Groundwater flow

Hydrogeology

Geology Dept., Anna University

Groundwater Flow ► ► ► ► ► ► ► ► ► ► ► ►

Pressure and pressure head Elevation head Total head Head gradient Discharge Darcy’s Law Hydraulic conductivity Permeability Transmissivity Lab / Field Methods of determination of K Unsaturated flow Flow nets

Pressure ► Pressure

is force per unit area

► Newton:

F = ma

F force (‘Newtons’ N or kg ms-2) m mass (kg) a acceleration (ms-2) ►P

Elango, L

= F/Area (Nm-2 or kg ms-2m-2 = kg s-2m-1 = Pa)

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Hydrogeology


Pressure and Pressure Head ► Pressure

relative to atmospheric, so P = 0

at water table

►P

= ρghp

ρ density g gravity hp depth

Elevation

Pressure Head P (increases w with depth below surfacce)

P = 0 (= Patm)

Head

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Hydrogeology


Elevation Head

Elevation datum

Elevation Head E (increases w with height above datum m)

wants to fall ► Potential P t ti l energy

Elevation

► Water

Head

Elango, L

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Hydrogeology


Total Head ► For

our purposes: ► Total T t l head h d = Pressure P head h d + Elevation El ti head ► Water flows down a total head gradient

Elevation datum

Total Head (constant: hydrostatic equilibrium m)

Elevation

P = 0 (= Patm)

Head

Elango, L

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Hydrogeology


Pressure and Elevation Heads - Laboratory

ψ = pressure head z = elevation head h = total head

9

Freeze and Cherry, 1979.

Pressure and Elevation Heads - Field

ψ = pressure head z = elevation head h = total head

10

Freeze and Cherry, 1979.

Elango, L

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Hydrogeology


Head Gradient ► Change

in head divided by distance in porous medium over which head change occurs ► dh/dx [unitless]

Head Gradient -General Concepts The water table is actually a sloping surface. Slope (gradient) is determined by the difference in water table elevation (h) over a specified distance (L). Direction of flow is downslope. Flow rate depends on the gradient and the properties of the aquifer.

Elango, L

S. Hughes, 2003

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Hydrogeology


Hydraulic Gradient Contour Map – Maps Topographic Gradient In hydrogeology, hydraulic head gradient is driving force Hydraulic Gradient = Change in Head / Length • rate at which water flows through is proportional to gradient

High Low h2 – h1 = 99 – 96 = 3’ L = 100’ Gradient = 3 / 100 = .03 NO UNITS

Discharge ►Q ►

Elango, L

(volume per time) Q VA volume/time Q= l /ti

m3/day /d

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Hydrogeology


Permeability Permeability: a measure of an earth material to transmit fluid, but only in terms of material properties, not fluid properties, depend on the size of pore spaces and to what degree the pore spaces are connected. Hydraulic conductivity: ability of material to allow water to move through it, it expressed in terms of m/day (distance/time). Function of the size and shape of particles, and the size, shape, and connectivity of pores. S. Hughes, 2003

Groundwater Movement -- Darcy’s Law Henry Darcy, 1856,- water flow through porous material. His equation describes groundwater flow. Darcy’s experiment: • Water is applied under pressure through end A, flows through the pipe, and discharges at end B. • Water pressure is measured using piezometer tubes Hydraulic head = dh (change in height between A and B) Flow length = dL (distance between the two tubes) Hydraulic gradient (I) = dh / dL S. Hughes, 2003

Elango, L

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Hydrogeology


Groundwater Movement -- Darcy’s Law The velocity of groundwater is based on hydraulic conductivity (K), as well as the hydraulic gradient (I). The equation to describe the relations between subsurface materials and the movement of water through them is

Q = KIA Q = Discharge = volumetric flow rate, volume of water flowing through an aquifer per unit time (m3/day) A = Area through which the groundwater is flowing, cross-sectional area of flow (aquifer width x thickness, in m2) S. Hughes, 2003

What is K? K = Hydraulic Conductivity = coefficient of permeability

Porous medium K is a function of both:

The Fluid

What are the units of K?

