Simulation and Catastrophe Detection of Spontaneous Combustion Processes in Sulfide Ores

Abstract

Spontaneous combustion of sulfide ores during mining can lead to severe fires. To explore the transformation of state in the whole process of spontaneous combustion of sulfide ores, the simulation experiment of the whole unsteady process of spontaneous combustion of sulfide ore heap was carried out, and the most appropriate wavelet function was selected, combined with catastrophe detection and other methods for data mining and processing. The results indicate spatial differences in the response of the ore heap to environmental temperature changes during the whole unsteady process of spontaneous combustion of the sulfide ore heap. The reaction in the area near the surface of the heap is more prominent and faster, and the response in the area near the center of the heap is longer in duration. Moreover, there must be at least one catastrophe point in this process, and the catastrophe temperature must be between 108.2 °C and 113.9 °C. Finally, the whole unsteady process of the spontaneous combustion of the sulfide ore heap can be divided into four regions. Among them, region (II) is in a stage of obvious self-heating/near spontaneous combustion, and it is the catastrophe stage.

1. Introduction

Pyrite, the most reactive sulfide ore, is present in large quantities in metal mines [1,2,3]. However, the spontaneous combustion of sulfide ores, a process that is highly susceptible to occur in sulfide-bearing mines, is one of the most critical factors affecting the safety of the mining process [4,5,6] and is also the most severe hazard in sulfide-bearing mines [7]. During the mining process, the exposed sulfide ore reacts with oxygen in the air, releasing a small amount of heat, which builds up and heats the sulfide ore, releasing more heat. When the temperature of the sulfide ore reaches a specific value for spontaneous combustion, spontaneous combustion can occur [8,9]. Compared to coal mine spontaneous combustion fires, spontaneous combustion fires of sulfide ores are rare, which has led to their dangers being somewhat overlooked. However, in reality, the spontaneous combustion of sulfide ores poses a significant threat to mine safety, and its safety issues urgently need to be addressed. Especially with the continuous development of the economy and society, the demand for non-coal mineral resources is increasing, and shallow resources are gradually depleting. Most non-coal mines have had to turn to deeper mining. Currently, the leading global mining depths range from 2500 to 4000 m, such as South Africa’s Rand gold mine with a maximum mining depth of 4530 m, and India’s Kolar gold mine on the Deccan Plateau, which reaches a depth of 3260 m. This trend inevitably highlights the issue of increased geothermal temperatures, and the rise in deep mine temperatures will significantly increase the probability of spontaneous combustion fires in sulfide ores. Therefore, the spontaneous combustion of sulfide ores is a critical issue that urgently needs to be addressed in metal mines [10,11].

To address the issue of the spontaneous combustion of sulfide ores, numerous scholars have conducted extensive research, primarily focusing on the spontaneous combustion mechanism, spontaneous combustion propensity evaluation, spontaneous combustion prediction and forecast, and spontaneous combustion prevention technology [12,13,14,15]. Regarding the mechanism of spontaneous combustion of sulfide ores, the academic community believes that various mechanical forces encountered during sulfide ore mining make their chemical properties more active, significantly increasing their susceptibility to spontaneous combustion [16]. To evaluate the tendency for spontaneous combustion and assess the associated risks, scholars have applied methods such as game theory, set pair analysis, comprehensive methods based on weights, Bayesian discriminant analysis, TOPSIS comprehensive evaluation method, and cloud models [17,18,19,20]. Additionally, some experts have developed various chemical inhibitors [21,22], including thermophilic agents [23], imidazole ionic liquids [24], and composite inhibitors [25], to prevent the spontaneous combustion of sulfide ores, but these solutions have not been entirely effective. Therefore, many scholars aim to identify potential locations of spontaneous combustion before any signs appear [26,27]. Predictive methods used include empirical statistical forecasting [28,29], mathematical modeling and numerical simulation [30,31], and landmark gas analysis and temperature measurement [32,33,34]. However, predicting surface combustion alone is not accurate and fails to provide timely warnings of danger. Scholars generally regard the spontaneous combustion of sulfide ores as a nonlinear multi-parameter coupled evolution process [35,36], which is simply divided into incubation, development, and approaching stages [4,37,38,39,40]. Currently, research on the entire process of the spontaneous combustion of sulfide ores is minimal, particularly under unsteady conditions. This research is crucial for accurately determining the timing and location of spontaneous combustion. Most scholars focus only on the temperature changes of sulfide ores during dynamic, spontaneous combustion, studying the generation and variation of temperature. However, the concentration of SO₂ gas monitored during the entire unsteady process of spontaneous combustion in sulfide ore heaps, like the time series of ore temperature, contains rich information about spontaneous combustion. Therefore, it is worth exploring the implicit details to reveal the hazardous characteristics of the complete spontaneous combustion process and achieve early warning of ore spontaneous combustion.

