Darcy’s Law

Q: volumetric flow rate [L3 T-1]h1: hydraulic head upstream [L]h2: hydraulic head downstream [L]K: hydraulic conductivity [L T-1]A: cross sectional area normal to flow direction [L2]L: distance between locations where h1 and h2 are measured [L]

Q

datum

h1 h2

LLhhKAQ 21 −=

LhhKAQ 12or −

−=

A

Review of the Fundamentals of Groundwater Flow

dldhKA

LhhKAQ −=

−−= 12

Darcy’s Law:

dldhK

AQq −==

Alternatively:

q is called specific discharge, also Darcy flux or Darcy velocity. However, q is not true flow velocity! The true flow velocity, v, also referred to as seepage velocity or pore water velocity is defined as

⎟⎠⎞

⎜⎝⎛−==

dldh

nK

nAQv

Bulk area is APorosity is nArea for flowis (nA)

Confining bed

Confining bed

Confined aquifer b

Potentiometric surfaceTransmissivity (T) forconfined aquiferT = K b

Water table or “free surface”

Confining bed

hUnconfined aquifer h

Transmissivity for unconfined aquiferT = K h

h: saturated thickness of aquifer

Darcy’s Law and Hydraulic Head1. Hydraulic Head

AL

hhKQ 12 −−=

h1 and h2 are hydraulic heads associated with points 1 and 2.The hydraulic head, or total head, is a measure of the potential of the water fluid at the measurement point.

“Potential of a fluid at a specific point is the work required totransform a unit of mass of fluid from an arbitrarily chosen stateto the state under consideration.”

Q

datum

h1 h2

hp1

hp2

z1z2

Three Types of Potentials

A. Pressure potentialwork required to raise the water pressure

Reference state

Current statez = 0P = 0v = 0

z = zP = Pv = v

V: volume

: density of water assumed to be independent of pressure

w

P

w

P PdPmm

dPVm

Wρρ

=== ∫∫ 00111

ρw

B. Elevation potentialwork required to raise the elevation

C. Kinetic potentialwork required to raise the velocity

Wm

mgdz gzZ

2 0

1= =∫

Wm

madzm

mdvdt

dz vdvvZ Z v

3 0 0 0

21 12

= = = =∫ ∫ ∫

Total [hydraulic] head:

hg

Pg

zvgw

= = + +Φ

ρ

2

2

Unit [L]

(dz = vdt)

Total potential:

Φ = + +P

gzv

wρ

2

2Unit [L2T-1]

hP

gz

vgw

= + +ρ

2

2

pressurehead [L]

elevation [L]

Kinetic term

Piezometer

datum

h1 h2

z1 z2

Pg1

ρ gPρ

2

A fluid moves from where the total head is higher to where it is lower. For an ideal fluid (frictionless and incompressible), the total head would stay constant.

Total head or hydraulic head:

gvz

gPhw 2

2

++=ρ Kinetic term

negligible

h = hydraulic head [L]= pressure head [L]

z = elevation head [L]Important: h is relative to datum (reference state)

For Groundwater Flow

datum

A B

h2

piezometers

h1

flowdirection?

gP wρ

Hydraulic gradient

Q Kh h

LA

Kh h

LA

KiA Kdhdl

A

=−

= −−

= − = −

1 2

2 1

datum

h1 h2

L

Gradient of h with respect to xat point A is:

dhdx

hx

h hx x

Ax

=

≅−−

→∆

∆∆0

2 1

2 1

lim

h2

h1

x1 x2

A

h

x

x

y

θ

xh∂∂

yh∂∂

dldh

h=4

32

1

Magnitude and direction of gradient vector in 2D

22

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+⎟⎠⎞

⎜⎝⎛∂∂

=yh

xh

dldh

xhyh∂∂∂∂

= arctanθ

Extension of Darcy’s LawDarcy’s Law in 1D:

dldhKq −=

( )

kzhKj

yhKi

xhKkqjqiqq

kzhj

yhi

xhhhgrad

dldh

zyx ∂∂

−∂∂

−∂∂

−=++=

∂∂

+∂∂

+∂∂

=∇==

zhKq

yhKq

xhKq zyx ∂

∂−=

∂∂

−=∂∂

−=∴ ;;

i, j, k: unit vectors

NOTE: above equations are valid only if K is scalar, i.e., aquifer is isotropic. In other words, K does not vary with direction at any location.

