Stochastic Analysis of Groundwater Flow Processes

1. Stochastic Analysis of Groundwater Flow Processes CWR 6536 Stochastic Subsurface Hydrology

2. Methods for deriving moments for groundwater flow processes • Exact analytic solutions • possible only if analytical solution to governing equation available. Not very realistic • Monte Carlo simulations • Delhomme, 1979 • Smith and Freeze,1979 • Approximate Analytical/Numerical Solutions • Gelhar, 1993; Hoeksema and Kitanidis, 1984;Dagan, 1989; McLaughlin and Wood,1988; James and Graham 1998.

3. 3-D saturated groundwater flow • K=K(x,y,z;z) random hydraulic conductivity field (assumed geologically isotropic) • f=f(x,y,z;z) random head field • would like to derive pdfs/moments of random head field given pdfs/moments of random hydraulic conductivity field

4. Monte Carlo Simulation • 1-D Flow Problem Smith and Freeze, 1979a • Domain discretized in x-direction • K(x) generated for each block in x-direction for multiple realizations (200) • f(x) solved for each realization • Statistics of f(x) calculated at each x over 200 realizations

5. 1-D Monte Carlo Simulation Results • Mean head field uniform • Head variance increases with Ln K variance • Head variance increases with Ln K correlation scale • Head variance increases with ratio of Ln K correlation scale to length of domain • Normality of head field only approached in interior of domain and interval of normality decreases with head variance • Flux through system random due to finite domain and selection of BCs

6. 2-D Monte Carlo Simulation Results • Uniform flow field • same trends as for 1-D case • anisotropy of Ln K field affects head variance • deterministic layering and proximinty to boundaries affects head variance • 2-D head variance reduced from 1-D head variance by ~ 50% • region of normality larger for 2-D than for 1-D

7. 2-D Monte Carlo Simulation Results • Non-uniform flow field • head variance increases with mean head gradient • head variance is greater than for uniform flow case • Uncertainty of model predictions dpeends on both hydraulic conductivity and flow configuration (governed by dimensionality and BCs)

8. Concept of Effective Hydraulic Conductivity • Uniform Keff reproduces the ensemble mean head when inserted into the deterministic model everywhere in domain • The Darcy flux calculated using the uniform Keff in the deterministic model reproduces the ensemble mean Darcy flux • For 2-D steady unidirectional flow in an unbounded domain, effective conductivity is the geometric mean • For bounded non-uniform gradient fields effective conductivity not easily defined