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Article The Modelling of Process in Saturated Based on the Thermal-Hydro-Mechanical Multi-Physics Field Coupling Theory

Dawei Lei 1,2, Yugui Yang 1,2,* , Chengzheng Cai 1,2, Yong Chen 3 and Songhe Wang 4 1 State Key Laboratory for Geomechanics and Deep Underground Engineering, China University of and Technology, Xuzhou 221008, China; [email protected] (D.L.); [email protected] (C.C.) 2 School of Mechanics and Civil Engineering, China University of Mining and Technology, Xuzhou 221116, China 3 State Key Laboratory of Coal and Safe Mining, China University of Mining and Technology, Xuzhou 221116, China; [email protected] 4 Institute of , Xi’an University of Technology, Xi’an 710048, China; [email protected] * Correspondence: [email protected]

 Received: 2 September 2020; Accepted: 22 September 2020; Published: 25 September 2020 

Abstract: The freezing process of saturated soil is studied under the condition of water replenishment. The process of soil freezing was simulated based on the theory of the and mass conservation equations and the equation of mechanical equilibrium. The accuracy of the model was verified by comparison with the experimental results of soil freezing. One-side freezing of a saturated 10-cm-high soil column in an open system with different parameters was simulated, and the effects of the initial , , and of soil particles on soil heave, freezing depth, and lenses distribution during soil freezing were explored. During the freezing process, water migrates from the warm end to the frozen fringe under the actions of the temperature gradient and pore pressure. During the initial period of freezing, the frozen front quickly moves downward, the freezing depth is about 5 cm after freezing for 30 h, and the final freezing depth remains about 6 cm. The freezing depth of the soil column is affected by soil and thermal conductivity, but the final freezing depth mainly depends on the temperatures of the top and lower surfaces. The frost heave is mainly related to the amount of water migration. The relationship between the amount of frost heave and the hydraulic conductivity is positively correlated, and the thickness of the stable is greatly affected by the hydraulic conductivity. With the increase of the hydraulic conductivity and initial void ratio, the formation of ice lenses in the soil become easier. With the increase of the initial void ratio and thermal conductivity of soil particles, the frost heave of the soil column also increases. With high-thermal-conductivity soil, the formation of ice lenses become difficult.

Keywords: frost heave; ice lens; THM coupling process; ice pressure

1. Introduction in China is mainly distributed in high latitudes. With the change of temperature, water and ice in the pores of frozen soil can transform into each other. Frozen soil is a dynamic system related to temperature, and the dynamic system involves transfer, material exchange, and soil deformation. Therefore, the structure, composition, and mechanical properties of frozen are more complicated than unfrozen soil [1,2]. The most important problem in permafrost engineering is frost heave, especially when the of soil is high. This has caused great damage to the

Water 2020, 12, 2684; doi:10.3390/w12102684 www.mdpi.com/journal/water Water 2020, 12, 2684 2 of 19 infrastructure in regions. For example, frost heave can cause railroad undulation, the rupture of oil pipeline, and instability of the electric tower [3–5]. Therefore, exploring the mechanism of the freezing process in soils is of great significance to permafrost engineering. Many scholars have carried out a lot of studies on the soil freezing process by experimental and numerical simulation methods. Taber [6] studied the soil freezing process by an experimental method to explain the frost heave of soil and the formation of ice lenses. Later, Everett [7] established the hydrothermal model of frozen soils through capillary theory, which is the first frost theory. This theory was used to explain the frost heave phenomenon and estimates frost heave. However, Everett did not verify the model through experimental data, and the theory cannot explain the formation of discontinuous ice lenses. When the soil freezes, the heat of soil migrates from the high-temperature region to thenlow-temperature region, and the content of unfrozen water in frozen soils drops sharply [8]. The soil freezing is a dynamic coupling process under the influence of the temperature gradient and hydraulic gradient [9]. Harlan [8] established a hydrothermal coupling model for partially frozen soil based on the analogy of the mechanism of water transport in partially frozen soils and unsaturated soils. However, this model cannot explain the formation of ice lenses. Kay and Groenevelt [10] proposed the appropriate energy equation and combined it with the Clapeyron equation to find a better transport coefficient. Accordingly, Kay and Groenevelt [10] proposed a theory of water transfer under a temperature gradient and under a water pressure gradient. These early theories about soil freezing focused on the temperature field and the seepage field, and ignored the role of the stress field during the freezing process. Additionally, the formation of the ice lens has not been explained . Assuming that there is a region of low moisture content and low moisture conductivity between the bottom of the ice lens and the frozen front, Miller [11] established the second frost theory. With a large amount of studies about coarse-textured soil, Gilpin [12] developed a model of frost heave and ice lenses. The model assumed that the latency of pore water release at the isotherms where most of the pore water freezes or an ice lens is forming, and the water in the unfrozen water film is driven entirely by normal driven pressure. According to the experimental results, Konrad and Morgenstern [13] presented that seepage velocity is continuous across the region of the fringe under steady-state conditions, and used a model that distinguishes the passive and active regions to explain the frost heave characteristics of fine-textured soils. The model confirms the correctness of the Clausius–Clapeyron equation at the bottom of ice lenses. Nishimura et al. [14] used a thermo-hydro-mechanical finite element formulation to simulate the pipeline heave. Tan et al. [15] established a thermo-hydro coupling model and focused on the evolution of the temperature field and simulated the freezing process of the soil column without water replenishment. Wu et al. [16] believes that water activity is the inducement of change between water and ice. According to the coupling relationship between the formation and development of ice and the flow of water and heat, Wu et al. [16] established a kinetic model of ice development. Song et al. [17] and Xu et al. [18] used the lattice Boltzmann method (LBM) model to predict the distribution of water content and the unfrozen water content and hydraulic conductivity during freezing, respectively. The theory of soil freezing has been relatively well developed, and some models have also considered the ice lenses. However, most models do not consider the effects of soil deformation on heat transfer and water migration, and the effect of the horizontal displacement limit, which caused the model to describe the actual physical process of soil freezing inaccurately. Soil freezing is a thermo-hydro-mechanical coupling process caused by a temperature gradient, and the coupling relationship of soil freezing can be shown in Figure1. Due to the adsorption of the soil particles, the water film between the particles is kept in a liquid state when the temperature is lower than the freezing temperature. When the temperature continues to decrease, the water film begins to freeze, and with the ice wedging between the particles, the soil particles are separated. Under the appropriate condition of water replenishment and temperature, the ice layer between the soil particles becomes thicker to form the ice lens [19]. It has been widely accepted by scholars that phase change of the pore water and the formation of ice lens occur in the so-called ‘frozen fringe’ and the stress criterion is considered as the criteria for the formation of a new ice lens [12,20–22]. Thomas et al. [21] thought Water 2020, 12, x FOR PEER REVIEW 3 of 19 et al. [21] thought that a new ice lens forms when the pore pressure exceeds the sum of the separation strength and the overburden stress of the freezing soil. Cao and Liu [23] found that if the pore pressure criterion is used, the distribution of ice pressure will be discontinuous, and there are errors between the simulation results and experimental data. The simulation results using the ice pressure criterion are closer to the experimental data. In this study, a new thermo-hydro-mechanical coupling model is established to describe the freezing process in soil. The coupling relationship between the physical fields is shown in Figure 1. The effects of the initial void ratio, hydraulic conductivity, and thermal conductivity of soil particles on soil freezing process are explored. A new condition for the formation of ice lenses is proposed. In thisWater model,2020, 12 the, 2684 following items are assumed: 3 of 19 (1) The pores of soil are filled with water and ice during freezing, and the soil is regarded as thatisotropic a new ice and lens elastic forms mediums. when the pore pressure exceeds the sum of the separation strength and the (2)overburden The pore stress ice is of immobile the freezing relative soil. to Cao the and soil Liu . [23] found that if the pore pressure criterion is used, (3)the The distribution soil particles, of ice pore pressure ice, and will water be discontinuous, are incompressible. and there are errors between the simulation (4)results A weighting and experimental algorithm data. is used The to simulation convert the results soil usingof the thethree-phase ice pressure system criterion into area single-phase closer to the experimentalsystem for data. the calculation of heat transfer.