K = Q / IA

L3/ TxL /2

=

L T

The larger the K, the greater the flow rate (Q)

Elango, L

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Hydrogeology


Groundwater Flow Darcy’s Law Discharge through porous media is p proportional to the p hydraulic gradient and area of cross section

Q=KAi

19

www.twdb.state.tx.us/gam

Darcy velocity Rearrange the equation to Q/A = KI known as the flux (v), (v) which is an apparent velocity (Darcy velocity) “apparent velocity” –velocity of water through an aquifer if it were an open conduit Flux doesn't account for the water molecules actually following a tortuous path in and out of the pore spaces. spaces They travel quite a bit farther and faster in reality than the flux would indicate. Not a true velocity as part of the column is filled with sediment

Elango, L

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Hydrogeology


Linear/pore velocity Actual groundwater velocity is higher than that determined by Darcy’s Law.

True Velocity – Average Mean Linear Velocity? y account for area through g which flow is occurring g Only DOES account for tortuosity of flow paths by including porosity (n) in the calculation. Flow area = porosity (n) x area

v Average linear/pore velocity = n

How K value vary? WELL SORTED POORLY SORTED Coarse (sand-gravel) Coarse - Fine

WELL SORTED Fine (silt-clay)

y and Hydraulic y y Permeability Conductivity

High

Low

Sorting of material affects groundwater movement. Poorly sorted (well graded) material is less porous than well-sorted material. S. Hughes, 2003

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Hydrogeology


The Smaller the Pore Size The Larger the Surface Area The Higher the Frictional Resistance The Lower the Permeability

High

Low

Less n High K

High n Less K

Freeze and Cherry 2000

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Hydrogeology


Porosity and hydraulic conductivity

Porosity (%)

Material Unconsolidated Clay Sand Gravel Gravel and sand

45 35 25 20

Rock Sandstone Dense limestone or shale Granite

15 5 1

Hydraulic Conductivity (m/day) 0.041 32.8 205.0 82.0 28.7 0.041 0.0041 (Keller, 2000)

Validity of Darcy Law ► Valid

only for slow movement(ie laminar). Type of flow is controlled by the "Reynolds Number". ► This Thi dimensionless di i l quantity tit is i the th ratio ti off inertial forces and viscous forces ► The utility of the Reynolds Number is that we can learn the general nature of the flow with no additional calculations at all NR = ρ vD/ μ ρ density μ viscosity D dia of pipe v velocity

Elango, L

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Hydrogeology


►

► ► ►

►

Darcy's law is only valid for slow, viscous flow; fortunately, most groundwater flow cases fall in this category. Typically any flow with a Reynolds number less than one is clearly laminar, and it would be valid to apply Darcy's law. Experimental tests have shown that for flow regimes with values of Reynolds number up to 10 may still be Darcian. Reynolds number for porous media flow is expressed as

where ρ is the density of water (units of mass per volume), v is the specific discharge (not the pore velocity — with units of length per time), d30 is a representative grain diameter for the porous medium (often taken as the 30% passing size from a grain size analysis using sieves), and μ is the viscosity of the fluid. http://en.wikipedia.org/wiki/Darcy's_law

Darcy Law is valid – NR less than 1 and up to 10 Reynold’s No., Re 0.01 0.1 1

Q

Laminar Region

10 100 1000

Laminar Turbulent but non-linearFlow

D Darcy’s ’ L Law OK

after Freeze & Cherry

Grad h

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Hydrogeology


Transmissivity Need Mechanism to Compare Aquifers

Measure of the amount of water that can be transmitted horizontally through a unit width by the full saturated thickness of the aquifer under a hydraulic gradient of 1

Assume Unit Width Assume Hydraulic Gradient of 1 Using Darcy’s Law

Q = -KA (Δh / L) Transmissivity T = KB

T = transmissivity (L2/T) K = hydraulic conductivity (L/T) B = saturated thickness of aquifer (L)

Elango, L

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Hydrogeology


Thus the TWO important parameters are ►S ►T ►

known as aquifer parameters

Homogeneous vs Heterogenous Variation as a function of Space Homogeneity – same properties in all locations

Heterogeneity properties change spatially

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Hydrogeology


Isotropy vs Anisotropy Variation as a function of direction Isotropic same in direction Anisotropic changes with direction

