An implicit algorithm for the propagation of a hydraulic fracture with a fluid lag

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Propagation of a hydraulic fracture parallel to a free surface (2005, background): “It can be shown that the deep fracture solution evolves from an early time similarity solution [29] characterized by the time exponents aO 1⁄4 13; aP 1⁄4 1 3 ; ag 1⁄4 23; aj 1⁄4 2 3 and a finite lag, to a large time similarity solution with same time exponents but with zero lag [7, 11, 35].”
A fixed grid algorithm for simulating the propagation of a shallow hydraulic fracture with a fluid lag (2011, methodology): “Numerical algorithms, based either on a fixed grid [18] or on a stretching mesh [17, 33], have recently been developed for modeling planar hydraulic fractures with lag, in an infinite or a semiinfinite medium.”