Research on the Development Height Prediction Model of Water-Conduction Fracture Zones under Conditions of Extremely Thin Coal Seam Mining

Abstract

Addressing the difficult problem of predicting the height of water-conducting fracture zones in shallow and thin coal seams, a prediction model of water-conduction fracture zones based on a backpropagation (BP) neural network was developed by integrating theoretical analysis, field measurements, and algorithmic advancements. Firstly, through overburden migration analysis and correlation tests, the height index system of the water-conducting fracture zone was determined. This system includes mining height, buried depth, dip angle, working face width, and overburden rock lithology, with five groups of characteristic parameters. Then, 35 pairs of minefield-measured data were collected to establish the measured height data set of the water-conducting fracture zone. Secondly, a BP neural network prediction model and a traditional support vector regression (SVR) prediction model were constructed based on a Pytorch framework, and the models were trained and tested by selecting data sets. Thirdly, the optimal prediction model was determined by comparing the model with the empirical model and multiple regression model of mining regulations for coal pillar maintenance and pressure in buildings, water bodies, railways, and main shafts. Finally, a typical mine was selected for application to verify the suitability of the optimal model. The results show that: (1) the predicted value of the neural network model is consistent with the change trend of the measured value, which accords with the theoretical law; (2) compared with traditional forecasting methods, the error of the BP neural network prediction model is stable and the prediction effect is the best; (3) dropout can effectively mitigate mitigation training overfitting, achieve regularization, and improve prediction accuracy; (4) the field application further verified that the BP neural network model is the best for predicting the height of water-conducting fracture zones of extremely thin coal seams, and the research results can provide technical guidance for similar fragile coal seams.

1. Introduction

Coal mining disrupts the initial equilibrium state of the rock mass, causing bending deformation in the overlying strata of the coal seam. This process forms mining fractures, known as water-conducting fracture zones [1]. The formation of water-conducting fracture zones provides an efficient channel for groundwater resources but also results in the degradation and destruction of surface ecology after coal mining [2]. Therefore, in order to regulate the coal industry, the State Administration of Coal Industry formulated the Regulations on Coal Pillar Maintenance and Coal Pressure Mining in Buildings, Water Bodies, Railways, and Main Shafts and Lanes [3], which is a standard to predict the height of water-conducting fracture zones by an empirical formula. However, since the cumulative height of a single layer involved in this empirical formula ranges from 1 to 3 m, its applicability is relatively limited in the case of extremely thin coal seams. Looking at the existing literature, few scholars have conducted in-depth research on this topic, and a targeted prediction method for extremely thin coal seams is urgently needed in the field of mining engineering.

The development height of water-conducting fracture zones is influenced by numerous factors. The main challenge in predictions is to determine the degree of influence of each factor on the water-conducting fracture zone to ensure the accuracy of the height prediction. Currently, scholars both domestically and internationally have conducted studies on predicting water-conducting fracture zones through theoretical analysis, engineering analogies, numerical simulations, physical simulation analyses, and field measurements. Many scholars have conducted research and exploration using masonry beam theory [1], key layer theory of overlying rock [4,5,6,7,8,9,10,11], transfer rock beam theory [12,13], etc. They have employed physical similarity simulation methods [14,15,16], numerical analysis methods [17], and numerical simulation software [18,19,20,21,22,23,24] to simulate the development laws of mining overlying rock in order to enhance prediction accuracy. The development, law, and mechanical mechanism of water fracturing induced by mining in various overburden structures are disclosed. Compared with theoretical analysis, numerical simulation, and other methods, field measurement is the most intuitive method to study the development of overlying rock fractures. Guo et al. [25], Lin et al. [26], Zhu et al. [27], and others have observed the height of water-conducting fracture zones in different areas through borehole observations. The analysis of the field-measured data provides a robust reference index for determining the depth of the water-conducting fracture zone in the mining area. In addition, due to the advancements in computer science, machine learning methods have been successfully applied to predict the development height of water-conducting fracture zones. Many domestic scholars have started to investigate more precise prediction models. Li et al. [28] established a prediction model for water-conducting fracture zones using a backpropagation (BP) neural network. The BP neural network prediction model has gradually gained attention from many scholars. Chai et al. [29] used a genetic algorithm–support vector regression (GA-SVR) to establish a prediction model for predicting the height of water-conducting fracture zones. Lou and Tan [30] used a particle swarm optimization–BP (PSO-BP) neural network to perform global optimization predictions for the height of water-conducting fracture zones. Shi et al. [31] adopted the PCA-GA-Elman optimization model to effectively mitigate the interaction between factors, enhancing the accuracy of the prediction results. However, a relatively large amount of data was needed. Wu et al. [32] utilized the radial basis function neural network (RBFNN) method to predict the development height of water-conducting fracture zones under fully mechanized mining conditions. These machine learning methods provide a new way to predict the height of water-conducting fracture zones more accurately. The scholars mentioned above have conducted a comprehensive and in-depth analysis of the development height of water-conducting fracture zones under various overlying rock structures. This study provides crucial support for predicting the development height of water-conducting fracture zones. However, previous studies have mainly focused on the general situation, and research on extremely thin coal seams is relatively limited.

