A mixed finite element method for Darcy flow in fractured porous media with non-matching grids^∗

¹ MOX-Dip. di Matematica “F. Brioschi” Politecnico di Milano via Bonardi 9, 20133 Milano, Italy
carlo.dangelo@polimi.it
² MOX-Dip. di Matematica “F. Brioschi” Politecnico di Milano via Bonardi 9, 20133 Milano, Italy
anna.scotti@mail.polimi.it

Received: 27 October 2010
Revised: 6 July 2011

Abstract

We consider an incompressible flow problem in a N-dimensional fractured porous domain (Darcy’s problem). The fracture is represented by a (N − 1)-dimensional interface, exchanging fluid with the surrounding media. In this paper we consider the lowest-order (ℝ T₀, ℙ₀) Raviart-Thomas mixed finite element method for the approximation of the coupled Darcy’s flows in the porous media and within the fracture, with independent meshes for the respective domains. This is achieved thanks to an enrichment with discontinuous basis functions on triangles crossed by the fracture and a weak imposition of interface conditions. First, we study the stability and convergence properties of the resulting numerical scheme in the uncoupled case, when the known solution of the fracture problem provides an immersed boundary condition. We detail the implementation issues and discuss the algebraic properties of the associated linear system. Next, we focus on the coupled problem and propose an iterative porous domain/fracture domain iterative method to solve for fluid flow in both the porous media and the fracture and compare the results with those of a traditional monolithic approach. Numerical results are provided confirming convergence rates and algebraic properties predicted by the theory. In particular, we discuss preconditioning and equilibration techniques to make the condition number of the discrete problem independent of the position of the immersed interface. Finally, two and three dimensional simulations of Darcy’s flow in different configurations (highly and poorly permeable fracture) are analyzed and discussed.