## 5 Results and Discussions
### Temperature Distribution within the Freezing Wall at Two Characteristic Time Points
On the contrary, as illustrated by Figures 11 and 12, except for that corresponding to the freezing wall with a thickness of 1 m, thaw settlement distribution along the X-axis has a similar size to that along the Y-axis, which represents that the fluctuation of the thaw settlement along the longitudinal direction of the tunnel is approximately as large as that along the transverse direction of the tunnel. Furthermore, Figure 11 conveys that the settlement is distributed uniformly among different directions. In addition, as presented by Figures 10 and 12, with the increase in the thickness of the freezing wall, the frost heave and the thaw settlement grow accordingly. In addition, to qualify the effects of the thickness of the freezing wall on the maximum ground surface displacements, an attempt was made to fit the simulation results into mathematical equations. It was found that the exponential equation (Equation (11)) can best represent the trends with respect to the thickness of the freezing wall, which consequently implies that the increase in the  
Figure 7: The temperature distribution within freezing walls of different thicknesses at the cross-section of the Y-axis at the starting point of the fast-freezing stage: (**a**) 1.0 m on the 8 th day; (**b**) 1.5 m on the 12 th day; (**c**) 2.0 m on the 16 th day; (**d**) 2.5 m on the 20 th day; (**e**) 3.0 m on the 25 th day. Figure 10: The displacement distribution curves of the frost heave of the ground surface (**a**) along the X-axis and (**b**) along the Y-axis. Figure 9: The displacement contours of the frost heave induced by freezing walls of different thicknesses: (**a**) 1.0 m; (**b**) 1.5 m; (**c**) 2.0 m; (**d**) 2.5 m; (**e**) 3.0 m.  
To evaluate the displacement distribution of frost heave and thaw settlement quantitatively, the absolute values of the displacement data of the frost heave and thaw settlement gained from the numerical simulation were fitted to the amplitude version of the Gaussian peak function in the form of Equation (12). The results are summarized in Table 5. As illustrated in Table 5, the parameter \(w\), governing the width of the distribution curve, is larger for the curves of the X-axis than those of the Y-axis, which indicates that the curves along the X-axis are wider than those along the X-axis. This may stem from the geometrical shape of the freezing wall, which is longer in the direction of the X-axis than the Y-axis. The longer side might give rise to a wider settlement, while the shorter side causes a narrower one. However, the difference between them is rather sharp in the curves of frost heave, yet relatively minor in those of thaw settlement, which is consistent with what is shown in Figures 9 and 10. It is also noteworthy that the values of the \(w\) of the thaw settlement curves are larger than those of the frost heave curves, which indicates that the thaw settle  
Figure 11: The displacement contours of the thaw settlement induced by freezing walls of different thicknesses: (**a**) 1.0 m; (**b**) 1.5 m; (**c**) 2.0 m; (**d**) 2.5 m; (**e**) 3.0 m.  
For the maximum absolute values of the displacement curve, which are indicated by \(y_{0}+A\), although the thaw settlement rate is slightly larger than the frost heave rate, those of the frost heave are larger than those corresponding to the thaw settlement. This may be attributed to the excavation of the cross passage and the installation of the liners. When freezing the surrounding soil of the cross passage, part of the soil within the cross passage area will also freeze. However, when excavating the cross passage, this part of the soil will be removed, which will reduce the volume of the soil which has the potential to settle, along with the restriction of the settlement imposed by the liners. These two factors might work together to lead to a smaller displacement of the thawing process.