Chapter7 FLOOD ROUTING ________________________________________________________ ____________________________________
7.1
Introduction
The flood hydrograph discussed in many hydrology and hydraulics textbooks is in fact a wave. The stage and discharge discharge hydrographs of unsteady and non-uniform non-uniform flows flows represent the passage of waves of either through a river or a reservoir. As this wave moves down the river, the shape of the wave gets modified due to various factors, such as storage, frictional resistance, lateral addition or withdrawal of flows flows etc. When a flood wave passes through a reservoir, its peak is attenuated and the time base is enlarged due to the effects of storage. lood waves passing down a river have their peaks attenuated due to frictional resistance if there is no lateral inflow. The addition of lateral inflows can cause a reduction of attenuation or even an amplification of the flood wave. The aspects of the changes in a flood wave passing through a channel system form the basis of flood routing. !n general, flood routing is the techni"ue of determining the characteristics of a flood hydrograph at the outlet of a reservoir or at a downstream section of a river channel by utili#ing the data of flood flow at one or more upstream sections. The hydrologic analysis of problems such as flood forecasting, flood protection, reservoir design and spillway design invariably re"uire inputs from flood routing analysis. !n these applications two broad categories of routing can be recogni#ed. These are$ %. &eservoi &eservoirr or storage storage routing routing and '. chann channel el routing routing !n reservoir routing, which is also occasionally referred to as lumped routing, the effect of a flood wave entering entering a reservoi reservoirr is studied. studied. (nowing (nowing the volumevolume-ele elevati vation on characteri characteristi stics cs of the reservoir and the outflow-elevation relationship for the spillways and other outlet structures in the reservoir, the effect of a flood wave entering the reservoir is studied to predict the variations of reservoir elevation and discharge outflow with time. This form of reservoir routing is essential in$ %. the design of the capacity of spillwa spillways ys and other reservoir outlet structures and '. the location and si#ing si#ing of the capacity of reservoirs reservoirs to meet specific re"uirements. re"uirements. !n channel routing, which is also referred to as distributed routing, the change in the shape of a hydrograph as it travels down a channel is studied. )y considering a channel reach and an input hydrograph at the upstream end, this form of routing aims to predict the flood hydrograph at various sections of the reach. !nformation on the flood peak attenuation and the duration of high water levels obtained by channel routing is of utmost importance in flood forecasting operations and flood protection works.
A variety of routing methods are available and they can be broadly classified into two categories namely$
*-%
i. hydro hydrolog logic ic routing routing and ii. ii. hydraul hydraulic ic routing routing +ydrologic routing methods employ essentially the e"uation of continuity. +ydraulic routing methods, on the other hand, utili#e both the continuity e"uation and the e"uation of motion derived derived from ewtons ewtons second second law of motion/ motion/ for unsteady unsteady flow. flow. The limi limitati tations ons of data unavailabi unavailabilit lity y notwithstanding, the basic basic differe differential ntial e"uations used in the hydraulic hydraulic routing, known as the 0t. 1enant e"uations afford a better description of unsteady flow, than hydrologic methods.
7.1. 7.1.1 1
Thee Basi Th Basicc Eq Equa uati tion onss
The passage of a flood through a reservoir or a channel reach reach is an unsteady flow phenomenon. phenomenon. !t is classified in open channel hydraulics as gradually varied unsteady flow. The lumped form of the e"uation of continuity, which is used in all hydrologic lumped/ routing as primary primary e"uation states that, the differenc differencee between the inflow inflow rate !/ and the outflow rate 2/ is e"ual to the rate of change of storage 0/, i.e. I
−O =
dS
------- *.%
dt
2n the other hand, the hydraulic or distributed/ routing re"uires the distributed form of the continuit continuity y and momentu momentum m conservat conservation ion diffe different rential ial e"uations. e"uations. The diffe different rential ial e"uation e"uation of continuity for unsteady flow in a reach no lateral inflow is given by$ ∂
V
onconservative onconservative form$
∂
y
+
x
y
∂
4onservative form$
∂
∂
V
∂
Q x
x
+
+
∂
y
∂
t
∂
A
∂
t
=
=
3
3
------*.'a
------*.'b
The momentum conservation e"uation e"uation of motion/ for a flood wave is derived from the application of the ewtons second law of motion and is given as$ onconservative form$ form$
∂
y
∂
x
+
V ∂ V
g ∂
∂
4onservative form$
% ∂
Q
A ∂
t
+
%
A
% ∂ + + x g ∂
Q ' A ∂
x
+ g
∂
y
∂
x
V
= S f − S 3
------ *.5a
− g , S 3 − S f / = 3
------*.5b
t
where 1 is the velocity of flow at a given section, 0 3 is the channel bed slope, 0f is the slope of the total energy line, A is the cross-sectional area of flow, 6 is the discharge and g is the acceleration due to gravity. The first to the sixth terms in the flow e"uation either *.5a or *.5b/ represent the local acceleration term, convective acceleration term, pressure gradient force term, gravitational
*-'
force term and the frictional force term respectively. When all the forces relevant to all six terms in either *.5a/ or *.5b/ are significant, then the corresponding flow is unsteady and non-uniform commonly referred to as the dynamic wave/. The flow is steady and non-uniform in flow where the local acceleration term is negligible. 2n the other hand, the flow is said to be steady and uniform when the gravitational forces tend to balance the frictional forces. !n this case, the flow characteri#es a kinematic wave. A diffusion wave exists in flows where there is no acceleration. The continuity e"uations *.'a and *.'b and the e"uations of motion *.5a and *.5b are believed to have been first developed by A.7.4 )arr è de 0aint 1enant around %8*% and these e"uations are commonly known as the 0t. 1enant e"uations. +ydraulic flood routing involves the numerical solution of the 0t. 1enant e"uations. 9etails about these e"uations can be found in many standard hydrology and hydraulics textbooks.