Considering this, the study conducted small-scale simulation experiments of the whole unsteady process of spontaneous combustion of sulfide ore heap through a custom experimental setup indoors, and the trends of temperature and SO₂ gas concentration at each measurement point of the heap were investigated to find out the distribution of high-temperature regions in sulfide ore heap and the migration pattern of high-temperature areas over time. The wavelet transform and fuzzy integrated evaluation methods were also used to extract the ore spontaneous combustion information from the measured ore temperature and SO₂ gas concentration data series. Finally, catastrophe detection and kinetic feature parameter extraction methods are used to detect the hidden catastrophe points in the whole unsteady process of the spontaneous combustion of sulfide ore heaps, providing some insight into the implementation of sulfide ore spontaneous combustion warning.

2. Materials and Methods

2.1. Materials

The ore samples used in the experiment were obtained from the copper mine in Tongshan, Anhui Province. The composition of the ore samples was analyzed using X-ray fluorescence spectroscopy (XRF), a technique widely used for material composition analysis in fields such as mineralogy and metallurgy. XRF determines the elemental composition and concentration of a sample by measuring the fluorescent X-rays emitted from the sample when it is irradiated with X-rays. Before the analysis, the XRF equipment was calibrated using standard samples with known compositions, covering the entire concentration range of the elements to be measured. Additionally, the ore samples were meticulously ground and surface-treated to ensure uniform particle size, reduce errors due to sample heterogeneity, and maintain a clean and flat measurement surface. These steps were taken to ensure the accuracy of the analysis results. Ultimately, the primary elements in the ore samples, from highest to lowest concentration, were iron, sulfur, oxygen, silicon, calcium, barium, magnesium, aluminum, and copper. The main chemical composition of the ore samples is shown in

Table 1. Main chemical composition and content of the ore samples used.

Element	Fe	S	O	Si	Ca	Ba
Content (%)	31.632	27.606	24.129	9.822	2.28	4.328
Element	Mg	Al	Ce	Cu	As	Zn
Content (%)	0.939	0.766	0.53	0.344	0.26	0.093
Element	K	Mn	Ti	Na	Sr	Mo
Content (%)	0.082	0.072	0.059	0.032	0.021	0.006

.

During the process of crumbling sulfide ore into a heap, large particles tend to roll down to the bottom while fine particles accumulate at the top, forming a cone with a specific slope. For the experiment, the simulated ore heap is designed as a cone with a base diameter of 300 mm, a height of 160 mm, a slope angle of 46.8°, and a mass of approximately 9.1 kg. The particle size composition of the ore heap is detailed in

Table 2. Simulating the particle size composition of ore heap.

Grain size/mesh	>100	50~100	20~50	10~20	<10
Mass fraction%	5	10	15	15	55
Mass/kg	0.455	0.91	1.365	1.365	5.005

below.

Eight measurement points (labeled A through H) are strategically arranged within the ore heap to simulate spatial distribution and monitor temperature variations in different regions. The schematic diagram of the measurement points is illustrated in Figure 1, the yellow points in the figure indicate the measurement points A to H. The coordinates for each temperature measurement point are detailed in

Table 3. Temperature measurement points distribution.

measurement point	A	B	C	D
Coordinates/mm	( 0,0,150 )	( −75,−75,80 )	( −32,−32,120 )	( −140,−140,10 )
measurement point	E	F	G	H
Coordinates/mm	( 113,113,25 )	( 5,5,80 )	( 38,38,40 )	( 135,135,5 )

.

