We use molecular dynamics simulations to investigate whether two different structural quantities, free volume per particle and a metric for local translational order, can be related to the shear rate-dependent viscosity of a model colloidal fluid. Our results suggest that while the characteristic auto correlation time of the free volumes (a dynamic quantity) qualitatively tracks the shear viscosity of the fluid, the shear rate-dependent average free volume (a static quantity) does not. On the other hand, the translational order metric does appear to be strongly correlated to the shear viscosity over a wide range of conditions.