Mixed-dimensional modeling of flow in porous media

Abstract

Modeling flow in dynamically fracturing porous media is of high interest for a wide range of natural and technical applications, for instance, geothermal energy production or carbon capture and storage. In this work, we present new mixed-dimensional models for flow in porous media including fractures with time- and space-dependent geometries. The models are implemented using our new grid implementation Dune-MMesh which is tailored for the discretization of mixed-dimensional partial differential equations with fully conforming interface of codimension one. First, we propose a mixed-dimensional model for capillarity-free two-phase flow in dynamically fracturing porous media. The model is discretized by a fully conforming finite-volume moving-mesh algorithm that explicitly tracks the fracture geometry. Further, generalizing an earlier model for single-phase flow in fractured porous media, we derive a dimensionally reduced model including spatially varying apertures. In several numerical examples, using a mixed-dimensional discontinuous Galerkin discretization, the model demonstrates significant improvements for curvilinear fracture geometries. Finally, we propose a mixed-dimensional phase-field model for fracture propagation in poro-elastic media combining discrete fracture and phase-field modeling approaches. The corresponding discontinuous Galerkin discretization tracks the fracture geometry by adding facets to the fracture triangulation according to the phase-field indicator and is validated with results known from literature.