# Experimental Study on Geysers Induced by the Release of Trapped Air in Storage Tunnel Systems
## 4 Data Analysis and Discussion  
A surge-type geyser is caused mainly by instantaneous surge pressure. Referring to the working principle of the water hammer pump, when there is instantaneous extreme pressure in the system, the pressure can send water to a higher position. If considering the water-air mixture, the lifting height will increase due to the density difference. Surge-type geyser height mainly depends on the value of instantaneous extreme pressure, and the instantaneous pressure normally can be analyzed using the classical water hammer theory. Previous studies [5; 8] have carried out some detailed analysis for this kind of geyser, so it is not discussed in this paper. In the process of gas-flow geyser development, it is difficult to analyze the instantaneous velocity feature. However, through analyzing the average free surface velocity, \(\overline{v}_{1}\), and average interface net  
Figure 5: Level variation of free surface and interface in the process of a gas-flow geyser. (**a**) Operation condition: mode A, \(D\) = 0.05 m, \(H_{0}\) = 0.675 m, \(L_{0}\) = 1.2 m; (**b**) Operation condition: mode B, \(D\) = 0.05 m, \(H_{0}\) = 0.675 m, \(L_{0}\) = 2.4 m.  
Figure 6: Level variation of free surface in the process of a surge-type geyser. Operation condition: mode B, \(D\) = 0.05 m, \(H_{0}\) = 0.675 m, \(L_{0}\) = 2.4 m.  
velocity, \(\overline{v}_{net}\), from a certain height of the water column, \(h_{w}\), to 0 (\(h_{w}=0\) means the air column is broken), we can effectively estimate the intensity of gas-flow geyser and its occurrence conditions. Gas-flow geyser development is mainly dominated by buoyancy, external pressure, and air compression and expansion. The influencing factors are mainly the physical constants and other parameters, including the water kinematic viscosity (\(v_{w}\)), water density (\(\rho_{w}\)), surface tension (\(\sigma\)), acceleration of gravity (\(g\)), external pressure (\(H_{0}\)), air volume (\(V\)), diameter of shaft (\(D\)), diameter of tunnel (\(D_{0}\)), height of water column (\(h_{w}\)), etc. The average free surface velocity can be expressed as:  
\[\overline{v}_{1}=\varphi(\rho_{w},v_{w},\sigma,g,H_{0},V,D,D_{0},h_{w}) \tag{1}\]  
There are 10 variables in Equation (1) and three basic dimensions (mass, length, and time). Thus, seven dimensionless terms can be obtained, making length variables dimensionless, and the average free surface velocity can be expressed as:  
\[f\Bigg{(}\frac{\overline{v}_{1}}{\sqrt{gD}},\frac{\big{(}\sqrt{gD}\big{)}D}{v _{w}},\frac{\rho_{w}\big{(}\sqrt{gD}\big{)}^{2}D}{\sigma},\frac{H_{0}}{D_{0}},\frac{V}{\pi{D_{0}}^{3}/4},\frac{D}{D_{0}},\frac{h_{w}}{D_{0}}\Bigg{)}=0 \tag{2}\]  
The first dimensionless term on the left side of Equation (2) is the Froude number (\(Fr\)), the second and third dimensionless terms are equivalent to the Reynolds number (\(Re\)) and the Weber number (\(We\)), the term \(\sqrt{gD}\) can be thought as a characteristic velocity. As indicated by [PERSON] and [PERSON] [23], the influence of kinematic viscosity and surface tension can be ignored when \(Re>3\times 10^{5}\) and \(We>1.2\times 10^{4}\). In this paper, the free surface velocity at the moment of air column breaking under typical condition (e.g., \(D=0.05\) m, \(H_{0}=0.675\) m, \(L_{0}=2.4\) m) is approximately 6.8 m/s; this can represent the characteristic velocity of a geyser, \(v_{w}\approx 10^{-6}\) m\({}^{2}\)/s and \(\sigma\approx 0.072\) N/ m, the corresponding \(Re\) and \(We\) are \(3.4\times 10^{5}\) and \(3.2\times 10^{4}\), which are larger than the above limiting values. Thus, the second and third dimensionless terms can be neglected, and the relevant dimensionless terms for the Froude number \(Fr\) of the free surface can be reduced to:  
\[Fr=\frac{\overline{v}_{1}}{\sqrt{gD}}=\alpha(\frac{H_{0}}{D_{0}})^{\alpha 1} \big{(}\frac{V}{\pi{D_{0}}^{3}/4}\big{)}^{\alpha 2}(\frac{D}{D_{0}})^{\alpha 3}( \frac{h_{w}}{D_{0}})^{\alpha 4} \tag{3}\]  
where \(\alpha\), \(\alpha\)1, \(\alpha\)2, \(\alpha\)3, and \(\alpha\)4 are empirical constants.