Mathematical and numerical analysis of the mixed formulation of single phase flow in three-dimensional fractured porous media

Three-dimensional fractured-porous media are typically modeled using a reduced approach known as the Discrete Fracture Matrix (DFM) model. In this model, the rock matrix is represented in three dimensions, while the fracture network is represented as a two-dimensional (co-dimension one) structure. The system is governed by a set of partial differential equations (PDEs) that couple the flow in the 3D rock matrix with the flow in the 2D fracture network. Mixed finite element methods are particularly well-suited for discretizing this system because they ensure local mass conservation. This paper presents an existence and uniqueness result for the PDEs in mixed form, assuming that the hydraulic head is continuous across fractures (i.e., the fractures are conductive). Additionally, error estimates for the mixed finite element discretization are derived under a regularity assumption, and a mixed-hybrid approximation for the coupled problem is developed. The theoretical findings are validated through comparison with an analytical solution.