Sequential Fully Implicit Methods for Multiscale Modeling of Compositional Flows
Arthur Moncorgé · Repository for Publications and Research Data (ETH Zurich) · 2018
The fully implicit (FI) method is widely used for numerical modeling of multiphase flow and transport in porous media.This method is unconditionally stable, but that comes at the cost of a high computational cost.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).The most common sequential schemes are: the implicit pressure explicit saturations and compositions (IMPES), and the sequential fully implicit (SFI) schemes.Problems of practical interest often involve tightly coupled nonlinear interactions between the multiphase flow and the multi-component transport.For such problems, the IMPES approach usually suffers from prohibitively small timesteps in order to obtain useful solutions.The SFI method, on the other hand, does not suffer from a temporal stability limit, but experience indicates that complex interphase mass transfer behaviors often lead to large numbers of SFI outer iterations compared with the FI method.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 demonstrate that the convergence difficulties are directly related to the treatment of the coupling between the flow and transport problems, and 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.A Newton-based scheme is used to iterate out all the pressure dependent nonlinearities in both the accumulation and flux terms of the overall-volume balance equation.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-offreedom, 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.The proposed compositional SFI approach employs an overall balance for the pressure equation; however, unlike existing volume-balance sequential implicit (SI) schemes (Acs et al., SPEJ 1985; Watts, SPEJ 1986; Trangenstein and Bell, SIAM 1989; Pau et al., Comp.Geosci.2012; Faigle et al., Comp.Meth.App.Mech.Eng.2014; and Doster et al., CRC 2014), which use overall compositions, this SFI formulation is well suited for the natural variables.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.For a wide parameter range, we show that the proposed SFI algorithm is as robust as the FI method with a comparable convergence rate.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. I would like to express my sincere gratitude to my research advisors Prof.Patrick Jenny and Prof. Hamdi Tchelepi without whom it would not have been possible for me to work on an external thesis project at ETH Zürich while still being based at Total.This is clearly not their usual way of working with their students and organizing calls and discussing ideas through Skype with a mature