Numerical simulation of two-phase flows in 2-D petroleum reservoirs using a very high-order CPR method coupled to the MPFA-D finite volume scheme

Abstract The focus of the present work is to model the two-phase flow of oil and water in 2-D anisotropic petroleum reservoirs using a novel appropriate combination of a non-orthodox cell-centered Multipoint Flux Approximation with a Diamond Stencil (MPFA-D) finite volume method to discretize the pressure equation coupled to the very high-resolution Correction Procedure via Reconstruction (CPR) scheme for the discretization of the saturation equation. To suppress numerical oscillations (under/overshoots) near shocks, that are typical in higher-order schemes, and to deliver high accuracy in smooth regions of the solution, a hierarchical Multidimensional Limiting Process (MLP) is used in the reconstruction stage. The integration in time is carried out using a third-order Runge–Kutta method. The coupling of the pressure-saturation set of equations is carried out using a classical Implicit Pressure Explicit Saturation (IMPES) procedure. To properly couple the MPFA-D method with the CPR formulation, it is necessary to obtain an adequate velocity reconstruction throughout the control volumes of the mesh. Because the cell-centered finite volume method naturally delivers fluxes across control surfaces of the primal grid, then, we use a reconstruction operator based on the lowest order Raviart–Thomas interpolation functions and the Piola transformation, to get the complete knowledge of the velocity field throughout the domain. Finally, some representative problems were solved to evaluate the accuracy, efficiency, and shock-capturing capabilities of our new numerical methodology.