Transcript Approximate Analytical/Numerical Solutions to the Groundwater

Approximate Analytical Solutions to the
Groundwater Flow Problem
CWR 6536
Stochastic Subsurface Hydrology
3-D Steady Saturated
Groundwater Flow
  Kf    Kf    Kf 
0 
 
 

x  x  y  y  z  z 
• K(x,y,z) random hydraulic conductivity field
• f (x,y,z) random hydraulic head field
• want approximate analytical solutions to the 1st and
2nd ensemble moments of the head field
System of Approximate Moment Eqns
to order e2
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 sf2=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
Possible Solution Techniques
• Fourier Transform Methods (Gelhar et al.)
• Greens Function Methods (Dagan et al.)
• Numerical Techniques (McLaughlin and
Wood, James and Graham)
Fourier Transform Methods Require
• A solution that applies over an infinite domain
• Coefficients in equations that are constant, or can
be approximated as constants or simple functions
• Stationarity of the input and output covariance
functions (guaranteed for constant coefficients)
• All Gelhar solutions use a special form of Fourier
transform called the Fourier-Stieltjes transform.
Recall Properties of the Fourier
Transform
• In N-Dimensions (where N=1,3):

F (k ) 
  

1
ik 
f
(

)
e
d

N
2  

 ik  
f ( )   F (k )e
dk



• Important properties:




F (af ( x )  bg ( x )  aF (k )  bG (k )

S ff (k ) 
 ik  
P ( )e
d
N  ff
2  
1

 n f 

n
F  n    ik j  F k 
 x j 
Recall properties of the spectral
density function
• Spectral density function describes the distribution
of the variation in the process over all frequencies:

 
s f   S ff (k )dk
2

• Eg
Look at equation for Pff(x,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' )
•
•
•
•
Are coefficients constant?
Can input statistics be assumed stationary?
If so output statistics will be stationary.
Assume  x' F ( x' )  0 ; substitute   x  x' ;   x'
0   2 Pff1 ( )  Pff ( )   x 'f0
Solve equation for Pff(x,x’)
• Expand equation
 2 P ff1 ( )  2 P ff1 ( )  2 P ff1 ( )
0


2
2
 1
 2
 3 2
P ff ( ) f 0 P ff ( ) f 0 P ff ( ) f 0






 1
 1
 2  2
 3
 3
• Take Fourier Transform



2
2
0  (ik1 ) S ff 1 (k )  (ik 2 ) S ff 1 (k )  (ik3 ) S ff1 (k )
2
  0
  0
  0
 (ik1 ) S ff (k )
 (ik 2 ) S ff (k )
 (ik3 ) S ff (k )
 1
 2
 3
Solve equation for Pff(x,x’)
• Rearrange
  0
0
0 
iS ff (k ) k1
 k2
 k3









1
2
3
2
S ff1 (k ) 
k
df 0
• Align axes with mean flow direction and let J  
d1

  ik1 JS ff (k )
2
S ff1 (k ) 
k
Look at equation for Pff(x,x’)
0   2 Pf1f1 ( x, x' )  F ( x)  Pf1f1 ( x, x' )  P ff1 ( x, x' )  f 0 ( x)
•
•
•
•
Are coefficients constant?
Can input statistics be assumed stationary?
If so output statistics will be stationary.
Assume F ( x)  0 ; substitute   x  x' ;
0   2 Pf1f1 ( )  P ff1 ( )  f 0
  x
Solve equation for Pff(x,x’)
• Expand equation
0
 2 Pf1f1 ( )
 1
2

 2 Pf1f1 ( )
 2
2

 2 Pf1f1 ( )
 3 2
P ff1 ( ) f 0 P ff1 ( ) f 0 P ff1 ( ) f 0






 1
 1
 2
 2
 3
 3
• Take Fourier Transform



2
2
0  (ik1 ) S f1f1 (k )  (ik 2 ) S f1f1 (k )  (ik3 ) S f1f1 (k )
2
  0
  0
  0
 (ik1 ) S ff1 (k )
 (ik 2 ) S ff1 (k )
 (ik3 ) S ff1 (k )
 1
 2
 3
Solve equation for Pff(x,x’)
• Rearrange
  0
0
0 
 iS ff1 (k ) k1
 k2
 k3









1
2
3
2
Sf1f1 (k ) 
k
• Align axes with mean flow direction and let J  



2 2
 ik1 JS ff1 (k ) ik1 J   ik1 JS ff (k )  k1 J S ff (k )
 
2
2
4
Sf1f1 (k ) 
  2 
k
k 
k
k

df 0
d1
Procedure
• Given Pff() Fourier transform to get Sff(k)
 
e.g. P ff ( )  exp 
 

 

3
S ff  2

2 2
 1  k


• Use algebraic relationships to get Sff1(k) and Sf1f1(k)
  ik1 J
3

S ff1 (k )   2
2
2 2
k  1  k


 k12 J 2
3

Sf1f1 (k )   4
2
2 2
k  1  k


• Inverse Fourier transform to get Pff1() and Pf1f1()
• Then multiply each by e2=slnK2 to get Pff() and Pff()
Results
• Head Variance:
s h2 
s ln k 2 J 2 2
3
• Head Covariance


  
  

2




cos


1
e

2
e

1




 

2 2 2




s ln K J  

Pff ( ,  ) 


2
2 



2

 3 cos 2   1  1  e    2   e   1  4  8  
 

2  

 






 





