A fixed grid algorithm for simulating the propagation of a shallow hydraulic fracture with a fluid lag

This paper describes a fixed grid algorithm to simulate the propagation of shallow hydraulic fractures, under plane strain or axisymmetric conditions. Because of the low stress environment that exists near a free surface, these fractures are generally characterized by a fluid front that lags behind the fracture edge. The simulation of shallow hydraulic fractures requires therefore the tracking of two distinct fronts. The proposed algorithm, which traces its roots to the one described by Zhang et al. (Int. J. Numer. Anal. Methods Geomech. 2005; 29:1317-1340), advances the fracture by a constant step and computes the corresponding time required to reach this new fracture configuration as well as the matching location of the fluid front within the fixed grid. For any trial value of the time and of the front position, a non-linear system of algebraic equations is solved using either Newton's method or a fixed point iteration scheme, with preconditioning. The non-linear system is formulated by combining an implicit finite volume method for solving the lubrication equation and the displacement discontinuity method for solving the elasticity equation. The critical difference with the algorithm of Zhang et al. lays in the approach chosen to update the fluid front position. Rather than using a filling fraction method, the new algorithm adopts a velocity treatment of the fluid-front location that is akin to a one-dimensional implementation of a level set algorithm for front tracking. This conversion from a fluid volume to a fluid velocity-based approach to update the front position increases the degree of implicitness of the algorithm, which is responsible for a significant reduction of the CPU time by about two orders of magnitude.