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Heat and Mass Transfer

, Volume 54, Issue 9, pp 2845–2852 | Cite as

Stochastic analysis of uncertain thermal parameters for random thermal regime of frozen soil around a single freezing pipe

  • Tao Wang
  • Guoqing Zhou
  • Jianzhou Wang
  • Lei Zhou
Original

Abstract

The artificial ground freezing method (AGF) is widely used in civil and mining engineering, and the thermal regime of frozen soil around the freezing pipe affects the safety of design and construction. The thermal parameters can be truly random due to heterogeneity of the soil properties, which lead to the randomness of thermal regime of frozen soil around the freezing pipe. The purpose of this paper is to study the one-dimensional (1D) random thermal regime problem on the basis of a stochastic analysis model and the Monte Carlo (MC) method. Considering the uncertain thermal parameters of frozen soil as random variables, stochastic processes and random fields, the corresponding stochastic thermal regime of frozen soil around a single freezing pipe are obtained and analyzed. Taking the variability of each stochastic parameter into account individually, the influences of each stochastic thermal parameter on stochastic thermal regime are investigated. The results show that the mean temperatures of frozen soil around the single freezing pipe with three analogy method are the same while the standard deviations are different. The distributions of standard deviation have a great difference at different radial coordinate location and the larger standard deviations are mainly at the phase change area. The computed data with random variable method and stochastic process method have a great difference from the measured data while the computed data with random field method well agree with the measured data. Each uncertain thermal parameter has a different effect on the standard deviation of frozen soil temperature around the single freezing pipe. These results can provide a theoretical basis for the design and construction of AGF.

Nomenclature

Cf

Heat capacity in the frozen area, [J/(m3·°C)]

Cu

Heat capacity in the unfrozen area, [J/(m3·°C)]

L

Latent heat of melting, [J/m3]

r0

Outer diameter of freezing pipe, [m]

T

Temperature, [oC]

Tm

Temperature in the frontal freezing surface, [°C]

ΔT

Phase transition temperature range, [°C]

Tc

Temperature for the outside surface of freezing pipe, [°C]

T0

Initial temperature for the soil around freezing pipe, [°C]

t

Time, [s]

Li, Lj

Length of local average element, [m]

xi

Center of local average element, [m]

Greeks

λf

Thermal conductivity in the frozen area, [W/(m·°C)]

λu

Thermal conductivity in the unfrozen area, [W/(m·°C)]

θ

Scale of fluctuation for random field, [m]

1 Introduction

Artificial ground freezing (AGF) is a technique that converts the water (in the soil) into the ice by artificial refrigeration technology. Then a watertight frozen soil wall can be obtained, which can be used as the temporary support. As the method of refrigeration technology improvement, AGF has been widely adopted in the civil engineering (tunneling, landslide stabilization, and underpinning), the mining engineering (shaft sinking) and the environmental engineering (containment of hazardous waste) [1]. There are two advantages for AGF. Firstly, it reduces the ground permeability, which ensures the safety of design and construction for the underground workings. Secondly, it improves the mechanical properties (strength and stiffness) of the frozen soil and increases the stability of the future excavations. The temperature is important to determine the stability of the freezing wall, and the temperature change will lead to a series of mechanical behavior variations of the frozen soil, which have an adversely impact on the mechanical state of the freezing wall. This will seriously endanger the safety of the AGF [2, 3]. Therefore, it is very important to understand the thermal regime of frozen soil around a single freezing pipe.

Previous studies on the thermal regime of frozen soil around freezing pipe have been focused on the deterministic temperature characteristics [4, 5, 6, 7]. These thermal analyses are developed under the assumption that the soil properties are deterministic. But in fact, the soil properties are variable because of the complex geological processes [8, 9, 10, 11, 12, 13, 14]. In the AGF technique, the randomness the soil properties can truly make the states of the frozen soil become stochastic, which is bad for the stability of the freezing wall. Some researchers had tried to consider the random aspects as random variables [15, 16]. Others had tried to consider the random aspects as stochastic processes [17, 18]. But both the random variables and the stochastic processes cannot consider the spatial variability of the soil properties. Random field method can quantify the correlation between any two observations in a field, and a few people have paid their attention on the random thermal regime of foundation soils in permafrost regions [19, 20, 21, 22, 23, 24]. Up to now, no researchers have studied the random thermal regime of frozen soil around a single freezing pipe. The variability and analogy method of uncertain thermal parameters is important and more analysis is necessary to clarify how they affect the random thermal regime.

