# A 3D Two-Phase Landslide Dynamical Model on GIS Platform
## 2 Methods
### Numerical Calculation  
The Navier-Stokes equations form the cornerstone for modeling insoluble multiphase systems and serve as the fundamental basis for all Computational Fluid Dynamics (CFD) numerical simulations. In accordance with the pivotal assumptions of [PERSON] and [PERSON] [14, 40], landslides are regarded as continuous fluids with isothermal mixtures. Thus, from the Eulerian perspective, the continuity equation is expressed as follows:  
\[\frac{\partial\rho}{\partial t}+\
abla\cdot(\rho\mathbf{v})=\mathbf{0} \tag{8}\]  
Figure 2: DEM to STL schematic diagram. where \(\rho\) is the mass density, \(t\) is time, \(\mathbf{v}\) is the velocity, and the momentum conservation equation is given as:  
\[\frac{\partial\rho\mathbf{v}}{\partial t}+\rho\mathbf{v}\cdot\
abla\mathbf{v}=S _{\mathbf{f}} \tag{9}\]  
where \(S_{\mathbf{f}}\) is the field which includes gravity, friction, and other forces. Taking into account the gravitational and fluid surface tension effects, the momentum equation of an insoluble multiphase system is as follows:  
\[\frac{\partial\rho\mathbf{v}}{\partial t}+\rho\mathbf{v}\cdot\
abla\mathbf{v}= -\
abla p+\
abla\cdot\mathbf{\tau}+\rho\mathbf{g}+\mathbf{F} \tag{10}\]  
where \(p\)(x, t) is the pressure filed, \(\mathbf{\tau}\) is the shear stress, \(\mathbf{g}\) is the gravitational acceleration, and \(\mathbf{F}\) is the surface tension. We posit that a landslide behaves as an incompressible fluid, thereby maintaining a constant mass density. Subsequently, from Equation (8), we can deduce:  
\[\
abla\cdot\mathbf{v}=\mathbf{0} \tag{11}\]  
We have opted for OpenFOAM v9(Open-source Field Operation And Manipulation) software as the computational tool for conducting numerical simulations of the model. It is a renowned open-source CFD software developed using the C++ programming language. It relies on a suite of partial differential equations to elucidate fluid motion phenomena. Each fluid flow instance can be portrayed through a set of partial differential equations, with a multitude of integrated solvers to address these equations. Its capabilities encompass grid generation, case simulation, and the processing of output readings and post-processing [41]. To account for the influence of air during the landslide motion process, we opted for the interFoam solver among the multiphase flow solvers to simulate the movement of the landslide. OpenFOAM employs the interface capturing method to solve the control equation, with interface delineation achieved via the Volume of Fluid (VOF) approach. The VOF method introduces a volume fraction variable \(\alpha\) to demarcate the presence of two distinct fluids [42]. Considering a scenario where each grid cell encompasses two immiscible media (air and soil), the VOF method employs the parameter \(\alpha\) as the ratio of the soil volume within an individual grid cell to the overall grid cell volume. Specifically, \(\alpha=1\) signifies complete saturation with soil within the grid cell, while \(\alpha=0\) indicates complete air occupancy. When \(\alpha\) falls within the interval \((0,1)\), it indicates that the cell is situated at the interface between the two phases. We assume that a landslide is an incompressible Newtonian fluid. Then, we can get the momentum conservation equation of the VOF model:  
\[\frac{\partial\rho\mathbf{v}}{\partial t}+\
abla\cdot(\rho\mathbf{v}\mathbf{ v})-\
abla\cdot(v\
abla\mathbf{v})-\
abla\mathbf{v}\cdot\
abla v=-\
abla p_{ rgh}-\mathbf{g}\cdot\mathbf{h}\
abla\rho+\sigma\kappa\
abla\alpha \tag{12}\]  
\[p_{rgh}=p-\rho\mathbf{g}\cdot\mathbf{h} \tag{13}\]  
where \(v\) is the viscosity, and \(\mathbf{h}\) is the position vector of the center of the grid cell, \(\sigma\) is the surface tension coefficient, \(\kappa\) is the curvature at the interface, its value is \(\
abla\cdot\mathbf{n}\), \(\mathbf{n}\) is the unit normal phase vector of the curved surface element at the two-phase interface.