Mathematical study of a petroleum-engineering scheme

Abstract

Abstract. Models of two phase flows in porous media, used in petroleum engineering, lead to a system of two coupled equations with elliptic and parabolic degenerate terms, and two unknowns, the saturation and the pressure. For the purpose of their approximation, a coupled scheme, consisting in a finite volume method together with a phase-by-phase upstream weighting scheme, is used in the industrial setting. This paper presents a mathematical analysis of this coupled scheme, first showing that it satisfies some a priori estimates: the saturation is shown to remain in a fixed interval, and a discrete L 2 (0, T ; H 1 (Ω)) estimate is proved for both the pressure and a function of the saturation. Thanks to these properties, a subsequence of the sequence of approximate solutions is shown to converge to a weak solution of the continuous equations as the size of the discretization tends to zero.Mathematics Subject Classification. 35K65, 76S05, 65M12.

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

“…Gathering the above results shows (19). The existence of at least one solution is then easily proved by considering the function…”

“…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.…”

“…Since (19) holds for all α ∈ [0, 1], and since the problem is linear for α = 0, a classical topological degree argument provides the existence of at least one solution to Scheme (14).…”

A posteriori error estimates, stopping criteria, and adaptivity for two-phase flows

“…The analysis of (1.1a)-(1.4b) including the existence, uniqueness, and well-posedness results has been carried out in [54,21,6,7,23,24,25,18,51,4], see also [3,60,34] for degenerate problems, and the references therein. For the use and analysis of mixed finite element methods for the numerical approximation of (1.1a)-(1.4b) we refer to, e.g., [33,9,73] and the references therein, for discontinuous Galerkin methods, to, e.g., [38,39,40,8] and the references therein, for cell-centered finite volume methods to, e.g., [45] and the references therein, and for vertexcentered finite volume methods to, e.g., [48] and the references therein. Multiscale and mortar techniques, efficient parallelization, and multinumerics and multiphysics formulations have been investigated in [63].…”

Study of a numerical scheme for miscible two‐phase flow in porous media

“…Although this does not provide particular difficulties in the continuous case (thanks to Stampacchia's results), it prevents from using general schemes for the discretization of the space terms (like it is done in [9], concerning a wide class of discretization methods, namely the gradient schemes). On the contrary, we are led to use two-point flux approximation for the space terms, in the same spirit of [10] (and further works like [1] in the case of one compressible phase). Note that such a difficulty also arises in [7] or [2], or more generally in some elliptic problems with irregular data [6] where nonlinear test functions of the unknown must be used.…”