Field Monitoring and Numerical Study of an Artificial Ground Freezing Reinforcement Project for Cross Passage

Abstract

Artificial ground freezing (AGF), recognized for its environmental sustainability and safety, is commonly used in underground construction projects within water-saturated soils. This study presents the design scheme and monitoring results of an AGF reinforcement project for a cross passage located in strata with low seepage velocity on Hohhot Metro Line 2. A transient heat transfer model, based on the assumption of no seepage, was developed, incorporating phase transitions and nonlinear changes in thermal parameters. In the model, soil thermal parameters are treated as variables dependent on unfrozen water content, which is represented by the soil freezing characteristic curve (SFCC). To derive the SFCC expressions, a semi-empirical approach was employed. This approach avoids the complexity of obtaining SFCCs experimentally and mitigates the arbitrariness inherent in the commonly used traditional apparent heat capacity method. The model was subsequently validated using experimental data from the literature and field monitoring results. The development and key indicators, including the thickness and average temperature of the frozen curtain in a single stratum without seepage, were investigated. The results show that the central and slightly right areas of the cross-passage axis exhibit a thinner frozen curtain and higher average temperature, especially in the pump room area, where the effective thickness of the curtain is at its minimum. Therefore, it is recommended to closely monitor the development of the frozen curtain in these areas and optimize the layout of freezing pipes. This study may serve as a reference for similar projects.

1. Introduction

Artificial ground freezing (AGF) has become a crucial construction technology in geotechnical engineering due to its distinct advantages, including environmental sustainability and compliance with safety standards. During the AGF process, multiple freezing pipes are rapidly cooled, leading to the freezing of the surrounding soil. The typical refrigerant used is low-temperature brine (e.g., calcium chloride solution), which can reach temperatures ranging from −25 °C to −40 °C. Compared to its natural state, frozen soil exhibits enhanced strength and reduced water permeability. The continuous, closed frozen soil curtain that forms provides a safe, dry environment for subsequent excavation. AGF was first employed in a coal mine in South Wales 160 years ago and was patented by Poetsch in Germany in 1883 [1]. In recent decades, with the expansion of urban rail transit construction, AGF has been successfully applied in subway tunnel construction, shield tunnel entrance reinforcement, underground passage construction, and related projects due to its ability to meet the stringent safety and environmental requirements of urban environments [2,3,4,5,6]. AGF is also widely employed in reinforcement projects for cross passages, which are crucial for firefighting, evacuation, and ventilation, and are considered essential components of subway systems [7,8].

In AGF projects, the temperature field provides key indicators that directly influence the stability of the excavated chamber, such as closure time, thickness, and the average temperature of the frozen curtain. The temperature field within the strata is influenced by various factors, including the fundamental properties of the soil (e.g., porosity, permeability coefficient, specific heat capacity, and thermal conductivity), characteristics of groundwater flow, specifications and layout of refrigeration equipment, and construction conditions. Common methods for studying the AGF temperature field include analytical approaches, indoor experiments, and numerical simulations.

Ground freezing is a complex, transient process influenced by the ice-water phase transition and nonlinear changes in thermal parameters. Therefore, deriving analytical solutions for the temperature field is challenging. However, as freezing progresses beyond a certain point, the cooling rate of the ground decreases. When the cooling rate slows sufficiently, the temperature field becomes nearly constant over time and is often approximated as a steady-state field. Based on this assumption, Trubak and Barkhokin derived analytical solutions for temperature fields in the ground for single, single-row, and double-row pipe configurations as early as the mid-20th century [9]. In recent years, many researchers have developed analytical solutions for more complex scenarios, building upon previous studies [9,10,11]. The solution to the temperature field represents a complex moving boundary problem involving multi-field coupling and can only be derived under specific assumptions, such as regularly arranged freezing pipes and simplified boundary conditions. Indoor experiments accurately and intuitively reflect key indicators of the freezing temperature field, making them widely used. To evaluate the accuracy of earlier analytical solutions, Pimentel et al. developed an experimental setup to investigate the temperature field characteristics under different seepage velocities [12]. The comprehensive experimental data provide a valuable reference for subsequent analytical calculations and simulation studies. Researchers have developed more advanced experimental devices to investigate changes in soil properties and temperature fields during freezing [13,14,15]. Although reliable, experiments typically require expensive, complex equipment and are time-consuming, limiting their widespread adoption.

Recent advancements in numerical simulation research have been significant. Researchers have employed finite element methods that account for thermo-hydraulic coupling, thermo-mechanical coupling, and combinations with Monte Carlo simulations [3,16,17,18,19]. In the context of AGF projects, Yang et al. examined the asymmetry of frozen walls under the influence of seepage, using a coupled hydro-thermal model [20]. Mauro et al. employed a coupled thermo-hydraulic model that incorporates changes in unfrozen water content during freezing, based on a tunnel freezing project for the Naples subway. This model was applied to study ground temperature changes during the freezing process and to evaluate the thickness of the frozen curtain [21]. Currently, in terms of numerical modeling, the traditional apparent heat capacity modeling method remains widely used. This method assumes that a designated temperature range around 0 °C fully encompasses the entire phase transition process, which introduces some degree of arbitrariness. As a result, researchers have proposed using the soil freezing characteristic curve (SFCC), which represents the real relationship between unfrozen water content in soil and temperature, for modeling [22]. SFCC-based models can more accurately capture dynamic changes in unfrozen water content and soil thermal parameters. Pimentel et al. [5] and Li et al. [6] investigated temperature fields in urban AGF projects using coupled thermo-hydraulic models based on power function-type SFCCs. Casini et al. developed a coupled thermo-hydraulic model incorporating a van Genuchten model-type SFCC to study the AGF temperature field in the Isarco River underpass tunnel and proposed a new standard for evaluating frozen wall thickness [23]. Liu et al. obtained SFCC parameters from soil column freezing tests and developed a coupled thermo-hydraulic model. The model was validated through a model experiment and a cross-passage reinforcement project in the Beijing Metro [14,24]. Although SFCC-based modeling is more accurate, obtaining the SFCC often requires time-consuming and costly soil freezing tests, which present challenges for AGF research. A more convenient method capable of reasonably predicting the SFCC would be highly beneficial. Moreover, existing simulation research primarily focuses on improving control equations and modeling methods, with limited attention given to key indicators of the frozen curtain, which are closely linked to the safety of the project. Therefore, a three-dimensional transient heat transfer model, based on a convenient method for SFCC prediction and nonlinear thermal parameters, is essential for studying the development and key indicators of the frozen curtain in AGF engineering.