FigureFigure 1. 1. CouplingCoupling relationship relationship of of physical physical fields fields in in soil soil freezing. freezing.

2. TheoreticalIn this study, Model a of new Soil thermo-hydro-mechanical Freezing coupling model is established to describe the freezing process in soil. The coupling relationship between the physical fields is shown in Figure1. According to the theory of adsorption, the pore water is not frozen completely when the The effects of the initial void ratio, hydraulic conductivity, and thermal conductivity of soil particles on temperature is below the freezing temperature, and the unfrozen water adsorbs on the surface of the soil freezing process are explored. A new condition for the formation of ice lenses is proposed. In this soil particles, and the thickness of the water film mainly depends on the temperature [24]. This model, the following items are assumed: process can be illustrated in Figure 2. (1) The pores of soil are filled with water and ice during freezing, and the soil is regarded as isotropic and elastic mediums. (2) The pore ice is immobile relative to the soil skeleton. (3) The soil particles, pore ice, and water are incompressible. (4) A weighting algorithm is used to convert the soil of the three-phase system into a single-phase system for the calculation of heat transfer.

2. Theoretical Model of Soil Freezing

According to the theory of adsorption, the pore water is not frozen completely when the temperature is below the freezing temperature, and the unfrozen water adsorbs on the surface of the soil particles, and the thickness of the water film mainly depends on the temperature [24]. This process can be illustrated in Figure2. Water 2020, 12, 2684 4 of 19 Water 2020, 12, x FOR PEER REVIEW 4 of 19

FigureFigure 2. Microscopic schematic schematic of of soil soil freezing. freezing. 2.1. Heat Transfer 2.1. Heat Transfer InIn the the process process ofof soilsoil freezing,freezing, heat transfer transfer and and heat heat convection convection occur occur according according to tothe the thermodynamicsthermodynamics theory. theory. The The evolution evolution of of the the temperature temperature fieldfield conformsconforms to the law of of conservation conservation of energy.of energy. The The heat heat conduction conduction equation equation for for soil soil freezing freezing can can be be expressed expressed as: as:

𝜕(Φ𝑑𝑉∂(ΦdV) ) +∇+(−𝜆∇𝑇( λ )T𝑑𝑉)dV + ρ +𝜌 𝑐c∇𝑇T →v ∙⃗𝑑𝑉 dV 𝑣 = 0 = 0 (1) (1) 𝜕𝑡 ∂t ∇ − ∇ l l∇ · where 𝑡 is time; T is temperature; 𝑐 and 𝜌 are the specific heat capacities and the of pore where t is time; T is temperature; cl and ρl are the specific heat capacities and the density of pore water, respectively;water, respectively;is the Hamiltonian∇ is the Hamiltonian operator operator and means and means= ∂ + ∇ ∂= (in+ two dimensions); (in two dimensions);Φ represents Φ ∇ ∇ ∂x ∂y therepresents heat content the heat of soil content per unit of soil volume; per unitd stands volume; for 𝑑 the stands differential for the operator;differentialdV operator;refers to 𝑑𝑉 the refers volume elementto the volume of the soils; elementλ refers of the the soils; thermal 𝜆 refers conductivity; the thermal and conductivity;→v is the seepage and 𝑣⃗ velocity is the seepage of pore velocity water: of pore water: Φ = H(T Tr) LnSiρi (2) Φ = 𝐻(𝑇−𝑇− )−−𝐿𝑛𝑆𝜌 (2) wherewhereL 𝐿is is the the latent heat of of waterwater phase change; change; 𝑛n representsrepresents the the porosity porosity of ofthe the soil; soil; 𝑆 Sisi theis the ice ice volumevolume percentage percentage in in pores; poresTr; is𝑇 the is referencethe reference temperature; temperature;ρi is the𝜌 is density the density of ice; Hof isice; the 𝐻 volumetric is the heatvolumetric capacity heat of soil, capacity which of cansoil, be which expressed can be as:expressed as: 𝐻 = (1−𝑛)𝜌 𝑐 + (1−𝑆)𝑛𝜌 𝑐 +𝑆𝑛𝜌 𝑐 (3) H = (1 n)ρ scs + (1 S )nρ c + S nρ c (3) − − i l l i i i where 𝜌 is the density of the soil particles, and 𝑐 and 𝑐 are the specific heat capacities of the soil whereparticlesρs is and the ice, density respectively. of the soilTice particles,et al. [25] es andtablishedcs and ac relationshipi are the specific between heat the capacities ice percentage of the 𝑆 soil particlesand temperature and ice, respectively. based on the Ticeresults et al.of experiments [25] established as follows: a relationship between the ice percentage Si and temperature based on the results of experiments as follows: 1−1−(𝑇−𝑇) 𝑇≤𝑇 𝑆 = (4)  0𝑇>𝑇α 1 [1 (T T0)] T T0 S =  − − − ≤ (4) i   0 T > T0 Water 2020, 12, 2684 5 of 19

where T0 is the freezing point temperature (273.15 K) of water in pores ignoring the effect of the sorption isotherm [21,22,25–27], and here α is a parameter depending on the pore size of soil:

dV = (1 + e)dVs (5) where e and dVs are the porosity ratio and the soil particle volume of soil element, respectively:

