Numerical analysis of coupled water, heat and stress in saturated freezing soil

Cold Regions Science and Technology 72 (2012) 43–49

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Numerical analysis of coupled water, heat and stress in saturated freezing soil Jiazuo Zhou a, b, Dongqing Li a,⁎ a b

State Key Laboratory of Frozen Soil Engineering, Cold and Arid Regions Environmental and Engineering Research Institute, Chinese Academy of Sciences, Lanzhou 730000, China Graduate University of Chinese Academy of Sciences, Beijing 100049, China

a r t i c l e

i n f o

Article history: Received 14 July 2011 Accepted 19 November 2011 Keywords: Saturated freezing soil Coupled water Heat and stress Ice lens Numerical analysis COMSOL Multiphysics

a b s t r a c t Clapeyron equation can be applied in freezing soil to describe the relationship among temperature, water pressure and ice pressure when ice and water coexist in phase equilibrium. The mathematical deduction shows that the driving force that makes the unfrozen water in soil moves from high temperature area to low temperature area is determined by gravity, temperature and pore pressure. Upon proposing the concept of separating void ratio as a judge criterion for the formation of ice lenses, adjusting the hydraulic conductivity to describe the unfrozen water gathering at the front of ice lenses and the growth of ice lens, a mathematical model of coupled water, heat and stress is established. A typical process of coupled water, heat and stress that happens in a saturated freezing soil column is simulated by COMSOL Multiphysics simulation software. The amount of frost heave is calculated, and the result of simulation gives the distribution bar graph of ice lenses and distribution curves of temperature, equivalent water content and pore pressure, and shows how they change. © 2011 Elsevier B.V. All rights reserved.

1. Introduction Temperature water and stress interact in freezing soil. Under the effect of temperature gradient in the soil, the unfrozen water migrates from the high temperature area to the low temperature area and gathers at the freezing front, so that soil becomes deformed and frost heave happens. The stress field also changes when unfrozen water migrates in freezing soil. Because of the phase change during soil freezing, water migration affect heat transfer in return. At the same time, stress change makes the change of void ratio and pore pressure, and then affects the process of water migration. Sill and Skapski (1956) studied on the water migration and its driving force based on the capillary theory and analyzed the factors that can affect frost heave. According to unfrozen water dynamics, Harlan (1973) derived the equations of coupled water and heat, by solving which the temperature and water fields can be obtained. Konrad and Morgenstern (1982) proposed the concept of segregation potential which is defined as the ratio of the velocity of water migration to temperature gradient, and predicted the amount of migrated water. O'Neill and Miller (1985) established the rigid ice model that can explain the formation and development of ice lenses. Shen and Ladanyi (1987) gave a simplified model of coupled heat, moisture and stress, and executed numerical analysis on computer. Upon applying Clapeyron equation to derive the relationship between temperature and water-driving force, on the basis of Darcy's

⁎ Corresponding author. Tel.: + 86 931 4967278; fax: + 86 931 8271054. E-mail address: [email protected] (D. Li). 0165-232X/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.coldregions.2011.11.006

law and unfrozen water dynamics, this paper studies the process of water migration under the effect of gravity, temperature and pore pressure. This paper also proposes the separating void ratio as a judge criterion for the formation of ice lenses, describing the gathering of water and the development of ice lenses by adjusting the hydraulic conductivity. Upon considering the interaction of water, temperature and pore pressure, a mathematical model of coupled water, heat and stress is established and checked by a numeric example. 2. Mathematical model 2.1. Equation of static balance Fig. 1 is a three phase diagram of saturated freezing soil. In the figure, e is void ratio, Si is the ratio of the volume of pore ice to the volume of pore. The relative volume is with respect to the soil grain, so in the diagram the relative volume of soil grain is 1. In freezing soil, Si is a function of temperature (Tice et al., 1976), which can be written as


Si ¼

1−½1−ðT−T 0 Þα 0

T ≤ T0 T > T0

ð1Þ

where T is temperature; α is experimental parameter; T0 is the freezing temperature of pore water in Celsius temperature.

