Gaussian Finite Element Closure of Steady State Unsaturated Flow in Randomly Heterogeneous Soils

Abstract

In this study, I develop a Gaussian Closure method to simulate steady state unsaturated flow in randomly heterogeneous soils. I predict pressure heads and fluxes and evaluate uncertainties associated with these predictions, without resorting to Monte Carlo simulation, upscaling, or linearization of the governing flow equations and the constitutive relationship between unsaturated hydraulic conductivity and pressure head. Upon treating dimensionless pressure head as a multivariate Gaussian function in the manner of Amir and Neuman [2001], I obtain a closed system of coupled non-linear differential equations for the first and second moments of pressure head and flux for both spatially uncorrelated Y (log saturated hydraulic conductivity) and spatially correlated Y. Computational examples for unsaturated flow in a vertical plane, subject to deterministic forcing terms including a point source, show a good agreement between my Gaussian closure solution and a more general Monte Carlo solution. The computational examples include a uniform domain, eight subdomains, spatially uncorrelated non-uniform Y cases, spatially correlated Y cases, and conditional Y cases. Though the computational examples treat the random pore size parameter a as being uniform across the entire flow domain, I show theoretically that the Gaussian closure method could apply to spatially variable a statistics.