7.
!"dro#o$ic stora$e routin$
There are essentially two types of storage routing. These are$ reservoir hydrologic-routing and channel hydrologic-routing. !n both these methods, the characteristics of an outflow 2/, either at a given channel section or at the outlet of a reservoir are estimated by utilising the information on inflow !/ at a given upstream section or sections.
7..1
!"dro#o$ic reser%oir routin$
A flood wave !t/, enters a reservoir provided with an outlet flow control such as a spillway. 9ue to the passage of the flood wave through the reservoir, the water level in the reservoir changes with time. Thus, the discharge and storage also change with time. )ut the outflow is a function of the reservoir elevation only, i.e. 2:2+/. !n an uncontrolled reservoir, typically, O
=
' 5
Cd
' g Le H 5 ; '
= O ,h /
------*.<
where + is the depth of flow over the spillway, = e is the effective length of the spillway crest and 4d is the coefficient of discharge characteristic to the spillway. 0imilarly, for other forms of outlets, such as gated spillways, sluice gates, etc. other relations for 2+/ are available. 0imilarly, the storage in the reservoir is also a function of the reservoir elevation and thus 0:0h/. >nlike in an ideal channel for which storage is a function of both inflow and outflow, in an ideal reservoir storage is a function of outflow only. The relationship between reservoir storage and outflow can be expressed in the following general form$ S
=
f , O/
------*.?
A common relationship between outflow and reservoir storage is the following power function$ S
= KO
n
------*.@
in which ( : storage coefficient and n : exponent. or n = %, ". *.@ reduces to the linear form
*-5
S = KO
------*.*
&eservoir storage routing involves finding the variation of 0, +, and 2 with time, i.e. finding 0:0t/, 2:2t/ and +:+t/ given !:!t/. or the reservoir storage routing, the following data have to be known$ i. a relationship between the reservoir volume and the water level above the crest of the spillway ii. a relationship between the reservoir water-surface elevation and the outflow and hence a relationship between the reservoir storage and the outflow discharge. iii. the temporal distribution of the inflow hydrograph !:!t/ iv. The initial values of 0, ! and 6 at time t:3 There are a variety of methods available for the routing of floods through a reservoir. All of them use the finite difference form of the lumped continuity e"uation mass balance/ ".*.%. The initial reservoir water elevation is often assumed to be hori#ontal. Thus, the storage routing is also known as Level Pool Routing . Bany methods have been developed for =evel Cool routing. Two of the most commonly used level pool routing methods are$ • The 4oefficient method • The storage indication Culs/ method These methods are described in the following sections.
7..1.1 The coe&&icient routin$ 'ethod
The coefficient routing method, which is commonly referred to as the general reservoir routing method, assumes that the reservoir or channel storage is directly proportional to the outflow from the reservoir. The storage concept is well established in flow-routing theory and practice. 0torage routing is used not only in reservoir routing but also in stream channel and catchment routing. Techni"ues for storage routing are invariably based on the differential e"uation of continuity " *.%/. "uation *.% can be solved by analytical or numerical means. The numerical approach is usually preferred because it can account for an arbitrary inflow hydrograph and because it lends itself readily to computer solution. The approach to solving this e"uation can be visualised by discreti#ing ". *.% on the x-t plane ig. *.%/. The x-t plane is a graph showing the values of a certain variable in discrete points in time and space. igure *.% shows two consecutive time levels, % and ' separated between them by an interval ∆t , and two spatial locations or sections depicting inflow section %/ and outflow section '/, with the reservoir located between them. The discreti#ation of ". *.% on the x-t plane leads to$ I %
+I '
'
−
O%
+O
'
'
=
S'
∆
− S
%
t
---------*.8
*-<
in which !% : inflow section %/ at time level %D ! ' : inflow section %/ at time level 'D 2 % : outflow section '/ at time level %D 3 ' : outflow section '/ at time level 'D 0 % : storage at time level %D 0 ' : storage at time level 'D and ∆t : time interval. "uation *.8 states that between two time-levels % and ' separated by a time interval ∆t, average inflow minus average outflow is e"ual to change in storage.
ti!e t "i!e level 2=t#∆t
0'
!'
2'
∆t
"i!e level 1= t
0% !%
2%
!nflow section %
Fi$ure 7.1
2utflow section '
x
Discreti(ation o& stora$e equation in the )*t p#ane
or linear reservoirs, the storage-outflow relationship for time-level % can be written as$ 0%: (2%
------*.E
And for time-level ', the e"uation can be written as$ 0' : (2'
-------*.%3
0ubstituting ". *.E into *.%3, and solving for 2' we get$ 2' : 43!' F 4%!% F 4'2%
---------*.%%
in which C O , C 1 and 42 are routing coefficients defined as follows$ C 3
=
∆t ; K ' + , ∆t ; K /
4l : 43 C '
=
' − , ∆t ; K / ' + , ∆t ; K /
--------*.%' --------*.%5 --------*.%<
The reader can verify that 43 F 4% F 4' : %. Thus, the routing coefficients are interpreted as weighting coefficients. These routing coefficients are a function of ∆tK , the ratio of time interval to storage constant. The components 43!' , 4%!% and 4'2% of ". *.%% are called partial outflows.
