Practical models describing hysteresis behavior of unfrozen water in frozen soil based on similarity analysis

Cold Regions Science and Technology 157 (2019) 215–223

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Practical models describing hysteresis behavior of unfrozen water in frozen soil based on similarity analysis

T

⁎

Yang Zhoua,b,c, , Jian Zhoua, Xiang-you Shia, Guo-qing Zhoua,b a

State Key Laboratory for Geomechanics and Deep Underground Engineering, China University of Mining & Technology, Xuzhou, Jiangsu 221116, PR China School of Mechanics & Civil Engineering, China University of Mining & Technology, Xuzhou, Jiangsu 221116, PR China c JiangSu Collaborative Innovation Center for Building Energy Saving and Construct Technology, Xuzhou, Jiangsu 221116, PR China b

A R T I C LE I N FO

A B S T R A C T

Keywords: Unfrozen water Hysteresis model Frozen soil Unsaturated soil Similarity theory

Hysteresis behavior of unfrozen water in frozen soil is investigated, and practical models which predict the soil thawing characteristic (STC) curve from the soil freezing characteristic (SFC) curve are developed. By comparing the Gibbs-Thomson equation in frozen soil with the Young-Laplace equation in unsaturated soil, a similarity theory is established, which indicates that formulas and models describing liquid water in unsaturated soil can be applied to describe unfrozen water in frozen soil using the same form. Five models describing hysteresis behavior of unfrozen water are then developed from theories of unsaturated soil. Comparisons between model predictions and measured results in literature are conducted. The predicted results of three models are in good agreement with measured data, and these three models are derived from the formulas of Pham et al. (2003), Van Genuchten (1980) and Fredlund and Xing (1994), respectively. The concept of average relative error is introduced to investigate the accuracy of different models in a quantitative way, and the results show that the model derived from the formula of Fredlund and Xing (1994) has the best performance in describing hysteresis behavior of unfrozen water in frozen soil.

1. Introduction Soil freezing phenomenon is frequently encountered for engineering both in cold regions and in the application of artificial ground freezing (AGF) technique. During the engineering design process, the temperature calculation of frozen ground is of great significance since the mechanical behavior of frozen soil is mainly controlled by the soil temperature. For simplified calculation, soil can be treated as a two-phase material like water, and methods used in Stefan problems can be applied to obtain the temperature field of frozen ground (Zhou et al., 2014, 2018a; Zhou and Xia, 2015). However, for practical calculation, the effect of unfrozen water should be considered. It is well-known that unfrozen water exists in frozen soil at temperatures below soil freezing point (Williams, 1964; Anderson and Morgenstern, 1973). The presence of unfrozen water has a great impact not only on the development of temperature field but also on the redistribution of water and the frost heave of soil. From another perspective, the formula describing the soil freezing characteristic (SFC) also provides an important equation for mathematical models of soil freezing process (Harlan, 1973; Jame and Norum, 1980; O'Neill and Miller, 1985; Zhou and Zhou, 2010, 2012;

Zhou et al., 2018b; Li et al., 2018). Since the unfrozen water is important to mechanical and physical behaviors of frozen soil, many researches including the measurement, the empirical formula and the physical theory of unfrozen water have been conducted. Methods for the measurement of unfrozen water can be classified into four categories, which are the dilatometer method (Patterson and Smith, 1981; Spaans and Baker, 1995), the calorimetric method (Kolaian and Low, 1963; Williams, 1964; Kozlowski, 2003a,b), the time domain reflectometry (TDR) method (Patterson and Smith, 1981; Oliphant, 1985; Smith and Tice, 1988) and the nuclear magnetic resonance (NMR) method (Yoshikawa and Overduin, 2005). Through analyzing experimental data, several empirical formulas have been presented. The most-widely-used formula is the power-type formula (Anderson and Tice, 1971; Xu et al., 2001); Xu et al. (2001), Nixon (1991) presented fitting parameters of the power-type formula for many soil samples. There are also other types of empirical formulas that can be used to describe the SFC curve (Dillon and Andersland, 1966; Anderson and Tice, 1972). These formulas are summarized in the introduction section of Liu and Yu (2013), and there is no need to reintroduce them here.

⁎ Corresponding author at: State Key Laboratory for Geomechanics and Deep Underground Engineering, China University of Mining & Technology, Xuzhou, Jiangsu 221116, PR China. E-mail address: [email protected] (Y. Zhou).

https://doi.org/10.1016/j.coldregions.2018.11.002 Received 17 February 2018; Received in revised form 31 October 2018; Accepted 1 November 2018 Available online 02 November 2018 0165-232X/ © 2018 Elsevier B.V. All rights reserved.

