Optimization of Emergency Evacuation Paths for Personnel in Tailings Dam Failure Disasters

Abstract

The tailings dam failure poses a significant threat to the life and property security of downstream residents, with emergency evacuation serving as a crucial measure to mitigate disaster losses. Herein, a multiobjective evacuation path model is developed to address the key issue of emergency evacuation path planning following a tailings dam failure. The model is based on multiobjective evacuation mathematical theory, with the goals of minimizing evacuation time and reducing personnel risk, while considering the downstream road network structure and the inundation area of the dam break as constraints. To further solve the multisource point problem within the model, a Dijkstra’s algorithm with high compatibility to GIS road networks is employed. Additionally, the algorithm is improved using a circular search range limitation method, significantly enhancing the solution efficiency of the evacuation path model. An evacuation route‐planning system for tailings‐dam breach emergencies was developed using the ArcGIS Engine and Visual Studio, with data stored and managed in a geodatabase. Benchmark results show that, compared with the standard Dijkstra’s algorithm, the improved algorithm reduced the number of nodes and arcs traversed by 168 and 106, respectively, and achieved a 49.6% speedup (runtime reduced from 876.47 to 441.23 ms). Then, the system is successfully applied to the emergency evacuation of downstream residents following the tailings dam failure at the Tongshankou tailings dam in Hubei Province. Dam break simulation for Tongshankou indicated four downstream villages were threatened, with peak depths of ~ 4.5–7.5 m and flow velocities > 6 m/s. Computed total evacuation times were 4.09 min (Chengongjin village), 7.92 min (Chentingjie village), 10.21 min (Chenxizhen village), and 9.76 min (Wangjia village), all shorter than the corresponding inundation arrival times of 6.35 min, 10.91 min, 12.24 min, and 17.44 min, respectively, yielding positive safety margins (Δt = 2.03 to 7.68 min).

Summary

• •

  A novel multiobjective evacuation path model is developed for tailings dam failure disasters.

• •

  The Dijkstra’s algorithm is enhanced with a circular restricted search range method.

• •

  An evacuation path planning system is constructed using ArcGIS Engine and Geodatabase.

• •

  The proposed model and system are successfully applied to the Tongshankou tailings dam in Hubei Province.

1. Introduction

Tailings dams, as structures used by mining enterprises to store tailings and wastewater, primarily function to contain waste materials generated during the ore processing [1, 2]. However, overtime, improper management or extreme unforeseen events may lead to dam failures, posing a significant threat to the safety of downstream populations [3–5]. According to data from the past 100 years, the global failure rate of tailings dams is estimated at 1.2%, resulting in ~2650 fatalities and impacting about 317,000 individuals through displacement, property damage, and health risks [4–7]. Emergency evacuation is critical to reducing casualties and mitigating losses in tailings dam failures [8–10]. Previous studies have suggested that establishing and optimizing scientific evacuation path models can effectively reduce the life and property losses caused by such disasters, particularly in preventing mass casualties and postdisaster chaos [11–14]. Concurrently, advances in dam-slope monitoring, such as automated delineation of erosion areas on tailings-dam slopes, offer hazard inputs directly relevant to evacuation modeling [15], and broader geohazard syntheses on submarine landslides highlight the need for integrated hazard–evacuation frameworks [16]. Thus, the timely and effective evacuation of people from disaster-stricken areas to shelters, using scientifically grounded methods, holds immense practical value and significance in disaster response.

Effective emergency evacuation path planning requires not only the consideration of the shortest evacuation time and route but also the path’s safety and risk control to avoid evacuation failures during the critical evacuation phase [14, 17]. To date, some progress has been made by researchers both domestically and internationally in areas such as evacuation path model construction, risk assessment of evacuation paths, and evacuation behavior of individuals [18–23]. For example, Lovreglio et al. [18] and Ye et al. [19] conducted emergency evacuation studies based on toxic gas leakage in industrial parks. Lovreglio integrated a toxic gas diffusion model with an emergency evacuation model to determine the shortest evacuation path. Ye, on the other hand, used the Floyd algorithm to analyze the impact of dynamic toxic gas diffusion on the safety of evacuation paths and designed an evacuation plan based on minimizing the exposure to toxic gas. Moreover, emergency evacuation planning in fire scenarios has been extensively studied. Yan et al. [20] used the Pyrosim program to simulate fire scenarios and developed an evacuation plan for super high-rise buildings. Li et al. [21] combined the fire dynamic simulator (FDS) and building information model (BIM) to create evacuation plans for underground commercial streets under three different fire scenarios. Yang et al. [24] proposed a path planning method based on multiobjective robust optimization, solving the model using the NSGA-II algorithm and obtaining the optimal Pareto solution under certain robust control parameters based on the principle of minimizing the total cost function. Zhang et al. [22] introduced a dynamic path optimization algorithm that uses real-time information to update evacuation routes based on incident conditions. Another study developed a multisource multichain evacuation model based on dynamic risk assessment, which mitigates path conflicts caused by multiple evacuees [23]. In the context of tailings facilities, research on dam-slope erosion detection and rainfall–excavation–angle controls on fractured-slope stability and fracture-network evolution should be integrated into evacuation route optimization practice [25, 26]. However, to the best of our knowledge, there is limited comprehensive research in the fields of environment and safety on the inundation areas caused by tailings dam failure and the evacuation of people after such incidents. More importantly, for tailings dam failure disasters, factors such as the evolution of sand flow after the dam break, the inundation area, tailings flow velocity, and surrounding terrain can significantly impact evacuation route planning. In particular, failure to assess the timing of water flow reaching different affected points can result in certain evacuation paths becoming impassable at specific times, leading to evacuation failure [14, 27]. Therefore, incorporating the effects of sand flow after a dam failure, as well as the constraints posed by the road network and inundation areas, is a critical direction to enhance the effectiveness of emergency evacuation route planning.

Herein, a novel emergency evacuation route planning method based on the characteristics of tailings dam failure was proposed based on simulating the evolution of sand flow after the dam break, the downstream potential disaster points and affected personnel are identified. The evacuation route model is then constructed with the objectives of minimizing evacuation time and risk. Unlike traditional route planning methods, this study introduces the Dijkstra’s algorithm, which is further improved to leverage its efficient path search capabilities and strong adaptability to address the dynamic and multisource evacuation route problems. Additionally, considering the influence of sand flow after the dam break, appropriate model improvements are made to ensure the feasibility of the route, avoiding the potential issues of path infeasibility in traditional algorithms. This method allows for the rapid and accurate planning of optimal evacuation routes during a tailings dam failure and the timely relocation of affected personnel to safe areas, thus effectively reducing casualties and property losses caused by the disaster.

