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We introduce a coupled system of PDEs for the modeling of the fluid-fluid andfluid-solid interaction in a poroelastic material with a single staticfracture. The fluid flow in the fracture is modeled by a lower-dimensionalDarcy equation, which interacts with the surrounding rock matrix and the fluidit contains. We explicitly allow the fracture to end within the domain, and thefracture width is an unknown of the problem. The resulting weak problem isnonlinear, elliptic and symmetric, and can be given the structure of afixed-point problem. We show that the coupled fluid-fluid problem has asolution in a specially crafted Sobolev space, even though the fracture widthcannot be bounded away from zero near the crack tip.  For numerical simulations, we combine XFEM discretizations for the rockmatrix deformation and pore pressure with a standard lower-dimensional finiteelement method for the fracture flow problem. The resulting coupled discretesystem consists of linear subdomain problems coupled by nonlinear couplingconditions. We solve the coupled system with a substructuring solver andobserve very fast convergence. We also observe optimal mesh dependence of thediscretization errors even in the presence of crack tips.

Simulation of Deformation and Flow in Fractured, Poroelastic Materials