44

J. Zhou, D. Li / Cold Regions Science and Technology 72 (2012) 43–49

Component

2.2. Pore pressure

Relative volume

Ppor is pore pressure, which can be expressed as

Ice

P por ¼ σ −σ ′

eSi

e Water Soil grain

Consequently, we can express Ppor in another way uniting Eqs. (6), (7), (9), (10) and (11)

e(1– Si )

1

l

P por ¼ ∫x ðγ−γ0 Þdx0 þ Es

1

According to Fig. 1, the unit weight of soil can be expressed as ð2Þ

es ¼

h i 1 l ð1 þ e0 Þ σ −∫x ðγ−γ 0 Þdx0 þ e0 Es

( P por ¼

where σ is the total stress of soil; x is the height of a certain point in soil mass. The height of soil mass is l, and the bottom of the soil mass is defined as origin. There is a load of P* at the top of the soil mass, so the boundary condition of stress is   σ x¼l ¼ P

ð5Þ



l

σ ¼ P þ ∫x γdx0

∫lx ðγ−γ 0 Þdx0 þ Es σ

ð6Þ

σ0 ¼ P þ

l ∫x γ0 dx0

ð7Þ

du ¼ε dx

ð16Þ

The bottom of soil mass is restricted, so the boundary value of displacement is

ð8Þ

where σ′ is effective stress; Es is modulus of compression; ε is strain. At the beginning of freezing, the strain is zero, and the effective stress equals to the initial total stress σ0, so we can obtain effective stress by solving Eq. (8) σ ′ ¼ −Es ε þ σ 0

ð9Þ

In the undeformed status, the unit height of soil is 1 + e0, and that in the deformed status is 1 + e. The relationship between strain and void ratio is

ε¼

e−e0 1 þ e0

ð10Þ

ð17Þ

Solving Eqs. (16) and (17) gives u ¼ ∫0 εdx0

ð18Þ

Substituting Eq. (10) into Eq. (18), displacement becomes a function of void ratio x

′

ð15Þ

The relationship between displacement u and strain ε is

u ¼ ∫0

The compression curve of soil gives dσ ¼ −Es dε

e b es e ≥ es

2.3. Displacement and amount of frost-heaving

x

The initial total stress is σ0. According to Eq. (6), σ0 can be expressed as 

e−e0 1 þ e0

ujx¼0 ¼ 0

Solving Eq. (4) and the boundary condition (5), we obtain

ð14Þ

From the above, pore pressure can be expressed as

ð3Þ

ð4Þ

ð13Þ

On the basis of Eqs. (12) and (13), we can obtain es, which refers to the void ratio matching Ppor(max)

The equation of static balance (one-dimensional) is dσ þγ ¼0 dx

ð12Þ

P por ð maxÞ ¼ σ

where γ is the unit weight of soil; g is the gravity acceleration; ρs , ρi and ρw are the density of soil grain, ice and water respectively. At the beginning of freezing, T > 0, Si = 0, e = e0, so the initial unit weight of soil γ0 is g γ0 ¼ ðρ þ e0 ρw Þ 1þe s

e−e0 1 þ e0

Ppor(max) is the maximum value of Ppor, which equals to the total stress.

Fig. 1. Three phase diagram of saturated freezing soil.

g ½ρ þ eSi ρi þ eð1−Si Þρw  γ¼ 1þe s

ð11Þ

e−e0 dx 1 þ e0 0

ð19Þ

The amount of frost heave equals to the displacement of the top surface. ΔH ¼ uðlÞ

ð20Þ

2.4. Ice lens There have been some scholars who studied the formation of ice lenses. Konrad and Morgenstern (1980)thought that when temperature lowers to θsf an ice lens begins to form, and when temperature lowers to θsm, the ice lens stops growing. This judge criterion for the formation of ice lenses can be expressed as

θsm ≤ T ≤ θsf

ð21Þ

J. Zhou, D. Li / Cold Regions Science and Technology 72 (2012) 43–49

45

O'Neill (1983) thought when the pore pressure exceeds the total pressure, the ice lens forms. P por ≥ σ

