Physical Model Test of Artificial Freezing-Inclined Shaft

Based on the vertical straight artificial freezing engineering in Northern Shaanxi, a three-dimensional (3D) physical simulation test system was developed, consisting of six parts, which are simulation box, shaft model, loading system, freezing system, external environment simulation system, and data acquisition system.*e physical model and actual test results show that the 3D physical simulation test system is reasonable and reliable. *e test model results show that the distance from the freezing pipe significantly affects the freezing wall temperature. For the case of four adjacent, two adjacent tangential freezing, and two adjacent axial freezing pipes, the cooling rates were 1.37, 2.79, and 1.96°C/h, respectively. *e field measurement showed that the proximity to the freezing pipe increases the cooling rates. *e cooling rates of points 1k#, 2k#, and 3k# were 25.61, 25.32, and 25.35mm/d, respectively. *e increment rates of vertical and horizontal freezing pressures with temperature were 8.78 and 2.97 kPa/°C, respectively. Furthermore, the freezing pressure time fitting formula was given. *e calculated results of temperature and freezing pressure are consistent with the measured results, indicating the reasonability and reliability of the 3D physical simulation test scheme of the artificial freezing-inclined shaft in this work.


Introduction
Vertical straight-line artificial frozen walls are often used to excavate shafts in areas with large water content, shallow coal seam, and soft soil [1]. However, if the strength of the frozen wall is not enough during the sinking, water gushing and sand flooding are likely to occur, which will pose a threat to the safety of human life and bring about substantial economic losses [2][3][4]. To ensure that the inclined shaft can safely cross the water-rich soft soil stratum, the formation speed and thickness of the frozen wall must be studied. However, only a few studies of the freezing projects in the inclined shaft can be referenced at present. Such study is an effective and economical method to analyze the distribution law of the freezing temperature and stress in the inclined shaft using a three-dimensional (3D) physical simulation test.
Presently, several studies have investigated the temperature field of frozen walls. Wang and Wei [5] considered the phase change of water in the soil layer and the variation of the freezing temperature with time. e formation of the frozen wall and its temperature field characteristics under oblique conditions were analyzed in detail, using large-scale finite element (FE) software. Yang et al. [6] studied the characteristics of the artificial freezing tunnel test model and prediction of the surrounding rock temperature field. Wang et al. [7] studied the influence of the temperature field of the frozen wall on that of the hydration heat of the outer wall and obtained the basic law of temperature change of the outer and freezing walls during sinking in extra-thick alluvium. Based on the exponential integral function, Li and Xia [8] studied the temperature field of artificially frozen soils and obtained the theoretical expression of the thickness, velocity, and freezing time of the frozen soil curtain. Lü [9] analyzed the temperature variation law thermodynamicbased theory and the phase change law of frozen soils. Cao et al. [10] studied the adaptability of the freezing method under the condition of water-rich sandy pebble stratum and the rationality of the freezing design scheme. Gianpiero et al. [11] studied the artificial ground freezing (AGF) technique to excavate a tunnel below the groundwater table. Li [12] studied the freezing temperature field and thermal conductivities of two representative soil samples and analyzed the numerical simulation of the freezing temperature with the equivalent thermal conductivity tested, using ANSYS software. Yue et al. [13] simulated the freezing process of artificially frozen soils of Dalian Road tunnel connecting passage with FE analysis (FEA) method and determined the existing varying laws of temperature, displacement, and stress fields. Hu and Zhao [14] studied the precision of the back folding model for the temperature field of AGF. Fan and Yang [15] presented a detailed case study of a cross passage, including the temperature variations in brine, and selected monitoring points around the passage based on field testing results during freezing and frozen wall maintenance. From the above studies mainly about the temperature field distribution of freezing-vertical shaft, the temperature field distribution of freezing-inclined shaft still needs further study.