*-?
The coefficient linear reservoir/ routing procedure is illustrated by the following example. E)a'p#e 7.1
A linear reservoir has a storage constant ( : ' h, and it is initially at e"uilibrium with inflow and outflow e"ual to %33 mG;s. &oute the following inflow hydrograph through the reservoir. Ta+#e 7.1
In&#o, h"dro$raph &or E)a'p#e 7.1
Timehr/ !nflowm5;s/
3 %33
% %?3
' '?3
5 <33
< 833
? %333
Timehr/ !nflowm5;s/
8 ??3
E <33
%3 533
%% '?3
%' '33
%5 %?3
@ E33 %< %'3
* *33 %? %33
irst it is necessary to select an appropriate time interval. An examination of the inflow hydrogaph reveals that the time-to-peak is t $ : ? h. A rule-of-thumb for ade"uate temporal resolution is to make the ratio t $ ∆t at least e"ual to ?. 0etting ∆t : % hr assures that t $ ∆t : ?. With ∆t :% h, the ratio ∆tK = %;'. rom "s. *.%' to *.%<, 43 : 4%:%;?, and 4' : 5;?. The routing calculations are shown in Table *.'. 4olumn % shows the time and 4olumn ' shows the inflow hydrograph ordinates. 4olumns 5 to ? gives the partial flows while 4ol. @ shows the computed outflow, which is the sum of the partial flows at each time-level. The recursive procedure continues until the calculated outflow 4ol. @/ is within ? percent of baseflow %33 m5;s/. Clotted inflow and outflow hydrographs 4ols. ' and @/ are shown in ig. *.'. The calculated peak outflow *?8 m5;s/ occurs at t : * hr. +owever, the shape of the outflow hydrograph reveals that the true peak outflow occurs somewhere between @ and * hr. The true peak-outflow is approximated graphically at *@? m5;s, occurring at about @.@ hr. The peak outflow is substantially less than the peak inflow %333 m 5;s/, showing the attenuating effect of the reservoir. Also, the time elapsed between the occurrences of peak inflow and peak outflow %.@ hr/ is approximately e"ual to the storage constant. The reservoir exerts a diffusive action on the flow, with the net result that peak flow is attenuated and time base is increased. !n the linear reservoir case, the amount of attenuation is a function of ∆tK . The smaller this ratio, the greater the amount of attenuation exerted by the reservoir. 4onversely, large values of ∆tK cause less attenuation. 1alues of ∆tK greater than ' can lead to negative attenuation. This amounts to amplificationD therefore, values of ∆tK greater than ' are not used in reservoir routing. Ta+#e 7. Co#. 1 Ti'e -hr 3 % '
The coe&&icient 'ethod routin$ &or E)a'p#e 7.1 Co#. Co#. / / In&#o,-' 0s CI %33 %?3 53 '?3 ?3
Co#. 2 C# I#
'3 53
Co#. 3 C2O1
Co#. 4 Out&#o,-'/0s %33.3 @3 %%3.3 @@ %<@.3
*-@
% < ? @ * 8 E %3 %% %' %5 %< %? %@ %* %8 %E '3 '%
<33 833 %333 E33 *33 ??3 <33 533 '?3 '33 1&' %'3 %33 %33 %33 %33 %33 %33 %33
?3 %@3 '33 %83 %<3 %%3 ?3 @3 ?3 <3 53 '< '3 '3 '3 '3 '3 '3 '3
?3 83 %@3 '33 %83 %<3 %%3 ?3 @3 ?3 <3 53 '< '3 '3 '3 '3 '3 '3
8*.@ %53.@ '''.< 5
'%*.@ 5*3.@ ?8'.< *'E.< *?*.@ *3<.@ @%'.8 ?3*.* <%<.@ 558.8 '*5.5 '%8.3 %*<.8 %<<.E %'@.E %%@.% %3E.* %3?.8 %35.?
1200 Inflow Hydrograph ) 1000 s / 3 ^ 800 m ( e 600 g r a h 400 c s i D
Outflow Hydrograph
200
0 0
2
4
6
8 10 12 14 16 18 20 Time (Hrs)
Fi$ure 7.
5pp#ication o& the coe&&icient routin$ 'ethod6 E)a'p#e 7.1
A distinct characteristic of reservoir routing is the occurrence of peak outflow at the time when inflow e"uals outflow see ig. ()2/. 0ince outflow is proportional to storage ". *-*/, peak outflow corresponds to maximum storage. 0ince storage ceases to increase when outflow e"uals inflow, maximum storage and peak outflow must occur at the time when inflow and outflow coincide. Another characteristic of reservoir routing is the immediate outflow response, with no apparent lag between the start of inflow and the start of outflow see ig. ()2/. rom a mathematical
*-*
standpoint, this property is attributed to the infinite propagation velocity of surface waves in an ideal reservoir.
