Vanishing permeability limit of convection in multilayer porous media

TL;DR

This paper rigorously analyzes the zero-permeability limit in layered porous media convection, showing convergence to a model with an impermeable layer and addressing the associated mathematical degeneracies.

Contribution

It provides a rigorous mathematical framework for the zero-permeability limit in layered porous media convection models, including convergence and attractor analysis.

Findings

Solutions converge in L2 norm over finite time intervals.

The global attractors of the system converge as permeability tends to zero.

The pressure degeneracy issue is resolved through refined estimates.

Abstract

We analyze the asymptotic behavior of the Boussinesq-Darcy system describing convection in layered porous media in the limit where the permeability of one layer tends to zero. We show that the limiting dynamics are governed by the Boussinesq-Darcy model with an impermeable layer, both in terms of convergence of solutions in L2 on finite time intervals and convergence of the corresponding global attractors. This limit is singular, as the pressure equation becomes degenerate when the permeability vanishes in part of the domain, resulting in a loss of uniqueness of the pressure in the impermeable layer. This difficulty is resolved by combining uniform estimates in the permeable layers with refined control of the pressure equation in the vanishing-permeability layer. The results provide a rigorous description of the zero-permeability limit in layered porous-media convection models.