2.1 Analysis of temperature differences around drainage structures under different insulation conditions
After installing insulation measures in the tunnel, the surrounding rock around the drainage structure will be improved to varying degrees compared to non insulation conditions.
Taking the cross-section at a depth of 1/60 of the tunnel as an example, the temperature difference at each buried depth under different insulation conditions are shown in Table 1.
Cross section position
Burial depth
|
15 cm on the left side
|
10 cm on the left side
|
5 cm on the left side
|
Centerline position
|
5 cm on the right side
|
10 cm on the right side
|
15 cm on the right side
|
5/13 a
|
0.1
|
0.7
|
0.5
|
0.4
|
0.4
|
0.7
|
0.2
|
10/13 a
|
0.3
|
0.8
|
0.6
|
0.6
|
0.3
|
0.6
|
0.2
|
16/13 a
|
0.2
|
0.6
|
0.5
|
0.6
|
0.4
|
0.5
|
0.3
|
21/13 a
|
0.2
|
0.7
|
0.6
|
0.5
|
0.4
|
0.9
|
0.3
|
28/13 a
|
|
|
0.4
|
0.4
|
0.5
|
|
|
33/13 a
|
|
|
0.3
|
0.4
|
0.4
|
|
|
38/13 a
|
|
|
0.3
|
0.5
|
0.4
|
|
|
Table 1. The temperature difference at each position of the inverted arch under insulation and no insulation conditions.
Notes: a: Burial depth of central drainage ditch.
Based on the data presented in Tables 1–3 and Fig. 7 and Fig. 8, it is evident that different insulation measures implemented in the tunnel lead to varying degrees of increase in temperature values at each location compared to the conditions without insulation. At a depth of 5/13–21/13 times the burial depth of the central drainage ditch, the temperature difference at each measuring point is relatively small at 15 cm on both sides of the centerline, while the temperature difference at other positions is slightly larger compared to each other. At the position of 5/13 − 10/13 times the central drainage ditch, the temperature difference between each measuring point with and without insulation on the inverted arch is about 0.2 ℃-0.8 ℃. The temperature difference between each measuring point when the inverted arch is insulated and the central drainage ditch is insulated and without insulation is about 0.2 ℃-1.1 ℃. The temperature difference between the insulation of the inverted arch, the insulation of the central drainage ditch, and the insulation of the anti-cold drainage tunnel is about 0.3 ℃-1.1 ℃ at each measuring point when there is no insulation. At the location of 16/13–21/13 times the burial depth of the central drainage ditch, without insulation, the temperature value is about 0.3 ℃-0.9 ℃ lower than that of the inverted arch insulation, about 0.5 ℃-0.9 ℃ lower than that of the inverted arch insulation and central drainage ditch insulation, and about 0.4 ℃-1.3 ℃ lower than that of the inverted arch insulation, central drainage ditch insulation and cold proof drainage tunnel insulation. At the location of 28/13–38/13 times the burial depth of the central drainage ditch, the temperature differences under the three insulation conditions compared to those without insulation are about 0.3 ℃-0.5 ℃, 0.3 ℃-0.6 ℃, and 0.4 ℃-0.8 ℃, respectively.
Table 2
Temperature difference at different positions of inverted arch and central drainage ditch under insulation and no insulation conditions.
Cross section position Burial depth | 15 cm on the left side | 10 cm on the left side | 5 cm on the left side | Centerline position | 5 cm on the right side | 10 cm on the right side | 15 cm on the right side |
5/13 a | 0.2 | 0.8 | 0.7 | 0.8 | 0.7 | 0.8 | 0.2 |
10/13 a | 0.6 | 1.1 | 0.9 | 0.9 | 0.8 | 0.8 | 0.6 |
16/13 a | 0.7 | 0.9 | 0.8 | 0.8 | 0.8 | 0.8 | 0.6 |
21/13 a | 0.5 | 0.9 | 0.7 | 0.7 | 0.8 | 0.9 | 0.6 |
28/13 a | | | 0.6 | 0.4 | 0.6 | | |
33/13 a | | | 0.4 | 0.4 | 0.4 | | |
38/13 a | | | 0.3 | 0.4 | 0.5 | | |
Notes: a: Burial depth of central drainage ditch. |
Table 3
Temperature differences at various positions of tunnel inverts, central drainage ditches, and cold relief tunnels under insulation and no insulation conditions.
