1. Approximate Analytical Solutions to the Groundwater Flow Problem CWR 6536 Stochastic Subsurface Hydrology

2. 3-D Steady Saturated Groundwater Flow K(x,y,z) random hydraulic conductivity field  (x,y,z) random hydraulic head field want approximate analytical solutions to the 1st and 2nd ensemble moments of the head field

3. System of Approximate Moment Eqns to order  2 Use  0 (x), as best estimate of  (x) Use   2 =P  (x,x) as measure of uncertainty Use P  (x,x’) and P f  (x,x’) for cokriging to optimally estimate f or  based on field observations

4. Possible Solution Techniques Fourier Transform Methods (Gelhar et al.) Greens Function Methods (Dagan et al.) Numerical Techniques (McLaughlin and Wood, James and Graham)

5. 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.

6. Recall Properties of the Fourier Transform In N-Dimensions (where N=1,3): Important properties:

7. Recall properties of the spectral density function Spectral density function describes the distribution of the variation in the process over all frequencies: Eg

8. Look at equation for P f  (x,x’) Are coefficients constant? Can input statistics be assumed stationary? If so output statistics will be stationary. Assume ; substitute

9. Solve equation for P f  (x,x’) Expand equation Take Fourier Transform

10. Solve equation for P f  (x,x’) Rearrange Align axes with mean flow direction and let

11. Look at equation for P  (x,x’) Are coefficients constant? Can input statistics be assumed stationary? If so output statistics will be stationary. Assume ; substitute

12. Solve equation for P  (x,x’) Expand equation Take Fourier Transform

13. Solve equation for P  (x,x’) Rearrange Align axes with mean flow direction and let

14. Procedure Given P ff (  ) Fourier transform to get S ff (k) Use algebraic relationships to get S f   (k) and S     (k) Inverse Fourier transform to get P f   (  ) and P     (  ) Then multiply each by  2 =  lnK 2 to get P f  (  ) and P  (  )

15. Results Head Variance: Head Covariance