Response Surface Analysis on Multiple Parameter Effects on Borehole Gas Extraction Efficiency

Abstract

To explore the impact of different factors on the effectiveness of borehole gas extraction, in situ stress tests were conducted in a test mining area. A theoretical model of gas migration within the coal matrix–fracture system was established. Based on field data, a numerical model was constructed to study the variation patterns of the effective extraction radius under different extraction conditions. Using the response surface methodology, the interactions of different factors and their impact on the effective extraction radius were analyzed, resulting in a response surface model for each factor and the effective extraction radius. The results indicate that the initial permeability of the coal seam has the greatest impact on the extraction radius, with a maximum range of 2.027 m. The influence of extraction time, extraction negative pressure, and borehole diameter decreases sequentially. The borehole diameter has the least impact, with a range of 0.608 m. The response surface model has good significance, with a coefficient of determination (R²) of 0.9957, and it can explain over 99.57% of the response values. The response surface between the initial permeability of the coal seam and extraction time shows the greatest degree of distortion, indicating a significant interaction effect on the extraction radius. In contrast, the response surface between extraction time and extraction negative pressure shows the least degree of distortion, indicating that their interaction effect is the least significant. These findings can provide a theoretical reference for improving borehole design and enhancing gas extraction efficiency.

1. Introduction

Coal accounts for 56.0% of China’s total energy consumption. Compared with previous years, although the proportion of coal production and consumption has decreased, it still occupies a dominant position in China’s energy structure and will continue to play an important role in China’s future economic development [1,2]. With the increase in coal mining depth in China, geological conditions have become more and more complex, and the probability of coal mine disasters is also increasing [3]. Compared with other coal mine accidents, gas accidents have always been one of the largest types of accidents, with the largest danger and the highest death rate in coal mines [4], so the containment of gas accidents cannot be ignored. Gas extraction is one of the important methods to prevent gas accidents, and the effect of gas extraction is affected by many factors.

Based on theories of gas flow and elastic mechanics, Shang et al. [5] conducted numerical simulations to study the effects of extraction negative pressure, borehole diameter, extraction time, and the initial permeability of the coal seam on the effective extraction radius. A multiple linear regression model of the influencing factors on the effective extraction radius was established. Chen et al. [6] used COMSOL Multiphysics numerical software to establish a three-dimensional numerical model of gas extraction boreholes. They analyzed the evolution pattern of gas pressure in the coal body and quantitatively studied the influence of borehole spacing on gas extraction effectiveness. Wang et al. [7] conducted qualitative and quantitative analyses of the extraction efficiency by examining three main factors influencing gas extraction: coal seam permeability, gas pressure, and burial depth. Fang et al. [8] used COMSOL software (version 6.2) to analyze the factors affecting gas occurrence and further investigated the impact of differences in gas pressure distribution on the effective extraction radius. Wang et al. [9] established a gas flow–solid coupling model based on stress and transport equations, used COMSOL software for numerical simulations of various factors affecting gas extraction efficiency, and obtained the distribution patterns of gas pressure. Wei et al. [10], addressing the creep characteristics of soft coal, established a flow–solid coupling mathematical model considering the creep characteristics of coal based on the permeability dynamic evolution equation that accounts for matrix shrinkage and effective stress. Using COMSOL Multiphysics software, they calculated the gas extraction volumes for single and multiple boreholes, optimized the gas extraction parameters, and conducted field trial applications. Zhang et al. [11] studied the relationship between gas geology (initial gas pressure, initial permeability, and burial depth) and the effective extraction radius, achieving precise borehole placement for gas extraction. Gao et al. [12] established a gas–air mixed flow model to study the effects of extraction time, negative extraction pressure, and extraction leakage on gas concentration, revealing the mechanism of gas extraction leakage. Xu et al. [13] established a gas seepage unit model around the borehole, derived the theoretical mathematical expression for calculating the effective extraction radius, conducted numerical simulations of gas extraction using computational fluid dynamics software, and obtained the effective extraction radius. To study the gas seepage characteristics in highly anisotropic coal seams, Nian et al. [14] established an anisotropic dual-porosity model and analyzed the impact of permeability anisotropy on gas pressure, gas extraction volume, and changes in the effective extraction radius. Additionally, they investigated the mechanisms by which ground stress, initial gas pressure, ultimate adsorption strain, and the Langmuir volume constant affect permeability anisotropy and gas extraction volume. Based on the gray correlation analysis of factors affecting gas extraction, such as borehole diameter, borehole spacing, and negative extraction pressure, Wang et al. [15] used orthogonal experimental design and COMSOL to simulate the effective extraction radius under different conditions. Zhang et al. [16] combined similarity simulation experiments and numerical simulations to analyze the impact of negative pressure changes on the gas seepage characteristics of coal. Kong et al. [17] analyzed the influence of multiple factors on the effective extraction radius of boreholes under hydraulic punching conditions by constructing a fluid–solid coupling numerical model.

