Flow-type landslide is one type of landslide that generally exhibits characteristics of high flow velocities, long jump distances, and poor predictability. Simulation of its propagation process can provide solutions for risk assessment and mitigation design. The smoothed particle hydrodynamics (SPH) method has been successfully applied to the simulation of two-dimensional (2D) and three-dimensional (3D) flow-like landslides. However, the influence of boundary resistance on the whole process of landslide failure is rarely discussed. In this study, a boundary condition considering friction is proposed and integrated into the SPH method, and its accuracy is verified. Moreover, the Navier-Stokes equation combined with the non-Newtonian fluid rheology model was utilized to solve the dynamic behavior of the flow-like landslide. To verify its performance, the Shuicheng landslide event, which occurred in Guizhou, China, was taken as a case study. In the 2D simulation, a sensitivity analysis was conducted, and the results showed that the shearing strength parameters have more influence on the computation accuracy than the coefficient of viscosity. Afterwards, the dynamic characteristics of the landslide, such as the velocity and the impact area, were analyzed in the 3D simulation. The simulation results are in good agreement with the field investigations. The simulation results demonstrate that the SPH method performs well in reproducing the landslide process, and facilitates the analysis of landslide characteristics as well as the affected areas, which provides a scientific basis for conducting the risk assessment and disaster mitigation design.

The occurrence of landslides is usually introduced by earthquakes or heavy rainfall, and is always accompanied by a large number of casualties and extensive damage [

For physical experiments, it is a very effective and straightforward way of observing landslide characteristics. However, large-scale landslide or debris flow experiments are very expensive and time-consuming. Recently, numerical simulations based on various numerical methods have been widely employed in the simulation of landslides [

Among the recent advanced numerical methods, SPH is a mesh-free method, a completely Lagrangian method in which the particles carry field variables, (such as mass, pressure, energy, etc.) and move with material velocity [

Boundary treatment has been the main focus and difficulty of the SPH method, which affects the accuracy and precision of the calculation. Many different approaches have been proposed in the literature to enforce the solid boundary conditions to prevent particle penetration [

In this study, a boundary treatment method is proposed that considers friction based on the SPH method. The boundary particle viscous force term is obtained by interpolating the values of the fluid particles near the boundary. In addition, the boundary treatment is verified by comparing the simulation results to the experimental data. The Cross model was employed in the SPH method to simulate the landslides. The rheological parameters were derived from the Bingham model and the Mohr-Coulomb criterion. As a case study, the effectiveness and stability of the application were verified by reproducing the Shuicheng flow-like landslide event in both 2D and 3D that occurred on 23 July 2019, in Guizhou Province, China. The simulation results were compared to the field survey data, showing that the SPH model can provide an accurate analysis of the kinetic characteristics of flow-like landslides, including the flow path, movement velocity, run-out distance and deposition.

For the SPH method, the entire domain is represented as a set of randomly distributed particles with no connection between them. Two main steps are required to obtain the fluid governing equations in SPH form, namely, the kernel approximation and the particle approximation. Firstly, approximation of the field function and its derivative are derived from an integral representation method by using a smoothing kernel function. Secondly, these approximations are replaced by the sum of all neighboring particles in the support domain. It is the kernel approximation and particle discretization that make the SPH method work well for free surface flows and large deformations [

The integral representation of a function

There are several kernel functions that are widely utilized. Typically, higher-order kernel functions often enhance the computing accuracy, and substantial time is required. The cubic spline kernel, which belongs to the B-spline family, is one of the kernels that is commonly employed in SPH method [

Due to the density approximation, the pressure field in the SPH formulation for weakly compressible fluids generally exhibits instability and numerical noise. Particularly, in the present case, the noise of the pressure field may be critical. In this study, the delta-SPH method [

Regarding the momentum

In this work, the temperature of landslides is considered as a constant. Therefore, the energy equation is not considered. In the typical SPH method for resolving the compressible flows, the particle motion is governed by the pressure gradient, whereas the particle pressure is determined using an appropriate Equation of State (EoS).