Wang and Yang [16] established a plane FE model to simulate the whole process of ground frozen-shaft excavation-thawing. rough the sequential coupling of temperature and stress fields, the stress law of the frozen wall during thawing in the surface soil section was obtained. Liu et al. [17] used scale test models to numerically analyze artificial freezing at different temperatures in complicated strata. Vitel et al. [18] modeled the heat transfer between a frozen pipe and the surrounding ground during AGF activities. Zhang et al. [19] studied the transient temperature field of frozen soil with distributed optical fiber. e results showed an inverse correlation and the nonmonotonic correlation between the rate of temperature change of normal frozen soil and its initial water content. Liu and Yan [20] studied the formation and development of frozen walls under the vertical straight-line frozen condition of water-bearing sand stratum and studied the distribution law of frozen wall temperature field in the inclined shaft. Ji et al. [21] presented a model based on 2D heat conduction and calculated the temperature distribution in the frozen wall. Wang et al. [22] used a self-made test device to study the laws of the temperature field and obtain information for predicting the frost heave and numerical simulation of artificially frozen soil. He et al. [23], through numerical simulation analysis and field monitoring data, studied the development law of the temperature field and frozen wall during freezing. Zhang and Yu [24] used the FE method to analyze the formation and temperature distribution of the freezing wall in the artificial freezing method. Hu et al. [25] established AGF temperature field models to study the effect of the soil freezing point on the frozen wall thickness. Shen et al. [26] studied the influence of temperature and moisture content on sandstone thermal conductivity from a case using the AGF method. Kim et al. [27] studied the FE modeling (FEM) and FEA for AGF in egress shafts. Roman et al. [28] studied the AGF of fully saturated soils in a thermal problem. Hou et al. [29] studied the freezing temperature field distribution law of multirow pipes in a large cross-sectional inclined shaft construction. Yao et al. [30] established a calculation model of artificial frozen soil temperature field, based on the support vector machine algorithm, and presented a kernel function suitable for the temperature field. Ren et al. [31] studied the mechanical properties and temperature field of the frozen wall in the water-rich sand layer by combining indoor physical mechanics test, field measurement, and finite element numerical simulation. Sheng and Wei [32] measured the multicircle temperature field of frozen walls and analyzed the problems of the shallow section, deep freezing, cold energy waste, and slow excavation speed of the frozen shaft in the early stage. However, these studies mainly focus on the freezing pressure, temperature field distribution, and the interaction mechanism between freezing wall and shaft wall in freezing-vertical shaft, but the studies on the distribution law of temperature and stress in a freezinginclined shaft are still rare.
us, this work focuses on the freezing engineering of a main inclined shaft in Northern Shaanxi. Based on the characteristics of the sinking freezing-inclined shaft, a set of 3D physical simulation test systems were developed and used for the completed 3D physical simulation test. e results can provide some references for similar projects.  Table 1. e water inflow from the main inclined shaft was estimated as 1250 m 3 /h from the pumping test. Vertical straight-line artificial frozen method was used to sink the inclined shaft. Five frozen holes with a spacing of 2.35 m from each were made in each row. e thicknesses of the frozen wall on the roof, both sides, and bottom plate were 6.0, 3.0, and 5.0 m, respectively. e layout of frozen holes is shown in Figure 1.

Physical Model System
Based on the characteristics of the inclined frozen method, a 3D physical model system for the freezing-inclined shaft was developed. e system consisted of external environment simulation, freezing, model box, and loading; and data acquisition systems are shown in  e external environment simulation system in Figure 2 consists of a high-and low-temperature environment laboratory and temperature control system, which can simulate the ambient temperature from 30 to −30°C. e system can keep the laboratory temperature constant and also isolate the heat exchange between the laboratory and external environments to reduce the impact of the external environment on physical test models.
e freezing system in Figure 3 consists of a low-and constant-temperature reaction bath and freezing pipe, and the minimum temperature was 2 Advances in Civil Engineering recorded as −30°C. e loading system in Figure 4 consists of a horizontal loading plate and reaction steel channel. e loading plate was placed directly in the box, and the two loading plates were connected by a rubber plate. e reaction steel channel was bolted onto the bottom plate of the box, and the horizontal loading plate was pressed by a jack. e data acquisition system in Figure 5 consists of a static strain gauge, vibrating wire reader, and fiber grating sensor regulator, which can be selected according to different test contents.

