Analysis of the Model Used to Predict Continuous Deformations of the Land Surface in Areas Subject to Discontinuous Deformations—A Case Study

Abstract

This article analyses and evaluates the model used by the Jastrzebie Coal Company in Poland to forecast the values of continuous deformations of land surfaces caused by underground mining. Particular attention is paid to the values of terrain inclinations, which cause significant deflections of buildings located in mining areas affected by exploitation. The inclination forecasts were made using Bialek’s model, with the values of its parameters determined in situ and after completion of the exploitation of a longwall. The obtained results were compared with the values of the terrain inclinations calculated from the results of geodetic measurements carried out at observation points located near the buildings and discontinuous linear deformations (i.e., the ground steps). The conducted research shows that the correlations between the absolute values of the practical inclinations and the values of the theoretical inclinations were rather weak, as the values of their correlation coefficients did not exceed 0.24. The tested model underestimated 45.5% of the values of the inclinations observed in situ by an average of −53.5%. The model values of the inclinations for the remaining observed inclinations (54.5%) were overestimated by an average of 461.9%. The largest anomalies were obtained in the case of inclination values from points located near the ground steps. The effectiveness of predicting inclination values with a commonly used model is strongly reduced when discontinuous linear deformations appear. Nonetheless, it reflects well the mining area categories.

1. Introduction

The forecasting of continuous deformations of a land surface (i.e., a type of terrain deformation that does not cause a break in the terrain surface, occupies large areas, and occurs slowly, from several to several dozen years [1], e.g., subsidence—$W$, inclinations—$T$, curvatures—$K$, horizontal strains—${\epsilon }_x$, and horizontal displacements—$U_x$) caused by underground operations on hard coal deposits is a very important issue for many reasons. The results of predictions provide information about the expected condition of the subsoil and buildings during and after a deposit’s extraction, and they are useful for urban planners, architects, builders, constructors of engineering structures, surveyors, and many others. For example, urban planners decide on land use based on the categories of a mining terrain that are separated based on inclinations, curvatures, and horizontal strain values [2,3]; buildings constructors choose the proper load-bearing structure for an object and the building’s reinforcement against mining impacts based on the categories of a building’s resistance connected with a land category [4,5].

Similar issues in predicting ground behavior and problems related to the occurrence of land surface deformations may occur during the construction and operation of large-diameter underground pipelines, as well as tunnels, which may resemble mining corridors located at relatively shallow depths [6,7,8].

Scientists around the world and especially in countries where the underground mining of hard coal deposits is carried out (Ukraine, Poland, Germany, France, Spain, Great Britain, Brazil, Chile, Columbia, the United States of America, Canada, North and South Korea, Russian Federation, China, Malaysia, Indonesia, Australia, India, the Republic of South Africa, and Turkey) have been conducting research for several decades on creating and improving models describing the impact of underground mining on the land surfaces.

The first attempts to describe the vertical movement of the ground toward voids created in rock masses due to the mining of hard coal seams were made at the University of Science and Technology in Cracow (Poland) by Stanislaw Knothe in the 1950s. This researcher described the profile of a subsidence basin that formed on the land surface after the end of exploitation using the Gaussian function [9]. The profile of the subsidence trough was symmetrical to the point located above the center of the exploitation field, and the $W$ subsidence reached its maximum value there ($W_{max}$). At points located above the edges of the exploitation, the subsidence was half of the maximum value. Then, the scientist developed a possible method for calculating the value of the subsidence at any time during exploitation and after its completion [10]. The model proposed calculating the subsidence values as linear and dependent on the values of the parameters of operation (exploitation coefficient—$a$) and the main influence’s ranges (tangent of the $\beta$ angle of the main influences range—tan $\beta$). The value of the coefficient depends on the method of mining concerning the goaf’s liquidation, the $\beta$ tangent of the geological-mining conditions of an operation, and the mechanical properties of the rocks constituting the rock mass.

Knothe’s research was continued by Jerzy Litwiniszyn, who assumed that the rock mass was a stochastic medium [11,12,13,14,15]. He used probability theory to determine the vertical displacements of the rock mass [16].

The issue of the horizontal displacement of rock masses as a result of underground mining was dealt with by Witold Budryk [17] and Tadeusz Kochmanski [18].

