







































 
 

 

40 
© 2023 by the authors; licensee Asian Online Journal Publishing Group 
 

Asian Review of Environmental and Earth Sciences 
Vol. 10, No. 1, 40-49, 2023 

ISSN(E) 2313-8173 / ISSN(P) 2518-0134 
DOI: 10.20448/arees.v10i1.4539 

© 2023 by the authors; licensee Asian Online Journal Publishing Group 

 
 

 
 
 
Experimental determination of the boundaries of the influence of a stope working 
on the earth's surface 

 
Filatieva Elvira1   

Olha Fursova2   

Filatiev Mikhail3   

  
( Corresponding Author) 

 
1Department of Fire Safety, Lugansk State University named after Vladimir Dahl, Lugansk Region, Ukraine. 
1Email: elafilatyeva@gmail.com   
2,3Department of Technosphere Safety, Lugansk State University named after Vladimir Dahl, Lugansk Region, 
Ukraine. 
2Email: metmashdtu@gmail.com  
3Email: Mfilatev@gmail.com  

 
Abstract 

The theoretical part of the research methodology is developed according to the scheme of 
subsidence of points on the earth's surface relative to the projection of the face. The curve of the 
trajectory of the subsidence of the earth's surface is divided by characteristic points at different 
stages of subsidence of the earth's surface. Such stages include: the beginning of the displacement 
of the earth's surface, the active stage of displacement, the end of the active stage and the 
attenuation of the processes of subsidence of the earth's surface. According to the goal and the 
design scheme, on the basis of experimental data, we determined the parameters corresponding to 
the location of a point on the earth's surface where it began to settle. In relation to the scheme 
under consideration, three well-known dependencies were analyzed to describe the subsidence 
curve of the earth's surface: the exponential equation, the hyperbolic tangent function, and the 
logistic curve. Based on them, it was established that the main influencing factor determining the 
boundary of the dynamic half-mold is the depth of mining operations, and the boundary angles are 
practically independent of this parameter. 

 
Keywords: Beginning of subsidence, Boundary angles, Boundary, Depth, Dynamic semi-trough, Earth surface, Face, Movement, Point. 

 
Citation | Elvira, F., Fursova, O., & Mikhail, F. (2023). 
Experimental determination of the boundaries of the influence of a 
stope working on the earth’s surface. Asian Review of Environmental 
and Earth Sciences, 10(1), 40–49. 10.20448/arees.v10i1.4539 
History:  
Received: 3 January 2023 
Revised: 22 February 2023 
Accepted: 6 March 2023 
Published: 20 March 2023 
Licensed: This work is licensed under a Creative Commons 

Attribution 4.0 License  
Publisher:  Asian Online Journal Publishing Group 
 

Funding: This study received no specific financial support. 
Authors’ Contributions: All authors contributed equally to the conception 
and design of the study. 
Competing Interests: The authors declare that they have no conflict of 
interest. 
Transparency: The authors confirm that the manuscript is an honest, 
accurate, and transparent account of the study; that no vital features of the 
study have been omitted; and that any discrepancies from the study as planned 
have been explained. 
Ethical: This study followed all ethical practices during writing. 

 

 

Contents 
1. Introduction ...................................................................................................................................................................................... 41 
2. Materials and Methods ................................................................................................................................................................... 41 
3. Results and Discussion ................................................................................................................................................................... 41 
4. Conclusions ....................................................................................................................................................................................... 49 
References .............................................................................................................................................................................................. 49 
 

 
 
 

 

 

 

mailto:elafilatyeva@gmail.com
mailto:metmashdtu@gmail.com
mailto:Mfilatev@gmail.com
https://creativecommons.org/licenses/by/4.0/
https://creativecommons.org/licenses/by/4.0/
https://www.doi.org/10.20448/arees.v10i1.4539
https://orcid.org/0000-0002-1041-0535
https://orcid.org/0000-0002-9622-2622
https://orcid.org/0000-0001-5608-6737


Asian Review of Environmental and Earth Sciences, 2023, 10(1): 40-49 

41 
© 2023 by the authors; licensee Asian Online Journal Publishing Group 

 

 

Contribution of this paper to the literature 
This experimental and theoretical studies have made it possible to draw important conclusions 
for science and industrial activity about the stages of subsidence of the earth's surface. The 
characteristic points and the curve of dependence of rock subsidence on the degree of 
development of clearing operations were determined. 

