Numerical Analysis of Temperature Field Distribution Characteristics of Surrounding Rock in Cross-Line Subway Tunnels

1  Introduction

Metro systems, characterized by their advantages of “high capacity, traffic congestion relief, and rapid punctuality,” are increasingly pivotal in facilitating rapid urban economic growth. While offering convenient transportation options for commuters, metro operations also face significant thermal environmental challenges within tunnels. During operation, metro trains generate considerable amounts of heat; a portion of this heat is vented outdoors through piston airflow, but the remaining heat accumulates within the tunnel. Research shows that surrounding rock absorbs 25%~40% of the total heat dissipation during subway operations [1]. When tunnel temperature decreases, the heat stored in the rock is released, resulting in increased energy consumption for the subway air conditioning system. Field monitoring conducted by Cockram and Birnie [2] on the London Underground found that over five years of operation, the surrounding rock developed a 5-m-thick layer of high-temperature heat storage. Initially, Japan’s subway tunnels were designed without adequately considering thermal environmental issues, which later required retrofitting of air conditioning systems, leading to a temporary shutdown [3]. In the U. S. subway system, nearly half of the annual expenditures are devoted to environmental control [4]. Similarly, the initial designs for Beijing Subway Lines 1 and 2 also displayed shortcomings in addressing thermal hazards due to heat accumulation in the tunnel rock mass during prolonged operation [5]. The continuous rise in temperature within the tunnel rock mass not only elevates environmental control costs but may also jeopardize train operation.

Research on heat transfer in subway tunnels primarily utilizes field measurements, scaled-down model experiments, and numerical simulations [6–10]. Field measurements and scaled-down model experiments are instrumental in monitoring the temperature distribution characteristics of tunnel rock mass. A weighted analysis of this data elucidates the impact of soil properties and the thermal characteristics of the tunnel lining on heat transfer within the rock mass. Additionally, some researchers have employed simulation software to model temperature field distributions in the tunnel air and surrounding rock under various boundary conditions. These simulations offer theoretical insights that inform the establishment of subway source heat pumps and the design of air conditioning systems, contributing to more efficient thermal management in the metro environment.

Zhu et al. [11] investigated the thermophysical parameters of the surrounding rock, the distribution patterns of ground temperature, and the temporal evolution of the temperature field in Lanzhou Metro tunnels through a combination of field monitoring and numerical simulations. Their results indicate that the temperature and its gradient, as well as the thermal penetration depth of the surrounding rock, are positively correlated with the ambient temperature inside the tunnel and the duration of heat exchange, while negatively correlated with the distance from the tunnel wall. Ren et al. [12] performed long-term observations of ground temperatures across four typical landform areas along the Xi’an Metro line, investigating the location of the thermostatic layer and ground temperature distribution patterns in different landform areas during the spring. Their conclusions indicated that ground temperature is influenced by atmospheric temperature; however, this effect exhibits a lag and limited scope, exerting minimal impact on soil located below the initial water table. Wang et al. [13] constructed a scaled model to test the distribution of surrounding rock temperature in subway tunnels over 17 years under periodic air temperature variations. This model allowed them to determine the heat storage and release characteristics of the surrounding rock in subway tunnels during long-term operation. Zhao et al. [14] analyzed the heat transfer characteristics of subway tunnel walls through field monitoring and calculated the specific heat absorption ratio (SHAR) of the surrounding rock and soil. They identified the influence patterns of train operational conditions, tunnel air temperature, and surrounding rock temperature fields on the tunnel wall temperature field. Zhao et al. [15] developed a convective heat transfer model for deep-buried tunnels with non-steady wall temperatures based on measured tunnel temperature data. This model predicts changes in tunnel air temperature, indicating that during the first 30 days of subway operation, the range of affected surrounding rock extends approximately 3 m due to heat transfer. Furthermore, they found that heat transfer in the surrounding rock of the tunnel is a non-steady-state process, with its heat transfer capacity stabilizing over time. Li et al. [16] experimentally determined the fracture parameters of tunnel rock mass and calculated the equivalent thermal conductivity of the surrounding heterogeneous rock, providing a theoretical reference for simulating temperature field distributions in irregular subway tunnel rock masses.

