## 3 Methodology
### Geochemical Modelling  
#### Code Descriptions and Capabilities  
The simulations were performed using CrunchFlow [45; 46], a multicomponent reactive flow and transport code for studying fluid rock interactions in porous media. The reactive transport code has many advantages such as a complete equilibrium thermodynamic treatment, flexible kinetic rate law formulations for each mineral depending on the reaction mechanism, and a range of 3D flow capabilities. The reactive transport code [47] numerically solves the mass balance of solutes, as shown in Equation (7):  
\[\frac{\partial\left(\phi C_{j}\right)}{\partial t}=\
abla\cdot\left(D\
abla C _{j}\right)-\
abla\left(qC_{j}\right)+R_{j}\left(j=1,\;2,\;3\ldots n\right) \tag{7}\]  
where \(\phi\) is the updated porosity, \(C_{j}\) is the concentration of component \(j\) (mol m\({}^{-3}\)), q is the Darcy velocity (m s\({}^{-1}\)), \(R_{j}\) is the total reaction rate affecting component \(j\) (mol m\({}^{-3}\) s\({}^{-1}\)) and \(D\) is the combined dispersion-diffusion coefficient (m\({}^{2}\) s\({}^{-1}\)). The carbonation experiments were conducted in closed batch conditions, that is, without flow, so that Equation (7) can be simplified as Equation (8):  
\[\frac{\partial\Big{(}\phi C_{j}\Big{)}}{\partial t}=R_{j}\;(j=1,\;2,\;3\ldots n) \tag{8}\]  
The reaction rate is only described as function of time. The mineral dissolution to aqueous phases and precipitation of secondary phases are kinetically controlled, thus a reaction rate is given in term of primary species. The reaction kinetic is treated based on the rate law types including the transition state theory (TST), irreversible, monod, dissolution only and precipitation only [46]. In this work, a common theoretical framework provided by the TST rate law [48; 49; 50] is adopted. The dissolution and precipitation are treated as reversible at equilibrium, by explicitly including a dependence on Gibbs energy or saturation state (Equation (9)):  
\[R_{m}=\pi A_{m}k_{m}\Big{(}\prod a_{n}\Big{)}\Bigg{|}\bigg{(}\frac{Q_{m}}{K_{ \alpha}}\bigg{)}^{n}-1\Bigg{|}^{m} \tag{9}\]  
where \(R_{m}\) is the rate of precipitation (rate \(>0\)) or dissolution (rate \(<0\)) of mineral m in mol L\({}^{-1}\) s\({}^{-1}\), \(A_{m}\) is the reactive surface area and \(k_{m}\) the kinetic constant of mineral m (in mol.m\({}^{-2}\).s\({}^{-1}\)). The sign of the saturation index \(\Omega=\log\Big{(}\frac{Q_{m}}{K_{\alpha}}\Big{)}\) determines the sign of the reaction rate. Negative means dissolution and positive means precipitation. The kinetic constant at given temperature \(T\) (\(K\)) is calculated from Equation (10):  
\[k=k_{25}\text{Exp}\bigg{[}\frac{E_{a}}{R}\bigg{(}\frac{1}{T}+\frac{1}{298.15} \bigg{)}\bigg{]} \tag{10}\]  
where \(k_{25}\) is the kinetic constant at 25 \({}^{\circ}\)C, \(E_{a}\) is the apparent activation energy (KJ mol\({}^{-1}\)) and \(R\) is the gas constant (J mol\({}^{-1}\)K\({}^{-1}\)). The change in initial porosity (\(\phi_{i}\)) and mineral bulk surface area (\(A_{m}\)) owing to dissolution is computed from Equation (11):  
\[A_{m}=A^{\text{initial}}\Bigg{(}\frac{\phi_{m}}{\phi_{(i)m}}\Bigg{)}^{\frac{2} {3}}\Bigg{(}\frac{\phi}{\phi_{i}}\Bigg{)} \tag{11}\]  
and the change due to precipitation is computed from Equation (12):  
\[A_{m}=A^{\text{initial}}\Bigg{(}\frac{\phi}{\phi_{i}}\Bigg{)}. \tag{12}\]