## 5 Thermal Damage
### Discussions  
Since P-wave velocity and tensile strength showed a good correlation to the TD\({}_{\text{in}}\), these two parameters could be solely used to evaluate the thermal damage. However, granite has different physical, mechanical, and mineralogical properties. It is known that these factors impact the thermal damage process. Therefore, additional parameters need to be included in the TD model for a more comprehensive assessment. As a result, ten parameters were considered to determine TD empirically. Equation (2) indicates that TD\({}_{\text{emp}}\) is directly proportional to the temperature, mineral content, and porosity of granite specimens, while higher UCS and tensile strength have an opposing effect. Hence, the proposed empirical equation contains not only physico-mechanical parameters but mineral content as well. The correlation between the TD\({}_{\text{in}}\) and TD\({}_{\text{emp}}\) is provided in Figure 4.  
Since a fair agreement of R = 0.77 was reached, machine learning algorithms were implemented to enhance the results. As a result, among the eight proposed models, the Ensemble Bagged Trees model (Figure 5a) and the Regression Tree (Fine Tree) model (Figure 5b) were considered the best prediction models and outperformed the Gaussian Process Regression model (Figure 5c). The reasons behind this better performance are (1) they are more robust to noise and (2) they make fewer assumptions about the data distribution. They could do even better if the dataset implemented were larger [44; 45]. The order of importance of the ten input parameters to evaluate TD in predictive models was assessed, and the results are presented in Figure 6. The three most important parameters were temperature, tensile strength, and porosity. Hence, predictive models with these data can perform well. However, this process indicated that mineralogical properties and UCS exhibited negative weights, being influenced by TD rather than causing it. Therefore, Equation (2) was updated to Equation (3), and Figure 7 was provided, where the updated correlation between the actual and predicted thermal damage is shown. Moreover, Equation (4) was also proposed, where the porosity term is eliminated. Finally, Figure 8 shows the new correlation between the actual and predicted thermal damage considering temperature, UCS, and tensile strength only. To sum up, three new equations were proposed to determine TD in granites in this study. The main difference is that the new equations do not contain the P-wave velocity parameter as in Equation (1). Figure 3 shows the good correlation between the predicted  
Figure 8: The updated correlation between the actual and predicted thermal damage considering temperature, UCS, and tensile strength only. thermal damage using Equation (1) and tensile strength. However, tensile strength data in the graph are clustered into 4-5 groups (not continuous); therefore, it alone cannot be used to determine the TD of rocks. Figures 4, 7 and 8 show the correlation between the actual and predicted thermal damage based on Equations (2)-(4), respectively. Based on the order of importance of the TD input parameters shown in Figure 6, Equation (3) is better suited to determine the TD. The novelty of this work is the ability to determine the TD at high temperatures by using temperature, porosity, UCS, and tensile strength data, These findings can be very helpful in geothermal energy extraction, deep mining, and nuclear waste disposal applications where rock is exposed to very high temperatures. As limitations of this work, the samples in the published literature were heated at different heating rates and then cooled either in an oven or in the open air. The duration of the heating process ranged from half an hour to 24 h. Therefore, variations in the heating rate, heating time, and cooling method could affect the test outcomes. Also, the diversity of the collected data could be an issue. Since the experiments were conducted in different environments, the experimental results could be inconsistent.