Subsequent researchers refined the Knothe model, determined the values of its parameters, and examined the influence of time on the development of the subsidence basin and the exploitation speed on the values of deformation indicators [19,20,21].

However, Knothe’s model proved to be suitable only for simple mining operations carried out in horizontal seams, the impacts of which are added up and calculated linearly. Therefore, subsequent scientists from the Silesian University of Technology in Gliwice (Poland) developed this model which has the ability to evaluate the impacts of the exploitation of multiple inclined seams. Desymmetrization of the vertical profiles of subsidence troughs along the edges of exploitations was introduced by Jan Zych [22] and Jan Bialek [23,24], and the delinearization of influence values was introduced by Bernard Dzeniuk [25] and Jan Bialek [26].

In addition, the introduced mining margin involved shifting the actual exploitation edge toward the goaves due to the octahedral strain occurring in the roof directly above the workings [27].

After many years of improving Knothe’s formula for subsidence, Jan Bialek obtained a form of this formula that takes the following into account: nonlinear summation of influences resulting from the exploitation of an unlimited number of inclined seams, desymmetrization of the profile of the subsidence trough with the edges of the exploitation field, which has a complicated shape due to the existence of the $d$ mining margin, distant influences [28], reactivation of old goaves [29], and the time factor [30,31].

Jan Bialek’s formula for the subsidence of a mining area, as well as the inclination and curvature (sequentially calculated as the first and second derivatives of the subsidence) together with Awierszyn’s hypothesis [32] (which allows for the calculation of the values of the horizontal displacement and horizontal strain), because of their universality and the possibility of being implemented in computer software, are commonly used to forecast continuous deformations of land surfaces caused by underground mining of hard coal carried out in Central and Eastern Europe, the south of Poland, and the Upper Silesian Coal Basin by Jastrzebie Coal Company.

Bialek’s formula was implemented in the EDN–OPN computer program [33]. It allows for calculating the values of deformation indicators at a specific point (EDBJ1 program) and the creation of maps of the distribution of deformation indicators for the whole surface of the mining area (EDBJ2 program) [34]. There are also other computer programs enabling deformation forecasts based on Knothe’s geometric integral theory of mining influences, e.g., MODEZ [35,36,37], DEFK [38,39,40,41], and on the finite element method [42,43,44,45,46,47] or neural networks [48], but in commercial activity in Poland the most common is the EDN-OPN. The effectiveness and accuracy of this program have been repeatedly checked for various geological-mining conditions [49,50,51,52,53]. This program is characterized by very good reproductions of actual values of deformation indicators by their predicted values but after determining the appropriate values of the parameters of Bialek’s formula [54]. However, this program may not work properly when, in addition to continuous deformations, discontinuous deformations [55,56,57,58,59,60] also unexpectedly appear on the terrain’s surface (terrain surface deformations that consist of a break in the continuity and relative, macroscopically visible displacement of the subsurface layers of the ground, and the clear destruction of the terrain’s surface. Deformations of this type occupy relatively small areas and have clear boundaries. They occur quickly within a few hours or days, or even suddenly—within a few minutes [1]). Incorrect operation of the program was shown in this article on examples of inclination values, as follows: theoretical, calculated in the program and real, observed in the terrain using geodetic measurements made on 51 observation points located around nine objects.

The influence of discontinuous linear deformations on buildings and their deflection values [61], as well as on the values of continuous deformations of the terrain surface [56], has already been investigated. However, the influence of their occurrence on the effectiveness of predicting continuous deformations of the terrain surface using a commonly used model (especially in Poland) has not been determined yet, which makes this research unique.

2. Materials and Methods

Below are descriptions of the research area, the underground hard coal mining which caused the terrain inclinations, the observation network and geodetic measurements carried out, and the EDN-OPN program and Bialek’s formula, which were used to calculate the predicted values of the terrain inclinations and the parameter values of this formula.

2.1. Research Area

The material comes from the research area located in the south of Poland, in the Upper Silesian Coal Basin, in the city of Rybnik, where hard coal is mined by the Jastrzebie Coal Company (Figure 1).

