An analysis of vapour transfer in unsaturated freezing soils

Journal Pre-proof An analysis of vapour transfer in unsaturated freezing soils

Zuoyue He, Jidong Teng, Zhijun Yang, Linong Liang, Hongzhong Li, Sheng Zhang PII:

S0165-232X(18)30315-X

DOI:

https://doi.org/10.1016/j.coldregions.2019.102914

Reference:

COLTEC 102914

To appear in:

Cold Regions Science and Technology

Received date:

18 July 2018

Revised date:

15 March 2019

Accepted date:

4 October 2019

Please cite this article as: Z. He, J. Teng, Z. Yang, et al., An analysis of vapour transfer in unsaturated freezing soils, Cold Regions Science and Technology(2018), https://doi.org/ 10.1016/j.coldregions.2019.102914

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© 2018 Published by Elsevier.

Journal Pre-proof

An analysis of vapour transfer in unsaturated freezing soils Zuoyue He1, 2 , Jidong Teng3 , Zhijun Yang1 , Linong Liang2 , Hongzhong Li2 , Sheng Zhang*3 1

Guangdong Province Communications Planning & Design Institute Co., Ltd , Guangzhou 510507, China 3

School of Civil Engineering, Central South University, Changsha 410075, China

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2

School of Earth Sciences and Engineering, Sun Yat-sen University, Guangzhou 510275, China

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*Correspondence to: Sheng Zhang

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Address: Railway Campus of Central South University, Shaoshan South Road No.68, Changsha, Hunan Province, 410075, China.

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Tel.: (+86) 13907315427 Fax: (+86)

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ABSTRACT

The effects of vapour on water content in different unsaturated frozen soils have not been

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systematically analyzed in the literature. Based on the thermodynamic equilibrium theory and

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coupled water-heat theory, a new method for calculating unfrozen water content and ice content is obtained. A new model is then built by importing this method to the coupled heat and mass transfer theory. Unfrozen water content and ice content in this new model are only related to hydraulic parameters and temperature, which means specific physical meaning. The comparisons between simulations and test results of sandy loam validate the new model. Simulated results also show that temperature is the major factor to vapour transfer instead of suction. And vapour transfer in silt and sand cannot be neglected with freezing except clay. Initial water content, freezing temperature, freezing time and ground water table can all influence vapour transfer in freezing soils. In a word, even though the water content increment is low, the total mass of the 1

Journal Pre-proof increased amount and the average increase of the total water content in the frozen area are about 40kg and 0.033 within the unit area in susceptible frost heaving soils such as silt, respectively. Remarkable frost heave could then occur due to vapour. Therefore, the vapour in unsaturated frozen soils must be paid more attention to in practical engineering. This paper strengthens the understanding of canopy effect and also validates that canopy effect usually occurs in covered

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freezing silt instead of sand or clay.

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Keywords: Vapour transfer; total water content; moisture flux; ice content; canopy effect

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1. Introduction

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Frost damage, especially frost heave is an important issue for the safety in cold and arid regions,

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which has attracted much attention in the world (Sheng et al., 2014). Traditional experimental and theoretical research on moisture transfer mainly focuses on the liquid water transfer driven

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by suction and temperature in freezing soils (Hoekstra, 1966; Harlan, 1973; Guymon and Luthin,

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1974; Taylor and Luthin, 1978; Jame and Norum, 1980; Mageau and Morgenstern, 1980; Kung

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and Steenhuis, 1986; Gray and Granger, 1986; Stähli et al., 1999; Kane and Stein, 2010; Zhou et al., 2014). Classical frost heave models also have the same characteristics, such as the rigid ice model (Miller, 1972; O’Neill and Miller, 1985; Sheng et al., 1995a, 1995b) and the segregation potential theory (Konrad and Morgenstern, 1980, 1981, 2011). And these studies all suppose that the effect of vapour on moisture transfer is negligible. However, the effect of vapour transfer on the total water content cannot be ignored when initial water content is low, which can induce frost damage in cold regions as well (Milly, 1984; Teng et al., 2019). Li et al. (2014) found that water content in the subgrade soil was very high after examining an airport in northwestern China and then named this phenomenon canopy effect. 2

Journal Pre-proof In this region, the groundwater table is more than 20m deep, which means that capillary water cannot reach a high region where canopy effect is observed. The annual precipitation is limited, while the annual evaporation is significant, so water can also not remarkably accumulate in soils. In fact, the hydraulic conductivity and vapour diffusivity of the airport runway are all very low. Liquid water or vapour unlikely passes through this almost impervious cover. Therefore, Li et al. (2014) concluded that liquid water can not cause canopy effect, while vapour in subgrade soils

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may be a rational reason.

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Experimental study in the literature can support this conclusion. The temperature in this

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airport is below 0 o C during the whole winter and the minimum value even reaches -20 o C. Test

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results of Nakano et al. (1984), Eigenbrod and Kennepohl (1996) and Guthirie (2006) show that

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when the initial water content is low, vapour transfer in freezing soils is not negligible and even plays a major role. Vapour transfers towards freezing front continuously and promotes the

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formation of ice crystals simultaneously, even causing frost heave. From a one-dimensional freezing test of a calcareous sand conducted by Zhang et al. (2016), a similar result was also

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obtained. Temperatures at the top and bottom boundary are -10 o C and 10 o C, respectively. The

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freezing time is 7 days. A void exists between the specimen and water table, which can ensure that only vapour transfers upward and then enter the specimen. The result shows that the total water content below the impervious cover increases remarkably even though the initial water content is 0, while the increase is very limited when the specimen is not frozen. Similar tests conducted by Teng et al. (2018) also showed that significant ice accumulation in an unsaturated clean siliceous sand below an impervious cover was observed. In addition, the initial conditions and boundary conditions of these tests are quite similar to those of canopy effect. Therefore, it can be concluded that vapour transfer is the major reason for this remarkable increase of water

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Journal Pre-proof content in freezing soils when the initial water content is low, which is the same as the main characteristic of canopy effect. This conclusion is consistent with the viewpoint of Li et al. (2014) and also suggests a reasonable theoretical approach for canopy effect. Only a few coupled models for liquid water- vapour-heat transfer in freezing soils are proposed in the literature (Kennedy and Lielmezs, 1973; Nassar and Horton, 1989, 1992; Hansson et al., 2004). However, the results predicted by these models show that the total water