7..1. tora$e indication 'ethod
The storage indication method is also known as the modified Puls routing method. !n some references, it is also called the Hoodrich routing method. !t is used to route flood wave through actual reservoirs, for which the relationship between outflow and storage is usually of a non-linear nature. The method is based on the differential e"uation of storage, ". *.%. The discreti#ation of this e"uation on the x-t plane ig. *.%/ leads to ". *.8. !n the storage indication method, ". *.8 is transformed to its e"uivalent form$ 'S '
∆t
+ O = I + I + '
%
'
' S %
∆t
−O
%
--------*.%?
in which the unknown values 0' and 3'/ are on the left side of the e"uation and the known values inflows, initial outflows and storage/ are on the right hand side. The left side of ". *.%? is known as the *to+ge indi-tion .untity) !n the storage indication method, it is first necessary to assemble geometric and hydraulic reservoir data in suitable form. or this purpose, the following curves or tables/ are prepared$ %/ elevation-storage, '/ elevation-outflow, 5/ storage-outflow, and
*-8
%. '. 5. <. ?.
0et the counter at n = % to start. >se ". *.%? to calculate the storage indication "uantity I'0nF%;∆t/ #On#1 J at time level n#%. >se the storage indication "uantity versus outflow relation to determine the outflow 2nF% at time level n # %. >se the storage indication "uantity and outflow at time leveln # % to calculate /02S n#1 ∆t 2 n#1J = /02S n#1 ∆t # 2 n#1J-'2 n#1/. !ncrement the counter by %, go back to step ' and repeat. The recursive procedure is terminated either when the inflow ceases or when the outflow hydrograph has substantially receded to baseflow discharge.
The procedure is illustrated in xamples *.' and *.5 using the same data as in xample *.%, not only to confirm that the storage indication method is not only applicable to a linear but also to a non-linear reservoir.
E)a'p#e 7.
>se the data of example *.% to perform a reservoir routing by the storage indication method. The reservoir storage 0/ and outflow from the reservoir 2/ are related by an e"uation 0:'2/. >se ∆t : % hr. The re"uired table for outflow 2/, the storage 0/ and the storage indication "uantity I'K0; ∆t/ F2J is shown below. Ta+#e 7./
The out&#o,*stora$e re#ationship &or E)a'p#e 7. Out&#o, -'/0s
100 200 300 400 500 600 700 800 900 1000 1100
tora$e - '/s*1*h
200 400 600 800 1000 1200 1400 1600 1800 2000 2200
tora$e indication quantit" 8-90 t:O; '/s*1
500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500
This table can also be replaced by a graph of the storage indication "uantity versus outflow. !n this example, this graph is as given below$
*-E
6000 5000
n a u ( 4000 n o & t 1 a # " ' 3 & 3000 d ! n & % g 2000 a r o t $
1000 0 0 0 1
0 0 2
0 0 3
0 0 4
0 0 5
0 0 6
0 0 7 3
Outflow (O ! "
Fi$ure 7./
0 0 8
0 0 9
0 0 0 1
0 0 1 1
#1
The out&#o,*stora$e re#ationship &or E)a'p#e 7.
The calculations are shown in Table *.<. At time t:3 the counter is at n:%. The outflow is %33 m5;s the baseflow/ and the corresponding storage indication "uantity from the table or graph above/ is ?33 m5;s. The entries in column 5 of the calculations table represent I'0;∆t/-2J n which is e"ual to I'0;∆t/F2J-'2Jn the indication "uantity minus twice the output at calculation time step n. At t:% this is therefore e"ual to ?33-'K%33:533 m 5;s. or t:' and other subse"uent time steps, the indication "uantity is computed from ". *.%? using the current and previous inflows as well as the previous entry of I'0; ∆t/-2J in column 5. or each step-computation of the storage indication "uantity, the estimates of the corresponding outflow are computed by interpolation using Table *.5 or the graph on ig. *.5) This recursive procedure is continued until the outflow hydrograph ordinate column ?/ is within ?L of the e"uilibrium inflow. The complete routing solution is summarised in Table *.< below. The solution of this problem as summari#ed in column ? of Table *.< can be confirmed to be the same as that was obtained in Table *.' in xample *.%. This must be expected because we have used the same outflow-storage relationship in both examples. xample *.5 below illustrates the use of the indication method for a non-linear reservoir. E)a'p#e 7./
>se the data of example *.% to perform a reservoir routing by the storage indication method if the reservoir has an outflow-storage relationship 0:k2n where k:3.3' and n:'. >se ∆t :% hr. Croceeding as in the above example, it is important to construct the storage-outflow relationship table and;or graph. The corresponding table and graph for this example are given in Table *.? and igure *.< respectively.
Ta+#e 7.2
o#ution o& E)a'p#e 7.
Col. 1 Time t (hr)
Col. 2 Col. 3 3 Inflow (I) m /s [(2S/elt)!"# 0 100 300.0
Col. $ Col. 5 [(2S/elt)%"# "&tflow (") m 3 /s 500.0 100.0
*-%3
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
Ta+#e 7.3
150 250 400 800 1000 900 700 550 400 300 250 200 150 120 100 100 100 100 100 100 100
330.0 438.0 652.8 1111.7 1747.0 2188.2 2272.9 2113.8 1838.3 1523.0 1243.8 1016.3 819.8 653.9 524.3 434.6 380.8 348.5 329.1 317.4 310.5
550.0 730.0 1088.0 1852.8 2911.7 3647.0 3788.2 3522.9 3063.8 2538.3 2073.0 1693.8 1366.3 1089.8 873.9 724.3 634.6 580.8 548.5 529.1 517.4
110.0 146.0 217.6 370.6 582.3 729.4 757.6 704.6 612.8 507.7 414.6 338.8 273.3 218.0 174.8 144.9 126.9 116.2 109.7 105.8 103.5
The out&#o,*stora$e re#ationship &or E)a'p#e 7./ Outflo3 Sto+ge 0S Sto+ge indi-tion Outflo3 Sto+ge 0S Sto+ge indi-tion % % 1 % % 1 0! * ! * h .untity 0! * ! * h .untity /024S ∆t#O5 !% *1 /024S ∆t#O5 !% *1 1100.0 540 1161.9 100 500.0 2863.8 1323.2 580 1204.2 140 591.6 2988.3 1521.6 620 1245.0 180 670.8 3110.0 1703.2 660 1284.5 220 741.6 3229.0 1872.5 700 1322.9 260 806.2 3345.8 2032.1 740 1360.1 300 866.0 3460.3 2183.9 780 1396.4 340 922.0 3572.8 2329.4 820 1431.8 380 974.7 3683.6 2469.4 860 1466.3 420 1024.7 3792.6 2604.8 900 1500.0 460 1072.4 3900.0 2736.1 500 1118.0
*-%%
4000
2
3500
$)0.02*O
1 !
