Mathematical equations for pressure relief angle of protective seam inclination in outburst coalmine

Protective seam mining is one kind of most effective measure to reduce coal and gas outburst risk. The pressure relief angles along inclination (δ m ) are key parameters for evaluating the effect of protective seam mining. However, the numerical relation between δ m and coal seam dip (a) is dened by discrete data and is dicult to determine δ m accurately. In this study, the variations of δ m with respect to seam dips are analyzed to derive analytical equations that can be used to accurately calculate δ m . The relationship between δ m and seam dip (a) can be expressed as parabolic or inverted parabolic curves. Mathematical equations for δ m are derived by curve tting technique. Furthermore, polynomial equations are determined as the most appropriate for δ m calculation when the polynomial order is selected as 7, 6, 4 and 5 respectively. These derived equations are computationally solved and veried using actual and eld test data of δ m. with satisfactory consistency and accuracy. The equations are suggested as supplement and improvement for Detailed Rules on Prevention of Coal and Gas Outburst.


Introduction
Coal and gas outbursts are one of the greatest natural hazards in underground coalmining, which continue to threaten safety and limit production. Many countries have experienced coal and gas outbursts in underground mining (Aguado and Nicieza, 2007;Black, 2019;Hedlund, 2012;Saleh and Cummings, 2011;Skoczylas et al., 2014;Sobczyk, 2014;Wang et al., 2014). Coal is a major energy resource and plays a critical role in the development of the national economy and society. This situation is expected to remain for a considerable time in China (Feng et al., 2018;Xie et al., 2011;Zhang et al., 2018). However, the main coal-bearing strata in China had undergone complex and multi-stage tectonic movement superposition and evolution, which leads to the formation of widely distributed tectonically deformed coal deposits. The tectonically deformed coal is characterized by low permeability, hard to drain and prone to coal and gas outburst. In-situ stress, gas pressure and gas content typically increase with mining depth in Chinese coal mines, and as a result, the risk of coal and gas outburst also increases. Gas related hazards are still the primary risks affecting coal mine safety (Chen et al., 2019a;Chen et al., 2019b;Han et al., 2012;He et al., 2010;Liang, 2018;Wang et al., 2018;Wang et al., 2012;Wu et al., 2011;Xie et al., 2015;Yin et al., 2017;Zhang et al., 2019).
Two outburst prevention measures, protective seam mining and pre-drainage of coal seam gas, are typically implemented before mining in outburst-prone coal seams (Chen et al., 2019b;Safety, 2019;Skoczylas et al., 2014).
Research and engineering practices show that protective coal seam mining is one of the most effective measures to prevent coal and gas outburst. Applications of protective seam mining have clearly demonstrated to signi cantly reduce risks associated with coal and gas outburst in coal mines in China (Jin et al., 2016;Li et al., 2014;Liu and Cheng, 2015;Wang et al., 2013b;Yang et al., 2011). Furthermore, protective seam mining and pressure-relief gas drainage were applied in low permeability and outburst-prone coal seams to achieve coextraction of coal and seam gas (Shang et al., 2019;Tu and Cheng, 2019;Wang et al., 2013a;Yuan, 2016;Zhang et al., 2020).
Detailed Rules on Prevention of Coal and Gas Outburst (DRPCGO), which is promulgated by China's State Administration of Work Safety (Safety, 2019), speci es that protective seam mining must be given priority in outburst-prone coal seams when the relevant conditions are met. DRPCGO is currently the most authoritative management and technical legal document on the prevention of coal and gas outburst in China. Moreover, protective seam mining has also been introduced to prevent rock burst in underground coalmine as a China National Standard (GB/T 25217.12-2019)(State administration of market ragulation, 2019; Xu et al., 2019).
The pressure relief angle along inclination is one of key parameters de ning the effective protected area. Two methods can be employed to determine the protected zone along an inclination, including eld-testing and data reference according to Appendix E in DRPCGO (Safety, 2019). Generally, adopting eld-testing to determine the protection range along an inclination requires drilling boreholes for measurement of parameters including gas pressure, gas content and maximum expansion deformation rate of the protected seam, etc. (Cao et al., 2018;Jin et al., 2016;Li et al., 2014;Liu et al., 2017;Xue and Yuan, 2017;Yuan and Xue, 2014). DRPCGO also emphasizes that historical data reference can be applied to estimate the effective protected zone when there is no eld-testing data or the rst time to exploit the protective seam in a coal mine (Safety, 2019). Several studies reported the protected zone along inclination can be evaluated by using numerical stimulation to investigate the characteristic of deformation of coal seam, pressure relief rule and gas ow etc. (Jia et al., 2013;Wang et al., 2010;Zhang et al., 2016). The method of numerical stimulation presents a potential approach for estimating the effective protected zone. However, it is not be applied widely as a criterion because it has not yet established a set of general simulation method and parameters.
Issues will typically be encountered in estimating the protected zone along inclination by using data reference according to Appendix E in DRPCGO. As the relationship between pressure relief angles and coal seam dip is de ned by discrete data, as shown in Table 1, where it is di cult to determinate the precise values of pressure relief angles directly. To solve this issue, this study analyzed the variation of pressure relief angles with seam dips to establish mathematical models for pressure relief angles calculation for protective seam mining. The models have been demonstrated to be of high precision and ease of use for engineering application. The protective seam refers to the coal seam or strata that is mined out rst to eliminate or weaken the outburst hazard of the adjacent coal seams. Figure 1 illustrates that the upper and lower seams undergo deformation after mining of the protective seam, and due to this in uence, the in-situ stress decreases and the permeability is typically enhanced. Additionally, a large volume of gas is liberated and pre-drained, decreasing the gas pressure and gas content in adjacent coal seams so that the outburst risk of the protected seam is reduced (Wang et al., 2017b;Wang et al., 2013b;Xie and Xu, 2017;Yang et al., 2011;Yin et al., 2015;Yuan, 2016).
For dipping seams, as shown in Fig. 2, the boundary of the pressure relief zone is elliptical after the protective seam is mined, and the minor axis is approximately equal to the width of the mined-out area. Moreover, the length of two half major axes are not equal, and the length in the roof is longer than that in the oor. Because the upper strata will cave after mining, and the in uence range in the roof is also larger than that in the oor (Cheng, 2010;Yu, 1986).

