Characterizing the permeation performance of nanoporous material is an initial step toward predicting microflows and achieving acceptable designs in sealing and filtration applications. This study deals with analytical, numerical, and experimental studies of gaseous leaks through soft packing materials subjected to nonuniform axial compression in valve stuffing boxes. A new analytical model that accurately predicts gaseous leak rates through nanoporous packing materials assumed made of capillaries having an exponentially varying section. Based on Navier–Stokes equations with the first-order velocity slip condition for tapered cylinder capillaries, the analytical model is used to estimate gas flow through soft packing materials. In addition, computational fluid dynamic modeling using cfx software is used to test its capacity to estimate the permeation of compression packing ring materials assuming the fluid flow to follow Darcy's law. Helium gas is used as a reference gas in the experiments to characterize the porosity parameters. The analytical and cfx numerical leak predictions are compared to leak rates measured experimentally using different gas types (helium, nitrogen, air, and argon) at different pressures and gland stresses. The analytical and numerical models account for the porosity change with the stem axial distance because the packing ring set is subjected to an exponentially varying radial compression. The predictions from analytical model are in close agreement with the cfx model and in better agreement with experimental measurements.