Influence of hydration heat on stochastic thermal regime of frozen soil foundation considering spatial variability of thermal parameters

Applied Thermal Engineering 142 (2018) 1–9

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Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng

Research Paper

Influence of hydration heat on stochastic thermal regime of frozen soil foundation considering spatial variability of thermal parameters

T

⁎

Tao Wanga,b, , Guoqing Zhoua, Dongyue Chaob, Leijian Yinb a b

State Key Laboratory for Geomechanics and Deep Underground Engineering, China University of Mining and Technology, Xuzhou, Jiangsu 221116, China School of Mechanics and Civil Engineering, China University of Mining and Technology, Xuzhou, Jiangsu 221116, China

H I GH L IG H T S

3D stochastic thermal model of frozen soil foundation with internal heat source is presented. • AInfluences hydration heat on stochastic thermal regime of frozen soil foundation are estimated. • A sensitivityof analysis of coefficient of variation and autocorrelation distance is carried out. •

A R T I C LE I N FO

A B S T R A C T

Keywords: Frozen soil foundation Hydration heat Stochastic temperature spatial Spatial variability Thermal parameters

Construction method of cast-in-situ concrete is very popular for the foundation engineering in permafrost regions. But the hydration heat of concrete can lead to the melt of ice in frozen soil zone, and then the temperature variation can lower the bearing capacity of frozen soil foundation. In this study, a three-dimensional (3D) stochastic thermal model of frozen soil foundation with phase change and internal heat source is proposed, and the influences of the hydration heat on the stochastic thermal regime of frozen soil foundation are estimated. Considering the different variability of thermal conductivity, heat capacity and latent heat, a sensitivity analysis for the stochastic thermal regime is presented. The results show that the hydration heat of concrete has a great influence on the stochastic thermal regime of frozen soil foundation in the early days after the base constructed. The variability of temperature is more obvious when the distance from the footing centerline is close. Different variability of the thermal parameters has a different effect on the stochastic thermal regime. These results can improve our understanding of the influences of hydration heat on the stochastic thermal regime and provide a theoretical basis for the safety of cast-in-place construction method.

1. Introduction Nowadays, a lot of engineering constructions, such as QinghaiTibetan Railway, Road, Oil Pipeline and Power Transmission Line, have been carried out in permafrost regions. With the development of economy, more engineering constructions will be implemented in the future [1]. The frozen soil is one of the key factors for ensuring the safety of the engineering constructions in permafrost regions, and its properties are closely related to the temperature, especially for the frozen soil with a high percentage of ice. The temperature variation can leads to corresponding changes in the mechanical properties of the frozen soil. The bearing capacity of the frozen soil foundation is low in the melting state and it will affect the stability of upper engineering [24]. For the viaduct pier and bases of transmission line tower in permafrost regions, the cast-in-situ concrete method is more popular than

⁎

the precast concrete method because of the environmental complexity and transportation difficulty. However, in the cast-in-situ concrete method, the hydration heat of concrete can accelerate the melt of ice in frozen soil and then the temperature variation can lower the bearing capacity of frozen soil foundation [5-7]. Therefore, the Influence of hydration heat on the thermal regime of frozen soil foundation needs to be studied before the engineering constructions are carried out in permafrost regions. Thus far, some researchers had focused on the influence of hydration heat on the thermal regime of frozen soil foundation by numerical methods [5,8,9], whereas others had employed in-situ monitoring methods [10-12]. However, most of the thermal analysis of frozen soil foundation with phase change and internal heat source were developed under the assumption that the thermal parameters and boundaries were deterministic. In fact, the thermodynamic parameters of frozen soil and

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

https://doi.org/10.1016/j.applthermaleng.2018.06.069 Received 11 May 2018; Received in revised form 21 June 2018; Accepted 22 June 2018 Available online 23 June 2018 1359-4311/ © 2018 Elsevier Ltd. All rights reserved.

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(Qt) is given by

concrete are uncertain [13-16]. Also, the stochastic climate can lead to the uncertainty of upper boundary conditions of foundation soils [1719]. The upper engineering in warm permafrost regions will be become more dangerous because of these uncertainties. Therefore, study on the influence of hydration heat on the stochastic thermal stability is very important. A few researchers had paid their attention to these uncertainties in permafrost regions [20-22]. Although these scientific papers can analyze the two-dimensional stochastic thermodynamic issues for the permafrost engineering, they cannot solve the three-dimensional problems. Especially for cast-in-situ concrete method, the uncertain hydration heat will further affect the thermal properties of frozen soil foundation. In this paper, considering the heat generation of cement hydration, a 3D stochastic thermal model of frozen soil foundation with phase change and internal heat source is proposed, and the influences of the hydration heat on the stochastic thermal regime of the frozen soil foundation are estimated. Taking the different variability of thermal conductivity, heat capacity and latent heat into account, a sensitivity analysis for the uncertain temperature of the foundation soils is presented. The distributions of mean temperature and standard deviation are obtained, and the rules of stochastic heat influence are analyzed. These results can provide a theoretical basis for the safety of cast-inplace construction method.

