Research on the Dynamic Model of Emergency Rescue Resource-Allocation Systems for Mine-Fire Accidents, Taking Liquid CO₂ Transportation as an Example

Abstract

After a mine-fire accident occurs, a large number of emergency resources need to be allocated to rescue those involved in the mine-fire accident. The allocation of emergency resources for mine-fire accidents has the characteristic of being a complex system with strong uncertainty. To investigate the impact of various variables on the allocation of emergency resources in mine-fire situations, this paper analyzes the relevant factors that influence the process of allocating emergency resources during mine fires. It defines the variables of the mine-fire emergency resource-allocation system based on relevant assumptions. Causal loop and stock flow diagrams are drawn to illustrate the relationships between the variables and the system dynamics equation. Finally, a system dynamics model for mine-fire emergency resource allocation is established. The Vensim software was used to simulate the model of a mine-fire emergency rescue. The simulation produced curves for the evolution rate of the fire, the arrival rate, the demand for emergency resources, in-transit resources, arrival, and the usage of resources during the emergency. The results indicate a positive correlation between the quantity of emergency resources in-transit and the arrival rate of emergency resources: they are positively correlated with the amount of emergency-management investment. Additionally, the duration of the maximum quantity of emergency resources in-transit is positively correlated with the length of the emergency resource-allocation route. On the other hand, the evolution rate of the mine fire and the arrival rate of its emergency resources are negatively correlated with the level of emergency management. The evolution rate of the mine fire becomes larger and the damage caused by the mine-fire accident is greater when the decision-making ability of commanders is at a low level.

1. Introduction

Mine fires are a serious global issue [1] and pose a significant threat to the safe production of coal mines [2]. They are characterized by a sudden onset, the rapid development of the fire, and a difficulty in fire extinguishing and rescuing. When a mine fire burns, it produces a large number of toxic and harmful gases and heat, which forms fire wind pressure; then the wind flow is affected by fire power, which easily causes the disorder of its state and chaos for the whole ventilation system. In serious cases, it may cause gas and coal-dust explosions, resulting in a vicious accidents of mass casualties and injuries, which will cause irreparable losses and harm to the mine [3]. Therefore, when a fire occurs in a coal mine, effective rescue measures must be taken in the shortest time to prevent the spread of fire and minimize economic losses and casualties effectively caused by fire accidents. In the process of the emergency rescue of those involved in mine-fire accidents, a large number of emergency resources are needed, such as emergency rescue experts, medical rescue forces, emergency materials, etc. The supply of sufficient emergency resources is crucial for the emergency rescue of those involved in mine-fire accidents. Additionally, the timely and effective allocation of emergency resources is one of the key factors determining the success or failure of the rescue operation. Once a mine-fire accident has happened, emergency resources must be transported to the accident mine in the shortest time, otherwise it will lead to the further spread of the fire, and even cause secondary accidents such as gas explosions and coal-dust explosions, resulting in greater casualties and property losses. Therefore, it is of great significance to improve the allocation efficiency and arrival rate of emergency resources for mine-fire rescue after the occurrence of mine-fire accidents.

As a variety of fire accidents and secondary accidents are caused by uncertainty, different mine-fire accidents require specific rescue measures and emergency rescue resources. Therefore, the allocation of mine-fire emergency resources is a complex system with strong uncertainty. In addition, the demand quantity of emergency resources is a time-varying complex system that is multi-objective, non-linear, and multi-feedback [4]. Conventional mathematical methods are insufficient to solve this problem. Instead, the system dynamics theory searches for the root cause of problems by analyzing the internal structure of the system and its feedback characteristics. This approach considers the internal components of the system that are mutually causal [5]. System dynamics have a strong advantage in solving the simulation of complex systems, and a large number of complex scientific problems in various fields have been solved by many scholars with system dynamics. Songtang He et al. [6] studied sustainable development during the utilization of debris flow on shoal land using the theory of system dynamics. TY Loh et al. [7] developed a model for the loss-risk variable of underwater vehicles from a system dynamics perspective, proposing risk-control decisions. KT Gebre et al. [8] evaluated the increase in feed supply and different fattening strategies for the Ethiopian sheep population using a system dynamics model. Liu XR et al. [9] proposed a micro-dynamic feedback model for predicting demand based on the system dynamics theory, taking into account the differences in urban evacuation demand during an earthquake, providing a new method for evacuation-demand prediction.

System dynamics has also been widely applied and practiced in accident and emergency rescue in recent years. Li X et al. [10] investigated the emergency-evacuation behavior of large crowds in urban underground complexes during fires using system dynamics, and a simplified model of a large crowd evacuation in the urban underground complexes was established. Rafael Diaz et al. [11] studied the material supply levels for post-disaster reconstruction in the affected areas by creating a system dynamics model to provide decision support for post-disaster reconstruction. Xu S et al. [12] established a system dynamics model using Anylogic software to simulate the spread of crowd panic in a chemical industry park during a disaster. The model considers the evolution of people’s emotions and the emergency disaster environment of the chemical industry park. The authors conducted simulation experiments for different disaster severities, visibilities, and group sizes. Li Jian et al. [13] studied the factors that influence the speed of emergency-material allocation and transportation using system dynamics. They developed a general system dynamics model for emergency material allocation and transportation, which primarily analyzes the factors affecting the speed of emergency material allocation from the government’s perspective. The study also examines the impact of the government’s emergency-response capacity for the allocation of emergency materials.

