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Approximate Analytical/Numerical Solutions to the Groundwater Transport Problem

Approximate Analytical/Numerical Solutions to the Groundwater Transport Problem. CWR 6536 Stochastic Subsurface Hydrology. 3-D Saturated Groundwater Transport. v i (x,y,z) spatial random velocity field c(x,y,z, t) spatiotemporal random concentration field

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Approximate Analytical/Numerical Solutions to the Groundwater Transport Problem

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

  2. 3-D Saturated Groundwater Transport • vi(x,y,z) spatial random velocity field • c(x,y,z, t) spatiotemporal random concentration 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 conc field

  3. 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 approximated as a converging power series in the small parameter e.

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

  5. Derive Mean Concentration Equation • Evaluate mean conc distribution to order e2 • Solve equations for E[ci(x,t)] • Therefore • To solve for second order mean concentration need E[ni’(x)c1(x,t)].

  6. Derive Velocity-Conc Cross-Covariance • Pre-Multiply equation for c1(x) by nk’(x’) and take nk’(x’) inside derivatives with respect to x: • Take expected values: • Need velocity auto-covariance to get • After solve , evaluate at x=x’ to get second order approx to mean concentration

  7. Derive Concentration Covariance Function • Evaluate concentration covariance to order e2 • Need to determine

  8. Solve for Concentration Covariance • Post-Multiply equation for c1(x,t) by c1(x’,t’), and take c1(x’,t’) inside derivatives with respect to x and t: • Take expected values: • Need velocity-conc cross-covariance to get

  9. System of Approximate Moment Eqns • Use c0(x)+e2 E[c2(x)] as best estimate of c(x,t) • Use sc2(x,t)=Pcc(x,t;x,t) as measure of uncertainty • Use Pcv(x,t;x’) and Pcc(x,t;x’,t’) to optimally estimate c or v based on field observations

  10. Alternative Formulation of Mean Concentration Eqns • Series of mean equations • , thus add these linear equations

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