Transcript Approximate Analytical/Numerical Solutions to the

Approximate Analytical/Numerical
Solutions to the Groundwater Flow
Problem
CWR 6536
Stochastic Subsurface Hydrology
3-D Saturated Groundwater Flow
  Kf    Kf    Kf 
0 
 
 

x  x  y  y  z  z 
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K(x,y,z) random hydraulic conductivity field
f (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
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 2f
K f
 2f K f
 2f K f
0

K

K

x x
x 2
y 2 y y
z 2 z z
 2f
1 K f  2f 1 K f  2f 1 K f
0





2 K x x
2 K y y
2 K z z
x
y
z
 2f
 ln K f  2f  ln K f  2f  ln K f
0





x x y 2
y y z 2
z z
x 2
0   2f   ln K  f
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.
LnK ( x)  F ( x)  ef ( x)
F ( x)  ELnK ( x)
E f ( x)  0 Var  f ( x)  1 e   lnK
• Assume solution of the output random variable
can be approximate as a converging power series
in the small parameter e.
f ( x)   e ifi f0 ( x)  ef1( x)  e 2f2 ( x)  ....
Small Perturbation Methods
• Insert expansion into governing equation

0   2 f0 ( x)  ef1 ( x)  e 2f2 ( x)  ...


  F ( x)  ef ( x)    f0 ( x)  ef1 ( x)  e 2f2 ( x)  ...
• Collect terms of similar order
0   2f0 ( x)  F ( x)  f0 ( x)


0  e  2f1 ( x)  F ( x)  f1 ( x)  f ( x)  f0 ( x)


0  e 2  2f2 ( x)  F ( x)  f2 ( x)  f ( x)  f1 ( x)

Solve Mean Head Distribution
• Evaluate mean head distribution to order e2


Ef ( x)  E  e ifi  Ef0 ( x)  eEf1( x)  e 2 Ef2 ( x)  ....
• Solve equations for E[fi(x)]
0   2f0 ( x)  F ( x)  f0 ( x)
0   2 E f1 ( x)  F ( x)  E f1 ( x)  E  f ( x) f0 ( x)
0   2 E f 2 ( x)  F ( x)  E f 2 ( x)  E f ( x)  f1 ( x)
• Therefore to first order
E f ( x)  f0 ( x)
Solve Head Covariance Function
• Evaluate head covariance to order e2
Pff ( x, x' )  E f ( x)  E f ( x)f ( x' )  E f ( x' )



 f0 ( x)  ef1 ( x)  e 2f2 ( x)  ...  f0 ( x) 
 E

2
 f0 ( x' )  ef1 ( x' )  e f2 ( x' )  ...  f0 ( x' ) 



 ef1 ( x)  e 2f2 ( x)  ... ef1 ( x' )  e 2f2 ( x' )  ...
 e 2 E f1 ( x)f1 ( x' )  ....
• Need to determine Ef1( x)f1( x' )

Solve for Head Covariance
• Post-Multiply equation for f1(x) by f1(x’):


0  2f1( x)  F ( x)  f1( x)  f ( x)  f0 ( x) f1( x' )
• Take f1(x’) inside derivatives with respect to x:
0  2f1( x)f1( x' )  F ( x)  f1( x)f1( x' )  f ( x)f1( x' )  f0 ( x)
• Take expected values:
0   2 Ef1 ( x)f1 ( x' )  F ( x)  Ef1 ( x)f1 ( x' )  E f ( x)f1 ( x' )  f0 ( x)
  2 Pf1f1 ( x, x' )  F ( x)  Pf1f1 ( x, x' )  Pff1 ( x, x' )  f0 ( x)
• Need Head-Log Conductivity Crosscovariance Pff1 ( x, x' )
Solve for Head-Log Conductivity
Cross-Covariance
• Pre-Multiply equation for f1(x’) by f(x):


0  f ( x)  x' 2f1 ( x' )   x' F ( x' )   x'f1 ( x' )   x' f ( x' )   x'f0 ( x' )
• Take f(x) inside derivatives with respect to x’:
0   x' 2 f ( x)f1 ( x' )   x' F ( x' )   x' f ( x)f1 ( x' )   x' f ( x) f ( x' )   x'f0 ( x' )
• Take expected values:
0   x' 2 E f ( x)f1 ( x' )   x' F ( x' )   x' E f ( x)f1 ( x' )   x' E f ( x) f ( x' )  x'f0 ( x' )
  x' 2 P ff1 ( x, x' )   x' F ( x' )   x' P ff1 ( x, x' )   x' P ff ( x, x' )   x'f0 ( x' )
• Need log-conductivity auto-covariance Pff ( x, x' )
System of Approximate Moment Eqns
0   2f 0 ( x)  F ( x)  f 0 ( x)
0   x' 2 P ff1 ( x, x' )   x' F ( x' )   x' P ff1 ( x, x' )   x' P ff ( x, x' )   x'f 0 ( x' )
0   2 Pf1f1 ( x, x' )  F ( x)  Pf1f1 ( x, x' )  P ff1 ( x, x' )  f 0 ( x)
• Use f0(x), as best estimate of f(x)
• Use f2=Pff(x,x) as measure of uncertainty
• Use Pff(x,x’) and Pff(x,x’) for cokriging to
optimally estimate f or f based on field observations