Fractal Dimension Warning via Microseismic Time–Energy Data During Rock Mass Failure

Abstract

The early warning of disasters such as ground pressure in deep hard rock mines has long constrained the safe and efficient development of mining activities. Based on fractal theory and fractal dimension interpretation, this study constructs a microseismic monitoring system for mining areas, extracting key elements, particularly time and energy elements. Using the box-counting method of fractal theory, the study investigates the fractal dimensions of microseismic time–energy elements, data interpretation, and disaster source early warning. Through parameter analysis, events related to local potential failure are identified and extracted, and disaster characteristics are revealed based on microseismic activity. A time–energy fractal dimension-based analysis method is developed for preliminary fractal analysis and prediction of regional damage. A time–energy-centered early warning model is constructed, narrowing the prediction range to a scale of 10 m. Based on the fractal interpretation of time–energy data, the prediction and early warning of rock mass failure in mining areas are achieved, with the reliability of nested energy warnings ranging between 91.7% and 96.2%. A comprehensive evaluation criterion for fractal dimension values is established, enabling accurate delineation of warning zones and providing scientific decision-making support for mine safety promotion.

1. Introduction

The stability of mining rock engineering structures is closely related to the rock occurrence conditions and mechanical change mechanisms [1,2,3]. The monitoring and prediction of its disaster process has always been a common challenge in fields such as mining geotechnical engineering [4]. At present, commonly used methods for analyzing and preventing disasters include on-site basic investigations, theoretical-basis analysis, indoor experimental analysis, numerical simulation, on-site monitoring experiments, and on-site industrial experiments [5,6]. In the on-site inspection and monitoring process, routine stress and displacement monitoring of the stability of the rock mass structure in mines is relatively well developed, and can quickly grasp the deformation and movement data of local points, and thus judge and evaluate the possibility and development trends of significant damage [7,8]. However, efforts to master the structural stability of mining development and production have not been satisfactory, mainly due to the inability to comprehensively and macroscopically monitor and provide feedback on global fracture sources [9,10]. For this reason, many scholars and engineering technicians have introduced real-time online monitoring systems, such as microseismic monitoring. By analyzing and interpreting the data set of microseismic events, the possibility and location of regional damage can be inferred and interpreted.

The data elements of microseismic monitoring systems are complex, comprehensive, and rich [11,12,13,14], and are often used in the field of geotechnical engineering, such as in mining, to study the stability of surrounding rocks, mining sites, and main tunnels. Microseismic monitoring technology is widely used worldwide in mining and other engineering projects. It is a convenient method for real-time, continuous, and multi-dimensional monitoring of rock structure failure processes [15,16,17,18]. Rapid positioning in real-time microseismic observation is a method that has emerged to address the suddenness, uncertainty, and emergency nature of mining disasters. Compared with traditional monitoring techniques, microseismic monitoring technology has the characteristics of real-time, dynamic, three-dimensional, global, and visual monitoring. It can quickly identify and locate the precise location of potential damage areas in the surrounding rock mass [19,20]. From a mathematical perspective, the essence of earthquake source localization lies in finding the minimum value of the objective function constructed by the difference between the observed time and the theoretical time, with the imaginary earthquake source position as the function [21]. Therefore, parameters such as time, space, energy, moment magnitude, apparent volume, and energy rate ratio in microseismic systems provide possibilities for studying and analyzing the process of seismic source changes.

Rock mechanics and fractal mathematics can be closely combined, and applying fractal theory can more effectively solve engineering practical problems in rock mechanics [22,23,24,25]. From B B. Mandelbrot [22,26] (1967, 1983) founded the theory of fractal geometry, domestic and foreign scholars, including Xie [27], Chen [23], Takayuki Hirata [28], and Hong [29,30], have ingeniously studied rock mechanics using the fractal branch of mathematics combined with microseismicity in seismology [31]. At present, fractal theory has been successfully applied in the field of rock mechanics, and has achieved many innovative results, but there is still relatively little research on the expansion of the application of rock-based engineering to different fields [32]. Limited data indicate that rock fracture sources also follow certain fractal laws in time and space. Therefore, fractal theory may be used to obtain the essential laws of rock fracture through the chaotic and disordered information of fracture sources. However, fractal research in this field is still in its infancy, and the interpretation of complex information lacks systematicity. The fractal dimension distribution during the destruction process is still unknown [33]. Meanwhile, unlike spatial distribution, which only presents the location information of the rupture source, the degree of damage caused by the rupture source is largely related to its energy release [34]. Studying the fractal characteristics of energy during rock failure can also provide a basis for judgment in the analysis of failure instability in the engineering field [35].

To provide early warning and prediction of regional disasters in mining rock engineering from the perspectives of time and energy, this paper combines microseismic monitoring, fractal theory, and time–energy factors for classification and research. It mainly relies on carrying out interpretation and analysis of data from a deep second-phase project of a copper mine in Xinjiang and a microseismic monitoring system to conduct early warning research. The overall method involves on-site microseismic testing, data simulation, and on-site comparative verification. The aim is to predict the intensity of regional disasters through data interpretation, to promote the trend control and prevention of the stability of mining rock engineering structures, and to ultimately provide theoretical and technical support at the industrial test level for regional disaster analysis and interpretation.

