Asynchronous Time Integration of Flux-conservative Transport

We investigate the potential of a flux-conservative, asynchronous method for the explicit time integration of subsurface transport equations. This method updates each discrete unknown using a local time step, chosen either in accordance with local stability conditions, or based on predicted change of the solution, as in Omelchenko and Karimabadi (Self-adaptive time integration of flux-conservative equations with sources. J. Comput. Phys. 216 (2006)). We show that the scheme offers the advantage of avoiding the overly-restrictive global CFL conditions. This makes it attractive for transport problems with localized time scales, such as those encountered in reservoir simulation where localized time scales can arise due to well singularities, spatial heterogeneities, moving fronts, or localized kinetics, amongst others. We conduct an analysis of the accuracy properties of the method for one-dimensional linear transport and first order discretization in space. The method is found to be locally inconsistent when the temporal step sizes in two adjacent cells differ significantly. We show numerically, however, that these errors do not destroy the order of accuracy when localized. The asynchronous time stepping compares favorably with a traditional first order explicit method for both linear and nonlinear problems, giving similar accuracy for much reduced computational costs. The computational advantage is even more striking in two dimensions where we combine our integration strategy with an IMPES treatment of multiphase flow and transport. Global time steps between pressure updates are determined using a strategy often used for adaptive implicit methods. Numerical results are given for immiscible and miscible displacements using a structured grid with nested local refinements around wells.