1 Introduction

As an environmentally friendly technique in geotechnical engineering, the artificial ground freezing (AGF) is commonly used in tunneling for groundwater control and/or temporary excavation support [15, 23, 44], as shown in Fig. 1a. Besides, it is also reported as an effective method to deal with contaminated sites [6, 18, 25]. The AGF technique can convert pore water into ice to prevent groundwater from flowing into the working area and enhance the strength and stiffness of soils [20, 48]. During the freezing process, the required freezing time to create a completely closed frozen curtain not only depends on the number, size and spacing of freeze pipes, but also can be remarkably influenced by groundwater seepage flow [28]. The whole freezing process is complicated due to the coupled thermo-hydraulic interaction at low temperatures. First, the ice-water phase change occurs, while the temperature drops to a negative value. Second, groundwater can take away the heat around a freeze pipe and slow down the cooling process, and the formation of a frozen curtain can in turn affect the seepage field by changing groundwater direction and velocity.

Fig. 1
figure 1

Schematic diagrams of a artificial ground freezing converts natural soil into frozen soil via freeze pipes; b pipe inclination of an AGF system

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Construction accidents are likely to occur when the AGF technique fails to serve as waterproof barriers owing to the existence of an unfrozen path (see [35, 43]). Pipe inclination inevitably exists when installing the freeze pipes (Fig. 1b), which can increase the occurrence possibility of an unfrozen zone and thereby undermine the integrity of a completely frozen curtain. In practice, the pipe inclination in both vertical and horizontal directions is often encountered, which can be even pronounced in cases with long freeze pipes. Papakonstantinou et al. [32] reported that pipe inclination of 1.2% is likely to occur when installing a freeze pipe along the horizontal direction. The cooling plan has to be extended to minimize the occurrence possibility of unfrozen paths, resulting in an extension in project duration and the corresponding expenses.

Various studies can be found in assessing the effect of seepage velocity on the freezing efficacy and finding approaches to reduce the energy consumption of the AGF system. Table 1 summarizes typical existing publications relating to the AGF. Natural soils are often spatially variable [34, 46]; however, few investigations accounted for the spatial variability in the thermal conductivity of soil solid (ks). Therefore, the results based on the assumption of homogenous porous media could be inconsistent with the actual situations and thus increase potential risks. Furthermore, extensive engineering experience reveals that seepage has a remarkable influence on the AGF process [17, 35, 40, 41]. A closed frozen wall is unlikely to form when the seepage velocity exceeds a certain value (termed critical seepage velocity vc). Vitel et al. [43] pointed that the area between two freeze pipes cannot be frozen under 1.0–2.0 m/d groundwater conditions, while Jessberge and Jabow (1996) indicated that vc is 2.0 m/d [16]. Other reports [2, 8] also recommended different values of vc. A universal vc value is difficult to identify since it depends on the specific operating conditions, such as soil properties, groundwater condition, coolant temperature and so on [38].

Table 1 Typical existing literature on the AGF technique
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The current study mainly focuses on the influences of pipe inclination and spatial variation of ks on the efficacy of AGF and aims at evaluating the critical seepage velocity (vc) quantitatively. As an effective approach to investigate the coupled thermo-hydraulic problem, a finite element model is adopted and validated with a laboratory test under seepage flow conditions. Since the temperature evolution of a point is mainly affected by two closest freeze pipes, a unit cell model including two freeze pipes is considered by a coupled thermo-hydraulic finite element method. Furthermore, the pipe inclination is evaluated in the unit cell model by prescribing freeze pipe spacing (Lp). Spatial variation of ks is represented by a three-dimensional (3D) lognormal random field. Different stochastic models involving various factors (e.g., variation in ks, Lp and v) are conducted by Monte Carlo simulations to estimate the statistical characteristics of freezing time. The results are compared to cases where ks is constant; the latter is more familiar with practitioners. The additional freezing time is tabulated by a series of parametric analyses to account for the influence of different sources of uncertainty. This work provides a rule of thumb for estimating a reasonable spacing of freeze pipes (Lp) and the corresponding critical seepage velocity (vc). The freeze pipe spacings used in practice are discussed to account for the occurrence of inevitable pipe inclination.

