Solution of hydraulic fracture problem accounting for lag
  The paper presents a method for solving hydraulic fracture problems accounting for the lag. The method consists in matching the outer (basic) solution neglecting the lag, with the inner (auxiliary) solution of the derived 1D integral equation with conditions, accounting for the lag and asymptotic behavior of the opening and the net-pressure. The method refers to practically important cases, when the influence of the local perturbation, caused by the lag, becomes insignificant at a distance, where the leading plane-state asymptotics near the fracture front is still applicable. The universal asymptotics are used for finding the matching constants of the basic (outer) solution and for formulation of matching condition for the solution of inner (auxiliary) problem. The method is illustrated by the solution of the Spence and Sharp plane-strain problem for a fracture propagating symmetrically from the inlet, where a Newtonian fluid is pumped at a constant rate. It is stated that the method developed for deep fractures may also serve in the intermediate range between deep and shallow fractures. 
Alexander M. Linkov
physics.flu-dyn