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Approximate Analytical Solutions to the Groundwater Flow Problem

Approximate Analytical Solutions to the Groundwater Flow Problem. CWR 6536 Stochastic Subsurface Hydrology. System of Approximate Moment Eqns to order e 2. Use f 0 (x), as best estimate of f (x) Use s f 2 =P ff (x,x) as measure of uncertainty

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Approximate Analytical Solutions to the Groundwater Flow Problem

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

  2. System of Approximate Moment Eqns to order e2 • 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

  3. Fourier Transform Techniques • Require an infinite domain • Require coefficients in pdes for Pff1 and Pf1f1to be constant • Require input covariance function to be stationary. • Convert pdes for covariance functions Pff1 and Pf1f1 into algebraic expressions for Sff1 and Sf1f1 .

  4. Spectral Relationships • Assume a form for Sff , inverse Fourier transform to get Pff1 and Pf1f 1 ; Multiply by e2=slnK2 to get Pff and Pff

  5. 3-D Results • Assume 3-D exponential input covariance for Pff. • 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 Pff. • 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 Pff. • 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 e2:

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

  15. Effective Hydraulic Conductivity • Therefore, recalling sf2=1, e=slnK

  16. Results for isotropic exponential Sff • 1-D F11=1, therefore: • 2-D F11= F22 =slnK2/2, therefore: • 3-D F11= F22 = F33 = slnK2 /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.

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