Qt = θ0 Cc (1−e−mt )

where θ0 is the ultimate adiabatic temperature for the concrete; Cc is the heat capacity for the concrete; m is a constant; t is the time. Taking the derivative of the Eq. (4), the internal heat source for the cast-in-situ concrete method can be written as

qv =

Xe =

Cf ⎧ ⎪ Cu + Cf + ⎨ 2 ⎪ Cu ⎩

=

σ2 8Ve Ve′

(6)

3

3

3

∑ ∑ ∑ (−1) j (−1)k (−1)l (T1j T2k T3l)2Γ2 (T1j, T2k, T3l) (7)

j=0 k=0 l=0

(1)

Γ 2 (T1j, T2k , T3l ) =

8 T1j T2k T3l −

Tm−ΔT ⩽ T ⩽ Tm + ΔTu

λf T < Tm−ΔT ⎧ ⎪ λ = λ f + λu − λf [T −(Tm−ΔT )] Tm−ΔT ⩽ T ⩽ Tm + ΔT 2ΔT ⎨ ⎪ λu T > Tm + ΔT ⎩

X (x , y, z ) dxdydz e

where σ is the standard deviation of the 3D random field; Ve′ is the volume of e′; T1j, T2k , T3l are the distances of the relative location for the two 3D local average elements and the detailed description is in Appendix A. Γ 2 (T1j, T2k , T3l ) is the 3D variance function and it is decided by the standard correlation function. The detailed calculation formulas of Γ 2 (T1j, T2k , T3l ) is

⎟

T > Tm + ΔTu

∫Ω

Cov (Xe , Xe′)

T < Tm−ΔT L 2ΔT

1 Ve

where Ve is the volume of e and Ωe is the domain of integration of X. According to Eq.(6), The covariance of two 3D local average elements can be expressed as

In permafrost regions, 3D thermal analysis without the internal heat source for the frozen soil engineering had been developed [23,24]. Because of the hydration heat of concrete, the proposed thermal model need take the internal heat source into account. The method of sensible heat capacity can deal with the phase change [25]. Therefore, the differential equations of this problem are given by

C=

(5)

The random field method could be used for considering the uncertainty of the thermodynamic parameters of frozen soil and concrete [13,30]. In this study, the thermal conductivity, volumetric heat capacity and latent heat of the frozen soil and concrete are modeled as a 3D random field, respectively. The 3D random field can be divided by cube elements and the 3D local average element is defined as

2.1. Governing equations of transient temperature field

⎜

dQt = θ0 Cc me−mt dt

2.3. Analysis methods of 3D stochastic thermal regime

2. Mathematical model and equations

∂ ⎛ ∂T ⎞ ∂ ⎛ ∂T ⎞ ∂ ⎛ ∂T ⎞ ∂T + + λ λ λ + qv = C ∂x ⎝ ∂x ⎠ ∂y ⎝ ∂y ⎠ ∂z ⎝ ∂z ⎠ ∂t

(4)

(2)

T1j

T2k

∫0 ∫0 ∫0

ζ ) ρ (ξ , η , ζ ) dξdηdζ T3l

T3l

(1−

ξ η )(1− )(1 T1j T2k (8)

where ρ (ξ , η , ζ ) is the standard correlation function of the 3D random field. In this paper, the uncertain thermodynamic parameters of the frozen soil and concrete are taken as different independent random fields. According to the research results of Zhu and Zhang [31], taking the heterogeneity of the thermodynamic parameters into account, the standard correlation function in the 3D parameter space can be expressed as

(3)

where C represents the volumetric heat capacity; λ represents the thermal conductivity; Parameters with subscript f and u represent the frozen and the unfrozen states, respectively; qv and L represent the internal heat source and latent heat, respectively; Tm and ΔT represent the phase transition temperature and temperature interval, respectively. In this paper, the value of parameter Tm and ΔT are −0.5 °C and 1 °C, respectively [26,27].