Cold Regions Science and Technology 157 (2019) 215–223

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However, there are controversial issues concerning the application of Clapeyron equation in frozen soil and assumptions about the pore-water pressure and the pore-ice pressure, which are used during the development of these similarity theories. Therefore, this section abandons the Clapeyron equation along with these assumptions, and establishes a new similarity theory using the Gibbs-Thomson equation. The soil water characteristic (SWC) of unsaturated soil is a relation between the liquid water content and the soil matric suction during the drying or the wetting process of soil. The matric suction of unsaturated soil is defined as (Fredlund and Xing, 1994)

For theoretical analysis of unfrozen water, there are three types of theories reported in literature. The first type is the liquid film theory developed by Gilpin (1980). Gilpin (1980) assumed that the liquid water near the surface of solid substrate experiences an adsorption force acting towards the solid. Through a thermodynamic analysis, a mathematical equation was derived relating the thickness of unfrozen film with the temperature, the disjoining pressure and the average surface curvature. However, the theory does not present any formulas for the unfrozen water content. The second type is named as the similarity theory, because the similarity between the saturated frozen soil and the unsaturated soil is utilized. The similarity theory obtains the formula describing the SFC curve through two steps. First, the matric suction of frozen soil is linked to the matric suction of unsaturated soil. Secondly, the relation between the soil temperature and the matric suction of frozen soil is developed, and it is usually done by introducing the Clapeyron equation. Two existing similarity theories developed by Koopmans and Miller (1966) and Liu and Yu (2013, 2014) can be found in literature. The third type of theory is developed utilizing the GibbsThomson equation (Wang et al., 2017; Bai et al., 2018). For a given subfreezing temperature, the Gibbs-Thomson equation determines the critical pore radius below which the water is at a liquid state. Therefore, the unfrozen water content in the frozen soil can be calculated through an integration process once the function describing the soil pore distribution is specified. During the thawing process of frozen soil, the relation between the unfrozen water content and the temperature is usually different from the SFC. This relation is defined as the soil thawing characteristic (STC) hereafter, and the difference between the STC and the SFC is wellknown as hysteresis. The mechanism for the hysteresis behavior of unfrozen water can be explained from many aspects. The first aspect comes from the pore-blocking effect, in which the shape of ink-bottle is often used to describe the soils pores. The freezing point of the inkbottle pore, which is controlled by the size of bottleneck, is lower than the melting point, which is determined by the size of bottle. The second aspect is the effect of free-energy barrier (Petrov and Furo, 2006); due to this effect, the freezing point depression for a given pore is larger than the melting point depression of the pore, and can be described by the freezing-thawing hysteresis coefficient (Wang et al., 2018). The third aspect is the effect of contact angle, the advancing contact angle during the soil freezing process is different from the retreating contact angle during the thawing process, which also leads to hysteresis behavior. Other aspects such as the supercooling phenomenon may influence the hysteresis behavior of unfrozen water as well. Although a large amount of research work has been conducted on the unfrozen water in frozen soil, the investigation on the hysteresis behavior of unfrozen water is insufficient. There are experimental results reported in literature (Koopmans and Miller, 1966; Wang et al., 2007; Tian et al., 2014; Kruse and Darrow, 2017; Zhang et al., 2018); however, no mathematical model describing the hysteresis behavior of unfrozen water is developed to the best of authors' knowledge. In this paper, the main objective is to develop practical hysteresis models which predict the STC curve from the SFC curve. A similarity theory is first developed, which indicates that formulas and models describing liquid water in unsaturated soil can be applied to describe unfrozen water in frozen soil using the same form. Five models describing the hysteresis behavior of unfrozen water in frozen soil are then established, and experimental data reported in literature are applied to evaluate the accuracy of these models.

ϕaw = ua − u w

(1)

in which ua is the pore-air pressure and uw is the pore-water pressure. To establish a theoretical formula for the SWC, the structure of soil pores needs to be considered, and it is usually simplified as a bundle of interconnected and randomly distributed cylindrical capillaries. A probability density function f(r), which represents the relative volume of capillaries of radius r to r + dr, is used to describe the radius distribution of these capillaries. The Young-Laplace equation for unsaturated soil can be written as

ϕaw =

2σaw cos φ r

(2)

in which σaw is the air-water interfacial tension, φ is the contact angle between the water and the capillary surface, r is the radius of the capillary. For a drying process or a wetting process, the contact angle is constant, thus Eq. (2) presents a unique reciprocal relation between ϕaw and r. For a given ϕaw, the radius determined by Eq. (2) is denoted as rc, and capillaries with radius larger than rc are occupied by air. The liquid water content for unsaturated soil at a matric suction of ϕaw can then be written as (Fredlund and Xing, 1994)

