Irregular and stepwise behaviour of hydraulic fracturing: insights from linear cohesive crack modelling with maximum stress criterion

There is growing evidence that fracture advancement in saturated and dry porous media may be smooth or stepwise. In the stepwise behaviour, there are also pressure oscillations in saturated porous materials, as predicted by Biot's theory. The type of behavior depends on the specifications of the problem and material properties. Not all the adopted numerical models are capable of capturing stepwise behaviour. While non-local fracturing models are perfectly adapted to capture the stepwise behaviour, it is shown that cohesive models are also capable to model such a behaviour. In fact, it is necessary to satisfy a consistency condition for the numerical solution; in other words, the fracture advancement/time-stepping algorithm must not impose a constraint on the tip advancement speed. This is peculiar to cohesive models because there the unit of fracture advancement is specified beforehand, while in other models, such as peridynamics, it is an outcome of the analysis. In inhomogeneous media, it can be expected to capture irregular results from hydraulic fracturing (HF); however, homogeneous media can also show such behaviour as presented herein. In this study, the eXtended Finite Element Method (XFEM) is used in conjunction with a cohesive crack model to investigate the stepwise or irregular behaviour of HF in homogeneous saturated porous media both under quasistatic and dynamic conditions. The results clearly show that the extent of irregularity depends directly on the intensity of dynamic effects in the problem. Moreover, fracture forerunning is one of the reasons of the stepwise fracture growth in the solutions. Although the results are obtained by XFEM, the conclusions are not restricted to this particular numerical method, and the findings can provide useful insights into the necessary aspects of the numerical algorithms for dynamic HF modelling.