Introduction

In recent decades, the rapid urbanization and population growth in China have led to an increasing need to exploit underground resources to accommodate the growing urban population1,2,3. The development of underground railway systems, particularly through the use of shield machines for construction and excavation, has been rapid and significant4,5,6. However, traditional underground excavation methods, especially in areas with loose clay and fine sand soil layers that are highly permeable to water, can lead to collapse issues. In such cases, the application of artificial freezing technology (AGF) to strengthen the surrounding soil is a preferred method. AGF involves cooling the water in the soil to freeze it, absorbing heat from the surrounding soil to create a permafrost layer, which effectively isolates groundwater and provides a seal and strength advantage7,8,9,10,11. Consequently, AGF has been extensively utilized in tunnel construction and has garnered attention within the geotechnical engineering community for its effectiveness in such applications12,13,14. In artificial freezing technology, permafrost curtain thickness to design thickness is an important guarantee of safe construction15. In view of this, many scholars at home and abroad have carried out a lot of research on the freezing temperature field in recent years by means of model tests, field measurements and numerical simulations. Wu Tao derived an analytical formula for the transient temperature rise in frozen soil based on the development and closure laws of the frozen wall within seepage strata. This work elucidates the deterministic relationship between the geometrical dimensions of the frozen wall. Subsequently, a theoretical framework for longitudinal temperature measurement was established. The findings indicate that, consistent with grey correlation theory, the correlation coefficients of freezing holes No. 7–10 and No. 18–22 were identified as weak points in the frozen wall. These predicted weak points were subsequently confirmed during the process of vertical shaft excavation16. Yao, Zhishu conducted numerical simulations and measured analyses of the temperature field during the wellbore sinking process using the artificial freezing method in Cretaceous formations. A numerical model was established to investigate the temperature field associated with the freezing project, predicting and analyzing the effective thickness, average temperature, and well-wall temperature of the frozen wall at varying excavation depths. Comparative analyses of the field temperature measurements from inspection holes and well walls were subsequently performed, revealing a strong agreement between the numerical simulation results and the field measurements. The field measurements indicated that the cooling rates for fine-, medium-, and coarse-grained sandstones were 0.364 °C/d, 0.397 °C/d, and 0.440 °C/d, respectively, at the onset of freezing17. Zequn et al. fully integrated the pipe shed method and artificial freezing technique, employing numerical simulation to elucidate the development of the freezing wall thickness and temperature field distribution during the active freezing period under three distinct pipe filling methods. Their findings indicate that the concrete intermittent filling method enables the formation of a reliable freezing curtain within the stipulated time frame. It takes approximately 15 days from the onset of permafrost contact with the pipe wall until the designed thickness is achieved, with the freezing effect of empty pipes lagging behind that of filled pipes by about 28 days. Therefore, targeted enhanced measures are required in field applications18. Jian et al. investigate the issue of challenging freezing wall closure in coal mine shafts, which is attributed to the elevated seepage velocity of aquifers during artificial freezing methods. They employed finite element software COMSOL to construct a hydrothermal coupling model to analyze the impact of various groundwater seepage velocities on the formation of the artificial freezing wall under the combined effects of seepage and temperature fields. The analysis results provide the following recommended principles for optimizing the freezing plan design: Firstly, closing the downstream area to effect water isolation; secondly, expediting the closure of the upstream area to reduce the overall time required for freezing wall closure19. Yuanhao conducted research on the Gongbei Tunnel in Zhuhai, where the pre-support technology’s freeze-sealed pipe shed method was employed. Utilizing field temperature observation data, the study investigated the water resistance of the freeze-sealed pipe shed in water-rich soil. The findings indicate that the closure of the frozen soil wall can be categorized into two distinct modes. The frozen soil between the tubes prior to excavation has met the design specifications, and the freeze-sealed pipe shed maintains excellent water resistance performance20. Tao-Wu et al. tackled the issue of prolonged freezing wall closure due to high-velocity groundwater in the freezing of artificial strata, developing a comprehensive framework for monitoring the freezing temperature field under rapid groundwater flow. The findings indicated that the proposed theory and method were effective in identifying weak points of the freezing wall and preventing incidents of freezing wall closure under high-velocity groundwater conditions17. Linjie et al. conducted research to address the issue of preventing frozen pipe fracture during artificial freezing construction. The study was conducted in the context of the freezing project for the main shaft of Yuncheng mine in Shandong Province. Based on similarity theory, they performed tests on frozen pipes in composite strata within the active freezing section. The test results indicated that the frozen pipe experiences constant vertical compression in the clay layer, whereas it undergoes vertical compression followed by tension in the sand layer 21. To investigate the factors contributing to the slow advancement of freezing walls in permeable strata with high percolation velocities, Wang Bin et al. conducted freezing tests in such conditions using a hydro-thermal coupling model test system. Building upon the outcomes of the model tests, they employed the segmental equivalence method and derived a formula for calculating the steady-state temperature field of a single row of pipes subjected to seepage22. Liu Peng conducted a numerical simulation to investigate the changes in the temperature field, moisture field, and vertical displacement of the most unfavorable soil layer resulting from freezing and expansion during the freezing process. This study clearly illustrates the migration pattern of the moisture field throughout the entire freezing process. The strong freezing zone is characterized by a high cooling rate and a rapid decrease in moisture content, whereas the weak freezing zone exhibits the opposite trend23.

