Heat Mass Transfer DOI 10.1007/s00231-016-1883-5
ORIGINAL
Estimation on the influence of uncertain parameters on stochastic thermal regime of embankment in permafrost regions Tao Wang1,2 · Guoqing Zhou1 · Jianzhou Wang1 · Xiaodong Zhao1 · Xing Chen2
Received: 17 March 2016 / Accepted: 25 July 2016 © Springer-Verlag Berlin Heidelberg 2016
Abstract For embankments in permafrost regions, the soil properties and the upper boundary conditions are stochastic because of complex geological processes and changeable atmospheric environment. These stochastic parameters lead to the fact that conventional deterministic temperature field of embankment become stochastic. In order to estimate the influence of stochastic parameters on random temperature field for embankment in permafrost regions, a series of simulated tests are conducted in this study. We consider the soil properties as random fields and the upper boundary conditions as stochastic processes. Taking the variability of each stochastic parameter into account individually or concurrently, the corresponding random temperature fields are investigated by Neumann stochastic finite element method. The results show that both of the standard deviation under the embankment and the boundary increase with time when considering the stochastic effect of soil properties and boundary conditions. Stochastic boundary conditions and soil properties play a different role in random temperature field of embankment at different times. Each stochastic parameter has a different effect on random temperature field. These results can improve our understanding of the influence of stochastic parameters on random temperature field for embankment in permafrost regions.
* Tao Wang
[email protected] 1
State Key Laboratory for Geomechanics and Deep Underground Engineering, China University of Mining and Technology, Xuzhou 221116, Jiangsu, China
2
School of Mechanics and Civil Engineering, China University of Mining and Technology, Xuzhou 221116, Jiangsu, China
1 Introduction In China, the permafrost regions are mainly in the west and about 22.4 % of the country [1]. At present, a lot of engineering constructions have been built in these cold regions. Such as Qinghai–Tibetan Railway, it passes across more than 550km permafrost regions, and the railway embankments have a significant influence on the thermal regime of the ground. Many researches have indicated that thaw settlement of permafrost can result in instability and failure of construction [2–4]. Under the global warming, these constructions of cold regions will face many severe problems from permafrost degradation [5]. It is very important to ensure the thermal stability of permafrost under the embankment because the permafrost could not be recovered once destroyed. Because of the complexity of the analytical solution for the temperature field, it is very popular to apply numerical methods, such as finite element method, to analyze the thermal stability of embankment [6–8]. It is well known that the correct value of soil properties and boundary conditions directly affect the results. Up to now, most of the thermal analysis of embankment is developed under the assumption that the soil properties and boundary conditions are deterministic; therefore, their results are deterministic. In fact, the property parameters of soil are variable because of the complex geological processes [9–12]. Also, the upper boundary conditions of an expressway embankment are stochastic due to the changeable atmospheric environment [13–15]. The randomness of soil properties and boundary conditions will lead to the randomness of temperature field. Therefore, it is significant important to consider the random aspects of the parameters and conditions. Some literatures had investigated the random temperature fields of embankment in cold regions by stochastic finite element method [16–19]. However, these studies make some simple
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Heat Mass Transfer
assumptions for the stochastic parameters of soil properties and boundary conditions. The variability of random variables is important and more analysis is necessary to clarify how they affect the random temperature field for embankment in permafrost regions. After we know the influence of different stochastic parameters on the random temperature field, some protection measures can be adopted. In this paper, based on theories of random field and heat transfer, we consider the soil properties as random fields and the upper boundary conditions as stochastic processes, the effect of random soil properties and boundary condition on the random temperature field of an embankment in a cold region are analyzed by NSFEM. The influence of different stochastic variables on thermal regime is evaluated. The results can provide a theoretical basis for the engineering reliability analysis and design of actual embankment in permafrost regions.