2. Methods

2.1. Emergency Evacuation Route Planning

Figure 1 illustrates the core of the personnel evacuation problem downstream of the dam break, which involves planning evacuation paths from multiple disaster sites to emergency shelters based on the downstream road network structure, using the sand flow range from the dam break as the trigger condition. In Figure 1, S_i, V_n, and P_k represent the sets of multiple disaster sites, network structure nodes, and emergency shelters, respectively, while the dotted arrows represent the road network structure. The goal of evacuation path planning is to design a road network structure that facilitates efficient evacuation.

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2.2. Study Region

Tongshankou tailings pond (Figure 2), located at 1.5 km southwest of Chengui Town, Daye City, Hubei Province, China, is the result of the expansion of the former Hongyan reservoir. It is nestled against mountains in the south and surrounded by dams on three sides: east, west, and north. It is a valley-type tailings pond. The tailings pond has a catchment area of 1.82 km², an inner reservoir area of 0.69 km², a dam height of 37 m, and the flood discharge capacity of the overflow well can reach 35.04 m³/s. The total storage capacity of the tailings pond is 13.12 million m³, which is a third-class reservoir and belongs to the A-class danger source of the mine. Eight villages, a highway, and a middle school are located within 2 km downstream of the tailings dam (Figure 3).

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2.3. Construction of Multiobjective Evacuation Path Model

2.3.1. Model Objective Function

The construction of the multiobjective evacuation path model of the tailings pond dam break accident has two objectives: speed and safety, with the corresponding objective functions being the shortest evacuation time and the lowest risk.

(1) The shortest evacuation time is as follows: 1 $\begin{array}{}\min T=\max \left\{\left({\left(t_{S_i{P}_j}\right)}_m+t_{S_i{P}_j,\text{wait}}\right)x_{n{S}_i}{y}_{n{P}_j}\right\},\end{array}$ 2 $\begin{array}{}{\left(t_{S_i{P}_j}\right)}_m\in \left\{{\left(t_{S_i{P}_j}\right)}_1,{\left(t_{S_i{P}_j}\right)}_2,\dots ,{\left(t_{S_i{P}_j}\right)}_k\right\},\end{array}$ where ${\left(t_{S_i,P_j}\right)}_m$ (represents the time required for the disaster point to arrive at the m(m ≤ k) path of the emergency shelter, and $t_{S_i,P_j,\text{wait}}$ represents the waiting time for the disaster point to arrive at the emergency shelter; x_nSi is a binary decision variable equal to 1 if the n-th road is assigned to the evacuation originating from source S_i, and 0 otherwise; y_nPj is a binary decision variable equal to 1 if the n-th road is connected to shelter P_j, and 0 otherwise.

The shortest-time evacuation model comprehensively considers the time required to reach each road node as well as the waiting time in the evacuation process at each disaster point, thereby improving the overall efficiency of disaster victim evacuation.

(2) The lowest risk of potential disaster victims is as follows: 3 $\begin{array}{}\min \text{Risk}=\sum _{n=1}^{\left|N\right|}{\left(s_n\right)}_{\min }\times t_{n,\text{wait}},\end{array}$ 4 $\begin{array}{}{S}_n=\min \left\{{\left(S_n\right)}_1,{\left(S_n\right)}_2,\dots ,{\left(S_n\right)}_k\right\},\end{array}$ where ${\left(S_n\right)}_{\min }$ represents that in the evacuation process, the lowest risk should be taken as the target, and the safety of the disaster victims should be fully considered.

2.3.2. Model Construction

The evacuation targets of the dam break accident evacuation path model are the shortest evacuation time and the lowest risk during evacuation, with the former being the primary goal and the latter being human-centered. Therefore, by combining the above evacuation models, the multiobjective evacuation path model for a tailings pond dam break accident is constructed as follows: 5 $\begin{array}{}\min D=\min T+\min \text{Risk},\end{array}$ 6 $\begin{array}{}\min T=\min \left\{\left(\sum _{i=1}^n\sum _{j=1}^n{\left(t_{S_i,P_j}\right)}_m+t_{n,\text{wait}}\right)x_{n{S}_i}{y}_{n{P}_j}\right\}\forall n=121212,,\dots ,\left|N\right|\forall i=,,\dots ,\left|S\right|\forall j=,,\dots ,\left|P\right|\left.\right),\end{array}$ 7 $\begin{array}{}\min \text{Risk}=\sum _{n=1}^{\left|N\right|}{S}_n\times t_{n,\text{wait}},\end{array}$ 8 $\begin{array}{}\sum _{j=1}^{\left|P\right|}{y}_n{P}_j=1\left(\forall n=12,,\dots ,\left|N\right|\right),\end{array}$ 9 $\begin{array}{}C{v}_i{v}_j\left(t\right)\le C{v}_i{v}_j,\max ,\end{array}$ 10 $\begin{array}{}{VoIP}_j\left(t\right)\le {VoIP}_{j,\max },\end{array}$ 11 $\begin{array}{}\sum _{\begin{array}{c}j=1\\ j\ne i\end{array}}^n{x}_{ij}-\sum _{\begin{array}{c}j=1\\ j\ne i\end{array}}^n{x}_{ji}=\left\{\begin{array}{cc}1& i=1\\ -1& i=n\\ 0& \text{other}\end{array}\right.,\end{array}$ 12 $\begin{array}{}\sum _{\begin{array}{c}j=1\\ j\ne i\end{array}}^n{x}_{ij}=\left\{\begin{array}{cc}\le 1& i\ne n\\ =0& i=n\end{array}\right.,\end{array}$

Equation (5) states the overall objective of the evacuation routing model for a tailings-dam breach scenario. Equation (6) minimizes the total evacuation time for all affected sites to reach emergency shelters, while Equation (7) minimizes the inundation risk posed by the tailings flow during evacuation. Equations (8)–(12) specify the model constraints: Equation (8) enforces complete evacuation from each affected site; Equation (9) imposes node-capacity limits such that the number of evacuees traversing any node does not exceed its maximum service capacity; Equation (10) enforces shelter-capacity limits; Equation (11) guarantees connectivity by requiring at least one feasible path from each affected site to a shelter; and Equation (12) imposes a no-backtracking constraint, prohibiting U-turns and unnecessary repetition of nodes or edges.

2.4. Path Planning Method Based on Improved Dijkstra’s Algorithm

2.4.1. Basic Dijkstra’s Algorithm

The Dijkstra’s algorithm is a labeled shortest-path algorithm developed by Dutch computer scientist Edsger Dijkstra in the 1950s. This algorithm can find the shortest path between any source point and the remaining node in the road network data structure [28]. Therefore, the Dijkstra’s algorithm has been extensively studied in the fields of logistics, transportation, surveying and mapping science, disaster science, and other disciplines, and it has become one of the most well-known shortest path-solving algorithms [29].

