Research Papers

The Tip Region of a Near-Surface Hydraulic Fracture

[+] Author and Article Information
Zhi-Qiao Wang

School of Engineering and Technology,
China University of Geosciences,
Beijing 100086, China

Emmanuel Detournay

Department of Civil, Environmental, and
University of Minnesota,
Minneapolis, MN 55455
e-mail: detou001@umn.edu

1Corresponding author.

Manuscript received January 8, 2018; final manuscript received January 10, 2018; published online February 15, 2018. Editor: Yonggang Huang.

J. Appl. Mech 85(4), 041010 (Feb 15, 2018) (11 pages) Paper No: JAM-18-1014; doi: 10.1115/1.4039044 History: Received January 08, 2018; Revised January 10, 2018

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.

Copyright © 2018 by ASME
Your Session has timed out. Please sign back in to continue.


Young, C. , 1999, “Controlled-Foam Injection for Hard Rock Excavation,” 37th U.S. Rock Mechanics Symposium Rock Mechanics for Industry (USRMS), Vail, CO, June 7–9, pp. 115–122. https://www.onepetro.org/conference-paper/ARMA-99-0115
Jeffrey, R. G. , and Mills, K. W. , 2000, “Hydraulic Fracturing Applied to Inducing Longwall Coal Mine Goaf Falls,” 4th North American Rock Mechanics Symposium, July 31–Aug. 3, Seattle, WA, pp. 423–430. https://www.onepetro.org/conference-paper/ARMA-2000-0423
Murdoch, L. C. , 2002, “Mechanical Analysis of Idealized Shallow Hydraulic Fracture,” J. Geotech. Geoenviron., 128(6), pp. 488–495. [CrossRef]
Bunger, A. P. , and Cruden, A. R. , 2011, “Modeling the Growth of Laccoliths and Large Mafic Sills: Role of Magma Body Forces,” J. Geophys. Res.: Solid Earth, 116(B2), p. B02203.
Michaut, C. , 2011, “Dynamics of Magmatic Intrusions in the Upper Crust: Theory and Applications to Laccoliths on Earth and the Moon,” J. Geophys. Res.: Solid Earth, 116(B5), p. B05205.
Tayler, A. B. , and King, J. R. , 1987, “Free Boundaries in Semi-Conductor Fabrication,” Free Boundary Problems: Theory and Applications (Pitman Research Notes in Mathematics, Vol. 1), Longman Scientific & Technical, Harlow, UK, pp. 243–259.
Hosoi, A. E. , and Mahadevan, L. , 2004, “Peeling, Healing, and Bursting in a Lubricated Elastic Sheet,” Phys. Rev. Lett., 93(13), p. 137802. [CrossRef] [PubMed]
Lister, J. R. , Peng, G. G. , and Neufeld, J. A. , 2013, “Viscous Control of Peeling an Elastic Sheet by Bending and Pulling,” Phys. Rev. Lett., 111(15), p. 154501. [CrossRef] [PubMed]
Zhang, X. , Detournay, E. , and Jeffrey, R. G. , 2002, “Propagation of a Penny-Shaped Hydraulic Fracture Parallel to the Free-Surface of an Elastic Half-Space,” Int. J. Fract., 115(2), pp. 125–158. [CrossRef]
Zhang, X. , Jeffrey, R. G. , and Detournay, E. , 2005, “Propagation of a Hydraulic Fracture Parallel to a Free Surface,” Int. J. Numer. Anal. Methods Geomech., 29(13), pp. 1317–1340. [CrossRef]
Gordeliy, E. , and Detournay, E. , 2011, “A Fixed Grid Algorithm for Simulating the Propagation of a Shallow Hydraulic Fracture With a Fluid Lag,” Int. J. Numer. Anal. Methods Geomech., 35(5), pp. 602–629. [CrossRef]
Bunger, A. P. , and Detournay, E. , 2005, “Asymptotic Solution for a Penny-Shaped Near-Surface Hydraulic Fracture,” Eng. Fract. Mech., 72(16), pp. 2468–2486. [CrossRef]
Hewitt, I. J. , Balmforth, N. J. , and De Bruyn, J. R. , 2015, “Elastic-Plated Gravity Currents,” Eur. J. Appl. Math., 26(1), pp. 1–31. [CrossRef]
Flitton, J. C. , and King, J. R. , 2004, “Moving-Boundary and Fixed-Domain Problems for a Sixth-Order Thin-Film Equation,” Eur. J. Appl. Math., 15(6), pp. 713–754. [CrossRef]
Lister, J. R. , 1990, “Buoyancy-Driven Fluid Fracture: The Effects of Material Toughness and of Low-Viscosity Precursors,” J. Fluid Mech., 210, pp. 263–280. [CrossRef]
