Method for establishing soil freezing characteristic curve based on thermodynamic principle

Abstract

The invention discloses a method for establishing a soil body freezing characteristic curve based on a thermodynamic principle. The method comprises the following steps: step 1, determining zoning characteristics of the soil body freezing characteristic curve; 2, determining an ice water chemical potential balance equation in the freezing process; 3, determining an ice water mole entropy equation and a phase change latent heat equation; 4, determining a soil mechanical equilibrium equation and a freezing temperature equation; 5, determining a capillary effect equation and a freezing temperature equation of capillary pores; and 6, determining a pore size distribution equation, and determining a soil body freezing characteristic curve. According to the thermodynamic principle and the microscopic physical structure of the soil body pores, a theoretical model capable of representing the change rule of the unfrozen water in the soil body freezing process with high precision is established, the quantitative relation among the freezing temperature, the content of the unfrozen water and the pore structure parameters is defined, the application range is wide, and the method is suitable for large-scale popularization and application. And theoretical support is provided for related research of the soil freezing process.

Description

Technical Field

[0001] The present invention relates to a method for establishing a soil freezing characteristic curve, and in particular to a method for establishing a soil freezing characteristic curve based on thermodynamic principles. Background Art

[0002] With the continuous advancement of infrastructure construction in permafrost regions, freeze-thaw damage is becoming increasingly prominent in cold-region projects, significantly impacting the safe operation of key projects such as roads, pipelines, and tunnels. During the freezing process, soil pore water does not completely transform into ice. Instead, a certain proportion of unfrozen water remains, controlled by temperature and in thermodynamic equilibrium with the ice phase. Its changing behavior significantly regulates important physical properties of the soil, such as thermal conductivity, strength, and deformation. Numerous studies have shown that the relationship between unfrozen water and temperature determines the hydrothermal-mechanical coupling mechanism of frozen soils and is a key factor in inducing engineering problems such as foundation frost heave, pavement cracking, tunnel frost damage, and slope slippage in cold-region projects. Therefore, accurately characterizing the temperature-dependent behavior of unfrozen water is essential for revealing the response mechanisms of frozen soils and establishing freeze-thaw stability assessment models.

[0003] During the freezing process, the soil freezing characteristic curve (SFCC) is an important parameter that describes the variation of the unfrozen water content of the soil with temperature. In previous studies, this characteristic was mainly determined by low-temperature tests to determine the unfrozen water content, and empirical models were used for fitting. This type of method relies on experimental acquisition and has problems such as long testing cycles, high costs, and poor applicability. In addition, most models lack a clear physical basis and are difficult to apply to multiple types of soils or complex working conditions. The existing technology has not yet established a general method that can predict SFCC based on thermodynamic principles and combined with microphysical structural characteristics. Therefore, it is urgent to propose a freezing characteristic curve modeling method with a clear theoretical basis, a wide range of applicability, and strong parameter accessibility to meet the needs of cold-region engineering design and numerical simulation for expressing the variation of frozen soil moisture.

[0004] This invention aims to accurately predict the freezing characteristic curve of frozen soil. Specifically, based on thermodynamic principles and the microphysical structure of soil, this invention establishes a model for accurately predicting the freezing characteristic curve. This invention will help to reveal the freezing mechanism of soil and provide a theoretical basis for numerical simulation of geotechnical engineering in cold regions. Summary of the Invention

[0005] In order to solve the above problems, the present invention discloses a method for establishing a soil freezing characteristic curve based on thermodynamic principles. Compared with other measurement methods, the method of the present invention is simple to calculate, and the physical meaning of each parameter is clear and easy to obtain.

[0006] The object of the present invention is achieved through the following technical solutions:

[0007] A method for establishing a soil freezing characteristic curve based on thermodynamic principles includes the following steps:

[0008] Step 1: Determine the partition characteristics of the soil freezing characteristic curve;

[0009] The step 1 includes the following sub-steps:

[0010] a. Determine the microscopic phase transition mechanism during soil freezing. During the initial freezing phase, water molecules aggregate locally within the pore system, forming ice nuclei. Surrounding water molecules continue to adsorb and grow, ultimately forming various types of ice crystals. The ice crystal formation process, consisting of two stages: ice nucleation and crystal growth, is the physical basis for the transformation of unfrozen water.

[0011] b. Determine the segments of the temperature-time curve during the freezing process of salt-free soil. The freezing process can be divided into a uniform cooling phase, an endothermic nucleation phase (AB), an accelerated phase transition phase (BC), a dynamic equilibrium phase (CD), and a cooling phase (DE).

[0012] c. Determine the zones of microscopic phase change during soil cooling and freezing. The microscopic behavior of the freezing process can be divided into five regions: Zone I is the pure cooling stage, where no phase change occurs; Zone II is the initial nucleation stage, where ice nuclei are initially formed; Zone III is the accelerated phase change stage, where latent heat is released; Zone IV is the dynamic equilibrium stage, where latent heat release and cooling are balanced; and Zone V is the final cooling stage, where crystallization is essentially complete.

[0013] d. Determine the segmentation of the unfrozen water content-time curve during the soil freezing process. These segments include the pre-freezing stable content A′B′, the rapid decline B′C′, the large-scale latent heat release C′D′, the slow decline D′E′, and the final stable segment. Each stage corresponds to the transformation process of different frozen water types, reflecting the freezing order of free water, capillary water, and adsorbed water in the soil.