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content below the impervious cover increases hardly. The effect of vapour on moisture transfer is

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quite small, which opposes to the above freezing test results. Therefore, these models are

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incapable of explaining the mechanism for canopy effect. Zhang et al. (2016a, 2016b) proposed a

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new model in which three phase changes of vapour were taken into account, namely evaporation,

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condensation and desublimation. The simulated results show that vapour transfer can lead to a remarkable increase of water content in covered freezing soils. Although this model can

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reproduce canopy effect and is also validated by freezing tests, some questionable or unsolved

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points still exist. For example, the maximum unfrozen water content established by the generalized Clapeyron equation is not validated by test, so the correctness and applicability of

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this formula are questionable. More importantly, this model fails to explain the observed phenomenon that canopy effect is prone to occur in silt instead of sand or clay. Coupled liquid water-vapour- heat transfer in freezing soils is a pretty complicated process. Studies on vapour transfer and the mechanism for canopy effect are not perfect yet and further theoretical research is still pressingly needed. In this paper, the determinations of the ice content and the maximum unfrozen water content are firstly discussed. A new model for coupled liquid water- vapour- heat transfer in unsaturated freezing soils is then proposed. Test results in the literature are used to validate this model. The 4

Journal Pre-proof effects of vapour transfer on the total water contents in different kinds of soils are analyzed using this model. Lastly, the total water content and its increment due to vapour transfer are a lso predicted with taking into account several influencing factors. This research method is fully reasonable and reliable. The research results will be also credible and attract more attentions to vapour transfer in unsaturated freezing soils, which can then provide a theoretical reference for practical engineering.

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2. Mathematical model

In unsaturated freezing soils, moisture transfers due to suction, temperature and gravity. The

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mass conservation equation is expressed as follows (Saito et al., 2006; Hansson et al., 2004)

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 w  v i i   '  h  T h T      K wh   1  K wT  K vh  K vT (1) t t w t z  z z z   z 

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where θ w, θv and θi are the liquid water content, the equivalent vapour content and the pore ice content, respectively. ρw (=1000 kg/m3 ) and ρi (=916 kg/m3 ) are the liquid water density and the

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ice density, respectively. h (m) and T (K) are the water pressure head and the temperature,

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' respectively. K wh (m/s) and KwT (m2 /K/s) are the isothermal and thermal hydraulic conductivities

of liquid water due to the water pressure head and the temperature, respectively. Kvh (m/s) and KvT (m2 /K/s) are the isothermal and thermal hydraulic conductivities of vapour due to the the water pressure head and the temperature, respectively. z (m) is the spatial coordinate positive upward. t (s) is the time. The energy conservation equation considering evaporation, condensation and desublimation is given by (Nassar and Horton, 1989, 1992; Hansson et al., 2004)

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  qwT    qvT   q  T   T   Li i i  Lv w v   '   Cw  Cv  Lv w v (2) t t t t  z  z z z

where Cp (J/m3 /K) and  ' (W/m/K) are the equivalent volumetric heat capacity and the thermal conductivity considering soil skeleton, liquid water, vapour and ice, respectively. Cw (=4.18×106 J/m3 /K) and Cv (=6.3×103 J/m3 /K) are the heat capacities of liquid water and vapour, respectively. Li (=3.34×105 J/kg) and Lv (J/kg) are the latent heats of water freezing and vaporization,

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respectively. qw (m/s) and qv (m/s) are the liquid water flux and vapour flux, respectively.

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It is noted that there are three independent variables in the two governing equations, namely

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θw, θi and T, so one additional equation is required. The ice content is usually only assumed as a

 di  d w  w (3)  di    d w  i

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2013; Kelleners et al., 2016), as follows

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function of the unfrozen water content (Fuchs et al., 1978; Newman and Wilson, 1997; Kelleners,

It is shown that the ice content is implicit and can be calculated by iteration. In fact, it is ture

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only in mass in the first formula in Eq. (3). In the second formula, the volumetric moisture content is correct. In addition, besides the unfrozen water content, temperature can also influence the ice content. It is obvious that the ice content will increases with temperature decreasing if the total water content remains unchanged, which is obtained from test and field monitoring results (Patterson and Smith, 1981; Stein and Kane, 1983; Liu and Li, 2012; Zhou et al., 2014). Tice et al. (1976) then proposed a new formula for the saturation of pore ice (Si), as follows 0; T  Tf  Si   (4) α 1  1  T  Tf   ; T  Tf

6

Journal Pre-proof where T (o C) and Tf (=0 o C) are the temperature and the freezing temperature of pure liquid water, respectively. And α is an experimental parameter depending on soil category. The relationship between the saturation of pore ice and ice content can be expressed as Si=θi/n. And n is the porosity. Each of the two equations for pore ice only takes into account one factor. In fact, no matter how low the temperature is, unfrozen water content in freezing soils always exists. And pore ice can generate only if the unfrozen water content is greater than the maximum unfrozen

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water content (Li et al., 2010; Wang et al., 2014). Therefore, the ice content can be determined as

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0  i   (5)  w  u (T  Tf and w  u )

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where θu is the maximum unfrozen water content.

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The maximum unfrozen water content in the literature is usually obtained from freezing tests (Sheng et al., 1995, 2013). An exponential function for θu is then proposed by measuring the

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liquid water content at different temperatures. The fitting parameters in this determined method

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will change with soil properties. When the soil type changes, the test must be performed again,

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which means that the application of this method is quite limited. Zhang et al. (2016a, 2016b)then proposed a theoretical formula based on the generalized Clapeyron equation and the van Genuchten model. However, this formula is not validated by test, so its correctness and applicability are questionable. Williams (1967) proposed another formula named the freezingpoint depression equation from test, as follows hu  Li

T  Tf (T  Tf ) (6) gTf

where g (=9.8 m/s2 ) is the gravitational acceleration. And hu (m) is the maximum water pressure head which is consistent with the maximum unfrozen water content. Eq. (6) is based on the 7

Journal Pre-proof derivation using thermodynamic equilibrium theory and establishes a relationship between the maximum water pressure head and the temperature. Importantly, Eq. (6) has been validated by tests conducted by Williams (1967) and is also widely used by researchers such as Zhang et al. (2007), Li and Sun (2008), Li et al. (2009) and Li et al. (2010). This equation is adopted in this paper as well. Substituting Eq. (6) into a model for the soil water characteristic curve (SWCC), the maximum unfrozen water content can be then obtained. It can be known that the maximum

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unfrozen water content is only related to the hydraulic parameters for SWCC and temperature,

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without increasing the number of the total parameters, so is the ice content calculated by Eq. (5).

