# Experimental Study on Influencing Factors of Characteristic Index of Local Horizontal Frozen Body of Double-Row Pipe under Seepage

Authors: Renliang Shan; Weijun Liu; Gaojun Chai; Shengchao Xiao
Journal: Advances in Materials Science and Engineering
 (2020)

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## Abstract

In order to deeply understand the influence of different factors on the characteristic indicators of the double-row-pipe partial horizontal frozen body in the sand-gravel stratum under the effect of seepage flow, several sets of sand-gravel stratum frozen model tests were carried out based on the similarity theory. The research showed that (1) under the effect of seepage flow, the backwater surface of the horizontal frozen body is in the shape of a quadratic parabolic and the development velocity of the backwater surface is inversely proportional to the square of the freezing time. The influence of seepage velocity on the downstream length and the freezing front development velocity of the backwater surface is stronger than the refrigerant temperature; (2) overlap time of the second row of frozen soil column increases with the increase of the distance between frozen pipes and seepage velocity and decreases with the decrease of the refrigerant temperature. The influence of various factors on the overlap time follows the order as the distance between freezing pipes > seepage velocity > refrigerant temperature. The average development velocity of the upstream surface decreases with the increase of the seepage velocity and the refrigerant temperature and first increases and then decreases with the increase of freezing pipe spacing. The influence of various factors on the average development velocity follows the order as seepage velocity > the distance between freezing pipes > refrigerant temperature. In addition, through the regression analysis, the quantitative relationship between the maximum arrangement spacing of the frozen pipes, the seepage velocity, and the refrigerant temperature was obtained. The research results can provide a basis for large-area partial horizontal freezing construction in Beijing sand-gravel stratum.


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## Body

## 1. Introduction

Freezing technology is a construction technique that sends refrigerant into the stratum to form an impervious structure by freezing the water [1]. Because of its adaptability to complex stratum, good water-sealing ability, no pollution, technical maturity, and reliability, it has been widely used in construction [2–5]. The effects of different factors such as seepage velocity, distance between freezing pipes, refrigerant temperature, and diameter of the frozen pipe were studied by Zhou et al. [6]. It was concluded that the distance between freezing pipes is the primary factor to influence the frozen soil column overlap; however, the interaction between various factors was not summarized. A coupled thermohydraulic finite model of frozen soil with an optimization algorithm combined with ant colony was proposed by Marwan et al. [1]. After verifying the model through experimental data, the optimal placement of frozen pipes was obtained. Based on the Bakholdin formula, Hu and He [7, 8] derived the equivalent trapezoidal algorithm to calculate the average temperature of the double-row-pipe or even the multirow-pipe frozen wall. Based on the potential superposition theory, multipipe steady-state freezing temperature field was achieved. According to the boundary separability of the harmonic equation, the straight linear arrangement model can be decomposed into two special single-row-pipe freezing problems, and the temperature field analytical solution to the generalized circular double-loop pipe is thus obtained [9]. Wang et al. [10] simulated the artificial shaft freezing process under the effect of large-flow groundwater by using the numerical calculation model and concluded that the artificial freezing temperature field is significantly affected when the flow velocity is greater than 5 m/d. Lai et al. [11] deduced the governing differential equations for the coupling problem of phase transition temperature field and seepage field. Then, the finite element calculation formula of the problem was obtained by the Galerkin method. Yang and Pi [12] simulated the single-pipe freezing process by numerical simulation and obtained the quantitative relationship between the maximum arrangement spacing of the frozen pipe and the thermal conductivity of the stratum and the temperature of the outer wall surface of the frozen pipe and other factors. The simulation correctness was verified by the measurement data by Lin et al. [13]; then, the influence of seepage velocity on the frozen wall formation was analyzed by numerical simulation. Pimentel et al. [14] simulated the formation process of single-row-pipe frozen wall under high seepage velocity through large-scale physical model test and compared test results with the existing artificial ground freezing closed-form solution.Until now, a lot of research studies have been performed on the stratum freezing technology [1, 6–19], but some limitations still exist. Firstly, when they are used to study the development law of the frozen wall under the seepage, most of the model tests are the single-row-pipe frozen one. The development laws of the temperature fields of the single-row-pipe, double-row-pipe, and even multitube-pipe frozen wall under the effect of seepage flow are significantly different from each other. Secondly, the frozen pipes are full-length freezing, which is significantly different from the partial freezing technique in temperature field under the effect of seepage flow. In addition, the model test mostly uses homogeneous sand as the soil heat transfer medium, but rarely uses sand gravel. In order to understand the effects of various factors such as the seepage velocity, distance between freezing pipes, and refrigerant temperature on the characteristic indexes of the horizontal frozen body, multiple groups of model test were also applied in sand-gravel stratum. The research results can provide a basis for large-area partial freezing construction in Beijing sand-gravel stratum (Beijing has been marked with a red star in the map of China in Figure 1).Figure 1
Location of Beijing (from the Internet).


