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# SPH-based numerical modeling for the post-failure behavior of the landslides triggered by the 2016 Kumamoto earthquake

- Zili Dai
^{1, 3}Email author, - Fawu Wang
^{1}, - Yu Huang
^{2}, - Kun Song
^{3}and - Akinori Iio
^{1}

**Received:**21 September 2016**Accepted:**16 November 2016**Published:**29 November 2016

## Abstract

### Background

The 2016 Kumamoto earthquake triggered many fast landslides in Kyushu Region, Japan, which caused great damages and increased difficulty of rescue after the earthquake. The post-failure behavior analysis of the earthquake-triggered landslides is of great importance to determine the hazardous areas and estimate the extent of damage.

### Results

The features of the landslides triggered by the Kumamoto earthquake were presented and the damages caused were described based on the field investigation. Then, a mesh-free particle method called smoothed particle hydrodynamics (SPH) was introduced and a 3D numerical model was established based on a Bingham flow model and the equations of continuity and motion. Applying this model, two typical fast landslides in the Kumamoto earthquake zone were simulated. Their propagations were reproduced and the time histories of the velocity and run-out distance were obtained.

### Conclusions

The run-out distances simulated matched the in-situ evidences, thus verifying the calculation accuracy of the SPH model. The SPH-based approach presented in this work for the post-failure behavior analysis should contribute to the prevention and mitigation of fast landslide disaster.

## Keywords

- Fast landslides
- Post-failure behavior
- Kumamoto earthquake
- Smoothed particle hydrodynamics
- Numerical simulation

## Background

_{S}7.3 (Mw 7.0 according to the US Geological Survey). About 28 h before the main shock, a foreshock earthquake with a magnitude 6.2 had shook the same region severely at 21:26 JST on April 14. According to Okumura (2016), this series of earthquake shocks were triggered by the reactivation of the Futagawa and Hinagu Faults. Figure 1 shows the location of the earthquake shocks and fault zones in the earthquake region, which exactly proves Okumura’s idea.

As always, analytical investigation of landslide stability through limit equilibrium method (LEM) and finite element method (FEM) is a hot topic in geology engineering and many promising results have been obtained (Collins and Sitar, 2011; Zhou and Cheng, 2013; Xu et al., 2013). However, for earthquake-triggered landslides, the destructive potential often cannot, in practice, be reduced by stabilization of the source areas. To minimize the degree of destruction, not only the timely forecast of a potential landslide is of great significance, but also the reasonable estimation of its post-failure behavior and consequences. The information regarding the evolution processes, the velocity distribution, as well as the possible hazard zone can be valuable for determining the hazardous areas, estimating the intensity of the hazard and setting up appropriate protection measures in practical engineering.

For run-out prediction, empirical formulations (derived from a large sample of landslides statistical data) and analytical approaches (i.e. simple sliding block models) indeed provide valuable information. They are usually of high approximation and disregard crucial local topography effects such as diversions or bifurcations. To better predict the consequence of landslides, a robust computational tool that can capture the whole process would offer additional information of significance. For instance, Andersen and Andersen (2010) presented a numerical model based on the material point method (MPM) to simulate the dynamic evolution of landslides; Zhang et al. (2015a) adopted discontinuous deformation analysis (DDA) to study the run-out distance of landslides induced by the 2008 Wenchuan earthquake; Zhang et al. (2015b) simulated a landslide process occurred in Southern China by the particle finite element method (PFEM). Discrete element method (DEM) is a possible approach to simulate soil flow or landslides. For example, Lin and Lin (2015) investigated the evolution of a large landslide in the Butangbunasi River, southern Taiwan using the DEM. This method treats the geomaterial as an assembly of rigid grains that are governed by the Newton’s laws of motion. Nevertheless, the utilization of the DEM in practice is restricted due to its undesirable nature of simulating individual soil grains. To reduce the computation, the number of grains used in typical DEM simulations is usually much less than that in real problems.

