SimilarityCriterion of FreezingModel Test consideringNonlinear Variation of Thermal Parameters with Temperature

*e significant differences in specific heat and thermal conductivity of ice and water lead to the changes of specific heat and thermal conductivity of soil during the freezing process. *is makes it hard for the temperature field similarity criterion based on constant thermal parameters to accurately reflect the temperature field evolution of soil mass caused by nonlinearity of thermal parameters in the process. Based on heat conduction differential equation considering nonlinear changes of thermal parameters, this paper uses similarity transformation method to derive the similarity criterion of the temperature field in the frozen soil model test and arrives at the conclusion that the prototype soil and model soil should meet when the original soil is used for the model test. At the same time, given the impact of the third boundary condition on the similarity criterion, the thermal physical similarity conditions for the model soil are derived. On this basis, ABAQUS finite element software is used to numerically simulate the linear and nonlinear prototype andmodel temperature fields.*e third boundary condition considered the temperature evolution of the characteristic points during the freezing process is analyzed. *e calculation results indicate that the nonlinear thermal conductivity similarity criterion established herein can correctly reflect the evolution process of the prototype frozen soil temperature field. It is also suggested that the model soil thermal parameters are reasonably calculated. At the same time, it shows that the nonlinear freezing similarity criterion of the soil, when the third boundary condition is satisfied, has clear physical meaning and higher practical value. *e research results provide a practical and reasonable parameter calculation method for the model soil preparation in the frozen soil model test and a theoretical basis and technical support for the design and implementation of the water-heat-force coupling model test on frozen soil.


Introduction
Model test is a science in which, on the basis of a certain geometric-physical relationship, a model is made and tested in place of a prototype so as to have results which can be used to predict the development of the prototype according to the corresponding similarity criterion [1][2][3]. Model test is an important means to study the occurrence mechanism and development law of complex physical phenomena, while similarity criterion is the theoretical basis for model test design [4,5]. Derivation of the model test similarity criterion based on engineering conditions is the basis and necessary condition for improving the model test prediction accuracy.
Freezing process of soil is a complicated problem involving coupling of temperature field, moisture field, and stress field, and the evolution process of temperature field provides the basis on and the key to the study of water migration and stress changes [6,7]. Harlan proposed a temperature field model that considers water migration and pore water phase transitions in soil.
is model carries important theoretical significance for studying the variation law of water migration with time during the movement of soil freezing temperature gradient and freezing front [8]. Bao et al. established an enthalpy-based heat conduction calculation model for frozen soil based on the heat diffusion equation and then solved and analyzed the heat conduction process between soil layers [9]. Based on the transient heat source method, Trigui et al. studied composite material with phase transition characteristics on the basis of the transient heat source method and derived its sensible heat, heat storage, and latent heat through inverse calculation [10]. From the perspective of frozen soil temperature field model test, the applicable similarity criterion is derived. Zhu et al. [11] analyzed the similarity criterion of the freezing and thawing process of frozen soil when there is no external load and carried out corresponding freezing and thawing model test based on this. Zhang et al. [12] took temperature field of single-tube frozen soil as the research object and deduced the similarity criterion of the temperature field of single-tube frozen soil. Zhang et al. [13] conducted frost heave centrifugal model tests of undisturbed soil with different water contents of 20 g and 30 g and discussed the heat transfer process of the canal centrifuge model and the impact of the moisture content of canal foundation soil in the closed system on the normal frost heave displacement of the canal. e existing similarity criterion for the temperature field model test of frozen soil is established based on constant thermal parameters, that is, the nonlinear changes of thermal parameters such as thermal conductivity coefficient and specific heat with temperature are not considered. During the freezing process of the soil, thermal parameters such as thermal conductivity coefficient and specific heat change significantly with temperature [14][15][16]. Hence, the temperature field similarity criterion established based on constant thermal parameters obviously cannot accurately reflect the nonlinear freezing process of soil. In cold area engineering and artificial freezing method construction, engineering accidents caused by temperature field calculation errors are not uncommon [17,18]. Based on the nonlinear change law of thermal parameters with negative temperature, similarity criterion for frozen soil model tests is established which not only has important theoretical significance but also means huge economic value and social benefits.
In addition, in actual engineering, as the frozen boundary of the soil is not adiabatic, there is a nonnegligible heat exchange between the frozen area and the surrounding environment or adjacent media [19,20]. When the original soil is used for temperature field freezing model test, according to the third boundary condition, the geometric ratio of the model and the prototype must be 1 : 1, i.e., the model has to be of the same size as that of the prototype. is makes it difficult to model the prototype project, and model test conducted in the scale mode obviously looses the theoretical basis and application conditions [21][22][23]. erefore, it is necessary to consider the similarity criterion of temperature field and the boundary and then derive the similarity conditions for the model soil thermal physical parameters.
Based on the nonlinear heat conduction differential equation, this paper, using similarity transformation method derives the similarity criteria for temperature field similarity in the frozen soil model test and has established the similarity relationship between the prototype and the model when original soil is used for the model test. At the same time, considering the impact of the third boundary condition on similarity criterion of the temperature field model test, the paper also derives the conditions that the model soil should meet after material transformation in terms of thermal physical property parameters. Based on the deduced similarity criterion and the thermal property parameters of similar materials, the paper simulates numerically the linear and nonlinear frozen soil temperature fields and the third boundary condition by means of ABAQUS finite element software and then analyzes temperature evolution of characteristic points.

