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

TL;DR

This paper introduces a novel IMPES numerical scheme employing immersed finite element and mixed finite volume methods on structured grids for simulating two-phase flow in heterogeneous porous media, achieving optimal convergence for pressure and velocity.

Contribution

The paper develops a new IMPES scheme combining IFEM and mixed finite volume methods on structured grids for efficient two-phase flow simulation in heterogeneous media.

Findings

Achieves optimal convergence rates for pressure and velocity.

Demonstrates suboptimal convergence for saturation.

Efficient computation independent of media heterogeneity.

Abstract

We develop a numerical scheme for a two-phase immiscible flow in heterogeneous porous media using a structured grid finite element method, which have been successfully used for the computation of various physical applications involving elliptic equations \cite{li2003new, li2004immersed, chang2011discontinuous, chou2010optimal, kwak2010analysis}. 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 \cite{rannacher1992simple} nonconforming space, which is a modification of the work in \cite{kwak2010analysis} where `broken' $P_1$ nonconforming element of Crouzeix-Raviart \cite{crouzeix1973conforming} was developed. For the Darcy velocity, we apply the mixed finite volume method studied in \cite{chou2003mixed, kwak2010analysis} 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. Thus the whole procedure can be implemented in an efficient way using a structured grid which is independent of the underlying heterogeneous porous media. Numerical results show that our method exhibits optimal order convergence rates for the pressure and velocity variables, and suboptimal rate for saturation.