Most hydraulic fracturing treatments are in the viscosity-dominated regime. Hence, fracture growth does not depend on the rock toughness and it can be shown that the fracture aperture near the fracture front, when viewed at the scale of the whole fracture, is not characterized by the classical square root behavior predicted by linear elastic fracture mechanics, where is the distance from the tip. Instead, the asymptotic tip aperture that reflects the predominance of viscous dissipation is of the form, under conditions of large efficiency and small fluid lag. After demonstrating the intimate connection between the tip aperture and the fracture propagation regime, we report the results of hydraulic fracturing laboratory experiments in PMMA and glass blocks that employ a novel optical technique to measure the fracture opening. These experiments provide incontrovertible evidence that the power law index, characterizing the fracture aperture near the tip, depends on the propagation regime in accordance with theoretical findings. Finally, we demonstrate that a coarsely-meshed planar hydraulic fracture simulator can produce accurate results relative to benchmark solutions provided that the appropriate tip behavior is embedded in the algorithm. Through theoretical, experimental, and computational considerations, these results make it clear that advances in the accuracy and efficiency of fracture simulators critically depend on a sophisticated treatment of the near-tip aperture that goes beyond basic linear elastic fracture considerations.


Fluid-driven fractures represent a particular class of tensile fractures that propagate in solid media, typically under preexisting compressive stresses, as a result of internal pressurization by an injected viscous fluid. Hydraulic fractures are most commonly engineered for the stimulation of hydrocarbon-bearing rock strata to increase the production of oil and gas wells [1–3], but there are other industrial applications such as remediation projects in contaminated soils [4–6], waste disposal [7,8], excavation of hard rocks [9], preconditioning and cave inducement in mining [10,11]. Furthermore, hydraulic fractures manifest at the geological scale as kilometer-long vertical dikes bringing magma from deep underground chambers to the earth's surface [12–14], or as subhorizontal fractures known as sills that divert magma from dikes [15–17].

Since the pioneering work by Kristianovitch and Zheltov [18], there have been numerous contributions on the modeling of fluid-driven fractures that have been mainly motivated by the application of hydraulic fracturing to the stimulation of oil and gas wells. The early efforts naturally focused on analytical solutions for fractures having simple geometries, either along straight lines in plane strain or penny-shaped in situations of radial symmetry [18–25]. However, all these solutions were approximate as they contain strong assumptions about either the opening or the pressure field. In recent years, the limitations of these solutions have shifted the focus of research towards the development of numerical algorithms, to model the three-dimensional propagation of hydraulic fractures in layered strata characterized by different mechanical properties and/or in-situ stresses [26–33].

Most of the hydraulic fracture simulators that are freed of a priori constraints on the fracture shape and of the approximations associated with models commonly referred to as "Pseudo-3D," are based on linear elastic fracture mechanics (LEFM); this is reflected by the imposition of a square root asymptotic behavior on the fracture aperture, (where is the distance from the crack front) in the tip region. As it is well known, the square root asymptote is intimately linked to the energy dissipated in the creation of new fracture surfaces in the rock [34]. However, it was progressively realized in the late 1980's and early 1990's [35–37] that another tip asymptote of the form (for a Newtonian fluid and in the absence of leak-off) arises under conditions where the energy in the tip region of a propagating fracture is essentially dissipated in viscous flow. These results then motivated a systematic reexamination of the classical KGD and penny-shaped fractures [38–49] as well as the construction of comprehensive tip asymptotics that incorporate toughness, leak-off, viscous flow and the existence of a lag between the fluid front and the crack edge [50–54].

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