The tip region of a fluid-Drivenfracture in an elastic medium

The focus of this paper is on constructing the solution for a semi-infinite hydraulic crack for arbitrary toughness, which accounts for the presence of a lag of a priori unknownlength between the f.uid front and the crack tip. First we formulate the governing equa-tions for a semi-infinite fluid-driven fracture propagating steadily in an impermeablelinear elastic medium. Then, since the pressure in the lag zone is known, we suggest anew inversion of the integral equation from elasticity theory to express the opening interms of the pressure. We then calculate explicitly the contribution to the opening fromthe loading in the lag zone, and reformulate the problem over the fluid-filled portion of thecrack. Tne asymptotic fbrms of the solution near and away from the tip are then dis-cussed. It is shown that the solution is not only consistent with the square root singularity-of linear elastic fracture mechanics, but that its asymptotic behavior at infinity is actuallygiven by the singular solution of a semi-infinite hydraulic fracture constructed on theassumption that the fluid flows to the tip of the fracture and that the solid has zerotoughness. Further, the asymptotic solution for large dimensionless toughness is derived, including the explicit dependence of the solution on the toughness. The intermediate partof the solution fin the region where the solution evolves flom the near up to the far fromthe tip asymptote) of the problem in the general case is obtained numerically and relevantresults are discussed, including the universal relation between the fluid lag and thetoughness.