Advances in Civil Engineering
Volume 2019 (2019), Article ID 6435060, 12 pages
https://doi.org/10.1155/2019/6435060

Research Article

Application of Analytical Solution to Steady-State Temperature Field by Double-Row-Pipe Freezing and Verification with Field Measurement: A Case Study of Connected Aisle

Zequn Hong¹ and Xiangdong Hu²

¹Department of Geotechnical Engineering, Tongji University, Shanghai 200092, China
²Key Laboratory of Geotechnical and Engineering of Ministry of Education, Tongji University, Shanghai 200092, China

Correspondence should be addressed to Zequn Hong; 1432310@tongji.edu.cn

Received 27 June 2019; Accepted 16 September 2019; Published 30 September 2019

Academic Editor: Castorina S. Vieira

Copyright © 2019 Zequn Hong and Xiangdong Hu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

In order to solve the problem of sealing water and bearing capacity of a connected aisle in an underwater shield tunnel, a double-circle horizontal freezing method was adopted for ground reinforcement in the connected aisle of Maliuzhou Tunnel, which is China’s first shield tunnel with superlarge diameter built in a composite stratum. This paper proposed a new double-row-pipe freezing model for the calculation of frozen wall thickness based on analytical solution to steady-state temperature field. Besides, field measurement and transient numerical studies of the active freezing period were also carried out to study the freeze-sealing effect. The results show that frozen wall thickness obtained by analytical solutions agrees well with numerical simulation results, which verifies the applicability of the newly proposed calculation method. Field analysis indicates that soil temperature gradually approaches a stable value which is far below the freezing point, and a reliable water-sealing curtain can be formed around the designed connected aisle. Maximum impact of soil excavation on the frozen wall is about 10°C, and reducing exposure time of excavation surface can effectively alleviate the weakening of frozen wall. To obtain comprehensive analysis for freezing wall thickness, a more reasonable arrangement of temperature-measuring holes is expected in future freezing engineering.

1. Introduction

With the development of urbanization, building boundary has gradually expanded to surrounding undeveloped periphery, and construction of subway has become a good strategy for many cities to improve the regional traffic environment. Due to the influence of ground roads, buildings, and underground pipelines, the use of traditional soil reinforcement methods is challenging in subway construction. As a mature technology for temporary support and foundation reinforcement in soft soil areas, the artificial ground freezing (AGF) method is increasingly used in urban underground space, such as tunnel engineering [1, 2], foundation pit engineering [3], departure and reception of shield [4].

It is generally known that groundwater has always been an intractable problem in underground works. For underwater tunnels, Qian Fang analyzed the distribution of seepage pressure and found the law of circumferential effective stress around underwater tunnel [5]. Fuming Huang studied the steady seepage into a circular tunnel by analytical calculations based on the conformal mapping method [6]. In terms of research on the AGF method, the first case of freezing construction for municipal engineering is south railway tunnel in Da-Re area of Beijing underground railway system [7], and it fills the technical gap of horizontal freezing in tunnel engineering and provides a new way to improve shallow excavation technology. Since then, more and more freezing schemes emerged in urban underground projects, especially in connected aisles [8–10]. The most up-to-date technique and theory in AGF field is the freeze-sealing pipe roof (FSPR) method, which was first used in the Gongbei tunnel of Hong Kong-Zhuhai-Macao Bridge in China [11, 12]. The FSPR method utilizes three different types of freezing pipes to overcome the problem of curve horizontal freezing under complex geological conditions, which is of great significance for construction of shallow-buried tunnels [13, 14].

Freezing effect is particularly important to ensure safety of soil excavation when it is applied to connected aisles of underwater tunnel because groundwater may be connected with rivers or seas. Any weak position of frozen curtain may cause a large amount of water to flow into the channel being excavated and thereby submerge the entire shield tunnel. Many studies have introduced engineering examples of the AGF method from Beijing, Shanghai, and Nanjing in China, mainly focusing on construction process and its impact on the environment [8, 15–18]. However, analysis of global characteristics of temperature field of frozen soil curtain before excavation is rarely reported in previous literatures. Ground freezing is a dynamic process that changes with time [19], and the overall development of temperature field often determines the success or failure in the construction of connected passage. It is necessary to grasp main influencing factors of frozen curtain properties and temperature distribution of frozen soil at different times before excavation.

