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What good is Linear Elastic Fracture Mechanics in Hydraulic Fracturing Garagash, Dmitry

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Fluid-driven fracture presents an interesting case of crack elasticity and fracture propagation nonlinearly coupled to fluid flow. With the exceptions of a few numerical studies, previous hydraulic fracture modeling efforts have been based on the premise of Linear Elastic Fracture Mechanics (LEFM): specifically, that the damage (aka cohesive) zone associated with the rock breakage near the advancing fracture front is lumped into a singular point, under the tacit assumption that the extent of the cohesive zone is small compared to lengthscales of other physical processes relevant in the HF propagation. The latter include the dissipation in the viscous fluid flow in the fracture channel, of which the fluid lag - a region adjacent to the fracture tip filled with fracturing fluid volatiles and/or infiltrated formation pore fluid - is the extreme manifestation. In this work, we address the validity of the LEFM approach in hydraulic fracturing by considering the solution in the near tip region of a cohesive fracture driven by Newtonian fluid in an impermeable linearelastic rock. First, we show that the solution in general possesses an intricate structure supported by a number of nested lengthscales (a general sentiment for HF), on which different dissipation processes are realized (or are dominant). The latter processes can be cataloged as (1) dissipation in the fracture cohesive zone, â câ , parameterized by the fracture energy Gc (cohesive energy release per unit fracture advance), the peak cohesive stress $\sigma_c$ destroyed by fracturing, and corresponding fracture aperture scale $w_c = G_c/\sigma_c$; (2) the LEFM â reductionâ of the cohesive zone process, â $k$â , quantified by $G_c$, but with the cohesive zone replaced by a singularity ($\sigma_c \rightarrow \infty$ and $w_c \rightarrow 0$); (3) viscous fluid dissipation associated with the fluid lag region, â oâ , parametrized by an equivalent fracture energy $G_o = \sigma_o w_o$, where $\sigma_o$ is the ${ in \ situ}$ confining stress (signifying the fracturing fluid pressure drop in the lag region from value $\sim\sigma_o$ to near zero) and $w_o$ is the corresponding fracture aperture scale given previously by Garagash and Detournay (2000); and (4) the viscous dissipation along the rest of the fracture (away from the fluid lag), â mâ (Desroches et al, 1994). Furthermore, each of the above limiting processes corresponds to distinct solution asymptotes. The HF tip solution structure is bookended by the â câ or â oâ (solid or fluid process zones) asymptotes near the tip and by the â mâ asymptote away from the tip, while the LEFM â kâ asymptote may emerge within the transitional region, as an intermediate asymptote, depending on values of the two governing parameters: the cohesive-to-lag fracture energy ratio $G_c=G_o$ and the cohesive-to-in-situ stress ratio $\sigma_c=\sigma_o$. For typical sets of parameters representative of both field hydraulic fractures and their lab siblings, $G_c/G_o$ is either $\sim 1$ (low-viscosity frac. fluid) or $ 1$ is shown to be required for appearance of the LEFM intermediate asymptote near the HF tip. Since $\sigma_c \sim 1$ MPa for most rocks, it can be easily recognized that the latter condition for the relevance of the LEFM to hydraulic fracturing is mostly realized in laboratory experiments conducted under reduced levels of confining stress, and would almost never occur in the field (with the exception of very-near-surface fracturing and/or highly overpressured permeable formations).

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