Doubly nonlinear parabolic equations for a general class of Forchheimer gas flows in porous media

TL;DR

This paper develops a mathematical model using doubly nonlinear parabolic equations to describe generalized Forchheimer gas flows in porous media, incorporating gravity and nonlinear source terms, with analysis of solution estimates.

Contribution

It introduces a new doubly nonlinear parabolic equation for pseudo-pressure in Forchheimer flows, including gravity and nonlinear effects, and provides maximum and gradient estimates for solutions.

Findings

Maximum estimates of solutions established using trace theorem and Moser's iteration.

Gradient estimates derived under relevant theoretical conditions.

Model applicable to isentropic, ideal, and slightly compressible gases.

Abstract

This paper is focused on the generalized Forchheimer flows of compressible fluids in porous media. The gravity effect and other general nonlinear forms of the source terms and boundary fluxes are integrated into the model. It covers isentropic gas flows, ideal gases and slightly compressible fluids. We derive a doubly nonlinear parabolic equation for the so-called pseudo-pressure, and study the corresponding initial boundary value problem. The maximum estimates of the solution are established by using suitable trace theorem and adapting appropriately the Moser's iteration. The gradient estimates are obtained under a theoretical condition which, indeed, is relevant to the fluid flows in applications.