Cross section position Burial depth | 15 cm on the left side | 10 cm on the left side | 5 cm on the left side | Centerline position | 5 cm on the right side | 10 cm on the right side |
5/13 a | 0.2 | 0.5 | 0.6 | 0.6 | 0.5 | 0.7 |
10/13 a | 0.5 | 1.1 | 0.7 | 0.8 | 0.7 | 0.8 |
16/13 a | 0.4 | 0.9 | 0.7 | 0.8 | 0.9 | 0.9 |
21/13 a | 0.8 | 1.3 | 0.8 | 0.9 | 1.1 | 0.8 |
28/13 a | | | 0.8 | 0.6 | 0.8 | |
33/13 a | | | 0.6 | 0.8 | 0.7 | |
38/13 a | | | 0.4 | 0.7 | 0.6 | |
Notes: a: Burial depth of central drainage ditch. |
After conducting a comparative analysis, it is notable that the temperature difference between inverted arch insulation and central drainage ditch insulation is relatively significant at positions 16/13–21/13 times the burial depth of the central drainage ditch, ranging from 0.3 ℃-0.4 ℃. However, the temperature difference at other burial depths is relatively small. When implementing both inverted arch insulation and central drainage ditch insulation, the temperature difference around the anti-cold drainage tunnel is relatively large, reaching 0.3 ℃-0.6 ℃, compared to implementing only inverted arch insulation or central drainage ditch insulation with anti-cold drainage tunnel insulation. The temperature difference at other buried depths is relatively small.
After comprehensively comparing the experimental results under the three different insulation conditions, it was found that all three schemes can achieve certain insulation and anti-freeze effects, but there are slight differences in their insulation effects. Regarding the insulation effect of the surrounding rocks around the central drainage ditch, the latter two insulation measures have a more prominent insulation effect than only the inverted arch insulation. However, the three insulation measures cannot effectively solve the problem of freezing damage in the central drainage ditch. Therefore, it may be necessary to consider thickening the insulation layer of the central drainage ditch or increasing its burial depth and implementing insulation measures. Alternatively, the central drainage ditch could be eliminated, and instead, a cold-proof drainage hole could be set directly below the freezing depth with an insulation outlet.
2.2 Tunnel drainage and flow velocity characteristics
To measure the characteristics of tunnel drainage and flow velocity, a plastic cylindrical cylinder with a diameter of d = 10 cm and a height of h = 65 cm was utilized to collect the seepage water discharged from the tunnel via both the central drainage ditch and the anti-cold drainage tunnel. By measuring the time required to collect the same volume of water flow, the flow velocity characteristics were determined.
The drainage test results of the central drainage ditch and the anti-cold drainage tunnel without insulation in the tunnel are presented in Tables 4 and 5. Based on the test results, the relationship curve between water flow rate and freezing time can be depicted, as illustrated in Fig. 9.
Table 4
Drainage test results of the central drainage ditch.
Freezing time/ h | 0 | 6 | 14 | 20 | 33 | 40 | 46 | 53 | 65 | 78 |
Amount of water /cm3 | 510 | 510 | 510 | 510 | 510 | 510 | 510 | 510 | 510 | 510 |
Time /s | 142 | 253 | 395 | 562 | 708 | 859 | 1021 | 1184 | 1329 | 1573 |
Discharge of water /cm3·s− 1 | 3.60 | 2.02 | 1.29 | 0.91 | 0.72 | 0.59 | 0.50 | 0.43 | 0.38 | 0.32 |
Table 5
Drainage test results of cold proof drainage tunnel.
Freezing time/ h | 0 | 8 | 18 | 24 | 35 | 48 | 59 | 72 |
Amount of water /cm3 | 510 | 510 | 510 | 510 | 510 | 510 | 510 | 510 |
Time /s | 121 | 215 | 337 | 449 | 573 | 718 | 824 | 912 |
Discharge of water /cm3·s− 1 | 4.21 | 2.37 | 1.51 | 1.14 | 0.89 | 0.71 | 0.62 | 0.56 |
From Fig. 9, it is apparent that during the early stage of freezing, the water flow velocity in both the central drainage ditch and anti-cold drainage tunnel is relatively high, with a water flow rate of 3.60 cm3·s-1 in the central drainage ditch. However, as the freezing time increases, the water flow velocity gradually decreases. Within the freezing time range of 0–20 hours, the water flow velocity decreases faster, and the reduction amplitude is significant. Once the freezing time exceeds 20 hours, the rate and amplitude of the decrease in water flow rate with the increase of freezing time are relatively small. At a freezing time of 78 hours, the water flow rate decreases to 0.32 cm3·s-1. The initial water flow rate of the anti-cold drainage tunnel is 4.21 cm3·s-1. A significant decrease in water flow rate occurs within the freezing time range of 0–24 hours. Subsequently, as the freezing time continues to increase, the change in water flow rate becomes relatively small and tends to be gentle. At a freezing time of 72 hours, the water flow rate is 0.56 cm3·s-1.