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It is notable that SWCC is usually obtained in isothermal condition. The effect of temperature on

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SWCC is a new research point in unsaturated soils. And some significant results have been obtained. However, these research usually focus on positive temperature. There are few

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pressingly needed on this issue.

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documents about the effect of negative temperature on SWCC. Therefore, further research is

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In this paper, Eqs. (1), (2), (5) and (6) together form a new model for liquid water-vapour- heat transfer in unsaturated freezing soils. The auxiliary equations for this model are given in the

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Appendix. It is known only two independent variables exists in this model, namely θw and T. New approaches for the pore ice content and the maximum unfrozen water content are also established, respectively. Therefore, the effects of vapour on total moisture transfer can be analyzed using this model. 3. Numerical implementation The governing Eqs. (1) and (2) are highly non-linear because the coefficients in the two equations vary with time and they will affect each other as well. The solution procedure is facilitated by the COMSOL Multiphysics package (5.1) which is a commercial platform for 8

Journal Pre-proof solving partial differential equations using the finite element method. In COMSOL Multiphysics, the two governing equations can be written in the following general form

da

    Γ  f (7) t

where κ={θ w, T} is the independent variable in the governing equations. da is damping

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coefficient. Γ and f are the flux vector and the source term, respectively.

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The generalized solution to Eq. (7) is given as

   ,      Γ,   da  t 



   f ,  



(8)

   ,    Γ,   da  t 



   f ,   n  Γ,  





(9)

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where  is the virtual displacement. And Ω is the calculating domain. Eq. (8) can be rewritten as

the boundary.

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where ∂Ω is the boundary of the calculating domain. And n is the outward normal unit vector of

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The time terms are discretized by using the backward difference as following   i   i 1  (10) t t

where the superscript refers time. Substituting Eq. (10) into Eq. (9), the following equation can be obtained   i   i 1  ,    Γ,   da t  Ω



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   f ,    n  Γ,  Ω

Ω

Ω

(11)

Journal Pre-proof Eq. (11) can be rewritten as

 d  ,  i

a

Ω



 t Γ, 





 t f , 

Ω

  d  a

Ω

i 1

,



Ω



 t n  Γ, 



Ω

(12)

Stable scheme is employed for variable κ, that is

Lastly, the following formula can be obtained





 t Γ, 



Ω



 2t f , 



Ω



 2 da i 1 , 



Ω







 i   i 1 (13) 2

 t da i 1 ,   2t n  Γ, 



Ω

(14)

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Ω

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

2 da  i , 

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

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Then appropriate initial and boundary conditions are required to solve these equations. The

u  z,0  u0 (15)

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initial condition can be expressed as

where u0 is a known quantity of variable u when time is 0.

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The first type of condition named Dirichlet boundary condition is given by

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u |s  us (16)

where us is the prescribed value of variable u at the boundary. The second type of condition named Neumann boundary condition is given by    nΓ     (17)  u  T

where Λ is a source term at the boundary. And   / u  and λ are the matrices for the constraint T

and Lagrange multiplier, respectively. 10

Journal Pre-proof 4. Experimental validation Mizoguchi (1990) conducted a freezing test using Kanagawa sandy loam which was packed in a 20 cm long with an internal diameter of 8 cm cylinder. The samples were prepared with the same initial state involving a uniform temperature of 279.85K and close water content of 0.33. The side and bottom of the samples were thermally isolated. The top of the cylinder was exposed to a circulating fluid with a temperature of 267.15K. Using a variable heat flux upper boundary

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condition: qh =-hc(TTop -TCoolant ), the heat transfer was simulated. qh (W/m2 ) is the heat flux. hc=28

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(W/m2 /K) is the convective heat transfer coefficient. TTop and TCoolant are the temperatures at the

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soil surface and of the circulating fluid, respectively. These samples were tested with 12h, 24h

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and 50h, respectively. After test, these samples were divided into 1 cm slices for measuring the

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water content distributions.

The Soil Water Characteristic Curve was measured by Ishida (1985) and the fitting parameters

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are showed in the following: θs=0.535, θs=0.05, α=1.11 1/m, n=1.48, m=0.2 and l=0.5 (Hansson

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et al., 2004). The saturated hydraulic conductivity Ks (=3.2×10-6 m/s) was measured directly. The

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comparisons of the measured and simulated total volumetric water content and temperature profiles are shown in Fig. 1 and Fig. 2, respectively. It can be observed that the simulated results of the total water content and temperature agree fairly well with the test data in which temperature at 50hr was lacking. The rapid decrease of the total water content at freezing front and slow recover in deeper are also well simulated. It is notable that the initial water content in this test is quite large, which means that the saturation is also large and then the channel for vapour transfer is limit. The both top and bottom of the soil column are closed, resulting in no moisture including liquid water and vapour transferring into the interior. These conditions do not well match the background in this paper. 11

Journal Pre-proof However, the simulated results show that the proposed model can pretty simulate the coupled moisture and heat transfer in freezing soils, although the further experimental research is needed. 0

0

(a) 0 hr

(b) 12 hr -0.05

Depth, m

-0.1

-0.15

-0.1

-0.15

Test Simulated

-0.2

-0.2

0.5

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0.1 0.2 0.3 0.4 Total volumetric water content

0

0.1 0.2 0.3 0.4 Total volumetric water content

0.5

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0

0

0

(d) 50 hr

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(c) 24 hr -0.05

-0.15

-0.2 0

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-0.1

0.1 0.2 0.3 0.4 Total volumetric water content

Depth, m

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-0.05

-0.1

-0.15

-0.2

0.5

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Depth, m

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Depth, m

-0.05

0

0.1 0.2 0.3 0.4 Total volumetric water content

0.5

Fig. 1 Measured and simulated total volumetric water content at (a) 0h, (b) 12h, (c) 24h and (d) 50h.

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Depth, m

-0.05

-0.1 Test: 12 hr Simulated: 12 hr Test: 24 hr Simulated: 24 hr Simulated: 50 hr

-0.15

-0.2 -6

-2 0 2 o Temperature, C

4

6

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-4

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Fig. 2 Measured and simulated temperatures at 12h, 24h and 50h.

