# Deformation and Stability Characteristics of Layered Rock Slope Affected by Rainfall Based on Anisotropy of Strength and Hydraulic Conductivity

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Control Differential Equation

_{x}is the hydraulic conductivity in the x-direction, the unit is m/s; k

_{y}is the hydraulic conductivity in the y-direction, the unit is m/s; Q is the applied boundary flux, m

_{w}is the slope of the storage curve, t is the time, the unit is s; γ

_{w}is the unit weight of water, the unit is N/m

^{3}.

_{w}γ

_{w}; A is a designation for summation over the area of an element; and L is a designation for summation over the edge of an element. In an abbreviated form, the finite element seepage equation can be expressed as:

#### 2.2. Theory of Fluid–Solid Coupling

_{a}is the pore-air pressure, the unit is kPa; u

_{w}is the pore water pressure, the unit is kPa; [D] is the drained constitutive matrix; and {m

_{H}} can be expressed as:

_{a}− u

_{w}).

#### 2.3. Establishment of the Factor of Safety for Unsaturated Rock

_{0}= 0 and E

_{n}= 0, according to the balance of the force in the normal and tangential directions of the slip surface of the rock slice, it can be obtained:

_{i}is the effective normal force on the slip surface of the rock slice and the unit is kN; S

_{i}is the shear force on the slip surface of the rock slice and the unit is kN, and the safety factor Fs can be derived as:

_{i}is the resistance force, ${R}_{i}=[{W}_{i}\mathrm{cos}{\alpha}_{i}+{Q}_{i}\mathrm{cos}({\omega}_{i}-{\alpha}_{i})+{J}_{i}\mathrm{sin}({\beta}_{i}-{\alpha}_{i})]\mathrm{tan}{\phi}_{i}+{c}_{i}{b}_{i}\mathrm{sec}{\alpha}_{i}$ and the unit is kPa; T

_{i}is the sliding force, ${T}_{i}={W}_{i}\mathrm{sin}\alpha -{Q}_{i}\mathrm{sin}({\omega}_{i}-{\alpha}_{i})+{J}_{i}\mathrm{cos}({\beta}_{i}-{\alpha}_{i})$; ψ

_{i}is the transfer coefficient, ${\psi}_{i-1}=\frac{(\mathrm{sin}{\alpha}_{i}-\lambda {f}_{i-1}\mathrm{cos}{\alpha}_{i})\mathrm{tan}{\phi}_{i}+(\mathrm{cos}{\alpha}_{i}+\lambda {f}_{i-1}\mathrm{sin}{\alpha}_{i})Fs}{(\mathrm{sin}{\alpha}_{i-1}-\lambda {f}_{i-1}\mathrm{cos}{\alpha}_{i-1})\mathrm{tan}{\phi}_{i-1}+(\mathrm{cos}{\alpha}_{i-1}+\lambda {f}_{i-1}\mathrm{sin}{\alpha}_{i-1})Fs}$; c

_{i}is the effective cohesion for every rock slice and the unit is kPa; φ

_{i}represents the shear strength angle for every rock slice, the unit is the degree. J

_{i}is the seepage force of the rock slip which adopts a simplified seepage field treatment method, assuming that its position of action is located at the center of gravity of the rock below the underwater line, and the distance from the center of the slip surface of the rock strip is h

_{wi}/2, and h

_{wi}is the distance from the underwater line to the slip surface. E

_{i}and E

_{i−1}are the horizontal effective forces between the two sides of the rock slice and the unit is kN; λf

_{i}E

_{i}and λf

_{i−1}E

_{i−1}are the shear forces between the two sides of the rock strip, f

_{i}is the function of the inter-strip force, and λ is the proportional coefficient. The distances between the action positions on both sides and the center of the slip surface of the rock strip are Z

_{i}and Z

_{i−1}, respectively; W

_{i}is divided into two parts based on the underwater line: W

_{i}= W

_{i}

_{1}+ W

_{i}

_{2}, where W

_{i}is the gravity of the rock above the underwater line and W

_{i}

_{2}is floating weight of the rock below the underwater line. Q

_{i}is an external force on the rock surface; ω

_{i}is the angle between the direction of the external force and the normal direction; α

_{i}is the angle of the slip surface.