 
1. Introduction 

One of the little-studied issues in the development of coal seams is the reliable determination of the boundaries 
of the influence of stope workings on the earth's surface. This is confirmed by the results [1] of comparing the 
experimentally determined sizes of the earth's surface displacement troughs with their parameters calculated 
according to the normative document [2]. For example, in the conditions of the "Stepnaya" mine, the calculated 
values of the dimensions of the troughs of the earth's surface displacement were significantly less than the 
experimentally determined parameters. In the conditions of the mine named after P.L. Voikov, on the contrary, the 
experimental values of the dimensions of the displacement troughs of the earth's surface were several times higher 
than their calculated values. This situation indicates the relevance of work related to the study of the impact of 
stopes on the earth's surface. 

The results of such work determine the successful solution of engineering problems on the manifestation of 
rock pressure on the lining of workings, the establishment of possible water and gas inflows from the undermined 
coal-rock stratum and environmental consequences. The purpose of the work is to establish the factors that 
determine the boundary of the dynamic semi-trough on the earth's surface in front of the projection of a moving 
stope. 
 

2. Materials and Methods 
The theoretical part of the research methodology was developed taking into account the scheme of subsidence 

of the earth's surface [3] relative to the projection of the stope Figure 1. 
In this scheme, instead of time along the abscissa axis, the distances (L) from the projection of the stope line 

onto the earth's surface to the observation points were plotted. The characteristic points of the curve of subsidence 
dynamics are: A- corresponds to the beginning displacement, O - is located in the alignment with a stope and 
serves as a reference point along the abscissa axis; B - the beginning of the active stage; C is the maximum settling 
rate and the inflection point of the curve, D is the end of the active stage and the beginning of its decay, F is the 
beginning of the residual effect. The decay stage in the scheme under consideration is limited by the point F. Its 

subsidence ( o ) is approximately 0.97 0.99 of the final ( к ) at the end of the processes of rock 
compaction [3]. 
 

 
Figure 1. Scheme of subsidence of the earth's surface relative to the projection of the production face. 

Note:  1 - curve of the trajectory of the points of subsidence of the earth's surface; 2 - reservoir under development; 3 - the position of the stope relative 

to the curve of the dynamics of subsidence of the earth's surface at the initial moment of influence on point A ; 
н , 

к - respectively, the initial 

and final subsidence of the earth's surface; 0 - depth of the flat bottom of the shear trough to the compaction of the rocks; Ln , Lk - the distance 

between the projection of the stope and points A and F , respectively, at the beginning and end of the displacement; δo , γo , βo - boundary angles 
(depending on the direction of mining of the extraction column) that determine the position of point A (the beginning of the earth's surface 

displacement); - stope advancing direction. 
 

 
According to the goal and the design scheme Figure 1, on the basis of experimental data, it is necessary to 

determine for each specific case the parameters that determine the position of point A. These include boundary 

angles (δo - when mining seams along strike, γo - along rise, βo - along dip), as well as LH - the distance from the 
projection of the production face to point A. 
 

3. Results and Discussion 
The division of the process of displacement of undermined rocks and the earth's surface into separate stages 

was carried out using the recommended [3-5] functions. 



Asian Review of Environmental and Earth Sciences, 2023, 10(1): 40-49 

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As applied to the scheme under consideration Figure 1, the dynamics of subsidence of the earth's surface is 
described by an exponential equation [1]: 

( )
( )

2
11 exp ,

L Lн
L к


 

 − +
 = −
 
 

                                                                 (1) 

Where   - is the subsidence of the observation point on the earth's surface when its projection is removed at a 

distance L from the production face, mm; 

1
  - empirical coefficient determined from experimental data. 

The first three derivatives of the function (1) are the equations of the settling rate, acceleration and acceleration 
change. The extrema of the obtained dependencies are used to determine the coordinates of the characteristic 
points, which are used as the boundaries of the displacement stages [3]. 

In a similar way, the boundaries of the stages of subsidence of the earth's surface are determined based on the 
extrema of the first two derivatives of the functions of the hyperbolic tangent [4]: 

( )  ,)(tan1
321

nLnhnL ++=                     (2) 

Where n1, n2, n3 - are empirical coefficients determined by the least squares method. 
When constructing the curve of the dynamics of subsidence of the earth's surface from the absolute values of 

the experimental data according to the function of the hyperbolic tangent, it was found that the coefficient n1 is 
numerically equal to half of the final subsidence ( 2

1 кn = ). 