2. Engineering Background and Microseismic Monitoring

2.1. Engineering Background

The copper mine is located in the northwest of China (as shown in Figure 1a), where there are abundant resources. As the mining depth of the ore body gradually deepens, the problems of mining ground pressure and other disasters become increasingly prominent. Mines have adopted various methods for detection and prevention. Still, they are mostly limited to local points, and the overall and macroscopic distribution of ground pressure is “unclear” and “difficult to see through”. To this end, the mine under study commissioned relevant research units to construct, operate, analyze, and interpret real-time, online, uninterrupted remote monitoring systems. This lays the foundation for analyzing and researching disaster warnings in mining operations. In particular, this article combines fractal theory with microseismic time and energy data to monitor and warn of mine ground pressure, which provides a research foundation and application scenarios for ensuring safety.

To reproduce and analyze the pattern of ground pressure manifestation in key areas of the mine, the technical department of the mine constructed the second phase of the microseismic monitoring system(Zhong Ke microseismic monitoring, SSS, Wuhan, China) in the deep part of the mine. Figure 1b shows an overview of the main underground monitoring layout in the mine, Figure 1c shows the layout of the surface industrial square and the above-ground monitoring center, and Figure 1d shows the topology diagram of the microseismic monitoring system, which is connected to the above-ground and underground communication network as a whole.

2.2. Microseismic Monitoring and Experimental Methods

(1)

Experimental background:

Based on the analysis of the characteristics of regional microseismic data in the mine, combined with the mining technology conditions and current situation of the copper mine, real-time dynamic online microseismic monitoring of ground pressure and rock burst in the mine was carried out to address issues such as regional disasters. Real-time identification and analysis of disasters through microseismic systems were carried out to conduct research on ground pressure activity information characterization and disaster warning technology, providing a scientific basis and technical support for the safety promotion and management of mines.

(2)

Sensor layout

The entire Phase II project installed a total of 22 microseismic sensors, including 19 single-axis sensors and 3 three-axis sensors (Figure 2). To monitor and warn of large-scale regional damage during the mining process of the Phase II project, fractal theory was mainly used to analyze and interpret the microseismic monitoring data.

(3)

Overall distribution characteristics of microseismic events

During the research period, the main communication system for microseismic monitoring was basically intact, and the overall monitoring data integrity was relatively good. The total number of monitoring events, the number of effective microseismic monitoring events, and the distribution of effective microseismic monitoring event rates within the study area are presented in Figure 3.

As shown in the figure, over the time series of the research period, there were relatively high event rates in July 2019, August 2019, November 2019, and January 2020, which macroscopically reflects the relatively high frequency of mine ground pressure activities in the time series. Especially in November 2019 and January 2020, there were not only higher event rates, but also more prominent and significantly higher total and effective event numbers.

(4)

Extraction and spatial distribution of effective rock fracture events

Full data analysis of the above-mentioned microseismic events was conducted to grasp the basic situation of the entire mine, facilitating the determination of research scope and the selection of research objects. According to the actual mining situation and cycle planning of the mine, there were very few working mining faces in the first phase of the project, with a small number of events and low microseismic activity. The mining intensity of the four middle sections related to Phase II, during which the main mining activities and filling processes of the mine were undertaken, is relatively high, showing strong event activity and relatively severe damage. We used the relationship between the location and the relative energy of microseismic events to establish an effective microseismic event database (Figure 4).

3. Early Warning Methods for Microseismic Events’ Time and Energy

3.1. Fundamental Principles of Time–Energy Fractal Warning

Time–energy fractal dimension warning is a method for predicting and warning of mining area disasters based on time–energy fractal dimension values and their trends. Time–energy fractal warning is based on the analysis and comparison of the fractal dimension of time–energy information before and after the occurrence of disasters in mining areas, to identify warning signals and predict the possibility of disasters. This method divides the mining area into multiple sub-regions in equal proportions, and collects and analyzes relevant time–energy parameters and fractal basic elements for seismic mining sources in each sub-region. Then, by comparing the fractal dimensions of time–energy at different times and spatial positions, the time–energy patterns and anomalies related to the disaster can be determined.

By analyzing time series data in terms of dimensionality, such as trend analysis, periodicity analysis, mutation analysis, etc., the changing trends and abnormal situations of time series data can be observed. By analyzing the statistical characteristics and patterns of time series data, possible disasters or abnormal events can be predicted. In addition, by calculating the fractal dimension of time series data, it can observed whether there are significant differences in fractal dimensions at different time scales. By comparing fractal dimensions at different time scales, the possibility of anomalies or disasters can be determined. Meanwhile, attention should be paid to the distribution characteristics of microseismic events in the time dimension. By calculating the time fractal dimension of microseismic events and predicting the trend of changes in the time fractal dimension, the potential movement state and evolution law of faults and other damages can be predicted. Overall, the occurrence of disasters is a complex situation of certainty and uncertainty, regularity and irregularity, and complexity and simplicity.

