- Methodology
- Open Access

# Numerical simulation of landslide over erodible surface

- Wei Liu
^{1, 2}, - Siming He
^{1, 2, 3}Email author and - Xinpo Li
^{1, 2}

**Received:**13 October 2014**Accepted:**29 July 2015**Published:**6 August 2015

## Abstract

### Background

Estimating the magnitude and intensity of landslides is a fundamental requirement in quantitatively evaluating the risks involved, and preparing a mitigation strategy. Though the physics-based dynamic model of landslide can predict the travel distance, kinematic velocity, and hazard zone, the effects of erosion and the excess pore water pressure during the dynamic process of landslide are often ignored.

### Results

In order to study these factors, a physics-based dynamic model of landslide considering erosion and excess pore water pressure is presented in this paper. A high-precision numerical method based on the finite volume method is proposed to solve the model equations. Several numerical tests are performed to verify the numerical method and the model. The effects of erosion and excess pore water pressure on the dynamic process of landslide are also analyzed.

### Conclusions

The numerical results indicate that the scale and mobility of a landslide are influenced by the effect of erosion and excess pore water pressure. The excess pore water pressure can reduce the resistance to shear stress from the erodible bed and lead to a higher erosion amount and longer moving distance of the landslide. It also affects the degree of erosion and further affects the dynamic process of the landslide. The sensitivity analysis of the parameters that influence excess pore water pressure indicate that these parameters have a significant impact on the evolution of excess pore water pressure, and that the degree of saturation of bed sediment has the highest influence on excess pore water pressure.

## Keywords

- Landslide
- Erosion
- Excess pore water pressure
- Numerical simulation

## Background

Landslides are a natural phenomenon that can strike human settlements in mountainous regions resulting in serious consequences, including untold number of deaths and injuries and massive economic losses (Wang and Sassa 2010; Luna and Remaitre 2012). Because of the huge destructive power of landslides, understanding how to prevent them is becoming more and more important for hazard evaluation, risk assessment, and the preparation of mitigation measures (Savage and Hutter 1989; Chen and Lee 2000; Iverson and Denlinger 2001; Hungr et al. 2005; Pudasaini and Hutter 2007). Predicting the maximum extension and velocity of landslide is one of the ultimate goals.

In recent years, several physics-based dynamic models for landslides have been developed based on the constitutive law of fluid mechanics (Pitman et al. 2003; McDougall and Hungr 2004, 2005; Goren and Aharonov 2007; Goren and Aharonov 2009; George and Iverson 2011; Luna and Remaitre 2012). Most of them assume a constant landslide volume for the duration of motion, neglecting the important role of entrainment found along the landslide path. However, landslide paths are typically covered by surficial deposits such as colluvium, residual soil, and organics (McDougall and Hungr 2005). These deposits may be loose and have high water content, and may be mobilized by the rapid loading of the moving landslide. The phenomena of entrainment is frequently observed on landslides in fields, and the deposited materials may accumulate several times in volume with respect to the initially mobilized mass (Vandine and Bovis 2002). Erosion action not only increases the volume of landslide, but also enhances the landslide mobility significantly, including the travel distance and area covered by the landslide hazard (Crosta et al. 2003; McDougall and Hungr 2005; Iverson 2012). Therefore, it is important for the landslide dynamic model to consider the entrainment effect. In addition, pore water pressure changes caused by the failure of path material can occur at the base of a rapid landslide. The pore water pressure also plays a crucial role in the dynamic process of landslide, because it counteracts the normal stresses at the grain contact, and thereby reduces inter-granular friction (George and Iverson 2011). Generally speaking, the pore water pressure includes a hydrostatic component that balances the pore-fluid weight, and a non-hydrostatic or “excess” component (Iverson 2009). The excess pore water pressure at the base of the landslide may be produced by grain-crushing (Gerolymos and Gazetas 2007), dilatancy (Savage and Iverson 2009), effective stress (Luna and Remaitre 2012), and thermal pressurization (Vardoulakis 2000; Goren and Aharonov 2007; De Blasio and Elverhøi 2008; Goren and Aharonov 2009).

In this study, considering the effects of entrainment and excess pore water pressure, a physics-based dynamic model for landslide is proposed in order to further study the dynamic mechanism of the landslide. For solving the model equations, a high-precision numerical method based on the finite volume method is also proposed. The paper is organized as follows: The two-dimensional Savage–Hutter type model is presented in section 2. The entrainment model and the excess pore water pressure model are introduced in section 3 and 4, respectively. The full model is covered in section 5. The numerical method is presented in section 6, and a series of numerical tests are performed in section 7. Finally, section 8 summarizes the results with concluding remarks.

