An IMPES scheme for a two-phase flow in heterogeneous porous media using a structured grid

Abstract

We develop a numerical scheme for a two-phase immiscible flow in heterogeneous porous media using a structured grid finite element method, which has been successfully used for the computation of various physical applications involving elliptic interface equations (Li et al., 2003, 2004; Chou et al., 2010; Kwak et al., 2010; Chang and Kwak, 2011). The proposed method is based on the implicit pressure-explicit saturation procedure. To solve the pressure equation, we use an IFEM based on the Rannacher–Turek Rannacher and Turek (1992) nonconforming space, which is a modification of the work in Kwak et al. (2010) where ‘broken’ P1 nonconforming element of Crouzeix–Raviart (1973) [1] was developed.For the Darcy velocity, we apply the mixed finite volume method studied in Chou et al. (2003) and Kwak et al. (2010) on the basis of immersed finite element method (IFEM). In this way, the Darcy velocity of the flow can be computed cheaply (locally) after we solve the pressure equation. The computed Darcy velocity is used to solve the saturation equation explicitly. The whole procedure can be implemented on a structured grid which is independent of the underlying heterogeneous porous media. In fact, we apply a new version of multigrid algorithm to solve the pressure equation for which the CPU time grows like O(N). Numerical tests for analytic problems show that our method is almost optimal in all variables. Problem with sources/sinks is also computed and it seems to work well.