Engineering method for predicting the displacement of the earth’s surface under the influence of stope workings of coal mines

The development of a sufficiently reliable forecast of the values of trough parameters on the earth’s surface and the displacement of undermined rocks using mathematical modeling methods is an urgent problem for mining. Not only the protection of objects on the earth’s surface depends on its successful solution, but also the choice of the location of mine workings and rational ways to protect them from the influence of rock pressure, the forecast of gas release from undermined sources, the occurrence of gas-dynamic phenomena, the rationale for rational ventilation schemes for excavation areas, the calculation of the bearing capacity supports and many other engineering tasks. The study is based on the use of experimental data obtained during the operation of a separate extraction area, respectively, at the first stage when the working face is removed from the split furnace and at the second stage above the working face when it is sufficiently removed from the open working. This makes it possible to obtain all the necessary experimental data for constructing a general scheme for the development of clearing operations and the formation of a trough for the displacement of the earth’s surface in two stages. On the basis of the studies carried out, it was found that the maximum subsidence of the earth’s surface above the cut working with a sufficient distance from the production face is equal to the subsidence above the production face at the stage of attenuation of the processes of displacement of undermined rocks and the earth’s surface. This gives grounds to make an assumption about the closeness of the displacement parameters of the stationary and dynamic semitroughs.