The novelty of this study lies in the derivation of a semi-empirical SFCC directly from the soil grain-size distribution and index properties. This approach mitigates the complexity of experimentally obtaining SFCCs and reduces the arbitrariness inherent in the traditional apparent heat capacity method. The proposed SFCCs were implemented in finite element heat transfer simulations, and the predictions were compared with experimental data from the literature and field monitoring results. The comparative analysis demonstrates good agreement, even when groundwater flow is neglected. The model was subsequently employed to quantitatively examine the key indicators of the frozen curtain within a single stratum from a three-dimensional perspective, with a particular focus on the weak positions of the curtain.

2. Project Overview and Field Monitoring

2.1. Project Profile

The AGF method played an important role in the construction of Hohhot Metro Line 2. One of the cross passages, located between Genghis Khan Park Station and Yijiacun Station in the northern section of Line 2, was reinforced using this method.

In this interval, the center-to-center distance between the twin-tube shield tunnels is 14.00 m. The outer diameter of the tunnel is 6.20 m, and the concrete segment thickness is 0.35 m. The cross passage and pump room are located within saturated silty clay and fine sand layers, at a burial depth of 25.0 m. Drilling revealed that the groundwater is phreatic, with the water table depth varying from 17.9 m to 23.5 m. Due to the poor stability of the strata and the presence of groundwater, the AGF method was used to reinforce the cross passage and meet safety and environmental protection requirements. The first lining of the cross passage consists of a grid steel frame and shotcrete, while the second lining is composed of reinforced concrete. In addition, a flexible waterproof layer was installed between the first and second linings. The cross-passage structure and the frozen curtain outline are shown in Figure 1.

2.2. Design Scheme

To meet safety and quality standards, the frozen curtain must maintain a minimum effective thickness of 2.0 m, and its average temperature must remain below −10 °C. In this project, 73 freezing holes were drilled, with depths ranging from 3.60 m to 12.00 m, and all freezing pipes were specified as φ89 × 8 mm in diameter. Figure 2 shows the arrangement of the freezing holes and other holes. Most freezing holes were drilled from the left tunnel (designated N and Z), with a total of 16 rows, while three rows were drilled from the right tunnel (designated W and Z). The horizontal spacing between the freezing holes at the top and bottom (N1, N2, N14, N15, W1, and W2) is 800 mm, with local spacings of 700 mm and 750 mm, respectively. Additionally, eight thermometer holes (designated C) were placed to monitor and record temperatures at various ground positions, while four pressure-release vents (designated X) were installed to measure and release pore water pressure in the unfrozen zone after the frozen curtain had closed.

A calcium chloride solution was selected as the refrigerant due to its low freezing point. To prevent airflow from affecting the freezing process, an insulation layer composed of soft, flame-retardant foam material with a thickness exceeding 30 mm was placed on the segments corresponding to the freezing zone, covering the designed frozen curtain. The on-site photos are shown in Figure 3.

During the active freezing stage, frozen columns began to form around the freezing pipes in the soil mass. The frozen columns gradually expanded outward, ultimately forming a closed frozen curtain. Once the frozen curtain reached its designed thickness, the project transitioned to the maintenance freezing stage. In this stage, the cross passage was excavated, and the brine was used only to maintain the stability of the frozen curtain. During the active freezing stage, the brine temperature must remain stable below −28 °C, and it must be kept below −25 °C during the maintenance stage. The planned duration of the active freezing stage was 45 days; however, a thorough evaluation of the freezing effect was conducted before excavation.

2.3. Field Monitoring

Field monitoring primarily targets three indicators: brine temperature, soil temperature, and tunnel section convergence. Monitoring of tunnel section convergence reflects the frost heave effect. Field monitoring indicated minimal frost heave, as most of the frozen zone in this project is located in a fine sand layer. This phenomenon is consistent with the well-established understanding that sand exhibits a low frost heave rate. As this paper focuses on the temperature field, frost heave will not be discussed in further detail. Additionally, field monitoring included pressure monitoring, which provided a reference for determining the closure time of the frozen curtain via pressure-release vents. The active freezing stage is critical to the overall freezing process; therefore, only the monitoring records from this stage (45 days) are presented.

Since the freezing station began operation, brine temperatures at both the outlet and inlet were monitored hourly using thermometers installed in the pipeline. Figure 4 illustrates the recorded brine temperatures. According to the records, the brine temperature decreased rapidly to below −18 °C within the first five days of freezing and stabilized below −24 °C by the 14th day. Prior to excavation, the brine temperature remained stable and met the design requirements, with the difference not exceeding 2 °C, indicating that the frozen curtain had reached a stable stage.

Eight thermometer holes were arranged near the freezing zone at various locations. With the exception of the C7 hole, which was longer and had additional temperature monitoring points, the remaining seven holes had monitoring points at depths of 0.5 m, 1.5 m, and 3.0 m. An electronic temperature monitoring system was developed to record temperatures at each point every hour, with automatically generated data tables reflecting continuous temperature changes. In this paper, the temperature recorded at 8 a.m. daily is used as the representative daily temperature. Figure 5 presents the temperature values recorded from six representative holes.

Due to equipment failure, the data from the C3 thermometer hole were incomplete, and the C7 hole, positioned separately, could not reflect the symmetry of the frozen curtain’s development. Therefore, thermometer holes C1, C2, C4, C5, C6, and C8 were selected for further analysis. As shown in Figure 5 and similar to the brine temperature trend, temperatures at all points exhibit three stages: rapid decline, slow decline, and stabilization. At the onset of AGF, the temperature difference between the soil and freezing pipe surfaces is significant, resulting in intense heat exchange and a rapid decrease in ground temperature. With the release of latent heat and the gradual decrease in the temperature difference between the soil and the freezing pipes, the cooling rate of the ground at most monitoring points significantly reduces after 25 days. As the frozen walls continue to expand, heat conduction within the strata gradually reaches equilibrium, causing the temperature to stabilize.

