# Study on Temperature Field Evolution and Frozen Wall Closure Judgment Criteria of Inclined Shaft under Inclined Holes Freezing Condition

Authors: Chi Zhang; Yu Zhang; Weihao Yang; Haipeng Li; Quan Lu
Journal: Geofluids
 (2022)

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## Abstract

A similarity experiment study is conducted to investigate the evolution characteristics of temperature field and frozen wall closure judgment criteria for the inclined shaft, under inclined holes freezing condition using an adjustable-angle freeze sinking analogous instrument. The changes of temperature and frozen wall thickness with the freezing time in the inclined shaft section, are recorded in real time, and the water pressure of axial and radial hydrological holes are measured. The results demonstrate that the soil temperature change can be divided into three stages under the conditions of salt-water temperature of -32°C and flow rate of 3.18 m/s: rapid cooling before reaching the freezing point, slow cooling in the process of freezing latent heat release, and accelerated cooling and stabilization after freezing latent heat release. The distance from the freezing pipe is the main factor affecting the freezing wall temperature. Affected by the arched arrangement of frozen pipes, the freezing speed in the position of arch crown and arch baseline is slower than the position of wall corner and base plate. A new judgment criterion of frozen wall closure based on the water pressure of the axial hydrological pipe is recommended. In all, this paper has an enlightenment significance for understanding the evolution of the soil temperature field, and predicting the frozen wall closure time in applications of the inclined holes freezing method for the inclined shaft.


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## Body

## 1. Introduction

The artificial freezing method is a reliable method for shaft sinking when the shaft passes through the water-rich soft stratum [1, 2]. By embedding the freezing pipe in the water-rich unstable soil layer [3], the artificial freezing method can freeze the natural soil into artificial frozen soil, so as to increase its strength and stability [4]. This ensures the normal progress of the drilling and construction through the protection of the freezing wall (FW) [5]. Thus, this method is also widely used in engineering applications such as mining [6], landslide stabilization, subway [7, 8], and water conservancy project [9], for example.The prediction of temperature field evolution and the determination of frozen wall closure time are the core technologies of freezing method used to solve the contradiction between the strength of frozen wall and engineering investment [10]. Although the artificial freezing method has a wide range of applications, it is still difficult to judge the evolution of temperature field, which is affected by multiple factors such as the geological conditions, wellbore geometry and layout of freezing system [11], for example. Thus, researchers have carried out extensive parametric studies on the operating conditions, refrigerant type, system geometry and ground properties [12, 13]. For instance, Vitel and Rouabhi (2015) develop a coupled model between frozen pipe and surrounding soil. They deduce that the frozen zone is negatively correlated with the refrigerant temperature, and the temperature of the wall is more similar to the temperature in the annular space with a higher coolant flow rate [6]. In order to study the temperature distribution of flowing groundwater as well as the deformations of the ground over time in the application of freezing method, Pimentel et al. (2007) develop a 3D thermal-hydraulic-mechanical (THM) model. Their simulation results are coherent with the experimental results [14]. At present, the studies on the temperature field of freezing method mainly focus on vertical shaft, and the used methods are mainly based on numerical simulation, which lacks physical experiments to reveal the actual freezing situation [15].The application of the freezing method in vertical shaft has tended to be mature [16, 17], and the current technical difficulties mainly focus on the construction of inclined shaft. Because the shaft is inclined, the frozen wall around the shaft unevenly grows in the freezing process, which makes the determination of the freezing time more difficult. For the inclined shaft drilling by the freezing method, the vertical freezing hole is the most commonly used freezing method. In order to assess the temperature field distribution of a freezing inclined shaft, Sun and Ren (2021) develop a three-dimensional physical simulation test system. They deduce that the heat capacity of sand first decreases, and then increases. Afterwards, it decreases and finally tends to be stable during cooling in the range of 25°C to -20°C [15]. He and Du et al. (2017) conduct an analysis for the development of temperature field and frozen wall in water rich silty stratum, based on the field detection and numerical simulation. The results of their study demonstrate that the turbulence of groundwater has a certain weakening effect on the freezing [18], which makes the calculation of the actual project by numerical means difficult. In fact, the arrangement of vertical pipes in the freezing process of inclined shaft greatly increases the project cost and causes lot of material waste, which is the main problem restricting the application of the freezing method in inclined shaft engineering. If the inclined freezing hole can be applied to inclined shaft excavation, these problems will be well solved.The current study mainly focuses on the vertical freezing hole. In addition, there is a lack of understanding of temperature field for the inclined freezing hole method, which is the main limitation of the development of this method [15]. Fan (2014) compares the formation characteristics of the freezing curtain when the inclined shaft is frozen along the axis with the vertical freezing method. He shows that the axis freezing can reduce the maximum stress borne by the shaft wall in the excavation process, because the frozen wall is more uniform [19]. Chen et al. (2013) conduct a systematic study on the development law of freezing temperature field in inclined shaft under the inclined freezing hole using physical simulation test. The results of the study demonstrate that the cooling rate of the arch and side wall is clearly faster than other positions at the initial stage of freezing [20]. Although some studies have been performed to assess the development characteristics of frozen wall under the background of inclined hole freezing, the discussion on the evolution and statistical analysis of temperature field is not detailed enough [21, 22], and the judgment criteria for frozen wall closure is not provided. Therefore, further experiments and systematic data analysis should be performed to understand the evolution of temperature field, and to investigate an accuracy determination means of frozen wall closure in the inclined hole freezing method [23, 24].In this paper, a physical simulation experiment is developed according to the practical inclined shaft parameters, in order to investigate the temperature field evolution under the inclined freezing hole. The freezing system, freezing time and formation strictly comply with similar criteria. Based on the experimental results, the evolution law of axial and radial temperature field of shaft section is discussed in detail, and a novel judgment criterion of frozen wall closure based on the water pressure of hydrological pipe is proposed. It is expected that this study can guide a further understanding for the evolution characteristics of temperature field for the inclined shaft, under inclined holes freezing condition.


