Fluid Flows of Mixed Regimes in Porous Media

TL;DR

This paper develops a unified nonlinear parabolic equation to model fluid flows in porous media across pre-Darcy, Darcy, and post-Darcy regimes, providing bounds, stability, and dependence results for the pressure and its gradient.

Contribution

It introduces a single equation of motion that unifies all three flow regimes in porous media, enabling comprehensive analysis of complex flow scenarios.

Findings

Derived a nonlinear degenerate parabolic pressure equation.

Established bounds for pressure and gradient over time.

Proved continuous dependence and structural stability of solutions.

Abstract

In porous media, there are three known regimes of fluid flows, namely, pre-Darcy, Darcy and post-Darcy. Because of their different natures, these are usually treated separately in literature. To study complex flows when all three regimes may be present in different portions of a same domain, we use a single equation of motion to unify them. Several scenarios and models are then considered for slightly compressible fluids. A nonlinear parabolic equation for the pressure is derived, which is degenerate when the pressure gradient is either small or large. We estimate the pressure and its gradient for all time in terms of initial and boundary data. We also obtain their particular bounds for large time which depend on the asymptotic behavior of the boundary data but not on the initial one. Moreover, the continuous dependence of the solutions on initial and boundary data, and the structural stability for the equation are established.