In a review paper in this same volume, we present the state of the art on modeling of compressible viscous flows ranging from single-phase to two-phase systems. It focuses on mathematical properties related to weak stability because they are important for numerical resolution and on the homogenization process that leads from a microscopic description of two separate phases to an averaged two-phase model. This review serves as the foundation for Parts I and II, which present averaged two-phase models with phase exchange applicable to magma flow during volcanic eruptions. Part I establishes a two-phase transient conduit flow model ensuring: 1) mass and volatile species conservation, 2) disequilibrium degassing considering both viscous relaxation and volatile diffusion, and 3) dissipation of total energy. The relaxation limit of this model is then used to obtain a drift-flux system amenable to simplification. Here, in Part II, we summarize this model and propose a 1.5D simplification of it that alleviates three issues causing difficulties in its numerical implementation. We compare our model outputs to those of another steady-state, equilibrium degassing, isothermal model under conditions typical of an effusive eruption at an andesitic volcano. Perfect equilibrium degassing is unreachable with a realistic water diffusion coefficient because conduit extremities always contain melt supersaturated with water. Such supersaturation has minor consequences because results show that assuming equilibrium degassing has a small influence on mass discharge rate. In contrast, releasing the isothermal assumption reduces significantly mass discharge rate by cooling due to gas expansion, which in turn increases liquid viscosity. We propose a simplified system using Darcy's law and omitting several processes such as shear heating and liquid inertia. This minimal system is not dissipative but approximates the steady-state mass discharge rate of the full system within 10%. A regime diagram valid under a limited set of conditions indicates when this minimal system captures the ascent dynamics of effusive eruptions. Interestingly, the two novel aspects of the full model, diffusive degassing and heat balance, cannot be neglected. In some cases with high diffusion coefficients, a shallow region where porosity and velocities tend towards zero develops initially, possibly blocking an eventual steady state. This local porosity loss also occurs when a steady-state solution is subjected to a change in shallow permeability. The resulting shallow porosity loss features many characteristics of a plug developing prior to a Vulcanian eruption.