Based on previous studies, we collected drilling data from the Niuwu Mining area in Fuxian County, China, focusing on the No. 3 extremely thin coal seam background. Firstly, the primary influencing factors were identified through correlation analysis. Secondly, IBMSPSS Statistics 27 software was used to construct both a multiple linear regression model and a multiple nonlinear regression model, while Python was utilized to develop a BP neural network model. By comparing the measured and predicted values, the evolution law of water-conducting fracture zones in extremely thin coal seams was analyzed. The applicability of the optimal model was verified to enhance the prediction of the development height of water-conducting fracture zones in very thin coal seams. This study provides a reference for further research on predicting the development height of water-conducting fracture zones.

2. Project Overview

The Fuxian Niuwu Mining area is located in Niuwu Town, Fuxian County, within the administrative division of Yan’an City, Shaanxi Province, China. The regional reserves of resources are 819 Wt, mining thickness is 0.5~0.69 m, and the average thickness of coal is 0.6 m. The mining area is located in the Yan’an slope of the Ordos Depression basin and displays typical stable platform characteristics. The strata are inclined to the northwest, and the structure is monoclinal with a gentle dip angle. The columnar shape of the rock strata at the top and bottom of the working face is illustrated in Figure 1.

The coal layer position is stable, and the structure is simple. There is a layer of 0.3–0.8 m mudstone between the No. 3 coal seam and the immediate roof. The direct base composition is fine-grained sandstone. The working face is arranged along an inclined direction for mining, and the goaf roof is managed using comprehensive mechanized mining and full caving methods. The inclination of the coal seam is 0°~3°, and the average is 1°. The occurrence characteristics of the No. 3 coal seam are illustrated in Figure 2.

3. Construction of Data Set on the Height of the Water-Conduction Fracture Zone

3.1. Construction of the Index System of the Height of the Water-Conducting Fracture Zone

Figure 3 illustrates that after coal seam mining, the original stress balance is disrupted, leading to subsequent damage to the overlying rock. According to the failure characteristics, the overlying rock is divided into a caving zone, a fracture zone, and a bending subsidence zone from top to bottom, with the first two being referred to as water-conducting fracture zones. With the advancement of the working face, the water-conducting fracture zone undergoes a developmental process from occurrence, rise, reaching maximum height, falling back, and eventually stabilizing. According to the final shape of the overburden failure range formed after the advancement of the working face to a certain extent, the vertical distance between the highest point of the development of the water-conducting fracture zone and the upper mining boundary is considered as the maximum height of the water-conducting fracture zone, referred to as the guiding height [33].

The water-conducting fracture zone is constrained by multiple factors, but the empirical “Three Rules” take into account only a few influencing factors. Based on previous research results [21], this paper conducts a quantitative analysis of multiple factors.

(1)

Mining thickness

Mining thickness affects the stress redistribution of roof strata and the failure range of overburden rock. It is the most critical factor for the failure of overburden rock and the development height of water-conducting fracture zones. In general, the greater the mining thickness, the higher is the height of the caving zone and water-conducting fracture zone, and the ratio is approximately linear [34].

(2)

Width of the working face

The width of the working face, which refers to the inclined layout distance of the working face, is an influential factor in determining the size of the mining space affecting the water-conducting fracture zone and the area of the mining space. In the initial stage of coal seam mining, the height of the water-conducting fracture zone gradually increases with the increase in working face width [35].

(3)

Mining Depth

The buried depth of a coal seam alters the initial stress of overlying rock, leading to the failure of the overlying rock under compression, tension, and shear stress. The greater the buried depth of a coal seam, the greater is the stress on the overlying rock, leading to a corresponding increase in the degree of damage to the overlying rock [34].

(4)

Overburden Lithology

The development of mining fractures in overlying rock is closely related to the mechanical properties of the overlying rock, which determine the strength of the rock layer. Brittle rock layers are prone to fracturing, while plastic rock layers are more likely to bend and deform without developing fractures easily. The harder the rock layer, the greater is the height of overlying rock failure and the water-conducting fracture zone [34,35,36,37].

(5)

Coal Seam Inclination

The change in the inclination angle of a coal seam mainly affects the failure pattern of overlying rock. The water-conducting fracture zone of a horizontal coal seam has a saddle shape with high ends and a low middle. In contrast, the caving rock of an inclined coal seam fills the lower goaf due to gravity, resulting in the water-conducting fissure zone taking on an upper and lower shape [34].

(6)

Mining Methods

The influence of the mining method on the height of the water-conducting fracture zone mainly depends on the size of the mining space and the movement form of caving rock in the goaf [37].

In the study of water-conducting fracture zones under fully mechanized mining, both domestic and foreign scholars have identified mining thickness, mining depth, working face width, coal seam inclination angle, and overlying rock lithology as the primary factors influencing the height of water-conducting fracture zones.

3.2. Data Set Construction

By collecting a large number of field-measured data, we summarized a total of 35 examples of the height of water-conducting fracture zones in extremely thin coal seams, as shown in

Table 1. Measured values and influencing factors of the mining overburden height.