In the fractal theory warning method for microseismic events, different time–energy elements can be used to analyze and predict the evolution trends of underground rock masses and faults during the disaster process in mining areas. The time and energy fractal dimension warning method for mining area disasters can be carried out by comparing different time scales and energy fractal dimensions. The key element of the time–energy fractal warning method is the “time–energy fractal dimension”, which calculates and divides the fractal dimension of time and energy elements in the mining area. The time dimension includes the collection and analysis of time series data in the mining area, including a review of historical data and monitoring of current data. The energy dimension includes the energy distribution of microseismic events during the disaster evolution process in mining areas, especially the sensor data and related changes in geological environmental conditions at various locations in the monitoring network. Early warning analysis mainly relies on fractal analysis, which compares and analyzes the energy fractal data of the mining area at different times and locations to reveal abnormal behavior and risk signals.

The comprehensive use of time and energy fractal warning methods can provide a more comprehensive understanding of the disaster risks in mining areas. At the same time, it is necessary to combine other indicators, models, or professional knowledge for comprehensive analysis and judgment to improve the accuracy and reliability of early warning. In addition, the effectiveness of warning methods also needs to be verified and adjusted in practical applications.

3.2. Key Procedures for Time–Energy Fractal Warning Implementation

The temporal and spatial dimension warning method can help mine management personnel to monitor and predict the level of disaster risk in real time, and take corresponding measures to reduce the damage and impact of disasters. The main steps include data collection, data processing, feature extraction, model construction, and prediction and warning. The specific logic and process of time and energy element dimension analysis and warning are shown in Figure 5 [36].

The specific time dimension warning method includes the following steps [36]:

By preliminary screening and processing of the microseismic monitoring data obtained from the mining site, and the removal of abnormal data, the three-dimensional spatial coordinates and energy values of the monitored microseismic monitoring rupture source events are obtained, as shown in Figure 6.

The three-dimensional spatial coordinate values of the microseismic monitoring events are obtained in chronological order, and the energy values corresponding to the rupture source events and their three-dimensional spatial coordinate values are selected to preliminarily analyze the trends and relative size relationships of the monitored events in the three-dimensional spatial coordinates.

Based on the above steps, after calculating the maximum value of the energy value, the required data set is constructed, and the three-dimensional spatial coordinates of the monitored microseismic rupture source event and the associated energy value data set K are selected, $\mathrm{K}=\{x,y,z,t,E_E\}$, The construction process is as follows:

The maximum values of the three-dimensional positioning coordinates for the X, Y, and Z directions and the energy values, denoted as x_min, x_max, y_min, y_max, z_min, z_max, E_min and E_max, are calculated;

The boundary dimensions of the point cloud structure framework model are constructed, with the coordinates of the three directions as the boundary;

Based on the maximum values of the coordinates in each direction of the three-dimensional space, and the specific requirements for grid division accuracy, according to the principle of evenly distributed division proportional to the entire research area, the sets corresponding to each direction are counted and denoted as XX, YY, ZZ, respectively.

The calculation process is as follows:

The spacing between the grid divisions in the X-axis direction is denoted as LX:

$$LX={\displaystyle \frac{(x_{max}-x_{min})}{(N-1)}}$$

(1)

From this, a linear spacing set XX is generated, with the maximum and minimum values in the X-axis direction as the boundary:

$$\begin{gathered} \mathrm{X}\mathrm{X}=(x_{min},x_{min}+\mathrm{L}\mathrm{X}\ast 1,x_{min}+\mathrm{L}\mathrm{X}\ast 2,···, \\ {x}_{max}-\mathrm{L}\mathrm{X}\ast 2,x_{min}-\mathrm{L}\mathrm{X}\ast 1,x_{max}) \end{gathered}$$

(2)

The spacing between grid divisions in the Y-axis direction is denoted as LY:

$$LY={\displaystyle \frac{(y_{max}-y_{min})}{(N-1)}}$$

(3)

From this, a linear spacing set YY is generated, with the maximum and minimum values in the Y-axis direction as the boundary:

$$\begin{gathered} \mathrm{Y}\mathrm{Y}=(y_{min},y_{min}+\mathrm{L}\mathrm{Y}\ast 1,y_{min}+\mathrm{L}\mathrm{Y}\ast 2,···, \\ {y}_{max}-\mathrm{L}\mathrm{Y}\ast 2,y_{min}-\mathrm{L}\mathrm{Y}\ast 1,y_{max}) \end{gathered}$$

(4)

The spacing between grid divisions in the Z-axis direction is denoted as LZ:

$$LZ={\displaystyle \frac{(z_{max}-z_{min})}{(N-1)}}$$

(5)

From this, a linear spacing set ZZ is generated, with the maximum and minimum values in the Z-axis direction as the boundary:

$$\begin{gathered} \mathrm{Z}\mathrm{Z}=(z_{min},z_{min}+\mathrm{L}\mathrm{Z}\ast 1,z_{min}+\mathrm{L}\mathrm{Z}\ast 2,···, \\ {z}_{max}-\mathrm{L}\mathrm{Z}\ast 2,z_{min}-\mathrm{L}\mathrm{Z}\ast 1,z_{max}) \end{gathered}$$

(6)

The energy values corresponding to only the coordinate data points are classified into levels, and the data set for the spatial distribution energy level within the entire point cloud structure framework model is obtained, denoted as Ee.

Based on the maximum and minimum values of the three-dimensional spatial coordinates X, Y, and Z, or specified actual boundary values, mesh nodes are divided to obtain a point cloud structure framework model consisting of several uniformly distributed points within the research space.

