The tip region of a fluid-driven fracture 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 o f a lag of a priori unknown length between the fluid front and the crack ti p. First, we formulate the governing equations for a semi-infinite fluid-dr iven fracture propagating steadily in an impermeable linear elastic medium. Then. since the pressure in the lag zone is known, we suggest a new invers ion of the integral equation from elasticity theory to express the opening in terms of the pressure. We then calculate explicitly the contribution to the opening from the loading in the lag zone, and reformulate the problem o ver the fluid-filled portion of the crack. The asymptotic forms of the solu tion near and away from the tip are then discussed It is shown that the sol ution is not only consistent with the square root singularity of linear ela stic fracture mechanics, bur that its asymptotic behavior at infinity is ac tually given by the singular solution of a semi-infinite hydraulic fracture constructed on the assumption that the fluid flaws to the tip of the fract ure and that the solid has zero toughness. Further; the asymptotic solution for large dimensionless toughness is derived, including the explicit depen dence of the solution on the toughness. The intermediate part of the soluti on (in the region where the solution evolves from the near tip to the far f rom the tip asymptote) of the problem in the general case is obtained numer ically and relevant results are discussed including the universal relation between the fluid lag and the roughness. [S0021-8936(00)02401-6].