ξ2 η2 ζ2 ⎞ ⎛ ρ (ξ , η , ζ ) = exp ⎜−2 2 + 2 + 2 ⎟ θx θy θz ⎠ ⎝

(9)

where θx is the autocorrelation distance for x-direction; θy is the autocorrelation distance for y-direction and θz is the autocorrelation distance for z-direction. After obtaining the covariance matrices for the 3D local average random field by Eq. (7), the uncorrelated random variables can be calculated by orthogonal transformation method, and the stochastic temperatures of the frozen soil foundation can be calculated by Neumann expansion method [32]. The statistical properties of the stochastic thermal regime can be obtained by mathematical statistics approach. Based on the governing equations of transient temperature field, calculation of cement hydration heat and analysis methods of 3D stochastic thermal regime, a stochastic finite element program was compiled in MATLAB 7.0. The compiled program can directly output

2.2. Calculation of cement hydration heat Cast-in-situ concrete method is very popular for the foundation engineering because of the environmental complexity and transportation difficulty in permafrost regions. One component of the concrete is the cement. The hydration heat of the cement is very obvious after the cast-in-situ concrete constructed. It will raise the thermal effect of the frozen soil and then affect the security and stability of the foundation soil. The exothermic process of the cement hydration heat for the ordinary Portland cement can be expressed as exponential formula [28,29]. Therefore, the computational formula of the hydration heat 2

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6.0m

20.0m

C

D

2.5m

8.0m

ĉ Ċ

1.5m

26.0m

Concrete base 26.0m 30.0m

3.0m

ċ

Concrete base Y

Z

(b)

X

X

(c) B

A

Fig. 1. The calculation model of the concrete base and frozen soil foundation. Part I is fine sand; Part II is loam and part III is gravel and clay.

[1], in Fenghuoshan region, A and B follow a normal distribution and the mean values are −5.2 °C and 12 °C, respectively. Their coefficients of variation are about 0.2. The lateral boundaries of the model are adiabatic and the ground temperature gradient at a 30 m depth is 0.038 °C/m according to the drill hole data in this area. With the adverse conditions taking into consideration, all calculations are chosen to begin in the summer when the temperatures on the ground surface are at the highest of the year, that is α0 = 0 and th = 0. The initial temperature distributions in the ground are obtained through a long-term transient solution with the natural conditions without considering the effect of climate warming and surface disturbances. It took about 100 years to get a steady-state before the base constructed. After that, the construction of the concrete base is assumed to be finished in a short time, and the heat influences caused by ground excavation time and ground excavation processes are negligible. Initial temperatures of concrete base are determined by the temperature of natural ground surface on July 15. Finally, the stochastic thermal regime of frozen soil foundation can be calculated on the basis of the stochastic thermal model of this paper.

the statistical results for the stochastic thermal regime. 3. Description of the computational model and parameters Fig. 1 is the analysis process of the physical model for a transmission line tower. Fig. 1(c) shows the geometric dimensioning of the calculation section and three different layers for the calculation model. The strength of the concrete base is C30. According to field testing data and related documents [26-29], in the cast-in-place construction method, the constant (m) is 0.295 when the cast-in temperature of concrete is 5 °C, and the ultimate adiabatic temperature (θ0) is 40.1 °C when the cement dosage is 350 kg/m3. Table 1 is the details of the thermal parameters for soil layers and concrete. The objective of this paper is to estimate the influences of the hydration heat on the thermal regime after the base constructed, so the long-term climate warming does not considered. According to the principle of adherent layer and stochastic processes method [33,34], the temperature boundaries of natural ground face (Tg) and concrete face (Tc) can be expressed as

2π π Tg = (A + 4) + B sin ⎛ th + + α 0⎞ 2 ⎝ 8760 ⎠

4. Results and analyses

(10)

4.1. Distribution of stochastic thermal regime for frozen soil foundation

2π π Tc = (B + 4) sin ⎛ th + + α 0⎞ 8760 2 ⎝ ⎠

(11)