## 2. Test Design

### 2.1. Test Equipment and Similarity Theory

Based on the principle of energy conservation and Darcy’s law in porous media, the coupled equation of water and heat in freezing process can be written as follows:(1)Ceq∂T∂t=∇λeq∇T−ρwCwu⟶∇T+Lρi∂θi∂t,(2)μs∂Hp∂t=∇KT∇H,where Cw represents the hydrothermal capacity; Ceq denotes the equivalent volume heat capacity (kJ/(m3·°C)); λeq indicates the equivalent thermal conductivity (kJ/(m·s·°C)); ρw and ρi are water density and ice density, respectively (kg/m3); L represents latent heat (kJ/m3); T is soil temperature (°C); t denotes freezing time (s); u⟶ indicates seepage velocity vector (m/d); μs represents specific storativity; Hp is waterhead (m); and KT denotes permeability coefficient (m/d).If the porosity of soil isθs and the content of water and ice in the pores is θw and θi, respectively, then(3)Ceq=θsρsCs+θwρwCw+θiρiCi,(4)λeq=θsλs+θwλw+θiλi,where λs, λw, and λi represent the thermal conductivity of soil particles, water, and ice.In the process of freezing, the soil temperatureT not only affects the content of water and ice in the pores but also blocks the original pore passage due to the formation of ice. Therefore, the permeability coefficient KT is also a variable affected by the temperature.


### 2.2. Test Equipment and Similarity Theory

Figure2 shows a schematic diagram of the test device. The seepage velocity was controlled by the pump pressure, and the refrigerant at the set temperature was outputted by the refrigeration compressor. Moreover, the local frozen sections of the two-row pipe were considered as the freezing system.Figure 2
Schematic diagram of the freezing model test system.The diameter of the frozen pipes was 16 mm, which simulated the diameter of 159 mm in the project. The geometrical reduction ratioCd is 10. List π terms and similarity criterion according to dimensions, the physical property similarity ratio of all materials is 1 (Cαsp=Cλ=CB=1). Meanwhile, the temperature reduction ratio CT = 1, the time reduction ratio Ct = Cd2 = 100, and the seepage velocity reduction velocity Cv = 1/Cd = 1/10. The sand-gravel soil for test was taken from the construction site. The physical parameters are as follows: particle density of 2.499 g/cm3, saturated water content of 2.7%, and porosity of 18.57.


### 2.3. Measuring Point Arrangement

The layout diagram of the frozen pipe and the temperature measuring point is shown in Figures2 and 3. In Figure 3, S means the spacing between the freezing pipes, and the unit standard of measurement is mm. A total of two rows and four columns of frozen pipes were arranged. Only 0.32 m in the middle was a local frozen section, and nonfrozen sections were insulated by thermal insulation cotton. 36 temperature measuring points were arranged. In this test, the platinum resistance PT100 was used to measure the temperature of the measuring point with a measurement accuracy of 0.1°C. When the temperature of the measuring point was below 0°C, it was considered that the frozen front had expanded to the location of the measuring point.Figure 3
Model design diagram.