## 2. Material and Method

### 2.1. Engineering Background

In this study, a practical inclined shaft sinking under vertical holes freezing condition is used as the basis of indoor similarity experiment design. As shown in Figure1, this inclined shaft sinking engineering is located in Yulin City, Shanxi Province, China, and the engineering difficulty lies in the stratum of saturated sand sandwiched by the clay layers. Therefore, the intercalated layers of saturated sand and clay are considered as the simulation stratum of this study.Figure 1
Location of the inclined shaft sinking engineering and the photos of the site engineering.According to the inclined shaft size and formation characteristics, the freezing hole layout scheme is designed under the condition of inclined hole freezing method, which is used as the prototype of laboratory simulation, as shown in Figure2. The single lap of inclined holes freezing is used as the freezing process. The specific design parameters of this freezing method are shown in Table 1.Figure 2
Schematic diagram of the inclined shaft section and layout parameters of the inclined freezing holes: (a) from the transverse section view, (b) from the longitudinal section view.
(a)(b)Table 1
Prototype parameters of the freezing system.
Classification:Salt waterFreezing pipeParameter:TemperatureSpeedSizeAngleHole spacingValue:-32°C0.12 m/sφ159 mm ×6 mm10°1.2 m


### 2.2. Similarity Experiment Design and Establishment

#### 2.2.1. Similarity Criterion

For freezing through the freezing pipe, the temperature field at the cross section of the freezing pipe and surrounding soil can be expressed as:(1)∂θn∂τ=an∂2θn∂r2+1r∂θn∂r(2)an=λn∙Cnwhere r is radial coordinate (m), r0 is the freezing pipe diametral coordinate (m) such that 0 < r0 < r < ∞, τ is the time of freezing (s) such that τ >0, θn is the temperature of point r (°C), n =1 represents the non-frozen zone, n =2 represents the frozen zone, an is the thermal diffusivity (m2∙s−1), λn is the thermal conductivity of soil (W∙m−1∙°C−1) and Cn is the volumetric heat capacity (J∙m3∙°C−1).The boundary condition is given by:(3)θr,0=θ0θ∞,τ=θ0θρ,τ=θDθr0,τ=θywhere θ0 is the original temperature of soil (°C), ρ is the coordinate of the phase transition interface, θD is the freezing temperature (°C) and θy is the temperature of salt-water (°C).At the phase transition interface (r=ρ), the heat balance equation can be expressed as:
(4)λ2∂θ2∂rr=ρ−λ1∂θ1∂rr=ρ=Bdρdτwhere B is the freezing latent heat per unit volume of soil.According to the equation analysis method and Equations (1)-(4), the similarity criterion can be expressed as:
(5)FF0,K0,R,θ=0where F0=a∙τ/r2 is the Fourier criterion, K0=B/c∙t is the Kosovich criterion, R is the Geometric criterion and θ is the rule of the temperature.If the simulation experiment is performed based on the original soil,Ca = 1 and Cc= 1. When the moisture content and the freezing latent heat of soil are similar, the time similarity ratio of the simulation experiment is the square of the geometric similarity ratio, and the temperature of each point in the model is consistent with the prototype, which can be expressed in equations (6) and (7).
(6)Cτ=Cr2(7)Ct=1where C is the similarity coefficient, subscripts a, c, τ and t represent the thermal conductivity, volumetric specific heat, time and temperature, respectively.


#### 2.2.2. Similarity Experiment Design and Establishment

The specific parameters of practical inclined shaft and the freezing holes are shown in Figure2 and Table 1, respectively. Based on the similarity criterion, accuracy requirement and the dimensions of the practical engineering, the freezing pipe uses a precision seamless metal tube having a diameter of 6 mm. Thus, the geometric contraction is 159/6 =26.5. Accordingly, the wall thickness of the freezing pipe calculated by the similarity criterion is 6/26.5 = 0.226 mm. Due to the limitation of machining accuracy, the calculated size of the freezing pipe is adjusted in the actual experiment, and the final used size is Φ6 × 1 mm. In addition, the time scale of similarity experiment is set to 26.52 = 702.25. The inclination of the freezing pipe is set to 10° according to the inclination of the original inclined shaft. The positive freezing period temperature of the simulated test is -32°C and the speed of salt-water is 0.12 × 26.5 = 3.18 m/s.According to the stratigraphic distribution of the original inclined shaft, the stratum of the similarity model is the saturated sand sandwiched by the clay layers. The detailed layout and parameters are presented in Figure3(a)–3(b). The sand used in the experiment is the river sand produced in Xuzhou, China, which is pretreated by the immersion method. In addition, the clay is taken from Lingshou County, Hebei Province, China, and prepared with 15% water content. In order to facilitate the filling of soil and the placement of sensors such as temperature measuring lines, the test bench is layered, filled and tamped in a vertical state. Each clay layer is filled in three times, with a filling of 0.1 m for each time, then manually compacted and measured for the rate of water content. The saturated sand is infilled 0.15 m each time and tapped by an electric flat vibrator (Figure 3(b)). In order to isolate heat dissipation, the outer surface of the test bench is wrapped with foam board and polyurethane insulation cotton. Moreover, the soft polyurethane foam is used as external thermal insulation material in the salt-water pipeline, as shown in Figure 3(a).Figure 3
(a) Site photos of similar experiments and (b) parameters of stratum layout.