Fig. 2 shows the thawing and freezing processes around the concrete base in the early days. As indicated in Fig. 2(a)−(c), the concrete base was encircled by the positive isothermal, and the melting area is come into being around the base. The reason for this phenomenon is mainly that the strong cement hydration process in the initial stage after the base constructed. It is obvious that these thawed soils can decrease the thermal stability of the frozen soil foundation. After the concrete base constructed 3 month (Fig. 2(d)), the mean temperature of foundation soils surrounding the concrete base has a big reduction because of the heat dissipation of cement hydration heat. From the four plots of the mean temperature, we can conclude that the cement hydration heat has a great influence on the stochastic thermal regime of frozen soil foundation in about 1 month after the base constructed. The ground temperatures on the 15th day and the 3rd month of frozen soil foundation for 5.5 m away from the footing centerline are obtained by

where A is the yearly average temperature; B is the yearly variation temperature; α0 is phase angle; th is time, and its unit is h According to the meteorological information and measured data Table 1 Physical parameters of soil layers and concrete for the transmission line tower. Physical parameters

λf (W/ (m·°C))

Cf (J/(m3 · oC))

λu (W/ (m · oC))

Cu (J/(m3 · °C))

L (J/m3)

Fine sand Loam Gravel and clay Concrete

2.210 1.220 1.820

1.580 × 106 2.073 × 106 2.032 × 106

1.686 1.093 1.600

2.059 × 106 2.513 × 106 2.635 × 106

2.04 × 107 6.01 × 107 3.69 × 107

3.583

2.375 × 106

3.583

2.375 × 106

0

3

Applied Thermal Engineering 142 (2018) 1–9

T. Wang et al. 0

0

0 -2

-2

-4

-4

-4

-4

-6

-6

-6

-6 -8

-8

-10

-10

-14

(a) -6

-4

-2

0

2

4

(b) -4

-2

0

2

4

6

-12

-14 -6

(c) -4

-2

0

2

4

-14 -6

6

(d) -4

-2

0

2

4

6

X (m)

X (m)

X (m)

X (m)

-8 -10

-12

-14 -6

6

-8

-10

-12

-12

Y (m)

-2

Y (m)

-2

Y (m)

Y (m)

0

Fig. 2. Distributions of mean temperature surrounding the concrete base after construction: (a) on the 7th day; (b) on the 28th day; (c) on the 3rd month; (d) on the 1st year.

0

0

-2

-2

-2

-2

-4

-4

-4

-4

-6

-6

-6

-6

-8

-10

-4

-2

0

X (m)

2

4

6

-14 -6

-12

(c)

(b) -4

-2

0

2

4

6

-14 -6

-4

-2

0

X (m)

X (m)

-8

-10

-12

-12

(a)

-8

-10

-10

-12 -14 -6

-8

Y (m)

0

Y (m)

0

Y (m)

Y (m)

Fig. 3. Comparison between the computed and measured temperatures of frozen soil foundation for 3.0 m away from the footing centerline: (a) on the 15th day; (b) on the 3rd month.

2

4

6

-14 -6

(d) -4

-2

0

2

4

6

X (m)

Fig. 4. Distributions of standard deviation surrounding the concrete base after construction: (a) on the 7th day; (b) on the 28th day; (c) on the 3rd month; (d) on the 1st year.

base constructed. As shown in Fig. 1(c), the depth of concrete base is 8 m and the thermal conditions are discontinuous at the depth of -8.0 m. Therefore, in the Fig. 3(a), for the simulated temperature profile at the depth from −7.5 m to −8.0 m, the line is not smooth. This result is the combined action of hydration heat and boundary conditions. When the coefficients of variation is 0.2 and autocorrelation distance is 2.0 m, Fig. 4 shows the standard deviation of foundation soils surrounding the

measurement method [35], and a comparison is given in Fig. 3. It can be seen from Fig. 3(a) and (b) that the computed mean temperature is roughly same as the measured temperature on the two plots. Both the hydration heat and the upper boundary conditions have an effect on the stochastic thermal regime of frozen soil foundation. As the depth increases, the effect of upper boundary conditions is smaller and smaller. The hydration heat of concrete is obvious in the early days after the 4

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volumetric heat capacity, and the volumetric heat capacity has a greater impact than the latent heat. The standard deviation increases with the increase of coefficient of variation. As indicated in Fig. 5(b)−(d), the combined action of each coefficient of variation on the standard deviation are obtained. The standard deviations increase slightly with the construction time. It can be seen from Fig. 5(a)−(d) that the standard deviation for the stochastic temperature increases with the increase of M. This means that the thermal analysis, especially in the phase change zone, varies greatly because of the variability of thermal parameters. Therefore, the traditional thermal analysis does not reflect the variability of the stochastic temperature and the stochastic thermal model is necessary.

Table 2 Different combinations for coefficients of variation. No.