Many scholars have achieved significant results in the sensitivity analysis of gas extraction efficiency. However, most previous studies have focused on the qualitative analysis of the impact of single factors on gas extraction efficiency. The variation patterns of gas extraction efficiency under the interaction of multiple factors require further quantitative analysis. Coal mine gas extraction is a complex system engineering project, where the effectiveness of its implementation is often influenced by multiple factors rather than a single factor. Therefore, studying only a single factor cannot fully reveal the mechanisms affecting gas extraction, nor can it reliably guide on-site gas extraction in coal mines. Based on previous research in this field and practical experience in coal mine gas extraction, this paper mainly selects initial coal seam permeability, extraction time, extraction negative pressure, and borehole diameter as the primary factors for study. This study is based on the Liuzhuang Mine of the China Coal Xinji Company. It employs the hole stress relief method for in situ stress testing in the mining area. Using a gas desorption–diffusion–seepage model and COMSOL Multiphysics numerical simulation software, the gas extraction process in boreholes is analyzed. Additionally, Minitab software (https://www.minitab.com) is used to design response surface experiments to investigate the impact of multi-factor interactions on borehole gas extraction efficiency. This research identifies the primary and secondary influencing factors and establishes the relationship between the effective extraction radius of boreholes and multiple factors. The findings provide significant reference value for the design of gas extraction in underground coal mines. Specifically, through this study, the quantitative relationships between coal seam permeability, extraction time, extraction negative pressure, borehole diameter, and the effective extraction radius were established. This research reveals the key factors affecting gas extraction efficiency, making on-site gas extraction in coal mines more targeted and effective. For example, in the practical implementation of gas extraction projects in coal mines, measures such as enhancing coal seam permeability through hydraulic fracturing, reasonably increasing extraction time or negative pressure, and appropriately enlarging the borehole diameter can maximize gas extraction efficiency. The findings of this study can provide a reference for weighting these measures and guide the practical work on-site in coal mines.

2. In Situ Stress Testing

Permeability is a crucial factor affecting the coal seam gas extraction process, and it is closely related to the magnitude of in situ stress. Conducting in situ stress testing in the mining area provides essential stress data for constructing numerical models. Our field test employed the hole stress relief method to ensure the accuracy and reliability of the data. The advantage of the stress relief method for boreholes is that it is one of the most applicable and reliable methods for determining absolute stress. This method is not only technically mature but also provides accurate and reliable measurement results. Therefore, it has widespread application in the fields of rock mechanics and engineering. Two in situ stress measurement boreholes, K1 and K2, were arranged at the test site. Both boreholes were located on the south side of the roadway, approximately 1.5 m from the floor, with a spacing of about 1.2 m between them. The layout of the measurement boreholes is shown in Figure 1.

Due to the fragmented nature of the core samples obtained during the in situ stress measurements, it was challenging to obtain the rock mechanical properties parameters for stress calculations through confining pressure tests. Therefore, rock blocks from boreholes K1 and K2 that were suitable for laboratory rock mechanics tests were processed into standard rock samples (φ50 mm × 100 mm). Uniaxial compression tests were conducted on these samples using the RMT-150C rock mechanics testing machine (Wuhan, China) (Figure 2) to obtain the rock’s elastic modulus, Poisson’s ratio, and strength parameters. The test results are shown in

Table 1. Uniaxial compression test results of cores obtained from boreholes K1 and K2.