Problem statement
Conducting cleaning operations in coal mines, even at depths of more than 1000 m, has an impact on changes in the state of the earth's surface.Initially, the study of the processes of formation of shear troughs on the earth's surface was aimed at solving one problem -of protecting buildings, structures and other objects from destruction and the harmful effects of mine workings [1][2][3][4].Based on the results of these studies, a regulatory document [5] was developed, which regulated the rules for protecting structures and natural objects from the harmful effects of underground mine workings.Currently, in the coal mines of Ukraine, the requirements of the current rules [6] are mandatory for use, which determine only the conditions for underworking earth's surface.The calculation of displacements and deformations of the earth's surface and rock mass is a complex problem that has not yet received its fundamental solution.For this reason, empirical calculation methods and separate analytical dependencies are used in practice.In recent years, taking into account the possibilities of computer technology, mathematical modeling of the processes of shifting underworked rocks and the earth's surface.This direction of scientific research is the most promising and relevant, since its implementation does not require long and laborious observations both on the earth's surface and in mine workings.
In addition, such an approach to solving the issue under consideration makes it possible to significantly expand the range of engineering tasks to be solved, related not only to the protection of objects on the earth's surface, but also to the safety of mining operations and the manifestation of rock pressure on the lining of workings.The development of a sufficiently reliable forecast of the values of trough parameters on the earth's surface and the displacement of undermined rocks using mathematical modeling methods is an urgent problem for mining.Not only the protection of objects on the earth's surface depends on its successful solution, but also the choice of the location of mine workings and rational ways to protect them from the influence of rock pressure, the forecast of gas release from undermined sources, the occurrence of gas-dynamic phenomena, the rationale for rational ventilation schemes for excavation areas, the calculation of the bearing capacity supports and many other engineering tasks.
In the general case, the formation of a shear trough on the earth's surface consists of two stages [5,6].The first is related to the start of operation of the excavation area and the development of stope work when separating the stope from the open cut.At this stage, the process of shifting the undermined rocks of the earth's surface is achieved and a stationary half-trough is formed above the open cut.The second characterizes the shift of the undermined rocks and the earth's surface above the moving stope.The relationship between the stages of the formation of a shear trough on the earth's surface and the development of stopping operations in the excavation area is characterized [7] by a graph (figure 1).
Normative documents [5,6] do not consider the logical transition from the first stage of the formation of the trough of the earth's surface displacement to the second.In addition, there are ambiguous recommendations for determining individual parameters.It is envisaged that the stage of attenuation of the process of shifting the earth's surface ends when the stope is removed from the projection considered point at a distance that exceeds the depth of work (H ) by 1.2÷1.4times.There is also no exact definition of the location of the point of the beginning of the displacement of the earth's surface.
By a similar principle, mathematical models have been constructed that separately describe the processes of displacement of the earth's surface at different stages of the development of clearing operations.At the initial stage of operation of the excavation area, the maximum subsidence of the earth's surface (η m ) is considered when the stope is removed from the open cut [8].In parallel with this, mathematical models [9][10][11] make it possible to predict the subsidence of points on the earth's surface above the stope in time, when its advance no longer has a significant effect on the maximum subsidence of the earth's surface.
Preliminary analysis [12,13] based on the processing of experimental data [4,8,[14][15][16][17][18] showed that in some mining and geological conditions, with little changing values of the thickness of the developed seam (m), the depth of mining operations (H ) and strength properties undermined rocks, maximum subsidence of points on the earth's surface η m (Figure 1, curve 8) is close to functional dependence on the distance (L p ) of the stope from the cut working: where a p , b p , c p are empirical coefficients typical for some mining-geological and mining-technical conditions.In addition to the least squares method, to determine the coefficients a p , b p , c p of equation 1, the following equations were proposed [12,13]: where v ox -speed of advancement of the stope, m/month; r and R -respectively, the correlation coefficient and the correlation ratio.
For an analytical description of the dynamics of subsidence of the earth's surface under the influence of a moving stope, it was previously proposed to use logistic [7], exponential [9], or hyperbolic tangent [10] dependences.Comparative analysis of these dependences showed approximately the same, almost functional, convergence with experimental data.
Above the stope, the subsidence of the earth's surface (η ox ) in the case of applying the logistic curve, corresponds to the equation: where L ox -is the distance from the projection of a point on the earth's surface to the production face, m.The coefficients a p , b p , c p of equation ( 5) are also determined in two ways -by the least squares method and using the results of the correlation analysis between the displacement trough parameters and the mining-geological and mining-technical operating conditions of the extraction areas: where L l is the lava length.
In addition to equation ( 5), dependencies were proposed [7] to determine the coordinates of the characteristic points of the earth's surface displacement (figure 1) using the coefficients a p , b p , c p .The coordinates of point A (the beginning of the earth's surface displacement) in front of the stope, respectively, along the abscissa and ordinate axes: Similarly, for other characteristic points: Point O (origin) Point B (the beginning of the active stage of the earth's surface displacement) Point C (maximum settling rate and curve inflection) Point D (completion of the active stage and the beginning of the attenuation of the displacement processes) Point F (beginning of the residual influence of the side job) In the scheme under consideration (figure 1), the decay stage is limited by the point F. Her settling η 0 is approximately 0.97÷0.99 of the final value η at the end of rock compaction processes [9].
Approximate values of the duration of the active stage of the earth's surface displacement are given in the regulatory documents [5,6] depending on the depth of the mining operation (H ) and the speed of advance of the mining faces.According to mathematical models [9][10][11], the characteristic features of the processes of displacement of rocks and the earth's surface are also considered in time.This approach does not allow, in the general case, to link the process of displacement of the earth's surface with the development of clearing operations in the excavation area.For this reason, in the general scheme (figure 1), instead of time along the abscissa, we considered the distance L ox from the projection of the stope lines onto the earth's surface to the observation points.Using the method [9] with the use of derivatives, the coordinates of the characteristic points were determined from the extrema of the obtained dependences (equations 9 -14).The coefficients (a p , b p , c p and a o x, b o x, c o x) of equations 1 and 5 are determined by the most reliable least squares method.For a separate excavation area, when using these coefficients, dependencies 1 and 5 are inherently are functional.
The initial data for calculating the coefficients according to equations 2, 3, 4 and 6, 7, 8 are calculated using the known values of m, H, v ox and m, H, v ox , L respectively.This greatly simplifies the necessary calculations of the parameters of the displacement of the earth's surface troughs, but these dependencies are not functional.
If we prove the closeness of the results obtained by the least squares method, and using empirical equations established on the basis of correlation analysis, then with high reliability it is possible to predict the processes of displacement of undermined rocks and the parameters of the trough displacement of the earth's surface.In this case, there is no need to simultaneously conduct long-term and labor-intensive observations, both on the earth's surface and in mine workings.

Purpose, idea and research methodology 2.1. The goal of the research
Given the state of the issue under consideration, the purpose of this work is: • on the basis of theoretical assumptions and experimental data, to establish a logical relationship between the parameters of the shear half-trough above the cut working and above the stope at its sufficient distance from the cut furnace; • to consider the convergence of the results between the parameters of the trough of the earth's surface displacement, determined respectively on the basis of the least squares method and the method based on correlation dependencies on mining-geological and mining-technical conditions for mining operations.

