A method for determining the unfrozen water content of frozen soil based on the premelting theory

Abstract

The present invention relates to a method for determining the unfrozen water content of frozen soil based on the pre-melting theory, comprising the following steps: S1. Based on the pre-melting theory and considering soil properties, establish the relationship between the water film thickness and the supercooling temperature; S2. Establish an unfrozen water content model based on the arrangement of soil particles; S3. Collect soil samples and measure the particle size distribution, salt concentration and porosity ratio of the soil; S4. Characterize the particle size distribution of soil particles based on the equivalent particle size, and reduce the equivalent particle size of silty clay, loess and sand by 0.28, 0.3 and 0.36 respectively to correct the influence of the particle shape or surface roughness; calculate the impurity density on the particle surface using the maximum water film thickness and the measured salt concentration. S5. Calculate the unfrozen water content of the frozen soil. The parameters involved in the present invention are few, and the value ranges have been given, and the expression is simple.

Description

Technical Field

[0001] The present invention relates to the field of calculation of unfrozen water in frozen soil in cold regions, and particularly to a method for determining the unfrozen water content of frozen soil based on the pre-melting theory. Background Art

[0002] The content of unfrozen water affects the structure of pore ice and the bonding strength between soil particles and ice skeletons in frozen soil. The internal friction angle of the soil increases with the decrease of temperature. At negative temperatures, the strength of the soil is greater. The volume of water and ice in the soil changes due to the phase change of water and ice, and the change of the volume fraction of each component in the soil also causes changes in the thermal conductivity and volumetric heat capacity of the soil. These related variables have a significant impact on the thermal state of geological structures and geotechnical engineering structures. When predicting the thermal stability of cold region engineering, it helps to determine the heat transfer rate and freeze-thaw depth. The unfrozen water content controls the permeability of frozen soil and plays an important role in the water transport process of the soil-water system in cold environments. In saline-alkali areas, unfrozen water carries solutes and migrates from the unfrozen zone to the frozen zone due to low-temperature suction, and then ice lenses are formed at the cold end of the freezing front. At the same time, due to the recharge of groundwater, the ice lenses thicken, hindering water migration and solute transport.

[0003] Unfrozen water models can be divided into empirical models and theoretical models. Empirical models obtain data through experiments and fit curves. The physical meaning of the parameters in the models is not clear, and the parameters need to be re-calibrated in the laboratory according to different working conditions. There is a specific mathematical derivation process in theoretical models, but the model expressions are complex and difficult to apply to numerical simulation calculations. The parameters in the expressions are more and more difficult to obtain. In summary, the expression form of the models, the number of parameters, and the difficulty of obtaining them limit the application of unfrozen water models in numerical simulation and cold region engineering. Summary of the Invention

[0004] To overcome the problems existing in the above-mentioned prior art, it is particularly important to study a model and a determination method with fewer parameters, a given value range, and a simple expression.

[0005] The purpose of the present invention is to provide a method for determining the unfrozen water content of frozen soil based on the pre-melting theory with a simple model expression and accurate calculation. The calculation method of the unfrozen water content is given, and the variation law of the water film thickness in the soil with the doping level and temperature is determined. Only by measuring the common physical indexes of the soil to be measured, such as salt concentration, particle size distribution, and void ratio, the unfrozen water content of the soil at different temperatures can be calculated.

[0006] The purpose of the present invention is achieved by the following technical solutions: A method for determining the unfrozen water content of frozen soil based on the pre-melting theory, comprising the following steps:

[0007] S1. Establish the relationship between the water film thickness and the supercooling temperature.

[0008]

[0009] Wherein: △T represents the degree of supercooling, that is, the decrease value of the initial freezing temperature, that is, the deviation value from 273.15K; R g is the gas constant; T m = 273.15K; N im is the impurity concentration on the surface of soil particles; A H is the Hamaker constant.

[0010] S2. Establish a model for the relationship between the unfrozen water content f l and the water film thickness.

[0011]

[0012]

[0013] Wherein, R e represents the equivalent radius of the particles; d p represents the water film thickness between ice particles and solid particles; d gb represents the water film thickness between ice particles; SC and FCC represent two common packings, simple cubic packing (abbreviated as SC) and face-centered cubic packing (abbreviated as FCC), f p (SC) and f p (FCC) respectively represent the percentage (filling rate) of particles in the total volume under SC and FCC arrangements, which can be obtained by transforming the void ratio e (f p = 1 / (1 + e)); r represents the radius of the ice-liquid contact surface, which is determined by the Gibbs-Thomson relationship.