(1 n) (1 Si)n nSi λ = λs − λl − λi (6)

Substituting Equations (2), (3), and (5) into Equation (1) and assuming that the particles are incompressible, the equations of heat transfer can be written as:

∂T 1 ∂e C + C + ( λ T) + ρ c T →v = 0 11 ∂t 12 1 + e ∂t ∇ − ∇ l l∇ · " # e ∂Si e ∂Si C = H + (T Tr)(ρ c ρ c ) Lρ (7) 11 1 + e − i i − l l ∂T − 1 + e i ∂T

C = [(1 S )ρ c + S ρ c ](T Tr) LS ρ 12 − i l l i i i − − i i 2.2. Mechanical Equilibrium During the freezing process of the soil, the skeleton of the soil is filled with pore ice and water. The total stress of frozen soil is equal the sum of the skeleton stress and pore pressure. According to Biot’s effective stress law [28], the equilibrium equation can be expressed as:

e σij,j + α∗P,i + fi = 0 (8)

e where σij is the effective stress tensor; α∗ is the Biot coefficient; P is the pore pressure; and fi is the body force in the ith direction: K α∗ = 1 (9) − Ks where K is the bulk modulus of the soil skeleton, and Ks is the bulk modulus of soil particles. The total stress σ11 in the vertical direction can be expressed as:

Z h σ11 = Pob + γdx∗ (10) x where Pob is the of the soil column; h is the height of the soil column; and γ is the unit weight. According to the composition of frozen soil, the unit weight γ can be calculated by the following equation: γ = g[(1 n)ρs + nS ρ + n(1 S )ρ ], (11) − i i − i l where g is the gravitational acceleration. 0 Therefore, the initial total stress σ11 can be written as:

Z h 0 = + σ11 Pob γ0dx∗ (12) x and the initial unit weight γ0 can be given as:

γ = g[(1 n )ρs + n ρ ] (13) 0 − 0 0 l where n is the initial porosity, n = e0 , and e is the initial porosity ratio. 0 0 1+e0 0 Water 2020, 12, 2684 6 of 19

Based on the principle of effective stress, the relationship of the total stress σ11, the effective stress e σ11, and pore pressure P can be written as:

Water 2020, 12, x FOR PEER REVIEW e 6 of 19 σ11 = σ11 + P (14)

For isotropic porous media materials,𝜎 the = relationship 𝜎 +𝑃 between the effective stress tensor and(14) the strain tensor can be expressed as: For isotropic porous media materials, the relationship between the tensor and   the strain tensor can be expressed as:e 0 σij = λ∗εkkδij + 2µ∗εij + σij P0 (15) ∗ ∗ − 𝜎 = 𝜆 𝜀𝛿 +2𝜇 𝜀 +(𝜎 −𝑃) (15) where λ∗∗ and µ∗ are∗ the Lame constant; εkk is the volume strain and εkk = ε11 + ε22 + ε33; ε11, ε22, ε33 where 𝜆 and 𝜇 are the Lame constant; 𝜀 is0 the volume strain and 𝜀 = 𝜀 +𝜀 +𝜀 ; are the normal strain; δij is the Kronecker symbol; σ is the initial total stress tensor; and P0 is the initial 𝜀 ,𝜀 ,𝜀 are the normal strain; 𝛿 is the Kroneckerij symbol; 𝜎 is the initial total stress tensor; and pore pressure. 𝑃 is the initial pore pressure. The freezing process of the soil layer is simulated, and the schematic diagram of the soil column The freezing process of the soil layer is simulated, and the schematic diagram of the soil column model is shown in Figure3. When shallow soil freezes in the , frost heave does model is shown in Figure 3. When shallow soil freezes in the natural environment, frost heave does not occur in the horizontal direction. From Figure3, it can be seen that the horizontal strains of the not occur in the horizontal direction. From Figure 3, it can be seen that the horizontal strains of the model meet the condition ε22 = ε33 = 0 and εkk = ε11. Equation (15) can be written as: model meet the condition 𝜀 = 𝜀 = 0 and 𝜀 = 𝜀. Equation (15) can be written as: e 0 σ = λ∗ ∗ε + 2µ∗∗ε + σ (16) 𝜎 11= 𝜆 𝜀11+2𝜇 𝜀11 +𝜎11 (16)

Figure 3. Schematic diagram of the soil column mechanical model. Figure 3. Schematic diagram of the soil column mechanical model. Soil particles are assumed to be undeformed, and the frost heave is derived from the expansion Soil particles are assumed to be undeformed, and the frost heave is derived from the expansion of the pores. Before the deformation, the void ratio was e0; the volume of soil particles is A; and the of the pores. Before the deformation, the void ratio was 𝑒; the volume of soil particles is A; and the volume of pores is (1 − A). After the deformation, the void ratio is 𝑒; the volume of soil particles is A; and the volume of pores is Q. The following relationships can be obtained: 1−𝐴 𝑒 = (17) 𝐴 𝑄 𝑒 = (18) 𝐴 The volumetric strain can be expressed as:

Water 2020, 12, 2684 7 of 19 volume of pores is (1 A). After the deformation, the void ratio is e; the volume of soil particles is A; − and the volume of pores is Q. The following relationships can be obtained:

1 A e = − (17) 0 A Q e = (18) A The volumetric strain can be expressed as:

Q (1 A) ε = ε = − − (19) kk 11 1

Substituting Equations (17) and (18) into Equation (19), the ε11 can be written as:

e e0 εkk = ε11 = − (20) 1 + e0

P = σ σe (21) 11 − 11 Substituting Equations (10), (12), (16), and (20) into Equation (21), the pore pressure equation can be rewritten as: Z h E(1 ν)(e e0) P = (γ γ0)dx∗ + − − (22) − (1 + e )(1 + ν)(1 2ν) x 0 − where E is the compressive modulus and ν is Poisson’s ratio. From the soil damage criterion, we can get the following formula:

