The results of lattice-Boltzmann simulations of the flow field through a random packing of spheres are compared with NMR velocimetry and propagator measurements. Flows are investigated for Péclet (Pe) and Reynolds (Re) numbers in the range 182<Pe<350 and 0.4<Re<0.77, respectively. An MRI visualization of the 3-D packing of spheres for which NMR data are obtained is used as the matrix for the simulation, thereby enabling a direct assessment of the ability of the lattice-Boltzmann method to quantitatively predict flow phenomena within porous media of complex geometry. By introducing normalized parameters, hydrodynamical dispersion over normalized displacement lengthscales extending over nearly four orders of magnitude is investigated and is shown to be dominated by mechanical dispersion over most of this range. At the largest lengthscales studied holdup may play a significant role in the hydrodynamics characteristic of the porous medium. Quantitative agreement between NMR measurements and the predictions of the lattice-Boltzmann simulation is obtained in all cases.