*4.3. Accumulation Area (Area III)*

The accumulation area had a fan shape, with a length of up to 110 m along the sliding direction and a maximum width of 100 m in the vertical sliding direction. The lithology of the accumulation area is dominated by Quaternary loess, which contains moderately weathered coarse sandstone scraped off the opposite mountain. The landslide struck the opposite side of the mountain and accumulated in the Piliqinghe Gully. The area was 1.46 <sup>×</sup> 104 m2, the thickness of the sliding body was 3–6 m, and the volume was about 6 <sup>×</sup> 104 m3.

#### **5. Dynamic Analysis**

## *5.1. Theoretical Basis*

DAN-W is numerical simulation software developed by Hungr to simulate the whole process of landslide movement and to study the dynamics of landslides [23]. The 3D numerical model was set up according to the two-dimensional simulation conditions provided by the calculation profile in Figure 3b. Based on the aerial views, the path widths of landslide were confirmed. In DAN-W, the Lagrangian analytical solution of the Saint–Venant equation is mainly used to treat the sliding body with the rheological features that are formed by a combination of several blocks with certain materials (Figure 6). In the curve coordinates, the corresponding physical equations and equilibrium equations are established for each block (Figure 6), as in Equations (1)–(7) [23].

$$\mathbf{F} = \mathbf{\gamma} \mathbf{H}\_{\mathbf{i}} \mathbf{B}\_{\mathbf{i}} \text{ds}\sin\alpha + \mathbf{P} - \mathbf{T} \tag{1}$$

Here, F is the sliding force (N); γ is the unit weight (KN/m3); H is the block height (m); B is the block width (m); ds is the nominal length of the block(m); α is the slope foot (◦); P is the internal tangential pressure (N); and T is the base resistance (N); i is the block index.

$$V\_i = v\_i' + \frac{\mathbf{g}(\mathbf{F}\Delta\mathbf{t} - \mathbf{M})}{\gamma \mathbf{H}\_i \mathbf{B}\_i \mathbf{d}s} \tag{2}$$

Here, *V* is the new speed when sliding body movement. The new velocity at the end of a time step is obtained from the old velocity, *v* (m/s); g is the gravitational acceleration(m/s2); Δt is the time step interval(s); M is momentum flux; and the other parameters are the same as in Equation (1).

$$\mathbf{h}\_{\mathbf{j}} = \frac{2\mathbf{v}\_{\mathbf{j}}}{(\mathbf{S}\_{\mathbf{i}+1} - \mathbf{S}\_{\mathbf{i}})(\mathbf{B}\_{\mathbf{i}+1} + \mathbf{B}\_{\mathbf{i}})} \tag{3}$$

Here, h is the average depth of the slip mass; j is block boundary index; i is block index; S is the curve displacement (m); and the other parameters are the same as in Equation (1).

$$\mathbf{V} = \mathbf{V}\_{\text{R}} + \sum \mathbf{V}\_{\text{point}} + \sum\_{i=1}^{n} \mathbf{Y}\_{i} \mathbf{L}\_{i} \tag{4}$$

Here, V is the entire volume of the loess landslide deposits(m3); VR is the volume of the initial landslide (m3); Vpoint is the volume of the unstable body (m3); Y is the yield rate; L is the length of the i block; and i is the block index.

**Figure 6.** The 3D numerical DAN-W model of the landslide. The parameters are the same as in Equations (1)–(7) (It is modified from [23]).

Momentum and mass during the entrainment of the path material could influence landslide kinematics. To describe the entrainment process quantitatively, an entrainment ratio (ER) could be offered to calculate the increase of the landslide volume for a specific entrainment zone in the DAN model [49].

$$\text{ER} = \frac{\text{V}\_{\text{Entrained}}}{\text{V}\_{\text{Fragmented}}} = \frac{\text{V}\_{\text{E}}}{\text{V}\_{\text{R}}(1 + \text{F}\_{\text{F}})} \tag{5}$$

where VE (i.e., VEntrained) is the volume of the entrained path material (m3); VFragmented is the volume of the fragmented material in the sliding source area(m3). VR is the volume of the initial loess landslide (m3); and FF is the fractional amount of volume expansion due to fragmentation (0.25). The entire volume of the loess landslide deposits is equal to VR(1 + FF) + VE [40]. In this study, VR equals 5.0 <sup>×</sup> 104 m3 and VE equals 25.0 <sup>×</sup> <sup>10</sup><sup>4</sup> <sup>m</sup>3. The length of the entrainment area was approximately 240 m. To simulate the phenomenon of entrainment, an ER equal to 4.0 was used in the DAN model of the loess landslide. According to the pore-water pressure increased and soil saturation during the sliding process because of the snow infiltration. The landslide was transformed into debris flow, showing a flow state, so the scraping volume was huge.

The movement speed of the sliding body and the thickness of the landslide accumulation body are calculated using Equations (1)–(5). In addition, the amount of resistance encountered during the movement of the sliding body is determined by different types of rheological models. In the DAN-W software, the resistance is mainly controlled by different base rheological models. DAN-W provides a range of rheological models. According to the existing research results and the trial-error method [50,51], the Voellmy model (V) and the Frictional model (F) are more suitable for landslide dynamic hazard research. The Frictional model is mainly used for landslides when the particle sizes of the residual body are large. The Frictional model is also used for mountains with open hillside cracks where the turbulent flow is not developed. The Voellmy model is suitable for the simulation of a landslide with fractured particles where there is a visible liquidized layer in the sliding mass. From Equation, it is evident that the rheological model is proportional to the velocity of the sliding body, so it could simulate the energy damage of the turbulent flow. This was caused by the liquefied material that has high moisture content, including the loose soil covering the flow path and a spring appearing in the path. This opinion has been accepted by Geotechnical Engineering Office (GEO) of Hong Kong [14,52].

Voellmy model: The expression of base resistance is as follows

$$
\pi = \text{f}\sigma + \gamma \frac{\nu^2}{\xi'} \tag{6}
$$

where f is the friction coefficient of the sliding body, σ is total stress perpendicular to the direction of the sliding path, γ is the material unit weight, ν is the moving speed of the sliding body, ξ is the turbulence coefficient, and τ is the resistance at the bottom of the sliding body. The constant friction coefficient (f) is a parameter that should be determined using the Voellmy model. The friction coefficient was modified by the pore pressure and could reach much smaller values when the path material shows wet features.

Frictional model: Assume that the flow of the sliding body is controlled by the effective normal stress acting on each block. The expression of resistance τ is as follows

$$
\pi = \sigma (1 - \gamma\_{\mu}) \tan \varphi \tag{7}
$$

where γ<sup>μ</sup> is the pore pressure coefficient (specifically, the ratio of pore pressure to total stress); ϕ is the internal friction angle; σ is the total stress perpendicular to the direction of the sliding path; and τ is the resistance at the bottom of the sliding body.
