## 4 Discussion  
Accurately predicting the equivalent circulating density (ECD) during tripping in/out and drilling operations is crucial in ensuring safe and cost-effective well drilling. There are several empirical and physics-based hydraulics models available in the literature [3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20]. However, the application of the models is limited to the considered assumptions and model controlling parameters. Therefore, it is common to practice calibrating the model with a measured dataset ([PERSON] et al., 2008) [23]. Recent research has focused on using data-driven modeling techniques applied in diverse fields including the petroleum industry. This study explored the performance of multivariable regression (MVR), the Group Method of Data Handling (GMDH), a standard Multilayered Perceptron (ANN), and the proposed GMDH-featured ANN model to predict ECD based on drilling parameters. Before implementing the machine learning algorithms, the field drilling dataset was preprocessed to make it clean and to select the appropriate features. The data used in this study included surface and downhole parameters. Appropriate Python libraries were employed for data preprocessing and feature selection. The first part of the study compared multivariable regression with the GMDH regression. The multivariable regression related two or more independent variables with the target variable. The nature of the regression was an independent linear combination of the variables. However, input features may have nonlinearly varied with the target parameter. Moreover, the input parameters may have had an interaction effect on the target parameters. To study these effects, the GMDH regression was considered and compared with the multivariable regression. The application of the GMDH network has been implemented in several fields [34; 35; 36; 37; 38; 39; 40; 41; 42; 43; 44]. The GMDH algorithm, as described in several references [34; 35; 46; 53], utilizes multiple inputs to identify the best combination and generates a quadratic polynomial. An external criterion determines the selection of this optimal combination of two features. Two input features were selected to compare the GMDH method (Equation (8) with MVR (Equation (1)). A total of 18 experiments were designed (Table 3 in Section 3). Figures 5-7 offer a comparison of the results obtained from the experiments. The results showed that all the GMDH models predicted a higher R\({}^{2}\) compared to the multivariable regression (MVR). This indicated that the nonlinear and interaction terms had a significant effect on the ECD prediction. The degree of the model accuracy performance depended on the correlation of the features with the target variable. The second part of this study involved comparing the performance of the standard ANN and the proposed GMDH-featured ANN. To ensure consistency in the comparison, both the proposed GMDH-featured ANN (Figure 3) and the fully connected ANN models  
Figure 13: (**a**) Experiment 5a: normal-featured ANN D3–D8 (R\({}^{2}\) = 0.788, MSE = 0.213). (**b**) Experiment 5b: GMDH-featured ANN D3–D8 (R\({}^{2}\) = 0.87, MSE = 0.121). (Figure 1) were developed, unlike the standard GMDH network where neurons are not fully connected (Figure 2b). The ANN used only two features, and the proposed GMDH-featured ANN had five inputs generated from the two selected features. Out of the eight experimental designs (Table 4), three of the designs showed that the proposed GMDH-featured ANN exhibited a higher model performance as compared with the ANN, whereas the remaining five designs were the same as shown in Table 5. This could have been due to the insignificant impact of the nonlinear and interacting terms. Table 6 displays the input features of the standard ANN and the modeling results obtained from the selected three experiments #1, #5, and #8.