On averaging of toughness heterogeneity when modelling hydraulic fracture evolution

In this paper we describe various approaches used to capture heterogeneity within the 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 and other crucial treatment parameters. Our focus is on studying several strategies to average fracture toughness and assess their suitability for use in advanced computational methods such as FE/BEM. In practice the well log and petrophysical data deduced from various measurements and observations are upscaled and/or homogenized to the spatial approximation size. The fracture toughness is one of the most delicate physical parameters and application of the homogenization techniques are rather uncertain, hence any proposed averaging will depend on process conditions and the toughness distribution. We propose and analyse a notion of an average toughness and show that it is a process dependent variable and provide some recommendations how to implement the defined measure into the numerical modelling. As an example, we use periodic distributions and consider model without leak off that allows us straightforward handling different regimes (toughness/viscosity).