# Reservoir Performance Prediction in Steam Huff and Puff Injection Using Proxy Modelling
## 2 Methodology
### Predictive Model Development
Then, to transfer the contained information, an activation (or transfer) function is used. This is also known as forward propagation and it is conducted for every connection in the network. After defining the model, the back propagation algorithm is used to readjust the weight and minimize the error between the output and the database. In this study, a multilayer neural network structure was implemented, where the input layer consisted of 28 neurons as input parameters (listed in Table 1), the output layer consisted of 1 neuron as output parameter, and there were multiple hidden layers in between, with each hidden layer consisting of a number of neurons. The activation function for all layers is the hyperbolic tangent function (tanh) presented in Equation (7). This activation function is used as it shows good performance when employed to build a multilayer neural network. \[f(x)=\tanh(x)=\frac{2}{1+e^{-2x}}-1 \tag{7}\]  
Figure 4: The artificial neural network structure for steam huff and puff injection proxy models used in this study. Three evaluation metrics were employed to assess the performance of the developed models, which were the coefficient of determination (R-square), the mean absolute error (_MAE_) and root mean square error (_RMSE_). The R-square is a statistical metric used to measure the regression prediction compared with the actual data point, where 1 indicates a perfect fit. The adjusted R-square is a modification of the R-square calculation, where it takes into account the number of input parameters (\(k\)). Its value is always lower than the R-square. Both are presented in Equations (8) and (9), respectively, where \(\hat{y}_{i}\) is the predicted value, \(y_{i}\) is the simulation data, and \(n\) is the number of experimental cases. In this study, the adjusted R-square will be used as the evaluation metric for the fitting coefficient, as it has the potential to be more accurate for a high number of input parameters. A scatter plot of simulated and predicted data is presented to visualize the performance of the model. In a good, representative model, both training and testing data points should be located close to the 45-degree baseline (\(y=x\)), as the baseline represents a perfect fit (\(R^{2}=1\)). Additionally, the _MAE_ and _RMSE_ are calculated for the training and testing results of each model. _MAE_ is the average difference of the predicted and actual data, which indicates the relative deviation between them. _RMSE_ is the square root of the mean of the square of all of the errors. It gives a relatively high weight to large errors and grows as the frequency of large errors increases. Both metrics can range from 0 to infinity and lower values are preferred. The _MAE_ and _RMSE_ are given in Equations (10) and (11), respectively. \[R^{2}=1-\frac{\sum_{1}^{n}(y_{i}-\hat{y}_{i})^{2}}{\sum_{1}^{n}(y_{i}-\overline {y}_{i})^{2}} \tag{8}\]  
\[R^{2}_{adj}=1-\frac{\big{(}1-R^{2}\big{)}(n-1)}{n-k-1} \tag{9}\]  
\[MAE=\frac{1}{N}\ \sum_{1}^{n}\lvert y_{i}-\hat{y}_{i}\rvert \tag{10}\]  
\[RMSE=\sqrt{\frac{\sum_{1}^{n}(y_{i}-\hat{y}_{i})^{2}}{n}} \tag{11}\]  
Lastly, Sobol analysis [51], a variance-based sensitivity analysis, was conducted for each developed proxy model to determine the significance of input parameters toward each model. This method aims to quantify the amount of variance that each input parameter contributes to the variance of output. It identifies the total effect of an input parameter, which can be separated into main effect and interaction effect [52]. The main effect is the contribution to model output variance based on the variation of the input parameter itself and the interaction effect is the contribution of particular combinations of model input. The sum of both main and interaction effects is called the total effect. This analysis is conducted for the model that has the better performance after evaluating the fitting metrics between the polynomial regression and the neural network model.