ð22Þ

According to Eq. (15) and inequality (22), a judge criterion for the formation of ice lenses in terms of void ratio can be expressed as e ≥ es

ð23Þ

Another opinion was that an ice lens forms when the pore pressure exceeds the sum of the total stress and the separation strength (Gilpin, 1980; Nixon, 1991). P por ≥ σ þ P sep

ð24Þ

where Psep is separation strength. According to Eqs. (12), (14) and inequality (24), we can obtain e ≥ es þ

P sep ð1 þ e0 Þ Es

ð25Þ

Comparing inequalities (23) and (25), the difference of them is that inequality (25) introduces the separation strength which leads P inequality (25) to gain the term Eseps ð1 þ e0 Þ. To be able to judge the formation of ice lenses by void ratio directly, we propose a concept of separating void ratio esep. When the void ratio is less than esep, the pore ice grains cannot connect each other, even if the soil grains are separated, as is shown in Fig. 2. As the void ratio is greater than or equals to esep, ice grains connect each other to become integrated and ice lenses begin to form, as is shown in Fig. 3. As for silt which is liable to form ice lenses, the separating void ratio is approximately equals the maximas void ratio when it is in the most incompact status. From the above the judge criterion for the formation of ice lenses can be expressed as e ≥ esep

ð26Þ

Because esep is greater than es, inequality (26) can be expressed in another way   e ≥ es þ esep −es

ð27Þ

soil grain

ice

water

Fig. 3. Diagram of freezing soil whose void ratio is greater than or equals to esep.

Comparing inequalities (25) and (27), the term esep − es in inequality (27) can be viewed as the result of the separation strength Psep enlarging the separating void ratio εsep in inequality (24). 2.5. Driving force of water migration Pore pressure equals to the weighted sum of pore ice pressure and pore water pressure (O'Neill and Miller, 1985): P por ¼ χP w þ ð1−χ ÞP i

ð28Þ

where χ is expressed as 1:5

χ ¼ ð1−Si Þ

ð29Þ

Clapeyron equation gives the relationship among temperature, water pressure and ice pressure when ice and water coexists in phase equilibrium (Black, 1995; Chen et al., 2006; Kay and Groenevelt, 1974) ! ′ Pw Pi′ T′ − ¼ L ln ρw ρi T 0′

ð30Þ

where L is the latent heat of fusion; P′w and P′i are the absolute pressure of pore water and pore ice respectively; T′ is Kelvin temperature; T0′ is the freezing temperature in Kelvin temperature. Rearranging Eq. (30), we obtain ′ ¼ Pw

! ρw ′ T′ P i þ Lρw ln ρi T 0′

ð31Þ

Expressing the Eq. (31) in terms of gauge pressure and Celsius temperature, we obtain ðP w þ P a Þ ¼

  ρw T þ 273 ðP i þ P a Þ þ Lρw ln T 0 þ 273 ρi

ð32Þ

where Pw and Pi are the gauge pressures of pore water and ice respectively; Pa is atmospheric pressure; T is Celsius temperature; T0 is the freezing temperature in Celsius temperature. Rearranging Eq. (32) gives

soil grain

ice

water

Fig. 2. Diagram of freezing soil whose void ratio is less than esep.

Pw ¼

    ρw ρw T þ 273 Pi þ −1 P a þ Lρw ln T 0 þ 273 ρi ρi

ð33Þ

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J. Zhou, D. Li / Cold Regions Science and Technology 72 (2012) 43–49

Uniting Eqs. (28) and (33), Pi can be eliminated, so pore pressure is expressed as

Pw ¼

  ð1−χ Þðρw −ρi ÞP a þ ð1−χ Þρi ρw L ln TTþ273 þ273 þ ρw P por 0

ð1−χ Þρi þ χρw

ð34Þ

Water head ψ which drives the unfrozen water to migrate can be expressed as ψ¼xþ

Pw ; ρw g

ð35Þ

Substituting Eqs. (38)–(41) into Eq. (42), we obtain   ∂ ∂ ∂ψ ½ ρi eSi dxs þ ρw eð1−Si Þdxs  ¼ ρw ð1 þ eÞ k dxs ∂t ∂x ∂x

Because the volume of soil grain is constant, Eq. (43) can be divided by dxs   ∂ ∂ ∂ψ ½ ρi eSi þ ρw eð1−Si Þ ¼ ρw ð1 þ eÞ k ∂t ∂x ∂x

2.6. Hydraulic conductivity In freezing soil, hydraulic conductivity is a function of temperature (Gilpin, 1980; Nixon, 1991).