Number	Face Drilling	Mining Thickness/m	Mining Depth/m	Dip Angle/°	Width/m	Rock Structure	Height of Water-Conduction Fracture Zone/m
1	GJW1303	0.57	67	2	119	hard–soft	9.7
2	HH3201	0.64	50	1	110	hard–soft	7.3
3	AZ1201	0.55	30	3	100	hard–soft	10.2
4	HC1107	0.84	50	3	142	hard–soft	9.5
5	HC2	0.85	76	2	142	hard–soft	10.6
6	AZ1	0.61	70	1	100	soft–hard	7.3
7	AZ2	0.55	82	3	100	soft–hard	6.9
8	N1	0.60	60	1	120	hard–soft	8
9	N3	0.70	59	3	128	soft–soft	8.7
10	N5	0.40	79	2	128	soft–soft	6.1
11	KN1	0.70	145	0	130	hard–soft	8.5
12	M1	0.58	123	3	119	hard–soft	6.6
13	M7	0.54	106	1	120	soft–hard	6.7
14	M9	0.57	95	2	125	soft–hard	7.2
15	X1	0.55	58	2	150	hard–soft	6.9
16	N2	0.62	90	2	140	hard–soft	7.7
17	C1	0.63	67	1	170	soft–hard	8.2
18	C2	0.59	72	3	170	soft–hard	7.4
19	N2	0.51	65	3	120	hard–soft	6.5
20	C3	0.66	95	3	170	soft–hard	8.6
21	N6	0.50	75	2	140	hard–soft	9.2
22	F5	0.67	100	1	135	soft–soft	8.4
23	G1	0.53	103	2	121	soft –soft	6.5
24	G2	0.69	85	1	123	soft–hard	8.8
25	G3	0.49	90	3	125	hard–soft	8.4
26	G4	0.61	65	2	120	hard–hard	7.9
27	G5	0.66	71	2	140	hard–hard	8.2
28	A1	0.58	78	1	145	soft–hard	7.6
29	A2	0.72	80	1	100	soft–hard	9.1
30	A3	0.64	87	1	142	soft–soft	8.8
31	H1	0.55	90	1	125	hard–soft	8.4
32	H2	0.67	108	2	155	soft–hard	9.1
33	Q5	0.59	82	3	145	soft–soft	7.9
34	X5	0.49	75	2	145	hard–hard	8.7
35	H3	0.69	120	2	160	hard–soft	10

. According to the examples summarized in Table 1, the maximum mining depth (H) is 145 m, and the minimum mining depth (H) is 50 m. The maximum value of coal seam inclination, A, is 3°, and the minimum value is 0°. The maximum thickness (M) of the coal seam is 0.85 m, while the minimum thickness is 0.4 m. The maximum value of the working face width (L) is 170 m, while the minimum value is 100 m. The hardness of a coal seam is classified according to the Platts coefficient. The hardness of the No. 3 coal seam in the Niuwu Mining area is soft coal, with a value of 0.4. Previous studies have categorized the rock structure based on the lithology characteristics of the overlying rock into hard–hard, soft–hard, hard–soft, and soft–soft, with values of 1, 0.75, 0.5, and 0.25, respectively. The above data are normalized according to the following formula, and the input and output data are converted into interval (−1,1) dimensional values. The normalization processing formula x_n [38] is:

$$x_n={\displaystyle \frac{2(X_n-X_{\min })}{X_{\max }-X_{\min }}}-1,$$

(1)

where: x_n is the n^th normalized data of various parameters; X_n is the n^th data of each parameter. X_max and X_min represent the maximum and minimum values of each parameter.

Normalization is a common data preprocessing method whose main purpose is to scale the data to the same range and eliminate dimensional differences between features. Dimensional difference refers to the difference in the value range and units of different features, which affects the performance and stability of the model. Through normalization processing, the value range of different features can be scaled to the same interval, so as to eliminate the impact of dimensional differences and improve the performance and stability of the model.

Normalization is widely used in machine learning to improve the performance and stability of models. In feature engineering, normalization can be used to scale the value range of different features to the same interval to improve the performance and stability of the model. In image processing, normalization can be used to scale the pixel value of the image to the range of (0,1), which is convenient for subsequent processing and analysis. In natural language processing, normalization can be used to convert text data into numerical vectors for easy processing and analysis by machine learning algorithms.

The data in Table 1 were normalized, and the processing results are presented in

Table 2. Sample data processing results.