θ (ϕaw ) =

∫r

rc

f (r ) dr

min

(3)

in which rmin is the minimum capillary radius. On the other hand, the Gibbs-Thomson equation describing the depression of freezing point in capillaries is (Wang et al., 2017)

T0 − T =

T0 γiw Lρi r

(4)

in which T is the freezing point (K), T0 = 273.15 K, and γiw is the icewater interfacial energy. From Eq. (4), there is a unique reciprocal relation between the depression of freezing point and the radius of the capillary (there is a similar reciprocal relation for the melting point as well). For a given temperature T, the capillary radius determined from Eq. (4) is denoted as rc, and capillaries with radius larger than rc are at frozen state. The unfrozen water content can then be expressed as

θ (T ) =

∫r

rc

f (r ) dr

min

(5)

Defining the primitive function of f(r) as F(r), Eq. (5) can be rewritten as.

K θ (T ) = F ⎛ c ⎞ − C , C = F (rmin ) ⎝ ∆T ⎠

(6)

in which Kc is T0γiw/Lρi, ΔT is T0-T. Using the primitive function F(r), the liquid water content in unsaturated soil presented by Eq. (3) can be rewritten as

2. A new similarity theory between saturated frozen soil and unsaturated soil

K θ (ϕaw ) = F ⎛⎜ u ⎞⎟ − C ϕ ⎝ aw ⎠

The unfrozen water in saturated frozen soil is similar to the liquid water in unsaturated soil, and some similarity theories have been developed using the Young-Laplace equation and the Clapeyron equation (Koopmans and Miller, 1966; Black and Tice, 1989; Liu and Yu, 2013).

(7)

in which Ku is 2σaw cos φ. Comparing Eq. (6) with Eq. (7), the unfrozen water content in frozen soil can be obtained from the SWC formula of unsaturated soil 216

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using

ϕaw =

Ku ∆T Kc

(8)

Therefore, this new similarity theory presents a linear relation between ϕaw and ΔT, and it is similar with the theory of Koopmans and Miller (1966). However, this similarity theory is more complete since the Clapeyron equation and corresponding assumptions are avoided. To some extent, the experimental validation of the theory of Koopmans & Miller (Koopmans and Miller, 1966; Black and Tice, 1989) also confirms this similarity theory. Linear transformation usually does not change the form of equation. Therefore, formulas used to fit the SWC in unsaturated soil can also be used to fit the phase composition curve (PCC) of frozen soil in the same form; models describing the hysteresis behavior of the SWC in unsaturated soil can also be applied to describe the hysteresis behavior of PCC in frozen soil. This is a practical application of this new similarity theory. In Eq.(4), the ice-water interfacial energy is used, which means the effect of air is excluded, therefore strict application of above similarity theory requires the condition of saturated frozen soil. However, as long as the air does not alter the shape of the PCC curve essentially, above similarity theory may still be applied as an approximation. Another well-known issue is that the presence of salinity has great effect on the unfrozen water; nevertheless, above similarity theory may still be applicable if the effect of salinity does not change the shape of the PCC curve essentially.

Fig. 1. Two points suggested by Pham et al. (2003) (BDC: boundary drying curve; BWC: boundary wetting curve; SWC*: scanning wetting curve).

3. Hysteresis models obtained from theories in unsaturated soil From theories in unsaturated soil, five models describing the hysteresis behavior of unfrozen water in frozen soil are developed in this section. Fig. 2. Calibration of model I using the data in Kruse and Darrow (2017).

3.1. Model I ϕ λ ⎡ 1 + λ − λ we ⎤ ⎧ ϕ ⎪ θmax ⎜⎛ ϕae ⎟⎞ ⎢ ⎥ ϕ > ϕae ϕwe θd (ϕ) = ϕ ⎢ ⎥ 1 λ λ + − ⎝ ⎠ ⎨ ϕ ae ⎦ ⎣ ⎪ θmax ϕ ≤ ϕae ⎩

The first model is denoted as model I, and it is obtained from the hysteresis model in unsaturated soil presented by Mualem (1977). Mualem (1977) derived the following formula that predicts the SWC of the wetting process from the SWC of the drying process

Swe (ϕaw ) = 1 − [1 − Sde (ϕaw )]1/2

in which ϕaw is simplified as ϕ, ϕae is the air entry value, ϕwe is the water entry value. The parameter ϕwe is usually estimated from the measured data, and other parameters θmax,ϕae, and λ are used as curvefitting parameters. After the curve-fitting formula for the SWC of the drying process is obtained, the formula for the SWC of the wetting process can be written as

(9)

in which the subscripts ‘w’ and ‘d’ represent the wetting process and the drying process, respectively. The effective saturation is defined as

Se =

θ − θmin θmax − θmin

(10)

in which θmin, θmax are the minimum and the maximum water contents experienced during the wetting and the drying processes. From the similarity theory, the following equation that predicts the STC from the SFC may immediately come to mind