  ρi Si þ ρw ð1−Si Þ ∂e eðρi −ρw Þ ∂Si ∂T ∂ ∂ψ þ ¼ k ρw ð1 þ eÞ ∂t ρw ð1 þ eÞ ∂T ∂t ∂x ∂x

k0 ½1−ðT−T 0 Þβ k0

T ≤ T0 T > T0

T ≤ T 0 ; x b xsep T > T 0 ; x b xsep x ≥ xsep

ð37Þ

The x axis points to the frozen zone from the unfrozen zone, and xsep is the position where an ice lens forms. As the Finite Element Method is applied, according to Eq. (26) we check whether the void ratio e at each node is greater than or equal to separating void ratio esep. If inequality (26) is met at the position of xsep, we change the hydraulic conductivity to zero in the zone of x ≥ xsep. 2.7. Equation of water migration According to Fig. 1, the mass of ice mi and water mw in the unit volume of soildxare

eSi dx, so in unit time The ice content in the unite volume of soil is 1þe the latent heat generated by the increment of ice content is   eSi Lρi ∂t∂ 1þe dx . In unit time the heat that was taken away by water

dx. The law of conservation of energy gives the equation flow is C w v ∂T ∂x of thermal diffusion (one-dimensional)

ð38Þ

ρ eð1−Si Þ dx mw ¼ w 1þe

ð39Þ

where C, Cs, Cw, and Ci are the volumetric heat capacities of soil , soil grain, water and ice respectively, and they are united in the equation C¼

1 ½C þ eð1−Si ÞC w þ eSi C i  1þe s

ð48Þ

λ, λs, λw and λi are the coefficients of heat conductivity of soil, soil grain, water and ice respectively, and they are united in the equation(An et al., 1989; Nicolsky et al., 2009) λ ¼ λs 1þe λw

C

ð40Þ

The relationship between the volume of soil dxand the volume soil grain dxs is ð41Þ

The law of conservation of mass gives the equation of water migration (one-dimensional) ∂ ∂v ðm þ mw Þ ¼ −ρw dx ∂t i ∂x

ð47Þ

eð1−Si Þ 1þe

eSi

λi 1þe

ð49Þ

Substituting Eqs. (40) and (41) into Eq. (47) gives

Darcy's law gives the relationship between the velocity of water flow v and water head ψ

dx ¼ ð1 þ eÞdxs

    ∂T ∂ eSi ∂ ∂T ∂T dx ¼ dx−Lρi λ dx−C w v dx ∂t ∂t 1 þ e ∂x ∂x ∂x

1

ρ eS mi ¼ i i dx 1þe

∂ψ ∂x

ð46Þ

2.8. Equation of thermal diffusion

C

v ¼ −k

ρi eSi þ ρw eð1−Si Þ ρs

ð36Þ

where k0 is the hydraulic conductivity of unfrozen soil; β is an experimental parameter. After an ice lens forms, the unfrozen water cannot flow through the ice lens, so we adjust the hydraulic conductivity as 8 < k0 ½1−ðT−T 0 Þβ k ¼ k0 : 0

ð45Þ

Water content w is the mass ratio of the sum of pore water and pore ice to the soil grains. w¼

k¼

ð44Þ

Rearranging Eq. (44) gives

Eqs. (34) and (35) explain that the driving force of unfrozen water migration is determined by gravity, temperature and pore pressure.