Song and Chen [17] solved the thermal equilibrium equation of the soil temperature field using a dimensionless mathematical model. Their results indicate that the dimensionless temperature of the tunnel surrounding rock gradually decreases with increasing Fourier number and approaches a steady-state value. A higher Biot number (BiS) for tunnel airflow signifies more intense convective heat transfer between the tunnel wall and the air inside, resulting in a shorter time for the dimensionless wall temperature to reach a steady-state value. Huang et al. [18] conducted a study combining wind tunnel tests with numerical simulations to investigate piston wind speed variations during different subway operation phases, providing a theoretical foundation for predicting temperature distribution in subway tunnel surrounding rock. Zhang and Li [19] developed a two-dimensional transient heat transfer model utilizing the Green’s function method and finite element method. This model accommodates the heterogeneity of thermal-physical properties in the surrounding rock of subways, enabling effective analysis of heat transfer issues and accurate prediction of tunnel temperatures. Vasilyev et al. [20] noted through numerical simulations of the Moscow Metro that a heat storage zone forms around subway tunnels during long-term operation, exhibiting temperatures higher than those of natural soil. This zone is influenced by tunnel air temperatures, with minimum temperatures recorded at 4°C above natural soil temperatures and maximum temperatures ranging from 17.3°C to 20.3°C. Hu et al. [21] numerically analyzed the influence of factors including the thermal-physical properties of the surrounding rock, initial ground temperature, and tunnel temperature on the heat transfer range within the surrounding rock. Their results indicated that the heat transfer range in the subway surrounding rock is significantly affected by these thermal-physical properties and initial conditions, while the heat transfer coefficient between air and the lining has a relatively minor impact on the surrounding soil’s temperature distribution. Sun et al. [22] examined the effects of subway tunnel wall temperature, surrounding soil temperature, and wall roughness on tunnel air temperature. They developed an analytical heat transfer model for single-track tunnels that accurately predicts air temperatures within deeply buried subway tunnels. Wei et al. [23] conducted a numerical analysis on how train speed, train running intervals, and external ambient temperature affect the temperature field within subway tunnels. Their findings demonstrated that train-induced airflow significantly influences the tunnel temperature field, showing a positive correlation with train speed and a negative correlation with train intervals, with effects becoming more pronounced as ambient temperature decreases. Zhang and Li [24] utilized numerical simulation and response surface modeling to analyze five key factors impacting subway tunnel temperature: passenger volume, car weight, regenerative braking system efficiency, soil thermal conductivity, and soil heat capacity. They established a mathematical model for predicting subway tunnel temperature, providing valuable references for ventilation system design and operational strategy formulation. Cao et al. [25,26] conducted numerical analyses of heat and moisture transfer characteristics in rock masses of monorail subway tunnels under periodic boundary conditions across different regions. Results indicated that the surrounding rock undergoes a dynamic expansion phase in the heat storage zone during the first 12 years, followed by a dynamic stabilization phase. During this period, the thickness of the tunnel affected by heat transfer expands from 14.36 m to 48.84 m, while the thickness affected by moisture transfer increases from 6.81 to 23.51 m. Xu et al. [27,28] investigated the ventilation and heat exchange characteristics in single-track high-temperature underground tunnels through numerical simulations and analytical methods, proposing mathematical models capable of effectively predicting rock mass temperatures in deeply buried, high-temperature tunnels. Increasing research attention has also been directed toward the development of energy tunnels and subway source heat pumps, proposing methods to effectively utilize heat from the tunnel rock mass by modifying lining structures [29–32]. Magdy et al. [33,34] investigated the thermal performance of parallel energy tunnels operating simultaneously. The study investigated the influence of groundwater flow through surrounding sandy soil on tunnel heat transfer when the energy tunnel gap ranges from 0.5D to 4D. The results indicated that as the gap increases, the difference in heat exchange power between the upstream and downstream tunnels gradually diminishes.

In summary, research on predicting the temperature field of surrounding rock in subway tunnels has reached a relatively mature stage. Numerous scholars have elucidated the heat transfer characteristics within the tunnel surrounding rock through field measurements, experimental studies, and numerical simulations. However, existing research on tunnel rock mass heat transfer primarily focuses on single-tunnel structures, with limited studies addressing the heat transfer characteristics of intersecting subway tunnels. With the rapid advancement of the global economy, the underground rail sector has also experienced swift development. The design of intersecting underground rail lines has become increasingly commonplace in practical engineering projects, presenting ever more complex challenges for research into heat transfer processes and thermal environments both within and outside tunnels. This paper utilizes COMSOL Multiphysics to establish a three-dimensional heat transfer model. Based on the finite element method, it compares and analyzes the impact characteristics of different vertical spacing and intersecting angles on tunnel heat transfer properties. It reveals the variation patterns of the temperature field in the tunnel rock mass under line-crossing conditions during the long-term operation of the railway system. The growth in subway lines inevitably corresponds to more complex climatic and geological conditions. Therefore, this study also comparatively investigates the effects of climatic and geological conditions on the heat transfer characteristics of tunnel surrounding rock, revealing the temperature field distribution characteristics under varying climatic and geological scenarios. This research provides a theoretical basis for addressing the temperature rise of the surrounding rock caused by long-term tunnel operation and the ensuing thermal environmental issues. It holds significant reference value for optimizing subway air-conditioning system operation plans and subway line design.

2  Physical and Mathematical Models

2.1 Physical Model and Assumptions

To avoid inaccuracies in predicting tunnel heat transfer performance resulting from excessive simplification in similarity experiments, a three-dimensional geometric model was established using COMSOL finite element analysis software, based on the actual dimensions and burial depth of the subway cross-section [25]. The model dimensions were scaled 1:1 relative to the actual tunnel, with the soil domain set at 1000 m × 1000 m × 70 m. The shape of the tunnel exerts a negligible influence on heat transfer within it. Based on actual engineering data, the tunnel domain is simplified as two circular passages, each with a radius of 3.3 m, and the tunnel lining structure has a thickness of 0.3 m [32]. The tunnel interior is assumed to be filled with air, with the airflow direction parallel to the tunnel axis. The impact of auxiliary equipment within the tunnel on ventilation is disregarded. The vertical distance between the shallow-buried tunnel K1 and the deep-buried tunnel K2 is defined as “h”. The shallowly buried tunnel crown is located 10 m below ground level. A simplified representation of the cross-line subway tunnel model is illustrated in Fig. 1.

Figure 1: Schematic diagram of intersecting tunnel model.

The platform screen doors in the subway system significantly reduce air heat exchange between the tunnel and the station platform. Therefore, heat transfer through the tunnel rock mass is simplified to a convective and conductive heat transfer problem within a three-dimensional enclosed space. The following assumptions were made during model development: (1) The specific heat capacity and thermal conductivity of soil are isotropic, remaining constant regardless of temperature or location; (2) Temperature changes due to migration and phase transitions in the liquid phase of porous media are neglected; (3) Porosity and density of rock and soil are constant values, and soil does not deform due to moisture migration.

2.2 Mathematical Model

2.2.1 Governing Equations

During train operation, systems such as air conditioning and braking equipment generate substantial heat. A portion of this heat accumulates within the tunnel and cannot be dissipated immediately. This heat is absorbed by the tunnel rock mass primarily through thermal conduction at the tunnel walls and is subsequently released back from the rock mass when the tunnel air temperature decreases. Due to the complexity of train operations—including acceleration, deceleration, and varying traction conditions that correspond to different air conditioning modes—and the challenges involved in accurately quantifying heat generation within the tunnel and the effects of piston airflow, the tunnel heat transfer model is simplified. Under this simplification, heat transfer in the tunnel intersection section is primarily governed by convective heat exchange between the tunnel air and the tunnel walls, coupled with conductive heat transfer through the tunnel rock mass. The governing equations employed in this study are as follows:

The study employs the continuity equation, momentum equation, and energy equation for incompressible fluids to solve for air flow and heat transfer processes within the tunnel.