2 Stochastic analysis methods of random temperature field 2.1 Deterministic governing equations According to the deterministic analytical method of the soil zone of embankment in permafrost regions [3], the temperature fields of embankment are considered as a nonlinear problem of heat transfer with phase change. Based on the method of sensible heat capacity [20], the differential equations of this problem are given by ∂T ∂ ∂T ∂T ∂ + =C (1) ∂x ∂x ∂y ∂y ∂t
C=
Cf
Cu +Cf 2 Cu
f = f + u
+
u − f 2�T [T
L 2�T
T < Tm − �T Tm − �T ≤ T ≤ Tm + �Tu T > Tm + �Tu
− (Tm − �T )]
(2)
T < Tm − �T Tm − �T ≤ T ≤ Tm + �T T > Tm + �T
(3)
where f and u represent the frozen and the unfrozen states, respectively; Cf and λf are the volumetric heat capacity and thermal conductivity of embankment in the frozen area, respectively; Parameters with subscript u are the corresponding physical components in the unfrozen area; L is the latent heat per unit volume; Tm is the freezing point of soil; ΔT is temperature range of the phase transition; t is time and x, y are distances. It is very difficult to obtain the analytical solution for this problem. We obtain a solution by the Galerkin method [5] and the backward difference method [21]. The following finite element formulae are obtained.
13
[C] [C] {T }t− t + {F}t {T }t = [K] + t t
(4)
where [K] is the stiffness matrix; [C] is the capacity matrix; {T}t is the column vector of temperature; {F}t is the column vector of load; Δt is the time step; and t is the time. Both [K], [C] and {F}t are deterministic variables in the conventional deterministic finite element analysis, so {T}t of Eq. (4) is a deterministic result. In this paper, [K] and [C] are not deterministic because soil properties are stochastic, and {F}t is not deterministic because boundary conditions are random. Therefore, {T}t of Eq. (4) is a random result. 2.2 Stochastic finite element methods Based on the earlier studies of stochastic analysis model for uncertain temperature characteristics for embankment in warm permafrost regions [19], the upper boundary conditions are modeled as stochastic processes. According to the meteorological information and the regression analysis, we can reduce the boundary temperature to an annual sine function [4]. Considering the climate warming and its randomness in the next 50 years [22], the air temperature is assumed as following formulation T = A + B sin
π 2π th + + α0 8760 2
+
C t 365 × 24 × 50 h
(5)
where A is the yearly average temperature; B is the yearly variation temperature; C is the rise rate of air temperature; α0 is phase angle; th is time, and its unit is h. In this paper, the mean values of A, B and C are −4, 11.5 and 2.6 °C, respectively [7]. Based on the random field theory [9, 10], we modeled the thermal conductivity, volumetric heat capacity and latent heat as a 2D continuous random field, respectively. When the 2D random field is divided by triangular elements [23], the local average random field of an element is defined as 1 X(x, y)dxdy Xe = (6) Ae Ωe where Ae is the area of e and Ωe is the possessive section of e. The covariance of two local average elements is
Cov(Xe , Xe′ ) = σ 2
M M
ω(K) ω′(R)
K=1 R=1 (K) (K) (K) ′(R) ′(R) ′(R) g(Ni , Nj , Nk , Ni , Nj , Nk )
(7)
where Ni and Nj are the shape functions of e; Ni′ and Nj′ are the shape functions of e′; M is the number of basis points; ω′(K) is the weighted coefficient of e and ω(R) is the weighted coefficient of e′.
Heat Mass Transfer
T = T (0) − T (1) + T (2) − T (3) + · · · T (0) = K −1 0 R T (m) = K −1 �KT (m−1) (m = 1, 2, . . . )
(8)
11.1 m 3.6 m C
5.0 m
.5
B
30.0 m
A
D
1:1
30.0 m
3.0 m
Equation (7) is the calculating formula of the covariance for two local average elements. If the standard correlation function is known, we can obtain the result of covariance matrix. Because the covariance matrices obtained from Eq. (7) are full-rank matrices, calculating the covariance matrices is inefficient. Therefore, a set of uncorrelated random variables is obtained by orthogonal transformation method [24]. After considering the randomness of soil properties and boundary conditions, the temperature field of an embankment can be obtained by NSFEM [25]. For the random temperature fields of an embankment, the following formulas can be employed [24].