ð43Þ

ð42Þ

  ∂T Lρi ∂ ∂ ∂T ∂ψ ∂T ðeSi dxs Þ ¼ dxs − λ dxs þ C w k dx 1 þ e ∂t ∂t ∂x ∂x ∂x ∂x s

ð50Þ

Dividing Eq. (50) by dxs and rearranging it we obtain     Lρ e ∂Si ∂T Lρi Si ∂e ∂ ∂T ∂ψ ∂T C− i − ¼ λ þ Cw k 1 þ e ∂T ∂t 1 þ e ∂t ∂x ∂x ∂x ∂x

ð51Þ

3. Numerical example and analysis This paper considers a typical process of coupled water, heat and stress that happens in a 12 cm high soil column. The soil column whose initial void ratio was 0.6 was put in a thermotank at the constant temperature of 1 °C for enough time. When the freezing process begins (initial state), the top surface temperature was kept at − 3 °C and the bottom temperature was kept at 1 °C; Its side wall was heat-isolated and flow-isolated, and a load of 100 kPa was put on

J. Zhou, D. Li / Cold Regions Science and Technology 72 (2012) 43–49

16 14 12 Height (cm)

the top surface. Because the top surface is frozen instantaneous without water migration, considering the volume of water increases by 9% when freezing happens, we set the top boundary value of void ratio as 1.09 × 0.6 = 0.654. At the bottom, there is a tube which is used to supply water from the outside and the void ratio at bottom does not change; we set bottom boundary value of void ratio as 0.6. Upon ignoring the affection of confining pressure, the process of coupled water, heat and stress can be viewed as one-dimensional. From the deduction above, this problem can be described in mathematics as follow: Governing equations are

47

10 frozen zone

8 6 4 2

dσ ¼0 dx

ð52Þ

du e−0:6 ¼ dx 1 þ 0:6

0

unfrozen zone 0

40

20

100

80

60

Time (h)

ð53Þ

Fig. 4. Frozen zone and unfrozen zone.

  ρi Si þ ρw ð1−Si Þ ∂e eðρi −ρw Þ ∂Si ∂T ∂ ∂ψ þ ¼ k ρw ð1 þ eÞ ∂t ρw ð1 þ eÞ ∂T ∂t ∂x ∂x

ð54Þ

    Lρ e ∂Si ∂T Lρi Si ∂e ∂ ∂T ∂ψ ∂T − ¼ λ þ Cw k C− i 1 þ e ∂T ∂t 1 þ e ∂t ∂x ∂x ∂x ∂x

ð55Þ

Boundary values are σ jx¼0:12 ¼ 100 kPa

ð56Þ

ujx¼0 ¼ 0

ð57Þ

ejx¼0 ¼ 0:6

ð58Þ

ejx¼0:12 ¼ 0:654

ð59Þ

T jx¼0 ¼ 1

ð60Þ

T jx¼0:12 ¼ −3

ð61Þ

Initial values are ejt¼0 ¼ 0:6

ð62Þ

T jt¼0 ¼ 1

ð63Þ

Uniting and solving Eqs. (1), (12), (13), (29), (34), (35), (37), (46), (48), (49) and (52)–(63), we can obtain the fields of temperature, water content, stress, displacement and ice lenses. This problem is modeled by the COMSOL Multiphysics simulation software. 120 quadric-Lagrange elements are generated, and the time step of 6 min is adopted. The parameters and their values are shown in Table 1. Fig. 4 consists of two curves, the upper curve refers to the height of the top surface versus time, and the other one refers to the height of

the freezing front versus time. As is shown in this figure, the amount of frost heave increases almost linearly with time goes by. At the time of 10 h, 20 h, 40 h and 80 h, the values of frost heave ratio are 2.53%, 4.92%, 11.78% and 26.82% respectively. The freezing front descends very quickly at the beginning and then descends slowly with a steady change rate. Above the freezing front curve is the frozen zone and below it is the unfrozen zone. Xu and Deng (1991) had the similar curves by experiments, and Thomas et al. (2009) had the similar curves by numerical analysis. Fig. 5 shows the distribution of ice lenses at different times. Ice lenses are distributed discontinuously layer by layer, which are confirmed in laboratory (Xu et al., 2010). At the time of 39.4 h, an ice lens begins to form, and with time goes by, more ice lenses form and grow to increase the frost heave. Temperature is distributed linearly along the soil column in both the frozen zone and unfrozen zone, but temperature gradient is different in these two zones (Fig. 6). As time increases, the length of frozen zone increases while the length of unfrozen zone decreases (Fig. 4), so the temperature gradient in the frozen zone decreases and the temperature gradient in the unfrozen zone increases in Fig. 6. Fig. 7 shows the temperature curves at five points versus time. The point of 1 cm is near the bottom, and at this point temperature decreases rapidly and then to a steady level. The further the distance between a certain point and the bottom is, the more time the temperature at the point will take to decrease to a steady value. As is shown in the curves of 6 cm and 8 cm, at the beginning, temperature decreases rapidly and then decreasing rate of temperature