(1) Continuity equation:

∇⋅(Aρf uf)=0(1)

(2) Momentum equation:

ρf∂uf∂τ=−∇Pf−12fDρfdh|uf|uf(2)

(3) Energy equation:

ρfACP,f∂Tf∂τ+ρfACp,fuf∇Tf=(AλW∇T)+12fDρfAdh|uf|uf2+Qwall(3)

where A is the cross-sectional area of the subway tunnel; ρf is the density of the air in the tunnel, kg/m3; uf is the velocity of the air in the tunnel, m/s; Cp,f is the constant-pressure heat capacity of the air in the tunnel, J/(kg·°C); Tf is the temperature of the air in the tunnel, °C; τ is the time, s; λf is the thermal conductivity of the air in the tunnel, W/(m·K); ∇ is the Hamiltonian operator.

The heat conduction equation for the tunnel rock mass is as follows:

ρsCp,s∂Ts∂τ=∇⋅(λs∇Ts)(4)

where ρs is the density of the tunnel surrounding rock, kg/m3; Cp,s is the constant pressure heat capacity of the tunnel surrounding rock, J/(kg·°C); Ts is the temperature of the tunnel surrounding rock, °C; λs is the heat conduction coefficient of the tunnel surrounding rock, W/(m·K).

2.2.2 Initial and Boundary Conditions

To enable a more precise analysis of the relationship between temperature and soil depth, the initial temperature of the soil domain was uniformly set to the constant-temperature layer temperature of 15.6°C [17]. The heat transfer between tunnel air and tunnel wall involves the third type (convection) heat transfer boundary condition. This study employs the Fluid-Solid Coupling Module in COMSOL to model the air domain and lining domain, fully accounting for their mutual interaction. Train operation dynamics exhibit complexity, and tunnel temperatures are influenced by factors such as piston airflow generated by moving trains, making precise calculations challenging. This study estimates tunnel temperatures based on Reference [35] and typical annual meteorological data for the region, with results shown in Fig. 2. The upstream temperature calculated by this method comprehensively considers factors including heat generated by train operation, heat dissipation via piston airflow, and wall heat exchange. The convective heat transfer coefficient correlates with the Reynolds number, and varying train operating conditions cause non-steady airflow within the tunnel. For simplification purposes, an equivalent convective heat transfer coefficient of 8.38 W/(m2·K) is adopted [36]. The governing heat transfer equation is expressed as follows:

q=−λ(∂t∂y)w=h(tw−tf)(5)

where q is the heat flux of the surrounding rock, kW/m2; (∂t∂y)w is the temperature gradient of the fluid on the wall surface of the tunnel paste, °C/m; tw is the average temperature of the tunnel wall, °C; tf is the average air temperature inside the tunnel, °C.

Figure 2: Air temperature inside and outside the tunnel.

The upper boundary of the soil domain is subjected to multiple boundary conditions, encompassing both natural convective heat transfer and forced convective heat transfer. Xi’an’s near-surface wind speed averages approximately 1.3 m/s, the near-surface convective heat transfer coefficient is set at 9.56 W/(m2·K). Considering that the other boundaries of the soil domain are located far from the tunnel, boundary effects associated with adiabatic conditions can be reasonably disregarded. Therefore, these distant soil domain boundaries are modeled as adiabatic. Additionally, the tunnel wall surface is assigned a no-slip velocity boundary condition to accurately represent the interaction between the tunnel wall and the air flow within the computational domain.

2.3 Model Parameters

To investigate the impact of intersecting tunnel alignments on rock mass heat transfer, the soil in the tunnel heat transfer model is treated as a homogeneous soil domain. Based on monitoring data from existing literature and geological survey data, soil properties are approximated as constant values [35], as shown in Table 1.

2.4 Mathematical Model Validation

Reference [14] documents experimental measurement data of rock mass temperatures within Shanghai Metro tunnels. The tunnel’s burial depth and geometric configuration in this model bear considerable similarity to the Xi’an operational conditions. Based upon the boundary and initial conditions defined in this reference, a decade-long simulation study of Shanghai Metro tunnel rock mass temperatures was conducted to validate the mathematical model’s accuracy. The comparison between the simulated data and experimental measurements is presented in Fig. 3. It can be observed that during the first year of operation, the maximum relative error between the simulated rock temperature values within the 0~20 m zone from the tunnel wall and the experimental results reported in the literature was 2.9%, occurring at a distance of 1 m from the tunnel wall. The overall average relative error of the tunnel rock temperature was 1.13%, as illustrated in Fig. 3a. By the seventh year, the comparison between the simulated rock temperatures in the 0~20 m zone and the measured values from the literature experiments is shown in Fig. 3b. The overall average relative error for tunnel rock temperatures increased slightly to 3.47%. Part of this discrepancy can be attributed to the fact that the measured values were derived from scaled-down experiments, where the limited testing space may have introduced boundary effects. Additionally, the simulation idealistically neglected the contact thermal resistance between the lining structure and the soil. Nonetheless, the comparison indicates that the results obtained through the three-dimensional numerical simulation exhibit an error of less than 7% relative to the experimental values reported in the literature, demonstrating excellent agreement and satisfying the accuracy requirements for engineering calculations.

Figure 3: Model validation [14].

2.5 Mesh Independence Verification and Time Step Determination

Generally, higher grid density and shorter time steps yield more accurate computational results. However, once both the time step and grid density reach certain critical thresholds, the model’s computational accuracy becomes effectively independent of further increases in these parameters. Additionally, using shorter time steps alongside larger grid sizes can cause computational time to increase exponentially, impacting efficiency. To balance accuracy and computational efficiency, multiple numerical experiments were conducted on the model under the 7 m spacing scenario for the cross-city subway tunnel. A free tetrahedral mesh was employed for discretisation. The ground surface and inner/outer walls of the concrete lining were refined based on the physical field and boundary conditions. Five different mesh configurations were tested, and the temperature data at the intersection center point (coordinates x = 500 m, y = 500 m, z = −20.1 m) were extracted at the 10th year for comparative analysis, as shown in Fig. 4. The results indicate that when the number of grid cells exceeds 4 million and the time step is less than 5 days, the computed temperature values stabilize and remain essentially unchanged. Therefore, subsequent calculations adopt a grid size of 5 million cells and a time step of 5 days to ensure a reliable balance between computational accuracy and efficiency, as shown in Fig. 5.