P σ σsep (23) ≤ 11 − where σsep is the separation strength of the freezing soil (negative value for tensile strength). Substituting Equation (22) into Equation (23) and assuming that when e is equal to en, Equation (23) takes the equal sign, the following relationships can be obtained:

Z h E(1 ν)(en e0) (γ γ0)dx∗ + − − = σ11 σsep (24) − (1 + e )(1 + ν)(1 2ν) − x 0 − R ( + h )( + )( )( + ) Pob x γ0dx∗ σsep 1 ν 1 2ν 1 e0 en = − − + e0 (25) E(1 ν) − Therefore, the Equation (22) can be expressed as:  R h E(1 ν)(e e0)  (γ γ0)dx∗ + − − e < en P =  x − (1+e0)(1+ν)(1 2ν) (26)  R h −  P + γdx σsep e en ob x ∗ − ≥ Frost heave can be calculated by integrating the strain in height and can be written as:

Z h Z h e e0 Hi = ε11dx = − dx (27) 0 0 1 + e0 Water 2020, 12, 2684 8 of 19

2.3. Water Migration The pore ice is assumed to be fixed during the freezing process, and the pore water is movable relative to the skeleton of soil. According to the law of mass conservation, the governing equation of seepage can be expressed as: ∂(MadV) + ρ →v dV = 0 (28) ∂t l∇ where Ma is the mass of pore water and ice in the unit volume of soils; →v is the velocity relative to the soil particles: Ma = n[S ρ + (1 S )ρ ] (29) i i − i l k →v = (ϕ + γlz) (30) −γl ∇ where k is the hydraulic conductivity; γl is the weight of unit water; ϕ is the seepage pressure that consists of the pore pressure P and the cryogenic suction PT (due to the ice/water interface tension) [29], as ϕ = (P + PT):  β k [1 (T T0)] T T0 =  − − ≤ (31)  k0  1 T > T0 where k0 is the hydraulic conductivity when the soil is unfrozen, T0 is the freezing point temperature (273.15 K) of water in pore ignoring the effect of the water vapor sorption isotherm [21,22,26,27], and β is a parameter depending on the size and structure of pore. During the soil freezing process, the experimental results showed that the temperature gradient will cause the pore water to migrate from the high-temperature region to the low-temperature region [9]. The Clapeyron equation is presented to describe the relationship between ice pressure, water pressure, and temperature [10]: P P T l + i = L ln (32) ρl ρi T0 where Pl and Pi are the water pressure and ice pressure, respectively. The cryogenic suction PT can be expressed as [12]:

T T0 PT = Pl Pi ρlL − (33) − ≈ T0

Substituting Equations (29), (30), and (33) into Equation (28), the governing equation of water transfer can be written as: ! ! ∂T 1 ∂e k kρlL C21 + C22 + ρl P + ρl T + ρl ( k z) = 0 ∂t 1 + e ∂t ∇ −γl ∇ ∇ −γlT0 ∇ ∇ − ∇

∂S C = n(ρ ρ ) i (34) 21 i − l ∂T C = S ρ + (1 S )ρ . 22 i i − i l 2.4. According to Miller’s second theory of frost heave, there is a frozen fringe in the process of soil freezing, and a new ice lens is formed in the frozen fringe. The development of the mechanical method is relatively complete because the mechanical method is relatively visualized and has a clear physical meaning. In 1961, Bishop gave an expression for pore pressure in unsaturated soils. Miller [30] applied the expression to saturated freezing soils as:

P = χP + (1 χ)P (35) l − i Water 2020, 12, 2684 9 of 19 where χ is a weighting coefficient, and can be expressed as the following Equation [31]:

χ = (1 S )1.5 (36) − i According to Equation (32), the can be obtained as: ! ρl T Pl = Pi + ρlL ln (37) ρi T0

Substituting Equation (37) into Equation (35), the relationship between P and Pi can be obtained as: " # ! ρl T P = χ + (1 χ) Pi + χρlL ln (38) ρi − T0

T P Pi + χρlL ln (39) ≈ T0 The formation of ice lenses occurs when the ice pressure exceeds the sum of the external load and the soil separation strength: P σ σsep (40) i ≥ 11 − Substituting Equation (39) into Equation (40), we have:

T P χρlL ln σ11 σsep (41) − T0 ≥ −

According to Equations (10), (22), and (25), the equivalent form of Equation (41) can be obtained   1.5 T R h (1 Si) ρlL ln + Pob + γ0dx∗ σsep (1 + ν)(1 2ν)(1 + e0) − T0 x − − e + e0 (42) ≥ E(1 ν) − Equation (42) can be used as the criterion for the formation of ice lenses. It can be seen from Equation (42) that the formation of ice lenses is related to the temperature, the overburden pressure, and the separation strength of the freezing soil.

3. Model Validation and Numerical Analysis During the freezing process of soil, moisture will be replenished to the frozen fringe. In this section, the thermo-hydro-mechanical coupling model is used to describe the soil freezing process. After appropriate initial values and boundary conditions are given, the finite element method within the platform of COMSOL Multiphysics can be used to solve the equations to obtain the soil temperature field and displacement field during the freezing process.

3.1. Model Validation Lai et al. [26] conducted one-side freezing experiments for silty column in an open system. The soil column was frozen from top to bottom and water was freely available at the base. The initial water content of the 20 cm soil column was 20.5%; the temperatures of the top and lower surfaces were controlled to 268.15 and 274.15 K; the initial temperature of the soil column was 274.15 K; and the temperature gradient of the soil column was 0.30 K/cm. The lower surface of the soil column can be used for water replenishment. Other parameters related to soil column from Lai et al. [26] are listed in Table1. The simulation results and experimental data are compared in Figure4. The frost heave and freezing depth of the soil column are shown in Figure4a; the relationship between temperatures at the heights of 5, 10, 14, and 18 cm and time are shown in Figure4b. Water 2020, 12, x FOR PEER REVIEW 10 of 19 Water 2020, 12, 2684 10 of 19

