A 3D Two-Phase Landslide Dynamical Model on GIS Platform

Abstract

The landslide dynamics model is one of the methods for evaluating landslide motion processes, contributing to disaster prevention and mitigation. With the advancement of science and technology, GIS has become the mainstream platform for landslide simulation. However, the three-dimensional movement of landslides is intricate, leading to a lack of methods for three-dimensional landslide numerical simulation on GIS platforms. In this paper, we propose a three-dimensional, two-phase landslide dynamics model. Through the proposed solution, three-dimensional modeling and numerical simulation of landslides can be achieved on GIS platforms. Simultaneously, drawing inspiration from the SPH kernel functions, we visualize the results of the three-dimensional model on the GIS platform. Simulation of the Yigong landslide demonstrates that our solution can realize three-dimensional landslide simulation on the GIS platform. Our model adeptly captures numerous details in the landslide motion process. However, constrained by the inherent limitations of the three-dimensional model, the model results are susceptible to numerical oscillations and diffusion, with the accuracy of the model being controlled by grid partitioning.

1. Introduction

Thousands of landslides occur every year in the world. Landslides represent one of the most detrimental geological hazards on a global scale. Owing to their abrupt occurrence, high velocity, and extensive magnitude, they pose a formidable risk to the safety of adjacent communities, leading to substantial infrastructural and economic losses [1,2,3,4]. Landslides seriously hamper regional economic development, so governments of all countries are highly concerned by them. In recent years, the increasing frequency of extreme weather events has resulted in a higher occurrence of such disasters, implying a rising magnitude of associated losses [5,6]. Therefore, it is imperative to implement reliable approaches for the prevention and mitigation of the impacts of these disasters. Landslide dynamics constitutes a fundamental theory in the study of landslides, providing insights into crucial parameters such as scale, velocity, sliding distance, and the extent of the resulting disasters. Understanding the dynamic behavior of landslides is essential for predicting their final distribution and plays a vital role in serving disaster mitigation and engineering endeavors. Presently, the application of landslide dynamics in landslide hazard assessment has attained a level of practicality, making it an indispensable tool in the field of geotechnical research and natural disaster management.

Landslide dynamics models within the framework of fluid dynamics have made significant strides. In fluid dynamics, landslides are considered to be continuous mediums. The primary theoretical foundation of the fluid dynamics model is the Navier-Stokes equation, which expresses the motion characteristics of the block through three equations: energy conservation, momentum conservation, and mass conservation [7]. The solution of the Navier-Stokes equations provides fundamental information such as the velocity, acceleration, and pressure of landslides at different time intervals, enabling the simulation of landslide motion. However, these governing equations are all partial differential equations. A significant challenge lies in solving complicated time-dependent non-linear hyperbolic model equations, thus developing a comprehensive representation of the entire runoff process, where the solutions of these equations should capture the fundamental physical principles underlying the entire process. Regrettably, the development of a comprehensive analytical solution is an arduous task, particularly in intricate scenarios such as geophysical flows. Even under certain simplified conditions, the exact solutions to these equations tend to be either exceedingly intricate to be ascertained in a closed-form manner or, in numerous instances, nonexistent [8,9,10,11]. This predicament stems from the intricate nature of the nonlinear partial differential equations and the dynamic evolution of boundary conditions, collectively delineating the dynamic behavior of landslide flow, spanning the spectrum from simple to complex.

In order to solve this equation and simulate the landslide motion process, some scholars have resorted to simplifications. Firstly, Voellmy was a trailblazer in mass flow modeling. He reshaped avalanche dynamics based on fluid dynamics, derived the model equations and their integrals, and developed the Voellmy model [12]. However, due to the assumption of a mass point, the theoretical formulations of this model are overly simplified, making it incapable of accurately predicting the run-out distances and mass distribution information in the deposition area of geophysical flows [13]. Furthermore, this model is unable to account for the rheological and mechanical properties of materials, is constrained to one-dimensional scenarios, and depends on topographical characteristics and the duration of long-run periods. Subsequently, Savage and Hutter established the depth-average theory on this foundation. They assumed that the granular material is an incompressible, frictional Coulomb-like continuum with finite volume similar to Coulomb’s friction law and thus derived a one-dimensional depth-averaged mass conservation equation and momentum conservation equation. In the depth-averaged model, it is assumed that changes in porosity result in variations in density, and there is no systematic dependence of friction on shear strain rate [14]. The model’s governing equations have undergone a significant simplification through depth averaging. However, it comes at the cost of losing some details of the flow field, such as intense fluidization and high shear rates.

