## 2 Methodology
### Model Setup
The oxidation of hydrogen by SO\({}_{4}\)2\({}^{-}\) and/or CO\({}_{2}\) in the model is kinetically controlled. The reaction kinetics describe the time-dependent changes of reaction products and educts of a chemical reaction. To calculate the irreversible reactions, catalyzed by microorganisms like sulfate-reducing bacteria and methanogenic bacteria, the Monod equation (Equation (4)) is used. Bacterial concentrations are kept constant in the model with the aim to model the \"worst-case\" scenario. \[\psi^{\text{growth}}=\psi^{\text{growth}}_{\text{max}}\left(\frac{c^{8}}{ \alpha+c^{8}}\right) \tag{4}\]  
where \(\psi^{\text{growth}}_{\text{max}}\) is the maximum specific growth rate (mol L\({}^{-1}\) s\({}^{-1}\)), \(c^{8}\) is the concentration of the limiting substrates (mol L\({}^{-1}\)), and \(\alpha\) is the half-velocity constant. The maximum specific growth rate that is assumed in the model is 2.30 \(\times\) 10\({}^{-9}\) mol kgw\({}^{-1}\) s\({}^{-1}\) for methanogenesis (at 37 \({}^{\circ}\)C) [22] and 9.26 \(\times\) 10\({}^{-8}\) mol kgw\({}^{-1}\) s\({}^{-1}\) for BSR (at 30 \({}^{\circ}\)C) [52]. For bacterial sulfate reduction, the limiting substrate is sulfate and for methanogenesis it is carbon dioxide. [PERSON] [5] defined a new model to describe the substrate-limited growth models (Equation (5)), where \(t_{\text{e}}\) is the \"individual time of eating\" (s) and \(n_{\text{max}}\) is the maximum population size (m\({}^{-3}\)). Due to \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Parameter**} & \multirow{2}{*}{**Cap Rock Brine**} & \multicolumn{2}{c}{**Irreducible Water in the**} & \multirow{2}{*}{**Underlying Rock Brine**} \\  & & & **Reservoir Rock** & \\ \hline pH & 6.4 & 6.4 & 5.9 \\ Temperature (\({}^{\circ}\)C) & 37.0 & 40.0 & 60.0 \\ Elements & Concentration (mol kgw\({}^{-1}\)) & Concentration (mol kgw\({}^{-1}\)) & Concentration (mol kgw\({}^{-1}\)) \\ Al & 1.31 \(\times\) 10\({}^{-7}\) & 2.209 \(\times\) 10\({}^{-8}\) & 1.776 \(\times\) 10\({}^{-8}\) \\ Ba & 5.85 \(\times\) 10\({}^{-7}\) & 3.922 \(\times\) 10\({}^{-7}\) & 2.206 \(\times\) 10\({}^{-5}\) \\ C\({}_{\text{tot}}\)1  & 3.11 \(\times\) 10\({}^{-2}\) & 1.762 \(\times\) 10\({}^{-2}\) & 7.405 \(\times\) 10\({}^{-3}\) \\ Ca & 3.63 \(\times\) 10\({}^{-2}\) & 1.186 \(\times\) 10\({}^{-2}\) & 1.562 \(\times\) 10\({}^{-2}\) \\ Cl & 1.27 & 1.123 & 5.396 \\ Fe & 5.69 \(\times\) 10\({}^{-2}\) & 6.572 \(\times\) 10\({}^{-2}\) & 4.263 \(\times\) 10\({}^{-11}\) \\ K & 5.28 \(\times\) 10\({}^{-1}\) & 6.151 \(\times\) 10\({}^{-1}\) & 4.604 \(\times\) 10\({}^{-1}\) \\ Mg & 8.73 \(\times\) 10\({}^{-4}\) & 2.579 \(\times\) 10\({}^{-3}\) & 1.142 \(\times\) 10\({}^{-2}\) \\ N & 5.61 \(\times\) 10\({}^{-2}\) & 6.485 \(\times\) 10\({}^{-2}\) & 5.081 \(\times\) 10\({}^{-2}\) \\ Na & 1.27 & 1.123 & 5.396 \\ S\({}_{\text{tot}}\)2  & 1.01 & 1.143 & 7.540 \(\times\) 10\({}^{-1}\) \\ Si & 8.73 \(\times\) 10\({}^{-5}\) & 9.878 \(\times\) 10\({}^{-5}\) & 1.509 \(\times\) 10\({}^{-4}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Composition of the initial irreducible water in the reservoir rock, the cap rock brine, and the underlying rock brine. the lack of data regarding the characteristic time of eating and the maximum population size, the Panfilov model is not considered. \[\psi^{\rm growth}=\frac{1}{t_{\rm e}}\frac{n}{1+\frac{n^{2}}{n_{\rm max}^{2}}} \bigg{(}\frac{c^{\rm s}}{\alpha+c^{\rm s}}\bigg{)} \tag{5}\]  
The lower boundary of the column (in the underlying rock) and the upper boundary of the column (in the cap rock) are defined as diffusive flux. To check the numerical accuracy of the results, discretization studies for one scenario are performed, in which the number of shifts and the number of cells are refined. The model calculation for one scenario is rerun and results are compared [38].