Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

A Reexamination of the Classical PKN Model of Hydraulic Fracture


This article reexamines the classical PKN model of hydraulic fracture Perkins and Kern (J. Pet. Tech. Trans. AIME, 222:937–949 (1961)) and Nordgren (J. Pet. Tech. 253:306–314 (1972)) using novel approaches, which have recently been developed to tackle this class of problems that are characterized by a moving boundary and strong non-linearities in the governing equations. First, we demonstrate, using scaling arguments only, that a PKN hydraulic fracture has two limiting time asymptotic behaviors: storage-dominated at small time, and leak-off-dominated at large time. Next, we investigate the multiscale nature of the tip asymptotics and its implication for the construction of a robust and efficient numerical algorithm. In particular, we show that in the storage-dominated regime the tip aperture w behaves according to w ~ x 1/3 (where x is the distance from the tip), and in the leak-off-dominated regime according to w ~ x 3/8. However, the solution in the leak-off-dominated regime has a boundary layer structure at the tip, with w ~ x 3/8 acting as an intermediate asymptote that matches an inner solution with a w ~ x 1/3 asymptote to the outer (global) solution. Finally, we describe an efficient numerical algorithm with the multiscale tip asymptotics embedded in the tip element, which is used to compute the transient solution that connects the small- and large-time asymptotes.

This is a preview of subscription content, log in to check access.


  1. Adachi J., Detournay E.: Self-similar solution of a plane-strain fracture driven by a power-law fluid. Int. J. Numer. Anal. Methods Geomech. 26, 579–604 (2002)

  2. Adachi J., Detournay E.: Plane strain propagation of a hydraulic fracture in a permeable rock. Eng. Fract. Mech. 75(16), 4666–4694 (2008)

  3. Adachi J.I., Peirce A.P.: Asymptotic analysis of an elasticity equation for a finger-like hydraulic fracture. J. Elast. 90(1), 43–69 (2007)

  4. Barenblatt G.: Scaling, Self-Similarity, and Intermediate Asymptotics, vol. 14 of Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, UK (1996)

  5. Carter E.: Optimum fluid characteristics for fracture extension. In: Howard, G., Fast, C. (eds) Drilling and Production Practices, pp. 261–270. American Petroleum Institute, Tulsa, OK (1957)

  6. Desroches J., Detournay E., Lenoach B., Papanastasiou P., Pearson J., Thiercelin M., Cheng A.-D.: The crack tip region in hydraulic fracturing. Proc. Roy. Soc. Lond. Ser. A 447, 39–48 (1994)

  7. Detournay E.: Propagation regimes of fluid-driven fractures in impermeable rocks. Int. J. Geomech. 4(1), 1–11 (2004)

  8. Economides, M., Nolte, K. (eds.): Reservoir Stimulation. 3rd edn. Wiley, Chichester, UK (2000)

  9. Economides M., Mikhailov D., Nikolaevskiy V.: On the problem of fluid leakoff during hydraulic fracturing. Transp. Porous Media 67(3), 487–499 (2007)

  10. Garagash D., Detournay E.: The tip region of a fluid-driven fracture in an elastic medium. ASME J. Appl. Mech. 67(1), 183–192 (2000)

  11. Gordeyev Y.N., Zazovsky A.F.: Self-similar solution for deep-penetrating hydraulic fracture propagation. Transp. Porous Media 7(3), 283–304 (1992)

  12. Kemp, L.F.: Study of Nordgren’s equation of hydraulic fracturing. SPE Production Eng. 5(SPE 19959), 311–314 (1990)

  13. Mendelsohn D.: A review of hydraulic fracture modeling—part I: general concepts, 2D models, motivation for 3D modeling. J. Energy Resour. Technol. 106, 369–376 (1984)

  14. Mitchell S.L., Kuske R., Peirce A.: An asymptotic framework for finite hydraulic fractures including leak-off. SIAM J. Appl. Math. 67(2), 364–386 (2007)

  15. Mitchell S.L., Kuske R., Peirce A.P.: An asymptotic framework for the analysis of hydraulic fractures: the impermeable case. ASME J. Appl. Mech. 74(2), 365–372 (2007)

  16. Nordgren, R.: Propagation of vertical hydraulic fractures. J. Pet. Tech. 253(SPE 3009), 306–314 (1972)

  17. Perkins T., Kern L.: Widths of hydraulic fractures. J. Pet. Tech. Trans. AIME 222, 937–949 (1961)

  18. Valkó P., Economides M.: Hydraulic Fracture Mechanics. Wiley, Chichester, UK (1995)

  19. Weisstein E.W.: Faá di Bruno’s Formula. From MathWorld – A Wolfram Web Resource

Download references

Author information

Correspondence to E. Detournay.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Kovalyshen, Y., Detournay, E. A Reexamination of the Classical PKN Model of Hydraulic Fracture. Transp Porous Med 81, 317–339 (2010). https://doi.org/10.1007/s11242-009-9403-4

Download citation


  • Hydraulic fracture
  • PKN model