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A coupled thermal-hydraulic-mechanical application for assessment of slope stability
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A coupled thermal-hydraulic-mechanical application for assessment of slope stability Di Wu a,b,⇑, Tengfei Deng a, Weiping Duan c, Wudi Zhang a a b
School of Energy and Mining Engineering, China University of Mining and Technology-Beijing, Beijing, China School of Civil, Environmental and Mining Engineering, The University of Western Australia, Perth, Australia c Sinosteel Maanshan Institute of Mining Research Co., Ltd, Anhui, China Received 20 October 2018; received in revised form 5 November 2019; accepted 20 December 2019
Abstract The stability of a slope is subjected to thermal (T), hydraulic (H), and mechanical (M) loadings and their coupling effects. Modeling the coupled THM processes that occur in the slope is important for reliably assessing and predicting the slope performance and stability. Therefore, a numerical model, which can consider the full coupling among the thermal (temperature variation), hydraulic (pore water pressure), and mechanical (stress and displacement) processes, is developed in this study. The developed model is employed to analyze slope stability, and the simulated results are seen to coincide well with the results obtained by traditional limit equilibrium calculation. A comparison of the results verifies the validity of the developed model for slope stability analyses under THM coupled effects. Furthermore, the capability of the developed THM model for predicting the slope performance is validated through comparisons of three case studies in terms of both laboratory experiments and numerical simulations. A favorable agreement between the modeling results and the compared data confirms the capability of the developed model to accurately describe the behavior of a slope affected by THM coupled processes. The modeling results can also contribute to a better understanding of slope failure induced by the THM couplings. Ó 2020 Production and hosting by Elsevier B.V. on behalf of The Japanese Geotechnical Society. This is an open access article under the CC BYNC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Keywords: Slope; THM; Coupled model; Numerical simulation; Stability analysis
1. Introduction Landslide events often occur at roadsides or open pit mines, negatively affecting the traffic and the mining operations, respectively, or even causing the loss of property and human lives (Basahel and Mitri, 2017). Therefore, it is essential and also meaningful to investigate and analyze slope stability, which can provide significant information for early warning and the control of landslides. The problems related to unstable slopes are due to variations in the
Peer review under responsibility of The Japanese Geotechnical Society. ⇑ Corresponding author at: School of Energy and Mining Engineering, China University of Mining and Technology-Beijing, Beijing, China. E-mail address: [email protected] (D. Wu).
rock and soil mass conditions (e.g., geometry and dimensions, development of joints and cracks, surface drainage and groundwater conditions), to external factors including natural factors (e.g., earthquakes and rainfall), and to manmade activities (e.g., excavations) (Pantelidis, 2010; Turrini and Visintainer, 1998; Malamud et al., 2004; Keefer, 1989; Wang and Niu, 2009). In summary, slope stability is influenced by the mechanical (M, e.g., action of gravity, stress redistribution after the slope excavation, and dynamic loading induced by seismic activities) and hydraulic (H, e.g., groundwater seepage and rainwater infiltration) processes and their interactions. In addition, both the mechanical and hydraulic processes in a slope are affected by the variation in temperature:
https://doi.org/10.1016/j.sandf.2019.12.007 0038-0806/Ó 2020 Production and hosting by Elsevier B.V. on behalf of The Japanese Geotechnical Society. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article as: D. Wu, T. Deng, W. Duan et al., A coupled thermal-hydraulic-mechanical application for assessment of slope stability, Soils and Foundations, https://doi.org/10.1016/j.sandf.2019.12.007
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i) Changes in temperature lead to the development of thermal stress, and thus, strain in the rock and the soil mass of a slope. ii) The variation in temperature influences the properties (e.g., density and viscosity) of the water flowing through the rock and the soil mass of a slope, and the freezing and thawing of water are also dependent on the environmental temperature. As discussed above, slope stability can be significantly affected by the mechanical, hydraulic, and thermal (T, e.g., temperature variations affected by the seasonal changes) processes and their couplings. In terms of analyzing slope stability, numerous researchers have conducted significant studies. For instance, Basahel and Mitri employed empirical methods based on rock mass classification systems to assess rock slope stability (Basahel and Mitri, 2017). Karaman et al. used kinematic methods to conduct a stability analysis for a slope (Karaman et al., 2013). Tanyas and Ulusay carried out 2D limit equilibrium back-analyses to assess the slope stability of the Alipasa feldspar open pit mine in western Turkey, and the calculated results agreed well with the movement monitoring data (Tanyas and Ulusay, 2013). In addition to conventional approaches, an increasing number of researchers have adopted numerical methods to evaluate slope stability. For instance, Napoli et al. investigated the slope stability in bimrocks using both finite element (FEM) and limit equilibrium (LEM) methods (Napoli et al., 2018). Tang et al. proposed a modified limit equilibrium method with variable water pressure and shear strength to estimate the stability coefficient of a sloping mass of stratified rock, and thus, to identify the potential sliding surface (Tang et al., 2017). Shamekhi and Tannant presented a methodology for evaluating the probabilistic rock slope stability using numerical modelling that incorporates a statistical analysis of the variability of the joint set’s geometric parameters (Shamekhi and Tannant, 2015). Since slope stability can be notably influenced by the THM coupled processes, some studies have been carried out to assess slope stability under multi-field coupling effects. In terms of evaluating the stability of an embankment on a sloping ground under the T-M coupled effects, several researchers have studied sloping embankments in permafrost considering the geo-temperature and deformation (Niu et al., 2005; Pei et al., 2014; Li et al., 2015). These studies only focused on the influence of the subzero temperatures. Therefore, the obtained results are not universally applicable. Some studies have also been reported with regard to assessing slope stability in consideration of the H-M coupled effects. For instance, Chen et al. analyzed the causative mechanisms of rainfall-induced fill slope failures (Chen et al., 2004). They believe that rainfall can significantly influence the flux boundary condition across the ground surface, and thus, reduce the effective stress and the shear strength of the soil that may result in the failure of
the slope. In addition, some other researchers have also investigated slope stability under the coupled H-M effects (Rahardjo et al., 2010; Rahimi et al., 2011). Although some recent papers have provided fully coupled THM models for simulating the behavior of geotechnical materials, such as studies (Gawin and Schrefler, 1996; Neaupane et al., 1999; Franc¸ois et al., 2009; Zhou and Ng, 2015), they were not focused particularly on slope issues. In general, current studies on slope stability assessments only consider the coupled H-M or T-M effects. To date, few studies have been conducted to evaluate slope stability in consideration of the coupled THM effects, especially those incorporating THM couplings in the development of a numerical model for slope stability assessments. It is vitally important to consider the thermal effect in such slope stability analyses. This is because the instability and failure of a slope are often caused by the changes in temperature that occur during the excavation and maintenance of the slope. For instance, under the long-term effect of thermal expansion and contraction, the seepage water in a slope may cause the breakthrough of the potential slip surface of the slope, which may eventually lead to the instability of the slope. As a result, there was an urgent need to conduct the present study, whose aim is to investigate the evolution of evaluation indicators for slope stability in response to coupled THM processes. This paper is organized as follows: 1) Firstly, a THM coupling model is presented. The model combines equations to describe the thermal, hydraulic, and mechanical processes (Section 2). 2) Secondly, the validity of the developed model is verified by comparing the model simulation results with the data of traditional limit equilibrium calculation (Section 3). 3) Thirdly, the availability of the developed model is further validated by comparing the model simulation results with the results from other case studies, and the validated model is used for some applications (Section 4). 4) Finally, the conclusions are presented (Section 5).
2. Governing equations for the coupled THM model 2.1. Basic assumptions Before developing the coupled THM model mathematically, the following assumptions should be made: (1) The slope is assumed to be a continuous porous medium in which the solid, water, and dry air are independent of each other and the dry air is regarded as ideal gas. The solid particles are incompressible, but the porous medium skeleton can be deformed. (2) The slope is assumed to be an isotropic elastoplastic structure.
Please cite this article as: D. Wu, T. Deng, W. Duan et al., A coupled thermal-hydraulic-mechanical application for assessment of slope stability, Soils and Foundations, https://doi.org/10.1016/j.sandf.2019.12.007
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(3) The density of the water varies with the temperature, while the viscosity of the water is assumed to be independent of the temperature.
2.2. Mechanical equations 2.2.1. Effective stress The effective stress in an unsaturated and formable porous medium (e.g., a slope) can be expressed as follows (Schrefler and Zhan, 1993; Nuth and Laloui, 2008): rij ¼ re þ adij p
p ¼ S w pw þ S a pa
ð2Þ
where Sw and Sa are the degrees of the saturation of water and air in the porous medium, respectively, and pw and pa are the pore water pressure and air pressure, respectively. 2.2.2. Thermo-elastic-plastic constitutive relation The thermo-elastic-plastic constitutive equations can be written in the following form: re ¼ Dee ¼ D e - ep - et ð3Þ where D is the elasticity matrix, and e, ee, ep, and et are the total strain, elastic strain, plastic strain, and thermal strain, respectively. Elasticity matrix D can be expressed in terms of Lame´ parameters k and l in the following form: 3 2 k þ 2l k k 0 0 0 7 6 6 k k þ 2l k 0 0 07 7 6 7 6 7 6 k k k þ 2l 0 0 0 7 6 ð4Þ D ¼6 7 6 0 0 0 l 0 07 7 6 7 6 6 0 0 0 0 l 07 5 4 0
0
0
0
l
where k and l can be calculated, respectively, by the following expressions: Et k¼ ð1 þ tÞð1 - 2tÞ
ð5Þ
E l¼ 2ð 1 þ t Þ
ð6Þ
where E is the elastic modulus and t is Poisson’s ratio. The thermal strain in the porous medium can be obtained by the following equation: det ¼ bt dij dT
where bt is the coefficient of linear thermal expansion and T is the temperature of the solid matrix of the porous medium. 2.2.3. Geometric equation of deformation For small deformations, the geometric equation of deformation is expressed as follows: 1 ð8Þ eij ¼ ui;j þ uj;i 2 where u is the displacement in the skeleton of the porous medium.
ð1Þ
where rij is the total stress vector, re is the effective stress, a is Biot’s coefficient, dij is Kronecker’s delta (dii = 1; di–j = 0), and p is the average pressure of the mixture of water and air.