Number	Face Drilling	Mining Thickness/m	Mining Depth/m	Dip Angle/°	Width/m	Rock Structure	Height of Water-Conduction Fracture Zone/m
1	GJW1303	−0.244	−0.642	0.333	−0.457	−0.333	0.600
2	HH3201	0.067	−1.000	−0.333	−0.714	−0.333	−0.467
3	AZ1201	−0.333	0.684	1.000	−1.000	−0.333	0.822
4	HC1107	0.956	−1.000	1.000	0.200	−0.333	0.511
5	HC2	1.000	−0.453	0.333	0.200	−0.333	1.000
6	AZ1	−0.067	−0.579	−0.333	−1.000	0.333	−0.467
7	AZ2	−0.333	−0.326	1.000	−1.000	0.333	−0.644
8	N1	−0.111	−0.789	−0.333	−0.429	−0.333	−0.156
9	N3	0.333	−0.811	1.000	−0.200	−1.000	0.156
10	N5	−1.000	−0.389	0.333	−0.200	−1.000	−1.000
11	KN1	0.333	1.000	−1.000	−0.143	−0.333	0.067
12	M1	−0.200	0.537	1.000	−0.457	−0.333	−0.778
13	M7	−0.378	0.179	−0.333	−0.429	0.333	−0.733
14	M9	−0.244	−0.053	0.333	−0.286	0.333	−0.511
15	X1	−0.333	−0.832	0.333	0.429	−0.333	−0.644
16	N2	−0.022	−0.158	0.333	0.143	−0.333	−0.289
17	C1	0.022	−0.642	−0.333	1.000	0.333	−0.067
18	C2	−0.156	−0.537	1.000	1.000	0.333	−0.422
19	N2	−0.511	−0.684	1.000	−0.429	−0.333	−0.822
20	C3	0.156	−0.053	1.000	1.000	0.333	0.111
21	N6	−0.556	−0.474	0.333	0.143	−0.333	0.378
22	F5	0.200	0.053	−0.333	0.000	−1.000	0.022
23	G1	−0.422	0.116	0.333	−0.400	−1.000	−0.822
24	G2	0.289	−0.263	−0.333	−0.343	0.333	0.200
25	G3	−0.600	−0.158	1.000	−0.286	−0.333	0.022
26	G4	−0.067	−0.684	0.333	−0.429	1.000	−0.200
27	G5	0.156	−0.558	0.333	0.143	1.000	−0.067
28	A1	−0.200	−0.411	−0.333	0.286	0.333	−0.333
29	A2	0.422	−0.368	−0.333	−1.000	0.333	0.333
30	A3	0.067	−0.221	−0.333	0.200	−1.000	0.244
31	H1	−0.333	−0.158	−0.333	−0.286	−0.333	0.022
32	H2	0.200	0.221	0.333	0.571	0.333	0.333
33	Q5	−0.156	−0.326	1.000	0.286	−1.000	−0.200
34	X5	−0.600	−0.474	0.333	0.286	1.000	0.156
35	H3	0.289	0.474	0.333	0.714	−0.333	0.733

.

4. Multiple Regression Prediction Model

Based on previous research results [39,40,41], it has been found that the development of water-conducting fracture zones in mining overlying rock is not solely influenced by a single factor. The development of conductivity height is restricted by multiple factors, and each factor exhibits a linear relationship with the water-conducting fracture zone. The empirical formula of the “Three Rules” considers only a few factors, leading to limited adaptability and ineffective application of the prediction model. Based on correlation analysis, we selected mining thickness, mining depth, dip angle, working face width, and overburden lithology as multiple regression indicators, as described in

Table 3. Analysis of the correlation between each factor and the water-conducting fracture zone.

		Rock Structure	Mining Thickness/m	Mining Depth/m	Dip Angle/°	Width/m
Height of water-conduction fracture zone/m	Pearson correlation	0.395 *	0.607 **	0.419 *	0.376 *	0.391 *
	Significance (double tail)	0.031	0.000	0.021	0.041	0.033
	N	30	30	30	30	30

Notes: * The correlation is significant when the confidence (double test) is 0.05; ** The correlation is significant when the confidence (double measure) is 0.01.

. The model was established, and the prediction results were analyzed.

Table 3 illustrates that at a significance level of 0.05, the height of the water-conducting fracture zone is significantly positively correlated with the rock structure, mining depth, dip angle, and width. The correlation coefficients are 0.395, 0.419, 0.376, and 0.391, respectively, with corresponding significance levels of 0.031, 0.021, 0.041, and 0.033, all of which are less than 0.05. When the significance level is 0.01, there is a significant positive correlation between the height of the water-conducting fracture zone and the thickness of the coal seam. The correlation coefficient is 0.607, and the significance is 0, which is less than 0.01.

According to the correlation analysis of the above five factors, mining thickness has the greatest influence on the water-conducting fracture zone, followed by mining depth, overlying rock lithology, working face width, and finally the dip angle. It can be seen that the influence of the five factors on the water-conducting fracture zone is also strong and weak.

4.1. Modeling Analysis of Multiple Linear Regression Model

Multiple linear regression involves two or more influential factors as independent variables to explain changes in the dependent variable. When there is a positive correlation between the five factors and the height of the water-conducting fracture zone, the correlation coefficient is positive, and the regression coefficient is also positive, indicating that an increase in the independent variable will lead to an increase in the dependent variable. When there is a negative correlation between the independent variable and the dependent variable, the correlation coefficient is negative, and the regression coefficient is also negative, indicating that an increase in the independent variable will lead to a decrease in the dependent variable. Based on the results obtained in Table 3, the correlation of various factors was analyzed. The five factors were all positively correlated with the height of the water-conducting fracture zone; therefore, the five factors, namely mining thickness, mining depth, dip angle, working face width, and overlying rock lithology, were adopted for modeling.

The five factors were taken as horizontal coordinates, while the development height of the water-conducting fracture zone was taken as the vertical coordinate to create scatter plots (refer to Figure 4).

This method is used to determine whether the five factors have a linear relationship with the height of the water-conducting fracture zone. The collected 30 sample data are in good agreement with the trend line in Figure 4, indicating a linear positive correlation between the various factors and the development height of the water-conducting fracture zone.