The discussed mining area is specific for two reasons. Firstly, it is an area with specific geometric conditions (relatively short longwalls with overlapping mining edges) and mining conditions (relatively small depths of longwalls compared to currently observed average exploitation depths, and the exploitation of the thick hard coal seams is divided into several layers and inclined at large angles) of the conducted exploitation, as well as the geological structure of the subsurface ground layers, which consist of loosely sands and clays, as already described in [57]. Secondly, this area is specific for social reasons, as it is inhabited by a local community that protects its property from damage caused by the mining plant, which is (or was) often the source of income for the residents and their maintenance. On the one hand, it is about the welfare and protection of the interests of the residents and, on the other hand, of the Jastrzebie Coal Company.

The landform is rather uniform here. Escarpments are localized in the middle part of the area. The land elevation (~265 m) shows that the area is lowland. Development of the terrain consists of meadows, agricultural fields, and pastures in the east, allotment gardens in the southwest, apartment buildings and utility buildings in the north (Figure 2).

Single-family houses are freestanding and have two floors. The load-bearing construction is based on the walls. The buildings have resistance category ratings in the 2nd to 4th categories [4,5] concerning the mining exploitation influences.

2.2. Underground Exploitation of Hard Coal Seams

The underground operation of the 404/3, 404/5, and 405/1 seams of hard coal took place in the years 2013–2019, using a system involving the cave-in of roof rocks into the void formed after extraction of the deposit from the longwalls. The mining workings are to a medium depth of 505 m and a medium height of 3.2 m (Figure 3a).

Two longwalls (1/II and 2/II) were exploited in the 404/3 hard coal seam. The heights of the longwalls were 4.0 m and 3.4 m, respectively. The depth of the operation was from 410 m (east) to 575 m (west). The seam declined at an angle of 17°. The exploitation in the 1/II longwall took place during the time period of May–December 2013 and in the 2/II longwall in August 2016–February 2017.

The extraction from the 404/5 hard coal seam started in November 2015 in the 1/II longwall. It had a depth of 470 m and a height of 3.3 m. Exploitation of this longwall ended in June 2016. Exploitation of the 2/II longwall was conducted for less than a year (October 2017–April 2018). It was 3.5 m high and had a depth of 550 m. The declination of the deposit was 18.5°.

Operations on the 405/1 seam of hard coal started in February 2019 in the 1/II longwall. The average depth of the exploitation was 550 m and its height was 1.7 m. The declination of the hard coal seam was 18°. The exploitation of this longwall was finished in August 2019.

The characteristics of the operations carried out in the 404/3, 404/5, and 405/1 seams of the hard coal are presented in

Table 1. Underground operation of the 404/3, 404/5, 405/1 seams of a hard coal in analyzed area.

Feature	The 404/3 Seam		The 404/5 Seam		The 405/1 Seam
Longwall	1/II	2/II	1/II	2/II	1/II
Depth (m)	410–500	500–575	430–510	500–600	495–605
Height (m)	4.0	3.4	3.3	3.5	1.7
Declination (°)	18.8	15.8	16.9	21.1	18.0
Run (m)	460	245	500	300	443
Length (m)	220	243	230	233	240
Time	May 2013– December 2013	August 2016– February 2017	November 2015– June 2016	October 2017– April 2018	February 2019– August 2019

.

The overburden is approximately 220 m thick and is formed by layers of sands, gravels, Quaternary clays, Miocene clays, and sands. Thus, the rock mass here is not the most compact, and therefore, to a very large extent, it can be treated as a loose medium. Such a structure also partially explains the appearance of discontinuous linear deformations on the ground surface, which is described in the next section.

2.3. Discontinuous Deformations of the Land Surface

The underground mining operations on the 404 and 405 seams of hard coal, in addition to continuous deformations of the terrain’s surface (e.g., subsidence and inclinations), caused many linear discontinuous deformations, mostly ground steps, faults, and cracks.

Discontinuous deformations of the land surface occurred in the zone located in the northwest. They run near the northern edges of the 2/II longwalls in the 404/3 seam and the 404/5 seam, parallel to the main road, under and next to the residential buildings numbered B1, B2, and B3, and a garage marked by the G letter. The ground steps and faults have heights from 5 cm to 30 cm and are from 10.5 m to 224.8 m long. Their direction is southwestern (Figure 3b).