The drainage test results of the central drainage ditch and anti-cold drainage tunnel under insulation conditions for the inverted arch of the tunnel are shown in Tables 6 and 7.
Table 6
Drainage test results of the central drainage ditch.
Freezing time/ h | 0 | 6 | 14 | 20 | 33 | 40 | 46 | 53 |
Amount of water /cm3 | 510 | 510 | 510 | 510 | 510 | 510 | 510 | 510 |
Time /s | 182 | 328 | 479 | 615 | 723 | 846 | 987 | 1103 |
Discharge of water /cm3·s− 1 | 2.80 | 1.55 | 1.06 | 0.83 | 0.71 | 0.60 | 0.52 | 0.46 |
Table 7
Drainage test results of cold proof drainage tunnel.
Freezing time/ h | 0 | 8 | 18 | 24 | 35 | 48 |
Amount of water /cm3 | 510 | 510 | 510 | 510 | 510 | 510 |
Time /s | 110 | 215 | 342 | 457 | 571 | 704 |
Discharge of water /cm3·s− 1 | 4.64 | 2.37 | 1.49 | 1.12 | 0.89 | 0.72 |
After insulation is installed on the inverted arch of the tunnel and the central drainage ditch, record the drainage results of the same volume of water collected by the central drainage ditch as the freezing time increases, as shown in Table 8; The test results of installing insulation in the cold proof drainage tunnel are shown in Table 9.
Table 8
Drainage test results of the central drainage ditch.
Freezing time/ h | 0 | 6 | 14 | 20 | 33 | 40 | 46 | 53 |
Amount of water /cm3 | 510 | 510 | 510 | 510 | 510 | 510 | 510 | 510 |
Time /s | 166 | 266 | 346 | 440 | 536 | 601 | 674 | 735 |
Discharge of water /cm3·s− 1 | 3.0 | 1.92 | 1.47 | 1.16 | 0.95 | 0.85 | 0.76 | 0.69 |
Table 9
Drainage test results of cold proof drainage tunnel.
Freezing time/ h | 0 | 8 | 18 | 24 | 35 | 48 |
Amount of water /cm3 | 510 | 510 | 510 | 510 | 510 | 510 |
Time /s | 106 | 197 | 335 | 401 | 457 | 486 |
Discharge of water /cm3·s− 1 | 4.81 | 2.59 | 1.52 | 1.27 | 1.12 | 1.05 |
According to Tables 6–9, the relationship curve between the outlet flow rate and freezing time is plotted based on the drainage test results of the central drainage ditch and the drainage test results of the anti-cold drainage tunnel, as shown in Fig. 10.
From Fig. 10, it can be observed that during the early stage of freezing, under different insulation conditions, the water flow velocity of both the central drainage ditch and anti-cold drainage tunnel is relatively high. When the inverted arch is insulated, the water flow rate of the central drainage ditch is 2.80 cm3/s. Whereas, when both the inverted arch and central drainage ditch are insulated, the water flow rate of the central drainage ditch is 3.07 cm3/s. As the freezing time increases, the water flow rate under both insulation conditions gradually decreases. Within the freezing time range of 0–20 hours, the water flow rate decreases rapidly, followed by a gradual slowdown. At a freezing time of 53 hours, the water flow rate under the inverted arch insulation and inverted arch and central drainage ditch insulation conditions is 0.46 cm3/s and 0.69 cm3/s, respectively. The initial water flow rates of the inverted arch, the central drainage ditch, and the anti-cold drainage tunnel during insulation are 4.64 cm3/s and 4.81 cm3/s, respectively. As the freezing time increases, the water flow rate gradually decreases. Within the freezing time range of 0–20 hours, the water flow rate decreases significantly, followed by a smaller change and gradual flattening. When the freezing time reaches 48 hours, the water flow rates are 0.72 cm3/s and 1.05 cm3/s, respectively.
