## 2 Materials and Methods
### Experimental Design  
To design the experiments and model the correlation between the independent variables and the response, a three-factor, three-level Box-Behnken experimental design (BBD) was employed. Olivine content (A), carbonation time (B), and carbonation pressure (C) were the independent variables since olivine CO\({}_{2}\) mineralization is significantly influenced by CO\({}_{2}\) exposure time and pressure [45]. The response variable was the hydraulic conductivity. To explore the full capacity of the carbonation setup, the high and low levels of the carbonation pressure was selected to cover the operational capacity of the apparatus. The independent variables were set at three levels (+1, -1, 0). Table 3 display the independent variables and their levels in actual and coded values. Fifteen (15) runs, including the three replicated centre points (runs 13-15), were conducted for the randomized order design. The centre points were replicated three times to determine experimental reproducibility. The complete experimental design matrix developed utilizing the software package Design-Expert Version 12 (Stat-Ease, Inc., Minneapolis, MN, USA), along with their obtained experimental responses, are given in Table 4. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Standard Run No.**} & **A: Olivine Content (\%)** & **B: Carbonation Time (Hours)** & **C: Carbonation Pressure (kPa)** & **Hydraulic Conductivity (m/s)** \\ \hline
1 & 10 & 6 & 150 & \(5.24\times 10^{-10}\) \\
2 & 30 & 6 & 150 & \(2.21\times 10^{-10}\) \\
3 & 10 & 23.5 & 150 & \(1.09\times 10^{-10}\) \\
4 & 30 & 23.5 & 150 & \(6.34\times 10^{-11}\) \\
5 & 10 & 14.8 & 100 & \(3.21\times 10^{-10}\) \\
6 & 30 & 14.8 & 100 & \(9.34\times 10^{-11}\) \\
7 & 10 & 14.8 & 200 & \(2.65\times 10^{-10}\) \\
8 & 30 & 14.8 & 200 & \(8.07\times 10^{-11}\) \\
9 & 20 & 6 & 100 & \(4.07\times 10^{-10}\) \\
10 & 20 & 23.5 & 100 & \(7.79\times 10^{-11}\) \\
11 & 20 & 6 & 200 & \(3.45\times 10^{-10}\) \\
12 & 20 & 23.5 & 200 & \(7.06\times 10^{-11}\) \\
13 & 20 & 14.8 & 150 & \(9.05\times 10^{-11}\) \\
14 & 20 & 14.8 & 150 & \(9.08\times 10^{-11}\) \\
15 & 20 & 14.8 & 150 & \(9.03\times 10^{-11}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Experimental design matrix with independent variables expressed in actual units and experimental response. \begin{table}
\begin{tabular}{l c c c} \hline \hline \multirow{2}{*}{**Independent Variable**} & **Low** & **Mid** & **High** \\  & **Coded Value** & \(-\)**1** & **0** & **+1** \\ \hline Olivine content (wt.\%) & 10 & 20 & 30 \\ Carbonation time (hours) & 6 & 14.8 & 23.5 \\ Carbonation pressure (kPa) & 100 & 150 & 200 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Actual and coded values of the independent variables. Model Development  
To determine the relationship between the independent variables and the response, RSM requires a mathematical model. In the current study, the response is well modeled by the following quadratic equation given in Equation (1) [46]. \[Y=\beta_{O}+\sum_{i=1}^{n}\beta_{i}X_{i}+\sum_{i=1}^{n}\beta_{ii}X_{i}^{2}+\sum _{i=1}^{n}\sum_{j=1}^{n}\beta_{ij}X_{i}X_{j}+e \tag{1}\]  
where; \(Y\) is the response, \(n\) is the number of factors studied, \(i\) and \(j\) are the index numbers of the factors, \(\beta_{O}\) is the intercept term, \(\beta_{i}\) is the linear coefficient, \(X_{i}\) and \(X_{j}\) are the coded independent variables, \(\beta_{ii}\) is the quadratic coefficient, \(\beta_{ij}\) is the interaction coefficient, and \(e\) is the random error between the predicted and the real experimental values. Optimization and Validation of Optimized Conditions  
Using the desirability function (\(D\)) method, numerical optimization was carried out, as shown in Equation (2). \[D=(d_{1}\times d_{2}\times\ldots d_{m})^{1/m}=\left(\prod_{i=1}^{m}d_{i}\right) ^{1/m} \tag{2}\]  
where \(m\) is the number of responses used in the optimization analysis, and \(di\) indicates the desirability of the response.