s
3 3000
m # 2500 " % ) t ( l e /2000 S 2 [ 1500
1000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 8 2 6 0 4 8 2 6 0 4 8 2 6 0 4 8 2 6 0 1 1 1 2 2 3 3 3 4 4 5 5 5 6 6 7 7 7 8 8 9 3 !1
"&tflow ["# m s
Fi$ure 7.2
The stora$e indication quantit" %ersus out&#o, &or E)a'p#e 7./
>sing the same procedure as in xample *.', we obtain the solution as summarised in Table *.@ below.
Ta+#e 7.4
o#ution o& E)a'p#e 7./ Col. 1 Time t (hr) 0 1 2 3 4 5 6 7 8 9 10 11 12
Col. 2 Col. 3 Inflow (I) m 3 /s [(2S/elt)!"# 100 900 150 934 250 1056 400 1270 800 1634 1000 1968 900 2092 700 2046 550 1930 400 1790 300 1638 250 1506 200 1394
Col. $ Col. 5 [(2S/elt)%"# "&tflow (") m 3 /s 1100 100 1150 108 1334 139 1706 218 2470 418 3434 733 3868 888 3692 823 3296 683 2880 545 2490 426 2188 341 1956 281
*-%'
13 14 15 16 17 18 19 20 21
150 120 100 100 100 100 100 100 100
1284 1180 1090 1022 980 954 936 924 916
1744 1554 1400 1290 1222 1180 1154 1136 1124
230 187 155 134 121 113 109 106 104
2ccasionally, the reservoir elevation-outflow-storage information is also provided especially when the reservoir water-elevation is pertinent in the reservoir regulation plans. !n such cases, the storage indication method can be extended to provide this re"uired reservoir elevation information. This is illustrated in the example given below. E)a'p#e 7.2
A reservoir has an elevation-storage-outflow relationship as shown on Table *.* below$ Ta+#e 7.7
E#e%ation*Out&#o,*stora$e re#ationship &or E)a'p#e 7.2 'leation (m) 100.00 100.50 101.00 101.50 102.00 102.50 102.75 103.00
S1*+ m3 3.35 3.47 3.88 4.38 4.88 5.37 5.53 5.86
(2S/ t%") m3 /s (") m3 /s 0.00 310.19 10.00 331.48 26.00 385.26 46.00 451.83 72.00 524.04 100.00 597.22 116.00 627.76 130.00 672.22
&oute the flood wave given on Table *.8 through this reservoir if the initial water level in the reservoir is %33.? meters. >se a ∆t:@ hrs:'%@33 sec.
Ta+#e 7.<
F#ood h"dro$raph &or routin$ in E)a'p#e 7.2
+&!% Hr" Inflow !3,"
0 6 12 18 24 30 36 42 48 54 60 66 10 30 85 140 125 96 75 60 46 35 25 20
*-%5
700 650 600
t 550 i t n s / a 3 500 & , m n o i 450 t a c i n 400 i e g 350 a r o t S 300 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 3 4 5 6 7 8 9 0 0 1 2 3 4 1 1 1 1 1
3
"&tflow m /s Fi$ure 7.3
tora$e indication quantit" %ersus out&#o, &or E)a'p#e 7.2.
As it has been shown in the previous examples, it is important to construct a storage indication versus outflow graph if any of these types. !t has been stressed that such graph is very useful in these problems for interpolation purposes. The graph for this example is given on igure *.&. With the information provided so far, it is now possible to estimate the re"uired outflow hydrograph. This has been done using the usual procedure. The computations are shown in column ? of Table *.E. The reservoir elevations corresponding to these outflows are shown in column @ of Table *.E. These were estimated by interpolation from igure *.@.
*-%<
140.00 130.00 120.00 110.00
s100.00 /
3
90.00
m80.00 w70.00 o60.00 l f t 50.00 &40.00 " 30.00 20.00 10.00 0.00 0 . 0 0 1
5 . 0 0 1
0 . 1 0 1
5 . 1 0 1
0 . 2 0 1
'leation (m)
Fi$ure 7.4
E#e%ation*Out&#o, re#ationship &or E)a'p#e 7.2.