Based on the comprehensive review of relevant literature, the artificial freezing method has been widely applied in various special projects, including contact channels, tunnel construction, shield tunnel end reinforcement, and others. Experts and scholars have, through numerical simulation, model testing, and field measurements, elucidated the mechanisms of freezing reinforcement, the changes in the temperature field due to freezing and thawing, and have developed a mature body of freezing theory. Currently, artificial freezing technology primarily targets traditional single-lane tunnels, with limited research conducted on double-lane tunnels. In comparison to single-lane tunnels, double-lane tunnels feature a larger construction section, a broader freezing range, and more intricate thermal disturbances. These factors contribute to more complex issues, such as the distribution of the temperature field during freezing, the formation of freezing walls, and soil deformation post-thawing. The existing research findings remain incomplete; thus, further investigation into the artificial freezing methods applicable to double-lane tunnels is essential. This paper aims to explore the development and change laws of the freezing temperature field in double-line tunnels, providing a basis for future engineering construction. The research is based on the coastal city metro double-line tunnel freezing reinforcement project. A three-dimensional water-heat coupling model is constructed to study the development and distribution of the frozen soil curtain and temperature field, and to simulate the dynamic evolution of the frozen soil curtain. The findings of this project can enhance the theoretical reference basis for the subsequent design of similar projects.

Freeze programme design

Project overview

According to the relevant investigation report, the tunnel’s central line is located at a depth of 24.1 m, where the ground pressure is significant. The entire excavation section extends to a depth of 9.1 m. Above the station, the proposed tunnel necessitates the continued passage of ground vehicles, and some underground piles remain undismantled. Given that these conditions do not impede the normal passage of ground vehicles, the cut-and-cover method is not deemed suitable. Moreover, the cost-effectiveness of using shield machines for tunnelling is not high in this context. This section of the two-lane tunnel is situated in a unique geographical setting, traversing various unfavorable soil layers, including silty clay and highly permeable fine sand. Consequently, the application of artificial freezing technology to reinforce the surrounding soil is imperative, ensuring the soil achieves a certain strength prior to further excavation.

Arrangement of freezing pipes

The proposed freezing pipe design features two rows of holes drilled in parallel on both sides of the shaft, designated as the left and right working wells. The left working well, serving as the primary site for freezing construction, is primarily occupied by the inner circle of freezing pipes, while the right working well, acting as the secondary site, accommodates the outer circle of pipes. Both sets of holes are drilled to the opposite side of the working well structure, thereby minimizing the number of openings and pipes. This design, featuring two rings of freezing pipes drilled from opposite sides of the working shafts, is primarily due to the significant depth of the two-lane tunnel and the extensive excavation area, necessitating the use of freezing walls for support. Figure 1a, b illustrate the location of holes in the left and right working wells, with a diameter of 108 mm for the freezing pipes. Figure 1c, d present the top views of the inner and outer circles of freezing pipes, respectively, highlighting the length of each pipe’s hole. Figure 1e depicts the open hole position of the freezing tubes.

Fig. 1
Fig. 1
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Freeze tube design.

3D numerical simulation modelling

Underlying assumption

Numerical model for the hydrothermal coupling mechanism, considering the porous media in seepage, freezing and other conditions of the role of the mechanism is complex, with reference to the relevant basis and based on the actual engineering profile, based on the following basic assumptions to establish a numerical hydrothermal coupling model24,25,26.