Darcy’s Law in 3D:

In an anisotropic aquifer, K is a second-order tensor.It has 9 components in the Cartesian coordinate system:

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

zzzyzx

yzyyyx

xzxyxx

KKKKKKKKK

K

Darcy’s Law becomes

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

∂∂

∂∂

∂∂

zhyhxh

zzzyzx

yzyyyx

xzxyxx

z

y

x

KKKKKKKKK

qqq

formmatrix inor

( )hgradKq ⋅−=

Darcy’s Law with expanded terms:

zhK

yhK

xhKq

zhK

yhK

xhKq

zhK

yhK

xhKq

zzzyzxz

yzyyyxy

xzxyxxx

∂∂

−∂∂

−∂∂

−=

∂∂

−∂∂

−∂∂

−=

∂∂

−∂∂

−∂∂

−=

( )zyxjixhKq

jiji

,,3,2,1,

:notationtensor In

=

∂∂

−=

NOTE Kxx, Kyy, Kzz are principal components of the K tensor; parallel or normal to the direction of the maximum or minimum K.Kij (where i ≠ j) are cross terms of the K tensor, representing contributionsto the flow rate in direction i from the gradient in direction j.

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

zzzyzx

yzyyyx

xzxyxx

KKKKKKKKK

K

x

y

zKyy

Kzz

Kxx

xy

z

Kyy

Kzz

Kxx

If the Cartesian coordinateaxes are aligned with the principal directions of the K tensor,the cross terms are all equal to zero:

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

zz

yy

xx

KK

KK

000000

General K tensor: Aquifer formation

Use only one index for the K tensor:

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

z

y

x

KK

KK

000000

zhKq

yhKq

xhKq zzyyxx ∂

∂−=

∂∂

−=∂∂

−= ;;

We have components of specific discharge:

or in tensor notation:

( )zyxixhKq

iii

,,3,2,1=∂∂

−=

q

grad(h)

In isotropic media, K is scalar, specific discharge isparallel to hydraulic gradient

q

grad(h)

In anisotropic media, K is tensor or vector, specific discharge is NOT parallel to hydraulic gradient

GROUNDWATER STORAGEa) Storage Concept

V AhV A hw

w

==∆ ∆

if: Q1 = Q2

∆∆

Vh

w ==

00

Steady State

Storage Coefficient: Volume of water that an aquifer releases from or takes into storage per unit surface area of the aquifer per unit change in head.

if: Q1 Q2≠

∆∆

Vh

w ≠≠

00

Non-Steady State

SV

A hw=

∆∆

LL L

3

2

⎡⎣⎢

⎤⎦⎥

Q2

Q1

∆h

h

Storage tank

A

b) Unconfined Aquifer (Specific Yield)

n: Porosity; assume 100% drainable

nhA

VS wy =∆

∆=

hnAVnAhV

w

w

∆=∆=

Sy : specific yield

Typically, not 100% drainable

hfnAVfnAhV

w

w

∆=∆=

fnhA

VS wy =∆

∆=

(f < 1)

Sr specific retention

Storage from unconfinedaquifer involves physicaldewatering

A: area

h

aquifer

h∆

ry SSn +=

c) Confined Aquifer

∆∆

VA

hw ∝

∆∆

VA

S hw =

S Sbs = [L-1]

SV

A hw=

∆∆

water inputb

piezometer

pumpwaterlevel ∆h

∆Vw

No physical dewatering involved.Storage coefficient for confined aquifer

63 1010 −− −≈S usually << porosity

S : storage coefficient, dimensionless

Ss : specific storage

A

I. Derivation of the Groundwater Flow Equation

A. Continuity Principle (conservation of mass)inflow - outflow = storage

B. Darcy’s LawSpecify how inflow and outflow are calculated

∆

“Control Volume”

