Propagation of a penny-shaped hydraulic fracture parallel to the free-surface of an elastic half-space

Xi Zhang, Emmanuel Detournay, Rob Jeffrey

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In this paper we analyze the problem of a penny-shaped hydraulic fracture propagating parallel to the free-surface of an elastic half-space. The fracture is driven by an incompressible Newtonian fluid injected at a constant rate at the center of the fracture. The flow of viscous fluid in the fracture is governed by the lubrication equation, while the crack opening and the fluid pressure are related by singular integral equations. We construct two asymptotic solutions based on the assumption that either the solid has no toughness or that the fluid has no viscosity. These asymptotic solutions must be understood as corresponding to limiting cases when the energy expended in the creation of new fracture surfaces is either small or large compared to the energy dissipated in viscous flow. It is shown that the asymptotic solutions, when properly scaled, depend only on the dimensionless parameter R, the ratio of the fracture radius over the distance from the fracture to the free-surface. The scaled solutions can thus be tabulated once and for all and the dependence of the solution on time can be retrieved for specific parameters, through simple scaling and by solving an implicit equation.

Original languageEnglish (US)
Pages (from-to)125-158
Number of pages34
JournalInternational Journal of Fracture
Issue number2
StatePublished - May 2002

Bibliographical note

Funding Information:
The research reported in this paper was carrying out during the sabbatical stay of E.D. with CSIRO Petroleum. The financial support of CSIRO is gratefully acknowledged. Numerical simulations for the zero-toughness case relied on a numerical algorithm originally devised by Alexei Savitski during the course of his PhD research. The authors would like to thank him for making his computer program available.


  • Hydraulic fracturing
  • Radial fracture


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