Following this, scholars such as Hutter, Gray, Denlinger, Mangeney, Hungr, et al. further refined and extended the model [8,9,15,16,17,18,19]. As most of these models are singularly focused on single-phase behavior, researchers like Iverson, Savage, Pitman, and Pudasaini later expanded the depth-average model to incorporate the effects of pore fluid [20,21,22,23,24,25,26,27]. For example, Iverson and Denlinger introduced a fluid-solid mixture model to describe the motion of two-phase mixtures and extended it to two dimensions [22,28]. Pitman divided landslides into liquid and solid phases to depict the landslide motion process [23]. These models are all based on depth-average theory, which simplifies the motion in the Z-axis direction under the assumption of shallow mass flow. This simplification allows for the efficient solution of governing equations within a relatively short timeframe, thereby enhancing the capability to predict landslide movement processes. However, the actual movement of landslides is often more complicated. Therefore, Pudasaini unified previously employed single-phase models and introduced a versatile two-phase mass flow model. The solid phase stress was governed by the Mohr-Coulomb criterion, while the fluid phase stress was characterized using non-Newtonian viscous stresses. A novel generalized resistance term was defined to discern fluid phases as mixed or separated. This model was applicable to the movement of the majority of two-phase flows [26]. Building upon this, Pudasaini introduced a multiphase mass flow model comprising three distinct phases, described using viscoplastic rheology, in which fine debris can be regarded as the third phase [27].

The depth-averaged model assumes landslides as shallow mass flows; However, questions arise about its validity when abrupt changes occur in the terrain. When encountering steep terrain and strong nonlinearity, the most appropriate method is to minimize the simplification of the Navier-Stokes equation, which means that we need to solve the complete three-dimensional Navier-Stokes equation [29,30]. Some people combine 3D models with depth-averaged models to study the 3D motion of landslides. For instance, Domnik employed a depth-averaged model to reduce computational costs in regions with smooth changes of flow variables, while a three-dimensional model was utilized in areas such as the initiation and deposition regions to preserve the fundamental physical characteristics of the fluid [31]. However, the majority of studies involve localized three-dimensional simulations of landslides while adopting a depth-averaging approach for overall solution strategies. Few studies have conducted three-dimensional simulations of the entire landslide motion process.

With the continuous advancement of hardware and software, coupled with the rapid growth of numerical and scientific computing techniques, numerical simulations have become the primary method for analyzing landslide motion processes. In practical landslide simulations, which frequently involve extensive geographical regions and copious amounts of geospatial data, the complexity and intricacy of data handling and analysis are notable. GIS effectively addresses these challenges through its advantages in data processing and representation [32,33]. Notably, the emergence of three-dimensional GIS technology has further amplified the prowess of GIS in data acquisition, processing, modeling, visual representation, and data sharing. This has allowed for improved depiction of surface movement processes within the framework of landslide dynamics models on GIS platforms. For flow-like models on GIS platforms, employing a grid network of Digital Elevation Models (DEMs) is deemed the most appropriate approach. GIS has become the mainstream system for engineering design and urban planning. Many scholars have integrated models related to landslide motion processes into GIS platforms. Mergili developed the landslide motion simulation program r.avaflow on the GIS platform, which integrates a multiphase flow method based on the depth-averaged theory. The program is currently widely adopted globally [34]. Building upon previous research, Yu developed a three-dimensional landslide surge height calculation extension module on the GIS platform, thereby extending existing two-dimensional models to three dimensions. However, the majority of studies have primarily engaged in two-dimensional simulations of landslide motion processes or confined their investigations to localized three-dimensional simulations on GIS platforms [35,36]. There is limited research on conducting real three-dimensional simulations of landslide dynamics throughout the entire process based on DEM on GIS platforms. The challenge of constructing a three-dimensional landslide dynamic model based on DEM and visualizing the computational results on GIS platforms remains an unresolved issue.

Therefore, addressing the aforementioned issues, in this study, we propose a solution to construct a three-dimensional landslide dynamic model based on DEM. Drawing inspiration from Smoothed Particle Hydrodynamics (SPH) kernel functions, we transform the computational results of the three-dimensional model into two dimensions, enabling visualization on GIS platforms. This approach facilitates a comparative analysis between the results of the three-dimensional model and those of the two-dimensional model, thereby assessing their respective strengths and weaknesses.

2. Methods

As a result of advancements in remote sensing technology and computer capabilities, GIS has emerged as a predominant platform for simulating landslides and debris flows [34,37]. Within the GIS framework, expression formats primarily comprise vector data and grid data formats. Among these, the grid data format proves optimal for representing fields and aligns well with the Eulerian description approach. Particularly for flow-like models on GIS platforms, the grid networks of DEM facilitate streamlined and accurate calculations [38]. Therefore, we have devised a solution to construct a three-dimensional landslide dynamic model based on DEM. At the same time, we borrowed from the SPH kernel function to transform the simulation results into two dimensions, allowing for their visualization within a GIS platform. The comprehensive process is methodically segmented into five distinct phases, as shown in Figure 1: (1) DEM correction, (2) geometric modeling, (3) numerical calculation, and (4) post-processing.

