An Asymptotic Framework for the Analysis of Hydraulic Fractures: The Impermeable Case

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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].”
Numerical modeling of hydraulic fracture problem in permeable medium using cohesive zone model (2012, background): “[35] SL Mitchell, R Kuske, and AP Peirce.”
An extended finite element method for hydraulic fracture problems (2009, background): “Recent asymptotic analysis has demonstrated that these coupled equations have a complex multi-scale structure at the fracture tip in which competing physical process manifests at different time and length scales [2–4].”
An implicit level set method for modeling hydraulically driven fractures (2008, background): “The degenerate non-linear lubrication PDE, the hypersingular non-local elasticity operator, and the fracture propagation criterion combine to yield a multi-scale structure of the solution near the fracture tip [24–27].”
Coupling schemes for modeling hydraulic fracture propagation using the XFEM (2013, unknown): “This approach exploits the limiting behavior of the solution close to the crack tip, which is established by detailed asymptotic analysis [15,18,25,20].”