Nonlinear Heat Conduction Model and Similarity Criterion
Mathematical description of heat conduction problem involves heat conduction differential equations and boundary value conditions, and the boundary value conditions include initial conditions and boundary conditions [20]. Boundary value conditions and heat conduction mathematical model are the basis for establishing similarity criterion of nonlinear heat conduction.

Boundary Conditions of Heat Conduction Process.
Boundary conditions of temperature field [23,24] mainly fall into three categories. e first category describes the temperature of the soil boundary or the surrounding environment; the second category concerns the heat flux density on the soil boundary; the third category is about the heat exchange volume between the soil and the surrounding medium. e existing similarity criterion often takes temperature shrink ratio of 1 in the process of establishment, which requires that the boundary temperature must be constant in heat conduction. erefore, the criterion cannot be used to consider the heat exchange between the soil and the external environment [23]. In cold area engineering and artificial freezing method construction, freezing of the soil body is a heat conduction phenomenon under the coupling effect of the boundary value condition, so temperature field similarity criterion without considering the third boundary condition obviously cannot guarantee rationality and accuracy of the model test.

Mathematical Model of Nonlinear Heat Conduction.
At present, the conventional heat conduction equation describing freezing process dismisses the change of thermal parameters with negative temperature, which is expressed as follows [24]: where C is soil specific heat; ρ is soil density; T is soil temperature; t is freezing time; λ is soil thermal conductivity coefficient; and g v is heat flux intensity of the cold (hot) source in the soil. e temperature continuity and energy conservation conditions must be satisfied on the freezing front h of the soil, i.e., where T 0 is soil phase transition temperature; T f and T u , respectively, represent frozen soil temperature and melted soil temperature; λ f is frozen soil thermal conductivity coefficient; n is the external normal direction of the soil boundary; λ u is melted soil thermal conductivity coefficient; Q is the latent heat of phase change per unit volume of soil; and h is the freezing front location. e differential equation of the nonlinear heat conduction temperature field considering thermal parameter variation with temperature is [25] where C(T) and λ(T) are the specific heat and thermal conductivity coefficient of soil, respectively. For soil with certain dry density and water content, its thermal conductivity coefficient and specific heat involve a function related to temperature T [26]. Existing studies have shown that freezing of water in soil is concentrated in the high temperature freezing stage [27,28], and it is difficult for one linear fitting to reflect the evolution of thermal parameters with freezing temperature. To improve fitting accuracy of the test data, a quadratic function is hereby taken to describe the variation trend of thermal parameters with temperature, namely, where A, B, D, and E are fitting constants, whose units are kJ/(kg·°C 3 ), kJ/(kg·°C 2 ), W/(m·°C 3 ), and W/(m·°C 2 ), respectively; λ 0 is the thermal conductivity coefficient at 0°C, W/(m·°C); C 0 is the specific heat at 0°C, kJ/(kg·°C); and g v is 0 when there is no internal heat source in the soil. If frozen soil is isotropic body, i.e., the soil particles, unfrozen water, ice, and gas are evenly distributed in frozen soils, then the temperature continuity and energy conservation conditions must be met on the freezing front h of the soil, namely,