2.1. DEM Correction

Thanks to the advancements in remote sensing technology, we now have multiple avenues for acquiring remote sensing imagery of the landslide study area. It is important to emphasize that the obtained images must undergo rectification before their utilization. In this study, we employed remote sensing techniques to acquire Digital Elevation Model (DEM) images separately for the landslide body and the landslide surface. These two remote sensing images are subsequently utilized for the three-dimensional modeling of the landslide and the extraction of a three-dimensional model for the landslide body. Next, within a GIS platform, the aforementioned DEM datasets are rectified: any outliers are rectified using a raster calculator. Ultimately, these two images are overlaid to generate the DEM image encompassing the entire landslide area. The results are then exported in the GeoTIFF file format.

2.2. Geometric Modeling

By parsing the file information of DEM data files and three-dimensional model data files, we achieved the transformation of DEM data files into three-dimensional models.

We drew inspiration from Peter Carlson’s “tiff-to-stl” project on GitHub and utilized it to achieve the conversion of DEM to STL format (https://github.com/peterbcarlson41/tiff-to-stl, (accessed on 2 June 2023)).

A DEM can be characterized as a numerical or digital portrayal of altitude values across a planetary surface, whether in its entirety or specific regions, presented as a function of geographical coordinates. DEM is commonly depicted as a grid of elevation values and holds a pivotal role in geospatial analysis, hydrological modeling, landscape characterization, and more [39].

The STL file format is utilized to depict the geometry of three-dimensional objects by segmenting their surfaces into small triangular facets. Each individual triangular facet is defined by the three-dimensional coordinates of its vertices in conjunction with a unit normal vector corresponding to the triangular facet. The direction of the unit normal vector is determined utilizing the right-hand rule, oriented outward from the triangular facet. It is important to note that STL files exclusively capture the surface geometry of three-dimensional objects and do not encompass additional attributes such as color, material textures, and the like. The data storage format of STL files comes in two variants: binary format and ASCII format. Due to its succinct nature, the binary format is the more prevalent choice. In the context of the present discourse, we have opted to utilize the binary format as our selected method for data storage.

In the context of STL files’ binary format, data adheres to a standardized structure. The initial 84 bytes of the file are allocated to precisely outline the file-related information pertinent to the three-dimensional model. Among these bytes, the initial 80 constitute the file header, serving as a repository for storing critical particulars, including the file name, etc. In immediate succession, a 4-byte integer is employed to convey the count of triangular facets constituting the model.

Subsequently, a systematic presentation of the geometric particulars for each individual triangular facet follows. Each of these facets is allocated a consistent 50-byte space, where the initial 48 bytes meticulously capture crucial information: the facet’s unit normal vector and the coordinates of its three vertices. These values are consistently encoded as three sets of 4-byte floating-point numbers. The final 2 bytes are designated for encapsulating attributes unique to the respective triangular facet. This incorporation contributes to a comprehensive portrayal of the model’s surface characteristics, thereby enhancing the overall understanding of its geometric configuration (

Table 1. Binary format.

Bytes	Data Type	Description
80	ASCII	file header
4	unsigned integer	triangle facet count
4	float	i for unit normal vector
4	float	j for unit normal vector
4	float	k for unit normal vector
4	float	X for unit normal vector1
4	float	Y for unit normal vector1
4	float	Z for unit normal vector1
4	float	X for unit normal vector2
4	float	Y for unit normal vector2
4	float	Z for unit normal vector2
4	float	X for unit normal vector3
4	float	Y for unit normal vector3
4	float	Z for unit normal vector3
2	unsigned integer	attribute information

).

Taking triangle ABC as an illustrative example, we elucidate the methodology for defining the unit normal vector. Given the coordinates of the three vertices of triangle ABC, we can derive the following:

AB = B − A

(1)

AC = C − A

(2)

$$\mathbf{N}=\frac{\mathbf{A}\mathbf{B}\times \mathbf{A}\mathbf{C}}{\left|\mathbf{A}\mathbf{B}\times \mathbf{A}\mathbf{C}\right|}$$

(3)

where A = (X₁, Y₁, Z₁), B = (X₂, Y₂, Z₂), C = (X₃, Y₃, Z₃), and N represents the unit normal vector of triangle ABC, with the normal direction determined by the right-hand screw rule.

