Nitsche’s extended finite element method for a fracture model in porous media

Abstract

We study the numerical approximation of Darcy equations in a fractured domain, with highly discontinuous permeability tensors. In order to avoid the meshing of the fracture, we work with the asymptotic model obtained as the fracture’s thickness goes to and the fracture’s permeability goes to infinity in the weak formulation. This leads to treat a non-standard transmission condition between the two domains delimited by the fracture, instead of solving the equations in the fracture. Then, we propose an approximation of the asymptotic model based on Nitsche’s extended finite element method, which allows to consider fractures not aligned with the mesh. We prove consistency and stability of the discrete formulation and we present some numerical tests showing the relevance of the method.

Keywords:

• Darcy equations
• fracture
• asymptotic model
• finite elements
• NXFEM

AMS Subject Classifications:

• 65N30
• 65N12
• 35S15

Acknowledgements

The authors would like to thank both referees for their useful and relevant remarks.

Notes

No potential conflict of interest was reported by the authors.