The direct causes of the formation of faults and ground steps were the large thickness of the hard coal deposits (almost 10 m in total), shallow depth of the exploitation (an average depth of 500 m), large deviation in the hard coal seams (an average deviation of 19°) in the northwest direction and an overlapping of operational edges [57].

Discontinuous deformations cause changes in the inclinations of objects in a horizon and deviations from vertical. These objects are subject to the monitoring of deflections (for example, by geodetic measurements), and if their permissible values are exceeded, they are subject to rectification [62,63,64,65,66,67,68,69]. Living in tilted buildings is burdensome and problematic and, in the case of their renovation or rectification, even impossible. Moreover, the resulting mining damage contributes to a significant decline in the value of buildings on the real estate market. Therefore, an important issue is the prediction of correct values of inclinations in the event of discontinuous linear deformations.

2.4. Observation Network and Geodetic Measurements

As previously mentioned, in order to compare the theoretical values of inclinations calculated by Bialek’s model with their real measured values, an observation network was established near nine objects (eight residential buildings numbered from B1 to B8 and one garage marked with the G letter), on which appropriate geodetic measurements were made.

The residential buildings are single-family houses that usually have two floors (a ground floor and an aboveground floor), and their heights did not exceed 10 m. The horizontal projections of the buildings have the shapes of squares and rectangles with dimensions not exceeding 21 m × 15 m. Most of the objects have a wall-bearing structure, with ceilings made of concrete with steel girders. The walls are made of bricks or ceramic hollow bricks. The walls have a thickness from 0.22 m to 0.48 m. The buildings’ roofs serve as the ceilings, gables, or envelope roofs and are covered by bitumen felt layers or metal and ceramic roof tiles. The single garage is located near building no. 3 (B3) and has dimensions of 5.80 m × 10.35 m × 3.85 m (length × width × height). It is a three-car garage made of bricks. Pictures of the specific buildings are presented in Figure 4.

The observation network consisted of 51 measurement points stabilized in the ground (marked in brown with the $z$ letter) using metal pins (marked with the letter $p$), nails (marked with the letter $g$), or wooden pickets (marked with the letter $d$), as shown in Figure 5.

The methods of stabilizing the measurement points depended on the type of surface (asphalt—a metal pin 7.5 cm long, the 10LK model (Figure 6a); paving stones—a nail 10 cm long, the 10ZSO model (Figure 6b); lawn—a wooden picket 35 cm long (Figure 6c)).

Situational and altitude geodetic surveys after stabilization of measurement points were conducted.

Situational measurements took place on 2 March 2019 and were carried out by use of the GNSS receiver and the RTN (real-time network) method (Figure 7a). This method consists in sending correction data from several surrounding physical reference stations with known positions determined with the PL-ETRF2000 coordinate system. After receiving the approximate position of the GNSS receiver via the NMEA GGA message, the computing system creates a virtual reference station near the receiver for which the VRS surface corrections are calculated. They are then sent as if they came from a single reference station as in the RTK method (real-time kinematic). During the surveys, a Trimble R8s dual-frequency receiver with a TSC3 controller and a network of permanent stations forming the ASG-EUPOS system (exactly four stations: TAR1, KATO, WOD1, and ZYWI) were used. Data were transmitted in real time via the Internet and port 8080 (corrections RTCM 166, 161, 164, and 165, respectively). The measurements were performed in the vicinity of at least 6 satellites included in the GPS and GLONASS systems, whose orbits were at least 19° above the horizon line. The receiver initialization took over 60 s, a single measurement of a particular picket lasted an average of 4 s in 5 intervals of 10 s.

Each point was measured twice and its $x$ and $y$ coordinates were determined for the WGS 84 rotation ellipsoid as the average value of the two measures. Then, the coordinates obtained this way were transferred to the PL 2000 reference system applicable in Poland. The error ($m$) in the position of the point along the $x$ axis was $m_x=\pm 0.035 \left[m\right]$, and along the $y$ axis it was $m_y=\pm 0.020 \left[m\right]$.