1

2

3

4

5

6

7

8

9

λf and λu Cf and Cu L

0.1 M M

0.2 M M

0.3 M M

M 0.1 M

M 0.2 M

M 0.3 M

M M 0.1

M M 0.2

M M 0.3

concrete base for the 7th day, the 28th day, the 3rd month and the 1st year after the base constructed. From Fig. 4(a)−(b), the larger standard deviation is at the surface of the undisturbed ground and concrete base, and it generally decreases with the depth. The maximum standard deviation is 2.3 °C. After the concrete base constructed 3 month (Fig. 4(c)), the standard deviation has a slight decrease. The maximum standard deviation is 1.1 °C. This is the combined action of upper boundary conditions and cement hydration heat. On October 15, the air temperature has a reduction and the temperature of upper boundary for natural ground face and concrete face is −1.2 °C and 0 °C, respectively. Also, the cement hydration heat has a decrease. As shown in Fig. 4(d), the influence of cement hydration heat on the frozen soil foundation almost has been disappeared.

4.3. Influences of horizontal autocorrelation distance on standard deviation The autocorrelation distance of uncertain thermal properties of frozen soil foundation is an important parameter for stochastic thermal analysis. To analyze the influence of horizontal autocorrelation distance of thermal parameters on stochastic thermal regime of frozen soil foundation, six values of horizontal autocorrelation distance (5m, 10 m, 20 m, 50 m, 100 m and 200 m) for thermal conductivity, volumetric heat capacity and latent heat are taken into account, respectively. Table 3 is the details of the different combinations for horizontal autocorrelation distance. The variations of standard deviation with horizontal autocorrelation distance are obtained in Fig. 6. The overall influences of coefficient of variation are meaningful for the sensitivity analysis, so the standard deviation in Fig. 6 represents the mean standard deviation. From Fig. 6(a), the standard deviation decreases with the construction time and horizontal autocorrelation distance. In particular, when the horizontal autocorrelation distance is 5 m, the standard deviation decreases from 1.51 °C to 0.89 °C with the increase of construction time; when the horizontal autocorrelation distance is 200 m, the standard deviation decreases from 1.15 °C to 0.62 °C. As indicated in Fig. 6(b)-(c), the standard deviation decreases with the construction time and horizontal autocorrelation distance. It can be seen from Fig. 6(a)-(c) that the standard deviation for the stochastic temperature decreases with the increase of R. It is well known that the longer the autocorrelation distances are, the weaker the correlation of

4.2. Influences of coefficient of variation on standard deviation To analyze the influence of coefficient of variation of stochastic thermal parameters on stochastic thermal regime of frozen soil foundation, three values of coefficient of variation (0.1, 0.2 and 0.3) for thermal conductivity, volumetric heat capacity and latent heat are taken into account, respectively. Table 2 is the details of the different combinations for coefficients of variation. The variations of standard deviation of frozen soil foundation with coefficient of variation are obtained in Fig. 5. The overall influences of coefficient of variation are meaningful for the sensitivity analysis, so the standard deviation in Fig. 5 represents the mean standard deviation. Fig. 5(a) shows the separated influence of each coefficient of variation on the standard deviation. It can be seen that the influences between thermal conductivity, volumetric heat capacity and latent heat are obviously different. The thermal conductivity has a more obvious effect than the

and

u u u

= 0.1

Cf and Cu= 0.1

L=0.1

= 0.2

Cf and Cu= 0.2

L=0.2

Cf and Cu = 0.3

L=0.3

= 0.3

1.7

o

Standard deviation

1.1 1.0 0.9 0

50

100

150

200

250

300

350

400

f

(b)

1.5

and

= 0.1

Cf and Cu= 0.1

L=0.1

= 0.2 u

Cf and Cu= 0.2

L=0.2

= 0.3

Cf and Cu = 0.3

L=0.3

u

u

f

1.6

1.4 1.3 1.2 1.1

0

50

100

Operating time (d)

1.5 1.4

150

200

250

300

350

400

Operating time (d) 1.8 f f

1.7 o

C

f

Standard deviation

Standard deviation

1.2

and

and f

1.6

1.3

0.8

1.7 f

C

1.4

f

and

C

(a)

and

o

C

f

f

(c)

f

and and and

u u u

= 0.1

Cf and Cu= 0.1

= 0.2

Cf and Cu= 0.2

L=0.1 L=0.2

= 0.3

Cf and Cu = 0.3

L=0.3

o

f

Standard deviation

1.6 1.5

and and and

u u u

1.3 1.2 1.1 1.0

0

50

100

150

200

250

300

350

Operating time (d)

= 0.1

Cf and Cu= 0.1

= 0.2

Cf and Cu= 0.2

L=0.1 L=0.2

= 0.3

Cf and Cu = 0.3

L=0.3

(d)

1.6 1.5 1.4 1.3 1.2

0

50

100

150

200

250

300

350

400

Operating time (d)

Fig. 5. Variation of standard deviation of frozen soil foundation with coefficient of variation: (a) M = 0; (b) M = 0.1; (c) M = 0.2; (d) M = 0.3. 5

400

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T. Wang et al.