Physical Simulation Materials.
e physical simulation test is an effective method for solving complex problems, and similar materials are the key to such a test. Typical sand was    used as a similar material in the frozen section of the inclined shaft, and its thermophysical parameters at different temperatures are shown in Table 2 [33]. e density reduction, specific heat capacity reduction, thermal conductivity reduction, and freezing latent heat ratios of water are shown in the following equation: where ρ and ρ ′ are the densities of the prototype material and physical model, respectively, kg/m 3 ; Cand C ′ are heat capacities of the prototype material and physical model, respectively, J/(kg·°C); λ and λ ' are the thermal conductivities of the prototype material and physical model, respectively, W/(m·°C); Q and Q ' are the latent heat values of the freezing water of the prototype material and physical model, respectively, kJ/m 3 .

Geometric
Scaling. e model length, width, and height are 1.0, 0.6, and 1.0 m, respectively. Based on Saint-Venant's principle, the soil within 3-5 times of tunnel diameter was selected as the research sample. Also, based on the size of the 3D physical model table, the shaft wall was reversely pushed out and scaled using a geometric ratio of 1 : 50. us, the clear width, height, and arc height of the shaft wall model were 100, 25, and 50 mm. In the physical simulation test, a seamless copper pipe with a diameter of 6.35 × 0.7 mm was used for freezing the pipe, and row spacing was 42 × 200 mm.

Temperature Scaling.
According to the Kosovitch criterion, the temperature scaling can be expressed in the following equation: where T and T ′ are the temperatures of the prototype material and physical model, respectively,°C; C T is temperature scaling. Substituting equation (1) into (2) gives C T � T/T ′ � 1, which means that the temperature of the prototype is consistent with that of the physical simulation test.

Time Scaling.
Based on the Fourier criterion, time scaling can be expressed in the following equation: where t and t ' are times of the prototype material and physical model, respectively, s; C t is the time scale. Substituting equations (1) and (2) into equation (3) gives C t � 2500.

Layout of Temperature Measuring
Points. e measuring point arrangement is shown in Figure 6. A total of 21 temperature monitoring points were arranged in the test. e three red measuring points (1 # to 3 # ) are affected by four tangential and axially adjacent frozen pipes, which were recorded as monitoring section I. Six black points (4 # to 9 # ) were used to monitor the temperature change in the edge zone, which were recorded as monitoring section II. e four

Layout of Pressure Measuring
Points. e pressure sensor, which monitors the force change between the shaft and freezing wall during freezing, was mainly arranged on the shaft. e arrangement of the measuring points is shown in Figure 7.
e frozen system will be started after the stabilization of the data of each measuring point. Afterward, the data of each sensor will be recorded. e intersection of the frozen wall surface with different running times during the freezing process is shown in Figure 8.

