Galerkin finite element method for generalized Forchheimer equation of slightly compressible fluid in porous media

TL;DR

This paper develops and analyzes a Galerkin finite element method for solving a nonlinear degenerate parabolic equation modeling slightly compressible fluid flow in porous media, proving existence, uniqueness, and convergence.

Contribution

It introduces a finite element approach for the generalized Forchheimer equations, providing rigorous analysis and error estimates for the numerical solutions.

Findings

Proved existence and uniqueness of the finite element approximation.

Established a priori estimates in various function spaces.

Numerical experiments confirm theoretical convergence rates.

Abstract

We 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 the density. We study Galerkin finite elements method for the initial boundary value problem. The existence and uniqueness of the approximation are proved. The prior estimates for the solutions in $L^\infty(0,T;L^q(\Omega)), q\ge 2$, time derivative in $L^\infty(0,T;L^2(\Omega))$ and gradient in $L^\infty(0,T;W^{1,2-a}(\Omega)),$ with $a\in (0,1)$ are established. Error estimates for the density variable are derived in several norms for both continuous and discrete time procedures. Numerical experiments using backward Euler scheme confirm the theoretical analysis regarding convergence rates.