A simple formula for calculating porosity of magma in volcanic conduits during dome-forming eruptions

We present a simple formula for analyzing factors that govern porosity of magma in dome-forming eruptions. The formula is based on a 1-dimensional steady conduit flow model with vertical gas escape, and provides the value of the porosity as a function of magma flow rate, magma properties (viscosity and permeability), and pressure. The porosity for a given pressure depends on two non-dimensional numbers ε and θ. The parameter ε represents the ratio of wall friction force to liquid-gas interaction force, and is proportional to the magma viscosity. The parameter θ represents the ratio of gravitational load to liquid-gas interaction force and is inversely proportional to the magma flow rate. Gas escape is promoted and porosity decreases with increasing ε or θ. From the possible ranges of ε and θ for typical magmatic conditions, it is inferred that the porosity is primarily determined by ε at the atmospheric pressure (near the surface), and by θ at higher pressures (in the subsurface region inside the conduit). The porosity near the surface approaches 0 owing to high magma viscosity regardless of the magnitude of the magma flow rate, whereas the subsurface porosity increases to more than 0.5 with increasing magma flow rate.