Law of Temperature Change.
e temperature variation law of each measuring point is shown in Figure 9. e stability of the cold source is one of the important factors ensuring the stable development of the frozen wall. From Figure 9, it took 6 h for the temperature to drop from 27 to 0°C, with the cooling rate at about 4.5°C/h. e cold source temperature reaches −30°C at the 26 th hour and the test changes from the active to maintenance-freezing period. e maintenance-freezing temperature was set to −20°C. e cold source temperature began increasing within 36-42 h due to the change of the temperature set value and stabilized at the 56 th hour. Afterward, the test entered the maintenance-freezing period. e freezing was stopped at the 120 th hour, and the temperature difference of desalinated brine during freezing did not exceed 1°C, which ensured the smooth running of the test.
As shown in Figure 9, points 1 # , 2 # , and 3 # are affected by four tangential and axially adjacent frozen pipes. During the entire freezing process, the temperature of each measuring point was divided into four stages. e first is a rapid cooling stage, in which the temperature dropped from 27.4 to 0°C at 1.37°C/h, and the lowest temperature was −15.6°C at the 48 th hour. At this time, the temperature of each measuring point was higher than −10°C, and the test was changed from the active to the maintenance-freezing period. At the second stage, which is stationary, the temperature of each measuring point stabilized at the 56 th hour after a small temperature rise, and the freezing entered the maintenance period. For the third stage, which is the rapid heating stage, the temperature of each measuring point increased rapidly from −12.1 to 0°C in 34 h at 0.38°C/h. At the 210 th hour, the temperature of each measuring point culminated to 11.8°C. At the fourth stage, which is stable, the temperature of each measuring point was stable with an average value of 11.1°C. Points 4 # , 5 # , 6 # , 7 # , 8 # , and 9 # were used to monitor the temperature change of sand at the edge zone of the test model. e distance from point 4 # to the shaft axis was equal to that from point 6 # , while the distance from point 5 # to the shaft axis was equal to that from point 7 # . Points 8 # and 9 # were located at the ends of the shaft. e cooling speeds of  Advances in Civil Engineering measuring points 4 # and 6 # , 5 # and 7 # , and 8 # and 9 # were the same, with values of 0.92, 0.82, and 1.57°C/h, respectively. e excavation at the 76 th hour exposed measuring point 8 # and its temperature was abrupt after 76 h. Points 10 # , 11 # , 12 # , and 13 # were used to monitor the temperature change of sand between two adjacent tangential frozen pipes. e temperature variation of each measuring point was the same for points 1 # -9 # during freezing, and both had rapid cooling, stationary, rapid heating, and stable stages. e temperature variation of each measuring point was the same and the cooling rate was about 2.79°C/h. During the test change from the active to the maintenance-freezing period, the temperatures of points 10 # -13 # were smaller than those of points 4 # -9 # , due to the proximity of points 10 # -13 # to the freezing pipe.
us, a larger cooling capacity was obtained. Measuring points 14 # , 15 # , 16 # , 17 # , 18 # , 19 # , 20 # , and 21 # monitored the temperature change of sand between two adjacent frozen pipes in the axial direction. e temperature variation of each measuring point was the same for points 4 # -13 # during freezing, and the cooling rate was about 1.96°C/h.
In summary, the cooling rate of points 10 # -13 # was the fastest, followed by that of points 14 # -21 # and then points 1 # -3 # . e cooling rate of points 10 # -13 # was the slowest because of the smaller distance from points 10 # -13 # and 14 # -21 # to the cold source compared to that from points 14 # -21 # and points 1 # -3 # , respectively. Moreover, the distance from points 1 # -3 # to the cold source was smaller than that from points 4 # -9 # . e heat conduction process of frozen wall is shown in Figure 10; r 1 is the radius of freezing pipes, r 2 is the radius of frozen wall, r is the radius of a certain point in the frozen wall, and dr is the radius of the micro unit. T 1 and T 2 are temperatures of cold source and outer boundary of frozen wall. According to Fourier heat conduction law, the heat transfer rate of frozen wall is shown in the following equation: where L is freezing pipe length and λ represents the thermal conductivities. Equation (4) shows that heat transfer rate is proportional to the temperature difference and inversely proportional to the distance. In the freezing process, the time is equal and the temperature difference is constant, so the heat transfer rate is related to the distance. e larger the distance, the smaller the heat transfer rate and the slower the cooling rate.
Equation (4) and physical simulation test results show that the distance from the cold source significantly influences the cooling rate.

Distribution and Variation of Freezing Pressure.
e distribution and variation of freezing pressure are shown in Figure 11. Points 2 # , 4 # , 6 # , 8 # , 10 # , and 12 # monitor the vertical freezing pressure. e maximum vertical freezing pressure is 353.86 kPa at the 48 th hour and the rate of increment is 7.32 kPa/h. From Figure 9, the freezing wall temperature is lowest at the 48 th hour. Also, the minimum and initial temperatures are −13.2 and 27.1°C, respectively; thus the rate of increment of the vertical freezing pressure with temperature is 8.78 kPa/°C. Points 1 # , 3 # , 5 # , 7 # , 9 # , and 11 # monitor the horizontal freezing pressure. Furthermore, the maximum vertical freezing pressure is 119.55 kPa at the 48 th hour with an increment rate of 2.49 kPa/h and the increment rate of the horizontal freezing pressure with temperature is 2.97 kPa/°C. Besides, the vertical freezing pressure is 2.96 times the horizontal freezing pressure.
As shown in Figures 9 and 11, the temperature change significantly influences the freezing pressure. During the active-freezing period, the temperature decreases rapidly, while the freezing pressure increases rapidly. e freezing pressure tends to stabilize at the 56 th hour because of the temperature stabilization of each measuring point. Freezing stopped at the 120 th hour when the temperature of the freezing wall increased and the freezing pressure began to decrease.