#### 2.2.3. Arrangement of Freezing Equipment and Monitoring Probes

(1) Freezing Equipment. The arrangement of the freezing equipment is shown in Figure 4. The main freezing pipes use 26 copper pipes with a dimension of Φ6 × 1 mm, and the center distance of each copper pipe is 53 mm. The trunk pipe uses steel pipe with a dimension of Φ25 × 2 mm. The three-stage rotor anti-corrosion flowmeter is used to measure the salt-water flow, in order to ensure that the flow rate reaches the design value. The complete freezing experiment system consists of inclined shaft freezing simulation test rig, frozen pipe, low constant temperature refrigeration unit, boundary insulation panel and data acquisition system (Figure 3(a)).Figure 4
Layout of freezing pipes.(2) Hydrological Pressure Monitoring Probes. The BPR-40 pressure sensor is used to monitor the hydraulic pressure change in the hydrological pipes during the freezing process, which has a range of 0 ~ 1 MPa and a resolution of ±0.005 MPa. In order to ensure the measurement accuracy, the pressure sensor is calibrated by YJY-600A pressure gauge before the experiment. The layout of hydrological pipes is presented in Figure 5. Two axial hydrological pipes (S1、S3) and two radial hydrological pipes (S2、S4) are set at the center of the shaft of the saturated sand monitoring section D1 and D2.Figure 5
Layout of the hydrological pipes.(3) Temperature Monitoring Probes. The copper-constantan thermocouple is used as the temperature monitoring probes. It is calibrated by ice water mixture before the experiment. As shown in Figure 6, four groups of temperature sensors are set in radial direction, respectively, at sections D1 and D2. Each group has 25 temperature monitoring points with a spacing of 20 mm. The nomenclature of J1, J2, J3 and J4 represent arch crown, arch baseline, wall corner and base plate, respectively. The location of the intersection of the section arch baseline and the vertical axis of symmetry is denoted by No 1, and the other serial numbers radially increase outward.Figure 6
Radial temperature measurement arrangement: (a) Site photo and (b) Schematic diagram of specific parameters.As shown in Figures7, 91 measuring points are set in the circumferential circle of the sections D1 and D2 with a spacing of 15 mm. The location of the central position of vault is denoted by No 1, and the other serial numbers increase clockwise. The nomenclature of H and J represents the circumferential and radial direction, respectively. Several temperature measuring points are especially set on the outer surface of the model to measure the boundary temperature. In addition, two thermocouples are arranged in each of the outlets to measure the temperature of the returned salt-water.Figure 7
Circumferential temperature measurement arrangement: (a) Site photo and (b) Schematic diagram of specific parameters.


## 2.1. Engineering Background

In this study, a practical inclined shaft sinking under vertical holes freezing condition is used as the basis of indoor similarity experiment design. As shown in Figure1, this inclined shaft sinking engineering is located in Yulin City, Shanxi Province, China, and the engineering difficulty lies in the stratum of saturated sand sandwiched by the clay layers. Therefore, the intercalated layers of saturated sand and clay are considered as the simulation stratum of this study.Figure 1
Location of the inclined shaft sinking engineering and the photos of the site engineering.According to the inclined shaft size and formation characteristics, the freezing hole layout scheme is designed under the condition of inclined hole freezing method, which is used as the prototype of laboratory simulation, as shown in Figure2. The single lap of inclined holes freezing is used as the freezing process. The specific design parameters of this freezing method are shown in Table 1.Figure 2
Schematic diagram of the inclined shaft section and layout parameters of the inclined freezing holes: (a) from the transverse section view, (b) from the longitudinal section view.
(a)(b)Table 1
Prototype parameters of the freezing system.
Classification:Salt waterFreezing pipeParameter:TemperatureSpeedSizeAngleHole spacingValue:-32°C0.12 m/sφ159 mm ×6 mm10°1.2 m


## 2.2. Similarity Experiment Design and Establishment

### 2.2.1. Similarity Criterion

For freezing through the freezing pipe, the temperature field at the cross section of the freezing pipe and surrounding soil can be expressed as:(1)∂θn∂τ=an∂2θn∂r2+1r∂θn∂r(2)an=λn∙Cnwhere r is radial coordinate (m), r0 is the freezing pipe diametral coordinate (m) such that 0 < r0 < r < ∞, τ is the time of freezing (s) such that τ >0, θn is the temperature of point r (°C), n =1 represents the non-frozen zone, n =2 represents the frozen zone, an is the thermal diffusivity (m2∙s−1), λn is the thermal conductivity of soil (W∙m−1∙°C−1) and Cn is the volumetric heat capacity (J∙m3∙°C−1).The boundary condition is given by:(3)θr,0=θ0θ∞,τ=θ0θρ,τ=θDθr0,τ=θywhere θ0 is the original temperature of soil (°C), ρ is the coordinate of the phase transition interface, θD is the freezing temperature (°C) and θy is the temperature of salt-water (°C).At the phase transition interface (r=ρ), the heat balance equation can be expressed as:
(4)λ2∂θ2∂rr=ρ−λ1∂θ1∂rr=ρ=Bdρdτwhere B is the freezing latent heat per unit volume of soil.According to the equation analysis method and Equations (1)-(4), the similarity criterion can be expressed as:
(5)FF0,K0,R,θ=0where F0=a∙τ/r2 is the Fourier criterion, K0=B/c∙t is the Kosovich criterion, R is the Geometric criterion and θ is the rule of the temperature.If the simulation experiment is performed based on the original soil,Ca = 1 and Cc= 1. When the moisture content and the freezing latent heat of soil are similar, the time similarity ratio of the simulation experiment is the square of the geometric similarity ratio, and the temperature of each point in the model is consistent with the prototype, which can be expressed in equations (6) and (7).
(6)Cτ=Cr2(7)Ct=1where C is the similarity coefficient, subscripts a, c, τ and t represent the thermal conductivity, volumetric specific heat, time and temperature, respectively.


### 2.2.2. Similarity Experiment Design and Establishment

The specific parameters of practical inclined shaft and the freezing holes are shown in Figure2 and Table 1, respectively. Based on the similarity criterion, accuracy requirement and the dimensions of the practical engineering, the freezing pipe uses a precision seamless metal tube having a diameter of 6 mm. Thus, the geometric contraction is 159/6 =26.5. Accordingly, the wall thickness of the freezing pipe calculated by the similarity criterion is 6/26.5 = 0.226 mm. Due to the limitation of machining accuracy, the calculated size of the freezing pipe is adjusted in the actual experiment, and the final used size is Φ6 × 1 mm. In addition, the time scale of similarity experiment is set to 26.52 = 702.25. The inclination of the freezing pipe is set to 10° according to the inclination of the original inclined shaft. The positive freezing period temperature of the simulated test is -32°C and the speed of salt-water is 0.12 × 26.5 = 3.18 m/s.According to the stratigraphic distribution of the original inclined shaft, the stratum of the similarity model is the saturated sand sandwiched by the clay layers. The detailed layout and parameters are presented in Figure3(a)–3(b). The sand used in the experiment is the river sand produced in Xuzhou, China, which is pretreated by the immersion method. In addition, the clay is taken from Lingshou County, Hebei Province, China, and prepared with 15% water content. In order to facilitate the filling of soil and the placement of sensors such as temperature measuring lines, the test bench is layered, filled and tamped in a vertical state. Each clay layer is filled in three times, with a filling of 0.1 m for each time, then manually compacted and measured for the rate of water content. The saturated sand is infilled 0.15 m each time and tapped by an electric flat vibrator (Figure 3(b)). In order to isolate heat dissipation, the outer surface of the test bench is wrapped with foam board and polyurethane insulation cotton. Moreover, the soft polyurethane foam is used as external thermal insulation material in the salt-water pipeline, as shown in Figure 3(a).Figure 3
(a) Site photos of similar experiments and (b) parameters of stratum layout.