Table 1. List of parameters used in the model validation. Table 1. List of parameters used in the model validation. Parameter Value 3 Density of solidParameter particles, 𝜌(kg/m ) 2360 Value Density of water, 𝜌 (kg/m3) 1000 3 Density of solid particles, ρs (kg3 /m ) 2360 Density of ice, 𝜌(kg/m ) 917 Density of water, ρ (kg/m3) 1000 thermal conductivity of water,l 𝜆 (J·m−1·s−1·K−1) 0.58 Density of ice, ρ (kg/m 3) 917 thermal conductivity of ice,i 𝜆 (J·m−1·s−1·K−1) 2.22 thermal conductivity of water, λ (J m 1 s 1 K 1) 0.58 l − − −1 − −1 −1 thermal conductivity of solid particles,· 𝜆1(J··1m· ·1s ·K ) 1.5 thermal conductivity of ice, λi (J m s K ) 2.22 − − − −1 −1 Specific heat capacities of solid particles,· · 𝑐(J··1kg 1·K 1) 2360 thermal conductivity of solid particles, λs (J m− s− K− ) 1.5 −1 −1 Specific heat capacities of water, 𝑐(J··kg ··K1 ·) 1 4180 Specific heat capacities of solid particles, cs (J kg− K− ) 2360 Specific heat capacities of ice, 𝑐 (J·kg−1··K−11)· 1874 Specific heat capacities of water, cl (J kg− K− ) 4180 · 2 · SpecificGravitational heat capacities acceleration, of ice, c (J𝑔kg(m/s1 K) 1) 9.811874 i · − · − GravitationalHeight of acceleration,soil model, ℎg(cm)(m/s 2) 209.81 LatentHeight heat of of fusion soil model, for water,h (cm) 𝐿(J/kg) 334,560 20 Latent heat of fusion for water, L (J/kg) 334,560

20 Test: Frost heave 16 Calculated: Frost heave 12

8 4

20 0 /mm heave Frost Test: Frozen front position 16 Calculated: Frozen front position

12

Height /cm 8

4

0 0 20406080100120140 Time /h (a) 3 Test: 5cm 10cm 2 14cm 18cm

1 Calculated: 5cm 10cm 14cm 18cm

0 K

-1

-2 Temperature / Temperature -3

-4

-5 0 20 40 60 80 100 120 140 Time /h (b)

Figure 4. 4. ComparisonComparison between between simulation simulation re resultssults and and experimental experimental data. data. (a: Frost (a: Frost heave heave profiles profiles and frozenand frozen front front profiles profiles during during freezing freezing process; process; b: Temperatureb: Temperature profiles profiles at the at heights the heights of 5, of10, 5, 14, 10, and 14, 18and cm 18 during cm during freezing freezing process process)..).

Water 2020, 12, 2684 11 of 19

From Figure4a, the frost heave of the soil column continues to increase with the freezing time. The frozen front descends very quickly at the beginning and then descends slowly. From Figure4b, the temperature of the soil near the cold end descends quickly and then stabilizes. The simulation results of the frost heave, freezing depth, and temperature are almost consistent with the experimental data. As illustrated in Figure4, the simulation results are consistent with the experimental results.

3.2. The Effect of Soil Parameters on the Soil Freezing Process The types of soil are different in different regions or different depths. The porosity, thermal conductivity, and water conductivity of different soils are not identical. Previous scholars mainly studied the influence of external conditions on the freezing process of soils, such as the external temperature and overburden pressure [21,31]. There are few reports on the influence of soil properties on the freezing process. The effects of the initial void ratio (e0), thermal conductivity of soil particles (λs), and hydraulic conductivity (k0) on the freezing process of soil were analyzed in this study. The parameters of the model required for calculation are listed in Table2 and these parameters come from Yin et al. [27] and Zhou and Li [22]. In order to analyze the effect of soil properties on the freezing process of the soil column, e0, k0, and λs assumed the different values listed in Table3.

Table 2. List of parameters used in the calculation examples.

Parameter Value 3 Density of solid particles, ρs (kg/m ) 2700 3 Density of water, ρl (kg/m ) 1000 3 Density of ice, ρi (kg/m ) 917 Compressive modulus, E (MPa) 0.8 1 1 1 thermal conductivity of water, λl (J m− s− K− ) 0.58 · 1 · 1 · 1 thermal conductivity of ice, λi (J m− s− K− ) 2.22 · · · 1 1 Specific heat capacities of solid particles, cs (J kg K ) 2360 · − · − Specific heat capacities of water, c (J kg 1 K 1) 4180 l · − · − Specific heat capacities of ice, c (J kg 1 K 1) 1874 i · − · − Gravitational acceleration, g (m/s2) 9.8 Height of soil model, h (cm) 10 Latent heat of fusion for water, L (J/kg) 334,000 Overburden pressure, Pob (kPa) 200

Table 3. Values of the variables in the simulated cases.

Initial Hydraulic Temperatures of Thermal Conductivity Case No. Initial Void Ratio Conductivity Top and Lower of Soil Particles (10 11m s 1) Surfaces (K) (W m 1 K 1) − − · − − 1 0.4 1.0 271.65/274.15 0.70 2 0.6 1.0 271.65/274.15 0.70 3 0.8 1.0 271.65/274.15 0.70 4 0.4 1.5 271.65/274.15 0.70 5 0.4 2.0 271.65/274.15 0.70 6 0.4 2.0 271.65/274.15 1.20 7 0.4 2.0 271.65/274.15 1.70

The relationships between temperatures at heights of 9.5, 9, 8, 7, 6, 4.5, and 2 cm and time in case 2 are demonstrated in Figure5. The temperature drops rapidly near the cold end, which is mainly caused by a large temperature gradient. Near the temperature for 273.15 K, due to the release of latent heat by the of pore water, the decrease rate of temperature becomes smaller as illustrated in Figure5. Zhou and Li [22] also obtained similar results. Water 2020, 12, 2684 12 of 19 Water 2020,, 12,, xx FORFOR PEERPEER REVIEWREVIEW 12 of 19

274.5 9.5cm9.5cm 9.0cm9.0cm 8.0cm8.0cm 7.0cm7.0cm 6.0cm6.0cm 4.5cm4.5cm 2.0cm2.0cm 274.0

273.5 K K

273.0

272.5 Temperature / Temperature Temperature / Temperature

272.0

271.5 0 102030405060

Time /h Figure 5. The temperature evolution curve of the soil column with didifferentfferent heights.heights.

