A non-linear finite volume method coupled with a modified higher order MUSCL-type method for the numerical simulation of two-phase flows in non-homogeneous and non-isotropic oil reservoirs

Abstract

In this work, we propose a finite volume scheme to simulate two-phase flows in non-homogeneous and non-isotropic 2-D petroleum reservoirs. The governing equations are solved using the IMPES (IMplicit Pressure and Explicit Saturation) procedure, where the face fluxes from the pressure equation are approximated by a non-linear two-point flux approximation (NL-TPFA) that guarantees monotone solutions for the absolute pressure field. The scheme is based on the construction of one-sided fluxes on each adjacent cell independently and then, the unique edge flux is built as a linear combination of the one-sided fluxes. In our NL-TPFA finite volume scheme, we use auxiliary variables that are located on the vertices of the primal mesh. The nodal auxiliary unknowns are written as linear combinations of the neighboring cell-centered unknowns reducing our scheme to a fully cell-centered one. To solve the non-linear system of equations and to guarantee monotone solutions for arbitrarily anisotropic permeability tensors, we use the Picard iteration method with the Anderson acceleration technique to improve the computational efficiency. On the other hand, to solve the hyperbolic saturation equation, we propose a modified second-order finite volume method. The basic idea of our method is that the reconstructed saturation on the edge that violates the local Discrete Maximum Principle (DMP) is limited, otherwise, the reconstructed saturation is expressed as a convex combination of its unlimited and limited reconstructed values.