For the AGF method used in the connected aisle of shield tunnel, two common arrangement forms of freezing pipes are single-side jacking and double-side jacking. The first form is often used in metro tunnels, which have a smaller shield diameter, and single-side arrangement can ensure that the frozen soil curtain is closed [20]. The second form is usually used for long-distance connected channels that often occur in large-diameter highway shield tunnels [21], in which case single-side horizontal jacking is difficult to guarantee the accuracy of the freezing pipe arrangement, and then double-side jacking is necessary in this situation.

Based on double-side jacking arrangement, this paper introduced the field measurement of ground freezing with two-circle pipes applied to the connected aisle of Maliuzhou Tunnel, which is the first shield tunnel with large diameter built in the composite stratum. By analyzing this typical freezing case, this paper aims to explore the optimal freezing scheme of double-circle freezing methods and optimal monitoring program of measuring holes in future projects.

2. Engineering Background

2.1. Project Profile

Zhuhai is an important node city in Guangdong-Hong Kong-Macao Greater Bay Area, Hengqin new District, and Macau across the river, where Hengqin Port has become an important gateway connecting the mainland and Macau. At present, there are two main routes from Hengqin Island to Zhuhai main city, namely, Hengqin Bridge and Hengqin Second Bridge. These existing two routes cross river in the form of bridges above the surface of water. In view of the impact of bad weather such as rainstorm and typhoons, to ensure that Hengqin island and Zhuhai urban area can maintain smooth traffic all day under extreme conditions, it is necessary to build a third external channel—Maliuzhou Tunnel.

Maliuzhou Tunnel is located in Nanwan District and Hengqin New District, the submarine tunnel spans the Maliuzhou waterway, and it is 2300 meters east of Hengqin Bridge and 3500 meters west of Hengqin Second Bridge. The total length of Maliuzhou Tunnel is 2200 meters, of which the shield section is about 1082 meters. This is China’s first superlarge-diameter shield tunnel built in composite strata with an external diameter of 14.5 meters, about 5 stories high. One connected aisle is set up along the whole tunnel for escape in emergencies, and it is constructed in the middle of the shield section and just below the Maliuzhou waterway, shown in Figure 1.

Figure 1: The alignment of Maliuzhou Tunnel.

This connected aisle is composed of two bell mouths which are connected to tunnel steel lining at both ends and an intermediate standard horizontal channel. The total length is about 15.2 m, and the internal radius is 1.4 m. From the height direction, elevation of the channel center is roughly at −20 meters. According to geological data, the strata from riverbed surface to the bottom of Maliuzhou Tunnel are ②1, silt; ②2, clay; ②4, silty clay; and ④, medium-coarse sand. The connected aisle section is circular in structure and designed to be excavated in the strata ②1, ②2, and ②4, shown in Figure 2.

Figure 2: Geological profile of the connected aisle.

Distance between the bottom of bell mouth and medium-coarse sand is 3.36 m. The upper part of the connected aisle is Maliuzhou waterway with a maximum water level of 3.306 meters and a reappearance period of 50 years. Considering the stratigraphic characteristics and engineering features, a design scheme adopts the artificial freezing method for soil reinforcement and the mining method for excavation to ensure construction safety and alleviate influence on the surrounding hydrogeological environment.

2.2. Freezing Construction Scheme

As this connected aisle is located in silt, clay, and medium-coarse sand strata, the horizontal freezing method is adopted to reinforce the ground layer, that is, by constructing horizontal freezing holes in soil outside the channel to be excavated between east and west line tunnels, the heat in soil is taken away by circulating low temperature brine in prejacked freezing pipes so that the lateral soil around aisle forms a frozen wall with high strength and good sealing property. Then, excavation construction of the channel is carried out under the protection of the frozen curtain.

A double-circle horizontal freezing scheme is used in this project. A total of 42 freezing pipes are arranged, of which 18 are in the inner circle with radius 2790 mm (N1∼N18), the length ranges from 15.8 m to 17.4 m. 24 freezing tubes are in the outer circle with radius 4015 mm (W1∼W24), and the length of these tubes ranges from 15.8 m to 18.8 m. All freezing pipes are made of low carbon seamless steel with a diameter of 108 mm and a thickness of 8 mm. Arrangement of freezing pipes in two circles is shown in Figure 3.

Figure 3: Layout of freezing tubes in inner and outer circles.

