InterPore2024

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

14 May 2024, 14:45

15m

Oral Presentation (MS03) Flow, transport and mechanics in fractured porous media MS03

Speaker

Maximilian Hörl (University of Stuttgart)

Description

We consider single-phase flow in a fractured porous medium governed by Darcy's law with spatially varying matrix-valued hydraulic conductivities in both bulk and fractures. In particular, we account for general fracture geometries parameterized by aperture functions on a submanifold of codimension one. Given a fracture with a width-to-length ratio of the order of a small parameter $\epsilon$, we derive limit models as $\epsilon \to 0$. In the limit $\epsilon \to 0$, we obtain discrete fracture models where fractures are represented as submanifolds of codimension one. The limit models provide a computationally efficient description with explicit fracture representation, while avoiding thin equi-dimensional subdomains with a need for highly resolved meshes in numerical methods. 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 ${\epsilon }^{\alpha }$ for a parameter $\alpha \in \mathbb{R}$. Depending on the value of $\alpha$, we obtain five different limit models as $\epsilon \to 0$, for which we present rigorous convergence results. Additionally, preliminary results are also available for the case of two-phase flow.