Table 3 Different combinations for horizontal autocorrelation distance.

Table 4 Different combinations for vertical autocorrelation distance.

No.

1

2

3

4

5

6

No.

1

2

3

4

5

6

θx and θy θz

5 R

10 R

20 R

50 R

100 R

200 R

θx and θy θz

H 0.6

H 0.8

H 1.0

H 1.2

H 1.4

H 1.6

there is a mutual effect between moisture migration, heat conduction and deformation for the frozen soil foundation. Then, a stochastic coupled moisture-heat-deformation model would be more in accord with the physical process between the concrete base and the permafrost foundation. This mutual effect is very complex, especially for considering phase change and internal heat source. A deterministic coupling relationship has been developed [36,37], which could be referenced when study the 3D stochastic coupling process. In addition, there is not enough in-situ monitoring data at present, and the thorough evaluation of variability is very difficult. The evaluation of variability needs a large number of in-situ monitoring data. Only after obtaining enough statistical data (thermal parameter and thermal regime) did the stochastic results of uncertainty thermal characteristics (mean and standard deviation) can be calculated and evaluated. At present, some in-situ observed ground temperatures for 3.0 m away from the footing centerline can be found. The in-situ observed ground temperatures are measured by thermistors, the deterministic mean for measured ground temperatures can be calculated by statistical analysis method. A series of in-situ monitoring methods has been carried out [10–12], which could be referenced when evaluate the variability of thermal parameter and thermal regime. Although there are some disadvantages, this study can improve our understanding of the influences of the hydration heat on the stochastic thermal regime considering the spatial variability of thermal parameters and provide a theoretical basis for the safety of cast-in-place construction method.

thermal parameters. Therefore, the change rules of the standard deviation are reasonable. 4.4. Influences of vertical autocorrelation distance on standard deviation In general, the horizontal autocorrelation distance is far greater than the vertical autocorrelation distance because the frozen soils foundation is layered. In order to analyze the influence of vertical autocorrelation distance of thermal parameters on stochastic thermal regime of frozen soil foundation, six values of vertical autocorrelation distance (0.6 m, 0.8 m, 1.0 m, 1.2 m, 1.4 m and 1.6 m) for thermal conductivity, volumetric heat capacity and latent heat are taken into account, respectively. Table 4 is the details of the different combinations for vertical autocorrelation distance. The variations of standard deviation of frozen soil foundation with vertical autocorrelation distance are obtained in Fig. 7. The overall influences of coefficient of variation are meaningful for the sensitivity analysis, so the standard deviation in Fig. 7 represents the mean standard deviation. From Fig. 7(a), the standard deviation decreases with the construction time and vertical autocorrelation distance. In particular, when the vertical autocorrelation distance is 0.6 m, the standard deviation decreases from 1.52 °C to 0.81 °C with the increase of construction time; when the vertical autocorrelation distance is 1.6 m, the standard deviation decreases from 1.12 °C to 0.51 °C. As indicated in Fig. 7(b), a similar conclusion can be drawn. Especially for Fig. 7(c), H = ∞ mean that the horizontal spatial variability did not be considered. At this time, the standard deviation still decreases with the construction time and vertical autocorrelation distance.

6. Conclusions Frozen soil is easily disturbed by climate change and anthropogenic impact, and its thermal regime is one of the key factors determining the safety of cast-in-place construction method. This study focused on the stochastic thermal influence of concrete base on the permafrost area, specifically considering the cement hydration heat. A 3D stochastic analysis model is developed, which can solve 3D stochastic thermodynamic problems with internal heat source. Taking different spatial variability of thermal parameters into account, a sensitivity analysis for the stochastic thermal regime of permafrost foundation is presented. The simulation results show that the stochastic heat influence of the concrete base on the frozen soil foundation is significant because of heat generation of cement hydration. In the early days after the base constructed, the base is encircled by the thawing circle of frozen soil and the variability of temperature is very obvious. Especially for the frozen soil in phase change zone, the variability of temperature can