Ta+#e 7.=
u''ar" o& the co'putations &or E)a'p#e 7.2 Col. 1
Time (hr) 0 6 12 18 24 30 36 42 48 54 60 66
5 . 2 0 1
0 . 3 0 1
Col. 2 Col. 3 Col. $ Col. Col. + Inflow (2S/elt)!" " m3 /s m3 /s (2S/elt)%" m 3 /s m3 /s 'leation m 10 314 340 13 100.6 30 322 354 16 100.72 85 355 437 41 101.4 140 396 580 92 102.35 125 409 661 126 102.93 96 398 630 116 102.75 75 391 569 89 102.3 60 382 526 72 102 46 370 488 59 101.75 35 361 451 45 101.48 25 349 421 36 101.25 20 338 394 28 101.05
igure *.* shows the temporal distribution of the inflow flood wave, the estimated outflow hydrograph and the corresponding reservoir elevation for xample *.<.
*-%?
Inflow
Outflow
-l%at&on !
103.5 103.0 102.5 102.0 101.5 101.0 100.5 100.0 99.5 99.0
160 140 s120 / m100 w 80 o l f t 60 & "
3
40 20 0
0
6
12
18
24
30
36
42
48
54
60
m n o i t a ( e l '
66
Time (hrs)
Fi$ure 7.7
7..1./
5 co'parison o& the te'pora# distri+ution o& the in&#o, &#ood ,a%e> the out&#o, h"dro$raph and the reser%oir e#e%ation &or E)a'p#e 7.2. Genera# re'ar?s on stora$e routin$
Bost large reservoirs have some type of outflow control, wherein the amount of outflow is regulated by gated spillways. !n this case, both hydraulic conditions and operational rules determine the prescribed outflow. 2perational rules take into account the various uses of water. or instance, a multipurpose reservoir may be designed for hydropower generation, flood control, irrigation, and navigation. or hydropower generation, reservoir pool level is kept within a narrow range, usually close to the optimum operating level of the installation. 2n the other hand, flood-control operation may re"uire that a certain storage volume be kept empty during the floodseason in order to receive and attenuate the incoming floods. lood-control operations also re"uire that the reservoir releases be kept below a certain maximum, usually taken as the flow corresponding to bank-full stage. !rrigation re"uirements may vary from month to month depending on the consumptive needs and crop patterns. or navigation purposes, outflow should be a nearly constant value that will ensure a minimum draft downstream of the reservoir. &eservoir operational rules are designed to take into account the various water-demands. These are often conflicting and, therefore, compromises must be reached. Bultipurpose reservoirs allocate reservoir volumes to the different uses. !n this way, operational rules may be developed to take into account the re"uirements of each use. !n general, outflow from a reservoir with gated outlets is determined by prescribed operational policies. !n turn, the latter are based on the current level of storage, incoming flow, and downstream flow re"uirements. The differential e"uation of storage can be used to route flows through reservoirs with controlled outflow. !n general, the outflow can be either %/ uncontrolled ungauged/, '/ controlled gated/, or 5/ a combination of controlled and uncontrolled. The discreti#ed e"uation, including controlled outflow, is given by$ I %
+I '
'
+
O%
+O
'
'
−O = +
S'
− S ∆t
%
-------*.%@
*-%@
−
!n which Q is the mean regulated outflow during the time interval At. "uation *.%@ can be + expressed in storage indication form$ ' S '
∆t
+ O = I + I + '
%
'
' S %
∆t
− O − 'O %
+
--------*.%*
−
With Q known, the solution proceeds in the same way as with the uncontrolled out-flow case. + !n the case where the entire outflow is controlled, ". *.%* reduces to$ S'
= S % +
∆t '
, I %
+ I ' / − , ∆t /O +
---------*.%8
)y which the storage volume can be updated based on average inflows and mean regulated outflow. 2ther re"uirements, such as estimates of reservoir evaporation where warranted i.e., in semiarid and regions/ may be implemented properly to account for the storage volumes. 7..
!"dro#o$ic channe# routin$
!n reservoir routing presented in the previous sections, the storage was a uni"ue function of the outflow discharge. +owever, in channel routing, the storage is a function of both the inflow and the outflow discharges. +ence, a different routing approach is re"uired. !t is important to note that the flow in a river during a flood passage belongs to the category of gradually varied unsteady flow. The water surface is not only unparallel to the channel bottom but also varies with time. Thus, considering a channel reach having a flood flow, the total volume in storage can be considered under two categories as$ %. Crism storage '. Wedge storage Crism storage is the volume that would exist if uniform flow occurred at the downstream depth, i.e. the volume formed by an imaginary plane parallel to the channel bottom drawn at the outflow section of the flow. 2n the other hand, wedge storage is a wedge-like volume formed between the actual water surface profile and the surface of the prism storage. These storages are illustrated on igure *.8 below. Wedge storage !nflow !/
Crism storage
2utflow 2/
-Fi$. 7.
*-%*
Fve wedge storage
Crism
Crism Crism
-Fi$. 7.<+ Fi$ure 7.< 7...1
@ris' and ,ed$e stora$e in a ri%er channe# durin$ &#ood &#o, The Aus?in$u' channe# routin$ 'ethod
2ne of the most commonly used hydrologic channel routing procedure is the 6u*7ingu! !ethod . This method was developed during the period %E5<-%E5? for studies of the Buskingum 4onservation 9istrict lood 4ontrol CroMect B494C/ of the >.0 Army 4orps of ngineers. The method which utilises the concept of both wedge and prism storage is of the form S
=
K[ xI !