In this paper, the uncertain thermal parameters are considered as random variable, stochastic process and random field, respectively. A stochastic analysis method of uncertain thermal regime of frozen soil around a single freezing pipe is developed, and the mean and the standard deviation are obtained by Monte Carlo (MC) method. Taking the variability of each stochastic parameter into account individually, the effects of different uncertain thermal parameters on the random thermal regime is evaluated. The results can improve our understanding of the influence of uncertain soil parameters on the stochastic thermal regime of frozen soil around a single freezing pipe.

2 Mathematical model

2.1 Governing differential equations

In AGF project, when the thermal regime of frozen soil around freezing pipe was simulated and analyzed, only the heat conduction and the phase change of water in the soils were considered instead of water migration in soils [4]. The original three-dimensional (3D) heat conduction problem can be simplified as a one-dimensional (1D) one because the longitudinal length of a freezing pipe is far greater than its diameter and there is no heat transfer in the circumferential direction. Figure 1 is the computational model of the thermal regime of frozen soil around a single freezing pipe. Based on the method of sensible heat capacity [25], the differential equations of this problem are given by
$$ C\frac{\partial T}{\partial t}=\frac{1}{r}\frac{\partial }{\partial r}\left(\lambda r\frac{\partial T}{\partial r}\right)\kern4em {r}_0\le r<\infty $$
(1)
$$ C=\left\{\begin{array}{ll}{C}_{\mathrm{f}}& T<{T}_{\mathrm{m}}-\Delta T\\ {}\frac{C_{\mathrm{u}}+{C}_{\mathrm{f}}}{2}+\frac{L}{2\Delta T}& {T}_{\mathrm{m}}-\Delta T\le T\le {T}_{\mathrm{m}}+\Delta {T}_{\mathrm{u}}\\ {}{C}_{\mathrm{u}}& T>{T}_{\mathrm{m}}+\Delta {T}_{\mathrm{u}}\end{array}\right. $$
(2)
$$ \lambda =\left\{\begin{array}{ll}{\lambda}_{\mathrm{f}}& T<{T}_{\mathrm{m}}-\Delta T\\ {}{\lambda}_{\mathrm{f}}+\frac{\lambda_{\mathrm{u}}-{\lambda}_{\mathrm{f}}}{2\Delta T}\left[T-\left({T}_{\mathrm{m}}-\Delta T\right)\right]& {T}_{\mathrm{m}}-\Delta T\le T\le {T}_{\mathrm{m}}+\Delta T\\ {}{\lambda}_{\mathrm{u}}& T>{T}_{\mathrm{m}}+\Delta T\end{array}\right. $$
(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 soil in the frozen area, respectively; Parameters with subscript u are the corresponding physical components in the unfrozen area;r0 is the outer diameter of freezing pipe; L is the latent heat per unit volume; Tm is the freezing point of soil; ΔT is the temperature range of the phase transition; t is the time and r is the radius.
Fig. 1

Computational model of thermal regime of frozen soil around a single freezing pipe

2.2 Boundary conditions and initial conditions

There are three kinds of common boundary conditions about the heat transfer problems. For the computational model of the thermal regime of frozen soil around a single freezing pipe, the boundary conditions can be expressed as
$$ T\left({r}_0,t\right)={T}_c $$
(4)
$$ T\left(\infty, t\right)={T}_0 $$
(5)
where Tc is the temperature for the outside surface of freezing pipe;T0 is the initial temperature for the soil around freezing pipe.
The initial conditions are as follows
$$ T\left(r,0\right)={T}_0 $$
(6)

2.3 Finite element formulae

In this study, it is very difficult to obtain the analytical solution for Eqs. (1)–(6) after considering the randomness of the thermal parameters (especially for the spatial variability of soil properties). A solution can be obtained by the Galerkin method [26]. The following finite element formulae are obtained.
$$ \left[K\right]{\left\{T\right\}}_t+\left[C\right]{\left\{\frac{\partial T}{\partial t}\right\}}_t={\left\{F\right\}}_t $$
(7)
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; and t is the time.
Based on the backward difference method [27], Eq. (7) can be written as
$$ \left(\left[K\right]+\frac{\left[C\right]}{\Delta t}\right){\left\{T\right\}}_t=\frac{\left[C\right]}{\Delta t}{\left\{T\right\}}_{t\hbox{-} \Delta t}+{\left\{F\right\}}_t $$
(8)
where Δt is the time step.

