Numerical simulation of 1-D oil and water displacements in petroleum reservoirs using the correction procedure via reconstruction (CPR) method

Abstract

The focus of this work is to investigate and to apply, for the first time, a very high-resolution correction procedure via reconstruction (CPR) numerical discretization technique for the hyperbolic saturation equation that describes 1-D oil-water displacement through heterogeneous porous media. The CPR method can achieve high-order accuracy via a compact stencil consisting of the current cell and its immediate neighbors; in addition, the CPR recovers simplified versions of nodal discontinuous Galerkin (NDG), spectral volume (SV), and spectral difference (SD) methods using an adequate polynomial reconstruction function, whose coefficients are preprocessed and stored. Indeed the CPR versions of NDG and SV/SD are highly efficient. In order to suppress numerical oscillations near shocks, which are typical from higher-order schemes, a hierarchical MLP (multi-dimensional limiting process) is used in the reconstruction stage. The integration in time is carried out using a third-order RK (Runge-Kutta) method. A number of 1-D two-phase flow benchmark problems are solved, to verify the accuracy, efficiency, and shock-capturing capability of the adopted methodology.