0
3
ð7Þ
2.2.4. Equation of mechanical equilibrium According to the principle of momentum balance, and ignoring the influence of inertia, the following expression can be obtained for the mechanical equilibrium: Drij þ qeq gi ¼ 0
ð9Þ
where gi is the gravity vector and qeq is the equivalent bulk density. qeq ¼ ð1 nÞqs þ nS w qw þ nS a qa
ð10Þ
where qs, qw, and qa are the densities of the solid matrix, liquid water, and dry air, respectively, and n is the porosity of the porous medium. The densities of liquid water and dry air are temperature-dependent (Cui and Fall, 2015). n o0:55 qw ¼ 314:4 þ 685:6 1 ½ðT w 273:15Þ=374:141=0:55 ð11Þ qa ¼
Ma p Ra T a a
ð12Þ
where Ma is the molar mass of air, Tw and Ta are the temperatures of water and air, respectively, and Ra is the universal gas constant of air. 2.3. Hydraulic equations 2.3.1. Richards’ equation Without considering the compressibility of solid particles and liquid water, the modified Richards’ equation is employed to describe the unsaturated-saturated flow of water in the porous medium (Comsol, 2014): C
@H p þ r Kr H p þ Z ¼ Qm @t
ð13Þ
where C is the volumetric specific humidity, Hp is the pressure head, t is the time, K is the hydraulic conductivity, Z is the coordinate for the vertical elevation, and Qm is the water seepage source. 2.3.2. van Genuchten equations The van Genuchten (VG) equations are used to describe the changes in C and K (Comsol, 2014):
Please cite this article as: D. Wu, T. Deng, W. Duan et al., A coupled thermal-hydraulic-mechanical application for assessment of slope stability, Soils and Foundations, https://doi.org/10.1016/j.sandf.2019.12.007
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Table 1 Input parameters, boundary conditions, and initial values used in the three simulation schemes.
Solid material parameters Young’s modulus (MPa) Poisson’s ratio Density (kg/m3) Cohesion (kPa) Internal friction angle (°) Thermal conductivity [W/(mK)] Specific heat capacity [J/(kgK)] Thermal expansion coefficient (1/K) Hydraulic conductivity (m/s) Parameters of the VG model hs hr a m l Liquid material parameters Density (kg/m3) Reference density (kg/m3) Thermal conductivity [W/(mK)] Thermal expansion coefficient (1/K) Mechanical module Top of the slope Surface of the slope Toe of the slope Bottom side Lateral sides Volume force Hydraulic module Top of the slope Surface of the slope Toe of the slope Bottom side Lateral sides Thermal module Top of the slope Surface of the slope Toe of the slope Bottom side Lateral sides Initial temperature
M
H-M
THM
100 0.35 2000 12.38 20 N/A N/A N/A N/A
100 0.35 2000 12.38 20 N/A N/A N/A 8.25e-5
100 0.35 2000 12.38 20 1.16 840 5e-7 8.25e-5
N/A N/A N/A N/A N/A
0.5 0.05 14.5 2.68 0.5
0.5 0.05 14.5 2.68 0.5
N/A N/A N/A N/A
Based on Eq. (11) 1000 N/A 2e-5
Based on Eq. (11) 1000 0.6 2e-5
Free Free Free Fixed Roller Gravity
Free Free Free Fixed Roller Gravity
Free Free Free Fixed Roller Gravity
N/A N/A N/A N/A N/A
Mass flux (3e-5 kg/m2s) Open Open No flow No flow
Mass flux (3e-5 kg/m2s) Open Open No flow No flow
N/A N/A N/A N/A N/A N/A
N/A N/A N/A N/A N/A N/A
35 35 35 15 15 15
( C¼
8
45°
° 135
1 1 m hr ÞS e m 1 S e m Hp < 0 Hp P 0
0 8 < :
h i2 1 m Hp < 0 Sel 1 1 Sem Hp P 0
1
ð14Þ
ð15Þ
13
10
10
K¼
am ð hs 1m
°C °C °C °C °C °C
where a, m, and l are the constants of the porous medium, hs and hr are the saturated water content and the residual water content, respectively, and Se is the effective saturation that can be further expressed as ( 20
Fig. 1. Schematic diagram of the soil slope.
Se ¼
1
½1þjaH p j 1
n m
Hp < 0 Hp P 0
ð16Þ
Please cite this article as: D. Wu, T. Deng, W. Duan et al., A coupled thermal-hydraulic-mechanical application for assessment of slope stability, Soils and Foundations, https://doi.org/10.1016/j.sandf.2019.12.007
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Fig. 2. Mesh generation of the slope geometry. Fig. 4. Equivalent plastic zone distribution of the slope under the coupled H-M effect.
where cs and cw are the specific heat capacities of the solid matrix and water, respectively, and ks and kw are the thermal conductivities of the solid matrix and water, respectively. 3. Model validation against traditional limit equilibrium calculation The developed model is implemented into the software COMSOL Multiphysics to carry out the numerical computation, and the simulated results are compared with the results obtained by traditional limit equilibrium calculation in order to verify the validity of the developed model. The input parameters, boundary conditions, and initial values used for the numerical computation are listed in Table 1. Fig. 3. Equivalent plastic zone distribution of the slope under the M effect.
2.4. Thermal equations The heat convection through dry air is insignificant; hence, the heat transfer via dry air is ignored in this study. In addition, the viscous dissipation of the groundwater and the dissipative heat generated by the deformation of the porous medium are also ignored. In consideration of the thermal energy balance between the porous medium and water, and according to Fourier’s law, the following equation can be obtained (He et al., 2006): @T ð1 nÞqs cs @H þ nS w qw cw @T@tw ¼ D½ð1 nÞks DT þ nS w kw DT w p @ ðdij eij Þ ð1 nÞT bt @t þS w qw cw D T w KD H p þ Z
ð17Þ
Fig. 5. Equivalent plastic zone distribution in the slope under the coupled THM effect.
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3.1. Strength reduction method
the factor of safety (FOS) is just the critical trial factor and can be obtained as follows:
During the analysis of the slope stability by the finite element method (FEM), the technique for the shear strength reduction (SSR) is commonly employed. When performing the analysis of the slope stability with the SSR FEM, each set of shear strength parameters is obtained by dividing the actual strength parameters by trial factor Ftrial. Thus, the reduced cohesion, ctrial, and the internal friction angle, /trial, can be defined by (Xu et al., 2009)
FOS ¼ c=c0 ¼ tanu=tanu0
where c and / are the reduced cohesion and the internal friction angle at the critical limit state, respectively. In the present study, the critical limit state, which is designated as the deformation break criterion, corresponds to the results of the non-convergence in the FEM calculation.
ctrail ¼ c=F trail
ð18Þ
3.2. Model simulation with FEM
/trail ¼ arctanðtan/=F trail Þ
ð19Þ
where c and / are the actual cohesion and the internal friction angle, respectively. The process for the above trial calculation is continued until the slope reaches the critical limit state, for which
0
ð20Þ
0
A homogeneous soil slope, with a height of 10 m and a slope angle of 45°, shown in Fig. 1, is selected for the model simulation. The geometry of the soil slope is imported into COMSOL Multiphysics for generating the mesh, as shown in Fig. 2.
Fig. 6. Limit equilibrium analysis of the slope stability with different calculation schemes: (a) M, (b) H-M, and (c) THM. Please cite this article as: D. Wu, T. Deng, W. Duan et al., A coupled thermal-hydraulic-mechanical application for assessment of slope stability, Soils and Foundations, https://doi.org/10.1016/j.sandf.2019.12.007
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In this section, three schemes are selected for the model prediction of the slope stability: analyses under the mechanical (M) effect, the coupled hydro-mechanical (HM) effect, and the coupled thermo-hydro-mechanical (THM) effect. Table 1 shows the input parameters, boundary conditions, and initial values for the model simulations of each scheme. 3.2.1. Slope stability analysis under the M effect The mechanical equations of the developed model are implemented into the ‘‘Structural Mechanics” module of COMSOL Multiphysics for numerical calculation, and the finite element strength reduction method is used to reduce the soil internal friction angle and cohesion. When the calculation results are not able to converge, Ftrial is equal to 1.029. That is, the FOS of the soil slope under the M effect is 1.029. Fig. 3 shows the equivalent plastic zone distribution in the slope at this moment (it can be denoted as t0). It can be noticed from Fig. 3 that, when the calculation results are not convergent, the maximum plastic strain (equivalent plastic volumetric strain) in the slope is 0.07. It can also be detected from this figure that the distribution of the critical slip surface within the slope is at the critical limit state. 3.2.2. Slope stability analysis under the coupled H-M effect The hydraulic and mechanical equations of the developed model are implemented into the ‘‘Structural Mechanics” and the ‘‘Fluid Flow” modules of COMSOL, respectively, and a coupling between the two modules is proceeded in COMSOL. The input parameters and corresponding values listed in Table 1 are used for the coupled computation. After loading the H effect from the moment of t0 for 60 min, the stability state of the slope is investigated with the strength reduction method. When the calculation process does not converge, Ftrial is equal to 0.845. Therefore, the FOS of the slope is also 0.845, which is 0.184 less than the FOS of the slope when it is only under the M effect, indicating that the loading of the coupled HM effect reduces the slope safety factor by almost 17.9%. The plastic zone distribution in the slope is shown in Fig. 4. It can be seen from this figure that, when the slope reaches a critically steady state, the maximum plastic strain in the slope is 0.03, which is obviously smaller than that under the M effect (0.07). This indicates that the material properties of the slope were weakened by the influence of
Table 2 Contrast of the FOS between the model simulation and the limit equilibrium calculation results.
Model simulation Limit equilibrium calculation Deviation
M
H-M
THM
1.029 1.002 2.7%
0.845 0.817 3.4%
0.787 0.762 3.3%
7
Table 3 Input parameters, boundary conditions, and initial values used in Case Study 1.