Because of the required small error in measuring the heights of the ground points, altitude measurements were conducted using a Carl Zeiss Jena Ni007 (Koni007) precise leveler (Carl Zeiss, Jena, Germany) with a reading accuracy of 0.0001 m (Figure 7b) on 4 March 2019 (first measurement) and on 17 August 2019 (last measurement). There were differences determined in the altitudes of neighboring points ($\mathsf{\Delta }{h}_{i,i+1}$), which eliminated the need to refer measurements to the reference points of known altitudes. Changes in the height differences were observed between the initial ($s$) and final ($f$) measuring cycles. The error in determining the relative altitude difference was $m_{\mathsf{\Delta }h}=\pm 0.001 \left[m\right]$.

In addition to the situational and altitude surveys, double measurements of the horizontal distances between neighboring, observation points ($l_{i,i+1}$) were also carried out to determine the values of the inclinations occurring in individual sections ($T_{i,i+1}$). They were measured using the Leica Geosystems D510 electronic distance meter with a reading accuracy of 0.001 m (Figure 7c). The error in determining the length between subsequent points was $m{l}_{i,i+1}=\pm 0.0015 \left[m\right]$.

2.5. Bialek’s Model and the Values of Its Parameters

This article examines the efficiency of the most popular model (Bialek’s formula) used in forecasting continuous deformations of a land surface induced by underground mining in the Upper Silesian Coal Basin in Poland and on examples of land inclinations and where linear discontinuous deformations also occurred on the land surface.

Bialek’s model allows for the calculation of the subsidence values ($W_B$) in the mining area and its derivatives (first derivative—the inclinations ($T_B$) of the land surface; second derivative—the curvatures ($K_B$) of the land surface) induced by underground extraction of hard coal from exploitation fields, as follows:

• Of any shape and any number of vertices;
• Of any height;
• Located at various depths;
• Located in one or several hard coal seams lying horizontally or inclined at any angle;
• With a longwall front moving at any velocity;
• Exploited at any time;
• Located in old goaves;
• Containing mining margins;
• Causing distant influences;
• Causing asymmetry of the profile of the subsidence trough on its slopes.

Bialek’s formula for the subsidence of a mining area is based on the geometric integral theory of influences by Stanislaw Knothe [9,10].

Knothe’s theory describes the course of subsidence in a mining area using a two-parameter Gaussian function. Assuming that the exploitation field has the shape of a half-plane and the subsidence trough, created as a result of its excavation, is stabilized (static), the formula for the $W_K$ subsidence has the following form:

$$W_K\left(x;a, g, r\right)=-{\displaystyle \frac{a·g}{r}}{{\displaystyle \int }}_x^{\infty }{e}^{-\pi {\displaystyle \frac{x^2}{r^2}}}dx \left[\mathrm{m}\right],$$

(1)

where the variables mean the following:

• $a$—Exploitation coefficient;
• $g$—Thickness of hard coal seam (a height of longwall) [m];
• $r={\displaystyle \frac{h}{tan\beta }}$—Range radii of the main influences [m];
• $h$—Depth of the exploitation [m];
• $\beta$—Range angle of the main influences [°];
• $x$—Distance from the longwall front [m].

The profiles of the real subsidence troughs are characterized by certain deviations from the profiles calculated using Knothe’s formula. Because they are difficult to describe using a linear theory of influences, Jan Bialek significantly modified this formula by introducing additional parameters that allow for Knothe’s function to be more flexible. These modifications led to an improvement in the description of the asymmetry of the profile of the subsidence trough in relation to the edge of exploitation and the exploitation’s influences in the extreme parts of this trough.

Bialek developed a very general methodology for describing deformations, allowing for summing of the influences of an exploitation distributed in various seams. This methodology includes, in addition to lateral activation (influence of old goaves located in the same seam), the activation of old goaves in neighboring seams.

The operator method ($A_n$ operators) of modifying the description of the subsidence trough profile used by Bialek is related to the presence of the $d$ mining margin. It refers to the issue of the increase in the volume of the rock mass as a result of vertical strains caused by mining exploitation. The essence of this description is the assumption that the $W_B$ subsidence can be calculated as the sum of the $W_K$ linear influences calculated using Knothe’s formula and the $\mathsf{\Delta }{W}_K$ nonlinear correction, as follows:

$$W_B\left(x\right)=W_K\left(x\right)+f\left(\gamma ,W_K\right) \left[\mathrm{m}\right],$$

(2)

where the following holds:

• $\gamma$—A simplified form of the ${\gamma }_{oct}$ octahedral strain [${\displaystyle \frac{\mathrm{mm}}{\mathrm{m}}}$].

