A cell-centered multipoint flux approximation method with a diamond stencil coupled with a higher order finite volume method for the simulation of oil–water displacements in heterogeneous and anisotropic petroleum reservoirs

Abstract

The simulation of fluid flow and transport in heterogeneous and anisotropic oil and gas reservoirs poses a great challenge from the numerical point of view, due to the modeling of complex depositional environments, including inclined laminated layers, channels, fractures and faults and the modeling of deviated wells, making it difficult to build and handle the Reservoir Characterization Process (RCM), particularly by using structured meshes (cartesian or corner point), which is the current pattern in petroleum reservoir simulators. Under certain hypotheses, the mathematical model that describes the fluid flow in petroleum reservoirs includes an elliptic equation with heterogeneous, possibly discontinuous, coefficients for the pressure field and a non-linear hyperbolic equation for the saturation field. In the present paper, these equations are solved via an Implicit Pressure Explicit Saturation (IMPES) procedure. To solve these equations, we use a full cell-centered finite volume formulation. The pressure equation is discretized by a non-orthodox Multipoint Flux Approximation Method with a Diamond type stencil (MPFA-D) and used for the first time for the solution of two-phase flow problems in heterogeneous porous media. It is very robust and capable of reproducing piecewise linear solutions exactly by means of a linear preserving interpolation with explicit weights that avoids the solution of locally defined systems of equations. For the solution of the saturation equation, we use a Monotone Upstream Centered Scheme for Conservation Laws (MUSCL) method based on a gradient reconstruction obtained by a least square technique in which monotonicity is reinforced by an appropriate slope limiter. The method can be used with general polygonal meshes, even though we restrict ourselves to conforming triangular and quadrilateral grids. In order to validate and show the robustness of our formulation, we solve some problems including heterogeneous and anisotropic reservoirs and displacements with high mobility ratios. Our results compare well with others found in literature.