As shown, compared to the C2 hole, the cooling rate at the C1 hole is faster, and the stabilized temperature is lower, particularly at depths of 1.5 m and 3.0 m. This is partly because the C1 hole, located inside the curtain, was also influenced by other freezing pipes, such as N2-5 to N2-7, which were positioned closer to the interior of the cross passage. In contrast, C2, located outside the curtain, was farther from these pipes. Additionally, because the frozen curtain has a cage-like shape, “cooling energy” is more easily concentrated as the frozen soil expands inward, whereas outward expansion results in the dissipation of “cooling energy” into a larger volume of soil. This supports previous research in which Fan et al. found that in silty soil, the inward expansion rate of a frozen wall can be up to 1.43 times faster than the outward rate [25]. Similarly, the C4 and C5 holes located inside the frozen curtain exhibit lower temperatures and higher cooling rates compared to the C6 and C8 holes positioned outside. In addition, for most thermometer holes, temperatures at the 0.5 m depth are generally higher than those at deeper points, and their cooling rates are lower. It is possible that the insulation layer placed on the tunnel segments did not fully block the impact of air convection on the ground temperature.

Analyzing the temperature monitoring data can help estimate the development rate of the frozen walls. The calculation method involves dividing the distance between the selected temperature monitoring point and the nearest freezing pipe by the number of days it takes for the temperature at that point to reach 0 °C to determine the development rate. Calculations showed that the frozen soil near the C8 thermometer hole developed at the slowest rate, approximately 40 mm/day. Using the slowest rate and the drawing method, it was concluded that the estimated final effective thickness significantly exceeded the designed thickness of 2 m. However, due to differing thermal parameters between unfrozen and frozen soil and the release of latent heat, the development rate of the frozen soil is nonlinear, introducing some error into this method. To ensure excavation safety, two exploration holes were drilled along the left and right contour lines of the excavation section, 45 days after the freezing process commenced. These holes were positioned 1100 mm from the freezing pipes. The temperatures measured in the exploration holes were all below −6.4 °C, indicating that the frozen curtain had reached its designed thickness. Furthermore, according to the ice formation formula [4], the average temperature of the frozen curtain reached −11.41 °C, meeting the design requirements.

When the frozen curtain closes, the hydraulic connection between the sealed soil and the external environment is cut off, often resulting in the accumulation of significant frost heave pressure within the sealed soil. On one hand, frost heave pressure increases the additional stress on the tunnel segments and must be released; on the other hand, the rise in pressure indicates that the frozen curtain has closed, providing a valuable reference for construction. The initial pressure values of the four vents were approximately 0.1 MPa, and they increased steadily starting from day 15, indicating that the frozen curtain had closed. The maximum pressure, recorded on day 31, reached 0.46 MPa, meeting the designed pressure release standard, at which point the release operation was carried out.

3. Formulation of the Numerical Model

3.1. Basic Assumptions

Freezing is a process involving the coupling of multiple physical fields. Due to the close relationships among various physical fields under real conditions, it is challenging to fully describe the coupling process using existing methods. As a result, certain assumptions are made. In this study, to develop a numerical model accounting for phase transitions, the following assumptions are introduced:

(1)

The strata in the model are simplified as homogeneous, isotropic, and saturated porous media, with soil pores filled exclusively with liquid water (unfrozen) and solid ice (frozen), and no other media, such as air.

(2)

The model is only applicable in cases where there is no seepage or where seepage can be neglected. As the primary mode of heat transfer during freezing, only heat conduction is considered in the simulation, excluding air convection and thermal radiation at the ground surface.

(3)

Since the temperature field is the primary focus, changes in the stress field and soil porosity due to frost heave are excluded.

(4)

The average temperature of brine at the inlet and outlet is used as the boundary condition for freezing pipes, regardless of the heat exchange between the brine and the pipe wall.

3.2. Control Equations of Temperature Field

Heat transfer in the ground primarily follows Fourier’s law of heat conduction. According to the local thermal equilibrium hypothesis, it is assumed that the temperatures of soil particles, liquid water, and ice are equal within a very short time frame over the representative elementary volume (REV), allowing the energy conservation equations of the three phases to be combined. Considering the latent heat released during the ice-water phase transition and neglecting the seepage term, a unified energy conservation equation for a saturated porous medium during freezing is established [26], as shown in Equation (1). It should be noted that in the following text, the subscripts s, l, and i represent the soil solid matrix, liquid water, and ice, respectively, and the Einstein summation convention is not used in this paper.

$${\left(\rho c\right)}_{eq}{\displaystyle \frac{\partial T}{\partial t}}=\nabla \left({\lambda }_{eq}\nabla T\right)+L{\rho }_i{\displaystyle \frac{\partial {\theta }_i}{\partial t}}$$

(1)

where T is the temperature, K; θ is the volumetric content, %; t is the time, s; L is the latent heat of the ice-water phase transition, taking as 334 kJ/kg, c is the specific heat, J/(kg·K); $\rho$ denotes the density, kg/m³; ${\left(\rho c\right)}_{eq}$ represents the equivalent volumetric heat capacity, J/(m³·K); ${\lambda }_{eq}$ is the equivalent thermal conductivity, W/(m·K). The two equivalent thermal parameters are expressed using the local volume-averaged method, and both, along with the latent heat, are dependent on the ice-water content during freezing in porous media. According to the volume-averaged theory, the equivalent volumetric heat capacity, ${\left(\rho c\right)}_{eq}$, is expressed as follows [19]:

$${\left(\rho c\right)}_{eq}={\theta }_s{\rho }_s{c}_s+{\theta }_l{\rho }_l{c}_l+{\theta }_i{\rho }_i{c}_i$$

