Stochastic analysis of uncertain thermal parameters for random thermal regime of frozen soil around a single freezing pipe
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 onedimensional (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
 C_{f}
Heat capacity in the frozen area, [J/(m^{3}·°C)]
 C_{u}
Heat capacity in the unfrozen area, [J/(m^{3}·°C)]
 L
Latent heat of melting, [J/m^{3}]
 r_{0}
Outer diameter of freezing pipe, [m]
 T
Temperature, [^{o}C]
 T_{m}
Temperature in the frontal freezing surface, [°C]
 ΔT
Phase transition temperature range, [°C]
 T_{c}
Temperature for the outside surface of freezing pipe, [°C]
 T_{0}
Initial temperature for the soil around freezing pipe, [°C]
 t
Time, [s]
 L_{i}, L_{j}
Length of local average element, [m]
 x_{i}
Center of local average element, [m]
Greeks
 λ_{f}
Thermal conductivity in the frozen area, [W/(m·°C)]
 λ_{u}
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
2.2 Boundary conditions and initial conditions
2.3 Finite element formulae
3 Stochastic analysis methods of random thermal regime
3.1 Analogy method of uncertain thermal parameters
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.
In sampling test, x_{1}, x_{2}, ⋯, x_{n}, 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

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.
Physical parameters of freezing pipe and soil
Property  Value 

Outer diameter of freezing pipe, r_{0} (m)  0.0795 
Initial temperature for the soil around freezing pipe, T_{0} (°C)  15 
Temperature for the outside surface of freezing pipe, T_{c} (°C)  −30 
Freezing point of soil, T_{m} (°C)  −0.5 
Temperature range of the phase transition, ΔT (°C)  0.5 
Volumetric heat capacity of soil in the frozen area, C_{f} (J/m^{3}/^{o}C)  2.228 × 10^{6} 
Volumetric heat capacity of soil in the unfrozen area, C_{u} (J/m^{3}/°C))  2.822 × 10^{6} 
Thermal conductivity of soil in the frozen area, λ_{f} (W/m/°C)  1.578 
Thermal conductivity of soil in the unfrozen area, λ_{u} (W/m/°C)  1.125 
Latent heat per unit volume, L (J/m^{3})  1.026 × 10^{8} 
Scale of fluctuation for random field, θ (m)  0.25 
Constant, m  3 
5 Results and analyses
5.1 Influence of analogy method on mean temperature
5.2 Influence of analogy method on standard deviation
5.3 Validations with actual measurements and experimental data
5.4 Influence of uncertain thermal parameters on mean temperature
5.5 Influence of uncertain thermal parameters on standard deviation
6 Summaries and conclusions
 (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)
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)
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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