[0014] e. Determine the zones of the soil freezing characteristic curve. In the pre-freezing phase (Zone I), the unfrozen water content approaches the total moisture content; in the rapid freezing phase (Zone II), the unfrozen water content decreases dramatically; in the steady freezing phase (Zone III), the frozen water content approaches saturation, and the unfrozen water content changes slowly. The overall curve is an "S" shape, reflecting the staged transformation of pore water freezing.

[0015] Step 2, determine the ice-water chemical potential balance equation during the freezing process;

[0016] The step 2 includes the following sub-steps:

[0017] a. During the ice-water phase transition process when the soil gradually freezes, when the equilibrium state is reached, the chemical potentials of the ice and water phases are equal. According to the principles of thermodynamics, the chemical potential of the soil water phase at a certain equilibrium moment is expressed as:

[0018]

[0019] Where μ w is the water chemical potential, J / mol; T is the thermodynamic temperature, K; p is the pressure, Pa; is the chemical potential of water at a reference state of 273.15K and one standard atmospheric pressure, J / mol; R is the ideal gas constant, 8.314 J / (K·mol); a w is the water activity.

[0020] b. According to the principles of thermodynamics, the chemical potential of soil ice phase is expressed as:

[0021]

[0022] Where μ i is the ice chemical potential, J / mol; The chemical potential of ice at a reference state of 273.15 K and one standard atmospheric pressure, J / mol; a i is ice activity.

[0023] c. Determine the chemical potential equilibrium equation for the ice-water two-phase.

[0024] μ w (T,p w ,a w )=μ i (T,p i ,a i ) (3)

[0025] d. As the soil freezes further, the chemical potential of the ice and water phases changes. When the next equilibrium state is reached, both phases produce a chemical potential change of dμ. i and dμ w , and at this time the chemical potentials of the ice and water phases are equal.

[0026] μ w (T,p w ,a w )+dμ w =μ i (T,p i ,a i )+dμ i (4)

[0027] e. Based on the principles of thermodynamics, expand the differential of water chemical potential.

[0028]

[0029] f. Based on the principles of thermodynamics, differential expansion of the chemical potential of ice.

[0030]

[0031] g. Determine the differential equilibrium equation for the chemical potential of the ice-water two-phase.

[0032]

[0033] Step 3, determine the ice-water molar entropy equation and the phase change latent heat equation;

[0034] Step 3 includes the following sub-steps:

[0035] a. For a multi-component system, the chemical potential μ of component k k for:

[0036]

[0037] Where G is Gibbs free energy, J; n k is the amount of substance of component k, mol.

[0038] b. In thermodynamic systems, Gibbs free energy is a function of the system's state parameters (temperature, pressure, and the chemical potential of each component). For any component k in the system, the rate of change of its chemical potential with temperature can be expressed through the partial differential relationship:

[0039]

[0040] Where S k is the partial molar entropy of substance k, J / (K·mol).

[0041] c. The rate of change of chemical potential with pressure is expressed by the partial differential relationship:

[0042]

[0043] Where V k is the partial molar volume of substance k, cm 3 / mol.

[0044] d. Combine equations (9) and (10) with equation (7) to obtain a variant of the differential equilibrium equation for the chemical potential of the ice-water two-phase.

[0045] -s w dT+v w dp w +RTd(lna w )=-s i dT+v i dp i +RTd(lna i ) (11)

[0046] Where, v w is the molar volume of water, 18 cm 3 / mol;v i is the molar volume of ice, 20 cm 3 / mol; p w is the pore water pressure, Pa; p i is the pore ice pressure, Pa.

[0047] e. Determine the molar entropy equation for ice water. Rearranging the terms in equation (11) yields:

[0048] v w dp w -v i dp i +RTd(lna w )-RTd(lna i )=(s w -s i )dT (12)

[0049] f. According to the principles of thermodynamics, the difference in molar entropy between ice and water can be expressed as:

[0050]

[0051] Where H w and H i are the molar enthalpies of water and ice, J / mol; L wi is the molar latent heat of phase change of water, 6012 J / mol.

[0052] g. Determine the phase change latent heat equation. Combining equations (12) and (13), we can obtain the relationship between phase change latent heat and pressure.

[0053]

[0054] Step 4: Determine the soil mechanical equilibrium equation and freezing temperature equation;

[0055] The step 4 includes the following sub-steps:

[0056] a. Determine the mechanical equilibrium equation. Soil is a typical porous medium. According to the effective stress principle, we can derive:

[0057] σ=σ s +(1-χ)p i +χp w (15)

[0058] Where T0 is the total stress, Pa; σ s is the effective stress, Pa; χ is the pore water pressure coefficient.

[0059] b. Determine the soil pore water pressure coefficient. When water and ice are evenly distributed in the soil pores during freezing, the saturated soil pore water pressure coefficient χ is the ice saturation S ri function.

[0060] χ=(1-Sri ) 1.5 (16)

[0061] Where S ri is ice saturation.