Number	Strength (MPa)	Elastic Modulus (GPa)	Poisson’s Ratio
S1	69.789	24.865	0.208
S2	84.363	38.694	0.185
S3	64.322	34.477	0.183
Average value	74.158	34.012	0.192

.

Based on the formula for the hole-core stress relief method and the measured rock mechanics parameters, the three-dimensional in situ stress measurement results for the measurement points were calculated, as shown in Figure 3.

The results of the hole–core stress relief method for in situ stress testing at boreholes K1 and K2 indicate the following: The range of values for the maximum principal stress and intermediate principal stress at the measurement points on-site are 18.92 to 19.80 MPa and 14.52 to 14.57 MPa, respectively. The value for the minimum principal stress is 11.75 MPa. The azimuths are in the ranges of 265.9 to 266.4°, 74.6 to 88.6°, and 176.2 to 176.5°. The inclinations are in the ranges of 7.0 to 11.9°, 78.1 to 84.6°, and −0.43 to −4.44°. The horizontal stress exceeds the overburden stress, with the range of the maximum horizontal principal stress to overburden stress ratio being 1.38 to 1.41, indicating that the stress at the measurement points is predominantly tectonic. These in situ stress test results provide support for setting up stress conditions for subsequent numerical modeling.

3. Numerical Model Development

3.1. Coal Seam Gas Migration Model

To effectively conduct numerical simulation research, the gas migration governing equations for the numerical model must first be theoretically constructed. The fractures in the coal seam divide the coal body into individual matrix elements. Adsorbed gas in the coal matrix desorbs and diffuses into the fractures, where it then flows through the fractures by seepage. This process is described by the dual-porosity gas migration model. When using this model, the matrix permeability is defined as a constant, only considering the dynamic changes in fracture permeability, simplifying the gas migration in the coal seam to a sequential process.

During the coal seam gas extraction process, the adsorbed gas within the coal matrix elements desorbs outward as a mass source, leading to continuous diffusion and seepage. The mass exchange equation between the coal matrix and fracture system is expressed as follows [18,19,20]:

$$Q_s=aD\left(c_m-c_f\right)={\displaystyle \frac{1}{\tau }}\left(c_m-c_f\right)$$

(1)

In this equation, Q_s is the mass exchange rate between the unit volume of the coal matrix and the fracture system (kg/(m³·s)). a is the matrix shape factor (m⁻²). D is the gas diffusion coefficient (m²/s). c_m is the gas concentration in the coal matrix (kg/m³). c_f is the gas concentration in the coal fractures (kg/m³). τ is the adsorption time (s).

According to the ideal gas law, the relationship between gas concentration and pressure in the coal matrix and fractures is as follows:

$$c_m={\displaystyle \frac{M_g}{RT}}{p}_m$$

(2)

$$c_f={\displaystyle \frac{M_g}{RT}}{p}_f$$

(3)

In this equation, M_g is the molar mass of the gas (kg/mol). R is the universal gas constant, with a value of 8.314 J/(mol·K). T is the temperature of the coal seam (K). p_m is the gas pressure in the coal matrix (MPa). p_f is the gas pressure in the coal fractures (MPa).

Considering the adsorption time, the equation for gas diffusion from the coal matrix to the fractures can be modified to:

$$Q_s={\displaystyle \frac{M_g}{\tau RT}}\left(p_m-p_f\right)$$

(4)

During the coal seam gas extraction process, the matrix system serves as a positive mass source for the fracture system, while the fracture system acts as a negative mass source for the matrix system. Therefore, according to the law of mass conservation, the mass exchange rate between the coal matrix and the fracture system should equal the rate of change in the gas mass in the coal matrix over time, which can be expressed as:

$${\displaystyle \frac{\partial m_m}{\partial t}}=-{\displaystyle \frac{M_g}{\tau RT}}\left(p_m-p_f\right)$$

(5)

In this equation, m_m is the total gas content mass per unit volume in the coal matrix (kg). t is time (s).