Field Measurement and Rationality Verification of the Test Model
To verify the rationality of the 3D physical simulation test scheme proposed in this study, field measurements of freezing wall temperature and pressure were carried out. e temperature and freezing pressure monitoring points are shown in Figure 12.

Analysis of Field Measurement Results
7.1.1. Temperature. e measured results of the freezing wall temperature are shown in Figure 13. e cooling rate of sand during the active-freezing period is approximately linear with time, and the temperature difference among the three temperature measuring points is small. During the activefreezing period, the cooling rate of 1 k# is the fastest because of its proximity to the freezing pipes, while those of 2 k# and 3 k# are equal. When the freezing period enters the maintenance period, the temperature of point 1 k# is lower than those of points 2 k# and 3 k# , due to its proximity to the freezing pipes. e sand showed a faster cooling rate at this   Figure 14. Due to the large amount of hydration heat produced by the concrete pouring of the shaft lining, the local temperature field of the freezing wall increases to a maximum temperature of 16.7°C. When the hydration heat of concrete dissipates, the frozen wall begins to refreeze on the 6th day, and the freezing pressure increases. e freezing stops on the 17 th day and the frozen wall begins to thaw. us, the supporting effect of the frozen wall decreases, and the surrounding water and soil pressure are mainly borne by the shaft. e sensor pressure then culminates to 370.76 kPa.

Comparative Analysis
7.2.1. Temperature. Measuring points 11 # , 4 # , and 6 # correspond to measuring points 1 k# , 2 k# , and 3 k# , respectively. Based on the time scale, the inverse of the actual temperature of the measuring point corresponding to the temperature of each measuring point in the physical simulation test was calculated and plotted ( Figure 15). Figure 15 shows that, due to certain differences between the on-site and indoor physical simulation tests, the physical simulation temperature could not perfectly match with that of the on-site measured data, but the cooling laws of all measuring points were equal, which indicates that the 3D physical simulation test scheme of the artificial freezing-inclined shaft in this study is feasible and better reflects the general law of artificial freezing-inclined shaft cooling.
wherep 1 , p 2 , and p 3 are fitting coefficients.    Advances in Civil Engineering e monitoring instruments were embedded after excavation in the actual project because those of the test model were embedded in advance. is disparity resulted in different freezing pressure changes for both scenarios. However, after the concrete hydration heat dissipated, the freezing wall began freezing. During such a process, the relationship between the temperature change and freezing pressure is expected to satisfy equation (4), which is shown in Figure 16. e fitting coefficients are presented in Table 3.
It can be seen from Figure 16 that the calculated results of temperature and freezing pressure are consistent with the measured results, which further indicates that the 3D physical simulation test scheme of the artificial freezinginclined shaft in this work is reasonable and reliable.

Conclusions
In this study, a 3D physical simulation test system was developed to test the vertical straight-line artificial freezinginclined shaft and conduct field measurements of the freezing wall temperature and pressure. e results of this study can be summarized as follows: (1) A 3D physical simulation test system was developed; the comparison between the model and the actual test results showed the reasonability and reliability of the 3D physical simulation test system. (2) eoretical analysis, test model, and field measurement results show that the distance from the cold source significantly influences the cooling rate, and the proximity to the frozen pipe increases the cooling speed of the freezing wall.
(3) e temperature change significantly affects the freezing pressure; the relationship between the freezing pressure and temperature can be expressed by a quadratic polynomial. Furthermore, the freezing pressure time fitting formula was given, and the calculated results of temperature and freezing pressure are consistent with the measured results.

Data Availability
All data generated or analyzed during this study are included within this article.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper. Freezing pressure (kPa) 1 k# 2 k# 3 k# Figure 16: Calculated and measured temperature-freezing pressure curves.