The Tip Region of a Near-Surface Hydraulic Fracture

Abstract

This paper investigates the tip region of a hydraulic fracture propagating near a free-surface via the related problem of the steady fluid-driven peeling of a thin elastic layer from a rigid substrate. The solution of this problem requires accounting for the existence of a fluid lag, as the pressure singularity that would otherwise exist at the crack tip is incompatible with the underlying linear beam theory governing the deflection of the thin layer. These considerations lead to the formulation of a nonlinear traveling wave problem with a free boundary, which is solved numerically. The scaled solution depends only on one number K, which has the meaning of a dimensionless toughness. The asymptotic viscosity- and toughness-dominated regimes, respectively, corresponding to small and large K, represent the end members of a family of solutions. It is shown that the far-field curvature can be interpreted as an apparent toughness, which is a universal function of K. In the viscosity regime, the apparent toughness does not depend on K, while in the toughness regime, it is equal to K. By noting that the apparent toughness represents an intermediate asymptote for the layer curvature under certain conditions, the obtention of time-dependent solutions for propagating near-surface hydraulic fractures can be greatly simplified. Indeed, any such solutions can be constructed by a matched asymptotics approach, with the outer solution corresponding to a uniformly pressurized fracture and the inner solution to the tip solution derived in this paper.

Static and dynamic fluid-driven fracturing of adhered elastica

“…From our measurements of the adhesion energy and taking an estimate of the reduced tip pressure σ = 101 × 10 3 Pa (which we will justify later when describing the measured lag length) we can calculate the transition timescale T C . Table I lists (13), (16), the time of transition t C = ((24/π ) 1/4 /1.52) 20 6.0T C . Hence, we would anticipate experiments 1-6 to be in the viscosity dominant regime and experiments 7-12 to be in the adhesion dominant regime.…”

“…Equating these two expressions gives transitional timescale T C V = 12μ(QT ) 1/4 σ 1/2 /B 3/2 κ 15/4 . Hence, for a constant flux injection, if the injection is stopped at some time t = T < T C V the blister will continue to propagate slowly in the viscosity dominant regime with radial extent (19) and decreasing lag length (20). When t > T C V , there will be a transition to adhesion control where the fracture front becomes stationary.…”

“…We note that since submitting our manuscript for publication, we have been made aware of a recent publication that presents a comparable analysis of the problem of fluid-driven fracturing of 074101-2 adhered thin elastica in which two regimes of propagation are described in the context of near-surface hydraulic fractures [20]. This work complements the mathematical approach taken here, where in addition the results of these analyses have been tested experimentally, demonstrating the formation of a vapor tip and different regimes of propagation.…”

Propagation of a Plane Strain Hydraulic Fracture With a Fluid Lag in Permeable Rock

“…Based on the solution strategy described in [7], the propagation of deep-buried or shallow plane strain and penny-shaped hydraulic fracture as well as specific tip behaviors have been extensively studied [12,13]. The scope of this paper is restricted in the deep-buried plane strain hydraulic fracture.…”

Influence of geochemistry on toughening behavior of organic-rich shale

“…For instance, hydraulic fracturing operations rely on accurate theoretical solutions for the mechanics of hydraulic fracturing [26,49]. In the case of impermeable rocks, the near tip solution and the regimes of propagation are dictated by crucial invariants depending on the local elastic and fracture constants [26,83]. Therefore, a fundamental understanding of the fracture response at the nano-and microscopic length-scales is essential.…”