The adsorption of gas only occurs in the coal matrix. The gas mass in the coal matrix includes both adsorbed gas and free gas. Therefore, the gas content mass per unit volume in the coal matrix can be expressed as follows [21]:

$$m_m={\varphi }_m{\displaystyle \frac{M_g}{RT}}{p}_m+\left(1-{\varphi }_m\right){\rho }_n{\rho }_c{\displaystyle \frac{V_L{p}_m}{p_m+P_L}}$$

(6)

In this equation, ϕ_m is the porosity of the coal matrix (%). ρ_n is the gas density at standard conditions (kg/m³). ρ_c is the bulk density of coal (kg/m³). V_L is the Langmuir volume constant (m³/t). P_L is the Langmuir pressure constant (MPa).

The gas density at standard conditions can be calculated using the following equation:

$${\rho }_n={\displaystyle \frac{M_g}{V_m}}$$

(7)

In this equation, V_m is the molar volume of the ideal gas at standard conditions (m³/mol).

By rearranging the above equations, we can obtain the governing equation for the change in gas pressure in the coal matrix over time:

$$\left[{\varphi }_m+\left(1-{\varphi }_m\right){\displaystyle \frac{{\rho }_{c}RT{V}_L{P}_L}{V_m{\left(p_m+P_L\right)}^2}}\right]{\displaystyle \frac{\partial p_m}{\partial t}}=-{\displaystyle \frac{1}{\tau }}\left(p_m-p_f\right)$$

(8)

The mass balance equation for gas within the unit volume of the coal fracture system is:

$${\displaystyle \frac{\partial m_f}{\partial t}}+\nabla \left({\rho }_f\overrightarrow{q}\right)=\left(1-{\varphi }_f\right)Q_s$$

(9)

In this equation, m_f is the mass of free gas per unit volume in the coal fracture system (kg). ρ_f is the gas density in the coal fracture system (kg/m³). $\overrightarrow{q}$ is the velocity vector according to Darcy’s law (m/s). ϕ_f is the porosity of the coal fracture system (%).

The mass of free gas per unit volume in the coal fracture system is:

$$m_f={\rho }_f{\varphi }_f$$

(10)

Since the mass of gas is very small, the effect of gravity on gas diffusion and flow in the coal seam can be ignored. According to Darcy’s law, the velocity vector of the gas is given by:

$$\overrightarrow{q}=-{\displaystyle \frac{k}{\mu }}\nabla p_f$$

(11)

In this equation, k is the permeability of the coal (m²). μ is the dynamic viscosity of the gas (Pa·s).

Finally, the governing equation for gas flow in the fracture system can be expressed as:

$${\varphi }_f{\displaystyle \frac{\partial p_f}{\partial t}}+p_f{\displaystyle \frac{\partial {\varphi }_f}{\partial t}}-\nabla \left({\displaystyle \frac{k}{\mu }}{p}_f\nabla p_f\right)={\displaystyle \frac{1}{\tau }}\left(1-{\varphi }_f\right)\left(p_m-p_f\right)$$

(12)

Thus, the governing equation for gas migration in the dual-porosity medium model is:

$$\{\begin{array}{c}\left({\varphi }_m+\left(1-{\varphi }_m\right){\displaystyle \frac{{\rho }_{c}RT{V}_L{P}_L}{V_m{\left(p_m+P_L\right)}^2}}\right){\displaystyle \frac{\partial p_m}{\partial t}}=-{\displaystyle \frac{1}{\tau }}\left(p_m-p_f\right)\\ {\varphi }_f{\displaystyle \frac{\partial p_f}{\partial t}}+p_f{\displaystyle \frac{\partial {\varphi }_f}{\partial t}}-\nabla \left({\displaystyle \frac{k}{\mu }}{p}_f\nabla p_f\right)={\displaystyle \frac{1}{\tau }}\left(1-{\varphi }_f\right)\left(p_m-p_f\right)\end{array}$$

(13)

This model is used to control the gas flow process in numerical simulations of gas extraction, ensuring the accuracy of the sensitivity analysis of gas extraction parameters.