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5. Liquid water-vapour transfer in unsaturated freezing soils

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Based on the proposed model, liquid water- vapour transfer in unsaturated freezing soils is firstly

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studied. Three different kinds of soils are chosen, namely sand, clay and silt. Silt is a typical soil in Lanzhou which is a city in northwestern China, and the hydraulic parameters are obtained by

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Zhang et al. (2016) and are listed in Table 1. The other parameters in Table 1 are from the results

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obtained by Carse and Parrish (1988), which represents the other two typical types of soils. The

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boundary conditions and the initial conditions are determined according to the practical engineering in northwestern China (Zhang et al., 2016). The initial water content is low, which is a major feature in the cold and arid region. The ground water table is 20m and locates at the bottom. The boundary at the top is closed, which means that liquid water flux and vapour flux are all 0 m/s. The temperatures at the top and bottom are -10 o C and 15 o C, respectively. The initial temperature of the soil column is 15 o C. The freezing period is 90 day. Table 1 The hydraulic parameters of different soils

Sand

θs

θr

α(1/m)

n

Ksat (m/s)

Note

0.471

0.049

9.8

3.73

6.11e-6

Carse and Parrish (1988)

13

Journal Pre-proof Silt

0.49

0.065

0.546

2.32

2.55e-6

Zhang et al. (2016)

Clay

0.38

0.068

0.8

1.09

5.56e-7

Carse and Parrish (1988)

It is known that the the residual water content of clay is 0.068, which is the largest in these three kinds of soils. For a good convergence, the initial water content is then determined as 0.08 which is just slightly greater than the residual water content of clay. The simulated

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results are shown in Figs. 3, 4 and 5.

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The liquid water contents at the top all decrease as shown in Fig. 3. The decrease of sand is

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the largest, followed by silt, while there is almost no decrease in clay. The liquid water

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content at the bottom in silt increases remarkably but the increments in clay and silt are all

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very limited. Water holding capacity and permeability are the main reasons for the differe nce. For sand, the water holding capacity is the worst and the permeability is the best. Therefore,

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moisture will easily transfer and cannot then accumulate. However, for clay, the water holding capacity is the best and the permeability is the worst, which means that moisture cannot

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transfer almost. The water holding capacity and permeability in silt are all moderate, so

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moisture transferring from the unfrozen area to the freezing front can accumulate continuously. It should be noted that the decrease of the liquid water at the top has two parts of meanings in freezing soils. One part is that liquid water transfer downwards and the other part is that liquid water is a source for ice formation. Therefore, the ice content in sand is the largest, followed by silt, while there is almost no ice in clay, which is shown in Fig. 4. The total water contents in the three kinds of soils have a similar trend as shown in Fig. 5.

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Journal Pre-proof 20 19

Sand Silt Clay

18

Height, m

17 16 5 4 3 2 1 0 0.2 0.3 0.4 Liquid water content

0.5

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0.1

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Fig. 3 Liquid water content

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20

Sand Silt Clay

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Height, m

19.5

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19

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18.5

18

0.01

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0

0.02 0.03 Ice content

0.04

0.05

Fig. 4 Ice content

Height, m

20 19

Sand

18

Silt Clay

17 16 5 4 3 2 1 0 0.1

0.2 0.3 0.4 Total water content

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0.5

Journal Pre-proof Fig. 5 The total water content

The liquid water flux and the vapour flux are shown in Fig. 6. It can be seen that the liquid water flux is negative in the entire soil column of sand, indicating that the liquid water transfers downwards. α and n of sand are all large as shown in Table 1, so the suction is also low even though the initial water content is low. For example, the suction is just -0.285m when the water content is 0.08 in sand. Liquid water cannot easily transfer upwards from the

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ground water table by the suction. Therefore, the total water content at the bottom in the

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sandy column just increases slightly. The gravity potential plays a major role at this time, so

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liquid water will transfer downwards to the freezing front, resulting in an increase of the ice

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content. The liquid water flux at the freezing front is 0 m/s, which is caused by that the ice

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formation prevents liquid water transferring. The liquid water flux in clay is 0 m/s except that at the bottom. The liquid water flux at the bottom in silt is quite large, while the order of

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magnitude of the liquid water flux at the top is just 10 -14 m/s. However, it is notable that the order of magnitude of the liquid water flux at the top in sand or silt reaches 10 -11 m/s which is

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nearly 1000 times that of the liquid water flux. Therefore, it can be concluded that vapour

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transfer is the major reason for the ice formation at the frozen area, especially for silt. 20 (a) Sand Silt Clay

Height, m

15

10

5

0 -8 -1x10

-8

0 1x10 Liquid water flux, m/s

16

2x10

-8

Journal Pre-proof 20 (b)

Height, m

15

10

5

0

-11

1.75x10 Vapour flux, m/s

3.5x10

-11

of

0

ro

Fig. 6 The liquid water flux and the vapour flux

-p

A simulated example with a higher initial water content is also conducted. The initial water

re

contents of sand and silt are all 0.16. The initial water contents of clay are 0.16 and 0.31,

lP

respectively. The other conditions all remain unchanged. The simulated results are shown in Figs. 7, 8 and 9. In Fig. 7, the liquid water contents at the top in both sand and silt decrease,

na

but the maximum total water contents all increase about 0.05. The total water contents in both

ur

sand and silt are lower than the initial water content at a lower height, which is caused by that the permeability is large and then the liquid water will easily transfer downwards under the

Jo

effect of gravity potential. The ice contents in sand and silt are shown in Fig. 8. It can be seen that the maximum ice contents in sand and silt reach 0.07 and 0.08, respectively. However, the total water content change hardly and ice content is also 0 in clay as shown in Fig. 9. When the initial water content is 0.31, the ice content is just 0.04 as well. Therefore, the initial water content has a more significant effect on sand or silt instead of clay. It can be also concluded from this example that a larger initial water content can lead to a larger total water content and a larger ice content as well.

