Gaussian analysis of unsaturated flow in randomly heterogeneous porous media

Abstract

We propose a new method for the solution of stochastic unsaturated flow problems in randomly heterogeneous soils which avoids linearizing the governing flow equations or the soil constitutive relations, and places no theoretical limit on the variance of constitutive parameters. The proposed method applies to a broad class of soils with flow properties that scale according to a linearly separable model provided the dimensionless pressure head ψ has a near-Gaussian distribution. Upon treating ψ as a multivariate Gaussian function, we obtain a closed system of coupled nonlinear differential equations for the first and second moments of pressure head. We apply this Gaussian closure to one-dimensional steady state, transient, and two-dimensional unsaturated flow through randomly stratified soils with hydraulic conductivity that varies exponentially with aψ̅ where ψ̅=(¹/α)ψ is dimensional pressure head and α is a random field with given statistical properties. For flow in a one-dimensional steady state medium, we obtain good agreement between Gaussian closure and Monte Carlo results for the mean and variance of ψ over a wide range of parameters provided that the spatial variability of α is small. Our solution provides considerable insight into the analytical behavior of the stochastic flow problem. For transient flow in a one-dimensional unsaturated medium with randomly homogeneous soil, Gaussian closure and Monte Carlo results for mean dimensionless pressure are in excellent agreement. However, Gaussian closure and Monte Carlo results for variance, although qualitatively acceptable, are only quantitatively accurate for large times or for early times when the variance of the log hydraulic conductivity is small. Finally, we develop a finite element framework with which to solve the two-dimensional steady state Gaussian closure moment equations.