5. Discussion This study investigated the stochastic thermal regime of the permafrost foundation on the basis of a 3D stochastic analysis model, taking the heat generation of cement hydration into account. The distributions of stochastic thermal regime are obtained and the influences of spatial variability of thermal parameters on standard deviation are analyzed. However, some issues remain to be discussed. First, the ground excavation time and ground excavation processes have an effect on the thermal regime of the permafrost foundation. This study assumed that the time and processes are short, and the heat influences caused by ground excavation time and ground excavation processes are negligible. It is more reasonable to take the ground excavation time and ground excavation processes into account in the next work. Second, 1.8

1.2

x

and y= 50m

x

and y= 100m

x

and y= 200m

C

and y= 20m

1.0 0.8 0.6

0

50

100

150

200

250

Operating time (d)

300

350

1.4

x

and y= 20m

x

and y= 50m

400

1.2

x

and y= 100m

x

and y= 200m

1.0 0.8 0.6 0.4

0

50

100

150

200

250

Operating time (d)

300

350

(c)

1.6

and y= 10m x

o

1.4

x

and y= 5m x

1.6

and y= 10m x

Standard deviation

Standard deviation

o

C

1.6

1.8

(b)

C

and y= 5m x

1.4

x

and y= 5m

x

and y= 10m

x

and y= 20m

x

and y= 50m

x

and y= 100m

x

and y= 200m

o

(a)

Standard deviation

1.8

400

1.2 1.0 0.8 0.6 0.4

0

50

100

150

200

250

300

350

400

Operating time (d)

Fig. 6. Variation of standard deviation of frozen soil foundation with horizontal autocorrelation distance: (a) R = 0.8 m; (b) R = 1.0 m; (c) R = 1.2 m. 6

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T. Wang et al.

Standard deviation

z z

1.2

z

z

1.6

= 0.8 m

= 0.6 m

z

= 1.0 m = 1.2 m

z

1.4

z

= 1.4 m = 1.6 m

1.0 0.8 0.6

z

1.2

z

(c) z

1.6

= 0.8 m

= 0.6 m

z

= 1.0 m

C

z

1.4

1.8

(b)

= 1.2 m

1.4

z

o

o

C

z

C

1.6

= 0.6 m

o

z

z

= 1.4 m

Standard deviation

1.8

(a)

Standard deviation

1.8

= 1.6 m

1.0 0.8

1.2

z z

= 0.8 m = 1.0 m = 1.2 m = 1.4 m = 1.6 m

1.0 0.8 0.6

0.6 0

50

100

150

200

250

300

350

400

0

50

Operating time (d)

100

150

200

250

300

350

400

0.4

0

50

100

Operating time (d)

150

200

250

300

350

400

Operating time (d)

Fig. 7. Variation of standard deviation of frozen soil foundation with vertical autocorrelation distance: (a) H = 250 m; (b) H = 500 m; (c) H = ∞.

thermal regime and the influences of the spatial variability of thermal parameters on standard deviation can provide an important reference for the safety of cast-in-place construction method.

truly lead to the randomness of mechanical properties, which can decrease the stability of frozen soil foundation and make the upper construction dangerous. Taking different coefficients of variation into account, the standard deviations of the stochastic temperature increase with the increase of coefficient of variation, and the thermal conductivity has a more obvious effect than the volumetric heat capacity and latent heat. Taking different autocorrelation distance into account, the standard deviations of the stochastic temperature decrease with the increase of autocorrelation distance. The distributions of stochastic

Acknowledgements The authors thank the two anonymous reviewers for their comments and advice. This research was supported by the Fundamental Research Funds for the Central Universities(Grant No. 2017QNA30).

Appendix A. Detailed deduction of Eqs. (7) From Eq. (6), the mathematical expectation of the 3D local average element can be written as

1 E (Xe ) = E ⎛⎜ T T xi yi Tzi ⎝

xi + (Txi /2)

yi + (Tyi /2)

z i + (Tzi /2)

∫x −(T /2) ∫y −(T /2) ∫z −(T /2) i

xi

i

i

yi

zi

X (ξ , η , ζ ) dξdηdζ ⎞⎟ = m ⎠

(12)

where xi, yi and zi are the coordinates of the central point for the local average element; Txi, Tyi and Tzi are the edge lengths of the local average element; m is the mean of the 3D random field. From Eq. (6), the variance of the 3D local average element can be written as

1 Var (Xe ) = Var ⎜⎛ ⎝ Txi Tyi Tzi

xi + (Txi /2)

yi + (Tyi /2)

z i + (Tzi /2)