Desroches, J. , Detournay, E. , Lenoach, B. , Papanastasiou, P. , Pearson, J. R. A. , Thiercelin, M. , and Cheng, A. H.-D. , 1994, “The Crack Tip Region in Hydraulic Fracturing,” Proc. R. Soc. London, Ser. A, 447(1929), pp. 39–48. [CrossRef]
Garagash, D. I. , and Detournay, E. , 2000, “The Tip Region of a Fluid-Driven Fracture in an Elastic Medium,” ASME J. Appl. Mech., 67(1), pp. 183–192. [CrossRef]
Garagash, D. I. , Detournay, E. , and Adachi, J. I. , 2011, “Multiscale Tip Asymptotics in Hydraulic Fracture With Leak-Off,” J. Fluid Mech., 669, pp. 260–297. [CrossRef]
Dontsov, E. V. , and Peirce, A. P. , 2015, “A Non-Singular Integral Equation Formulation to Analyse Multiscale Behaviour in Semi-Infinite Hydraulic Fractures,” J. Fluid Mech., 781, p. R1.
Lecampion, B. , and Detournay, E. , 2007, “An Implicit Algorithm for the Propagation of a Hydraulic Fracture With a Fluid Lag,” Comput. Meth. Appl. Mech. Eng., 196(49–52), pp. 4863–4880. [CrossRef]
Detournay, E. , and Peirce, A. P. , 2014, “On the Moving Boundary Conditions for a Hydraulic Fracture,” Int. J. Eng. Sci., 84, pp. 147–155. [CrossRef]
Detournay, E. , 2016, “Mechanics of Hydraulic Fractures,” Annu. Rev. Fluid Mech., 48(1), pp. 311–339. [CrossRef]
Timoshenko, S. , 1958, Strength of Materials. Part1: Elementary Theory and Problems, 3rd ed., Van Nostrand Reinhold, New York.
Batchelor, G. K. , 1967, An Introduction to Fluid Dynamics, Cambridge University Press, Cambridge UK.
Hutchinson, J. W. , and Suo, Z. , 1991, “Mixed Mode Cracking in Layered Materials,” Adv. Appl. Mech., 29, pp. 63–191. [CrossRef]
Li, S. , Wang, J. , and Thouless, M. D. , 2004, “The Effects of Shear on Delamination in Layered Materials,” J. Mech. Phys. Solids, 52(1), pp. 193–214. [CrossRef]
Seifert, U. , 1991, “Adhesion of Vesicles in Two Dimensions,” Phys. Rev. A, 43(12), p. 6803. [CrossRef] [PubMed]
Majidi, C. , and Adams, G. G. , 2009, “A Simplified Formulation of Adhesion Problems With Elastic Plates,” Proc. R. Soc. London, Ser. A, 465(2107), pp. 2217–2230. [CrossRef]
Majidi, C. , O'Reilly, O. M. , and Williams, J. A. , 2012, “On the Stability of a Rod Adhering to a Rigid Surface: Shear-Induced Stable Adhesion and the Instability of Peeling,” J. Mech. Phys. Solids, 60(5), pp. 827–843. [CrossRef]
Dyskin, A. V. , Germanovich, L. N. , and Ustinov, K. B. , 2000, “Asymptotic Analysis of Crack Interaction With Free Boundary,” Int. J. Solids Struct., 37(6), pp. 857–886. [CrossRef]
Cook, R. D. , Malkus, D. S. , Plesha, M. E. , and Witt, R. J. , 2007, Concepts and Applications of Finite Element Analysis, Wiley, New York.


Grahic Jump Location
Fig. 1

Semi-infinite near-surface hydraulic fracture

Grahic Jump Location
Fig. 2

Lag Λms versus K in the ms-scaling

Grahic Jump Location
Fig. 3

Far-field curvature Ω″ms(∞) versus K in ms-scaling

Grahic Jump Location
Fig. 4

Aperture at the fluid front, Ωms(Λms,K) versus K in the ms-scaling

Grahic Jump Location
Fig. 5

Aperture profile in ms-scaling

Grahic Jump Location
Fig. 9

Bounds of limiting solutions

Grahic Jump Location
Fig. 6

Profiles of the curvature in Log-Log scales

Grahic Jump Location
Fig. 8

Bounds of limiting solutions

Grahic Jump Location
Fig. 7

Net pressure profile in ms-scaling

Grahic Jump Location
Fig. 10

Mesh and beam element for the numerical scheme

Grahic Jump Location
Fig. 11

Convergence of far-field curvature with length of channel

Grahic Jump Location
Fig. 12

Convergence of lag with discretization length Δξ¯



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In