The XFEM for a simplified model in hydraulic fracturing

Hydraulic fracturing is an important branch in geomechanics, where a fracture initiates and propagates in the host rock due to the induced hydraulic loading. The pressure exerted by the fracking fluid onto the surrounding solid, typically is obtained by a Reynolds equation which relates the crack width with the pressure. This poses severe difficulties for the numerical treatment, also in terms of robustness. In the presented work, the Reynolds equation is replaced by pre-defined pressure distributions which leads to a much simpler and robust coupled problem. For each assumed distribution, the critical pressure is determined where the critical energy release rate Gc is exceeded. The influence of the different distributions is investigated in a separate example. The energy release rates are determined based on stress intensity factors with the XFEM by using crack opening displacements. This happens by a comparison of an approximated state which represents the computed displacements in the solid, and a reference state which represents the expected openings for a pure mode I, II and III. This method is intuitive, computationally cheap and has the advantage that only displacements are fitted, wherefore no additional consideration of pressurized crack surfaces is required. The critical pressure is determined based on the superposition principle in linear elastic fracture mechanics. Herein, a separate observation of the external loadings by means of volume forces and tractions at the boundary, and the internal loadings by means of the pressure exerted by the fracking fluid is done. Based on a scaling factor of the internal state, the critical pressure is extracted from the energy release rate.