## 2 Materials and Methods
### Chemical and Isotopic Analysis of Water and Dissolved Gases  
The chemical composition of water was analyzed by (i) acidimetric titration (HCl 0.01 N) for the determination of HCO\({}_{3}\)\({}^{-}\) and (ii) ion chromatography, using Metrohn 761 and Metrohn 861 chromatography for the determination of major anions (Cl\({}^{-}\), SO\({}_{4}\)\({}^{-}\), NO\({}_{3}\)\({}^{-}\), Br\({}^{-}\) and F\({}^{-}\)) and cations (Na\({}^{+}\), K\({}^{+}\), Ca\({}^{2+}\) and Mg\({}^{2+}\)), respectively. The analytical error was \(<\)5%. The \({}^{18}\)O/\({}^{16}\)O and D/\({}^{1}\)H isotopic ratios of water (expressed as \(8^{18}\)O-H\({}_{2}\)O and \(8\)D-H\({}_{2}\)O in% vs. V-SMOW) were determined using a Picarro L2130-i analyzer based on cavity ring-down spectroscopy (CRDS). The analysis of the \(8^{13}\)C-TDIC (expressed in% vs. V-PDB) was performed by mass spectrometry (MS) using a Finnigan MAT252 instrument on CO\({}_{2}\) recovered after the reaction of SrCO\({}_{3}\) with anhydrous phosphoric acid following the procedure described by [PERSON] et al. [55] and a two-step extraction and purification procedure as described in [PERSON] et al. [56]. The analytical error and the reproducibility for \(8^{13}\)C-TDIC analysis were \(\pm\)0.05% and \(\pm\)0.1%, respectively.  
The chemical compositions of dissolved gases in the headspace of the sampling flask (CO\({}_{2}\), N\({}_{2}\), Ar + O\({}_{2}\), H\({}_{2}\) and He) were determined using a Shimadzu 15A gas chromatograph equipped with a 5 m long stainless steel column packed with Chromosorb PAW 80/100 mesh and a thermal conductivity detector (TCD), whereas CH\({}_{4}\) was analyzed using a Shimadzu 14A gas chromatograph equipped with a 10 m long stainless steel column packed with Chromosorb PAW 80/100 mesh coated with 23% SP 1700 and a flame ionization detector (FID). Argon and O\({}_{2}\) were analyzed using a Thermo Focus gas chromatograph equipped with a 30 m long capillary molecular sieve column and a TCD. The analytical error for GC analysis was \(<\)5% [54] and references therein. The concentrations of dissolved gas species were determined as the sum of the (i) number of moles of each gas in the headspace (n\({}_{i,g}\)), calculated according to the ideal gas law at 20 \({}^{\circ}\)C (i.e., laboratory temperature)considering the partial pressures determined by gas chromatography (P\({}_{\rm i}\)), and the volume of the flask headspace (V\({}_{\rm gas}\)) and (ii) the number of moles of each gas remaining in the liquid (n\({}_{\rm i,j}\)), calculated according to Henry's law, assuming that equilibrium was attained within the flask between the separated gas phase and the liquid, e.g., [54; 57; 58; 59]. The \({}^{13}\)C/\({}^{12}\)C isotopic ratios of CO\({}_{2}\) and CH\({}_{4}\) (expressed as \(\delta^{13}\)C-CO\({}_{2}\) and \(\delta^{13}\)C-CH\({}_{4}\), respectively, in% vs. V-PDB) in the headspace of the sampling flask were analyzed by CRDS using a Picarro G2201-i analyzer. The \(\delta^{13}\)C-CO\({}_{2}\) values were corrected considering the enrichment factor for gas-water isotope equilibrium between gaseous and dissolved CO\({}_{2}\)[56; 60].  
The chemical and isotopic analyses were carried out respectively by the Laboratory of Fluid Geochemistry and Laboratory of Stable Isotopes (Department of Earth Science, University of Florence), respectively, with the sole exception of the determination of \(\delta^{13}\)C-TDIC, which was carried out at the Laboratory of Fluids Geochemistry of the INGV, Sezione di Napoli, Osservatorio Vesuviano.