SPSS statistical software was used to conduct linear regression analysis on mining thickness, mining depth, dip angle, working face width, overburden lithology, and water-conducting fracture zone. The results of the regression analysis are as follows.

Table 4. Analysis of variance of sample data.

Model		Sum of Squares	Degrees of Freedom	Mean Square	F	Significance
	Regression	36.818	5	7.364	104.171	0.000
	Residual error	1.697	24	0.071
	Total	38.515	29

Notes: 1 Dependent variable: height of water-conducting fracture zone/m; 2 Predictive variables: (constant), working face width/m, dip angle/°, mining depth/m, mining thickness/m, rock structure.

presents the F-test results of the regression equation, and the significance value of p = 0.000 (<0.01) indicates that the linear regression equation is statistically significant. Table 4 also presents the T-test results of the regression coefficient, indicating a proportional relationship between various factors and the development height of the water-conducting fracture zone.

Multiple linear regression is a statistical method used to examine the relationship between a dependent variable (Y) and multiple independent variables (X₁, X₂… X_n) through a linear regression equation. It helps to understand the relationship between the variables. The expression is given by Equation (2):

$$Y=a_0+a_1{x}_1+a_2{x}_2+\cdots +a_k{x}_k+b_0,$$

(2)

In Equation (2), where b₀ is the constant term, a₁, a₂…… a_k are the partial regression coefficients for Y, and x₁, x₂… x_k are the arguments.

According to the results presented in

Table 5. Undetermined coefficient obtained by sample fitting.

Model		Unnormalized Coefficient		Standardization Coefficient	t	Significance
		B	Standard Error	Beta
1	(constant)	0.938	1.453		0.645	0.525
	Mining thickness/m	2.673	1.094	0.207	2.443	0.022
	Mining depth/m	0.009	0.004	0.149	2.153	0.042
	Dip Angle/°	0.491	0.129	0.303	3.818	0.001
	Width/m	0.036	0.012	0.237	3.069	0.005
	Rock structure	−1.132	0.369	−0.218	−3.067	0.005

Note: 1 Dependent variable: height of water-conducting fracture zone/m.

and the significance of each factor, the five multifactorial variables of mining thickness (M), mining depth (H), dip angle (A), working face width (L), rock structure (J), and the height of the water-conducting fracture zone were utilized to analyze the 30 data points collected in Table 1. SPSS was employed for conducting multivariate linear regression analysis on the aforementioned data. The fitting results are shown in Equation (3):

$$Y_1=2.673M+0.009H+0.491A+0.036L-1.132J;$$

(3)

where M represents the thickness in meters, H represents the mining depth in meters, A represents the angle in degrees, L represents the width in meters, and J represents the rock structure. According to relevant statistical knowledge, the closer the goodness-of-fit measure R is to 1, the higher is the degree of fit and the better is the model’s effectiveness. The model R = 0.629 indicates that the prediction model for water-conducting fracture zones has a general effect, and the prediction accuracy aligns well with the measured results. This alignment is primarily attributed to the limited amount of data available.

4.2. Modeling and Analysis of Multiple Regression Nonlinear Model

Multivariate nonlinear regression prediction models aim to fit a nonlinear function to sample data and then use the fitting function to predict the response value. Commonly used function forms include power functions, exponential functions, and logarithmic functions. Among them, the power function exhibits strong generalization ability, and the prediction model selects the power function as the form of its prediction model. The basic form is as follows:

$$Y=b_0+b_1{x_1}^{c_1}+b_2{x_2}^{c_2}+\cdots \cdots {\mathrm{b}}_3{x}_1{x}_2+\cdots \cdots .$$

(4)

Since the model is based on the formation conditions in Fuxian County, northern Shaanxi Province, it is analyzed according to the occurrence of coal seams in the study area. Based on the method of multivariate nonlinear regression analysis, SPSS software was used to establish the regression model, and the variables were modeled sequentially. The mathematical model constructed is as follows:

$$Y_2=a_{0}M+a_1{M}^2+b_{1}H+b_2{H}^2+b_3{H}^3+c_{1}J+c_2{J}^2+c_3{J}^3+b_0,$$

(5)

$$Y_3=a_{0}M+a_1{M}^2+b_{1}H+b_2{H}^2+b_3{H}^3+c_{1}J+c_2{J}^2+c_3{J}^3+b_0,$$

(6)

$$Y_4=a_{0}M+a_1{M}^2+b_{1}H+b_2{H}^2+b_3{H}^3+c_{1}J+c_2{J}^2+c_3{J}^3+b_0,$$

(7)

where M is the mining thickness, H is the mining depth, and J is the rock structure.

The model was calculated by SPSS software, and 27 parameters of Equations (5)–(7) above needed to be fitted. a₀, a₁, b₁, b₂, b₃, c₁, c₂, c₃, and b₀ in the Equations citation (5)–(7) above are the factor coefficients in the three formulas, respectively, and the parameter fitting results are shown in

Table 6. Fitting parameters of the prediction model of the water-conducting fracture zone.

