Sequential Fully Implicit Methods for Multiscale Modeling of Compositional Flows

Abstract

The fully implicit (FI) method is widely used for numerical modeling of multiphase flow and transport in porous media. It entails iterative linearization and solution of fully-coupled linear systems with mixed elliptic/hyperbolic character. However, in methods that treat the near-elliptic (flow) and hyperbolic (transport) parts separately, such as multiscale formulations (Jenny et al., JCP 2003, Møyner and Lie, JCP 2016), sequential solution strategies are used to couple the flow (pressures and velocities) and the transport (saturations / compositions). SFI schemes solve the fully coupled system in two steps: (1) Construct and solve the pressure equation (flow problem). (2) Solve the coupled species transport equations for the phase saturations and phase compositions. In SFI, each outer iteration involves this two-step sequence. Here, we propose a new SFI variant based on a nonlinear overall-volume balance equation. The first step consists of forming and solving a nonlinear pressure equation, which is a weighted sum of all the component mass conservation equations. The resulting pressure field is used to compute the total-velocity. The second step of the new SFI scheme entails introducing the overall-mass density as a degree-of-freedom, and solving the full set of component conservation equations cast in the natural-variables form (i.e., saturations and phase compositions). During the second step, the pressure and the total-velocity fields are fixed. The SFI scheme with a nonlinear pressure extends the SFI approach of Jenny et al. (JCP 2006) to multi-component compositional processes with interphase mass transfer. We analyze the `splitting errors' associated with the compositional SFI scheme, and we show how to control these errors in order to converge to the same solution as the FI method. 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 capillary pressure or gravity, it is possible to have flow reversals. These reversals can strongly affect the convergence rate of SFI methods. We employ phase upwinding (PU) as well as a new hybrid upwinding (HU) scheme. HU determines the upwinding direction differently for the viscous, capillary pressure and buoyancy terms in the phase velocity expression. The use of HU leads to a consistent SFI scheme in terms of both pressure and compositions, and it improves the SFI convergence significantly in settings with strong capillarity and/or buoyancy. Finally, we use the multiscale restriction-smoothed basis (MsRSB) method (Møyner and Lie, JCP 2016) for the parabolic pressure operator. This sequential scheme then allows the design of robust numerical methods that are optimized for the sub-problems of flow and transport. Thus, we strongly recommend using this SFI method for sequential formulations in general, and multiscale formulations in particular.