Ste (t ) = 1 − [1 − S fe (t )]1/2

(12)

θw (ϕ) λ λ ⎧ λ ⎛ ϕae ⎞ ⎡ ϕ ⎤ ϕ ≥ ϕae θae ⎢ ⎜⎛ ae ⎟⎞ + ⎜ ⎟ (ϕ − ϕwe ) ⎥ ⎪ ϕ ⎠ ϕmax ⎝ ϕmax ⎠ ⎪ ⎝ ⎣ ⎦ ⎪ λ = ⎨ θ ⎡ ⎛1 + λ − λ ϕ ⎞ + λ ⎛ ϕae ⎞ (ϕ − ϕ ) ⎤ ϕ < ϕ < ϕ ⎟ ⎜ ⎟ we ⎥ we ae ⎪ ae ⎢ ⎜ ϕae ⎠ ϕmax ⎝ ϕmax ⎠ ⎦ ⎪ ⎣⎝ ⎪ θmax ϕ ≤ ϕwe ⎩ (13)

(11)

in which the subscripts ‘t’ and ‘f’ represent the thawing process and the freezing process, respectively. The effective saturation is still defined by Eq. (10) with θmin, θmax the minimum and the maximum water contents experienced during the freezing and the thawing processes.

in which ϕmax is the soil suction at the meeting point of the two SWC curves at high suction, and it is also estimated from the measured data; the parameter θae can be calculated from

3.2. Model II

θae =

The second model is denoted as model II, and it originates from the method in unsaturated soil presented by Hogarth et al. (1988). Hogarth et al. (1988) applied the following formula to fit the SWC data of the drying process for unsaturated soil

θmax 1 + λ − λ (ϕwe / ϕae )

(14)

From the similarity theory, the method by Hogarth et al. (1988) can be transformed to a method describing the hysteresis behavior of 217

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unfrozen water in frozen soil. The formula used to fit the SFC can be written as tm ⎧ t λ⎡1 + λ − λ t ⎤ ⎪ θmax ⎛ f ⎞ ⎢ t ⎥ t < tf θf (t ) = ⎝ t ⎠ ⎢ 1 + λ − λ tm ⎥ ⎨ f ⎦ ⎣ ⎪ θmax t ≥ tf ⎩

(15)

in which tf is the soil freezing point (°C), tm is the soil melting point (°C); tm is estimated from the measured data, and other parameters tf, θmax and λ are used as curve-fitting parameters. After the curve-fitting formula for the SFC is obtained, the formula for the STC can be written as λ

λ ⎧ λ t ⎡ t ⎤ θie ⎢ ⎛ f ⎞ + ⎛ f ⎞ (t − tm ) ⎥ t ≤ tf ⎪ t t t ⎝ ⎠ r r ⎝ ⎠ ⎪ ⎣ ⎦ ⎪ λ θt (t ) = ⎨ θ ⎡ ⎛1 + λ − λ t ⎞ + λ ⎛ t f ⎞ (t − t ) ⎤ t < t < t ie ⎢ m ⎥ f m ⎪ t f ⎠ tr ⎝ tr ⎠ ⎦ ⎪ ⎣⎝ ⎪ θmax t ≥ tm ⎩ ⎜

⎜

⎟

⎟

⎜

⎟

(16)

in which tr is the temperature at the meeting point of the SFC and the STC curves at low temperature, and it is estimated from the measured data; the parameter θie can be calculated from

θie =

θmax 1 + λ − λ (tm/ t f )

(17)

3.3. Model III The third model is denoted as model III, and it originates from the hysteresis model in unsaturated soil presented by Pham et al. (2003). Pham et al. (2003) applied the following form of equation to describe the SWC for both the drying and the wetting processes

w (ϕ) =

ab + cϕd b + ϕd

(18)

in which w is gravimetric water content, and a, b, c, d are four curvefitting parameters. Eq. (18) is first used to fit the SWC data of the drying process, and the curve-fitting parameters are obtained and denoted as a, bd, c, dd. For the SWC curve of the wetting process, the two parameters a and c are assumed to be the same as that of the drying process; therefore, the subscripts for a and d are omitted. Another two parameters b and d for the wetting process are determined from two data points on the wetting SWC curve. The matric suctions of the two points suggested by Pham et al. (2003) are 1/ dd

b ϕ1w = ⎛ d ⎞ ⎝ 10 ⎠

(19) 1/ dd

⎧ b (a − w1 ) ⎤ ϕ2w = ϕ1w − 2 ⎡ d ⎥ ⎢ w1 − c ⎦ ⎨⎣ ⎩

− bd1/ dd

⎫ ⎬ ⎭

(20)