In the field of CFD, smoothed particle hydrodynamics (SPH) is a recently developed mesh-free particle method. The main advantage of this method is that it bypasses the need for a numerical grid and avoids the severe mesh distortions caused by large deformation. In contrast to the DEM, SPH predicts material behavior on the basis of the classical conservation equations of mass, momentum and energy, so that it is more suitable for landslide modeling. Recently, several preliminary applications of SPH to landslides have been undertaken with some promising results. For example, McDougall and Hungr (2004) developed an SPH model combined with depth-integrated equations to simulate the propagation stage of flow-like landslides. This model was then extended by Pastor et al. (2008; 2009; 2014), Haddad et al. (2010) and Cascini et al. (2014) to simulate the dynamic process of flow-like landslides, debris flows, lahars and avalanches. These models have many unique features, such as the ability to account for bed entrainment, rheology variation, nonhydrostatic and anisotropic internal stress states. Huang et al. (2011; 2012) presented an SPH model combined with Navier-Stokes equations to predict the run-out distance of the flow-like landslides triggered by the 2008 Wenchuan earthquake. This model was then extended to 3D by Dai et al. (2014). In this paper, the 3D SPH model is introduced to simulate the propagation of two typical landslides induced by the 2016 Kumamoto earthquake. Combining with the field investigation, the post-failure behavior of the landslides is analyzed and the performance of the SPH model is evaluated. The sliding path, run-out distance, velocities, and the distribution deposits obtained in the simulation could play important roles in mapping hazardous areas and estimating hazard intensity, and also the identification and design of appropriate protective measures.

## Methods

### Concepts of smoothed particle hydrodynamics

*x*

_{ i }represents the concerning particle, and

*x*

_{ j }is a neighboring particle in the support area;

*m*and

*ρ*are the mass and density of the particle,

*W*is a kernel function. A cubic B-spline function, originally used by Monaghan and Lattanzio (1985), is employed as the kernel function in this paper.

### Neighboring particle searching algorithm

SPH is a mesh-free method based on interactions with the neighboring particles. The creation of the neighbor list is one of the key points in terms of the high performance of the code. And the efficiency of the SPH model depends highly on how to build and use the neighbor list, especially in the 3D model. The Verlet neighbor list algorithm (Verlet, 1967), as one of the most efficient neighboring particle searching methods, was adopted in the presented model. First, the distance between each particle pair was calculated, then a potential neighbor list was constructed in which the particle pair distances were all within a “skin” layer radius, *r*
_{
s
}. In this list, only those pairs of particles within a distance cutoff radius, *r*
_{
c
}, will interact, resulting in a further neighbor list. Note that the neighbor list is updated every time step while the potential neighbor list is updated every *N* time steps. The value of *N* was suggested to be taken as 7, *r*
_{
c
} and *r*
_{
s
} can be taken as 2.5*r* and 3.2*r* respectively, where *r* is the radius of the interacting particles (Verlet, 1967; Dominguez et al., 2011).

### Governing equations

*ρ*is the density,

*t*is time,

*v*represents the velocity vector, the superscripts

*α*and

*β*denote the coordinate directions,

*σ*is the stress tensor,

*F*is the external force vector per unit mass.

*П*

_{ ij }, was incorporated into the pressure terms of the equation of motion (Monaghan and Gingold 1983). Therefore, the Eq. (6) can be rewritten as:

### Constitutive model

*τ*is the shear stress,

*τ*

_{ y }is the yield shear stress,

*η*represents the viscosity coefficient.

*D*donates the tensor of strain-rates which can be defined by:

*D*

_{ Π }is the second invariant of the tensor of strain-rates, and can be defined by:

### Boundary treatment

During the propagation, landslide mass receives contact forces from the local topography. In response to the repulsive force, the landslide may change direction, spread or contract, and split or join in while traveling across the 3D terrain. To truly represent the propagation of the landslide, it is therefore important to accurately consider the effect of the solid boundary on the slope movement. No-slip boundary condition is a widely used boundary treatment method when dealing with free surface flows (Monaghan, 1994; Morris et al., 1997). The performance of this boundary treatment has been validated by previous works (Lo and Shao, 2002; Dai et al., 2014). Therefore, the presented work incorporates a no-slip boundary condition to consider the contact forces from the local topography.

## Results and discussions

As shown in Fig. 2, a lot of landslides were triggered by the 2016 Kumamoto earthquake. In this section, two typical landslides were taken as the case studies to analyze their post-failure behavior. One is the landslide near the Aso Bridge and anther one is located in the northeast of the Aso Volcanological Laboratory of Kyoto University.