In the field of coal-mine safety management and emergency rescue, some scholars have conducted research using the system dynamics theory. M You et al. [14] simulated the evolutionary game among stakeholders in the internal safety inspection system of coal enterprises. They investigated the influence of different reward and punishment strategies on the game process and equilibrium state under different scenarios. Cooke and David L. et al. [15] had studied the system causality of the Westley coal mine and the causality among various factors leading to a mine gas explosion using system dynamics. Yan Junai et al. [16]. had studied the causal relationship between emergency rescue capability and various influencing factors using system dynamics. However, the model analyzed the influencing factors of emergency-response capability from a macro perspective and did not analyze or discuss the allocation of emergency materials after a coal mine accident. He Gang [17,18] analyzed the impact of coal-mine safety-management-level factors on safety-management levels using a system dynamics method. The model identified the actual contribution rate of each variable in the model to the safety-management level.

The above research results are of great significance for enhancing the application of system dynamics in coal-mine safety management. They can be used for emergency-rescue capability analysis, emergency resource allocation, and improving coal-mine safety management and emergency management. However, the existing research results of system dynamics mainly focuses on general emergencies, and there is no relevant research on the emergency resource allocation of coal-mine accidents, especially for the allocation of emergency resources in the process of emergency rescues for mine-fire accidents. In addition, the current study does not consider the management factors when studying the allocation of emergency resources. However, it is important to note that the management factors, as well as the command- and decision-making abilities of individuals, have a significant impact on emergency rescue operations. This paper utilizes the system dynamics theory to investigate the allocation and transportation of emergency resources during a mine-fire emergency rescue. The analysis of key factors affecting the allocation and transportation of emergency resources is presented. A system dynamics model for the allocation of emergency resources in mine-fire situations is established, followed by a dynamic simulation analysis. The results provide insight into the influence law of various factors on the allocation and transportation of emergency resources during mine-fire accidents. This study offers a new theoretical basis for improving the allocation and transportation efficiency of emergency resources in mine-fire accidents.

2. Influencing Factors of Emergency Resources’ Allocation

The main influencing factors in the allocation process of mine-fire emergency resources can be divided into management factors, personnel factors, equipment factors, and environmental factors. These factors influence the allocation of emergency resources for mine-fire accidents. Among them, management factors mainly refer to the level of emergency management, including the emergency-management investment, the investment–transportation conversion rate, and the effect of emergency rescue, etc. Personnel factors mainly include the emergency decision-making ability of commanders and the delay time of emergency response, etc. Equipment factors primarily include transportation time, capacity, and average speed. Environmental factors mainly include factors such as the mine-disaster-bearing capacity and the route length of emergency resources allocation, etc.

In summary, the factors affecting the allocation of emergency resources for mine-fire accidents can be summarized as follows: the rate of CO₂ production, the evolution rate of mine-fire accidents, the amount of evolutionary-level change, the mine-disaster-bearing capacity, the effect of the emergency rescue, the emergency decision-making ability of commanders, the demand quantity of emergency resources, the level of emergency management, the emergency-management investment, the investment–transportation conversion rate, the capacity of transportation, the quantity of emergency resources in-transit, the arrival quantity of emergency resources, the rate of arrival, the inventory quantity of emergency resources, the quantity of the initial inventory, the used quantity, the average speed of transportation, the route length of emergency resources; allocation, the time of transportation, the usage time, and the inventory quantity of emergency resources. The definitions of each influencing factor are given in

Table 1. Definition of influencing factors.

No.	Influencing Factors	Definition
1	the rate of CO2 production	The variable describing the change process of fire spread in mine-fire accidents, which are mainly determined by the burning rate of combustibles.
2	the evolution rate of mine-fire accidents	A variable describing the rate of change of fire spread in a mine-fire accident.
3	the amount of evolutionary-level change	The evolution level of the mine-fire accident is influenced by changes in certain influencing factors, resulting in fluctuations in its magnitude. Ultimately, these variations affect the rate of evolution for the mine-fire accident.
4	the mine-disaster-bearing capacity	The disaster resistance of a mine is directly related to its infrastructure, degree of automation, and safety-management system. A mine with a complete infrastructure, high degree of automation, and sound safety-management system has a stronger the disaster-bearing capacity and is less prone to accidents.
5	the effect of emergency rescue	The effectiveness of a series of rescue actions taken to extinguish a mine-fire accident is directly related to the commander’s decision-making ability.
6	the emergency decision-making ability of commanders	In emergency rescue operations during mine fires, the decision-making ability of the commander-in-chief is crucially dependent on their professional knowledge, rescue experience, personal characteristics, etc.
7	the demand quantity of emergency resources	The total amount of emergency resources required for the entire mine-fire emergency rescue process.
8	the level of emergency management	The emergency-management capability of coal-mining enterprises is linked to their training in emergency management, conducting emergency drills, etc.
9	the emergency-management investment	The investment scale of coal-mining enterprises in emergency management includes two aspects: not meeting the investment requirements and meeting the investment requirements for emergency management.
10	the investment–transportation conversion rate	The efficiency of converting funds into transportation capacity.
11	the capacity of transportation	After the occurrence of a mine-fire accident, the amount of rescue resources that can be deployed by the emergency rescue command center per unit of time and the size of the transport capacity are related to the size of the investment in emergency management.
12	the quantity of emergency resources in-transit	The total quantity of emergency resources being transported.
13	the quantity of emergency resources arriving	The quantity of emergency resources transported from the emergency resource store to the accident mine.
14	the rate of arrival	The arrival rate of emergency resources from the emergency resource pool to the coal mine, and the ratio of the arrival amount of emergency resources to the transportation time.
15	the inventory quantity of emergency resources	The inventory of each emergency resource depot at a specific point in time.
16	the quantity of initial inventory	The initial state of each emergency resources depot’s inventory level.
17	the used quantity of emergency resources	The amount of emergency resources that have been consumed in the process of emergency rescue for mine-fire accidents.
18	the average speed of transportation	The average speed at which emergency resources are deployed from the initial moment to the mine where the accident occurred.
19	the route length of emergency resources allocation	The transportation route of emergency resources and the transportation time required for their allocation routes vary, which will affect the emergency rescue of mine-fire accidents.
20	the time of transportation	The time elapsed between the initial allocation of emergency resources and the transportation of the final batch to the mine.
21	the usage time	The time it takes to transport emergency resources to the coal mine and place them in the emergency rescue station.
22	the delay time of emergency response	After the occurrence of a mine-fire accident, the total time required for the entire process of receiving an alarm, giving emergency notice, initiating emergency rescue, establishing an emergency rescue headquarters, and making decisions on emergency resource allocation.

