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Table 3 Symbols and descriptions of thermodynamic parameters used in this study

From: Quantitative relationship between plume emission and multiple deflations after the 2014 phreatic eruption at Ontake volcano, Japan

\(\Delta V\)Deflated volume7.0 × 105 m3Narita and Murakami (2018)
\(\rho\)Water bulk density\(\rho = S\rho_{\text{l}} + \left( {1 - S} \right)\rho_{\text{v}}\)
\(S\)Volumetric liquid fraction0–1
\(\rho_{\text{l}}\)Liquid water density688–777 kg/m3a
\(\rho_{\text{v}}\)Vapor density25–55 kg/m3a
\(\beta_{\text{f}}\)Water compressibilitySingle phase: \(\frac{1}{\rho }\left( {\frac{{{\text{d}}\rho }}{{{\text{d}}P}}} \right)_{\text{h}}\)b
Two phases: \(\frac{\rho c}{\phi } \left[ {1.92 \times 10^{ - 6} P^{ - 1.66} } \right]\)dGrant and Sorey (1979)
\(P\)Pressure5–10 MPa
\(T\)Temperature264–311 °C 
\(\left\langle {\rho c} \right\rangle\)Volumetric heat capacity of the wetted rock\(\left( {1 - \phi } \right)\rho_{\text{r}} C_{\text{r}} + \phi S\rho_{\text{l}} C_{\text{l}}\)Grant and Sorey (1979)
\(\rho_{\text{r}}\)Host rock density2000 kg/m3
\(C_{\text{r}}\)Isobaric specific heat of host rock1 kJ/kg/°C
\(C_{\text{l}}\)Isobaric specific heat of liquid water5.0–6.1 kJ/kg/ °Ca
\(\beta_{\text{c}}\)Compressibility of the pressure source1.4 × 10−8–10−9 Pa−1c
  1. a\(\rho_{\text{l}}\), \(\rho_{\text{v}}\) and \(C_{\text{l}}\) are calculated at boiling temperature (264–311 °C) at 5–10 MPa
  2. bWater bulk compressibility is calculated during decompression from lithostatic to hydrostatic pressure at constant enthalpy. For single-phase state, it is directly calculated using STEAM software (Harvey and Lemmon 2013), which allows for calculating equation of state of pure water
  3. cRefer to Additional file 1: Text S2 for details
  4. dThis approximation is validity for P in the range of 4–120 bars. Unit of P is bar