The specific grid partitioning process is as follows: first, the length, width, and height of the research area and their proportional relationship are calculated; then, in the three-dimensional space, the rectangular box of the reduced study area is obtained by proportionally reducing it by the same multiple in all directions, and numbered to form a micro-unit cell; further, the micro-unit cells are distributed in a uniform and nonoverlapping manner within the study area, and after partitioning, a point cloud structure framework model is obtained by combining the centroids of several micro-unit cells within the study area (as shown in Figure 7).

The energy values corresponding to the three-dimensional coordinate points in the point cloud structure framework model are interpolated (as shown in Figure 8) to obtain the global energy values of the three-dimensional spatial points and the energy field within the entire point cloud structure framework model. The specific interpolation operation of energy values is carried out by using the natural nearest neighbor interpolation method to interpolate the energy values in the point cloud structure framework model, searching for a subset of the dissipated energy values for the most recent event in the existing energy value data, and selecting and interpolating the relative energy values according to the relative proportion relationship within the region, based on weights, to obtain the energy values f(x) of uniform points in three-dimensional space.

The f(x)′ calculation formula is as follows:

$$f\left(x\right)=\sum _{i=1}^N\left(w_i\left(x\right)f_i\right)$$

(7)

In the formula, f(x) is the interpolation result at the point x to be interpolated;

$w_i\left(x\right)$ represents the weight of the sample point ${i=\mathrm{1,2},3,\dots ,n}_i$, involved in interpolation concerning interpolation point x;

And f_i is the value at sample point i.

The $w_i\left(x\right)$’ calculation formula is as follows:

$$w_i\left(x\right)={\displaystyle \frac{{\alpha }_i\cap \alpha \left(x\right)}{\alpha \left(x\right)}},0\le w_i\left(x\right)\le 1$$

(8)

In the formula, a_i represents the area of the Thiessen polygon where the sample points participating in interpolation are located;

a(x) is the area of the Thiessen polygon where point x is to be interpolated;

And ${\alpha }_i\cap \alpha \left(x\right)$ is the area where the two intersect.

The three-dimensional uniformly distributed point coordinates within the point cloud structure framework model, as well as the corresponding four-dimensional data of energy values after conducting the interpolation operation, are displayed in a data-driven manner. Here, we mainly use sample indicators and their fractal dimensions, and then use a normalization calculation for early warning.

$${DT}_{i\_new}={\displaystyle \frac{DT-{DT}_{i\_min}}{{DT}_{i\_max}-{DT}_{i\_min}}}$$

(9)

Based on the requirements, the energy dissipation evolution distribution map with preset accuracy, and the horizontal slice/vertical profile maps of each research focus area, are exported (as shown in Figure 9).

Based on the distribution of the energy dissipation field and energy value conversion, the data within the point cloud structure framework model are screened and classified into equal levels, and the filtered three-dimensional spatial coordinates or coordinate area range are exported to achieve disaster point target location warning. The classification method is as follows: based on the needs of the engineering site, multiple energy level warning levels are determined. In this section, safety warning colors are used as a reference for level and color selection, and four equal levels are determined. Red, orange, yellow, and blue correspond to Level I (particularly severe), Level II (severe), Level III (relatively severe), and Level IV (moderate), respectively. The specific method for warning of the target location and energy level of disaster points is as follows: within the set time step and accuracy rules and requirements, the energy values attached to the positions of each point in the point cloud structure framework model and their energy fractal value levels are comprehensively analyzed, and attention is paid to the red point location area corresponding to the time–energy fractal value. The coordinates in the research and analysis space are locked and predicted in a timely fashion for warning. After interpolating and analyzing the location and time–energy fractal dimension values of random rupture source events, the time–energy fractal dimension values corresponding to the uniformly distributed points with the required accuracy are obtained and divided according to the specified level. Taking the fourth-order equipartition as an example, the coordinate points corresponding to the top 25% energy levels were statistically calculated and predicted [36].

4. Interpretation and Analysis of Results of Time Dimension Data

4.1. Data Background

The process of temporal and spatial dimension interpretation uses almost identical data sources and macro-indicators to those used in the aforementioned spatiotemporal dimension interpretation process. It also applies a nested combination of multiple indicators, forms, and parameters to warn of regional damage. Then, using fractal theory, the chaotic and disordered unknown processes are digitized, digitized, and visualized. The main difference and refinement lies in the nested internal loop iteration of the energy index, which solves the time–energy fractal dimension values in the composite space. Therefore, this article mainly focuses on the nested fractal analysis of time and energy elements. The basic information sources are consistent, and the measurement forms of different indicators are unified. Then, early warning is achieved through abnormal information monitoring. Relatively speaking, overall large-scale destruction in a region is a probabilistic event. When composite predictions are made for samples and indicators on multiple timelines, the interval values with a small proportion of samples in the corresponding indicators have a high potential for danger.

A fractal interpretation model can mitigate the effects of geological heterogeneity and equipment limitations in various ways. Specifically, the model utilizes adaptive algorithms, multi-sensor data fusion, and noise filtering, as well as network optimization techniques, to optimize microseismic data processing and reduce errors caused by device constraints such as sensitivity and sampling frequency. In addition, through fractal feature extraction, multi-scale analysis, and signal correction, the model effectively solves the problem of geological heterogeneity, and ensures accurate interpretation of signals.

