## 6 Conclusions  
In this paper, based on the high-temperature and water-cooling experiments and some other experimental data collected from the references, the variation behaviors of cohesion, the internal friction angle and tensile strength have been systematically studied, and the failure criterion considering the effect of heating and cooling treatment have been proposed. Some conclusions can be drawn as follows:1. It is found that there are two types of variation behaviors of cohesion and the internal friction angle for the granite samples after heating and cooling treatment.
2. Unified empirical equations have been provided to describe the different types of variation behaviors of cohesion and the internal friction angle for the granite samples after heating and cooling treatment. The sensitivity analyses show that the fitting coefficients have quite clear physical meanings.
3. The tensile strength decreases in a similar way for the different granite samples with the increasing treatment temperature. A linear empirical equation has been given to describe this behavior.
4. A modified Mohr-Coulomb failure criterion with a \"tension cut-off\" has been established for the granite samples considering the effects of high-temperature and water-cooling treatment. The criterion has been validated by well describing the different types of granite behaviors after heating and cooling treatment.
5. A discussion has been supplied on the mechanism of the different granite behaviors. It should be closely related to the damage and cracking process during the heating and cooling treatment on the granite samples.  
The variation behaviors of cohesion, the internal friction angle, and tensile strength should be helpful for understanding the mechanical behavior of hot dry rock during the hydraulic fracturing in geothermal production. As a main rock type for the hot dry rock reservoirs, the mechanical properties of granite under high temperature will be affected by the injected water, which may have significant influences on the fracturing characteristics. The proposed failure criterion can be applied for numerical modelling on the reservoir. In addition, the strength parameters and failure criterion of granite after high-temperature and water-cooling treatment in this paper should also be helpful for understanding wellbore stability problems.  
In the future studies, more systematic experiments should be carried out for a better understanding on the mechanism of the different behaviors, and more experimental data are still required for improving the failure criterion.  
Conceptualization, C.C.; methodology, C.C.; software, Y.X., J.Y. and P.C.; validation, J.Y.; formal analysis, C.C., J.Y. and Y.X.; investigation, C.C., Y.X., J.Y. and P.C.; resources, C.C., J.Y. and Y.X.; data curation, J.Y. and Y.X.; writing--original draft preparation, J.Y. and Y.X.; writing--review and editing, C.C.; supervision, C.C.; funding acquisition, C.C. All authors have read and agreed to the published version of the manuscript.  
This research is funded by the Project for technical research (Z083612025097).  
Not applicable.  
The data presented in this study are available on request from the corresponding author.  
[PERSON] and [PERSON] are appreciated here for their help and support in the laboratory experimental studies.  
The authors declare no conflicts of interest.