Gradient discretization of Hybrid Dimensional Darcy Flows in Fractured Porous Media with discontinuous pressures at the matrix fracture interfaces

TL;DR

This paper develops and analyzes gradient discretization schemes for simulating Darcy flow in fractured porous media, effectively handling pressure discontinuities at matrix-fracture interfaces on general meshes.

Contribution

It extends the gradient discretization framework to hybrid dimensional models with complex fracture networks, introducing VAG and HFV schemes with proven convergence.

Findings

VAG and HFV schemes satisfy the gradient scheme framework

Numerical tests confirm convergence and accuracy

Model effectively captures pressure discontinuities at interfaces

Abstract

We investigate the discretization of Darcy flow through fractured porous media on general meshes. We consider a hybrid dimensional model, invoking a complex network of planar fractures. The model accounts for matrix-fracture interactions and fractures acting either as drains or as barriers, i.e. we have to deal with pressure discontinuities at matrix-fracture interfaces. The numerical analysis is performed in the general framework of gradient discretizations which is extended to the model under consideration. Two families of schemes namely the Vertex Approximate Gradient scheme (VAG) and the Hybrid Finite Volume scheme (HFV) are detailed and shown to satisfy the gradient scheme framework, which yields, in particular, convergence. Numerical tests confirm the theoretical results. Gradient Discretization; Darcy Flow, Discrete Fracture Networks, Finite Volume