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

TL;DR

This paper rigorously derives simplified discrete fracture models for Darcy flow in fractured porous media as the fracture aperture shrinks to zero, revealing five different limit models based on hydraulic conductivity scaling.

Contribution

It provides a rigorous mathematical derivation of discrete fracture models from Darcy flow equations considering various scaling regimes of hydraulic conductivity.

Findings

Five distinct limit models depending on conductivity scaling.

Rigorous convergence proofs for each derived model.

Framework applicable to complex fracture geometries.

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.