Advances in Civil Engineering

Volume 2018, Article ID 7219826, 10 pages

https://doi.org/10.1155/2018/7219826

## A Coupled Thermo-Hydromechanical Model of Soil Slope in Seasonally Frozen Regions under Freeze-Thaw Action

State Key Laboratory of Geomechanics and Geotechnical Engineering, Institute of Rock and Soil Mechanics, Chinese Academy of Sciences, Wuhan 430071, China

Correspondence should be addressed to Zheng Lu; moc.361@msrhwzl

Received 29 August 2018; Revised 29 October 2018; Accepted 13 November 2018; Published 2 December 2018

Academic Editor: Filippo Ubertini

Copyright © 2018 Yongxiang Zhan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Soil slope diseases in seasonally frozen regions are mostly related to water migration and frost heave deformation of the soil. Based on the partial differential equation defined using the COMSOL Multiphysics software, a thermo-hydromechanical coupling model considering water migration, ice-water phase change, ice impedance, and frost heave is constructed, and the variations in the temperature field, migration of liquid water, accumulation of solid ice, and deformation of frost heave in frozen soil slopes are analysed. The results show that the ambient temperature has a significant effect on the temperature and moisture field of the slope in the shallow area. In addition, the degree of influence gradually weakens from the outside to the inside of the slope, and the number of freeze-thaw cycles in deep soil is less than that in shallow soil. During the freezing period, water in the unfrozen area rapidly migrates to the frozen area, and the total moisture content abruptly changes at the vicinity of the freezing front. The maximum frozen depth is the largest at the slope top and the smallest at the slope foot. During the melting period, water is enriched at the melting front with the frozen layer melting; the slope is prone to shallow instability at this stage. The melting of the frozen layer is bidirectional, so the duration of slope melting is shorter than that of the freezing process. The slope displacement is closely related to the change in temperature—a relation that is in agreement with the phenomenon of thermal expansion and contraction in unfrozen areas and reflects the phenomenon of frost heave and thaw settlement in frozen areas.

#### 1. Introduction

Seasonally frozen soil is a special soil-water system wherein ice and water coexist. The seasonally frozen soil region in China accounts for 53.5% of the total area of China [1]. Given the development of China’s western region, the implementation of strategy for revitalizing the northeast, and the development strategy known as the Belt and Road Initiative, a large amount of infrastructure-related construction will be required in the seasonally frozen soil regions, e.g., the Sino-Russian oil pipeline project in Northeast Asia, the Harbin–Dalian high-speed railway project in China, and China’s National Highway 301. More than two-thirds traffic trunk lines in China are located in seasonally frozen soil regions, and freeze-thaw slope diseases along the lines are very serious. Most of these diseases are related to water migration and frost heave deformation of the soil [2, 3].

The coupling of water, heat, and force is a key problem in the study of frozen soil in seasonally frozen regions and is also at the frontier of international research in this field. Throughout the years, many studies have proposed various frozen soil models. Harlan [4] first established the water-heat coupling model based on the principles of unfrozen water dynamics and energy conservation, and the equations in this model had a clear physical meaning. Taylor and Luthin [5] calculated the water field and temperature field using the implicit finite-difference method based on the Harlan model and compared these calculations with the experimental data. O’Nell and Miller [6] studied the formation and growth of ice lenses in frozen soil and proposed and developed a rigid ice model. An et al. [7] analysed the coupling of water and heat during soil freezing. Shen and Ladanyi [8] addressed the coupling problem of water, heat, and force in three fields by proposing a simplified coupling model based on the Harlan hydrodynamic model. Lai et al. [9] first derived the mathematical mechanical model and solution for the coupled problem of temperature, seepage, and stress fields based on phase changes predicted by heat transfer theory, seepage theory, and frozen soil mechanics. Li et al. [10] established a heat, moisture, and deformation coupling model based on the theory of equilibrium, continuity, and energy principles of a multiphase porous medium. Gens et al. [11] established a new mechanical model that encompasses frozen and unfrozen behaviours within a unified framework based on effective stress. In recent years, numerous studies on thermo-hydromechanical (THM) coupling have been performed. In particular, with the development of computer technology, many studies [12–16] have addressed the multi-field coupling problem using the finite-element method.

In this study, a THM coupling model is constructed considering water migration, ice-water phase change, ice impedance, and frost heave. The coupling simulation is realized using the finite-element analysis software COMSOL Multiphysics, and the variations in the slope temperature field, migration of liquid water, accumulation of solid ice, and deformation of frost heave are analysed under freezing and thawing environments.