(2)

For saturated soil, frost heave deformation is not considered, and it is assumed that ${\theta }_l+{\theta }_i+{\theta }_s=1$. Multiple estimation methods exist for the equivalent thermal conductivity ${\lambda }_{eq}$, and in this study, the commonly used weighted geometric mean method is employed [16,27].

$${\lambda }_{eq}={{\lambda }_s}^{{\theta }_s}\cdot {{\lambda }_l}^{{\theta }_l}\cdot {{\lambda }_i}^{{\theta }_i}$$

(3)

3.3. Equations of Unfrozen Water Saturation

Based on the assumptions of mass conservation and constant porosity, the second term on the right-hand side of Equation (1) can be rewritten as

$$L{\rho }_i{\displaystyle \frac{\partial {\theta }_i}{\partial t}}=-L{\rho }_l{\displaystyle \frac{\partial {\theta }_l}{\partial t}}=-L{\rho }_l{\displaystyle \frac{\partial {\theta }_l}{\partial T}}{\displaystyle \frac{\partial T}{\partial t}}=-L{\rho }_{l}F\left(T\right){\displaystyle \frac{\partial T}{\partial t}}$$

(4)

where $F\left(T\right)$ represents the derivative of the unfrozen water volumetric content with respect to temperature, i.e., the derivative form of the SFCC [22]. Therefore, an appropriate SFCC expression must be incorporated into numerical simulations. Various expressions have been proposed by researchers to represent the SFCC. The first category includes empirical models. Commonly used models include the Anderson–Tice model, represented as a power function [28], and the Kozlowski model, represented as a piecewise function [29]. Although these models are simple and easy to apply, their parameters often lack physical significance, and obtaining function values at 0 °C is challenging. The second category is based on the soil-water characteristic curve (SWCC). Among these, an SFCC model is commonly developed based on the van Genuchten SWCC model or the Fredlund–Xing SWCC model [30,31,32]. The third category includes theoretical models, such as the SFCC model that distinguishes between capillary water and bound water [33], or one based on particle size distribution [34]. Compared to the first two categories, the third has a clearer physical basis; however, its complexity makes it difficult to apply in numerical calculations.

A prerequisite for using the first two types of SFCC models in numerical simulations is the accurate determination of the model parameters. Standard practice involves testing soil samples to obtain unfrozen water contents at various subzero temperatures and fitting the data to existing models to derive the corresponding expressions. However, this method relies on time-consuming and costly experiments. If there is a need to preliminarily estimate the ground temperature distribution, a relatively simple method should be employed. As mentioned earlier, some researchers bypass the actual SFCC and instead use the traditional apparent heat capacity method [22], which assumes that the phase transition occurs entirely within an artificially designated temperature range. Within this range, a linear or smooth step function is used to approximate the SFCC. However, it is difficult to ensure that the assumed phase transition process aligns with real conditions. For example, in plastic soils, unfrozen water still exists in large quantities even at very low temperatures due to the high proportion of bound water. In coarse-grained soils, the proportion of free water is high, and most of the pore water freezes into ice within a very narrow temperature range. The artificially designated temperature range may overestimate the unfrozen water content at subzero temperatures. This study intends to adopt a semi-empirical approach to estimate the SFCC using one or two readily accessible soil index properties, balancing convenience and rationality.

Based on the assumption that the drying-wetting process in unfrozen soil is analogous to the freezing-thawing process in frozen soil, the ice phase in frozen soil is typically used to replace the gas phase in unfrozen soil. Therefore, the SFCC model can be derived from the SWCC expression [35]. Among the widely used SWCC models, the Fredlund–Xing (F–X) model [31] provides a strong fit within the suction range of 0 to 10⁶ kPa, as shown in the following expression:

$${\theta }_w=C\left(h\right)\left[{\displaystyle \frac{{\theta }_{sat}}{{\left[\ln \left[\exp \left(1\right)+{\left(\frac{h}{a}\right)}^n\right]\right]}^m}}\right]$$

(5)

$$C\left(h\right)=\left[1-{\displaystyle \frac{\ln \left(1+\frac{h}{h_r}\right)}{\ln \left(1+\frac{{10}^6}{h_r}\right)}}\right]$$

(6)

where ${\theta }_w$ is the volumetric content of unfrozen water, %; ${\theta }_{sat}$ is the saturated volumetric water content, equal to the porosity, %; $h$ is the suction, kPa; $h_r$ denotes the suction corresponding to the residual water content, kPa; a is primarily a function of the air entry value of the soil in kPa; m and n are dimensionless parameters, and parameters a, m, n, and $h_r$ should be obtained through testing soil samples. Zapata et al. proposed a simple and effective method to determine the parameters of the F–X model by analyzing a substantial amount of soil test data [36]. For non-plastic soils, only the diameter $d_{60}$ (in mm) is required.

$$a=0.8627{\left(d_{60}\right)}^{-0.751}$$

(7)

$$m=0.1772\ln d_{60}+0.7734$$

(8)

$$n=7.5$$

(9)

$$h_r=a/\left(d_{60}+9.7e^{-4}\right)$$

(10)

For plastic soils, it is necessary to use the product $fPI$, which is the product of the plasticity index (PI, %) and f, the percentage of soil particles passing the #200 U.S. Standard Sieve are expressed as a decimal number.

$$a=0.00364{\left(fPI\right)}^{3.35}+4\left(fPI\right)+11$$

(11)

$$m=0.0514{\left(fPI\right)}^{0.465}+0.5$$

(12)

$$n=m\cdot \left[-2.313{\left(fPI\right)}^{0.14}+5\right]$$

(13)

$$h_r=a\cdot 32.44e^{0.0186\left(fPI\right)}$$

(14)

In the freezing process, assuming no solute is present, the chemical potential equilibrium relation between solid ice and liquid water can be described by the Clausius–Clapeyron equation:

$$\mathrm{d}{P}_i-{\displaystyle \frac{{\rho }_i}{{\rho }_l}}\mathrm{d}{P}_l+{\displaystyle \frac{{\rho }_{i}L}{T}}\mathrm{d}T=0$$