F
G
H
E
Fig. 1 The computational model. Part I is fill; Part II is silty clay and part III is weathered mudstone
0
where K0 is the mean of stiffness matrix, ΔK is the undulatory section, R is the stochastic load vector. The mean and standard deviation can be obtained by probabilistic analysis approach, and the computational formulas are N 1 E(T) = Ti N
(9)
i=1
S(T) = =
N
1 [T i − E(T)]2 N −1 i=1 N
N 1 [T i ]2 − [E(T)]2 N −1 N −1
(10)
i=1
We have made a program based on aforementioned procedure, which can consider the randomness of parameters and conditions simultaneously or separately.
3 Description of the computational models The embankment presented in this paper is located at an altitude of 4500 m in Qinghai–Tibetan railway and the computational domain is shown in Fig. 1. The width of pavement is 7.2 m; the bottom width of the embankment reaches 22.2 m on the grounds that embankment height is 5.0 m with a slope of 1:1.5; the computational domain is extended for 30 m from the side-slope foot of the embankment, and 30 m under the natural ground surface. In the embankment model (Fig. 1), part I is fill; part II is silty clay
and part III is weathered mudstone. Their thermal parameters are listed in Table 1 [3, 7]. Because of radiation effect [26] and the related data [27], the temperatures (Tn) at the native surfaces AB (Fig. 1) are changed according to the following expression π 2π th + + α 0 Tn = (A + 2.5) + (B + 0.5) sin 8760 2 C (11) + th 365 × 24 × 50 The temperatures (Ts) at the side slopes BC (Fig. 1) are varied according to the following function π 2π th + + α 0 Ts = (A + 4.7) + (B + 1.5) sin 8760 2 C (12) + th 365 × 24 × 50 The temperatures (Tp) at the ballast pavement surface CD (Fig. 1) is changed as the following π 2π th + + α 0 Tp = (A + 5.5) + (B + 2.5) sin 8760 2 C (13) + th 365 × 24 × 50 The lateral boundaries AF and DE are adiabatic, and the geothermal heat flux at boundary FE is 0.06 W m−2 in this paper [16]. Both the structure and the random fields were discretized into triangle elements. The finite element meshes and random field meshes is shown in Fig. 2. There are 943 triangular elements and 522 nodes in Fig. 2. The initial temperature distributions of parts II and III on July 15 are obtained through a long-term transient solution with
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Table 1 Thermal parameters of media in embankment
Heat Mass Transfer Physical parameters
λf [W/(m °C)]
Cf [J/(m3 °C)]
λu [W/(m °C)]
Cu [J/(m3 °C)]
L (J/m3)
Fill Silty clay
1.980 1.351
1.913 × 106 1.879 × 106
1.919 1.125
2.227 × 106 2.357 × 106
2.04 × 107 6.03 × 107
Weathered mudstone
1.824
1.846 × 106
1.474
2.099 × 106
3.77 × 107
In order to study the influence of stochastic parameters on random temperature field for embankment in a permafrost region, the numerical simulation for the uncertain temperature distribution and changes of embankment based on four cases have been performed. Case 1: Taking stochastic soil properties and boundary conditions into account. Namely, the upper boundary conditions are considered as random processes; the soil properties are considered as random fields. Case 2: Taking stochastic boundary conditions into account. Namely, the upper boundary conditions are considered as random processes; the soil properties are considered as deterministic values. Case 3: Taking stochastic soil properties into account. Namely, the soil properties are considered as random field; the upper boundary conditions are considered as deterministic value. Case 4: Taking the variability of each stochastic parameter into account individually. Namely, such parameters as yearly average temperature, yearly variation temperature and rise rate of air temperature are considered as random variables individually; thermal conductivity, volumetric heat capacity and latent heat are considered as random field individually.