[0014] S3. Collect soil samples, measure the soil particle size distribution and salt concentration, and determine the volume percentage of soil particles under each particle size in the soil.

[0015] S4. Characterize the particle size distribution of soil particles based on the equivalent particle size, and reduce the equivalent particle size by 0.21 - 0.23, 0.16 - 0.28, 0.25 - 0.3, 0.36 for silt, silty clay, loess, and sand respectively to correct the influence of soil particle shape or surface roughness; calculate the impurity density on the particle surface using the maximum water film thickness and the measured salt concentration.

[0016] S5. Calculate the unfrozen water content of frozen soil.

[0017] The parameters involved in the present invention are few, and the value ranges have been given, and the expression is simple. Description of the Drawings

[0018] The specific implementation manners of the present invention will be further described in detail below with reference to the accompanying drawings.

[0019] Figure 1 It is the method flow chart of the present invention.

[0020] Figure 2 It is for characterizing the particle size distribution of soil particles based on the equivalent particle size of the present invention.

[0021] Figure 3 It is the SC model of the embodiment;

[0022] Figure 4 It is in the FCC model of the embodiment;

[0023] Figure 5a It is one of the diagrams showing the relationship between the water film thickness and the supercooling temperature;

[0024] Figure 5b It is two of the diagrams showing the relationship between the water film thickness and the supercooling temperature;

[0025] Figure 5c It is three of the diagrams showing the relationship between the water film thickness and the supercooling temperature;

[0026] Figure 6a It is the result of the unfrozen water content model of the first type of soil in the embodiment;

[0027] Figure 6b It is the result of the unfrozen water content model of the second type of soil in the embodiment;

[0028] Figure 6c It is the result of the unfrozen water content model of the third type of soil in the embodiment;

[0029] Figure 6d It is the result of the unfrozen water content model of the fourth type of soil in the embodiment;

[0030] Figure 6e It is the result of the unfrozen water content model of the fifth type of soil in the embodiment;

[0031] Figure 6f It is the result of the unfrozen water content model of the sixth type of soil in the embodiment;

[0032] Figure 6g It is the result of the unfrozen water content model of the seventh type of soil in the embodiment;

[0033] Figure 6h It is the result of the unfrozen water content model of the eighth type of soil in the embodiment. Specific implementation manners

[0034] The specific technical solutions of the present invention will be described in conjunction with the embodiments. As Figure 1The process shown includes the following steps:

[0035] S1. Establish the relationship between the water film thickness and the subcooling temperature.

[0036]

[0037] Where: △T represents the degree of supercooling, that is, the decrease in the initial freezing temperature (the deviation value from 273.15K). R g is the gas constant; T m = 273.15K; N im is the impurity concentration on the surface of soil particles; the molar density ρ of the pore solution l can be considered a constant in the calculation; q m is the latent heat of fusion per mole.

[0038] The step S1 includes:

[0039] S101. Premelting is a common phenomenon in all solids. When the system is at ultra-low temperature, the interstitial spaces in the system are completely filled with ice crystals. When the temperature rises, surface melting occurs at the interface of the ice in contact with the matrix, and grain boundary melting occurs between ice crystals. The forms of water existence are film water and interstitial water. At this time, the system can be divided into three layers according to the medium form. The first layer is the spherical solid layer, the second layer is the premelted quasi-liquid layer (water film) with a width of d, and the last layer is the ice crystal layer. If the liquid in the system contains a certain amount of impurities, the Gibbs free energy per unit area of the system can be written as:

[0040] G(T,P,N s ,N l ,N im ) = μ s (T,P)N s + μ l (T,P)N l + μ im (T,P)N im + R g T(N s lna s + N l lna l + N im lna im ) + G interface (d) (1)

[0041] Where: T, P are temperature and pressure, μ s , μ l , μ im are the chemical potentials per mole of solid, liquid, and impurity respectively, μ s , μ l , μ imThe activities of the solid, liquid, and impurity, respectively. Under normal conditions, the activity of ice is 1, N s , N l , N im The number of moles per unit area of the solid, liquid, and impurity, respectively, R g is the gas constant, G interface is the interfacial free energy related to the water film. The solid (ice) and the liquid phase (water) are in equilibrium at the freezing point, and its thermodynamic equilibrium condition is:

