A variational inequality formulation to incorporate the fluid lag in fluid-driven fracture propagation

We have developed an efficient method to model the fluid lag in fluid-driven fracture propagation via a variational inequality formulation. The distinct feature of this method is that the configurations with and without a lag can be handled in a unified framework and no change of formulation is needed during the simulation at the time the fracturing liquid reaches the fracture tip. This is achieved by formulating the problem as solving for the nonnegative pressure field in the fracture via a time-dependent (parabolic) variational inequality. Without introducing extra assumptions but merely based on mass conservation, this method is able to predict whether a fluid lag is going to remain or completely disappear as the fracturing progresses.