Nanocomposites based on layered silicates offer a remarkable potential in improving mechanical properties with filler loadings lower than 5%. For such polymer nanocomposites it is important to improve intercalation and exfoliation of the silicate layers by finding the right balance between these effects. Previous works showed that intercalation and exfoliation can be influenced by the use of an additional, extensional flow which can be generated by hyperbolical dies used in the injection molding process. Since such hyperbolical dies account for high shear rates, additional pressure drop and viscous dissipation, it is of high practical relevance to correctly predict the injection pressure needed. Besides conical geometries, hyperbolical die geometries are of importance due to their uniform elongational rate. Using hyperbolical dies having different diameters, D, and length-to-diameter L/D ratios, a full rheological characterization has been carried out for a polypropylene-filled nanocomposite, and the experimental data have been fitted both with a viscous model (Cross) and a viscoelastic one (the Kaye - Bernstein, Kearsley, Zapas / Papanastasiou, Scriven, Macosko or K-BKZ/PSM model). Four injection molding dies have been also used to reach apparent shear rates up to 500,000 s-1. Particular emphasis has been given on the pressure-dependence of viscosity. It was found that only the viscoelastic simulations were capable of reproducing the experimental data well, while any viscous modeling always underestimates the pressures, especially at the higher apparent shear rates and L/D ratios.