#### 2. THM Coupling Theory

Based on the main physical processes in seasonally frozen soil, several hypotheses have been proposed: the freeze-thaw soil medium of the slope is incompressible, homogeneous, and isotropic; there is only movement of liquid water in frozen soil, while the ice remains stationary; the effect of water vapour migration on unfrozen water and heat flow migration is ignored; the effect of water migration caused by temperature gradients and convection heat transfer is ignored; the unfrozen water content in frozen soil is in dynamic equilibrium with the negative temperature of the soil. Based on these assumptions, the THM-coupled model in seasonally unsaturated frozen soils is established.

##### 2.1. Water Field Governing Equation

For plane problems, the law of water migration in unsaturated frozen soils can be expressed by Richards’ equation with a phase transition [17]:where is the volume content of unfrozen water in the soil; is the volume content of ice in the soil; is the time coordinate (s); *x*, *z* are the space coordinates (m); *x* is the horizontal direction, and *z* is the vertical direction, taking *z* as positive upward; is the density of ice (kg/m^{3}); is the density of water (kg/m^{3}); is the hydraulic conductivity of unsaturated soil (m/s), and is the diffusivity of seasonally frozen soil (m^{2}/s).

##### 2.2. Temperature Field Governing Equation

The heat conduction equation for the latent heat of the phase transition as the internal heat source is expressed as follows [17]:where is the temperature of the soil (°C), *C* is the volume heat capacity of the soil (J/m^{3}/°C), is the soil thermal conductivity (W/m/°C), and is the phase-change latent heat (J/kg).

##### 2.3. Stress Field Governing Equation

The equilibrium equation of isotropic linear thermoelastic materials can be expressed as follows:

The Cauchy equation is expressed as

The stress continuity equation can be expressed as follows [18]:where is the shear modulus (Pa), is the Lamé constant (Pa, ), *µ* is Poisson’s ratio, is the volume strain, and is the comprehensive coefficient of body thermal expansion and frost heave (°C^{−1}). When the soil temperature is lower than the freezing temperature (), is the frost heaving coefficient of the soil (°C^{−1}). When the soil temperature is higher than the freezing temperature (), is the thermal expansion coefficient of the soil (°C^{−1}). is the Kronecker delta notation, is the displacement component, and is the volume force component for the plane problem *i* = (*x*, *z*), and is the bulk elastic modulus of the soil (Pa).

By using the above static equilibrium equation and Cauchy equation, the THM-coupled stress-control equation can be expressed as

##### 2.4. Phase Transition Dynamic Equilibrium Relation

The content of unfrozen water in the pores of the soil can be expressed aswhere and represent the volume of water and ice, respectively.

The relation between the content of unfrozen water and the negative temperature is always in a dynamic equilibrium [19], which can be expressed aswhere is the empirical constant related to the properties of the soil and can be selected according to the type of soil: sand (0.61), silt (0.47), and clay (0.56).

The water in soil is composed of pore ice and pore water, the volume content of unfrozen water , the volume content of ice , and the total-volume water content , in which is the initial water content of the soil slope, taking .

##### 2.5. Impedance of Ice to the Formation of Water Flow in Frozen Soils

For frozen zones, because of the ice in the soil slope, the impedance coefficient *I* is introduced to describe the impedance of ice to the formation of water flow. The hydraulic conductivity and water diffusivity of frozen soils are reduced to 1/*I* of the thawed soil with the same water content of unfrozen soil.

The impedance coefficient is mainly determined by experience, Taylor and Luthin [5] and Jame and Norm [20] suggested that the impedance coefficient depends on the volume content of ice in the frozen soil and can be expressed as follows:

Using this treatment method, the value of the impedance coefficient is very arbitrary. Chen et al. [21] suggested that the impedance coefficient can be expressed aswhere is the unsaturated hydraulic conductivity of thawed soil under the condition of saturated unfrozen water content (), is the saturated hydraulic conductivity of thawed soil, and is the saturated water content of the soil.

The saturated unfrozen water content in the frozen soil is

Equation (10) can effectively avoid the arbitrariness of the value of the impedance coefficient.

##### 2.6. Heaviside Step Function

The two-dimensional Heaviside step function is introduced to characterise the ice-water transition process during freezing. The expression of the Heaviside step function is as follows:where is the slope temperature (°C); is the temperature of the phase transition point (°C), taking **=** −0.3; and dT is the transition gap (°C), taking dT **=** 0.3.

The Heaviside step function is shown in Figure 1.