### 2.4. Experimental Program

According to previous research studies [6, 19], the horizontal frozen body formation process under the effect of seepage can be divided into three stages. Firstly, the second row (last row) of the frozen soil column overlapped. Secondly, the first row of the frozen soil column overlapped and formed a continuous horizontal frozen body; meanwhile, the backwater surface developed. Thirdly, the upper water surface continued to develop. While the distance between freezing pipes S, seepage velocity u, and refrigerant temperature Tc are important factors to affect the development law of the frozen wall. All the test factors and corresponding levels are shown in Table 1. For ease of analysis, all of the test results below have been normalized according to similarity theory.Table 1
Test table.
Test level
S (m)
U (m)·d−1
T
c (°C)
1
0.08
5
−25
2
0.10
10
−27
3
0.14
15
−29
4
20Under hydrostatic conditions, the radius of the cooling zone is generally 3 to 5 times the thickness of the outer side of the frozen column [6]. The maximum distance of the frozen pipe in the model test is 140 mm, and the radius of the cooling zone is about 297 mm. Meanwhile, the distance between the frozen pipe and boundary of both sides is 390 mm. Given the inhibition effect of seepage water on the development of temperature field, the range of cooling zone is reduced.


## 2.1. Test Equipment and Similarity Theory

Based on the principle of energy conservation and Darcy’s law in porous media, the coupled equation of water and heat in freezing process can be written as follows:(1)Ceq∂T∂t=∇λeq∇T−ρwCwu⟶∇T+Lρi∂θi∂t,(2)μs∂Hp∂t=∇KT∇H,where Cw represents the hydrothermal capacity; Ceq denotes the equivalent volume heat capacity (kJ/(m3·°C)); λeq indicates the equivalent thermal conductivity (kJ/(m·s·°C)); ρw and ρi are water density and ice density, respectively (kg/m3); L represents latent heat (kJ/m3); T is soil temperature (°C); t denotes freezing time (s); u⟶ indicates seepage velocity vector (m/d); μs represents specific storativity; Hp is waterhead (m); and KT denotes permeability coefficient (m/d).If the porosity of soil isθs and the content of water and ice in the pores is θw and θi, respectively, then(3)Ceq=θsρsCs+θwρwCw+θiρiCi,(4)λeq=θsλs+θwλw+θiλi,where λs, λw, and λi represent the thermal conductivity of soil particles, water, and ice.In the process of freezing, the soil temperatureT not only affects the content of water and ice in the pores but also blocks the original pore passage due to the formation of ice. Therefore, the permeability coefficient KT is also a variable affected by the temperature.


## 2.2. Test Equipment and Similarity Theory

Figure2 shows a schematic diagram of the test device. The seepage velocity was controlled by the pump pressure, and the refrigerant at the set temperature was outputted by the refrigeration compressor. Moreover, the local frozen sections of the two-row pipe were considered as the freezing system.Figure 2
Schematic diagram of the freezing model test system.The diameter of the frozen pipes was 16 mm, which simulated the diameter of 159 mm in the project. The geometrical reduction ratioCd is 10. List π terms and similarity criterion according to dimensions, the physical property similarity ratio of all materials is 1 (Cαsp=Cλ=CB=1). Meanwhile, the temperature reduction ratio CT = 1, the time reduction ratio Ct = Cd2 = 100, and the seepage velocity reduction velocity Cv = 1/Cd = 1/10. The sand-gravel soil for test was taken from the construction site. The physical parameters are as follows: particle density of 2.499 g/cm3, saturated water content of 2.7%, and porosity of 18.57.


## 2.3. Measuring Point Arrangement

The layout diagram of the frozen pipe and the temperature measuring point is shown in Figures2 and 3. In Figure 3, S means the spacing between the freezing pipes, and the unit standard of measurement is mm. A total of two rows and four columns of frozen pipes were arranged. Only 0.32 m in the middle was a local frozen section, and nonfrozen sections were insulated by thermal insulation cotton. 36 temperature measuring points were arranged. In this test, the platinum resistance PT100 was used to measure the temperature of the measuring point with a measurement accuracy of 0.1°C. When the temperature of the measuring point was below 0°C, it was considered that the frozen front had expanded to the location of the measuring point.Figure 3
Model design diagram.