In order to extract information from each triangular face in the DEM, the method we employed is as follows:

We exemplify the method for acquiring fundamental information on triangular patches using a grid data configuration featuring three rows and two columns. Given the pixel values associated with raster cells A, B, C, D, E, and F, our approach involves traversing each row of the raster data. Upon encountering raster cell A, we initiate the retrieval of pixel values for its three neighboring cells—specifically, B, C, and D. Following this, we meticulously assess the validity of pixel values for cells B and C. Once both the B and C data are validated and provided that the pixel value of A is also verified, the raster cells A, B, and C coalesce to form the triangular facet ABC. Analogously, if the pixel value of D satisfies the validity criteria, raster cells B, C, and D collectively compose the triangular facet BCD. This sequence of operations is replicated as we proceed to traverse raster cell C, during which we employ the identical methodology to scrutinize the pixel values of cells C, D, E, and F. Assuming that the data for all the aforementioned cells are substantiated as valid, a total of four triangular facets can be synthesized (Figure 2). More precisely, a set of four adjacent raster cells constitutes the basis for generating two distinct triangular facets. Within the context of an m-row and n-column dataset, the theoretical potential exists to derive up to (m − 1) × (n − 1) × 2 triangular facets. It is of paramount importance to underscore that the term “valid pixel values” within the aforementioned context pertains to pixel values that are devoid of anomalies and conform to predetermined stipulations encompassing both the raster and data ranges.

Assuming that ‘t’ represents the affine transformation information extracted from the raster image, it constitutes a tuple comprising six elements. These elements respectively signify the X-coordinate of the top-left corner of the raster image, horizontal pixel resolution, horizontal rotation angle, Y-coordinate of the top-left corner, vertical rotation angle, and vertical pixel resolution. Consequently, the coordinates of raster cell ‘A’ can be represented as follows:

A_x = t(0) + ((X_min + X) × t(1) + (Y_min + Y) × t(2))

(4)

A_y = t(3) + ((X_min + X) × t(4) + (Y_min + Y) × t(5))

(5)

A_z = Z

(6)

where t(0) is the X-coordinate of the top-left corner of the raster image, t(1) is the horizontal pixel resolution, t(2) is the horizontal rotation angle, t(3) is the Y-coordinate of the top-left corner, t(4) is the vertical rotation angle, t(5) is the vertical pixel resolution, X represents the row number of raster A, Y represents the column number of raster A, X_min indicates the minimum value of row numbers within the specified window range, and Y_min indicates the minimum value of column numbers within the specified window range. The variable Z is contingent on the specified requirements, and in this context, we default to utilizing the pixel values from the raster.

By employing the aforementioned approach, the conversion from DEM to STL format can be achieved. Simultaneously, based on the analysis of the formats of the aforementioned two files, integration into a GIS platform can be achieved through Python programming language.

Due to the lack of shape definition in the acquired three-dimensional model, making it unsuitable for direct mesh generation, and the need for a three-dimensional model of the landslide body for subsequent initial condition setup, we addressed this issue using FreeCAD. Under the “Part” workbench within it, three-dimensional model shape definitions can be accomplished, and furthermore, landslide body three-dimensional models can be obtained through three-dimensional Boolean operations.

landslide body = landslide − landslide surface

(7)

By employing the aforementioned methods, it is feasible to achieve the three-dimensional modeling and shape definition of landslides based on DEM images of the landslides.

2.3. 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 Savage and Hutter [14,40], landslides are regarded as continuous fluids with isothermal mixtures. Thus, from the Eulerian perspective, the continuity equation is expressed as follows:

$$\frac{\partial \rho }{\partial t}+\nabla ·\left(\rho \mathbf{v}\right)=\mathbf{0}$$

(8)

where $\rho$ is the mass density, $t$ is time, $\mathbf{v}$ is the velocity, and the momentum conservation equation is given as:

$$\frac{\partial \rho \mathbf{v}}{\partial t}+\rho \mathbf{v}·\nabla \mathbf{v}={\mathit{S}}_{\mathbf{f}}$$

(9)

where ${\mathit{S}}_{\mathit{f}}$ 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:

$$\frac{\partial \rho \mathbf{v}}{\partial t}+\rho \mathbf{v}·\nabla \mathbf{v}=-\nabla p+\nabla ·\mathsf{\tau }+\rho \mathbf{g}+\mathbf{F}$$

(10)

where p(x, t) is the pressure filed, $\mathsf{\tau }$ 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:

$$\nabla ·\mathbf{v}=\mathbf{0}$$

(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 α 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 α as the ratio of the soil volume within an individual grid cell to the overall grid cell volume. Specifically, α = 1 signifies complete saturation with soil within the grid cell, while α = 0 indicates complete air occupancy. When α 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:

$$\frac{\partial \rho \mathbf{v}}{\partial t}+\nabla ·\left(\rho \mathbf{v}\mathbf{v}\right)-\nabla ·\left(v\nabla \mathbf{v}\right)-\nabla \mathbf{v}·\nabla v=-\nabla p_{rgh}-\mathbf{g}·\mathbf{h}\nabla \rho +\sigma \kappa \nabla \alpha$$

(12)

$$p_{rgh}=p-\rho \mathbf{g}·\mathbf{h}$$

(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 ∇∙n, n is the unit normal phase vector of the curved surface element at the two-phase interface.