The distance Ln Figure 1 is determined from the condition 
н = 0. The minimum value of the function (2) 

asymptotically approaches zero, therefore, to determine the beginning of the impact of cleaning operations on the 

earth's surface, an assumption was introduced 0,01
1

dн к =  =
к . The parameter Ln for this case is determined 

from Equation 2: 

2 3 3

2 2

arctan (2 1) 1.946
.н

h d n n
L

n n

 − − −
= =                (3) 

The derivatives of the hyperbolic tangent function correspond to the dependencies [6]: 

( ) ,tan1)(
32

2
21

nLnhnnL +−=
,
                     (4) 

 .)(tan1)(tan2)(
32

2
32

2
21

nLnhnLnhnnL +−+−=
.
        (5) 

Based on the values of the extrema of Equations 4, 5, the coordinates of the characteristic points of the 
curve of the dynamics of subsidence of the earth's surface are determined. 

The logistic curve equation for describing the dynamics of subsidence of the earth's surface has the form 
[5]: 

,
)exp(1

)(
Lcb

a
L

−+
=                                         (6) 

Where a - is an empirical coefficient corresponding to the final subsidence of the earth's surface ( к ); 

b, c - empirical coefficients that determine the position of the curve relative to the abscissa axis and the width of the 
middle section, i.e. the duration of the active stage of subsidence of the earth's surface. 
The first derivative of Equation 6 is characterized by the dependence: 

( )
( )

.
exp1

)exp(
2

Lcb

Lccba
L

−+

−
=                                         (7) 

Extreme value of the function ( )L  corresponds to the inflection point C of the logistic curve Figure 1 with 

coordinates ( ln
,
2

b a

c

). 

The second derivative of the original Equation 6: 

( )
( ) 

( )3

2

exp1

exp1)exp(

Lcb

LcbLccba
L

−+

−−−−
=                              (8) 

Has two extreme values. The values of these coordinates determine the position of the active stage of 
subsidence of the earth's surface (points B and D). 

The parameter Ln for the logistic curve (6) was determined from the conditions ккн d  == 01,01  and 

ка = : 

11 1
ln( )

4.595 ln
.н

d
bbL

c c

−
−

= − = −                (9) 

Parameter Lk  found from the condition ккd  )99,097,0(2 == : 

21 1
ln

3.892 ln
.к

d

bb
L

c c

− 
   − − = = −            (10) 

The coordinates of the characteristic points (A, O, B, C, D, F) of the curve of the dynamics of subsidence of the 
earth's surface, determined according to the initial dependencies (1, 2, 6), are summarized in Table 1. 



Asian Review of Environmental and Earth Sciences, 2023, 10(1): 40-49 

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Table 1. Dependencies for determining the coordinates of the characteristic points of the curve of the dynamics of subsidence of the earth's 
surface according to the exponential, hyperbolic tangent and logistic equations. 

Characteristic 
points of the curve 
of the dynamics of 
subsidence of the 
earth's surface  
Figure 1 

Exponential equation Hyperbolic tangent equation Logistic equation 

Abscissa 
L, m 

Ordinate  , mm Abscissa L, m 
Ordinate  , 

mm 

Abscissa L, 
m 

Ordinate 
 , mm 

A нL−  0 
2

3298,2

n

n+
−

 
0 

c

b

−

− ln595,4  
0 

O 0 ( ) 2
021 Lехрк −−   0  31 tanh1 nn +  0 

b

к

+1

  

B нL−
1

5246,0



 
0.241 к  

2

3658,0

n

n+
−

 
0.21 к  

( )
c

b

−

73,3ln  0.21 к  

C нL−
1

7071,0



 
0.393 к  

3

2

n

n
−

 
0.50 к  

c

bln  0.50 к  

D нL−
1

2247,1



 
0.777 к  

2

3658,0

n

n−  
0.80 к  

( )
c

b

−

268,0ln  0.79 к  

F 

( )
нL

d
−

−

−

1

1ln



 

(0.97÷0.99) к  
2

3946,1

n

n−  (0.97÷0.99)
к  

c

bln892,3 −  (0.97÷ 
0.99)

к  

 

The next stage of the work was to determine for each object of observation the empirical parameters 
included in the original equations. 