:

3. System Dynamics Simulation

3.1. Construction of the Model

3.1.1. Hypothesis and Explanation of the Model

The study focuses on the allocation of emergency resources for mine-fire accidents, taking into account the evolution rate of the fire, the demand quantity for emergency resources, and their transportation. Two state variables, arrival rate and demand quantity for emergency resources, are selected as they are strongly associated with the main link. Based on these two state variables, other variables are classified and arranged to select the most closely related auxiliary variables and constants to the demand quantity and arrival rate of emergency resources. Variables without a functional relationship, whether independent or dependent, are ignored to obtain the best model boundary.

The emergency resources studied in this paper mainly to refer the emergency supplies that are in high demand and easily consumed during mine-fire emergency operations. Other emergency resources that are in low demand and difficult to consume are not within the scope of this study. A mine-fire accident is typically a sudden occurrence in a specific area of a mine, and does not usually have a large-scale impact outside of the mine. In the event of an accident, the required emergency resources may not be extensive. This paper examines the allocation of emergency resources in the case of a fire accident, assuming that the emergency resource depot inventory is sufficient to meet the needs of the entire rescue operation.

3.1.2. Definition of Variables

In system dynamics, variables can be classified into four categories: constants, state variables, auxiliary variables, and rate variables. The same classification can be applied to variables in the emergency resource-allocation system for mine-fire accidents, as shown in

Table 2. Variables of the model.

No.	The Name of the Variable	Abbreviation for Variable	Types of Variables
1	the rate of CO2 production	VCO2	auxiliary variable
2	the evolution rate of mine-fire accidents	Vfire	auxiliary variable
3	the amount of evolutionary-level change	Δevo	auxiliary variable
4	the mine-disaster-bearing capacity	Cbear	constant
5	the effect of emergency rescue	Erescue	auxiliary variable
6	the emergency decision-making ability of commanders	Adecision	constant
7	the demand quantity of emergency resources	Qdemand	state variable
8	the level of emergency management	Lmanage	constant
9	the emergency-management investment	Invest	constant
10	the investment–transportation conversion rate	Ri-t	constant
11	the capacity of transportation	Ctrans	auxiliary variable
12	the quantity of emergency resources in-transit	Qtrans	auxiliary variable
13	the quantity of emergency resources arriving	Qarriving	auxiliary variable
14	the rate of arrival	Rarrival	rate variable
15	the inventory quantity of emergency resources	Qinventory	state variable
16	the quantity of initial inventory	Qinitial	constant
17	the used quantity of emergency resources	Qused	auxiliary variable
18	the average speed of transportation	Vtrans	constant
19	the route length of emergency resources’ allocation	Lroute	constant
20	the time of transportation	Ttrans	auxiliary variable
21	the usage time	Tusage	constant
22	the delay time of the emergency response	Tdelay	constant

.

3.1.3. Causal Loop Diagram

A causality loop diagram is utilized to illustrate the interconnection and interaction among various elements in the system, including definite causal relationships and feedback loops, and to effectively demonstrate the system’s structure. In the event of a mine-fire accident, the main department responsible for the accident will immediately establish an accident-emergency headquarters and various emergency rescue teams. During the emergency rescue-organization process, the general commander needs to predict the types and quantities of emergency resources required based on the scale of the mine-fire accident and the direction of the fire’s evolution. They should then use appropriate transportation methods to transfer the emergency resources from the depot to the coal mine incident site by allocating routes reasonably. When allocating emergency resources, various factors can affect the allocation rate. To analyze these factors more deeply, they are transformed into variables or constants. A causality loop diagram of the factors affecting emergency resources’ allocation for mine-fire accidents is then established, as shown in Figure 1. The diagram illustrates the relationship between variables, with each arrow indicating the direction of the causal effect. Solid arrows indicate a definite causal relationship between the two variables, which can be expressed on both sides of the equation. Dashed arrows represent a connection between the two variables, but with no clear positive or negative effect. Therefore, only the solid arrows are marked with “+” and “-” symbols, while the dashed arrows are not. The “+” symbol represents that the tail variable has a promoting effect on the variable to which it points, while the “-” symbol represents a inhibiting effect. Using the above symbols like this, the causal relationship between all the variables is marked out, and the causal loop diagram is formed.

