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

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 $\epsilon$ and the ratio $K_{\mathrm{f}}^{\star }\mathrm{/}{K}_{\mathrm{b}}^{\star }$ of the characteristic hydraulic conductivities in the fracture and bulk domains is assumed to scale with ${\epsilon }^{\alpha }$ 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 $\epsilon \to 0$. Depending on the value of $\alpha$, we obtain five different limit models as $\epsilon \to 0$, for which we present rigorous convergence results.