In the VOF model, the mass density of a two-phase mixed cell is defined as the weighted average of the two-phase mass densities:

$$\rho ={\alpha }_1{\rho }_1+(1-{\alpha }_1){\rho }_2$$

(14)

Since the mass density of an incompressible Newtonian fluid is constant, we can get:

$$\frac{\partial \alpha }{\partial t}+\mathbf{v}·\nabla \alpha =\mathbf{0}$$

(15)

Equations (11), (12) and (15) is the control equation of the VOF model.

The above method allows for the generation of a three-dimensional model grid and numerical solution.

2.4. Post-Processing

In practical applications, our primary focus lies in determining the extent of landslide distribution and the thickness of deposits in the deposition area. In order to visualize the results of the three-dimensional model on the GIS platform, we drew inspiration from the SPH kernel function and transformed the computational results of the three-dimensional model into a two-dimensional format.

Smoothed Particle Hydrodynamics (SPH) is an unstructured particle method based on the Lagrangian formulation, widely applied in engineering and scientific domains [43]. As the solution results of the three-dimensional model consist of a series of grid data, including the coordinates of points forming grid cells, the centroid coordinates of grid cells, and numerical values for various physical fields, we employ the following method to convert them into two dimensions.

$$f\left(x, y\right)=\sum _{i=1}^{n}W\left(x_i, y_i\right)\times V\left(x_i, y_i\right)\times \alpha (x_i, y_i)$$

(16)

$$W\left(x_i, y_i\right)=1/\pi h^2\times e^{-{\left(\frac{x_i^2+y_i^2}{h}\right)}^2}$$

(17)

$$h=\sqrt{2 }\sigma$$

(18)

where $f\left(x, y\right)$ represents the elevation value at the two-dimensional grid $\left(x, y\right)$, $V\left(x_i, y_i\right)$ represents the volume of the grid cell with centroid coordinates ($x_i, y_i$), $\alpha (x_i, y_i)$ denotes the volume fraction, indicating the proportion of the landslide body within the grid cell, $W\left(x_i, y_i\right)$ represents the kernel function indicating the contribution of this grid cell to the two-dimensional grid $\left(x, y\right)$, and $h=\sqrt{2 }\sigma$ denotes the smoothing radius of the kernel function.

Using the above method, the elevation values for the corresponding two-dimensional grid can be determined based on the known basic information of the three-dimensional grid cells.

3. Experimental Verification

In this section, we selected the Yigong landslide case to validate the viability of our methodology. The Yigong landslide, one of the most renowned long run-out landslides, occurred on 9 April 2000 in the rugged and steep terrain of the southern flank of the Nyainqentanglha Shan Mountains in southeast Tibet (94°58′03″; 30°12′11″) [44,45]. Originating at an altitude of approximately 5000 m, the landslide endured for 10 min, traversed approximately 10 km, and descended by around 3300 m. The volume of the region where the landslide originated exceeds 1 × 10⁸ m³, while the overall volume of deposited material from the landslide is approximately 3 × 10⁸ m³. Notably, the volume of the landslide deposits was roughly three times that of the initiation zone. Ultimately, the debris avalanche was funneled into the Yigong River, forming a dam of approximately 60 m in height [46,47].

We selected the OpenFOAM software as the computational tool for numerical simulations of the three-dimensional model. However, the grid generation tools within OpenFOAM require the input of the geometric surfaces of the model to perform three-dimensional mesh partitioning. Our proposed solution can address this issue.

Firstly, we acquired DEM data of the Yigong landslide surface and body. The DEM and source height were obtained by the case in the Massflow tutorial. Then, within a GIS platform, we replaced the No Data regions and anomalies with zero values through raster calculations. Subsequently, we overlaid the two remote-sensing images to derive an accurate DEM dataset of the landslide. Ultimately, we exported the remote sensing images of the landslide surface and the landslide in GeoTIFF file format.

Based on the methodology described earlier, we achieved the above-mentioned functionalities using the Python3 programming language. In order to extract information for each triangular facet from the DEM, which includes the coordinates of the three vertices forming the facet, we utilized modules such as sys, argparse, deque, pack, unpack, numpy, and gdal. These modules can extract essential information from the Yigong landslide dataset, including the dimensions of rows and columns, affine transformation parameters, and pixel values corresponding to the specified band. Building upon this foundation, we converted the raster data from its original geographic coordinates to pixel coordinates. Subsequently, we utilized the pixel values of raster cells to extract fundamental details of each triangular face formed by these cells. This information encompasses the pixel coordinates of the three vertices comprising each triangular face, along with the unit normal vector. Utilizing this information, we can generate STL files for the landslide surface and landslide body, thus achieving the transformation from DEM to a three-dimensional geometric model (Figure 3). Notably, according to the analysis of Geometric Modeling, it is evident that the number of generated triangular facets is determined by the initial resolution of the DEM. The three-dimensional geometric model is crafted to represent the shape of the landslide using these triangular facets. Therefore, the quality of the initial landslide dataset will determine the quality of the constructed three-dimensional geometric model. Under the support of hardware capabilities, a higher resolution in the initial landslide DEM results in a greater number of corresponding triangular facets. Consequently, the constructed three-dimensional geometric model closely approximates the actual topographical conditions of the landslide.