Solid material parameters Young’s modulus (MPa) Poisson’s ratio Density (kg/m3) Cohesion (kPa) Internal friction angle (°) Hydraulic conductivity (m/s) Parameters of the VG model hs hr Α M L Liquid material parameters Density (kg/m3) Mechanical module Top of the slope Surface of the slope Toe of the slope Bottom side Lateral sides Volume force Hydraulic module Top of the slope Surface of the slope Toe of the slope Lateral sides Bottom side
Unsaturated
Saturated
45 0.35 1900 10 29 1e-5
45 0.35 2040 5 27 1e-5
0.45 0.28 14.5 2.8 0.5
0.45 0.28 0 0 0
Based on Eq. (11)
Based on Eq. (11)
Free Free Free Fixed Roller Gravity
Free Free Free Fixed Roller Gravity
No No No No No
No No No No No
flow flow flow flow flow
flow flow flow flow flow
the H-M couplings. Consequently, a landslide may occur when there is a smaller strain in the slope under the coupled H-M effect, in comparison with that affected only by the M process. This also implies that the addition of H loading on the slope increases the possibility of instability, and thus, slope slide. The reason is ascribed to the fact that the H loading decreases the effective stress in the slope. 3.2.3. Slope stability analysis under the coupled THM effect The thermal, hydraulic, and mechanical equations of the developed model are introduced separately as the ‘‘Heat Transfer”, ‘‘Structural Mechanics”, and ‘‘Fluid Flow” modules of COMSOL Multiphysics, and the linkage of the three sets of equations with a coupling of three modules is fulfilled in COMSOL. After adding all the T, H, and M loads from the moment of t0 for 60 min, the stability of the slope is analyzed using the strength reduction FEM. When the calculation results are not convergent, Ftrial is calculated to be 0.787. Hence, the FOS of the slope affected by the coupled THM processes is also 0.787, which is 0.242 less than that affected by the M effect and 0.058 smaller than that impacted by the coupled H-M effect. This indicates that the addition of both T and H processes reduces the FOS of the slope by almost 23.5% in comparison with that under only the M effect. The plastic zone distribution in the slope influenced by the coupled THM processes is graphically demonstrated in Fig. 5.
Please cite this article as: D. Wu, T. Deng, W. Duan et al., A coupled thermal-hydraulic-mechanical application for assessment of slope stability, Soils and Foundations, https://doi.org/10.1016/j.sandf.2019.12.007
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Fig. 7. Total displacement distribution in the slope at (a) saturated and (b) unsaturated states.
Table 4 Comparison between the results of the current study and Case Study 1. Maximum total displacement
The current study Case Study 1 Deviation
FOS
Unsaturated
Saturated
Unsaturated
Saturated
158.2 160 1.1%
188.1 180 4.5%
2.615 2.547 2.7%
1.972 1.953 1.0%
From Fig. 5, it can be found that, the maximum plastic strain of the slope is 0.02 when it reaches a critical steady state, indicating that the material parameters of the slope
are further weakened under the impact of the THM couplings. As a result, a smaller strain in the slope may lead to a landslide under the coupled THM loadings rather than
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Table 5 Input parameters, boundary conditions, and initial values used in Case Study 2.
Based on Eq. (11)
Based on Eq. (11)
Free Free Fixed Roller Gravity
Free Free Fixed Roller Gravity 8 kPa 8 kPa
Surface of the slope Bottom side
8 kPa Inward mass flux (the value is based on Fig. 9a) 8 kPa No flow
Lateral sides
No flow
No.15 No.14 No.13
No.12 No.11
60cm 20cm
20cm
No.10
No.5
No.9
No.4
No.8
No.3
No.7
No.2
No.6
No.1
80cm
0.57 0.231 12 1.68 0.5
10cm
0.57 0.231 12 1.68 0.5
20cm
10cm
100 0.3 2450 12.38 20 4.38e-2
10cm
200 0.3 2450 12.38 20 4.38e-2
10cm
Condition 2
10cm
Condition 1 Solid material parameters Young’s modulus (MPa) Poisson’s ratio Density (kg/m3) Cohesion (kPa) Internal friction angle (°) Hydraulic conductivity (m/s) Parameters of the VG model hs hr a m l Liquid material parameters Density (kg/m3) Mechanical module Top of the slope Surface of the slope Bottom side Lateral side Volume force Hydraulic module Initial pressure value Top of the slope
Range of water injection (Condition 1)
45°
8 kPa Inward mass flux (the value is based on Fig. 9b) No flow
40cm 140cm
Range of water injection (Condition 2)
Fig. 8. Layouts of measuring units, water injection positions, and the slope dimensions.
ysis of the slope stability are shown in Fig. 6. It should be pointed out that the red zone in Fig. 6 means the set of potential slip surfaces in the slope. It can be seen in Fig. 6 that the slope safety factors obtained by the traditional limit equilibrium method under the M, coupled H-M, and THM effects are 1.002, 0.817, and 0.762, respectively. The limit equilibrium calculation results also indicate that the addition of T and H loadings to the M process reduces the FOS, which is similar to the results of the developed model simulation that are illustrated above. 3.4. Model verification
that under both the coupled H-M and M effects; namely, the slope influenced by the coupled THM processes is more prone to sliding. 3.3. Analysis with limit equilibrium method A slope stability analysis with the traditional limit equilibrium method is conducted with the help of the software GeoStudio. The SLOPE/W, SEEP/W, and TEMP/W modules of GeoStudio are selected for analyzing the slope stability in the current study. The SLOPE/W module is used for the limit equilibrium analysis of the slope stability under the M effect, and the SEEP/W and SLOPE/W modules are used for that under the coupled H-M effect. The limit equilibrium analysis of the slope stability under the impact of THM couplings is performed with the TEMP/W, SEEP/W, and SLOPE/W modules. The corresponding input parameters and values used for the limit equilibrium calculation can be found in Table 1. The detailed results for the limit equilibrium anal-
The contrast of the FOS between the model simulation and the limit equilibrium calculation results is demonstrated in Table 2. Table 2 indicates that the results of the three schemes obtained by the finite element simulation are 2.7%, 3.4%, and 3.3% higher than those calculated by the traditional limit equilibrium method, respectively. The deviations between the results of the two methods are acceptable, implying that the simulation results of the model are in good agreement with the results calculated by the limit equilibrium method. Therefore, it is concluded that the validity of the developed model has been verified. 4. Model validation against case studies and simulation results In order to further verify the applicability of the developed THM coupling model, three case studies are selected for the verification. The case studies include numerical studies and laboratory experiments.
Please cite this article as: D. Wu, T. Deng, W. Duan et al., A coupled thermal-hydraulic-mechanical application for assessment of slope stability, Soils and Foundations, https://doi.org/10.1016/j.sandf.2019.12.007
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Fig. 9. Water injection rate with time in the experiment and the model simulation.
4.1. Validation against Case Study 1 Case Study 1, which is a numerical study, is used currently for testing the capability of the developed model to describe the H-M coupled processes occurring in a slope. Rabie carried out this numerical study to evaluate a slope stability under the effects of two kinds of infiltrations (saturated and unsaturated) (Rabie, 2014). The simulation results of this study were used for a comparison with the results from the present study in order to verify the ability of the developed model for evaluating slope stability under the H-M coupled effects. Table 3 shows the input parameters, boundary conditions, and initial values used for the numerical simulation of Case Study 1.
The developed model is implemented to predict the total displacements for unsaturated and saturated slopes, respectively, and the simulated results are demonstrated in Fig. 7. From this figure it can be seen that the maximum total displacement of the unsaturated slope is lower than that of the saturated one. This is because suction in the unsaturated slope is higher than that in the saturated slope; and thus, the effective stress in the unsaturated slope is higher than that in the saturated slope. Therefore, the strength and rigidity of the unsaturated slope is higher, and the displacement generated in the unsaturated slope is smaller. The maximum total displacement and FOS obtained by the current study are compared with those from Case Study 1, as illustrated in Table 4. From this table, it can be seen
Please cite this article as: D. Wu, T. Deng, W. Duan et al., A coupled thermal-hydraulic-mechanical application for assessment of slope stability, Soils and Foundations, https://doi.org/10.1016/j.sandf.2019.12.007
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(a) Data from monitoring points No. 1 - No. 5
(b) Data from monitoring points No. 6 - No. 10
(c) Data from monitoring points No. 11 - No. 15 Fig. 10. Comparisons between the measured and the simulated results for the negative PWP at all the monitoring points under Condition 1.
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(a) Data from monitoring points No. 1 - No. 5
(b) Data from monitoring points No. 6 - No. 10
(c) Data from monitoring points No. 11 - No. 15 Fig. 11. Comparisons between the measured and the simulated results for the negative PWP at all the monitoring points under Condition 2.
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(a) Condition 1
(a) Condition 1
(b) Condition 2 Fig. 12. Predicted distributions of total displacement in the slope when it fails.
that the maximum total displacements in the saturated and unsaturated slopes predicted by the developed model are 4.5% higher and 1.1% lower, respectively, than those obtained from Case Study 1. Moreover, the safety factors of the unsaturated and saturated slopes calculated by the developed model are 2.7% and 1.0% higher, respectively, than those obtained from Case Study 1. The deviations between the results of the present and the case studies are acceptable, indicating that the developed model is capable of estimating well the slope stability under the influence of the H-M coupled processes. In addition, Table 4 also demonstrates that, when the slope transforms from the saturated state to the unsaturated state, the maximum total displacement in the slope decreases and the FOS increases. 4.2. Validation against Case Study 2 Case Study 2, which is a laboratory experiment, is used for further testing the capability of the developed model to describe the H-M coupled processes in a slope. Kitamura et al. conducted this laboratory experiment to investigate the slope stability under two different conditions: water injection from the top and water injection into the bottom, for simulating the effects of rainfall and the rise in groundwater level, respectively, on the slope stability (Kitamura
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(b) Condition 2 Fig. 13. Predicted distributions of equivalent plastic strain in the slope when it fails.
0 , 2.5
1.75 , 2.5
P1 3 , 1.66 P2 3.4,1.4
5.6 , 0.5 4.75 , 0.5 0,0
5.6 , 0
Fig. 14. Geometry of the slope used for the simulation.
et al., 2007). The case study is selected to verify the availability of the developed model for assessing the slope stability under the H-M couplings, by comparing the model prediction results of the present study with the data from the referred experiment investigation. Table 5 lists the input parameters, boundary conditions, and initial values for the model simulations of Case Study 2. Fig. 8 schematically shows the slope dimension and the arrangement of the measuring units in the tested slope. Fifteen tensiometers (labeled No. 1 to No. 15) are buried in the slope (with an angle of 45° and a height of 80 cm) for
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D. Wu et al. / Soils and Foundations xxx (xxxx) xxx Table 6 Input parameters, boundary conditions, and initial values used in Case Study 3.
