Abstract
Asymptotic solutions for fracture opening, volume and specific surface energy for small and large fracture radii are presented from the literature. By comparison to numerical simulation of static circular fractures subject to constant and uniform internal pressures, it is found that a good approximation for fracture opening, for intermediate fracture radii, is obtained from a power mean (with exponent 1/2) of the small and large fracture radii limiting cases. The power mean equation for fracture opening is used to derive corresponding equations for fracture volume, specific surface energy and mode I and II stress intensities. These are then combined to form an approximate solution to describe the propagation of circular toughness-dominated near-surface hydraulic fractures, suitable for small, large and intermediate fracture radii. The approximate solution is shown to closely approximate results from equivalent numerical simulation.
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