17

Journal Pre-proof 20 Sand Liquid water content Sand Total water content Silt Liquid water content Silt Total water content

Height, m

15

10

5

0 0.1

0.2 0.3 The fraction

0.4

0.5

of

0

ro

Fig. 7 Liquid water contents and total water contents in sand and silt

-p

20

re

19

lP

Height, m

19.5

na

18.5

Sand Silt

18

0.02

0.04 0.06 Ice content

0.08

0.1

ur

0

Jo

Fig. 8 Ice contents in sand and silt

20

Height, m

15

10 Ice content (0.16) Total water cotnent (0.16) Ice (0.31) Total water content (0.31)

5

0 0

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 The fraction

18

Journal Pre-proof Fig. 9 Ice contents and total water contents in clay

The vapour fluxes due to the suction and the temperature are shown in Fig. 10 and 11, respectively. The effects of the suction and the temperature on the vapour flux correspond to the third and the last terms in Eq. 1, respectively. It is found that the vapour flux is mainly caused by the temperature and the effect of the suction is almost negligible. In sand or silt, the vapour flux due to the temperature decreases with the increase of the initial water content,

of

but this decrease is quite limited, as shown in Fig. 11. The order of magnitude of the vapour

ro

flux at the top in sand or silt is also the same as that in Fig. 6. These results also show that the

-p

vapour transfer plays an important role in the ice formation even though the initial water

re

content increases. However, the vapour flux in clay cannot be ignored only when the initial

lP

water content is extremely large, but the quantity of the vapour flux is still small compared to that in sand or silt. A larger initial water content can reduce the channel for vapo ur

na

transferring while the liquid water also increases and can turn into vapour under the temperature, which is the reason for the increase of the vapour flux in clay in Fig. 11.

ur

Although the vapour flux is non- negligible in sand, moisture cannot accumulate for a better

Jo

permeability and a worse water holding capacity. Therefore, even if vapour turn into liquid water under a low temperature, moisture cannot accumulate and will then transfer downwards sequentially, resulting in few change of the total water co ntent. The test conducted by Zhang et al. (2016) shows that vapour can transfer upwards and then accumulates at the top in a sand column under some specific conditions, which seems to be contrary to the results simulated by the model proposed in this paper. Therefore, a further study for this difference is required in future.

19

Journal Pre-proof 20

10

Sand (0.08) Sand (0.16) Silt (0.08) Silt (0.16) Clay (0.08)

5

Clay (0.16) Clay (0.31)

Height, m

15

0

-12

-12

-12

-12

2x10 4x10 6x10 8x10 1x10 Vapour flux due to the suction, m/s

-11

of

0

ro

Fig. 10 Vapour flux due to the suction

-p

20

re

10

lP

Height, m

15

na

5

0

-12

-11

-11

-11

-11

7x10 1.4x10 2.1x10 2.8x10 3.5x10 Vapour flux due to the temperature, m/s

ur

0

Sand (0.08) Sand (0.16) Silt (0.08) Silt (0.16) Clay (0.08) Clay (0.16) Clay (0.31)

Jo

Fig. 11 Vapour flux due to the temperature

6. Analysis of the effects of factors on vapour transfer 6.1 The effects of the initial water content The effects of the initial water content on vapour transfer are shown in Fig. 12. No vapour means that vapour transfer is not considered at all in the governing equations. The initial conditions and boundary conditions are the same as those in Section 5. The total water content at the top is the most important for canopy effect, so the curves are shown only within the top 2m in Fig. 12. In fact, the total water content considering vapour has few differences to that without 20

Journal Pre-proof vapour in the other part of the soil column. It is seen that the effect of vapour transfer on the total water content is limited when the initial water content is low. However, the vapour transfer has a significant effect on the total water content in silt with a higher initial water content, but this effect is still few in sand or clay. The differences of the water holding capacity and the permeability are the major reason for those results, as discussed in Section 5.

of

20

ro

Height, m

19.5

(a)

19

-p

0.08 0.08; no vapour 0.16 0.16; no vapour

re

18.5

18 0.05

lP

0.2

na

20

0.1 0.15 Total water content

ur

Height, m

19.5

Jo

19

(b)

18.5

18 0.05

0.1

0.15 0.2 0.25 Total water content

21

0.3

Journal Pre-proof 20

19.5 Height, m

(c) 19

18.5

0.31 0.31; no vapour

18 0.05

0.15 0.2 0.25 Total water content

0.3

0.35

of

0.1

ro

Fig. 12 The effects of the initial water content: (a) sand, (b) silt and (c) clay

-p

6.2 The effects of the temperature

re

The effects of the temperature on vapour transfer are shown in Fig. 13.The temperatures at the

lP

top are -1 oC and -10 o C, respectively. And the initial water contents in sand, silt and clay are 0.16, 0.16 and 0.31, respectively. The other simulated conditions are also the same as those in

na

Section 5. It can be found that a lower temperature can lead to a larger total water content, a

ur

larger ice content and a deeper freezing front at the top in the same soil column. It is also seen that the effect of vapour transfer on the total water content in silt is the most remarkable. And

Jo

this effect will be more notable when the temperature at the top is lower. A similar trend of this effect is also shown in sand or clay, but this effect is quite limited.

22

Journal Pre-proof 20 (a)

Height, m

19.5

o

19

-1 C o

-10 C

18.5

o

-1 C; no vapour o

-10 C; no vapour 18 0.05

0.15 0.2 Total water content

0.25

of

0.1

20

ro

(b)

-p

Height, m

19.5

re

19

0.15 0.2 Total water content

na

18 0.1

lP

18.5

0.25

20

ur

(c)

Jo

Height, m

19.5

19

18.5

18 0.3

0.31

0.32 0.33 0.34 Total water content

0.35

Fig. 13 The effects of the temperature: (a) sand, (b) silt and (c) clay

6.3 The effects of the freezing period

23

Journal Pre-proof According to the above analyses, it can be concluded that silt is the most sensitive to vapour transfer. It is then rational and acceptant that silt is chosen for next analysis. The effects of the freezing period on vapour transfer are shown in Fig. 14. The simulated freezing period are 5d, 30d and 90d, respectively. And the other simulated conditions are also the same as those in Section 5. It is shown that a longer freezing period can lead to a larger total water content as well. And the effect of vapour transfer to the total water content is also more significant with the

of

freezing period increasing.

ro

20

-p

Height, m

19.5

na

18 0.1

lP

18.5

re

19

5d 5d; no vapour 30d 30d; na vapour 90d 90d; no vapour

0.15 0.2 Total water content

0.25

ur

Fig. 14 The effects of the freezing period

Jo

6.4 The effects of the ground water table It is found that the capillary water can transfer upwards about 4m as shown in Fig. 3 and the freezing depth is about 1.2m as shown in Fig. 4. Therefore, the height that the capillary water can reach is just greater than the freezing front when the ground water table is 5m. And this height is just lower than the freezing front when the ground water table is 6m. For simulating this situation, the ground water table are then set at the depths of 5m, 6m and 20m, respectively. The effects of the freezing period on vapour transfer are shown in Fig. 14 and silt is also chosen for analysis. The other simulated conditions are the same as those in Section 5 as well. It can be seen that the 24