As shown in Figure 4, the basic principle of the Dijkstra’s algorithm is as follows: given a certain source point u as the vertex, two sets, V and V–S are defined to store all nodes of the road network data structure. If there is a shortest path between two points u and v, then point v is added to set V, and vice versa to set V–S. After the algorithm is completed, all nodes with the shortest path between point u are stored in set V, and all the unreachable points of point u are stored in set V–S. According to its solving principle, the Dijkstra’s algorithm uses a greedy strategy to find the shortest path between the target source point and each node. This approach is useful for determining the shortest time path to each emergency shelter caused by the tailings dam break accident. Furthermore, the Dijkstra’s algorithm is well-integrated with the GIS platform, allowing it to perform path-solving activities directly in the GIS-constructed road network data structure. In this article, the Dijkstra’s algorithm is used to solve the multiobjective evacuation path model of a tailings dam break accident.

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As shown in Figure 5, Dijkstra’s algorithm solves the shortest path problem by expanding outward layer by layer from a given source node, with the source point acting as the center. Starting from the source, the algorithm iteratively explores all remaining nodes in the network, ultimately determining the shortest (time or distance) path between the source and each of the other nodes. During the process, every node treated as a new source is assigned a distance of 0, while intermediate nodes without adjacent edges to the source point are assigned an infinite distance ∞. For a directed network graph with n nodes, the time complexity of Dijkstra’s algorithm is O(n²), and the space complexity is O(n).

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2.4.2. Improved Dijkstra’s Algorithm by Circular Restricted Search Range Method

The blind traversal of all nodes in Dijkstra’s algorithm leads to an excessively large search space, resulting in low search efficiency. This issue becomes particularly problematic as the solution area expands. If all nodes in the network data structure are traversed repeatedly, the computation becomes increasingly complex, and the time complexity becomes unacceptable. For tailings pond dam break accidents involving a rapid dam break process, a large sand flow inundation area, and a large number of victims [30, 31], the most important thing is to use the algorithm to quickly solve the appropriate evacuation path and safely transfer the crowd to the emergency shelter before the dam break sand flow arrives [32]. The road network data structure involved in the tailings dam break accident is extensive, and most of China’s tailings ponds are located in towns and villages where urban and rural areas intersect. The road network data structure is complex, and there are many road nodes. If the traditional Dijkstra’s algorithm is used to solve the evacuation path, all nodes in the road network data structure must be traversed, which reduces solution efficiency. This fails to meet the timeliness requirement of an efficient dam break evacuation path solution.

To improve the search range of the traditional Dijkstra’s algorithm and enhance computational efficiency, the concept of restricted search range was introduced in this work. The idea behind restricted search range calculation is to use the spatial location attribute of the road to restrict the large road network data structure to a small range. This range is likely to include the shortest path from the source point to the end point. By preferentially traversing all nodes within this range, the number of nodes is reduced, resulting in improved solving efficiency. Given that the common restricted search range method is insufficient for emergency evacuation in the event of a tailings dam break accident due to the large computation required, this article proposes a flexible circular restricted search range calculation method. As illustrated in Figure 6, the concept behind the method is as follows: using the spatial characteristics of GIS technology, draw a circle with the same radius from the disaster site (starting point) and the emergency shelter (ending point), determining their respective circular restricted ranges, and using the two circular restrictions as the internal tangent circle to draw the outer rectangular area, which is the restricted search area. In brief, the limiting of the range reduces the number of traversal nodes, lowering time complexity and improving computational efficiency. If the shortest path cannot be found within the obtained search area, the search area can be expanded by adjusting the radius of the two internal tangent circles until it covers the entire area.

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Buffer analysis is a buffer polygon area built around the geographic space element entity by setting a buffer radius. Any point within this polygon area has a distance smaller than or equal to the buffer radius. The mathematical formula for buffer analysis is as follows: 13 $\begin{array}{}B=\left\{x_i:d_{\min }\left(x_i,P_j\right)\le R,P_j\in C\right\}{d}_{\min }\left(x,y\right)=\sqrt{\sum _{i=1}^n{\left(x_i-y_i\right)}^2}.\end{array}$

The mathematical principle of the above formula is that the buffer radius R is calculated for the specified element entity P_j to generate buffer B; C = {P_j, j = 1, 2,…, n} is the set of buffer radius R less than or equal to the buffer; and d_min represents the minimum Euclidean distance between the center point of the buffer and other points. Buffer analysis is classified into three types based on the type of element entity: point, line, and plane. The buffer established by a point element is a circular area with a point at the center and a specified distance as the radius. In this article, the buffer analysis method is used to calculate the size of the circular restricted range of disaster sites and emergency shelters, thereby determining the restricted search range of the Dijkstra’s algorithm. Additionally, the circular buffer is established with an increasing sequence of r_i {i = 100, 200, 300,…, n} as the radius, ensuring that if the shortest path is not found within the current search range, the search range can be expanded with the next radius until the shortest path is discovered.

2.4.3. Solution Process for Personnel Evacuation Model

The improved Dijkstra’s algorithm is used to solve the evacuation model. The specific solution steps are as follows:

• 1.

  The representative road nodes are extracted from the weighted and directed road network diagram, and two node sets, A and B are constructed. The set A is the target source point of the shortest path to be found, while set B is the target end point of the shortest path to be found. The Cartesian product of two sets is C = A × B = {(a, b) | a ∈ A, b ∈ B}.

• 2.

  The minimum Euclidean distance d_min between each pair of disaster source points and emergency shelters is solved using Equation 13.

• 3.

  According to the specific disaster source point (x₀, y₀) and emergency shelter (x_i, y_i), a spatial coordinate system is established, with the origin in the lower-left corner of the road network diagram.

• 4.

  The search range is restricted, and the buffer radius r_i is used as the circular restricted range of two nodes. Using the two circular restricted ranges as an internal tangent circle, the outer rectangular region of the vertical space coordinate system is drawn, and the restricted search range is defined. Let x_min = min (x₀, x_i), x_max = max (x₀, x_i), y_min = min(y₀, y_i), and y_max = max (y₀, y_i), then, the coordinates of the four vertices S, T, E, D in the outer rectangular region are (x_min−r_i, y_max + r_i), (x_max + r_i, y_min + r_i), (x_min−r_i, y_max −r_i), and (x_max + r_i, y_min−r_i), respectively.

For any road node W(x_j, y_j), if the coordinate of the road node meets (x_min − r_i, y_max + r_i) ≤ x_j ≤ (x_max + r_i, y_min − r_i) ≤ y_i ≤ y_max + r_i, it indicates that the road node is within the restricted search range. In this case, the Dijkstras algorithm can be used to determine if there is a shortest path within the restricted range. Figure 7 shows the specific process for solving the personnel evacuation model.