(15)

where $P_i$ represents the pore ice pressure and $P_l$ represents the pore water pressure, kPa. By integrating between atmospheric pressure and the freezing point temperature, the following equation can be obtained [23,37]:

$$P_i={\displaystyle \frac{{\rho }_i}{{\rho }_w}}{P}_w-{\rho }_{i}L\ln \left({\displaystyle \frac{T}{T_0}}\right)$$

(16)

where $T_0$ is the freezing point (the default value is 273.15 K). Similarly to matrix suction h, the ice-water pressure difference can be defined as negative temperature suction $\psi$ (in kPa). Based on the common assumption in studies that the pore ice pressure is zero, $\psi$ can be expressed as

$$\psi =-{\rho }_{l}L\ln \left({\displaystyle \frac{T}{T_0}}\right)$$

(17)

Substituting $\psi$ for h and Equation (17) into Equations (5) and (6), the FX–Clapeyron model can be obtained [32]:

$${\theta }_w=\left\{1-\left[{\displaystyle \frac{\ln \left(1+\frac{-L{\rho }_l\ln \frac{T}{T_0}}{h_r}\right)}{\ln \left(1+\frac{{10}^6}{h_r}\right)}}\right]\right\}{\displaystyle \frac{{\theta }_{sat}}{{\left\{\ln \left[\exp \left(1\right)+{\left(\frac{-L{\rho }_l\ln \frac{T}{T_0}}{a}\right)}^n\right]\right\}}^m}}$$

(18)

For the initial saturated soil, the unfrozen water saturation S (1/1), can be expressed by ${\theta }_w$:

$$S={\displaystyle \frac{{\theta }_w}{{\theta }_{sat}}}$$

(19)

This semi-empirical approach allows the SWCC parameters to be derived from the soil grain-size distribution curve and index properties, enabling the estimation of the ice-water phase transition process at subzero temperatures. In addition, the freezing point, $T_0$, is a key parameter. Since saturated sand contains a high proportion of free water, its freezing temperature is close to 0 °C [38]. However, due to its high bound water content, the freezing temperature of clay is typically lower than that of sand. The expression for estimating the freezing point of clay is given as follows [29]:

$$T_0=-0.0729{w_p}^{2.462}{w}^{-2}$$

(20)

where $T_0$ is expressed in °C; $w_p$ is the plastic limit, %; w is the total water content, %. It should be noted that the empirical formulas (e.g., Equations (7) to (14) and (20)) are dimensionless in nature. These formulas are concerned solely with calculating numerical values and do not involve any units or dimensions.

3.4. Model Validation Against Physical Experiment

The experimental data from the literature were compared with the simulation results to validate the accuracy of the heat transfer model. Pimentel et al. developed a large-scale physical model with internal dimensions of 1.3 m × 1.0 m × 1.2 m. The exterior of the box was insulated, and three vertical freezing pipes were arranged in a straight line inside (Figure 6a). A series of temperature sensors was arranged at the middle height (ten representative sensors are shown in Figure 6b, represented by blue dots). The seepage velocity could be adjusted by altering the water heads on the left and right sides of the box. For further details on the experiment, refer to [12].

The soil used to fill the box in the experiment was medium sand. Based on the grain-size distribution curve and the semi-empirical approach described above, the SFCC for medium sand can be predicted. As shown in Figure 7a, the estimated SFCC closely aligns with the measured data points.

Pimentel et al. conducted experiments with six test groups, labeled V1 to V6, under different seepage conditions [12]. Since seepage is not incorporated into the heat transfer model, only the V2 test (with zero seepage velocity) was included in the simulation. Considering the uniformity of the physical model in the vertical direction, a two-dimensional numerical simulation was carried out using COMSOL Multiphysics 6.1. The initial temperature, T_ini, was set to 20.7 °C based on recorded data. Based on the experimental setup, the boundaries of the model were defined using non-flowing and adiabatic boundary conditions. The top temperatures (T_top) and bottom temperatures (T_bot) of the freezing pipes, recorded during the experiment, were averaged and applied as temperature boundary conditions for the pipe walls in the model. Free triangular elements were employed, and a mesh independence test was performed. Six different mesh sizes were tested: 2594, 4874, 9332, 16,926, 35,900, and 71,992 elements. To balance accuracy with computational efficiency, a mesh of 35,900 elements was ultimately selected for solving the model. The meshing scheme and boundary conditions are shown in Figure 7b.

According to the simulation results, the closure time of the frozen wall is 15 h, which is very close to the measured 14.4 h, thereby confirming the validity of the model. Figure 8 shows the distribution of frozen walls at different freezing times on a cross-section at a height of 0.6 m, with solid black lines representing the 0 °C isotherms. In the absence of seepage, the temperature field is symmetrically distributed with the freezing pipe connection line as the axis.

Due to the temperature curves of points equidistant from the freezing pipes being too close to distinguish in the figure, only ten representative temperature monitoring points were selected for comparative analysis (the positions of the points are shown in Figure 6a). The simulated and monitored values at different times are plotted in Figure 9.

Statistical analysis shows that the mean absolute error (MAE) between the simulated and monitored temperatures is 0.59 °C, calculated using Equation (21):

$$\mathrm{MAE}={\displaystyle \frac{1}{N}}{\displaystyle {\sum }_{j=1}^N\left|T_{sim,j}-T_{mon,j}\right|}$$

(21)

where $T_{sim,j}$ and $T_{mon,j}$ represent the simulated and monitored temperature values at the j-th point, respectively, and N is the total number of data points. The mean relative error (MRE) is 9%, calculated using Equation (22):

$$\mathrm{MRE}={\displaystyle \frac{1}{N}}{\displaystyle {\sum }_{j=1}^N\frac{\left|T_{sim,j}-T_{mon,j}\right|}{T_{mon,j}}}\times 100\%$$

(22)

The coefficient of determination (R²) is 0.994. In summary, the simulation results align well with the monitored values, indicating that the proposed numerical model can accurately predict the temperature field distribution in the soil during freezing.