2.2. Experimental Equipment and Conditions

The complete experimental setup, independently developed according to the practical plan, is shown in the diagram of Figure 2 and in the physical picture of Figure 3, respectively. It mainly consists of a programmable high-temperature test chamber, a simulated ore heap, an Agilent data acquisition system, temperature probes, and a portable sulfur dioxide (SO₂) detector.

Before the experiment, the ore samples are mixed with approximately 0.48 kg of water to achieve a moisture content of 5%. After mixing, the mixture is allowed to settle for 1 h. The ore is then evenly and steadily stacked on a tray within a programmable high-temperature test chamber. An Agilent data acquisition system and temperature probe are connected, and a portable sulfur dioxide (SO₂) detector is positioned at the outlet of the metal pipe. High-temperature-resistant materials are used to seal any gaps between the ventilation openings and the pipe to minimize temperature loss and gas leakage.

The temperature gradient heating process begins at an initial temperature of 25 °C, which is incrementally increased by 5–10 °C. Once each target temperature is reached, it is allowed to stabilize for 5–10 min. The maximum experimental temperature does not exceed 300 °C. SO₂ concentration data are recorded throughout the experiment. Subsequently, the temperature and SO₂ concentration data from the acquisition system are transferred to a computer for further analysis and processing.

2.3. Extracting the Information of the Whole Unsteady Process of Spontaneous Combustion of Sulfide Ore Heap

The temperature data obtained from the indoor simulation experiments are influenced by multiple factors, including the temperature gradient increment in the programmable high-temperature test chamber and the oxidative self-heating of the ore. The former primarily affects the temperature variations at different measurement points within the ore heap. Consequently, the experimental data alone are insufficient for exploring the nature of the entire unsteady process of spontaneous combustion in the sulfide ore heap. To address this, it is necessary to extract useful information from the entire unsteady process of spontaneous combustion while filtering out experimental noise and accounting for the influence of the temperature gradient increment in the test chamber.

Given this context, the study employs wavelet analysis to decompose the measured temperature sequence into low-frequency and high-frequency components. The low-frequency sequence captures the effects of temperature gradients in the constant temperature chamber, while the high-frequency sequence reflects the intrinsic spontaneous combustion process of the ore. The high-frequency component contains detailed information about the entire unsteady process of spontaneous combustion in the sulfide ore heap. However, the choice of wavelet function significantly impacts the results of wavelet decomposition and reconstruction. Different wavelet functions can lead to substantial variations in the analysis of the same temperature sequence. Therefore, selecting the most appropriate wavelet function is critical for accurate analysis. In the field of sulfide ore spontaneous combustion, research on selecting suitable wavelet functions for specific temperature sequences is still lacking. Consequently, this study developed a method for extracting comprehensive information about the unsteady process of spontaneous combustion in sulfide ore heaps from measured temperature sequences, utilizing wavelet transform, Fuzzy Comprehensive Evaluation, and other techniques.

2.3.1. Wavelet Technology

Wavelet technology is the result of the development of the Fourier transform. It can realize local transformation in time and frequency domains at the same time, which has a better effect on the analysis of nonlinear processes. From a mathematical point of view, the wavelet technique is the expansion and translation of a function, which represents an orthogonal sequence square-integrable function. In this case, the continuous wavelet transform function is given as [41]

$$W_f\left(a,b\right)={\displaystyle \frac{1}{\sqrt{a}}}{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}f\left(t\right)\overline{\phi \left({\displaystyle \frac{t-b}{a}}\right)}dt$$

(1)

$f\left(t\right)$ is the proposed transformed signal, $\overline{\phi \left(t\right)}$ is the complex conjugate of wavelet basis function, $a$ is the scale factor, and $b$ is the translation factor. When $a$ > 1, the wavelet basis function expands and becomes more expansive in the time domain. On the other hand, the wavelet basis function contracts and becomes narrower, and $b$ only affects the window’s position in the time domain.

