Fig. 9From: Quantitative relationship between plume emission and multiple deflations after the 2014 phreatic eruption at Ontake volcano, JapanCalculated ρβ values for the shallow reservoir. Plots are made for each of two end members of \(\beta_{\text{c}}\): a \(\beta_{\text{c}}\) = 1.4 × 10−8 Pa−1 (\(V\) = 107 m3) and at b \(\beta_{c}\) = 1.4 × 10−9 Pa−1 (\(V\) = 108 m3) (see also Additional file 1: Text S2). ρβ are computed as a function of S, (0.0 ≤ S ≤ 1.0) at each pressure. Details of these parameters setting are described in Additional file 1: Text S1. Solid and open symbols correspond to two-phase and single-phase state, respectively. Solid circles and triangles correspond to host rock porosity \(\phi\) = 0.1 and 0.5 in the two-phase state, respectively. Red and blue symbols correspond to conditions of lithostatic and hydrostatic pressure at the 500-m depth, respectively. A black dashed line shows an upper limit value of \(\rho \beta\), which is derived from observations. Gray-shaded area indicates possible range of allowable ρβ values, within which even calculated values with pressure drop should rangeBack to article page