Prior works have discussed the appearance of a ceiling flux for the generalized Darcy model, but there has been no such discussion on the Forchheimer model. Moreover, the earlier results were obtained through numerical computations. Here we employ a semi-inverse solution to get analytical expressions for the solution and the volumetric flux, and we demonstrate, by direct comparison, that for the flux computation, the semi-inverse approximation is as accurate as a numerical solution to the problem. It is shown that neither model (Darcy or Forchheimer) has the desirable property of attaining a limiting flux for large driving pressures. However, when the pressure-dependence of viscosity is incorporated suitably to generalize the classical models, the generalized Darcy and Forchheimer models show the desirable behaviour of a ceiling flux for large pressures. Moreover, the volumetric flux obtained from the generalized models is always lesser than that obtained from the classical models. Such generalized models are most appropriate to model the flow in applications involving large pressure gradients, like enhanced oil recovery and carbon sequestration.