Then, we defined the shapes for the two geometric models above in FreeCAD v0.19 and obtained the three-dimensional geometric model of the landslide body through three-dimensional Boolean operations. It is an open-source CAD/CAE tool built upon the OpenCASCADE framework. As a 3D CAD modeling tool, FreeCAD is a versatile program compatible with Windows, Linux/Unix, and macOS systems. Notably, FreeCAD can be imported as a Python module into other software applications or within a dedicated standalone Python console, accompanied by its full spectrum of modules and components.

So, we integrated FreeCAD as a module into the previously mentioned Python script. Within this script, the “Mesh” module of FreeCAD was utilized to import the two three-dimensional geometric models generated earlier. Subsequently, by utilizing functions within the “Part” module, we established geometric tolerance for the sewing shape, setting it at 0.1 for this study. Then, we transformed the models into solid entities. These operations correspond to the “Create shape from mesh” and “Convert to solid” tools available within the Part workbench of the FreeCAD software. Through these sequential steps, we achieved the establishment of geometric model shapes, forming the foundation for subsequent mesh generation. Ultimately, to obtain the geometric model of the landslide, body functions within the “Part” module were employed to execute a three-dimensional Boolean operation on the two previously defined geometric models (landslide surface and landslide) (Figure 4). This operation calculated their difference set, resulting in a geometric model of the landslide body with a well-defined shape for subsequent initialization of conditions.

In the end, we performed numerical simulations on the three-dimensional models using OpenFOAM. OpenFOAM incorporates a suite of mesh generation tools readily available for direct utilization, offering flexibility in accommodating diverse requisites by adjusting distinct dictionary files. In this context, we employ the blockMesh tool to create a hexahedral background grid capable of encompassing the entirety of the landslide. Notably, in this study, the boundary condition assigned to each face is designated as “wall”, and no unit scaling is applied. Subsequently, we employed the snappyHexMesh tool to generate a mesh for the landslide model by modifying parameters within the snappyHexMeshDict dictionary file. We then used the topoSet and setFields tools to extract grid cells from the area defined by the landslide body’s geometric model and set initial conditions. To account for the influence of air during the landslide movement, here we assign α = 1 to the extracted grid cells mentioned above, denoting them as representing the landslide body. For the remaining cells, we set α = 0, indicating their classification as air. These two fluids were treated as Newtonian fluids. Based on previous studies [48,49], we designate the mass density of the landslide body as 2150 kg/m³, viscosity as 100 Pa·s, and kinematic viscosity as 0.0465 m²/s. Simultaneously, we designated the mass density of air as 1 kg/m³ and kinematic viscosity as 1.48 × 10⁻⁵ m²/s. Furthermore, a surface tension of 0.7 N/m was assigned between the two phases (

Table 2. Mechanical parameters of the Yigong landslide in the OpenFOAM.

	Landslide Body	Air
Mass density	2150 kg/m3	1 kg/m3
Kinematic viscosity	0.0465 m2/s	1.48 × 10−5 m2/s
Surface tension	0.7 N/m

).

For three-dimensional models, numerical stability issues must be taken into consideration during the solving process. To enhance the stability and accuracy of the computational process, we employed adaptive mesh refinement (AMR) technology. This functionality was implemented by modifying parameters within the dynamicMeshDict dictionary file. Finally we employed the interFoam solver from the multiphase flow solvers to solve the model and generate a comprehensive simulation of the landslide process. In this study, all boundary conditions are designated as “wall”. The initial field conditions are uniform fields. Within the background field, the four boundary conditions are set to “zeroGradient”. In the pressure field, four boundary conditions are defined as “fixedFluxPressure”. For the velocity field, four boundary conditions are specified as “noSlip”. All other parameters remain at their default values.

4. Results

To validate the feasibility of the three-dimensional model, we processed the generated grid information using Python to obtain the visualization of the three-dimensional model results on a two-dimensional plane. Specifically, within Paraview, we extracted pertinent grid data, including cell centroid coordinates, cell volume, and volume fraction of grid cells. Subsequently, we employed a Python script to perform a two-dimensional transformation and visualization of the three-dimensional model results as per Equation (17). Ultimately, we present the results of the three-dimensional landslide model on the GIS platform.