3 Stochastic analysis methods of random thermal regime

3.1 Analogy method of uncertain thermal parameters

As mentioned above, there are three analogy methods for considering the randomness of the uncertain thermal parameter, i.e., random variable, stochastic process and random field. For the 1D heat conduction problem around a single freezing pipe, the uncertain thermal parameters of frozen soil include thermal conductivity, volumetric heat capacity and latent heat. Based on the computational model of thermal regime of frozen soil around a single freezing pipe (Fig. 1), the 1D finite element method can be used to divide the structure. Figure 2 is the 1D local average elements.
Fig. 2

One-dimensional local average element

For the random variable method, the uncertain thermal parameter are modeled as the random variable and the local average element defined as
$$ {X}_{L_i}=X $$
(9)
where X is the random variable.
The covariance of two local average elements is
$$ Cov\left({X}_{L_i},{X}_{L_j}\right)=E\left({X}^2\right)-{\left[E(X)\right]}^2={\sigma}_X^2 $$
(10)
where σX is the standard deviation of random variable X.
For the stochastic process method, the uncertain thermal parameter are modeled as the stochastic process and the local average element defined as
$$ {X}_{L_i}=X(t) $$
(11)
where X (t) is the stochastic process.
The covariance of two local average elements is
$$ Cov\left({X}_{L_i},{X}_{L_j}\right)=E\left({\left(X(t)\right)}^2\right)-{\left[E\left(X(t)\right)\right]}^2={\sigma}_X^2(t) $$
(12)
where σX(t) is the standard deviation of stochastic process X (t).
For the random field method, the uncertain thermal parameter are modeled as the random field and the local average element defined as
$$ {X}_i=\frac{1}{L_i}{\int}_{x_i-{L}_i/2}^{x_i+{L}_i/2}X\left(\xi \right) d\xi $$
(13)
where xi is the center of local average element; Li is the length of local average element; X (ξ) is the random field.
The covariance of two local average elements is
$$ {\displaystyle \begin{array}{ll} Cov\left({X}_{L_i},{X}_{L_j}\right)& =\frac{1}{L_i{L}_j} Cov\left({\int}_{x_i-{L}_i/2}^{x_i+{L}_i/2}X\left(\xi \right) d\xi, {\int}_{x_j-{L}_j/2}^{x_j+{L}_j/2}X\left(\xi \right) d\xi \right)\\ {}& =\frac{\sigma_X^2}{2{L}_i{L}_j}\sum \limits_{k=0}^3{\left(-1\right)}^k{l}_k^2{\Gamma}^2\left({l}_k\right)\end{array}} $$
(14)
where σX is the standard deviation of random field X (ξ); Γ2(·) is the variance function of random field X (ξ).
The relationship between variance function and standard correlation function can be expressed as
$$ {\displaystyle \begin{array}{ll}{\Gamma}^2(L)& =\frac{1}{L^2}{\int}_{-L}^L\left(L-\left|\xi \right|\right)\rho \left(\xi \right) d\xi \\ {}& =\frac{2}{L}{\int}_0^L\left(1-\frac{\xi }{L}\right)\rho \left(\xi \right) d\xi \end{array}} $$
(15)
where ρ(⋅) is the standard correlation function of random field X (ξ).
Equation (15) is complex because the standard correlation function of soil propertiesis various [28]. According to the research results of Vanmarcke [29], a computational formula for the variance function can be obtained.
$$ {\Gamma}^2(L)={\left[1+{\left(L/\theta \right)}^m\right]}^{-1/m} $$
(16)
where θ is the scale of fluctuation for random field X (ξ); m is a constant.

