![(a) Elongated, bi-wing hydraulic fracture of length 2ℓ(t) “tunneling” in a layer of height 2b, b≪ℓ, driven by fluid injection with cumulative volume 2V(t). (b and c) Relocated micro-seismicity in the top and side (EW) views during Stage 3 hydraulic fracturing injection in vertical well 21-10 in Carthage Cotton Valley Gas Field, Texas (modified after Refs. [11,12]). Microseismicity is shown by opaque dots, such that darker parts of the micro-seismicity “cloud” correspond to higher spatial density of events. Microseismicity, which is induced on natural fractures along the path of the propagating hydraulic fracture, highlights the east wing of an elongated hydraulic fracture with aspect ratio b/ℓ≈1/8 (c), aligned in the direction N80E of the maximum regional horizontal stress [17] (b). Lack of observed micro-seismicity to the west of the injection well can be due to a distant (eastward) location of the two observation wells in this study; thus, a symmetric bi-wing fracture is assumed here. Rectangle in (c) shows the inferred fracture footprint. Perforated well interval over which the fluid injection took place is shown by a thick blue line in (c).](https://asmedc.silverchair-cdn.com/asmedc/content_public/journal/appliedmechanics/92/6/10.1115_1.4068114/1/m_jam_92_6_061009_graphic_abstract.png?Expires=1747739851&Signature=rCzS~GT45ew4HZUSuOPby~MWIeQbsfClLZ9XxvTqL1KTcOcrU7ukL7BKrYIQ6~uGG1j20T3ZfVnNHDjbDFr6E3IrdDlXqId244SWlZdjsfegPCdKeNHVP1YJJdNfrMKo9E3HY2MSvYko84haU9dCeXW09Kx1yNLWufxBd50I~XFEhT~EufYSJY68Y7PEnDBTHd74trffLPQk~f6kwidjkV90~hpwfZVM5-U~fidOTaSGXFB-Bqadx3bEmucelgjdJRY2mFUSmOPXgruYD7zqNd6R5vU4xwZpGSfhVchcmwDLQe7UcJHFGJJWdhK~TlATpu2LmpMs5dPAYcNJmY3bAQ__&Key-Pair-Id=APKAIE5G5CRDK6RD3PGA)
(a) Elongated, bi-wing hydraulic fracture of length “tunneling” in a layer of height , , driven by fluid injection with cumulative volume . (b and c) Relocated micro-seismicity in the top and side (EW) views during Stage 3 hydraulic fracturing injection in vertical well 21-10 in Carthage Cotton Valley Gas Field, Texas (modified after Refs. [11,12]). Microseismicity is shown by opaque dots, such that darker parts of the micro-seismicity “cloud” correspond to higher spatial density of events. Microseismicity, which is induced on natural fractures along the path of the propagating hydraulic fracture, highlights the east wing of an elongated hydraulic fracture with aspect ratio (), aligned in the direction N80E of the maximum regional horizontal stress [17] (). Lack of observed micro-seismicity to the west of the injection well can be due to a distant (eastward) location of the two observation wells in this study; thus, a symmetric bi-wing fracture is assumed here. Rectangle in () shows the inferred fracture footprint. Perforated well interval over which the fluid injection took place is shown by a thick blue line in ().
(a) Elongated, bi-wing hydraulic fracture of length “tunneling” in a layer of height , , driven by fluid injection with cumulative volume . (b and c) Relocated micro-seismicity in the top and side (EW) views during Stage 3 hydraulic fracturing injection in vertical well 21-10 in Carthage Cotton Valley Gas Field, Texas (modified after Refs. [11,12]). Microseismicity is shown by opaque dots, such that darker parts of the micro-seismicity “cloud” correspond to higher spatial density of events. Microseismicity, which is induced on natural fractures along the path of the propagating hydraulic fracture, highlights the east wing of an elongated hydraulic fracture with aspect ratio (), aligned in the direction N80E of the maximum regional horizontal stress [17] (). Lack of observed micro-seismicity to the west of the injection well can be due to a distant (eastward) location of the two observation wells in this study; thus, a symmetric bi-wing fracture is assumed here. Rectangle in () shows the inferred fracture footprint. Perforated well interval over which the fluid injection took place is shown by a thick blue line in ().
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.
