High-order bound-preserving finite difference methods for incompressible two-phase flow in porous media

Abstract

In this paper, we develop high-order bound-preserving (BP) finite difference (FD) methods for solving the incompressible and immiscible two-phase flow problem with capillary pressure in porous media. We use the implicit pressure explicit saturation (IMPES) scheme to solve for the pressure, auxiliary variables, and saturations of each phase in the coupled system. The boundedness of the saturations of the two phases, $S_w$ and $S_n$, between 0 and 1 is an important physical characteristic. Applying non-physical numerical approximations may lead to significant oscillations in the numerical results and cause instability in the simulation. We apply high-order FD method and BP technique to maintain the high-order accuracy and the boundary of saturations. In the BP technique, the main idea is to choose an appropriate time step and apply positivity-preserving (PP) technique to $S_w$ and $S_n$, respectively, and ensure that $S_w+S_n=1$. In addition, the high-order accuracy is obtained by the parameterized flux limiter. Numerical examples are presented to demonstrate the high-order accuracy of the scheme and the effectiveness of the BP technique.