 1.21x 1.06 x 0.929x 0.828x 0.741X
10*5 10-s 10-5 105 10-s l}-s l0-5 10-5
psl
in.Hg
0.18 0.25 0.36 0.52 0.74 1.03 1.42 r.94
6.11 0.09 8.36 0.12 12.19 0.18 17.51 0.26 24.79 0.36 34.61 0.51 47.68 0.70 64.88 0.95
Sl Units Temperature Specific Density gravity (o/cm1 fC) 0 5 10 15 20 25 JU
35 40 50 60 70 80 90 100
0.99987 0.99999 0.99973 0.99913 0.99824 0.99708 0.99568 0.99407 0.99225 0.98807 0.98323 0.97780 0.97182 0.96534 0.95839
0.99984 0.99996 0.99970 0.99910 0.9982r 0.99705 0.99565 0.99404 0.99222 0.98804 0.98320 0.97777 0.97179 0.96531 0.95836
Heat of vaporization (cal/g)
597.3 594.5 591.7 588.9 586.0 583.2 580.4 577.6 574.7 569.0 563.2 551.4 545.3 539.1
Kinematic viscosity (cs)
1..790 .t.520 1.310 1.140 1.000 0.893 0.801 0.723 0.658 0.554 0,4'14 0.4r3 0.365 0.326 0.294
Vaporpressure (mmHg) 4.58 6.54 9.20 12.78 17.53 23.76 31.83 42.18 55.34 92.56 149.46 233.79 355.28 525.89 760.00
(mb)
(g/cm')
6.23 6.11 8.89 8.72 L:2.27 r2.s1 17.38 17.04 23.83 23.37 32.30 31.67 43.2.7 42.43 5'7.34 56.24 75.23 73.78 123.40 125.83 199.26 203.19 311.69 317.84 473.67 483.01 70r.13 714.95 1013.25 1033.23
B APPENDIX
753
B APPENDIX: TABLE 8.1 AREAS UNDERTHE NORMALCURVE
,<,t=1,f:*"*'"t'a, 01 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 r.0 1.1 r.2 r.3 r.4 1.5 r.6 r.7 1.8 r.9 2.0 2.1 2.2 z-5
2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 J.Z 5.5
3.4
4.0
.0000 .0398 .0793 .1179 .t554 .1915 .2257 .2580 .2881 .3159 .3413 .3643 .3849 .4032 .4192 .4332 .4452 .4554 .4641 .4713 .4772 .4821 .486r .4893 .4918 .4938 .4953 .4965 .4974 .498t .4987 .4990 .4993 .4995 .4997
.0040 .0438 .0832 .1217 .1591 .1950 .2291, .2611 .2910 .3186 .3438 .3665 .3869 .4049 .4207 .4345 .4463 .4564 .4649 .4719 .4778 .4826 .4865 .4896 .4920 .4940 .4955 .4966 .49:15 .4982 .4987 .4991 .499? .4995 .499't
.02 .0080 .0478 .0871 .1255 .1628 .1985 .2324 .2642 .2939 .3212 .3461 .3686 .3888 .4066 .4222 .4357 .44't4 .4573 .4656 .4' 126 .4783 .4830 .4868 .4898 .4922 .4941 .4956 .4967 .4976 .4983 .4987 .4991 .4994 .4996 .4997
.03 .0120 .0517 .0910 .1293 .1664 .2019 .2357 .2673 .2967 .3238 .3485 .3708 .390'7 .4082 .4236 .4370 .4485 .4582 .4664 .4732 .4788 .4834 .487r .490r .4925 .4943 .4957 .4968 .49'77 .4983 .4988 .499r .4994 .4996 .4997
.04 .0159 .0557 .0948 .1331 .1700 .2054 .2389 .2704 .2995 .3264 .3508 .3729 .3925 .4099 .4251 .4382 .4495 .459r .467r .4738 .4793 .4838 .4875 .4904 .4927 .4945 .4959. .4969 .4977 .4984 .4988 .4992 .4994 .4996 .4997
.o7
.05 .0199 .0596 .0987 .1368 .1736 .2088 .2422 .2734 .3023 .3289 .3531 .3749 .3944 .4115 .4265 .4394 .4505 .4599 .4678 .4744 .4798 .4842 .48"18 .4906 .4929 .4946 .4960 .4970 .49'78 .4984 .4989 .4992 .4994 .4996 .4997
.0239 .0636 .1026 .1406 .r772 .2123 .2454 .2'764 .3051 .33t5 .3s54 .3770 .3962 .4t3t .4279 .4406 .4515 .4608 .4686 .4750 .4803 .4846 .4881 .4909 .4931 .4948 .4961 .4971 .4979 .4985 .4989 .4992 .4994 .4996 .4997
.0279 .0675 .1064 .1443 .1808 .21,57 .2486 .2794 .3078 .3340 .3577 .3790 .3980 .4147 .4292 .4418 .4525 .4616 .4693 .4756 .4808 .4850 .4884 .4911 .4932 .4949 .4962 .49' 72 .4980 .4985 .4989 .4992 .4995 .4996 .4997