2 Coupled thermo-hydraulic numerical model

2.1 Basic assumptions

Heat transfer and seepage flow during the freezing process should be considered when examining the influences of groundwater and pipe space on the performance of AGF technique. In addition, the development of temperature field is a heat transfer process with phase change during artificial ground freezing. For simplicity, a numerical model is proposed and based on the following assumptions.

  1. (1)

    The mechanical aspect (i.e., stress and strain) is not taken into account.

  2. (2)

    Saturated condition is adopted for natural soil; that is, liquid water and/or solid ice occupy the entire void space of soil.

  3. (3)

    Temperature of coolant is loaded directly on the outer wall of the freeze pipes, and the heat exchange between the freeze pipes and coolant is ignored.

  4. (4)

    Only heat conduction is considered, and convection and radiation are negligible.

  5. (5)

    Fourier’s law is suitable for describing the heat transfer process and the groundwater flow in porous media satisfies Darcy’s law.

2.2 Governing equation

2.2.1 Heat conduction equation

The heat conduction equation with water/ice phase transition deduced from energy conservation in freezing porous media can be expressed as:

$$\left( {\rho C_{p} } \right)_{{\text{e}}} \frac{\partial T}{{\partial t}} + \rho C_{p} {{\varvec{\upnu}}}\nabla T + \nabla \left( { - k_{{\text{eff}}} \nabla T} \right) = Q$$
(1)

where ρ is fluid density; Cp is fluid heat capacity at constant pressure, which equals the heat capacity of water (Cw). (ρCp)e is effective volumetric heat capacity at constant pressure; ν is seepage velocity vector of water; T is temperature; Q is heat source (or sink); keff is effective thermal conductivity of porous media, which mainly depends on the internal micro-structure of materials [19]. The thermal conductivity of multi-phase materials can be determined by the thermal conductivity and volume fraction of each component (i.e., soil solid, water and ice), and the geometric mean weighted model is used to estimate the effective thermal conductivity.

$$k_{{\text{eff}}} = k_{{\text{s}}}^{1 - n} \times k_{{\text{w}}}^{{f_{{\text{w}}} }} \times k_{{\text{i}}}^{{f_{{\text{i}}} }}$$
(2)

where n is porosity; fw and fi are, respectively, the volume fractions of water and ice satisfying fw + fi = n; kw and ki are thermal conductivity of water and ice, respectively; ks is thermal conductivity of soil solid typically ranging from 1.2 to 7.5 W/(mK) [33].

In addition, the value of (ρCp)e is also defined by an averaging model to account for both solid matrix and fluid properties.

$$\left( {\rho C_{p} } \right)_{{\text{e}}} = \left( {1 - n} \right)\rho_{{\text{m}}} C_{{\text{m}}} + n\rho C_{p}$$
(3)

where n is porosity; ρm is the density of solid matrix in the porous media; Cm is the heat capacity of solid matrix in the porous media at constant pressure.

2.2.2 Continuity equation

The continuity equation of water and/or ice is expressed as [14]:

$$\frac{\partial }{\partial t}(n\rho ) + \nabla \cdot (\rho {{\varvec{\upnu}}}) = 0$$
(4)

where n is porosity; ρ is fluid density; ν is seepage velocity vector of water.

The water flow water seepage in porous media is assumed to satisfy the Darcy’s law as:

$${{\varvec{\upnu}}} = - \frac{\kappa }{\mu }\nabla p = - \frac{{\kappa_{0} \kappa_{{\text{r}}} }}{\mu }\nabla p$$
(5)

where κ is intrinsic permeability of the porous medium; κ0 is permeability before freezing; κr is relative permeability; p is fluid's pressure; μ is fluid's dynamic viscosity, which can be expressed as a function of freezing temperature [21].

$$\mu = 2.1 \times 10^{ - 6} \exp \left( {\frac{1808.5}{{273.15 + T}}} \right)$$
(6)

where T is temperature.