#### 2.4. Establishment of the Numerical Calculation Model

#### 2.5. Determination of the Maximum Initial Matric Suction

#### 2.6. Establishment of Unsaturated Permeability Coefficient

^{−1}; θ

_{r}is the residual volumetric water content, the unit is m

^{−1}; θ

_{s}indicates the saturated volumetric water content, the unit is m

^{−1}; a is the fitting parameter closely related to the air-entry value of the unsaturated rock mass and the unit is kPa; n and m (m = 1 − 1/n and n > 1) are fitting parameters that control the slope at the inflection point in the volumetric water content function [18]. k

_{w}is the saturated hydraulic conductivity, unit is m/s; k is the adjusted hydraulic conductivity, the unit is m/s.

_{r}are known to have major influences on the unsaturated flow of rock slope [9]. The slope in the present study is located on the west side of the slope in a previous study [9], and the difference of unsaturated parameters between the two rock slopes is negligible. Therefore, the parameters of the water retention curve of the present slope were referred from a previous study [9]. The unsaturated parameter values are shown in Table 1 [9]. Accordingly, the water retention curve of the rock mass is shown in Figure 6.

#### 2.7. Definition of Anisotropy and the Calculation Conditions

_{11}= k

_{x}cos

^{2}β + k

_{y}sin

^{2}β, C

_{22}= k

_{x}sin

^{2}β + k

_{y}cos

^{2}β, and C

_{12}= C

_{21}= k

_{x}sinβcosβ + k

_{y}sinβcosβ. The k

_{x}and k

_{y}, as well as the dip angle β, were defined according to Figure 3. In the present study, k

_{x}represents the horizontal hydraulic conductivity; k

_{y}is the vertical hydraulic conductivity; β is the direction between k

_{x}and x-axis. Therefore, if β = 0°, then [C] will be reduced to:

_{r}= k

_{y}/k

_{x}considered. However, for layered rock masses, due to the existence of dip angles of the bedding plane, the conditions that the anisotropy angle β was not equal to 0 were also considered. Therefore, in this study, to restore a reasonable rock formation state, anisotropy angles ranging between 0 and 90° in the layered rock slope in the Pulang area were taken into consideration. As shown in Figure 3, the anisotropic angle of hydraulic conductivity was equivalent to the dip angle of the layered rock slope.

^{d}and v

^{d}are the elastic modulus and Poisson’s ratio of the seriously weathered rock mass; E

_{max}, v

_{max}, E

_{min}, and v

_{min}are the maximum and minimum elastic modulus and Poisson’s ratio of the intact rock; E

^{d}

_{max}, v

^{d}

_{max}, E

^{d}

_{min}, and v

^{d}

_{min}are the maximum and minimum elastic modulus and Poisson’s ratio of the seriously weathered rock mass, respectively. E

^{d}

_{max}, v

^{d}

_{max}, E

^{d}

_{min}, and v

^{d}

_{min}were measured by the uniaxial compression experiment. According to Equations (13)–(15), the strength parameters of the seriously weathered rock mass for different dip angles in the present study are shown in Figure 8.

## 3. Results

#### 3.1. Effects of the Hydraulic Conductivity and Strength Anisotropy on the Deformation Characteristics

#### 3.1.1. Analysis of the Horizontal Displacement of Monitoring Line

_{r}= 0.01 and k

_{r}= 0.1 are shown in Figure 9 to illustrate the influence of anisotropy direction β on deformation characteristics. Different k

_{r}values with β = 0, β = 30, and β = 90° are also displayed in Figure 10 to show the impacts of anisotropy ratio k

_{r}.

_{r}increased, the HD decreased. When the k

_{r}was smaller than 0.01, the maximum HD of the slope top appeared at 10 m from the surface. When the anisotropy angle was greater than 30° and k

_{r}= 1, the HD was the smallest. Therefore, considering the layered rock slope as a homogeneous medium may underestimate its horizontal displacement.

_{r}decreased, the shallow PWP gradually increased, and at the slope bottom, as the k

_{r}decreased, the shallow PWP gradually decreased. The distribution of shallow HD was also similar to shallow PWP. At the top, middle, and bottom of the slope, as the β increased, the shallow PWP gradually increased. However, the shallow HD first increased, then decreased, and reached the maximum dip angle of 30°. Since the HD was not only affected by water pressure but also by its strength parameters (elastic modulus and Poisson’s ratio), the variations of HD were not synchronous with PWP.