Homogeneous, Isotropic

Homogeneous, Anisotropic

Kz (X2, Z2)

z

(X1, Z1)

Kx

x

Heterogeneous, Isotropic

Elango, L

Heterogeneous, Anisotropic

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Hydrogeology


Determination of Hydraulic Conductivity Empirical Methods Laboratory Method Constant Head Permeameter Falling Head Permeameter Field Methods Tracer tests Pumping tests

Empirical Methods Grain-Size Analysis Particle density Bulk density Porosity

Elango, L

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Hydrogeology


Constant Head Permeameter K = VL/Ath V – flow volume in time t A – area of cross section L- Length of sample h – head difference

Better for high permeable material

Falling Head Permeameter

K = ((rt2 L/r / c2t)) (ln ( h1//h2)

Better for less permeable material

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Hydrogeology


Procedure and Limitations Sample collection Loading and Compaction Seepage along the walls Saturation Soil core collection

Field methods Tracer method – dying reagent used as a solvent – note the time interval for reaching to adjacent well. Pumping test method – Pump out the water from well and note down the time for drawdown and recovery (Next Unit!)

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Hydrogeology


Tracer methods

► Results

are only approximate as - the holes has to be closer or travel time long - flow direction need to be known, otherwise many holes are necessary - stratification, first arrival may be thro f t route faster t ► Point dilution Tests

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Hydrogeology


Tracers 1. Chemical solutions ► ► ►

► ►

► ►

Salt solutions, such as sodium chloride, are frequently used. They have many disadvantages. Big quantities must be injected (tens or hundreds of kilos for each injection) in order to obtain a detectable signal (measured as an increase in conductivity, or by using selective electrodes). The charged waters are heavier and tend to descend to the bottom of the aquifers. Effect on environment, their toxicity (either of the ion itself or of the quantities injected) and their possible biological uptake is to be considered. The choice is therefore limited: Na+, Li+, K+ (for the cations) and Cl-, I-, Br- (for the anions). However, some use ions toxic for the environment, such as Cu, Zn, Co, Cr, or Pb !

2. Fluorescent tracers (organic dyes) ► ► ►

► ►

Elango, L

Fluorescent substances are most commonly used Some dyes are strongly suspected to be carcinogens mostly due to the impurities it contains Photo-decomposition, biodegradation by microorganisms and chemical reactions with other substances present in the aquifer result in degradation products that are dangerous for the environment Three tracers that are safe are: fluorescein, Tinopal CBSx and d Rhodamine Rh d i WT The most commonly used is sodium fluorescein (Uranine), because it is non- toxic in low concentrations (LD50 1700 mg/kg) and is relatively inexpensive [Parriaux et al., 1988]

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Hydrogeology


Pumping tests

Groundwater Flow Nets Water table contour lines - represent "elevations" of water table Water table contour lines - used to determine the direction groundwater will flow in a given region Water table contours (called equipotential lines) - constructed to join areas of equal head Groundwater flow lines, which represent the paths of groundwater down slope, are drawn perpendicular to the contour lines Map of groundwater contour lines with groundwater flow lines is called a flow net

S. Hughes, 2003

Elango, L

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Hydrogeology


Groundwater Flow Nets Water table contours Distorted contours may occur due to anisotropic conditions (changes in aquifer properties).

Area of high permeability (high conductivity) S. Hughes, 2003

Groundwater Flow Nets A simple flow net Cross-profile view 100

Qal

50

WT

Qal

Aquitard

Aquitard Qal well

• Effect of a producing well • approximate diameter of the cone of depression is seen S. Hughes, 2003

Elango, L

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Hydrogeology


Mapping Water Levels Topographic Maps Contours of Elevation

Water Table Maps Equipotentials Contours of Hydraulic Head

Groundwater Flow Nets DRAINAGE BASIN

Flow lines

Water table contours in drainage basins roughly follow the surface topography, but depend greatly on the properties of rock and soil that compose the aquifer: • Variations in mineralogy and texture

