The breakthrough time of a hydraulic fracture contained between two tough layers

Hydraulic fracturing is a widespread technology used to enhance reservoir production but also to measure the in-situ stress field. It consists of growing a tensile (mode I) fracture via the injection of a viscous fluid (usually at a constant rate) from a wellbore. Hydraulic fracturing treatments vary widely in scales, rates and volumes injected. For example, a well stimulation operation typically consists in injecting up to 100’000 liters at a rate of about 20-50 liters per second, while only ~5 liters are injected over the two-three minutes of the duration of an injection cycle of a micro-hydraulic fracturing stress measurement. The main risk associated with hydraulic fracturing treatment relates to vertical fracture growth above the desired formation of interest, toward environmentally sensitive layers. In sedimentary basins, characterised by rock strata at different scales, hydraulic fractures are typically observed to be contained at depth and propagate horizontally in a “finger-like” geometry (Economides & Nolte 2000, Bunger & Lecampion 2017). Such containment is usually explained by the increase of the horizontal confining stress in the layers above and below the stimulated reservoir. Of course, variation of material properties such as elastic modulus, permeability, fracture toughness can also contribute to containment (Simonson et al. 1978, Thiercelin et al. 1989). In this work, we consider the symmetric scenario where the injection point is located at the center of a layer of height H bounded by two layers of similar properties. We investigate the case where the material fracture toughness in the central layer is lower than in the bounding layers, assuming the other properties to be uniform (see Fig. 1). We focus on the so-called toughness dominated regime, where the fluid viscous energy is negligible compared to the energy spent in creating new fracture surfaces. By means of numerical simulations and scaling analysis, we establish whether, or not, a fluid-driven fracture can remain confined between the two tough layers. We show that a ratio of fracture toughness between the layers larger than √2 is sufficient to contain indefinitely the growing hydraulic fracture. For toughness ratios lower than that limit, we estimate the amount of time the propagating fracture remains contained before breaking through. Finally, we discuss how a finite amount of fluid viscous energy affects containment in that configuration.