### 2.2.3. Arrangement of Freezing Equipment and Monitoring Probes

(1) Freezing Equipment. The arrangement of the freezing equipment is shown in Figure 4. The main freezing pipes use 26 copper pipes with a dimension of Φ6 × 1 mm, and the center distance of each copper pipe is 53 mm. The trunk pipe uses steel pipe with a dimension of Φ25 × 2 mm. The three-stage rotor anti-corrosion flowmeter is used to measure the salt-water flow, in order to ensure that the flow rate reaches the design value. The complete freezing experiment system consists of inclined shaft freezing simulation test rig, frozen pipe, low constant temperature refrigeration unit, boundary insulation panel and data acquisition system (Figure 3(a)).Figure 4
Layout of freezing pipes.(2) Hydrological Pressure Monitoring Probes. The BPR-40 pressure sensor is used to monitor the hydraulic pressure change in the hydrological pipes during the freezing process, which has a range of 0 ~ 1 MPa and a resolution of ±0.005 MPa. In order to ensure the measurement accuracy, the pressure sensor is calibrated by YJY-600A pressure gauge before the experiment. The layout of hydrological pipes is presented in Figure 5. Two axial hydrological pipes (S1、S3) and two radial hydrological pipes (S2、S4) are set at the center of the shaft of the saturated sand monitoring section D1 and D2.Figure 5
Layout of the hydrological pipes.(3) Temperature Monitoring Probes. The copper-constantan thermocouple is used as the temperature monitoring probes. It is calibrated by ice water mixture before the experiment. As shown in Figure 6, four groups of temperature sensors are set in radial direction, respectively, at sections D1 and D2. Each group has 25 temperature monitoring points with a spacing of 20 mm. The nomenclature of J1, J2, J3 and J4 represent arch crown, arch baseline, wall corner and base plate, respectively. The location of the intersection of the section arch baseline and the vertical axis of symmetry is denoted by No 1, and the other serial numbers radially increase outward.Figure 6
Radial temperature measurement arrangement: (a) Site photo and (b) Schematic diagram of specific parameters.As shown in Figures7, 91 measuring points are set in the circumferential circle of the sections D1 and D2 with a spacing of 15 mm. The location of the central position of vault is denoted by No 1, and the other serial numbers increase clockwise. The nomenclature of H and J represents the circumferential and radial direction, respectively. Several temperature measuring points are especially set on the outer surface of the model to measure the boundary temperature. In addition, two thermocouples are arranged in each of the outlets to measure the temperature of the returned salt-water.Figure 7
Circumferential temperature measurement arrangement: (a) Site photo and (b) Schematic diagram of specific parameters.


## 2.2.1. Similarity Criterion

For freezing through the freezing pipe, the temperature field at the cross section of the freezing pipe and surrounding soil can be expressed as:(1)∂θn∂τ=an∂2θn∂r2+1r∂θn∂r(2)an=λn∙Cnwhere r is radial coordinate (m), r0 is the freezing pipe diametral coordinate (m) such that 0 < r0 < r < ∞, τ is the time of freezing (s) such that τ >0, θn is the temperature of point r (°C), n =1 represents the non-frozen zone, n =2 represents the frozen zone, an is the thermal diffusivity (m2∙s−1), λn is the thermal conductivity of soil (W∙m−1∙°C−1) and Cn is the volumetric heat capacity (J∙m3∙°C−1).The boundary condition is given by:(3)θr,0=θ0θ∞,τ=θ0θρ,τ=θDθr0,τ=θywhere θ0 is the original temperature of soil (°C), ρ is the coordinate of the phase transition interface, θD is the freezing temperature (°C) and θy is the temperature of salt-water (°C).At the phase transition interface (r=ρ), the heat balance equation can be expressed as:
(4)λ2∂θ2∂rr=ρ−λ1∂θ1∂rr=ρ=Bdρdτwhere B is the freezing latent heat per unit volume of soil.According to the equation analysis method and Equations (1)-(4), the similarity criterion can be expressed as:
(5)FF0,K0,R,θ=0where F0=a∙τ/r2 is the Fourier criterion, K0=B/c∙t is the Kosovich criterion, R is the Geometric criterion and θ is the rule of the temperature.If the simulation experiment is performed based on the original soil,Ca = 1 and Cc= 1. When the moisture content and the freezing latent heat of soil are similar, the time similarity ratio of the simulation experiment is the square of the geometric similarity ratio, and the temperature of each point in the model is consistent with the prototype, which can be expressed in equations (6) and (7).
(6)Cτ=Cr2(7)Ct=1where C is the similarity coefficient, subscripts a, c, τ and t represent the thermal conductivity, volumetric specific heat, time and temperature, respectively.