For the exponential Equation 1 we found the values 
к , 

1
 , LH, for the Equation 2 of the hyperbolic tangent - 

n1 , n2 , n3 and for the logistic dependence (6) - a, b, c. 
The processing of experimental data, in order to determine the empirical coefficients of Equations 1, 2, 3, was 

carried out by the least squares method. Using their numerical values and dependencies to determine the 
coordinates of characteristic points Table 1, we found the boundaries of the stages of subsidence of the earth's 
surface during the cleanup operations at ten sites Table 2. 
 
Table 2. The results of determining the empirical coefficients and correlation relationships (R) by the least squares method for the objects of 
observation. 

Mine, reservoir, literary 
source 

Math functions 

Logistics Exponential Hyperbolic tangent 

a =
к  b c R 1  

нL  
к  R n1 = 0.5

к  n 2 n 3 R 

"Belozerskaya",  [3] 810 4.1 0.016 0.998 2.0 105 810 0.999 405 0.008 -0.70 0.997 

№22 "Kommunarskaya", 
3К

, [4] 
900 9.3 0.010 0.998 1.0 70 900 0.976 450 0.005 -1.11 0.998 

"Gramoteinskaya", 
Sychevsky- III, [7] 

2375 13.0 0.028 0.999 5.0 Thirty 2420 0.987 1180 0.015 -1.28 0.995 

Appalachian basin mine, [8] 980 26.0 0.040 0.995 6.5 20 1010 0.991 490 0.019 -1.65 0.994 
"Anniversary", 

6С  , [9] 915 12.5 0.050 0.997 1.5 20 910 0.974 458 0.026 -1.27 0.996 

Them. A.F. Zasyadko, t 3 , 
[10] 

400 7.0 0.006 0.997 3.0 70 400 0.974 200 0.003 -1.20 0.996 

Staszic, 352, [10] 480 4.1 0.020 0.999 5.0 60 980 0.984 490 0.010 -0.70 0.999 
Ruhr basin mine, Grimberg 
2/3 , [10] 

1420 5.8 0.010 0.998 5.5 200 1420 0.979 710 0.005 -0.87 0.998 

Them. CM. Kirov, PO 
"Leninskugol", Boldyrevsky, 
[11] 

1300 7.1 0.070 0.996 3.3 21 1310 0.997 638 0.041 -0.98 0.994 

"Steppe", [12] 832 5.8 0.064 0.996 3.1 20 835 0.994 416 0.029 -0.95 0.995 

 

It should be noted that in most cases the values 
к , determined using the considered functions, were 

practically equal to each other. Differences, as a rule, did not exceed 1.0% and only in one case (the mine of the 
Appalachian basin), the maximum difference was 3.1%. This indicates the possibility of using any of the considered 
functions to determine the ordinates of characteristic points. 

A similar conclusion was reached about the possibility of using the analyzed functions to determine the 
abscissas of the characteristic points of subsidence of the earth's surface. Using the empirical coefficients of 

the equations Table 2 for all mines, we calculated the abscissa (
AL ) of the characteristic point A Table 3. 

On the basis of experimental data [7, 13] it was established that the parameters of the trough of 
displacement of the earth's surface by 80% or more can be determined by the depth of work. To test and 
confirm this assumption, on the basis of the data Table 3, we determined the dependences of the average 
values of the abscissas of the characteristic point A on the depth of the treatment. The results of these 
calculations show that the characteristic point abscissas are directly proportional to the mining depth 
Figure 2. In absolute value, the correlation coefficient (r) for different coal basins was equal to 0.89.  

 
 
 



Asian Review of Environmental and Earth Sciences, 2023, 10(1): 40-49 

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Table 3. The results of determining the coordinates of the characteristic point A of subsidence of the earth's surface along the 

abscissa and boundary angles (δo , γo , βo ). 