The parameter of κr ranges from 0 to 1 and depends on the water saturation, which is utilized to consider the effect of water–ice transition during freezing process [30]:

$$\kappa_{{\text{r}}} = \sqrt {S_{{\text{r}}} } \left[ {1 - \left( {1 - S_{{\text{r}}}^{1/m} } \right)^{m} } \right]^{2}$$
(7)

where m is material constant and Sr is water saturation that equals the ratio of volume of water to pore space. In addition, the saturation degree is also regarded as a function of temperature as [31]:

$$S_{{\text{r}}} = \left[ {1 + \left( {\frac{{T_{0} - T}}{\chi }} \right)^{{\frac{1}{1 - \lambda }}} } \right]^{\lambda }$$
(8)

where χ and λ are material constants; T0 is bulk freezing point; T is temperature.

2.2.3 Latent heat of phase change

The latent heat generates when the pore water converts into ice, and the release of energy can be solved by regarding the latent heat as part of material heat capacity [5].

$$C_{{\text{a}}} = (C_{{\text{i}}} + C_{{\text{w}}} )/2 + L/\Delta T$$
(9)

where Ca is apparent heat capacity; Ci and Cw are heat capacity of ice and water, respectively. L is latent heat during the water/ice phase change; ΔT is temperature transition between water and ice.

3 Numerical simulation

3.1 Model description

As illustrated in Fig. 2a, two freeze pipes (pipes I and II) are considered, which can be treated as a unit cell of one circle of freeze pipes surrounding a tunneling excavation zone. Two typical temperature monitoring points are set to inspect the temperature variation, as shown in Fig. 2b. Monitoring point A is arranged in the middle of two freeze pipes. A frozen wall with a thickness of about 1.0 m is often desired. For this reason, point B located on the upstream of freeze pipe II with a distance of 0.5 m is set to monitor the influence of seepage on the temperature changes.

Fig. 2
figure 2

Schematic illustrations of the simulation model. a cut view of the model; b two freeze pipes (I and II) in the unit cell and location of two monitoring points (A and B)

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Figure 3a, b depict a 3D numerical model of these two freeze pipes subject to different directions of seepage flow. The model is set up with a length of 15 m, a width of 10 m and a height of 10 m to ensure that the boundary effects are negligible. The initial temperature of natural soil is set at 18 °C. The adiabatic boundary condition is adopted for all faces of the model. The diameter of freeze pipes is 0.108 m, which is generally adopted in practice [12]. The hydraulic heads on the outlet boundaries are set at 0 m. Thus, the hydraulic difference between opposite surfaces is equal to the hydraulic head at inlet boundary that can be equivalently transformed from the groundwater seepage velocity.

Fig. 3
figure 3

a Groundwater flows along x-direction (vx > 0 and vy = 0); b groundwater flows along y-direction (vx = 0 and vy > 0). The light blue surface indicates a zero hydraulic head boundary condition

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3.2 Material properties and cooling plan

The sandy soils are regarded as saturated media, and the properties of soil and fluid are invariant with respect to time. Besides, Darcy’s law and Fourier’s law are adapted to describe the groundwater flow and heat conduction in porous media, respectively. Material properties are listed in Table 2. The coolant temperature is applied directly to the surface of freeze pipes as a boundary condition according to the cooling plan listed in Table 3.