08 .0319 .07t4 .1103 .1480 .1844 .2190 .2518 .2823 .3106 .3365 .3599 .3810 .3997 .4162 .4306 .4430 .4535 .4625 .4699 .4762 .4812 .4854 .4887 .4913 .4934 .4951 .4963 .4973 .4980 .4986 .4990 .4993 .4995 .4996 .4998
.09 .0359 .0753 .rr4r .1517 .1879 .2224 .2549 .2852 .3133 .3389 .3621 .3830 .4015 .4177 .4319 .444r .4545 .4633 .4606 .4767 .4817 .4857 .4890 .4916 .4936 .4952 .4964 .4974 .498r .4986 .4990 .4993 .4995 .4997 .4998
.499968
(for Source:AfrerC. E. Weatherburn,Mathematical Statistics.London: CambridgeUniversity Press,1957 z : 0 to z : 3,1); C. H. Richardson,An Intoduction to Statistical Analysis. Orlando, FL: Harcourt Brace Jovanovich.1994(for z = 3.2to z: 3.4); A. H. Bowker and G. J. Lieberman,EngineeringStatistics. Eagl€wooGCliffs,Nf: Prentice-Hall, 1959 (for z : 4.0 and 5.0)'
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759 Open channelflow, 131 Orographicprecipitation, 19 Overland flow, 561 Pan, classA, 129 Parshallflume, 131 Partial duration series,734 Partially penetratingwells, 473 Peakflow,3lI-343 Penmanmethod"97 Permeability,438 Piezometrichead,435 Point precipitation, 27 Potentialevapotranspiration,101 Potentialtheory, 463 Precipitablewater, 16 Precipitation,7, 15-39, 164 Probability, 67I-699 Probablemaximum flood. 373 Probablemaximum precipitation (PMP), 34,392 Rainfall. 281.727 Rain gauges,127 Rationalmethod,312 Recession, 574 Regressionequation,692 Relativehumidity, 16 Remotesensing,135 Reservoirs,245 Reynoldsnumber,437 Road ResearchLaboratory (RRL) Method,631 Routing,234-257 Runoff, 153-169, 302 Runoff curve number,73 Saltwaterintrusion. 474 SCS:seeSoil ConservationService SCSAtt-Kin Tr-20 Method,254 SCSmethod,73 SCSTR-55 Method, 320,445 SCSTP-149Method,331 S-hydrograph,198-201 Simulationmodels,548-59 1, 594-628,630
Snow,265-305 Snowmelt,271-284 Snyder'smethod,207 Soil ConservationService(SCS) runoff curve number,73 unit hydrographmethod,211 Soil moisture,25,55, 165 Specificyrelds,432 Standarddeviation,683 Standardproject storm (SPS),395 Stanford WatershedModel IV
(swM-IV),550 Statisticalanalysis,67| -699, 7O8-739 Steadyflow routing, 252 methods,508 Stochastic. Storm drainage,402 Storm Water ManagementModel (swMM), 604,650 111-118, l7 l, 574 Streamflow, StreamflowSynthesisand Reservoir RegulationModel (SSARR), 565 Streamlines,446 448 Surfaceof seepage, Surfacerunoff. 177
swMM,650
Syntheticunit hydrographs,205-227, 335 Temperatureindexes,291 Theis' nonequilibrium approach,467 Thiessenmethod. 30 Thunderstorms,20 Time of concentration,182-185 Time series,535-544 Transmissivity,444 Transpiration,95- 100 Unconfined aquifer, 443, 461 Uniform flow field,463 Unit hydrographs,188-227 U.S. GeologicalSurvey Index-Flood Method.336 U,S. GeologicalSurvey Rainfall-Runoff Model, 597
University of Cincinnati Urban Runoff Model (UCURM), 655-658 Unsteadyflow,442,467 Unsteadyflow routing, 252 Urban drainage,309 Urban runoff models.309-351 Velocity measurement,114 Velocity potential, 440
Water budget,86, 293, 5'l.I Water equivalent,268 : Water t'apor, 16 WatershedHydrology Model
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