Fractures and thin heterogeneities as Robin-Wentzell interface conditions

Abstract

We formally derive suitable conditions for modeling thin inclusions as interfaces within heterogeneous diffusion problems formulated in the form of the divergence of a flux. Integration of the governing equations across the aperture of the thin inclusion yields interface conditions of Wentzell type for the flux jump and Robin type for the flux average. The condition on the flux jump involves an unconventional tangential diffusion operator applied to the average of the solution across the interface. The corresponding weak formulation is introduced, providing a framework that is readily applicable to finite element discretizations.

Extensive numerical validation against well-established benchmark problems demonstrates that this novel modeling technique can handle strong contrasts in material properties between inclusions and their embedding background, as is typical of flow problems in fractured porous media. In addition, numerical tests not only show that the proposed approach can naturally accommodate complex networks of thin inclusions, without the need for their explicit geometric representation, but also that it can accurately describe fractures with variable aperture.