A particular class of fractures in the Earth develops as a result of internal pressurization by a viscous fluid. These fractures are either natural such as volcanic dikes driven by magma from beneath the Earth's crust, or man-made hydraulic fractures created by injecting a viscous fluid from a borehole in order to increase production from oil or gas reservoirs. The questions that arise in connection with these problems are simple, yet fundamental: How is the fracture evolving in shape and size? How is the fracturing pressure varying with time? How does the process depend on properties of the rock? These questions are still open due to the formidable challenge of constructing solutions for fluid-driven fractures using either analytical or numerical techniques.
The difficulty of solving this problem originates from the non-linearity of the equation governing the flow of fluid in the fracture, the non-local character of the elastic response of the fracture, and the time-dependence of the equation governing the exchange of fluid between the fracture and the rock.
At present we are concerned with the development and application of multiple scale and singular perturbation analysis to the system of governing non-linear integro-differential equations. In addition to variations on the different spatial and temporal scales, there are other scaling relationships between important physical parameters. We have developed a unifying scaling framework which allows analysis of simultaneous effects, namely viscosity, toughness and leak-off. This gives us an understanding of the changes in behaviour near the tip.
Preliminary calculations suggest that when viscosity and leak-off are dominant, a more general expansion is necessary, which allows additional time dependencies in the coefficients.