(1) Neglect the effect of the stress field on the temperature field and consider only the coupling of the seepage and temperature fields; (2) The soil is a saturated, homogeneous, isotropic porous medium with constant total porosity; (3) Neglecting evaporation of water during freezing, Darcy’s law applies to groundwater flow in porous media; (4) Heat transfer in frozen porous media satisfies Fourier’s law; (5) Assuming that the soil begins to freeze at − 1 °C, the area within the − 10 °C isotherm can form a stable and solid freeze.

Model calcuation theory

Temperature field control equations

It is stipulated that during the freezing process, the soil is a saturated homogeneous isotropic sandy soil, and the temperature field is affected by the heat transfer from the sand particles and the seepage heat transfer from the groundwater at the same time. When analysing the heat transfer problem in sand seepage freezing process in COMSOL software, the heat transfer module for porous media is selected and the equations included in the software are:

$$ (\rho C_{p} )_{eff} \frac{\alpha T}{{\alpha \tau }} + \rho C_{p} u\nabla T + \nabla q = Q + Q_{vd} $$
$$ q = - k_{eff} \nabla T $$
$$ \left( {\rho C_{p} } \right)_{eff} = C_{eq} $$
$$ k_{eff} = \lambda_{eq} $$

where \(\nabla \) is the Hamiltonian operator; \(\rho \) is the density of the porous medium; \({C}_{\text{p}}\) is the specific heat capacity of the porous medium; \((\rho {C}_{\text{p}}{)}_{\text{eff}}\), i.e., \({C}_{\text{eq}}\), is the equivalent volumetric heat capacity of the saturated sandy soil, and \({k}_{\text{eff}}\) is the equivalent thermal conductivity; \(Q\) is the source of heat; \({Q}_{\text{vd}}\) is a source term; \(u\) is the seepage velocity vector; and \(q\) is the source of conduction heat.

Seepage field control equations

In the process of freezing the soil, the seepage of groundwater in that soil layer conforms to Darcy’s Law. The formula is:

$$ Q^{\prime } = KA\frac{\Delta H}{L} $$

where \(Q^{\prime}\) is the seepage volume; \(A\) is the cross-sectional area perpendicular to the seepage direction; \({\Delta }H\) is the head difference; \(L\) is the seepage path; \(K\) is the permeability coefficient.

Selecting Darcy’s law block for analysing groundwater seepage in COMSOL software, the partial differential equation that comes with the software is:

$$ \frac{\partial }{{\partial_{t} }}\left( {\varepsilon_{p} \rho } \right) + \nabla \left( {\rho u} \right) = Q_{m} $$
$$ u = - \frac{K}{\eta }\nabla p $$

where: \(p\) is the void water pressure; \(\varepsilon\) is the porosity; \(Q_{{\text{m}}}\) is various source terms; \(\eta\) is the water viscosity.

Temperature field coupled with seepage field

The permeability coefficient K of the soil layer changes with temperature during the freezing process, i.e., the permeability coefficient can be expressed as a function of temperature, and thus the coupling between the temperature field and the seepage field can be expressed through a functional relationship. When the water–ice phase transition occurs during the freezing process of the soil, the Heaviside step function H(T) is used to describe the coupling relationship between the permeability field and the temperature field, i.e.:

$$ K\left( T \right) = K_{u} H\left( T \right) + K_{f} \left[ {1 - H\left( T \right)} \right] $$

where \({K}_{\text{f}}\) and \({K}_{\text{u}}\), are the permeability coefficients of unfrozen and frozen soil, respectively.

Geometric modelling, meshing and boundary conditions

The transient, three-dimensional hydrothermal coupling model was developed using COMSOL finite element software version 6.0, with the geometric model’s dimensions set based on the freezing range and the impact of groundwater flow on the permafrost curtain. The geometric model dimensions were thus defined as 30 m × 40 m × 25 m. The model was discretized using a free tetrahedral mesh, with parameters customized to achieve an appropriate mesh size for ensuring the calculation model’s convergence and accuracy. To facilitate calculation convergence and accuracy, a denser grid was employed at the model’s boundaries, simplifying the calculation process. Conversely, a coarser grid was used in the freezing region of the frozen pipe. The model’s grid division is illustrated in Fig. 2a. All model boundaries were set as adiabatic, with the initial soil layer temperature set at 18 °C. The model’s surfaces were adiabatic, with the front and rear side boundaries designated as permeable (with the rear side being upstream and the front side downstream), and the remaining boundaries as impermeable. The model’s seepage velocity was approximately 0.3 m/d, based on the engineering profile, as depicted in the schematic diagram of the model boundary in Fig. 2b.