Mine, reservoir, literary 
source 

Depth of 
cleaning 

operation, 
H, m 

The 
thickness of 

the 
developed 
reservoir, 

m, m 

H/m 
Seam dip 

angle, 

α, degrees 

The distance 
between the 

projection of the 
production face and 

point A, 

AL , m 

Boundary 
angles, 

δo , γo , βo , 
degrees 

"Belozerskaya", [3] 420 1.30 323 12 169 68 
No. 22 "Kommunarskaya", 
K 3 , [4] 

652 1.47 444 20 178 75 

"Gramoteinskaya", 
Sychevsky- III, [7] 

220 4.50 49 Four 57 75 

Appalachian Basin Mine, [8] 220 1.65 133 - 28 83 
"Anniversary", S 6 , [9] 150 1.00 150 3 34 77 
Them. A.F. Zasyadko, m 3 , 
[10] 

1195 2.10 569 10 292 76 

"Stashitz", 352, [10] 480 2.10 229 - 126 75 
Ruhr mine, Grimberg , 2/3, 
[10] 

920 2.20 418 - 253 75 

Them. CM. Kirov, PO 
"Leninskugol", Boldyrevsky, 
[11] 

205 1.70 121 6 Thirty 82 

"Steppe", [12] 106 0.91 116 Four 37 71 

 

 
Figure 2. Dependence of the abscissa LA of the characteristic point A on the depth of the cleaning operations H and the parameter H/m. 

   Note:  1, 2 - Averaging direct links LA respectively with H and H m; ×, ○ - Experimental data; r - Correlation coefficient. 
 

 
A fairly close correlation (r = 0.95) was established between the abscissas LA and the relative parameter H/m 

Figure 2. This indicates that the parameter H/m, along with the depth, can determine the boundary of the dynamic 
trough in front of the projection of the stope. 

Connection of boundary angles (δo , γo , βo) with H and H/m has not been established Figure 3. Correlation 

coefficients were respectively - 0.13 and - 0.25. Boundary angles (δo , γo , βo) were in the range of 68-83°, with an 
average value of 76°. 
 



Asian Review of Environmental and Earth Sciences, 2023, 10(1): 40-49 

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Figure 3. Dependence of the boundary angles (δo , γo , βo ) on the depth of treatment operations H and the parameter H/m . 
Note:  1 - Straight line corresponding to the average values of the angles (76°); ×, ○ - Experimental values, respectively, of the 

dependence on H and H/m. 

 
Mining of coal seams has, first of all, an impact on the displacement of host rocks. As a result, the rock pressure 

on the lining of the development workings changes, and displacement troughs form on the earth's surface. In one 
case, the lining of the workings is deformed and the conditions for their maintenance in the zone of influence of the 
mining operations become much more complicated. In the second, it is necessary to take measures to protect 
objects on the earth's surface. Based on the similar course of the processes of subsidence of the earth's surface and 
the displacement of rocks under the influence of a moving stope on the contour of sectional workings, an 
assumption was made about a possible connection between the phenomena under consideration. The establishment 
of general patterns or differences between the subsidence of the earth's surface and the conditions for maintaining 
section workings can contribute to the successful solution of many engineering problems. These tasks include the 
development of rational measures for the protection of objects on the earth's surface and the maintenance of 
sectional workings in the zone of influence of clearing operations. Research in this direction is very relevant. 

A characteristic feature that combines the subsidence of the earth's surface and the displacement of rocks on the 
contour of sectional workings is the same type of experimental dependences of the dynamics of the processes under 
consideration. The common features of these processes include a gradual increase in the subsidence of the 
undermined rock mass on the contour of the section workings. The intensity of subsidence of the earth's surface 
and the displacement of rocks increases as the stope approaches. After the passage of the lava, the subsequent 
attenuation of the processes occurs. The dynamics of subsidence of the earth's surface and the displacement of roof 
rocks and soil of a sectional mine under the influence of a stope can be described by a diagram Figure 4. In this 

scheme, the abscissa shows the distances ( ЗL ) from the projection of the stope line onto the earth's surface to the 

observation points, as well as the distances from the stope ( ПL and КL ) to the points of observation of the 

displacement of soil and roof rocks on the working contour. 

Characteristic points that determine the dynamics of ongoing processes are: ПА , КА , ЗА - correspond to the 

beginning of the shift of soil rocks and the roof of workings and the earth's surface; ПО , КО , ЗО - are located in 

the alignment with the stope and serve as the origin of the coordinate axes; ПВ , КВ , ЗВ - the beginning of the 

active stage of displacement of soil rocks and the roof of workings and subsidence of the earth's surface; 
ПС , 

КС , 

ЗС - correspond to the maximum rate of rock displacement and subsidence of the earth's surface and are the 

inflection points of the curves; 
ПD , 

КD , ЗD - end of active stages and beginning of attenuation stages; 
ПF , 

КF , 

ЗF - the beginning of the residual impact of the stope on the processes under consideration. 

