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Figure from Hornberger et al. (1998)

Darcys data for two different sands

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Range in hydraulic conductivity, K

13 orders of magnitude

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Generalization of Darcys column

h/L = hydraulic

gradient

q = Q/A

Q is proportionalto h/L

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q is a vector

q

x

z

qx1

qz2z

x z

hKq

y

hKq

x

hKq

zz

yy

xx

q = Q/A

In general: Kz< Kx, Ky

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z

hKq

y

hKq

x

hKq

zz

yy

xx

q= - Kgradh

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Vector Form of Darcys Law

q = - Kgrad h

q = specific discharge (L/T)

K= hydraulic conductivity (L/T)

grad h= hydraulic gradient (L/L)

h = head (L)

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q= - Kgradh

Kis a tensor with 9 components

(three of which are Kx, Ky, Kz)

q is a vector with 3 components

h is a scalar

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Scalar

1 component

Magnitude Head, concentration,

temperature

Vector

3 components

Magnitude and

direction

Specific discharge, (&

velocity), mass flux,

heat flux

Tensor

9 components

Magnitude,

direction and

magnitude

changing withdirection

Hydraulic conductivity,

Dispersion coefficient,

thermal conductivity

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q= - Kgradh

Darcys law

grad h

q equipotential line

grad hq

Isotropic

Kx = Ky = Kz = K

Anisotropic

Kx, Ky, Kz

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Linear flow

paths assumed

in Darcys law

True flow paths

Average linear velocity

v = Q/An= q/n

n = effective porosity

Specific discharge

q = Q/A

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Representative Elementary Volume

(REV)

REV

Equivalent Porous Medium

(epm)

q= - Kgradh

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Law of Mass Balance+ Darcys Law =Governing Equation for Groundwater Flow

---------------------------------------------------------------

divq= - Ss (ht) +R* (Law of Mass Balance)

q= - Kgrad h (Darcys Law)

div (Kgrad h) = Ss (ht)R*

Water balance equation

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Inflow = Outflow

Recharge

Discharge

Steady State Water Balance Equation

Transient Water Balance Equation

Inflow = Outflow +/- Change in Storage

Outflow - Inflow = Change in Storage

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Figures from Hornberger et al. (1998)

UnconfinedaquiferSpecific yield = Sy

ConfinedaquiferStorativity = S

b

hh

Storage Terms

S = V / A h

S = Ssb

Ss= specific storage

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)( Wz

q

y

q

x

q zyx

OUTIN =

thSs

Wt

hS

z

hK

zy

hK

yx

hK

xszyx

)()()(

z

hKq

yhKq

x

hKq

zz

yy

xx

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Law of Mass Balance+ Darcys Law =Governing Equation for Groundwater Flow

---------------------------------------------------------------

divq= - Ss (ht) +W (Law of Mass Balance)

q= - Kgrad h (Darcys Law)

div (Kgrad h) = Ss (ht)W

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W

t

hS

z

hK

zy

hK

yx

hK

x

szyx

)()()(

RthS

yhhK

yxhhK

xyyx

)()(2D unconfined:

R

t

hS

y

hT

yx

hT

x

yx

)()(2D confined:

(S = Ssb & T = K b)

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Figures from:

Hornberger et al., 1998. Elements of Physical Hydrology,

The Johns Hopkins Press, Baltimore, 302 p.

Documents

Week2b