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2.5. Numerical Simulation of Dam-Break Hydraulics

2.5.1. Introduction to Simulation Software

FLOW-3D (Flow Science, USA) is a mature and efficient three-dimensional CFD solver whose core advantages are the TruVOF free-surface scheme on a structured Cartesian grid and the proprietary FAVOR method (fractional area and volume obstacle representation). TruVOF robustly captures free-surface evolution and improves numerical efficiency, while FAVOR represents complex geometries on simple Cartesian meshes via local fractional open areas and volumes, mitigating the difficulty of irregular boundaries in finite-difference frameworks. In addition, FLOW-3D provides extensive physics models and boundary condition options (e.g., gravity, turbulence closures, and sediment transport modules), allowing users to tailor setups to site-specific scenarios and enhance predictive skill [33]. Owing to these features, FLOW-3D is well suited for simulating tailings-dam breach hydraulics and the evolution of sediment-laden flows, and it has been widely applied in this context [34–36]. For example, Ghahramani et al. [37] back-analyzed the 1985 Stava (Italy) and 1994 Merriespruit (South Africa) tailings-dam failures with FLOW-3D, reproducing observed inundation footprints and key runout metrics.

2.5.2. Governing Equations

The evolution of sediment-laden dam-break flows is computed using the multiphase continuity (mass-conservation) equation and the full Navier–Stokes momentum equations, with turbulence represented by the RNG k–ε model. In a Cartesian coordinate system, the continuity, Navier–Stokes, and RNG transport equations are given as follows.

The continuity equation is as follows: 14 $\begin{array}{}\frac{\partial \left({\text{uA}}_x\right)}{\partial x}+\frac{\partial \left({\text{vA}}_y\right)}{\partial y}+\frac{\partial \left({\text{wA}}_z\right)}{\partial z}=0.\end{array}$

Here, u, v, and w are the velocity components of the slurry in the x, y, and z directions; Ax, A_y, A_z are the fractional open areas of the cell faces normal to the x, y, and z directions.

Momentum equations are as follows: 15 $\begin{array}{}\begin{array}{c}\frac{\partial u}{\partial t}+\frac{1}{V_F}\left[{\text{uA}}_x\frac{\partial u}{\partial x}+{\text{vA}}_y\frac{\partial u}{\partial y}+{\text{wA}}_z\frac{\partial u}{\partial z}\right]=-\frac{1}{\rho }\frac{\partial p}{\partial x}+G_x+f_x\\ \frac{\partial v}{\partial t}+\frac{1}{V_F}\left[{\text{uA}}_x\frac{\partial v}{\partial x}+{\text{vA}}_y\frac{\partial v}{\partial y}+{\text{wA}}_z\frac{\partial v}{\partial z}\right]=-\frac{1}{\rho }\frac{\partial p}{\partial y}+G_y+f_y\\ \frac{\partial w}{\partial t}+\frac{1}{V_F}\left[{\text{uA}}_x\frac{\partial w}{\partial x}+{\text{vA}}_y\frac{\partial w}{\partial y}+{\text{wA}}_z\frac{\partial w}{\partial z}\right]=-\frac{1}{\rho }\frac{\partial p}{\partial z}+G_z+f_z\end{array},\end{array}$ where ρ denotes the fluid density; p is the pressure acting on a fluid element; G_x, G_y and G_z are the components of gravitational acceleration along the x, y and z axes; and f_x, f_y, and f_z are the viscous-acceleration components in the corresponding directions. For a spatially and temporally varying dynamic viscosity μ, the corresponding viscous acceleration is given by: 16 $\begin{array}{}\begin{array}{c}\rho V_F{f}_x=\text{wsx}-\left[\frac{\partial }{\partial x}\left(A_x{\tau }_{\text{xx}}\right)+\frac{\partial }{\partial y}\left(A_y{\tau }_{\text{xy}}\right)+\frac{\partial }{\partial z}\left(A_z{\tau }_{\text{xz}}\right)\right]\\ \rho V_F{f}_y=\text{wsy}-\left[\frac{\partial }{\partial x}\left(A_x{\tau }_{\text{xy}}\right)+\frac{\partial }{\partial y}\left(A_y{\tau }_{\text{yy}}\right)+\frac{\partial }{\partial z}\left(A_z{\tau }_{\text{yz}}\right)\right]\\ \rho V_F{f}_z=\text{wsz}-\left[\frac{\partial }{\partial x}\left(A_x{\tau }_{\text{xz}}\right)+\frac{\partial }{\partial y}\left(A_y{\tau }_{\text{yz}}\right)+\frac{\partial }{\partial z}\left(A_z{\tau }_{\text{zz}}\right)\right]\end{array}.\end{array}$

Here, wsx, wsy, and wsz denote the wall shear stresses; τ_ij is the shear stress acting on the fluid element; i indexes the acting face and j the acting direction.

RNG model transport equations are as follows: 17 $\begin{array}{}\begin{array}{c}\frac{\partial k_T}{\partial t}+\frac{1}{V_F}\left({\text{uA}}_x\frac{\partial k_T}{\partial x}+{\text{vA}}_y\frac{\partial k_T}{\partial y}+{\text{wA}}_z\frac{\partial k_T}{\partial z}\right)=P_T+G_T+{\text{DIff}}_T-{\epsilon }_T\\ \frac{\partial {\epsilon }_T}{\partial t}+\frac{1}{V_F}\left({\text{uA}}_x\frac{\partial {\epsilon }_T}{\partial x}+{\text{vA}}_y\frac{\partial {\epsilon }_T}{\partial y}+{\text{wA}}_z\frac{\partial {\epsilon }_T}{\partial z}\right)=\frac{\text{CDIS}1\cdot {\epsilon }_T}{k_T}\cdot \left(P_T+\text{CDIS}3\cdot G_T\right)+{\text{DIff}}_{\epsilon }-\text{CDIS}2\frac{{\epsilon }_T^2}{k_T},\end{array}\end{array}$

where k_T is the turbulent kinetic energy; ε_T is the turbulent kinetic-energy dissipation rate; P_T is the production of k_T due to mean velocity gradients; G_T is the buoyancy-induced production term; DIff_T is the turbulent diffusion term; CDIS1 = 1.42, CDIS2 is a model constant, CDIS3 is a shear-rate function, and DIff_Z represents a molecular diffusion term.

2.5.3. Simulation Parameter Settings

A three-dimensional digital terrain model was built in Autodesk Civil 3D. During overtopping-induced dam-break failure, the primary breach may initiate anywhere along the dam crest; to adopt a conservative risk-envelope assumption under this uncertainty, we impose an initial notch centered on the crest (preformed breach) to represent a worst-case scenario and use it to trigger breach development, while tailings material properties are as summarized in Table 1.

Table 1 Tailings material properties and sediment-transport coefficients used in the simulations.

