We present a posteriori error estimates and stopping criteria for global-in-time, nonoverlapping domain decomposition methods to model flow and transport problems in a porous medium. The method considered uses the optimized Schwarz waveform relaxation method with Robin transmission conditions, whose coefficients can be optimized to improve convergence rates. A mixed formulation of an interface problem on the space-time interface is derived, and different time grids are used to adapt to different time scales in the subdomains. We estimate separately the error of the space-time DD method, that of the spatial discretization as well as that of the time discretization. Consequently, an effective criterion to stop the DD iterations is developed with local time stepping. This work has been supported by ANDRA, the ANR project DEDALES and the European Research Council (ERC) project GATIPOR.