16 14

Parameter

Value

Parameter

Value

Parameter

Value

α

−5

0.917

−8

Es (MPa) k0 (10-10m ⋅ s-1) Pa (kPa) ρs (103kg ⋅ m-3)

1.2

Cs (kJ ⋅ m− 3 ⋅ K− 1) Ci (kJ ⋅ m− 3 ⋅ K− 1) Cw (kJ ⋅ m− 3 ⋅ K− 1) L (kJ ⋅ kg− 1) esep

2160

β

ρi (103kg ⋅ m-3) ρw (103kg ⋅ m-3) g (m ⋅ s− 2) λs (W ⋅ m-1 ⋅ K-1) λi (W ⋅ m-1 ⋅ K-1) λw (W ⋅ m-1 ⋅ K-1)

T0 (°C)

0

2.5 101 2.7

1.0 9.81 1.20 2.22 0.58

Height (cm)

12 Table 1 Parameters and values.

10 8 6

1874

4 4180 334.56 1.2

2 0

39.4 h

42 h

50 h

60 h

75 h

Fig. 5. Histogram of ice lenses distribution at different times (the white parts are ice lenses).

J. Zhou, D. Li / Cold Regions Science and Technology 72 (2012) 43–49

16

10h

16

14

20h

14

12

40h

12

10

80h

Height (cm)

Height (cm)

48

8 6

6 4 2

-2

-1

0

0 -200

1

-150

Temperature (°C)

4cm

8cm

11cm

50

100

4. Conclusions On the basis of the law of conservation of mass, the law of conservation of energy and the static balance condition, combining the relationship between effective stress and void ratio, Clapeyron equation

6cm

16

10h 20h 40h 80h

14 12

0.5 0 -0.5 -1 -1.5

10 8 6 4

-2

2

-2.5 -3

0

Flux rate in unit area on the bottom surface of the column is shown by Fig. 11. Flux rate becomes larger and larger and rises up to a peak value before the time of 40 h. Then flux decreases. Xu and Deng also observed the peak value existing in the curve of flux rate versus time in laboratory. They explained that before an ice lens forming, water migration just happens in a single medium. They thought the water potential at freezing front is minus while that is zero at bottom. As the length of unfrozen zone decreases (Fig. 4), the gradient of water potential becomes greater in unfrozen zone, so the flux increases with time. After an ice lens forms, water migration happens in two mediums, one of which is unfrozen soil, and another is a layer with very low hydraulic conductivity that is named as freezing fringe. They also thought the peak value of flux rate is the result of water migration in single medium transforming to water migration in two mediums. Because the curve of water content vibrates after ice lens forming, the flux rate decreases with irregular waves. Fig. 12 gives the distribution of displacement along the soil column at different times. Because of the boundary value (57), displacement is zero at the bottom. As is shown in Fig. 8, pore pressure at unfrozen zone is minus, but the total stress equals to the load, so that the effective stress increases and consequently unfrozen zone is compressed and displacement in unfrozen is minus. Because of water migration and phase change that can enlarge the volume, displacement enlarges in frozen zone. Fig. 11 has a similarity with the curve given by Thomas et al.