Fig. 2 Finite element meshes and random field meshes
the upper boundary condition [Eq. (11)] without considering the effect of climate warming, and the temperatures of Parts I are determined by the temperature of natural ground surface in that date. The initial thermal conditions of the permafrost embankment are different for different construction dates. The worst temperature distributions occur on July 15, hence we take July 15 as the initial time. According to the research results [28], we assumed the correlation lengths of soil properties are 5 m, and standard correlation function is �x 2 + �y2 ρ = exp −2 (14) θ
Table 2 Different simulated tests
13
Case
According to the description of four different cases, Table 2 is the details. In this paper, we assumed the coefficients of variation of stochastic parameters are 0.2, namely, ξ = 0.2.
Coefficient of variation A
B
C
λf
λu
Cf
Cu
L
1 2 3
ξ ξ 0
ξ ξ 0
ξ ξ 0
ξ 0 ξ
ξ 0 ξ
ξ 0 ξ
ξ 0 ξ
ξ 0 ξ
4
ξ 0 0 0 0
0 ξ 0 0 0
0 0 ξ 0 0
0 0 0 ξ 0
0 0 0 ξ 0
0 0 0 0 ξ
0 0 0 0 ξ
0 0 0 0 0
0
0
0
0
0
0
0
ξ
Heat Mass Transfer
According to the law of large numbers of Bernoulli, the mean temperature is roughly same as the deterministic temperature for different cases. Therefore, we will only analyze the results of standard deviations for the embankment. 4.1 Case 1: Taking stochastic soil properties and boundary conditions into account Considering the stochastic effect of soil properties and boundary conditions, Fig. 3 shows the distribution of standard deviations on July 15 and October 15 after 15 years of the construction, respectively. From Fig. 3a, it can be seen that the larger standard deviation on July 15 after 15 years are main at the top of the boundary. The maximum standard deviation is 1.94 °C. Figure 3b shows that the larger
4
ST15-7-15
(a)
1.9 1.7 1.5 1.4 1.2 1 0.8 0.7 0.6 0.5 0.4 0.3 0.2
2
Y (m)
0 -2 -4 -6 -8 -10 -20
-18
-16
-14
-12
-10
-8
-6
-4
-2
standard deviation on October 15 after 15 years is at the bottom of the fill. The maximum standard deviation is 1.98 °C. Figure 4 shows the distribution of standard deviations on July 15 and October 15 after 30 years of the construction, respectively. From Fig. 4a, we can see that the larger standard deviation on July 15 after 30 years is still main at the top of the boundary. The maximum standard deviation is 2.04 °C. Figure 4b shows the larger standard deviation on October 15 after 30 years is still at the bottom of the fill. The maximum standard deviation is 2.08 °C. From Figs. 3a, b and 4a, b, we can see that the standard deviations of some partial regions are great magnitude, i.e., the discreteness of standard deviations is subsistent. In addition, both of the standard deviation under the embankment and the boundary increase with time. Therefore, with the passage of time, the results of conventional deterministic analysis may be farther from the true value.