Solid material parameters Young’s modulus (MPa) Poisson’s ratio Density (kg/m3) Cohesion (kPa) Internal friction angle (°) Saturated hydraulic conductivity (m/s) Thermal conductivity [W/(mK)] Heat capacity at constant pressure [J/(kgK)] Coefficient of thermal expansion (1/K) Parameters of the VG model hs hr a m l Liquid material parameters Density (kg/m3) Thermal module Top of the slope (K) Surface of the slope (K) Toe of the slope (K) Bottom side (K) Lateral sides (K) Initial value (K) Mechanical module Top of the slope Surface of the slope Toe of the slope Bottom side Lateral side Volume force Hydraulic module Top of the slope Surface of the slope Toe of the slope Bottom side Lateral sides
Situation (1)
Situation (2)
30 0.3 1600 30 10 1e-9 0.1 500
Variable Variable Variable Variable Variable 1e-9 0.1 500
0.42 0.168 0.0498 0.779 0.5
0.42 0.168 0.0498 0.779 0.5
Based on Eq. (11)
Based on Eq. (11)
373.15 373.15 373.15 273.15 273.15 273.15
373.15 373.15 373.15 273.15 273.15 273.15
Free Free Free Fixed Roller Gravity
Free Free Free Fixed Roller Gravity
Inward mass flux (1.5e-3 kg/m2s) Inward mass flux (1.5e-3 kg/m2s) Inward mass flux (1.5e-3 kg/m2s) No flow No flow
Inward mass flux (1.5e-3 kg/m2s) Inward mass flux (1.5e-3 kg/m2s) Inward mass flux (1.5e-3 kg/m2s) No flow No flow
Table 7 Mechanical parameters of the expansive soil for the numerical analysis in Case Study 3.
Initial condition Saturated condition
Young’s modulus (MPa)
Poisson’s ratio
Density (kg/m3)
Cohesion (kPa)
Internal friction angle (°)
30 10
0.3 0.4
1600 1400
30 20
10 6
The data in Table 7 are based on the referred study (Ding et al., 2015).
investigating the development of negative pore water pressure (PWP). The water injection positions under the two conditions are also shown in this figure. Fig. 9a and b show the evolution of the water injection rate with elapsed time under the two conditions, respectively. It can be seen in Fig. 9a that, under Condition 1, the water injection rate with time simulated by the model is consistent with that in the experiment. Moreover, in the
experiment, the slope fails after about 113 min, while in the model simulation, the slope fails after 120 min. In other words, the time point for the slope failure predicted by the developed model is fairly close to that monitored by the referred experiment. It can also be noticed in Fig. 9b that, under Condition 2, the water injection rate with time, set up in the model simulation, agrees well with that designed in the experiment. Moreover, in the experimental investigation, the slope fails
Please cite this article as: D. Wu, T. Deng, W. Duan et al., A coupled thermal-hydraulic-mechanical application for assessment of slope stability, Soils and Foundations, https://doi.org/10.1016/j.sandf.2019.12.007
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(a) Values of P1
(b) Values of P2 Fig. 15. Comparisons between measured and simulated results of the total displacement at monitoring points P1 and P2.
after about 258 min, while in the model simulation, the slope fails after 250 min. This indicates that the time point for the slope failure predicted by the developed model is relatively close to that recorded in the referred experiment. Therefore, it can be concluded that, the developed model can effectively evaluate the slope stability by accurately reporting its state. Fig. 10 illustrates comparisons of the negative PWP between the predicted results by the model simulation and the data from the experimental investigation at all the monitored points for Condition 1, while Fig. 11 illustrates comparisons of the negative PWP between the results simulated by the developed model and the data from the experiment at all the measured points for Condition 2. From these figures, it can be seen that the devel-
oped model can predict and analyze well the development of PWP within the slope. The evolution of PWP versus the elapsed time obtained by the model prediction basically agrees with the development of PWP with time obtained by the experimental measurement, except for some differences that are due to the tacit assumptions in the model simulation and the varied boundary conditions in the experimental investigation. Figs. 12 and 13 display the simulated distributions of the total displacement and the equivalent plastic strain, respectively, in the slope when it reaches the state of failure. It is seen that the results predicted by the developed model also agree well with those obtained in the referred study. The above comparison and simulation results indicate that the developed model is able to predict the PWP development, displacement, and plastic strain distributions in a
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slope, and that it is also capable of assessing slope stability under the coupled H-M effects. 4.3. Validation against Case Study 3 and model applications Based on a series of physical model tests, Ding et al. investigated the thermal expansion, humidity, and displacement of a soil slope (Ding et al., 2015). Their research is selected as Case Study 3 to examine the capability of the developed model to describe the THM coupled processes in a slope. Fig. 14 illustrates the geometry of the slope used for the simulation. During the simulation, the displacements of P1 and P2 are investigated. Table 6 lists the input parameters, boundary conditions, and initial values for the model simulations of Case Study 3 in two situations: (1) without consideration of soil softening (2) consideration of soil softening. Table 7 further demonstrates the mechanical parameters of the expansive soil for the numerical analysis.
t=50 h
t=200 h
Fig. 15a and b demonstrate the total displacements of P1 and P2, respectively, in the slope with time. From these figures, it can be noticed that the evolution of displacement versus time predicted by the developed model under Situation (2) agrees well with that measured in the referred study. This indicates that the numerical analysis that considers soil softening induced by THM couplings can match practical conditions. Fig. 15 also reveals that the developed model is capable of analyzing and predicting the response of the slope to the THM coupled effects. Consequently, the developed model is further used to simulate the THM-affected stress distribution in the slope when considering soil softening. The data on Tables 6 and 7 are used for simulating the application of the developed model. Fig. 16 illustrates the evolution of the stress distribution in the slope versus time. It can be observed from these figures that the stress development in the slope is induced by the coupled influence of thermal stress, pore water pressure, and gravity. Due to the thermal (100 °C) and hydraulic (rainfall) loads, a stress concentration zone is formed at the shallow layer along the top, surface, and toe of the slope, and with the lapse of time, this stress concentration zone extends to the deep areas of the slope.
t=100 h
t=300 h
Fig. 16. Evolution of the stress distribution in the slope versus time under the THM coupled effects. Please cite this article as: D. Wu, T. Deng, W. Duan et al., A coupled thermal-hydraulic-mechanical application for assessment of slope stability, Soils and Foundations, https://doi.org/10.1016/j.sandf.2019.12.007
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5. Conclusions Based on the results obtained in this study, the following conclusions can be drawn. (1) A THM coupled model has been developed to analyze the stability of slopes, and the model prediction results have been compared with the data of traditional limit equilibrium calculation. A good agreement was found between the two sets of outcomes, verifying the validity of the developed model for slope stability analyses. (2) The safety factor of the slope and the critical slip surface distribution within the slope were seen to be significantly influenced by the thermal, hydraulic, and mechanical processes and their couplings. (3) The modeling results coincided with the data from the referred case studies in terms of both numerical simulations and laboratory experiments, demonstrating the capability of the developed model to predict well the evolution of slope behaviors under the THM coupled processes. (4) It was confirmed that the developed model for slopes can provide accurate and abundant prediction results for assessing the slope performance, and thus, can be used to analyze slope stability under THM couplings. In addition, the developed model can also be applied to provide useful information for the design of slope reinforcement and the analyses of landslides. This paper provided model validation with LEM and three case studies. However, only Case Study 3 consisted of THM coupled effects and it had limited results (i.e., only displacement). With the limited slope data from the literature, the authors may consider a future study for validating the THM model with element tests. Acknowledgements The authors would like to acknowledge the support from the China Scholarship Council and Yue Qi Young Scholar Project, China University of Mining and Technology, Beijing, as well as the National Natural Science Foundation of China (Grant No. 51539006). References Basahel, H.M., Mitri, H., 2017. Application of rock mass classification systems to rock slope stability assessment: A case study. J. Rock Mech. Geotech. Eng. 9, 993–1009. Chen, H., Lee, C.F., Law, K.T., 2004. Causative mechanisms of rainfallinduced fill slope failures. J. Geotech. Geoenviron. 130 (6), 593–602. Comsol, Comsol Multiphysics 5.0, 2014,. Cui, L., Fall, M., 2015. A coupled thermo-hydro-mechanical-chemical model for underground cemented tailings backfill. Tunn. Undergr. Sp. Tech. 50, 396–414.