However, the continuous wavelet transform is computed with a large amount of redundancy. Therefore, in practical applications, the scale factor $a$ and translation factor $b$ are often discretized. The discrete wavelet transform function in the time domain, $t$, is given as

$$W_f\left(j,k\right)=a_0^{{\displaystyle \frac{-j}{2}}}{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}f\left(t\right)\overline{\phi (a_0^{-j}t-k)}dt$$

(2)

The Wavelet decomposition diagram is shown in Figure 4. As can be seen from the figure, an arbitrary sequence S can be decomposed into low-frequency components (cA₁) and high-frequency components (cD₁), and according to the demand, the low-frequency components can be decomposed even further to obtain the low-frequency components (cA₂ and cA₃) and high-frequency components (cD₂ and cD₃) on an arbitrary scale. Finally, the wavelet coefficients obtained are selectively reconstructed, and reconstruction sequences at different frequencies (cD_n) can be obtained.

2.3.2. Wavelet Function Optimization

The research combines Fuzzy Comprehensive Evaluation and the Entropy Weight Method to optimize the wavelet function for the temperature series of different measurement points.

The spontaneous combustion of sulfide ore is a complex phenomenon influenced by multiple factors and fields, making it a quintessential nonlinear process. Consequently, this study focuses on the temperature increment sequence of sulfide ore spontaneous combustion, which underscores the nonlinear characteristics of the process. To analyze this sequence, 55 wavelet functions from seven commonly used wavelet families are optimized to fit various sequences. The wavelet functions are detailed in

Table 4. List of wavelet functions.

Wavelet Families	Abridge	Wavelet Functions
Haar	haar
Daubechies	db	db1	db2	db3	db4	db5
		db6	db7	db8	db9	db10
Symlets	sym	sym1	sym2	sym3	sym4	sym5
		sym6	sym7	sym8
Coiflets	coif	coif1	coif2	coif3	coif4	coif5
BiorSplines	bior	bior1.1	bior1.3	bior1.5	bior2.2	bior2.4
		bior2.6	bior2.8	bior3.1	bior3.3	bior3.5
		bior3.7	bior3.9	bior4.4	bior5.5	bior6.8
Dmeyer	dmey
ReverseBior	rbio	rbio1.1	rbio1.3	rbio1.5	rbio2.2	rbio2.4
		rbio2.6	rbio2.8	rbio3.1	rbio3.3	rbio3.5
		rbio3.7	rbio3.9	rbio4.4	rbio5.5	rbio6.8

[42].

The temperature increment sequences of 8 measurement points are reconstructed by 55 wavelet function decomposition to obtain 8 × 55 low-frequency reconstruction sequences, which are used as the evaluation set of Fuzzy Comprehensive Evaluation. Each index value of the low-frequency reconstruction sequence is constructed from its statistical eigenvalues, which include mean ($\overline{X}$), coefficient of variation (C_v), first-order autocorrelation coefficient (r₁), and skewness coefficient (C_s). The statistical eigenvalues must satisfy the following conditions:

(1) The temperature change of the ore heap in the programmable high-temperature test chamber plays a dominant role, so the low-frequency reconstruction sequence should not be too large a difference from the $\overline{X}$ of the original sequence.

(2) The low-frequency reconstruction sequence does not contain the complex information of the whole process of spontaneous combustion in the ore heap, so the C_v should be smaller than that of the original sequence, and the r₁ should be larger than that of the original sequence.

(3) In the original sequence, the low-frequency components account for the main, so the low-frequency reconstruction sequence and the original sequence should maintain the same kurtosis and skewness, so the C_s should not be too large.

The evaluation indicators of the selected wavelet function are constructed through the above conditions, and the weight of each evaluation indicator is obtained through the Entropy Weight Method. The indicators are used as the factor set, the low-frequency reconstructed sequences obtained through 55 wavelet functions are used as the evaluation set, and the wavelet functions suitable for different measurement points are obtained through Fuzzy Comprehensive Evaluation.

2.4. Catastrophe Detection by Sample Entropy

Catastrophe theory is a method used to analyze time series data to identify significant changes and estimate critical points, such as mutation points. In the context of the spontaneous combustion of sulfide ore, we employ catastrophe detection techniques to understand the temperature changes throughout the entire unsteady process. This approach involves analyzing data sequences obtained through wavelet decomposition and reconstruction from a system dynamics perspective. By applying methods such as time series catastrophe detection and nonlinear feature parameter extraction, we can identify hidden catastrophe points in the transition zones of different stages and determine the corresponding temperatures at these critical points.

