Multiscale Hyperbolic-Parabolic Models for Nonlinear Reactive Transport in Heterogeneously Fractured Porous Media

TL;DR

This paper develops a multiscale homogenized model for nonlinear reactive transport in fractured porous media, combining parabolic and hyperbolic equations, and provides rigorous analysis and error estimates.

Contribution

It introduces a novel coupled hyperbolic-parabolic model for fractured media and establishes its well-posedness and approximation accuracy.

Findings

Derivation of a new homogenized model with interface conditions

Proof of well-posedness and regularity of the limit system

Quantitative error estimates for the multiscale approximation

Abstract

We study nonlinear reactive transport in a layered porous medium separated by an $\varepsilon$-thin, highly heterogeneous fracture whose aperture and obstacle pattern vary periodically. Species transport in the bulk is governed by parabolic reaction--diffusion equations, coupled to a convection-diffusion-reaction problem in the fracture with nonlinear wall and obstacle reactions and Peclet number of order $O(\varepsilon^{-1})$. Via multiscale analysis as $\varepsilon \to 0$, when the fracture collapses to a flat interface, we derive a new type of homogenized model consisting of bulk diffusion--reaction equations coupled through nonlinear interface conditions and a first-order semilinear hyperbolic system on the interface. We prove well-posedness and regularity of the limit system, construct a multiscale approximation with boundary-layer correctors, and derive quantitative error estimates in suitable energy norms.