A New Two-dimensional Second Order Non-oscillatory Central Scheme Applied to Multiphase Flows in Heterogeneous Porous Media

We are concerned with central differencing schemes for solving scalar hyperbolic conservation laws. We compare the Kurganov-Tadmor (KT) two-dimensional (Kurganov and Tadmor, 2000) second order semi-discrete central scheme in dimension by dimension formulation with a new two-dimensional approach introduced here and applied in numerical simulations for two-phase, two-dimensional flows in heterogeneous formations. This semi-discrete central scheme is based on the ideas of Rusanov’s method (Rusanov (1961)) using a more precise information about the local speeds of wave propagation computed at each Riemann Problem in two-space dimensions. We find the KT dimension by dimension has a much simpler mathematical description than the genuinely two-dimensional one with a little more numerical diffusion, particularly in the presence of viscous fingers. Unfortunately, as one can see in Abreu et al. (2007), the KT with the dimension by dimension approach might produce incorrect boundary behavior in a typical geometry used in the study of porous media flows: the quarter of a five spot. This problem has been corrected by the authors with the new semi-discrete scheme proposed here. We conclude with numerical examples of two-dimensional, two-phase flow associated with two distinct flooding problems: a two-dimensional flow in a rectangular heterogeneous reservoir (called slab geometry) and a two-dimensional flow in a 5-spot geometry homogeneous reservoir.