Reference Parameter
1°	a0	a1	b0	b1	b2	b3	c1	c2	c3
	10.108	0	−17.385	0.508	−0.005	1.305 × 10−5	12.47	−13.195	0
2°	a0	a1	b0	b1	b2	b3	c1	c2	c3
	5.79	0	−17.659	0.381	−0.002	0	44.046	−73.461	36.749
3°	a0	a1	b0	b1	b2	b3	c1	c2	c3
	−30.162	29.343	12.781	0.055	0	0	−1.664	0	0

below.

The obtained multivariate nonlinear regression fitting formula is shown in the following Equations (8)–(10):

$$Y_2=10.108M-17.385+0.508H-0.005H^2+1.305E-5H^3+12.47J-13.195J^2$$

(8)

$$Y_3=5.79M+0.381H-0.002H^2+44.046J-73.461J^2+36.749J^3-17.659$$

(9)

$$Y_4=12.781-30.162M+29.343M^2+0.055H-1.664J$$

(10)

R² was used to evaluate the model’s effectiveness after fitting. After calculations, the corrected R² values of 0.923, 0.759, and 0.435 under three different inclination angles are all high, indicating a strong fitting effect.

4.3. Verification of Multiple Regression Prediction Model

According to the multiple linear regression prediction model and multiple nonlinear regression prediction model established above, the predicted values of Y₁, Y₂, Y₃, and Y₄ water-conducting fracture zones (Y₁, Y₂, Y₃, and Y₄) in overlying rocks can be obtained under specific mining conditions.

The predicted value obtained based on the multiple linear regression prediction model was compared with the empirical formula of the Three Rules. Figure 5 illustrates that the trend change of the multiple linear regression prediction model is consistent with the measured value of the water-conducting fracture zone. The maximum absolute error is 4.5 m, and the maximum relative error is 58%. The multiple nonlinear regression prediction model remains consistent with the measured value of the water-conducting fracture zone. The maximum absolute error is 2.1 m, and the maximum relative error is 23%, and the overall trend is the same. The error of the predicted value obtained by the three-part procedure is relatively large, and the practicability for extremely thin coal seams is low. Although the prediction accuracy of the multiple linear regression model is lower than that of the multi-part nonlinear regression model, its accuracy is still higher than that of the three-part procedure. It shows that both the multiple linear regression model and the multiple nonlinear regression model can effectively predict the height of the water-conducting fracture zone in very thin coal seams. The above prediction model can obtain the corresponding prediction value of the development height of the water-conducting fracture zone by inputting the formation parameters of very thin coal seams, and more accurate prediction value can be obtained by the multi-factor index, which indicates that the prediction method has high universality.

5. BP Neural Network Model

5.1. Construction of BP Neural Network Model

The BP neural network is one of the widely used neural network models at present. The neural network model creates a multi-dimensional input and output function based on the training sample and utilizes this function to make predictions. The training process of the network involves adjusting the parameters of the function to enhance prediction accuracy. Due to its strong nonlinear mapping capability, self-learning, and generalization, artificial intelligence is extensively utilized in data processing, machine control, and pattern recognition.

According to the principle of sample selection for BP neural network construction, it is necessary to quantify the factors affecting the development height of water-conducting fracture zones. Based on the measured data on the development height of the water-conducting fracture zone of an extremely thin coal seam in Fuxian County, northern Shaanxi Province, 35 field measured data were selected as the learning training samples and test samples of the BP neural network (see Table 1), of which 30 were trained samples. Five test samples were used to evaluate the generalization ability of the final prediction model. When building the BP neural network model, we used Python language to write programs for simulation training and build a Pytorch framework to realize the BP neural network prediction model, and we used Pytorch to build a framework to further use Python language for quantitative data processing and intended to use mean squared error (MSE) as a loss function. The rectified linear unit (ReLu) was processed as an incentive function. The general steps of the BP neural network algorithm are shown in Figure 6.

This paper aims to utilize the enhanced three-layer BP model with one hidden layer. The basic structure is as follows: the input layer consists of five nodes, namely, mining thickness (M), mining depth (H), coal seam inclination (A), working face width (L), and rock structure (J). The output layer comprises one node, indicating the height (H_f) of the water-conducting fracture zone. For the hidden layer, the number of nodes is selected as 128 based on multiple training and experience calculations. The structure of the BP neural network model is shown in Figure 7. Through the continuous use of “signal forward calculation” and “error back propagation” reciprocating until the global error approaches the minimum value, the purpose of the prediction is finally achieved.

5.2. Network Learning and Training

The parameters of the model were conservatively selected to prioritize the stability of the model over training speed, so the three- and four-layer BP neural network were used as the training models. The loss function selected was the MSE function, and the activation function chosen was the ReLu function.

MSE is a commonly used loss function in regression tasks. It measures the average squared error between the predicted value and the actual value of the model. The specific formula is as follows:

$$MSE={\displaystyle \frac{1}{n}}{\displaystyle \sum {(x_i-y_i)}^2},$$

(11)

where n is the number of samples, x_i is the true value of the i^th sample, and y_i is the predicted value of the model for the i^th sample.

The ReLu function is a type of activation function commonly utilized in artificial neural networks. In a mathematical sense, it refers to the slope function, namely:

$$f(x)=\max (0,x),$$

(12)

The corresponding function image is shown in Figure 8.

