# Optimization of Offshore Saline Aquifer CO2 Storage in Smeaheia Using Surrogate Reservoir Models
## 3 Methodology
### Proxy Modeling
\[k_{ave}=\sum_{j=1}^{n}Z_{j}\Bigg{/}\sum_{j=1}^{n}\frac{Z_{j}}{k_{j}} \tag{5}\]  
This study assumes that the boundaries are completely sealed, resulting in a layer above the first layer and a layer below the last layer with negligible permeability and porosity. Consequently, the saturation of CO\({}_{2}\) in these layers will be zero. Moreover, in the event of a closed boundary, the reservoir will exhibit a pseudo-steady state condition. The pressure above the topmost layer and below the bottommost layer is assumed to be equal to the pressure of the current layer due to insufficient data. Figure 9: Proxy modeling workflow. The reservoir model contains 1.5105 \(\times\) 10\({}^{6}\) grid blocks, which makes it unusable for the input dataset. Consequently, the input grids were decreased by implementing sampling techniques and criteria that rely on the model's output. The input dataset of the saturation model was sampled using the parameter variation that specifically chooses cells that are more influenced by CO\({}_{2}\) injection. Performing an injection at the maximum rate has a greater impact on a bigger number of cells and spreads CO\({}_{2}\) over a wider area. Thus, only the grids that had a saturation value greater than zero after 50 years of injection at the maximum rate were selected for sampling. Nevertheless, this sampling technique resulted in overfitting for the pressure model input as a consequence of its spread across the whole storage model, rather than being limited to a specific region surrounding the injection well. The pressure model, which was trained using data from cells that showed the highest level of pressure change, failed to generalize in the blind evaluation and was consequently abandoned. Thereby, the grids were randomly sampled in order to train the pressure model; 1.485 \(\times\) 10\({}^{6}\) cells were selected randomly and excluded from all cases to ultimately keep 2.55 \(\times\) 10\({}^{4}\) random cells. Figure 10: Bottom hole pressure of numerical simulation cases used for SRM training. The final dataset consists of 31 features that exhibit variations across diverse ranges of real numbers, which could potentially result in instability throughout the training process. Based on the physics of the problem, the activation functions ReLU [69] and Sigmoid [70] have been chosen which produce output values ranging from 0 to 1. Thus, the dataset requires normalization between 0 and 1 by the min-max method (Equation (6)). The global nature of each property's minimum and maximum values allows for their application in denormalizing the ultimate forecast of the pressure model and facilitating comparison with the actual value. \[x_{norm}=\frac{x-x_{min}}{x_{max}-x_{min}} \tag{6}\]  
The final stage involves constructing the architecture of the ANN, which ensures an adequate degree of accuracy during the validation, testing, and blind evaluation processes. An attempt was made to train a single ANN for both qualities, which encountered a significant convergence challenge. Furthermore, the sampling technique for saturation and pressure models differs, resulting in distinct training data. Consequently, a distinct ANN was developed for each parameter. To attain this, hyperparameters must be tuned; they include the number of hidden layers, the number of units in each layer, the activation function, and the learning rate, in addition to features and amount of data. The features and examples of input data were determined based on the reciprocal relationship between accuracy and computer facilities tolerance. The optimal hyperparameters for the artificial neural network (ANN) structure (as shown in Table 2) were determined by multiple trial-and-error iterations during the training, validation, and blind assessment stages, following the SRM development flowchart (refer to Figure 9). Different optimization algorithms can be implemented to minimize the loss function and train an ANN. The SGD, the root mean squared propagation (RMSprop), and Adam are a group of available optimizers.