Propagation of Elongated Fluid-Driven Fractures: Rock Toughness Versus Fluid Viscosity

Abstract This article studies the effect of the rock fracture toughness on the propagation of elongated fluid-driven fractures. We use the “tough PKN” model (Sarvaramini and Garagash, 2015, “Breakdown of a Pressurized Finger-Like Crack in a Permeable Rock,” J. Appl. Mech., 82(6), p. 061006), an extension of the classical PKN model (Perkins and Kern, 1961, “Widths of Hydraulic Fractures,” J. Pet. Tech., 222, pp. 937–949; Nordgren, 1972, “Propagation of Vertical Hydraulic Fractures,” J. Pet. Tech., 253, pp. 306–314), which allows for a nonzero energy release rate into the advancing fracture front(s). We provide a self-consistent analysis of a “tough” elongated fracture driven by arbitrary fluid injection law under the assumption of the negligible fluid leak-off. We use scaling considerations to identify the nondimensional parameters governing the propagation regimes and their succession in time, provide a number of analytical solutions in the limiting regimes for an arbitrary power-law injection, and also posit a simplified, equation-of-motion, approach to solve a general elongated fracture propagation problem during the injection and shut-in periods. Finally, we use the developed solutions for a tough elongated fracture to surmise the relative importance of the viscous- and toughness-related dissipation on the fracture dynamics and broach the implications of the possible toughness scale dependence.