∫x −(T /2) ∫y −(T /2) ∫z −(T /2) i

xi

i

yi

i

zi

X (ξ , η , ζ ) dξdηdζ ⎟⎞ = σ 2Γ 2 (Txi, Tyi, Tzi ) ⎠

(13)

Tyi, Tzi ) is the variance function of the 3D random field where σ is the variance of the 3D random field; From Eq. (6), the covariance of two 3D local average elements can be expressed as can be expressed as 2

Γ 2 (Txi,

x + (T /2) y + (T /2) z + (T /2) x + (T /2) y + (T /2) z + (T /2) 1 1 Cov (Xe , Xe′) = Cov ⎛ V ∫x −i (T xi/2) ∫y −i (T yi/2) ∫z −i (T zi/2) X (ξ , η , ζ ) dξdηdζ , V ∫x −j (T xj/2) ∫y −j (T yj/2) ∫z −j (T zj/2) X (ξ , η , ζ ) dξdηdζ ⎞ yi i zi j xj yj j zj i j e′ ⎝ e i xi ⎠ 1 = V V Cov (YVi, YVj ) e e′

(14)

where xi + (Txi /2)

yi + (Tyi /2)

xi

yi

z i + (Tzi /2)

YVi =

∫x −(T /2) ∫y −(T /2) ∫z −(T /2)

YVj =

∫x −(T /2) ∫y −(T /2) ∫z −(T /2)

i

i

xj + (Txj /2)

j

i

yj + (Tyj /2)

xj

zj + (Tzj /2)

j

yj

j

X (ξ , η , ζ ) dξdηdζ

(15)

zi

X (ξ , η , ζ ) dξdηdζ

(16)

zj

Based on the random field theory, the covariance of YVi and YVj is

Cov (YVi, YVj ) =

1 8

3

3

3

∑ ∑ ∑ (−1) j (−1)k (−1)l (T1j T2k T3l)2Var (YVjkl)

(17)

j=0 k=0 l=0

where T10, T20 and T30 are the distances from the end of the first interval to the beginning of the second interval along the x, y and z coordinate axis, respectively; T11, T21 and T31 are the distances from the beginning of the first interval to the beginning of the second interval along the x, y and z coordinate axis, respectively; T12, T22 and T32 are the distances from the beginning of the first interval to the end of the second interval along the x, y and z coordinate axis, respectively; T13, T23 and T33 are the distances from the end of the first interval to the end of the second interval along the x, y and z coordinate axis, respectively. From Eq. (13), The covariance of YVi and YVj can be rewritten as

Cov (YVi, YVj ) =

σ2 8

3

3

3

∑ ∑ ∑ (−1) j (−1)k (−1)l (T1j T2k T3l)2Γ2 (T1j, T2k, T3l) (18)

j=0 k=0 l=0

Substitution of Eq. (18) into Eq. (14), Eq. (7) can be obtained directly. 7

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T. Wang et al.

Appendix B. Detailed process of stochastic analysis From Eqs. (1)–(3), the following finite element formulae can be obtained (19)

KT = R Where

C Δt

(20)

C Tt − Δt + Pt Δt

(21)

K=K+ R=

Where K is the stiffness matrix; C is the capacity matrix; Tt is the column vector of temperature; Pt is the column vector of load; Δt is the time step, and t is the time. The random stiffness matrix of Eqs. (19) can be broken down into (22)

K = K 0 + ΔK where K0 is the average temperature stiffness matrix; ΔK is the undulatory section Based on the Neumann expansion method, the inverse matrix can be written as

K−1 = (K 0 + ΔK )−1 = (E −P + P 2−P 3 + ⋯) K 0−1

(23)

where E is the unit matrix, and P = K0 ΔK. From Eqs. (19), the column vector of temperature can be expressed as -1

T = K−1R = (E −P + P 2−P 3 + ⋯) K 0−1R = T (0)−PT (0) + P 2T (0)−P 3T (0)+⋯ = T (0)−T (1) + T (2)−T (3) + ⋯

(24)

where T = K0 R, T = P T . From Eqs. (24), T(m) can be expressed as (0)

(i)

-1

i

(0)

T (m) = PT (m − 1) = K 0−1ΔKT (m − 1) (m = 1, 2, ⋯) Therefore, the T

(0)

can be obtained by T

(0)

(25)

= K0 R, the T -1

(1)

,T

(2)

, T ,… can be obtained from Eq. (25), and the T can be obtained from Eq. (24) (3)

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