Remarks on dealing with Toughness Heterogeneity in Modelling of Hydraulic Fracture

This paper aims to discuss recent studies on methods used to capture heterogeneity within a reservoir undergoing hydraulic fracturing treatment and their implication on modelling of fracture propagation. In highly laminated reservoirs with soft and/or weak layers capturing heterogeneity at an appropriate resolution is the key for successful prediction fracture growth, landing depth and in maximizing effective stimulated rock volume (eSRV). Typically, when using computational methods such as Finite Element Method, the well log and petrophysical data deduced from various measurements and observations are upscaled and/or homogenized to the element size. Naturally, toughness is one of the most delicate physical parameters to handle as application of the homogenization techniques are rather questionable here (Caiulo and Kachanov, 2010, Kachanov, 1994) and proposed averaging (if possible) depends on the problem under consideration and the toughness distribution. Fortunately, since hydraulic fracturing is probably the only stable crack propagation problem, some estimates can be provided here. In particular, in this paper we try to estimate possible errors introduced by application of two strategies how to incorporate heterogeneous fracture toughness into numerical modelling. The first is more classical one (Richards et al. 2020) and the second one is recent analysis (Dontsov et al., 2021). We present series of numerical tests utilising a few periodic toughness distributions. For the simulations, we use an extremely effective in house built time - space adaptive solver utilizing main ideas from those reported in (Mishuris and Wrobel, 2015). It is capable to compute rather arbitrary distribution of the toughness. We restrict ourselves to the KGD model without leak off only as it allows us easy handling different regimes (toughness/viscosity) that would be crucial for the analysis as follows from this research.