Use of a speed equation for numerical simulation of hydraulic fractures
  The paper treats the propagation of a hydraulically driven crack. We explicitly write the local speed equation, which facilitates using the theory of propagating interfaces. It is shown that when neglecting the lag between the liquid front and the crack tip, the lubrication PDE yields that a solution satisfies the speed equation identically. This implies that for zero or small lag, the boundary value problem appears ill-posed when solved numerically. We suggest e - regularization, which consists in employing the speed equation together with a prescribed BC on the front to obtain a new BC formulated at a small distance behind the front rather than on the front itself. It is shown that - regularization provides accurate and stable results with reasonable time expense. It is also shown that the speed equation gives a key to proper choice of unknown functions when solving a hydraulic fracture problem numerically. 
Alexander M. Linkov (Institute for Problems of Mechanical Engineering,   Saint Petersburg, Russia, Presently at Rzeszow Univ...
physics.flu-dyn cs.CE