The idea
The idea of the work is to determine the dependencies obtained on the basis of experimental data for a separate excavation area and processed by the least squares method, and compare them with the dependencies established by the general results of the correlation analysis for different mining, geological and mining conditions.

Methodology
The methodology is based on the use of experimental data obtained during the operation of a separate extraction area, respectively, at the first stage when the stope is removed from the split furnace and at the second stage above the stope with its sufficient distance from the open cut.This makes it possible to obtain all the necessary experimental data for constructing a general scheme (figure 1) for the development of stope work (removal of the stope from the split furnace) and the formation of a trough of earth surface displacement in two stages.The necessary amount of experimental observations to achieve the goals was carried out at the Stepnaya mine when the seam was mined with longwall No. 604 [17] and at the Yubileynaya mine when the seam was mined with longwall No. 530 [8].[17].ψ 1 , ψ 2angles of total displacements [6]; β 0 , γ 0 -boundary angles [6]; + -direction of advance of the stope; •, × -experimental data, respectively, of the maximum subsidence of points on the earth's surface above the cut working and subsidence above the stope [17].
According to the general design scheme, the influence of stopes on the earth's surface (figure 1) was calculated in two ways using the equation of curves for the maximum subsidence of points on the earth's surface.In the first case, the coefficients of equation ( 1) are determined from the experimental data [17] and [8] by the least squares method (a k p , b k p , c k p ).In the second, using the operating conditions of the excavation sites (m, H, v ox ), according to equations (2,3,4), the empirical coefficients (a a p , b a p , c a p ) equations 1, respectively) were calculated.

The research results
When determining the empirical coefficient calculated according to equation ( 2), it is necessary to take into account the value of the reservoir thickness m.The relation ≤ m must always hold between them a a p .If according to the calculations received a a p > m, then its value is taken equal to the thickness of the developed reservoir.of points when the stope is removed from the split furnace, calculated respectively by the least squares method and according to the results of the correlation analysis; 3, 4 -curves of subsidence of points of the earth's surface above the stope, respectively, calculated by the least squares method and according to the results of the correlation analysis; 5, 6 -respectively, the earth's surface and the reservoir being developed; 7 -surface of the coal massif of the open cut; 8 -conditional position of the stope as it moves relative to points on the earth's surface [8]; η -subsidence of the earth's surface; H -depth of cleaning operations; A, B, C, D, F -the position of the characteristic points calculated according to equations (9-14) using the coefficients a k , b k , c k ; A , B , C , D , F -positions of characteristic points calculated according to equations (9-14) using the coefficients a k , b k , c k ; A , B , C , D , F -the position of the abscissas of the characteristic points of subsidence of the earth's surface according to their dependence on the depth of the clearing operations [19]; ψ -angle of full movements [6]; δ a -boundary angle; −→ -direction of advance of the stope; •, × -experimental data, respectively, the maximum subsidence of points on the earth's surface above the cut working and subsidence above the stope [8].
It was also established that when the stope advance rate is less than 50 m/month, the value of the empirical coefficient b a p is approximately equal to 1330.Similarly, according to equations (2,3,4), the empirical coefficients a a p , b a p , c a p were calculated for the operating conditions of longwall No. 530 C ' 6 of the Yubileynaya mine seam (table 1).The coefficients of equation ( 5) were also determined in two ways -by the least squares method and using dependencies (6,7,8) obtained from the results of correlation analysis (table 2).
Having determined the empirical coefficients in two ways (a k ox , c k ox , c k ox and a a ox , c a ox , c a ox )we calculated, using equations (9)(10)(11)(12)(13)(14), the coordinates of the characteristic points of subsidence of the earth's surface above the stopes of the considered longwalls (table 3 and table 4).In addition to these data indicated in the tables, there are also the average values of the characteristic points abscissas.They are calculated depending on the depth of the mining operations according to the method [19], which combines the exponential, hyperbolic tangent and logistic equations recommended respectively [7,[9][10][11], to describe the curve of subsidence of points on the earth's surface above the moving stope.
On the basis of experimental data [17] and empirical coefficients, graphs of possible changes in the trajectories of maximum subsidence of points were plotted when the longwall of longwall No. 604 of the Stepnaya mine was removed from the open cut and subsidence of points on the earth's surface above the stope (figure 2).
To assess the convergence of the results, we analyzed the relationship between the coordinates of the characteristic points of subsidence of the earth's surface above the production faces, The most reliable are the results obtained on the basis of experimental data and their processing by the least squares method.
These results include the coordinates of the characteristic points of subsidence of the earth's surface above the stope when they are calculated according to equations 9-14 using the empirical coefficients of equation 5 calculated by the least squares method (L 1k ox , η 1k ox ).For this case, the determination of the coordinates of the characteristic points (A, B, C, D, F ) used the original equation 5 and its derivatives.For this reason, a practical coincidence of the results of determination with (table 3 and table 4) was obtained.Values close to them were also obtained using the results of correlation analysis (figure 4a), as evidenced by the close location of the bisector of the coordinate grid (1) and the averaging line (2).
Similar results were obtained when determining the values of the coordinates of characteristic points along the abscissa axis (figure 4b).The values L 1k ox are near the bisector of the coordinate grid (1), as is the averaging straight line (2).
Close to the coordinates along the abscissa axis (L 1k ox ) are and their average values (L ox ) calculated [19] depending on the depth of treatment (table 3 and table 4).