Sample Entropy (SampEn) is a parameter that characterizes the complexity of a time series, representing the probability of new pattern generation in a nonlinear dynamical system, thereby quantitatively assessing the system’s complexity. The specific calculation steps are as follows [43]:

Assume the original time series is [x₍₁₎, x₍₂₎, ……, x_(N)]. To perform phase space reconstruction with an embedding dimension of mmm, a set of mmm-dimensional vectors can be constructed as follows:

$$X\left(i\right)=\left[x\left(i\right),x\left(i+1\right),\dots \dots ,x\left(i+m-1\right)\right],i=\mathrm{1,2},\dots \dots ,N-m+1$$

(3)

The Euclidean distance $\mathrm{d}\left[X\right(i),X(j\left)\right]$ is defined as the maximum difference among the corresponding elements of X(i) and X(j), as shown in Equation (4).

$$\mathrm{d}\left[X\right(i),X(j\left)\right]=\underset{0\le k\le m-1}{\max }\left[\left|x\left(i+k\right)-x(j+k)\right|\right]$$

(4)

Assume a tolerance $r$ and count the number of vectors X(i) for which d[X(i), X(j)] < $r$, denoted as $n_i^m\left(r\right)$. Calculate the ratio of this count to the total number of vectors n-m, denoted as $C_i^m\left(r\right)$, as shown in Equation (5).

$$C_i^m\left(r\right)={\displaystyle \frac{n_i^m\left(r\right)}{n-m+1}},i=\mathrm{1,2},\dots \dots ,n-m+1$$

(5)

Calculate the average of $C_i^m\left(r\right)$ over all i, denoted as $C^m\left(r\right)$, as shown in Equation (6).

$$C^m\left(r\right)={\displaystyle \frac{1}{N-m+1}}\sum _{i=1}^{N-m+1}{C}_i^m\left(r\right)$$

(6)

Let the embedding dimension $m$ increase to $m$ = $m$ + 1, and repeat steps (3)–(6) to obtain $C^{m+1}\left(r\right)$. Finally, define sample entropy (SampEn) as shown in Equation (7).

$$\mathrm{S}\mathrm{a}\mathrm{m}\mathrm{p}\mathrm{E}\mathrm{n}\left(m,r\right)=\mathrm{l}\mathrm{n}\left[{\displaystyle \frac{C^m\left(r\right)}{C^{m+1}\left(r\right)}}\right]$$

(7)

Catastrophe detection by sample entropy combines sample entropy with the sliding window technique. It utilizes a window to extract subsequences from a time series and calculates their SampEn values. Then, new subsequences are obtained by gradually sliding the window with a chosen step size, and their SampEn values are calculated. This process continues until the sliding window traverses the entire sequence. Finally, a sequence of SampEn values is obtained, and the curve of this sequence is plotted. One can identify the interval or point of catastrophe by analyzing the curve.

To improve the accuracy of catastrophe point localization and facilitate comparative analysis, preprocessing the high-frequency reconstructed sequence is required before applying the catastrophe detection method for temperature detection. This preprocessing includes sequence extension and standardization. This study employs the cubic spline interpolation method to expand the high-frequency reconstructed sequence at each measurement point, and the expanded sequence length after the expansion is 1500. The extended sequence is then standardized through the range normalization method, which allows for a temperature time series appropriate for catastrophe detection.

3. Results

3.1. Experimental Results

Figure 5 illustrates the temperature changes at various measurement points within the ore heap. As observed, the temperatures at measurement points D and H, located at the bottom edge of the ore heap, are higher than those at other points. At 224 min, during a sudden temperature rise, the temperature at point D reached 135.302 °C; at 284.5 min, it reached a maximum of 201.491 °C. In contrast, the temperature at point F, located in the middle layer of the ore heap, was slightly lower. This temperature distribution can be attributed to the fact that the ore near the surface reacts more readily with the surrounding air, causing heat to conduct primarily to the surface layer.

Furthermore, this phenomenon indicates that the response of the ore heap to changes in environmental temperature varies spatially. The areas closer to the surface of the ore heap demonstrated a more pronounced and rapid response, while the areas in the middle layer of the ore heap exhibited a longer response duration. For example, measurement point F showed a noticeable lag in response time compared to measurement point D, with the temperature stabilizing around 100 °C.

