Fractured Poroelastic Media in the Limit of Vanishing Aperture

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 $\epsilon$ of the fracture. Based on a priori estimates, we rigorously derive limit models as $\epsilon \to 0$ 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.