Propagation of a plane-strain hydraulic fracture accounting for the presence of a cohesive zone and a fluid lag

We revisit the problem of the propagation of a plane-strain fluid-driven fracture in a quasi-brittle impermeable medium accounting for the presence of a fluid lag. The fracture process zone is simulated using a linear-softening cohesive model while lubrication flow accounts for the possible occurence of a fluid lag. The solution is obtained numerically via a fully implicit scheme based on a boundary element method for the fracture deformation and finite difference for fluid flow. The fluid lag is first automatically captured using the Elrod-Adams lubrication cavitation model during the initiation and early stage of fracture growth. We then switch to an algorithm tracking the fluid front for computational efficiency. Using dimensional analysis, we show that the propagation is governed by a dimensionless toughness and a time scale characterizing the disappearance of the fluid lag, (both similar to the linear elastic fracture mechanics case) and a ratio between the in-situ minimum confining stress and the material tensile strength. The cohesive forces reinforce the suction effect associated with a fluid lag, and leads to the further localization of the fluid pressure drop near the tip. This ultimately results in a slight increase of fracture opening and net pressure.