Journal Pre-proof total water content at the top increases with the ground water table rising. The increment is more notable at the freezing front, especially when the ground water table is 5m because the liquid water dominates. However, the effect of vapour transfer will decreases with the ground water table rising. A rational reason is that the liquid water at the bottom will increase with the ground water table rising, which leads to fewer channels for vapour transferring. 20

of

Line: vapour Symbol: no vapour

ro

Height, m

19.5

-p

19

18.5

GWT5m

0.15 0.2 0.25 Total water content

0.3

lP

18 0.1

GWT6m

re

GWT20m

na

Fig. 15 The effects of the ground water table

ur

6.5 The increased amount of the total water content due to vapour transfer

Jo

Vapour transfer has the most effect on the total water content in silt, which is k nown from above analysis. The conditions of the simulated examples in Fig. 12 (b), Fig. 13 (b), Fig. 14 in the case of 90 days of the freezing period and Fig. 15 in the case of 20m of the ground water table are all the same. These conditions correspond to those of the canopy effect observed in northwestern China. And the result of these examples shows that the increased amount of the total water content due to vapour transfer is the largest. The increased amount is defined as that the difference in the total water content with and without considering the vapour transfer. The increased amount is shown in Fig. 16. It can be seen that this increased amount usually exists at the top, corresponding to the results shown in Fig. 6 (b) and Fig. 11. The increased 25

Journal Pre-proof amount in the all soil column and in the frozen area are 0.431 and 0.4, respectively. And the total mass of this increased amount and the average increase of the total water content in the frozen area are about 40kg and 0.033 within the unit area, respectively. It can be then concluded that the increase amount due to vapour transfer is quite remarkable though the increase of the total water content at the top is limited. Due to the difference of the temperature between day and night or the change of the season, the temperature will fluctuate around 0 o C. This freezing- thawing effect

of

can cause that ice in soils cannot completely melt even at a high temperature. Therefore, ice at

ro

the freezing front still exists, blocking the liquid water to transfer downwards. Vapour can

-p

continually transfer upwards meanwhile, increasing the total water content. Silt is usually

re

considered susceptible to frost heave. Notable frost heave will be observed and increase with the freezing period increasing even if the water content in silt is quite low, leading to serious

lP

engineering diseases (Zhang et al., 2015). In a word, although this new model proposed in this

na

paper cannot completely reproduce the canopy effect, the result of the significant increased amount in the frozen area has been obtained. Therefore, vapour transfer should be paid more

Height, m

Jo

ur

attention to, especially in freezing silt. 20 15 10 5 0

0

0.01 0.02 0.03 0.04 0.05 Total increment due to vapour transfer

Fig. 16 Total increment due to vapour transfer

26

Journal Pre-proof 7 Conclusions A new model for coupled liquid water-vapour- heat transfer in unsaturated freezing soils is proposed in this paper. The ice content and the maximum unfrozen water content are newly determined. The effects of vapour transfer on the total water contents in different kinds of soils are then analyzed using this model. Some conclusions can be obtained as follow.

of

(1) Vapour transfer in freezing silt or sand is significant and cannot be ignored when the

ro

initial water content is quite low. However, there are few vapour transfer in freezing clay. And vapour transfer has a remarkable effect on the total water content in silt instead of sand and clay.

-p

The water holding capacity and the permeability are the major reason for this difference. Besides,

re

the effects of the temperature, the freezing period and the ground water table on the total water

lP

content and vapour transfer are also outstanding.

na

(2) The increased amount of the total water content due to vapour transfer is the largest in silt which is usually considered susceptible to frost heave. And the total mass of this increased

ur

amount and the average increase of the total water content in the frozen area are about 40kg and

Jo

0.033 within the unit area, respectively. Therefore, vapour transfer should be paid more attention to in practical engineering. The simulated results suggest that vapour isolated measures should be deeply studied in future as well. Lastly, although this new model proposed in this paper cannot completely reproduce the canopy effect, the result of the significant increased amount in the frozen area has been obtained, which deepens the understanding of the canopy effect.

Acknowledgements This research was supported by the National Natural Science Foundation of China (No. 51722812), the Excellent Youth Foundation of Hu' nan Scientific Committee (No. 2017JJ1033), 27

Journal Pre-proof Open Fund of State Key Laboratory of Frozen Soil Engineering (No. SKLFSE201712) and Scientific Research Project of Guangzhou Science and Technology Project (No. 201707010479).

References Carsel R F, Parrish R S. Developing joint probability distributions of soil water retention characteristics[J].

of

Water Resources Research, 1988, 24(5): 755-769.

ro

Eigenbrod K, Kennepohl G. Moisture accumulation and pore water pressures at base of pavements[J].

-p

Transportation Research Record: Journal of the Transportation Research Board, 1996 (1546): 151-161. Fuchs M, Campbell G S, Papendick R I. An Analysis of Sensible and Latent Heat Flow in a Partially Frozen

re

Unsaturated Soil[J]. Soil Science Society of America Journal, 1978, 42(3): 379-385.

lP

Gray D M, Granger R J. In situ measurements of moisture and salt movement in freezing soils[J]. Canadian

na

Journal of Earth Sciences, 1986, 23(5): 696-704.

Guthrie W S, Hermansson Å, Woffinden K H. Saturation of granular base material due to water vapor flow

ur

during freezing: laboratory experimentation and numerical modeling[M]//Current Practices in Cold

Jo

Regions Engineering. 2006: 1-12.

Guymon G L, Luthin J N. A coupled heat and moisture transport model for arctic soils[J]. Water Resources Research, 1974, 10(5): 995-1001. Hansson K, Šimůnek J, Mizoguchi M, et al. Water flow and heat transport in frozen soil[J]. Vadose Zone Journal, 2004, 3(2): 693-704. Harlan R L. Analys is of coupled heat-fluid transport in partially frozen soil[J]. Water Resources Research, 1973, 9(5): 1314-1323. Hoekstra P. Moisture Movement in Soils under Temperature Gradients with the Cold-Side Temperature Below Freezing[J]. Water Resources Research, 1966, 2(2): 241-250.

