Rigorous Derivation of Discrete Fracture Models for Darcy Flow in the Limit of Vanishing Aperture

Abstract

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.