Nonlocal finite element solutions for steady state unsaturated flow in bounded randomly heterogeneous porous media using the Kirchhoff Transformation

摘要

We consider steady state unsaturated flow in bounded randomly heterogeneous soils under influence of random forcing terms. Our purpose is to predict pressure heads and fluxes and evaluate uncertainties associated with these predictions, without resorting to Monte Carlo simulation, upscaling or linearization of the constitutive relationship between unsaturated hydraulic conductivity and pressure head. Following Tartakovsky et al. [1999], by assuming that the Gardner model is valid and treating the corresponding exponent a as a random constant, the steady-state unsaturated flow equations can be linearized by means of the Kirchhoff transformation. This allows us develop exact integro-differential equations for the conditional first and second moments of transformed pressure head and flux. The conditional first moments are unbiased predictions of the transformed pressure head and flux, and the conditional second moments provide the variance and covariance associated with these predictions. The moment equations are exact, but they cannot be solved without closure approximations. We developed their recursive closure approximations through expansion in powers of σᵧ and σᵦ, the standard deviations of Y = lnK(s), and β = ln α, respectively, where K(s), is saturated hydraulic conductivity. Finally, we solve these recursive conditional moment equations to second-order in σᵧ and σᵦ, as well as second-order in standard deviations of forcing terms by finite element methods. Computational examples for unsaturated flow in a vertical plane, subject to deterministic forcing terms including a point source, show an excellent agreement between our nonlocal solutions and the Monte Carlo solution of the original stochastic equations using finite elements on the same grid, even for strongly heterogeneous soils.