Darcy velocities for lognormal hydraulic conductivity with small variance and finite correlation length were approximated by periodic random fields. Accurate simulations of two-dimensional advection-dispersion processes were achieved with the global random walk algorithm, using 1010 particles in every transport realization. Reliable statistical estimations were obtained by averaging over 256 realizations. The main result is a numerical evidence for the mean square convergence of the actual concentrations to the macrodispersion process predicted by a known limit theorem. For small initial plumes the ergodic behavior can be expected after thousands of advection timescales, when the deviation from the theoretical prediction of the cross-section space-averaged concentration monotonously decays and falls under 20%. The increase of the transverse dimension of the initial plume slows down the approach to the quasi-ergodic state and has a nonlinear effect on the variability of the actual concentrations and dispersivities.