3.2. Stock and Flow Diagram and Equations

3.2.1. Stock and Flow Diagram

The stock and flow diagram is a graphical representation of the feedback form and control law of the system. It distinguishes the nature of variables on the based on causal loop diagrams and describes the logical relationship among morphological elements with more intuitive symbols. The relationship between the stock and flow of the system can be represented by the vivid symbols. The stock and flow diagram provides a clear overview of the material flow, information flow, and feedback function of the system. It can provide the system analysis with a blueprint for establishing the system dynamics equation and the basis for further data collection, system analysis, and strategic conception. This paper presents a stock and flow diagram for emergency resources’ allocation in the event of a mine-fire accident. The diagram was created using Vensim PLE software (Versin 9.3.5 x64) and is based on the causality loop diagram and properties of variables in the emergency resource-allocation system for mine-fire accidents, as shown in Figure 2.

3.2.2. System Dynamics Equations

The system dynamics equation with a quantified response is a special computer language that represents the quantitative relationship between interaction variables when a feedback loop is constructed using system dynamics, based on the local relationship between system elements. The resulting system image changes over time after computer simulation. The system dynamics’ equations are established based on the relationship between variables and references [13,16], as follows:

$$E\mathrm{re}scue=\left\{\begin{array}{c}{k}_1\times A_{decision}(R_{arrival}\le c)\\ k_2\times A_{decision}(R_{arrival}>c)\end{array}\right.$$

(1)

The decision ability coefficient (unit: dimensionless) is represented by constants k₁ and k₂, while the arrival rate coefficient (unit: ton/h) is represented by constant c.

$${\Delta }_{evo}=(l_1\times C_{bear}+l_2\times L_{manage}+l_3\times E_{rescue})/3$$

(2)

where l₁ is the coefficient of mine-disaster-bearing ability, l₂ is the coefficient of emergency-management level, and l₃ is the coefficient of emergency rescue effect, and the three units are dimensionless.

$$V_{fire}=V_{CO2}/{\Delta }_{evo}$$

(3)

$$Q_{demand}={\displaystyle {\int }_0^h{V}_{fire}dx}$$

(4)

$$C_{trans}=Invest\times R_{i-t}$$

(5)

$$Q_{trans}=\left\{\begin{array}{c}0 (T\le T_{delay})\hfill \\ Q_{demand}-Q_{arriving} (T>T_{delay}\&Q_{demand}-Q_{arriving}<C_{trans})\hfill \\ C_{trans} (T>T_{delay}\&Q_{demand}-Q_{arriving}\ge C_{trans})\hfill \end{array}\right.$$

(6)

$$Q_{arriving}=\left\{\begin{array}{cc}0\hfill & (T\le T_{delay})\hfill \\ {delay}_1(Q_{demand},T_{trans})& (T>T_{delay})\end{array}\right.$$

(7)

$$Q_{demand}=Q_{initial}-Q_{arrving}-Q_{used}$$

(8)

$$Q_{used}=\left\{\begin{array}{cc}0& (T\le T_{delay}+T_{trans})\\ {delay}_1(Q_{arriving},T_{usage})& (T>T_{delay}+T_{trans})\end{array}\right.$$

(9)

$$T_{trans}=L_{route}/V_{trans}$$

(10)

$$R_{arrival}=\left\{\begin{array}{cc}0& (Q_{arriving}\le 0)\\ Q_{arriving}/Time& (Q_{arriving}>0)\end{array}\right.$$

(11)

3.3. Simulation and Numerical Simulation

3.3.1. Case Background Information

A mine in the Yangquan Coal Industry Group is a high-gas-outburst mine, and the coal dust created during the mining operation possesses explosive qualities. The 15 # coal seam has a high tendency for spontaneous combustion, meaning it can easily oxidize at room temperature and ignite without an external ignition source. The natural ignition period is a key indicator of the risk of fire in a mine. It is the time elapsed from when the coal seam first comes into contact with air until it ignites naturally. The seam has a natural ignition period of only 33 days, indicating a high risk of spontaneous combustion. The production conditions at the mine are complex and threatened by various mining disasters. On 26 December 2007, the coal-mining team was organizing coal-cutting work at the 15,102 working face of the mine. At 15:17, due to the coal-mining machine colliding with pyrite nodules during the coal cutting, the collision produced sparks and the loose coal was ignited, resulting in an open fire on the upper part of the front beam of the 12th hydraulic support of the working face, as shown in Figure 3; then, the fire developed and spread rapidly, and the fire could not be controlled after on-site rescue, so the coal-mining face was closed.

At this point, extinguishing the fire in the working face is the top priority in order to expedite the continuation of mining operations. After the research and decision by the mine-fire emergency rescue headquarters, the ground drilling is used to locate the fire source directly in the working face, and liquid CO₂ is injected into the working face through the ground drilling hole to extinguish the fire. Liquid CO₂ is an excellent cooling agent. The temperature of liquid CO₂ is approximately −56.57 °C. When the temperature of liquid CO₂ is converted into the gaseous state and reaches 20 °C, the cooling capacity is 411.94 KJ/kg. It has the advantages of a fast gasification speed and a large cooling capacity, which can effectively improve the cooling efficiency, and has a beneficial effect on extinguishing the fire source in the closed fire zone.

