Numerical analysis for generalized Forchheimer flows of slightly compressible fluids in porous media

TL;DR

This paper analyzes the numerical approximation of generalized Forchheimer flows of slightly compressible fluids in porous media, focusing on stability, error estimates, and convergence of schemes for a nonlinear degenerate parabolic equation.

Contribution

It introduces a comprehensive numerical analysis framework for long-term approximation of Forchheimer flows, including stability, error bounds, and convergence results for both continuous and discrete schemes.

Findings

Stability established for all positive time in continuous and discrete schemes.

Error estimates derived for both time procedures.

Numerical experiments confirm theoretical convergence rates.

Abstract

In this paper, we will consider the generalized Forchheimer flows for slightly compressible fluids. Using Muskat's and Ward's general form of Forchheimer equations, we describe the fluid dynamics by a nonlinear degenerate parabolic equation for density. The long time numerical approximation of the nonlinear degenerate parabolic equation with time dependent boundary conditions is studied. The stability for all positive time is established in both a continuous time scheme and a discrete backward Euler scheme. A Gronwall's inequality-type is used to study the asymptotic behavior of the solution. Error estimates for the solution are derived for both continuous and discrete time procedures. Numerical experiments confirm the theoretical analysis regarding convergence rates.