Toward Application of Higher Order Finite Volume Schemes to Porous Media Flow

摘要

Multi-phase flow in porous media is governed by a non-linear parabolic equation for the pressure field and degenerate parabolic equations for the saturation fields. Although it is well understood that the pressure and saturation fields vary on different spatial and temporal scales, most conventional reservoir simulators apply the same discretization to both. Substantial computational advantages can be realized by using a coarse mesh, which can effectively capture the pressure variation, but does not provide sufficient accuracy for calculating variations in saturation. Hence, in this paper we investigate the use of higher-order methods to capture the sub-grid fine scale variations in the saturation profiles. Specifically, the moving least squares (MLS) technique [1], which is conceptually similar to multi-point-flux-approximation (MPFA) [2] techniques, is used to approximate higher-order hyperbolic and viscous fluxes in a multi-point fashion for finite-volume methods applied to porous media flow. The primary focus of this study is to examine the feasibility of higher-order schemes for the purposes of reservoir simulation. To this end, two test cases are considered; (1) 1D strongly hyperbolic tracer flow (2) 2D standard quarter-five spot problem to illustrate the advantages of the MLS scheme over traditional two-point flux approximation (TPFA) [2] schemes. Predictions show that the MLS scheme allows for computation of high-order derivatives of field variables for a Godunov-type approach to hyperbolic problems in an efficient manner. Unlike standard piecewise continuous reconstruction, interpolation of field variables in MLS is continuous across interfaces. This facilitates direct and accurate reconstruction of fluxes of viscous nature. In addition, because the MLS scheme has the desirable attributes of a MPFA scheme, K-orthogonality of the underlying finite-volume mesh is enforced by default.