3.2. Geometric Model and Boundary Conditions

Based on the actual geological conditions of the mine, a numerical model was established with dimensions of 40 m in length, 10 m in height, and a distance of 10 m between boreholes. Regarding boundary conditions, roller supports were set on both sides of the model, and fixed constraints were applied to the bottom boundary of the model, allowing overall settlement. For the convenience of numerical model establishment and analysis, it was assumed that the stress on the model is uniformly distributed, and the initial gas pressure and porosity within the model are also uniformly distributed. Since this study mainly focuses on the impact of coal seam permeability, extraction time, extraction negative pressure, and borehole diameter on the effectiveness of gas extraction, the influence of these assumptions, as well as the model grid division and boundary condition settings, on the study’s error was within an acceptable range. According to collected coal seam parameter data, a uniformly distributed load pressure of 19.8 MPa was applied to the surface of the model. The initial gas pressure in the coal seam was 0.6 MPa, the porosity of the coal seam was 0.01, the gas density was 0.716 kg/m³, and the gas dynamic viscosity was 1.8 × 10⁻⁵ Pa·s. At the beginning of the simulation experiment, gas extraction boreholes were set up. Once the model reached stress equilibrium, extraction negative pressure was applied for the calculation, and the simulation results under the influence of various parameters were obtained. A schematic diagram of the numerical model is shown in Figure 4.

4. Numerical Simulation Analysis and Discussion

4.1. Screening of Influencing Factors

In the 150,804 working faces of the 8th coal seam at Liuzhuang Mine, the coal seam gas pressure is 0.6 MPa. The effective extraction radius cannot be determined by the criterion of the gas pressure dropping below 0.74 MPa. According to the relative pressure index method, if the pre-extraction rate of the coal seam is 30%, the gas pressure reduction in the coal seam needs to reach 51%. When the simulated gas pressure at a certain point in the coal seam drops to 0.294 MPa, indicating a 51% reduction, the gas extraction at that point meets the standard. The distance from that point to the borehole is then considered the effective extraction radius.

In the multivariable orthogonal experiment method, range analysis is typically used to explore the influence of orthogonal experiment results. Range analysis can intuitively determine the strength of each factor’s impact on the experimental data. This study uses an orthogonal experimental design to analyze the influence of borehole gas extraction parameters on extraction effectiveness, identifying the least significant factor. Using the effective extraction radius R as the response indicator, the study examines the effects of extraction time, the initial permeability of the coal seam, negative extraction pressure, and borehole diameter on the effective extraction radius. Each of these four factors was set at three levels, and a three-level, four-factor orthogonal design table L9(34) was used, resulting in nine experimental schemes. The parameters in each experimental scheme were mainly set reasonably based on the actual conditions of the coal mine site and relevant industry standards, as shown in

Table 2. Orthogonal test scheme.

Test Serial Number	Initial Permeability of the Coal Seam (10−15 m2)	Borehole Diameter (mm)	Extraction Time (d)	Negative Pressure for Gas Extraction (kPa)	Effective Extraction Radius (m)
1	0.01	64	60	13	0.328
2	0.01	94	120	23	0.596
3	0.01	124	180	33	0.850
4	0.10	64	120	33	0.630
5	0.10	94	180	13	0.956
6	0.10	124	60	23	1.034
7	1.00	64	180	23	3.322
8	1.00	94	60	33	1.482
9	1.00	124	120	13	3.050

:

The results of the orthogonal experiments were processed and analyzed by calculating the mean and range for each factor at each level, followed by normalization. Each factor was then ranked to understand the degree of influence on the response indicator. The calculated results are shown in

Table 3. Range analysis of effective extraction radius.

