Probabilistic analysis of flow in random porous media by stochastic boundary elements

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 nonhomogeneous 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 dual reciprocity (DRBEM). 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.