![]() ![]() Here, the non-dimensional number ε represents the ratio of effects of the wall friction force to the liquid-gas interaction force, and the non-dimensional number θ represents the ratio of effects of the gravitational load to the liquid-gas interaction force. The constitutive equation of Flw is given by the form for a Poiseuille flow of the liquid adjacent to the cylindrical conduit wall (Wilson et al., 1980): We also assume that temperature change due to expansion is negligible because of the large heat capacity of the liquid magma therefore, the energy equation is not solved. (6) represents the mass-flow-rate fraction of the gas when equilibrium gas exsolution on the basis of the solubility curve of H2O in a magma (Burnham and Davis, 1974) is assumed. Equation (5) is the equation of state for the gas phase, and Eq. (3) and (4) the momentum conservations of the liquid and the gas respectively. Where u l and u g are the vertical velocities of the liquid and the gas respectively, ρ l (= 2500 kg m −3) and ρ g are the densities of the liquid and the gas respectively, Φ is the gas volume fraction (i.e., the porosity), n is the gas mass-flow-rate fraction, q is the mass flow rate (per unit of cross-section area kg m −2 s ρ1), z is the vertical coordinate measured positive upwards, P is the pressure of the magma, g is the acceleration due to gravity, Flg is the interaction force between the liquid and the gas, Flw is the friction force between the liquid and the conduit wall, R is the gas constant (462 J kg −1 K −1 for H2O gas), T is the magma temperature, n 0 is the initial H2O content, and s is the saturation constant (4.11×10 −6 Pa −1/2 for silicic magmas).Įquations (1) and (2) describe the mass conservations of the liquid and the gas respectively, and Eqs. This formula enables us to systematically investigate how porosity changes in response to changes in viscosity and permeability during magma ascent and also to identify the essential effects controlling the porosity profile in the conduit. This formula is based on a 1-dimensional steady conduit flow model that considers vertical gas escape from magma. In this study, we derive a simple formula for calculating the porosity in dome-forming eruptions as a function of the magma properties and geological conditions. However, the relationships between porosity and viscosity and between porosity and permeability are still unclear, which makes it difficult to understand the mechanism through which the complex porosity profiles are formed. Recent numerical studies have revealed that the porosity critically depends on magma properties such as viscosity and permeability in dome-forming eruptions and that complex porosity profiles may result as viscosity, permeability, or both change with depth the porosity increases in the subsurface region, and then decreases near the surface (e.g., Melnik and Sparks, 1999 Diller et al., 2006). When gas escape occurs efficiently, the porosity decreases, which may lead to an effusion of a lava dome with a low porosity (Eichelberger et al., 1986 Jaupart and Allegre, 1991 Woods and Koyaguchi, 1994). The porosity changes with depth owing to the competition between the vesiculation and escape of gas from the magma. 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.Īs silicic volatile-rich magma ascends to the surface and decompresses in volcanic conduits, the magma vesiculates and its porosity (i.e., gas volume fraction) increases. 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). ![]() Gas escape is promoted and porosity decreases with increasing ε or θ. The parameter θ represents the ratio of gravitational load to liquid-gas interaction force and is inversely proportional to the magma flow rate. The parameter ε represents the ratio of wall friction force to liquid-gas interaction force, and is proportional to the magma viscosity. The porosity for a given pressure depends on two non-dimensional numbers ε and θ. 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. We present a simple formula for analyzing factors that govern porosity of magma in dome-forming eruptions. ![]()
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