Level Group Number	Initial Permeability of the Coal Seam (10−15 m2)	Borehole Diameter (mm)	Extraction Time (d)	Negative Pressure for Gas Extraction (kPa)
1	1.774	4.280	2.844	4.334
2	2.620	3.112	4.276	4.952
3	7.854	4.934	5.128	2.962
Mean value 1	0.591	1.427	0.948	1.445
Mean value 2	0.873	1.037	1.425	1.651
Mean value 3	2.618	1.645	1.709	0.987
Range	2.027	0.608	0.761	0.664
Patch	1	4	2	3

:

Figure 5 shows the mean effective extraction radius under different levels of various factors. It is evident that the initial permeability of the coal seam has the largest mean difference across its three levels, indicating it has the greatest influence. The mean differences for extraction time and extraction negative pressure at the three levels are secondary, indicating that their impact is less significant than the initial permeability of the coal seam. In contrast, the borehole diameter has almost the same mean values across its three levels, indicating its influence is minimal. Additionally, combining the results from Table 3, it can be concluded that the four factors have varying degrees of influence on the effective extraction radius of the borehole. The factors ranked in descending order of influence are the initial permeability of the coal seam, extraction time, negative extraction pressure, and borehole diameter.

For the initial permeability of the coal seam, the maximum mean effective extraction radius is 2.618 m, and the minimum mean is 0.591 m, resulting in a range of 2.027 m, indicating its highest impact on the effective extraction radius. For the borehole diameter, the maximum mean effective extraction radius is 1.645 m, and the minimum mean is 1.037 m, resulting in a range of 0.608 m, indicating its lowest impact. Among the three levels of the extraction time factor, the maximum average effective extraction radius of the borehole is 1.709 m, and the minimum average is 0.948 m. The range of the effective extraction radius for extraction time is 0.761 m, indicating that its influence on the effective extraction radius of the borehole is less than that of the initial permeability of the coal seam. Among the three levels of the extraction negative pressure factor, the maximum average effective extraction radius of the borehole is 1.445 m, and the minimum average is 0.987 m. The range of the effective extraction radius for extraction negative pressure is 0.664 m, indicating that its influence on the effective extraction radius of the borehole is less than that of extraction time.

Therefore, in the response surface experiments, the borehole diameter factor was excluded, and the focus was on analyzing the effects of initial coal seam permeability, extraction time, and negative extraction pressure on borehole gas extraction efficiency.

4.2. Response Surface Model

The response surface method (RSM) is a technique for parameter optimization using mathematical and statistical analysis. By fitting a first-order or second-order model between the response function and influencing factors as an approximation of the true response function, the RSM actively collects data based on multiple linear regressions to obtain a well-behaved regression equation. The established complex multi-dimensional space surface closely approximates actual conditions, requiring relatively few experimental runs. Commonly used design methods in the RSM are the central composite design (CCD) and the Box–Behnken design (BBD).

When the factors are the same, the Box–Behnken design requires fewer experimental runs than central composite design, has approximate rotatability, lacks sequentiality, and avoids experimental combinations where factors are simultaneously at high levels. Moreover, the Box–Behnken design has been successfully applied in similar research scenarios in recent years [22,23]. Therefore, the Box–Behnken design was adopted. The Box-Behnken design includes 15 experimental schemes, as shown in

Table 4. Response surface experimental design scheme and results.

Test Serial Number	Initial Permeability of the Coal Seam (10−15 m2)	Extraction Time (d)	Negative Pressure for Gas Extraction (kPa)	Effective Extraction Radius (m)
1	1.00	120	33	2.316
2	0.01	60	23	0.222
3	0.10	120	23	0.654
4	0.01	120	33	0.273
5	1.00	180	23	3.703
6	1.00	120	13	2.158
7	0.10	180	13	0.761
8	0.10	120	23	0.651
9	0.10	60	13	0.450
10	0.10	120	23	0.652
11	0.01	180	23	0.308
12	0.10	180	33	0.810
13	1.00	60	23	1.453
14	0.10	60	33	0.480
15	0.01	120	13	0.259

. The effective extraction radius of boreholes under different extraction parameter conditions was calculated using COMSOL software, and the results are presented in Table 4.

Based on the results, an effective extraction radius and multi-factor coupling relationship model can be established to assist in guiding the dynamic optimization and adjustment of extraction parameters. Using a response surface model with polynomial degrees higher than two would increase the number of higher-order coefficients and significantly increase the computational load. A quadratic polynomial is more flexible and simple, offers high fitting accuracy, and is widely used. Therefore, this study employs a quadratic polynomial to express the relationship.

