Mixed finite volume element-upwind mixed volume element of compressible two-phase displacement and its numerical analysis

Abstract

A fundamental topic in numerical simulation of two-phase displacement is discussed in this paper. The mathematical model for the compressible problem is defined mainly by two nonlinear partial differential equations: a parabolic equation for the pressure and a convection–diffusion equation for the saturation. The Darcy velocity is determined by the pressure, and affects the whole physical process. The system is solved by a composite numerical scheme. The conservative mixed volume element is used for the first equation. The computational accuracy is improved for the Darcy velocity. The second equation is solved by a conservative upwind mixed volume element, where the mixed volume element and upwind approximation treat the diffusion and convection, respectively. The upwind method preserves the high computational accuracy, and numerical dispersion and nonphysical oscillation are eliminated. The saturation and its adjoint vector function are obtained simultaneously. An important feature in numerical scheme, the conservation of mass, is proved. By the traditional theoretical work of numerical analysis such as a priori estimates of differential equations, the optimal order error estimate is obtained. Finally, numerical tests show the effectiveness and practicability, then the present method possibly solves the challenging problems as a powerful tool.