Effective interface laws for Darcy--Boussinesq convection across a vanishing low-permeability layer
Abstract
We study Darcy--Boussinesq convection in a layered porous medium containing a thin layer whose thickness h0 and permeability K0 vanish simultaneously. The competition between these two small parameters is measured by the effective hydraulic conductance α= lim K0/h0 ∈ [0,∞]. We show that the collapsing layer is replaced by an effective interface whose transmission law depends qualitatively on this ratio. When α= 0, the limiting interface is impermeable; when $ 0 <α<\infty$, the normal Darcy flux is proportional to the pressure jump, yielding a finite-resistance transmission law; and when α=∞, the pressure is continuous and the interface becomes hydraulically transparent. We establish estimates uniform in the layer thickness and permeability, quantitative convergence of solutions on every finite time interval, and upper semicontinuity of the associated global attractors. The analysis has to accommodate both a degenerating geometry and a loss of coercivity in the pressure equation. In particular, we use a generalized Poincaré inequality for piecewise H1 pressures, a pressure extension across the collapsing layer, and uniform thin-strip estimates. The results provide a rigorous finite- and long-time validation of effective interface models for convection through thin low-permeability interphases.
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BibTeX
@misc{sha2026effective,
title = {{Effective interface laws for Darcy--Boussinesq convection across a vanishing low-permeability layer}},
author = {Kaijian Sha and Xiaoming Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.39421},
eprint = {2610.39421},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.39421},
}
RIS
TY - UNPB
TI - Effective interface laws for Darcy--Boussinesq convection across a vanishing low-permeability layer
AU - Kaijian Sha
AU - Xiaoming Wang
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.39421
UR - https://arxiv.org/abs/2610.39421
AB - We study Darcy--Boussinesq convection in a layered porous medium containing a thin layer whose thickness h_0 and permeability K_0 vanish simultaneously. The competition between these two small parameters is measured by the effective hydraulic conductance α= lim K_0/h_0 ∈ [0,∞]. We show that the collapsing layer is replaced by an effective interface whose transmission law depends qualitatively on this ratio. When α= 0, the limiting interface is impermeable; when $ 0 <α<\infty$, the normal Darcy flux is proportional to the pressure jump, yielding a finite-resistance transmission law; and when α=∞, the pressure is continuous and the interface becomes hydraulically transparent. We establish estimates uniform in the layer thickness and permeability, quantitative convergence of solutions on every finite time interval, and upper semicontinuity of the associated global attractors. The analysis has to accommodate both a degenerating geometry and a loss of coercivity in the pressure equation. In particular, we use a generalized Poincaré inequality for piecewise H^1 pressures, a pressure extension across the collapsing layer, and uniform thin-strip estimates. The results provide a rigorous finite- and long-time validation of effective interface models for convection through thin low-permeability interphases.
ER -