Properties of Generalized Forchheimer Flows in Porous Media

TL;DR

This paper analyzes the properties of generalized Forchheimer flows in porous media, providing estimates for pressure and its derivatives, and demonstrating continuous dependence on data and coefficients for both finite and infinite time horizons.

Contribution

It introduces new analytical techniques to estimate pressure norms and shows continuous dependence of solutions on initial data, boundary conditions, and coefficients in generalized Forchheimer flows.

Findings

Established $L^ abla$-norm estimates for pressure and derivatives.

Proved continuous dependence of solutions on data and coefficients.

Applied combined iterative techniques for degenerate parabolic equations.

Abstract

The nonlinear Forchheimer equations are used to describe the dynamics of fluid flows in porous media when Darcy's law is not applicable. In this article, we consider the generalized Forchheimer flows for slightly compressible fluids and study the initial boundary value problem for the resulting degenerate parabolic equation for pressure with the time-dependent flux boundary condition. We estimate $L^\infty$-norm for pressure and its time derivative, as well as other Lebesgue norms for its gradient and second spatial derivatives. The asymptotic estimates as time tends to infinity are emphasized. We then show that the solution (in interior $L^\infty$-norms) and its gradient (in interior $L^{2-\delta}$-norms) depend continuously on the initial and boundary data, and coefficients of the Forchheimer polynomials. These are proved for both finite time intervals and time infinity. The De Giorgi and Ladyzhenskaya-Uraltseva iteration techniques are combined with uniform Gronwall-type estimates, specific monotonicity properties, suitable parabolic Sobolev embeddings and a new fast geometric convergence result.