28

Journal Pre-proof Ishida T. Water Transport Through Soil-Plant System[J]. Bulletin of the Yamagata University Agricultural Science, 1985, 9:573-707. Jame Y W, Norum D I. Heat and mass transfer in a freezing unsaturated porous medium[J]. Water Resources Research, 1980, 16(4): 811-819. Kane D L, Stein J. Water movement into seasonally frozen soils[J]. Water Resources Research, 2010, 19(19): 1547-1557.

of

Kelleners T J. Coupled water flow and heat transport in seasonally frozen soils with snow accumulation[J].

ro

Vadose Zone Journal, 2013, 12(4): 108-118.

-p

Kelleners T J, Koonce J, Shillito R, et al. Numerical modeling of coupled water flow and heat transport in soil and snow[J]. Soil Science Society of America Journal, 2016, 80(2): 247-263.

re

Kennedy G F, Lielmezs J. Heat and Mass Transfer of Freezing Water-Soil System[J]. Water Resources

lP

Research, 1973, 9(2): 395-400.

Kung S K J, Steenhuis T S. Heat and moisture transfer in a partly frozen non heaving soil[J]. Soil Science

na

Society of America Journal, 1986, 50(5): 1114-1122.

ur

Konrad J M, Morgenstern N R. A mechanistic theory of ic e lens formation in fine-grained soils[J]. Canadian

Jo

Geotechnical Journal, 1980, 17(4): 473-486. Konrad J M, Morgenstern N R. Prediction of frost heave in the laboratory during transient freezing[J]. Canadian Geotechnical Journal, 2011, 19(3): 250-259. Konrad J M, Morgenstern N R. The segregation potential of a freezing soil[J]. Canadian Geotechnical Journal, 1981, 18(4): 482-491. Lai Y, Pei W, Zhang M, et al. Study on theory model of hydro-thermal–mechanical interaction process in saturated freezing silty soil[J]. International Journal of Heat and Mass Transfer, 2014, 78: 805-819. Li Q, Sun S, Dai Q. The numerical scheme development of a simplified frozen soil model[J]. Advances in Atmospheric Sciences, 2009, 26(5): 940-950.

29

Journal Pre-proof Li Q, Sun S. Development of the universal and simplified soil model coupling heat and water transport[J]. Sciencein China Series D: Earth Sciences, 2008, 51(1): 88-102. Li Q, Sun S, Xue Y. Analyses and development of a hierarchy of frozen soil models for cold region study[J]. Journal of Geophysical Research: Atmospheres, 2010, 115(D3). Li Q, Yao Y, Han L, et al. Pot-cover effect of soil[J]. Industrial Construction, 2014, 44(2): 69-71. (In Chinese) Liu B, Li D. A simple test method to measure unfrozen water content in clay–water systems[J]. Cold Regions

of

Science and Technology, 2012, 78: 97-106.

ro

Mageau D W, Morgenstern N R. Observations on moisture migration in frozen soils[J]. Canadian

-p

Geotechnical Journal, 1980, 17(17): 54-60.

McKenzie J M, Voss C I, Siegel D I. Groundwater flow with energy transport and water–ice phase change:

re

numerical simulations, benchmarks, and application to freezing in peat bogs[J]. Advances in water

lP

resources, 2007, 30(4): 966-983.

na

Miller R D. Freezing and heaving of saturated and unsaturated soil[J]. Highway Research Record, 1972. Milly P C D. A simulation analys is of thermal effects on evaporation from soil[J]. Water Resources Research,

ur

1984, 20(8): 1087-1098.

Jo

Mizoguchi M. Water, heat and salt transport in freezing soil[D]. University of Tokyo, Tokyo, 1990. Mualem Y. A new model for predicting the hydraulic conductivity of unsaturated porous media[J]. Water resources research, 1976, 12(3): 513-522. Nakano Y, Tice A, Oliphant J. Transport of water in frozen soil IV. Analysis of experimental results on the effects of ice content[J]. Advances in water resources, 1984, 7(2): 58-66. Nassar I N, Horton R. Simultaneous transfer of heat, water, and solute in porous media: I. Theoretical development[J]. Soil Science Society of America Journal, 1992, 56(5): 1350-1356. Nassar I N, Horton R. Water transport in unsaturated nonisothermal salty soil: II. Theoretical development[J]. Soil Science Society of America Journal, 1989, 53(5): 1330-1337.

30

Journal Pre-proof Newman G P, Wilson G W. Heat and mass transfer in unsaturated soils during freezing[J]. Canadian Geotechnical Journal, 1997, 34(1): 63-70. O'Neill K, Miller R D. Exploration of a Rigid Ice Model of Frost Heave[J]. Water Resources Research, 1985, 21(3): 281-296. Patterson D E, Smith M W. The measurement of unfrozen water content by time domain reflectometry: Results from laboratory tests[J]. Canadian Geotechnical Journal, 1981, 18(1): 131-144.

of

Sakai M, Toride N, S̆ImůNek J. Water and vapor movement with condensation and evaporation in a sandy

ro

column.[J]. Soil Science Society of America Journal, 2009, 73(3):707-717.

-p

Saito H, Šimůnek J, Mohanty B P. Numerical analysis of coupled water, vapor, and heat transport in the vadose zone[J]. Vadose Zone Journal, 2006, 5(2): 784-800.

re

Sheng D, Axelsson K, Knutsson S. Frost heave due to ice lens formation in freezing soils: 1. Theory and

lP

verification[J]. Hydrology Research, 1995, 26(2): 125-146. Sheng D, Axelsson K, Knutsson S. Frost heave due to ice lens formation in freezing soils: 2. Field

na

application[J]. Hydrology Research, 1995, 26(2): 147-168.

ur

Sheng D, Zhang S, Niu F J, et al. A potential new frost heave mechanism in high-speed railway

Jo

embankments[J]. Géotechnique, 2014, 64(2): 144-154. Sheng D, Zhang S, Yu Z, et al. Assessing frost susceptibility of soils using PCHeave[J]. Cold Regions Scienc e and Technology, 2013, 95: 27-38. Stähli M, Jansson P E, Lundin L C. Soil moisture redistribution and infiltration in frozen sandy soils[J]. Water Resources Research, 1999, 35(1): 95-104. Stein J, Kane D L. Monitoring the unfrozen water content of soil and snow using time domain reflectometry[J]. Water Resources Research, 1983, 19(6): 1573-1584. Taylor G S, Luthin J N. A model for coupled heat and moisture transfer during soil freezing[J]. Canadian Geotechnical Journal, 1978, 15(4): 548-555.

