Fluid-driven cracks in an elastic matrix in the toughness-dominated limit

The dynamics of fluid-driven cracks in an elastic matrix is studied experimentally. We report the crack radius R(t) as a function of time, as well as the crack shapes w(r, t) as a function of space and time. A dimensionless parameter, the pressure ratio Delta pf/Delta pv, is identified to gauge the relative importance between the toughness (Delta pf) and viscous (Delta pv) effects. In our previous paper (Lai et al. 2015 Proc. R. Soc. A 471, 20150255. (doi: 10.1098/rspa. 2015.0255)), we investigated the viscous limit experimentally when the toughness-related stresses are negligible for the crack propagation. In this paper, the experimental parameters, i. e. Young's modulus E of the gelatin, viscosity mu of the fracturing liquid and the injection flow rate Q, were chosen so that the viscous effects in the flow are negligible compared with the toughness effects, i. e. Delta pf/Delta pv1. In this limit, the crack dynamics can be described by the toughnessdominated scaling laws, which give the crack radius R(t). t2/5 and the half maximum crack thickness W(t). t1/5. The experimental results are in good agreement with the predictions of the toughness scaling laws: the experimental data for crack radius R(t) for a wide range of parameters (E, Q) collapse after being rescaled by the toughness scaling laws, and the rescaled crack shapes w(r, t) also collapse to a dimensionless shape, which demonstrates the selfsimilarity of the crack shape. The appropriate choice of the viscous or toughness scaling laws is important to accurately describe the crack dynamics. This article is part of the themed issue ` Energy and the subsurface'.