# An Integrated Approach for Simulating Debris-Flow Dynamic Process Embedded with Physically Based Initiation and Entrainment Models

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## Abstract

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## 1. Introduction

## 2. Physically Based Initiation Model of the Debris Flow Sources

#### 2.1. Limit Equilibrium Method for Slope Stability Analysis

#### 2.2. Critical Slip Surface Determination Incorporated with the Monte Carlo Method

- (1)
- Composite critical slip surface determination.

- The slip surface should be located entirely within the slope body range, and the shear inlet and shear outlet are on the slope surface;
- There should be smooth transitions between adjacent slip slice segments and no sharp corners. The angle between two adjacent slip slice segments should be greater than 90°;
- The horizontal spacing of adjacent nodes should be greater than the minimum horizontal spacing to avoid overlap of the slice division nodes.

- (2)
- Considering the spatial variability of the soil, calculate the range of the critical slip surface and the initiation source volume.

## 3. Entrainment-Incorporated Numerical Model

## 4. Wet/Dry Front Treatment over Complex Topography

- Dry edge. The flow depths of both neighboring cells are smaller than ${h}_{min}$, which is mathematically expressed as ${h}_{L}<{h}_{min},{h}_{R}{h}_{min}$, as shown in Figure 5a.
- Wet edge: in contrast to the dry edge, the flow depths of both neighboring cells are greater than ${h}_{min}$, which is ${h}_{L}\ge {h}_{min,}{h}_{R}\ge {h}_{min}$, as shown in Figure 5b.
- Semi-dry edge: the flow depth of one neighboring cell is greater than ${h}_{min}$ while the other one is less than ${h}_{min}$, and the flow surface level of the dry side is higher than that of the wet side. For example, ${h}_{L}\ge {h}_{min},{h}_{R}{h}_{min}$, and ${H}_{L}\le {H}_{R}$, as shown in Figure 5c;
- Semi-wet edge: the flow depth condition of the cells on both sides is the same as that of the semi-dry edge, but the flow surface level of the wet side is higher than that of the dry side—for example, ${h}_{L}>{h}_{min},{h}_{R}\le {h}_{min}$, and ${H}_{L}>{H}_{R}$, as shown in Figure 5d.

## 5. Case Study: The 2010 Hongchun Debris-Flow Event in the Yingxiu Town, China

#### 5.1. The Overview of the 2010 Hongchun Debris-Flow Event

#### 5.2. The Physically Based Estimation of the Slope Initiation Source

#### 5.3. Numerical Simulation of the Debris-Flow Dynamic Process

## 6. Discussion

#### 6.1. Advantages

#### 6.2. Limitations and Future Works

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The schematic diagram of the slope source initiation model. (

**a**) Potential source area in the debris-flow gully. (

**b**) The average slope along the gully direction.

**Figure 6.**Overview of the 14 August 2010 debris-flow event in Hongchun catchment, Yingxiu town, Sichuan province, China. (

**a**) Overview of the 2010 debris-flow event. (

**b**) The alluvial fan before the event. (

**c**) The alluvial fan after the event.

**Figure 10.**Simulation results of the 2010 Hongchun debris-flow process. (

**a**) Time $=50\mathrm{s}$. (

**b**) Time $=100\mathrm{s}$. (

**c**) Time $=200\mathrm{s}$. (

**d**) The final erosion depth contour.

Parameter | Unit Weight | Effective Cohesion | Pore Pressure Coefficient | Internal Friction Angle |
---|---|---|---|---|

Notation | $\gamma $ | c | $\lambda $ | $\phi $ |

Unit | $\mathrm{kN}/{\mathrm{m}}^{3}$ | $\mathrm{Pa}$ | $/$ | $\xb0$ |

Value range | $20.2\pm 1.0$ | $2900\pm 300$ | $0.4\pm 0.1$ | $28\pm 5$ |

Coefficient of variation | $0.05$ | $0.1$0 | 0.25 | $0.18$ |

Module | Parameter | Notation | Units | Value |
---|---|---|---|---|

Debris-flow material (rheology) | Debris flow density | ${\rho}_{d}$ | $\mathrm{kg}/{\mathrm{m}}^{3}$ | 2020 |

Dynamic viscosity | $\mu $ | $\mathrm{kPa}\xb7\mathrm{s}$ | 0.15 | |

Bulk basal friction angle of the flowing mass | ${\phi}_{d}\left({\phi}_{voellmy}\right)$ | ° | 12 | |

Chezy coefficient | ${C}_{z}$ | / | 12 | |

Control parameters (simulation) | Fitting parameter of the velocity profile | ${s}_{1}$ | / | 0.50 |

Grid size | $\Delta x\left(\Delta y\right)$ | $\mathrm{m}$ | 2.5 | |

Time increment | $\Delta t$ | $\mathrm{s}$ | 0.001 | |

The minimum depth | ${h}_{min}$ | $\mathrm{m}$ | 0.005 |

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**MDPI and ACS Style**

Han, Z.; Li, M.; Li, Y.; Zhao, M.; Li, C.; Xie, W.; Ding, H.; Ma, Y.
An Integrated Approach for Simulating Debris-Flow Dynamic Process Embedded with Physically Based Initiation and Entrainment Models. *Water* **2023**, *15*, 1592.
https://doi.org/10.3390/w15081592

**AMA Style**

Han Z, Li M, Li Y, Zhao M, Li C, Xie W, Ding H, Ma Y.
An Integrated Approach for Simulating Debris-Flow Dynamic Process Embedded with Physically Based Initiation and Entrainment Models. *Water*. 2023; 15(8):1592.
https://doi.org/10.3390/w15081592

**Chicago/Turabian Style**

Han, Zheng, Ming Li, Yange Li, Mingyue Zhao, Changli Li, Wendu Xie, Haohui Ding, and Yangfan Ma.
2023. "An Integrated Approach for Simulating Debris-Flow Dynamic Process Embedded with Physically Based Initiation and Entrainment Models" *Water* 15, no. 8: 1592.
https://doi.org/10.3390/w15081592