[0042]

[0043] At constant pressure, the chemical potential difference per mole of the solid and the liquid can be determined as:

[0044]

[0045] In the formula: S l , S s are the entropies of the liquid and the solid, respectively. At T m = 273.15 K, the latent heat of phase change q m is a constant, ΔT is T m - T. G interface has two contributions:

[0046]

[0047]

[0048] In the formula: F dis (d) is the contribution of the dispersion force to the interfacial free energy, F elec (d) is the contribution due to the capture of surface charges by ions in the liquid layer. The Hamaker constant A H is a common index for comparing the strength of van der Waals force interactions, q s is the surface charge density. For the solid and liquid phases of a single material, the van der Waals interaction is weak. When there are electrostatics forces, it dominates the contribution. ε is the relative dielectric constant of liquid water, ε0 is the vacuum permittivity, e is the electron charge, k B is the Boltzmann constant, N A is the Avogadro constant, κ is a constant, taking 7.237×10 7 m -1 / 2 ·mol -1 / 2 .

[0049] For an ideal dilute solution, -lna l = ρ i / ρ l Combining equations (2), (3), and (4) gives:

[0050]

[0051] In the formula: q m is the latent heat of fusion per mole, ρ l is the molar concentration of the liquid, N im is the number of impurities per unit area. Since the segregation coefficient of impurities in ice is very small (about 10 -6 ), it is considered that the impurity distribution in the pre-melted liquid is uniform.

[0052] S102. To obtain the influence of the ice-water interface curvature on the freezing temperature, in this embodiment, the anisotropy of the ice surface energy is ignored, and it is assumed that a spherical ice crystal with a radius of r is completely surrounded by liquid water. Considering that the area of the interface is no longer a constant, Equation (1) is rewritten as:

[0053] G(T,P,N s ,N l ,N im ) = μ s (T,P)N s + μ l (T,P)N l + μ im (T,P)N im + R g T(N s lna s + N l lna l + N im lna im ) + 4πr 2 γ sl (7)

[0055] For the case of an interface with principal radii of curvature r1 and r2, the temperature drop caused by the bending of the liquid surface at thermodynamic equilibrium can be calculated using Equation (2). Their relationship is shown in Equation (8):

[0056]

[0057] In the formula: The radius r of the ice crystal is related to N s (N s = 4πr 3 ρ s / 3), ρ s is the molar density of ice, γ sl is the interfacial free energy.

[0058] From Figures 5a to 5cIt can be seen that for a highly doped system, the thickness of the water film is almost completely affected by the impurity effect. In subsequent calculations, soil is a porous medium with a high doping level. Therefore, Equation (6) can be simplified to Equation (9):

[0059]

[0060] S2. Establish a model for the relationship between the unfrozen water content f l and the thickness of the water film.

[0061]

[0062]

[0063] In the formula, R e represents the equivalent radius of the particles; d p represents the thickness of the water film between the ice particles and the solid particles; d gb represents the thickness of the water film between the ice particles; f p is the proportion of spherical particles in the total volume, which can be obtained by transforming the void ratio e; r represents the radius of the ice-liquid contact surface, which can be determined by the Gibbs-Thomson relationship. To simplify the Gibbs-Thomson equation, the larger curvature radius at the ice-water interface can be excluded. Therefore, Equation (8) is expressed as:

[0064]

[0065] The step S2 includes:

[0066] Based on two common packing methods: simple cubic packing (abbreviated as SC) and face-centered cubic packing (abbreviated as FCC), establish the relationship between the water content and the water film on the particle surface and the ice crystal surface, the cracks at the particle contact, and the particle boundaries.

