  In this paper we are interested in the "fast path" fracture and we aim to use
global-in-time, nonoverlapping domain decomposition methods to model flow and
transport problems in a porous medium containing such a fracture. We consider a
reduced model in which the fracture is treated as an interface between the two
subdomains. Two domain decomposition methods are considered: one uses the
time-dependent SteklovPoincar{\'e} operator and the other uses optimized
Schwarz waveform relaxation (OSWR) based on Ventcell transmission conditions.
For each method, a mixed formulation of an interface problem on the space-time
interface is derived, and different time grids are employed to adapt to
different time scales in the subdomains and in the fracture. Demonstrations of
the well-posedness of the Ventcell subdomain problems is given for the mixed
formulation. An analysis for the convergence factor of the OSWR algorithm is
given in the case with fractures to compute the optimized parameters. Numerical
results for two-dimensional problems with strong heterogeneities are presented
to illustrate the performance of the two methods.