Nonlinear Heat Conduction Similarity
Criterion. e similarity criterion is derived based on the similarity transformation method. First, define the similar transformation of each physical quantity in formulas (3) and (7), namely, In the formula, the subscript "p" represents the physical quantity of the prototype; the subscript "m" represents the physical quantity of the model; C A , C B , C D , and C E are shrink ratios of the corresponding coefficients in formulas (4) and (5); C C0 is the soil specific heat shrink ratio; C λ0 is the thermal conductivity shrink ratio; C t is the time shrink ratio; C qv is the heat flux intensity shrink ratio of cold (hot) source; C l is the geometric shrink ratio; C T is the temperature shrink ratio; C Q is the latent heat shrink ratio; C ρ is the soil density shrink ratio. Substitute equation (8) into equations (3) and (7), then Mathematical Problems in Engineering According to principle of similarity, we describe similar physical phenomena with the same differential equation, and their single-valued conditions are similar [14]. By comparing the balance equations of the prototype and the model, 7 similarity criteria are derived after sorting, namely,

Similarity Condition Analysis
Based on the 7 derived similarity criteria, the similarity index formula can be further derived, namely, In the existing soil freezing model test, the original soil is often used as the model material, that is, the temperature shrinkage ratio C T is 1. at is, the initial temperature and the first boundary condition of each point in the model are the same as those of the corresponding point in the prototype. From it can be known that 4 Mathematical Problems in Engineering where C t is the time shrink ratio; C l is the geometric shrink ratio; and C ρ is the soil density.
In actual engineering, due to fragmentation of the soil and the cold source conditions during the test, the soil heat transfer process cannot be strictly carried out with temperature shrink ratio C T equal to 1. at is, the impact of the third boundary condition on temperature field should be taken into account. If the temperature shrink ratio C T is not 1, according to the index formula of the similarity ratio, there is at is, it is unreasonable to use original soil for the model test. Hence, it is necessary to seek materials with similar characteristics as the thermophysical properties of the original soil for the model test by material transformation. e relationship between the specific heat, thermal conductivity coefficient, and temperature of the prototype soil as indicated by the quadratic function is where A p , B p , D p , and E p are constants; T p (°C) is the prototype soil temperature; C 0p (kJ/(kg·°C)) is the specific heat of the prototype soil at 0°C; and λ 0p (W/(m·°C)) is the thermal conductivity coefficient of the prototype soil at 0°C. If it is assumed that the temperature shrink ratio is C T , the specific heat shrink ratio of soil is C C0 and the thermal conductivity coefficient shrink ratio of soil is C λ0 ; then, erefore, the specific heat and thermal conductivity coefficient of the model soil are where T m (°C) is the model soil temperature.
According to (C C 0 C T C ρ /C Q ) � 1, model soil latent heat per unit volume should meet Q m � (Q p /C C 0 C T C ρ ).
Similarity criterion for the third boundary condition is where α is the convection heat transfer coefficient between the soil and the environment. e corresponding similarity index formula is where C α is the convection heat transfer coefficient shrink ratio.
After considering the impact of the third boundary condition, the specific heat and thermal conductivity of the model soil should be According to (C C 0 C T C ρ /C Q ) � 1, model soil latent heat per unit volume should meet Q m � (Q p /C C 0 C ρ C T ). According to (C λ 0 C t /(C l ) 2 C ρ C C 0 ) � 1 and (C α C l /C λ 0 ) � 1, time shrink ratio is C t � (C C 0 C ρ C l /C α ).