Name	Mean grain size (mm)	Bulk density (kg /m3)	Critical shields number	Drag coefficient	Entrainment coefficient	Bed-load coefficient	Angle of repose (°)
Tailings	0.085	1350	Automatically computed	0.05	0.01	8	38

3. Results

3.1. Validation Results of Improving Dijkstra’s Algorithm

To validate the effectiveness of the improved algorithm, both Dijkstra’s algorithm and the improved Dijkstra’s algorithm were used to compute the shortest path between the starting point U and the endpoint V (Figure 6), based on the GIS platform. By calculating the shortest path between these two points, the differences in node traversal, arc segments, and computation time between the two algorithms were compared. The shortest path results obtained by both algorithms are shown in Figure 8, with detailed data presented in Table 2. As depicted, although both algorithms yield the same shortest path results, there are significant differences in the search range, the number of nodes and arc segments traversed, and the computation time. Unlike the traditional Dijkstra’s algorithm, which often traverses all nodes and arc segments in the network to find the shortest path between two points, the improved Dijkstra’s algorithm reduces the number of traversed road nodes by 168 and arc segments by 106, resulting in a 49.6% improvement in search efficiency and a decrease in runtime from 876.47 to 441.23 ms. The validation results indicate that the improved Dijkstra’s algorithm offers higher efficiency in shortest path computation compared to the traditional algorithm and is suitable for evacuation path calculation in larger network structures.

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Table 2 Comparison of path optimization results between Dijkstra’s algorithm and improved Dijkstra’s algorithm.

Algorithm	Number of arcs searched	Number of nodes searched	Running time (ms)
Dijkstra algorithm	380	223	876.47
Improved Dijkstra algorithm	212	107	441.23

3.2. Simulation Results of Dam Break of Tailings Pond

Given the longer dam structure on the northern side of the tailings reservoir and the greater number of downstream villages, the breach risk is more significant. The breach evolution process of the northern dam of the Tongshankou tailings reservoir was simulated and analyzed using FLOW-3D fluid software, as shown in Figure 9. As illustrated, 6 min after the dam breach (t = 360 s), a breach opening formed, and the sand flow rapidly progressed downstream, inundating Chengongjin village with a flooding depth of 7 m and a flow velocity exceeding 5 m/s. 10 min after the breach (t = 600 s), the breach widened significantly, and the main dam was completely destroyed by the immense impact force. The sand flow continued to advance rapidly toward the northwest, flooding Chentingjie village. 12 min after the breach (t = 720 s), Chenxizhen village was submerged, and the flooding depth at Chentingjie village exceeded 7.5 m, with a flow velocity exceeding 6 m/s. Chenxizhen village was submerged with a depth of more than 3 m and a flow velocity greater than 4.5 m/s. Yangtang village, located to the east and situated on the opposite side of the valley, remained unaffected as the sand flow accumulated within the valley. 17 min after the breach (t = 1020 s), the water level in the reservoir had decreased substantially, and the sand flow slowed. The sand flow advanced into flatter areas, inundating Wangjia village with a depth of 4 m and a flow velocity of 3.5 m/s. The northern dam body break lasted 17 min from the formation of the breach to the end of the breach. According to the simulation analysis results, when the dam break accident occurred in Tongshankou tailings pond, a total of four villages were flooded at different times, namely Chengongjin, Chentingjie, Chenxizhen, and Wangjia villages, with 1398 people expected to be affected. Based on the sand flow simulation results of the tailings dam breach, the severity of the disaster in the four villages was analyzed to determine the evacuation sequence of the affected populations (Table 3). The most severely impacted areas were Chengongjin and Chentingjie villages. The southernmost buildings in Chengongjin village are only 0.13 km away from the dam’s toe, and the sand flow front reaches the village within 6.35 min, completely submerging it. Chentingjie village, located at a bend in the valley, is 0.76 km away from the dam’s toe, and due to its lower elevation, it is completely inundated 10.91 min after the breach. After flooding Chentingjie village, the sand flow continues downstream toward Chenxizhen village, which has a large population of ~580 people and is located 0.89 km from the dam’s toe, making it a high-risk area. Wangjia village is affected by the sand flow later, at 17.44 min, and is located 0.97 km from the dam’s toe, placing it in the sand flow zone.

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Table 3 The basic situation of disaster areas affected by dam break of the tailings pond.

Disaster site	The time the sand flow flooded the village (min)	Distance between village and flood outlet (km)	Maximum water depth (m)	Maximum flow rate (m/s)	Severity
Chengongjin village	6.35	0.31	5.6	3.5	Severe disaster area
Chentingjie village	10.91	0.76	4.5	2.9	Severe disaster area
Chenxizhen village	12.24	0.89	4.1	1.7	Danger zone
Wangjia village	17.44	0.97	1.8	1.3	Sand flow zone

After determining the affected villages and people based on dam break simulation results, it is necessary to identify an emergency shelter for evacuation and transfer [38]. Based on the numerical simulation analysis of the dam break at Tongshankou tailings pond, combined with downstream DEM data and remote sensing images of Tongshankou tailings pond, the emergency shelters that can be used for the evacuation of downstream disaster victims were studied, and reasonable emergency shelters were selected based on the three indicators of safety, accessibility, and spatial property (Figure 10).

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3.3. Road Network Model

The dam break evacuation path planning system is built using the secondary development software ArcGIS Engine and the Geodatabase; the main interface of the system is shown in Figure 11.

[IMAGE OMITTED. SEE PDF]

The construction of the downstream road network serves as the foundation for planning the evacuation path in the event of a tailings pond dam break. The construction of the downstream road network of the Tongshankou tailings pond requires basic DEM data, remote sensing images, road data, population data, and so on. In this article, we collected basic data from the tailings pond’s downstream using UAV photography, Google Maps, field surveys, a local demographic yearbook, network resources, and so on. GIS technology is used to unify image and road attribute coordinates, as well as for georeferencing. The processed data is stored in a Geodatabase and loaded into the tailings pond dam break evacuation path planning system, laying the data foundation for the subsequent path planning (Figure 12).

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After determining the disaster area and emergency shelter based on the simulation results of tailings pond dam break sand flow inundation and the emergency shelter location principle, the identified disaster area and emergency shelter are loaded into the downstream road network of Tongshankou as points, and the downstream road network of Tongshankou is invoked in the road network construction module. Based on an improved Dijkstra’s algorithm, the shortest evacuation time and lowest risk are used to calculate the shortest path from each disaster point to the emergency shelter.

3.4. Evacuation Result of Tailings Dam Break

Most evacuation strategies for disasters involve the simultaneous evacuation of affected individuals. However, for sudden incidents such as tailings dam breaches, a simultaneous evacuation approach may result in severe flooding in heavily affected villages before evacuation is completed, leading to the inundation of these villages by tailings flows. In response to the personnel evacuation issue caused by tailings dam breaches, it is necessary to alter the traditional simultaneous evacuation strategy. This study proposes a sequential evacuation strategy based on the severity of the breach simulation analysis. The evacuation sequence is as follows: Chengongjin, Chengtingjie, Chenxizhen, and Wangjia villages.

