Consistent upwinding for sequential fully implicit multiscale compositional simulation

Abstract

There is strong interest to design sequential fully implicit (SFI) methods for compositional flow simulations with convergence properties that are comparable to fully implicit (FI) methods. SFI methods decompose the fully coupled system into a pressure equation and a transport system of the components. During the pressure update, the compositions are frozen, and during the transport calculations, both the pressure and total velocity are kept constant. The two systems are solved sequentially, and the solution, which is a fully implicit one, is obtained by controlling the splitting errors due to the decoupling. Having an SFI scheme that enjoys a convergence rate similar to FI makes it possible to design specialized numerical methods optimized for the different parabolic and the hyperbolic operators, as well as the use of high-order spatial and temporal discretization schemes. Here, we use the multiscale restriction-smoothed basis (MsRSB) method for the parabolic operator. We also show that phase-potential upwinding is incompatible with the total velocity formulation of the fluxes, which is common in SFI schemes. We observe that in cases with strong gravity or capillary pressure, it is possible to have flow reversals. These reversals can strongly affect the convergence rate of SFI methods. In this work, we employ phase upwinding (PU) as well as implicit hybrid upwinding (IHU) with a SFI method. IHU determines the upwinding direction differently for the viscous, buoyancy, and capillary pressure terms in the phase velocity expressions. The use of IHU leads to a consistent SFI scheme in terms of both pressure and compositions, and it improves the SFI convergence significantly in settings with strong buoyancy or capillarity. We demonstrate the robustness of the IHU-based SFI algorithm across a wide parameter range. Realistic compositional models with gas and water injection are presented and discussed.