This paper is concerned with an analysis of the near-tip region of a fluid-driven fracture propagating in a permeable saturated rock. The analysis is carried out by considering the stationary problem of a semi-infinite fracture moving at constant speed V. Two basic dissipative processes are taken into account: fracturing of the rock and viscous flow in the fracture, and two fluid balance mechanisms are considered - leak-off and storage of the fracturing fluid in the fracture. It is shown that the solution is characterized by a multiscale singular behaviour at the tip, and that the nature of the dominant singularity depends both on the relative importance of the dissipative processes and on the scale of reference. This solution provides a framework to understand the interaction of representative physical processes near the fracture tip, as well as to track the changing nature of the dominant tip process(es) with the tip velocity and its impact on the global fracture response. Furthermore, it gives a universal scaling of the near-tip processes on the scale of the entire fracture and sets the foundation for developing efficient numerical algorithms relying on accurate modelling of the tip region.
Bibliographical noteFunding Information:
We acknowledge partial support from the National Science and Engineering Research Council of Canada to D.G. under Discovery grant no. 371606 and from the National Science Foundation to E.D. under grant no. 0600058. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. The authors also acknowledge the contribution of Olga Kresse to the development of the numerical solution.
- boundary layer structure
- magma and lava flow
- porous media