To reduce the influence of high-temperature airflow on frozen soil during freezing process, in addition to horizontal freezing tubes, curved surface freezing pipes are also attached to the inner walls of tunnel segments on both sides of the connecting aisle. The size of the curved surface freezing pipe is , smaller than that of the horizontal freezing pipe. These two kinds of freezing pipes share the same brine circulation system, which is arranged within the west tunnel, and supplementary brine of pipes in the east tunnel is carried out through four penetrating holes along the contact channel.

3. Calculating Method of Frozen Soil Thickness

In freezing engineering, the basic parameters of the criterion for judging the safety state of frozen curtain are the thickness and average temperature. As far as the thickness index concerned, two methods are commonly used in the practical calculation. In this section, we proposed a new model for calculating this index in Maliuzhou Tunnel based on applicability evaluation of existing methods.

3.1. Single-Pipe Freezing Model

Since it is usually difficult to derive analytical solutions for temperature field generated by interaction of multiple freezing pipes in practical projects, the approximate calculation method with the single-pipe model is often used. In the middle and late stages of the freezing period, development speed of frozen soil gradually slows down, and Trubak first obtained the analytical solution of single-pipe freezing model under the assumption that freezing process is regarded as a steady state [22]. The schematic diagram of this model is shown in Figure 4, where r₀ is the radius of the freezing pipe and ξ is the radius of the frozen curtain.

Figure 4: Schematic diagram of the single-pipe freezing model.

Analytical solution of the temperature field for the single-pipe freezing model obtained by Trubak can be expressed as follows [22]:

where T is the distribution of the temperature field, Tf is the temperature of the freezing pipe wall, and r is the polar radius from any one point to the origin.

In the process of on-site freezing monitoring, we cannot get the location of frozen soil boundary beforehand, so we must calculate it by monitoring data. According to the temperature value at preset measuring point M, assuming the coordinates of M is (x₀, y₀) and the measured temperature is T₀ and then substituting these variables in equation (1) to obtain

Equation (2) is a new method for calculating the thickness of single-pipe frozen soil. By arranging measuring points in soil, we can get lots of monitoring data at a particular time and then put coordinates and temperature of these points into this formula, and a series of ξ are obtained. Select the smallest one to be the control value of frozen wall thickness at that time. With this thickness as the radius, draw circles on each freezing pipe, and then the freezing wall can be judged whether it is closed or not by the superposition or separation of adjacent circles.

3.2. Speed of Freezing Development

In the process of freezing, estimation of freezing speed is also a common method for predicting the development of frozen soil curtain in practical engineering. Assuming the distance from a monitoring point to freezing pipe is l₀, the time when temperature of this point drops to 0°C is t₀. Then, the moving speed of frozen boundary can be approximated as

with equation (3), the thickness of the frozen wall at any subsequent moment t’ after the closure of the curtain can be predicted according to the following formula:

Equation (4) introduces a useful method to calculate thickness of the frozen wall after the temperature of the measuring point dropped to 0°C. Because of the simplicity of this formula, it is currently adopted in most practical freezing projects, although it has not been introduced in any public literature.

3.3. Double-Row Freezing Model

Although the above two methods are simple and easy to use, they have great error compared with field freezing process. Almost all freezing pipes used in engineering are not only one. Due to the superposition effect of temperature field, simple linear addition of the single-pipe freezing model cannot reflect the actual distribution of temperature field. Secondly, a large number of field tests show that the speed of artificial ground freezing is a process that is from fast to slow. If the time spent in the early stage of dropping to the freezing point is used to estimate freezing speed, it undoubtedly enlarges this value, which is not safe to predict the development of frozen soil thickness later.

In order to calculate the thickness of frozen soil more reasonably, considering that the circle-pipe freezing with large diameter can be equivalent to the row-pipe freezing at local position of the temperature-measuring point, this paper proposes a new double-row-pipe freezing model instead of the single-pipe model for a better field application, and the diagram of this model is shown in Figure 5.

Figure 5: Schematic diagram of the double-row-pipe freezing model.

Similar to the single-pipe freezing model described by equation (1), in order to calculate frozen soil thickness by measuring point temperature, it is necessary to obtain the analytical solution to the temperature field of this double-row-pipe model. For the row arrangement of freezing pipes, based on the solution to dealing interference problem with multiholes, Bakholdin proposed the analytical results of single-row pipe model, which can be expressed as [23]

Equation (5) shows temperature distribution of single-row pipe freezing model, and it still differs from the diagram shown in Figure 5. For better engineering applications, the authors’ team has derived analytical solution to different kinds of row-pipe freezing models in recent years, including double-row-pipe model, triple-row-pipe model, and multi-row-pipe model [24, 25]. In terms of the double-row-pipe freezing model shown in Figure 5, analytical solution of the temperature field considering the temperature of the soil freezing point can be written as [25]

where m is a variable expressed as

where l is the distance between two adjacent freezing pipes, L is the distance between two rows, and TS represents the freezing point of soil.