### Model equations

#### Two-dimensional Savage–Hutter type dynamic model

*O-xyz*, where

*z*denotes the normal direction, the model equations can be written as,

*h*is the landslide height;

*u*and

*v*are the velocities in the

*x*and

*y*directions, respectively;

*g*

_{ x },

*g*

_{ y,}and

*g*

_{ z }are components of gravitational acceleration in the

*x*,

*y*, and

*z*directions, respectively;

*E*represents the erosion;

*u(z*

_{ b }

*)*and

*v(z*

_{ b }

*)*are the velocities at the bottom of the landslide in the

*x*and

*y*directions, respectively;

*φ*

_{int}is internal friction angle of the landslide;

*ρ*= (1 −

*c*)

*ρ*

_{ f }+

*cρ*

_{ s }is the density of the landslide, in which

*ρ*

_{ f }is fluid density,

*ρ*

_{ s }is solid density, and

*c*is solid volume fraction;

*τ*

_{ b }represents the basal shear stress of the landslide;

*k*

_{ ap }is the lateral pressure coefficient, and can be described as

where *φ*
_{
bed
} is the bed friction angle; “–” and “+” correspond to the active state (∂*u*/∂*x* + ∂*v*/∂*y* ≥ 0) and passive state (∂*u*/∂*x* + ∂*v*/∂*y* ≤ 0), respectively.

Eq. (1) represents the mass conservation for the landslide. Eqs. (2) and (3) represent the momentum conservation equation for unit volume of the landslide. The first and second terms on the right hand side of Eqs. (2) and (3) indicate the basal erosion effect and gravity force, respectively. The third term on the right hand side of Eqs. (2) and (3) represents the transverse shear stress. The fourth term on the right hand side of Eqs. (2) and (3) represents the frictional resistance in each direction.

### Entrainment model

*τ*

_{ b }is total basal traction from the landslide flow;

*τ*

_{ s }is the total resistance shear stress from the erodible surface; and the formula should also satisfy the Coulomb failure criterion. The quadratic velocity-dependent model proposed by Fraccarollo and Capart (2002) is applied to

*τ*

_{ b }, and is expressed as

*C*

_{ f }is a dimensionless coefficient which is typically less than 0.1. However, in order to avoid unusually lower basal traction when flow velocity is low, this formula is coupled with Coulomb failure criterion, and can be written as

*r*=

*ρ*

_{ f }/

*ρ*is density ratio;

*τ*

_{ s }can be expressed as

*φ*

_{ sat }is the friction angle of the saturation bed;

*P*

_{ e }is the excess pore water pressure generated by the moving mass flow on top of the bed deposits. However, it is found that, when the moving velocity approximates to zero, the entrainment rate approaches infinity. In order to prevent this, the entrainment rate formula is modified as

where *ξ* is a constant coefficient. The value of *ξ* is sensitive to the erosion rate. In order to obtain a reasonable value of *ξ*, the USGS experiment has been simulated. Simulation results indicate that the computed data agreed well with the experimental data by taking *ξ* = 0.06. The computed erosion rate (about 0.07 ~ 0.1 *m*
^{
3
}
*/s*) is also consistent with the experiment range (about 0.05 ~ 0.1 *m*
^{
3
}
*/s*). Hence, *ξ* = 0.06 is used as a reasonable parameter. At the speed of 10 *m/s*, \( {e}^{-\xi \left({u}^2+{v}^2\right)} \) approaches 0, and Eq. (9) reduces to the original entrainment rate formula.

### Excess pore water pressure model

*A*

_{ D }and

*B*

_{ D }are excess pore pressure parameters in the undrained direct shear state.

*A*

_{ D }changes with the strain value, and

*B*

_{ D }is affected by the loaded stress level, and is very sensitive to the degree of saturation (Luna and Remaitre 2012). The normal stress and the shear strength caused by the landslide can be expressed as,

where \( {g}_c=\sqrt{g_x^2+{g}_y^2} \) represents the component of gravitational acceleration along the tangential direction.

### The full model equations

The model equations (13)-(16) constitute a system of five equations with the variables *h*, *u*, *v*, and *E* for landslide, as well as for mobile surface of landslide path *z*. Together with the initial and boundary conditions, the system constitutes a well posed set of equations describing landslide flow over erodible surface.

