# Laboratory Tests on Swelling Properties of Field-Coring Gypsum Rock in Tunnels
## 5 An S-Shaped Swelling-Time Model  
Based on the previous analysis of the test results, the swelling ratio and swelling pressure with the immersion time within 2880 min has two main stages, which can be seen in Figures 8-11. The first stage takes place during the first 120 min of the OA segment. This stage is the rapid-growth stage. The second stage takes place between the 120 th and the 2880 th minute of the AB segment and it is a slow-growth and stable stage.  
Around 120 min, at point A at the end of the first stage, the free-swelling ratio, the lateral restricted-swelling ratio and the lateral restricted-swelling pressure reach more than 70% of those after 2880 min. This explains the rapid growth of the OA segment. After expanding the OA segment, it can be seen from Figures 8b, 9b, 10b and 11b that the swelling ratio or swelling pressure has three main growth stages in the first 120 min. There are two obvious turning points in these three stages. They are point C, at about the 10 th minute, and point D, at about the 50 th minute. The first stage covers from minutes 0 to 10 of the OC segment. During this period, the swelling ratio grows slowly. During the second stage, from minutes 10 to 50 of the CD segment, the swelling ratio grows rapidly. The third stage, takes place from minutes 50 to 120 of the DA segment. During this period, the swelling ratio grows slowly and gradually flattens.  
Gypsum rock often takes a long time to fully swell. However, in the early stage of immersion, the swelling ratio and swelling pressure increase rapidly. In view of the \"swelling-time\" characteristics and \"S\"-shaped growth characteristics of gypsum rock in the early stage of water immersion, a model was proposed to describe the early swelling behavior of gypsum rock [22]. The relationship between the swelling ratio and the swelling pressure with the immersion time can be expressed by  
\[S(t)=A_{2}-\frac{A_{1}}{1+\left(\frac{t}{A_{3}}\right)^{A_{4}}}, \tag{6}\]  
Figure 11: Lateral restricted-swelling pressure. Variation in axial lateral restricted-swelling pressure with time for (**a**) 2880 min and (**b**) during the first 120 min.  
where \(t\) is the immersion time, \(S(t)\) can be the maximum radial-swelling ratio, axial free-swelling ratio, lateral restricted-swelling ratio and lateral restricted-swelling pressure, and \(A_{1}\), \(A_{2}\), \(A_{3}\) and \(A_{4}\) are the parameters.  
The S-shaped model was used to predict the swelling behavior of gypsum rock. Figure 12a-d were used to predict the maximum radial free-swelling ratio, axial free-swelling ratio, lateral restricted-swelling ratio and lateral restricted-swelling pressure, respectively. The fitting parameters are shown in detail in Table 6. It can be observed from Figure 12 and Table 6 that the S-shaped model can describe the swelling behavior of gypsum rock with a high accuracy. The correlation coefficients are as large as 0.99 in the present study.  
For gypsum rock composed of gypsum and other non-swelling minerals, the S-shape model can provide some reference for preventing early swelling disasters.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \(A_{1}\) & \(A_{2}\) & \(A_{3}\) & \(A_{4}\) & \(R^{2}\) \\ \hline Maximum radial free-swelling ratio \(V_{D_{f}}\) & 3.88 & 3.86 & 37.77 & 2.74 & 0.99 \\ Axial free-swelling ratio \(V_{H_{f}}\) & 2.54 & 2.53 & 34.96 & 2.05 & 0.99 \\ Lateral restricted-swelling ratio \(V_{H_{L}}\) & 9.42 & 9.43 & 35.69 & 2.12 & 0.99 \\ Lateral restricted-swelling pressure \(p\) & 4.85 & 4.85 & 35.74 & 2.17 & 0.99 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Parameters of S-model.  
Figure 12: Swelling-time model. (**a**) Maximum radial free-swelling ratio; (**b**) axial free-swelling ratio; (**c**) lateral restricted-swelling ratio; (**d**) lateral restricted-swelling pressure.