Rigorous derivation of discrete fracture models for Darcy flow in the limit of vanishing aperture

Abstract

<abstract><p>We consider single-phase flow in a fractured porous medium governed by Darcy's law with spatially varying hydraulic conductivity matrices in both bulk and fractures. The width-to-length ratio of a fracture is of the order of a small parameter $ \varepsilon $ and the ratio $ {{K_\mathrm{f}}}^\star / {{K_\mathrm{b}}}^\star $ of the characteristic hydraulic conductivities in the fracture and bulk domains is assumed to scale with $ \varepsilon^\alpha $ for a parameter $ \alpha \in \mathbb{R} $. The fracture geometry is parameterized by aperture functions on a submanifold of codimension one. Given a fracture, we derive the limit models as $ \varepsilon \rightarrow 0 $. Depending on the value of $ \alpha $, we obtain five different limit models as $ \varepsilon \rightarrow 0 $, for which we present rigorous convergence results.</p></abstract>