## 2.2.2. Similarity Experiment Design and Establishment

The specific parameters of practical inclined shaft and the freezing holes are shown in Figure2 and Table 1, respectively. Based on the similarity criterion, accuracy requirement and the dimensions of the practical engineering, the freezing pipe uses a precision seamless metal tube having a diameter of 6 mm. Thus, the geometric contraction is 159/6 =26.5. Accordingly, the wall thickness of the freezing pipe calculated by the similarity criterion is 6/26.5 = 0.226 mm. Due to the limitation of machining accuracy, the calculated size of the freezing pipe is adjusted in the actual experiment, and the final used size is Φ6 × 1 mm. In addition, the time scale of similarity experiment is set to 26.52 = 702.25. The inclination of the freezing pipe is set to 10° according to the inclination of the original inclined shaft. The positive freezing period temperature of the simulated test is -32°C and the speed of salt-water is 0.12 × 26.5 = 3.18 m/s.According to the stratigraphic distribution of the original inclined shaft, the stratum of the similarity model is the saturated sand sandwiched by the clay layers. The detailed layout and parameters are presented in Figure3(a)–3(b). The sand used in the experiment is the river sand produced in Xuzhou, China, which is pretreated by the immersion method. In addition, the clay is taken from Lingshou County, Hebei Province, China, and prepared with 15% water content. In order to facilitate the filling of soil and the placement of sensors such as temperature measuring lines, the test bench is layered, filled and tamped in a vertical state. Each clay layer is filled in three times, with a filling of 0.1 m for each time, then manually compacted and measured for the rate of water content. The saturated sand is infilled 0.15 m each time and tapped by an electric flat vibrator (Figure 3(b)). In order to isolate heat dissipation, the outer surface of the test bench is wrapped with foam board and polyurethane insulation cotton. Moreover, the soft polyurethane foam is used as external thermal insulation material in the salt-water pipeline, as shown in Figure 3(a).Figure 3
(a) Site photos of similar experiments and (b) parameters of stratum layout.


## 2.2.3. Arrangement of Freezing Equipment and Monitoring Probes

(1) Freezing Equipment. The arrangement of the freezing equipment is shown in Figure 4. The main freezing pipes use 26 copper pipes with a dimension of Φ6 × 1 mm, and the center distance of each copper pipe is 53 mm. The trunk pipe uses steel pipe with a dimension of Φ25 × 2 mm. The three-stage rotor anti-corrosion flowmeter is used to measure the salt-water flow, in order to ensure that the flow rate reaches the design value. The complete freezing experiment system consists of inclined shaft freezing simulation test rig, frozen pipe, low constant temperature refrigeration unit, boundary insulation panel and data acquisition system (Figure 3(a)).Figure 4
Layout of freezing pipes.(2) Hydrological Pressure Monitoring Probes. The BPR-40 pressure sensor is used to monitor the hydraulic pressure change in the hydrological pipes during the freezing process, which has a range of 0 ~ 1 MPa and a resolution of ±0.005 MPa. In order to ensure the measurement accuracy, the pressure sensor is calibrated by YJY-600A pressure gauge before the experiment. The layout of hydrological pipes is presented in Figure 5. Two axial hydrological pipes (S1、S3) and two radial hydrological pipes (S2、S4) are set at the center of the shaft of the saturated sand monitoring section D1 and D2.Figure 5
Layout of the hydrological pipes.(3) Temperature Monitoring Probes. The copper-constantan thermocouple is used as the temperature monitoring probes. It is calibrated by ice water mixture before the experiment. As shown in Figure 6, four groups of temperature sensors are set in radial direction, respectively, at sections D1 and D2. Each group has 25 temperature monitoring points with a spacing of 20 mm. The nomenclature of J1, J2, J3 and J4 represent arch crown, arch baseline, wall corner and base plate, respectively. The location of the intersection of the section arch baseline and the vertical axis of symmetry is denoted by No 1, and the other serial numbers radially increase outward.Figure 6
Radial temperature measurement arrangement: (a) Site photo and (b) Schematic diagram of specific parameters.As shown in Figures7, 91 measuring points are set in the circumferential circle of the sections D1 and D2 with a spacing of 15 mm. The location of the central position of vault is denoted by No 1, and the other serial numbers increase clockwise. The nomenclature of H and J represents the circumferential and radial direction, respectively. Several temperature measuring points are especially set on the outer surface of the model to measure the boundary temperature. In addition, two thermocouples are arranged in each of the outlets to measure the temperature of the returned salt-water.Figure 7
Circumferential temperature measurement arrangement: (a) Site photo and (b) Schematic diagram of specific parameters.


## 3. Results and Discussion

### 3.1. Evolution Characteristics of Temperature Field

The evolution characteristics of temperature field are consistent on the two monitoring sections D1 and D2. Thus, only the temperature field of section D1 is discussed in this paper. Figure8 shows the evolution of the radial temperature field with the freezing time at arch crown-J1, arch baseline-J2, wall corner-J3 and base plate-J4. It can be seen that the average initial soil temperature is close to 26.46°C, which is close to the room temperature of 26.5°C. This verifies the accuracy of the temperature monitoring in this similarity experiment model.Figure 8
Evolution of the radial temperature field in monitoring section D1: (a) at arch crown-J1, (b) at arch baseline-J2, (c) at wall corner-J3, and (d) at base plate- J4.
(a)(b)(c)(d)


#### 3.1.1. Radial Temperature Field

As the frozen salt-water starts to circulate, the temperature of each point in the model gradually starts to decrease, and the drop rate of soil temperature near the freezing pipe becomes significantly higher than that far away. Considering the evolution of radial temperature field in monitoring section D1 at arch crown-J1 as example (Figure8(a)), the temperature drop rate at point 13 nearest to the freezing pipe is the fastest with a rate of 1.75°C/min, and the temperature drop rate at point 25 farthest to the freezing pipe is the slowest with a rate of 0.029°C/min. More precisely, the temperature variation trend of each position has a strong similarity in the whole freezing period, which can be divided into three stages according to the temperature drop rate: (1) soil cools rapidly before reaching the freezing point, (2) slow cooling during ice latent heat release, and (3) the temperature rapidly decreases after the release of ice latent heat, and then gradually tends to be stable.In the radial direction, the temperature difference of each point first increases to the maximum, and then gradually decreases (Figure8). At the 43rd hour, the temperature of each measuring point tends to be stable. More precisely, the maximum temperature difference in radial direction increases from J1 to J4 in a clockwise direction in the freezing process, as shown as Figure 8(a)–8(d). This is different from the cross-section temperature evolution of vertical shaft, which has a uniform maximum temperature difference in all the radial directions. In addition, the temperature difference of each point at the wall corner and base plate is significantly greater than that at the arch crown and arch baseline, when the temperature tends to be stable. This is due to the arched arrangement of the freezing pipe. Moreover, the corresponding freezing pipe density at each radial position is different, which is also confirmed in the later circumferential temperature monitoring (Figure 9).Figure 9
Circumferential temperature at the: (a) arch crown-J1, (b) arch baseline-J2, (c) wall corner-J3, and (d) base plate- J4.
(a)(b)(c)(d)