Fig. 2
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Schematic diagram of the model (COMSOL Multiphysics 6.0).

Selection of relevant parameters and brine cooling programme

Selection of relevant parameters

Combined with the engineering design scheme and, the investigation report and relevant permafrost tests27, the relevant parameters for the temperature field and seepage field calculation of this model are shown in Table 1.

Table 1 Relevant parameter values.
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Salt water cooling programme

The brine cooling process is executed according to a predefined cooling schedule, strictly prohibiting the direct reduction of brine to low temperatures for recycling. Regarding the freezing technology of the freezing tubes, the critical parameters include: The active freezing time cycle of the freezing tubes is 60 days; the brine temperature is designed to be lowered to − 28 °C by the 10th day of active freezing, and must be maintained at this temperature during the excavation of the twin tunnels to ensure smooth excavation; the average temperature of the internal permafrost curtains is designed to be set at no more than − 10 °C. This is done to ensure that the thickness of the permafrost curtain meets the intended design criteria. Table 2 presents the brine cooling programme.

Table 2 Saline cooling programme.
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For the accuracy of the numerical analysis of the temperature field of the freezing method, in many related studies, this group through a number of on-site measurements and numerical analyses to do a comparison of the relevant data basically coincide with the reference to the relevant literature28.

Analysis of numerical simulation results

Temperature field cloud and isotherm map analysis

To comprehend the overall alteration in the 3D temperature field under seepage conditions using the freezing method, profiles at X = 15 m and Z = 12.5 m are chosen for temperature field cloud and − 1 °C, − 10 °C isotherm analysis, as the Y-direction profiles are deemed less relevant to the study. Figure 3 illustrates the locations of the X = 15 m and Z = 12.5 m profiles, as indicated.

Fig. 3
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Sectional location diagram (COMSOL Multiphysics 6.0).

X-direction profile temperature field analysis

Figure 4 illustrates the evolution and distribution of temperature fields and isotherms in various profiles at different times (1d, 5d, 14d, 21d, 40d, 60d). Initially, at 1d, the freezing pipe begins to freeze, and the outline of the permafrost curtain has not yet formed. At this stage, groundwater flow is horizontal and uniform, from high to low heads, through the entire freezing area. The − 1 °C isotherm is closely encircled by the freezing pipe. By 5d, the outline of the frozen soil curtain gradually forms, and it begins to obstruct groundwater flow, causing some water to bypass the curtain. The − 1 °C isotherm partially completes its circular intersection, while the − 10 °C isotherm forms around the freezing pipe. At 14d, the − 1 °C isotherm has fully completed its circular intersection, indicating the curtain’s basic formation and significant impediment to groundwater flow. The formation of the curtain is a complex process influenced by multiple factors, with groundwater flow carrying away cold, leading to higher temperatures upstream than in the already formed curtain. By 21d, the − 10 °C isotherm also completes its circular intersection, and the outer circle of the permafrost curtain continues to expand outward, becoming thicker. At this point, groundwater flow is completely isolated, with only a small amount still present upstream but unable to enter the inner permafrost curtain. Between 40 and 60d, the permafrost curtain enters a stable development phase, spreading both inward and outward. Eventually, groundwater flows completely around the curtain, indicating that the soil on the waterfront has frozen over and that groundwater is effectively blocked. However, the development of the permafrost curtain upstream and downstream is uneven, leading to a ‘cogwheel-shaped’ distribution at the water surface and expanding the frozen area downstream. This process highlights the role of the permafrost curtain in halting groundwater flow.

Fig. 4
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X-direction temperature field cloud and isotherm map.

Z-direction profile temperature field analysis

The Z-direction profile allows for a visual examination of the isothermal intersection pattern and the progression of the freezing wall in the upstream and downstream sections of both left and right tunnels, as well as at the junction between them. Given the symmetry of the Z-axis direction at the top and bottom, a single profile can provide a comprehensive view. Consequently, the Z = 12.5 m profile is chosen for analyzing the evolution of the temperature field at various times (2d, 5d, 13d, 22d, 40d, 60d). The alterations in the temperature field cloud and isotherm map over the freezing period are depicted in Fig. 5.

Fig. 5
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Z-direction temperature field cloud and isotherm map.

