Modified Picard with multigrid method for two‐phase flow problems in rigid porous media

Abstract

Two‐phase flow problems in porous media can be found in several areas, such as Geomechanics, Hydrogeology, Engineering and Biomedicine, for example. Typically, these processes are mathematically modeled by a highly nonlinear system of coupled partial differential equations. The nonlinearity of the system makes the design and implementation of robust numerical solvers a challenging task. In this work we consider the flow of two immiscible and incompressible fluids within a non‐deformable porous medium. A mixed pressure‐saturation formulation is adopted, allowing the transition from the unsaturated to saturated zones and maintaining numerical mass conservation. A cell‐centered finite volume method and an implicit Euler scheme are considered for the spatial and time discretization of the problem. In this work, we propose a solution method for two‐phase flow problems which is based on the combination of the modified Picard linearization method and a very simple cell‐centered multigrid algorithm that performs efficiently even for heterogeneous random media. This is shown in the numerical experiments, where two test problems are presented to demonstrate the robustness of the proposed solver.