4. Modeling of the AGF Project

4.1. Model Description and Boundary Conditions

To quantitatively analyze the development of the ground temperature field and frozen curtain, a three-dimensional transient heat transfer model was developed based on the AGF project described above on Hohhot Line 2. In this model, the tunnel location, segment specifications, freezing pipe specifications, and layout were defined based on the design drawings.

The strata in which the cross passage is located consist mainly of silty clay and fine sand, both simplified as homogeneous and isotropic materials in the model. According to the hydrogeological data, the hydraulic gradient of groundwater in this area is less than 2‰. Given the low permeability of fine sand and silty clay, the influence of seepage was neglected. The soil domain dimensions of the model were set to 40 m, 30 m, and 30 m in length, width, and height, respectively, to minimize boundary effects on the temperature field (Figure 10). The sides, top, and bottom of the model were set with Dirichlet boundary conditions at a temperature of 13 °C (286.15 K), based on the monitored initial ground temperature. The surfaces of the tunnel segments in contact with air were defined as adiabatic boundaries, expressed as $\overrightarrow{n}\cdot \lambda \nabla T=0$. The heat transfer between the brine and freezing pipe walls was neglected to reduce computational costs. The freezing pipe walls were defined with Dirichlet boundary conditions for temperature, with the temperature load represented as an interpolation function based on the recorded brine temperature curve.

The numerical simulation was performed using COMSOL Multiphysics 6.1 (COMSOL AB, Stockholm, Sweden) and a tetrahedral meshing scheme was selected, with a dense mesh near the freezing pipes and a sparse mesh farther from the freezing area. A mesh independence test was performed by comparing temperatures at two fixed points on day 45 of freezing. The number of elements was set to five levels: 2.64 × 10⁵, 4.31 × 10⁵, 5.78 × 10⁵, 8.66 × 10⁵, and 12.65 × 10⁵, with a growth rate of approximately 1.5. The temperature values at one point were −24.74 °C, −23.23 °C, −22.88 °C, −22.78 °C, and −22.71 °C, while those at the other point were −14.61 °C, −13.73 °C, −13.26 °C, −13.11 °C, and −12.99 °C. The calculation results stabilized when the number of elements reached 5.78 × 10⁵. For the sake of accuracy, the number of elements was ultimately set to 8.66 × 10⁵. The material property parameters used in the model are shown in

Table 1. Material property parameters.

Material	Density (kg/m3)	Thermal Conductivity (W/m·K)	Specific Heat (J/kg·K)	Porosity
Silty Clay	1905	1.80	1400	0.48
Fine Sand	2160	1.51	1420	0.33
Water	1000	0.60	4185	-
Ice	918	2.22	2050	-

. The density and thermal parameters of silty clay and fine sand were obtained through on-site sampling and laboratory geotechnical testing, while the parameters of water and ice are obtained from [24]. It should be noted that the parameters of the soil solid matrix need to be determined through inverse calculation using Equations (2) and (3).

4.2. Unfrozen Water Saturations

The soil grain-size distribution curves and index properties of both soils were also obtained through geotechnical tests. Using the semi-empirical approach described in Section 3.3, the SFCCs were estimated. The freezing temperature of silty clay was calculated using Equation (20). The parameters required for the simulation are listed in

Table 2. The parameters required for estimating SFCC in the model.

Soil Type	d60 (mm)	fPI	a (kPa)	m	n	hr (kPa)	T0 (K)
Silty Clay	-	8.13	47.59	0.64	1.21	1795.85	272.98
Fine Sand	0.264	-	2.35	0.54	7.5	5.34	273.15

.

Considering the difference in saturated volumetric water content, unfrozen water saturation is used to represent the phase transition process in both soils. In Figure 11, along the negative direction of the horizontal axis (cooling direction), the estimated unfrozen water saturations of both soils exhibit a rapid decline, a slow decline, and eventual stabilization. Since the soils are saturated, once the freezing temperature is reached, a significant amount of free water freezes, causing a substantial reduction in unfrozen water saturation during the initial stage. Bound water begins to freeze as the temperature continues to decrease. The surface energy of soil particles reduces the free energy of bound water, thereby slowing down the freezing rate. Compared to fine sand, silty clay has a larger specific surface area and higher bound water content, causing the unfrozen water saturation to decrease more slowly, with the saturation remaining higher after reaching stability. The characteristics of the estimated SFCCs align with experimental findings as well [39].

4.3. Validation with Field Monitoring Results

By comparing the monitored data from the cross-passage project with the numerical simulation results, the reliability of the model in AGF engineering can be further validated. Figure 12 presents the monitored and simulated temperature values at twelve representative monitoring points during the active freezing stage. Although the tunnel segments in the freezing area were insulated, most of the 0.5 m monitoring points were still influenced by airflow in the tunnel, leading to a slight temperature increase. Therefore, only the temperature curves at 1.5 m and 3.0 m are shown.

As observed, the simulated temperature curves are smoother than the monitored ones due to the idealized assumptions of the model. The simulation shows that temperatures at the C1 hole (inside the curtain) are lower than those at the C2 hole (outside) once freezing reaches the stable stage. For all six holes, temperatures at the 3.0 m monitoring points are lower than those at the 1.5 m points for most of the time. These observations align with the monitoring results.

Statistical analysis shows that the MAE between the simulated and monitored temperatures is 1.41 °C, the MRE is 30%, and the R² is 0.962. Among all the data, only five pairs of points exhibit a temperature difference greater than 5 °C, with the maximum difference being 5.75 °C. These points are all concentrated during the rapid cooling stage at the C1-3.0 m monitoring point. If only the temperature data after day 25 are analyzed, the maximum temperature difference is only 3.15 °C, the MAE decreases to 1.24 °C, and the MRE reduces to 9%, indicating that the model exhibits higher prediction accuracy after the ground cooling rate slows down. Despite individual deviations, the model demonstrates good engineering applicability overall.