## Method

in which,

where *L*
_{
x
} and *L*
_{
y
} represent the operators in the *x* and *y* directions, respectively (Liang et al. 2006; Ouyang et al. 2013). The fractional step method is used to improve calculation stability. Taking *L*
_{
x
} as an example, the discretization scheme is given by the following steps:

**U**

_{ L }on the left side, and state

**U**

_{ R }on the right side. In addition, we couple the MUSCL approach with Roe scheme to reconstruct the interface data U

_{ L }and U

_{ R }for obtaining a high level of accuracy, and avoiding spurious oscillations (Liang et al. 2006; Ouyang et al. 2013). Hence, the numerical flux can be expressed as

**F**

_{ L }and

**F**

_{ R }are calculated from

**U**

_{ L }and

**U**

_{ R }, repetitively;

**J**represents the Jacobian matrix of

**F**. With Roe’s approximation, Eq. (20) can be expressed as,

*γ*is the eigenvector of

**J**;

*λ*represents the corresponding eigenvalues. The symbol

*α*represents the wave strength. All of them are evaluated at the average state

*h*

_{ ag },

*u*

_{ ag }and

*c*

_{ ag }. The expreesion for

*h*

_{ ag },

*u*

_{ ag }and

*c*

_{ ag }can be obtained from,

where *cfl* is Courant number, and is less than 1; *dx* is distance from the centroid of the cell; and *cm* can be expressed as \( cm=\sqrt{k_{ap}{g}_zh} \).

*L*

_{ x }and

*L*

_{ y }have the same treatment. Each operator is operated twice to obtain the solution at the next step. Each step uses the state of

**U**obtained from the previous step.

## Results and discussion

The purpose of this section is to analyse the numerical results of four tests in detail, and to show the effects of excess pore water pressure and erosion on the dynamic process of landslide. We assumed that the composition of the bed sediment is the same as that of the landslide, and that the eroded material and landslide can mix together rapidly. Hence, *c* is kept constant in the whole computation. The bottom velocities *u(z*
_{
b
}
*)* and *v(z*
_{
b
}
*)* are considered as one-tenth of the velocity of the landslide. Moreover, free boundary conditions are imposed on each side of the computational domain, and the Courant number is set as *cfl* = 0.7. The gravitational acceleration is *g* = 9.8 m/s^{2}.

### Test 1: Numerical comparisons with USGS flume experiments

*A*

_{ D }= 0.9 and

*B*

_{ D }= 1 for the numerical simulation. The values of other parameters are set to be the same as the measured values from Iverson et al. (2011). Comparison of numerical solutions and experimental results of flow height versus time is shown in Fig. 2. It shows that the numerically obtained result of flow height versus time basically agrees well with the experimental results. The erosion rate calculated by the current model is about 0.07 ~ 0.1

*m*

^{ 3 }

*/s*, which is generally consistent with the range obtained in the experiment, which is 0.05 ~ 0.1

*m*

^{ 3 }

*/s.*

### Test 2: Simulation of the dynamic process of landslide over erodible bed

_{L}= 30° and ψ

_{R}= 60°, and the maximum height of the landslide is set at H

_{max}= 0.5

*m*. The channel is 10

*m*long, and the inclined component of channel (β = 30°) lies in the range

*x*< 4

*m*; the horizontal component of channel lies in the range

*x*> 5

*m;*a circular arc transition zone smoothly joins the two regions. The distance from the centroid of the cell is set as

*dx*= 0.05

*m*. Values of the relevant parameters are listed in Table 1. The pore pressure parameters used were A

_{D}= 0.6, and B

_{D}= 0.9. These values correspond to the bed sediment that has a high degree of saturation. Four specific moments of the simulation results, at

*t*= 0.3

*s*, 0.7

*s*, 1.2

*s*, and 5

*s*are shown in Fig. 4. As the landslide accelerates and spreads out rapidly in the downslope direction, the erosion gradually becomes significant. An interesting phenomenon that could be noticed is that the velocity of falling is higher at the corner. Due to the increasing frictional resistance when the landslide is passing around the corner into the ground, the front part of landslide enters the stage of accumulation, and the latter part of landslide is subjected to extra resistance from the former part.

Parameters used in numerical experiments

Symbol | Values | |
---|---|---|

| Density of solid | 2700 |

| Density of fluid | 1000 |

| Solid volume fraction | 0.6 |

| The angle of internal friction | 40 |

| The angle of base friction | 40 |

| The angle of friction of saturation bed | 35° |

| Excess pore water pressure parameter | 0.6 |

| Excess pore water pressure parameter | 1 |

| Dimensionless coefficient | 0.015 |

### Test 3: Analysis of the effect of excess pore water pressure

*R,*which represents the degree of erosion, as

*R*=

*Et/Vol*, where

*Et*is the amount of erosion, and

*Vol*is initial volume of landslide; and

*Vol =*0.29

*m*

^{ 3 }. Different values of the coefficients

*B*

_{ D }and

*A*

_{ D }have been chosen, and are shown in Table 2. The relationship between the evolution of maximum of excess pore water pressure and

*R*is shown in Fig. 6. Similar behaviours are exhibited during the slide process. It shows that the excess pore water pressure decreases at first because of the deformation of landslide and the decrease in height of the landslide. With the accumulation of landslide, the height of the landslide increases, and this leads to a rise in excess pore water pressure. With the higher values of

*A*

_{ D }and

*B*

_{ D }, the excess pore water pressure is rising faster, and this reduces the resistance to shear stress from the erodible bed, and leads to a higher erosion amount and longer moving distance of the landslide.

