Adaptive Conservative Local Time-stepping Scheme for Multiphase Flow and Transport In Porous Media

Accurate and efficient simulation of the fluid flow and transport in the subsurface remains a challenging task. In implicit pressure explicit saturation (IMPES) formulation, the elliptic flow equation is discretized implicitly, whereas the hyperbolic transport equation is explicit. To ensure stability, the latter requires the choice of a sufficiently small timestep size, such that the local Courant-Friedrichs-Lewy (CFL) condition is not violated. This restriction leads to a large number of small global timestep sizes which drives the simulation very expensive. In order to take advantage of the local nature of the CFL condition, many local time-stepping methods were proposed. In this thesis we extend the use of the Adaptive Conservative Time Integration (ACTI) scheme devised by Jenny [1] for applications in reservoir simulation. We eliminate the most expensive step of the original algorithm and describe the details of local timestep size assignment to ensure stability for non-convex non-monotone flux functions. We investigate the performance of the scheme for a number of model problems with various physics including the tracer problem, immiscible two-phase flow in presence of buoyancy and the black oil model. We show that with modified ACTI, the size of the global timestep for explicit update of the transport can be chosen several orders of magnitude larger than that of standard IMPES. As a result, a speed-up of an order of magnitude is achieved in the number of flux and cell updates while maintaining negligible discrepancies in the solution when compared to conventional IMPES.

[1] P. Jenny, Time adaptive conservative finite volume method," Journal of Computational Physics, vol. 403, p. 109067, 2020.