The article presents systematic numerical simulations of the temporal behavior of a passive solute in a saturated three-dimensional heterogeneous medium. The groundwater flow is derived from the linearized solution of the Darcy equation with Gauss-distributed log hydraulic conductivity. The transport of a passive solute is studied by a random-walk method, which allows for a systematic study of the temporal behavior of the effective and ensemble dispersion coefficients. The numerical results are compared to the second-order perturbation theory expressions given in two companion papers <Dentz et al., 2000a, 2000b> and to nonperturbative results which follow from Corrsin's conjecture. The low-order perturbation theory is intrinsically based on the assumption of small heterogeneity, while Corrsin's conjecture does not take into account certain contributions due to higher-order terms of the perturbation series. The simulations yield, for the first time, systematic quantitative information on the validity and the limitations of these analytic approximations. For increasing heterogeneities, considerable deviations from the theoretically predicted transport behavior are observed.