Plane-strain propagation of a fluid-driven fracture during injection and shut-in: Asymptotics of large toughness

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Hydraulic fracture revisited: Particle velocity based simulation (2015, methodology): “The relative difference between our numerical solution and the one reconstructed from the data given in Garagash (2006) (based on 13-terms approximation) is shown in Fig.”
An Asymptotic Framework for the Analysis of Hydraulic Fractures: The Impermeable Case (2007, methodology): “Some of the terms in (46)-(47) have also been found in previous work: the terms in (46) were given in [16, 26, 31], and also in [ 28 ] in the asymptotic limit of large toughness, with the second given for x l, i.e.”
Numerical modeling of hydraulic fracture problem in permeable medium using cohesive zone model (2012, background): “To these two sets of mechanisms are associated the four following combined asymptotic regimes: storage-toughness [21], storage-viscosity [41], leak-off-toughness [7] and leak-off-viscosity [3].”
Modeling multi-scale processes in hydraulic fracture propagation using the implicit level set algorithm (2015, background): “A number of studies on the propagation of HF in the special case of plane strain have clearly established the pivotal role of the asymptotic behavior of the solution in the vicinity of the fracture tip [6,7,5,8–10].”