The allocation process of emergency resources for mine-fire accidents is studied by establishing the allocation model of emergency resources for mine-fire accidents based on the case of the fire rescue accident in the mine. The liquid CO₂ is transported by six tank trucks, with an average speed of 60 km per hour. Additionally, the distance from the coal mine to the liquid CO₂ storage site is 480 km.

3.3.2. Setting of Parameters

The basic parameters of the system dynamics model of mine-fire emergency resources; allocation in the simulation process are obtained through collecting and organizing key information on safety-production materials, emergency-management information, emergency investment, and the relevant research conclusions in references [17,18,19]. The specific parameters of the system dynamics model are set in

Table 3. Model parameter setting.

No.	Name of Parameter	Parameter Values	Unit
1	the investment of emergency management (Invest)	50	ten thousand yuan
2	the mine-disaster-bearing capacity (Cbear)	1	dimensionless
3	the emergency decision-making ability of commanders (Adecision)	1.1	dimensionless
4	the level of emergency management (Lmanage)	0.9	dimensionless
5	the investment–transportation conversion rate (Ri-t)	2	ton/ten thousand yuan
6	the quantity of initial inventory (Qinitial)	3000	ton
7	the average speed of transportation (Vtrans)	60	km/hour
8	the route length of emergency resources’ allocation (Lroute)	480	kilometer
9	the usage time (Tusage)	5	hour
10	the delay time of emergency response (Tdelay)	3	hour
11	the coefficient of decision-making ability (k1)	0.5	dimensionless
12	the coefficient of decision-making ability (k2)	1.5	dimensionless
13	the coefficient of arrival rate (c)	20	ton/hour
14	the coefficient of mine-disaster-bearing capacity (l1)	0.7	dimensionless
15	the coefficient of emergency-management level (l2)	0.5	dimensionless
16	the coefficient of emergency-rescue effect (l3)	0.8	dimensionless
17	simulation time (final time)	100	hour
18	simulation step size (time step)	1	hour

.

3.3.3. Fire-Source Combustion Experiment

In the mine-fire accidents emergency resource-allocation model, the mine-fire evolution is a crucial factor determining the emergency resources requirements and directly affects the accuracy of the simulation results. It is important to ensure consistency by comparing the simulated mine-fire evolution rate with the actual rate. To determine the actual mine-fire evolution rate, similar simulation experiments were conducted in the laboratory to study this evolution given the high cost and difficulty in controlling real mine-fire experiments. The experimental device used in the experimental process is described in reference [20], as shown in Figure 4.

The coal samples were collected from the 15 # coal seam in the mine. As the coal quality of the coal seam is anthracite, the combustion process mainly produces CO₂, which is easy to measure and serves as a better indicator of the combustion state of coal samples. The concentration of CO₂ can be used to analyze the evolution rate of mine fires. Based of the findings in reference [21], the evolution rate of mine fires can be determined by the generation rate of CO₂. Therefore, the generation rate of CO₂ during coal-sample combustion in the experiment was input into the model to obtain the evolution rate of the mine fire in this paper. The steps for measuring the CO₂ change curve in the coal-combustion process with experimental equipment are shown in reference [20].

The CO₂-emission curve was obtained from the laboratory mine-fire experiment during the combustion of the coal sample, as shown in Figure 5. As illustrated in Figure 5, the concentration of CO₂ generated during the combustion process of the coal sample rapidly increases with time, reaches a maximum value, and then gradually decreases before finally returning to the concentration of CO₂ in the air.

The experiment’s CO₂ generation rate curve was input into Vensim PLE simulation software (Versin 9.3.5 x64). The model’s parameters of were set according to the initial values of the relevant parameters in Table 3 to simulate the model of emergency resource allocation for mine-fire accidents. The resulting evolution curve of the mine fire is shown in Figure 6. Figure 6 shows that the evolution curve of the mine fire increases rapidly with time, reaching its maximum value at the eighth hour. After that, it gradually decreases until the evolution rate decreases to zero after 20 h. Thus, the simulation results for the evolution rate of mine fires in the emergency resource-allocation model for mine-fire accidents are accurate. This model can better simulate the allocation process of emergency resources for mine-fire accidents, making it applicable for simulating and predicting the allocation process of emergency resources for mine-fire accidents.

3.3.4. Model Checking

• Dimensional Consistency Test

The purpose of the dimensional consistency test is to determine whether the dimension of each variable is reasonable and conforms to the actual situation, and each variable must have the correct dimension. It is also essential to check whether the calculation results of the dimensions of variables at the left and right ends of each dynamic simulation equation are consistent. All variables in the model must have a reasonable meaning in the actual system [22]. The dimensions of all variables in the model and the dimensions of the left and right sides of each equation have been checked. The results indicate that the dimensions of all variables and equations in the model meet the requirement of consistency.

2.

Sensitivity Test

The sensitivity test of the system dynamics model involves analyzing the impact of the parameters on the output by modifying key parameters in the model and determining the degree of influence of each parameter on the model [18].

