Approximate Analytical/Numerical Solutions to the Groundwater Flow

Problem

CWR 6536

Stochastic Subsurface Hydrology

3-D Saturated Groundwater Flow

• K(x,y,z) random hydraulic conductivity field• (x,y,z) random hydraulic head field• No analytic solution exists to this problem• 3-D Monte Carlo very CPU intensive• Look for approximate analytical/numerical solutions

to the 1st and 2nd ensemble moments of the head field

zK

zyK

yxK

x

0

First-order Perturbation Methods• Bakr et al. Water Resources Research 14(2) p. 263-271,

April 1978• Mizell et al. Water Resources Research 18(4) p. 1053-

1067, August 1982• Gelhar, Stochastic Subsurface Hydrology Ch. 4 Sections

4.1-4.4• McLaughlin and Wood Water Resources Research 24(7)

p. 1037-1060, July 1988• James and Graham, Advances in Water Resources,

22(7),711-728, 1999.

Re-write equation in terms of Ln K

K

zz

K

zyy

K

yxx

K

x

zz

K

Kzyy

K

Kyxx

K

Kx

zz

K

zK

yy

K

yK

xx

K

x

K

ln0

lnlnln0

1110

0

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

Small Perturbation Methods• Expand input random variables into the sum of a

potentially spatially variable mean and a small perturbation around this mean, i.e.

• Assume solution of the output random variable can be approximate as a converging power series in the small parameter .

lnK 1)(Var 0)(E )()(

)()()(

xfxfxLnKExF

xfxFxLnK

....)()()()( 22

10 xxxx ii

Small Perturbation Methods

• Insert expansion into governing equation

• Collect terms of similar order

...)()()()()(

...)()()(0

22

10

22

102

xxxxfxF

xxx

)()()()()(0

)()()()()(0

)()()(0

12222

0112

002

xxfxxFx

xxfxxFx

xxFx

Solve Mean Head Distribution• Evaluate mean head distribution to order 2

• Solve equations for E[i(x)]

• Therefore to first order

....)()()()( 22

10 xExExEExE ii

)()()()()(0

)()()()()(0

)()()(0

1222

0112

002

xxfExExFxE

xxfExExFxE

xxFx

)()( 0 xxE

Solve Head Covariance Function

• Evaluate head covariance to order 2

• Need to determine

....)'()(

...)'()'(...)()(

)'(...)'()'()'(

)(...)()()(

)'()'()()()',(

112

22

122

1

022

10

022

10

xxE

xxxx

xxxx

xxxxE

xExxExExxP

)'()( 11 xxE

Solve for Head Covariance

• Post-Multiply equation for (x) by (x’):

• Take (x’) inside derivatives with respect to x:

• Take expected values:

• Need Head-Log Conductivity Crosscovariance

)'()()()()()(0 10112 xxxfxxFx

)()',()',()()',(

)()'()()'()()()'()(0

02

0111112

11111xxxPxxPxFxxP

xxxfExxExFxxE

f

)()'()()'()()()'()(0 0111112 xxxfxxxFxx

)',(1

xxPf

Solve for Head-Log Conductivity Cross-Covariance

• Pre-Multiply equation for (x’) by f(x):

• Take f(x) inside derivatives with respect to x’:

• Take expected values:

• Need log-conductivity auto-covariance

)'()'()'()'()'()(0 0''1''12' xxfxxFxxf xxxxx

)'()',()',()'()',(

)'()'()()'()()'()'()(0

0''''2'

0''1''12'

11xxxPxxPxFxxP

xxfxfExxfExFxxfE

xffxfxxfx

xxxxx

)'()'()()'()()'()'()(0 0''1''12' xxfxfxxfxFxxf xxxxx

)',( xxPff

System of Approximate Moment Eqns

• Use 0(x), as best estimate of (x)

• Use 2=P(x,x) as measure of uncertainty

• Use P(x,x’) and Pf(x,x’) for cokriging to optimally estimate f or based on field observations

)()',()',()()',(0

)'()',()',()'()',(0

)()()(0

02

0''''2'

002

11111

11

xxxPxxPxFxxP

xxxPxxPxFxxP

xxFx

f

xffxfxxfx