Challenges in tracing the crack front of a hydraulic fracture in inhomogeneous media

This talk will discuss the peculiarities of modelling hydraulic fracture propagating within a media with variable toughness. Numerous simulations have been performed utilizing our extremely accurate and effective in-house built time-space adaptive solver, which can obtain solutions for any of the 1D HF models (PKN, KGD, Radial, 3PD, non-local PKN) with arbitrary (fixed) fluid rheology, leak off law and pumping regime. The solver uses the crack opening and the fluid velocity as the basic unknowns in contrast to the conventional crack opening and fluid pressure pair [1,2]. We analyse the KGD and Radial HF models in an elastic material characterised by a periodic toughness distribution. Recently [3, 4], we have proposed an averaging-based approach that is dependent not only on the material but also on process dependent parameters. Their definition comes from temporal averaging (in contrast to the spacial one). As a result, all introduced measures rely also on the instantaneous crack speed. Such temporal average approach is general in its nature (not specific to HF), and can be used in analysis of any stable fracture propagation process. The simulations performed for both KGD and radial crack settings have allowed us to validate the temporal-averaging concept showing, among others, how the effective (averaged) toughness approaches its maximum value when the crack is sufficiently long, as was elegantly claimed by Dontsov and Suarez-Rivera [5].