Physics-based models of volcanic eruptions track conduit processes as functions of depth and time. When used in inversions, these models permit integration of diverse geological and geophysical data sets to constrain important parameters of magmatic systems. We develop a 1-D steady state conduit model for effusive eruptions including equilibrium crystallization and gas transport through the conduit and compare with the quasi-steady dome growth phase of Mount St. Helens in 2005. Viscosity increase resulting from pressure-dependent crystallization leads to a natural transition from viscous flow to frictional sliding on the conduit margin. Erupted mass flux depends strongly on wall rock and magma permeabilities due to their impact on magma density. Including both lateral and vertical gas transport reveals competing effects that produce nonmonotonic behavior in the mass flux when increasing magma permeability. Using this physics-based model in a Bayesian inversion, we link data sets from Mount St. Helens such as extrusion flux and earthquake depths with petrological data to estimate unknown model parameters, including magma chamber pressure and water content, magma permeability constants, conduit radius, and friction along the conduit walls. Even with this relatively simple model and limited data, we obtain improved constraints on important model parameters. We find that the magma chamber had low (<5wt%) total volatiles and that the magma permeability scale is well constrained at ~10-11.4 m2 to reproduce observed dome rock porosities. Compared with previous results, higher magma overpressure and lower wall friction are required to compensate for increased viscous resistance while keeping extrusion rate at the observed value.