The states of 20 s, 40 s, and 60 s using our three-dimensional model are given in Figure 5. Specifically, within the initial 20 s of motion, the landslide body was influenced by internal lateral pressure, and diffusion dominated the landslide’s behavior, with lateral pressure being the primary driving force. Between 20 s and 40 s, the landslide body was primarily influenced by gravitational forces entering the channel, where gravity became the main driving force during this stage. Ultimately, the landslide body reached the foothill at 60 s. Throughout the entire motion process, the simulation results of the three-dimensional model exhibited greater dispersion, with a decreasing trend in the maximum accumulation thickness of the landslide body. This can be attributed to the influence of numerical oscillations and numerical diffusion. Meanwhile, regarding the distribution of the landslide deposit thickness, typically, regions with greater thickness should exhibit more concentration. However, influenced by the grid partitioning of the three-dimensional model, the distribution of the landslide deposit in the three-dimensional model appears relatively divergent, deviating somewhat from the actual scenario (Figure 5A,B). Nevertheless, the overall distribution range remains essentially consistent.

According to the results of previous studies, Zhuang simulated the Yigong landslide, which reached the mouth of the valley between 60 and 90 s [49]. Wu, employing the Eulerian method and depth-averaged SPH, achieved arrival at the mouth of the valley between 50 and 100 s [38]. Compared to these studies, the three-dimensional landslide dynamic model developed in this research exhibits a faster motion speed. The landslide reaches the mouth of the valley at around 60 s, a result attributed to the interFoam solver employed in our model. This solver treats all internal friction within the landslide as fluid friction, as opposed to solid friction such as Coulomb friction. Fluid motion resistance is smaller compared to solid friction, resulting in higher motion speed. This discrepancy is acknowledged as a limitation of our model.

Figure 6 illustrates the variation in the deposit thickness of landslide deposits along profiles AB and CD. In profile AB, which extends along the channel direction, landslides are predominantly concentrated in the central portion of the channel. The thickness exhibits a trend of decreasing on both sides before increasing and then decreasing again. On the other hand, profile CD, oriented perpendicular to the channel direction, reveals that the thickness of landslide deposits increases as it approaches the central part of the channel. The maximum deposit thickness for both profiles is situated in the central part of the channel, approximately 70 m. Along profile AB, the thickness of landslide deposits exhibits a variation along the frontal region. Commencing from 72 m at the central channel, it decreases to 16 m, subsequently increases to 27 m, and eventually decreases to 10 m. Similarly, along the rear edge of the landslide, the deposit thickness decreases from 72 m to 16 m, then increases to 22 m before finally decreasing to 0 m. At profile CD, the thickness of landslide deposits decreases from 72 m at the central channel to 0 m on both sides. Compared to the depth-averaged model, the landslide deposit thickness in the three-dimensional model is obtained by cumulatively summing the volumes of individual landslide cells within the three-dimensional grid. In contrast, the depth-averaged model calculates the landslide thickness by solving the governing equations. This implies that our three-dimensional model can attain a more detailed distribution of landslide deposit thickness, taking into account the influence of topographical factors.

Figure 7 illustrates the velocity distribution of landslides and air at profile AB. The lower portion represents the motion of landslides and air, while the upper portion represents the motion of air. During the deposition phase of the landslide, the lower part of the fluid exhibits laminar flow as its primary mode of motion, while the upper part of the fluid, primarily composed of air, experiences turbulent flow. The landslide moves along the channel direction, with the velocity being minimal at the bottom and maximal at the top. Above the landslide, the air fluid generates a series of vortices through turbulent motion, with some being small and concentrated and others being large and dispersed. The interaction between these vortices contributes to the complexity of the air motion in this region.

In the direction perpendicular to the channel, specifically along the CD profile, the fluid moves from the left side of the channel to the right side, i.e., from a relatively higher altitude to a lower one. In the central part of the channel, where the landslide deposits have the maximum thickness, the air exhibits turbulent flow, while laminar flow predominates in the remaining areas. The movement of the landslide is primarily characterized by laminar flow, with the velocity decreasing closer to the bottom (Figure 8).

The pressure distribution along the AB profile from 20 s to 60 s is illustrated in Figure 9. Throughout the landslide motion, the pressure distribution corresponds to the extent of the landslide, with the bottom being consistently the area of maximum pressure and the top being the area of minimum pressure. Simultaneously, the three-dimensional model can capture more details during the landslide motion, such as fluid surges.

Our model effortlessly captures these details in the landslide motion process, indicating its capability for three-dimensional simulation of the intricate dynamics of landslides on the GIS platform.

5. Discussion

At present, the mainstream models for landslide numerical simulations on GIS platforms are two-dimensional models based on the depth-averaged theory. This model simplifies the complex fluid motion in the Z-direction through depth integration, enabling fast computation. Due to its simplicity and relatively low computational requirements, this model has gained widespread application worldwide. However, its success is limited to predicting the depth, average velocity, and distribution range of the fluid. This model cannot predict the velocity variations along the flow depth direction; it can only derive internal pressure through depth-averaged velocity and cannot obtain complete dynamic pressure and internal pressure profiles. The model fails to capture phenomena such as laminar flow, turbulence, and surges during the fluid motion process. At the same time, in areas where the landslide initiates motion, where it eventually deposits, and where significant terrain changes occur, the assumptions of shallow mass flow are no longer met. The reliability of the depth-averaged model will significantly decrease in such regions. In order to provide a comprehensive physical description of the entire process of landslide motion and enhance our understanding of the phenomenon, a three-dimensional landslide dynamics model that takes into account the influence of complex terrain is required. Therefore, we propose an alternative solution: constructing a three-dimensional landslide dynamics model on the GIS platforms.

