A reexamination of the classical PKN model of hydraulic fracture

This article reexamines the classical PKN model of hydraulic fracture Perkins and Kern (J. Pet. Tech. Trans. AIME, 222:937-949 (1961)) and Nordgren (J. Pet. Tech. 253:306-314 (1972)) using novel approaches, which have recently been developed to tackle this class of problems that are characterized by a moving boundary and strong non-linearities in the governing equations. First, we demonstrate, using scaling arguments only, that a PKN hydraulic fracture has two limiting time asymptotic behaviors: storage-dominated at small time, and leak-off-dominated at large time. Next, we investigate the multiscale nature of the tip asymptotics and its implication for the construction of a robust and efficient numerical algorithm. In particular, we show that in the storage-dominated regime the tip aperture w behaves according to w ~ x^1/3 (where x is the distance from the tip), and in the leak-off-dominated regime according to w ~ x^3/8. However, the solution in the leak-off-dominated regime has a boundary layer structure at the tip, with w ~ x^3/8 acting as an intermediate asymptote that matches an inner solution with a w ~ x^1/3 asymptote to the outer (global) solution. Finally, we describe an efficient numerical algorithm with the multiscale tip asymptotics embedded in the tip element, which is used to compute the transient solution that connects the small- and large-time asymptotes.