Fractured Poroelastic Media in the Limit of Vanishing Aperture

TL;DR

This paper rigorously derives limit models for poroelastic media with thin fractures as the fracture width approaches zero, revealing various regimes for flow and deformation behaviors.

Contribution

It introduces a systematic derivation of limit models for fractured poroelastic media, identifying multiple regimes based on parameter scaling.

Findings

Five regimes for hydraulic conductivity.

Two regimes for elasticity.

Different limit models including discrete fractures and two-scale problems.

Abstract

We consider a poroelastic medium with a thin heterogeneity, also referred to as a fracture. Fluid flow and mechanical deformation inside both bulk and fracture are governed by the quasi-static Biot equations. The fracture's material parameters, such as hydraulic conductivity and elasticity, are assumed to scale with powers of the width-to-length ratio $\varepsilon$ of the fracture. Based on a priori estimates, we rigorously derive limit models as $\varepsilon \rightarrow 0$ and identify different limit regimes. We obtain five regimes for the hydraulic conductivity and two for the elasticity. While many cases yield discrete fracture models, others result in two-scale limit problems dominated by normal flow or deformation.