Verification of Similarity Criterion and Model Soil Similarity Condition
To test the derived similarity criterion of the nonlinear freezing temperature field of the soil and the conditions that the model soil should meet in terms of thermal physical properties after considering the third boundary condition, a series of soil freezing model tests were designed and implemented. e original soil thermal conductivity coefficient and change law of specific heat with temperature were measured, and then by establishing soil temperature field models with different shrink ratios, the derived similarity criterion and similarity relationship between the model soil and the original soil considering the third boundary condition were tested.

Determination of ermal Conductivity Coefficient and
Specific Heat. Take a silty clay sample with remodeling density of 1910 kg/m 3 and water content of 33%. e specific heat and thermal conductivity coefficient of clay samples are measured by means of calorimetry and probe method under different negative temperatures. e resulting specific heat and thermal conductivity coefficient changes with temperature are shown in Figure 1.

Mathematical Problems in Engineering
It can be seen from Figure 1 that, as the temperature decreases, the soil sample specific heat gradually decreases and the thermal conductivity coefficient increases gradually, which is consistent with laws in the existing research [27,28]. According to the previous fitting formulas (4) and (5), the change law of specific heat and thermal conductivity coefficients with negative temperature are quadratic fitted, and the resulting expression is e latent heat calculation method provided by literature [24] is where Q (kJ/m 3 ) is the latent heat per unit volume of soil; ρ d (kg/m 3 ) and ρ (kg/m 3 )) represent soil dry density and soil density, respectively; L (kJ/kg) is the water latent heat of phase change; w T1 and w T2 are the unfrozen water content in frozen soil under negative temperature of T 1 and T 2 , respectively; and w 0 indicates the initial water content. According to the unfrozen water content test method provided in literature [29], unfrozen water content of frozen soil samples was obtained under different negative temperatures. According to equation (21), phase change latent heat of frozen soil can be obtained for different negative    Parameter Similarity constant Mathematical Problems in Engineering temperature intervals. e calculation results are shown in Table 1.

Freezing Test When the Temperature Shrink Ratio Is 1.
To test the validity of the established nonlinear heat conduction similarity criterion and the soil model established with the third boundary condition considered, the corresponding numerical model is established by ABAQUS finite element software. e adopted thermal parameter model and boundary conditions are shown in Table 2.
It can be seen from Table 2 that Model 1, Model 2, and Model 3 are soil freezing test plans with a temperature shrink ratio C T at 1, that is, the impact of the third boundary condition is excluded. e difference between Model 1 and Model 3 is that Model 3 is linear, that is, it does not consider thermal parameter variation with negative temperature. According to the similarity criterion, the similarity constants of Model 1 and Model 2 can be determined, as shown in Table 3. e dimensions and test parameters of each model are shown in Table 4.
e numerical simulation of one-dimensional freezing temperature field is carried out by means of ABAQUS finite element software according to the parameters determined in Table 3. Model 1 is a 2 m × 0.5 m rectangle. At the lower boundary of the rectangular model, a constant temperature cold source of −20°C is set. As can be seen from Table 1, when the temperature is reduced to −5°C, the phase change latent heat of the frozen soil sample has very small increment. To optimize the model calculation while guaranteeing calculation accuracy, the method adopted in [28] sets the latent heat release range of frozen soil at [0°C∼−2°C], and the phase change latent heat value is set at 139430 kJ/m 3 . e specific heat and thermal conductivity coefficient of the soil are set according to the values given in Figure 1. Model 2 is a 1 m × 0.25 m rectangle. e same soil body as Model 1 is taken, and it has the same boundary and cold source conditions as Model 1. Model 3 has the same size and material with Model 1, but when setting the soil thermal parameters, the variation of specific heat and thermal conductivity coefficient with temperature is not considered, and the average value in Figure 1 is taken as its thermal conductivity coefficient and specific heat, that is, in Model 3, the thermal conductivity coefficient is 1.82 W/(m·°C) and the specific heat is 1.58 kJ/(kg·°C).

