The paper discusses numerical simulations of dispersion processes of inert solutes in two-dimensional random log-permeability fields Y(x). A suitable number of realizations is generated preserving a given spatial correlation structure. For each realization a finite element model solves the flow equation and a particle-tracking method solves the transport equation. Ensemble averaging yields then the statistics. The range of σY2 investigated is 0.2÷2.0. The results show that Dagan's linear theory yields acceptable results in an unexpectedly broad range of σY2. Interestingly, a linearization of the flow equation yields larger deviations from the linear theory than the corresponding fully nonlinear model, thereby suggesting that previous conclusions drawn on the limitations of the linear theory might be restrictive.