[0062] c. When the soil is saturated, the ice saturation S ri and water saturation S r The sum is 1, and formula (16) is transformed into

[0063] χ=S r 1.5 (17)

[0064] d. Combining equations (15) and (17), we obtain:

[0065]

[0066] e. Assuming that the total stress change during freezing is approximately zero and there is no water migration and ice separation, Equation (18) can be differentiated to obtain:

[0067]

[0068] f. Determine the relationship between water content, porosity and void ratio:

[0069]

[0070] Where, θ is the volumetric water content; is the porosity; e is the porosity ratio.

[0071] g. Determine the differential equation for ice-water pressure. Combining equations (19) and (20), we obtain:

[0072]

[0073] h. Combining equations (14) and (21), we can obtain:

[0074]

[0075] i. The difference between the initial temperature T0 (273.15K) and the freezing temperature T in thermodynamic terms is very small. The temperature change in the second term on the left side of equation (21) (i.e., the differential term of water activity) can be ignored, and equation (21) can be written in integral form.

[0076]

[0077] j. Integrate equation (23) to obtain:

[0078]

[0079] k. Determine the freezing temperature equation.

[0080]

[0081] Step 5, determine the capillary effect equation and the freezing temperature equation of the capillary pores;

[0082] The step 5 includes the following sub-steps:

[0083] a. Soil is a porous medium, and the shape and size of the pores will affect ice crystallization. Since the ice-water interface in the soil pores is not flat, the free energy of the ice-water interface must be considered. Ice growth in the soil pores must overcome the chemical potential of the already formed block ice. Reaching a higher chemical potential μ i .

[0084]

[0085] Where A j is the jth ice-water interface area, cm 2 ; V j is the volume of ice crystals formed, cm 3 ; γ iw The constant 3.2×10 -6 J / cm 2 .

[0086] b. Determine the chemical potential change equation. i =p i The chemical potential change at -p0 is expressed as:

[0087]

[0088] c. Combining equations (26) and (27), we obtain:

[0089]

[0090] d. Determine the capillary effect equation. The soil pores are generalized into a number of cylindrical capillaries, and the ice-water interface is a sphere. When the radius is r, the pore water in the capillary pores is the crystallization radius r. c When freezing, the thickness of the water film is very small and can be ignored.

[0091]

[0092] e. Combining equations (24) and (28) we get the radius r j The freezing temperature of the capillary pores.

[0093]

[0094] Step 6: Determine the pore size distribution equation and the soil freezing characteristic curve;

[0095] Step 6 includes the following sub-steps:

[0096] a. The relationship between pore size and particle size is nonlinear, so the pore size distribution can be converted from the particle size distribution using this nonlinear relationship. The soil is divided into n particle size units. Assuming that the solid particles are arranged in a specific spatial configuration, the pore structure is simplified to a cylindrical capillary with equal diameter. The porosity ratio of each unit remains consistent, and the pore volume fraction is equal to the corresponding particle mass fraction. During freezing, large pores freeze first, leaving water in small pores. The unfrozen water content corresponding to each freezing stage can be expressed as the cumulative distribution of the unfrozen pore volume:

[0097]

[0098] Where θ j is the unfrozen water content, cm 3 / cm 3 or %; θ s is the saturated moisture content, cm 3 / cm 3 or%, assuming the soil is saturated, θ s Equivalent to the initial moisture content θ0; ω j is the mass fraction of particles in the jth part.

[0099] b. Based on the particle size distribution, the particle size is converted into pore size. The pore volume can be calculated from the number of particles and the pore radius r. j , pore length H j The equivalent relationship between them is expressed as:

[0100]

[0101] Where V vj is the pore volume, cm 3 ; V sj is the total volume of particles, cm 3 ; R j is the particle radius, cm; e is the porosity ratio; n j is the number of equivalent spherical particles in the jth unit.

[0102] c. Determine the void ratio.

[0103]

[0104] Where, ρ s is the density of soil particles, g / cm 3 ρ b is the dry density of soil sample, g / cm 3

[0105] d. Determine the number of particles.

[0106]

[0107] e. Determine the total volume of the particles. Based on the concept of "virtual" particles in the study by Campos-Guereta et al., define the shape factor β, transform the particles into "virtual" particles with equal particle density, and make the volume and mass of the particles and "virtual particles" equal.

[0108]

[0109] Where, is the number of “virtual” equivalent spherical particles corresponding to the j-th particle size unit.

[0110] f. Determine the length of the pores formed by the packing of “virtual” particles.

[0111]

[0112] g. Determine the pore volume.

[0113]

[0114] h. Determine the pore radius.

[0115]

[0116] i. Determine the soil freezing characteristic curve. Combining equations (30), (31), and (38), we can obtain:

[0117]

[0118] Beneficial effects of the present invention:

[0119] Based on thermodynamic principles and the microphysical structure of soil pores, the present invention establishes a model capable of accurately predicting soil freezing characteristic curves. This model derives the relationship between unfrozen water content and temperature during soil freezing, and has broad applications. The model considers the actual conditions of soil freezing, and the physical meaning of each parameter is clear and easily accessible. In addition to the aforementioned objects, features, and advantages, the present invention also has further objects, features, and advantages. The present invention will be further described in detail below with reference to the accompanying drawings. BRIEF DESCRIPTION OF THE DRAWINGS

[0120] Figure 1 Schematic diagram of the temperature-time curve and the unfrozen water content-time curve during the soil freezing process in the present invention.