+
( % − x) O ! ]
---------*.%E
Where K is a channel coefficient usually called the storage-time constant. !t has the dimension of time. !t is approximately e"ual to the time of travel of a flood wave through the channel reach. The coefficient x is a weighting factor that takes values between 3 and 3.?. When x:3 ". *.%E reduces to the linear storage e"uation ". *.*/. When x:3.?, then the inflow and outflow are e"ually important in determining the storage. 2n the other hand, ! is a positive exponent which varies from 3.@ for rectangular channels to a value of about %.3 for natural channels. When !:%.3 ". *.%E reduces to the linear form$ S=K/xI#01xO5
---------*.'3
"uation *.'3 is commonly referred to as the Buskingum "uation. Esti'ation o& and ) in the Aus?in$u' equation
igure *.Ea/ shows a typical inflow and outflow hydrographs through a channel reach. ote that unlike the case with the reservoir routing, the outflow peak does not occur at the point of intersection of the inflow and the outflow hydrographs. >sing the continuity e"uation ". *.8 /, 0I 1#I 2 ∆t2 8 0O1#O2 ∆t2 = ∆S The increment in storage at time t and the time increment ∆t can be calculated. 0ummation of the various incremental storage-values enables one to find the channel storage S versus time relationship ig *.Ea//. !f the inflow and outflow data is available for a given reach, values of S at various time intervals can be determined by the above approach. )y choosing a trial value of x, values of S at time t are plotted against the /xI#01xO5 values. !f the value of x is chosen correctly, a straight-line relationship as given by ". *.'3 is obtained. An incorrect value of x gives a plot of 0 versus /xI#01xO5 values which trace a looping curve. )y trial and error, a value of x can be chosen for which the plot very nearly describes a straight line. The inverse of the slope of *-%8
such line gives the value of K . !n a given channel reach the values of x and K are assumed to be constant. This procedure is demonstrated in example *.? below. E)a'p#e 7.3
stimate the values of K and x for a channel reach with an inflow and outflow discharge hydrographs as given in columns % to 5 in Table *.%3 below. Ta+#e 7.1
Ca#i+ration o& Aus?in$u' routin$ para'eters &or E)a'p#e 7.3
+&!% Inflow #I Outflow#O I#O (I#O,2 hr" (! 3," (!3," (!3," (! 3," (1 0 6 12 18 24 30 36 42 48 54 60 66
(2
(3 5 20 50 50 32 22 15 10 7 5 5 5
(4 5 6 12 29 38 35 29 23 17 13 9 7
(5
0 14 38 21 #6 #13 #14 #13 #10 #8 #4 #2
7 26 29.5 7.5 #9.5 #13.5 #13.5 #11.5 #9 #6 #3
/I(1#/O (! 3," $)Σ∆$ 3 3 ∆t (! ,"# (! ,"#hr /)0.35 x )0.3 x )0.2 hr (6 (7 (8 (9 (10 0 0 5 5 5 42 42 10.9 10.2 8.8 156 198 25.3 23.4 19.6 177 375 36.35 35.3 33.2 45 420 35.9 36.2 36.8 #57 363 30.45 31.1 32.4 #81 282 24.1 24.8 26.2 #81 201 18.45 19.1 20.4 #69 132 13.5 14 15 #54 78 10.2 10.6 11.4 #36 42 7.6 7.8 8.2 #18 24 6.3 6.4 6.6
∆$)Col5*
60 50 % g r 40 a h30 ' " 20 & 2 10 0
500 400 % a r o t $
300 g 200 100 0 0
6 12 18 24 30 36 42 48 54 60 66 +&!% (Hr"
Inf low !3,"
Outf low !3,"
$tor ag% !3," .hr
ig *.Ea/
*-%E
40 35 " 30 , 3 !25 O 20 / # 1 15 ( 0 I / 10 4
/)0.20
/)0.3
/)0.35
5 0 0
100
200
300
400
500
$torag% $ !3,"#hr
Fi$ure 7.=
ig *.Eb/ Esti'ation o& the and ) %a#ues in the Aus?in$u' equation
As a first trial, x:3.5? is selected and the value of /xI#01xO5 evaluated column 8/ and plotted against 0 in igure *.Eb/. 0ince a looped curve is obtained, further trials are performed with x:3.5 and 3.'3. !t is seen that from igure *.Eb/ that for x:3.', the data very nearly describes a straight line and as such, x:3.'3 is taken as an appropriate value for the channel reach. The value of K is then computed from the slope of the x:3.' line. !n this case K :%5.' hrs. Aus?in$u' 'ethod o& routin$
or a given channel reach, by selecting a routing interval ∆t and using the Buskingum e"uation, the change in storage is S 2 8 S 1 = K /x !2 8 I 1 # % x 0O2 O 5 l
-------------*.'%a
where suffixes % and ' refer to the conditions before and after the time interval ∆t . The continuity e"uation ". *.8/ for the reach is S '
I + I O + O ∆t − S = ∆t − ' ' '
%
'
%
%
----------*.'%b
rom the above e"uation, 2' is evaluated as O2 = C 9 I 2#C 1 I 1#C 2O1
------------*.''
