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 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  c 2 (x,t)=P cc (x,t;x,t) as measure of uncertainty Use P cv (x,t;x’) and P cc (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 P cvi and P cc to be constant Require input covariance function to be stationary Convert pdes for covariance functions P cvi and P cc into algebraic expressions for S cvi and S cc.

Spectral Solution for Steady-State Macrodispersive Flux (P cvi (x,x) )

Therefore mean equation looks like A ij determined from spectral relationship between S vic and S vivj S vivj is the inverse Fourier transform of P vivj determined from P ff, flow equation and Darcy’s law.

Results Assuming 3-D isotropic negative exponential for P ff, and  l,  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 Steady-State Concentration Variance

Results Assuming hole-type isotropic negative exponential for P ff, and  l,  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

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 P cvi and P cc 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