[0121] Figure 2 It is a schematic diagram of the soil freezing characteristic curve in the present invention.

[0122] Figure 3 Schematic diagram of the frozen soil components in the present invention.

[0123] Figure 4 Schematic diagram of the columnar capillary bundle model in the present invention.

[0124] Figure 5 Schematic diagram of the calculation process of the SFCC model in the present invention.

[0125] Figure 6 This is a comparison chart of the SFCC measured points and the predicted curve of the sample in the present invention.

[0126] Figure 7 This is a comparison chart between the measured and predicted unfrozen water content of the samples in the present invention.

[0127] Specific implementation instructions

[0128] Step 1: Determine the partition characteristics of the soil freezing characteristic curve;

[0129] The step 1 includes the following sub-steps:

[0130] a. Determine the microscopic phase transition mechanism during soil freezing. During the initial freezing phase, water molecules aggregate locally within the pore system, forming ice nuclei. Surrounding water molecules continue to adsorb and grow, ultimately forming various types of ice crystals. The ice crystal formation process, consisting of two stages: ice nucleation and crystal growth, is the physical basis for the transformation of unfrozen water.

[0131] b. Figure 1 Figure 2 shows the temperature-time curve and unfrozen water content-time curve during the soil freezing process according to the present invention. The temperature-time curve segmentation during the freezing process of salt-free soil is determined. The freezing process can be divided into a uniform cooling phase, an endothermic nucleation phase (AB), an accelerated phase transition phase (BC), a dynamic equilibrium phase (CD), and a cooling phase (DE).

[0132] c. Determine the zones of microscopic phase change during soil cooling and freezing. The microscopic behavior of the freezing process can be divided into five regions: Zone I is the pure cooling stage, where no phase change occurs; Zone II is the initial nucleation stage, where ice nuclei are initially formed; Zone III is the accelerated phase change stage, where latent heat is released; Zone IV is the dynamic equilibrium stage, where latent heat release and cooling are balanced; and Zone V is the final cooling stage, where crystallization is essentially complete.

[0133] d. Determine the segmentation of the unfrozen water content-time curve during the soil freezing process. These segments include the pre-freezing stable content A′B′, the rapid decline B′C′, the large-scale latent heat release C′D′, the slow decline D′E′, and the final stable segment. Each stage corresponds to the transformation process of different frozen water types, reflecting the freezing order of free water, capillary water, and adsorbed water in the soil.

[0134] e. Figure 2Figure 2 shows a schematic diagram of the soil freezing characteristic curve according to the present invention. It depicts the microscopic changes in the ice-water phase in the pores during the freezing process. The soil freezing characteristic curve is divided into zones. In the pre-freezing stage (Zone I), the unfrozen water content approaches the total moisture content; in the rapid freezing stage (Zone II), the unfrozen water content decreases sharply; and in the steady freezing stage (Zone III), the frozen water approaches saturation, and the unfrozen water content changes slowly. The overall curve is "S"-shaped, reflecting the staged transformation of pore water freezing.

[0135] Step 2, determine the ice-water chemical potential balance equation during the freezing process;

[0136] The step 2 includes the following sub-steps:

[0137] a. Figure 3 Figure 1 shows a schematic diagram of the frozen soil components of the present invention. During the ice-water phase transition process as the soil gradually freezes, when equilibrium is reached, the chemical potentials of the ice and water phases are equal. According to the principles of thermodynamics, the chemical potential of the soil water phase at a certain equilibrium moment is expressed as:

[0138]

[0139] Where μ w is the water chemical potential, J / mol; T is the thermodynamic temperature, K; p is the pressure, Pa; is the chemical potential of water at a reference state of 273.15K and one standard atmospheric pressure, J / mol; R is the ideal gas constant, 8.314 J / (K·mol); a w is the water activity.

[0140] b. According to the principles of thermodynamics, the chemical potential of soil ice phase is expressed as:

[0141]

[0142] Where μ i is the ice chemical potential, J / mol; The chemical potential of ice at a reference state of 273.15 K and one standard atmospheric pressure, J / mol; a i is ice activity.

[0143] c. Determine the chemical potential equilibrium equation for the ice-water two-phase.

[0144] μ w (T,p w ,a w )=μ i (T,p i ,a i ) (3)

[0145] d. As the soil freezes further, the chemical potential of the ice and water phases changes. When the next equilibrium state is reached, both phases produce a chemical potential change of dμ. i and dμ w , and at this time the chemical potentials of the ice and water phases are equal.

[0146] μ w (T,p w ,a w )+dμ w =μ i (T,p i ,a i )+dμ i (4)

[0147] e. Based on the principles of thermodynamics, expand the differential of water chemical potential.

[0148]

[0149] f. Based on the principles of thermodynamics, differential expansion of the chemical potential of ice.

[0150]

[0151] g. Determine the differential equilibrium equation for the chemical potential of the ice-water two-phase.

[0152]

[0153] Step 3, determine the ice-water molar entropy equation and the phase change latent heat equation;

[0154] Step 3 includes the following sub-steps:

[0155] a. For a multi-component system, the chemical potential μ of component k k for:

[0156]

[0157] Where G is Gibbs free energy, J; n k is the amount of substance of component k, mol.

