This paper describes the solution of the plane strain problem of a hydraulic fracture propagating in a permeable, linear elastic medium. The fracture propagation is driven by injection of an incompressible Newtonian fluid at a constant rate. The fracture opening and the fluid pressure are related through an elastic singular integral equation, and the flow of fluid within the fracture is modeled using lubrication theory. The leak-off or infiltration of fracturing fluid into the surrounding medium is treated as a one-dimensional diffusion process. The solution of this problem is restricted to cases where the toughness of the medium and the lag between the fluid front and the fracture tip are both zero. These particular conditions are taken to correspond to limiting cases where the energy rate dissipated in fracturing the medium is negligible compared to viscous dissipation (zero toughness) and the far-field stress perpendicular to the fracture is large (zero lag). The problem is solved numerically, using an explicit time-marching algorithm. A description of the near-tip asymptotic behavior, which is of fundamental importance for the successful convergence of the algorithm, is also included. We obtain the semi-analytic asymptotic solutions corresponding to small and large time, and compare them with the numerical solution, in order to delineate the limits of the propagation regimes.
Bibliographical noteFunding Information:
This research was conducted over a period of several years and was partially funded by Schlumberger, the Department of Civil Engineering of the University of Minnesota, through the Sommerfeld Fellowship awarded to JIA in 1999–2000, and the Donors of The Petroleum Research Fund administered by the American Chemical Society (Grant No ACS-PRF 43081-AC8). This support is gratefully acknowledged. We would also like to thank Dr. Dmitry Garagash from Dalhousie University (Halifax, NS) for the benefits of many interesting discussions. JIA thanks Schlumberger for permission to publish. Finally, the authors would like to thank an anonymous reviewer for his careful critique of the manuscript.
- Asymptotic analysis
- Hydraulic fracture
- Stable crack growth