Denoting these two data points as (ϕ1w,w1), (ϕ2w,w2), another two parameters for the wetting process can be determined as (w − c )(a − w )

dw =

Fig. 3. (a) Calibration of model II using the data in Koopmans and Miller (1966). (b) Calibration of model II using the data in Wang et al. (2007). (c) Calibration of model II using the data in Tian et al. (2014). (d) Calibration of model II using the data in Kruse and Darrow (2017).

bw =

ln ⎡ (a 1− w )(w − 2c ) ⎤ 1 2 ⎣ ⎦ ln(ϕ2w / ϕ1w ) (w1 −

(21)

c ) ϕ1dww

a − w1

(22)

Fig. 1 shows the positions of above two points (points P, R); the concepts of the BDC (boundary drying curve), the BWC (boundary wetting curve) and the SWC* (scanning wetting curve) can be found in Mualem (1977) and Pham et al. (2003). The distances from above two points to the characteristic line are the same 218

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Table 1 Parameters used in model II. Data source

Curve

tf

θmax

λ

tm

tr

Koopmans and Miller (1966) Koopmans and Miller (1966) Wang et al. (2007) Wang et al. (2007) Tian et al. (2014) Tian et al. (2014) Kruse and Darrow (2017) Kruse and Darrow (2017)

SFC STC SFC STC SFC STC SFC STC

−0.0901 −0.0901 −2.8270 −2.8270 −4.5499 −4.5499 −1.3511 −1.3511

41.0121 41.0121 26.2675 26.2675 99.0720 99.0720 43.4500 43.4500

4.0621 4.0621 0.7092 0.7092 13.6824 13.6824 0.2240 0.2240

−0.0180 −0.0180 −0.3000 −0.3000 −2.2900 −2.2900 −0.3900 −0.3900

Na −0.1500 Na −20.2100 Na −22.7600 Na −19.9000

Na = Not applicable.

(|ϕ1w − ϕO ∣ = |ϕ2w − ϕO|); the characteristic line is perpendicular to the ϕ axis and passes through the presumed inflection point of the BWC curve (point O, determined by |ϕ1w − ϕO ∣ = ∣ ϕQ − ϕI∣). These two points help to restrict the general shape of the BWC effectively. From the similarity theory, the method by Pham et al. (2003) can be transformed to a method describing the hysteresis behavior of unfrozen water in frozen soil. The following formula is applied to describe the SFC and the STC

w (t ) =

ab + c |t|d b + |t|d

θ =

θ − θr θs − θr

(29)

From the similarity theory, the formula describing the unfrozen water content in frozen soil can be written as (for both SFC and STC) m

1 ⎤ θ =⎡ n ⎢ ⎣ 1 + (α | t |) ⎥ ⎦

(30)

Following the procedure of model III, model IV is developed using Eq. (30) through two steps. First, Eq. (30) is used to fit the SFC data, and then the parameters θs, θr, m in the SFC formula are also applied for the STC formula. Secondly, the parameters α, n are determined from two measured points on the STC curve; these two points are chosen based on Eqs. (24)–(25), and they are denoted as (t1, θ1), (t2, θ2 ). Using these two measured points, the parameters α, n for the STC formula can be determined explicitly

(23)

in which w is now the gravimetric unfrozen water content. Eq. (23) is first used to fit the SFC data, and the curve-fitting parameters are obtained and denoted as a, bf, c, df. For the STC curve, the parameters a and c are also assumed to be the same as that of the SFC curve; therefore, the subscripts for a and c are omitted. Another two parameters b, d for the STC curve are determined from two data points on the STC curve. The temperatures for these two data points are

−1/ m

n=

ln[(θ1

−1/ m

− 1)/(θ2 ln(| t1 |/| t2 |)

− 1)] (31)

−1/ m

1/ df

b t1 = −⎛ f ⎞ 10 ⎝ ⎠

α= (24) 1/ df

⎧ b (a − w1 ) ⎤ t2 = t1 + 2 ⎡ f ⎨⎢ w1 − c ⎥ ⎦ ⎩⎣

−

⎫ bf1/ df ⎬ ⎭

Denoting these two data points as (t1, w1), (t2, w2), another two parameters of the STC curve can be determined as (w − c )(a − w )

ln ⎡ (a 1− w )(w − 2c ) ⎤ 1 2 ⎣ ⎦ ln(| t2 |/| t1 |)

(26)

bt =

(w1 − c )|t1 |dt a − w1

(27)

− 1)1/ n ∣t1 ∣

(32)

The parameters θs, θr represent the water content for soil in unfrozen state and the residue water content for soil at extremely low temperature, respectively. Therefore, these two parameters should be the same for both the SFC curve and the STC curve. For parameters α, m, n, we can also assume that α (or n) is the same for both curves, and m, n (or α, m) are determined from the two measured points. However, there will be no explicit formula for m, n (or α, m) in this situation. Thus, we assume that the parameter m is the same for both curves, and α, n can be determined explicitly from the two measured points using Eqs. (31)–(32).