[0067] (1) The relationship between the water content and the water film on the particle surface and the ice crystal surface

[0068] Figure 3 b and Figure 4 The cross-section enclosed by the black solid line in b is the cross-section of the unit cells SC and FCC. Take the corresponding cube of the cross-section as the unit cell. n p,i is the number of soil particles per unit volume of soil with a spherical radius of R i . For the SC model and the FCC model, n p,i are 1 / (8R i 3 ) and √2 / (8R i 3 ), respectively, and f p，iThe proportion of spherical particles in the total volume has a conversion relationship with n p,i That is, f p，i = 4πR i 3 n p,i . Then the surface area of soil particles can be expressed as:

[0069]

[0070] For the SC model, the contact area between two adjacent ice crystals is equal to the difference between the square area and the soil particle area. Considering that r i is much smaller than R i , the calculation formula can be written as Equation (12a):

[0071] s gb,i (SC) ≈ (2R i + 2r i ) 2 - π(R i + 2r i ) 2 (12a)

[0072] For the FCC model, the area of each particle boundary is equal to the difference between the triangular area and the soil particle area. It can be approximately calculated by Equation (12b):

[0073]

[0074] Each soil particle in the SC model has 3 grain boundaries ( Figure 3 a), and each soil particle in the FCC model has 8 grain boundaries ( Figure 4 a). Therefore, the total surface area contributed by the grain boundaries of ice crystals per unit volume can be estimated as:

[0075] S gb,i (SC) = 3n p,i s gb,i (SC) (13a)

[0076] S gb,i (FCC) = 8n p,i s gb,i (FCC) (13b)

[0077] The unfrozen water content per unit volume of soil particles and particle (ice) boundaries can be written as:

[0078]

[0079] (2) Changes in water content caused by curvature

[0080] There are two types of regions with curvature. One exists between two adjacent spheres ( Figure 3b), which is caused by the formation of ice crystals in pores composed of multiple particles. The other is the edge between the ice crystal boundary and the sphere ( Figure 3 c), which is formed when adjacent ice crystals come into contact on the particle surface.

[0081] For the SC model, one sphere has six adjacent spheres, and for the FCC model, one sphere has 12 adjacent spheres. Therefore, each sphere of the SC packing has 3 cracks, and each sphere of the FCC packing has 6 cracks. When r i is much smaller than R i , the volume of each crack is approximately 2πr i R i 2 , so:

[0082]

[0083]

[0084] If the curvature of the adjacent spheres and the ice-water interface is ignored, the liquid water on the cross-section ( Figure 3 b) is: s edge，i = 2(r i 2 - πr i 2 / 4). For the SC and FCC models, the number of pores corresponding to the unit cell soil particles ( Figure 3 b and Figure 4 c) are 1 and 8 respectively. In the SC model, the total length of each ice crystal boundary is equal to the arc length of the contact soil particles ( Figure 3 b), 4×2πR i / 4 = 2πR i . In the FCC model, the total length of each neck is equal to the arc length of three contact soil particles ( Figure 4 c), 3×2πR i / 6 = πR i . Therefore, the total water content of the neck per unit soil volume can be calculated by the following formula.

[0085]

[0086]

[0087] The total volume fraction of the liquid composed of soil particles with radius R i can be attributed to the sum of four contributions:

[0088]

[0089]

[0090] Considering the soil composition with different particle radii, the total liquid fraction in the soil can be obtained as follows:

[0091]

[0092] where: f i is the percentage of the volume of soil particles with particle size i in the total volume of soil particles, and R i are soil particles with particle size i. The particle size distribution of soil particles is characterized by the equivalent particle size, and the soil mass is simplified into an equivalent particle size spherical packing system. The specific calculation is shown in formula (19):

[0093]

[0094] Therefore, combining formula (18) and formula (19) gives formula (20), which is used to calculate the unfrozen water content. According to the discussion in S3, the applicable void ratio of the following formula can be calibrated.

[0095]

[0096]

[0097] S3. Collect soil samples, measure the soil particle size distribution and salt concentration, and determine the volume percentage of soil particles at each particle size in the soil.

[0098] S4. Based on the equivalent particle size to characterize the particle size distribution of soil particles, for silt, silty clay, loess, and sand, the equivalent particle size is reduced by 0.21 - 0.23, 0.16 - 0.28, 0.25 - 0.3, and 0.36 respectively to correct the influence of soil particle shape or surface roughness; calculate the particle surface impurity density using the maximum water film thickness and the measured salt concentration.

[0099] The step S4 includes:

[0100] S401. As Figure 1 shown, first, the equivalent particle size is used to characterize the particle size distribution of soil particles, and the soil mass is simplified into an equivalent particle size spherical packing system. The specific calculation is shown in formula (19).

