## 4 ANN Modeling
### Data Correlation Analysis and Data Preprocessing  
In the data correlation analysis, the dependency between the input parameters was identified to enhance the ANN model performance. If parameters that were highly dependent on each other are incorporated into input layers, the model performance would not be effective. Thus, in this study, the Pearson correlation method was adopted for data correlation analysis to identify the dependency between parameters. The Pearson's correlation coefficient, \(r_{xy}\), can be determined by,  
\[r_{xy}=\frac{\sqrt{\overline{x_{i}y_{i}}-n\ \overline{x}\ \overline{y}}}{\sqrt{ \big{(}\sum x_{i}^{2}-n\overline{x}^{2}\big{)}}\sqrt{\big{(}\sum y_{i}^{2}-n \overline{y}^{2}\big{)}}} \tag{3}\]  
where, \(n\) is the sample size, \(x_{i}\) and \(y_{i}\) are the individual sample points indexed with \(\overline{x}\) and \(\overline{y}\) are the mean values of target parameters \(x\) and \(y\), respectively. The range of Pearson correlation coefficient \(r_{xy}\) lies between \(-1\) and \(1\). The coefficient value \(-1\), \(0\), and \(1\) indicate perfectly negative correlation, no correlation, and perfectly positive correlation, respectively. The input parameters with negligible effects on the output parameters need to be excluded from the analysis for enhancing the performance of the ANN model. Since the primary purpose of this study is to determine optimal relief well design, variables specifying the storage capacity, storage stability, and storage efficiency were chosen as the output parameters, i.e., the CO\({}_{2}\) injection rate, the injection well bottomhole pressure, the RTI, and STI. The most influential input parameters on the output parameters are chosen according to previous studies, which are the relief well x and y coordinates, distance between the injection and relief wells, the injection period, the reservoir average porosity, and the reservoir average permeability [25; 56; 57; 58]. The range of selected input parameters is in Table 2.  
Figure 9: Flowchart for optimal relief well design using the developed ANN model. The calculated Pearson correlation coefficients between the input parameters are described as a heatmap in Figure 10. The correlation magnitudes between the parameters appear to be distributed within the range from \(-0.74\) to \(0.69\). It shows that the highest correlation level (\(r_{xy}=-0.74\)) was found between the x and y coordinates of the relief well, while the distance between the wells and the relief well y coordinates have the smallest dependency (\(r_{xy}=0.01\)). Although no perfect correlation was found, there are 3 pairs of parameters that show relatively high dependency with the absolute Pearson's correlation coefficient higher than \(0.7\): the porosity and the relief well x coordinates (\(r_{xy}=-0.70\)), the injection period, and the relief well y coordinates (\(r_{xy}=-0.72\)). Meanwhile, little dependencies were found over 2 pairs of parameters, the distance between the wells and the porosity (\(r_{xy}=-0.07\)), and the permeability and the porosity (\(r_{xy}=-0.07\)). The 300,000 samples extracted from the optimization results should be normalized to avoid numerical overflows induced by various weights due to the differences in the units and ranges. For all the input data, the StandardScaler in the scikit-learn package was used to normalize the mean of each parameter to zero and the variance to one. In general, an ANN operates with an input layer, an optimal number of hidden layers, an optimal number of nodes in each hidden layer, a suitable activation function, and an output layer.