#### 3.1.2. Analysis of the Horizontal Displacement of Monitoring Point

_{r}and β, the HDs calculated by numerical simulations all increased abruptly on 4 July 2019, indicating that the rock slope model could accurately predict the occurrence time of the landslide. It is worth noting that when k

_{r}= 0.01 and β = 30°, the calculated HD was consistent with the field monitoring data.

#### 3.1.3. Analysis of the Maximum Horizontal Displacement

_{r}and β conditions more prominently, the maximum horizontal displacement (MHD) of each monitoring line was defined. Figure 14a–c shows that MHD at the different positions of the slope is greatly affected by k

_{r}and β. When β = 30°, the MHD reached the maximum, and the elastic modulus and Poisson’s ratio of the rock slope were the smallest in this case. When the anisotropy angle was close to 60°, and the MHD reached its minimum. When the rock slope was considered to be isotropic (k

_{r}= 1), the simulated MHD was almost negligible. Previous studies only took k

_{r}or β into consideration, but ignored their collective effect. Table 3 shows the MHD of the differences between only considering k

_{r}or β and considering both k

_{r}and β compared with the isotropy condition.

_{r}or β and when considering both k

_{r}and β. Therefore, the k

_{r}and β must be considered in the deformation analysis of rock slope, and MHD is more affected by β than k

_{r.}

#### 3.2. Effects of the Hydraulic Conductivity and Strength Anisotropy on the Stability of the Rock Slope

#### 3.2.1. Analysis of the Factor of Safety

_{r}. In the horizontal rock bedding plane, the hydraulic conductivity in the horizontal direction reached the maximum, but as k

_{r}increased, the hydraulic conductivity of the vertical direction increased, and the wet front of rain could reach the deep part of the slope. During the period of heavy rainfall, when β = 90°, the smaller the anisotropy ratio k

_{r}, the more notable the decrease of the FS was. In the vertical rock bedding plane, the hydraulic conductivity of the vertical direction reached the maximum, and as k

_{r}decreased, the hydraulic conductivity of the horizontal direction decreased, and the rainwater infiltration capacity was much greater than the horizontal drainage capacity. When the anisotropy angle β was close to 30°, with the change of the anisotropy ratio k

_{r}, the change law of FS was not notable. However, when rock slope was treated as an isotropic medium, the FS was overestimated.

#### 3.2.2. Analysis of the Minimum Factor of Safety

_{r}and β conditions more prominently, the minimum factor of safety (MFS) and the factor of safety of landslide (FSL) on 4 July 2019 were defined. Variations in the factor of safety under varied k

_{r}and β are shown in Figure 17. When the k

_{r}is between 0.002 and 0.02, the MFS and FSL reached the minimum and maximum at β = 30 and β = 0°, respectively. It is worth noting that when the β was greater than 75°, the MFS reached the minimum, and the reason is discussed in Section 3.2.1. Table 4 shows the MFS and FSL of the differences between only considering k

_{r}or β and considering both k

_{r}and β compared with the isotropy condition.

_{r}and β. Therefore, we can consider only the effect of β but ignore k

_{r}to simplify the calculation.

## 4. Discussion

_{r}= 0.01 and β = 30°, the simulated landslide scope almost coincided with the field landslide scope. When the rock slope was regarded as a homogeneous medium (k

_{r}= 1), the distance between the simulated landslide scope and the field landslide scope was about 45 m, and the horizontal displacement was significantly small, which made it difficult to predict the occurrence of the landslide. According to the field survey, the field dip angle of the rock slope was 32°, as shown in Figure 19, which was similar to the dip angle of 30° in the simulation. Thus, in general, the landslide scope, horizontal displacement, dip angle, and landslide occurrence time of the field investigation were consistent with the results of the simulation. Therefore, the feasibility of the conversion Equations (11)–(14) of elastic modulus and Poisson’s ratio was also verified.

