On Averaging of Toughness Heterogeneity When Modelling Hydraulic Fracture Evolution

ABSTRACT: 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). 1. INTRODUCTION Typically, when using computational methods such as Finite Element Method, the well log and petrophysical data obtained from various measurements and observations are upscaled and/or "homogenized" to the element size. From all in-situ parameters, toughness is one of the most delicate physical parameters to handle, as application of the homogenization techniques are rather questionable here as commented on by Caiulo and Kachanov (2010) and Kachanos (1994). Compared to other types of fracture evolution, hydraulic fracturing is probably the most stable crack propagation process and thus, fortunately, some estimates can be provided here. We estimate errors introduced by various strategies to incorporate heterogeneous fracture toughness into numerical modelling (Dontsov and Suarez-Rivera (2021), Da Fies et al. (2021)). We use periodic distributions and consider model without leak off that allows us straightforward handling different regimes (toughness/viscosity). For the simulations, we use an extremely effective in house-built time – space adaptive solver utilizing main ideas from those reported by Wrobel and Mishuris (2015). The solver is capable to compute rather arbitrary distribution of the toughness.