4
ST15-10-15
(b)
1.9 1.7 1.6 1.4 1.2 1 0.8 0.7 0.6 0.5 0.4 0.3 0.2
2 0
Y (m)
4 Results and analyses
-2 -4 -6 -8 -10 -20
0
-18
-16
-14
-12
-10
-8
-6
-4
-2
0
X (m)
X (m)
Fig. 3 Distributions of standard deviation for the 15th year after construction: a on July 15, case 1 and b on October 15, case 1
ST30-7-15
(a)
2 1.8 1.6 1.5 1.4 1.2 1 0.8 0.7 0.6 0.5 0.4 0.3
2
Y (m)
0 -2 -4 -6 -8 -10 -20
-18
-16
-14
-12
-10
X (m)
-8
-6
-4
-2
0
4
ST30-10-15
(b)
2.1 2 1.9 1.8 1.6 1.4 1.2 1 0.8 0.7 0.6 0.5 0.4
2 0
Y (m)
4
-2 -4 -6 -8 -10 -20
-18
-16
-14
-12
-10
-8
-6
-4
-2
0
X (m)
Fig. 4 Distributions of standard deviation for the 30th year after construction: a on July 15, case 1 and b on October 15, case 1
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Heat Mass Transfer
4.2 Case 2: Taking stochastic boundary conditions into account Considering the stochastic effect of boundary conditions, Figs. 5 and 6 shows the distribution of standard deviations on July 15 and October 15 in the 15th and 30th year, respectively. From Figs. 5a and 6a, it can be seen that the larger standard deviation on July 15 is at the top of the boundary and the standard deviation generally decreases with the depth. The maximum standard deviation is 2.01 °C and 2.11 °C, respectively. Figures 5b and 6b shows that the larger standard deviation on October 15 is at the bottom of the fill. The maximum standard deviation is 1.24 °C and 1.45 °C, respectively. Comparing Figs. 3a and 5a, we can find that the standard deviation at the surface of embankment is roughly same on July 15 in the 15th year. Therefore, we can conclude that the random boundary conditions play an important role on July 15 for the random
ST15-7-15
(a)
Considering the stochastic effect of soil properties, Figs. 7 and 8 shows the distribution of standard deviations on July 15 and October 15 in the 15th and 30th year, respectively. From Figs. 7a and 8a, it can be seen that the larger standard deviation on July 15 isn’t at the top of the boundary. The standard deviations of some partial regions are great
4
2 1.8
2
1.4
-2
0.8
-4
0.6
-6
0.4
-8
0.2
ST15-10-15
1.2 1.1 1 0.9
0
1.2 1
(b)
2
1.6
0
Y (m)
4.3 Case 3: Taking stochastic soil properties into account
Y (m)
4
temperature field. For Figs. 4a and 6a, a similar conclusion can be made. Comparing Figs. 3b and 5b, we can find that the standard deviation is much bigger for same locations when considering the stochastic effect of soil properties and boundary conditions. Therefore, we can conclude that the random boundary conditions don’t play an important role on October 15 for the random temperature field. For Figs. 4b and 6b, a similar conclusion can be made.
0.5 0.3
0.8 0.7
-2
0.6
-4
0.5
-6
0.3
-8
0.2
0.4 0.25 0.1
0.1
-10 -20
-18
-16
-14
-12
-10
-8
-6
-4
-2
-10 -20
0
-18
-16
-14
-12
X (m)
-10
-8
-6
-4
-2
0
X (m)
Fig. 5 Distributions of standard deviation for the 15th year after construction: a on July 15, case 2 and b on October 15, case 2
ST30-7-15
(a)
1.8
2
1.4
-2
0.9 0.8
-4
0.6
-6
0.5
-8
0.3
0.4
1.4 1.3 1.2 1.1
0
1.2 1
ST30-10-15
(b)
2
1.6
0
Y (m)
4
2
Y (m)
4
1 0.9
-2
0.8 0.7
-4
0.6
-6
0.5
-8
0.3
0.4 0.2
0.2
-10 -20
-18
-16
-14
-12
-10
X (m)
-8
-6
-4
-2
0
-10 -20
-18
-16
-14
-12
-10
-8
-6
-4
X (m)
Fig. 6 Distributions of standard deviation for the 30th year after construction: a on July 15, case 2 and b on October 15, case 2
13
-2
0
Heat Mass Transfer ST15-7-15
(a)
1.2
2
1
-2
0.7
-4
0.6
-6
0.4
-8
0.2
1.6 1.5 1.4 1.2
0
0.9 0.8
ST15-10-15
(b)
2
1.1
0
Y (m)
4
1.3
Y (m)
4
0.5 0.3
1 0.8
-2
0.7 0.6
-4
0.5
-6
0.4
-8
0.2
0.3 0.1
0.1
-10 -20
-18
-16
-14
-12
-10
-8
-6
-4
-2
-10 -20
0
-18
-16
-14
-12
X (m)
-10
-8
-6
-4
-2
0
X (m)
Fig. 7 Distributions of standard deviation for the 15th year after construction: a on July 15, case 3 and b on October 15, case 3
ST30-7-15
(a)
0
0.9
-2
0.7
-4
0.6
-6
0.4
-8
0.2
0.5 0.3
1.5 1.4
1 0.8
1.7
2
1.1
0
ST30-10-15
(b)
1.6
1.2
2
Y (m)
4
1.4
Y (m)
4
1.2 1
-2
0.8 0.7
-4
0.6 0.5
-6
0.4 0.3
-8
0.2
0.1
-10 -20
-18
-16
-14
-12
-10
-8
-6
-4
-2
0
X (m)
-10 -20
0.1
-18
-16
-14
-12
-10
-8
-6
-4
-2
0
X (m)
Fig. 8 Distributions of standard deviation for the 30th year after construction: a on July 15, case 3 and b on October 15, case 3
