Treatment of relative permeability term in two-phase flow problems in porous media

This paper presents a control volume finite element model (CVFEM) to simulate simultaneous flow of two immiscible fluids in non-deformable porous media. The method is fully conservative at the local and global level. It keeps the data structure of the common finite element method (FEM). A pressure-based formulation is presented in this paper. The proper choice of primary unknown variables is a critical step in developing an efficient solution of the multiphase subsurface flow problems. Pressure-based models are one of the common choices to this end. This type of models consists of strong nonlinear terms and encounters convergence difficulties when the Jacobian matrix are poorly approximated. The most severe problem is related to the relative permeability term that appears as a function of volume fraction (or degree of saturation) of the wetting phase. Since water saturation is not a primary unknown variable, the relative permeability terms become a function of two primary unknowns, i.e. wetting and non-wetting pressures, together. A fully implicit first order accurate finite difference scheme is employed for temporal discretization of the equations. A full Newton method with exact Jacobian is considered in this work, and a rapid convergence has been achieved. The model is used to simulating a five-spot problem in a block heterogenous porous medium.