De nition of pressure relief angles along seam inclination
Based on Appendix E in DRPCGO, as shown in Fig. 3, A represents the protective seam, and B 1 and B 2 represent the protected seams. A is the lower protective seam relative to B 1 and is the upper protective seam relative to B 2 . The pressure relief range is determined by the seam dip a, the pressure relief angles δ 1 , δ 2 for the upper seam B 1, and δ 3, and δ 4 for the lower seam B 2 . C is the protective boundary.
Based on underground coal mining theory, the caving zone of the inclined coal seam is quite different from that of the horizontal coal seam in long wall mining. As shown in Figs. 4-5, after mining of a at coal seam, the caving zone will typically exhibit a symmetrical structure. For mining of an inclined coal seam, the caving zone will form an asymmetrical shell structure. When the dip angle of coal seam α exceeds 70°, the shell structure will not exist, and the caving zone will be more complex (LIU and YANG, 2013;Yongqi, 2004). Therefore, the pressure relief angles are related to coal seam dip and the mechanical properties of coal and adjacent strata, but primarily depend on seam dips (Safety, 2019;Yu, 1986).
According to Appendix D in DRPCGO, the protection zone of the protective seam along the inclination can be delineated according to the pressure relief angle (δ m , m = 1, 2, 3 and 4), as shown in Fig. 3. If the pressure relief angles cannot be measured, historical data can be referred to Table 1, which was rst established by experimental simulation from the Institute of Mining Survey in the former Soviet Union (Yu, 1986).
As shown in Table 1, the relationship between the pressure relief angles and the coal seam dip is de ned by discontinuous discrete data sets. This makes it di cult to infer values of relief angles for coal seam dips fall between the discrete datasets. Therefore, a mathematical function for the relationship between pressure relief angle and coal seam dip is required to solve this problem.  Table 1, each coal seam dip value (set to x) has a relief angle value (set to y) corresponding to it, which can be represented by a data array (x i , y i ) (i = 1,2 …n). For the convenience of calculation and use, an analytical equation y = f (x, c) is needed to re ect the numerical relationship between the quantity x and y. y = f (x, c) is called the tting model, and c= (c 1 , c 2 , •••, c n ) is the parameter to be solved in the equation.
For solving the functional relationship between coordinates represented by discrete point arrays, interpolation and curve tting are one of the most common data processing methods. The difference is that the interpolation requires all data points to be on the curve, while the curve tting just requires the curve to re ect the varying trend of the data, and all data points being on the curve are not necessary.
Interpolation methods rely more on measured interpolation reference data points. When there are only a few data points, a simple polynomial equation may be established. However, for the case of more data points, it is not easy to use the interpolation method. Typically, in this scenario, the order of the polynomial needs to be high, the calculation is complex, and the result is not reliable. In contrast, piecewise interpolation may be applied by constructing a linear or polynomial equation in each interval (Tibshirani, 2014). However, it is still not the best selection with complex mathematical equations limiting its application. As a primary estimation, there would not be less 2 or 3 intervals if using the piecewise interpolation for any δ m , it is estimated that every model includes 2 or 3 equations, by which the pressure relief angles can be estimated, nevertheless, the complex expression of a model with several equations is an obvious shortcoming. It is better to consider curve tting that can obtain a certain functional relationship, which is convenient for calculation and application (Guest, 2012). Curve tting was applied as the tool to establish the mathematical function in this research.
The general steps to establish the mathematical model by curve tting are as follows.
1. Draw the scatter plot; 2. Choose a suitable curve type based on the distribution of scatter points; 3. Fit the equation based on the principle of least squares; 4. Solve the function expression about the original variables x and y; 5. Model error analysis and accuracy testing.