### The landslide near the Aso Bridge

To reproduce the landslide propagation, 3D numerical simulation was conducted using the SPH model. The landslide mass, Aso Bridge and boundary were discretized into a series of SPH particles with a diameter of 7 m. The representative volume of each particle is the third power of the diameter, 343 m^{3} in this case. The landslide particles were evenly spaced with an initial velocity of 0 m/s in the source area. The number of particles along depth is dependent on the depth of the sliding surface, and varies in different position. Note that the source area, the depth of the sliding surface, and the volume of the landslide mass could be estimated according to the topographical maps before and after the earthquake. The earthquake shake is one of the key factors influencing the initiation of landslides. However, this work aims to analyze the post-failure of the landslide mass. During the propagation of the landslides, it is difficult to make sure whether the earthquake shake still continues or not. Therefore, to simplify the calculation, the ground motion due to the earthquake was not included in the simulation of the landslide propagation.

The parameters used in the propagation simulation of the landslide near Aso Bridge

Density |
| 1975 |

Yield strength |
| 20 |

Viscosity coefficient |
| 100 |

Acceleration of gravity |
| 9.8 |

Unit time step | ⊿ | 2.0 × 10 |

Time step |
| 4.0 × 10 |

Based on the SPH modeling, the propagation of the landslide was reproduced. The trajectory, sliding velocity and finally run-out distance were calculated. From the numerical results, it can be observed that the sliding velocity of the landslide was quite large. Therefore, the impact force of the landslide mass was huge. The structures on its trajectory were all damaged including the railway, road and the Aso Bridge.

### The landslide in the northeast of the Aso Volcanological Laboratory

^{2}. The total volume of the displaced mass was about 5.0 × 10

^{4}m

^{3}. The source area of the landslide was gentle (about 23°) and the surrounding terrain was also not steep. The elevation difference between the landslide toe and top was only 60 m. The run-out distance of the landslide was about 65 m, therefore the distance-to-length ratio of this landslide was about 0.59. On the landslide mass, the vegetation basically remained intact which indicate the structural integrity of the landslide mass. Figure 15 shows the right flank of the landslide. It can be observed that the average depth of the rupture surface was shallow, about 6.5 m, with a depth-to-length ratio of 0.06. The soil of the slope was distributed as layers paralleling to the surface of rupture, which was caused by the deposition of volcanic ash from the Aso Volcano nearby.

The location of these two landslides are very close, only about 2180 m according to Google earth. The soil property are similar, the main composition is the cohesive soil from the Aso volcano nearby. Therefore, the soil strength parameters used in this case was the same as listed in Table 1. Relative to the landslide near Aso Bridge, the slope gradient of the landslide near Volcanological Laboratory is more gentle. Therefore, the sliding velocity of this landslide was smaller, as well as the run-out distance. It shows that the slope gradient and the surrounding terrain influence the post-failure behavior of the landslide mass to some extent.

## Conclusion

The 2016 Kumamoto earthquake triggered a lot of fast landslides in Kyushu Region, and caused serious loss of life and property. Predicting the post-failure behavior of these landslides is of importance to determine the hazardous areas and estimate the extent of damage.

To analyze the post-behavior of the landslide, a 3D numerical model was presented based on smoothed particle hydrodynamics. A Bingham flow model and the equations of continuity and motion were used to describe the motion of the SPH particles.

Two typical landslides triggered by the Kumamoto earthquake were taken as the case studies. Field investigations were carried out to obtain the landslide features, soil properties and 3D terrain landscapes. The damages caused by these two landslides were described.

The propagation of these two landslides were simulated using the presented SPH model. The moving path of the landslide mass was reproduced. The time history of the velocity and run-out distance were calculated. The simulated run-out distance was consistent with the measured data in the field work, thus verifying the calculation accuracy of the SPH model.

## Declarations

### Acknowledgement

This work was partially supported by a fund for exploratory research from Shimane University (principal researcher: Atsushi Kamei), Open Fund of State Key laboratory of Geohazard Prevention and Geoenvironmental Protection (Chengdu University of Technology) (Grant No. SKLGP2017K005), Open Fund of Hubei Key laboratory of Disaster Prevention and Reduction, China (Three Gorges University) (Grant No. 2016KJZ06) and the National Natural Science Foundation of China (Grant No. 41530639). The authors would like to sincerely thank Mr. Osamu Osaka from Nihonkai Technical Consultants Co., Ltd and Mr. Seiji Sakata from Shimaken Consultant Co., Ltd for their work in field investigation in Kumamoto.

### Authors’ contributions

ZD, FW, KS and AI carried out the field investigation in Kumamoto earthquake zone. ZD conducted the numerical simulations and YH provided technical support at the stage of SPH modeling. ZD drafted the manuscript and all authors read and approved the manuscript.

### Competing interests

The authors declare that they have no competing interests.

**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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