Can one use averaging when modelling toughness heterogeneity in hydraulic fracture?

We discuss the various approaches used to capture heterogeneity within a reservoir undergoing hydraulic fracturing treatment and their implications for modelling fracture propagation. In highly laminated reservoirs with soft and/or weak layers, capturing heterogeneity at an appropriate resolution is the key for successful prediction of fracture growth and other crucial treatment parameters [1,2]. Typically, when using computational methods such as the 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 the effectiveness of homogenization techniques utilized are questionable [3,4]. 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. We estimate errors introduced by various strategies to incorporate heterogeneous fracture toughness into numerical modelling [5,6]. We restrict ourselves to periodic distributions and consider only the KGD model without leak-off, allowing us to easily handle different regimes (toughness/viscosity). For the simulations, we use an extremely effective (in house-built) time – space adaptive solver incorporating ideas reported in [7]. The solver is capable of computing rather arbitrary distribution(s) of the toughness. We propose and analyse the notion of an “effective” toughness and show that it is a process dependent variable. Comparisons with the average toughness over the domain are performed. Finally, we provide some recommendations on implementing the defined measure for numerically modelling hydraulic fracture.