Specifically, based on the microseismic parameters and characteristics of the main data of the second-phase project of the mine, the data sources required for the interpretation process of the temporal and spatial data are shown in

Table 1. Table of seismic source parameters (near-fault).

No.	Date	Time	Microseismic Event Coordinates (m)			Radiant Energy (J)	PS Wave Radiation Energy Ratio	Richter Magnitude	Seismic Moment (N·M)	Apparent Stress (MPa)	Apparent Volume (m3)	Corner Frequency (Hz)	Source Radius (m)	Number of Triggered Sensors (I)
			X	Y	Z
S	17 July 2019	11:23:42	217	536	432	125.89	0.58	1.2	7.9 × 108	0.04	3.4 × 105	240	29.2	4
2	18 July 2019	7:59:04	303	427	430	15.85	1.26	1.7	3.2 × 108	0.02	3.7 × 105	254	30.1	5
3	20 July 2019	16:31:26	419	391	384	2511.89	1.12	0.6	2.0 × 109	0.34	9.6 × 104	294	19.2	5
4	21 July 2019	13:09:23	325	449	411	1.26	1.11	2.4	1.3 × 108	0	5.3 × 105	199	33.9	4
5	21 July 2019	23:48:24	308	431	373	251.19	1.38	0.8	4.0 × 109	0.02	3.6 × 106	284	64.5	5
6	3 October 2019	18:06:39	454	412	251	1995.26	1.16	0	2.5 × 1010	0.02	2.0 × 107	205	113.4	6
7	13 October 2019	16:46:13	269	429	374	251.19	0.64	0.1	2.0 × 109	0.04	8.8 × 105	26	40.3	4
8	18 October 2019	18:35:30	264	468	378	3162.28	1.97	0.4	5.0 × 109	0.17	5.0 × 105	240	33.4	7
9	11 November 2019	12:51:03	361	453	345	398.11	1.17	0.5	1.3 × 1010	0.01	5.7 × 106	91	111.1	6
10	29 December 2019	18:39:33	418	415	311	19,952.62	1.17	0.4	5.0 × 1010	0.12	1.9 × 106	156	77	8
:
46	12 April 2020	8:13:36	319	440	368	31.62	1.71	1.6	5.0 × 108	0.02	5.2 × 105	358	33.7	3
47	12 April 2020	10:36:36	319	447	332	10	1.54	1.6	1.0 × 109	0	4.9 × 106	26	71.1	3
48	12 April 2020	11:38:42	387	385	283	3.98	1.11	1.8	6.3 × 108	0	6.2 × 106	269	77.2	3
49	22 April 2020	3:55:10	344	426	307	79.43	1.62	1.4	6.3 × 108	0.04	2.6 × 105	90	26.7	3
50	22 April 2020	12:14:21	320	463	335	1258.93	0.85	0.4	7.9 × 109	0.04	3.2 × 106	51	62.1	5
51	29 April 2020	21:27:44	298	465	308	31,622.78	1.09	0.5	6.3 × 1010	0.13	8.4 × 106	171	85.4	9
52	4 May 2020	22:50:23	305	458	351	100	1.06	−1.2	1.6 × 109	0.02	1.2 × 106	385	45.2	3

Note: excerpts from all development events near fault in target monitoring area.

. Based on the data and spatiotemporal representation analysis provided in

Table 2. Count of events at different stages and energy levels during process of fractal dimension calculation.

No.	Different Energy Levels
	Stage 1				Stage 2				Stage 3				Stage 4
	1	2	3	4	1	2	3	4	1	2	3	4	1	2	3	4
1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
2	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
3	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
5	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
6	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
7	1	0	0	0	1	0	0	0	1	0	0	0	1	0	0	0
8	0	0	0	0	0	0	0	0	1	0	0	0	1	0	0	0
9	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
10	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
11	1	0	0	0	1	0	0	0	1	0	0	0	1	0	0	0
12	0	0	0	0	0	0	0	0	2	0	0	0	2	0	0	0
13	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
14	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
15	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
16	1	0	0	0	1	0	0	0	1	0	0	0	1	0	0	0
17	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
18	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
19	0	0	0	0	1	0	0	0	0	0	0	0	0	0	0	0
20	1	0	0	0	2	0	0	0	0	0	0	0	0	0	0	0
21	0	0	0	0	1	0	0	0	1	0	0	0	1	0	0	0
22	0	0	0	0	5	0	0	0	12	0	0	0	13	0	0	0
23	1	0	0	0	4	0	0	0	5	0	0	0	6	0	0	0
24	1	0	0	0	1	0	0	0	2	0	0	0	2	0	0	0
25	0	0	0	0	0	0	0	0	1	0	0	0	1	0	0	0
26	0	0	0	0	0	0	0	0	2	0	0	0	4	0	0	1
27	0	0	0	0	1	0	0	0	3	0	0	0	3	0	0	0
28	0	0	0	0	0	0	0	0	1	0	0	0	1	0	0	0
29	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
30	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
31	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
32	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
33	0	0	0	0	0	0	0	0	0	0	0	0	1	0	0	0
34	0	0	0	0	1	0	0	0	0	0	0	0	0	0	0	0
35	0	0	0	0	0	0	0	0	1	0	0	0	1	0	0	0
36	1	0	0	0	1	0	0	0	1	0	0	0	1	0	0	0
37	0	0	0	0	1	0	0	0	1	0	0	1	1	0	1	0
38	0	0	0	0	0	0	0	0	4	0	0	0	5	0	0	0
39	2	0	0	0	2	0	0	0	0	0	0	0	0	0	0	0
40	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
41	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
42	0	0	0	0	0	0	0	0	1	0	0	0	1	0	0	0
43	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
44	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
45	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
46	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
47	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
48	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
49	1	0	0	0	1	0	0	0	1	0	0	0	1	0	0	0
50	0	0	0	1	0	0	0	1	0	0	0	0	0	0	0	0
51	1	0	0	0	1	0	0	0	1	0	0	0	1	0	0	0
52	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
53	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
54	0	0	0	0	0	0	0	0	0	0	0	1	0	0	1	0
55	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
56	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
57	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
58	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
59	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
60	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
61	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
62	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
63	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
64	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
Totals	11	0	0	1	25	0	0	1	43	0	0	2	49	0	2	1