The analysis of Fig. 5 reveals that the blue section at the lower temperature end of the figure corresponds to the location of the freezing tube. At the initial stage of freezing (2d), the freezing tube has just been activated, and the cold generated has not yet been transferred to the surrounding soil. Consequently, the soil temperature has not reached the freezing point, which is above 0 °C, with − 1 °C isotherms appearing sporadically and unevenly distributed. By 5d of freezing, the cold produced by the freezing tube is continuously transferred to the surrounding soil, leading to a further decrease in temperature and a more regular development of the − 1 °C isotherms. At 13d of freezing, the blue area shows a tendency to spread, and the − 1 °C isotherms in the upstream and downstream of the two-lane tunnel and at the tunnel junction have completed their intersection circle. Similarly, the − 10 °C isotherms in some areas have also completed their intersection circle. By 22d of freezing, the blue color of the frozen area further deepens, and the − 10 °C isotherms also complete their intersection circle. By the final freezing stage of 60d, the brine freezing is complete, and the temperature field cloud map of the three freezing areas displays an uneven development trend, primarily due to the influence of groundwater flow, which, however, does not affect the overall freezing effect. The development of the permafrost curtain in the three freezing areas tends to stabilize, forming a stable and solid structure that has reached the design thickness.

Isosurface maps to analyse the overall permafrost curtain thickness development pattern

To more intuitively analyze the variation in the thickness of the − 1 °C frozen soil curtain with freezing time, the changes in the 3D view of the − 1 °C equivalent surface at different times (2d, 5d, 10d, 20d, 40d, 60d) were selected for examination. As illustrated in Fig. 6, it is evident that to mitigate the impact of freezing induced by the intense heat exchange under rapid cooling, the freezing pipe maintains a temperature identical to the initial ground temperature at the onset of cooling. During the first day of the pre-freezing period, the temperature development is unidirectional, in accordance with the cooling plan, and the surrounding soils have all cooled below 18 °C. At this juncture, the cold generated by the freezing pipe is concentrated around it. By the 5th day of freezing, it is observed that there are columns of frozen soil with smaller cross-sections surrounding the freezing tubes, and the complete − 1 °C frozen soil curtain has not yet formed. The contour exhibits a distinct jagged shape, and the temperature distribution is uneven. On the 10th day of freezing, the − 1 °C permafrost curtain shows signs of expansion, with large areas of permafrost joints appearing around the left and right tunnels. The outline of the permafrost curtain is nearly complete, and the curtain at the junction of the left tunnel and the right tunnel lacks joints. After 20 days of freezing, a complete and stable − 1 °C permafrost curtain is formed, exhibiting a certain water-proofing effect. The outline of the permafrost curtain is almost toothless. Between 40 and 60d of freezing, the − 1 °C permafrost curtain develops steadily, spreading to both the inner and outer sides, growing thicker, and the surface of the contour becomes smoother.

Fig. 6
Fig. 6
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− 1 °C isosurface map (COMSOL Multiphysics 6.0).

Development patterns of effective permafrost curtain thicknesses in different directions for two-lane tunnels

Taking the centre of the left and right tunnels as the origin respectively, the left tunnel takes every 45° on the left half of the tunnel as a monitoring direction, and selects 5 directions from 0 to 180° as the monitoring direction in turn; similarly, the right tunnel takes every 45° on the right half of the tunnel as a monitoring direction, and selects 5 directions in turn. When the freezing front does not invade the excavation radius, the thickness of the frozen soil body is taken as the effective frozen soil curtain thickness; when the freezing front invades the excavation area, the thickness of the frozen soil body outside the excavation range is taken as the effective frozen soil curtain thickness. Under the condition of 0.3 m/d seepage velocity, the variation curves of effective permafrost curtain thickness in five directions of the left and right tunnels with time are shown in Fig. The monitoring directions of the left and right tunnels are shown in Fig. 7.

Fig. 7
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Schematic of monitoring orientation.

