Plane-Strain Propagation of a Fluid-Driven Fracture: Small Toughness Solution

The paper considers the problem of a plane-strain fluid-driven fracture propagating in an impermeable elastic solid, under condition of small (relative) solid toughness or high (relative) fracturing fluid viscosity. This condition typically applies in hydraulic fracturing treatments used to stimulate hydrocarbons-bearing rock layers, and in the transport of magma in the lithosphere. We show that for small values of a dimensionless toughness $K$⁠, the solution outside of the immediate vicinity of the fracture tips is given to $O(1)$ by the zero-toughness solution, which, if extended to the tips, is characterized by an opening varying as the $(2∕3)$ power of the distance from the tip. This near tip behavior of the zero-toughness solution is incompatible with the Linear Elastic Fracture Mechanics (LEFM) tip asymptote characterized by an opening varying as the $(1∕2)$ power of the distance from the tip, for any nonzero toughness. This gives rise to a LEFM boundary layer at the fracture tips where the influence of material toughness is localized. We establish the boundary layer solution and the condition of matching of the latter with the outer zero-toughness solution over a lengthscale intermediate to the boundary layer thickness and the fracture length. This matching condition, expressed as a smallness condition on $K$⁠, and the corresponding structure of the overall solution ensures that the fracture propagates in the viscosity-dominated regime, i.e., that the solution away from the tip is approximately independent of toughness. The solution involving the next order correction in $K$ to the outer zero-toughness solution yields the range of problem parameters corresponding to the viscosity-dominated regime.