Table 2 Material properties involved in numerical simulations
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Table 3 Cooling plan of the AGF system
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Geomaterials often have distinct spatial variability due to sedimentary processes and prompt larger variability in the vertical direction than that in the horizontal plane [9]. Herein, the spatial variability of soil solid thermal conductivity (ks) is characterized by a 3D lognormal random field, where the underlying normal random field is simulated by the modified linear estimation method [24] due to its competitive computational efficiency for 3D random field generation. The mean value of ks is selected as 3 W/(mK), and the coefficient of variation (COV) is 0.25 [45]. Since the autocorrelation distance in the horizontal direction is around 10 times that in the vertical direction [7], the scales of fluctuation (SOF) along the horizontal and vertical directions are set to 10 m and 1 m, respectively. Figure 4a shows a typical realization of the thermal conductivity random field. The mesh and boundary conditions of the numerical model are plotted in Fig. 4b.

Fig. 4
figure 4

a A typical realization of thermal conductivity random field (μks = 3.0 W/(mK) and COV = 0.25). The SOFs along the horizontal and vertical directions are 10 m and 1 m, respectively; b mesh size, hydraulic boundary conditions and temperature field of finite element model at 40 days. All the surfaces are adiabatic

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To explore the effects of freeze pipe spacing Lp and seepage velocity v, a series of simulation models are established. Five different freeze pipe spacings Lp (namely, 0.5 m, 0.8 m, 1.0 m, 1.2 m and 1.5 m) are considered. Regarding the seepage velocity, Locatelli et al. [27] pointed out that a mean velocity of groundwater is about 0.345 m/d. Besides, some scholars summarized that one hundred meters per year are a typical horizontal seepage velocity, which is roughly 0.274 m/d [37]. Therefore, the range of seepage velocity is considered from 0 to 1.5 m/d as indicated in Table 4. The model with 1.0 m freeze pipe spacing under 0.3 m/d seepage velocity is selected as the reference case.

Table 4 Range of influence factors considered and the selection of a reference case
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3.3 Model validation

The numerical results are compared with the data measured by laboratory tests [36] to validate the correctness of the coupled thermo-hydraulic freezing model. Pimentel et al.’s (2012b) model was characterized by an insulated box with inner dimensions of L × W × H = 1.3 m × 1.0 m × 1.2 m, as shown in Fig. 5a. To examine the influence of groundwater, the seepage flow (1.4 m/d) was investigated, and two opposite surfaces in the x-direction are set as the inlet and outlet boundaries. Due to the existence of inflow of water, the temperature at the inlet surface was kept at a constant value (18.5 °C). The temporal temperature, approximated by \({\text{T}}_{pipe} \left( t \right) = - 21 + 15\left( {t{ - }\frac{15}{{{ - }21{\text{ - T}}_{{\text{i}}} }}} \right)^{ - 1}\), was applied as the boundary condition of freeze pipes (see Fig. 5b). In addition, two paths (ML1 and ML2) were placed along x- and y-direction in z = 0.6 m plane. More details about the materials parameter involved in the tests can be referred to [36]. The numerical analysis is carried out using the proposed coupled thermo-hydraulic model. Temperatures at five typical time points (namely, 0 h, 1 h, 5 h, 20 h, 40 h) are selected to compare the calculated values and experimental data, as shown in Fig. 6 and Table 5. It can be observed that the simulated results generally agree with the experimental data apart from the deviation at 5 h where the simulated results overestimated the freezing effect especially at x = 1.2 m, which is likely resulted from the inaccurate boundary condition (i.e., insulated and impervious performance) imposed on the outer surface in the numerical model, whereas the outer surface of the physical model in the lab is neither completely insulated nor truly watertight, since this surface is exposed in air and thus the temperature varies with time (more details can be found in [39]). Table 5 lists the relative errors of the above two sets of temperature data along the ML1 and ML2 at different moments, showing that 82.4% of the points have a discrepancy being lower than 30%. The consistency between the simulation results and the laboratory data serves to validate the proposed numerical model.