[0158] b. In thermodynamic systems, Gibbs free energy is a function of the system's state parameters (temperature, pressure, and the chemical potential of each component). For any component k in the system, the rate of change of its chemical potential with temperature can be expressed through the partial differential relationship:

[0159]

[0160] Where S k is the partial molar entropy of substance k, J / (K·mol).

[0161] c. The rate of change of chemical potential with pressure is expressed by the partial differential relationship:

[0162]

[0163] Where V k is the partial molar volume of substance k, cm 3 / mol.

[0164] d. Combine equations (9) and (10) with equation (7) to obtain a variant of the differential equilibrium equation for the chemical potential of the ice-water two-phase.

[0165] -s w dT+v w dp w +RTd(lna w )=-s i dT+v i dp i +RTd(lna i ) (11)

[0166] Where, v w is the molar volume of water, 18 cm 3 / mol;v i is the molar volume of ice, 20 cm 3 / mol; p w is the pore water pressure, Pa; p i is the pore ice pressure, Pa.

[0167] e. Determine the molar entropy equation for ice water. Rearranging the terms in equation (11) yields:

[0168] v w dp w -v i dp i +RTd(lna w )-RTd(lna i )=(s w -s i )dT (12)

[0169] f. According to the principles of thermodynamics, the difference in molar entropy between ice and water can be expressed as:

[0170]

[0171] Where H w and H i are the molar enthalpies of water and ice, J / mol; L wi is the molar latent heat of phase change of water, 6012 J / mol.

[0172] g. Determine the phase change latent heat equation. Combining equations (12) and (13), we can obtain the relationship between phase change latent heat and pressure.

[0173]

[0174] Step 4: Determine the soil mechanical equilibrium equation and freezing temperature equation;

[0175] The step 4 includes the following sub-steps:

[0176] a. Determine the mechanical equilibrium equation. Soil is a typical porous medium. According to the effective stress principle, we can derive:

[0177] σ=σ s +(1-χ)p i +χp w (15)

[0178] Where T0 is the total stress, Pa; σ s is the effective stress, Pa; χ is the pore water pressure coefficient.

[0179] b. Determine the soil pore water pressure coefficient. When water and ice are evenly distributed in the soil pores during freezing, the saturated soil pore water pressure coefficient χ is the ice saturation S ri function.

[0180] χ=(1-S ri ) 1.5 (16)

[0181] Where S ri is ice saturation.

[0182] c. When the soil is saturated, the ice saturation S ri and water saturation S r The sum is 1, and formula (16) is transformed into

[0183] χ=S r 1.5 (17)

[0184] d. Combining equations (15) and (17), we obtain:

[0185]

[0186] e. Assuming that the total stress change during freezing is approximately zero and there is no water migration and ice separation, Equation (18) can be differentiated to obtain:

[0187]

[0188] f. Determine the relationship between water content, porosity and void ratio:

[0189]

[0190] Where, θ is the volumetric water content; is the porosity; e is the porosity ratio.

[0191] g. Determine the differential equation for ice-water pressure. Combining equations (19) and (20), we obtain:

[0192]

[0193] h. Combining equations (14) and (21), we can obtain:

[0194]

[0195] i. The difference between the initial temperature T0 (273.15K) and the freezing temperature T in thermodynamic terms is very small. The temperature change in the second term on the left side of equation (21) (i.e., the differential term of water activity) can be ignored, and equation (21) can be written in integral form.

[0196]

[0197] j. Integrate equation (23) to obtain:

[0198]

[0199] k. Determine the freezing temperature equation.

[0200]

[0201] Step 5, determine the capillary effect equation and the freezing temperature equation of the capillary pores;

[0202] The step 5 includes the following sub-steps:

[0203] a. Soil is a porous medium, and the shape and size of the pores will affect ice crystallization. Since the ice-water interface in the soil pores is not flat, the free energy of the ice-water interface must be considered. Ice growth in the soil pores must overcome the chemical potential of the already formed block ice. Reaching a higher chemical potential μ i .

[0204]

[0205] Where A j is the jth ice-water interface area, cm 2 ; V j is the volume of ice crystals formed, cm 3 ; γ iw is the ice-water interface free energy, which is taken as a constant of 3.2×10 -6 J / cm 2 .

[0206] b. Determine the chemical potential change equation. i =p i The chemical potential change at -p0 is expressed as:

[0207]

[0208] c. Combining equations (26) and (27), we obtain:

[0209]

[0210] d. Figure 4 The capillary bundle model of the present invention is shown in FIG. The capillary effect equation is determined. The soil pores are generalized into a number of cylindrical capillaries, and the ice-water interface is a sphere. When the radius is r, the pore water in the capillary pores is crystalline with a radius r. c When freezing, the thickness of the water film is very small and can be ignored.

[0211]

[0212] e. Combining equations (24) and (28) we get the radius r j The freezing temperature of the capillary pores.