magnitude and the discreteness is very obvious. The maximum standard deviation is 1.39 and 1.44 °C, respectively. Figures 7b and 8b shows that the larger standard deviation on October 15 is at the bottom of the fill. The maximum standard deviation is 1.67 and 1.73 °C, respectively. Comparing Figs. 3a and 7a, we can find that the standard deviation at the surface of embankment is different on July 15 in the 15th year. Therefore, we can conclude that the random soil properties don’t play an important role on July 15 for the random temperature field. For Figs. 4a and 8a, a similar conclusion can be made. Comparing Figs. 3b and 7b, we can find that the standard deviation is a little bigger for same locations when considering the stochastic effect of soil properties and boundary conditions. Therefore, we can conclude that the random soil properties play an important role on October 15 for the random temperature field. For Figs. 4b and 8b, a similar conclusion can be made. It can be seen from Figs. 5 and 7 that the stochastic boundary conditions and soil properties play a different role in random temperature field for
the different time. The stochastic boundary conditions have a great influence on July 15 while the stochastic soil properties have a great influence on October 15. From Figs. 6 and 8, a similar conclusion can be made. 4.4 Case 4: Taking the variability of each stochastic parameter into account individually Considering the stochastic effect of each stochastic parameter individually, we can obtain the mean and standard deviation for embankment at different times. In order to study the overall influence of each stochastic parameter on random temperature field, we calculate and analyze the average standard deviation of the embankment for each stochastic parameter individually, shown in Fig. 9. It can be also found from Fig. 9 that the influence of each stochastic parameter is different. From Fig. 9a, c, the yearly variation temperature is the most influential factor for random temperature field on January 15 and July
13
Heat Mass Transfer
Average standard deviation(°C)
0.4 yearly average temperature yearly variation temperature rise rate of air temperature thermal conductivity volumetric heat capacity latent heat
0.3
0.2
0.1
0.0
0
0.5
Average standard deviation(°C)
0.5
(a)
5
10
15
20
operating time (a)
25
yearly average temperature yearly variation temperature rise rate of air temperature thermal conductivity volumetric heat capacity latent heat
0.3
0.2
0.1
0
yearly average temperature yearly variation temperature rise rate of air temperature thermal conductivity volumetric heat capacity latent heat
0.3
0.2
0.1
0
0.5
(c)
0.4
0.0
(b)
0.4
0.0
30
5
10
15
20
25
30
operating time (a)
Average standard deviation(°C)
Average standard deviation(°C)
0.5
5
10
15
20
operating time (a)
25
30
(d)
0.4
0.3
yearly average temperature yearly variation temperature rise rate of air temperature thermal conductivity volumetric heat capacity latent heat
0.2
0.1
0.0
0
5
10
15
20
25
30
operating time (a)
Fig. 9 Average standard deviation of the embankment for each stochastic parameter individually: a on January 15, b on April 15, c on July 15 and d on October 15
15; the influences of yearly average temperature and thermal conductivity are roughly same and their standard deviations generally increase with the time; rise rate of air temperature, volumetric heat capacity and latent heat have little influence. From Fig. 9b, d, the yearly average temperature and thermal conductivity have a relatively greater impact on random temperature field on April 15 and October 15; the influence of yearly variation temperature is smaller; rise rate of air temperature, volumetric heat capacity and latent heat have little influence. In order to evaluate the greatest influence of each stochastic parameter on random temperature field, we calculate and analyze the maximum standard deviation of the embankment for each stochastic parameter individually, shown in Fig. 10. It can be also found from Fig. 10 that the influence of each stochastic parameter is different. From Fig. 10a, c, the yearly variation temperature is the most influential factor for random temperature field on January 15 and
13
July 15. From Fig. 10b, d, the yearly average temperature and thermal conductivity have a relatively greater impact on random temperature field on April 15 and October 15. Figure 10a–d show that the stochastic influences of rise rate of air temperature, volumetric heat capacity and latent heat are unobvious. Actually, the thermal conductivity of embankment in the frozen area is a very complicated problem. It is related to temperature, moisture and spatial location. In this paper, we made some assumptions for the spatial variability in x and y directions, and the standard deviation in x directions may be bigger than normal because the permafrost is layered in general. As a preliminary study, according to the results of this paper, we can understand that each stochastic parameter has a different effect on random temperature field and it is expected to provide theoretical basis for the reliability analysis and design of actual embankment in permafrost regions.