Figure 4: Grid independence verification and time step determination.

Figure 5: Detailed grid division diagram.

3  Temperature Field Analysis of the Intersecting Metro Tunnels

3.1 Annual Variation of Rock Mass Temperature Field in a Single Metro Tunnel

Based on the aforementioned boundary conditions, and incorporating climate data and soil physical parameters specific to the Xi’an region, a three-dimensional heat transfer model of a single tunnel was established. The burial depth of the tunnel crown is located 10 m below the soil surface. Temperature monitoring points are arranged at depths of 1, 2, 4, and 6 m below the tunnel floor slab. This configuration facilitated the observation of seasonal periodic variation patterns and spatial distribution characteristics of the tunnel rock mass temperature, as illustrated in Fig. 6.

Figure 6: Monitoring point layout of single tunnel.

Fig. 7 reveals that continuous train operations lead to a gradual increase in tunnel wall temperatures. Throughout the operation, the tunnel walls experience repeated annual cycles of heat absorption and release. In summer, when the tunnel air temperature exceeds the wall surface temperature, the tunnel walls absorb heat from the air, causing the temperature of the surrounding rock to gradually rise until the adjacent soil temperature reaches its peak. Conversely, in winter, when the tunnel air temperature is lower than the wall surface temperature, the tunnel walls release heat to the air, resulting in a decrease in surrounding rock temperature that eventually stabilizes after a certain period. During these cycles, heat is primarily concentrated near the tunnel, with surrounding rock temperatures steadily decreasing as the distance from the tunnel wall increases. By the 10th year of operation, the soil temperature near the tunnel stabilizes at approximately 1 m from the wall, reaching a maximum temperature of 26.27°C at this location.

Figure 7: Temperature profile of single tunnel surrounding rock.

3.2 Annual Variation of Rock Mass Temperature Field in Intersecting Subway Tunnels

A reference line, designated as Line A, is defined to connect the points (500, 500, 0) and (500, 500, −37). This line is perpendicular to the x-y plane and intersects both tunnel walls at their midpoints. Temperature observation points are established along Line A to analyze the distribution characteristics of rock mass temperatures. Vertically, Line A is divided into three distinct zones for the placement of temperature measurement points. Along the outer sides of both tunnels, longitudinal observation points are arranged at 5-m intervals—for example, points labeled a-c on one side and i-k on the other. Between the two tunnels, observation points are positioned 1 m away from each tunnel wall as well as at the midpoint between the tunnels; typical examples include points e, g, and f. Horizontally, detection points are distributed at horizontal distances of 0, 5, 10, and 15 m away from the tunnel side walls, such as points l-m and p-s, to enable detailed observation of seasonal variations in horizontal temperature transfer within the surrounding rock mass. The spatial arrangement of these temperature detection points is illustrated in Fig. 8.

Figure 8: Monitoring point layout for intersecting tunnel.

To investigate the impact of intersecting subway tunnel layouts on the heat transfer characteristics of the surrounding rock mass, monitoring points were installed either at the base of the single tunnel or at a depth of 6 m below the base of the shallow tunnel when the vertical spacing between intersecting tunnels was 12 m. The temperature trends of the surrounding rock over the subway operation period were monitored, with the results presented in Fig. 9. For the single-track tunnel, the temperature measured at 6 m below the tunnel floor exhibited periodic fluctuations over the 10 years, with an overall increase of 3.94°C during that decade. After the 10th year, the temperature increase trend plateaued, with the peak temperature rising by only 0.11°C in the 10th year. In contrast, for the intersecting tunnel, the temperature at the central monitoring point also displayed periodic fluctuations over the 10 years, but with a more pronounced increase of 7.67°C. Similarly, after the 10th year, the rock mass temperature increase trend flattened, with the peak temperature rising by 0.1°C in that year. Comparison of temperature change trends at both monitoring points indicates that intersecting tunnel layouts lead to greater heat accumulation in the rock mass, affecting the heat transfer characteristics and elevating the peak temperature of the tunnel rock mass. Additionally, the figure shows that the peak of the periodic temperature variation in the intersecting tunnel occurs approximately 70 days earlier than in the single-track tunnel. This suggests that, at an equivalent distance, the intersecting tunnel arrangement reduces the lag effect in heat transfer within the surrounding rock.

Figure 9: Rock mass temperature in single-line and crossing tunnels.

By establishing observation points within the tunnel intersection zone, the temporal and spatial distribution characteristics of the surrounding rock temperature field throughout the operational period can be effectively monitored, as depicted in Fig. 10. The data reveal that the extent of elevated temperatures in the surrounding rock continuously expands over time. After approximately 10 years of operation, the rock temperature approaches a dynamic equilibrium, accompanied by a notable slowdown in the expansion rate of the heat-affected zone. This indicates that the heat absorption and release mechanisms within the tunnel rock mass have attained a balance, leading to temperatures within this region fluctuating seasonally without further increases in their peak values. During the first five years, the configuration of the intersecting tunnels does not significantly alter the spatial reach of the heat transfer influence. Nevertheless, a localized high-temperature zone emerges at the center of the rock mass where the tunnels intersect, with temperatures exceeding those of the adjacent soil due to intensified heat transfer effects. After this initial period, temperatures within this zone continue to rise, and the affected area expands steadily. By around the tenth year, the peak temperature in the intersection zone reaches its maximum, and the rate of expansion of the temperature-affected area begins to diminish. These observations suggest that the horizontal temperature distribution of the rock mass in intersecting tunnel regions is strongly influenced by tunnel alignment configurations. In the short term, both the temperature within the rock mass at the intersection and the spatial extent of the heat-affected zone increase. However, by the tenth year of operation, the heat transfer process stabilizes, with peak temperatures plateauing and the growth of the affected area slowing, indicating a state of dynamic thermal equilibrium within the rock mass.

