Solving a model for 1-D, three-phase flow vertical equilibrium processes in a homogeneous porous medium by means of a Weighted Essentially Non Oscillatory numerical scheme

Mathematical models of multi-phase flow are useful in some engineering applications like enhanced oil recovery, filtration of pollutants into subsurface, etc. In this work, we derive a mathematical model for the motion of one-dimensional three-phase flow in a porous medium under the condition of vertical equilibrium, which can be viewed as an extension of some two-phase flow models described in the literature.Our model involves a system of two partial differential equations in the form of viscous conservation laws, whose solutions may contain very sharp transitions. We show that a high-order/high resolution Weighted Essentially Non Oscillatory scheme is an appropriate tool to discretize the buoyancy flux and obtain a well resolved representation of the solution of the model. In addition, we show that the efficiency of the scheme may be improved by using Implicit–Explicit (IMEX) strategies, where the parabolic terms are handled by an implicit discretization.