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Ding, J., Chen, R., Tong, J., Gong, B., 2015. Research on failure mechanism of shallow instability of expansive soil slope based on multifield coupling numerical analysis. Rock Soil Mech. 36 (Supp. 1), 159–168 (in Chinese). Franc¸ois, B., Laloui, L., Laurent, C., 2009. Thermo-hydro-mechanical simulation of ATLAS in situ large scale test in Boom Clay. Comput. Geotech. 36 (4), 626–640. Gawin, D., Schrefler, B.A., 1996. Thermo-hydro-mechanical analysis of partially saturated porous materials. Eng. Comput. 13 (7), 113–143. He, Y., Yang, L., Yang, J., 2006. Governing equation for coupled themohydro-mechanical behaviors in unsaturated rock mass. J. Southwest Jiaotong Uni. 41 (4), 419–423 (in Chinese). Karaman, K., Ercikdi, B., Kesimal, A., 2013. The assessment of slope stability and rock excavatability in a limestone quarry. Earth Sci. Res. J. 17 (2), 169–181. Keefer, D.V., 1989. Statistical analysis of an earthquake-induced landslide distribution-the 1989 Loma Prieta, California event. Eng. Geol. 58, 231–249. Kitamura, R., Sako, K., Kato, S., Mizushima, T., Imanishi, H., 2007. Soil tank test on seepage and failure behaviors of Shirasu slope during rainfall. Jpn. Geotech. J. 2 (3), 149–168 (in Japanese). Li, S., Lai, Y., Zhang, M., Yu, W., 2015. Seasonal differences in seismic responses of embankment on a sloping ground in permafrost regions. Soil Dynam. Earthquake Eng. 76, 122–135. Malamud, B.D., Turcotte, D.L., Guzzetti, F., Reichenbach, P., 2004. Landslide inventories and their statistical properties. Earth Surf. Process. Landforms 29, 687–711. Napoli, M.L., Barbero, M., Ravera, E., Scavia, C., 2018. A stochastic approach to slope stability analysis in bimrocks. Int. J. Rock Mech. Min. Sci. 101, 41–49. Neaupane, K.M., Yamabe, T., Yoshinaka, R., 1999. Simulation of a fully coupled thermo-hydro-mechanical system in freezing and thawing rock. Int. J. Rock Mech. Min. Sci. 36 (5), 563–580. Niu, F., Cheng, G., Ni, W., Jin, D., 2005. Engineering-related slope failure in permafrost regions of the Qinghai-Tibet Plateau. Cold Reg. Sci. Technol. 42 (3), 215–225. Nuth, M., Laloui, L., 2008. Effective stress concept in unsaturated soils: Clarification and validation of a unified framework. Int. J. Numer. Anal. Met. Geomech. 32, 771–801. Pantelidis, L., 2010. An alternative rock mass classification system for rock slopes. B. Eng. Geol. Environ. 69 (1), 29–39. Pei, W., Zhang, M., Lai, Y., Jin, L., Harbor, J., 2014. Thermal stability analysis of crushed-rock embankments on a slope in permafrost regions. Cold Reg. Sci. Technol. 106, 175–182. Rabie, M., 2014. Comparison study between traditional and finite element methods for slopes under heavy rainfall. HBRC J. 10, 160–168. Rahardjo, H., Satyanaga, A., Leong, E.C., Ng, Y.S., 2010. Effects of groundwater table position and soil properties on stability of slope during rainfall. J. Geotech. Geoenviron. 136 (11), 1555–1564. Rahimi, A., Rahardjo, H., Leong, E.C., 2011. Effect of antecedent rainfall patterns on rainfall-induced slope failure. J. Geotech. Geoenviron. 137 (5), 483–491. Schrefler, B.A., Zhan, X., 1993. A fully coupled model for water and airflow in deformable porous media. Water Resour. Res. 29 (1), 155– 167. Shamekhi, E., Tannant, D.D., 2015. Probabilistic assessment of rock slope stability using response surfaces determined from finite element models of geometric realizations. Comput. Geotech. 69, 70–81. Tang, H., Yong, R., Ez Eldin, M.A.M., 2017. Stability analysis of stratified rock slopes with spatially variable strength parameters: the case of Qianjiangping landslide. Bull. Eng. Geol. Environ. 76, 839–853. Tanyas, H., Ulusay, R., 2013. Assessment of structurally-controlled slope failure mechanisms and remedial design considerations at a feldspar open pit mine, Western Turkey. Eng. Geol. 155, 54–68.
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Turrini, C.T., Visintainer, P., 1998. Proposal of a method to define areas of landslide hazard and application to an area of the Dolomites, Italy. Eng. Geol. 50, 255–265. Wang, X., Niu, R., 2009. Spatial forecast of landslides in three gorges based on spatial data mining. Sensors 9, 2035–2061.
Xu, Q., Yin, H., Cao, X., Li, Z., 2009. A temperature-driven strength reduction method for slope stability analysis. Mech. Res. Commun. 36, 224–231. Zhou, C., Ng, C.W.W., 2015. A thermomechanical model for saturated soil at small and large strains. Can. Geotech. J. 52 (8), 1101–1110.
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Technical Paper
A coupled thermal-hydraulic-mechanical application for assessment of slope stability Di Wu a,b,⇑, Tengfei Deng a, Weiping Duan c, Wudi Zhang a a b
School of Energy and Mining Engineering, China University of Mining and Technology-Beijing, Beijing, China School of Civil, Environmental and Mining Engineering, The University of Western Australia, Perth, Australia c Sinosteel Maanshan Institute of Mining Research Co., Ltd, Anhui, China Received 20 October 2018; received in revised form 5 November 2019; accepted 20 December 2019
Abstract The stability of a slope is subjected to thermal (T), hydraulic (H), and mechanical (M) loadings and their coupling effects. Modeling the coupled THM processes that occur in the slope is important for reliably assessing and predicting the slope performance and stability. Therefore, a numerical model, which can consider the full coupling among the thermal (temperature variation), hydraulic (pore water pressure), and mechanical (stress and displacement) processes, is developed in this study. The developed model is employed to analyze slope stability, and the simulated results are seen to coincide well with the results obtained by traditional limit equilibrium calculation. A comparison of the results verifies the validity of the developed model for slope stability analyses under THM coupled effects. Furthermore, the capability of the developed THM model for predicting the slope performance is validated through comparisons of three case studies in terms of both laboratory experiments and numerical simulations. A favorable agreement between the modeling results and the compared data confirms the capability of the developed model to accurately describe the behavior of a slope affected by THM coupled processes. The modeling results can also contribute to a better understanding of slope failure induced by the THM couplings. Ó 2020 Production and hosting by Elsevier B.V. on behalf of The Japanese Geotechnical Society. This is an open access article under the CC BYNC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Keywords: Slope; THM; Coupled model; Numerical simulation; Stability analysis
1. Introduction Landslide events often occur at roadsides or open pit mines, negatively affecting the traffic and the mining operations, respectively, or even causing the loss of property and human lives (Basahel and Mitri, 2017). Therefore, it is essential and also meaningful to investigate and analyze slope stability, which can provide significant information for early warning and the control of landslides. The problems related to unstable slopes are due to variations in the
Peer review under responsibility of The Japanese Geotechnical Society. ⇑ Corresponding author at: School of Energy and Mining Engineering, China University of Mining and Technology-Beijing, Beijing, China. E-mail address: [email protected] (D. Wu).
rock and soil mass conditions (e.g., geometry and dimensions, development of joints and cracks, surface drainage and groundwater conditions), to external factors including natural factors (e.g., earthquakes and rainfall), and to manmade activities (e.g., excavations) (Pantelidis, 2010; Turrini and Visintainer, 1998; Malamud et al., 2004; Keefer, 1989; Wang and Niu, 2009). In summary, slope stability is influenced by the mechanical (M, e.g., action of gravity, stress redistribution after the slope excavation, and dynamic loading induced by seismic activities) and hydraulic (H, e.g., groundwater seepage and rainwater infiltration) processes and their interactions. In addition, both the mechanical and hydraulic processes in a slope are affected by the variation in temperature:
https://doi.org/10.1016/j.sandf.2019.12.007 0038-0806/Ó 2020 Production and hosting by Elsevier B.V. on behalf of The Japanese Geotechnical Society. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article as: D. Wu, T. Deng, W. Duan et al., A coupled thermal-hydraulic-mechanical application for assessment of slope stability, Soils and Foundations, https://doi.org/10.1016/j.sandf.2019.12.007
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i) Changes in temperature lead to the development of thermal stress, and thus, strain in the rock and the soil mass of a slope. ii) The variation in temperature influences the properties (e.g., density and viscosity) of the water flowing through the rock and the soil mass of a slope, and the freezing and thawing of water are also dependent on the environmental temperature. As discussed above, slope stability can be significantly affected by the mechanical, hydraulic, and thermal (T, e.g., temperature variations affected by the seasonal changes) processes and their couplings. In terms of analyzing slope stability, numerous researchers have conducted significant studies. For instance, Basahel and Mitri employed empirical methods based on rock mass classification systems to assess rock slope stability (Basahel and Mitri, 2017). Karaman et al. used kinematic methods to conduct a stability analysis for a slope (Karaman et al., 2013). Tanyas and Ulusay carried out 2D limit equilibrium back-analyses to assess the slope stability of the Alipasa feldspar open pit mine in western Turkey, and the calculated results agreed well with the movement monitoring data (Tanyas and Ulusay, 2013). In addition to conventional approaches, an increasing number of researchers have adopted numerical methods to evaluate slope stability. For instance, Napoli et al. investigated the slope stability in bimrocks using both finite element (FEM) and limit equilibrium (LEM) methods (Napoli et al., 2018). Tang et al. proposed a modified limit equilibrium method with variable water pressure and shear strength to estimate the stability coefficient of a sloping mass of stratified rock, and thus, to identify the potential sliding surface (Tang et al., 2017). Shamekhi and Tannant presented a methodology for evaluating the probabilistic rock slope stability using numerical modelling that incorporates a statistical analysis of the variability of the joint set’s geometric parameters (Shamekhi and Tannant, 2015). Since slope stability can be notably influenced by the THM coupled processes, some studies have been carried out to assess slope stability under multi-field coupling effects. In terms of evaluating the stability of an embankment on a sloping ground under the T-M coupled effects, several researchers have studied sloping embankments in permafrost considering the geo-temperature and deformation (Niu et al., 2005; Pei et al., 2014; Li et al., 2015). These studies only focused on the influence of the subzero temperatures. Therefore, the obtained results are not universally applicable. Some studies have also been reported with regard to assessing slope stability in consideration of the H-M coupled effects. For instance, Chen et al. analyzed the causative mechanisms of rainfall-induced fill slope failures (Chen et al., 2004). They believe that rainfall can significantly influence the flux boundary condition across the ground surface, and thus, reduce the effective stress and the shear strength of the soil that may result in the failure of
the slope. In addition, some other researchers have also investigated slope stability under the coupled H-M effects (Rahardjo et al., 2010; Rahimi et al., 2011). Although some recent papers have provided fully coupled THM models for simulating the behavior of geotechnical materials, such as studies (Gawin and Schrefler, 1996; Neaupane et al., 1999; Franc¸ois et al., 2009; Zhou and Ng, 2015), they were not focused particularly on slope issues. In general, current studies on slope stability assessments only consider the coupled H-M or T-M effects. To date, few studies have been conducted to evaluate slope stability in consideration of the coupled THM effects, especially those incorporating THM couplings in the development of a numerical model for slope stability assessments. It is vitally important to consider the thermal effect in such slope stability analyses. This is because the instability and failure of a slope are often caused by the changes in temperature that occur during the excavation and maintenance of the slope. For instance, under the long-term effect of thermal expansion and contraction, the seepage water in a slope may cause the breakthrough of the potential slip surface of the slope, which may eventually lead to the instability of the slope. As a result, there was an urgent need to conduct the present study, whose aim is to investigate the evolution of evaluation indicators for slope stability in response to coupled THM processes. This paper is organized as follows: 1) Firstly, a THM coupling model is presented. The model combines equations to describe the thermal, hydraulic, and mechanical processes (Section 2). 2) Secondly, the validity of the developed model is verified by comparing the model simulation results with the data of traditional limit equilibrium calculation (Section 3). 3) Thirdly, the availability of the developed model is further validated by comparing the model simulation results with the results from other case studies, and the validated model is used for some applications (Section 4). 4) Finally, the conclusions are presented (Section 5).
