Finite volume approximation for an immiscible two-phase flow in porous media with discontinuous capillary pressure

Abstract

We consider an immiscible incompressible two-phase flow in a porous medium composed of two different rocks so that the capillary pressure field is discontinuous at the interface between the rocks. This leads us to apply a concept of multi-valued phase pressures and a notion of weak solution for the flow which have been introduced in [Cancès & Pierre, SIAM J. Math. Anal, 44(2):966-992, 2012]. We discretize the problem by means of a numerical algorithm which reduces to a standard finite volume scheme in each rock and prove the convergence of the approximate solution to a weak solution of the two-phase flow problem. The numerical experiments show in particular that this scheme permits to reproduce the oil trapping phenomenon.

Gradient schemes for two‐phase flow in heterogeneous porous media and Richards equation

“…It is important to underline that the proof of convergence of general gradient schemes remains an open problem in the case where the estimates require the multiplication of the equations by nonlinear functions of the term involved in the discrete gradient: this is the case for instance when dealing with discontinuous capillary forces, or even for the standard two-phase flow problem but without assuming a lower bound on the relative permeabilities, or, equivalently, on the range of the saturation function. In these latter cases, the convergence proof [6,7,15,19] is known for two point flux approximations, but it relies on the maximum principle which does not hold for gradient schemes.…”

“…Then the corresponding weak formulation of the problem is obtained by replacing in (5) the first equation by (6) or the second one by (7). Note that this sense is an extension of the condition classically used in hydrogeological studies, which prescribes a constant condition with respect to time and space for the air pressure.…”

Travelling wave solutions for the Richards equation incorporating non-equilibrium effects in the capillarity pressure

“…Such an approach is also being used for defining extended pressure conditions in the case of porous media with block-type heterogeneities, and if models involving an entry pressure are adopted (see e.g. [40,9]). Further, for mathematical purpose k is extended in such a way that G(S) = G(1) > 0 for all S ≥ 1.…”

Design and Analysis of a Finite Volume Scheme for a Concrete Carbonation Model

“…For the sequence (s m ) m , we use Theorem 1 and the Ascoli theorem to show that (s m ) m converges to some function s ∈ W 1,∞ (0, T ). We also prove the convergence of the traces following [5]. Finally passing to the limit in the numerical scheme as in [4] we obtain that (s, u, v) is the weak solution to (2).…”