## 5. Conclusions

- (1)
- The strength conversion equations of elastic modulus and Poisson’s ratio were feasible, and the rock slope model could accurately predict the occurrence time, horizontal displacement, and scope of the landslide;
- (2)
- The different anisotropy ratios and dip angles of the bedding plane were found to have major impacts on the deformation and stability of the layered rock slope;
- (3)
- The horizontal displacement (HD) and maximum horizontal displacement (MHD) were determined to characterize the deformation characteristics of the rock slope. Considering the layered rock slope as a homogeneous medium could underestimate its HD and MHD. When the dip angle was 30°, the MHD reached the maximum. When the anisotropy angle was close to 60°, and the MHD reached its minimum;
- (4)
- The factor of safety (FS), the minimum factor of safety (MFS), and the factor of safety of landslide (FSL) were determined to characterize the stability characteristics of the rock slope. When the dip angle was 30°, the FS, MFS, and FSL of the rock slope reached the minimum. However, when the rock slope was treated as an isotropic medium, the FS, MFS, and FSL were overestimated;
- (5)
- The changing law of the anisotropy ratio was not obvious, and it was difficult to verify by field data. This could be a focus of future research work. The obtained results are likely to provide a theoretical basis for the prediction and monitoring of layered rock landslide.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Morgenstern–Price (M–P) method calculation principle and model. (modified after Morgenstern et al. [31]).

**Figure 2.**Geographic location of the study area: (

**a**) Study area location; (

**b**) contour map of rock slope; (

**c**) topography of rock slope; (

**d**) right elevation of rock slope; (

**e**) left elevation of rock slope.

**Figure 3.**Grid model: (

**a**) Rock landslides of rock slopes; (

**b**) displacement monitoring station (BPK2-1, BPK2-2, and BPK2-3).

**Figure 8.**The anisotropic strength parameters for different dip angles: (

**a**) Elastic modulus; (

**b**) Poisson’s ratio.

**Figure 9.**Horizontal displacement levels at different positions on the layered rock slope under different β values with k

_{r}= 0.01 and k

_{r}= 0.1: (

**a**) Top of the slope with k

_{r}= 0.01; (

**b**) middle of the slope with k

_{r}= 0.01; (

**c**) middle of the slope with k

_{r}= 0.01; (

**d**) top of the slope with k

_{r}= 0.1; (

**e**) middle of the slope with k

_{r}= 0.1; (

**f**) bottom of the slope with k

_{r}= 0.1.

**Figure 10.**Horizontal displacement levels at different positions on the layered rock slope under different k

_{r}values with β = 0, β = 45, and β = 90°: (

**a**) Top of the slope with β = 0°; (

**b**) middle of the slope with β = 0°; (

**c**) bottom of the slope with β = 0°; (

**d**) top of the slope with β = 30°; (

**e**) middle of the slope with β = 30°; (

**f**) bottom of the slope with β = 30°; (

**g**) top of the slope with β = 90°; (

**h**) middle of the slope with β = 90°; (

**i**) bottom of the slope with β = 90°.

**Figure 11.**Pore water pressure levels at different positions on the layered rock slope under different β values with k

_{r}= 0.01 and k

_{r}= 0.1: (

**a**) Top of the slope with k

_{r}= 0.01; (

**b**) middle of the slope with k

_{r}= 0.01; (

**c**) middle of the slope with k

_{r}= 0.01; (

**d**) top of the slope with k

_{r}= 0.1; (

**e**) middle of the slope with k

_{r}= 0.1; (

**f**) bottom of the slope with k

_{r}= 0.1.

**Figure 12.**Pore water pressure levels at different positions on the layered rock slope under different k

_{r}values with β = 0, β = 30, and β = 90°: (

**a**) Top of the slope with β = 0°; (

**b**) middle of the slope with β = 0°; (

**c**) bottom of the slope with β = 0°; (

**d**) top of the slope with β = 30°; (

**e**) middle of the slope with β = 30°; (

**f**) bottom of the slope with β = 30°; (

**g**) top of the slope with β = 90°; (

**h**) middle of the slope with β = 90°; (

**i**) bottom of the slope with β = 90°.

**Figure 13.**Variations in the horizon displacement of numerical simulation and field monitoring at different monitoring points during rainfall: (

**a**) BPK2-1; (

**b**) BPK2-2; (

**c**) BPK2-3.

**Figure 14.**Variations in the maximum horizon displacement (MHD) at the different positions of the slope: (

**a**) MHD of the top of the slope; (

**b**) MHD of the middle of the slope; (

**c**) MHD of the bottom of the slope.

**Figure 15.**Variations in the factor of safety during rainfall under different β values with k

_{r}= 0.01 and k

_{r}= 0.1: (

**a**) k

_{r}= 0.01; (

**b**) k

_{r}= 0.1.