Similarly, to use this result to calculate the thickness of frozen soil curtain, assuming the temperature of the measuring point M (x₀, y₀) is T₀ and then substituting (x₀, y₀) and T₀ into equations (6) and (7), we can get the expression of ξ:

As shown in Figure 5, the frozen wall consists of three parts: the distance from the upper boundary to the first row of freezing pipes ξ, the distance between two rows L, and the distance from the second row of freezing pipes to the lower boundary ξ. Therefore, the thickness of the frozen wall d can be expressed as

Equations (8) and (9) are the final calculation equations to be obtained in this paper. With these two equations, we can get frozen wall thickness by using the temperature value of measuring points based on analytical solution to temperature field of double-row-pipe freezing model. It is worth noting that although the analytical solution is steady state, the temperature of the measuring point is variable, which is a reaction to transient freezing process. The variation of the thickness value with time can be obtained by using the temperature value of the measuring point under a series of times, and the spatial distribution of thickness can be obtained by measuring points at different positions.

4. Field Measurement Analyses

The total monitoring time is over 70 days from the beginning of freezing to the end of soil excavation in Maliuzhou Tunnel, and a large amount of data was obtained. Based on data analysis, a series of results have been obtained in the aspects of building up the frozen soil curtain. This part will introduce the temperature monitoring program first and then analyze the main results of the time-temperature curve and thickness of frozen soil curtain.

4.1. Temperature Monitoring Program

The main contents of freezing monitoring in this project are divided into two parts, including freezing system monitoring and freezing wall monitoring. Among them, temperature of circulating brine and freezing soil are the main monitoring objects.

Two monitoring sensors for brine temperature in the freezing system are set up on the input and output roads of the trunk tube, G-Q indicates input temperature of brine and G-H indicates output temperature. By analyzing the temperature difference of brine between input and output roads, the volume of heat exchange between brine and soil can be regarded as an indicator to the freezing development in general.

Since the temperature of the freezing system is relatively simple, the focus of monitoring is on soil temperature. A total of 9 temperature-measuring holes with 45 sensor points are arranged in this project, and the specific opening position of the chosen holes in this paper can be seen from Figure 3. Arrangement of each temperature-measuring sensor from the horizontal and vertical sections along the connected aisle is intuitively shown in Figure 6.

(a)

(b)

Figure 6: Layout of soil temperature sensors: (a) vertical section; (b) horizontal section.

As shown in Figure 6, nine temperature-measuring holes are marked C1∼C9, respectively. C1 and C7 are located in the frozen soil curtain above the aisle to judge the development of the upper frozen curtain. Similarly, C4 and C9 are used to determine the freezing effect below. C2, C3, and C6 are designed as short holes to monitor the temperature at inner and outer boundaries of freezing curtain because frozen soil near the tunnel segments is most vulnerable to weakening of the external environment. C5 and C8 are two long holes arranged on both sides, in which C8 extends from the inner boundary of frozen soil to the outer boundary and C5 enters into the excavation face with the most temperature sensors that can reflect the characteristics of frozen walls in different locations more comprehensively.

4.2. Brine Temperature in Freezing System

Brine temperature is an important index, and cooling time of brine in the trunk pipe is also clearly specified in the design scheme. In this project, normal stable temperature monitoring began on the second day of freezing, by installing two sensors on input and output roads of the trunk tube, and the curve of brine temperature changing with time during the whole process is shown in Figure 7.

Figure 7: Time-temperature curve of brine.

G-Q represents the temperature of the input brine, as can be seen from Figure 7, and this value has been reduced to −25°C on the 6th day after the operation of refrigerator, which has reached the design requirements of −20°C. The overall curve trend indicates that the brine temperature is reduced at a faster rate in the early stage of freezing, and to the middle and late stages of freezing, this temperature is approximately maintained in a stable state.

