Generalized Forchheimer flows in heterogeneous porous media

TL;DR

This paper analyzes complex flow equations in heterogeneous porous media, establishing estimates and stability results for pressure and its gradient in generalized Forchheimer flows of compressible fluids.

Contribution

It introduces weighted norm estimates and continuous dependence results for pressure and gradient in generalized Forchheimer flows with spatially varying coefficients.

Findings

Weighted Lebesgue norm estimates for pressure and gradient

Continuous dependence on initial and boundary data

Asymptotic behavior for unbounded boundary data

Abstract

We study the generalized Forchheimer flows of slightly compressible fluids in heterogeneous porous media. The media's porosity and coefficients of the Forchheimer equation are functions of the spatial variables. The partial differential equation for the pressure is degenerate in its gradient and can be both singular and degenerate in the spatial variables. Suitable weighted Lebesgue norms for the pressure, its gradient and time derivative are estimated. The continuous dependence on the initial and boundary data is established for the pressure and its gradient with respect to those corresponding norms. Asymptotic estimates are derived even for unbounded boundary data as time tends to infinity.