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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. [ABSTRACT FROM AUTHOR]

We investigate the deflation dynamics of a fluid-driven fracture in a permeable elastic medium in a state of plane strain after fluid injection has ceased. Depending on the leak-off characteristics of the porous medium and the volume of injected fluid retained in the fracture at the time of shut-in, the fracture may start to recede almost immediately or continue to propagate till it arrests when the stress intensity factor drops below the fracture toughness. While occupying the arrest footprint, the fracture continues to deflate while the stress intensity factor decreases due to fluid loss to the porous medium. When the stress intensity factor drops to zero, the fracture starts the process of recession, which continues until it finally collapses. To establish a rigorous numerical scheme to explore the deflation dynamics of plane strain hydraulic fractures, we use recently established vertex and multiscale tip asymptotes for arrested and receding hydraulic fractures (Peirce and Detournay, 2022), including the r -vertex linear tip aperture asymptote w ˆ ∼ x ˆ for a receding hydraulic fracture and the stationary g -vertex asymptote w ˆ ∼ x ˆ 3 / 4 . Numerical experiments demonstrate that the multiscale asymptotes are required in order to achieve solutions that remain smooth through the arrest–recession transition point. In contrast, numerical solutions, obtained by only using vertex solutions to model the arrest and recession, exhibit jump discontinuities through this transition point. However, once the transients from these jump discontinuities have decayed the numerical schemes that use vertex and multiscale asymptotes yield almost identical solutions. A scaling analysis shows the existence of asymptotic power law behaviour for various quantities, such as the arrest time, in terms of two new dimensionless parameters. Finally, numerical solutions explore the dependence on the two dimensionless parameters of the arrest time, the duration of the arrest period, and the duration of the recession phase and confirm their asymptotic power law behaviours. [ABSTRACT FROM AUTHOR]