Figure 10: Distribution characteristics of rock mass temperature field over operating time (z = −20.1 m).

An observation surface was established at x = 500 to monitor changes in the temperature distribution of the tunnel rock mass along the depth direction during subway operation, as illustrated in Fig. 11. It can be observed that during the long-term operation of the underground railway system, the area of soil affected by heat transfer within the intersecting tunnels continues to expand. During the initial five years, temperature variations in the rock mass surrounding the two tunnels remained relatively independent, with fluctuations confined to a small radius and influenced by seasonal cycles. During this period, the temperature of the rock mass near the intersection area increased by only 2~5°C. By the fifth year, the two high-temperature zones formed by tunnel heat conduction began to merge and expand continuously. By the tenth year, a local dynamic thermal equilibrium had formed within this high-temperature zone, indicating that peak temperatures within it would no longer rise. Throughout the heat transfer process, the affected zone in the surrounding rock expanded uniformly at an initial stage, with the expansion rate gradually slowing over time. Taking the 18°C isotherm as an example, it expanded from approximately −39 to −47.5 m between the fifth and fifteenth years, whereas it only extended from −47.5 m to around −49.5 m between the fifteenth and twenty-fifth years. These findings confirm that tunnel heat conduction significantly influences the depth-wise temperature distribution within the surrounding rock. Thermal interactions between tunnels cause rock temperatures in the intersecting zones to rise continuously until dynamic equilibrium is reached. The expansion rate of the affected zone gradually slows over time.

Figure 11: Distribution characteristics of rock mass temperature field over operational time (x = 500 m).

3.3 Influence of Vertical Spacing between Intersecting Subway Tunnels on Rock Mass Temperature Field

Vertical clearances of 2, 7, 12, and 15 m were selected for intersecting subway tunnels, corresponding to actual operational clearances observed in existing tunnel systems. A comparative analysis of the rock temperature distribution characteristics associated with each vertical spacing was conducted to provide valuable references for operational strategies and environmental control design in intersecting subway sections. Using the previously established air temperature conditions as boundary inputs to the soil domain, the rock temperature field distribution within the subway tunnel sections was simulated after 20 years of continuous operation for each specified vertical clearance. This long-term simulation enabled an assessment of the influence of vertical tunnel spacing on the thermal behavior of the surrounding rock mass under realistic operational scenarios.

A 1000 m × 1000 m observation grid was established within the tunnel intersection zone to analyze the horizontal temperature distribution at the peak of dynamic thermal equilibrium resulting from rock mass heat transfer for vertical clearances of 2, 7, 12, and 15 m, as illustrated in Fig. 12. Taking the rock temperatures at the 20th year as a representative example, it can be observed that once the tunnel heat transfer reaches dynamic equilibrium, the spatial extent of the rock temperature-affected zones across different vertical spacing is remarkably similar, with only minor variations in temperature magnitudes within these zones. At a vertical spacing of 2 m, temperatures in the intersection zone range between 26°C and 30°C, and the high-temperature area is significantly larger compared to other scenarios. Conversely, at spacing of 12 and 15 m, temperatures generally remain around 20°C, with elevated temperatures concentrated primarily at the center of the intersection zone. Under these conditions, the vertical spacing of the tunnels exerts negligible influence on both the surrounding rock temperature and the spatial extent of the affected zone. From these observations, it can be concluded that while variations in vertical tunnel spacing impact the temperature values on the horizontal plane of the tunnel rock mass to varying degrees, the changes in the overall affected thermal range are minimal. Moreover, when the vertical clearance exceeds 12 m, the temperature distribution within the intersection zone is no longer significantly influenced by further increases in vertical spacing.

Figure 12: Temperature field distribution in the intersecting area at different vertical spacing (x-y plane).

An observation plane at x = 500 was used to examine the vertical distribution of tunnel rock mass temperature upon reaching dynamic thermal equilibrium under different vertical spacing between intersecting subway tunnels, as illustrated in Fig. 13. It can be observed that when the tunnel spacing is 2 m, the area of rock mass affected by heating is significantly smaller than other spacing scenarios. The 18°C isotherm extends only to approximately −42.5 m, which is about 15 m narrower than the range under the 15 m spacing condition. However, the temperature within this affected zone is markedly higher than that at larger spacings, with rock temperatures at the exact center of the two tunnels reaching up to 33°C. As the distance between tunnels increases, the heat-affected area expands, with temperatures within this zone elevated by 10°C~14°C above the initial temperature of the isothermal soil layer. This phenomenon occurs because increased tunnel spacing allows the heat-affected zone to spread over a larger volume of rock mass. Consequently, heat accumulation near the tunnels diminishes, as a portion of the thermal energy is transferred along the temperature gradient toward rock masses farther from the tunnels. This redistribution leads to a reduction in peak temperatures within approximately ten meters of the tunnels as spacing increases. From these observations, it can be concluded that the vertical temperature distribution within the tunnel rock mass is strongly influenced by the distance between tunnels. Specifically, a smaller vertical spacing between intersecting tunnels results in a smaller area of rock mass affected by heat transfer. This causes heat to concentrate and accumulate, leading to an increase in tunnel rock mass temperature. Conversely, a larger spacing promotes heat diffusion, expanding the heated area and thereby reducing peak temperatures near the tunnels.

Figure 13: Temperature field distribution in the rock mass of the intersection zone at different vertical spacing (x = 500 m).