## 2.4. Experimental Program

According to previous research studies [6, 19], the horizontal frozen body formation process under the effect of seepage can be divided into three stages. Firstly, the second row (last row) of the frozen soil column overlapped. Secondly, the first row of the frozen soil column overlapped and formed a continuous horizontal frozen body; meanwhile, the backwater surface developed. Thirdly, the upper water surface continued to develop. While the distance between freezing pipes S, seepage velocity u, and refrigerant temperature Tc are important factors to affect the development law of the frozen wall. All the test factors and corresponding levels are shown in Table 1. For ease of analysis, all of the test results below have been normalized according to similarity theory.Table 1
Test table.
Test level
S (m)
U (m)·d−1
T
c (°C)
1
0.08
5
−25
2
0.10
10
−27
3
0.14
15
−29
4
20Under hydrostatic conditions, the radius of the cooling zone is generally 3 to 5 times the thickness of the outer side of the frozen column [6]. The maximum distance of the frozen pipe in the model test is 140 mm, and the radius of the cooling zone is about 297 mm. Meanwhile, the distance between the frozen pipe and boundary of both sides is 390 mm. Given the inhibition effect of seepage water on the development of temperature field, the range of cooling zone is reduced.


## 3. Result Analysis

As shown in Figure4, the length of the horizontal frozen body along the flow direction can be divided into three parts: upstream length along the flow direction (hereinafter, referred to as upstream length Eu), middle length along the flow direction (hereinafter, referred to as middle length Em), and downstream length along the flow direction (hereinafter, referred to as downstream length Ed). This paper focuses on the influence of variable factors on the characteristic index of the horizontal frozen body, such as the length of downstream flow and the time of intersection (or the expansion speed of frozen front). Among them, the upstream length Eu is short due to the influence of seepage water, which is not analyzed in this paper.Figure 4
Schematic diagram of the horizontal frozen body. (a) Parameters of each stage. (b) Schematic diagram of horizontal frozen body [19].
(a)
(b)


### 3.1.Ed

Figure5 shows the temperature field distribution at the distance between frozen pipes of 0.8 m, the seepage velocity of 1 m/d, and the freezing temperature of −25°C for 40 d (unit:°C).Figure 5
Schematic diagram of temperature field distribution.It was observed in Figure5 that, in the late stage of freezing, the backwater surface is in the shape of arc. After many attempts and fitting by the author, it is considered that the downstream length has a quadratic parabola relationship with the x-axis (the coordinate axis is shown in Figure 2):(5)Ed=Atx2+Bt.Among them,At and Bt are approximately inversely proportional to the freezing time t, that is,(6)At=Ct+D,Bt=Et+F,where C, D, E, and F are related to the distance between freezing pipes S, the seepage velocity u, and the refrigerant temperature Tc. In the actual project, the length Ew along the water surface direction of the horizontal frozen body was fixed, while this experiment was arranged with two rows and four columns of frozen pipes, and the length Ew along the water surface direction was not fixed. Therefore, the influences of the seepage velocity u and the refrigerant temperature Tc on Ed were only analyzed by the test results of the freezing pipe spacing of 0.8 m. The influence of the freezing pipe spacing on the backwater surface is not considered, so the parameters can be obtained as follows:(7)C=cu,Tc,D=du,Tc,E=eu,Tc,F=fu,Tc,Ed=Ct+Dx2+Et+F.Whent⟶∞, the downstream limit length is obtained as Edlim = Dx2+F, while Ed¯lim was obtained by dividing 3S after [−3S/2, 3S/2] integration.In this paper, typical test results were selected to characterize the influence of various factors on the downstream limit average lengthEd¯lim and the quadratic coefficient D, as shown in Figure 6.Figure 6
Influences of various factors on downstream length. (a) Seepage velocityu vs. downstream length Ed. (b) Refrigerant temperature Tc vs. downstream length Ed.
(a)
(b)It was revealed from Figure6 that the refrigerant temperature Tc was approximately proportional to the downstream limit length in the test range. When the seepage velocity is small, the change of the seepage velocity has a significant effect on Ed¯lim. As the seepage velocity u increases, the refrigerant temperature Tc increases, but the downstream limit average length Ed¯lim decreases. Similarly, the larger the seepage velocity u, the lower the refrigerant temperature Tc, the smaller the absolute value of the quadratic coefficient D, and the flatter the backwater surface.In order to analyze the influence of different factors on each characteristic index of the horizontal frozen body, the range method was used in this paper. The range calculation results ofEd¯lim are shown in Table 2. It showed that the influence of the seepage velocity u on the downstream limit average length Edlim was greater than the refrigerant temperature Tc.Table 2
Calculation of range ofEd¯lim.
Seepage velocityu
Refrigerant temperatureTc
Range calculation (m)
0.41
0.11