TheThe temperaturetemperature distributions alon alongg the height of the soil column in case 2 are presentedpresented inin FigureFigure6 6.. TheThe dotteddotted lineline correspondscorresponds toto 273.15273.15 K.K. TheThe temperaturetemperature tendstends toto bebe linearlinear outsideoutside ofof thethe dotteddotted lineline region. region. Near Near the the temperature temperature for for 273.15 273.15 K, the K, changethe change of the of temperature the temperature gradient gradient is mainly is causedmainly bycaused the release by the ofrelease latent of heat latent from heat the from phase the transition phase transition of pore water. of pore The water. temperature The temperature fields at difieldsfferent at different times in casetimes 2 in are case shown 2 are in shown Figure in7. Figure 7.

10

8

6 1h1h 5h5h 4 10h10h Height /cm Height /cm 20h20h 35h35h 2 55h55h 273.15K

0 271.5 272.0 272.5 273.0 273.5 274.0 Temperature /K Temperature /K Figure 6. TemperatureTemperature distribution at didifferentfferent timestimes ofof casecase 2.2.

Water 2020, 12, x FOR PEER REVIEW 13 of 19 Water 2020, 12, 2684 13 of 19 Water 2020, 12, x FOR PEER REVIEW 13 of 19

Figure 7. Temperature distribution at 0, 5, 10, and 35 h of case 2.

The effects of the initialFigure void 7. TemperatureTemperature ratio (𝑒), distributionhydraulic conductivity at 0, 5, 10, and ( 𝑘 35), h and of case thermal 2. conductivity of soil particles (𝜆) on frost heave and frozen front during the freezing process of soil are illustrated in The e ects of the initial void ratio (e ), hydraulic conductivity (k ), and thermal conductivity of FiguresThe 8–10, effectsff respectively. of the initial It can void be ratioseen from(𝑒0), hydraulicFigure 8 that conductivity the soil column (𝑘0), and with thermal a high conductivityporosity ratio of orsoil a high particles particles water ( (𝜆 λcontents)) on on frost frost has heave heave a more and and obvious frozen frozen frontfrost front duringheave during andthe the freezinga freezingsmaller process processfreezing of ofsoildepth soil are areat illustrated the illustrated same in freezingFiguresin Figures time. 8–10,8– 10A respectively. ,high respectively. initial voidIt can It canratio be seen be means seen from froma highFigure Figure water 8 that8 content;that the thesoil therefore, soil column column with more with a highlatent a high porosity heat porosity in theratio soilorratio isa releasedhigh or a highwater and water content the content freezing has hasa depthmore a more obviousof obviousthe soil frost column frost heave heave is and also and a reduced. asmaller smaller freezingThe freezing final depth depthpositions at at the the of same the frozenfreezing front time. are Abasically A high high initial initial unchanged. void void ratio ratio means means a a high high water water content; content; therefore, therefore, more latent heat in the soil is released and the freezing depth of the soil column is also reduced.reduced. The final final positions of the frozen front are basically unchanged. frozen front are basically unchanged.

Case 1 Frost heave (0.4) 12 Case 2 Frost heave (0.6) Case 3 Frost heave (0.8) 8 Case 1 Frost heave (0.4) 12

Case 2 Frost heave (0.6) 4 Case 3 Frost heave (0.8) 8 Frost heave/mm 10 0 4 Case 1 Frozen front position (0.4)

9 Frost heave/mm 10 Case 2 Frozen front position (0.6) 0 Case 3 Frozen front position (0.8) Case 1 Frozen front position (0.4) 8 9 Case 2 Frozen front position (0.6) Case 3 Frozen front position (0.8) 7 8

6 7

5 6 Frozen front position/cm Frozen front 4 5

Frozen front position/cm Frozen front 0 102030405060 4 Time/h

0Figure 102030405060 8. Freezing process curves of cases 1, 2, and 3. Figure 8. Freezing processTime/h curves of cases 1, 2, and 3.

The frost heave increasesFigure by 4 8. mm Freezing for every process 0.5 curves ∙ 10 of casesm/s increase1, 2, and 3.in hydraulic conductivity from 1∙10 m/s to 2∙10 m/s after freezing for 60 h, as presented in Figure 9. The seepage of The frost heave increases by 4 mm for every 0.5 ∙ 10 m/s increase in hydraulic conductivity from 1∙10 m/s to 2∙10 m/s after freezing for 60 h, as presented in Figure 9. The seepage of

Water 2020, 12, x FOR PEER REVIEW 14 of 19 soil is directly affected by the hydraulic conductivity, which causes more water to migrate from the unfrozen region to the frozen front. This leads to a greater release of latent heat. The freezing depth of the soil with high permeability conductivity is slightly smaller, but the change is not obvious. The final positions of the frozen front are basically the same. Water 2020, 12, 2684 14 of 19

20 Case 1 Frost heave (1.0×10-11m s-1) Case 4 Frost heave (1.5×10-11m s-1) 16 Case 5 Frost heave (2.0×10-11m s-1) 12 8 4

10 0 heave/mm Frost × -11 -1 9 Case 1 Frozen front position (1.0 10 m s ) Case 4 Frozen front position (1.5×10-11m s-1) -11 -1 8 Case 5 Frozen front position (2.0×10 m s )

7

6

5 Frozen front position/cm 4

0 102030405060 Water 2020, 12, x FOR PEER REVIEW 15 of 19 Time/h

Figure 9. Freezing process curves of cases 1, 4, and 5. Figure 9. Freezing process curves of cases 1, 4, and 5.

