A Nonlinear Mixed Finite Element Method for a Degenerate Parabolic Equation Arising in Flow in Porous Media

Abstract

We study a model nonlinear, degenerate, advection-diffusion equation having application in petroleum reservoir and groundwater aquifer simulation. The main difficulty is that the true solution is typically lacking in regularity; therefore, we consider the problem from the point of view of optimal approximation. Through time integration, we develop a mixed variational form that respects the known minimal regularity, and then we develop and analyze two versions of a mixed finite element approximation, a simpler semidiscrete (time-continuous) version and a fully discrete version. Our error bounds are optimal in the sense that all but one of the bounding terms reduce to standard approximation error. The exceptional term is a nonstandard approximation error term. We also consider our new formulation for the nondegenerate problem, showing the usual optimal L2-error bounds; moreover, superconvergence is obtained under special circumstances.

Isothermal water flows in low porosity porous media in presence of vapor–liquid phase change

“…Concerning the situation with non-negligible capillary pressure, there are many references when the diffusive F vanishes after a threshold valued. We mention the reference [6,17,18,22,33,36,37,38,39]. In our approach we allow both degenerate parabolic and hyperbolic situations.…”

Analysis of an Euler implicit-mixed finite element scheme for reactive solute transport in porous media

“…An alternative approach is to combine the two nonlinearities in (1.3) into a single one and use the Kirchhoff transformation [2,5,31,37]. For this approach, assuming that the saturation is a Lipschitz continuous function of the pressure, as well as ∂ t Θ ∈ L 2 (J; L 2 (Ω)), an τ 2 + h 2 convergence order is obtained in [5,31,37]. The regularity assumption on ∂ t Θ is given u p in [31], resulting in a reduction of the convergence order to τ + h 2 .…”

“…Before dealing with the flux equation (4.2) we mention that the analysis in [5,37] carried out for the Richards equation leaves the flux equation unchanged, and considers the time integrated variant of the balance equation (4.1). Here we proceed as in [33] and integrate also (4.2) in time to obtain…”

“…in [11]. Due to their local mass conservation property, mixed finite element methods are a valuable discretization technique for problems involving flow in porous media [3,4,5,11,12,16,17,31,33,37,41,42].…”

“…In this paper we analyze the EI-MFE scheme for the transport equation (1.1), by employing techniques that are similar to those used in [5,31,37]. The main result shows the convergence of the fully discrete numerical scheme for (1.1).…”