From the tenth day to the end of channel excavation, brine temperature has been kept below −28°C, and temperature difference between input brine and output brine is kept within 2°C. This indicates that the heat exchange between soil and brine is relatively stable, which effectively guarantees the cold supply of the frozen wall and avoids the overweakening of the curtain caused by soil excavation.

4.3. Frozen Soil Temperature

As mentioned earlier, nine temperature-measuring holes have been arranged in this project, and different numbered holes are responsible for different monitoring tasks. In this section, we choose C1 and C5 to analyze the development of frozen soil. The time-temperature curve is shown in Figure 8.

(a)

(b)

Figure 8: Time-temperature curve of frozen soil: (a) C1 hole; (b) C5 hole.

Figure 8(a) shows the time-temperature curve of measurement points in C1, and C1 is a horizontal hole near the middle of the designed frozen soil curtain in which six temperature sensors are arranged. It can be seen that the cooling rate is also significantly higher in early time than that in the middle and late stages, which is similar to the brine temperature curve. However, compared with brine, the soil temperature changes more slowly and smoothly. From the absolute value of temperature, C1-1 has the lowest temperature among the six measuring points because it is in the middle of the depth of the connected aisle and is least affected by thermal disturbance of shield tunnels on both sides. After 40 days of freezing, temperature of C1-1 has been dropped to −25°C which is far below than the freezing point. Temperatures of other five sensors are slightly higher than C1-1, but they remained around −20°C. As shown in Figure 3, C1 hole is near the middle of the two freezing tubes in outer circle, and temperature data in this area dropping below freezing point indicate that frozen soil formed by the outer circle of freezing pipes has reached the state of overlap and the requirement for sealing of ground water can be initially reached.

Figure 8(b) shows the time-temperature curve of the measurement point in C5, and C5 is a long hole that all eight measuring points are within the inner circle of freezing pipes. From C5-8 to C5-1, as the depth increases, the farther the measuring point is from freezing tubes, the higher the temperature value obtained in general. However, due to the fact that C5-1 is close to the east line tunnel, it is affected by the curved surface freezing pipes, and its temperature value is lower than that of C5-2 and C5-3. Besides, this hole extends into the designed aisle which affects soil excavation; on the 55th day, the C5-1 to C5-5 points are removed. From the absolute value of temperature, the C5 hole indicates that temperature distribution within inner circle of freezing tubes is highly discrete, and the temperature difference is larger than other holes. The maximum temperature is around −5°C and is also below the freezing point. What is more, temperature of C5-1 and C5-2 shows that part of the soil in excavation surface has turned into frozen soil; although it is beneficial to ensure the safety of construction, considering that frozen soil is not easy to excavate, the distance between the inner freezing pipe and connected aisle may be properly controlled in future freezing design.

4.4. Influencing Factors on Frozen Wall

Factors influencing temperature of the frozen wall in connected aisle engineering mainly include soil excavation and external boundary conditions. In this section, C3 hole near the east tunnel is selected for analysis of excavation, and C4 hole is used for analysis of boundary conditions, shown in Figures 9 and 10.

Figure 9: Influence curve of soil excavation.

Figure 10: Spatial distribution curve of temperature at different times (C4).

Since the soil excavation is carried out from the west to the east, and C3 hole is opened from the east line tunnel, so temperature data are affected in the later stage of excavation. As shown in Figure 9, the temperature turning point of C3-1 appears on the 58th day, and the increase is about 10°C by the end of excavation. C3-2 and C3-3 showed a temperature rise on the 60th and 61st days, with the temperature increase of 12°C and 9.2°C, respectively. Similar turning time of these three measuring points shows that cooling effect of the curved surface freezing pipe inside tunnel segment is not obvious compared with thermal disturbance of high-temperature air after soil excavation. At the same time, it can be found that after soil excavation, the frozen wall and exposed environment in the channel would reach a new equilibrium temperature, which is still lower than the soil freezing point. Therefore, although the frozen wall during the maintenance freezing period is weakened, it can remain at a lower temperature to ensure overall stability and water-sealing performance.

The spatial distribution curve of temperature in C4 hole is chosen to react to boundary conditions, shown in Figure 10. The C4-5 point is close to the outer surface of the segment in the east line tunnel, and the C4-1 point is close to the boundary of the frozen soil curtain under the connected aisle. As the curve shows, temperature decreases first and then increases along the depth direction of the measuring hole, and low temperature in the middle position is because C4 hole passes through the cryogenic freezing pipe circle. In addition, we can find from this figure that the distance between adjacent curves from top to bottom is getting smaller and smaller, which also shows that with the progression of freezing, the amount of heat exchange between frozen soil curtain and brine decreases, and the temperature of frozen wall tends to be stable.