WT contours

N

• Fractures and cavities • Impervious layers • Climate

S. Hughes, 2003

Elango, L

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Hydrogeology

Elango, L


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Hydrogeology


Groundwater Flow Net 414

412

N

410

Water Flow Lines

408

Water Table Contours 406

404

Well 402

400

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Hydrogeology


More on gradients ► Three

point problems:

h

h

h


point problems:

h = 10m

(2 equall heads) h d ) h = 10m

► Gradient h = 9m

= (10 -9m)/CD (10m(10m 9 )/CD ► CD? Scale from map Compute

Elango, L

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Hydrogeology



h = 11m

point problems:

(3 unequall heads) h d ) Best guess for h = 10m h = 10m

► Gradient

= (10 -9m)/CD (10m(10m 9 )/CD ► CD?

h = 9m

Scale from map Compute

Groundwater Movement Determine flow direction from well data: Well #1 4252m elev depth to WT = 120m 4220 (WT elev = 4132m) 4180 4200

Well #2 4315m elev depth to WT = 78m ((WT elev = 4237m))

4240

1. Calculate WT elevations. 2. Interpolate contour intervals.

4260

4140 4160 4180

4280

4200

3 Connect 3. contours of equal elevation. 4. Draw flow lines perpendicular to contours.

Elango, L

4220 4240 4260

4280

Well #3 4397m elev depth to WT = 95m (WT elev = 4302m)

N

4300

S. Hughes, 2003

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Hydrogeology


Unsaturated Flow ►A

saturated porous medium – meaning all th voids the id are filled fill d with ith water. t ► Unsaturated zone - the pore space is only partially filled with water, the remainder of the pore space is taken up by air. ► We are interested in the hydraulics of the liquid phase transport of water in the unsaturated zone.

►Most

recharge of groundwater systems occurs during the percolation of water through the unsaturated zone. ►The movement of water in the unsaturated zone is controlled by both gravitational and capillary forces. ►Capillarity results from 2 forces: the mutual attraction between water molecules – cohesion, cohesion and the molecular attractions between water and different solid materials – adhesion.

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Hydrogeology


► Steady Steady--state

flow of water in the unsaturated zone can be determined from a modified form of Darcy’s law. Steady state -refers to a condition in which moisture content remains constant ► Steady Steady--state unsaturated flow (Q) is proportional to the effective hydraulic conductivity (Ke), the crosscross-sectional area (A) through which the flow occurs occurs, and gradient direction to both capillary forces and gravitational forces:

= KeA [(hc – z) / z] ± d h/d l ► Ke – the effective hydraulic conductivity is the K of material that is not completely saturated saturated. It is less than the saturated K for the material: ► (hc – z) / z is the gradient due to capillary forces (surface tension) and ± d h/d l is the gradient due to gravity. ►Q

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Hydrogeology


► Water

►

flow in the unsaturated zone:

Physical principles (potential theory and capillary forces)

►

Water retention and hydraulic conductivity

►

Parameterization of soil hydraulic y properties

What is a soil ? ► Soil

is the product of mechanical, chemical, and biological weathering of a parent material (mostly rocks)

Texture: Grain size distribution of mineral particles Structure: Spatial arrangement of soil particles

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Hydrogeology


How can we describe soils? ¾

general information color smell horizon structure etc. ¾ soil physical information

texture bulk density soil water retention hydraulic conductivity hydraulic parameters

Soil texture ► Grain

size distribution

Determined by: wet sieving (sand fraction) and pipette analyses (silt and clay)

USDA textural

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Hydrogeology


Soils as a 3 phase system Soil: 3 Phases: air + water + matrix (mineral and organic t i matrix

t water

i air

total volume

φM + φW + φ A = 1 φW + φ A = pore volume or porosity

Characteristics of 3 phase system I volumetric water content

matrix

water

air

[cm3

θv = water volume soil volume

cm-3]

mass of water *100 Gravimetric water content θ g = mass of dry soil [%] Volumetric air content [cm3

cm-3]

Porosity [cm3 cm-3]

Elango, L

air volume : φA = soil volume

φ=

pore volume soil volume

= φL + φW

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Hydrogeology


Characteristics of 3 phase system II matrix

water

air dry bulk density ρb =

dry soil mass soil volume

[g cm-3]

Soil texture

Bulk density

VWC

θv (cm³ cm−³)

ρb (g cm−3)