-1 -1 20 The soil with large thermal Case 5 Frost conductivity heave (0.7W m causes K ) a great amount of frost heave at the same -1 -1 freezing time. The heat transfer Case 6 Frostof soil heave is (1.2Waffected m K by) the thermal conductivity of the15 soil particles. -1 -1 Case 7 Frost heave (1.7W m K ) 10 With the increase of the thermal conductivity, the depth of freezing increases as demonstrated in Figure 10. The freezing depth of the soil column is obviously affected by the change5 of thermal conductivity. The10 greater the thermal conductivity of the soil column, the greater the freezing0 Frost heave/mm depth. The final positions of the frozen front are basicall Casey unchanged.5 Frozen front position The freezing (0.7W m -1depth K-1) of the soil column with better thermal9 conductivity reaches a stable Case position 6 Frozen frontfaster. position (1.2W m-1 K-1) Case 7 Frozen front position (1.7W m-1 K-1) As it is displayed8 in Figures 8–10 that the frozen front quickly moved downward during the initial period of freezing, the freezing depth is about 5 cm after freezing for 30 h, and the final freezing depth remains about7 6cm. The final temperature is linearly distributed along the height of the soil column in Figure 66, and the finial position of the frozen front depends on the temperatures of the cold and warm ends. Through these seven cases, it can be found that frost heave increases almost linearly 5 with freezing time. The formation4 and distribution of ice lenses is important to study the freezing process of soils, Frozen front position/cm the distribution of ice lenses of different cases after freezing 60 h is given in Figures 11–13. The 3 portions represent 0the ice lenses, 102030405060 and the black portions represent the soil regions. Due to the final stabilization of the frozen front, a thick ice lens, whichTime/h is called the ‘stable ice lens’ (region B in Figure

12), will be generated near the frozen front, and the ice lenses in region A is called as the ‘previous Figure 10. Freezing process curves of cases 5, 6, and 7. ice lenses’. Zhou and Li [22],Figure Sheng 10. Freezing et al. [32], process and curves Lai et of al. cases [26] 5, obtained 6, and 7. similar results about two types ofThe icefrost lens. heave increases by 4 mm for every 0.5 10 11 m/s increase in hydraulic conductivity · − fromAfter1 10 freezing11 m/s tofor2 6010 h,11 them/ resultss after freezingof the cases for 60under h, as different presented initial in Figure void9 .ratios The seepage are illustrated of soil is · − · − indirectly Figure a11.ffected The byinitial the hydraulicvoid ratio conductivity, has little effect which on causesthe stable more ice water lens. to This migrate is because from the the unfrozen final positionregion of to the the frozen frozen front front. will This not leads be affected to a greater by the releaseinitial void of latent ratio. heat.With Thethe increase freezing of depth the initial of the voidsoil ratio, with the high number permeability and thickness conductivity of previous is slightly ice lenses smaller, increase. but the change is notA obvious. The final positions of the frozen front are basically the same. The soil with large thermal conductivity causes a great amount of frost heave at the same freezing time. The heat transfer of soil is affected by the thermal conductivity of the soil particles. With the increase of the thermal conductivity, the depth of freezing increases as demonstrated in Figure 10. The freezing depth of the soil column is obviously affected by the change of thermalB conductivity. The greater the thermal conductivity of the soil column, the greater the freezing depth. The final

Case 1 Case 2 Case 3 Figure 11. Distribution of ice lenses at different initial void ratios. (Region A is called the ‘previous ice lenses’; Region B is called the ‘stable ice lens’.).

The formation of the ice lens is greatly affected by the hydraulic conductivity of soil, as presented in Figure 12. The number and thickness of previous ice lenses are increased in the case with large hydraulic conductivity, and the stable ice lens of soil becomes thicker. It is because the increase of the hydraulic conductivity causes the increase of the moisture of migration. However, there is no obvious change at the bottom of the stable ice lens. The reason is that the stable position of the frozen front is not affected by the hydraulic conductivity.

Water 2020, 12, x FOR PEER REVIEW 15 of 19

-1 -1 20 Case 5 Frost heave (0.7W m K ) Case 6 Frost heave (1.2W m-1 K-1) 15 Water 2020, 12, 2684 -1 -1 15 of 19 Case 7 Frost heave (1.7W m K ) 10

5

positions of the frozen front are basically unchanged. The freezing depth of the soil columnwithFrost heave/mm better 10 0 thermal conductivity reaches a stable position faster.Case 5 Frozen front position (0.7W m-1 K-1) As it is displayed9 in Figures8–10 that the frozen Case 6 frontFrozen quickly front position moved (1.2W downward m-1 K-1) during the initial Case 7 Frozen front position (1.7W m-1 K-1) period of freezing,8 the freezing depth is about 5 cm after freezing for 30 h, and the final freezing depth remains about 6 cm. The final temperature is linearly distributed along the height of the soil column in Figure6, and the7 finial position of the frozen front depends on the temperatures of the cold and warm ends. Through6 these seven cases, it can be found that frost heave increases almost linearly with freezing time. The formation5 and distribution of ice lenses is important to study the freezing process of soils, the distribution of4 ice lenses of different cases after freezing 60 h is given in Figures 11–13. The white Frozen front position/cm portions represent the ice lenses, and the black portions represent the soil regions. Due to the final 3 stabilization of the0 frozen front, 102030405060 a thick ice lens, which is called the ‘stable ice lens’ (region B in Figure 12), will be generated near the frozen front, and the ice lenses in region A is called as the Time/h ‘previous ice lenses’. Zhou and Li [22], Sheng et al. [32], and Lai et al. [26] obtained similar results about two types of ice lens.Figure 10. Freezing process curves of cases 5, 6, and 7.

A

B

Case 1 Case 2 Case 3 FigureFigure 11. Distribution of iceice lenseslenses atat didifferentfferent initial initial void void ratios. ratios. (Region (Region A A is calledis called the the ‘previous ‘previous ice icelenses’; lenses’; Region Region B is B called is called the the ‘stable ‘stable ice ice lens’.). lens’.).

TheAfter formation freezing of for the 60 ice h, thelens results is greatly of the affected cases underby the dihydraulicfferent initial conductivity void ratios of soil, are illustratedas presented in inFigure Figure 11 .12. The The initial number void ratioand thickness has little eofffect previous on the stableice lenses ice lens. are increased This is because in the the case final with position large hydraulicof the frozen conductivity, front will not and be the aff ectedstable by ice the lens initial of soil void becomes ratio.With thicker. the It increase is because of the the initial increase void of ratio, the hydraulicthe number conductivity and thickness causes of previous the increase ice lensesof the increase.moisture of migration. However, there is no obvious changeThe at formation the bottom of of the the ice stable lens isice greatly lens. The affected reason by is the that hydraulic the stable conductivity position of of the soil, frozen as presented front is notin Figure affected 12 by. The the numberhydraulic and conductivity. thickness of previous ice lenses are increased in the case with large hydraulic conductivity, and the stable ice lens of soil becomes thicker. It is because the increase of the hydraulic conductivity causes the increase of the moisture of migration. However, there is no obvious change at the bottom of the stable ice lens. The reason is that the stable position of the frozen front is not affected by the hydraulic conductivity. From Figure 13, the number of previous ice lenses is reduced, and stable ice lens has hardly changed with the increase in thermal conductivity. The large thermal conductivity of soil decreases the freezing time of frozen soil, which prevents further formation and increase of the ice lens in soil. Water 2020, 12, x FOR PEER REVIEW 16 of 19

Water 2020, 12, 2684 16 of 19 Water 2020, 12, x FOR PEER REVIEW 16 of 19

Case 1 Case 4 Case 5 Figure 12. Distribution of ice lenses at different hydraulic conductivity.