The initial learning rate was set to 0.0001, and MSE, R-squared determination coefficient, training duration, and model complexity were chosen as evaluation metrics. The results are shown in

Table 7. Parameters for training the model.

Model Name	Input Layer	Hidden Layer	Output Layer
3-layer BP neural network	1	1	1
4-layer BP neural network	1	2 (128 neurons/layer)	1
5-layer BP neural network	1	3 (128 neurons/layer)	1

below.

The core idea of support vector regression (SVR) is to find an optimal regression hyperplane in the high-level space by mapping the input space. The goal of SVR is to balance model complexity and error, and finally obtain a regression model with good generalization ability. Its mathematical expression is:

$$f(x)=\omega x+b$$

(13)

To ensure that the prediction error of all data points (xi, yi) is within ε, while minimizing ‖ω‖2, or expressed as $\min {\displaystyle \frac{1}{2}}{\Vert \omega \Vert }^2$, its constraint conditions are:

$$\begin{array}{l}\left|y_i-(\omega x_i+b)\right|\le \epsilon \\ {\xi }_i\ge 0,{{\xi }_i}^{※}\ge 0\end{array}$$

(14)

The relaxation variables of type ${\xi }_i$ and ${{\xi }_i}^{※}$ are used to process data points that cannot fall within ε.

By introducing the concept of kernel function $K(v1,v2)$, the complex inner product calculation of high dimensional space is transformed into the substitution calculation of a general function. Then, according to the principle of minimum classification interval, a regression model is established:

$$f(x)={\displaystyle \sum }_i^j({\alpha }_i^{※}-{\alpha }_i)K(x_i,x)+b,$$

(15)

$$b=y_j-{\displaystyle \sum }_i^j({\alpha }_i-{\alpha }_i^{※})(x_i,x_j)+\epsilon ,$$

(16)

where ε is the precision coefficient. The training samples whose Lagrange coefficient $({\alpha }^{※}-\alpha )$ is not 0 are the “support vectors”. The regression model is then used for data fitting and prediction. The diagram is shown in Figure 9.

As shown in

Table 8. Evaluation results of the training model.

Model Name	MSE	R2 Coefficient of Determination	Training Duration	Model Volume
3-layer BP neural network	0.15	0.8	35 min	180 k
4-layer BP neural network	0.12	0.9	43 min	203 k
5-layer BP neural network	0.09	0.95	65 min	230 k

, the coefficients of determination (R²) for the three-, four-, and five-layer BP neural network prediction models are all greater than or equal to 0.8. Additionally, the MSE is less than 0.2, suggesting a high level of prediction accuracy for this neural network. Therefore, it is feasible to use a BP neural network to predict the height of water-conducting fracture zones.

5.3. Analysis of Prediction Results

The closer the predicted value of the training sample is to the measured value, the better is the degree of fit of the model. This indicates the model’s stronger ability to describe the sample, proving the algorithm’s effectiveness in predicting the height of the water-conducting fracture zone.

The BP algorithm was used to train the established neural network model. Figure 10 shows the comparison between the predicted value and the measured value of the sample obtained by the BP neural network algorithm, the traditional SVR algorithm, and the linear model. According to the accuracy of model fitting, the prediction accuracy of BP neural network reached 93%, and that of SVR algorithm reached 83%. The lowest prediction accuracy of linear regression is only 63%. By comparison, it can be seen that SVR algorithm and BP neural network prediction model have a better fit with the sample data, and the fitting error of the linear model is large, which is due to the different manifestations of the three prediction methods. As shown in Figure 10, the BP neural network model has the most accurate prediction value. Moreover, the training error obtained after iterative training is the smallest, which indicates the best fit to the measured results. Compared with other prediction methods, the BP neural network has good self-learning and adaptive ability, which can automatically extract “reasonable rules” between input and output data through learning, and adaptive memory of the learned content in the network weight. In addition, the BP neural network also has good generalization ability and fault tolerance ability, which are more comprehensive compared to other prediction methods.

Generally speaking, BP neural networks have high accuracy and stability in sample prediction fitting.

5.4. Model Test

To verify the accuracy of the modified model, five groups of data (No. 31 to 35) from Table 1 were selected as test samples. These test samples were used to calculate and validate the trained neural network. The results obtained from the neural network calculations were then compared with the empirical formula from literature [3] and the field-measured results, as presented in

Table 9. Comparison between the calculated and measured values of the water-conduction fracture zone.

Number	Water-Conduction Fracture Zone Height /m				Comparison of Theoretical and Measured Values		Comparison between the Calculated Value and the Measured Value of Neural Network		SVR Algorithm Predicted Value Compared with the Measured Value
	Measured Value	BP Algorithm Predicted Value	SVR Regression Algorithm Predicted Value	Rule Formula Value	Absolute Error/m	Relative Error/%	Absolute Error/m	Relative Error/%	Absolute Error/m	Relative Error/%
31	10.2	10.12	8.90	12.28	2.28	22	0.88	8	1.3	13
32	8.0	7.98	7.62	5.5	2.5	31	0.02	0	0.38	5
33	8.7	8.55	8.30	6.1	2.6	30	0.15	2	0.4	5
34	8.4	8.3	7.90	5.9	2.5	30	0.1	1	0.5	6
35	9.1	9.1	8.90	15.2	6.1	67	0	0	0.2	2

.