(25)

dt =

(θ1

3.5. Model V

The procedure of model III contains two steps. The first step is to fit the SFC data using Eq. (23), and the parameters a, c of the SFC curve are also applied in the STC formula. The second step is to compute the parameters b, d from the two measured points on the STC curve directly. In sections 3.4–3.5, the procedure of model III is incorporated into another two formulas that are widely used in describing the SWC of unsaturated soil, and models IV-V are developed.

Fredlund and Xing (1994) presented the following formula m

1 ⎤ θ =⎡ n ⎢ ⎣ ln[e + (ϕ/ a) ] ⎥ ⎦

(33)

in which θ is also the normalized water content defined by Eq. (29), and θs, θr, a, m, n are curve-fitting parameters. From the similarity theory, the formula describing the unfrozen water content in frozen soil can be written as (for both the SFC and the STC)

3.4. Model IV

m

1 ⎤ θ=⎡ ⎢ ln[ e + (| t |/ a)n] ⎥ ⎣ ⎦

Van Genuchten (1980) presented the following formula m

1 ⎤ θ =⎡ n ⎢ ⎣ 1 + (αϕ) ⎥ ⎦

(34)

Similar to model IV, model V is developed through two steps. First, Eq. (34) is used to fit the SFC data, and the parameters θs, θr, m in the SFC formula are also used for the STC formula. Secondly, the parameters a, n are obtained from two measured points on the STC curve; these two points are chosen based on Eqs. (24)–(25), and they are

(28)

in which θ is a normalized water content defined by Eq. (29), and θs, θr,α, m, n are curve-fitting parameters 219

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Table 2 Parameters used in model III. Data source

Curve

Koopmans and Miller (1966) Koopmans and Miller (1966) Wang et al. (2007) Wang et al. (2007) Tian et al. (2014) Tian et al. (2014) Kruse and Darrow (2017) Kruse and Darrow (2017)

SFC

a

c

41.4200

b

5.3856

d

2.9675e

−10

−05

STC

41.4200

5.3856

1.3175e

SFC STC SFC STC SFC STC

26.5324 26.5324 100.1371 100.1371 46.0058 46.0058

9.6569 9.6569 2.7848 2.7848 −4.4547 −4.4547

3.9162e+3 1.3534 1.3376e+14 206.8541 12.4878 7.15825

9.7230 4.1449 5.3107 1.6817 20.2018 6.00035 0.7651 0.6164

Table 3 AREs for models II-V. Data source

Curve

II

III

IV

V

Koopmans and Miller (1966) Koopmans and Miller (1966) Wang et al. (2007) Wang et al. (2007) Tian et al. (2014) Tian et al. (2014) Kruse and Darrow (2017) Kruse and Darrow (2017)

SFC STC SFC STC SFC STC SFC STC

0.0377 0.0979 0.0671 0.1531 0.4643 1.5081 0.0290 0.0423

0.0236 0.0186 0.0711 0.1571 0.2169 0.2824 0.0077 0.0174

0.0315 0.0236 0.0406 0.1170 0.1813 0.2604 0.0077 0.0173

0.0305 0.0237 0.0307 0.1228 0.0724 0.1228 0.0075 0.0168

denoted as (t1, θ1), (t2, θ2 ). Using these two points, the parameters a, n can be determined explicitly as −1/ m

n=

a=

ln{[exp(θ1

∣t1 ∣ −1/ m

[exp(θ1

−1/ m

) − e]/[exp(θ2 ln(| t1 |/| t2 |)

) − e]1/ n

) − e]} (35)

(36)

4. Calibration of the models Four groups of experimental data reported in literature are used to calibrate the five models, which are results for the SS (solid-to-solid contact) soil shown in Fig.4 of Koopmans and Miller (1966), the fluvoaquic brown soil shown in Fig. 7 of Wang et al. (2007), the soil A shown in Fig. 7 of Tian et al. (2014), and the montmorillonite (untreated with cation) shown in Fig. 9 of Kruse and Darrow (2017). The soil used by Wang et al. (2007) is mixed with 5% saline, while other soils are salinefree, and details can be found in original literature. There are two issues need to be addressed before the calibration. First, the gravimetric water content, the volumetric water content and the saturation can be used in models II-V alike, since there is a linear relation between every two of these water contents and linear transformation usually does not change the form of the curve-fitting formula. Secondly, the two points given by Eqs. (24)–(25) may not be measured in these experiments, thus the measured data close to these two points are chosen; however, the two meeting points of the SFC curve and the STC curve are avoided, since they may not contain much information about the shape of the STC curve. Fig. 4. (a) Calibration of model III using the data in Koopmans and Miller (1966). (b) Calibration of model III using the data in Wang et al. (2007). (c) Calibration of model III using the data in Tian et al. (2014). (d) Calibration of model III using the data in Kruse and Darrow (2017).