, further progressive analysis of the spatiotemporal fractal dimension will be conducted. This article mainly relies on these data to interpret the spatiotemporal fractal dimension, improve the warning method, and ultimately investigate spatiotemporal fractal dimension warning, constructing a warning system based on spatiotemporal indicators and fractal dimensions.

By analyzing energy elements, the fractal dimension of energy elements at different time periods and regional positions can be calculated to observe their differences and changes. The fractal dimension of different energy elements may reflect their characteristics, relationships, and mutual influences. By comparing the fractal dimensions of energy elements in different regions, it is possible to predict the behavioral status, changing trends, and potential high-level risks of regional disasters. By analyzing the energy fractal dimension system of the mining area, and observing the structure, stability, and dynamic changes of the energy system (such as the distribution of microseismic energy release, the spatial distribution of energy accumulation, etc.), as well as the inter-relationships between different energy fractal dimensions in different regions, possible disasters or abnormal situations can be predicted. Thus, the complexity of energy fractal dimension distribution, such as in rock mass failure and fault migration, can be revealed, and potential regional disaster energy release patterns can be predicted.

4.2. Time Dimension Early Warning Process

The fractal warning system for mining disasters based on microseismic monitoring of energy elements is a system that uses microseismic monitoring data for fractal analysis and warning. Its fractal warning system mainly includes a microseismic monitoring system, data analysis, model construction, risk assessment, and warning level classification, as well as disaster emergency response and decision support. The display of Figure 5 generally presents a clustering trend in macro-analysis. The fractures that cause damage in the rock mass area are extremely significant. Specifically in the aforementioned Figure 10a and aggregation and nucleation are significantly displayed in Figure 10b. In addition, within the entire space, sliding based on a certain time window, the entire spatial domain is filled with C-bodies of a certain size. After the time window and C-body size are determined, the process as shown in Figure 10a was followed. The spatiotemporal fractal dimension measurement was calculated as shown in Figure 10b. The preliminary delineation of the time span and event cycle is shown in Figure 11.

After calculating the spatiotemporal fractal dimension as described above, the dynamic fractal dimension is calculated based on the partitioned spatial rectangular grid and time elements, as shown in Figure 12a. For the distribution formation, first, the fractal dimension values are presented in sequence from the row and column dimensions in three-dimensional space, and they are divided into corresponding large class intervals. Then, the fractal dimension values of the energy of the series of events at each stage are statistically calculated within the divided large class units, as shown in Figure 12a. As shown in Figure 12b, the energy fractal dimension values are classified again and divided into quadratic units. According to the nested warning classification method mentioned above, the spatiotemporal and temporal energy dimension warning identification of regional damage is carried out. At this point, the construction process of microseismic information spatiotemporal and temporal energy dimension analysis and warning is completed. Next, this is verified with specific data and analysis.

Based on the preliminary analysis of the time and space of events in the research area mentioned above, as well as the analysis process of the constructed framework model, detailed data statistics were conducted on a number of partitions ranging from 2 to 50 (as Table 3-3) [37]. In terms of the amount of microseismic event data and the model boundary size within the study area, the maximum number of events per microseismic unit uniformly fluctuated between 2 and 3 from the 23rd onwards. In view of this, if it were further subdivided into grids, this would increase the computational complexity and difficulty of numerical simulation, and more importantly, it would have no practical physical significance. It can be seen that in the existing analysis, the spatial span of the database is divided into 23 equal parts with the maximum side length, and the length, width, and height parameters of the unit sizes are 13.04 m, 8.70 m, and 10.87 m, respectively. Among them, the number of events in the smallest unit remains stable, at around 4. On the basis of determining the distribution of time and space windows and their spans, we completed the calculation of their spatiotemporal fractal dimensions. Meanwhile, in this article, the idea of fractal dimension analysis continues to be used to classify the energy fractal dimension levels within each unit.