Possible reasons for deviations at certain monitoring points include: (1) Some formulas and parameters used in the numerical calculations are empirical, meaning that actual soil parameters and thermal processes cannot be precisely modeled. (2) Soil is known to be heterogeneous and anisotropic; however, the model assumes it to be homogeneous and isotropic. Furthermore, the initial temperature distribution in the ground is non-uniform. Temperature monitoring shows that initial temperatures range from 11.50 °C to 13.93 °C, whereas the model assumes a uniform initial temperature of 13 °C. (3) During drilling, the actual angles and spacings of the freezing holes may deviate from the designed values, potentially influencing the temperature field. (4) In the simulation, only the impact of unfrozen water content was considered, excluding seepage. Hydrogeological data show groundwater seepage in the freezing zone, with the flow direction angled relative to the axis of the cross passage. Although the seepage velocity is relatively low, it may still disturb the freezing temperature field. The C8 thermometer hole is located upstream of the freezing pipes, where the soil continuously receives heat from warm groundwater flow, leading most monitored temperatures to be slightly higher than the calculated values.

According to Casini et al., it is more appropriate to set the target temperature of the frozen wall as the temperature at which unfrozen water reaches its residual state [23]. At this temperature, only a small amount of residual unfrozen water remains, ensuring the continuity of the ice phase, significantly reducing the soil’s permeability, and enhancing its strength. According to Equations (5) and (6), this temperature corresponds to the residual suction $h_r$. Calculations show that the target temperature for fine sand is very close to 0 °C, while for silty clay, it is approximately −1.65 °C. To ensure complete sealing and optimal waterproof performance of the frozen curtain, the lower value is chosen. Therefore, the outer boundaries of the frozen walls in the model are set as −1.65 °C isothermal surfaces. Analysis of the isothermal surface development shows that the entire frozen curtain closed on day 14, consistent with field monitoring.

Figure 13 illustrates the variation in the shape of the frozen curtain and the cumulative frozen volume over time. As depicted in Figure 13a, within the first 15 days, the development of the frozen curtain is substantial. After the curtain closes, its surface gradually becomes flat and smooth, with no significant change in shape. Figure 13b quantitatively represents this phenomenon. During the initial stage, particularly from day 5 to day 10, the frozen volume increases sharply by approximately 226 m³, with an average daily increase of 45.2 m³. However, in the last five days, only 38 m³ have been added, with an average daily increase of just 7.6 m³, indicating that the freezing process is gradually stabilizing.

5. Analysis of Frozen Curtain Indicators

5.1. Frozen Curtain Development

The comparative analysis of the numerical simulation and field monitoring further demonstrates that the numerical model, based on the semi-empirical approach for estimating the SFCC, is both reasonable and accurate. To derive general conclusions, the two strata in the numerical model from the previous chapter were unified as fine sand (since fine sand has a lower thermal conductivity and most of the curtain is located in this stratum), and the simulation was rerun.

As shown in Figure 14, five vertical cross-sections (SA to SE) are arranged in parallel at 1.80 m intervals between the tunnels to investigate the development and key indicators of the curtain from a three-dimensional perspective and to identify relatively weak areas. Section SC is located in the middle of the line connecting the centers of the two tunnels, where the cross-sectional area of the frozen curtain is the largest. Sections SA and SE are located at the junctions of the passage and the tunnels. Therefore, the development of the curtain at these locations is critical for excavation. Sections SB and SD are auxiliary sections located on either side of SC.

The dynamic evolution of the frozen curtain on each section was extracted from the simulation. In Figure 15, the areas enclosed by solid black lines represent the frozen curtain. In the initial stage, known as the single-pipe freezing stage, independent frozen columns gradually form around the freezing pipes. As the columns expand and the frozen walls form, the freezing process transitions to a plate freezing mode. For a single row of freezing pipes, the frozen wall exhibits asymmetry between the inside and outside of the curtain. Since more energy is required for the frozen soil to expand outward, the frozen wall expands more quickly inward, making it thicker on the inside.

It is evident that the larger the spacing between freezing pipes, the longer it takes to form a continuous frozen wall. Since two rows of freezing pipes intersect at the bottom of section SC, and the spacing is half that of the other four sections, a flat frozen wall appears here on day 5. Similarly, on section SA, continuous frozen walls form on both the left and right sides, as the radially arranged side freezing pipes are relatively close to one another. Due to the increased spacing between side freezing pipes, the continuous frozen walls appear later on both sides of the remaining four sections.

5.2. Thickness and Average Temperature of Frozen Curtain

In engineering, the effective thickness of the frozen wall, defined as the remaining thickness after deducting the excavated portion, directly affects the safety of subsequent excavation work. For clarity, in this paper, the thickness of the frozen wall, excluding the excavation scope, is referred to as the original thickness. Figure 16 presents a schematic diagram of the original and effective thicknesses, using the frozen curtain on section SD on day 45 as an example. It is evident that the effective thickness must be less than or equal to the original thickness.

Figure 17a illustrates the original and effective thicknesses of the top frozen wall across five sections. The figure displays the original thicknesses from day 15 (when the entire frozen curtain is closed) to day 45, and the effective thicknesses on day 45, as they are primarily associated with excavation. Since there is only a single row of pipes at the top, the original thickness observed in section SD is the smallest at all times. It is somewhat unexpected that, on day 45, section SA has the smallest effective thickness (2.74 m). This occurs because, on this section, the spacing between the N1 and N2 rows is small, and they are located very close to the excavation area, leading to significant encroachment of the frozen zone into the excavation area. The difference between the original thickness and the effective thickness on day 45 represents the thickness of frozen soil encroaching into the excavation area. It can be seen that nearly 1.30 m of frozen soil needs to be excavated at the top of section SA.

Figure 17b illustrates the original and effective thicknesses of the bottom frozen wall across five sections. On section SC, due to the presence of only a single row of pipes at the bottom and minimal influence from the N15 and W2 pipes, the final original thickness is the smallest, at 3.63 m. On day 45, the effective thicknesses on sections SB and SD are the smallest, at 3.16 m and 3.07 m, respectively, because the freezing pipes at the bottom of these two sections are close to the excavation area of the pump room, leading to the excavation of nearly 1.70 m of frozen soil. In addition, nearly 1.30 m of frozen soil at the bottom of section SA needs to be excavated. Therefore, it is recommended to optimize the layout of the freezing pipes near section SA and in the areas near the bottom of sections SB and SD, to avoid increasing freezing costs and excavation difficulties. Fortunately, as shown in Figure 17, the top and bottom effective thicknesses on each section meet the design requirements, with sufficient margins.