Sand

1.2 - 1.7

0.35 - 0.55

Sandy clay

1.2 - 1.7

0.25 - 0.55

Silt clay

1.2 - 1.6

0.40 - 0.50

clay

0.7 - 1.3

0.50 - 0.65

peat

0.1 - 0.5

0.60 - 0.90

volcanic ash

0.1 - 0.5

0.60 - 0.90

Measurement of soil water content (SWC) Direct methods: S Sampling li off undisturbed di t b d / di disturbed t b d soil il probes b

‘destructive’ Indirect methods: Measurement of physical parameters which are related to soil water content (e.g. dielectric permittivity or bulk conductivity)

‘non-destructive’

Elango, L

Time Domain Reflectometry (TDR) Frequency Domain Reflectometry (FDR) Ground Penetrating Radar (GPR) Electrical Resistivity Tomography (ERT)

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Hydrogeology


Direct measurement of SWC Procedure:

Sampling with known volume Kopeckyring

1.

take net weight of empty Kopeckyring

2 2.

ttake k weight ight off ri g fill d with ith fr h ring filled fresh wet soil

3.

dry probe at 104° for at least 48 hours

4.

determine weight loss of dried soil Example:

volume Kopecky-ring = 100 cm³ net weight ring = 90 g weight ring & wet soil = 275 g weight ring & dry soil = 240 g weight wet soil = 185 g weight dry soil = 150 g mass lost by drying = 35 g or 35 cm³ volumetric water content = 35cm3 100 cm-3 = 0.35 cm3cm-3 bulk density and volumetric water content can be determined at the same probe

Adhesion and cohesion

Adhesion and cohesion are responsible for the binding of water within capillaries adhesion is the physical property of different substances to attract each other by molecular forces (e.g. soil particles and water) cohesion is the physical property of a substance to attract each other by molecular forces (e.g. surface tension)

γ < 0 = hydrophilic

Elango, L

γ > 0 = hydrophobic

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Hydrogeology


Capillarity Driving force of water movement in unsaturated soils is the capillary force Capillary rise depends on pore diameter and surface tension z

Example of a clean glass-capillary P = 1 atm

ΔP =

2σ R

R

with: σ = surface tension [J m-2]

R = capillary radius [m] ΔP = pressure difference to free atmosphere [J kg-1 or m]

pressure P

Within soils surface tension depends on curvature radius of the meniscus, which depends on the pore properties and the liquid phase!

Capillary forces

water within the capillaries has a lower pressure compared to the free atmosphere tto ttake k water t r outt off th ill ri th r as th ill r fforce r the capillaries, the same fforce the capillary has to be applied

In general, soil do not consist of only one capillary. Moreover soils consist of a bundle of different capillaries with different pore diameters. diameters

The potential at which water will be held by the soil matrix due to capillary forces will be called matric potential ψm [cm]. Per definition the matric potential will be assigned negative in the vadose

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Hydrogeology


Water retention ►Water retention (θ(ψ) or θ(h) ) is the dependency between volumetric water content (θ (θ) and the matric potential or pressure head (ψ (ψ or h) residual water content (log10 h = 6) = θr 6

sand

adsorption region

clay loam 5

log10 h [cm]

4 capillary region

3 air entry value

2

or bubble point region

1 0 0

0,1

0,2

0,3

w ater conte nt [cm ³ cm -³]

¾

0,4

0,5

saturated water content = θs equals porosity

each soil is explicitly described by the retention curve

►Model

bundles

Water retention

of capillary

simplified model of the soils relationship between θ(h) and pore size distribution capillary rise full saturation

h = height above water level

θi = water filled cross section; Δθi = water filled cross section of pore class i

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Hydrogeology


Forschungszentrum Jülich Germany (Research Centre, Julich)

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Hydrogeology


Parameterisation of water retention θ(h) characteristics can be described by a mathematical function ► θ(h) parameterization = selection of a suitable function and its parameters ¾ commonly used function is the Rien van Genuchten function ►

θs − θr

(van Genuchten, 1980)

θ (h ) = θ r +

(1 + (α h ) )

n m

θs = saturated water content [cm3 cm-3] θr = residual water content [cm3 cm-3] α

= air entrance value [cm-1]

n & m = shape parameters [-]