Fig. 5
figure 5

a Geometric size and monitoring points information of laboratory test model [36]; b mesh size, boundary conditions and temperature field of validation model at 40 h. Ti is the initial temperature, which equals 18.5 °C

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Fig. 6
figure 6

Comparison of simulated results with experimental data along ML1 and ML2 at 1.4 m/d seepage flow. Dots denote the experimental data and the lines are the simulation results

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Table 5 Relative errors at five typical freezing periods under 1.4 m/d velocity of groundwater
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The validation is mainly used to check the effectiveness of the numerical model in solving the issue of coupled thermo-hydraulic AGF. For investigating the effects of three sources of uncertainty (variation in ks, Lp and v) on the performance of freeze process, different stochastic models with various parameters listed in Table 4 are conducted by Monte Carlo simulation with 100 times and the statistical responses are obtained. Each simulation represents one possible working field where ks spatially varies. For comparison, deterministic analyses on these cases without variation in ks are also analyzed in the next section.

4 Results and discussion

The proposed finite element model is utilized to illustrate the effects of spatial variability of ks and pipe inclination on the process of AGF, and to determine the critical seepage velocity (vc). This section discussed the behaviors of the temperature development and variation on the freezing time when the monitoring points reach a target temperature (usually set at 0 °C or − 10 °C in a real AGF project). Figure 7a, b illustrate the simulated temperature field of the reference case with deterministic and random thermal conductivity, respectively.

Fig. 7
figure 7

Temperature contours of the reference case (vx = 0.3 m/d and Lp = 1.0 m) at a 40-day freezing duration. a deterministic thermal conductivity, and b random thermal conductivity

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4.1 Effect of variation in k s

Deterministic and random models where two freeze pipes with Lp = 1.0 m and zero-seepage velocity are discussed in this section. The recorded temperature against freezing time is shown in Fig. 8, which shows the development trend of temperature of two monitoring points (A and B) among 100 simulations. Besides, the results of the deterministic model where ks remains constant are also plotted. One can find that the spatial variability of ks affects the freezing performance, presenting the results of random cases lie in both sides of the deterministic data. The temperature trends of the simulations are similar. Some variation of temperature at points A and B can also be observed in Fig. 9 as histograms. After 40 days of freezing, the temperature of point A dropped below the target temperature, while the temperature of point B ranges from − 4 to − 9 °C. The curves in Fig. 8a show that the temperatures of point A dropped rapidly within the earlier stage of the freezing days and a stair in the curves appears at around zero degrees due to the existence of latent heat in phase change. Figure 10 illustrates the 0 °C and − 10 °C temperature contours at three typical freeze durations. The thickness of frozen curtain gradually grows with the extension of the freeze period, and an elliptical frozen wall is eventually formed.

Fig. 8
figure 8

Temperature versus freezing time. a monitoring point A; b monitoring point B. Solid lines depict 100 simulation results of one random model (Lp = 1.0 m) where ks possess spatial variability (μks = 3 W/(mK) and COV = 0.25) under the zero-seepage condition. A target temperature is set at − 10 °C; black dashed lines are the results of deterministic model where ks is constant at 3 W/(mK)

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Fig. 9
figure 9

Histograms of temperature of monitoring points at 40 days. a point A; b point B. The dashed line represents deterministic results

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Fig. 10
figure 10

Temperature contours at different freezing periods under the zero-seepage condition. a 15 days; b 25 days; c 40 days

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4.2 Effect of seepage velocity

Figures 11 and 12 show the temperature distribution from deterministic simulation under different seepage velocities and directions at 40 days. It is clear that when the groundwater flows along the direction perpendicular to the pipe axis (Fig. 11), the formation of the closed frozen body around freeze pipes delays with the increment of velocity. The freezing process is accordingly affected under a relatively high seepage velocity that can carry more energy. As the seepage velocity increases, the energy of the freeze pipes is not sufficient to overcome the high-velocity of groundwater, which could impede the whole freezing process and reduce their efficiencies. By contrast, the difference among the results is quite modest when groundwater flows along the axis of pipes, as shown in Fig. 12. It merely exhibits a slight difference around the inlet zone (Fig. 12c). This phenomenon can be attributed to the fact that the water flow direction is consistent in the heat conduction direction when groundwater flows along x-direction, while the direction of seepage is perpendicular to the heat transfer direction under y-direction groundwater seepage.