The decay stages in the scheme are limited by the points 
ПF , 

КF and ЗF . The subsidence of the earth's surface 

at a point ЗF is approximately 0.97 ÷ 0.99 of the final ( 
К ) at the end of the rock compaction processes. The end 

of the processes at the points 
ПF and 

КF can be established experimentally by comparing their shift with a change 

in the working contour outside the influence of the stope. According to, when workings are located outside the 
zone of mining operations, the difference between the roof-soil convergence does not exceed 10%, and the decrease 

in the design sections of workings is 0.8%. When calculating the coordinates of the points 
ПF and 

КF , taking into 

account the above ratios, we used the recommendations. 
The purpose of the work is to establish, on the basis of experimental data, the characteristic stages of 

subsidence of the earth's surface and the displacement of soil rocks and the roof of a sectional mine under the 
influence of a moving stope. 

 
 



Asian Review of Environmental and Earth Sciences, 2023, 10(1): 40-49 

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Figure 4. The scheme of subsidence of undermined rocks and the earth's surface and soil relative to the face. 

    Note:  
 

1 – reservoir under development; 2 - stope; 3,4,5 - curves characterizing, respectively, the displacement of the soil and the roof of the sectional 

working and the subsidence of the earth's surface; ПL , КL , ЗL - abscissa axes, respectively, for the soil and the roof of the working and the 

earth's surface; П , К , З - y-axis for the soil and the roof of the working and the earth's surface, respectively; ПА , ПВ , ПС , ПD , ПF - 

characteristic points of the soil displacement curve; КА , КВ , КС , КD , КF - characteristic points of the roof displacement curve; ЗА , ЗВ , 

ЗС , ЗD , ЗF - characteristic points of subsidence of the earth's surface. 

 
The method of work included several stages: 

• Development of a general scheme for the subsidence of the earth's surface and the displacement of soil rocks 
and the roof of a sectional mine Figure 4; 

• Analysis of the available experimental data on subsidence of the earth's surface and displacement of rocks on 
the contour of sectional workings; 

• Selection of mathematical dependencies that most accurately describe the processes under consideration and 
reflect their physical essence; 

• Study of empirical equations obtained on the basis of the accepted mathematical dependence using derivatives 
to establish the characteristic points of subsidence of the earth's surface and the displacement of rocks on the 
contour of a sectional working; 

• Determination of the distances from the characteristic points of subsidence of the earth's surface to the 

projection of the production face ( 
З

AL , 
З

ВL , З

СL , 
З

DL , 
З

FL ) and the removal of characteristic points of 

displacement of soil and roof rocks from the production face ( 
П

AL , 
П

ВL , П

СL , 
П

DL , 
П

FL and 
К

AL , 
К

ВL , К

СL , 
К

DL , 

К

FL ); 

• Comparison of the established parameters and conclusions about the general or distinctive patterns of the 
processes under consideration. 

For the practical implementation of the scheme Figure 4, as the initial function describing the processes under 
consideration, based on the results of work, we took the logistic curve of the form: 

( )
)exp(1 Lcb

a
L

−+
= ,                                            (11) 

Where  is the subsidence of the observation point on the earth's surface or the displacement of rocks on the 

contour of the working; a is an empirical coefficient corresponding to the final value of subsidence ( К ); b , c are 

empirical coefficients that determine the characteristic stages of the processes; L - distances characterizing the 
position of points relative to the production face along the abscissa axis. 

Empirical coefficients of Equation 11 a, b, c, which correspond to the parameters of subsidence of the earth's 
surface in the conditions of the Stepnaya mine, were determined according to empirical dependencies. It was found 



Asian Review of Environmental and Earth Sciences, 2023, 10(1): 40-49 

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that the coefficients a, b And c depend on the thickness of the developed seam ( t ), the depth of work ( H ), the speed 

of advancing the stope ( оч ) and the length of the lava ( лL ): 

)1054,11064,2(

1
34 −− +−

=
m

a ,                                     (12) 

)19,014,0(

1

+−
=

H
b

оч
,                                            (13) 

)
1

ln(0148,0205,0
HLm

c
л 

+= .                                  (14) 

Correlation relationships ( R ) for empirical dependences (12,13,14) were respectively 0.881, 0.884 and 0.986. 
This indicates the possibility of a fairly accurate determination of the coefficients of dependence (11) according to 
Equations 12,13,14. For their calculation, we used the parameters characterizing the operating conditions of the 

157th and 161st longwalls of the Stepnaya mine ( t = 1.04m, H = 395m, оч =122m/month, лL = 300m). For the 

specified conditions, the values of the coefficients a, b and c characterizing the subsidence of the earth's surface, 
respectively, amounted to 809, 6.81 and 0.032. 