Results And Discussion
3.1. Selection of curve tting type According to the reference data in Table 1, we can draw scatter plots of the pressure relief angles with the coal seam dip by Origin 8 software, as shown in Fig. 6.
As shown in Fig. 6, δ 1 and δ 4 decrease rst, then remain stable and nally increase with , and the curves are like the inverted parabolic. By contrast, δ 2 and δ 3 increase rst, then remain stable and lately decrease with , and the curves are like the parabolic. As widely accepted, it is suitable for parabolic or inverted parabolic to use the polynomial curve tting (Guest, 2012). Therefore, the polynomial tting was severed as the tool for modelling in the research.

Parameter solving
For n sets of measured data (δ mi , α i ), the characteristic curve δ m -a can be approximately expressed by n-1 orders polynomial of α i by curve tting (Deboeverie et al., 2010). The polynomial function can be expressed: δ mn = c m1 + c m2 α m + c m3 α m 2 + … + c mn α m n−1 (1) Where, δ mn is the pressure relief angle, m = 1, 2, 3 and 4; c m1 , c m2 , c m3 , ••, c mn are the tting parameters; n is the total quantity of samples.
Its matrix expression is: 2 The curve tting parameters (c m1 , c m2 , c m3 , ••, c mn ) can be easily solved by using the Origin 9.0 software based on the principle of least squares (Seifert, 2014).

Polynomial tting equations
As shown in Table1, there are ten data arrays for each δ m . Based on the polynomial tting theory, eight polynomial equations, of orders from 2 to 9, can be derived by polynomial tting for δ 1 , δ 2 , δ 3 and δ 4 , respectively. In principle, if the order is too low, the tting accuracy will be too low to meet the requirement. However, if the order is too high, the tting curve will be locally oscillatory known as "Runge" phenomenon, which will result in low precision (Deboeverie et al., 2010;Fornberg and Zuev, 2007). Therefore, the rational selection of polynomial order is crucial for equation.
In general, the following principles and steps can be employed to determine the appropriate order polynomial equations.
Step 1: a general principle can be used to evaluate the accuracy of tting result by Determination Coe cient (R 2 ) and Residual Sum of Squares (RSS). The smaller the RSS is and the closer R 2 is to 1, the better is the tting result. However, this principle is not applicable for higher order equation which will usually incur Runge's phenomenon (Boyd, 2010;Deboeverie et al.,2010).
The expressions of R 2 and RSS are as follows: Where, δ mi is the actual value of pressure relief angle, m=1, 2, 3 and 4; i is the number of samples, i = 1, 2, 3, …, n; is the mean value of δ m ; a i is the value of coal seam dip; is the mean value of coal seam dip; is the predicted value of pressure relief angle.
Using Origin 9.0 software, the polynomial equations from order 2 to order 8 and the RSS and R 2 of four pressure relief angles can be determined respectively, as shown in Table 2.
Based on the data in Table 2, the curves of RSS and R 2 with the polynomial order for δ m are plotted in Fig.7.  Table 2 and Fig.7, RSSs decrease dramatically with the order increasing for δ m , but when the order reaches or exceeds a certain value that is 5 for δ 1 , δ 2 , and 3 for δ 3 , δ 4 , this decreasing trend becomes gentle. On the contrary, R 2 generally increases with the order, but it uctuates locally. Besides, the value of R 2 for δ 1 , δ 2 , δ 3 , and δ 4 reaches the maximum to 0.89, 1.00, 0.97 and 0.98 at order 7, 8, 7 and 8, respectively.
Step 2: the high orders with Runge's phenomenon need to be eliminated. In this study, it was found that the tting curves for δ 1 , δ 2 , δ 3 and δ 4 all appeared "Runge" phenomenon when the order reached as 8, 9, 7 and 8, respectively.
The results can be seen in Fig.8.
According to the above analysis, the low orders (from 2 to 4 for δ 1 and δ 2 ; 2 for δ 3 and δ 4 ) and the high orders (from 8 to 9 for δ 1 and δ 4 , 9 for δ 2 , and from 7 to 9 for δ 3 , respectively) can be eliminated. Based on step1 and step 2, the following orders (from 5 to 7 for δ 1 , from 5 to 8 for δ 2 , from 3 to 6 for δ 3 , and from 3 to 7 for δ 4 , respectively) can be further selected by step 3 and step 4.
Step 3: in some conditions, RSS of an equation may be rather small, but it only represents the overall error of the equation. The orders need to be eliminated if the predicted values seriously deviate from the actual values in local points because it is unable to meet the practical precision requirement (Deboeverie et al., 2010;Guest, 2012;Karakus, 2013). To solve this problem, the maximum absolute error (MAE) was compared between different polynomial order equations, and the minimum value of MAE is used to determine a high-precision equation. As shown in Fig.9, the minimum value of MAE can be derived for δ 1 , δ 2 , δ 3 and δ 4 when the orders are 7, 8, 5 and 6, respectively. Based on this analysis, the equations with the highest accuracy can be determined.