Fig. 11
figure 11

Temperature contours of the simulation homogenous model (Lp = 1.0 m) under various seepage velocity along x-directions at 40 days of freezing. a zero-seepage condition; b vx = 0.3 m/d; c vx = 1.0 m/d

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Fig. 12
figure 12

Temperature contours of the simulation homogenous model (Lp = 1.0 m) under various seepage velocity along y-directions at 40 days. a vy = 0.3 m/d; b vy = 1.0 m/d; c vy = 1.0 m/d

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Figure 13a illustrates the temperature development trends from deterministic models where seepage velocity ranges from 0 to 1.5 m/d along the x-direction. The temperature drops rapidly at the early stage of freezing time under a low-seepage velocity condition. Under the 0.1 m/d seepage velocity, the temperature trend is close to that of a zero-seepage condition. This is because the heat flux through freeze pipes is large enough to overcome the groundwater energy. On the contrary, when the groundwater flows along y-direction, the differences among the temperature development with different seepage velocities are slight (see Fig. 13b), which echoes the temperature contours in Fig. 12. Besides, the isotherm contours of deterministic model under 0.3 m/d seepage velocity are plotted in Fig. 14. When groundwater flows along x-direction, the frozen body gradually enlarges over freezing time (Fig. 14a). The water flow rate increases through the gap between the adjacent freeze pipes as the frozen arch grows. Once a closed frozen arch is formed, no groundwater can flow within the interior of the frozen arch, and the impact of the seepage flow reduces. As the freezing time continues, the isotherms increasingly move to the downstream. By contrast, the formation of frozen body tends to be symmetrical if the inflows of water are parallel to the axis of pipe (see Fig. 14b). The comparison between Fig. 14a, b also shows that the effect of seepage along x-direction is more evident than that along y-direction. Therefore, the groundwater that flows perpendicular to the pipe axis becomes the major concern in the AGF system.

Fig. 13
figure 13

Temperature evolution trends at point A from deterministic homogenous models under various seepage velocity. a Groundwater flows along x-direction; b groundwater flows along y-direction. The straight lines (y = 0 °C and y = − 10 °C) represent the target temperature

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Fig. 14
figure 14

Isotherm contours of the simulation homogenous model with 0.3 m/d seepage flow at 10, 15, 25 and 40 days. a Groundwater flows along x-direction; b groundwater flows along y-direction

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For the sake of identifying the value of critical seepage velocity vc, we consider the results in our designated case that the temperature of main monitoring points reaches the target temperature within 40 days and define the allowable maximum seepage velocity as vc. Figure 15 demonstrates the relationship between seepage velocity and the temperature obtained from simulation models at three typical freeze durations. The deterministic and random results are also compared to account for the variation in material’s thermal properties. For random cases, the mean values are calculated out of 100 repetitive simulations, which represent the average level of freezing time. In addition, the most unfavorable scenario of temperature should be prioritized considering the high operating costs and safety. Therefore, the maximum temperatures are also plotted in Fig. 15.

Fig. 15
figure 15

Temperatures of three freezing periods (t) under different seepage velocities (Lp = 1.0 m). The straight lines (y = 0 °C and y = − 10 °C) represent the target temperature

