## 5 Discussion and Conclusions  
In order to reveal the frost-heave mechanism of void water behind the lining of high-speed railway tunnels in cold areas, we first innovatively proposed the definition of two types of void water, classified the discharge conditions of these two types of void water during the freezing process, and experimentally analyzed the effects of the different discharge conditions on the frost-heave force. Then, based on the short-period temperature fluctuations and vault void morphological conditions, the circular wedge-shaped and flying-saucer-shaped void models were proposed, and numerical simulation was used to analyze the freezing process and characteristics of the two types of void water. Then, we comprehensively considered the relative position of the void and the drainage channel, introduced the drainage coefficient, and proposed an analytical model for calculating the frost-heave force in circular wedge-shaped voids, so as to understand the formation mechanism of frost heaving due to void accumulation. Most existing research focuses on the effects of temperature, surrounding rock, and lining on frost-heave force. On this basis, this paper considers the influence of accumulated water discharge conditions, short-period temperature fluctuation conditions, and void formation on frost-heave force. In addition to discovering that the magnitude of the tunnel water frost-heave force is influenced by the elastic resistance coefficient of the surrounding rock, the elastic resistance coefficient of the lining, and the elastic resistance coefficient of the ice, we also found that blockage of the drainage channel leads to the generation of a large frost-heave force. The process of frost heaving, melting, water supply, and further frost heaving, caused by short-term temperature fluctuations, leads to the frost-heave force becoming greater and greater. The freezing and melting process of the accumulated water develops from the thinner area to the thicker area. The closer the drainage channel is to the relative position of the void bottom, the greater the drainage coefficient, the faster the drainage channel is frozen, and the greater the frost-heave force. Figure 15: Value curve of groundwater excretion coefficient. \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \(\beta_{0}/\zeta_{0}\) & 0 & 0.2 & 0.4 & 0.47 & 0.53 & 0.6 & 0.67 & 0.73 & 0.8 & 0.87 & 0.93 & 1 \\ \hline \(\beta\) & 1 & 0.85 & 0.69 & 0.63 & 0.57 & 0.51 & 0.44 & 0.37 & 0.3 & 0.21 & 0.12 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Elastic resistance coefficient of the surrounding rock at all levels. The above research helps to propose an early warning and prediction system based on temperature field fluctuations, water leakage in the tunnel, and tunnel voids. Due to the limitations of monitoring and measurement technology, it is impossible to accurately measure the frost-heave force and voids in the original tunnel. Therefore, it will be necessary to conduct further research on the relative positional relationship between voids and the excretory channel in the future. In addition, we can start from aspects such as the freezing situation of the drainage ditch, the degree of ice hanging on the lining surface, and the monitoring of void patterns to achieve early warning and prediction of macroscopic freezing damage for an entire tunnel. Conceptualization, Y.W. ; Methodology, W.D. ; Software, W.D. ; Validation, Y.W. ; Formal analysis, W.D. ; Investigation, W.D. and Y.W. ; Data curation, W.D. ; Writing--original draft, W.D. ; Writing--review & editing, W.D. ; Visualization, K.H. ; Supervision, Y.W. ; Project administration, P.X. ; Funding acquisition, P.X. All authors have read and agreed to the published version of the manuscript. This research was funded by [National Key Research and Development Program of China] grant number [2022 YFB2603301] and [National Natural Science Foundation of China] grant number [51478473]. Not applicable.