Generalized cell-centered finite volume methods: application to two-phase flow in porous media

Finite volume methods and finite elements methods are generally opposed as competing approximation techniques. However there are ways to look at them which make these methods closer than what is usually thought, and this leads to the construction of generalized finite volume methods. Here we will restrict ourselves to cell-centered finite volumes. On one hand, cell-centered finite volume methods are widely used by engineers and scientists who have to perform numerical simulations. When building such methods, one discretizes the domain, usually with rectangles, but not always, and equations are written inside each cell of the discretization. Then relations between cells are written. To do so, the solution of the problem is approximated by piecewise constants. Most often, before being eliminated, flux unknowns are also introduced to write the intercell relations which describe conservation. This procedure is very close to the physics and, in this fashion, even complicated physical laws can be implemented easily. On the other hand, finite element methods have their advantages: they are based on a rigorous mathematical analysis, higher order methods can be obtained by increasing the degrees of the polynomials, and structured as well as unstructured meshes can be used depending on the application that is under study. In this paper we show how to combine the advantages of these two approximation techniques into one numerical procedure. We retain the point of view of finite volume methods, but we call them generalized since, to approximate the unknowns, we use general discontinuous piecewise polynomial approximations instead of just piecewise constants. This will be achieved using mixed-hybrid finite element and Godunov’s methods. This paper can be viewed as a sequence to a paper by Chavent-Roberts [CR91] where the method was described for linear elliptic and parabolic equations. To illustrate our point we consider as an example a model for incompressible two-phase flow in a porous medium which results in a nonlinear system of two equations. One equation, the saturation equation, is parabolic and represents conservation of one of the phases which implies continuity of the normal components of the Darcy velocity of this phase. The other equation, the pressure equation, is elliptic and represents conservation of both phases which implies continuity of the normal components of the total Darcy velocity. Our numerical procedure will follow closely these physical requirements. Even in the homogeneous case the