## 5 Discussion
The involvement of a deeper fluid having an increasing amount of CO\({}_{2}\) with a geogenic signature is also confirmed by the significant regression between the Na-Li temperature and the CO\({}_{2}\) partial pressure parameter logPCO\({}_{2}\) calculated under sampling conditions (Figure 5). The SO\({}_{4}\)-F geothermometer was adopted to verify the reliability of the results obtained by the Na-Li geothermometer. However, as already found in other geothermal systems [98; 99], the SO\({}_{4}\)-F geothermometer gave systematically negative temperature values using the equation calibrated for the temperature range of 75-150 \({}^{\circ}\)C [95]. Otherwise, the extrapolation of the equation calibrated for the range 150-300 \({}^{\circ}\)C [94] offers consistent results with those from Na-Li, but only for the springs showing fewer negative values of the saturation index for gypsum/anhydrite and fluorite. For example, Antrodoco: T(SO\({}_{4}\)-F) = 112 \(\pm\) 13 \({}^{\circ}\)C and T(Na-Li) = 119 \(\pm\) 14 \({}^{\circ}\)C (N = 18). In this study, using the Na-Li temperatures as the most likely temperature at depth and the equations of [94], for S11 (which has a mean T(Na-Li) = 196 \(\pm\) 25 \({}^{\circ}\)C, N = 8), we obtained the highest logPCO\({}_{2}\) = 3.5 value (logPCO\({}_{2}\) = 0.7 \(\pm\) 0.2 for Antrodoco). Such a relatively high temperature in S11 was also supposed to explain the boron isotope composition of this water that resembles a touralaine-equilibrated fluid [100]. In contrast, the obtained logPCO\({}_{2}\) values are higher than those obtained with the PHREEQCi code under sampling conditions (Figure 5) but are in the order of magnitude with those of other deep crustal fluids in the Apennine [101]. Therefore, the adoption of the Na-Li geothermometer allowed us to evaluate the temperatures of the San Vittorino system (up to 150-200 \({}^{\circ}\)C), where evidence of springs with significant deep contributions was found. Applying the geothermal gradient of the area (66 \({}^{\circ}\)C at 3 km depth in the Antrodoco 1 borehole) [102] at these temperatures, an estimation of the depth of fluids ranging between 2.2 and 8.8 km was carried out. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline  & **Li** & **Rb** & **Cs** & \(\mathbf{\delta^{13}}\)**Cext(g)** & **Na/Li** & **logP(CO\({}_{2}\))** & \(\mathbf{\delta D}\) & \(\mathbf{\delta^{18}}\)**O** & **R/Ra** & **He/Ne** & **R/Ra** & **He/Ne** \\ \hline  & **ppb** & **ppb** & **ppb** & ** \begin{tabular}{c} **\% (vs.** \\ **V-PDB**) \\ \end{tabular} ** & \({}^{\circ}\)C** & \multicolumn{4}{c}{**\% (vs.**} & \multicolumn{4}{c}{**\% (vs.**} & \\  & & & **V-VDB**) & \({}^{\circ}\)C** & \multicolumn{4}{c}{**\% (  
Figure 4: \(\delta^{13}\)Cext versus T, where T = temperature in \({}^{\circ}\)C obtained from an Na-Li geothermometer [96]. \(\delta^{13}\)Cext was calculated only for some groundwater samples from the San Vittorino Plain (displayed with circles), whose (\(\delta^{13}\)C\({}_{\rm DIC}\)) was measured. Historical data from the S10 Peschiera “Pe” spring are also shown for comparison (see also Figure S2). See Figure 1 for spring location. Given the correlation between the temperatures inferred by the Na-Li geothermometer, the isotopic composition of CO\({}_{2}\) gas (\(\delta^{13}\)C\({}_{\text{ext}}\)), logPCO\({}_{2}\), and some dissolved element concentrations (e.g., Li, As, Cs, and Rb), the Li-Rb-Cs ternary diagram was also developed [103] (Figure 6). Indeed, these elements generally act as less reactive and conservative elements in thermal waters, and they are often used to identify common origins or common deep processes starting from the chemical composition of waters [104; 105]. Li is incorporated in secondary quartz and chlorite minerals and can be acquired directly in the dissolution process [106]. Rb and Cs are vicariants of K; therefore, they move into water solution following the leaching of minerals or alteration products rich in K (e.g., zeolites or clays).