This study’s three-dimensional model assumes the landslide to be an incompressible Newtonian fluid. The model employs methods such as the VOF method to achieve the three-dimensional solution of the Navier-Stokes equations. In comparison to the two-dimensional depth-averaged model, this model is constructed based on the three-dimensional geometry of the landslide. Therefore, it can provide a more realistic representation of the terrain, unrestricted by complex topography. Additionally, the model considers fluid motion in all directions, providing details such as the three-dimensional velocity field and pressure field of the landslide. Consequently, it captures physical phenomena such as surges and turbulence during the landslide motion. When dealing with complex boundary conditions, this model excels in simulating interactions with the external environment more effectively.

Being an Eulerian-based model, the numerical stability poses a challenge for three-dimensional models of this type. Ignoring higher-order terms in the equations can lead to numerical dispersion. While higher-order linear numerical schemes provide more accurate solutions for the linearized equations, they tend to introduce spurious oscillations in regions with discontinuities. Therefore, achieving a balance between numerical diffusion and numerical oscillations is crucial for obtaining an appropriate physical solution. Although this study has alleviated these effects through adaptive grid refinement techniques, these issues remain prominent and cannot be overlooked. For Eulerian-based models, the accuracy of the model depends on the grid resolution. A higher resolution, meaning smaller grid cells, results in a three-dimensional landslide model that closely approximates real topographical conditions and better reflects actual motion. However, the solution process for this model is conducted on each grid cell. With more grid cells, the model’s solving time increases, consuming more computational resources. The balance between the computational cost of model solving and model accuracy is a challenging issue that needs to be addressed. In this scenario, when our focus is on the distribution range of landslides, the depth-averaged model with faster solving speed may be a better choice. However, when attention is directed towards more details in the landslide motion process, the three-dimensional model proposed in this study becomes a preferable option [38].

The three-dimensional model treats the friction between the earth’s mass flows as fluid friction. The interFoam solver in OpenFOAM is built upon this foundation. However, the actual process of landslide motion is more complex. Different stages and regions of landslide motion exhibit varying levels of friction. For instance, the initial stage involves larger block movements, while the deposition phase features smaller blocks. The upper part of the landslide body contains larger blocks compared to the smaller blocks in the lower part. Size effects are present among the landslide blocks [50], where larger particles exhibit rigid-body motion, and smaller particles behave more like fluid motion [51]. Frictional behavior typically involves Coulomb friction for the flow of large solid particles, such as large blocks. In contrast, the flow of smaller particles often exhibits fluid friction. Accurately describing the process of transitioning from Coulomb friction to fluid friction and specifying the friction types in different parts of a landslide is a challenging task. Our model, which treats the friction between landslide blocks as fluid friction, has limitations that result in discrepancies with the results of two-dimensional models. To address this issue, future work will involve the development of new solvers. Future research can involve coupling Coulomb friction with fluid friction during the landslide motion process. Coulomb friction may be applied to larger blocks, while fluid friction can be utilized for smaller particles. The key technological challenges lie in understanding the transition between the two friction types and determining appropriate boundary conditions.

6. Conclusions

The proposed real three-dimensional landslide dynamic model under the GIS platform in this study represents a significant advancement in simulating landslides in three dimensions. This model lays the foundation for more authentic simulations of actual landslide movements, thereby offering an opportunity to enhance our understanding of landslide dynamics. Through the three-dimensional simulation of landslide motion, we gain insights into its detailed dynamics.

In this study, we propose a methodology for three-dimensional modeling, numerical simulation, and two-dimensional visualization of landslides on the GIS platforms. Specifically, we proposed a solution to achieve a three-dimensional simulation of landslide motion on the GIS platform. Simultaneously, drawing inspiration from SPH kernel functions, we convert the simulation results into two dimensions. To demonstrate the feasibility of our approach, we select the Yigong landslide case for three-dimensional landslide modeling, numerical simulation, and two-dimensional visualization of results.

The case study results indicate that our model can capture numerous details in the landslide motion process, enabling numerical simulation of complex three-dimensional landslide dynamics on the GIS platform. However, constrained by the inherent limitations of the three-dimensional model, the model-solving process is susceptible to numerical oscillations and diffusion. The model’s accuracy is contingent on grid partitioning, and striking a balance between model precision, solving time, and expediting the solving process poses a challenging problem. Furthermore, the model considers the friction between mass flows as fluid friction, resulting in faster landslide motion speeds compared to the depth-averaged model. Future developments should focus on devising new solvers to achieve more precise simulations of landslide motion processes.