Shear-induced migration of 2 $\mu$m diameter spherical colloidal particles flowing through rectangular channels (50 $\mu$m $\times$ 500 $\mu$m cross-section) is studied by confocal microscopy. The confocal microscope allows imaging of the flowing particles far from the walls of the channel, at particle velocities up to 8000 $\mu$m s$^{-1}$. The particle volume fraction is varied from $\phi = 0.05$ to 0.34, and the flow rate is also varied, which results in a bulk Péclet number (${\hbox{\it Pe}}_{\rm B}$) which varies by two orders of magnitude. Concentration profiles are measured across the narrow dimension of the channel; particles at the larger volume fractions migrate toward the centreline, with the migration progressively stronger as ${\hbox{\it Pe}}_{\rm B}$ increases. The flow has been analysed using an existing mixture flow model under the assumption of fully developed flow and a proposed constitutive law which describes the suspension normal stresses as a function of both $\phi$ and the local Péclet number, ${\hbox{\it Pe}}$, the latter being defined as a variable quantity through the local shear rate. Shear thinning and shear thickening are not included. Comparisons made with the experimental data indicate that the dependence of the extent of migration upon ${\hbox{\it Pe}}_{\rm B}$ is captured well but discrepancies arise, at least in part because the assumption of full development is not valid for these experiments.