Based on the data in Tables 4 to 9, when there is no insulation in the tunnel, the water flow rates of the central drainage ditch and cold water release tunnel are 0.32 cm3/s and 0.56 cm3/s, respectively. However, when insulation is installed on the inverted arch, the water flow rate of the central drainage ditch and the anti-cold water outlet tunnel increases to varying degrees compared to when there is no insulation, with differences of 0.14 cm3/s and 0.16 cm3/s, respectively. This indicates that the insulation measures installed on the inverted arch have a certain insulation and anti-freezing effect on the drainage structure. When both the inverted arch and drainage structure are equipped with insulation, the water flow rates of the central drainage ditch and the cold proof drainage tunnel are 0.69 cm3/s and 1.05 cm3/s, respectively, which are 0.37 cm3/s and 0.49 cm3/s higher than when there is no insulation. The difference in insulation conditions compared to the inverted arch is 0.23 cm3/s and 0.33 cm3/s. Through comparative analysis of water flow differences, it can be seen that setting insulation at the same time for the inverted arch and drainage structure can achieve good insulation and antifreeze effects.
2.3 Prediction of on-site freezing depth based on model test results
2.3.1 Model Test Results
By conducting model tests to measure the freezing depth of the overlying mountain in the tunnel under the influence of environmental temperature and the freezing depth of the surrounding rock below the tunnel arch, the correlation between the model test and the tunnel site is established to provide support for the layout of drainage structures. The schematic diagram of the freezing depth of the overlying mountain and lower surrounding rock of the tunnel is shown in Fig. 11.
The model test results are sorted out and analyzed to obtain the ground temperature distribution characteristic curve at the portal or outside the tunnel, the ambient temperature change curve in the model test tunnel, the temperature distribution characteristics of the surrounding rock under the model test, and the freezing depth change curve under the tunnel invert in the model test.
1. Model test ground temperature test results
Analyze the ground temperature at the measuring point 5 cm outside the tunnel entrance, and obtain the variation curve of the temperature of the strata outside the tunnel with depth, as shown in Fig. 12.
Through model experiments, the ground temperature change curve at 5 cm outside the tunnel entrance was tested and fitted. It was found that the ground temperature change curve conforms to a quadratic polynomial T = Ax2 + Bx + C. The parameters of the fitted curve are presented in Fig. 12. The variance R2 of the fitting equation, as depicted in Fig. 12 is 0.996. This high value suggests that the quadratic polynomial can accurately represent the variation pattern of ground temperature outside the tunnel. Further analysis of the curve reveals that as depth increases, the temperature gradually rises with a substantial increase near the ground. However, as depth continues to increase, the temperature variation decreases gradually.
2. Test results of environmental temperature inside the model test tunnel
In the model experiment, the air temperature inside the tunnel was tested to obtain the air temperature values at different depths inside the tunnel, and the variation curve of the air temperature inside the tunnel with the depth of the tunnel was plotted, as shown in Fig. 13.
According to Fig. 13, the air temperature inside the tunnel measured in the model test was fitted, and the fitting results showed that the variation of the air temperature inside the tunnel with the depth of the tunnel showed a pattern of T = Ax2 + Bx + C (T is the air temperature inside the tunnel, x is the depth of the tunnel).
3. Temperature test results of the surrounding rock (at the location of the drainage structure under the inverted arch of the model test).
The temperature test results of the surrounding rock in the lower part of the tunnel are shown in Fig. 14.
The temperature distribution at the top of the central drainage ditch in the tunnel shows that within a depth range of about 0–12 cm, the temperature value is less than 0℃. However, for all other positions, the temperature values are positive. Along the axial direction of the tunnel, it can be observed that as the tunnel depth increases, the temperature at the top of the central drainage ditch gradually increases.
4. The relationship between the freezing depth of the model test and the environmental temperature inside the model test tunnel and the temperature of the surrounding rock under the inverted arch
In the model test, the relationship curve between the freezing depth of the surrounding rock below the inverted arch and the air temperature inside the tunnel is shown in Fig. 15 (a), and the relationship curve between the freezing depth and the temperature of the surrounding rock below the inverted arch (top of the central drainage ditch) is shown in Fig. 15 (b). The relationship curve between the freezing depth of the surrounding rock in the lower part of the tunnel and the depth of the tunnel in the model test is shown in Fig. 15 (c).
