Scaling of Physical Processes in Fluid-Driven Fracture: Perspective from the Tip

A particular class of fractures driven in a solid by pressurized viscous fluids is considered. These fractures could be either tens or hundreds meters long man-made hydraulic fractures in oil and gas reservoirs, or natural fractures, such as kilometers-long volcanic dikes driven by magma coming from upper mantle beneath the Earth's crust. Different physical mechanisms governing propagation of a fluid-driven fracture include (i) dissipation in the viscous fluid flow along the fracture, (ii) dissipation in the solid due to fracturing, (iii) lagging of the fluid front behind the fracture front, (iv) fluid leak-off (into the permeable solid), and others. Dissipation in the viscous fluid flow is often considered to be the dominant mechanism on fracture length and time scales of practical interest. Universal scaling of the non-dominant mechanisms (dissipation in the solid, fluid lag, etc.) in the global solution of fluid-driven fracture is derived in this paper based on the analysis of the boundary layer structure near the fracture leading edge. This scaling may be particularly important in guiding numerical solution of fractures when non-trivial fracture geometry or/and spatially varying properties of the solid prevent analytical investigation of the global solution.