  The sequential fully implicit (SFI) scheme was introduced (Jenny et al. 2006)
for solving coupled flow and transport problems. Each time step for SFI
consists of an outer loop, in which there are inner Newton loops to implicitly
and sequentially solve the pressure and transport sub-problems. In standard
SFI, the sub-problems are usually solved with tight tolerances at every outer
iteration. This can result in wasted computations that contribute little
progress towards the coupled solution. The issue is known as `over-solving'.
Our objective is to minimize the cost of inner solvers while maintaining the
convergence rate of SFI. We first extended a nonlinear-acceleration (NA)
framework (Jiang and Tchelepi 2019) to multi-component compositional models,
for ensuring robust outer-loop convergence. We then developed inexact-type
methods that alleviate `over-solving'. It is found that there is no need for
one sub-problem to strive for perfection, while the coupled (outer) residual
remains high due to the other sub-problem. The new SFI solver was tested using
several complex cases. The problems involve multi-phase and EoS-based
compositional fluid systems. We compared different strategies such as fixed
relaxations on absolute and relative tolerances for the inner solvers, as well
as an adaptive approach. The results show that the basic SFI method is quite
inefficient. Away from a coupled solution, additional accuracy achieved in
inner solvers is wasted, contributing to little or no reduction of the overall
outer residual. By comparison, the adaptive inexact method provides relative
tolerances adequate for the current convergence state of the sub-problems. We
show across a wide range of flow conditions that the new solver can effectively
resolve the over-solving issue, and thus greatly improve the overall
efficiency.