4.1. Calibration of model I Fig. 2 shows the calibration of model I using the data in Kruse and Darrow (2017), and the predicted results are in general agreement with the experimental data. For other experiments, model I fails to predict a reasonable shape for the STC curve, and there is no need to present these negative comparisons here. 220

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Table 4 Parameters used in model IV. Data source

Curve

α

n

θs

θr

m

Koopmans and Miller (1966) Koopmans and Miller (1966) Wang et al. (2007) Wang et al. (2007) Tian et al. (2014) Tian et al. (2014) Kruse and Darrow (2017) Kruse and Darrow (2017)

SFC

7.6435

7.5793

41.6703

6.7672

4.2608

STC

8.6709

3.0961

41.6703

6.7672

4.2608

SFC STC SFC STC SFC

0.3237 2.1604 0.2195 0.4550 0.0270

354.8361 158.3150 515.2915 174.7823 0.7596

26.2679 26.2679 99.0721 99.0721 46.0155

7.6731 7.6731 3.6325 3.6325 −0.0957

0.0045 0.0045 0.0320 0.0320 1.3589

STC

0.0279

0.6118

46.0155

−0.0957

1.3589

4.2. Calibration of model II Fig. 3(a)-(d) show the calibration of model II using four groups of experimental data, and Table 1 presents the parameters used in the model. The calibration of model II using the data of Koopmans and Miller (1966) is shown in Fig. 3(a); the results indicate that the fitted SFC curve is sound, and the predicted STC curve is in good agreement with the measured STC data. The parameters tr, tf and tm are all marked in the figure; tr and tm are the temperatures at the two meeting points of the SFC curve and the STC curve. The structure of the predicted STC curve is clear. There is a constant segment for temperature greater than tm, a linear segment for temperature between tf and tm, and a nonlinear segment for temperature smaller than tf. For model I, no information of the STC curve is used, and the model fails to predict a reasonable shape for the STC curve in most cases. For model II, the information about the two meeting points of the SFC curve and the STC curve is used; the fitted SFC curve is sound and the predicted STC curve is in a reasonable shape for all cases. However, the two meeting points may not contain much information about the actual shape of the STC curve. Therefore, obvious deviation between the predicted STC curve and several measured data points exhibits in Fig. 3(b)-(d), and the deviation is especially pronounced in Fig. 3(c), in which the unfrozen water content decreases steeply with the decreasing temperature near the phase-transition point. 4.3. Calibration of model III Fig. 4(a)-(d) show the calibration of model III using four groups of measured data, and the two points adopted for predicting the STC curve are marked in the figure. Table 2 presents the parameters used in model III. From Fig. 4(a)-(d), the predicted results of model III are in good agreement with the measured data; the comparison between Fig. 4(a)(d) and Fig. 3(a)-(d) shows that the performance of model III is better than that of model II. To discuss the accuracy of the prediction in a quantitative way, the average relative error is introduced N

ARE =

∑ abs ⎛

θpre, i − θmes, i ⎞

⎝

⎠

⎜

i=1

θmes, i

⎟

/N (37)

in which N is the number of the measured data points, θmes, i is the observed unfrozen water content for the ith measured point, and θpre, i is the predicted unfrozen water content at the temperature of the ith measured point. The AREs for models II–III are presented in Table 3, and it also confirms a better performance of model III over model II in both the fitting process of the SFC curve and the prediction of the STC curve. For the calibration of model III using the data in Wang et al. (2007),

Fig. 5. (a) Calibration of model IV using the data in Koopmans and Miller (1966). (b) Calibration of model IV using the data in Wang et al. (2007). (c) Calibration of model IV using the data in Tian et al. (2014). (d) Calibration of model IV using the data in Kruse and Darrow (2017).