(1) The influence law of different influencing factors on the quantity of resources in transit was analyzed in this paper: the initial value of emergency-management investments are adjusted to CNY 25,000 and CNY 75,000, respectively, and the initial value of the route lengths of emergency resources’ allocation are adjusted to 240 km and 720 km, respectively, under the condition of the effective rescue operation, as shown in Figure 6 and Figure 7.

The curve of the quantity of liquid CO₂ in transit can be divided into five stages, as shown in Figure 7 and Figure 8. In the first stage, the quantity of liquid CO₂ in transit is zero, due to the delay time of the emergency response between the occurrence of the mine-fire accident and the start of transportation. During the second stage, the quantity of liquid CO₂ in transit increases linearly over time, but stops increasing after reaching the maximum value. This is because the demand for liquid CO₂ during this stage is not large enough to reach the maximum transport capacity of the tankers. If the demand for liquid CO₂ continues to increase and the transport capacity cannot meet the demand, the quantity of liquid CO₂ in transit will no longer increase. In the third stage, the quantity of liquid CO₂ in transit remains constant after reaching its maximum. In the fourth stage, the quantity of liquid CO₂ in transit decreases linearly over time as demand for liquid CO₂ decreases and the maximum transport capacity of the tankers can meet the demand. Finally, in the fifth stage, the quantity of the liquid CO₂ in-transit is zero as demand for liquid CO₂ is not increasing and there are no instantaneous demands for it.

Figure 7 shows that increasing emergency-management investment by 0.5 times the initial value (CNY 750 thousand) leads to a significant increase in transportation capacity, which is maintained for a shorter time. On the other hand, reducing emergency-management investment by 0.5 times the initial value (CNY 250 thousand) results in a reduction in maximum transportation capacity, which is maintained for a longer time. The time required to reach the fourth stage is also longer in this case. The maximum transportation capacity is reduced and maintained for the longest time, and the time required to reach the fourth stage is longer when this this is the case. It is evident that the quantity of emergency resources in transit increases with higher investment in emergency management. There is a positive correlation between the quantity of emergency resources in transit, transportation capacity, and the investment of emergency management.

As can be seen from Figure 8, the changes in emergency resource-allocation routes have almost no influence on the maximum transportation capacity. However, the time required for the quantity of emergency resources in transit to enter the fourth stage also changes with the change in emergency resource-allocation route. The transportation capacity’s maximum value is maintained for a longer time when the length of the emergency resource-allocation route is increased by 0.5 times (720 km), resulting in a longer time to reach the fourth stage. Conversely, the maximum value of transportation capacity is maintained for the shorter time when the length of the emergency resource allocation route is decreased by 0.5 times (240 km), resulting in a shorter time to reach the fourth stage. The correlation between the duration of time after the maximum quantity of emergency resources in transit and the emergency resource allocation routes is positive. Changes in the length of the emergency resource-allocation routes do not affect transportation capacity.

The change curves of the quantity of resources in transit are shown in Figure 9 when the lengths of the emergency resource-allocation route and the emergency-management investment are twice their initial value (CNY 1000 thousand and 960 km, respectively). Figure 9 shows that increasing investments in emergency management leads to an increase in the maximum quantity of emergency resources in transit. If emergency-management investments is increased, the longer the emergency resource-allocation routes, the longer the duration after the maximum quantity of emergency resources in transit is reached, and the longer the transport time required. To efficiently allocate emergency resources after a mine-fire accident, it is essential to strategically position emergency resource depots near coal mines. This will minimize transportation distances and reduce response times. Moreover, increasing investment in emergency management is crucial to ensure adequate transportation capacity for resource allocation.

(2) This paper analyses the degree of influence that different factors have on the arrival rate of emergency resources. During the simulation process, the emergency resources’ allocation route is initially set to 240 and 720 km, respectively, and the initial value of the emergency-management level is set to 0.6 and 1.2, as shown in Figure 10 and Figure 11.

The arrival rate of liquid CO₂ can be observed from the curve in Figure 10 and Figure 11. It is assumed that the arrival rate of liquid CO₂ is zero in the first stage due to the deployment and transportation process, as well as the delay time of the emergency response. Therefore, the total time spent in this stage is T = T_delay + T_trans. In the second stage, the arrival rate of liquid CO₂ rapidly increases as it is continuously transported to the incident mine. In the third stage, the amount of liquid CO₂ remains constant after reaching its maximum value. This is due to the effective measures taken to prevent mine-fire accidents, resulting in a decreased demand for liquid CO₂. As a result, there is no need to transport liquid CO₂ to the affected mine, thus maintaining a steady arrival rate of liquid CO₂.

Figure 10 shows that the arrival rate of liquid CO₂ increases significantly when the route lengths of emergency resources’ allocation are two times that of the initial value, with the maximum arrival rate occurring the earliest. Conversely, the arrival rate of liquid CO₂ decreases significantly when the route lengths of emergency resources’ allocation are 0.5 times the initial value, with the maximum arrival rate occurring latest. In summary, the shorter the length of the emergency resource-allocation route, the higher the maximum arrival rate of emergency resources and transportation efficiency. There is a significant positive correlation between the arrival rate of emergency resources and the lengths of emergency resource-allocation routes.