where 43 : -( x # 9)& ∆t0 KKx#9)& ∆t 4% : Kx#9)&∆t KKx#9)&∆t C 2 = 0KKx 9)& ∆t 0KKx#9)&∆t ote that C 9 # C 1 # C 2 :%.3, ". *.'' can be written in a general form for the n-th time step as
*-'3
On =C 9 I n#C 1 I n1 #C 2On1
()2%
". *.'5 is known as 6u*7ingu! Routing :.ution and provides a simple linear e"uation for channel routing. !t has been found that for best results the routing interval ∆t should be so chosen that ( ; ∆t ; 2Kx) If ∆t < '(x, the coefficient 43 will be negative. Henerally, negative values of coefficients are avoided by choosing appropriate values of ∆t To use the Buskingum e"uation to route a given inflow hydrograph through a reach, the values of ( and x for the reach and the value of the outflow, 2 l, from the reach at the start are needed. The procedure is indeed simple. a/ (nowing ( and x, select an appropriate value of ∆t b/ 4alculate C 9 , C 1 , and 4'. c/ 0tarting from the initial conditions !% , O1 , and known !' at the end of the first time step ∆t calculate 2' by ". *.'' or ". *.'5 d/ The outflow that is calculated in step c/ becomes the known initial outflow that is used for the next time step. &epeat the calculations for the entire inflow hydrograph. The calculations are best done row by row in a tabular form. xample *.@ illustrates the computation procedure. Any computer 0pread0heet is ideally suited to perform the routing calculations and to view the inflow and outflow hydrographs. E)a'p#e 7.4
&oute the following hydrograph through a river reach for which ( : %'.3 hr and x : 3.'3. At the start of the inflow flood, the outflow discharge is %3 m5;s. Ta+#e 7.11 Ti'e -hr In&#o, -'/0s
The in&#o, h"dro$raph &or E)a'p#e 7.4
3 %3
@ '3
%' ?3
%8 @3
'< ??
53
5@ 5?
<' '*
<8 '3
?< %?
o#ution $ 0ince ( = %' hr and 2Kx = ' x %' x 3.' : <.8 hr, ∆t should be such that %' h+ ; ∆t N <.8 h+) !n the present case ∆t : @ hr is selected to suit the given inflow hydrograph ordinate interval.
>sing "s. *.'' or ". *.'5 the coefficients 43, 4% and 4' are calculated as 43 : - %'K 3.'3 F 3.? K @/; %' - %' K O)2# 9 ?K@ : 3.@;%'.@ : 3.3<8 4%: %' K 3.' F 3.? K @/;%'.@ : 3.<'E 4' : %' - %' K 3.' - 9)&4 @/;%'.@ : 3.?'5 or the first time interval, 3 to @ hr, !% : %3.3 4%!% : <.'E !' : '3.3 43!' : 3.E@ 2% : %3.3 4'2% : ?.'5 rom ". *.'' 2' : 43%'F4llF4'2% : %3.<8 m5;s
*-'%
or the next time step, @ to %' hr, 2 % = %3.<8 m5;s. The procedure is repeated for the entire duration of the inflow hydrograph. The computations are done in a tabular form as shown in Table *.%'. )y plotting the inflow and outflow hydrographs the attenuation and peak lag are found to be %3 m5;s and %' hr respectively. Ta+#e 7.1
Time h/ %/ 3 @
Aus?in$u' 'ethod o& routin$*E)a'p#e 7.4.- t 4 hrs
! m5;s/ '/ %3
3.3<8!' 5/
3.<'E!%
3.?'52% ?/
3.E@
<.'E
?.'5
'3
%3.<8 '.<3
%'
'5.@3
'5.8?
'?.E?
5?
<@.E5 %?.3'
'<.??
'*
<3.8* %%.?8
'%.58
'3
55.E' 3.*'
?<
%*.'5
3.E@ <8
'?.*<
??
%.53 <'
8.@% 5'.E<
%.@8 5@
'%.
@3
'.%@ 53
?.<8 %@.<@
'.@< '<
8.?8
?3 '.88
%8
2 m5;s/ @/ %3.33
8.?8
%*.*<
1&
'*.3<
"uations *.'% and *.8 can be combined in an alternative form of the routing e"uation as 2' : 2% F )% !% O 2%/ F )' ! ' O !%/ Where )% : ∆t;I( % - x/ F 3.?
---------*.'<
∆tJ
)' : 3.?∆t-(x/;I( % - x/ F 3.?
∆tJ
The use of ". *.'< is essentially the same as that of ". *.''.
7...
Aus?in$u'*Cun$e routin$ 'ethod
2ne of the modified methods that has worked well in river routing is the Buskingum-4unge method. As shown earlier, the Buskimgum method is based on the storage e"uation with
*-''
coefficients K and x derived by trial and error. 4unge showed that K and x could be derived by considering the hydraulics of the flow in the channel. &ecall that K was shown to be approximately e"ual to the time of travel of a flood wave through the reach. This assumption was used by 4unge to give K=∆ L-, where - is the average speed of the flood peak and ∆ L is the length of the river reach. 0ubstituting for K=∆ L- in". *.'%a and *.'%b gives$
∆ Lx , − / + ∆ L ,% − /, − / + % , − + − / = 3 I ' I % x O' O% O' I ' O% I % ' -∆t -∆t
----*.'?
Bultiplying through ". *.'? by - ∆ L and rearranging gives$
− I /% ,% − x/,O' − O% / I,O' − I ' / + ,O% − I % /J = 3 + + '∆ L -∆t ∆t
x, I '
-------*.'@
With K==∆ L- as an acceptable approximation, a means is re"uired for obtaining the x factor. 4unge derived the following expression for x from the channel properties$ x
=
% '
−
Q$ ' *=-∆ L
----------*.'*
where Q $ is the mean peak discharge, * is the average bed-slope and is the mean channel width and ∆ L is the channel reach) !n practice, the Buskingum coefficients are evaluated according to each reach forming the subdivisions of the total length of the river reach being considered. Then the routing of the inflow could proceed as shown earlier to obtain the outflow by recurrent application of the Buskingum routing e"uation.
*-'5
