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

Maximilian Hörl; Christian Rohde; Maximilian Hörl; Christian Rohde

doi:10.3934/nhm.2024006

Networks and Heterogeneous Media

2024, Volume 19, Issue 1: 114-156. doi: 10.3934/nhm.2024006

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Research article

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

Maximilian Hörl ^,,
Christian Rohde

Institute of Applied Analysis and Numerical Simulation, University of Stuttgart, Germany

Received: 04 August 2023 Revised: 18 December 2023 Accepted: 08 January 2024 Published: 30 January 2024

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.
- Fractured porous media,
- discrete fracture model,
- weak compactness,
- vanishing aperture,
- Darcy flow
Citation: Maximilian Hörl, Christian Rohde. Rigorous derivation of discrete fracture models for Darcy flow in the limit of vanishing aperture[J]. Networks and Heterogeneous Media, 2024, 19(1): 114-156. doi: 10.3934/nhm.2024006

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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.

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