Both BP neural networks and SVR algorithms are typical methods in prediction models, but the goal of BP neural networks is to ensure that the training error between the network output and the ideal output is the smallest and the closest to reality. According to Table 8, the predicted value of the BP neural network is basically consistent with the actual value. In general, the BP neural network has high accuracy in the prediction of test sample data and is suitable for the prediction of water-conducting fracture zones under this condition. The relative error and absolute error of the BP neural network are smaller than those of the SVR algorithm and empirical formula. The maximum absolute error and the maximum relative error of the theoretical value calculated by the rule formula in reference [3] are 6.1 m and 67%, respectively. The maximum absolute error and maximum relative error of the fracture zone height calculated by the neural network model using the BP algorithm are 0.88 m and 8%, respectively. The maximum absolute error and the maximum relative error between the predicted and measured values by SVR algorithm are 1.3 m and 13%, respectively. This is because the BP neural network has good fault tolerance by using “error backpropagation”, reducing the correlation between factors and improving the degree of model prediction. It shows that the neural network model constructed by the BP algorithm is closer to the results calculated by the SVR algorithm and empirical formula, and the prediction effect is better, which meets the needs of engineering. The main reason is that the traditional formula takes few factors into account, and there is no special explanation for the extremely thin coal seam. The BP neural network model involves more comprehensive factors affecting the water-conducting fracture zone, so it can achieve a higher degree of fitting.

6. Application Example

Based on previous research, we constructed a multiple linear regression model, multiple nonlinear regression model, SVR algorithm prediction model, and a BP neural network prediction model. With the No. 11503 working face in the first pan area of Luzegou Mine, Longzhen Coal Mine, Mizhi County, Yulin City, as the background, the measurement method is illustrated in Figure 11. Absolute error and relative error were chosen as indicators to evaluate the above models and the optimal model was then selected.

Based on the methods mentioned above, the analysis indicates that the height of the water-conducting fissure zone in the No. 11503 working face of the first pan area of Luzegou mine is 9.2 m. Additionally, the the BP neural network model was employed to forecast the height of the water-conducting fissure zone in the No. 11503 working face of Luzegou mine. Firstly, the quantitative data indicate that the mining depth of the working face is 60 m, the coal seam has a 3° inclination, the coal thickness is 0.75 m, the working face width is 100 m, the coal seam consists of soft coal, and the overlying rock structure is classified as hard–soft structure.

The flow chart depicted in Figure 12 serves as the input sample for BP neural network prediction. All variables of the working surface are normalized before being inserted into the four prediction models mentioned above. After multiple calculations, the height of the water-conducting fracture zone of the working surface is 9.08 m.

By calculating the absolute error and relative error, the prediction results of the multiple linear regression model, multiple nonlinear regression model, SVR algorithm prediction model, and BP neural network model are compared with the calculation results of the formula. The results are shown in Figure 13.

For the No. 11503 working face of Luzegou mine, the absolute error between the predicted value and the measured value by the multiple linear regression model is 0.15 m, and the relative error is 1.6%. The absolute error between the predicted value and the measured value is 0.17 m, and the relative error is 1.8%. The absolute error between the predicted value and the measured value using the BP neural network model is 0.12 m, and the relative error is 1%. The absolute error between the calculated value and the measured value is 1.0 m, and the relative error is 11%. The absolute error between the predicted value and the measured value using the SVR algorithm is 0.3 m, and the relative error is 3%. Therefore, the BP neural network model demonstrates the highest accuracy in predicting the results of this working surface.

7. Conclusions

(1)

The multiple regression linear model and the multiple regression nonlinear model of extremely thin coal seam show that the trend change between the measured value and the predicted value is consistent, and the error range is obviously reduced with the formula in the Three Gorges regulation, indicating that the regression prediction model is effective in predicting the height of the water-conducting fracture zone. It provides theoretical guidance for the field measurement of water-conducting fracture zones.

(2)

Through comparative analysis, multiple linear regression, multi-distance nonlinear regression, and BP neural network can all improve the prediction accuracy of the height of the water-conducting fracture zone. Among several prediction methods, the BP neural network had the best effect.

(3)

The BP neural network, multiple linear regression model, multi-distance nonlinear regression model, and traditional three-order empirical formula were compared to analyze their relative error and absolute error. The BP neural network has good fault tolerance by using “error backpropagation”, reducing the correlation between factors and improving the degree of model prediction. Moreover, the prediction accuracy of the BP neural network is as high as 93%, and the predicted result is basically consistent with the real value, which proves the accuracy of the model, and the application effect is good. This method can be used to predict the height of mine water-conduction fracture zones and guide mine safety production.

(4)

Since there is limited research on mining extremely thin coal seams both domestically and internationally, the height of the water-conducting fracture zone will be influenced by various factors during coal seam mining. Since our research focuses on a single mining area, the considered factors are relatively limited. In the future, it is advisable to explore additional influencing factors and conduct quantitative analysis using various modified optimal models. By doing so, the accuracy of the optimal model will be improved, providing a reference value for predicting the development height of water-conducting fracture zones under mining conditions of extremely thin coal seams.