The evolution of the effective thickness of the frozen body over time is depicted in Fig. 8. As freezing time increases, the effective thickness rises until the entire frozen area is solidified. Once the freezing front intersects the wall, the region solidifies rapidly and completely, leading to a unified frozen area within a short period. The development of the effective permafrost curtain thickness in the 90° direction of the left and right tunnels is opposite. The 90° direction of the left tunnel exhibits the slowest growth, with an effective permafrost curtain thickness of 2.3246 m at the end. Conversely, the 90° direction of the right tunnel shows the fastest development, with an effective permafrost curtain thickness of 4.0639 m, a difference of 1.0639 m from the left tunnel. This is primarily due to the influence of groundwater flow. The 90° direction of the left tunnel is upstream, where the freezing front is perpendicular to the flow velocity, making it more susceptible to the flow’s impact. In contrast, the 90° direction of the right tunnel, being downstream, is largely unaffected by the flow. Additionally, groundwater flow can carry coldness downstream, which is less conducive to further lowering the temperature of the frozen wall of the left tunnel. The cold brought to the right tunnel results in a ‘tailing’ effect in the downstream 90° direction, reducing the stratum temperature of the right tunnel and the difficulty of forming a freezing wall, thereby expanding the freezing area downstream. In the five directions of the right and left tunnel, the effective permafrost curtain thickness development is similar in the first 5 days of freezing, with the fastest development, primarily due to the cooling program that lowers the temperature from the initial 18 to − 20 °C by the 5th day. After the 5th day of freezing, the development of the effective permafrost curtain thickness in the left and right tunnels slows down. The fastest-growing directions in the left tunnel were 0° and 180°, reaching 3.096 m and 3.20724 m of effective permafrost curtain thickness, respectively, at 60 days of freezing, exceeding the predetermined design thickness of the frozen wall. The three directions of 45°, 90°, and 135° developed more slowly, primarily due to groundwater flow. At 60 days of freezing, they all reached more than 2.3 m. The overall trend of the right tunnel freezing wall development is faster than that of the left tunnel, with the five directions showing similar development and speed in the first 35 days. After 35 days of freezing, the 45°, 90°, and 135° directions developed faster compared to the other two directions, eventually forming the thickest effective permafrost curtain. The effective thicknesses of the 45° and 135° directions reached 3.717 m and 3.889 m, respectively, while the 0° and 180° directions developed more slowly, with final effective thicknesses reaching 3.294 m and 3.377 m. In summary, seepage significantly affects the effective permafrost curtain thickness, with the 45°, 90°, and 135° directions on the upstream side of the left tunnel being most affected, and the downstream side of the right tunnel being less affected. Under this seepage rate, the effective permafrost curtain thickness in each direction of the left and right tunnels meets the design criteria.

Fig. 8
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Variation curve of effective thickness of permafrost.

Route analysis

Route selection

To gain a profound understanding of the cooling law governing the temperature field, two primary surface paths are selected for analysis, as illustrated in Fig. 9. These paths are designed to elucidate the spatiotemporal variation in freezing temperatures. The first path originates from the X = 15 m profile and is parallel to the X-axis. It traverses the entire freezing area from upstream, with a 1 m interval between selected path observation points, resulting in a total of 27 points. The second path is parallel to the Z-axis and extends from the upper side of the left tunnel through the entirety of the left tunnel’s freezing area. It also features 1 m intervals between path observation points, amounting to 16 points in total. To facilitate the analysis, the freezing area of the main face 1 is divided into five sections: A, B, C, D, and E, while the freezing area of the main face 2 is divided into three sections: F, H, and G. The selection of these paths and the division of the freezing areas are depicted in Fig. 9.

Fig. 9
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Path location map.

Temporal and spatial evolution of the path temperature field

To gain a profound understanding of the cooling law governing the temperature field, two primary surface paths are selected for analysis, as illustrated in Fig. 10. These paths are designed to elucidate the spatiotemporal variation in freezing temperatures. To facilitate the analysis, the freezing region of the main surface 1 is divided into five sections: A, B, C, D, and E, while the freezing region of the main surface 2 is divided into three sections: F, H, and G.

Fig. 10
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Temperature change law of different main surfaces in space and time.