Taking the Chengongjin village, which was evacuated first, as an example, the user clicks the evacuation path planning option in the system and adds the village as the disaster point and the three emergency shelters as the shelter point. After clicking the solution path, the system interface displays the shortest path from Chengongjin village to three emergency shelters, as shown in Figure 13a. The shortest times for this village to reach the three shelter points are 4.09 min, 8.71 min, and 12.84 min, respectively. According to the risk assessment results for the disaster area affected by the tailings pond dam break, the dam break sand flow floods the road near Chentingjie village 7 min after it occurs. If Paths 2 and 3 are selected, the evacuation time will be longer than the arrival time of the dam break sand flow, which will pose a threat to the safety of the disaster victims. Therefore, for Chengongjin village, only Path 1 can be selected to evacuate to emergency shelter 1. Next, load the remaining three disaster points into the system one at a time, and calculate the time it takes for each to reach the three emergency shelters (Figure 10). Table 4 shows the statistical results for the length of the path to the three emergency shelters, evacuation time, and whether there are risks in the evacuation process of all disaster sites.

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Table 4 Statistics of evacuation path planning information of disaster-affected villages.

Disaster site	Path length (km)	Evacuation time (min)	Flooding time
Path 1	Path 2	Path 3	Emergency shelter 1	Emergency shelter 2	Emergency shelter 3
Chengongjin village	0.935	1.379	2.259	4.09	8.71	12.84	6.35
Chentingjie village	0.67	0.977	1.865	3.71	5.54	10.35	10.91
Chenxizhen village	1.071	0.548	1.422	5.94	5.05	7.89	12.24
Wangjia village	1.217	0.456	1.36	6.76	2.53	7.55	17.44

Table 5 depicts the path length and minimum evacuation time from each disaster site to three emergency shelters. Combined with the simulation results of the dam break of the tailings pond, the safe evacuation path is selected based on the risk of dam break sand flow during the evacuation process. The table shows that there are multiple safe evacuation paths for some disaster sites. Therefore, the distance between the disaster point and the emergency shelter is further considered, and the emergency shelter with the closest distance is selected. A comparison between the total evacuation time (Table 5) and the sand flow inundation times of the villages (Table 3) reveals that the evacuation route planning method proposed in this study can safely evacuate affected individuals to emergency shelters in the shortest time, before the sand flow inundates the affected villages. Therefore, the tailings dam breach evacuation route planning system developed in this study can be effectively applied to the field of tailings dam breaches, providing a valuable reference for formulating evacuation route plans for downstream affected populations following a tailings dam breach.

Table 5 Final results of the evacuation path planning.

Order	Disaster site	Number of evacuees	Shelter	Evacuation distance (km)	Evacuation time (min)	Waiting time (min)	Total evacuation time (min)
1	Chengongjin village	200	1	0.935	4.09	0	4.09
2	Chentingjie village	368	1	0.67	3.71	4.21	7.92
3	Chenxizhen village	580	2	0.548	5.05	5.16	10.21
4	Wangjia village	335	3	0.791	2.53	7.23	9.76

4. Discussion

This study offers significant contributions to the development and application of an effective multiobjective evacuation path model for tailings dam failure scenarios. The proposed model not only addresses the complexities of emergency evacuation but also provides practical solutions to mitigate the associated risks.

4.1. Effectiveness of the Improved Dijkstra’s Algorithm

The improved Dijkstra’s algorithm demonstrated substantial improvements in computational efficiency, particularly in shortest-path calculation. By minimizing the number of traversed road nodes and arc segments, the algorithm achieved a 49.6% increase in search efficiency when compared to the traditional approach. This enhancement is particularly critical in emergency evacuation scenarios, where rapid decision-making and timely responses can significantly impact the outcome. In the case of tailings dam break evacuations, where time is a vital factor, the improved algorithm’s efficiency could potentially reduce evacuation times and, consequently, save lives. Furthermore, the algorithm’s compatibility with GIS road networks adds to its practicality for real-world applications, especially in disaster management. The circular search range limitation method employed in this study effectively narrows the search area, ensuring that the algorithm focuses on the most relevant routes, reducing redundant calculations and improving the overall performance of the evacuation model.

4.2. Significance of Dam Break Simulation Results

The simulation results of the Tongshankou tailings pond failure provided invaluable insights into the progression of the disaster. By accurately predicting the time, depth, and velocity of the sand-flow inundation in the affected villages, we were able to identify critical areas at risk. This predictive capability is crucial in assessing the severity of the disaster and planning evacuation routes accordingly. Moreover, the integration of these simulation results into the evacuation path model allows for more precise and dynamic evacuation planning, ensuring that vulnerable populations are prioritized for evacuation. The dam break simulation also contributed to a better understanding of the flood dynamics, enabling the identification of high-risk zones and refining the optimization process in evacuation path selection. Ultimately, this approach enhances both the efficiency and safety of emergency responses following a tailings dam failure.

4.3. Road Network Model and Evacuation Path Planning

The construction of the road network model using GIS technology and the integration of various data sources such as DEM data, remote sensing images, and population data laid a solid foundation for evacuation path planning. Loading the disaster areas and emergency shelters into the road network and calculating the shortest paths based on the improved Dijkstra’s algorithm enabled us to plan safe and efficient evacuation routes. The consideration of both the shortest evacuation time and the lowest risk in the path calculation is a key factor in ensuring the safety of evacuees. The results showed that by carefully selecting evacuation paths based on the risk of dam-break sand flow and the distance to emergency shelters, the proposed evacuation route planning method can evacuate affected individuals to emergency shelters in the shortest time before the sand flow inundates the villages.

4.4. Practical Implications and Future Directions

The developed tailings dam breach evacuation route planning system demonstrates substantial practical significance in disaster management applications. This methodology can be effectively implemented for emergency response planning in tailings dam failure scenarios, providing a scientific basis for developing optimized evacuation strategies for downstream populations. Nevertheless, its current implementation is bounded by several assumptions and data constraints. First, uncertainties in breach timing, hydrograph generation, and tailings rheology may shift inundation extents and node level arrival times, shrinking or enlarging safety margins and, especially in sparse networks, occasionally switching the optimal route. Second, terrain, roughness and road-network layers can contain resolution or completeness errors that propagate to travel-time estimates and route feasibility. Third, the routing framework does not yet explicitly account for congestion, shelter capacity and queuing, or heterogeneous evacuee behavior, which could alter effective travel times and route priorities during mass evacuations. Despite these uncertainties, the framework is model-agnostic and can be generalized beyond the case study wherever node-level inundation-arrival times and a GIS road network are available; practical transfer requires site-specific calibration of the hazard-to-risk mapping, verification of network completeness and key attributes, and simple sensitivity checks to confirm robust recommendations.