∆

∆

y

∆xz

Qxleft side

Qz top

Qz bottom

Qx right side

Qyfront

Qy back

(x,y,z)

yxzzyxqQ

yxzzyxqQ

zxzyyxqQ

zxzyyxqQ

zyzyxxqQ

zyzyxxqQ

yxthzS

-

ZZ

ZZ

YY

YY

XX

XX

S

∆∆⎟⎠⎞

⎜⎝⎛ ∆

+

∆∆⎟⎠⎞

⎜⎝⎛ ∆

−

∆∆⎟⎠⎞

⎜⎝⎛ ∆

+

∆∆⎟⎠⎞

⎜⎝⎛ ∆

−

∆∆⎟⎠⎞

⎜⎝⎛ ∆

+

∆∆⎟⎠⎞

⎜⎝⎛ ∆

−

∆∆∆∆

∆∆

∆

2,, = top

2,, =bottom

,2

, =front

,2

, =back

,,2

=right

,,2

=left

)( =storage

sink Q + topQ+front Q +right Q =outflowsource Q + bottom Q+back Q +left Q =inflow

storage=outflow inflow

ZYX

ZYX

+

q x x y z y z q x x y z y z

q x y y z x z q x y y z x z

q x y z z x y q x y z z x y

Q S ht

x y z

X X

Y Y

Z Z

SS S

−⎛⎝⎜

⎞⎠⎟

− +⎛⎝⎜

⎞⎠⎟

+ −⎛⎝⎜

⎞⎠⎟

− +⎛⎝⎜

⎞⎠⎟

−⎛⎝⎜

⎞⎠⎟

− +⎛⎝⎜

⎞⎠⎟

+ =

∆∆ ∆

∆∆ ∆

∆∆ ∆

∆∆ ∆

∆∆ ∆

∆∆ ∆

∆∆

∆ ∆ ∆

2 2

2 2

2 2

, , , ,

, , , ,

, , , ,

Divide both sides by ∆ ∆ ∆x y z

zyxQq

thSq

zq

yq

xq

thS

zyxQ

z

zzyxqzzyxq

y

zyyxqzyyxq

x

zyxxqzyxxq

SSSSSSS

ZYX

SSS

ZZ

YYXX

∆∆∆=

∂∂

=+∂∂

−∂∂

−∂∂

−

∆∆

=∆∆∆

+∆

⎟⎠⎞

⎜⎝⎛ ∆−−⎟

⎠⎞

⎜⎝⎛ ∆+

−

∆

⎟⎠⎞

⎜⎝⎛ ∆−−⎟

⎠⎞

⎜⎝⎛ ∆+

−∆

⎟⎠⎞

⎜⎝⎛ ∆−−⎟

⎠⎞

⎜⎝⎛ ∆+

−

where

2,,

2,,

,2

,,2

,,,2

,,2

zhKq

yhKq

xhKq

zz

yy

xx

∂∂

−=

∂∂

−=

∂∂

−=

thSq

zhK

zyhK

yxhK

x SSSzyx ∂∂

=+⎟⎠⎞

⎜⎝⎛

∂∂

∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

+⎟⎠⎞

⎜⎝⎛

∂∂

∂∂

Heterogeneous/anisotropic with sink/sourcetransient 3D 0≠

∂∂

th

General Flow EquationThree dimensional, transient, heterogeneous, anisotropic

Laplace’s EquationSteady-state condition in homogeneous, isotropic medium without sink/source