For the simulation results obtained in Table 4, a response surface experiment was conducted to perform a multi-factor regression fitting analysis. The polynomial response surface regression equation between the effective extraction radius and multiple factors was established as follows:

R = −0.093 + 2.069K − 0.00420T + 0.0454Q − 2.204K² + 0.000020T² − 0.000985Q² + 0.01802KT + 0.00689K × Q + 0.000008T × Q

In this equation, R represents the effective extraction radius, K represents the coal seam permeability, T represents the extraction time, and Q represents the negative extraction pressure.

4.3. Response Surface Analysis of Effective Extraction Radius

The approximate function of the quadratic polynomial response surface model was transformed into a linear function in form through variable substitution. Then, the function values based on the response surface model were obtained through the parameter matrix of the experimental sample space. This allowed for the calculation of the error between the response values and the experimental values. The coefficients of the quadratic polynomial in the above equation were determined using the least squares method.

In the table, the p-value is a key analysis value for the significance of each term. The smaller the p-value, the less likely an extreme hypothetical situation occurs, indicating that the result is more significant.

The variance analysis of the regression equation is shown in

Table 5. Variance analysis.

Source	Degree of Freedom	Adj SS	Adj MS	F-Value	p-Value
Model	9	13.7142	1.52380	128.96	0.000
Linear	3	11.4241	3.80802	322.28	0.000
Initial permeability	1	9.1763	9.17633	776.62	0.000
The extraction time	1	2.2346	2.23458	189.12	0.000
Negative pressure for gas extraction	1	0.0131	0.01314	1.11	0.340
square	3	0.1478	0.04927	4.17	0.079
Initial permeability and Initial permeability	1	0.0899	0.08994	7.61	0.040
Extraction time and Extraction time	1	0.0189	0.01885	1.60	0.262
Negative pressure and Negative pressure	1	0.0359	0.03585	3.03	0.142
Two-factor interaction	3	1.5356	0.51187	43.32	0.001
Initial permeabilit and Extraction time	1	1.5293	1.52932	129.43	0.000
Initial permeability and Negative pressure	1	0.0062	0.00621	0.53	0.501
Extraction time and Negative pressure	1	0.0001	0.00009	0.01	0.934
Error	5	0.0591	0.01182	⋯	⋯
Total	14	13.7732	⋯	⋯	⋯

Note: Significance is indicated by p < 0.05, and high significance by p < 0.01. The coefficient of determination is R² = 0.9957, and the adjusted coefficient of determination is R²(adjust) = 0.988.

. It can be observed that the p-value of the response surface model for the target function R is much less than 0.01, indicating excellent model significance. This means the polynomial regression equation accurately reflects the influence of various factors on the response value (effective extraction radius). The p-values for the initial permeability term, extraction time term, the square of the initial permeability term, and the interaction term between initial permeability and extraction time are all less than 0.05, indicating that these terms are significant in the model. The p-values for the other terms are not much greater than 0.05, suggesting that their significance is also acceptable. Although the significance of extraction negative pressure in the model is not high, considering that it is a key indicator for coal mine gas extraction and other potential uncertainties, extraction negative pressure was included in the construction and further analysis of this model. The coefficient of determination is 0.9957, indicating that over 99.57% of the response values can be explained by this model.

Based on the quadratic polynomial regression equation, response surfaces were plotted, as shown in Figure 6, Figure 7 and Figure 8. These are three-dimensional surfaces resulting from the interaction effects of various independent variables on the response value. They allow for the analysis of the interaction between any two factors.

The response surface plots between pairs of the initial permeability of the coal seam, extraction time, and extraction negative pressure show that the effective extraction radius of the borehole changes the fastest along the initial permeability axis and the slowest along the extraction negative pressure axis. Among these, the response surface in Figure 6 has the most distortion, indicating a significant interaction between initial permeability and extraction time. The greater the initial permeability of the coal seam, the shorter the extraction time required to achieve the same extraction volume. The response surface in Figure 7 shows some distortion, suggesting a slightly significant interaction between initial permeability and extraction negative pressure. The greater the initial permeability of the coal seam, the lower the extraction negative pressure required to achieve the same extraction capacity. The response surface in Figure 8 shows no obvious distortion, indicating that the interaction between extraction time and extraction negative pressure is not significant. There is no clear interrelationship between extraction time and extraction negative pressure.

