Vesiculation of hydrous melts at 1 atm was studied in situ by synchrotron X-ray tomographic microscopy at the TOMCAT beamline of the Swiss Light Source (Villigen, Switzerland). Water-undersaturated basaltic, andesitic, trachyandesitic, and dacitic glasses were synthesized at high pressures and then laser heated at 1 atm. on the beamline, causing vesiculation. The porosity, bubble number density, size distributions of bubbles, and pore throats, as well as their tortuosity and connectivity, were measured in three-dimensional tomographic reconstructions of sample volumes, which were also used for lattice Boltzmann simulations of viscous permeabilities. Connectivity of bubbles by pore throats varied from ~ 100 to 10⁵ mm^-3, and for each sample correlated with porosity and permeability. Consideration of the results of this and previous studies of the viscous permeabilities of aphyric and crystal-poor magmatic samples demonstrated that at similar porosities permeability can vary by orders of magnitude, even for similar compositions. Comparison of the permeability relationships from this study with previous models (Degruyter et al., Bull Vulcanol 72:63-74, 2010; Burgisser et al., Earth Planet Sci Lett 470:37-47, 2017) relating porosity, characteristic pore-throat diameters, and tortuosity demonstrated good agreement. Modifying the Burgisser et al. model by using the maximum pore-throat diameter, instead of the average diameter, as the characteristic diameter reproduced the lattice Boltzmann permeabilities to within 1 order of magnitude. Correlations between average bubble diameters and maximum pore-throat diameters, and between porosity and tortuosity, in our experiments produced relationships that allow application of the modified Burgisser et al. model to predict permeability based only upon the average bubble diameter and porosity. These experimental results are consistent with previous studies suggesting that increasing bubble growth rates result in decreasing permeability of equivalent porosity foams. This effect of growth rate substantially contributes to the multiple orders of magnitude variations in the permeabilities of vesicular magmas at similar porosities.