3.2 Calculation method of random thermal regime

After considering the randomness of the uncertain thermal parameter, the random thermal regime of frozen soil around a single freezing pipe can be obtained by MC method. MC method can perfectly matches with deterministic FE method and avoids complex theoretical analysis. The MC simulation uses computer to research the random variable and its principal task is to obtain random sample according to the certain probability distribution.

Based on the Wiener-khinchin law of large Numbers, The following formulae can be obtained
$$ \underset{N\to \infty }{\lim }P\left(\left|\frac{1}{n}\sum \limits_{i=1}^n{X}_i-\mu \right|<\varepsilon \right)=1 $$
(17)
where X1, X2, ⋯, Xn, ⋯ are a series of random variables with the same distribution; μ is the mathematical expectation of the random variables.
Based on the Bernoulli law of large Numbers, The following formulae can be obtained
$$ \underset{N\to \infty }{\lim }P\left(\left|\frac{n}{N}-P(A)\right|<\varepsilon \right)=1 $$
(18)

In sampling test, x1, x2, ⋯, xn, are a series of random variables with the same distribution. According to Eq. (17), the average of the sampling test will converge to the mathematical expectation when n is large enough. According to Eq. (18), the frequency of events n/N will converge to the probability when n is large enough. After MC simulation, the mean and the standard deviation of the random thermal regime can be obtained by the statistical analysis approach. We have made a stochastic program based on aforementioned procedure, which can consider the randomness of uncertain thermal parameters simultaneously or separately. After each stochastic simulation, both mean and standard deviation need to be calculated and saved. When the mean and standard deviation are basically stable, the stochastic simulation is end.

4 Description of the computational models and parameters

According to the actual frozen data, as well as the thermal physical properties of the frozen soil and the unfrozen soil [4, 5, 6], the physical parameters are given in Table 1. The computational model of the thermal regime of frozen soil around a single freezing pipe showed in Fig. 1. For the numerical calculations, the model in Fig. 1 is partitioned into planar 2-node elements, explained as follows. Each element has two nodes, and each node has one degree of freedom. This type of element can be used to analyze the unsteady-state thermal process. The temperature inside the elements is assumed to remain constant. Taking the Fourier Number, heat transfer intensity and computational efficiency into account, these meshes near thermal boundary of the freezing pipe and soil surface are smaller. The 1D finite element meshes are shown in Fig. 3. There are 500 elements and 501 nodes in Fig. 3. Taking uncertain thermal conductivity, volumetric heat capacity and latent heat into account, the random thermal regime of the frozen soil around a single freezing pipe with three analogy method can be obtained by MC method. In order to study the influence of uncertain thermal parameters on the random thermal regime of frozen soil around a single freezing pipe, the numerical simulation for the uncertain temperature distribution and changes of the frozen soil based on three cases have been performed.
  • Case 1: Taking uncertain thermal conductivity into account. Namely, the thermal conductivity is considered as random field; the volumetric heat capacity and latent heat are considered as deterministic values.

  • Case 2: Taking uncertain latent heat into account. Namely, the latent heat is considered as random field; the thermal conductivity and volumetric heat capacity are considered as deterministic value.

  • Case 3: Taking uncertain volumetric heat capacity into account. Namely, the volumetric heat capacity is considered as random field; the thermal conductivity and latent heat are considered as deterministic values.

Table 1

Physical parameters of freezing pipe and soil

Property

Value

Outer diameter of freezing pipe, r0 (m)

0.0795

Initial temperature for the soil around freezing pipe, T0 (°C)

15

Temperature for the outside surface of freezing pipe, Tc (°C)

−30

Freezing point of soil, Tm (°C)

−0.5

Temperature range of the phase transition, ΔT (°C)

0.5

Volumetric heat capacity of soil in the frozen area, Cf (J/m3/oC)

2.228 × 106

Volumetric heat capacity of soil in the unfrozen area, Cu (J/m3/°C))

2.822 × 106

Thermal conductivity of soil in the frozen area, λf (W/m/°C)

1.578

Thermal conductivity of soil in the unfrozen area, λu (W/m/°C)

