Stochastic Galerkin Method for the Buckley-Leverett Problem in Heterogeneous Formations

Abstract

Summary A generalized polynomial chaos (gPC) and stochastic Galerkin method is presented for the Buckley-Leverett equation. The gPC framework is general and not limited to small perturbations in the input log-permeability variance. We assume a stochastic model for the permeability or the total velocity. The analysis illustrates some characteristic properties that need to be addressed for more complex problems of fractional flow flux functions. We show that the stochastic Galerkin formulation of the Buckley-Leverett equations is hyperbolic. We also elaborate on the consequences on hyperbolicity of the system after pseudo-spectral approximation, a common strategy to alleviate the cost of evaluating the flux function. The stochastic Buckley-Leverett problem is solved with a Riemann solver with HLL (Harten, Lax and van Leer) flux. Our stochastic Galerkin method is compared with standard Monte Carlo simulation. For sufficient spatial resolution, the numerical solutions of our gPC based stochastic Galerkin method display multiple discontinuities depending on the order of the stochastic truncation. In large-scale problems, discontinuities may not be visible due to lack of resolution and smoothing effects from stochastic averaging. Nevertheless, only a robust numerical scheme with shock-capturing properties can produce a solution that converges with refinement in space, time, and stochastic space.