#### 3.1.2. Circumferential Temperature Field

The circumferential temperature field between adjacent freezing tubes at the intersections of four observational radial and freezing-pipe rings (Figure6), is shown in Figure 9. The x-axis represents the relative position between the temperature measuring point and the two freezing pipes, and the central position of the adjacent freezing pipe is considered as the coordinate origin of the x-axis. Five minutes after freezing, the circumferential temperature of each position in the model remains relatively similar, which has an average value of 25.95~ 26.82°C and variances of 0.044~ 0.71. It can be seen that the initial average soil temperature is close to the room temperature of 26.5°C, which also verifies the accuracy of the temperature monitoring in this similarity experiment model. Ten minutes after freezing, the soil temperature near the freezing pipe significantly decreases, the original straight temperature curve starts to bend. The closer to the freezing pipe, the faster the temperature decreases, and the maximum difference between two points on the curve reaches 18.4°C. The temperature difference at each point continues to gradually decrease, and then the temperature curve gradually develops to a straight line again.It can be seen from Figure9(a)–9(d) that the characteristics of temperature field distribution are different in the four positions due to the asymmetric geometry of the frozen pipe arrangement. The temperature field at the arch crown is symmetrically distributed, and gradually tends to asymmetry with the down position. The closure time of the frozen wall in each position is summarized in Table 2. It can be seen that the closing time is almost 43.1 ~ 56 min (prototype 21~ 27.3d), and is gradually shortened from top to bottom. The freezing speed of the top position (arch crown and arch baseline) is slower than that of the bottom position (wall corner and base plate), under the condition of arched arrangement of frozen pipes. Therefore, more attention should be paid to the monitoring of the top position during the construction process.Table 2
Closure time of frozen wall.
Monitoring sectionSoil layerLocation of measuring pointsClosure time of frozen wallScale model (min)Real model (d)D1SandySoilArch crown5627.3Arch baseline55.627.1Wall corner52.725.7Base plate44.921.9


#### 3.1.3. Thickness of the Frozen Wall

The 0°C is used as the freezing point to judge the thickness of the frozen wall in radial direction. The maximum judgment time is 5.14 h, which is converted into the actual engineering time of 150.4 days. For each radial direction of section D1, the evolution and average growth rate of frozen wall thickness function of time, is shown in Figure10. It can be seen that the growth rate of frozen wall thickness greatly varies in the different radial directions. In addition, the growth rate of frozen wall at the arch baseline is significantly lower than that at other positions, which is not consistent with the statistical results for frozen wall closure speed at each position (Table 2). In fact, the radial temperature field of arch baseline has a sudden change process and this process greatly delays the temperature decrease, as shown as Figure 8(b). It is preliminarily speculated that this phenomenon is caused by local water movement during freezing. This inference should be further verified in the nest experiments. In summary, it is necessary to arrange more monitoring in the top position of inclined shaft during the construction process.Figure 10
Frozen wall thickness in monitoring section D1.


### 3.2. Determination of Frozen Wall Closure Time

The monitoring values of the water pressure in the hydrological pipes are shown in Figure11. It can be clearly seen that there is no significant change in the monitoring value of section D2-S3 and D2-S4. The water pressure vibrates near the average value of -0.035 MPa and -0.032 MPa, and the corresponding variances are, respectively, 0.031 and 0.0009, which is not coherent with the law of water pressure change in soil during freezing. After the experiment, it is deduced during the removal of the model that the insulation cotton of the water pipe at section D2 is damaged in the model establishment process. Therefore, the water pressure sensor cannot collect data due to the fact that the hydrological pipe is partially frozen during the experiment.Figure 11
Water pressure in the hydrological pipes.For the monitoring section D1, the water pressure of axial hydrological pipe S1 rapidly increases to the minimum value of -0.11 MPa after freezing for 54.8 min. This time is very close to the maximum intersection time of freezing wall calculated by temperature monitoring, which indicates that the intersection of frozen wall can be judged by the change of water pressure of axial hydrological hole under inclined holes freezing condition. More precisely, the water pressure in the hydrological pipe rapidly decreases after reaching the peak value and there is no gentle section, which is inconsistent with the water pressure change law of the pressure relief hole beside the subway tunnel. In the view of engineering scale, the model is small and the soil is rapidly frozen after the frozen wall is surrounded. However, the thickness of the freezing curtain slowly develops from the intersection of the freezing curtain to the maintenance freezing of soil excavation under background of large engineering scale, and eventually leads the water pressure to flatten. Affected by the damage of the sensor during soil freezing and expansion, the water pressure of radial hydrological pipe S2 rapidly increases to -0.07 MPa when frozen for 6.3 h, which contradicts the fact that the shaft core has been frozen judged by temperature monitoring (Figure8). Therefore, it is necessary to protect the monitoring probe in the hydrological observation hole against pressure in the freezing experiment.


## 3.1. Evolution Characteristics of Temperature Field

The evolution characteristics of temperature field are consistent on the two monitoring sections D1 and D2. Thus, only the temperature field of section D1 is discussed in this paper. Figure8 shows the evolution of the radial temperature field with the freezing time at arch crown-J1, arch baseline-J2, wall corner-J3 and base plate-J4. It can be seen that the average initial soil temperature is close to 26.46°C, which is close to the room temperature of 26.5°C. This verifies the accuracy of the temperature monitoring in this similarity experiment model.Figure 8
Evolution of the radial temperature field in monitoring section D1: (a) at arch crown-J1, (b) at arch baseline-J2, (c) at wall corner-J3, and (d) at base plate- J4.
(a)(b)(c)(d)


