# A Fractional-Order Creep Model of Water-Immersed Coal
## 5 Conclusions  
The long-term stability of coal pillar dams is an essential issue for the safety operation of underground reservoirs. In order to develop a proper creep model for coal rock considering both the stress level and water-immersion time, this paper implemented creep experiments and analyzed the creep properties of water-immersed coal, then a fractional-order creep model was developed considering the immersion time and discussed based on the experiment data. The main conclusions are as follows:  
1. Water-immersion time has a significant influence on the elastoplastic performance of coal, exhibiting the longer water-immersion time, the lower elastic modulus and strength, and the degradation of the elastic modulus and strength, which is nonlinear to the increment of the water-immersion time.
2. Both the stress level and water-immersion time reduce the elastic viscosity and plastic viscosity of coal with a slightly nonlinear relationship, thereby increasing its creep rate. Such a promoting effect is limited when the water-immersion time reaches to a certain period.
3. A fractional-order derivative-based creep model was constructed to meet the deformation law of coal under the conditions of the variable stress level and water-immersion time. A verification of the creep model associated with the creep experiment suggests good agreement with the experiment curves, the stress levels, and the water-immersion time, showing the controlling role of the creep deformation of coal. It can well reflect the creep characteristics of the water-immersed coal at each stage of the stress levels.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline
**Parameters** & E\({}^{*}\) & \(a\) & \(b\) & \(\eta_{1}^{*}\) & \(k_{1}\) & \(k_{2}\) \\
**Value** & 5.015 GPa & 0.3126 & 11.18947 & 0.0005946 & 0.00978 & 0.00419 \\ \hline
**Parameters** & \(\eta_{2}^{*}\) & \(m_{1}\) & \(m_{2}\) & \(\sigma_{s}^{0}\) & \(c\) & \(d\) \\
**Value** & 0.000604637 & 0.01953 & 0.0783 & 16.56 & 0.1154 & 8.604 \\ \hline
**Fitting** & R\({}_{B}^{2}\) & R\({}_{C}^{2}\) & R\({}_{D}^{2}\) & & & \\
**Value** & 0.85 & 0.86 & 0.83 & & & \\ \hline \end{tabular}
\end{table}
Table 1: Table of fitted parameters.  
Figure 12: Strain–time fitting curve for (**a**) specimen immersed for 7 days; (**b**) specimen immersed for 14 days; and (**c**) and specimen immersed for 30 days.  
**Author Contributions:** Conceptualization, G.L. and Q.W.; methodology, H.Y.; software, Z.L.; validation, G.L. and H.Y.; formal analysis, H.Y.; investigation, G.L.; resources, F.R. and Z.C.; data curation, F.R. and Z.C.; writing--original draft preparation, G.L.; writing--review and editing, H.Y.; visualization, Y.L.; funding acquisition, Z.L. and H.Y. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research was funded by the National Natural Science Foundation of China (Grant No.52004090) and Hebei Natural Science Foundation (Grant No.E2021508026).  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** The data supporting this study are included in the paper.  
**Conflicts of Interest:** Author [PERSON] was employed by the company China National Petroleum Corporation. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.