This also confirms that the initial permeability of the coal seam has a highly significant effect on the effective extraction radius of the borehole during gas extraction. The extraction time has a moderately significant effect, while the extraction negative pressure has a less significant effect. The most critical factor influencing borehole gas extraction is the permeability of the coal seam. When the permeability is high, the coal seam fractures are well-developed, allowing for higher permeability and making gas flow through the coal seam easier and more efficiently extracted by the borehole. The longer the extraction time, the more gas is extracted from the coal seam. However, due to the limitation of the initial permeability of the coal seam, the potential for improvement in gas extraction efficiency by extending the extraction time is limited. In actual coal mine production scenarios, an unreasonable increase in extraction time can negatively impact coal production. Increasing the extraction negative pressure can enhance the gas extraction capacity to some extent and accelerate the gas extraction rate from the coal seam. However, excessively high extraction negative pressure can lead to borehole leakage, adversely affecting gas extraction efficiency.

In summary, the factors affecting the effective extraction radius of the borehole, in order of significance, are initial permeability of the coal seam > extraction time > extraction negative pressure. This is consistent with the range analysis results of the orthogonal experiments. By comparing these results with other similar studies, it is found that for low-permeability coal seams, the primary factor affecting gas extraction efficiency is coal seam permeability. However, for coal seams with better permeability, the impact of coal seam permeability is not significant, and factors such as extraction time, borehole sealing quality, appropriate extraction negative pressure, and borehole diameter become the main factors influencing gas extraction efficiency. Therefore, for low-permeability coal seams, it is crucial to implement measures to increase permeability before extraction, such as hydraulic slotting, protective layer mining, and loosening blasting. These measures can increase the permeability of low-permeability coal seams and improve gas extraction efficiency. When coal mines have conditions for protective seam mining, priority should be given to adopting protective seam extraction measures to increase coal seam permeability. For single low-permeability coal seams, further measures should be selected based on the characteristics of the coal seam, such as hydraulic fracturing, liquid carbon dioxide phase transition fracturing, or gas injection displacement, to enhance coal seam permeability and extraction efficiency and fundamentally improve gas extraction effectiveness.

It should be noted that different coal mines have varying geological conditions and gas extraction conditions. The main factors and their weights affecting gas extraction efficiency will also differ. In practical coal mine applications, further characteristic analysis should be conducted based on specific conditions to obtain research results that are more applicable to specific scenarios.

5. Conclusions

(1) Significance of Initial Permeability: Based on the orthogonal experiment results, the initial permeability of the coal seam has the greatest impact on the effective extraction radius, with a maximum range of 2.027 m. In contrast, the borehole diameter has the least impact, with a range of 0.608 m. The influence of the factors on the effective extraction radius of the borehole decreases in the following order: initial permeability of the coal seam, extraction time, extraction negative pressure, and borehole diameter;

(2) Response Surface Model: Using the response surface method, the relationship model between the initial permeability of the coal seam, extraction time, extraction negative pressure, and effective extraction radius was obtained. The p-value of this response surface model is less than 0.05, indicating its high significance. The model’s determination coefficient is 0.9957, meaning it can explain over 99.57% of the response values;

(3) Interaction Effects: The interaction of multiple factors introduces new influences on the changes in the effective extraction radius. The initial permeability of the coal seam remains the most significant factor. The response surface between initial permeability and extraction time shows the greatest distortion, indicating the strongest interaction effect on the extraction radius. Conversely, the response surface between extraction time and extraction negative pressure shows the least distortion, indicating a negligible interaction effect. Therefore, increasing the permeability of the coal seam is one of the primary methods to enhance gas extraction efficiency;

(4) For coal seam gas extraction, especially in low-permeability coal seams, conducting pre-extraction coal seam fracturing to increase permeability can effectively enlarge the gas extraction radius, thereby significantly enhancing gas extraction efficiency.