## 2 Materials and Methods
### Mathematical Model
#### 2.2.1 Transport Processes  
The transport of species and CO\({}_{2}\) in porous material could be advective/dispersive and diffusive in both (liquid and gaseous) phases. It was assumed that during the ventilation phase, there will be no gas pressure gradient imposed, and thus the CO\({}_{2}\) transport in the gaseous form will be governed by diffusion and the dissolved CO\({}_{2}\) will be transported by advection, molecular diffusion, and dispersion. No other volatile species were considered in the analysis. Mathematically, species transport could be expressed by a partial differential equation describing \(i\) species mass (\(m_{i}\)) conservation over time in space [24]:  
\[\frac{\partial m_{i}}{\partial t}+\mathbf{q_{L}}\cdot\
abla c_{i}=\
abla\cdot [(D_{D,i}+D_{e,i})\
abla c_{i}]+R_{i}, \tag{1}\]  
where \(c_{i}\) is the dissolved species concentration (mol/m\({}^{3}\)), \(c_{i,g}\) is the species concentration in the gaseous form (mol/m\({}^{3}\)), \(\mathbf{q_{L}}\) is [PERSON] velocity (m/s), \(D_{D,i}\) is the dispersion tensor (m\({}^{2}\)/s), and \(D_{e,i}\) is the effective diffusion coefficient (m\({}^{2}\)/s). \(R_{i}\) is a term describing the source and/or sink of the species (mol/m\({}^{3}\)s) due to chemical reactions, decay, etc. Chemical processes in the bentonite were simulated using mass conservation law (mass action law), and all the reactions were assumed at chemical equilibrium, where activities of the species are related though the equilibrium constant \(K\)[25]. \[K=\frac{\prod_{products}a_{p}^{v_{p}}}{\prod_{reactants}a_{r}^{v_{r}}}, \tag{2}\]  
where \(a_{p}\) and \(a_{r}\) are activities and \(v_{p}\) and \(v_{r}\) are stoichiometric coefficients of the products and reactants, accordingly. With a chemical solver, the evaluation of mineral volume changes could be conducted. Changes in mineral volumes were converted into porosity changes:  
\[n=1-V_{minerals}-V_{inert}, \tag{3}\]  
where \(V_{minerals}\) is the volume of minerals 1 m\({}^{3}\) of the porous medium and \(V_{inert}\) is the volume of inert materials in 1 m\({}^{3}\) of the porous medium. Components of the dispersion tensor are [24]:  
\[D_{Dii}=\alpha_{L}\frac{q_{Li}^{2}}{|\mathbf{q_{L}}|}+\alpha_{T}\frac{q_{Li} ^{2}}{|\mathbf{q_{L}}|} \tag{4}\]  
\[D_{Dij}=(\alpha_{L}-\alpha_{T})\frac{\alpha_{Li}\cdot q_{Lj}}{|\mathbf{q_{L} }|}\]  
where \(D_{Dii}\) is the diagonal components of the dispersion tensor (m\({}^{2}\)/s); \(D_{Dij}\) is off-diagonal components of the dispersion tensor (m\({}^{2}\)/s); and \(\alpha_{L}\), \(\alpha_{T}\) are longitudinal and transverse dispersivities (m). The effective diffusion coefficient \(D_{e,i}\) for solutes in porous media was described as a function of liquid volume fraction and tortuosity [24]:  
\[D_{e,i}=D_{L,i}=\theta_{l}\tau D_{o}, \tag{5}\]  
where \(\tau\) is the medium tortuosity and \(\theta_{l}\) is volumetric water content (dimensionless). For partially saturated porous media, tortuosity is related to water content through relationships such as [26]:  
\[\tau=\frac{\theta_{l}^{7/3}}{n^{2}}, \tag{6}\]In a porous medium in which pores are fully water-filled, the liquid volume fraction \(\theta_{l}\) is equal to porosity \(n\). Meanwhile for an unsaturated medium, it is related to water saturation \(S_{w}\) (\(\theta_{l}=nS_{w}\)). At the same time, the gas volume fraction in the case of an unsaturated porous medium is:  
\[\theta_{g}=n-\theta_{l}=(1-S_{w})\cdot n \tag{7}\]  
In case of CO2 gas diffusion, mass is expressed as:  
\[\frac{\partial m_{CO2}}{\partial t}=\frac{\partial(\theta_{lCO2,d})}{\partial t }+\frac{\partial\left(\theta_{g}c_{CO2,g}\right)}{\partial t} \tag{8}\]  
According to [PERSON]'s law, the concentration of CO2 in gas phase (\(c_{CO2,g}\pmod{\text{m}^{3}}\)) is linearly related to dissolved CO2 concentration (\(c_{CO2,d}\pmod{\text{m}^{3}}\)):  
\[c_{CO2,g}=\frac{c_{CO2,d}}{H}, \tag{9}\]  
where \(H\) is [PERSON]'s constant for CO2 (dimensionless).