Based on microseismic monitoring data and time–energy fractal warning models, a preliminary assessment of the development trend and relative scale level of this type of damage can be conducted, and accurate prediction of micro-unit locations can be made according to the classification level. Specifically, firstly, based on the calculation of fractal dimension values in the aforementioned time and space partitioning, the fractal dimension values within each block and the fractal dimension values of different energy partitioning levels within each micro-unit block are calculated; then, through multiple iterations, the fitted fractal dimension values are obtained and classified into categories and warning levels; finally, according to the number of grids of different levels and numbers, the data are divided into grids for statistical, computational, and iterative verification, followed by accurate analysis of the entire data set.

4.3. Time Dimension Warning Results

Table 2 shows the statistical results of the number of events in each stage and energy level before the calculation of the temporal energy dimension. Specifically, based on the on-site situation, the microseismic event data are divided into four stages according to time, namely the first stage, the second stage, the third stage, and the fourth stage. The spatial division is reflected in the division results and data statistics of each micro-unit. The important factor is the corresponding division of different energy levels, which relies on temporal and spatial elements, while also being independent of spatiotemporal elements. Furthermore, by iteratively calculating the fractal dimension step by step, the goal of early warning can be achieved.

The statistical data of each energy level in each stage are shown in Figure 13. Overall, with the passage of time, the amount of data in each stage increases step by step, and shows a characteristic of rapid increase followed by slow increase. The number of low-level events in each stage is relatively high, accounting for 91.7%, 96.2%, 93.3%, and 94.2%, respectively; the proportion of intermediate-level events is extremely low, with the fourth stage accounting for about 3.8%; and high-energy events account for a relatively small proportion, with 4.4% and 1.9% in the third and fourth stages, respectively. On the other hand, although the total number of high-energy events is small, they are relatively concentrated in stages three and four, indicating that energy accumulation and damage accumulation have reached a certain peak, which is completely consistent with the characteristics of conventional rock and rock mass failure.

As for specific warnings, this section mainly combines energy partitioning elements with time and space elements for argumentation and analysis. The method is basically the same as the aforementioned method for spatiotemporal fractal warning, with the main difference being that it is based on the spatiotemporal warning location and conducts grading according to the spatiotemporal fractal dimension for dual regional disaster warning. In fact, the 23 grids analyzed above are used as the optimal framework model for calculating the fractal dimensions. Through calculation and numerical simulation iteration, data with fractal dimension values ranging from large to small were obtained, and corresponding fractal dimension classification categories were obtained, from which the precise location of the warning unit was selected.

After conducting fractal dimension analysis based on spatiotemporal elements, the warning micro-unit numbers within the entire sequence were obtained. The corresponding coordinates were calculated through programming and digitized. The specific warning location units and the number of rows, columns, and layers of their numbers are shown in Figure 14. The detailed unit based warning locations are shown in

Table 3. Spatiotemporal element fractal dimension warning parameter table.

No.	Warning Unit Number	Classification of Fractal Dimension Values	Number of Rectangular Units per Layer	Layers	Number	Rows	Columns
1	529	D	529	1	0	23	23
2	2409	D	529	4	293	12	17
3	2799	D	529	5	154	6	16
4	4370	D	529	8	138	6	0
5	4386	D	529	8	154	6	16
6	5518	D	529	10	228	9	21
7	5906	D	529	11	87	3	18
8	7096	D	529	13	219	9	12
9	8707	D	529	16	243	10	13
10	10068	D	529	19	17	1	17
11	10128	D	529	19	77	3	8
12	11708	D	529	22	70	3	1

. The overall spatial domain is divided into 23 × 23 × 23, with unit sizes of 13.04 m, 8.70 m, and 10.87 m for the length, width, and height parameters. It can be specific to one-fifth of one mining area of a general mine, with the required accuracy meeting the requirements, while the level of regional disaster prediction is relatively obvious. There are multiple highly clustered warning units adjacent to each other within a local layer, indicating continuity within the layer and continuity between layers. Further warning analysis can be conducted by categorizing the following energy elements and their fractal dimensions.

Consistently with the spatiotemporal warning method, temporal nested warning divides energy parameters into appropriate segment levels based on the initial determination of spatiotemporal points. The energy fractal dimension is iteratively calculated and graded based on this, and the higher level, d, is selected as the warning. The specific high-level warning location information of energy elements is shown in

Table 4. Parameter table of energy element fractal dimension warning levels.

No.	Warning Unit Number	Classification of Fractal Dimension Values	Number of Units per Layer	Layers	Number	Rows	Columns
1	3986	d	529	7	283	12	7
2	8699	d	529	16	235	10	5
3	10173	d	529	19	122	5	7
4	10128	d	529	19	77	3	8

. The detailed unit-based warning location is shown in Figure 15, in the 7th, 16th, and 19th layers, respectively; the 19th layer has two consecutive intervals with higher fractal dimension levels related to energy cutting.

Based on the fractal dimension analysis of the aforementioned spatiotemporal elements and the analysis of the fractal dimensions of energy elements, the warning unit groups for two locations were preliminarily determined. The specific data of warning sequence numbers and row column layers are shown in

Table 5. Fractal dimension warning parameters for spatiotemporal energy elements.