Fit the relationship curves between the freezing depth of the surrounding rock below the inverted arch and the air temperature inside the tunnel, as well as the temperature of the surrounding rock (at the top of the central drainage ditch). The relationship curve between freezing depth and air temperature inside the tunnel presents a quadratic polynomial form, and the freezing depth decreases in a parabolic form as the air temperature increases during testing. The fitting equation is y = AT2 + BT + C, and the fitting parameters are shown in Table 10.
Table 10
Table of fitting parameters between the freezing depth of tunnel surrounding rock and the air and surrounding rock temperature inside the tunnel.
Parameter name | A | B | C | R2 |
The relationship between freezing depth and air inside the tunnel | -0.04026 | -2.56736 | 1.36619 | 0.99737 |
The relationship between freezing depth and lower surrounding rock | -0.15991 | -2.30957 | 14.27015 | 0.99675 |
According to the fitting results, the fitting variances of the relationship curve between the freezing depth of the surrounding rock at the lower part of the tunnel arch and the air temperature inside the tunnel, as well as the relationship curve with the temperature of the surrounding rock at the lower part, are 0.99737 and 0.99675, respectively, indicating a high fitting correlation.
5. Relationship between the ambient temperature inside the model test tunnel and the temperature of the surrounding rock under the inverted arch
Through model experiments, the temperature changes of the air inside the tunnel along the axial direction and the temperature changes of the surrounding rock (at the top of the central drainage ditch) in the lower part of the tunnel along the axial direction were tested, and the relationship curves between the two were established, as shown in Fig. 16. The fitting equation is T=-0.02807x2 + 0.61524x + 4.21509 (R2 = 0.99787).
2.3.2 Analysis of on-site test results
Figure 17 shows the variation curve of air temperature inside the tunnel along the depth of entry.
According to Fig. 17, the temperature distribution fitting equation is T = 3.68×10-6x2+0.012x + 14.122, the air temperature values in the tunnel at different depths can be obtained. The parameters of the fitting equation are shown in Figure, and the calculated temperature results are shown in Table 11.
Table 11
Calculate the air temperature values inside different tunnel depths based on the measured temperature curve.
Cross section position Burial depth | 15 cm on the left side | 10 cm on the left side | 5 cm on the left side | Centerline position | 5 cm on the right side | 10 cm on the right side | 15 cm on the right side |
5/13 a | 0.1 | 0.7 | 0.5 | 0.4 | 0.4 | 0.7 | 0.2 |
10/13 a | 0.3 | 0.8 | 0.6 | 0.6 | 0.3 | 0.6 | 0.2 |
16/13 a | 0.2 | 0.6 | 0.5 | 0.6 | 0.4 | 0.5 | 0.3 |
21/13 a | 0.2 | 0.7 | 0.6 | 0.5 | 0.4 | 0.9 | 0.3 |
28/13 a | | | 0.4 | 0.4 | 0.5 | | |
33/13 a | | | 0.3 | 0.4 | 0.4 | | |
38/13 a | | | 0.3 | 0.5 | 0.4 | | |
The relationship equation between the air temperature in the model test tunnel and the freezing depth of the surrounding rock in the lower part of the tunnel is y=-0.040267T2-2.56736T + 1.36619. Due to the consistency and similarity between the relationship curve between the freezing depth of the surrounding rock in the lower part of the tunnel and the air temperature in the tunnel obtained from the model test and the curve between the freezing depth of the surrounding rock in the lower part of the tunnel and the air temperature in the tunnel, The temperature equation T=(-3.51057E-6)x2 + 0.01202x-14.12215 was measured at different depths inside the Hongtushan Tunnel on site. Combined with the relationship equation between the freezing depth and the temperature inside the tunnel obtained from model tests, the freezing depth curve of the Hongtushan Tunnel was obtained, as shown in Fig. 18 and Fig. 19.
$${y_1}=m{y_2}{\text{+}}n$$
In the formula: y1 is the on-site freezing depth; y2 represents the predicted similarity ratio of model experiments; m. n is the correction coefficient.
Through calculation, the correction coefficients m and n are taken as 0.72367 and 0.03726, respectively. The Pearson correlation analysis was performed using SPSS software to infer the revised model data and on-site data, and the correlation coefficient between the two sets of data was 0.904, greater than 0.85, indicating a high correlation. The model test can accurately reflect the freezing depth of the tunnel after correcting the frozen depth extrapolated to the site.