Figure 11 shows that increasing the emergency-management level by 0.5 times the initial value (high level: 1.2) significantly decreases the arrival rate of emergency resources, while decreasing the emergency-management level by 0.5 times the initial value (low level: 0.6) significantly increases the arrival rate of emergency resources. Therefore, the analysis suggests a negative correlation between the emergency-management level and the arrival rate. Contrary to general knowledge, it has been found that the improving the level emergency management reduces the evolution rate of mine fires, resulting in a decreased demand for emergency resources and a reduced the final arrival rate. Conversely, reducing the level of emergency management increases the evolution rate of mine fires, leading to an increased demand for emergency resources and an increased final arrival rate, as shown in Figure 12 and Figure 13. It is recommended that the emergency-management level of the mine be improved in daily mine management to reduce the rate of mine-fire incidents and minimize economic losses and casualties.

The model’s output results exhibit significant changes when adjusting key parameters in the simulation model, as shown in Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12 and Figure 13. In conclusion, this paper’s emergency resource-allocation model for mine-fire accidents has passed the sensitivity test and holds important practical significance for the allocation of emergency resources during mine-fire emergency rescue.

3.3.5. Simulation Result

The values of each constant are set in the Vensim PLE software (Versin 9.3.5 x64) based on the initial values of each parameter in Table 2. The system dynamics model for emergency resources’ allocation in mine-fire accidents is simulated. The changing trends of each state variable in the model, such as the demand quantity of liquid CO₂, the quantity of liquid CO₂ arriving, the inventory quantity of liquid CO₂, and the used quantity of liquid CO₂, are obtained and shown in Figure 14, Figure 15, Figure 16 and Figure 17.

The demand for liquid CO₂ refers to the total amount required in the entire emergency rescue process. The change curve of liquid CO₂ can be divided into four stages, as shown in Figure 14. In the first stage, the demand for liquid CO₂ is zero. This is due to the response time required from when a mine’s fire alarm is received until the establishment of an emergency rescue headquarters’ decision making on control methods and materials, specifically the delay time of the emergency response (T_delay). In the second stage, the demand for liquid CO₂ increases rapidly. The acceleration of the demand for liquid CO₂ is greater than zero as the rate of mine-fire evolution speeds up, and the intensity of the fire evolution continues to rise, necessitating more liquid CO₂ for fire control. In the third stage, the demand for liquid CO₂ increases at a slow pace, that is, the acceleration of the demand for liquid CO₂ is less than zero as the fire is effectively controlled and the evolution rate decreases with the application of liquid CO₂, resulting in a reduction in the required quantity. In the fourth stage, the demand for liquid CO₂ is not increasing. At this time, the evolution rate of the mine fire is almost zero, and mine fires have been extinguished, so the total quantity of liquid CO₂ required remains unchanged.

The inventory quantity of liquid CO₂ refers to the total quantity of liquid CO₂ remaining in the emergency resources depot. The change curve of the liquid CO₂ inventory quantity can be divided into four stages, as demonstrated in Figure 15. In the first stage, the arrival quantity of liquid CO₂ is zero and the duration is T = T_delay + T_trans. This is because there is a delay in the reaction time before the liquid CO₂ can be transported after the mine-fire accident. Additionally it takes time for the liquid CO₂ to be transported from the emergency resource depot to the affected coal mine. During the second stage, the arrival quantity of liquid CO₂ delivered to the mine increased rapidly, leading to an acceleration of mine fires. This resulted in an increased demand for liquid CO₂, which was continuously transported to the affected mine, causing a rapid increase in the quantity of liquid CO₂ delivered. In the third stage, the arrival quantity of liquid CO₂ increased, but at a decelerating rate. This was due to the gradual increase in the rate of mine fires, which resulted in a decreasing demand for liquid CO₂. As a result, the arrival quantity of liquid CO₂ decreased. However, it eventually increased again at a slow rate. In the fourth stage, the quantity of liquid CO₂ being transported to the incident coal mine no longer increases, indicating that the mine fire is under control.

The inventory quantity of liquid CO₂ refers to the total quantity of liquid CO₂ remaining in the emergency resources depot. The change curve of the inventory quantity of liquid CO₂ can be divided into four stages, as shown in Figure 16. In the first stage, the total inventory quantity of liquid CO₂ remains unchanged due to the delay time of the emergency response. The liquid CO₂ has not yet begun to be transported. The duration of this stage is T = T_delay + T_trans. During the second stage, the demand for liquid CO₂ increased, requiring continuous transportation from the emergency resources depot. As a result, the inventory of liquid CO₂ decreased rapidly over time. In the third stage, the evolution rate of the mine fire gradually decreased, and the use of liquid CO₂ slowed down due to the development of rescue operations. Consequently, the inventory of liquid CO₂ decreased slowly during this stage. In the fourth stage, the mine fire was extinguished due to the timely and effective application of liquid CO₂ to the working face. As a result, the evolution rate of the mine fire was reduced to zero, and there was no need to transport additional liquid CO₂ from the emergency resources depot. Therefore, the total quantity of liquid CO₂ in the emergency resources depot remained unchanged.