Equation testing and evaluation
The proposed polynomial equations were tested using error analysis, tting curve comparison and case veri cation to evaluate the accuracy and reliability.

Accuracy analysis
Based on the data in Table 1, the formulas in Table 3 are used to calculate the predicted value of δ m , and the absolute error and relative error for δ m are also calculated and summarized in Table 4. Table 4 shows the maximum value, minimum value and mean of the absolute error of the equations for δ m are 1.56°, 0.01° and 0.47°, respectively. The maximum value, minimum value and mean of the relative error are 2.08%, 0.01% and 0.63%, respectively. For the purpose of comparison, the actual value and the predicted value of δ m were plotted in Fig.10. It intuitively shows the curves of predicted value are consistent with the actual results, and veri es the validity of the equations for δ 1 , δ 2 , δ 3 and δ 4 . reported by previous studies, as shown in Table 5.  Table 5 also shows the maximum value, minimum value and mean of the absolute error for δ m are of 5.73°, 0.09°a nd 1.41°, and the maximum value, minimum value and mean of the relative error are 6.55%, 0.12% and 0.73%, respectively. Fig.11 provides a comparison bar between the eld measurements and the predicted values of δ m , according to the data in Table 5. Table 5 and Fig.11 show the predicted values are mostly consistent with the eld measurements, which also veri ed the validity of the equaitons for δ 1 , δ 2 , δ 3 and δ 4 .

Implication
The above analysis demonstrate that these equations can be computationally solved to calculated pressure relief angles for any given seam dips and have satisfactory accuracy, which can provide convenience for estimating the pressure relief angles along inclination as a kind of supplement and improvement for Detailed Rules on Prevention of Coal and Gas Outburst.
It should also be acknowledged that the pressure relief angles along inclination for protective seam mining are related to multiple geological and other factors besides coal seam dip angle. Further work needs be conducted to investigate the impact of geological and mining conditions on the pressure relief angles and develop a model with multi-factor parameters that are more appropriate for predicting the pressure relief angles in protective coal seam mining. It is more reliable to determine the pressure relief angles by eld measurement because of the variation in geological conditions at different coalmines.
Nevertheless, when the eld-testing method is adopted, a xed number of testing boreholes must be drilled for measurement of the outburst prediction parameters (gas content or gas pressure), to determinate the pressure relief angles along inclination as shown in Fig.12 (Undergroundcoal, 2020;Wang et al., 2017a). Before eld drilling, it would be useful to estimate the pressure relief angles by the proposed functions, as the predicted data can be used as reference for borehole designs to improve the accuracy of borehole positioning and reduce the complexity of drilling and testing engineering.

Conclusions
Traditionally, pressure relief angles along seam inclination can be determined by discontinuous discrete data sets.
However, it is di cult to infer its value for seam dips fall between the discrete datasets. Based on polynomial curve tting method, mathematical equations for pressure relief angle and seam dip are derived to solve this problem.
The polynomial equations for δ 1 , δ 2 , δ 3 and δ 4 when the orders are selected as 7, 6, 4 and 5 respectively are recommended as the most suitable prediction functions for the pressure relief angles along inclination on protective seam mining.
These equations can be computationally solved to calculated pressure relief angles for any given seam dips, which have satis ed precision requirements and are convenient to use. The calculated results can provide a theoretical reference for the design and evaluation of protection zones in protective seam mining for eliminating coal and gas Comparison of MAE with model order for δm Page 24/25

Figure 10
Comparison curves between the actual value and the predicted value Figure 11 Comparison chart of the eld-testing value and the predicted value Figure 12 Diagram of eld-testing to determinate the pressure relief angles