A mixed finite element approximation for Darcy-Forchheimer flows of slightly compressible fluids

TL;DR

This paper introduces a mixed finite element method for simulating Darcy-Forchheimer flows of slightly compressible fluids in porous media, providing stability, error estimates, and insights into parameter dependence.

Contribution

It develops a novel mixed finite element approach for nonlinear degenerate systems modeling Forchheimer flows, with proven stability and error bounds.

Findings

The method is stable under certain conditions.

Error estimates are derived for both continuous and discrete time.

Numerical experiments confirm convergence and parameter sensitivity.

Abstract

In this paper, we consider the generalized Forchheimer flows for slightly compressible fluids in porous media. Using Muskat's and Ward's general form of Forchheimer equations, we describe the flow of a single-phase fluid in $\mathbb R^d, d\ge 2$ by a nonlinear degenerate system of density and momentum. A mixed finite element method is proposed for the approximation of the solution of the above system. The stability of the approximations are proved; the error estimates are derived for the numerical approximations for both continuous and discrete time procedures. The continuous dependence of numerical solutions on physical parameters are demonstrated. Experimental studies are presented regarding convergence rates and showing the dependence of the solution on the physical parameters.