The empirical coefficients of Equation 11 for the analytical description of the displacement of roof and soil 
rocks on the contour of the 159th and 163rd drifts were determined from the results of processing the experimental 
data using the least squares method Figure 5. The established dependencies practically functionally describe the 
dynamics of rock displacement on the contour of sectional workings (R = 0.964÷0.986). This indicates the 
possibility of their application in engineering calculations. Thus, on the basis of the analysis of the available 
experimental data, the empirical coefficients of the logistic dependence (1) were determined, characterizing both the 
subsidence of the earth's surface and the displacement of roof rocks and soil of sectional workings. The use of one 
initial dependence allows you to establish the degree of closeness or difference between the parameters of 
subsidence of the earth's surface and the displacement of rocks on the contour of sectional workings. 
 

 

 
Figure 5. Dependence of rock displacement (  ) on the contour of the 159th (a) and 163rd (b) drifts on the distance to the stope ( L ) during 

the development of the 157th 161st lava by the Stepnaya mine. 
 Note: 1,2 - averaging curves for the displacement of rocks, respectively, of the roof and soil; ▲, ■ - experimental data; R is the correlation ratio. 

 
To establish the stages of the processes of subsidence of the earth's surface and the displacement of rocks on the 

contour of sectional workings (determining the coordinates of characteristic points), methodological approaches 
were used to study functions using their derivatives. The results of the study of the logistic curve and the general 
equations for determining the coordinates of the characteristic points are shown in Table 4. Substituting the values 
of the empirical coefficients ( a, b , c ) into these equations, we determined the coordinates of the characteristic 
points of subsidence of the earth's surface and the displacement of rocks on the contour of sectional workings 
during the development of the 157th and the 161st lava by the Stepnaya mine. Based on the numerical values of the 
coordinates of the characteristic points Table 4 and the location of the curves relative to the stope Figure 6, a 
comparative analysis was made of the processes of subsidence of the earth's surface and the displacement of rocks 
on the contour of sectional workings. 



Asian Review of Environmental and Earth Sciences, 2023, 10(1): 40-49 

48 
© 2023 by the authors; licensee Asian Online Journal Publishing Group 

 

 

Table 4. The results of determining the coordinates of the characteristic points of subsidence of the earth's surface and the displacement of 
rocks on the contour of the excavation 159th and 163rd drifts of the Stepnaya mine. 

Characteristic 
points of the 
logistic curve 

Equations for determining the 
coordinates of the characteristic 

points of the logistic curve 

Values of empirical coefficients (а, b , c ) of logistic dependence and 

coordinates of characteristic points ( L ,  ) 

Earth surface 
159th Drift 163rd Drift 

Roof The soil Roof The soil 

Abscissa, L , m Ordinate,  , mm 

a=809, b 
=6.81, c 
=0.032 

a=712, b 
=0.812, c 
=0.020 

a=497, b 
=0.631, c 
=0.021 

a=542, b 
=1.203, c 
=0.025 

a=517, b 
=1.427, c 
=0.028 

L , m  , mm L , m 


,mm 
L , m 


,mm 

L , m 


,mm 
L , m 


,mm 

A 
c

b

−

− ln595,4
 0 -84 0 -240 0 -241 0 -176 0 -151 0 

O 0 
b

К

+1


 0 104 0 393 0 305 0 246 0 213 

B 
c

b

−

)73,3ln(
 0.21∙

К  19 170 -76 150 -85 104 -45 114 -34 109 

C 
c

bln
 0.50∙

К  60 405 -10 356 -22 249 7 271 13 259 

D 
c

b

−

)286,0ln(
 0.79∙

К  101 639 17 562 20 393 60 428 60 408 

F 
c

bln892,3 −
 0.99∙

К  182 801 184 705 163 492 163 537 152 512 

 

Figure 6. Dependences of the subsidence of points on the earth's surface and the displacement of rocks on the contour of development 

workings ( ) on their position relative to the stope ( L ) in the conditions of the Stepnaya mine. 

