The expanded mixed finite element method for generalized Forchheimer flows in porous media

TL;DR

This paper develops and analyzes an expanded mixed finite element method for nonlinear Forchheimer flows in porous media, providing error estimates and numerical validation for pressure and flow divergence.

Contribution

It introduces an expanded mixed finite element approach for Forchheimer equations, establishing optimal error bounds and demonstrating numerical accuracy with Raviart-Thomas elements.

Findings

Optimal error estimates in $L^2$-norm for solutions.

Numerical results confirm theoretical error bounds.

Method effectively handles nonlinear Forchheimer flows.

Abstract

We study the expanded mixed finite element method applied to degenerate parabolic equations with the Dirichlet boundary condition. The equation is considered a prototype of the nonlinear Forchheimer equation, a inverted to the nonlinear Darcy equation with permeability coefficient depending on pressure gradient, for slightly compressible fluid flow in porous media. The bounds for the solutions are established. In both continuous and discrete time procedures, utilizing the monotonicity properties of Forchheimer equation and boundedness of solutions we prove the optimal error estimates in $L^2$-norm for solution. The error bounds are established for the solution and divergence of the vector variable in Lebesgue norms and Sobolev norms under some additional regularity assumptions. A numerical example using the lowest order Raviart-Thomas ($RT_0$) mixed element are provided agreement with our theoretical analysis.