1.125

Latent heat per unit volume, L (J/m3)

1.026 × 108

Scale of fluctuation for random field, θ (m)

0.25

Constant, m

3

Fig. 3

One-dimensional finite element meshes

5 Results and analyses

5.1 Influence of analogy method on mean temperature

Considering the uncertain thermal conductivity, volumetric heat capacity and latent heat into account, Fig. 4 shows the distribution of the mean temperature for the frozen soil around the single freezing pipe with three analogy methods, i.e., random variable method (Fig. 4a), stochastic process method (Fig. 4b) and random field method (Fig. 4c). It can be seen that the mean temperatures with three analogy methods are roughly same after 5, 10, 15, 20, 30, 40, 50 and 60 days of operation, which imply that three analogy methods of uncertain thermal parameters have no different influence on the mean temperature of frozen soil around the single freezing pipe. From Fig. 4a–c, the distributions of mean temperature are approximate to logarithmic functions, and the rate of change for the mean temperature decrease with the increase of the radial coordinate. The mean temperature changes in the frozen area are dramatic while the changes in the unfrozen area are slight. More than 3.5 m away from the freezing pipe center, the rate of change for mean temperature goes to zero. Therefore, we can conclude that the single freezing pipe have an impact on the mean temperature within 3.5 m for the pipe diameter and liquid temperature tested in this paper.
Fig. 4

Mean temperature of frozen soil around the single freezing pipe with three analogy method: (a) Random variable method; (b) Stochastic process method; (c) Random field method

5.2 Influence of analogy method on standard deviation

Considering the uncertain thermal conductivity, volumetric heat capacity and latent heat into account, Fig. 5 shows the distribution of the standard deviation for the frozen soil temperature around the single freezing pipe with three analogy methods, i.e., random variable method, stochastic process method and random field method. It can be seen that the standard deviations with random variable method and stochastic process method are roughly same after 5, 10, 15, 20, 30, 40, 50 and 60 days of operation, and the standard deviations with random field method are smaller than them. These results indicate that different analogy methods of uncertain thermal parameters have different influence on the standard deviation of the frozen soil temperature around the single freezing pipe. From Fig. 5a, b and c, the distributions of the standard deviation have a great difference at different radial coordinate location. The larger standard deviations are mainly at the phase change area while the smaller standard deviations are near the freezing pipe and away from the freezing pipe. For the random variable and the stochastic process method, the maximum standard deviation is 3.14 and 2.75 °C after 5 and 60 days of operation, respectively. For the random field method, the maximum standard deviation is 2.84 and 2.58 °C after 5 and 60 days of operation, respectively.
Fig. 5

Standard deviation of frozen soil temperature around the single freezing pipe with three analogy method: (a) Random variable method; (b) Stochastic process method; (c) Random field method

5.3 Validations with actual measurements and experimental data

In order to estimate the influence of three analogy methods of uncertain thermal parameters on random thermal regime of frozen soil around a single freezing pipe, the mean temperature and the standard deviation are obtained by field data and experimental data [30, 31]. A comparison among calculated temperatures, measured temperatures and experimental temperatures is given in Fig. 6. From Fig. 6a, Obviously, the computed mean temperatures with three analogy method well agree with the measured mean temperatures and experimental mean temperatures. Therefore, the stochastic analysis model used in this study can describe the stochastic temperature changes of the frozen soil around a single freezing pipe. From Fig. 6b, the computed standard deviations with random variable method and stochastic process method have a great difference from the measured standard deviations and experimental standard deviations. But the computed standard deviations with random field method well agree with the measured standard deviations and experimental standard deviations, the maximum difference is only 0.13 °C. Therefore, we can conclude that taking the uncertain thermal parameters as random fields is more reasonable.
Fig. 6

Comparison among calculated temperatures, measured temperatures and experimental temperatures of frozen soil around the single freezing pipe: (a) Mean temperature; (b) Standard deviation

