We present a theory for dynamic longitudinal dispersion coefficient (D) for transport by Poiseuille flow, the foundation for models of many natural systems, such as in fractures or rivers. Our theory describes the mixing and spreading process from molecular diffusion, through anomalous transport, and until Taylor dispersion. D is a sixth order function of fracture aperture (b) or river width (W). The time (T) and length (L) scales that separate preasymptotic and asymptotic dispersive transport behavior are T = b2/(4Dm), where Dm is the molecular diffusion coefficient, and L = , where p is pressure and μ is viscosity. In the case of some major rivers, we found that L is ∼150W. Therefore, transport has to occur over a relatively long domain or long time for the classical advection-dispersion equation to be valid.