# An unfitted finite element method for the Darcy problem in a fracture network

## TL;DR

This paper introduces an unfitted finite element method for simulating Darcy flow in fracture networks, allowing arbitrary fracture cuts through the mesh and simplifying junction coupling while maintaining optimal convergence.

## Contribution

It extends the Hughes--Masud stabilized formulation with penalty terms for interface conditions, enabling accurate and efficient modeling of complex fracture networks without mesh fitting.

## Key findings

- Achieves optimal convergence order in numerical experiments.
- Handles both planar and curvilinear fractures seamlessly.
- Simplifies junction coupling by over-penalizing pressure continuity.

## Abstract

The paper develops an unfitted finite element method for solving the Darcy system of equations posed in a network of fractures embedded in a porous matrix. The approach builds on the Hughes--Masud stabilized formulation of the Darcy problem and the trace finite element method. The system of fractures is allowed to cut through the background mesh in an arbitrary way. Moreover, the fractures are not triangulated in the common sense and the junctions of fractures are not fitted by the mesh. To couple the flow variables at multiple fracture junctions, we extend the Hughes--Masud formulation by including penalty terms to handle interface conditions. One observation made here is that by over-penalizing the pressure continuity interface condition one can avoid including additional jump terms along the fracture junctions. This simplifies the formulation while ensuring the optimal convergence order of the method. The application of the trace finite element allows to treat both planar and curvilinear fractures with the same ease. The paper presents convergence analysis and assesses the performance of the method in a series of numerical experiments. For the background mesh we use an octree grid with cubic cells. The flow in the fracture can be easily coupled with the flow in matrix, but we do not pursue the topic of discretizing such coupled system here.