Based on the fundamental principle of thermal equilibrium between tunnel air and surrounding soil, the annual variation in rock mass temperature within the intersection zone of cross-tunneling metro systems at different vertical spacing can be predicted, as illustrated in Fig. 14. The data clearly indicate that smaller vertical spacing between tunnels corresponds to significantly higher rock mass temperatures in the intersection zone, with thermal transfer processes reaching dynamic equilibrium earlier compared to larger spacing. For example, at a vertical spacing of 7 m, the peak temperature at the midpoint of the tunnel intersection is first observed on day 340, reaching 23.13°C. By the seventh year of subway operation, heat transfer approaches dynamic equilibrium, with annual increases in rock temperature diminishing to less than 0.1°C and stabilizing at a final peak temperature of 25.9°C. In contrast, at a 15 m spacing, the midpoint temperature peaks later, on day 465, reaching a lower maximum of 17.7°C. Dynamic equilibrium is not attained until the 16th year, with the final peak temperature stabilizing around 23°C. These findings demonstrate that under conditions of intersecting tunnels, the vertical spacing between tunnels significantly influences both the magnitude of the surrounding rock temperature and the time required to achieve thermal equilibrium. Specifically, the closer the spacing between intersecting tunnels, the higher the peak temperature attained in the surrounding rock mass and the shorter the time needed for the heat transfer process to reach a dynamic equilibrium state.

Figure 14: Trend of rock mass temperature at the intersection center over time under different vertical spacing.

The distribution of tunnel rock mass temperature on the horizontal plane is illustrated in Fig. 15. Fig. 15a,b shows observation results at points located 5 m and 15 m horizontally from the sidewall of the shallow tunnel, respectively. At point m (5 m from the shallow tunnel sidewall), when the vertical spacing between tunnels is 2 and 7 m, the rock mass temperature exhibits periodic increases, ultimately reaching dynamic equilibrium at approximately 22°C and 21.5°C, respectively. When the vertical spacing exceeds 12 m, both the magnitude and the rate of increase in rock temperature are no longer significantly influenced by the vertical spacing. Additionally, due to seasonal variations in tunnel air temperature, smaller vertical spacing corresponds to greater temperature fluctuations at this location. At point o (15 m from the shallow tunnel sidewall), the rock temperature is generally lower than at point m. Being farther from the intersection center, less heat is transferred to point o, and thus the rock temperature has not yet reached dynamic equilibrium even after 25 years, continuing to exhibit an upward trend. Furthermore, when the vertical spacing is large, the temperature at this point fluctuates significantly due to the influence of surface temperatures. Fig. 15c,d presents observation results at points 5 and 15 m horizontally from the sidewall of the deep-buried tunnel, respectively. At point q (5 m from the deep-buried tunnel sidewall), the temporal temperature trend is similar to that observed at point m. Located close to the deep-buried tunnel and largely unaffected by surface convective heat exchange, the temperature at point q is approximately 0.5°C higher overall than at point m. Similarly, when the tunnel spacing is 12 or 15 m, the rock mass temperature variation pattern remains consistent and appears unaffected by the vertical spacing. At point s (15 m from the deep-buried tunnel sidewall), the surrounding rock temperature is generally higher than at point o. Specifically, when the tunnel spacing is 2 m, the surrounding rock temperature exhibits the greatest fluctuation and reaches the highest peak temperature upon reaching dynamic equilibrium. When the tunnel spacing is 12 or 15 m, the temperature rises steadily without pronounced periodic variation, as temperature fluctuations at this point are primarily driven by the tunnel air temperature. In summary, both the magnitude and fluctuation amplitude of rock surface temperature values are correlated with the vertical spacing between intersecting tunnels. A smaller vertical spacing results in larger temperature fluctuations and higher peak temperatures within the rock mass. However, when the vertical spacing exceeds 12 m, temperature magnitude and fluctuations no longer exhibit significant variation with respect to tunnel spacing.

Figure 15: Trend of surrounding rock temperature variation with operating time at different vertical spacing.

3.4 Influence of Intersecting Angle on Rock Mass Temperature Field in Metro Tunnels

Under the condition of a 7-m vertical spacing between intersecting subway tunnels, intersection angles of 45°, 60°, and 90° were selected, corresponding to the actual operating angles observed in existing domestic tunnels. This configuration was established to facilitate a systematic analysis of the impact of intersection angles on heat transfer characteristics within the surrounding rock mass. Employing the previously defined air temperature profiles as boundary conditions for the soil domain, numerical simulations were conducted to model the temperature distribution of the surrounding rock in the tunnel sections over a continuous 20-year operational period for each specified intersection angle. This approach enabled a comparative assessment of how varying tunnel crossing geometries influence the thermal response and evolution of the rock mass temperature field under realistic operational conditions.

A 1000 m × 1000 m observation grid was established centered at the tunnel intersection point to analyze and compare the temperature distribution characteristics of the rock mass at dynamic thermal equilibrium for intersection angles of 45°, 60°, and 90°. The simulation results are presented in Fig. 16a–c. The findings indicate that when the intersection angle is 45°, the horizontal extent of the rock mass affected by heat transfer is substantially larger than that observed at a 90° intersection. Furthermore, the temperature within this expanded zone is approximately 2°C~3°C higher compared to the 90° case. These results demonstrate that the intersection angle between subway tunnels significantly influences both the spatial extent of heat-affected rock mass and the temperature magnitude within the intersection zone. Specifically, as the crossing angle decreases, the area of rock mass impacted by heat transfer progressively enlarges, accompanied by a steady increase in the rock mass temperature within this region. This suggests that smaller intersection angles lead to more pronounced heat accumulation and broader thermal influence in the surrounding rock mass.

Figure 16: Rock mass temperature distribution at different intersection angles (z = −20.1 m).

To obtain quantitative measurements of tunnel rock mass temperature at different intersection angles, temperature observation points were established within the intersection zone. Taking the specific case of a 90° intersection angle with a vertical spacing of 7 m between intersecting tunnels as an example, a reference plane was defined at a depth of z = −20.1 m. On this plane, the bisector of the intersection angle-projected from both tunnel axes-was designated as the reference line. Along this reference line, four temperature detection points, labeled t, u, v, and w, were positioned at distances of 0, 5, 10, and 15 m from the intersection point, respectively. The spatial arrangement of these points is illustrated in Fig. 17. This configuration enables systematic monitoring of temperature variations along the critical zone of tunnel intersection, facilitating detailed analysis of thermal behavior under specified geometric conditions.