2. Governing equations for the coupled THM model 2.1. Basic assumptions Before developing the coupled THM model mathematically, the following assumptions should be made: (1) The slope is assumed to be a continuous porous medium in which the solid, water, and dry air are independent of each other and the dry air is regarded as ideal gas. The solid particles are incompressible, but the porous medium skeleton can be deformed. (2) The slope is assumed to be an isotropic elastoplastic structure.
Please cite this article as: D. Wu, T. Deng, W. Duan et al., A coupled thermal-hydraulic-mechanical application for assessment of slope stability, Soils and Foundations, https://doi.org/10.1016/j.sandf.2019.12.007
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(3) The density of the water varies with the temperature, while the viscosity of the water is assumed to be independent of the temperature.
2.2. Mechanical equations 2.2.1. Effective stress The effective stress in an unsaturated and formable porous medium (e.g., a slope) can be expressed as follows (Schrefler and Zhan, 1993; Nuth and Laloui, 2008): rij ¼ re þ adij p
p ¼ S w pw þ S a pa
ð2Þ
where Sw and Sa are the degrees of the saturation of water and air in the porous medium, respectively, and pw and pa are the pore water pressure and air pressure, respectively. 2.2.2. Thermo-elastic-plastic constitutive relation The thermo-elastic-plastic constitutive equations can be written in the following form: re ¼ Dee ¼ D e - ep - et ð3Þ where D is the elasticity matrix, and e, ee, ep, and et are the total strain, elastic strain, plastic strain, and thermal strain, respectively. Elasticity matrix D can be expressed in terms of Lame´ parameters k and l in the following form: 3 2 k þ 2l k k 0 0 0 7 6 6 k k þ 2l k 0 0 07 7 6 7 6 7 6 k k k þ 2l 0 0 0 7 6 ð4Þ D ¼6 7 6 0 0 0 l 0 07 7 6 7 6 6 0 0 0 0 l 07 5 4 0
0
0
0
l
where k and l can be calculated, respectively, by the following expressions: Et k¼ ð1 þ tÞð1 - 2tÞ
ð5Þ
E l¼ 2ð 1 þ t Þ
ð6Þ
where E is the elastic modulus and t is Poisson’s ratio. The thermal strain in the porous medium can be obtained by the following equation: det ¼ bt dij dT
where bt is the coefficient of linear thermal expansion and T is the temperature of the solid matrix of the porous medium. 2.2.3. Geometric equation of deformation For small deformations, the geometric equation of deformation is expressed as follows: 1 ð8Þ eij ¼ ui;j þ uj;i 2 where u is the displacement in the skeleton of the porous medium.
ð1Þ
where rij is the total stress vector, re is the effective stress, a is Biot’s coefficient, dij is Kronecker’s delta (dii = 1; di–j = 0), and p is the average pressure of the mixture of water and air.
0
3
ð7Þ
2.2.4. Equation of mechanical equilibrium According to the principle of momentum balance, and ignoring the influence of inertia, the following expression can be obtained for the mechanical equilibrium: Drij þ qeq gi ¼ 0
ð9Þ
where gi is the gravity vector and qeq is the equivalent bulk density. qeq ¼ ð1 nÞqs þ nS w qw þ nS a qa
ð10Þ
where qs, qw, and qa are the densities of the solid matrix, liquid water, and dry air, respectively, and n is the porosity of the porous medium. The densities of liquid water and dry air are temperature-dependent (Cui and Fall, 2015). n o0:55 qw ¼ 314:4 þ 685:6 1 ½ðT w 273:15Þ=374:141=0:55 ð11Þ qa ¼
Ma p Ra T a a
ð12Þ
where Ma is the molar mass of air, Tw and Ta are the temperatures of water and air, respectively, and Ra is the universal gas constant of air. 2.3. Hydraulic equations 2.3.1. Richards’ equation Without considering the compressibility of solid particles and liquid water, the modified Richards’ equation is employed to describe the unsaturated-saturated flow of water in the porous medium (Comsol, 2014): C
@H p þ r Kr H p þ Z ¼ Qm @t
ð13Þ
where C is the volumetric specific humidity, Hp is the pressure head, t is the time, K is the hydraulic conductivity, Z is the coordinate for the vertical elevation, and Qm is the water seepage source. 2.3.2. van Genuchten equations The van Genuchten (VG) equations are used to describe the changes in C and K (Comsol, 2014):
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Table 1 Input parameters, boundary conditions, and initial values used in the three simulation schemes.
Solid material parameters Young’s modulus (MPa) Poisson’s ratio Density (kg/m3) Cohesion (kPa) Internal friction angle (°) Thermal conductivity [W/(mK)] Specific heat capacity [J/(kgK)] Thermal expansion coefficient (1/K) Hydraulic conductivity (m/s) Parameters of the VG model hs hr a m l Liquid material parameters Density (kg/m3) Reference density (kg/m3) Thermal conductivity [W/(mK)] Thermal expansion coefficient (1/K) Mechanical module Top of the slope Surface of the slope Toe of the slope Bottom side Lateral sides Volume force Hydraulic module Top of the slope Surface of the slope Toe of the slope Bottom side Lateral sides Thermal module Top of the slope Surface of the slope Toe of the slope Bottom side Lateral sides Initial temperature
M
H-M
THM
100 0.35 2000 12.38 20 N/A N/A N/A N/A
100 0.35 2000 12.38 20 N/A N/A N/A 8.25e-5
100 0.35 2000 12.38 20 1.16 840 5e-7 8.25e-5
N/A N/A N/A N/A N/A
0.5 0.05 14.5 2.68 0.5
0.5 0.05 14.5 2.68 0.5
N/A N/A N/A N/A
Based on Eq. (11) 1000 N/A 2e-5
Based on Eq. (11) 1000 0.6 2e-5
Free Free Free Fixed Roller Gravity
Free Free Free Fixed Roller Gravity
Free Free Free Fixed Roller Gravity
N/A N/A N/A N/A N/A
Mass flux (3e-5 kg/m2s) Open Open No flow No flow
Mass flux (3e-5 kg/m2s) Open Open No flow No flow
N/A N/A N/A N/A N/A N/A
N/A N/A N/A N/A N/A N/A
35 35 35 15 15 15
( C¼
8
45°
° 135
1 1 m hr ÞS e m 1 S e m Hp < 0 Hp P 0
0 8 < :
h i2 1 m Hp < 0 Sel 1 1 Sem Hp P 0
1
ð14Þ
ð15Þ
13
10
10
K¼
am ð hs 1m
°C °C °C °C °C °C
where a, m, and l are the constants of the porous medium, hs and hr are the saturated water content and the residual water content, respectively, and Se is the effective saturation that can be further expressed as ( 20
Fig. 1. Schematic diagram of the soil slope.
Se ¼
1
½1þjaH p j 1
n m
Hp < 0 Hp P 0
ð16Þ
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5
Fig. 2. Mesh generation of the slope geometry. Fig. 4. Equivalent plastic zone distribution of the slope under the coupled H-M effect.
where cs and cw are the specific heat capacities of the solid matrix and water, respectively, and ks and kw are the thermal conductivities of the solid matrix and water, respectively. 3. Model validation against traditional limit equilibrium calculation The developed model is implemented into the software COMSOL Multiphysics to carry out the numerical computation, and the simulated results are compared with the results obtained by traditional limit equilibrium calculation in order to verify the validity of the developed model. The input parameters, boundary conditions, and initial values used for the numerical computation are listed in Table 1. Fig. 3. Equivalent plastic zone distribution of the slope under the M effect.
2.4. Thermal equations The heat convection through dry air is insignificant; hence, the heat transfer via dry air is ignored in this study. In addition, the viscous dissipation of the groundwater and the dissipative heat generated by the deformation of the porous medium are also ignored. In consideration of the thermal energy balance between the porous medium and water, and according to Fourier’s law, the following equation can be obtained (He et al., 2006): @T ð1 nÞqs cs @H þ nS w qw cw @T@tw ¼ D½ð1 nÞks DT þ nS w kw DT w p @ ðdij eij Þ ð1 nÞT bt @t þS w qw cw D T w KD H p þ Z
ð17Þ
Fig. 5. Equivalent plastic zone distribution in the slope under the coupled THM effect.
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3.1. Strength reduction method
the factor of safety (FOS) is just the critical trial factor and can be obtained as follows:
During the analysis of the slope stability by the finite element method (FEM), the technique for the shear strength reduction (SSR) is commonly employed. When performing the analysis of the slope stability with the SSR FEM, each set of shear strength parameters is obtained by dividing the actual strength parameters by trial factor Ftrial. Thus, the reduced cohesion, ctrial, and the internal friction angle, /trial, can be defined by (Xu et al., 2009)
FOS ¼ c=c0 ¼ tanu=tanu0
where c and / are the reduced cohesion and the internal friction angle at the critical limit state, respectively. In the present study, the critical limit state, which is designated as the deformation break criterion, corresponds to the results of the non-convergence in the FEM calculation.
ctrail ¼ c=F trail
ð18Þ
3.2. Model simulation with FEM
/trail ¼ arctanðtan/=F trail Þ
ð19Þ
where c and / are the actual cohesion and the internal friction angle, respectively. The process for the above trial calculation is continued until the slope reaches the critical limit state, for which
0
ð20Þ
0
A homogeneous soil slope, with a height of 10 m and a slope angle of 45°, shown in Fig. 1, is selected for the model simulation. The geometry of the soil slope is imported into COMSOL Multiphysics for generating the mesh, as shown in Fig. 2.