From Figure 13, the number of previous ice lenses is reduced, and stable ice lens has hardly changed with the increaseCase in 1 thermal conductivity. Cas eThe 4 large thermal conductivity Case 5 of soil decreases the freezing time ofFigureFigure frozen 12. 12. soil, DistributionDistribution which prevents of of ice ice lenses lenses further at at different di ffformationerent hydraulic hydraulic and increase conductivity. of the ice lens in soil.

From Figure 13, the number of previous ice lenses is reduced, and stable ice lens has hardly changed with the increase in thermal conductivity. The large thermal conductivity of soil decreases the freezing time of frozen soil, which prevents further formation and increase of the ice lens in soil.

Case 5 Case 6 Case 7 FigureFigure 13. 13. DistributionDistribution of of ice ice lenses lenses at at different different thermal thermal conductivity conductivity of of solid solid particles. particles.

4.4. Conclusions Conclusions and and Discussion Discussion Case 5 Case 6 Case 7 InIn thisthis study,study, a newa new thermo-hydro-mechanical thermo-hydro-mechanical coupling coupling model model describing describing the soil the freezing soil freezing process processwas established wasFigure established considering13. Distribution considering the of formation ice lensesthe formation at of different ice lenses. of thermal ice Thelenses. conductivity model The consideredmodel of solid considered particles. the heat the transfer, heat transfer,heat convection, heat convection, phase change phase ofchange pore of water, poremechanical water, mechanical equilibrium, equilibrium, and formation and formation of ice lenses.of ice 4.A Conclusions new criterion and for the Discussion formation of ice lenses was derived. Seven cases of soil freezing were simulated to study the effects of the initial void ratio, hydraulic conductivity, and thermal conductivity of soil In this study, a new thermo-hydro-mechanical coupling model describing the soil freezing particles on the freezing process of soils. The following conclusion is made: process was established considering the formation of ice lenses. The model considered the heat transfer, heat convection, phase change of pore water, mechanical equilibrium, and formation of ice

Water 2020, 12, 2684 17 of 19

The final stable position of the frozen front depends mainly on the temperature of the cold and warm ends, but the freezing front of the soil with a lower initial void ratio and hydraulic conductivity and a large thermal conductivity reaches the stable position faster, and there are fewer previous ice lenses in this case. There was a positive correlation between the frost heave and initial void ratio, hydraulic conductivity, and thermal conductivity. The stable ice lens is mainly affected by the hydraulic conductivity, and as it increases, the thickness of the stable ice lens becomes larger. Because of the compressibility of soil particles, pore ice and pore water are ignored, and the applicability of the model will be reduced when used to study the freezing of deep soil. The research object of the theory proposed in this paper was saturated soil. For unsaturated soil, the degree of saturation needs to be introduced into the theory to adjust the model.

Author Contributions: D.L. Methodology, Investigation, Software, Writing—original draft. Y.Y.Conceptualization, Supervision, Writing—review and editing. C.C. Conceptualization, Writing—review and editing. Y.C. Data curation, Formal analysis. S.W. , Formal analysis. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the Fundamental Research Funds for the Central Universities, grant number 2019ZDPY18. Acknowledgments: The authors are grateful to the editor and reviewers for providing constructive and positive comments for this paper. Conflicts of Interest: The authors of this manuscript declare not to have any conflict of interest regarding this manuscript. None of the authors have any financial interests with any commercial entity that has interest in the subject or outcome of this manuscript including consultancy, paid expert consultancy, or honoraria, patent application, as well as other forms of conflict of interest, including personal and academic issues. The authors to the best of their knowledge conducted the study and reported the conclusions independently without any interference from partial or full funding sources or other entities.

Abbreviations e pore ratio n porosity P pore pressure, Pa t time, s T temperature, K e σij, σij total stress tensor and effective stress tensor, Pa σ11 normal stress in vertical direction, Pa εij strain tensor ε11, ε22, ε33 normal strain Constants and symbols 1 1 cs, c , c specific heat capacities of solid particles, pore water and pore ice, respectively, J kg K l i · − · − dV unit volume element of soils, m3 e0 initial porosity ratio E compressive modulus, Pa f body force in the ith direction, Pa m 1 i · − g gravitational acceleration, m s 2 · − h height of soil model, cm H volumetric heat capacity of the soils, J K 1 m 3 · − · − Hi frost heave, m K, KS the bulk modulus of dry soil and soil particles, respectively, Pa 1 k, k0 hydraulic conductivity of freezing soils and unfrozen soils, respectively, m s− 1 · L latent heat of fusion for water, J kg− · 3 Ma mass of pore water and ice in the unit volume of soils, kg m · − Water 2020, 12, 2684 18 of 19

Pi, Pl ice and water pressures, respectively, Pa PT cryogenic suction due to the ice/water interface tension, Pa Pob overburden pressure, Pa Si volume percentage of ice in pores T0 freezing point temperature of water in pore, K Tr reference temperature, K →v seepage velocity of pore water, m s 1 · − α parameter depending on the size of pore α∗ Biot coefficient β parameter dependent on the size and structure of pore 3 Φ heat content of soil per unit volume, J m− · 1 1 1 λ thermal conductivity of freezing soils, J m− s− K− · · · 1 1 1 λs, λl, λi thermal conductivity of soil particles, water and ice, respectively, J m− s− K− 3 · · · ρs, ρ , ρ density of soil particles, water and ice, respectively, kg m l i · − γ unit weight of soil, kg m 2 s 2 · − · − γ unit weight of water, kg m 2 s 2 l · − · − σsep separation strength of the freezing soil, Pa 0 σ11 initial normal stress in vertical direction, Pa λ∗, µ∗ Lame constant ν Poisson’s ratio of soil Hamiltonian operator ∇ d differential operator ϕ driving force that causes seepage χ weighting coefficient δij Kronecker symbol

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