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

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