Numerical investigation of the convergence of fixed-stress split scheme for phase-field hydraulic fracture problem

The coupling between subsurface flow and reservoir geomechanics plays a critical role in modeling reservoir de-formation, hydraulic fracturing, CO2 sequestration, and hydrocarbon recovery. Mikelić and Wheeler (2013); Mikelić et al. (2014) have analyzed different iterative coupling schemes and have shown that the fixed-stress split outperforming the undrained-split coupling algorithm is unconditionally stable. The fundamental assumptions followed by the iterative coupling schemes include nonzero stiffness and Biot parameters. However, when a crack nucleates and propagates, stiffness degrades to zero locally, which might cause numerical singularities. In this paper, we show, through several numerical examples, the problem of the standard fixed-stress split iterative coupling algorithm when considering natural fractures and propose to mitigate this problem by using block approximations of the inverse of the stiffness matrix.