Figure 6 illustrates the variation of SO₂ gas concentration during the whole unsteady process of spontaneous combustion of the sulfide ore heap. The figure indicates that there is SO₂ gas generation around 230–240 min into the experiment, though the gas concentration is not high. This suggests that some ores have entered a noticeable heat generation stage. This portion may be due to a few scattered ores around the sulfide ore heap during the experiment, which is not the primary focus of this study. At approximately 255 min, the SO₂ concentration suddenly increases, indicating that most of the sulfide ore heap had entered a high-speed oxidation stage, approaching the spontaneous combustion stage. When the SO₂ concentration reaches 20 ppm, the portable sulfur dioxide detector’s alarm goes off, and the experiment must be stopped.

The experimental results indicate that in the whole unsteady process of spontaneous combustion of sulfide ore heap, there is inevitably at least one catastrophe point corresponding to a catastrophe temperature. Once this catastrophe temperature is exceeded, it can be considered that the state of the ore heap has transitioned from safe to unsafe, and further reactions will lead to the combustion stage, resulting in a spontaneous combustion fire of the sulfide ore heap.

3.2. Temperature Field Reconstruction in Sulfide Ore Heap

The measured temperature variations are analyzed in MATLAB (version R2016a), with different colors representing different rates of temperature increase. The red indicates regions with higher temperature changes, while the blue represents regions with lower temperature changes. The simulated period for all cases is 40 min. The temperature field reconstruction, which depicts the temperature variation across the sulfide ore heap, is shown in Figure 7.

By analyzing the measured temperature changes in MATLAB, the temperature field of the sulfide ore heap can be roughly divided into four stages based on the temperature reconstruction:

Stage 1: Before 160 min, the temperature rise at the surface of the sulfide ore heap is more significant than the temperature rise in the interior.

Stage 2: From 160 to 200 min, the temperature rise in the sulfide ore heap’s interior is more significant than at the surface.

Stage 3: From 200 to 240 min, the temperature rise at the surface of the sulfide ore heap increases significantly, surpassing the temperature rise in the interior.

Stage 4: After 240 min, the temperature rise in the interior of the sulfide ore heap begins to increase significantly and exceeds the temperature rise at the surface.

These divisions help understand the changing temperature dynamics within the whole unsteady process of the spontaneous combustion of sulfide ore heap.

3.3. Optimization for Wavelet Functions

Figure 8 shows the reconstruction temperature increment sequence at point (A) after wavelet function (demy) reconstruction. The blue line represents the original temperature sequence at measurement point A, depicting minute-by-minute changes in the raw temperature of the sulfide ore. The temperature changes remain relatively stable before 200 min; however, after 200 min, significant fluctuations and peaks are observed; these peaks represent the maximum increments in the sulfide ore temperature. These changes are heavily influenced by the high-temperature test chamber, which obscures information about the ore’s spontaneous combustion, making it unsuitable for direct prediction.

The red line represents the low-frequency reconstruction sequence at measurement point A, illustrating the changes attributable to the test chamber. It closely matches the trends in the original temperature increment sequence, confirming that the high-temperature test chamber mainly affects the ore’s temperature changes.

The green line represents the high-frequency reconstruction sequence, excluding the influence of the high-temperature test chamber and reflecting the intrinsic changes in the ore’s self-heating. Before 200 min, the ore’s temperature changes are relatively stable; however, at 200 min, there is a noticeable peak corresponding to the rapid temperature increase observed in Figure 5 at the same time point. Unlike the original temperature sequence, the high-frequency reconstruction sequence captures intricate details, encompassing sub-microscopic changes throughout the oxidation and spontaneous combustion process of the entire ore heap. Therefore, these peaks represent the maximum temperature variations generated solely by the self-heating of the sulfide ore, indicating significant temperature fluctuations.

These high-frequency details are crucial for understanding the dynamic changes in the ore’s self-heating and combustion, laying the foundation for catastrophe detection.