### 3.2.te

Typical test results were selected to characterize the influence of various factors on overlap timete of the second row of permafrost columns, as shown in Figure 7.Figure 7
Curves of the effect of various factors on te. (a) Distance between frozen pipes S vs. te. (b) Seepage velocity u vs. te. (c) Refrigerant temperature Tc vs. te.
(a)
(b)
(c)For the comprehensive regression of each parameter, the equation ofte is as follows:(8)te=111.44−Tc−1.36exp1.37S2×u0.2+2.12,R2=0.977.As mentioned above, the range method was used for both the calculation and analysis. Table3 shows the results of the range calculation of each factor for te.Table 3
Calculation of variance ofte.
S
u
T
c
Range calculation (d)
16.6
4.4
2.4It can be seen from Table3 that the primary factor to affect te was the distance between frozen pipes S, followed by the seepage velocity u, and finally the refrigerant temperature Tc. Therefore, when the frozen soil column does not overlap or the freezing formation time is too long, it is recommended to shorten the spacing of the frozen pipe firstly. Then, it is the second choice to reduce the groundwater seepage velocity by means of grouting, and finally to reduce the temperature of the refrigerant will be considered.In the actual project, when the overlap time exceeds 50 d, it can be considered that the frozen soil column does not overlap.te = 50 d is brought into formula (4); thereby, the maximum distance of the frozen pipe can be obtained as below:(9)Smax=ln−Tc−0.61u0.2,u>0,Tc<0.Figure8 shows the relationship between Smax and u, Tc. It can be seen from Figure 8 and formula (5) that Smax decreased with the increase of both seepage velocity u and refrigerant temperature Tc, respectively. However, with the increase of seepage velocity u, the decrease rate of Smax tends to be gentle. It is known from Section 3.2 that the influence of the seepage velocity u on te is greater than the refrigerant temperature Tc, and then it was postulated that the primary factor to affect Smax is the seepage velocity u, followed by the refrigerant temperature Tc. Therefore, basic tests such as groundwater seepage velocity and soil porosity test should be performed in the actual project to provide the basis for the rational design of the freezing project.Figure 8
The relation diagram ofSmax with u and Tc.


### 3.3.vm

The average development velocityvm of the freezing front of the water surface is one of the important characteristic index of the horizontal frozen body:(10)vm=Stm,where tm is the time from the intersection of the second row of permafrost columns to the formation of continuous horizontal frozen body (that is, the overlap time of the first row of permafrost columns minus the overlap time of the second row of permafrost columns). Similarly, some typical test results were selected to characterize the influence of various factors on vm, as shown in Figure 9.Figure 9
Influences of various factors onvm. (a) Distance between frozen pipes S vs. vm. (b) Seepage velocity u vs. vm. (c) Refrigerant temperature Tc vs. vm.
(a)
(b)
(c)It was demonstrated in Figure9 that vm decreased with the increase of both u and Tc, respectively. In addition, vm increased first and then decreased with S.As mentioned above, the range method was used for calculation and analysis. Table4 shows the results of the range calculation of each factor for vm. The influences of various factors on vm were as follows: seepage velocity u > the distance between freezing pipes S > refrigerant temperature Tc.Table 4
Calculation of variance ofvm.
S
u
T
c
Range calculation (mm·d−1)
68.8
167.2
65.4