4.5. Thickness of Frozen Wall

The above analysis is all based on field temperature of frozen soil, but the frozen wall not only needs to meet the requirements of water sealing, it also serves as an effective bearing structure in the process of channel excavation. In this section, the double-row freezing model proposed in Section 3 will be adopted to calculate the thickness of frozen wall based on field temperature obtained by measuring holes introduced above.

From east to west along the axis of passage, five calculation sections are selected and marked as A–E to calculate the thickness. In addition, length of two sealing paths is calculated on the contact surfaces between two bell mouths and tunnel segment, shown in Figure 6(a). Temperature values at each calculated section are obtained by interpolating data from nearby temperature-measuring points. For the Maliuzhou connected aisle, the design scheme gave a requirement of the 40–45 days’ active freezing period, and the freeze effect assessment meeting before soil excavation was scheduled to be held on the 44th day of freezing, so we chose the temperature data one day before the meeting, i.e., the thickness of the 43rd day is calculated with equation (9) as shown in Table 1.

Table 1: Thickness of the frozen wall in each section calculated by analytical solutions.

Section
West bell mouth
E
D
C
B
A
East bell mouth

Single thickness (ξ/m)
0.9
0.9
1.01
0.97
0.82
0.85
0.92

Row spacing (L/m)
1.22
1.22
1.22
1.22
1.22
1.22
1.22

Total thickness (d/m)
3.02
3.02
3.24
3.16
2.86
2.92
3.06

Where freezing parameters used in the calculation are as follows: radius of freezing pipe, r₀ = 0.108 m; average distance between adjacent freezing pipes, l = 1.78 m; distance between two rows, L = 1.22 m; freezing point, Ts = −0.65°C, according to results of frozen soil test; temperature of the freezing pipe wall, Tf = −30°C.

As the results are shown in Table 1, the smallest value of single thickness of frozen wall (ξ) on the 43rd day is 0.82 m, while the distance between inner circle of freezing pipes and designed excavation boundary is 0.8 m. This comparison indicates that frozen soil is formed in the excavation face of all selected calculation sections. In addition, section D has the largest calculated thickness, and inner boundary of the frozen soil enters the excavation face of about 0.21 m. This conclusion has also been confirmed by the frost ring from site excavation of the Maliuzhou aisle, and the field freezing effect is shown in Figure 11.

Figure 11: Field freezing effect—frost ring in excavation face.

5. Validation and Simulation Studies

As the boundary of the frozen wall was considered as a straight line boundary in derivation of analytical solutions, it does not match the real situation. Such simplification may cause some errors in the calculation of frozen wall thickness. In this chapter, finite element software ANSYS is adopted to establish a transient thermal conduction model for the connected aisle of Maliuzhou Tunnel, considering phase transition and initial ground temperature. The variation characteristics of frozen soil curtain during the active freezing period are simulated.

5.1. Parameters and Boundary Conditions

For simulation of transient heat conduction, the influence of soil parameters is particularly significant. They mainly include soil freezing temperature, thermal conductivity, specific heat capacity, latent heat of phase change, and enthalpy value.

Referring to the experimental results of physical and mechanical properties of frozen soil in the ②1 soil layer, parameters used in the numerical simulation are shown in Table 2.

Table 2: Parameters used in numerical studies.

Parameters
Value
Unit

Soil freezing temperature
−0.65
°C

Thermal conductivity at positive temperature
98.50
kJ/(m·d°C)

Thermal conductivity at negative temperature
195.26
kJ/(m·d°C)

Specific heat capacity
1.21
kJ/(kg°C)

For heat conduction with phase change, since the curve of temperature versus time on phase transition interface is discontinuous, a variable H which represents enthalpy is introduced in the simulation, and these values at different temperatures are calculated, as shown in Figure 12.

Figure 12: Enthalpy curve with various temperatures.

Boundary condition of initial ground temperature is determined by field monitoring, and taking section C as an example, 18°C is used as the constant value for outer boundary of simulation area. The boundary condition of freezing pipes is determined according to the brine temperature of output trunk roads at different times. During the active freezing period, brine temperature of first 43 days in Figure 7 was selected for numerical analysis.

