Poroelastic Medium with Non-Penetrating Crack Driven by Hydraulic Fracture: FEM Approximation Using HHT-α and Semi-Smooth Newton Methods

A new class of poroelastic dynamic contact problems stemming from hydraulic fracture theory is introduced and studied. The two-phase medium consists of a solid phase and pores which are saturated with a Newtonian fluid. The porous body contains a fluid-driven crack endowed with non-penetration conditions for the opposite crack surfaces. The poroelastic model is described by a coupled system of hyperbolic–parabolic partial differential equations under the unilateral constraint imposed on displacement. After full discretization using finite-element and Hilber–Hughes–Taylor methods, the well-posedness of the resulting variational inequality is established. Formulation of the complementarity conditions with the help of a minimum-based merit function is used for the semi-smooth Newton method of solution presented in the form of a primal–dual active set algorithm which is tested numerically.