Numerical Approach of a Coupled Pressure-Saturation Model Describing Oil-Water Flow in Porous Media

Abstract

Two-phase flow in porous media is a very active field of research, due to its important applications in groundwater pollution, ${\text {CO}}_2$ sequestration, or oil and gas production from petroleum reservoirs, just to name a few of them. Fractional flow equations, which make use of Darcy’s law, for describing the movement of two immiscible fluids in a porous medium, are among the most relevant mathematical models in reservoir simulation. This work aims to solve a fractional flow model formed by an elliptic equation, representing the spatial distribution of the pressure, and a hyperbolic equation describing the space-time evolution of water saturation. The numerical solution of the elliptic part is obtained using a finite-element (FE) scheme, while the hyperbolic equation is solved by means of two different numerical approaches, both in the finite-volume (FV) framework. One is based on a monotonic upstream-centered scheme for conservation laws (MUSCL)-Hancock scheme, whereas the other makes use of a weighted essentially non-oscillatory (ENO) reconstruction. In both cases, a first-order centered (FORCE)-$\alpha$ numerical scheme is applied for intercell flux reconstruction, which constitutes a new contribution in the field of fractional flow models describing oil-water movement. A relevant feature of this work is the study of the effect of the parameter $\alpha$ on the numerical solution of the models considered. We also show that, in the FORCE-$\alpha$ method, when the parameter $\alpha$ increases, the errors diminish and the order of accuracy is more properly attained, as verified using a manufactured solution technique.

Funding

Repsol; Universidad Politécnica de Madrid