Keen interest has been shown in exploiting the transport properties of nonlinear photonic crystals for modulation, switching and routing applications at telecommunication frequencies. This is due to the rich anisotropy evidenced by a highly-textured dispersion surface, and its dependence on the permittivity profile of the structure. We consider a two-dimensional photonic crystal, presume an instantaneous change in profile due to an optical Kerr effect, and show how the beam's refraction angle depends on its intensity. For computational simplicity, we employ a self-consistent approach and the linear eigenvalue equation, which we solve using a finite element method. Previous studies have extracted directions of optical power flow from equifrequency contour gradients. Such gradients, more formally the group velocity, lack local definition and are only meaningful in the context of a spatial average over the unit cell. As a result, the relationship between the local character of the optical transport and the induced permittivity profile is obscured. By contrast, we explicitly consider the spatial dependence of the Poynting's vector, from which we also extract mean transport directions with much greater computational efficiency. We thereby demonstrate the interrelationship between optical transport and nonlinear response at the nanoscale. A consequent analysis of refraction in the context of the superprism effect reveals new aspects to the optical transport in such nonlinear systems.