### 3.4. vd

In this paper, the influence of the seepage velocityu and the refrigerant temperature Tc on the development velocity vd of the freezing front of the backwater surface was only analyzed by the test data of the distance between frozen pipes of 0.8 m. The reason is the same as in Section 3.1. By making derivative frozen time t of Ed, vd can be achieved:(11)vd=∂Ed∂t=−E+Cx2t2,t≥te.Figure10 reveals the variation of vd with frozen time under the condition with the distance between frozen pipes of 0.8 m, the seepage velocity of 1.5 m/d, and the refrigerant temperature of −25°C.Figure 10
v
d.Firstly, when influenced by the flow around the groundwater, the frozen front in the middle of the backwater surface develops faster. Because the groundwater flow velocity at both sides of the backwater surface is faster and the water temperature is higher, the frozen front develops slower than that in the middle. Secondly, the development velocity of the frozen front surface is inversely proportional to the square of the frozen timet. At the initial stage after the second row of frozen soil columns overlaps, the frozen front of the backwater surface develops faster, but the decline trend of speed is also faster. With the extension of the freezing time, vd gradually decreases. When the freezing time is long enough, the backwater surface reaches the cold and heat balance, and the frozen front development speed is zero.In order to facilitate the analysis of the influence of the seepage velocityu and the refrigerant temperature Tc on the development velocity of the frozen front surface, let k=−E+Cx2. Figure 11 shows the effect of the seepage velocity u and the refrigerant temperature Tc on k.Figure 11
Curves of the effect of various factors onk. (a) Seepage velocity u vs. k. (b) Refrigerant temperature Tc vs. k.
(a)
(b)Figure11 and formula (6) demonstrated that the influence law of the seepage velocity u and the refrigerant temperature Tc on vd is similar to that on ve, and vd decreased with the increase of both the seepage velocity u and the refrigerant temperature Tc.As mentioned above, the range method was used for calculation and analysis. Table5 shows the results of the range calculation of each factor for C and E. The effect of the seepage velocity u on vd was more remarkable than the refrigerant temperature Tc.Table 5
Calculation of variance of C and E.
Seepage velocityu
Refrigerant temperatureTc
C
0.74
0.10
E
1.16
0.32


### 3.5. ts

If either upstreamEu or downstream lengths Ed is ignored, the total freezing formation time ts can be divided into te and tm according to the time history:(12)ts=te+tm=S2ve+Emvm.In actual engineering, ifts is too long, it can be divided into two cases:(1)
The overlap timete of the last row of frozen soil columns is too long(2)
The development speedvm is too small, and the extension time tm is too longThe influences of various factors onte and vm according to the range calculate analysis were obtained in Sections 3.2 and 3.3. The influence of the distance between frozen pipes S and the seepage velocity u on te and vm is greater than the refrigerant temperature Tc. The influence of S on te is greater than u, while in the case of vm, opposite results were observed that the effect of S is weaker than u. Therefore, when result (1) occurs, it is recommended to shorten the distance between the frozen pipes. When result (2) occurs, in order to save the construction cost, it is recommended to use grouting and other measures to reduce the groundwater seepage velocity.


## 3.1.Ed

Figure5 shows the temperature field distribution at the distance between frozen pipes of 0.8 m, the seepage velocity of 1 m/d, and the freezing temperature of −25°C for 40 d (unit:°C).Figure 5
Schematic diagram of temperature field distribution.It was observed in Figure5 that, in the late stage of freezing, the backwater surface is in the shape of arc. After many attempts and fitting by the author, it is considered that the downstream length has a quadratic parabola relationship with the x-axis (the coordinate axis is shown in Figure 2):(5)Ed=Atx2+Bt.Among them,At and Bt are approximately inversely proportional to the freezing time t, that is,(6)At=Ct+D,Bt=Et+F,where C, D, E, and F are related to the distance between freezing pipes S, the seepage velocity u, and the refrigerant temperature Tc. In the actual project, the length Ew along the water surface direction of the horizontal frozen body was fixed, while this experiment was arranged with two rows and four columns of frozen pipes, and the length Ew along the water surface direction was not fixed. Therefore, the influences of the seepage velocity u and the refrigerant temperature Tc on Ed were only analyzed by the test results of the freezing pipe spacing of 0.8 m. The influence of the freezing pipe spacing on the backwater surface is not considered, so the parameters can be obtained as follows:(7)C=cu,Tc,D=du,Tc,E=eu,Tc,F=fu,Tc,Ed=Ct+Dx2+Et+F.Whent⟶∞, the downstream limit length is obtained as Edlim = Dx2+F, while Ed¯lim was obtained by dividing 3S after [−3S/2, 3S/2] integration.In this paper, typical test results were selected to characterize the influence of various factors on the downstream limit average lengthEd¯lim and the quadratic coefficient D, as shown in Figure 6.Figure 6
Influences of various factors on downstream length. (a) Seepage velocityu vs. downstream length Ed. (b) Refrigerant temperature Tc vs. downstream length Ed.
(a)
(b)It was revealed from Figure6 that the refrigerant temperature Tc was approximately proportional to the downstream limit length in the test range. When the seepage velocity is small, the change of the seepage velocity has a significant effect on Ed¯lim. As the seepage velocity u increases, the refrigerant temperature Tc increases, but the downstream limit average length Ed¯lim decreases. Similarly, the larger the seepage velocity u, the lower the refrigerant temperature Tc, the smaller the absolute value of the quadratic coefficient D, and the flatter the backwater surface.In order to analyze the influence of different factors on each characteristic index of the horizontal frozen body, the range method was used in this paper. The range calculation results ofEd¯lim are shown in Table 2. It showed that the influence of the seepage velocity u on the downstream limit average length Edlim was greater than the refrigerant temperature Tc.Table 2
Calculation of range ofEd¯lim.
Seepage velocityu
Refrigerant temperatureTc
Range calculation (m)
0.41
0.11