Height (cm)

Temperature (°C)

1

1cm

-50

Fig. 8. Distribution of pore pressure at different times.

become smaller until temperature decreases to 0 °C. Then temperature decreases with a bigger decreasing rate, at last temperature decreases to a steady value. The step shape curves of temperature versus time are the result of phase change of water that happens in the freezing soil. Zhao et al. (2009) had the similar tendency of temperature change in laboratory. Fig. 8 gives the pore pressure distribution along the column at 10 h, 20 h, 40 h and 80 h. Because water migrates from unfrozen zone to frozen zone and the water becomes ice in frozen zone, so the pore pressure in frozen zone increases. The load is 100 kPa and pore pressure in frozen zone is also 100 kPa, so it indicates that in frozen zone the stress generated by the load is totally balanced by pore pressure. In unfrozen zone pore pressure is minus so that the soil column can absorb water from the outside. The great gradient of pore pressure between frozen zone and unfrozen zone can resist water migration that caused by temperature gradient. Thomas et al. gave the similar pore pressure distribution. By the effect of temperature gradient, water migrates from unfrozen zone to frozen zone, so the water content at frozen zone increases and that in unfrozen zone decreases (Fig. 9). According to Eq. (1), Si is i not a smooth function of T, so there is a pulse in the curve of ∂S versus ∂T T (Fig. 10). The second term at the left hand side of water migration i Eq. (45) contains ∂S , but pore pressure in frozen zone is constant ∂T without gradient (Fig. 8), so that the pore pressure cannot resist the water migration driven by temperature gradient with a pulse. Consequently water content is not distributed smoothly along the column, but is distributed with vibration (Fig. 9). After the time of 40 h, the water content in frozen zone increases extremely. This is because the freezing front moves down slowly (Fig. 4) so that the water in unfrozen zone has enough time to migrate to frozen zone and form ice lenses. Ice lenses are discontinuous, so the water content vibrates intensively at the region that lasts from 4 cm to 8 cm.

2

-100

Pore pressure (kPa)

Fig. 6. Distribution of temperature at different times.

1.5

80h

8

2 -3

20h

40h

10

4 0

10h

0 0

20

40

60

80

Time (h) Fig. 7. Temperature versus time at different position.

100

0

30

60

90

120

Water content (%) Fig. 9. Distribution of water content at different times.

150

J. Zhou, D. Li / Cold Regions Science and Technology 72 (2012) 43–49

0

49

16 14

-1

∂Si ∂T

Height (cm)

12 -2 -3

10

10h

8

20h

6

40h

4

-4

80h

2 -5 -3

-2

-1

0

0

1

-1

T Fig. 10.

∂Si ∂T

12

Flux rate (10-6cm/s)

10 8 6 4 2

40

3

4

Fig. 12. Distribution of displacement at different times.

This work is supported by the Western Project Program of the Chinese Academy of Sciences (KZCX2-XB2-10), the Funding of the State Key Laboratory Frozen Soil Engineering (SKLFSEZY-03) and the open fund of Qinghai Research and Observation Base, Key Laboratory of Highway Construction & Maintenance Technology in Permafrost Region Ministry of Transport, PRC (2011-4-1).

20

2

versus T.

Acknowledgements

0

1

displacement (cm)

and the relationship between the content of unfrozen water and temperature, we establish a mathematical model of coupled water, heat and stress. The concept of separating void ratio is introduced to judge the formation of ice lenses. The result of the numerical analysis shows temperature distributes linearly in both of the frozen zone and unfrozen zone. The temperature gradient in frozen zone decreases with time while that in unfrozen zone increases. The temperature decreases with time goes by, and the further the distance between a certain point in the column and the bottom is, the more time the temperature of the point will take to decrease to a steady value. Water content in frozen zone increases while that in unfrozen zone decreases. After the ice lenses forms, water content increases extremely. The flux rate increases with time to a peak value, and then decreases with irregular waves. Water migrates from unfrozen zone to frozen zone, so that frost heave occurs in frozen zone and pore pressure in frozen zone achieves the value of the load, whereas in unfrozen zone soil is compressed and pore pressure is minus. The amount of frost heave of the soil column almost increases linearly with time.

0

0

60

Time (h) Fig. 11. Flux rate versus time.

80

100

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