Sequential fully implicit formulation for compositional simulation using natural variables

Abstract

The Sequential Fully Implicit (SFI) method was proposed (Jenny et al., JCP 2006), in the context of a Multiscale Finite Volume (MSFV) formulation, to simulate coupled immiscible multiphase fluid flow in porous media. Later, Lee et al. (Comp. Geosci. 2008) extended the SFI formulation to the black-oil model, whereby the gas component is allowed to dissolve in the oil phase. Most recently, the SFI approach was extended to fully compositional isothermal displacements by Moncorgé et al., (JCP 2017). 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. Experience indicates that complex interphase mass transfer behaviors often lead to large numbers of SFI outer iterations compared with the Fully Implicit (FI) method. 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 Darcy phase velocities and 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. 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.and Doster et al., CRC 2014), which use overall compositions, this SFI formulation is well suited for the natural variables (saturations and phase compositions). 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 Fully Implicit (FI) method. We then demonstrate that the compositional SFI has convergence properties that are very comparable to those of the FI approach. This robust sequential-implicit solution scheme allows for designing numerical methods and linear solvers that are optimized for the sub-problems of flow and transport. The SFI scheme with a nonlinear pressure formulation is well suited for multiscale formulations, and it promises to replace the widely...

A multiscale mixed finite element method applied to the simulation of two-phase flows

“…Thus, the resulting numerical model shares some similarities with the Sequential Fully Implicit (SFI) scheme of Multiscale Finite Volume methods [10,36]. The reader can also find good review and recent developments of the SFI scheme in [37,38], as well as of other developments on multiscale formulations in contrast to the approach considered in the current manuscript Users should be aware that sequential methods do not necessarily guarantee unconditional stabil-ity and convergence, even though each uncoupled subproblem is unconditionally stable and convergent. For instance, for the cases of coupled fluid flow and reservoir geomechanic investigated in [39], SFI simulations are only conditionally stable when fixed-strain split schemes are used, while those with fixed-stress split schemes are unconditionally stable.…”

“…For instance, for the cases of coupled fluid flow and reservoir geomechanic investigated in [39], SFI simulations are only conditionally stable when fixed-strain split schemes are used, while those with fixed-stress split schemes are unconditionally stable. For coupled flow and transport simulations (without mechanical influence), SFI strategies can be derived with convergence properties comparable with those of the fully implicit method, as discussed in [37,38]. It should also be remarked that, in strong coupling situations, where sequential algorithms may face stability constraints, and for which an unconditionally stable monolithic fully coupled method is the recommended strategy, stabilized finite element methods [40,41,42], using a unified finite element simulator for all subsystems evolved, could be a helpful option.…”

A numerical study of the additive Schwarz preconditioned exact Newton method (ASPEN) as a nonlinear preconditioner for immiscible and compositional porous media flow

“…The gas is usually injected in smaller volumes, with water injection in between the gas volumes to uphold a favorable mobility ratio. We revisit an example from [33] posed on the first layer of Model 2 from the 10th SPE Comparative Solution Project [7], which is initially filled with a mixture of carbon species (C1, C3, C6, C10, C15, and C20), all in the liquid phase. We use a three-phase model with six components plus water and assume that water is immiscible.…”

“…Domain decomposition is also very appealing as a nonlinear solution strategy in reservoir simulation: The system of nonlinear equations tends to be strongly coupled, highly nonlinear, and unbalanced (i.e., a safe step length for Newton's method is determined by a small subset of the full variable set [3]), so that unless we take very short timesteps, the initial guess is typically far from the solution. A particularly popular nonlinear variable-decomposition approach is sequential splitting of the full problem into a pressure subproblem and a set of transport subproblems [33,34,38,47,50], which also facilitates using efficient multiscale methods for the pressure subproblem [13,27,36].…”

Dynamic Coarsening and Local Reordered Nonlinear Solvers for Simulating Transport in Porous Media

“…Choice of Refinement Indicator. In dynamic gridding, refinement and coarsening can be controlled by rigorous error estimates on the basis of the underlying discretization (Klöfkorn et al 2002;Ohlberger 2009) or the Hessian of the approximate solution (Mostaghimi et al 2016;Adam et al 2017). Other approaches use approximate first-and second-order spatial and temporal derivatives of selected transport quantities (Mulder and Meyling 1993;Suicmez et al 2011;Hoteit and Chawathé 2016) and also mixed derivatives (Batenburg et al 2011), possibly with a temporal correction to avoid refinement of regions where transport quantities are not changing .…”