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.

本工作的重点是使用非正则单元格为中心的多点通量近似与菱形模板（MPFA-D）的新型适当组合来模拟二维各向异性石油储层中油和水的两相流有限体积法将压力方程离散化，并通过高分辨率（CPR）方案对高分辨率方程进行修正，以使饱和度方程离散化。为了抑制高阶方案中常见的冲击附近的数值振荡（下冲/过冲），并在解决方案的平滑区域中提供高精度，在重构阶段使用了分层多维限制过程（MLP）。时间的积分是使用三阶Runge–Kutta方法进行的。使用经典的隐式压力显式饱和（IMPES）过程进行方程组的压力-饱和度的耦合。为了使MPFA-D方法与CPR公式正确耦合，必须在整个网格控制体积中获得足够的速度重建。由于以单元为中心的有限体积方法自然会在原始网格的控制面上传递通量，因此，我们使用基于最低阶Raviart-Thomas插值函数和Piola变换的重构算子来获得速度场的完整知识整个域。最后，解决了一些具有代表性的问题，以评估我们新数值方法的准确性，效率和震撼性。