# Simulation of Debris-Flow Runout Near a Construction Site in Korea
## 3 Method of Numerical Simulation of Debris Flows Based on Landslide Susceptibility Analysis
### Numerical Simulation of the Runout Distance of Debris Flows  
This study used the program Debris-2D [36; 37] to simulate debris flow. In Debris-2D the theory of viscoplastic fluid, which is a branch of non-Newtonian fluid mechanics, is applied to model debris motion. The bulk of debris flow is treated as a continuum. The governing equations are based on mass  
Figure 6: Landslide probability map of Mt Goobong (after [11]). and momentum conservation, with a shallow water assumption. The coordinate system is Cartesian with the average bed elevation as the \(x\)-axis. The governing equations involve conservation of mass  
\[\frac{\partial H}{\partial t}+\frac{\partial(uH)}{\partial x}+\frac{\partial(vH )}{\partial y}=S \tag{3}\]  
and conservation of momentum in the \(x\)-and \(y\)-directions  
\[\frac{\partial(uH)}{\partial t}+\frac{\partial\left(u^{2}H\right)}{\partial x }+\frac{\partial\left(uvH\right)}{\partial y}=-gH\cos\theta\frac{\partial B }{\partial x}-gH\cos\theta\frac{\partial H}{\partial x}+gHsin\theta-\frac{1} {\rho}\frac{\tau_{0}u}{\sqrt{u^{2}+v^{2}}} \tag{4}\]  
\[\frac{\partial(vH)}{\partial t}+\frac{\partial(uvH)}{\partial x}+\frac{ \partial\left(v^{2}H\right)}{\partial y}=-gH\cos\theta\frac{\partial B}{ \partial y}-gH\cos\theta\frac{\partial H}{\partial y}-\frac{1}{\rho}\frac{ \tau_{0}v}{\sqrt{u^{2}+v^{2}}} \tag{5}\]  
where \(H\)(\(x\),\(y\),\(t\)) is flow depth, \(B\)(\(x\),\(y\)) is constant bed topography, \(u\)(\(x\),\(y\),\(t\)) and \(v\)(\(x\),\(y\),\(t\)) are, respectively, depth-averaged velocities in the \(x\)-and \(y\)-directions, \(\theta\) is the bottom bed slope, \(\tau_{0}\) and \(\rho\) are debris-flow yield stress [Pa] and density, respectively, which are assumed to be constant here, \(g\) is gravitational acceleration, and \(S\) denotes the effect of bottom erosion. Because the bottom shear layer is ignored, yield stress becomes the dominant bottom stress. In applying Debris-2D, the finite difference method is used to discretize the governing equations, i.e., Equations (3)-(5). In spatial discretization, the first-order Upwind method is applied to discretize the convective term and the second-order central difference method is used for the remaining terms. The explicit third-order Adams-Bashforth method is used for time advancement. During computation, the time step \(\Delta t\) is fixed under the Courant-Friedrichs-Lewy condition, and the volume of debris-flow bulk is conserved. At the start of the computation, Debris-2D uses Equation (6) to determine where debris flow can be initialized. If Equation (4) is not satisfied, the mass stays stationary with zero velocity and unchanged flow depth; otherwise, debris starts to flow. During computation, if the maximum velocity in the whole computational domain is less than the numerical error, the computation terminates automatically. \[\left(\frac{\partial B}{\partial x}+\frac{\partial H}{\partial x}-tan\theta \right)^{2}+\left(\frac{\partial B}{\partial y}+\frac{\partial H}{\partial y }\right)^{2}>\left(\frac{\tau_{0}}{\rho gcos\theta H}\right)^{2} \tag{6}\]  
The derivatives of \(B\) and \(H\) represent pressure effects, and tan\(\theta\) denotes the gravitational effect. The right-hand side is the resistance from yield stress. The main inputs include the topography, the initial debris source distribution, and the material properties of debris flow. For Debris-2D, the topographic digital elevation model (DEM) must be in square grids. Man-made structures or engineering countermeasures, e.g., check dams, can be included in a modified DEM for more precise simulation. In contrast to flood route modeling, Debris-2D simulates the movement of the continuum bulk of debris to simulate real debris flows better [41]. The initial distribution of debris sources must be determined through field survey. From field survey or satellite image analysis, one can identify dry debris volumes \(V_{d}\) and their corresponding triggering locations. Another input is the material properties of potential debris flow.