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Approximate Analytical/Numerical Solutions to the Groundwater Flow Problem

Approximate Analytical/Numerical Solutions to the Groundwater Flow Problem. CWR 6536 Stochastic Subsurface Hydrology. 3-D Saturated Groundwater Flow. K(x,y,z) random hydraulic conductivity field f (x,y,z) random hydraulic head field No analytic solution exists to this problem

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Approximate Analytical/Numerical Solutions to the Groundwater Flow Problem

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

  2. 3-D Saturated Groundwater Flow • 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

  3. 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.

  4. Re-write equation in terms of Ln K

  5. 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. • Assume solution of the output random variable can be approximate as a converging power series in the small parameter e.

  6. Small Perturbation Methods • Insert expansion into governing equation • Collect terms of similar order

  7. Solve Mean Head Distribution • Evaluate mean head distribution to order e2 • Solve equations for E[fi(x)] • Therefore to first order

  8. Solve Head Covariance Function • Evaluate head covariance to order e2 • Need to determine

  9. Solve for Head Covariance • Post-Multiply equation for f1(x) by f1(x’): • Take f1(x’) inside derivatives with respect to x: • Take expected values: • Need Head-Log Conductivity Crosscovariance

  10. Solve for Head-Log Conductivity Cross-Covariance • Pre-Multiply equation for f1(x’) by f(x): • Take f(x) inside derivatives with respect to x’: • Take expected values: • Need log-conductivity auto-covariance

  11. System of Approximate Moment Eqns • 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

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