According to the method described in Section 2.3.2, sym2, rbio5.5, bior5.5, bior5.5, db2, db2, db2, and db2 are selected as the wavelet functions for wavelet decomposition and reconstruction of measurement points A, B, C, D, E, F, G, and H, respectively.

The high-frequency reconstruction sequences obtained by wavelet decomposition and reconstruction using the respective optimized wavelet functions at each measurement point, i.e., the sequences containing the information of spontaneous combustion of sulfide ore heap, are taken as examples at measurement points A, B, C, and D. The results are shown in Figure 9. By comparing the high-frequency reconstruction sequences of each measurement point, the result of temporal and spatial differences in the spontaneous combustion processes in different regions can be obtained.

Moreover, by comparing the high-frequency reconstruction sequence of point A reconstructed by the sym2 wavelet function and that by the demy wavelet function, a big difference between the two can be found. The former is finer and fuller than the latter and can better reflect the spontaneous combustion information. It also shows the importance of wavelet function optimization.

3.4. Catastrophe Detection of the Whole Unsteady Process of Spontaneous Combustion of Sulfide Ore Heap

Taking the high-frequency reconstruction sequences of measurement points A, B, C, and D as examples, Figure 10 illustrates the preprocessed high-frequency reconstruction sequences. Compared to Figure 9, Figure 10 shows that after standardization, the temperature increment values on the vertical axis are normalized to a range of 0 to 1, and the horizontal axis is transformed from time to data points. By converting both axes to dimensionless quantities, the subsequent calculation load for catastrophe detection is reduced, and potential errors are avoided. Additionally, the standardized high-frequency reconstructed sequence is more compact compared to Figure 9.

Catastrophe detection is performed on the high-frequency reconstruction sequences of each preprocessed measurement point, combining sample entropy with the sliding window technique. The sliding window technique involves using a fixed-size window that slides over the data sequence with a set step size, incrementally calculating the target value for the data within each window according to a specified formula. In sample entropy mutation detection, a sliding window width of 225 and a step size of 1 are selected. The sample entropy of the high-frequency reconstruction sequences of the preprocessed data from the eight measurement points is calculated. Specifically, starting from the first data point of each sequence, the sample entropy of 225 temperature increment values is calculated. After each calculation, the window moves forward by one step, and the sample entropy is calculated again for the next 225 temperature increment values, starting from the second data point. This process is repeated to calculate the sample entropy for the entire sequence. The results of the sample entropy detection are shown in Figure 11. In the figure, A to H correspond to the sample entropy catastrophe detection results of the preprocessed high-frequency reconstructed sequences for points A to H, respectively. I–IV represent the four stages of the spontaneous combustion of sulfide ore, divided into stages I through IV.

Figure 11 shows that measurement points A, D, E, and H exhibit significant catastrophe regions (II) among the eight measurement points. This indicates a dynamic structural catastrophe occurred during the spontaneous combustion process in the sulfide ore heap at these four measurement points. In the case of point A, the value of sample entropy in the region (I) varies but not by much. As the window slides, the value of the sample entropy in the region (II) increases abruptly, and the value fluctuation becomes larger. To detect a catastrophe, the complexity of the time series must increase significantly, and its dynamic state must change more obviously. According to the “Rheology-Catastrophe theory” [44], the occurrence of a catastrophe is likely to be a transition toward the stage of approaching spontaneous combustion. Additionally, the variation in sample entropy at each measurement point confirms significant differences in the spontaneous combustion process across different areas of the sulfide ore heap.

A preliminary explanation for the spatial differences in the whole unsteady process of spontaneous combustion of sulfide ore heap is as follows: The porosity of the loose surface and shallow layers of the ore heap is higher compared to the internal regions, which have lower porosity. Consequently, the surface and shallow layers facilitate better heat dissipation, whereas the interior regions, with reduced porosity, exhibit poorer heat dissipation and improved heat storage conditions. This leads to increased heat accumulation in the interior regions, promoting the progression of oxidation and spontaneous combustion reactions. However, as one moves towards the central region of the heap, the ore becomes denser, resulting in increased ventilation resistance and reduced oxygen supply. This reduction in oxygen availability impedes the spontaneous combustion effect. Therefore, the combination of ventilation and heat storage effects significantly contributes to the observed spatial differences in the spontaneous combustion process of the sulfide ore heap.