 Note:  1 - curve of subsidence of the earth's surface; 2,4 – roof and soil displacement curves on the contour of the 159th drift; 3.5 - roof and soil 

displacement curves on the contour of the 163rd drift; 
ПА , 

ПВ , 
ПС , 

ПD , 
ПF - characteristic points of displacement of soil rocks; 

КА , 

КВ , 
КС , 

КD , 
КF - characteristic points of displacement of roof rocks; ЗА , ЗВ , ЗС , ЗD , ЗF - characteristic points of subsidence of 

the earth's surface. 
 
Coefficient a characterizes the end of processes. Its maximum value corresponded to the subsidence of the 

earth's surface (809 mm), which is somewhat less than the recoverable thickness of the developed seam ( t = 1.04 
m). The displacement of the roof on the contour of the 159th and 163rd drifts, respectively, reached 705 and 537 
mm. The final displacement of the soil in these workings was 492 and 512 mm. The displacement of soil rocks, in 
terms of the nature of manifestation and the absolute values of the parameters, differs little from the displacement 
of roof rocks. This is obviously due to the low strength properties of the host rocks. Under the conditions of strong 



Asian Review of Environmental and Earth Sciences, 2023, 10(1): 40-49 

49 
© 2023 by the authors; licensee Asian Online Journal Publishing Group 

 

 

enclosing rocks, the differences in the displacement of the roof and soil of sectional workings should be expected to 
be more significant. Coefficients b and c define the coordinates of the characteristic points along the x-axis. The 
sizes of individual stages characterizing the intensity of the processes depend on their ratio. The coefficient b = 6.81 
for the earth's surface was 5–10 times higher than its value (0.631–1.422) for the circuit of section workings, and 
the value c = 0.032 was comparable with similar coefficients for workings ( c = 0.020–0.028). 

Different values of the empirical coefficients caused the unequal location of the characteristic points of the shear 
trough on the day surface and the rocks on the contour of the sectional workings relative to the stope. The 
beginning of the shift of the roof and soil rocks occurred at a distance of -151 ÷ -241 m (Figure 6, Table 4), which 

is much more than the distance from the projection of a point on the earth's surface ЗА to the stope alignment ( 
АL

= -84 m). 
The beginning of subsidence of the earth's surface corresponds to the beginning of active stages of rock 

displacement (points
ПВ  and 

КВ ) on the contour of workings. In all cases, the active stage of rock movement 

began ahead of the stope. This indicates that the location of the point ЗА on the earth's surface determines the 

beginning of the active manifestation of high rock pressure in front of the stope (the zone of HRP). The obtained 
results indicate the participation of the entire rock mass from the developed reservoir to the earth's surface in the 
formation of the HRP zones. They are confirmed by directly proportional experimental dependences of rock 
displacements in workings on the depth of mining. The maximum intensity of rock displacement on the contour of 

workings was observed Figure 6 in the immediate vicinity of the stope at points 
ПС and 

КС . The active stage of 

subsidence of the earth's surface occurred (points ЗА , ЗВ , ЗС ) after the stope passed over the goaf. 

Approximately the same distances (152÷184 m) from the stope (points 
ПF , 

КF , ЗF ) corresponded to the end 

of the processes of subsidence of the earth's surface and the displacement of rocks in the contour of workings. 
The above experimental and theoretical studies made it possible to draw the following important conclusions 

for science and production activities: 

• The beginning of subsidence of the earth's surface corresponds to the beginning of the stage of intensive 
displacement of the roof and soil rocks ahead of the stope; 

• The active stage of displacement of rocks in the contour of development workings begins in front of the 
stope, and ends after its passage; 

• The active stage of subsidence of the earth's surface occurs above the worked-out space behind the stope; 

• The processes of subsidence of the earth's surface and the displacement of rocks on the contour of sectional 
workings stop at approximately the same distance after the passage of the stope. 

 

4. Conclusions  
The conducted research allowed to establish the following: 

• The main influencing factor that determines the boundary (LA) of the dynamic half-trough on the earth's 
surface in front of the projection of the moving stope is the depth of mining (H). This dependence is 

directly proportional LA=0.263·H ; 

• Boundary angles (δo , γo , βo) practically do not depend on the depth of treatment operations. Their values 
were in the range of 68÷83°, with an average value of 76°. 

 

References 
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Ukraine: GSTU, 2004. 
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