Figure 17: Detection point layout within intersection angle (z = −20.1 m).

Based on the principle of heat transfer equilibrium between tunnel air and surrounding soil, annual variation predictions for the temperature of the rock mass within subway tunnel intersection zones were obtained for different tunnel intersection angles. The temporal evolution of surrounding rock temperature within the intersection area during the operational period is presented in Fig. 18. The results indicate that, except for observation points t and u, smaller intersection angles correspond to higher surrounding rock temperatures at all other monitoring locations. At points t and u, the influence of the intersection angle on rock temperature is negligible; however, the temperature fluctuations at these points exhibit greater amplitude across different angle conditions. Notably, point u shows temperatures approximately 1°C lower on average than point t. This difference arises because both points are situated closer to the tunnel wall surface, where the rock mass rapidly attains dynamic thermal equilibrium due to steep temperature gradients. This rapid equilibrium prevents further increases in temperature peaks and facilitates heat transfer to more distant soil layers. At point v, the effect of the intersection angle on rock mass temperature is more pronounced. The highest temperatures occur at the 45° intersection angle, while the lowest temperatures are observed at 90°. This variability results from differences in heat transfer paths and distances from the tunnels to point v under varying intersection angles, which modulate the amount of heat reaching this location. Consequently, distinct temperature disparities are evident across different crossing angles at this point. Compared to point v, point w exhibits an even greater sensitivity to intersection angle variations. Although the absolute temperature differences between angles increase at point w, the overall temperature variations remain relatively modest due to its greater distance from the tunnel walls. At larger intersection angles, the majority of the heat within the surrounding rock is absorbed by nearby soil or dissipated externally, with limited heat transmission beyond 15 m from the tunnels. In summary, for tunnel crossings with equal vertical spacing, both the magnitude and growth rate of surrounding rock temperature increase as the tunnel intersection angle decreases. Near the intersection center and close to the tunnels, the crossing angle exerts minimal influence on rock mass temperatures. At locations farther from the intersection, however, the temperature growth rate at a given point varies significantly under different crossing angle conditions, with smaller angles corresponding to higher achievable peak temperatures once heat transfer reaches dynamic equilibrium.

Figure 18: Trend of rock mass temperature variation over time at different crossing angles.

3.5 Influence of Climatic Conditions on the Temperature Field of Rock Mass Surrounding Cross-Tunnel Sections

To evaluate the adaptability of the numerical model developed in this study under varying climatic conditions, analyses were performed on the temporal variation of rock mass temperature fields during subway tunnel operation using geological and meteorological parameters representative of Shenzhen (hot summers and warm winters), Beijing (cold climate), and Harbin (severe cold climate). These cases were compared against the climate profile of Xi’an (characterized by hot summers and cold winters), thereby encompassing a range of climatic zones, including hot-summer/warm-winter, hot-summer/cold-winter, cold, and severe cold environments. The representative thermal properties of soils corresponding to each city are summarized in Table 2, providing key parameters for the model’s boundary and initial conditions. Additionally, the monthly average outdoor air temperatures for these cities, which serve as crucial boundary conditions for the simulation, are illustrated in Fig. 19a. This comparative analysis facilitates an assessment of how different climate regimes influence the thermal behavior of rock masses surrounding intersecting metro tunnels over extended operational periods.

Figure 19: Monthly average air temperature.

The predicted annual variations in tunnel air temperature during long-term subway operation can be effectively approximated by applying the tunnel air balance principle combined with equivalent algorithmic approaches [36]. The resulting monthly average tunnel air temperatures for each representative city are depicted in Fig. 19b. Utilizing these tunnel air temperature profiles as boundary conditions for the soil heat transfer simulations, the temperature fields of the surrounding rock mass in subway tunnel sections across different climatic regions can be modeled over a continuous operational period of 20 years. This methodology enables a comprehensive evaluation of regional climatic impacts on the thermal behavior of tunnel rock masses over extended service duration.

An observation point established at x = 500 m was used to examine the vertical distribution of surrounding rock temperature along the intersecting subway tunnel under different climatic conditions. The corresponding temperature profiles are presented in Fig. 20. Taking the 25th year of operation as a representative example, it is observed that when the intersecting subway line is located in Harbin—a region characterized by severe cold climate—the overall temperature of the surrounding rock near the tunnel is approximately 5°C~7°C lower than that measured in the Xi’an region. Conversely, for the Shenzhen location, which experiences hot summers and warm winters, the surrounding rock temperature near the tunnel is about 3°C~5°C higher than in Xi’an. Importantly, the temperature of the tunnel’s surrounding rock is minimally influenced by surface conditions. Surface convective heat exchange affects only the shallow soil layers, and the vertical extent of this influence remains largely consistent regardless of the climatic zone. These findings indicate that while the geographical location of intersecting subway tunnels imposes varying degrees of impact on the absolute temperature values within the surrounding rock mass at the intersection zone, the overall spatial range of the temperature distribution remains largely unaffected. In other words, climate-dependent temperature differences influence thermal magnitudes but do not significantly alter the extent of the heat-affected rock mass around the tunnels.

Figure 20: Rock mass temperature distribution in different climate zones (x = 500 m).

