An implicit level set method for modeling hydraulically driven fractures

We describe a novel implicit level set algorithm to locate the free boundary for a propagating hydraulic fracture. A number of characteristics of the governing equations for hydraulic fractures and their coupling present considerable challenges for numerical modeling, namely: the degenerate lubrication equation; the hypersingular elastic integral equation; the indeterminate form of the velocity of the unknown fracture front, which precludes the implementation of established front evolution strategies that require an explicit velocity field; and the computationally prohibitive cost of resolving all the length scales. An implicit algorithm is also necessary for the efficient solution of the stiff evolution equations that involve fully populated matrices associated with the coupled non-local elasticity and degenerate lubrication equations. The implicit level set algorithm that we propose exploits the local tip asymptotic behavior, applicable at the computational length scale, in order to locate the free boundary. Local inversion of this tip asymptotic relation yields the boundary values for the Eikonal equation, whose solution gives the fracture front location as well as the front velocity field. The efficacy of the algorithm is tested by comparing the level set solution to analytic solutions for hydraulic fractures propagating in a number of distinct regimes. The level set algorithm is shown to resolve the free boundary problem with first order accuracy. Further it captures the field variables, such as the fracture width, with the first order accuracy that is consistent with the piecewise constant discretization that is used.