Gaussian finite element closure of steady state unsaturated flow in randomly heterogeneous soils

摘要

We present 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 method applies in principle to a broad class of soils with unsaturated properties that scale according to a linearly separable model, provided that pressure head has a near-Gaussian distribution. We apply the method to soils whose relative hydraulic conductivity varies exponentially with pressure head. The flow domain is a checkerboard of uniform subdomains within each of which the soil hydraulic properties are random constants. Across the flow domain, these properties are spatially auto- and cross-correlated. Flow is described by means of finite element equations with spatially-correlated random coefficients. Averaging the finite element equations in probability space, while treating dimensionless pressure head as a multivariate Gaussian function, yields a system of partially coupled nonlinear algebraic equations that can be solved numerically for the ensemble mean, variance and covariance of pressure head across the domain. We do so for two-dimensional flow in a bounded vertical domain under coupled mean uniform and convergent flows, and compare the results with those of standard Monte Carlo simulations.