## 5 Results and Discussion  
### Results  
In this section, we compare and analyze the performances of a single model and hybrid model on the Well X data from the test set. We verified that the hybrid model has better accuracy, robustness, and generalization ability, and we finally selected and evaluated the best hybrid model for ROP prediction.  
The model performances of two empirical models are shown in Figure 8, and the model evaluation indicators are shown in Table 3.  
Observing the prediction results, as shown in Figure 8, reveals that the predicted curves of the Motahhari model fit the original data better than the Hareland model. As shown in Table 3, the three evaluation indicators of the Motahhari model are much better than those of the Hareland model. Generally speaking, the performance of the Motahhari model is better than that of the Hareland model.  
In this paper, the main function of the empirical model is to make up for the loss of accuracy of the machine learning model in the case of low data similarity, and to improve the overall robustness and generalization ability. In this process, the empirical model is used to deal with data with low data similarity, which cannot be handled stably and accurately by the machine learning model. Therefore, before building the hybrid model, we need to choose the better empirical model to ensure the performance of the hybrid model. After the comparison, we chose the Motahhari (Mota) model with a better model performance as the representative empirical model, and we combined it with three machine learning models (SVM, ANN, RF) to establish three hybrid models: Mota-SVM, Mota-ANN, and Mota-RF.  
The depth of the test well X ranges from 2330 m to 3055 m. In this continuous well section, the drill bit is a PDC drill bit named KM1662.  
The model performances are shown in Figures 9-11.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Evaluation Indicators** & **Hareland** & **Motahhari** \\ \hline RMSE & 7.25 & 5.59 \\ MRE & 0.40 & 0.25 \\ PEV & 4.73 & 3.02 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Comparison of [PERSON]’s and [PERSON]’s model evaluation indicators.  
Figure 8: Comparison results of real and predicted ROP: (**a**) prediction results of Hareland model; (**b**) prediction results of Motahhari model.  
Figure 10: ROP prediction results of Mota-SVM hybrid model and single models: (**a**) ROP prediction results of SVM model; (**b**) ROP prediction results of Motahhari model; (**c**) ROP prediction results of Mota-SVM hybrid model.  
Figure 9: ROP prediction results of Mota-ANN hybrid model and single models: (**a**) ROP prediction results of ANN model; (**b**) ROP prediction results of Motahhari model; (**c**) ROP prediction results of Mota-ANN hybrid model.  
As shown in Figures 9-11, the prediction curves of the three hybrid models are closer to the real data curve than the single models. Specifically, the hybrid model can better predict with the help of an empirical model when the data similarity of the input data is low compared to the machine learning model. The hybrid model compensates for the shortcomings of the machine learning model with the help of the empirical model.