[0213]

[0214] Step 6: Determine the pore size distribution equation and the soil freezing characteristic curve;

[0215] Step 6 includes the following sub-steps:

[0216] a. The relationship between pore size and particle size is nonlinear, so the pore size distribution can be converted from the particle size distribution using this nonlinear relationship. The soil is divided into n particle size units. Assuming that the solid particles are arranged in a specific spatial configuration, the pore structure is simplified to a cylindrical capillary with equal diameter. The porosity ratio of each unit remains consistent, and the pore volume fraction is equal to the corresponding particle mass fraction. During freezing, large pores freeze first, leaving water in small pores. The unfrozen water content corresponding to each freezing stage can be expressed as the cumulative distribution of the unfrozen pore volume:

[0217]

[0218] Where θ j is the unfrozen water content, cm 3 / cm 3 or %; θ s is the saturated moisture content, cm 3 / cm 3 or%, assuming the soil is saturated, θ s Equivalent to the initial moisture content θ0; ω jis the mass fraction of particles in the jth part.

[0219] b. Based on the particle size distribution, the particle size is converted into pore size. The pore volume can be calculated from the number of particles and the pore radius r. j , pore length H j The equivalent relationship between them is expressed as:

[0220]

[0221] Where V vj is the pore volume, cm 3 ; V sj is the total volume of particles, cm 3 ; R j is the particle radius, cm; e is the porosity ratio; n j is the number of equivalent spherical particles in the jth unit.

[0222] c. Determine the void ratio.

[0223]

[0224] Where, ρ s is the density of soil particles, g / cm 3 ρ b is the dry density of soil sample, g / cm 3

[0225] d. Determine the number of particles.

[0226]

[0227] e. Determine the total volume of the particles. Based on the concept of "virtual" particles in the study by Campos-Guereta et al., define the shape factor β, transform the particles into "virtual" particles with equal particle density, and make the volume and mass of the particles and "virtual particles" equal.

[0228]

[0229] Where, is the number of “virtual” equivalent spherical particles corresponding to the j-th particle size unit.

[0230] f. Determine the length of the pores formed by the packing of “virtual” particles.

[0231]

[0232] g. Determine the pore volume.

[0233]

[0234] h. Determine the pore radius.

[0235]

[0236] i. Such as Figure 5 The calculation flow diagram of establishing the SFCC model of the present invention is shown in FIG. Determine the soil freezing characteristic curve. Combining equations (30) (31) (38) yields:

[0237]

[0238] Verification of soil freezing characteristic curve:

[0239] The present invention selects the determination coefficient R 2 , root mean square error RMSE and performance deviation ratio RPD, and the actual prediction effect was tested. The freezing test data of 13 groups of typical salt-free soil samples in authoritative literature were taken. First, the measured particle size distribution data were curve fitted based on the logistic growth model (Formula 40), and the cumulative distribution percentage corresponding to the specific particle size value was calculated using this expression. Then the particle size units were divided into 2, 5, 10, 15, 20, 30, 40, 50, 60, 70, 80, 90, 100, 200, 500, 1000, 2000 μm. After calculating the freezing temperature and the corresponding unfrozen water content according to the model, the model of the present invention was evaluated by comparing with the measured data under different soil types, dry density, particle density and initial volume moisture content.

[0240]

[0241] Where w i The diameter is smaller than d i is the cumulative mass fraction (μm), a, b and c are the model fitting parameters; θ mi ,θ pi and Respectively represent the average value of the measured value, predicted value and measured value of the unfrozen water content of the soil sample, and N is the number of measured points of the unfrozen water content of the soil sample. 2 The closer (≤1) is to 1, the smaller the RMSE and the larger the RPD (RPD < 2.5 indicates poor model accuracy, and RPD > 3 indicates high model prediction accuracy), indicating that the model prediction effect is better and the prediction accuracy is higher.

[0242] Table 1 Physical properties of soil samples

[0243]

[0244] Table 2 Statistical parameters for model evaluation

[0245]

[0246] Table 2 shows that the model shows good prediction performance for all 13 soil samples including silty sand, silt and silty clay, R2 The model has the best prediction effect on silty clay, followed by silt, and finally silty sand. Figure 6 、 Figure 7 Together with Table 2, it can be seen that the constructed SFCC model has excellent performance in accurately predicting the SFCC of frozen soil and is effective in quantifying the moisture characteristics of frozen soil.

[0247] like Figure 6 As shown, the SFCC curves calculated by the theoretical model closely match the measured unfrozen water and temperature data in terms of overall trends and key transition zones. These curves accurately reflect the typical three-stage pattern of unfrozen water content changing with temperature during soil freezing, including a pre-freezing plateau, a rapid freezing phase, and a gentle tail phase. Furthermore, the model's predictions agree well with the measured data, demonstrating a high degree of consistency, validating the model's reliability in predicting SFCC.

[0248] like Figure 7 As shown in the figure, all the measured and predicted comparison data points are basically around the 1:1 line and are within the range of ±10%, indicating that the prediction error of the model is small, further highlighting the applicability of the model to a variety of soil samples.

[0249] In summary, the prediction model of the soil freezing characteristic curve obtained by the present invention can effectively predict the changing relationship between the unfrozen water content and temperature for soils with different pore structures and different soil types. It is feasible to apply the method for establishing the soil freezing characteristic curve of the present invention to the actual soil freezing process.