5.2. Modeling and Meshing

The middle section (section C) is selected for modeling, the outer boundary of this model takes twice the radius of frozen area, and coordinates of freezing pipes are determined according to the field layout. The finite element model and meshing figure are shown in Figure 13.

(a)

(b)

Figure 13: (a) Simulation model and (b) meshing figure of section C.

Taking into account the irregularity of freezing pipe arrangement, four paths are set in different directions of section C to extract numerical results to obtain thickness of frozen soil curtain, and the specific schematic diagram of the four paths is shown in Figure 13(a).

5.3. Numerical Results

The thickness value of frozen wall along four paths based on the numerical results obtained from 39th day to 43rd day of freezing is shown in Table 3. Cloud figures of temperature development at different times of 5 days, 10 days, 25 days, and 43 days are shown in Figure 14.

Table 3: Thickness of the frozen wall along each path calculated by numerical results (m).

Time (day)
Path 1
Path 2
Path 3
Path 4
Average

39
3.149
2.749
3.01
3.039
2.986

40
3.196
2.819
3.057
3.068
3.035

41
3.236
2.842
3.097
3.144
3.079

42
3.26
2.854
3.149
3.19
3.113

43
3.289
2.877
3.19
3.219
3.143

(a)

(b)

(c)

(d)

Figure 14: Cloud figures of temperature distribution at different times: (a) 5 days; (b) 10 days; (c) 25 days; (d) 43 days.

From Table 3, the minimum thickness of frozen soil occurs on path 2, indicating that freezing effect above the connected aisle is relatively weak. However, the absolute value has also reached 2.8 m before excavation, which has met the design requirements. In addition, as far as the result on 43rd day is concerned, thickness of the frozen wall in section C obtained by the analytical solution proposed in Section 4.5 is 3.16 m, while the average thickness obtained by numerical calculations in Table 3 is 3.143 m. Errors between these two methods are completely within allowable range of practical engineering, which verifies the applicability of the proposed calculation method by the double-row-pipe freezing model. Besides, thickness error on different paths of a same section reminds us that when using the analytical method to calculate the thickness of frozen wall, it is expected that more temperature-measuring holes are set along the circumferential direction at the boundary of the designed frozen soil.

Cloud figures of temperature development in Figure 14 intuitively shows that with the increase of freezing time, blue area of low temperature gradually spreads around freezing pipes, and a curtain ring of frozen soil has been formed by 43rd day which can ensure safety of subsequent construction. Field excavation works of the connected aisle were formally carried out from west tunnel to east tunnel on 48th day of freezing. It is proud that after 28 days of efforts, the entire 15.2 meters long passage was successfully penetrated, which paved the way for full-line opening of Maliuzhou Tunnel. At present, this connected aisle is in good working condition and is ready to ensure effective communication between the east and west tunnels.

6. Conclusions

In this paper, based on the freezing project of the connected aisle in Maliuzhou Tunnel, a new double-row-pipe freezing model is proposed for the calculation of frozen wall thickness. Through field measurement and numerical studies, optimal freezing scheme of double-circle freezing methods and optimal monitoring program of measuring holes were put forward. Some main conclusions and suggestions for future freezing construction are also obtained.

Traditional single-pipe freezing model and freezing speed prediction model are not suitable for the calculation of frozen soil thickness in multipipe freezing. Analytical solution to double-row-pipe model has good applicability in calculating thickness in horizontal freezing projects. Analytical results fit well with numerical simulation results, and the error is within allowable range of engineering. Field measurement indicates that the decreasing speed of brine and soil temperature gradually slows down with time and finally tends to be stable during the whole freezing process. Temperature along the axis direction of the connected aisle shows the characteristics of high value at both ends and low value in the middle, but the difference is not large. Thermal disturbance in the excavated channel makes the temperature of frozen wall rise about 10°C. It is better to speed up soil excavation and reduce the exposure time of the frozen wall.

Under the freezing parameters of this project, a certain thickness of frozen soil will be formed in excavation surface, which is conducive to ensuring safety of construction. But accurate and comprehensive calculation of frozen soil thickness needs a more reasonable arrangement of temperature monitoring holes.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare no conflicts of interest.

Acknowledgments

The authors would also like to express their sincere gratitude to Lei Han and Yanghui Liu for their contribution to this field measurement. The authors are deeply indebted to financial support from National Natural Science Foundation of China (Grant no. 51478340) and Science and Technology Program of Ministry of Transport of China (Grant no. 2013318J11300).

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