Multiscale tip asymptotics for a deflating hydraulic fracture with leak-off

This paper deals with the construction of the tip asymptotes for a hydraulic fracture deflating in a permeable elastic medium. Specifically, the paper describes the changing nature of the asymptotic fields during the arrest and recession phases following propagation of the fracture after fluid injection has ended. It shows that as the fracture deflates in the arrest phase, the region of dominance of the linear elastic fracture mechanics tip asymptote $w\sim x^{1/2}$ of the fracture aperture $w$ with distance $x$ from the front shrinks to the benefit of an intermediate asymptote $w\sim x^{3/4}$ . Hence only the velocity-independent $3/4$ asymptote is left at the arrest–recession transition. Furthermore, with increasing receding velocity of the front, a linear asymptote $w\sim x$ develops progressively at the fracture tip, with $w\sim x^{3/4}$ again becoming an intermediate asymptote. These universal multiscale asymptotes for the arrest and recession phases are key to determining, in combination with a computational algorithm that can simulate the evolution of a finite fracture, the decaying stress intensity factor during arrest, the time at which the fracture transitions from arrest to recession, and the negative front velocity during recession.