02

2

2

2

2

2

=∂∂

+∂∂

+∂∂

zh

yh

xh

( ) ( ) ( ) ( )22

2

2

2

2

2

∇=∂∂

+∂∂

+∂∂

zyx

Laplacian Operator

∇ =2 0h

Two-dimensional

thbSbq

yhbK

yxhbK

x SSSyx ∂∂

=+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

+⎟⎠⎞

⎜⎝⎛

∂∂

∂∂

thSq

yhT

yxhT

x ssyx ∂∂

=+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

+⎟⎠⎞

⎜⎝⎛

∂∂

∂∂ *

Homogeneous & Anisotropic (in terms of transmissivity)

thSq

yhT

xhT ssyx ∂

∂∂∂

∂∂

=++ *2

2

2

2

Homogeneous & Isotropic

th

TS

Tq

yh

xh ss

∂∂

∂∂

∂∂

=++*

2

2

2

2

b: saturated thickness

II. Mathematical model of groundwater flow

A. Governing Equation

02 =∇ h (assumptions?)

th

KSh S

∂∂

=∇2 (assumptions?)

Simpler Examples:

thSq

zhK

zyhK

yxhK

x SSSzyx ∂∂

=+⎟⎠⎞

⎜⎝⎛

∂∂

∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

+⎟⎠⎞

⎜⎝⎛

∂∂

∂∂

Assumptions: 3-D, transient, heterogeneous, anisotropic,with sinks/sources(principal K components aligned with coordinate axes)

B. Initial ConditionsFor any transient problem, the initial head distributionmust be known in order to solve for head changes with time, i.e.,

Q

t = 0

t = t h0

( ) ),,(0,,, zyxfzyxh =

( ) 00,,, hzyxh =For this example,

where f(x,y,z) is known

C. Boundary ConditionsHeads, fluxes or some combination of the twomust be known at boundaries in order to solve for head changes with space in the interior of the flow field.

(Wang and Anderson, 1982)

h1 h2

1. Specified head (Dirichlet condition)

( ) 10 ),,,(,,, Γ∈= tzyxhtzyxhwhere h0(x,y,z,t) is known

For example,h=h1 on the left boundaryh=h2 on the right boundary

2. Specified flux (Neumann condition)

2),,,( Γ∈=∂η∂

− tzyxqhK o

where qo(x,y,z,t) is a known flux across the boundary.

A special case is zero flux boundary condition, i.e.,

20 Γ∈=∂η∂

−h

Γ

η

+++ + ++ +

+ +++

QUESTION: A river could be treatedas either a specified-head or specified flux boundary. What’s the difference between the two treatments?

3. Combination of 1 & 2 (Cauchy condition)also referred to as head-dependent boundary condition

b/ K/

h=hb

q/h

the flux into the aquiferthrough the aquitardis dependent on the headin the aquifer, i.e.,

3// Γ∈−

−=∂η∂

−b

hhKhK b

QUESTION: Could a river also be treated as a head-dependent boundary condition? If so, what’s the difference with the previous two treatments?

III. Solutions of Mathematical Model

Solutions to a mathematical model of groundwater flow can be obtained analytically or numerically.

Analytical Solution: Head h is expressed explicitly as mathematical formula (function) of x, y, z, t.

For example, steady-state flow in a 1-D confined aquifer:

Q

L

h1 h2

( )

slope!constant

check

12

121

⇒

−=

−+=

Lhh

dxdh

xL

hhhxh

Example of Analytical Solutions

0=∂∂

=⎟⎠⎞

⎜⎝⎛

∂∂

∂∂

thS

xhT

x

1. Governing equation: 2. Boundary conditions:

( )h x hX = =0 1

( )h x hX L= = 2

h1 h2b

Ldatum

x

( )

( )

Th

xd hdx

ddx

dhdx

dx dx

dhdx

addx

h dx adx

h x ax b

∂∂

2

2

2

20 0 0= → = ⎛⎝⎜

⎞⎠⎟ =

= =

= +

∫ ∫

∫ ∫

with boundary condition (1): x = 0, h(x) = h1we found: b = h1with boundary condition (2): x = L, h(x) = h2we found: a h h

L=

−2 1

∴ =−

+h x h hL

x h( ) 2 11

Numerical Solution:Head h is solved approximately at predefined “nodal points” as illustrated below. Numerical solution is typically obtained through a computer code.

(Wang and Anderson, 1982)