PySPH is an open-source framework for the simulation of the SPH method [

Assuming that under the condition of shear rate,

Typically,

The Mohr-Coulomb yield criterion with the cohesion

Notice that the advantage of the Cross model over the Bingham model is that the effective viscosity is a continuous variable, and the numerical instability is avoided, as shown in

The boundary treatment has been a main challenge in SPH simulation and affects the calculation accuracy as well as the efficiency. It is vital to select an adequate boundary condition that represents the effect of solid boundary in order to closely describe the dynamic mechanism of landslides, as shown in

A simple and convenient boundary condition was proposed that can consider the boundary friction. In this work, two types of boundary conditions need to be set. Firstly, when a particle approaches to a solid boundary, the kernel function will be truncated, and an error in the solution result will occur. To address this issue, a generalized wall boundary condition by Adami et al. [

Secondly, the discrete particles of landslides may slip at the boundary and thus suffer from the slippage resistance (see

Through the above two steps, the no-slip boundary conditions considering friction on the solid boundary was successfully set. When calculating the viscous forces on the boundary, both the dynamic viscosity coefficient can be obtained by interpolation from the fluid, or a fixed viscosity coefficient can be set.

Prediction-correction algorithm was adopted to execute the time integration. For continuity equations and momentum equations of the form, all the values of time-variant quantities are predicted at the time step

Afterwards, these values are updated at another half time step:

At the end of the time step, the values are calculated as follows:

The selection of the magnitude of time step

The flow chart of the SPH with proposed boundary condition is shown in

In this section, the proposed boundary condition coupled SPH model is utilized to simulate the granular flow model test conducted by Moriguchi et al. [

Two different flume inclinations were chosen for the two-dimensional simulation, and the parameters are given in

Parameters | Notation | Value |
---|---|---|

Density | 1379 | |

Equivalent viscosity coefficient | 1.0 | |

Cohesion | 0.0 | |

flume inclination | Case 1:45, case 2:55 | |

Angle of internal friction | 40 | |

Particle distance | 0.001 | |

No. of fluid particles | 1525 | |

No. of boundary particles | 1737 | |

Simulation duration | 2 |

The simulation process with the flume inclination of

To quantitatively analyze the impact of the boundary, the impact force of the flowing sand was measured for different flume inclinations and different boundary conditions. The simulated time histories of the impact force of the flowing sand are presented in

The Shuicheng flow-like landslide event was triggered by intense rainfall in July 2019 (

The Guizhou Provincial Geological Disaster Emergency Technical Guidance Center immediately carried out the field investigations following the occurrence of landslide. The primary characteristics of the flow-like landslides are depicted in ^{4} m^{3}. According to Zhou et al. [^{3}. Depending on the test results, the angle of internal friction and the cohesion of the flow-like landslide can be set approximately as 30° and 30 kPa, respectively. In the previous simulations, the Bingham model was widely applied to simulate the landslides or debris flows considering a range of dynamic viscosities from 20 to 500

According to the simulation results, the whole process duration was about 90 s.

Landslides usually have a long distance, and the measurement and selection of various parameters may vary greatly in different locations. Therefore, it is meaningful to conduct a sensitivity analysis to determine how various rheological parameters affect the simulation results.

Case | Rheological parameters of flow-like landslides | Consider boundary friction | Relative error norm | ||
---|---|---|---|---|---|

ϕ (°) | c (kPa) | Dynamic viscosity | |||

1 | 30 | 30 | 255 | True | |

2 | 30 | 30 | 255 | False | 0.4411 |

3 | 30 | 0 | 255 | True | 0.1828 |

4 | 30 | 15 | 255 | True | 0.1089 |

5 | 20 | 30 | 255 | True | 0.2928 |

6 | 10 | 30 | 255 | True | 0.6119 |

7 | 30 | 30 | 155 | True | 0.0133 |

8 | 30 | 30 | 55 | True | 0.0530 |

Since the 2D simulation cannot reflect the diffusion and lateral contraction, 3D simulation on the real complex terrain is necessary. In this section, the 3D terrain is generated from the original topographic map at a scale of 1/2000 and homogenizes the mesh. The previously created mesh is then transformed into a sequence of boundary particle particles. Similarly, the landslide is discretized into a succession of particles each with its own set of properties. In this simulation, the particle distance is set to 7.5 m, resulting in 12825 boundary particles and 7010 fluid particles. The strength parameters adopted in the 3D simulation are the same as in the 2D simulation (