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The adverse effect of seepage is more evident with freezing time when the groundwater flows along x-direction. Temperatures at point A and point B increase as vx increases, and the temperatures of point A are always lower than those of point B due to the enhancement effect of two adjacent pipes. At 40 days, the temperature of point A fails to descend to 0 °C and − 10 °C when vx exceeds 1.0 m/d and 0.5 m/d, respectively, regardless of the variation in ks. Similar observations can be found from the average values of 100 random field simulations. The critical seepage velocity for the worst case is relatively small. For safety, the value of vc can be recommended within the range of 0.3–0.5 m/d. If the velocity does not exceed vc, a closed frozen body should be able to form within 40 days in most cases. Once exceeds, some extra measurements (e.g., extending the freezing period or reducing the freeze pipe spacing) should be taken to avoid any connected unfrozen paths that may lead to the leakage of groundwater or support failure. However, when the water flows along y-direction that parallels to the pipe axis (see Fig. 15d–f), temperatures of monitoring points become less sensitive to the seepage velocity. The temperatures of points A and B are below zero degree at 40 days. Therefore, due attention would be diverted to the groundwater that flows along the direction perpendicular to the pipe axis.

4.3 Effect of freeze pipe spacing

In the AGF system, freeze pipe spacing is one of the main design parameters. However, inclination inevitably occurs during installing a freeze pipe. In order to determine the optional freeze pipe spacing with consideration of the pipe inclination, five typical freeze pipe spacings are selected (see Table 4) and the seepage velocity is limited to 0.3 m/d; the latter is a common seepage flow velocity of groundwater in reality [37]. Figure 16a–c reveal the effect of the freeze pipe spacing between two freeze pipes on the temperature of the monitoring points at vx = 0.3 m/d. These trends are similar under various seepage velocity. Deterministic results and the average levels of 100 simulation results are close and both of them are lower than data of the unfavorable scenarios (i.e., maximum temperature). After a 40-day continuous freezing period, the temperature at monitoring point A can decrease below − 10 °C if the freeze pipe spacing is controlled within 1.0 m. The corresponding results under y-direction groundwater are presented in Fig. 16d–e. In general, the temperature drops gradually with time. Little difference in temperature between freeze pipe spacing can be seen in Fig. 16d–e, showing that the freeze pipe spacing has a negligible effect on freeze performance when the groundwater flows along the direction parallel to the pipe axis. Under a common groundwater (vx = 0.3 m/d), the spacing Lp should be kept within around 1.1 m to ensure the temperature of point A can reach the target temperature (− 10 °C) at 40 days. Otherwise, freeze pipes require more time to generate a closed frozen body. The corresponding measures should also be adopted to ensure the reliability and safety of the AGF system.

Fig. 16
figure 16

Temperatures of three freezing periods (t) with at different freeze pipe spacings at vx = 0.3 m/d. The straight lines (y = 0 °C and y = − 10 °C) represent the target temperature

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4.4 Engineering implications

The interaction between groundwater seepage and freeze pipe spacing is considered in terms of the impact on the required freezing time that the temperatures at monitoring points drop to below a target temperature (0 °C or − 10 °C). The results of freezing time for the target temperature from deterministic and random models are calculated and listed in Table 6. The statistical characteristics of 100 Monte Carlo simulations, including the average freezing time and COV, are utilized to show the influence of the spatial variability of ks. It is worth mentioning that the most unfavorable scenario (i.e., upper bound of freezing time) should be considered. The upper bounds of the freezing time are also presented in Table 6, which could be estimated by the following equation [23].

$$T_{{\text{u}}} = T_{{{100}}} { + }\left( {T_{100} - T_{{{99}}} } \right)$$
(10)

where Tu is the upper bound of freezing time, and T100 and T99 are the largest and second-largest freezing time out of 100 Monte Carlo simulations, respectively. The term T100–T99 is adapted to correct the bias caused by the underestimation of T100.

Table 6 Freezing time for target temperature of deterministic and random analyses (vx = 0.3 m/d and vy = 0 m/d)
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Data in Table 6 show an increasing trend in freezing time with freeze pipe spacing. Deterministic results and average freezing time from stochastic models have no obvious difference. However, the upper bounds are greater than the average freezing time, which is more pronounced with the increase in freeze pipe spacing. For example, when Lp exceeds 1.2 m, the upper bound freezing time is approximately as large as 1.5 times of the average value. It is necessary to extend the freezing time when Lp is over 1.2 m to avoid the unfrozen path and ensure a certain safety margin. The variation in freezing time cannot be overlooked since any unfrozen path can threaten the safety of projects.

