An algorithm for cohesive crack propagation in variably saturated porous media via a time sub-stepping scheme

This study presents a novel approach for simulating hydraulic fracture propagation in variably saturated porous media using the time sub-stepping method. Based on the authors’ previous THM model, a variably saturated model is developed based on mixture theory using two independent variables of displacement and water pressure to describe the behavior of intact porous media. The proposed model adopts the crack as an interface element coupled to the general equilibrium and mass conservation equations. XFEM and β$$\beta$$ method are used to discretize the governing equations spatially and temporally, respectively. The time sub-stepping approach is used to unconstrain the crack tip advancement velocity to obtain a consistent solution. Three examples are solved to validate the model and show its robustness. The first two numerical examples are selected to demonstrate the capability of the proposed model to simulate consolidation and gravitational drainage in the variably saturated porous media. The third example studies crack propagation in an unsaturated porous plate. The time sub-stepping method finds when the crack propagates to an element ahead of the crack tip and smoothes the stepwise behavior for the quasi-static propagation of a single hydrofracture with longitudinal laminar flow.