Fig. 6. Limit equilibrium analysis of the slope stability with different calculation schemes: (a) M, (b) H-M, and (c) THM. Please cite this article as: D. Wu, T. Deng, W. Duan et al., A coupled thermal-hydraulic-mechanical application for assessment of slope stability, Soils and Foundations, https://doi.org/10.1016/j.sandf.2019.12.007
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In this section, three schemes are selected for the model prediction of the slope stability: analyses under the mechanical (M) effect, the coupled hydro-mechanical (HM) effect, and the coupled thermo-hydro-mechanical (THM) effect. Table 1 shows the input parameters, boundary conditions, and initial values for the model simulations of each scheme. 3.2.1. Slope stability analysis under the M effect The mechanical equations of the developed model are implemented into the ‘‘Structural Mechanics” module of COMSOL Multiphysics for numerical calculation, and the finite element strength reduction method is used to reduce the soil internal friction angle and cohesion. When the calculation results are not able to converge, Ftrial is equal to 1.029. That is, the FOS of the soil slope under the M effect is 1.029. Fig. 3 shows the equivalent plastic zone distribution in the slope at this moment (it can be denoted as t0). It can be noticed from Fig. 3 that, when the calculation results are not convergent, the maximum plastic strain (equivalent plastic volumetric strain) in the slope is 0.07. It can also be detected from this figure that the distribution of the critical slip surface within the slope is at the critical limit state. 3.2.2. Slope stability analysis under the coupled H-M effect The hydraulic and mechanical equations of the developed model are implemented into the ‘‘Structural Mechanics” and the ‘‘Fluid Flow” modules of COMSOL, respectively, and a coupling between the two modules is proceeded in COMSOL. The input parameters and corresponding values listed in Table 1 are used for the coupled computation. After loading the H effect from the moment of t0 for 60 min, the stability state of the slope is investigated with the strength reduction method. When the calculation process does not converge, Ftrial is equal to 0.845. Therefore, the FOS of the slope is also 0.845, which is 0.184 less than the FOS of the slope when it is only under the M effect, indicating that the loading of the coupled HM effect reduces the slope safety factor by almost 17.9%. The plastic zone distribution in the slope is shown in Fig. 4. It can be seen from this figure that, when the slope reaches a critically steady state, the maximum plastic strain in the slope is 0.03, which is obviously smaller than that under the M effect (0.07). This indicates that the material properties of the slope were weakened by the influence of
Table 2 Contrast of the FOS between the model simulation and the limit equilibrium calculation results.
Model simulation Limit equilibrium calculation Deviation
M
H-M
THM
1.029 1.002 2.7%
0.845 0.817 3.4%
0.787 0.762 3.3%
7
Table 3 Input parameters, boundary conditions, and initial values used in Case Study 1.
Solid material parameters Young’s modulus (MPa) Poisson’s ratio Density (kg/m3) Cohesion (kPa) Internal friction angle (°) Hydraulic conductivity (m/s) Parameters of the VG model hs hr Α M L Liquid material parameters Density (kg/m3) Mechanical module Top of the slope Surface of the slope Toe of the slope Bottom side Lateral sides Volume force Hydraulic module Top of the slope Surface of the slope Toe of the slope Lateral sides Bottom side
Unsaturated
Saturated
45 0.35 1900 10 29 1e-5
45 0.35 2040 5 27 1e-5
0.45 0.28 14.5 2.8 0.5
0.45 0.28 0 0 0
Based on Eq. (11)
Based on Eq. (11)
Free Free Free Fixed Roller Gravity
Free Free Free Fixed Roller Gravity
No No No No No
No No No No No
flow flow flow flow flow
flow flow flow flow flow
the H-M couplings. Consequently, a landslide may occur when there is a smaller strain in the slope under the coupled H-M effect, in comparison with that affected only by the M process. This also implies that the addition of H loading on the slope increases the possibility of instability, and thus, slope slide. The reason is ascribed to the fact that the H loading decreases the effective stress in the slope. 3.2.3. Slope stability analysis under the coupled THM effect The thermal, hydraulic, and mechanical equations of the developed model are introduced separately as the ‘‘Heat Transfer”, ‘‘Structural Mechanics”, and ‘‘Fluid Flow” modules of COMSOL Multiphysics, and the linkage of the three sets of equations with a coupling of three modules is fulfilled in COMSOL. After adding all the T, H, and M loads from the moment of t0 for 60 min, the stability of the slope is analyzed using the strength reduction FEM. When the calculation results are not convergent, Ftrial is calculated to be 0.787. Hence, the FOS of the slope affected by the coupled THM processes is also 0.787, which is 0.242 less than that affected by the M effect and 0.058 smaller than that impacted by the coupled H-M effect. This indicates that the addition of both T and H processes reduces the FOS of the slope by almost 23.5% in comparison with that under only the M effect. The plastic zone distribution in the slope influenced by the coupled THM processes is graphically demonstrated in Fig. 5.
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Fig. 7. Total displacement distribution in the slope at (a) saturated and (b) unsaturated states.
Table 4 Comparison between the results of the current study and Case Study 1. Maximum total displacement
The current study Case Study 1 Deviation
FOS
Unsaturated
Saturated
Unsaturated
Saturated
158.2 160 1.1%
188.1 180 4.5%
2.615 2.547 2.7%
1.972 1.953 1.0%
From Fig. 5, it can be found that, the maximum plastic strain of the slope is 0.02 when it reaches a critical steady state, indicating that the material parameters of the slope
are further weakened under the impact of the THM couplings. As a result, a smaller strain in the slope may lead to a landslide under the coupled THM loadings rather than
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Table 5 Input parameters, boundary conditions, and initial values used in Case Study 2.
Based on Eq. (11)
Based on Eq. (11)
Free Free Fixed Roller Gravity
Free Free Fixed Roller Gravity 8 kPa 8 kPa
Surface of the slope Bottom side
8 kPa Inward mass flux (the value is based on Fig. 9a) 8 kPa No flow
Lateral sides
No flow
No.15 No.14 No.13
No.12 No.11
60cm 20cm
20cm
No.10
No.5
No.9
No.4
No.8
No.3
No.7
No.2
No.6
No.1
80cm
0.57 0.231 12 1.68 0.5
10cm
0.57 0.231 12 1.68 0.5
20cm
10cm
100 0.3 2450 12.38 20 4.38e-2
10cm
200 0.3 2450 12.38 20 4.38e-2
10cm
Condition 2
10cm
Condition 1 Solid material parameters Young’s modulus (MPa) Poisson’s ratio Density (kg/m3) Cohesion (kPa) Internal friction angle (°) Hydraulic conductivity (m/s) Parameters of the VG model hs hr a m l Liquid material parameters Density (kg/m3) Mechanical module Top of the slope Surface of the slope Bottom side Lateral side Volume force Hydraulic module Initial pressure value Top of the slope
Range of water injection (Condition 1)
45°
8 kPa Inward mass flux (the value is based on Fig. 9b) No flow
40cm 140cm
Range of water injection (Condition 2)
Fig. 8. Layouts of measuring units, water injection positions, and the slope dimensions.
ysis of the slope stability are shown in Fig. 6. It should be pointed out that the red zone in Fig. 6 means the set of potential slip surfaces in the slope. It can be seen in Fig. 6 that the slope safety factors obtained by the traditional limit equilibrium method under the M, coupled H-M, and THM effects are 1.002, 0.817, and 0.762, respectively. The limit equilibrium calculation results also indicate that the addition of T and H loadings to the M process reduces the FOS, which is similar to the results of the developed model simulation that are illustrated above. 3.4. Model verification
that under both the coupled H-M and M effects; namely, the slope influenced by the coupled THM processes is more prone to sliding. 3.3. Analysis with limit equilibrium method A slope stability analysis with the traditional limit equilibrium method is conducted with the help of the software GeoStudio. The SLOPE/W, SEEP/W, and TEMP/W modules of GeoStudio are selected for analyzing the slope stability in the current study. The SLOPE/W module is used for the limit equilibrium analysis of the slope stability under the M effect, and the SEEP/W and SLOPE/W modules are used for that under the coupled H-M effect. The limit equilibrium analysis of the slope stability under the impact of THM couplings is performed with the TEMP/W, SEEP/W, and SLOPE/W modules. The corresponding input parameters and values used for the limit equilibrium calculation can be found in Table 1. The detailed results for the limit equilibrium anal-
The contrast of the FOS between the model simulation and the limit equilibrium calculation results is demonstrated in Table 2. Table 2 indicates that the results of the three schemes obtained by the finite element simulation are 2.7%, 3.4%, and 3.3% higher than those calculated by the traditional limit equilibrium method, respectively. The deviations between the results of the two methods are acceptable, implying that the simulation results of the model are in good agreement with the results calculated by the limit equilibrium method. Therefore, it is concluded that the validity of the developed model has been verified. 4. Model validation against case studies and simulation results In order to further verify the applicability of the developed THM coupling model, three case studies are selected for the verification. The case studies include numerical studies and laboratory experiments.
Please cite this article as: D. Wu, T. Deng, W. Duan et al., A coupled thermal-hydraulic-mechanical application for assessment of slope stability, Soils and Foundations, https://doi.org/10.1016/j.sandf.2019.12.007
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Fig. 9. Water injection rate with time in the experiment and the model simulation.
4.1. Validation against Case Study 1 Case Study 1, which is a numerical study, is used currently for testing the capability of the developed model to describe the H-M coupled processes occurring in a slope. Rabie carried out this numerical study to evaluate a slope stability under the effects of two kinds of infiltrations (saturated and unsaturated) (Rabie, 2014). The simulation results of this study were used for a comparison with the results from the present study in order to verify the ability of the developed model for evaluating slope stability under the H-M coupled effects. Table 3 shows the input parameters, boundary conditions, and initial values used for the numerical simulation of Case Study 1.
The developed model is implemented to predict the total displacements for unsaturated and saturated slopes, respectively, and the simulated results are demonstrated in Fig. 7. From this figure it can be seen that the maximum total displacement of the unsaturated slope is lower than that of the saturated one. This is because suction in the unsaturated slope is higher than that in the saturated slope; and thus, the effective stress in the unsaturated slope is higher than that in the saturated slope. Therefore, the strength and rigidity of the unsaturated slope is higher, and the displacement generated in the unsaturated slope is smaller. The maximum total displacement and FOS obtained by the current study are compared with those from Case Study 1, as illustrated in Table 4. From this table, it can be seen
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(a) Data from monitoring points No. 1 - No. 5
(b) Data from monitoring points No. 6 - No. 10
(c) Data from monitoring points No. 11 - No. 15 Fig. 10. Comparisons between the measured and the simulated results for the negative PWP at all the monitoring points under Condition 1.
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(a) Data from monitoring points No. 1 - No. 5
(b) Data from monitoring points No. 6 - No. 10
(c) Data from monitoring points No. 11 - No. 15 Fig. 11. Comparisons between the measured and the simulated results for the negative PWP at all the monitoring points under Condition 2.
