The purpose of this work is to model dispersion at the macroscale as the result of mechanisms acting at the microscale. We simulated advection and molecular diffusion of a passive tracer in networks of capillaries with different radius distributions. The solute transport process asymptotically reached a Fickian regime at the end of a transitory period of variable duration. One effect of heterogeneity was to increase the asymptotic dispersion coefficients and the transient time. Another, more important, result is the observation of a transition of the longitudinal dispersion from the Taylor-Aris dispersion in homogeneous networks to the so-called mechanical dispersion in highly heterogeneous ones. The existence of this transition is supported by previous numerical and experimental studies. In the case of transverse dispersion, the asymptotic transverse coefficient appeared very weakly related to the Peclet number at all heterogeneity levels. We propose an explanation to this apparent contradiction to experimental observations. ¿ 2001 American Geophysical Union