The plots in Fig. 10a, b illustrate that the freezing tube cooling persists over time, with temperatures in each region continuing to decline. Figure 10a reveals that after 5 days of freezing, the temperatures at observation points along the main surface 1 approach but do not reach the freezing temperature. By 10 days, the temperatures in the freezing regions A, C, and E surpass the freezing temperature of − 1 °C. The temperatures in A, C, and E have been reduced to the freezing temperature. As freezing time progresses, the temperatures at observation points in the freezing regions A, C, and E continue to decrease. The cold volume of the freezing tube, centered around the tube, spreads out, leading to the lowest temperature at observation points along the main face 1, which is approximately − 26 °C, passing through the freezing tube. Points in the freezing regions B and D, situated in the inner parts of the two-lane tunnels, maintain an initial temperature of about 18 °C due to their distance from the freezing tubes. The temperature distribution of the entire main surface 1 exhibits a ‘double W’ shape, with the cooling efficiency following the pattern C > A = E > B = D, where C has the highest efficiency, and B and D have the lowest, due to the distribution of freezing pipes and groundwater flow. The lower efficiency in A and E compared to C is attributed to their upstream and downstream positions relative to the freezing area, which are influenced by groundwater flow. The highest efficiency in C is due to its position at the intersection of the left and right tunnel freezing areas, where groundwater flow is minimal and freezing pipes can effectively deliver cold, resulting in the highest efficiency. Conversely, B and D, where freezing pipes cannot cool the soil, have the lowest efficiency. Figure 10b shows that the temperature distribution of the entire main surface 2 follows a ‘W’ shape, with F and H zones having the highest efficiency due to their proximity to freezing tubes, which generate cold to cool the soil. Thus, F = G > H, indicating the varying cooling efficiencies across the zones.

Effect of different seepage rates on the freezing temperature field

To investigate the impact of varying seepage velocities on the temperature field development of the freezing method in a two-lane tunnel, multiple numerical simulations are conducted, with initial seepage velocity set at 0.3 m/d and incrementally increased by 1 m/d. The analysis encompasses the temperature field generated by the freezing method, the intersection time of the tundish curtain, and its final formation, assessing how changes in seepage velocity influence the temperature field. For this study, seepage rates of 0.3 m/d, 1.3 m/d, 2.3 m/d, and 3.3 m/d were selected to examine the cloud maps and isotherm maps of the frozen curtain over 60 days of freezing, thereby allowing a comparison of the effects of different seepage rates on the frozen curtain. Figure 11 illustrates the cloud diagrams and isothermograms of the frozen temperature field after 60 days of freezing at varying seepage rates.

Fig. 11
Fig. 11
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Freezing temperature field at different seepage rates.

When the seepage velocity is at the initial value of 0.3 m/d, a slight jagged shape appears on the upstream − 1 °C freezing front, while the temperature field develops uniformly. This observation suggests that the seepage velocity is low, resulting in minimal influence on the frozen curtain. As the seepage velocity increases to 2.3 m/d, the freezing temperature field begins to be significantly affected by groundwater flow, leading to an uneven and unstable development of the permafrost curtain. At a seepage rate of 3.3 m/d, groundwater exerts a pronounced impact on the formation of the permafrost curtain via the freezing method; the upstream outer freezing front approaches the edge of the freezing pipe, and the disparity in the formation of the permafrost curtain between the upstream and downstream regions becomes markedly more pronounced. Ultimately, the thickness of the permafrost curtain formed during the entire freezing period does not meet the design requirement of 1.5 m.

The seepage velocity significantly influences the development of both upstream and downstream permafrost curtains. As the seepage velocity increases, the degree of unevenness between the upstream and downstream permafrost curtains also escalates. This phenomenon primarily occurs because, during the freezing process, the cold generated by the upstream freezing pipe is continuously transported downstream by the flow of groundwater. As seepage velocity increases, a greater amount of cold is conveyed downstream, leading to a continuous reduction in the freezing range of the upstream while simultaneously expanding the freezing range downstream. Consequently, this results in a pronounced discrepancy in the unevenness of the upstream and downstream permafrost curtains.

To assess the development of the permafrost curtain in the freezing project, the inter-circle time and the thickness of the permafrost curtain are key parameters to be examined. Figure 12 illustrates the change in intersection time of the − 1 °C and − 10 °C isotherms under varying seepage velocities. As depicted in Fig. 12, both the − 1 °C and − 10 °C isotherms increase with rising seepage velocity; notably, when the seepage velocity exceeds 1.3 m/d, the growth rate of the inter-circle time accelerates further. Furthermore, when the seepage velocity surpasses 2.3 m/d, the intersection time between the − 1 °C and − 10 °C isotherms experiences an ‘explosive’ growth.

Fig. 12
Fig. 12
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Plot of intersection time of isotherms at − 1 °C and − 10 °C versus seepage rate.

