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In the first part of this paper, a universal fluid velocity based algorithm for simulating hydraulic fracture with leak-off, previously demonstrated for the PKN and KGD models, is extended to obtain solutions for a penny-shaped crack. The numerical scheme is capable of dealing with both the viscosity and toughness dominated regimes, with the fracture being driven by a power-law fluid. The computational approach utilizes two dependent variables; the fracture aperture and the reduced fluid velocity. The latter allows for the application of a local condition of the Stefan type (the speed equation) to trace the fracture front. The obtained numerical solutions are carefully tested using various methods, and are shown to achieve a high level of accuracy.

Fluid velocity based simulation of hydraulic fracture: a penny shaped model—part I: the numerical algorithm