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the ARE of the predicted STC curve is 15.71%, and it has something to do with the obvious deviation between the prediction and the observation near −15 °C. The measured unfrozen water content decreases with the temperature increasing from −20.21 °C to −15.21 °C. This is an abnormal behavior, and it may be caused by the measurement error or the presence of salinity. However, most theories including the formula used in model III assume that the unfrozen water content increases with the increasing temperature; thus, this abnormal behavior cannot be predicted by model III. Except for this abnormality, the prediction of model III agrees well with the experimental observation. For the calibration of model III using the data in Tian et al. (2014), the AREs for the fitted SFC curve and the predicted STC curve are 21.69% and 28.24%, respectively. In fact, the saturations for most of the measured points are near 5%; even 1% absolute error in the predicted saturation will cause 20% relative error. It explains why the AREs for the two curves are a little pronounced. However, this will not limit the application of the model since the absolute error is insignificant, and the fitted SFC curve and the predicted STC curve are in good agreement with the measured results. The main objective of the model is to predict the STC curve from the SFC curve; however, the fitted SFC curve and the predicted STC curve may not form a closed loop, as shown in Fig. 4(a), (b), (d). In practical application, the STC curve is usually applied during the thawing process of the soil, and there is no requirement for the two curves to form a closed loop. Therefore, not forming a closed loop has no great effect on the practical application of the model. 4.4. Calibration of model IV Fig. 5(a)-(d) show the calibration of model IV using the measured data; the parameters used in model IV are listed in Table 4. The predicted results of model IV is in good agreement with the experimental data. From the AREs of model IV presented in Table 3, the overall performance of model IV is a little better than that of model III. 4.5. Calibration of model V Fig. 6(a)-(d) show the calibration of model V using the measured data; the parameters used in model V are listed in Table 5. The predicted results of model V is in good agreement with the experimental data. From the AREs of model V presented in Table 3, model V has the best performance in describing the hysteresis behavior, and the AREs of model V calibrated with the data of Tian et al. (2014) reduce to 7.24% for the fitted SFC curve and 12.28% for the predicted STC curve. In conclusion, models III-V can be used to describe the hysteresis behavior of unfrozen water in frozen soils. After the SFC curve is obtained through the fitting process of the measured data, only two points during the thawing process need to be measured so as to predict the STC curve, and this helps to reduce the time and effort spent to obtain the STC curve greatly. The agreement between the predicted results of models III-V and the measured data also indicates that the procedure used in model III is applicable in developing practical hysteresis models. For other formulas describing the SWC curve in unsaturated soil or the SFC curve in frozen soil, practical models describing the hysteresis behavior of unfrozen water can be developed following this procedure as well. The concepts of BDC, BWC, SWC* shown in Fig.1 can be introduced to frozen soil straightforwardly; the calibration of models III-V shows preliminarily that there is no strict requirement for the STC curve to be the boundary curve, and detailed investigations will be conducted in the future.

Fig. 6. (a) Calibration of model V using the data in Koopmans and Miller (1966). (b) Calibration of model V using the data in Wang et al. (2007). (c) Calibration of model V using the data in Tian et al. (2014). (d) Calibration of model V using the data in Kruse and Darrow (2017).

5. Conclusions In this paper, the hysteresis behavior of unfrozen water in frozen soil is investigated, and practical models which predict the STC curve 222

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Table 5 Parameters used in model V. Data source

Curve

a

n

θs

θr

m

Koopmans and Miller (1966) Koopmans and Miller (1966) Wang et al. (2007) Wang et al. (2007) Tian et al. (2014) Tian et al. (2014) Kruse and Darrow (2017) Kruse and Darrow (2017)

SFC STC SFC STC SFC STC SFC STC

0.1236 0.1003 3.3643 0.5389 4.5826 2.5318 961.2408 2053.4065

7.6839 3.1452 21.4050 9.4464 442.8576 280.2348 0.7328 0.5898

41.6522 41.6522 26.3033 26.3033 99.0705 99.0705 46.0836 46.0836

6.6533 6.6533 −426.9192 −426.9192 0.2701 0.2701 11.1329 11.1329

7.7247 7.7247 0.0110 0.0110 0.5731 0.5731 48.4301 48.4301

from the SFC curve are developed. A similarity theory is first established by comparing the Gibbs-Thomson equation with the YoungLaplace equation, which indicates that formulas and models describing the liquid water in unsaturated soil can also be applied to describe unfrozen water in frozen soil using the same form. Five models describing the hysteresis behavior of unfrozen water in frozen soil are then established using the theory in unsaturated soil. Models I-III are developed based on hysteresis models in unsaturated soil presented by Mualem (1977), Hogarth et al. (1988) and Pham et al. (2003), respectively; following the procedure of model III, models IV-V are also developed based on formulas presented by van Genuchten (1980) and Fredlund and Xing (1994), which are widely used in unsaturated soil. Four groups of experimental data reported in literature are used for model calibration, and the results show that the fitted SFC curve and the predicted STC curve obtained from models IIIV agree well with the experimental observation. Since only two points on the STC curve need to be measured in these models, the time and effort spent to obtain the STC curve will be greatly reduced. The average relative error is also defined to study the accuracies of different models further. The results show that the overall performance of model IV is a little better than that of model III, and model V has the best performance in predicting the hysteresis behavior of unfrozen water in frozen soil.

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