Unlike the regularly arranged top and bottom pipes, the side freezing pipes are arranged radially at varying angles. As shown in Figure 13 and Figure 15, the original thickness of the side frozen wall differs in both the horizontal and vertical directions. Therefore, 52 horizontal paths (blue dots in Figure 14) are arranged across the five sections at vertical intervals of 0.5 m to further investigate the thickness distribution of the side frozen wall and identify weak areas. Taking the horizontal height of the tunnel center as z = 0 m, sections SB to SD have 12 paths covering the range from z = 1 m (the cross-passage roof) to z = −4.5 m (the pump room floor), while the other two sections have 8 paths covering the range from z = 1 m to z = −2.5 m (the cross-passage floor). The original thicknesses of the side frozen wall at each path on day 45 are obtained and summarized in Figure 18a. The blank spaces in the figure represent non-excavation areas where no paths have been arranged. Each cell in the figure represents a path, with the numerical value indicating the original thickness (m). The darker the blue of the cell, the smaller the thickness. The average original thickness on section SC is the smallest, at only 3.23 m, while section SD follows closely with 3.45 m. The thinnest area is observed at a height of −1.5 m on section SD, with a thickness of only 2.94 m.

The effective thicknesses for all paths on day 45 are summarized in Figure 18b. To highlight the difference between smaller values, a nonlinear color-mapping method is employed. As shown in the figure, dark blue cells are still predominantly distributed on sections SC and SD, making the average effective thickness on section SD the smallest at 2.43 m, followed by section SC at 2.47 m. Additionally, on these two sections, the effective thicknesses of the side wall near the pump room are noticeably smaller. The path with the smallest thickness is located at −4.0 m on section SC (corresponding to the bottom of the pump room), which is just 0.25 m higher than the design value.

The average temperatures of the frozen curtain at various freezing times for each section were calculated using the integral method and plotted in Figure 19, along with the corresponding closure times. The development trend of the average temperature shows a clear inflection point around day 25. After day 25, the rate of decrease in the average temperature slows. Although the final average temperature on each section is below −10 °C, section SD has the highest temperature, followed by section SC. Furthermore, due to the large spacing between pipes and the greater cross-sectional area of the curtain, the frozen curtain closes last on section SC. Therefore, it is recommended to closely monitor the development of the frozen curtain at the center and slightly right areas of the cross-passage axis, and optimize the layout of side freezing pipes near the pump room by adding more pipes or reducing pipe spacing.

It should be noted that the semi-empirical approach used in this paper to predict SFCC has not been validated for various soil types or across a sufficient number of projects. Therefore, this method is recommended only for preliminary predictions of the freezing temperature field. The proposed numerical model assumes that the strata are homogeneous and isotropic, which obviously oversimplifies real-world conditions. Subsequently, the research will be extended to explore various influencing factors, including the spatial variability of natural strata, soil types, climatic conditions, and the arrangement of freezing pipes. In addition, the discussions in this chapter are applicable only to projects with similar freezing pipe layout schemes. Other freezing schemes should be examined separately. For example, in a freezing project where radially arranged freezing holes were drilled entirely from the left tunnel, the weakest part of the frozen curtain may be at the right bell mouth [8].

6. Conclusions

This study presents the design scheme and field monitoring results of an AGF project for a cross passage on Hohhot Metro Line 2, along with the development of a transient heat transfer numerical model based on the no-seepage assumption. To incorporate a realistic ice-water phase transition during soil freezing, a semi-empirical approach based on SWCC parameter estimation was employed to derive SFCC expressions for the two soils in the freezing zone. The model was validated using experimental data from existing literature and monitoring data from the project. Five typical sections and 52 horizontal paths were arranged to investigate the characteristics of the frozen curtain in a single stratum and examine weak areas of the entire curtain. The following conclusions can be drawn:

(1)

During the freezing process, with the expansion of the frozen area and the release of latent heat from the water-ice phase transition, the temperatures at all monitoring points decrease rapidly, then slowly, and ultimately stabilize. The development rate of the frozen soil towards the inner side of the frozen curtain exceeds the outward development rate.

(2)

In the absence of soil freezing test data, SFCC expressions can be estimated using readily available soil grain-size distribution curves and index properties. Comparative analysis demonstrates that the numerical model incorporating the semi-empirical method of estimating SFCC is valid and has good simulation accuracy. This approach also avoids the artificial assumptions about the water-ice phase transition process inherent in the traditional apparent heat capacity method.

(3)

Numerical simulations reveal that the cumulative volume of frozen soil varies nonlinearly over time. In the initial freezing stage, the average daily increase in frozen soil volume can reach 6 times that of the final freezing stage.

(4)

In the axial direction of the cross passage, distinct temperature field differences are observed across sections. The spacing between freezing pipes is negatively correlated with the original thickness of the frozen wall. The center and slightly right areas of the cross-passage axis exhibit a thinner frozen curtain, higher average temperature, and longer closure time. The minimum effective thickness is found at the side frozen wall near the bottom of the pump room, exceeding the design value by less than 0.3 m.

(5)

The spacing between the side freezing pipes near the pump room should be reduced to ensure the effective thickness of the frozen curtain. Additionally, the freezing pipes near section SA, as well as the bottom freezing pipes near sections SB and SD, should be maintained at a sufficient distance from the excavation area to prevent large-scale encroachment of frozen soil into the excavation areas, which could complicate excavation and reduce the effective thickness of the frozen wall.

(6)

If groundwater seepage is present in the freezing zone and hydrogeological data are available, it would be more appropriate to develop a coupled thermo-hydraulic model to account for seepage. Seepage may still disturb the temperature field, even at very low velocities.