These statistics listed in Table 6 can serve as a rule of thumb for predicting freezing time and assessing the risk in comparable AGF projects. It should be noted that there may be some other working conditions with other designated freeze pipe spacings under various grounder water conditions. Similarly, for a given groundwater velocity (v), its relationship with the pipe spacing (Lp) can be obtained via the approach mentioned in Sect. 4.3, which is expected to provide a reference for participators to reasonably design the pipe spacing value, as demonstrated in Fig. 17a. This recommendation may have practical implications. When layers with high-flow rate are encountered during underground construction, reducing the freeze pipe spacing is an effective strategy to overcome the high energy of groundwater. That is, different seepage velocities correspond to various freeze pipe spacing. More precisely, the ranges of optional freeze pipe spacing can be estimated via the results in Fig. 17a, once the local groundwater is already known. The ultimate freeze pipe spacing used in practice should be reduced for a safety margin given the existence of pipe inclination. The threshold of available freeze pipe spacing can be obtained by introducing the maximum pipe inclination:

$$L_{{{\text{pu}}}} =\,L_{{\text{p}}} - {2}\beta_{{\text{m}}} H_{{\text{p}}}$$
(11)

where Lpu is the ultimate freeze pipe spacing used in project; βm is the allowable maximum deflection of freeze pipe, such as 1.2%; Hp is the length of freeze pipe. The relationship between Lpu and Lp is plotted in Fig. 17b. To sum up, the preferred method is to use a long cooling plan and shorten the spacing between freeze pipes to ensure the integrity of the frozen ground and reduce potential risks.

Fig. 17
figure 17

a Relationship of freeze pipe spacing versus seepage velocity; b schematic diagram of two freeze pipes with inclination and the relationship between Lp and Lpu. The temperature recorded at point A is used in tabulating to evaluate a completely closed frozen curtain

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5 Conclusions

A coupled thermo-hydraulic numerical model is proposed to examine the performance of an AGF system that is subjected to uncertainties in pipe inclination and thermal conductivity. The model is validated by laboratory test results. The numerical analysis indicates that the coupled model is capable of simulating the freezing process of AGF under groundwater conditions. In addition, Monte Carlo simulations have been carried out to assess how the uncertainties involving variation in thermal conductivity, inclination of freeze pipes and seepage velocity impact upon the freezing efficacy. From this work, the following findings are observed.

  1. (1)

    The temperature fields from Monte Carlo simulations of the random models were compared to those of deterministic scenarios, which indicate the spatial variability of ks has some influence on the freezing performance. By contrast, the temperature development of AGF is more significantly influenced by seepage velocity, exhibiting that the freezing time should increase with the increment of seepage velocity. Besides, the pipe inclination during installation prescribed by freeze pipe spacing is also considered, which has an adverse effect on the freezing time. The resultant effect of these three sources of uncertainties can be accounted for by extending the freezing process. The results of additional freezing time are tabulated by a series of parametric studies.

  2. (2)

    Reducing the freeze pipe spacing is an effective strategy to overcome the high energy of groundwater when high-flow seepage rates are encountered. Different seepage velocities correspond to various freeze pipe spacing. The work is capable of determining the optional range of freeze pipe spacings and the corresponding critical seepage velocity, and can serve as a rule of thumb for estimating the ultimate freeze pipe spacing utilized in the AGF projects when considering the inevitable pipe inclination during installation.

These findings may be applicable in tunnel construction with AGF technique. It should be noted that the proposed method is limited to exploring the thermo-hydraulic process of AGF. The variation in hydraulic conductivity is ignored in this study since the seepage flow speed was considered separately. The mechanical behaviors need to be considered since the development of a multi-physics field model requires understanding the fundamental mechanisms of the AGF system and proper application of numerical techniques to obtain more reliable results.