**Figure 16.**Variations in the safety factors during rainfall under different k

_{r}values with β = 0, β = 30, and β = 90°: (

**a**) β = 0; (

**b**) β = 30; (

**c**) β = 90°.

**Figure 17.**Variations in the factor of safety: (

**a**) Factor of safety of landslide (FSL); (

**b**) minimum factor of safety (MFS).

**Figure 18.**Cloud map of horizontal displacement of rock slope when landslide occurs (day 91): (

**a**) Isotropic (k

_{r}= 1); (

**b**) k

_{r}= 0.01, β = 30°.

Layer | Materials | Fitting Parameters | Hydraulic Conduction Coefficient | ||||
---|---|---|---|---|---|---|---|

A (kPa) | m | n | θ_{s} | θ_{r} | k (m/s) | ||

I | Strongly weathered carbonaceous slate | 10 | 0.33 | 1.5 | 0.242 | 0.001 | 8.08 × 10^{−5} |

Rock Types | Anisotropy Ratio k _{r} = k_{y}/k_{x} | Anisotropic Angle/ Dip Angle β (°) | Elastic Modulus E^{d} (GPa) | Poisson Ratio v^{d} | Unit Weight (kN/m^{3}) | Cohesion (kPa) | Friction Angle (°) |
---|---|---|---|---|---|---|---|

seriously weathered carbonaceous slate | $\left[\begin{array}{c}0.002\\ 0.01\\ 0.02\\ 0.1\\ 1\end{array}\right]$ | $\left[\begin{array}{c}0\\ 15\\ 30\\ 45\\ 60\\ 75\\ 90\end{array}\right]$ | $\left[\begin{array}{c}2.902\\ 3.014\\ 2.645\\ 2.728\\ 2.953\\ 4.428\\ 5.561\end{array}\right]$ | $\left[\begin{array}{c}0.3569\\ 0.3546\\ 0.3500\\ 0.3731\\ 0.3638\\ 0.3800\\ 0.3615\end{array}\right]$ | 22.4 | 93.6 | 33.3 |

MHD | Only Considering k_{r} | Only Considering β | Considering Both k_{r} and β |
---|---|---|---|

Slope top | 23% | 118% | 2.25 × 10^{5}% |

Slope middle | 86% | 127% | 3.38 × 10^{5}% |

Slope bottom | 81% | 123% | 1.17 × 10^{5}% |

MHD | Only Considering k_{r} | Only Considering β | Considering Both k_{r} and β |
---|---|---|---|

FSL | 7% | 16% | 16.4% |

MFS | 6% | 17% | 17.6% |

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## Share and Cite

**MDPI and ACS Style**

Xia, C.; Lu, G.; Zhu, Z.; Wu, L.; Zhang, L.; Luo, S.; Dong, J.
Deformation and Stability Characteristics of Layered Rock Slope Affected by Rainfall Based on Anisotropy of Strength and Hydraulic Conductivity. *Water* **2020**, *12*, 3056.
https://doi.org/10.3390/w12113056

**AMA Style**

Xia C, Lu G, Zhu Z, Wu L, Zhang L, Luo S, Dong J.
Deformation and Stability Characteristics of Layered Rock Slope Affected by Rainfall Based on Anisotropy of Strength and Hydraulic Conductivity. *Water*. 2020; 12(11):3056.
https://doi.org/10.3390/w12113056

**Chicago/Turabian Style**

Xia, Chengzhi, Guangyin Lu, Ziqiang Zhu, Lianrong Wu, Liang Zhang, Shuai Luo, and Jie Dong.
2020. "Deformation and Stability Characteristics of Layered Rock Slope Affected by Rainfall Based on Anisotropy of Strength and Hydraulic Conductivity" *Water* 12, no. 11: 3056.
https://doi.org/10.3390/w12113056