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

Abstract

The gradient scheme family, which includes the conforming and mixed finite elements as well as the mimetic mixed hybrid family, is used for the approximation of Richards equation and the two-phase flow problem in heterogeneous porous media. We prove the convergence of the approximate saturation and of the approximate pressures and approximate pressure gradients thanks to monotony and compactness arguments under an assumption of non-degeneracy of the phase relative permeabilities. Strong convergence results stem from the convergence of the norms of the gradients of pressures, which demand handling the nonlinear time term. Numerical results show the efficiency on these problems of a particular gradient scheme, called the Vertex Approximate Gradient scheme.

Gradient Schemes for Stokes Problem

“…Now setting v = ID u − uD in (5) and using Ω divD uD χD q = 0 for all q ∈ YD , we can write ID u − uD Thanks to (6) and to the inequality ab ≤ b 2 , the above inequality yields the existence of C1, increasing function of 1/βD , CD and η, such that ID u − uD D ≤ C1εD (u, p). The conclusion follows, thanks to the definitions of ID u and ID p, to Definition 2.2, to the triangle inequality and to the use of (6).…”

“…This framework is given in Section 2, as an extension of the notion of gradient schemes provided for scalar elliptic problems [3,4,5,7,6]. Among the schemes which are included in this framework, we briefly present in Section 3 the Taylor-Hood scheme, an extended version [1] of the Marker-And-Cell (MAC) scheme [8,9,11] and the Crouzeix-Raviart scheme [2] (these three schemes are useful in many industrial applications).…”

Uniform-in-Time Convergence of Numerical Schemes for Richards’ and Stefan’s Models

“…-and of having already been studied for many models -elliptic, parabolic, linear or non-linear, possibly degenerate, etc. We refer the reader to [3, 4,5,10,8] for more details.…”

Vertex Approximate Gradient Scheme for Hybrid Dimensional Two-Phase Darcy Flows in Fractured Porous Media

“…In this paper, we propose to extend the results obtained in [17] in the particular case of the VAG scheme to the hybrid dimensional Darcy flow model. The convergence of the VAG scheme to a weak solution is obtained assuming that the relative permeabilities of both phases are bounded from below by a strictly positive constant, and assuming that the family of meshes is shape regular.…”