Values of coefficients B_{D} and A_{D}

Symbol | Group 1 | Group 2 | Group 3 | Group 4 |
---|---|---|---|---|

| 0 | 0.2 | 0.6 | 0.6 |

| 0 | 0.6 | 0.2 | 0.6 |

*A*

_{ D }and

*B*

_{ D }. The result of

*R*versus different values of

*A*

_{ D }and

*B*

_{ D }is shown in Fig. 7. It shows that the erosion amount changes with different values of

*A*

_{ D }and

*B*

_{ D }. The sensitivity difference between

*A*

_{ D }and

*B*

_{ D }is evident, and the erosion amount is more sensitive to

*B*

_{ D }. A significant phenomenon that should be noticed is that the erosion amount increases almost linearly with change in

*A*

_{ D }and

*B*

_{ D }. For example, with a constant value of

*B*

_{ D }= 0.6, the ratio of erosion amount increases by 0.022 % when

*A*

_{ D }increases by 0.1. With a constant value of

*A*

_{ D }= 0.2, the ratio of erosion amount increases by 0.1 % when

*B*

_{ D }increases by 0.1, for values of

*B*

_{ D }less than 0.5. The ratio of erosion amount increases faster (0.15 %) when

*B*

_{ D }increases by 0.1, for values of

*B*

_{ D }greater than 0.5.

### Test 4: Simulation of two-dimensional landslide over erodible bed

*x*< 215

*cm*), a horizontal run-out zone (

*x*> 255

*cm*), and a transition zone joining the two regions. Superimposed on the inclined section of the chute, is a shallow parabolic cross-slope topography (

*y*

^{2}

*/*2

*b*with

*b*= 110

*cm*). The granular material is released from rest on the parabolic inclined section of the chute by means of a Perspex cap that opens rapidly (

*t*= 0

*s*). The cap is fitted to the basal chute topography and has a spherical free surface. The major axis of the cap is 32

*cm*in length, and the maximum height of the cap above the reference surface is 22

*cm*. The values of the parameters used are the same as in test 2. The motion status of the landslide and the eroded degree of bed at

*t*= 1

*s*are shown in Fig. 9. The red lines at

*x*= 2.15 m and

*x*= 2.55 m indicate the position of the transition zone joining the two slope regions. The landslide shape and the erosion degree are well captured. A slight jump of erosion occurs around 2.15

*m*due to the continuous erosion by latter part of the landslide in the accumulation process, and the degree of erosion. The numerical results are in agreement with the reality, and thus, the applicability of the present model is proved.

## Conclusion

In this paper, a physics-based dynamic model based on Savage–Hutter theory for landslide has been presented. The effects of entrainment and excess pore water pressure on the dynamic process of landslide are considered. The entrainment rate model satisfying the boundary momentum conservation condition and the excess pore water pressure model based on the Skempton’s equations have been introduced in the present model. A high-precision calculation method was proposed to solve the model equations. Finally, four numerical tests were performed to simulate landslide dynamics over erodible surface. The evolution of entrainment volume, velocity, travel distance, and excess pore water pressure of landslide, as well as hazard zone, can be predicted simultaneously. Numerical results indicate that erosion enhances the destructive power, and enlarged the hazard area of a landslide, and the evolution of excess pore water pressure of bed material causes a change in the erosion amount, further influencing the dynamic process of landslide. It is seen that the relevant parameters have a significant impact on the evolution of excess pore water pressure; however, the degree of saturation of bed sediment has the highest influence on excess pore water pressure. These findings are consistent with observable phenomena in natural landslide. However, more precise and complete experiments are needed to verify and improve the theoretical model to reflect the process of landslide more realistically.

## Declarations

### Acknowledgment

The authors thank two anonymous reviewers for helpful suggestions. This research has received financial support from the NSFC (Grant No. 41272346, 41101008), the National Key Basic Research Program of China (2013CB733201) and the STS project of Chinese Academy of Sciences (project No. KFJ-EW-STS-094).

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

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