Conclusions
The conducted research allowed to draw the following conclusions: • the maximum subsidence of the earth's surface from the side of the split furnace at a ox , η 1a ox -coordinates set according to equations (9-14), for which empirical coefficients are calculated using correlation analysis and equation 6-8; L oxcertain distances from the characteristic points to the projection of the stope [19]; η k ox , η a oxsubsidence of the earth's surface according to equation 5 and empirical coefficients calculated respectively by the least squares method and on the principles of correlation analysis; ×,calculated values respectively for the conditions of mines 'Stepnaya' and 'Yubileynaya'; ×,calculated values, L 1a o x -respectively, for the conditions of the Stepnaya and Yubileinaya mines; x, y-calculated values, η ox o x, η k o x, η a o x respectively, for the conditions of the Stepnaya and Yubileinaya mines; • -averaged values for the location of characteristic points depending on the depth of work (table 3 and table 4).sufficient distance from the cutter furnace coincides with the subsidence above the stope at the stage of attenuation of the processes of displacement of undermined rocks, which gives reason to consider the displacement parameters of the stationary and dynamic semitrough close to each other; • dependencies established by the methodology using the results of correlation analysis are close in their accuracy to the results obtained by the least squares method, which greatly simplifies engineering calculations and obtaining initial data for their production; • the averaged abscissas of the characteristic points of subsidence of the earth's surface above the stope are determined quite accurately from empirical dependencies and data on the depth of the stope.

Figure 1 .
Figure1.Scheme of the formation of a trough of the earth's surface displacement (a) and its correspondence to the parameters of stopes and the displacement of rocks (b). 1 -the developed layer; 2 -split production; 3 -the position of the treatment face relatively to the split furnace at which the movement of the earth surface begins (η m = 0); 4 -i-e position of the treatment face after a complete study of the earth's surface (formation of a flat bottom of the mold); 5the earth's surface; 6 -dynamic half -mold over the treatment face; 7 -flat bottom of the mold movement; 8 -the trajectory of the maximum points of the earth surface subsidence; 9stationary half-mold from the side of the split furnace; β 0 , γ 0 -boundary angles, respectively, from the fall and rise of the formation; ψ 1 , ψ 2 -angles of complete displacements, respectively, from the fall and rise of the formation; -the angle of full displacements, corresponding to the beginning of the displacement of the earth's surface when removing the face from the split production at a distance; L H , θ 0 , θ k -the angles of maximum subsidence, respectively, when the processes of displacement of the earth's surface rocks (η m = 0) and the formation of a flat bottom (η m = η 0 ) ; H -the average depth of conducting cleaning works; α-the angle of the fall of the formation; η, η m -respectively, subsidence and maximum subsidence of the earth's surface; η 0 -the depth of the flat bottom of the mold ; A 0 , A -points of the beginning of the movement of the earth's surface, respectively, is a distance from the split furnace with leaning face and in front of it; B, C, D, F -the characteristic point displacement under the influence of clearing face movement; −→ -is a direction of clearing face movement.

