A simple formula for calculating porosity of magma in volcanic conduits during dome-forming eruptions
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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.
Key wordsConduit flow dome-forming eruptions magma porosity gas escape from magma
As 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. The porosity changes with depth owing to the competition between the vesiculation and escape of gas from the magma. 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). 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). 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.
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. This formula is based on a 1-dimensional steady conduit flow model that considers vertical gas escape from magma. 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.
2. A Formula for Calculating Porosity
Equations (1) and (2) describe the mass conservations of the liquid and the gas respectively, and Eqs. (3) and (4) the momentum conservations of the liquid and the gas respectively. Equation (5) is the equation of state for the gas phase, and Eq. (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. 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.
In the 1-dimensional steady conduit flow models, the porosity of magma is determined by solving the differential equations (i.e., Eqs. (1)–(8)) as a two-point boundary value problem (referred to as “DE-2BV”). The boundary condition at the bottom end of the conduit is that the pressure is equal to the pressure at the magma chamber and ug = ul, and the boundary condition at the vent is that the pressure is equal to the atmospheric pressure. The variations of physical quantities such as Φ, ul, ug and P with depth throughout the conduit and the value of the mass flow rate q are obtained such that the boundary conditions are satisfied. On the other hand, Eq. (10) provides an algebraic expression of Φ for a given ε, θ, and P. This formula cannot determine the value of q as DE-2BV does, but determines the relationships among Φ, µ, and k at a given pressure when the value of q is somehow known. Because Eq. (10) is valid for general forms of µ and k, it is useful for studying how Φ varies as µ and k change with depth in a complex way. In this section, we demonstrate that Eq. (10) can correctly estimate the porosity as a function of P for realistic forms of µ and k when the value of q is given as a parameter.
The above results show that if the value of q is known, Eq. (10) can correctly determine the porosity as a function of P for realistic forms of β and k. In actual eruptions, the value of q can be estimated from field observations. In addition, we know that P at the surface is atmospheric and that P in the subsurface region is greater than atmospheric. Therefore, we can evaluate how the porosity at the surface or in the subsurface region is controlled by the variations of β and k on the basis of Eq. (10).
4. Geological Implications
During dome-forming eruptions, magma porosity changes through the competing effects of the magma vesiculation and the gas escape from magma. According to field observations, the porosity of lava domes typically ranges from 0 to 0.5 (e.g., Melnik and Sparks, 2002; Kueppers et al., 2005; Mueller et al., 2005). On the other hand, Clarke et al.(2007) showed that the porosity in the subsurface region where the pressure is higher than about 10 MPa can be as large as 0.5–0.7, as was the case for the pre-explosion (dome growth) state of the 1997 events in Soufriere Hills Volcano, Montserrat (SHV). We discuss the mechanism for these observed porosity distributions to be generated on the basis of our simple formula, Eq. (10).
Figure 2 describes how the porosity near the surface (P = 0.1 MPa) and that in the subsurface region (P = 10 MPa) depend on ε and θ on the basis of Eq. (10). In this diagram, possible ranges of ε and θ estimated from the typical values of β, k, rc and q for actual dome-forming eruptions are shown. Generally, the value of ε increases dramatically with decreasing P in response to the increase in β (Fig. 3(c)). This effect is taken into consideration for the possible range of ε in Fig. 2. At pressures near the surface (P = 0.1 MPa), the porosity depends on ε and is unaffected by θ (Fig. 2(a)). On the other hand, at pressures in the subsurface region (P = 10 MPa), the porosity depends on θ rather than on ε (Fig. 2(b)).
The above results show that the increase in the magma viscosity plays an important role in explaining the low porosity (close to zero) near the surface observed in lava domes. The porosity near the surface decreases with increasing ε (Fig. 2(a)). Considering that the permeability decreases with decreasing porosity (e.g., Mueller et al., 2005) and that there is not a large variation of rc, the increase in ε is ascribed to the increase in the magma viscosity (see Eq. (11)). The viscosity drastically increases in the region near the surface because of volatile exsolution and crystallization. As a result, the ascent of the liquid is suppressed owing to large wall friction force, whereas the gas ascends easily (i.e., efficient gas escape). For example, when the viscosity near the surface increases up to 1014 Pa s, ε can become larger than 103, which leads to a porosity smaller than 0.2 (Fig. 2(a)). The porosity near the surface remains low even when the mass flow rate is high, because the porosity in this region is insensitive to θ (Fig. 2(a)).
In contrast, the porosity in the subsurface region is sensitive to θ (Fig. 2(b)). This indicates that the increase in the mass flow rate plays a major role in explaining the high porosity in the subsurface region estimated for SHV (0.5∓ 0.7). Considering again the permeability-porosity relationship, the increase in mass flow rate results in a decrease in θ (see Eq. (12)), which in turn leads to an increase in subsurface porosity (see Fig. 2(b)). For the possible ranges of ε and θ in Fig. 2(b), the porosity Φ is larger than 0.5 when θ is smaller than about 1. In the case of k ∼ 10-12 m2 (the value for SHV when Φ = 0.5–0.7; Melnik and Sparks, 2002), the mass flow rate q must exceed 5 kg m-2 s-1 for θ< 1. (e.g., compare Fig. 3(a–c) with Fig. 3(d–f)). This estimation is consistent with the observation that q at the pre-explosion state of the 1997 events in SHV reached about 28 kg m-2 s-1 (Sparks et al., 1998). We suggest that the high flow rate immediately prior to the explosive activity induced the increase in the subsurface porosity.
In conclusion, we have derived a simple formula for calculating the porosity of magma in dome-forming eruptions as a function of mass flow rate, magma properties such as the viscosity and the permeability, and pressure. On the basis of this formula, we have shown that the increase in the magma viscosity due to volatile exsolution and crystallization near the surface plays a key role in the formation of a porosity distribution in dome-forming eruptions. The porosity near the surface approaches 0 owing to the high magma viscosity regardless of the magnitude of the mass flow rate, whereas the subsurface porosity increases to more than 0.5 with increasing mass flow rate. In order to understand the mechanism of the porosity change in dome-forming eruptions, we need to quantitatively evaluate complex effects of the magma properties such as degassing-induced crystallization (e.g., Melnik and Sparks, 2005), non-Newtonian rheology of crystal-bearing magma (e.g., Caricchi et al., 2007), and various relationships between permeability and porosity (e.g., Eichelberger et al., 1986; Takeuchi et al., 2005). The simple formula obtained in this paper (Eq. (10)) will be useful for systematically analyzing the relationship between these complex effects of the magma properties and the conduit flow dynamics.
We thank Sebastian Mueller and Wim Degruyter for insightful comments in improving an earlier version of the manuscript. We are grateful to Alain Burgisser and Shigeo Yoshida for helpful reviews and suggestions that greatly improved the manuscript. This work was supported by Grant-in-Aid for Scientific Research (B) (No. 18340130, 21340123) and for Young Scientist (B) (No. 21740322) from MEXT, and the Earthquake Research Institute cooperative research program.
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