Claims

1. A method for establishing a soil freezing characteristic curve based on thermodynamic principles, characterized in that: For the following steps: Step 1: Determine the partition characteristics of the soil freezing characteristic curve; Step 2, determine the ice-water chemical potential balance equation during the freezing process; Step 3, determine the ice-water molar entropy equation and the phase change latent heat equation; Step 4: Determine the soil mechanical equilibrium equation and freezing temperature equation; Step 5: Determine the capillary effect equation and the freezing temperature equation of the capillary pores. Step 6: Determine the pore size distribution equation and the soil freezing characteristic curve.

2. The method for establishing a soil freezing characteristic curve based on thermodynamic principles according to claim 1, characterized in that: The step 1 comprises: a. Determine the microscopic phase transition mechanism during soil freezing. During the initial freezing phase, water molecules aggregate locally within the pore system, forming ice nuclei. Surrounding water molecules continue to adsorb and grow, ultimately forming various types of ice crystals. The ice crystal formation process, consisting of two stages: ice nucleation and crystal growth, is the physical basis for the transformation of unfrozen water. b. Determine the segments of the temperature-time curve during the freezing process of salt-free soil. The freezing process can be divided into a uniform cooling phase, an endothermic nucleation phase (AB), an accelerated phase transition phase (BC), a dynamic equilibrium phase (CD), and a cooling phase (DE). c. Determine the zones of microscopic phase change during soil cooling and freezing. The microscopic behavior of the freezing process can be divided into five regions: Zone I is the pure cooling stage, where no phase change occurs; Zone II is the initial nucleation stage, where ice nuclei are initially formed; Zone III is the accelerated phase change stage, where latent heat is released; Zone IV is the dynamic equilibrium stage, where latent heat release and cooling are balanced; and Zone V is the final cooling stage, where crystallization is essentially complete. d. Determine the segmentation of the unfrozen water content-time curve during the soil freezing process. These segments include the pre-freezing stable content A′B′, the rapid decline B′C′, the large-scale latent heat release C′D′, the slow decline D′E′, and the final stable segment. Each stage corresponds to the transformation process of different frozen water types, reflecting the freezing order of free water, capillary water, and adsorbed water in the soil. e. Determine the zones of the soil freezing characteristic curve. In the pre-freezing phase (Zone I), the unfrozen water content approaches the total moisture content; in the rapid freezing phase (Zone II), the unfrozen water content decreases dramatically; in the steady freezing phase (Zone III), the frozen water content approaches saturation, and the unfrozen water content changes slowly. The overall curve exhibits an S-shaped pattern, reflecting the staged transition of pore water freezing.

3. The method for establishing a soil freezing characteristic curve based on thermodynamic principles according to claim 1, characterized in that: The step 2 includes: a. During the ice-water phase transition process when the soil gradually freezes, when the equilibrium state is reached, the chemical potentials of the ice and water phases are equal. According to the principles of thermodynamics, the chemical potential of soil water phase at a certain equilibrium moment is expressed as: Where μ w is the water chemical potential, J / mol; T is the thermodynamic temperature, K; p is the pressure, Pa; is the chemical potential of water at a reference state of 273.15K and one standard atmospheric pressure, J / mol; R is the ideal gas constant, 8.314 J / (K·mol); a w is the water activity. b. According to the principles of thermodynamics, the chemical potential of soil ice phase is expressed as: Where μ i is the ice chemical potential, J / mol; The chemical potential of ice at a reference state of 273.15 K and one standard atmospheric pressure, J / mol; a i is ice activity. c. Determine the chemical potential equilibrium equation for the ice-water two-phase. m w (T,p w ,a w )=μ i (T,p i ,a i ) (3) d. As the soil freezes further, the chemical potential of the ice and water phases changes. When the next equilibrium state is reached, both phases produce a chemical potential change of dμ. i and dμ w , and at this time the chemical potentials of the ice and water phases are equal. μ w (T,p w ,has w )+dμ w =μ i (T,p i ,has i )+dμ i (4) e. Based on the principles of thermodynamics, expand the differential of water chemical potential. f. Based on the principles of thermodynamics, differential expansion of the chemical potential of ice. g. Determine the differential equilibrium equation for the chemical potential of the ice-water two-phase.

4. The method for high-precision quantification of soil phase transition temperature based on nucleation and pre-melting principles according to claim 1, wherein step 3 comprises: a. For a multi-component system, the chemical potential μ of component k k for: Where G is Gibbs free energy, J; n k is the amount of substance of component k, mol. b. In thermodynamic systems, Gibbs free energy is a function of the system's state parameters (temperature, pressure, and the chemical potential of each component). For any component k in the system, the rate of change of its chemical potential with temperature can be expressed through the partial differential relationship: Where S k is the partial molar entropy of substance k, J / (K·mol). c. The rate of change of chemical potential with pressure is expressed by the partial differential relationship: Where V k is the partial molar volume of substance k, cm 3 / mol. d. Combine equations (9) and (10) with equation (7) to obtain a variant of the differential equilibrium equation for the chemical potential of the ice-water two-phase. -s w dT+v w dp w +RTd(lna w )=-s i dT+v i dp i +RTd(lna i ) (11) Where, v w is the molar volume of water, 18 cm 3 / mol;v i is the molar volume of ice, 20 cm 3 / mol; p w is the pore water pressure, Pa; p i is the pore ice pressure, Pa. e. Determine the molar entropy equation for ice water. Rearranging the terms in equation (11) yields: v w dp w -v i dp i +RTd(lna w )-RTd(lna i )=(s w -s i )dT (12) f. According to the principles of thermodynamics, the difference in molar entropy between ice and water can be expressed as: Where H w and H i are the molar enthalpies of water and ice, J / mol; L wi is the molar latent heat of phase change of water, 6012 J / mol. g. Determine the phase change latent heat equation. Combining equations (12) and (13), we can obtain the relationship between phase change latent heat and pressure.