Figure 13 illustrates the variation in effective permafrost curtain thickness both upstream and downstream at different seepage velocities. As shown in the figure, seepage velocity significantly influences the development of the effective permafrost curtain in both directions. With increasing seepage velocity, the thickness of the upstream effective permafrost curtain diminishes; notably, at a seepage velocity of 3.3 m/d, the effective thickness of the upstream permafrost curtain falls below the design thickness by 1.5 m. In contrast, the effective thickness of the downstream permafrost curtain increases. This phenomenon occurs primarily because the cold generated by the upstream freezing pipe is transported downstream by groundwater flow, thereby enlarging the freezing range downstream.

Fig. 13
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Effective permafrost curtain thickness upstream and downstream of different seepage rates.

Under seepage conditions, the effective thickness of the downstream permafrost curtain is greater than that of the upstream. As seepage velocity increases, the thickness of the permafrost curtain upstream decreases. Groundwater flow transports the cold released by the freezing pipe beyond the freezing area, thereby enlarging the downstream freezing region. This results in a slight increase in the thickness of the downstream permafrost curtain while simultaneously weakening the overall freezing effect, thus leading to a decline in the total thickness of the permafrost curtain. Therefore, during future field construction, attention should be directed towards the upstream freezing area, and if necessary, an appropriate number of freezing pipes should be added upstream.

Conclude

In this study, the engineering scenario of horizontal freezing for reinforcing the soil body surrounding a two-lane tunnel is examined. Utilizing the finite element software COMSOL, a three-dimensional hydrothermal coupling model is developed for a detailed analysis. The study primarily concludes that:

  1. 1.

    Through the analysis of the temperature field cloud map and isotherm map of the two-direction profiles of the numerical simulation results, it is evident that during the pre-freezing period prior to the formation of the permafrost curtain, groundwater flowed normally. Initially, the freezing area extended into the surrounding region at 15 days into the freezing process, with the freezing area primarily diffusing around the circular freezing pipe. Once the − 1 °C isotherm completed its circular intersection, the outline of the permafrost curtain was essentially formed, which began to impede the flow of groundwater. Consequently, groundwater started to flow around the permafrost curtain. In the mid-freezing stage, the freezing area expanded, and at this juncture, there was no groundwater flow within the internal area of the two-lane tunnel. In the later stages of freezing, the permafrost curtain further developed towards the sides of the freezing tubes, and the freezing area continued to expand, becoming stable and firm.

  2. 2.

    The overall trend of − 1 °C permafrost can be visualized through the − 1 °C contour surface cloud map. During the pre-freezing period, the freezing tube is encircled by permafrost columns with a small cross-section. In the middle stage of freezing, the − 1 °C permafrost curtain exhibits an expanding trend, with large areas of permafrost associations emerging around the left and right tunnel lines, and the outline of the permafrost curtain becoming largely defined. In the late stage of freezing, a complete and stable − 1 °C permafrost curtain is established, which has a certain water-stopping effect. The permafrost curtain develops steadily, extending to both the inner and outer sides, growing thicker and thicker, and showing a tendency to expand. The surface of the contour is not ‘cogwheel-shaped’, and the surface appears smoother.

  3. 3.

    The effective thickness of the permafrost curtain is significantly affected by seepage, with the upstream sides of the left tunnel experiencing the greatest influence in the 45°, 90°, and 135° directions, and the downstream sides of the right tunnel showing less impact. Notably, the development trend of the effective permafrost curtain thickness in the 90° direction of both the left and right tunnels is opposite. Specifically, the effective permafrost curtain thickness in the 90° direction of the left tunnel grows the slowest, reaching a final thickness of 2.3246 m, whereas the thickness in the 90° direction of the right tunnel increases the fastest, culminating in a final thickness of 4.0639 m. This discrepancy amounts to 1.7393 m. Despite these variations, the effective permafrost curtain thicknesses in each direction of both tunnels meet the design criteria at this seepage rate.

  4. 4.

    As time progresses, the cold quantity from the freezing tubes continues to be released, leading to a decrease in the temperature of each freezing zone. The temperature distribution over the entire main surface 1 exhibits a ‘double W’ shape, following a temporal and spatial change pattern. At equal freezing times, the cooling efficiency of zone C is the highest, with zones A and E showing comparable efficiency, and zones B and D having the lowest efficiency, thus C > A = E > B = D. Conversely, the temperature distribution over the entire main surface 2 presents a ‘W’ shape at any given time and spatial change, with zones F and H exhibiting the highest and similar cooling efficiencies. Zone G, situated within the left line tunnel, remains unaffected by the cold, resulting in the lowest cooling efficiency among the zones, hence F = G > H.