Time–Energy Alert Number	Warning Unit Number	Layers	Number	Rows	Columns
1	8707	16	243	10	13
	8699	16	235	10	5
2	10128	19	77	3	8

. The first group consists of two units, which are extremely close in number, mainly due to them having similar numbers of layers, rows, and columns. Therefore, the potential damage caused by their composition is more obvious (as shown in the 16th layer of Figure 16). The second group consists of a single unit, but there is an overlapping unit of spatiotemporal warning and energy warning, indicating that the potential damage it may cause is not only highly concentrated in quantity, but also highly explosive in terms of energy release (as shown in the 19th layer of Figure 16). Therefore, analysis based on the comprehensive fractal dimensions of time, space, and energy can achieve better warning of potential location damage.

Overall, the warning system provides a nested analysis method for the spatial and temporal distribution statuses and trends of regional disasters in microseismic monitoring rupture sources, which is based on fractal theory box dimension. By utilizing the main spatial location and other parameter information of the massive microseismic monitoring events monitored, the macroscopic damage area and degree of damage were preliminarily determined; then, a fractal theory box dimension data statistical framework model was established. Finally, based on the clustering trends of event numbers, coordinate points within the research object, and the possibility and potential degree of energy explosion obtained from the box dimension data statistical framework model, the disaster occurrence locations in different regions and different danger levels were locked in, and prediction and warning based on fractal dimension were carried out.

This research on fractal warning for mining area disasters provides a microseismic monitoring method for predicting the energy dissipation trend of rupture sources. This method can globally simulate the distribution of rupture sources and the characteristics of energy release and dissipation in three-dimensional space, and lock in the area of local damage, thereby predicting and forecasting the damage area and degree. It can also predict the spatiotemporal evolution law of microseismic events, especially the aggregation/dissipation of energy in space. This method can simulate and obtain the fracture morphology and severity of the research object, especially the distribution of the energy fractal field, and predict the rock mass failure and instability state through the simulation results of micro-fracture source energy dissipation. It provides an accurate, reliable, and visual new method for studying the energy dissipation of rock mass fracture processes.

The time dimension early warning system utilizes micro-rupture source data and regulates different levels and quantities of rows and columns through programming code, overcoming the prediction situation of insufficient flexibility in existing spatial distribution and control variables. The data format is simple and highly applicable, with the capability of quickly simulating and displaying the distribution of attributes. By implementing grid partitioning and visual display through programming code, the accuracy-related and statistical difficulties in existing technologies have been overcome. This system is capable of automatically and quickly analyzing massive amounts of data, achieving the analysis of global trends and quantitative comparative analysis. According to this method, the energy release of mining rock engineering can be classified into levels to achieve early warning and prevention effects. The advantages of analyzing energy value data with scalar and scattered distributions are obvious, and the calculation speed is not limited, providing a foundation for the analysis of three-dimensional energy fields.

5. Discussion

The discussion regarding the limitations of this study is as follows:

Limitations in data sources: The study relies heavily on microseismic monitoring data, and does not fully incorporate other geological and stress field data, which may compromise the comprehensiveness of the early warning system. Limitations in model generalizability: The model is based on data from a specific mining area, and has not been validated for its applicability under different geological conditions, raising concerns about its generalizability. Insufficient dynamic adaptability: The model does not fully account for the dynamic evolution process of rock mass failure, which may affect the accuracy of long-term early warnings.

We compared the fractal dimension results of microseismic time–energy data in our study with the results of research by Chen G [38] et al. on the relationship between fractal characteristics and energy in coal–rock masses. We found consistency in the trends of failure progression and fractal dimension changes, although differences in rates were observed, likely due to variations in the physical and mechanical properties of different rock masses.

Additionally, we referenced the study by Mao H [39] et al. on the application of fractal dimensions of microseismic energy for the early warning of surrounding rock deformation, further validating the applicability of the fractal dimension warning model in different geological environments.

6. Conclusions

This article mainly studies an extraction method for time and energy elements and warning indicators in a monitoring system of mining area damage, establishes a methodical system for early warning of changes in the stability of mining rock structure and its regional damage, and proposes a comprehensive early warning fusion method. The main conclusions drawn are as follows:

(1)

Engineering background and data preprocessing: We summarized the intrinsic relationship between microseismic monitoring in mining sites and the stability of development production structures, clarified the actual engineering background and overview, identified and selected basic data, key elements, fractal data source sets, and proposed theories and methods.

(2)

Design of warning methods for the destruction of temporal energy elements: Based on the demand and activity analysis of the mining site, the macroscopic stability distribution characteristics of the rock mass structure in the mine were mastered. Combining existing data sets, time and energy elements were extracted, and fractal theory and fractal dimensions were used to interpret the fractal field for regional early warning. A practical early warning model was established, which was optimized and upgraded, forming a systematic method and system.

(3)

Fractal analysis and regional damage prediction based on time and space, with time and energy as the core elements, were achieved through the construction of a nested fractal dimension warning method. Mainly through methods such as data interpretation and scheme design, construction of nested analysis models for spatiotemporal and temporal energy dimensions, programming for fractal solution of temporal energy elements, classification of fractal dimension values, and division and identification of temporal energy fractal dimension warning forms, preliminary implementation of temporal energy fractal dimension warning prediction was achieved. After double nested dimension analysis, multiple regional potential clustering areas based on space–time were reasonably delineated, and then the high-level dimension values of adjacent or overlapping regions in the temporal energy dimension were comprehensively determined, to predict and warn of disasters in the area.