Freezing Test When the Temperature Shrink Ratio C T Is not 1.
According to the previous derivation, the similarity conditions established with consideration to the third boundary condition are verified, and the relevant parameters are shown in Table 5.
Model 4 and Model 5 consider the impact of the third boundary condition, that is, heat exchange between the soil interface and the external environment is allowed. Model 4 is the same as model 1 inters of size and material properties, but when the upper condition is set, No. 4 upper boundary condition is set to be convective heat transfer. Model 5 has a size shrink ratio C l � 2 and temperature shrink ratio C T � 2 compared with Model 4, and the remaining boundary settings are the same as in Model 4. Because there is reduced scale in size and cold source between Model 4 and Model 5 and convective heat transfer exists between the boundary and the environment, Model 5 needs to be tested using model soil in replace of original soil. If it is assumed that the convection heat transfer coefficient shrink ratio C α is 2, according to the similarity criterion of the third boundary condition, the thermal conductivity coefficient shrink ratio C λ0 is 4. If it is assumed that the specific heat shrink ratio C C0 is 4, the conditions for the model soil thermal physical parameters are deduced according to the above conclusion, and the specific heat and thermal conductivity coefficient of the model soil are derived as   Parameter Similarity constant According to the derived similarity criterion, the similarity constant between Model 4 and Model 5 is determined, as shown in Table 6. e dimensions and test parameters of each model are shown in Table 7.

Results Analysis
To test the validity of the established similarity criterion and model soil parameters, the stable development process of three characteristic points of the model test is extracted on the basis of the previous numerical simulation results. e temperature measurement points are arranged, as shown in  Table 8.
It can be seen from Figure 3 and Table 8 that certain errors exist in temperature curves of Model 1 and Model 3. Existing literature also indicates that prediction accuracy is higher when frozen soil temperature field considers thermal parameter variation with temperature [28][29][30][31].
at is, similarity criterion derived by nonlinear theory has higher prediction accuracy. By multiplying Model 2 by the time shrink ratio C t , it can be found that the corresponding time has equal temperature value, and the temperature shrink ratio between Model 1 and Model 2 is 1. is suggests that when theoretical similarity criterion and original soil are used for model test, the similarity relationship between the prototype and the model is  Table 9 for different time periods.    Mathematical Problems in Engineering It can be seen from Figure 4 and Table 9 that Model 4 and Model 5 have similar temperature fields, that is, the temperature value of corresponding time involves a twofold relationship.
is indicates that the model soil thermal parameters given herein are reasonable when the model soil after transformation is taken for the model test.
In actual engineering, heat exchange between the frozen soil and the outside world is inevitable, and the constancy of heat source form is also difficult to ensure. Hence, when original soil is taken for temperature field model test, certain errors are inevitable between the predicted value and that in engineering. It is more reasonable to consider the presence of similarity criterion in heat exchange between the model and the environment, which requires that the original soil and the model soil have similar thermal parameters under different material temperatures, that is, when performing temperature field model test, material transformation is a must for the model test of frozen soil temperature field.

Conclusions
After considering material nonlinearity and the third boundary condition, the similarity criterion of the freezing model test and the calculation method of the model soil thermal parameters are derived. e research results provide directions for the preparation of frozen model soil, which is expected to provide theoretical basis and technical support for the design and implementation of frozen soil model tests.
e main research conclusions are drawn as follows: (1) e nonlinear freezing of soil and the thermal parameter variation with negative temperature determine that the impact of material nonlinearity on the heat conduction equation and similarity criterion must be considered when designing the frozen soil model test.
(2) e heat exchange between the soil and the external environment during the freezing process determines that study on frozen soil heat conduction must consider impact of the third boundary condition, that is, model soil with a similar relationship with the original soil is required for the temperature field model test.
(3) Considering the similarity criterion of heat exchange between the model and the environment, material similarity under different temperatures should be met, that is, the similarity criterion and similarity materials of the frozen soil model test must be determined by material transformation.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.