  In the presence of strong heterogeneities, it is well known that the use of
explicit schemes for the transport of species in a porous medium suffers from
severe restrictions on the time step. This has led to the development of
implicit schemes that are increasingly favoured by practitioners for their
computational efficiency. The transport equation requires knowledge of the
velocity field, which results from an elliptic problem (Darcy problem) that is
the most expensive part of the computation. When considering large reservoirs,
a cost-effective way of approximating the Darcy problems is using multiscale
domain decomposition (MDD) methods. They allow for the pressure and velocity
fields to be computed on coarse meshes (large scale), while detailed basis
functions are defined locally, usually in parallel, in a much finer grid (small
scale). In this work we adopt the Multiscale Robin Coupled Method (MRCM,
[Guiraldello, et al., J. Comput. Phys., 355 (2018) pp. 1-21], [Rocha, et al.,
J. Comput. Phys., (2020) 109316]), which is a generalization of previous MDD
methods that allows for great flexibility in the choice of interface spaces. In
this article we investigate the combination of the MRCM with implicit transport
schemes. A sequentially implicit strategy is proposed, with different
trust-region algorithms ensuring the convergence of the transport solver. The
method is assessed on several very stringent 2D two-phase problems,
demonstrating its stability even for large time steps. It is also shown that
the best accuracy is achieved by considering recently introduced non-polynomial
interface spaces, since polynomial spaces are not optimal for high-contrast
channelized permeability fields.
