## 3 Modeling CO2 Entrapment for Sandstone and Shale Formations
\begin{table}
\begin{tabular}{l c c} \hline \hline  & **Homogeneous Saline Aquifer** & **Heterogeneous Saline Aquifer** \\ \hline Initial pressure & 11,800 kPa at 1000 m & 7159 kPa at 730 m \\ Temperature & 80 \({}^{\circ}\)C & 70 \({}^{\circ}\)C \\ Salinity & 50,000 ppm & 70,000 ppm \\ \(H_{i}^{*}\) & \(5.66\times 10^{5}\) & \(6.44\times 10^{5}\) \\ \hline \hline \end{tabular}  
* Solubility trapping in brine can be enhanced by diffusion. To model this effect, the diffusion coefficient (\(D\)) for super-critical CO2 in brine is applied to compute the effective CO2 diffusion (\(D_{\textit{eff}}\)) considering tortuosity \(\tau\)[35]: \[D_{\textit{eff}}=\frac{D}{\tau}\] (2) In shales, where diffusion is a relevant mechanism due to lower permeability, the tortuosity value ranges approximately from 40 to 70 [36]. An effective diffusion equal to 2.8 \(\times\) 10\({}^{-7}\) cm\({}^{2}\)/s [37] was used in the simulations.
* Chemical trapping by adsorption of CO2 in the gas phase was modeled using a Langmuir model [38], which is a widely accepted isotherm adsorption equation [39; 40]. With data for shales [41; 42] and sandstone [43], two isotherms were matched to model the adsorption. Table 3 summarizes Langmuir parameters (\({}^{B}CO_{2}\) and \({}^{\omega}CO_{2,max}\))  
\end{table} 
Table 2: Summary of the initial conditions of the reservoir models. obtained according to Equation (3), that is, the extended Langmuir isotherm for multicomponent adsorption [44; 45]:  
\[\omega_{CO_{2}}=\frac{\omega_{CO_{2},\ max}B_{CO_{2}}y_{CO_{2},\ gP}}{1+p\sum_{j}B _{j}y_{j,\ g}} \tag{3}\]  
where  
\({}^{B}CO_{2}\) is the parameter for Langmuir isotherm relation;  
\({}^{B}CO_{2}\) is the moles of adsorbed \(CO_{2}\) per unit mass of rock;  
\({}^{\omega}CO_{2,max}\) is the maximum moles of adsorbed \(CO_{2}\) per unit mass of rock;  
\({}^{Y}CO_{2,g}\) is the molar fraction of adsorbed \(CO_{2}\) in the gas phase. * The residual CO\({}_{2}\) trapping due to the relative permeability and capillarity hysteresis with the saturation changes was modeled with the maximum gas trapped (\(S_{gt}\)) converted to the Land's constant (C) [46] in the two-phase [PERSON]'s model [47], as recommended by [PERSON] et al. [48], according to Equation (4): \[S_{gt}=\frac{S_{g\ max}}{1+CS_{g\ max}}\] (4) where \(S_{g\ max}\) is the maximum gas saturation. [PERSON] and [PERSON] [49] obtained a \(S_{gt}\) distribution for sandstone, shale, and carbonates based on more than 30 published coreflood data for CO\({}_{2}\) and brine. The mean value for each lithology is presented in Table 4.  
* Ionic trapping due to acidic water reactions for bicarbonate and carbonate ions generation was modeled using kinetic parameters from the PHREEQC database [50; 51]. \[\text{OH}^{-}+\text{H}^{+}=\text{H}_{2}\text{O}\] (5) \[\text{CO}_{2}+\text{H}_{2}\text{O}=\text{H}^{+}+\text{HCO}_{3}{}^{-}\] (6) \[\text{CO}_{3}{}^{2-}+\text{H}^{+}=\text{HCO}_{3}{}^{-}\] (7)  
The synthetic water composition is given in Table 5 for the homogeneous case [52] and the heterogeneous case (Personal communication with PETROBRAS. 2023. Rio de Janeiro-RJ, Brazil). \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Sandstone** & **Shale** \\ \hline \(S_{gt}\) & 0.25 & 0.35 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Maximum gas trapped due to hysteresis of relative permeabilities and capillarity. \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Sandstone** & **Shale** \\ \hline \(B_{CO_{2}}\) & 0.00017 kPa\({}^{-1}\) & 0.00017 kPa\({}^{-1}\) \\ \(\omega_{CO_{2},\ max}\) & 0.07 pmol/kg & 0.18 pmol/kg \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameters of Langmuir isotherm to model the CO\({}_{2}\) adsorption. \begin{table}