5.4 Influence of uncertain thermal parameters on mean temperature

Considering the stochastic effect of uncertain thermal conductivity, uncertain latent heat and uncertain volumetric heat capacity individually, the mean temperature of the frozen soil around the single freezing pipe with aforementioned three cases can be obtained (Fig. 7a–c). It can be seen that the mean temperatures with the three cases are roughly same after 5, 10, 15, 20, 30, 40, 50 and 60 days of operation, which imply that the three cases have no different influence on the mean temperature of frozen soil around the single freezing pipe. Comparing Figs. 4 and 7, it can be found that the mean temperatures with the three cases are roughly same as the three analogy methods. Therefore, we can conclude that the analogy method and simulation cases have no different influence on the mean temperature of the frozen soil around the single freezing pipe.
Fig. 7

Mean temperature of frozen soil around the single freezing pipe with three cases: (a) Case 1; (b) Case 2; (c) Case 3

5.5 Influence of uncertain thermal parameters on standard deviation

In order to study the overall influence of each stochastic parameter on the stochastic thermal regime, we calculated and analyzed the average standard deviation of the frozen soil temperature around the single freezing pipe for each uncertain thermal parameter individually, shown in Fig. 8a. It can be found from Fig. 8a that the influence of each stochastic parameter on the average standard deviation is different. The thermal conductivity is the most influential factor; the latent heat is the middle; the volumetric heat capacity has minimum influence. The average standard deviations with three cases generally increase with the time. In order to evaluate the greatest influence of each stochastic parameter on the stochastic thermal regime, we calculated and analyzed the maximum standard deviation of the frozen soil temperature around the single freezing pipe for each uncertain thermal parameter individually, shown in Fig. 8b. It can be also found from Fig. 8b that the influence of each stochastic parameter is different. Similarly, the thermal conductivity is the most influential factor; the latent heat is the middle; the volumetric heat capacity has minimum influence. The maximum standard deviations with three cases generally decrease with the time.
Fig. 8

Standard deviation of frozen soil temperature around the single freezing pipe with three cases: (a) Average standard deviation; (b) Maximum standard deviation

6 Summaries and conclusions

The objectives of this study are to investigate the influence of uncertain thermal parameters on the random thermal regime of frozen soil around a single freezing pipe. The uncertain thermal parameters of the frozen soil are considered as random variable, stochastic process and random field, respectively. A stochastic analysis method of the uncertain thermal regime of the frozen soil around a single freezing pipe is developed, and the stochastic finite element program is compiled by MATLAB, which can consider the stochastic soil properties simultaneously or separately. The results can provide a theoretical basis for the design and construction of AGF. According to this paper, the following conclusions can be drawn:
  1. (1)

    Considering the uncertain thermal parameters of frozen soil as random variable, stochastic process and random field, the mean temperatures of the frozen soil around the single freezing pipe are the same while the standard deviations are different. The mean temperature changes in the frozen area are dramatic while the changes in the unfrozen area are slight. The distributions of standard deviation have a great difference at different radial coordinate location and the larger standard deviations are mainly at the phase change area.

     
  2. (2)

    For the mean temperatures, the computed data with three analogy method well agree with the measured data. For the standard deviations, the computed data with random variable method and stochastic process method have a great difference from the measured data. The computed data with random field method well agree with the measured data. Therefore, taking the uncertain thermal parameters of the frozen soil as random fields is more reasonable.

     
  3. (3)

    Each stochastic parameter has a different effect on the standard deviation. The thermal conductivity is the most influential factor; the latent heat is the middle; the volumetric heat capacity has minimum influence. The average standard deviations with three cases generally increase with the time while the maximum standard deviations with three cases generally decrease with the time.

     

Notes

Acknowledgments

The authors thank the three anonymous reviewers for their comments and advice. This research was supported by the National Natural Science Foundation of China (Grant No. 51604265 and 51323004), the 111 Project (Grant No. B14021) and the China Postdoctoral Science Foundation funded project (Grant No. 2017M620229).

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Tao Wang
    • 1
    • 2
  • Guoqing Zhou
    • 1
  • Jianzhou Wang
    • 1
  • Lei Zhou
    • 2
  1. 1.State Key Laboratory for Geomechanics and Deep Underground EngineeringChina University of Mining and TechnologyXuzhouChina
  2. 2.School of Mechanics and Civil EngineeringChina University of Mining and TechnologyXuzhouChina

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