The hydromechanical equilibrium state of poroelastic media with a static fracture: A dimension-reduced model with existence results in weighted Sobolev spaces and simulations with an XFEM discretization

Abstract

We introduce a coupled system of partial differential equations for the modeling of the fluid–fluid and fluid–solid interactions in a poroelastic material with a single static fracture. The fluid flow in the fracture is modeled by a lower-dimensional Darcy equation, which interacts with the surrounding rock matrix and the fluid it contains. We explicitly allow the fracture to end within the domain, and the fracture width is an unknown of the problem. The resulting weak problem is nonlinear, elliptic and symmetric, and can be given the structure of a fixed-point problem. We show that the coupled fluid–fluid problem has a solution in a specially crafted Sobolev space, even though the fracture width cannot be bounded away from zero near the crack tip. For numerical simulations, we combine XFEM discretizations for the rock matrix deformation and pore pressure with a standard lower-dimensional finite element method for the fracture flow problem. The resulting coupled discrete system consists of linear subdomain problems coupled by nonlinear coupling conditions. We solve the coupled system with a substructuring solver and observe very fast convergence. We also observe optimal mesh dependence of the discretization errors even in the presence of crack tips.