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(a) Condition 1
(a) Condition 1
(b) Condition 2 Fig. 12. Predicted distributions of total displacement in the slope when it fails.
that the maximum total displacements in the saturated and unsaturated slopes predicted by the developed model are 4.5% higher and 1.1% lower, respectively, than those obtained from Case Study 1. Moreover, the safety factors of the unsaturated and saturated slopes calculated by the developed model are 2.7% and 1.0% higher, respectively, than those obtained from Case Study 1. The deviations between the results of the present and the case studies are acceptable, indicating that the developed model is capable of estimating well the slope stability under the influence of the H-M coupled processes. In addition, Table 4 also demonstrates that, when the slope transforms from the saturated state to the unsaturated state, the maximum total displacement in the slope decreases and the FOS increases. 4.2. Validation against Case Study 2 Case Study 2, which is a laboratory experiment, is used for further testing the capability of the developed model to describe the H-M coupled processes in a slope. Kitamura et al. conducted this laboratory experiment to investigate the slope stability under two different conditions: water injection from the top and water injection into the bottom, for simulating the effects of rainfall and the rise in groundwater level, respectively, on the slope stability (Kitamura
13
(b) Condition 2 Fig. 13. Predicted distributions of equivalent plastic strain in the slope when it fails.
0 , 2.5
1.75 , 2.5
P1 3 , 1.66 P2 3.4,1.4
5.6 , 0.5 4.75 , 0.5 0,0
5.6 , 0
Fig. 14. Geometry of the slope used for the simulation.
et al., 2007). The case study is selected to verify the availability of the developed model for assessing the slope stability under the H-M couplings, by comparing the model prediction results of the present study with the data from the referred experiment investigation. Table 5 lists the input parameters, boundary conditions, and initial values for the model simulations of Case Study 2. Fig. 8 schematically shows the slope dimension and the arrangement of the measuring units in the tested slope. Fifteen tensiometers (labeled No. 1 to No. 15) are buried in the slope (with an angle of 45° and a height of 80 cm) for
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D. Wu et al. / Soils and Foundations xxx (xxxx) xxx Table 6 Input parameters, boundary conditions, and initial values used in Case Study 3.
Solid material parameters Young’s modulus (MPa) Poisson’s ratio Density (kg/m3) Cohesion (kPa) Internal friction angle (°) Saturated hydraulic conductivity (m/s) Thermal conductivity [W/(mK)] Heat capacity at constant pressure [J/(kgK)] Coefficient of thermal expansion (1/K) Parameters of the VG model hs hr a m l Liquid material parameters Density (kg/m3) Thermal module Top of the slope (K) Surface of the slope (K) Toe of the slope (K) Bottom side (K) Lateral sides (K) Initial value (K) Mechanical module Top of the slope Surface of the slope Toe of the slope Bottom side Lateral side Volume force Hydraulic module Top of the slope Surface of the slope Toe of the slope Bottom side Lateral sides
Situation (1)
Situation (2)
30 0.3 1600 30 10 1e-9 0.1 500
Variable Variable Variable Variable Variable 1e-9 0.1 500
0.42 0.168 0.0498 0.779 0.5
0.42 0.168 0.0498 0.779 0.5
Based on Eq. (11)
Based on Eq. (11)
373.15 373.15 373.15 273.15 273.15 273.15
373.15 373.15 373.15 273.15 273.15 273.15
Free Free Free Fixed Roller Gravity
Free Free Free Fixed Roller Gravity
Inward mass flux (1.5e-3 kg/m2s) Inward mass flux (1.5e-3 kg/m2s) Inward mass flux (1.5e-3 kg/m2s) No flow No flow
Inward mass flux (1.5e-3 kg/m2s) Inward mass flux (1.5e-3 kg/m2s) Inward mass flux (1.5e-3 kg/m2s) No flow No flow
Table 7 Mechanical parameters of the expansive soil for the numerical analysis in Case Study 3.
Initial condition Saturated condition
Young’s modulus (MPa)
Poisson’s ratio
Density (kg/m3)
Cohesion (kPa)
Internal friction angle (°)
30 10
0.3 0.4
1600 1400
30 20
10 6
The data in Table 7 are based on the referred study (Ding et al., 2015).
investigating the development of negative pore water pressure (PWP). The water injection positions under the two conditions are also shown in this figure. Fig. 9a and b show the evolution of the water injection rate with elapsed time under the two conditions, respectively. It can be seen in Fig. 9a that, under Condition 1, the water injection rate with time simulated by the model is consistent with that in the experiment. Moreover, in the
experiment, the slope fails after about 113 min, while in the model simulation, the slope fails after 120 min. In other words, the time point for the slope failure predicted by the developed model is fairly close to that monitored by the referred experiment. It can also be noticed in Fig. 9b that, under Condition 2, the water injection rate with time, set up in the model simulation, agrees well with that designed in the experiment. Moreover, in the experimental investigation, the slope fails
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(a) Values of P1
(b) Values of P2 Fig. 15. Comparisons between measured and simulated results of the total displacement at monitoring points P1 and P2.
after about 258 min, while in the model simulation, the slope fails after 250 min. This indicates that the time point for the slope failure predicted by the developed model is relatively close to that recorded in the referred experiment. Therefore, it can be concluded that, the developed model can effectively evaluate the slope stability by accurately reporting its state. Fig. 10 illustrates comparisons of the negative PWP between the predicted results by the model simulation and the data from the experimental investigation at all the monitored points for Condition 1, while Fig. 11 illustrates comparisons of the negative PWP between the results simulated by the developed model and the data from the experiment at all the measured points for Condition 2. From these figures, it can be seen that the devel-
oped model can predict and analyze well the development of PWP within the slope. The evolution of PWP versus the elapsed time obtained by the model prediction basically agrees with the development of PWP with time obtained by the experimental measurement, except for some differences that are due to the tacit assumptions in the model simulation and the varied boundary conditions in the experimental investigation. Figs. 12 and 13 display the simulated distributions of the total displacement and the equivalent plastic strain, respectively, in the slope when it reaches the state of failure. It is seen that the results predicted by the developed model also agree well with those obtained in the referred study. The above comparison and simulation results indicate that the developed model is able to predict the PWP development, displacement, and plastic strain distributions in a
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slope, and that it is also capable of assessing slope stability under the coupled H-M effects. 4.3. Validation against Case Study 3 and model applications Based on a series of physical model tests, Ding et al. investigated the thermal expansion, humidity, and displacement of a soil slope (Ding et al., 2015). Their research is selected as Case Study 3 to examine the capability of the developed model to describe the THM coupled processes in a slope. Fig. 14 illustrates the geometry of the slope used for the simulation. During the simulation, the displacements of P1 and P2 are investigated. Table 6 lists the input parameters, boundary conditions, and initial values for the model simulations of Case Study 3 in two situations: (1) without consideration of soil softening (2) consideration of soil softening. Table 7 further demonstrates the mechanical parameters of the expansive soil for the numerical analysis.
t=50 h
t=200 h
Fig. 15a and b demonstrate the total displacements of P1 and P2, respectively, in the slope with time. From these figures, it can be noticed that the evolution of displacement versus time predicted by the developed model under Situation (2) agrees well with that measured in the referred study. This indicates that the numerical analysis that considers soil softening induced by THM couplings can match practical conditions. Fig. 15 also reveals that the developed model is capable of analyzing and predicting the response of the slope to the THM coupled effects. Consequently, the developed model is further used to simulate the THM-affected stress distribution in the slope when considering soil softening. The data on Tables 6 and 7 are used for simulating the application of the developed model. Fig. 16 illustrates the evolution of the stress distribution in the slope versus time. It can be observed from these figures that the stress development in the slope is induced by the coupled influence of thermal stress, pore water pressure, and gravity. Due to the thermal (100 °C) and hydraulic (rainfall) loads, a stress concentration zone is formed at the shallow layer along the top, surface, and toe of the slope, and with the lapse of time, this stress concentration zone extends to the deep areas of the slope.
t=100 h
t=300 h
Fig. 16. Evolution of the stress distribution in the slope versus time under the THM coupled effects. Please cite this article as: D. Wu, T. Deng, W. Duan et al., A coupled thermal-hydraulic-mechanical application for assessment of slope stability, Soils and Foundations, https://doi.org/10.1016/j.sandf.2019.12.007
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5. Conclusions Based on the results obtained in this study, the following conclusions can be drawn. (1) A THM coupled model has been developed to analyze the stability of slopes, and the model prediction results have been compared with the data of traditional limit equilibrium calculation. A good agreement was found between the two sets of outcomes, verifying the validity of the developed model for slope stability analyses. (2) The safety factor of the slope and the critical slip surface distribution within the slope were seen to be significantly influenced by the thermal, hydraulic, and mechanical processes and their couplings. (3) The modeling results coincided with the data from the referred case studies in terms of both numerical simulations and laboratory experiments, demonstrating the capability of the developed model to predict well the evolution of slope behaviors under the THM coupled processes. (4) It was confirmed that the developed model for slopes can provide accurate and abundant prediction results for assessing the slope performance, and thus, can be used to analyze slope stability under THM couplings. In addition, the developed model can also be applied to provide useful information for the design of slope reinforcement and the analyses of landslides. This paper provided model validation with LEM and three case studies. However, only Case Study 3 consisted of THM coupled effects and it had limited results (i.e., only displacement). With the limited slope data from the literature, the authors may consider a future study for validating the THM model with element tests. Acknowledgements The authors would like to acknowledge the support from the China Scholarship Council and Yue Qi Young Scholar Project, China University of Mining and Technology, Beijing, as well as the National Natural Science Foundation of China (Grant No. 51539006). References Basahel, H.M., Mitri, H., 2017. Application of rock mass classification systems to rock slope stability assessment: A case study. J. Rock Mech. Geotech. Eng. 9, 993–1009. Chen, H., Lee, C.F., Law, K.T., 2004. Causative mechanisms of rainfallinduced fill slope failures. J. Geotech. Geoenviron. 130 (6), 593–602. Comsol, Comsol Multiphysics 5.0, 2014,
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Please cite this article as: D. Wu, T. Deng, W. Duan et al., A coupled thermal-hydraulic-mechanical application for assessment of slope stability, Soils and Foundations, https://doi.org/10.1016/j.sandf.2019.12.007
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Please cite this article as: D. Wu, T. Deng, W. Duan et al., A coupled thermal-hydraulic-mechanical application for assessment of slope stability, Soils and Foundations, https://doi.org/10.1016/j.sandf.2019.12.007