### 3.1.1. Radial Temperature Field

As the frozen salt-water starts to circulate, the temperature of each point in the model gradually starts to decrease, and the drop rate of soil temperature near the freezing pipe becomes significantly higher than that far away. Considering the evolution of radial temperature field in monitoring section D1 at arch crown-J1 as example (Figure8(a)), the temperature drop rate at point 13 nearest to the freezing pipe is the fastest with a rate of 1.75°C/min, and the temperature drop rate at point 25 farthest to the freezing pipe is the slowest with a rate of 0.029°C/min. More precisely, the temperature variation trend of each position has a strong similarity in the whole freezing period, which can be divided into three stages according to the temperature drop rate: (1) soil cools rapidly before reaching the freezing point, (2) slow cooling during ice latent heat release, and (3) the temperature rapidly decreases after the release of ice latent heat, and then gradually tends to be stable.In the radial direction, the temperature difference of each point first increases to the maximum, and then gradually decreases (Figure8). At the 43rd hour, the temperature of each measuring point tends to be stable. More precisely, the maximum temperature difference in radial direction increases from J1 to J4 in a clockwise direction in the freezing process, as shown as Figure 8(a)–8(d). This is different from the cross-section temperature evolution of vertical shaft, which has a uniform maximum temperature difference in all the radial directions. In addition, the temperature difference of each point at the wall corner and base plate is significantly greater than that at the arch crown and arch baseline, when the temperature tends to be stable. This is due to the arched arrangement of the freezing pipe. Moreover, the corresponding freezing pipe density at each radial position is different, which is also confirmed in the later circumferential temperature monitoring (Figure 9).Figure 9
Circumferential temperature at the: (a) arch crown-J1, (b) arch baseline-J2, (c) wall corner-J3, and (d) base plate- J4.
(a)(b)(c)(d)


### 3.1.2. Circumferential Temperature Field

The circumferential temperature field between adjacent freezing tubes at the intersections of four observational radial and freezing-pipe rings (Figure6), is shown in Figure 9. The x-axis represents the relative position between the temperature measuring point and the two freezing pipes, and the central position of the adjacent freezing pipe is considered as the coordinate origin of the x-axis. Five minutes after freezing, the circumferential temperature of each position in the model remains relatively similar, which has an average value of 25.95~ 26.82°C and variances of 0.044~ 0.71. It can be seen that the initial average soil temperature is close to the room temperature of 26.5°C, which also verifies the accuracy of the temperature monitoring in this similarity experiment model. Ten minutes after freezing, the soil temperature near the freezing pipe significantly decreases, the original straight temperature curve starts to bend. The closer to the freezing pipe, the faster the temperature decreases, and the maximum difference between two points on the curve reaches 18.4°C. The temperature difference at each point continues to gradually decrease, and then the temperature curve gradually develops to a straight line again.It can be seen from Figure9(a)–9(d) that the characteristics of temperature field distribution are different in the four positions due to the asymmetric geometry of the frozen pipe arrangement. The temperature field at the arch crown is symmetrically distributed, and gradually tends to asymmetry with the down position. The closure time of the frozen wall in each position is summarized in Table 2. It can be seen that the closing time is almost 43.1 ~ 56 min (prototype 21~ 27.3d), and is gradually shortened from top to bottom. The freezing speed of the top position (arch crown and arch baseline) is slower than that of the bottom position (wall corner and base plate), under the condition of arched arrangement of frozen pipes. Therefore, more attention should be paid to the monitoring of the top position during the construction process.Table 2
Closure time of frozen wall.
Monitoring sectionSoil layerLocation of measuring pointsClosure time of frozen wallScale model (min)Real model (d)D1SandySoilArch crown5627.3Arch baseline55.627.1Wall corner52.725.7Base plate44.921.9


### 3.1.3. Thickness of the Frozen Wall

The 0°C is used as the freezing point to judge the thickness of the frozen wall in radial direction. The maximum judgment time is 5.14 h, which is converted into the actual engineering time of 150.4 days. For each radial direction of section D1, the evolution and average growth rate of frozen wall thickness function of time, is shown in Figure10. It can be seen that the growth rate of frozen wall thickness greatly varies in the different radial directions. In addition, the growth rate of frozen wall at the arch baseline is significantly lower than that at other positions, which is not consistent with the statistical results for frozen wall closure speed at each position (Table 2). In fact, the radial temperature field of arch baseline has a sudden change process and this process greatly delays the temperature decrease, as shown as Figure 8(b). It is preliminarily speculated that this phenomenon is caused by local water movement during freezing. This inference should be further verified in the nest experiments. In summary, it is necessary to arrange more monitoring in the top position of inclined shaft during the construction process.Figure 10
Frozen wall thickness in monitoring section D1.


## 3.1.1. Radial Temperature Field

As the frozen salt-water starts to circulate, the temperature of each point in the model gradually starts to decrease, and the drop rate of soil temperature near the freezing pipe becomes significantly higher than that far away. Considering the evolution of radial temperature field in monitoring section D1 at arch crown-J1 as example (Figure8(a)), the temperature drop rate at point 13 nearest to the freezing pipe is the fastest with a rate of 1.75°C/min, and the temperature drop rate at point 25 farthest to the freezing pipe is the slowest with a rate of 0.029°C/min. More precisely, the temperature variation trend of each position has a strong similarity in the whole freezing period, which can be divided into three stages according to the temperature drop rate: (1) soil cools rapidly before reaching the freezing point, (2) slow cooling during ice latent heat release, and (3) the temperature rapidly decreases after the release of ice latent heat, and then gradually tends to be stable.In the radial direction, the temperature difference of each point first increases to the maximum, and then gradually decreases (Figure8). At the 43rd hour, the temperature of each measuring point tends to be stable. More precisely, the maximum temperature difference in radial direction increases from J1 to J4 in a clockwise direction in the freezing process, as shown as Figure 8(a)–8(d). This is different from the cross-section temperature evolution of vertical shaft, which has a uniform maximum temperature difference in all the radial directions. In addition, the temperature difference of each point at the wall corner and base plate is significantly greater than that at the arch crown and arch baseline, when the temperature tends to be stable. This is due to the arched arrangement of the freezing pipe. Moreover, the corresponding freezing pipe density at each radial position is different, which is also confirmed in the later circumferential temperature monitoring (Figure 9).Figure 9
Circumferential temperature at the: (a) arch crown-J1, (b) arch baseline-J2, (c) wall corner-J3, and (d) base plate- J4.
(a)(b)(c)(d)