5. The method for establishing a soil freezing characteristic curve based on thermodynamic principles according to claim 1, characterized in that: The step 4 comprises: a. Determine the mechanical equilibrium equation. Soil is a typical porous medium. According to the effective stress principle, we can derive: s = s s +(1-x)p i +xp w (15) Where T0 is the total stress, Pa; σ s is the effective stress, Pa; χ is the pore water pressure coefficient. b. Determine the soil pore water pressure coefficient. When water and ice are evenly distributed in the soil pores during freezing, the saturated soil pore water pressure coefficient χ is the ice saturation S ri function. x=(1-S ri ) 1.5 (16) Where S ri is ice saturation. c. When the soil is saturated, the ice saturation S ri and water saturation S r The sum is 1, and equation (16) is transformed into χ = S r 1.5 (17) d. Combining equations (15) and (17), we obtain: e. Assuming that the total stress change during freezing is approximately zero and there is no water migration and ice separation, Equation (18) can be differentiated to obtain: f. Determine the relationship between water content, porosity and void ratio: Where, θ is the volumetric water content; is the porosity; e is the porosity ratio. g. Determine the differential equation for ice-water pressure. Combining equations (19) and (20), we obtain: h. Combining equations (14) and (21), we can obtain: i. The difference between the initial temperature T0 (273.15K) and the freezing temperature T in thermodynamic terms is very small. The temperature change in the second term on the left side of equation (21) (i.e., the differential term of water activity) can be ignored, and equation (21) can be written in integral form. j. Integrate equation (23) to obtain: k. Determine the freezing temperature equation.

6. The method for establishing a soil freezing characteristic curve based on thermodynamic principles according to claim 1, characterized in that: The step 5 comprises: a. Soil is a porous medium, and the shape and size of the pores will affect ice crystallization. Since the ice-water interface in the soil pores is not flat, the free energy of the ice-water interface must be considered. Ice growth in the soil pores must overcome the chemical potential of the already formed block ice. Reaching a higher chemical potential μ i . Where A j is the jth ice-water interface area, cm 2 ; V j is the volume of ice crystals formed, cm 3 ; γ iw is the ice-water interface free energy, which is taken as a constant of 3.2×10 -6 J / cm 2 . b. Determine the chemical potential change equation. i =p i The chemical potential change at -p0 is expressed as: c. Combining equations (26) and (27), we obtain: d. Determine the capillary effect equation. The soil pores are generalized into a number of cylindrical capillaries, and the ice-water interface is a sphere. When the radius is r, the pore water in the capillary pores is the crystallization radius r. c When freezing, the thickness of the water film is very small and can be ignored. e. Combining equations (24) and (28) we get the radius r j The freezing temperature of the capillary pores.

7. The method for establishing a soil freezing characteristic curve based on thermodynamic principles according to claim 1, characterized in that: The step 6 comprises: a. The relationship between pore size and particle size is nonlinear, so the pore size distribution can be converted from the particle size distribution using this nonlinear relationship. The soil is divided into n particle size units. Assuming that the solid particles are arranged in a specific spatial configuration, the pore structure is simplified to a cylindrical capillary with equal diameter. The porosity ratio of each unit remains consistent, and the pore volume fraction is equal to the corresponding particle mass fraction. During freezing, large pores freeze first, leaving water in small pores. The unfrozen water content corresponding to each freezing stage can be expressed as the cumulative distribution of the unfrozen pore volume: Where θ j is the unfrozen water content, cm 3 / cm 3 or %; θ s is the saturated moisture content, cm 3 / cm 3 or%, assuming the soil is saturated, θ s Equivalent to the initial moisture content θ0; ω j is the mass fraction of particles in the jth part. b. Based on the particle size distribution, the particle size is converted into pore size. The pore volume can be calculated from the number of particles and the pore radius r. j , pore length H j The equivalent relationship between them is expressed as: Where V vj is the pore volume, cm 3 ; V sj is the total volume of particles, cm 3 ; R j is the particle radius, cm; e is the porosity ratio; n j is the number of equivalent spherical particles in the jth unit. c. Determine the void ratio. Where, ρ s is the density of soil particles, g / cm 3 ρ b is the dry density of soil sample, g / cm 3 d. Determine the number of particles. e. Determine the total volume of the particles. Based on the concept of "virtual" particles proposed by Campos-Guereta et al., define the shape factor β, transform the particles into "virtual" particles with equal particle density, and make the volume and mass of the particles and "virtual particles" equal. Where, is the number of "virtual" equivalent spherical particles corresponding to the j-th particle size unit. f. Determine the length of pores formed by the packing of "virtual" particles. g. Determine the pore volume. h. Determine the pore radius. i. Determine the soil freezing characteristic curve. Combining equations (30), (31), and (38), we can obtain:

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