# Optimization of Offshore Saline Aquifer CO2 Storage in Smeaheia Using Surrogate Reservoir Models
## 4 Results  
### Model Training  
The process of constructing an Artificial Neural Network (ANN) model consists of distinct stages: training, validation, and testing. Each stage necessitates the use of a distinct dataset. The dataset is divided into training, validation, and testing sets, in this study with ratios of 80%, 10%, and 10% respectively. During the fitting phase, the model undergoes training using the training dataset. After each epoch, the resulting model is evaluated using the validation dataset. After completing the training and validation of the model, it is evaluated using the testing dataset. A suitable model exhibits consistent loss values across all time steps. This fact also must be confirmed by metrics. The primary loss function utilized for this study was the Mean Absolute Error (_MAE_) (Equation (7)) due to its suitability as a metric for regression tasks. It enables the detection of errors in the initial set of characteristics during the training process. Moreover, the real MAE in bar units for the pressure, which has been normalized for model training, can be computed to offer a more accurate representation of the model's precision. The Mean Squared Error (_MSE_) (Equation (8)) is used as a supplementary metric during training to assess the effectiveness of the primary loss function. In addition, the absolute percentage error was used to accurately quantify and visually depict the extent of the problem.  
\[MSE=\frac{1}{n}{\sum_{i=1}^{n}}\Big{(}Y_{i_{true}}-Y_{i_{prediction}}\Big{)}^{2} \tag{7}\]  
\[MAE=\frac{1}{n}{\sum_{i=1}^{n}}\Big{|}Y_{i_{true}}-Y_{i_{prediction}}\Big{|} \tag{8}\]  
where Y is the output feature, \(i\) is the output index in the range from 1 to \(n\), and \(n\) is the number of instances. The true output represents the computed variable by the numerical simulation, while the predicted value indicates the proxy model output.  
In order to enhance the accuracy assessment of the pressure model, we compute the corresponding MAE using the denormalized data (i.e., pressure in bar):  
\[Y=Y_{norm}(Y_{max}-Y_{min})+Y_{min} \tag{9}\]  
By replacing the denormalized pressure data (Equation (9)) in the MAE formula (Equation (8)), Equation (10) will be obtained (in bar):  
\[Real\ MAE=(Y_{max}-Y_{min})\frac{1}{n}{\sum_{i=1}^{n}}\Big{|}Y_{i_{true-norm}}- Y_{i_{prediction-norm}}\Big{|} \tag{10}\]  
Tables 3 and 4 represent the performance metrics of CO\({}_{2}\) saturation and pressure models. The extremely low MAE and MSE values provide confirmation of the successful training of the models without underfitting. Furthermore, their consistent values for training, validation, and testing effectively eliminate any potential for overfitting. An MAE value below 0.05 demonstrates a high level of competence in pressure model development.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Model** & **Optimizer** & **Hidden Layer** & **Units** & **Activation** & **Initial Learning Rate** \\ \hline CO\({}_{2}\) Saturation & Adam & 5 & 128–512 & ReLU, Sigmoid & \(5\times 10^{-4}\) \\ Pressure & Adam & 3 & 128–512 & ReLU, Sigmoid & \(5\times 10^{-4}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Hyperparameters.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Performance Metrics** & **Training** & **Validation** & **Testing** \\ \hline MSE & \(7.91\times 10^{-5}\) & \(8.14\times 10^{-5}\) & \(8.13\times 10^{-5}\) \\ MAE & \(4.2\times 10^{-3}\) & \(4.2\times 10^{-3}\) & \(4.2\times 10^{-3}\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: CO\({}_{2}\) Saturation SRM Accuracy Evaluation.