Numerical Schemes for Modelling Two Phase Flow with Interfaces

Abstract

We discuss the numerical modelling of two phase flow in porous media (see [1]). We consider a one-dimensional problem describing flow of two incompressible and immiscible fluids through a porous medium where the non-wetting phase (oil) is displaced by the wetting fluid (water). The used model is based on Darcy’s law and we consider either horizontal (neglecting the influence of gravity) or vertical flow. In case of horizontal flow, we compare our solution with analytical solution published in [2]. In case of gravity driven vertical flow, there is no known analytical solution and we propose our solution as a benchmark solution.Our numerical model is based on the modelling of interface separating zones where the water is present and where it is not. We semidiscretize the problem in space and obtain a mass preserving system of ODEs. We have moving grid points only on region where the water is present, and these move accordingly to the evolution of the interface. We can choose to have non-equidistant grid with more grid points in the neighbourhood of the interface. This guarantees very good approximation of sharp fronts during infiltration. Results obtained by our methods are compared with well-known result in [2] which was obtained by semi-analytical method and they are in perfect agreement.