An unfitted finite element discrete fracture model for low-permeability barriers via local stiffness matrix modification

Abstract

Finite element methods are among the most widely used discretizations for flow in porous media, and their discrete fracture models (DFMs) for highly conductive fractures are well-established. Low-permeability barriers, by contrast, have long resisted this framework because they induce pressure discontinuities that continuous elements cannot represent directly. In this paper, we propose a simple extension of the linear finite element method for modeling low-permeability barriers through a closed-form modification of the local stiffness matrix of each barrier-cut element. The method retains exactly the same $H^1$-conforming $P^1$-finite element space, preserves the original sparsity pattern and symmetric positive definiteness, requires no mesh fitting, and coincides with the standard finite element method away from barriers. Moreover, an inexpensive, purely local post-processing step recovers the discontinuous pressure field with a sharp jump at the barrier interface, thereby removing the one-element-wide smearing present in the continuous solution. Convergence studies with manufactured solutions and two- and three-dimensional benchmark problems from the literature confirm the effectiveness of the method.