1/n

Elango, L

mostly m = 1-

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Hydrogeology


Typical hydraulic properties Te xtural class

θr 3

clay clay loam loam loam y san d san d san dy clay san dy clay loam san d loam silt silty clay silty clay loam silty loam

cm cm 0.068 0.095 0.078 0.057 0.045 0.100 0.100 0.065 0.034 0.070 0.089 0.067

α

θs -3 3

3

cm cm 0.38 0.41 0.43 0.41 0.43 0.38 0.39 0.41 0.46 0.36 0.43 0.45

-3 3

n -1

cm 0.008 0.019 0.036 0.124 0.145 0.027 0.059 0.075 0.016 0.005 0.010 0.020

1.09 1.31 1.56 2.28 2.68 1.23 1.48 1.89 1.37 1.09 1.23 1.41

Ks cm day da -1 4.80 6.24 24.96 350.20 712.80 2.88 31.44 106.10 6.00 0.48 1.68 10.80

Schaap, M.G., F.L. Leij, and M.Th. Van Genuchten. 2001. Rosetta: A computer program for estimating soil hydraulic parameters with hierarchical pedotransfer functions. Journal of Hydrology. 251:163-176.

Typical retention curves water retention

clay clay loam loam loamy sand sand sandy clay sandy clay loam sandy loam silt silty clay silty clay loam silty loam

0,5 ,

3 -3 θ [cm cm ]

0,4

0,3

0,2

0,1

0 0

1

2

3

4

5

6

log h [-cm]

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Hydrogeology


Darcy’s law Henry Darcy 1856

ΔH q = −K Δz water or Darcy-flux [m s-1]

hydraulic gradient

permeability [m s-1]

¾Darcy’s law is only valid in saturated media and if the pores are not too small or large ¾It is not valid at the pore scale

Buckingham Darcy’s law The Buckingham Darcy’s law describes the water flow in the unsaturated zone, whereby the permeability will be expressed as a function of water content. content saturated

Edgar Buckingham (1907)

-1

K [cm min ]

0,3

q = −K

∂H ∂z

unsaturated decrease of flow cross section

increase of tortuosity

¾ hydraulic conductivity function

0,2

Increase of flow resistance in smaller pores

0,1

0 0

1

2

3

4

log h [-cm]

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Hydrogeology


Richards’ Equation Continuity equation for water flow in combination with lamina flow equation of Buckingham-Darcy. continuity equation

∂θ ∂q =− ∂z ∂t

Lorenzo Richards (1931)

Darcy or Buckingham Darcy

q = −K

∂H ∂z

∂θ ∂ ⎡ ∂H ⎤ ∂ ⎡ ⎛ ∂h ⎞⎤ = ⎢ K (h ) = ⎢ K (h )⎜ − 1⎟⎥ ⎥ ∂t ∂z ⎣ ∂z ⎦ ∂z ⎣ ⎝ ∂z ⎠⎦

¾only valid if: (limited mobility of the air phase) air phase is continuous

no water transport within the gas phase no deformation of the solid phase (rigid medium = no swelling or shrinking)

Numerical solution of Richards’s equation non-linear partial differential equation Æ K and θ depend on h non time variable boundary conditions (e. (e.g. precipitation, evaporation, drainage) vertical layered profiled

functions h(z,t) and θ(z,t) will be estimated at discrete times ti and points zj spatial resolution: set of points zj (mostly fixed and user defined) time resolution: set of times ti (will be set internally by the program and depends on boundary conditions and convergency of the numerical solution)

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Hydrogeology


HYDRUS--1D/2D/3D HYDRUS numerical water, heat, and solute transport model ¾ solves numerically Richards’ Richards equation for the water flow

Jiri Šimůnek (2007) download at: http://www.pc-progress.cz/ Reference: Simunek, J., M. Th. van Genuchten and M. Sejna, The HYDRUS-1D Software Package for Simulating the Movement of Water, Heat, and Multiple Solutes in Variably Saturated Media, Version 3.0, HYDRUS Software Series 1, Department of Environmental Sciences, University of California Riverside, Riverside, California, USA, 270 pp., 2005.

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Groundwater flow

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