Future work will address these issues by incorporating multidimensional complexity factors, including: (1) traffic-flow dynamics and emergent congestion during mass evacuations, represented via agent-based and/or traffic-assignment coupling; (2) dynamic coupling between the real-time evolution of fluids and time-dependent routing, enabling on-the-fly rerouting under changing hazards; (3) assimilation of real-time observations (rainfall, cameras, UAV/remote sensing) to update states and routes; and (4) cross-site validation across diverse terrains, hydrological regimes, and infrastructure configurations, together with scalability tests on larger regional networks. Overall, this study represents a significant step toward operational evacuation planning for tailings-dam breach disasters, while highlighting clear pathways for strengthening robustness and generalizability.

5. Conclusions

This study investigates the key issues related to the evacuation route planning for downstream affected populations in the event of a tailings dam breach, and the following main conclusions have been drawn:

• 1.

  Evacuation path planning is a multiobjective and multiconstraint problem. We formulated a model minimizing evacuation time and risk under road-network and inundation constraints; in Tongshankou, dam-break simulation identified four villages at risk (peak depth ~ 4.5–7.5 m, velocity > 6 m/s).

• 2.

  To efficiently solve the multisource shortest-paths problem, we used Dijkstra’s algorithm augmented with a circular restricted search. Compared with the standard Dijkstra’s algorithm, this variant reduced the numbers of nodes and arcs traversed by 168 and 106, respectively; improved search efficiency by 49.6%; and shortened the runtime from 876.47 to 441.23 ms.

• 3.

  A dam-break evacuation modeling framework (ArcGIS Engine + Geodatabase) was implemented and evaluated via a case-study simulation of the Tongshankou pond. The total evacuation times (4.09, 7.92, 10.21, and 9.76 min) were lower than the corresponding inundation-arrival times (6.35, 10.91, 12.24, and 17.44 min), yielding safety margins of 2.26 min, 2.99 min, 2.03 min, and 7.68 min, respectively, and indicating timely evacuation under the modeled conditions.

Data Availability Statement

The data that support the findings of this study are available upon request from the corresponding author. The data are not publicly available due to privacy or ethical restrictions.

Disclosure

The authors certify that the submission of the manuscript is original work and is not under review at any other publication.

Conflicts of Interest

The authors declare no conflicts of interest.

Author Contributions

Meixian Qu: conceptualization, methodology, investigation, formal analysis, writing – original draft. Shu Zeng: methodology, investigation, date curation. Yongwei Zhang: methodology, investigation, date curation. Hongyan Zhou: conceptualization, methodology, investigation. Shaohua Hu: methodology, writing – review and editing, supervision, funding acquisition.

Funding

This study was supported by the financial supports from the National Natural Science Foundation of China (Grant 42271026).

Acknowledgments

This study was supported by the financial supports from the National Natural Science Foundation of China (No. 42271026).

References

1 Wang S., Mei G., Xie X., and Cui Y., Analysis on Influence of Tailing Pond Construction on Stability of Surrounding Loose Waste Dump, IOP Conference Series: Earth and Environmental Science, 2020, IOP Publishing, 012104.

2 Yu G., Yu Y., and Liu J., et al.An Integrated Management System of Man-Machine Communication for Tailings Pond Based on RSCIC, IOP Conference Series: Materials Science and Engineering, 2019, IOP Publishing, 012033.

3 Garcia F. F., Cotrim C. F. C., and Caramori S. S., et al.Mine Tailings Dams’ Failures: Serious Environmental Impacts, Remote Solutions, Environment, Development and Sustainability. (2025) 27, no. 8, 18179–18201, https://doi.org/10.1007/s10668-024-04628-z.

4 Hudson-Edwards K. A., Kemp D., and Torres-Cruz L. A., et al.Tailings Storage Facilities, Failures and Disaster Risk, Nature Reviews Earth & Environment. (2024) 5, no. 9, 612–630, https://doi.org/10.1038/s43017-024-00576-4.

5 Zare M., Nasategay F., Gomez J. A., Moayedi Far A., and Sattarvand J., A Review of Tailings Dam Safety Monitoring Guidelines and Systems, Minerals. (2024) 14, no. 6, https://doi.org/10.3390/min14060551, 551.

6 Yu H., Zahidi I., Fai C. M., Liang D., and Madsen D. Ø., Tailings Dam Failures: A Critical Evaluation of Current Policies and Practices, Results in Engineering. (2025) 25, https://doi.org/10.1016/j.rineng.2025.103973, 103973.

7 Dixon-Hardy D. W. and Engels J. M., Guidelines and Recommendations for the Safe Operation of Tailings Management Facilities, Environmental Engineering Science. (2007) 24, no. 5, 625–637, https://doi.org/10.1089/ees.2006.0133, 2-s2.0-34248592280.

8 Zhu Y., Li H., and Wang Z., et al.Optimal Evacuation Route Planning of Urban Personnel at Different Risk Levels of Flood Disasters Based on the Improved 3D Dijkstra’s Algorithm, Sustainability. (2022) 14, no. 16, https://doi.org/10.3390/su141610250, 10250.

9 Gai W.-M., Du Y., and Deng Y.-F., Evacuation Risk Assessment of Regional Evacuation for Major Accidents and Its Application in Emergency Planning: A Case Study, Safety Science. (2018) 106, 203–218, https://doi.org/10.1016/j.ssci.2018.03.021, 2-s2.0-85044769293.

10 Rozo K. R., Arellana J., Santander-Mercado A., and Jubiz-Diaz M., Modelling Building Emergency Evacuation Plans Considering the Dynamic Behaviour of Pedestrians Using Agent-Based Simulation, Safety Science. (2019) 113, 276–284, https://doi.org/10.1016/j.ssci.2018.11.028, 2-s2.0-85058033566.

11 Tamakloe R., Hong J., Tak J., and Park D., Finding Evacuation Routes Using Traffic and Network Structure Information, Transportation Research Part D: Transport and Environment. (2021) 95, https://doi.org/10.1016/j.trd.2021.102853, 102853.

12 Yang Q., Zhang X., Zhang Z., He L., Yan X., and Na J., Fire Scenario Zone Construction and Personnel Evacuation Planning Based on a Building Information Model and Geographical Information System, ISPRS International Journal of Geo-Information. (2022) 11, no. 2, https://doi.org/10.3390/ijgi11020110, 110.

13 Haris M., Mahmood I., Badar M., and Alvi M. S. Q., Modeling Safest and Optimal Emergency Evacuation Plan for Large-Scale Pedestrians Environments, Winter Simulation Conference (WSC), 2018, IEEE, 917–928.