## 3.1.2. Circumferential Temperature Field

The circumferential temperature field between adjacent freezing tubes at the intersections of four observational radial and freezing-pipe rings (Figure6), is shown in Figure 9. The x-axis represents the relative position between the temperature measuring point and the two freezing pipes, and the central position of the adjacent freezing pipe is considered as the coordinate origin of the x-axis. Five minutes after freezing, the circumferential temperature of each position in the model remains relatively similar, which has an average value of 25.95~ 26.82°C and variances of 0.044~ 0.71. It can be seen that the initial average soil temperature is close to the room temperature of 26.5°C, which also verifies the accuracy of the temperature monitoring in this similarity experiment model. Ten minutes after freezing, the soil temperature near the freezing pipe significantly decreases, the original straight temperature curve starts to bend. The closer to the freezing pipe, the faster the temperature decreases, and the maximum difference between two points on the curve reaches 18.4°C. The temperature difference at each point continues to gradually decrease, and then the temperature curve gradually develops to a straight line again.It can be seen from Figure9(a)–9(d) that the characteristics of temperature field distribution are different in the four positions due to the asymmetric geometry of the frozen pipe arrangement. The temperature field at the arch crown is symmetrically distributed, and gradually tends to asymmetry with the down position. The closure time of the frozen wall in each position is summarized in Table 2. It can be seen that the closing time is almost 43.1 ~ 56 min (prototype 21~ 27.3d), and is gradually shortened from top to bottom. The freezing speed of the top position (arch crown and arch baseline) is slower than that of the bottom position (wall corner and base plate), under the condition of arched arrangement of frozen pipes. Therefore, more attention should be paid to the monitoring of the top position during the construction process.Table 2
Closure time of frozen wall.
Monitoring sectionSoil layerLocation of measuring pointsClosure time of frozen wallScale model (min)Real model (d)D1SandySoilArch crown5627.3Arch baseline55.627.1Wall corner52.725.7Base plate44.921.9


## 3.1.3. Thickness of the Frozen Wall

The 0°C is used as the freezing point to judge the thickness of the frozen wall in radial direction. The maximum judgment time is 5.14 h, which is converted into the actual engineering time of 150.4 days. For each radial direction of section D1, the evolution and average growth rate of frozen wall thickness function of time, is shown in Figure10. It can be seen that the growth rate of frozen wall thickness greatly varies in the different radial directions. In addition, the growth rate of frozen wall at the arch baseline is significantly lower than that at other positions, which is not consistent with the statistical results for frozen wall closure speed at each position (Table 2). In fact, the radial temperature field of arch baseline has a sudden change process and this process greatly delays the temperature decrease, as shown as Figure 8(b). It is preliminarily speculated that this phenomenon is caused by local water movement during freezing. This inference should be further verified in the nest experiments. In summary, it is necessary to arrange more monitoring in the top position of inclined shaft during the construction process.Figure 10
Frozen wall thickness in monitoring section D1.


## 3.2. Determination of Frozen Wall Closure Time

The monitoring values of the water pressure in the hydrological pipes are shown in Figure11. It can be clearly seen that there is no significant change in the monitoring value of section D2-S3 and D2-S4. The water pressure vibrates near the average value of -0.035 MPa and -0.032 MPa, and the corresponding variances are, respectively, 0.031 and 0.0009, which is not coherent with the law of water pressure change in soil during freezing. After the experiment, it is deduced during the removal of the model that the insulation cotton of the water pipe at section D2 is damaged in the model establishment process. Therefore, the water pressure sensor cannot collect data due to the fact that the hydrological pipe is partially frozen during the experiment.Figure 11
Water pressure in the hydrological pipes.For the monitoring section D1, the water pressure of axial hydrological pipe S1 rapidly increases to the minimum value of -0.11 MPa after freezing for 54.8 min. This time is very close to the maximum intersection time of freezing wall calculated by temperature monitoring, which indicates that the intersection of frozen wall can be judged by the change of water pressure of axial hydrological hole under inclined holes freezing condition. More precisely, the water pressure in the hydrological pipe rapidly decreases after reaching the peak value and there is no gentle section, which is inconsistent with the water pressure change law of the pressure relief hole beside the subway tunnel. In the view of engineering scale, the model is small and the soil is rapidly frozen after the frozen wall is surrounded. However, the thickness of the freezing curtain slowly develops from the intersection of the freezing curtain to the maintenance freezing of soil excavation under background of large engineering scale, and eventually leads the water pressure to flatten. Affected by the damage of the sensor during soil freezing and expansion, the water pressure of radial hydrological pipe S2 rapidly increases to -0.07 MPa when frozen for 6.3 h, which contradicts the fact that the shaft core has been frozen judged by temperature monitoring (Figure8). Therefore, it is necessary to protect the monitoring probe in the hydrological observation hole against pressure in the freezing experiment.


## 4. Conclusion

Combined with the actual inclined shaft engineering, a similarity experiment is established to explore the evolution characteristics of temperature field and frozen wall closure judgment criteria for the inclined shaft, under inclined holes freezing condition. Based on the experimental results and analyses, the following main conclusions can be drawn:(1)
In the process of saturated sand freezing with inclined freezing holes, the soil temperature change can be divided into three stages: rapid cooling before reaching the freezing point, slow cooling in the process of freezing latent heat release, and accelerated cooling and stabilization after freezing latent heat release(2)
The distance from the freezing pipe is the main factor affecting the freezing wall temperature. Under the combination of salt-water temperature of -32°C and flow rate of 3.18 m/s, the average growth rate of freezing wall thickness in sandy soil is almost 27.2 ~ 42.8 mm/h. It completes intersection after freezing for almost 44.9 ~ 56 min, which corresponds to an actual engineering time of 21.9 ~ 27.3 days(3)
Affected by the arched arrangement of frozen pipes, the freezing speed in the position of arch crown and arch baseline is lower than the position of wall corner and base plate. More precisely, a sudden change exists in the radial temperature field of the arch baseline within five hours of the initial freezing, which greatly reduces the radial freezing speed. Therefore, more attention should be paid to the monitoring of the top position during the construction process(4)
The time for water pressure of axial hydrological pipe increasing to the minimum value is close to the maximum intersection time of freezing wall calculated by temperature monitoring. This indicates that the closure of frozen wall can be judged by the change of water pressure in the axial hydrological hole, under the inclined holes freezing condition.



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*Source: 8002667-2022-05-18.xml*