Based on the principle of thermal equilibrium between tunnel air and surrounding soil, the annual variation in rock mass temperature during long-term operation can be predicted for intersecting subway tunnels situated in different climatic regions, as illustrated in Fig. 21. The temperature evolution observed at points q and s—located 5 and 15 m horizontally from the tunnel sidewall, respectively, as depicted in Fig. 6—displays distinct characteristics influenced by regional climate. At point q, the effects of regional climate and initial soil temperature on rock mass temperature are particularly pronounced. A comparison of rock mass temperature variations between Beijing and Xi’an reveals only a minor absolute temperature difference between the two sites. However, Beijing experiences significantly greater temperature fluctuations than Xi’an. This disparity arises because seasonal variations of air temperature both inside and outside the Beijing tunnel are more extreme than those in Xi’an, resulting in amplified temperature oscillations in the tunnel rock mass due to heat exchange with tunnel air. When comparing Harbin and Shenzhen, it becomes evident that the tunnel rock mass temperature is strongly influenced by the initial soil temperature. Harbin, with an initial soil temperature of approximately 6.6°C, exhibits peak temperature increases of 4.1°C and 2.06°C during the first and second years, respectively, eventually stabilizing within the range of 16.78°C–17.56°C. In contrast, Shenzhen, with an initial soil temperature near 23.2°C, shows smaller initial temperature increases of 1.52°C and 0.72°C during the first two years, stabilizing later at approximately 26.34°C~27.01°C—about 9.5°C higher than Harbin. This difference is attributed to the lower initial soil temperature in Harbin; despite similar patterns of air temperature fluctuations inside the tunnels, the larger thermal gradient in Harbin accelerates heat transfer rates. Conversely, Shenzhen’s higher baseline air temperature contributes to a higher equilibrium temperature of the surrounding rock mass once dynamic thermal equilibrium is reached. The temperature variation pattern at point s parallels that observed at point q. However, due to its greater distance from the tunnel wall, point s receives less heat through the rock mass, resulting in overall temperatures that are approximately 0.5°C~1.5°C lower than those at point q. Notably, the rock mass in Harbin, with its lower initial temperature and steeper temperature gradient, experiences a significantly faster rate of heating compared to rock masses in other geographical locations. In summary, both the rate of temperature increase in the rock mass around intersecting subway tunnels and the magnitude of the peak temperature attained at dynamic thermal equilibrium are significantly modulated by geographic and climatic factors. Specifically, colder climatic regions with lower ambient temperatures tend to experience larger increases in rock mass temperature at tunnel intersections. However, the final peak temperature reached during dynamic thermal equilibrium is constrained by the thermal interactions between the tunnel air and the surrounding rock, resulting in comparatively lower equilibrium temperatures in colder regions.

Figure 21: Temperature variation trends in the surrounding rock of subway tunnels across different regions.

4  Conclusions

Based on the thermal equilibrium equation governing tunnel rock mass temperature and incorporating relevant meteorological data, this study systematically analyzed the effects of vertical spacing, intersection angle, climatic conditions, and geological parameters on the temperature distribution within rock masses surrounding intersecting subway tunnels. The following key conclusions were drawn:

(1)   Compared to single-line subway tunnels, intersecting tunnel configurations induce heat accumulation in the surrounding rock near the tunnels. This phenomenon not only elevates the temperature levels within the rock mass but also extends the time necessary for the tunnel system to achieve thermal equilibrium.

(2)   The magnitude and fluctuation range of surrounding rock temperature are closely correlated with tunnel spacing. Decreasing the vertical spacing between tunnels leads to intensified heat accumulation in the intersection area, resulting in higher temperatures within this region. However, when the vertical spacing between intersecting tunnels exceeds 12 m, the temperature values cease to exhibit significant variation with further increases in spacing. While reducing the vertical distance between tunnels decreases the horizontal extent of the rock mass affected by heat transfer, it concurrently causes the temperature within the affected area to continuously rise.

(3)   The intersection angle between tunnels influences not only the spatial distribution characteristics of rock mass temperature but also the temperature magnitude within the intersection zone. When the vertical spacing between intersecting tunnels is 7 m and the intersection angle is less than 60°, heat generated by the tunnels accumulates extensively near the intersection area. As a result, the rock mass temperature within the intersection zone continuously increases, accompanied by a progressive expansion of the heat-affected region.

(4)   Both the rate of increase in rock temperature at the tunnel intersection and the peak temperature attained during dynamic thermal equilibrium are significantly influenced by regional climatic factors. Colder ambient temperatures in a given climate zone tend to produce greater increases in rock temperature at the intersection. However, due to the moderating effects of air temperatures both inside and outside the tunnel, the ultimate peak temperature achieved during dynamic thermal equilibrium is comparatively lower in these cooler regions.

This study employs the finite element method to investigate the influence of vertical spacing, intersection angles, and regional climatic factors on the temperature distribution within tunnel rock masses under intersecting metro lines. The analysis incorporates heat transfer induced by air turbulence during fresh air intake as well as fluid-solid coupling effects between tunnel air and the tunnel walls. This study provides valuable theoretical support for the effective management of thermal hazards in high-temperature tunnels and offers significant scientific insights for the design of cross-city metro lines. However, this study evaluates these factors individually, without comprehensively addressing their combined impact on heat transfer within the rock mass. Building upon this work, subsequent research will focus on the combined influence of train operational conditions, geological characteristics, and regional climatic factors on the tunnel thermal environment. Concurrently, subsequent investigations will explore the thermodynamic responses arising from heat transfer effects in intersecting tunnels, specifically the stress variations and strain within the lining, as well as rock deformation resulting from elevated temperatures in the surrounding rock mass. These will provide essential theoretical references for optimized tunnel design, operational management strategies, and enhanced train air conditioning system performance.

Acknowledgement: Not applicable.

Funding Statement: The author sincerely acknowledges the support from Fundamental Research Funds of the National Natural Science Foundation of China (NSFC) [Grant Number 51476073], Gansu Province Natural Science Foundation (21JR7RA304, 24JRRA245), Gansu Province Higher Education Industry Support Plan Project (2023CYZC-38).

Author Contributions: Aoyu Zheng: Software, Investigation, Writing—original draft. Ye Wang: Methodology, Supervision, Writing—review & editing. Huanhuan Li: Writing—review & editing. Yuanfeng Lu: Writing—review & editing. All authors reviewed and approved the final version of the manuscript.

Availability of Data and Materials: The data that support the findings of this study are available from the corresponding authors upon reasonable request.

Ethics Approval: Not applicable.

Conflicts of Interest: The authors declare no conflicts of interest.

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