Probabilistic Analysis of Flow in Random Porous Media by Stochastic Boundary Elements

Engineering Analysis with Boundary Elements, 19(3), 239-255, 1997.

Abstract :  

The mathematical and numerical modeling of groundwater flows in random porous media is studied assuming that the formation's hydraulic log-transmissivity is a statistically homogeneous, Gaussian, random field with given mean and covariance function. In the model, log-transmissivity may be conditioned to take exact field values measured at a few locations. Our method first assumes that the log-transmissivity may be expanded in a Fourier-type series with random coefficients, known as the Karhunen-Loeve (KL) expansion. This expansion has optimal properties and is valid for both homogeneous and non-homogeneous fields. By combining the KL expansion with a small parameter perturbation expansion, we transform the original stochastic boundary value problem into a hierarchy of deterministic problems. To the first order of perturbation, the hydraulic head is expanded on the same set of random variables as in the KL representation of log-transmissivity. To solve for the corresponding coefficients of this expansion, we adopt a boundary integral formulation whose numerical solution is carried out by using boundary elements and the dual reciprocity (DR-BEM). To illustrate and validate our scheme, we solve three test problems and compare the numerical solutions against Monte Carlo simulations based on a finite difference formulation of the original flow problem. In all three cases we obtain good quantitative agreement and the present approach is shown to provide both a more efficient and accurate way of solving the problem.

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