[0101] S402. For silt, silty clay, loess, and sand, the equivalent particle size is reduced by 0.21 - 0.23, 0.16 - 0.28, 0.25 - 0.3, and 0.36 respectively to correct the influence of soil particle shape or surface roughness. That is, α in S2 takes 0.21 - 0.23, 0.16 - 0.28, 0.25 - 0.3, and 0.36 for silt, silty clay, loess, and sand respectively.

[0102] S403. In the natural state, the particles in the soil do not necessarily contact each other. Therefore, there is always a spacing 2δ between the media regardless of the arrangement method. In the unit packing system, there is:

[0103]

[0104]

[0105] The void ratio of soil represents the ratio of the total volume of voids in the soil to the total volume of soil particles, and there is a conversion relationship with f p : e = 1 / f p -1. When the soil particles are in contact with each other (i.e., δ = 0), the minimum values of e sc and e fcc can be calculated as 0.35 and 0.9 respectively using Equation (21).

[0106]

[0107]

[0108] As the temperature approaches the freezing point infinitely, assuming that the ice crystal is at the center of the pore and is infinitesimal, it can be approximately considered as a point. At this time, the ice crystal has the maximum water film thickness.

[0109]

[0110]

[0111] Based on the geometric relationship of particle arrangement, the correlation between the initial volume concentration of pores and the surface impurity concentration of the water film under different soil particle packings is established, as shown in Equation (24).

[0112] N im = ηc0d0 (24)

[0113] In the formula: η is the number of electrolytic ions of unit molecular impurities. For example, for sodium chloride, η = 2, and c0 is the concentration of the solution.

[0114] S5. Calculate the unfrozen water content of frozen soil.

[0115] The step S5 can refer to Figure 1 , and the specific steps are as follows:

[0116] S501. Combine the soil parameters measured in S3 and the calculation results in S4 and substitute them into the model of the relationship between the water film thickness and the supercooling temperature established in S1.

[0117] S502. Select a suitable packing model according to the measured void ratio of the soil and establish a model of the unfrozen water content of frozen soil in combination with the relationship between the water film thickness and the supercooling temperature.

[0118] Figures 6a to 6h are the results of the unfrozen water content model for 8 types of soil.

Claims

1. A method for determining the unfrozen water content of frozen soil based on the premelting theory, characterized in that Including the following steps: S1. Establish the relationship between the water film thickness and the subcooling temperature; S2. Establish a model for the relationship between the unfrozen water content f l and the water film thickness; In the formula, SC and FCC represent two common packings. Simple cubic packing is abbreviated as simple cubic packing, written as SC for short, and face-centered cubic packing is abbreviated as face-centered cubic packing, written as FCC for short; α is the particle correction factor; R e represents the equivalent radius of the particle; f p (SC) and f p (FCC) respectively represent the percentage of particles in the total volume under the SC and FCC arrangements, i.e., the packing fraction, which is obtained by transforming through the void ratio e. f p = 1 / (1 + e); d p represents the water film thickness between the ice particles and the solid particles; r represents the radius of the ice-liquid contact surface, which is determined by the Gibbs-Thomson relationship; d gb represents the thickness of the water film between ice particles; S3. Collect soil samples, measure the soil particle size distribution and salt concentration, and determine the volume percentage of soil particles at each particle size in the soil; S4. Characterize the particle size distribution of soil particles based on the equivalent particle size, and reduce the equivalent particle size for different soils respectively; calculate the impurity concentration on the surface of soil particles using the maximum water film thickness and the measured salt concentration; S5. Calculate the unfrozen water content of frozen soil.

2. The method for determining the unfrozen water content of frozen soil based on the premelting theory according to claim 1, wherein The relationship between the water film thickness and the subcooling temperature established in S1 is: Where: ρ l is the molar density of the pore solution; q m is the latent heat of fusion per mole; △T represents the degree of supercooling, that is, the decrease in the initial freezing temperature, i.e., the deviation value from 273.15K; T m = 273.15K; R g is the gas constant; N im is the impurity concentration on the surface of soil particles; d is the water film thickness.

3. The method for determining the unfrozen water content of frozen soil based on the premelting theory according to claim 1, wherein In S4, for silt, silty clay, loess, and sand, the equivalent particle size is reduced by 0.21 - 0.23, 0.16 - 0.28, 0.25 - 0.3, and 0.36 respectively to correct the influence of the soil particle shape or surface roughness.

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