The used quantity of liquid CO₂ refers to the total amount of liquid CO₂ that has been put into the working face for fire extinguishing, and the change curve of the used quantity of liquid CO₂ can be divided into three stages, as showed in Figure 17. In the first stage, the used quantity of liquid CO₂ is zero, and the total time is T = T_delay + T_trans + T_usage, as the delay time of emergency response and the time required for the transportation of liquid CO₂ from the emergency resources depot to the coal mine, liquid CO₂ has not yet been transported to the coal mine. In the second stage, liquid CO₂ is transported continuously to the incident coal mine with the increase in time, and liquid CO₂ is applied in the fire-extinguishing process of the mine fire, so the used quantity of liquid CO₂ increases. In the third stage, the mine fire has been extinguished, and the evolution rate of the mine fire becomes zero; it need not use liquid CO₂, so at this time, the used quantity of liquid CO₂ is no longer increased and remains unchanged.

The evolution rate of a mine fire is directly affected by the success of emergency rescue operations, which in turn is influenced by the decision-making ability of the commander. To investigate the impact of emergency rescue actions on the evolution rate of mine fires, we obtained the change in evolution rate under three conditions of high (1.6), medium (1.1), and low (0.6) command abilities by varying the quantitative value of the commander’s decision-making ability, as illustrated in Figure 18. As can be seen from Figure 18, the emergency rescue operations have a positive impact on mine-fire rescue when commanders have a high level of decision-making ability (1.6). This effectively controls the development and spread of the fire, resulting in a smaller evolution rate. When the commander’s decision-making ability is low (0.6) during emergency rescue operations, the action taken may not only fail to control the fire but also lead to its further expansion. This can cause the mine fire to spread rapidly, resulting in greater damage.

4. Discussion

After the occurrence of mine-fire accidents, effective emergency resource allocation is critical for the emergency rescue of those involved. To improve the allocation efficiency of emergency resources during mine-fire accidents, it is necessary to analyze the relevant influencing factors and their impact on the arrival rate of emergency resources. The current research lacks an in-depth study of the allocation process for emergency resources in mine-fire accidents, with little consideration given to management factors. However, it is important to note that management factors and human decision-making capabilities have a direct impact on emergency rescue operations.

To address the aforementioned issues, this paper deduced the system dynamics equations between variables and established a system dynamics model for emergency resource allocation in mine-fire accidents. The model was developed by analyzing the relevant factors of emergency resource allocation. The system dynamics model can analyze the changing variables over time during the emergency allocation of resources for mine-fire accidents. It identifies the variables that significantly influence the allocation rate of emergency resources for mine-fire accidents. This provides a theoretical basis for optimizing and improving the allocation process of emergency resources, resulting in increased allocation rates for mine-fire accidents. The system dynamics model of mine-fire emergency resource-allocation established in this paper takes into account not only environmental factors and equipment factors, but also human factors and management factors. Thus, the established system dynamics model can objectively and comprehensively reflect the entire process of allocating for mine-fire accidents. This provides a decision-making basis for emergency rescue commanders to formulate an effective emergency resource-allocation plan.

The research in this paper also has certain limitations. The system dynamics model established can only be applied to mine-fire accidents and not to other disaster accidents. Future research should aim to expand the application scope of the system dynamics model established in this paper. This will enable the model to simulate the emergency resource-allocation process for public safety and natural disaster accidents. By doing so, the emergency resource allocation rate for various disaster accidents can be improved, and the losses caused by accidents can be reduced.

5. Conclusions

(1)

According to relevant influencing factors of emergency resource allocation for mine-fire accidents, the factors are defined into four categories: constant, auxiliary variable, state variable, and rate variable. To establish a system dynamics model, a causality loop diagram and the stock flow diagram are created, which consist of each variable and constant. Additionally, the system dynamics equations of the emergency resource-allocation model are established. The dimensions of all variables in the model and the dimensions on the left and right sides of each equation meet the consistency requirements;

(2)

During the simulation of fire case, we analyzed the impact of various factors on the quantity of emergency resources in transit. By altering the value of emergency-management investment and the allocation route of emergency resources, it was observed that the quantity of emergency resources in transit is positively correlated with the emergency-management investment. Moreover, the duration of the maximum quantity of emergency resources in transit is positively correlated with the allocation route of emergency resources. The impact of various factors on the arrival rate of resources was analyzed by altering the length of the emergency-management level. The results showed a significant positive correlation between the arrival rate of emergency resources and emergency-management investment. However, the emergency-management level was found to be negatively correlated with the evolution rate of mine fires and the arrival rate of emergency resources. Therefore, the model passed the sensitivity test;

(3)

The change trend of emergency-resource demand, the quantity of emergency resources in transit, the arrival quantity of emergency resources, and the used quantity of emergency resources are determined through the simulation of a mine-fire case. The trend in emergency-resource demand is to accelerate the increase from zero, then decelerate the increase, and finally let it remain unchanged. Similarly, the trend for emergency resources’ arrival is to increase rapidly from zero, then increase slowly, and maintain the same value, finally. The quantity inventory of emergency resources remains unchanged initially, then it accelerates the decrease, then it decelerates the decrease, and finally it stabilizes. The trend for used emergency resources is to gradually increase from zero and then remain stable after reaching the maximum value;

(4)

The rules’ changes in the evolution rate of mine fires were obtained by analyzing the impact of emergency rescue actions under the three conditions of the commander’s decision-making ability: high, medium, and low levels. The rate of the evolution of mine fires decreases and they are controlled more effectively when emergency rescue commanders possess a high-level decision-making ability. Conversely, the rate of the evolution of mine fires increases and damage caused by them is greater when emergency rescue commanders have a low-level decision-making ability.