Figure 2 .
Figure 2. Dependence of the change in the parameters of the displacement trough on the earth's surface with distance from the stope of Chapter No. 604 (Stepnaya mine) on the open working (a) and above the stope (b): 1, 2 -curves of the trajectory of the maximum subsidence of points when the working face is removed from the split furnace, calculated respectively by the least squares method and according to the results of the correlation analysis; 3, 4 -curves of subsidence of points on the earth's surface above the stope, respectively, calculated by the least squares method and according to the results of correlation analysis (they coincide on the graph); η -subsidence of the earth's surface; H -depth of cleaning operations; A, B, C, D, F -the position of the characteristic points calculated according to equations (9-14) using the coefficients a, b, c; A , B , C , D , F -positions of characteristic points calculated according to equation (9-14) using coefficients a a , b a , c a ; A , B , C , D , F -the position of the abscissas of the characteristic points of subsidence of the earth's surface according to their dependence on the depth of the clearing operations [19]; 5, 6 -respectively, the earth's surface and the reservoir being developed; 7 -surface of the coal massif of the open cut; 8 -conditional position of the stope in relation to the reference points R25, K30, R35, R40 and R45 as it moves[17].ψ 1 , ψ 2angles of total displacements[6]; β 0 , γ 0 -boundary angles[6]; + -direction of advance of the stope; •, × -experimental data, respectively, of the maximum subsidence of points on the earth's surface above the cut working and subsidence above the stope[17].

Figure 3 .
Figure 3. Dependence of the change in the parameters of the trough of the earth's surface displacement as the stope of longwall No. 530 (Yubileynaya mine) is removed from the cut working (a) and above the stope (b): 1, 2 -curves of the trajectory of the maximum subsidence of points when the stope is removed from the split furnace, calculated respectively by the least squares method and according to the results of the correlation analysis; 3, 4 -curves of subsidence of points of the earth's surface above the stope, respectively, calculated by the least squares method and according to the results of the correlation analysis; 5, 6 -respectively, the earth's surface and the reservoir being developed; 7 -surface of the coal massif of the open cut; 8 -conditional position of the stope as it moves relative to points on the earth's surface [8]; η -subsidence of the earth's surface; H -depth of cleaning operations; A, B, C, D, F -the position of the characteristic points calculated according to equations (9-14) using the coefficients a k , b k , c k ; A , B , C , D , F -positions of characteristic points calculated according to equations (9-14) using the coefficients a k , b k , c k ; A , B , C , D , F -the position of the abscissas of the characteristic points of subsidence of the earth's surface according to their dependence on the depth of the clearing operations[19]; ψ -angle of full movements[6]; δ a -boundary angle; −→ -direction of advance of the stope; •, × -experimental data, respectively, the maximum subsidence of points on the earth's surface above the cut working and subsidence above the stope[8].

Figure 4 .
Figure 4. (a) -along the ordinate axis (subsidence of the earth's surface at the corresponding points); (b) -along the abscissa axis (distance from the projection of production faces to the considered points); 1 -bisectors of coordinate grids; L 1kox , η 1k ox -coordinates established according to equations(9)(10)(11)(12)(13)(14), for which the empirical coordinates are calculated by the least squares method according to equation 5; L 1a ox , η 1a ox -coordinates set according to equations (9-14), for which empirical coefficients are calculated using correlation analysis and equation 6-8; L oxcertain distances from the characteristic points to the projection of the stope[19]; η k ox , η a oxsubsidence of the earth's surface according to equation 5 and empirical coefficients calculated respectively by the least squares method and on the principles of correlation analysis; ×,calculated values respectively for the conditions of mines 'Stepnaya' and 'Yubileynaya'; ×,calculated values, L 1a o x -respectively, for the conditions of the Stepnaya and Yubileinaya mines; x, y-calculated values, η ox o x, η k o x, η a o x respectively, for the conditions of the Stepnaya and Yubileinaya mines; • -averaged values for the location of characteristic points depending on the depth of work (table3 and table 4).

Table 1 .
The results of determining the coefficients of equation 1 by the least squares method *note -accepted, a a p = m, according to the empirical equation (2) a a p = 1009mm, which is greater than the value of m = 910mm

Table 2 .
The results of determining the coefficients of equation 5 by the least squares method

Table 3 .
Estimated values of the coordinates (η o x, L o x) of the characteristic points of the curve of subsidence of the earth's surface above the stope when processing the 604th lava of the formation C 6 Stepnaya mine.

Table 4 .
Estimated values of the coordinates (η o x, L o x) of the characteristic points of the earth surface subsidence curve above the stope when processing the 530th lava of the seam C ' determined in different ways (table3 and table 4).