Gaussian analysis of one-dimensional unsaturated flow in randomly heterogeneous soils

We propose a new method for the solution of stochastic unsaturated flow problems in randomly heterogeneous soils that applies to both steady-state and transient regimes, avoids linearizing 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 the linearly separable model of L. Vogel and coworkers, provided that 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 unsaturated flow through a randomly stratified soil with hydraulic conductivity that varies exponentially with αψ where ψ=1/α ψ is dimensional pressure head and α is a random field with given statistical properties. Upon disregarding correlation between αand ψ, we obtain good agreement between Gaussian closure and Monte Carlo results for the mean of ψover a wide range of parameters; the agreement between variances is good far from the soil bottom where pressure head is prescribed, but not as good near the bottom boundary. We expect that the latter problem will be remedied once we account for correlation between α and ψ. Our solution provides considerable insight into the analytical behavior of the stochastic flow problem.