Asymptotic models for free boundary flow in porous media

TL;DR

This paper develops rigorous asymptotic models for free boundary flows in porous media, capturing nonlinear interactions up to quadratic terms in both 2D and 3D cases, including effects of bottom topography.

Contribution

It introduces new asymptotic models for free boundary Darcy and Forchheimer flows that account for weak nonlinear interactions and extend to various dimensions and topographies.

Findings

Models accurately capture quadratic nonlinear interactions.

Applicable to 2D and 3D flows with/without bottom topography.

Provides a rigorous mathematical framework for these asymptotic models.

Abstract

We provide rigorous asymptotic models for the free boundary Darcy and Forchheimer problem under the assumption of weak nonlinear interaction, in a regime in which the steepness parameter of the interface is considered to be very small. The models we derive capture the nonlinear interaction of the original free boundary Darcy and Forchheimer problem up to quadratic terms. Furthermore, we provide models that consider both the two-dimensional and three-dimensional cases, with and without bottom topography.