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

We present an implicit moving mesh algorithm for the study of the propagation of plane-strain hydraulic fracture in an impermeable medium. In particular, the fluid front is allowed to lag behind the fracture tip. The solution, expressed in the proper scaling, evolves, for a given value of a dimensionless toughness, from a zero stress/zero time self-similar solution characterized by a finite lag, to a large stress/large time self-similar solution with zero lag. A numerical solution for the transient problem is presented using two distinct meshes, stretching at different velocities, for the lag and fluid filled part of the crack. The lubrication and elasticity equations are solved in a coupled manner using an implicit scheme. The lag size, also coupled to the preceding equations, is obtained via the fracture propagation condition. The numerical results are discussed and compared with the known zero lag solution for large stress/large time.