Heat Mass Transfer
Maximum standard deviation(°C)
1.6 yearly average temperature yearly variation temperature rise rate of air temperature thermal conductivity volumetric heat capacity latent heat
1.2
0.8
0.4
0.0
0
5
10
15
20
operating time (a)
25
1.2
0.8
0.4
0.0
0
yearly average temperature yearly variation temperature rise rate of air temperature thermal conductivity volumetric heat capacity latent heat
1.2 0.9 0.6 0.3
0
1.8 yearly average temperature yearly variation temperature rise rate of air temperature thermal conductivity volumetric heat capacity latent heat
1.6
(b)
1.5
0.0
30
(c)
2.0
Maximum standard deviation(°C)
1.8
(a)
Maximum standard deviation(°C)
Maximum standard deviation(°C)
2.0
5
10
15
20
25
30
operating time (a)
5
10
15
20
operating time (a)
25
30
(d)
1.5 1.2
yearly average temperature yearly variation temperature rise rate of air temperature thermal conductivity volumetric heat capacity latent heat
0.9 0.6 0.3 0.0
0
5
10
15
20
25
30
operating time (a)
Fig. 10 Maximum standard deviation of the embankment for each stochastic parameter individually: a on January 15, b on April 15, c on July 15 and d on October 15
5 Summaries and conclusions The objectives of this study are to investigate the influence of stochastic parameters on stochastic thermal regime for embankment in permafrost regions. We mode the soil properties as random fields and the upper boundary conditions as stochastic processes. A stochastic finite element program is compiled by MATLAB, which can consider the stochastic soil properties and boundary conditions simultaneously or separately. The result can improve our understanding of the influence of different stochastic parameters on random temperature field for embankment. According to this paper, the following conclusions can be drawn: 1. The randomness of soil properties and boundary conditions lead to the randomness of temperature characteristics. When considering the stochastic effect of soil properties and boundary conditions, both of the stand-
ard deviation under the embankment and the boundary increase with time. Therefore, with the passage of time, the results of conventional deterministic analysis may be farther from the true value. 2. Stochastic boundary conditions and soil properties play a different role in random temperature field of embankment at different times. The stochastic boundary conditions have a greater influence on July 15 while the stochastic soil properties have a greater influence on October 15. 3. Each stochastic parameter has a different effect on random temperature field. The yearly variation temperature is the most influential factor on January 15 and July 15; the yearly average temperature and thermal conductivity have a relatively greater impact on April 15 and October 15; rise rate of air temperature, volumetric heat capacity and latent heat have little influence on random temperature field.
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Acknowledgments This research was supported by the Major State Basic Research Development Program (Grant No. 2012CB026103), the China Postdoctoral Science Foundation funded project (Grant No. 2016M591958), the National Natural Science Foundation of China (Grant No. 41271096) and the 111 Project (Grant No. B14021). We express our sincere thanks to the two reviewers for their valuable comments and suggestions on the content of the paper.
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