## 3.2.te

Typical test results were selected to characterize the influence of various factors on overlap timete of the second row of permafrost columns, as shown in Figure 7.Figure 7
Curves of the effect of various factors on te. (a) Distance between frozen pipes S vs. te. (b) Seepage velocity u vs. te. (c) Refrigerant temperature Tc vs. te.
(a)
(b)
(c)For the comprehensive regression of each parameter, the equation ofte is as follows:(8)te=111.44−Tc−1.36exp1.37S2×u0.2+2.12,R2=0.977.As mentioned above, the range method was used for both the calculation and analysis. Table3 shows the results of the range calculation of each factor for te.Table 3
Calculation of variance ofte.
S
u
T
c
Range calculation (d)
16.6
4.4
2.4It can be seen from Table3 that the primary factor to affect te was the distance between frozen pipes S, followed by the seepage velocity u, and finally the refrigerant temperature Tc. Therefore, when the frozen soil column does not overlap or the freezing formation time is too long, it is recommended to shorten the spacing of the frozen pipe firstly. Then, it is the second choice to reduce the groundwater seepage velocity by means of grouting, and finally to reduce the temperature of the refrigerant will be considered.In the actual project, when the overlap time exceeds 50 d, it can be considered that the frozen soil column does not overlap.te = 50 d is brought into formula (4); thereby, the maximum distance of the frozen pipe can be obtained as below:(9)Smax=ln−Tc−0.61u0.2,u>0,Tc<0.Figure8 shows the relationship between Smax and u, Tc. It can be seen from Figure 8 and formula (5) that Smax decreased with the increase of both seepage velocity u and refrigerant temperature Tc, respectively. However, with the increase of seepage velocity u, the decrease rate of Smax tends to be gentle. It is known from Section 3.2 that the influence of the seepage velocity u on te is greater than the refrigerant temperature Tc, and then it was postulated that the primary factor to affect Smax is the seepage velocity u, followed by the refrigerant temperature Tc. Therefore, basic tests such as groundwater seepage velocity and soil porosity test should be performed in the actual project to provide the basis for the rational design of the freezing project.Figure 8
The relation diagram ofSmax with u and Tc.


## 3.3.vm

The average development velocityvm of the freezing front of the water surface is one of the important characteristic index of the horizontal frozen body:(10)vm=Stm,where tm is the time from the intersection of the second row of permafrost columns to the formation of continuous horizontal frozen body (that is, the overlap time of the first row of permafrost columns minus the overlap time of the second row of permafrost columns). Similarly, some typical test results were selected to characterize the influence of various factors on vm, as shown in Figure 9.Figure 9
Influences of various factors onvm. (a) Distance between frozen pipes S vs. vm. (b) Seepage velocity u vs. vm. (c) Refrigerant temperature Tc vs. vm.
(a)
(b)
(c)It was demonstrated in Figure9 that vm decreased with the increase of both u and Tc, respectively. In addition, vm increased first and then decreased with S.As mentioned above, the range method was used for calculation and analysis. Table4 shows the results of the range calculation of each factor for vm. The influences of various factors on vm were as follows: seepage velocity u > the distance between freezing pipes S > refrigerant temperature Tc.Table 4
Calculation of variance ofvm.
S
u
T
c
Range calculation (mm·d−1)
68.8
167.2
65.4