14 Zhu Y., Zhou Z., Zhou J., Xu X., Wu X., and Nie W., An Improved 3D Dijkstra Algorithm of Evacuation Route Considering Tailings Dam Failure, Natural Hazards. (2024) 121, 1–23.

15 Li Q., Geng J., Song D., Nie W., Saffari P., and Liu J., Automatic Recognition of Erosion Area on the Slope of Tailings Dam Using Region Growing Segmentation Algorithm, Arabian Journal of Geosciences. (2022) 15, no. 5, https://doi.org/10.1007/s12517-022-09746-4, 438.

16 Li Z. and Li Q., Balancing Submarine Landslides and the Marine Economy for Sustainable Development: A Review and Future Prospects, Sustainability. (2024) 16, no. 15, https://doi.org/10.3390/su16156490, 6490.

17 Dong L., Deng S., and Wang F., Some Developments and New Insights for Environmental Sustainability and Disaster Control of Tailings Dam, Journal of Cleaner Production. (2020) 269, https://doi.org/10.1016/j.jclepro.2020.122270, 122270.

18 Lovreglio R., Ronchi E., Maragkos G., Beji T., and Merci B., A Dynamic Approach for the Impact of a Toxic Gas Dispersion Hazard Considering Human Behaviour and Dispersion Modelling, Journal of Hazardous Materials. (2016) 318, 758–771, https://doi.org/10.1016/j.jhazmat.2016.06.015, 2-s2.0-84989298917.

19 Ye R., Xu S., Liu J., Wang J., Wang Y., and Jiang J., Investigation and Simulation of Non-Adaptive Psychology of Crowd Evacuation Under Toxic Gas Leakage Accident in Chemical Park, Journal of Loss Prevention in the Process Industries. (2023) 85, https://doi.org/10.1016/j.jlp.2023.105160, 105160.

20 Yan Z., Wang Y., and Chao L., Simulation Study on Fire and Evacuation of Super High-Rise Commercial Building, Case Studies in Thermal Engineering. (2023) 52, https://doi.org/10.1016/j.csite.2023.103519, 103519.

21 Li X., Wang C., Kassem M. A., and Ali K. N., Emergency Evacuation of Urban Underground Commercial Street Based on BIM Approach, Ain Shams Engineering Journal. (2024) 15, no. 4, https://doi.org/10.1016/j.asej.2024.102633, 102633.

22 Zhang H., Zhao Q., Cheng Z., Liu L., and Su Y., Dynamic Path Optimization With Real-Time Information for Emergency Evacuation, Mathematical Problems in Engineering. (2021) 2021, no. 1, https://doi.org/10.1155/2021/3017607, 3017607.

23 Chen G., Liang X., Zhou L., and Men J., A Dynamic Risk-Based Routing Approach for Multi-Source and Multi-Sink Evacuation Problem in Chemical Industrial Parks, Journal of Loss Prevention in the Process Industries. (2022) 77, https://doi.org/10.1016/j.jlp.2022.104760, 104760.

24 Yang X., Zhang R., Li Y., and Pan F., Passenger Evacuation Path Planning in Subway Station Under Multiple Fires Based on Multiobjective Robust Optimization, IEEE Transactions on Intelligent Transportation Systems. (2022) 23, no. 11, 21915–21931.

25 Li X., Li Q., Wang Y., Liu W., Hou D., and Zhu C., Effect of Slope Angle on Fractured Rock Masses Under Combined Influence of Variable Rainfall Infiltration and Excavation Unloading, Journal of Rock Mechanics and Geotechnical Engineering. (2024) 16, no. 10, 4154–4176, https://doi.org/10.1016/j.jrmge.2024.08.019.

26 Li Q., Wang Y., Li X., Tang S., Gong B., and Jiang S., Evolutionary Characteristics of the Fracture Network in Rock Slopes Under the Combined Influence of Rainfall and Excavation, Bulletin of Engineering Geology and the Environment. (2025) 84, no. 1, https://doi.org/10.1007/s10064-025-04084-7, 84.

27 Casareale C., Bernardini G., Bartolucci A., Marincioni F., and D’Orazio M., Cruise Ships Like Buildings: Wayfinding Solutions to Improve Emergency Evacuation, Building Simulation, 2017, Springer, 989–1003.

28 Khakzad N., A Methodology Based on Dijkstra’s Algorithm and Mathematical Programming for Optimal Evacuation in Process Plants in the Event of Major Tank Fires, Reliability Engineering & System Safety. (2023) 236, https://doi.org/10.1016/j.ress.2023.109291, 109291.

29 Yang J., Cai B., Li X., and Ge R., Optimal Path Planning for Electric Vehicle Travel Time Based on Dijkstra, 35th Chinese Control and Decision Conference (CCDC), 2023, IEEE, 721–726.

30 Rizzo C., Maranzoni A., and D’Oria M., Probabilistic Mapping and Sensitivity Assessment of Dam-Break Flood Hazard, Hydrological Sciences Journal. (2023) 68, no. 5, 700–718, https://doi.org/10.1080/02626667.2023.2174026.

31 He G., Chai J., Qin Y., Xu Z., and Li S., Evaluation of Dam Break Social Impact Assessments Based on an Improved Variable Fuzzy Set Model, Water. (2020) 12, no. 4, https://doi.org/10.3390/w12040970, 970.

32 You L., Li C., Min X., and Xiaolei T., Review of Dam-Break Research of Earth-Rock Dam Combining With Dam Safety Management, Procedia Engineering. (2012) 28, 382–388, https://doi.org/10.1016/j.proeng.2012.01.737, 2-s2.0-84863176761.

33 Flow Science, Inc, FLOW-3D User Manual, Version 2021.0, 2021, Flow Science, Inc, Santa Fe, NM, USA..

34 Chácara D. and Filho W. O., Rheology of Mine Tailings Deposits for Dam Break Analyses, REM - International Engineering Journal. (2021) 74, no. 2, 235–243, https://doi.org/10.1590/0370-44672020740098.

35 Yang J., Su W., and Li B., Calculation of Natural Alluvial Separation of Sandy Tailings Slurry Based on FLOW-3D, Mechanics in Engineering. (2023) 45, no. 3, 559–564.

36 Qiu D., Xu J., and Lin H., Numerical Analysis of the Overtopping Failure of the Tailings Dam Model Based on Inception Similarity Optimization, Applied Sciences. (2024) 14, no. 3, https://doi.org/10.3390/app14030990, 990.

37 Ghahramani N., Improved Empirical and Numerical Predictive Modelling of Potential Tailings Dam Breaches and Their Downstream Impacts, 2023, University of British Columbia.

38 Bayram V. and Yaman H., A Stochastic Programming Approach for Shelter Location and Evacuation Planning, RAIRO - Operations Research. (2018) 52, no. 3, 779–805, https://doi.org/10.1051/ro/2017046, 2-s2.0-85047118854.