Simulation of Nonlinear Immiscible Displacement by an Operator Splitting High Order Godunov-Mixed Finite Element Method

Immiscible displacement can be modelled by a system of partial differential equations which includes a parabolic-elliptic equation for pressure and a non-linear advection-diffusion equation for saturation. The main issue of this paper concerns the numerical integration of the saturation equation, using an operator splitting (OS) approach to decouple diffusive from advective forces. In this framework, an hyperbolic conservation law is first integrated by an explicit high order Godunov method. Next, a degenerate parabolic equation for the diffusion step is approximated implicitly either on regular or unstructured grids using mixed finite elements (MFE) or multipoint flux approximation (MPFA), respectively. Numerical results are given to validate the effectiveness of the sequential formulation under a wide range of flow regimes.