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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. System of Approximate Moment Eqns • Use 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

  4. Solution Techniques • Fourier Transform Techniques (Gelhar et al) • Finite Difference/Finite Element Techniques (Graham and McLaughlin) • Greens Function or Impulse Response Techniques (Neuman et al, Cushman et al, Li and Graham)

  5. Fourier Transform Techniques(Gelhar et al) • Require an infinite domain • Require coefficients in pdes for Pcvi and Pcc to be constant • Require input covariance function to be stationary • Convert pdes for covariance functions Pcvi and Pcc into algebraic expressions for Scvi and Scc.

  6. Spectral Solution for Steady-State Macrodispersive Flux (Pcvi(x,x) )

  7. Spectral Solution for Steady-State Macrodispersive Flux (Pcvi(x,x) ) • Therefore mean equation looks like • Aij determined from spectral relationship between Svic and Svivj • Svivj is the inverse Fourier transform of Pvivj determined from Pff, flow equation and Darcy’s law.

  8. Results • Assuming 3-D isotropic negative exponential for Pff, and al, al<<l (Gelhar and Axness, 1981): • Longitudinal macrodispersivity increases with variance and correlation scale of log conductivity • Transverse macrodispersivity increase with variance of log conductivity, independent of correlation scale and depends on local dispersivity • Fickian relationship emerges as a result of constant conc. gradient assumption

  9. Spectral Solution for Steady-StateConcentration Variance

  10. Results • Assuming hole-type isotropic negative exponential for Pff, and al, al<<l (Vomvoris and Gelhar, 1990): • Simpler negative exponential spectrum gives infinite concentration variance (caused by small wave number energy, i.e. high wave length variations at a scale large than plume scale) • Concentration variance increases with increasing log hydraulic conductivity mean and variance, increasing mean concentration gradient, and decreasing local dispersivity

  11. Numerical Solution • Solve coupled pdes using finite element or finite difference technique • Does not require an infinite domain • Does not require coefficients in pdes for Pcvi and Pcc to be constant • Does not require input covariance functions to be stationary • Does not require any special form of the input covariance function • Requires lots of computer time and memory

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