## 3.4. vd

In this paper, the influence of the seepage velocityu and the refrigerant temperature Tc on the development velocity vd of the freezing front of the backwater surface was only analyzed by the test data of the distance between frozen pipes of 0.8 m. The reason is the same as in Section 3.1. By making derivative frozen time t of Ed, vd can be achieved:(11)vd=∂Ed∂t=−E+Cx2t2,t≥te.Figure10 reveals the variation of vd with frozen time under the condition with the distance between frozen pipes of 0.8 m, the seepage velocity of 1.5 m/d, and the refrigerant temperature of −25°C.Figure 10
v
d.Firstly, when influenced by the flow around the groundwater, the frozen front in the middle of the backwater surface develops faster. Because the groundwater flow velocity at both sides of the backwater surface is faster and the water temperature is higher, the frozen front develops slower than that in the middle. Secondly, the development velocity of the frozen front surface is inversely proportional to the square of the frozen timet. At the initial stage after the second row of frozen soil columns overlaps, the frozen front of the backwater surface develops faster, but the decline trend of speed is also faster. With the extension of the freezing time, vd gradually decreases. When the freezing time is long enough, the backwater surface reaches the cold and heat balance, and the frozen front development speed is zero.In order to facilitate the analysis of the influence of the seepage velocityu and the refrigerant temperature Tc on the development velocity of the frozen front surface, let k=−E+Cx2. Figure 11 shows the effect of the seepage velocity u and the refrigerant temperature Tc on k.Figure 11
Curves of the effect of various factors onk. (a) Seepage velocity u vs. k. (b) Refrigerant temperature Tc vs. k.
(a)
(b)Figure11 and formula (6) demonstrated that the influence law of the seepage velocity u and the refrigerant temperature Tc on vd is similar to that on ve, and vd decreased with the increase of both the seepage velocity u and the refrigerant temperature Tc.As mentioned above, the range method was used for calculation and analysis. Table5 shows the results of the range calculation of each factor for C and E. The effect of the seepage velocity u on vd was more remarkable than the refrigerant temperature Tc.Table 5
Calculation of variance of C and E.
Seepage velocityu
Refrigerant temperatureTc
C
0.74
0.10
E
1.16
0.32


## 3.5. ts

If either upstreamEu or downstream lengths Ed is ignored, the total freezing formation time ts can be divided into te and tm according to the time history:(12)ts=te+tm=S2ve+Emvm.In actual engineering, ifts is too long, it can be divided into two cases:(1)
The overlap timete of the last row of frozen soil columns is too long(2)
The development speedvm is too small, and the extension time tm is too longThe influences of various factors onte and vm according to the range calculate analysis were obtained in Sections 3.2 and 3.3. The influence of the distance between frozen pipes S and the seepage velocity u on te and vm is greater than the refrigerant temperature Tc. The influence of S on te is greater than u, while in the case of vm, opposite results were observed that the effect of S is weaker than u. Therefore, when result (1) occurs, it is recommended to shorten the distance between the frozen pipes. When result (2) occurs, in order to save the construction cost, it is recommended to use grouting and other measures to reduce the groundwater seepage velocity.


## 4. Conclusions

(1)
Under the effect of seepage flow, the backwater surface of the horizontal frozen body in the late stage of freezing is in the shape of quadratic parabolic, and the development velocity of the frozen front surface of the back water surface is inversely proportional to the square of the freezing time. When the length of the water-side surface is fixed, the influence ofu on the downstream limit length and the backwater surface development velocity is greater than the Tc.
(2)
The overlap timete of the second row of frozen soil columns increases with the increase of S and u and decreases with the decrease of Tc, while the average development velocity vm of the upstream surface decreases with the increase of u and increases with the decrease of Tc, and as S increases first and then decreases. The primary factor to affect te is S, followed by u, and finally is Tc. However, the primary factor to affect vm is u, followed by S, and finally is Tc.
(3)
Under the experimental conditions of this model, the regression formula between the overlap time of the second row of frozen soil columns and various factors was obtained by regression analysis, and the calculation formula of the maximum arrangement distance of the frozen pipes was obtained. It can provide theoretical basis for the parameter design of large-scale horizontal freezing construction of the Beijing subway.



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*Source: 8267692-2020-04-01.xml*