Mixed approach for coupled flow and mechanics in a fractured medium, 2023

This thesis presents a mixed finite element method of the Biot system of poroelasticity within a reservoir, coupled with a mixed formulation of flow within a fracture. The fracture is represented as a planar object of one dimension less than the domain of the reservoir. The spatial discretization is combined by a multipoint stress mixed finite element (MSMFE) method for elasticity and the multipoint flux mixed finite element (MFMFE) method for the Darcy flow within the reservoir matrix, coupled with a MFMFE method for the flow within the fracture. In the reservoir matrix, the lowest order Brezzi-Douglas-Marini mixed finite element spaces for the poroelastic stress and Darcy velocity, along with piecewise constant displacement, pressure and rotation are considered. Within the fracture, we consider compatible pairs of element spaces for the fracture pressure and flux, along with piecewise constant mortar displacement. A stability analysis is performed on both the continuous and semi-discrete problem. Furthermore, existence and uniqueness are shown for the semi-discrete and fully-discrete solutions. Different numerical simulations are presented in the end.

Link to the thesis.