31

Journal Pre-proof Teng J, Shan F, He Z, et al. Experimental study of ice accumulation in unsaturated clean sand[J]. Géotechnique, 2019, 69(3): 251-259. Teng J, Zhang X, Zhang S, et al. An analytical model for evaporation from unsaturated soil[J]. Computers and Geotechnics, 2019, 108: 107-116. Tice AR, Anderson DM, Banin A. The prediction of unfrozen water contents in frozen soils from liquid limit determinations. Cold Regions Research & Engineering Laboratory, U.S. Army Corps of Engineers, 1976.

ro

Soil science society of America journal, 1980, 44(5): 892-898.

of

Van Genuchten M T. A closed-form equation for predicting the hydraulic conductivity of unsaturated soils[J].

-p

Wang A W, Xie Z H, Feng X B, et al. A soil water and heat transfer model including changes in soil frost and thaw fronts[J]. Science China Earth Sciences, 2014, 57(6): 1325-1339.

re

Williams P J. Properties and behavior of freezing soils[J]. Norwegian Geotechnical Institute, 1967.

lP

Wu D, Lai Y, Zhang M. Heat and mass transfer effects of ice growth mechanisms in a fully satur ated soil[J].

na

International Journal of Heat and Mass Transfer, 2015, 86: 699-709. Zhang S, Sheng D, Zhao G, et al. Analysis of frost heave mechanisms in a high-speed railway embankment[J].

ur

Canadian Geotechnical Journal, 2015, 53(3): 520-529.

Jo

Zhang S, Teng J, He Z, et al. Canopy effect caused by vapour transfer in covered freezing soils[J]. Géotechnique, 2016, 66(11): 927-940. Zhang S, Teng J, He Z, et al. Importance of vapor flow in unsaturated freezing soil: a numerical study[J]. Cold Regions Science & Technology, 2016, 126:1-9. Zhang X, Sun S, Xue Y. Development and Testing of a Frozen Soil Parameterization for Cold Region Studies[J]. Journal of Hydrometeorology, 2007, 8(4): 852-861. Zhou J, Wei C, Li D, et al. A moving-pump model for water migration in unsaturated freezing soil[J]. Cold Regions Science and Technology, 2014, 104: 14-22.

32

Journal Pre-proof Zhou X, Zhou J, Kinzelbach W, et al. Simultaneous measurement of unfrozen water content and ice content in frozen soil using gamma ray attenuation and TDR[J]. Water Resources Research, 2014, 50(12): 9630-9655.

Appendix:

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A1: Hydraulic parameters

-p

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The soil water characteristic curve (SWCC) is written as follows (van Genuchten, 1980)

Se  1  ( h)n 

-m

(A1)

re

where, Se=(θ w-θr)/(θs-θr) is the effective degree of saturation. θs and θr are the saturated and

lP

residual water content. α (1/m), n and m (=1-1/n) are the fitting parameters of the SWCC.

na

' The isothermal hydraulic conductivities of liquid water due to the water pressure head K wh

m ' K wh  10i K wh  10i Ks Sel 1  1  Se1/m   (A2)   2

Jo

ur

(m/s) in freezing soils is expressed as (Mualem, 1976; Taylor and Luthin, 1978)

where, K wh (m/s) is the isothermal hydraulic conductivities of liquid water due to the water pressure head in unfrozen soils. Ks (m/s) is the saturated hydraulic conductivity. l (=0.5) is the fitting parameter in Mualem model (Mualem, 1976). Ω is an empirical parameter. The thermal hydraulic conductivities of liquid water due to the temperature K wT (m2 /K/s) is expressed as (Saito et al., 2006)

K wT  K wh hGwT

33

1 d (A3)  0 dT

Journal Pre-proof where, G wT is the gain factor. γ (=75.6-0.1425(T-273.15)-2.38*10-4 (T-273.15)2 g/s2 ) is the surface tension of soil water. γ0 (=71.89 g/s2 ) is the surface tension at 25 o C. The isothermal hydraulic conductivities of vapour due to the water pressure head Kvh (m/s) is given as (Saito et al., 2006)

K vh 

D

w

vs

Mg H r (A4) RT

of

where, D (=τηanaD0 m2 /s) is the vapour diffusion coefficient in soils. τ is the tortuosity factor.

ro

ηa is a strengthening factor. na is the fraction of air. D0 (m2 /s) is the vapour diffusion

-p

coefficient in air. M (=0.018 kg/mol) is the molecular weight of liquid water. g (m/s2 ) is the R (=8.341 J/mol/K) is the universal gas constant.

Hr

re

gravitational acceleration.

lP

(=exp(hMg/R/T)) is the relative humidity. The fraction of the vapour θv is equal to ρvsHr(θs-

3

6014.79   10 vs  exp  31.37   7.92 103 T   T   T

(A5)

ur

na

θr)/ρw and the saturated vapour density is given as

Jo

The thermal hydraulic conductivities of vapour due to the temperature KvT (m2 /K/s) is given as (Saito et al., 2006)

K vT 

D

w

 Hr

d vs (A6) dT

where, η is an enhancement factor. A2: Thermal parameters The effective heat capacity Cp (J/m3 /K) is written as (Lai et al., 2014; McKenzie et al., 2007)

Cp  Cnn  Cww  Cvv  Cii (A7) 34

Journal Pre-proof where, θn is the fraction of the soil skeleton. Cx (J/m3 /K) is the heat capacity of each phase and x=n, w, v and i. The effective thermal conductivity λ’(W/m/K) is expressed as (Wu et al., 2015)

 '   n 

n

 w   v   i  w

v

i

(A8)

where, λx (W/m/K) is the thermal conductivity of each phase and x=n, w, v and i.

of

The latent heats of water vaporization Lv (J/kg) can be expressed as (Sakai et al., 2009)

Jo

ur

na

lP

re

-p

ro

Lv  2.501106  2369.2  T (A9)

35

Journal Pre-proof 1. A new model for coupled liquid water-vapour-heat transfer in unsaturated freezing soils is proposed in this paper. The ice content and the maximum unfrozen water content are newly determined.

Jo

ur

na

lP

re

-p

ro

of

2. The increased amount of the total water content due to vapour transfer is the largest in silt which is usually considered susceptible to frost heave.

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