Parameters | Notation | Value |
---|---|---|

Density | 2150 | |

Dynamic viscosity | 255 | |

Angle of internal friction | 30 | |

Cohesion | 30 | |

Particle distance | 1.5 | |

No. of fluid particles | 7010 | |

No. of boundary particles | 12825 | |

Simulation duration | 90 |

The results suggest that the front flow takes about 28 s to travel nearly 520 m to reach the isolated island. The front flow velocity reaches a maximum with an average velocity of 29.8 m/s. Due to the good stability of the soil near the isolated island, it diverts the debris to the sides and forms a divergence. The diversion of the stable slope protects the lower residential houses and creates an “island of safety”. The existence of isolated islands has dramatically slowed down the landslides. Therefore, the front flow remains stable at about 90 s and is deposited in the lowlands. Eventually, the estimated total volume of deposition is 1.1 × 10^{6} m^{3}, which is less than the field investigation value 1.91 × 10^{6} m^{3}. The reason might be that the simulation process does not take into account the entrapment process. In addition, the simulated landslide paths and deposited areas matched well with the results of field investigations. The topographic data, in this case, were obtained from the 1/2000 scale topographic map, and it is considered that more accurate topographic data would help to obtain better simulation results.

The displacement and velocity time histories of the landslide front and rear were investigated to gain a better understanding of the landslide characteristics in the present simulation. ^{2}, and a landslide duration of 60 s using the empirical equations. Both values turned out to be slightly smaller than the simulated results. Quantitative validation of the landslide simulation based on the field measured data and empirical equations were challenging. The error may be triggered by a combination of natural and artificial reasons, which makes the simulation more challenging. In addition, it can be seen that the flow's pressure field is formed smoothly.

The landslide material is discretized into a succession of SPH particles of almost the same diameters in this model. The greater the number of particles and the narrower the particle spacing, the greater the computing time required. It should be mentioned that, in the current code, the SPH module is accelerated based on Open Multi-Processing (OpenMP). In theory, the higher the number of particles in the SPH, the more accurate the simulation will be, but a balance needs to be reached with the needed calculation time.

A rational simulation of flow-like landslide propagation process contributes to hazard analysis and disaster mitigation design. In this study, the Navier-Stokes equation combined with the non-Newtonian fluid rheology model was utilized to investigate the dynamic behavior of the flow-like landslide.

A simple and convenient boundary condition was proposed that can consider boundary friction. By using the proposed boundary condition, the simulation results are in good agreement with the experimental results. Moreover, since this operation is limited to boundary particles, it will not increase the computational overhead.

The Shuicheng flow-like landslide, which occurred in Guizhou in 2019, was studied as a case. The parameter sensitivity is analyzed under the 2D model, and the results show the shearing strength parameters have more influence on the computing accuracy than the coefficient of viscosity.

A 3D model of Shuicheng landslide was constructed and computed that corresponded to the site conditions. In terms of the solid volume and deposition area, the computed results are generally consistent with the field investigations. In addition, the characteristics of the front and the rear flow velocity and zone of influence of the flow-like landslide were analyzed, which will help the mitigation or countermeasure design work and offer evidence for the hazard assessment.

The approach provides an effective tool for studying the dynamic behaviors of landslides. However, the current study does not take into account the entrainment process of flow-like landslides. Large-scale practical problems usually require massive particles, and future research should be directed to GPU-accelerated modules.

The authors appreciate the valuable discussions with Zhang Weijie (College of Civil and Transportation Engineering, Hohai University, China) and Shi Wenbin (College of Civil Enginnering, Guizhou University, China) on topics of landslides.