Figure 11 shows that the sample entropy variation at the four measurement points exhibits a similar pattern: a transition from relative stability to a catastrophic state, followed by a period of brief stability and a final sharp decrease. Initially, during the early stages of spontaneous combustion in the sulfide ore heap, the system remains relatively stable. Heat accumulation and gradual temperature increases result in minimal changes in sample entropy. As the temperature rises significantly and spontaneous combustion becomes evident, the system transitions from a safe to an unsafe state, leading to a sharp increase in sample entropy. This rise in sample entropy reflects the growing complexity and unpredictability of the combustion process. During the peak of spontaneous combustion, the sample entropy remains high with significant fluctuations due to the chaotic nature of the reactions. In the final stage, as the ore particles within the heap reach full oxidation and the system approaches equilibrium, the sample entropy decreases, indicating a stabilization of the system.

The analysis of catastrophe points from measurement points A, D, E, and H in

Table 5. Results of catastrophe analysis for measurement points A, D, E, and H.

Measurement Point	Catastrophe Point	Mean of Sample Entropy		Catastrophe Temperature /°C
		I.	II.
A	1027	0.581	0.649	113.9
D	842	0.483	0.598	111.8
E	834	0.551	0.603	108.2
H	839	0.566	0.647	110.9

reveals that the critical temperatures at which catastrophic events occur during the unsteady process of spontaneous combustion differ across these points. This variability can be attributed to several factors, including the ore’s intrinsic thermal conductivity, particle size, and environmental conditions, such as ventilation and void ratios within the ore heap. Furthermore, the mean sample entropy values for regions (I) and (III)—which represent stages before and after the catastrophic region (II)—were calculated for these measurement points. It was observed that the mean sample entropy values for region (III) are consistently higher than those for region (I). This suggests that the high-rate oxidation or combustion stage (region III) is more complex and exhibits greater variability compared to the earlier stage of spontaneous combustion development (region I). The increased sample entropy in the high-rate oxidation stage reflects the heightened complexity and chaotic nature of the reactions occurring during this period.

4. Conclusions and Discussion

(1) Simulation experiments on the whole unsteady process of spontaneous combustion of sulfide ore heap have shown spatial differences in the response of sulfide ore heap to environmental temperature changes. The whole unsteady process of spontaneous combustion of sulfide ore heap inevitably has at least one catastrophe point corresponding to a critical temperature. Once this temperature is exceeded, the ore heap is considered to have transitioned to an unsafe state. This process can be divided into four stages by reconstructing the temperature field: surface temperature rise more significant than internal temperature rise, internal temperature rise more significant than surface temperature rise, a substantial increase in surface temperature rise exceeding internal temperature rise, and a significant increase in internal temperature rise exceeding surface temperature rise.

Experimental uncertainties include the structure of the ore and the singularity of ore selection. The structure of the ore is a multi-void medium with anisotropy, so the results of repeated experiments may be slightly different. The ore was selected from the same mine, and the results may differ for ores from other mines, which is the direction of the follow-up research in this study.

(2) The information on the non-stationary whole process of spontaneous combustion of sulfide ore heap is extracted from the measured temperature-time series by wavelet technique. A new quantitative method is developed to select wavelet functions appropriate for different measurement points by combining the Fuzzy Comprehensive Evaluation and Entropy Weight Method. The method can efficiently extract high-frequency reconstruction sequences containing information about the process of spontaneous combustion of sulfide ore heap.

(3) Catastrophe detection by sample entropy in the whole unsteady process of spontaneous combustion of sulfide ore heap indicates that not all areas within the ore heap exhibit significant catastrophe. Under the combined effects of ventilation and heat storage, there are noticeable differences in the spontaneous combustion process among different regions. The catastrophe temperature ranges from 108.2 °C to 113.9 °C. Based on the trend of sample entropy variation, the whole unsteady process of spontaneous combustion is divided into four stages: region (I) represents the relatively stable stage of oxidation and self-heating development, region (II) represents the stage of significant self-heating/approaching combustion, region (III) represents the stage of high-speed oxidation/combustion, and region (IV) represents the post-effect. Region (II) is the catastrophe stage.