Approximate AnalyticalNumerical Solutions to the 的近似analyticalnumerical解.pptVIP

Approximate Analytical/Numerical Solutions to the Groundwater Transport Problem CWR 6536 Stochastic Subsurface Hydrology 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 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 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) 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. Spectral Solution for Steady-State Macrodispersive Flux (Pcvi(x,x) ) 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. 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 Spectral Solution for St