Unified Convergence Analysis of Numerical Schemes for a Miscible Displacement Problem

Abstract

This article performs a unified convergence analysis of a variety of numerical methods for a model of the miscible displacement of one incompressible fluid by another through a porous medium. The unified analysis is enabled through the framework of the gradient discretisation method for diffusion operators on generic grids. We use it to establish a novel convergence result in L ∞ (0, T ; L 2 (Ω)) of the approximate concentration using minimal regularity assumptions on the solution to the continuous problem. The convection term in the concentration equation is discretised using a centred scheme. We present a variety of numerical tests from the literature, as well as a novel analytical test case. The performance of two schemes are compared on these tests; both are poor in the case of variable viscosity, small diffusion and medium to small time steps. We show that upstreaming is not a good option to recover stable and accurate solutions, and we propose a correction to recover stable and accurate schemes for all time steps and all ranges of diffusion.

A virtual element method for the miscible displacement of incompressible fluids in porous media

“…Moreover, one has div u n I = Π 0 k G n . Thus, setting δ n := U n −u n I , it holds that δ n ∈ K h ⊂ K, where K h and K were defined in (38) and (13), respectively, and therefore, δ n V h = δ n 0,Ω . Owing to the assumptions on a(•) in ( 6) together with (37), we have, further using (49) with v h = δ n ∈ K h and (10),…”

“…We consider the miscible displacement of one incompressible fluid by another in a porous medium. This problem can be formulated in terms of a system of partial differential equations, where a parabolic diffusion-convection-reaction type equation is nonlinearly coupled with an elliptic system, see also [33,37,38,52].…”

“…In the present contribution, we consider the miscible displacement of one incompressible fluid by another in a reservoir, described by a time-dependent coupled system of nonlinear partial differential equations, that is a basic (but meaningful) model instrumental to applications such as oil recovery and environmental pollution [33,37,38,52,55]. One must note that, on this and similar models, there already exists a large literature with many competitive schemes, adopting for instance finite elements [37,40,41], discontinuous Galerkin methods [4,49,54], and finite volumes [2,33,34].…”

Analysis of lowest-order characteristics-mixed FEMs for incompressible miscible flow in porous media

“…In the last several decades, numerical methods and analyses for the miscible displacement system (1.1)-(1.3) have been studied extensively, e.g., see [19,22,37,42] and references therein. Two review articles were written by Ewing and Wang [24] and Scovazzi et.…”

“…Some other type methods of characteristics can be found in [2,31,40]. In addition, the characteristics type methods have been applied and analyzed for many other linear and nonlinear parabolic PDEs from various engineering applications [2,4,19,27,28,38,46]. Numerical simulations show that the time-truncation errors of the MMOC are much smaller than those of standard schemes for convection-dominated models.…”