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

2. 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

3. Fourier Transform Techniques Require an infinite domain Require coefficients in pdes for P f    and P      to be constant Require input covariance function to be stationary. Convert pdes for covariance functions P f   and P       into algebraic expressions for S f   and S      

4. Spectral Relationships Assume a form for S ff, inverse Fourier transform to get P f    and P      Multiply by    lnK 2 to get P f  and P 

5. 3-D Results Assume 3-D exponential input covariance for P ff. 3-D Head-LnK Conductivity Cross-Covariance:

6. 3-D Results 3-D Head Covariance 3-D Head Variance:

7. 2-D Results Assume 2-D Whittle-A covariance for P ff. 2-D Head-LnK Conductivity Cross-Covariance:

8. 2-D Results 2-D Head Covariance Function 2-D Head Variance

9. 1-D Results Assume 1-D hole function covariance for P ff. 1-D Head-Covariance: 1-D Head Variance

10. Interpretation Head variance decreases with increasing problem dimensionality 1-D and 2-D infinite domain analyses require hole-type input functions for finite solutions Isotropic Ln K function produces anisotropic head covariance and head-lnK cross-covariance functions head correlated over longer distances than lnK head variance much smaller than lnK variance

11. Effective Hydraulic Conductivity Constant value of hydraulic conductivity which when inserted into deterministic flow equations reproduces ensemble mean behavior To use effective properties to describe single realization requires stationarity and ergodicity. If stationarity and ergodicity do not apply can only use effective hydraulic conductivity to describe expected value of behavior. Need head variance to quantify uncertainty around the expected value.

12. Effective Hydraulic Conductivity Assume that Insert expressions into Darcy’s law:

13. Effective Hydraulic Conductivity Take Expected Value To order   :

14. Effective Hydraulic Conductivity Need to evaluate Take Fourier transform Then inverse Fourier transform, setting  x-x’=0

15. Effective Hydraulic Conductivity Therefore, recalling  f 2 =1,  =  lnK

16. Results for isotropic exponential S ff 1-D F 11 =1, therefore: 2-D F 11 = F 22 =  lnK 2 /2, therefore: 3-D F 11 = F 22 = F 33 =  lnK 2 /3, therefore: Isotropic because coordinates aligned with mean flow direction and have isotropic input covariance

17. More Results For perfectly layered porous media: Note that all of these results require uniform mean flow, stationary isotropic input covariance functions, otherwise must solve numerically.