Bialek’s formula describing the $W_B$ subsidence and taking into account the asymmetry of the subsidence trough profile has the following form:

$$W_B=\left(1-a_w\right)W_K\left(r_1\right)+a_w{W}_K\left(r_2\right)-A_1\left(2+{\displaystyle \frac{A_3}{2}}\right){\displaystyle \frac{W_K\left(r_1\right){\left[r_1\gamma \left(r_1\right)\right]}^2}{A_3{\left[\frac{W_K\left(r_1\right)}{2}+\frac{W_K\left(r_2\right)}{2}\right]}^2+{\left[r_1\gamma \left(r_1\right)\right]}^2}}\left[\mathrm{m}\right],$$

(3)

where the following holds:

• $A_1$—A dimensionless multiplier taking into account the asymmetry of the subsidence trough profile caused by the $d$ mining margin;
• $W_K\left(r_1\right)$, $W_K\left(r_2\right)$—The subsidence calculated by Knothe’s formula using the two different $r_1$ and $r_2$ radii of the influences’ dispersion [m];
• $\gamma \left(r_1\right)$—A simplified form of the ${\gamma }_{oct}$ octahedral strain for $r=r_1$ $\left[{\displaystyle \frac{\mathrm{mm}}{\mathrm{m}}}\right]$;
• $a_w=0.4÷1.25 A_1$;
• $A_3=6.667$;
• $r_2=2r_1$ [m];
• $r_1={\displaystyle \frac{h}{tan\beta }} F\left(A_1\right)$[m] (Table 2).

Table 2. Values of the $F\left(A_1\right)$ function [24].

${\mathit{A}}_1$	0	0.050	0.100	0.150	0.200	0.250	0.300
$F\left(A_1\right)$	0.800	0.844	0.916	1.003	1.099	1.200	1.303

Table 2. Values of the $F\left(A_1\right)$ function [24].

${\mathit{A}}_1$	0	0.050	0.100	0.150	0.200	0.250	0.300
$F\left(A_1\right)$	0.800	0.844	0.916	1.003	1.099	1.200	1.303

The use of the 3 formulas requires the determination of the three parameters based on geodetic measurements, which can be clearly defined based on the profile of the full, established or quasistatic subsidence trough. They are as follows:

• $tan\beta$—The tangent of an angle of the main influences; a parameter of Knothe’s theory determined based on the maximum inclination $(T_{\max }$) and the maximum subsidence $(W_{\max }$) measured in the full subsidence trough with a known exploitation depth ($h)$:

  $$tan\beta ={\displaystyle \frac{h}{r}}={\displaystyle \frac{T_{max}}{\frac{W_{max}}{h}}};$$

  (4)
• a—The exploitation coefficient determined based on the maximum subsidence ($W_{\max }$) measured in the full subsidence trough and the thickness ($g$) of the hard coal seam (height of the exploitation), as follows:

  $$a={\displaystyle \frac{W_{max}}{g}};$$

  (5)
• $A_1$—The dimensionless multiplier taking into account the asymmetry of the subsidence trough profile caused by the $d$ mining margin; it can be assumed that the approximately size of the $d$ mining margin is proportional to the size of the $A_1$ parameter; the most frequently observed values are $A_1=0.10÷0.25$; to determine the $A_1$ parameter, it is necessary to know the value of the maximum subsidence ($W_{\max }$) in the full subsidence trough ($W_{\max }=a·g$) and the subsidence value above the exploitation edge in this trough ($W_{ed}$), as follows:

  $$A_1={\displaystyle \frac{0.5W_{max}-W_{ed}}{W_{max}}}.$$

  (6)

Value of the $T_B$ inclination of a land surface (the slope of the subsidence trough) in Bialek’s model is calculated as the first derivative of the $W_B$ subsidence calculated by Formula (3), as follows:

$$T_B={\left(W_B\right)}^{\prime }={\displaystyle \frac{\partial W_B}{\partial x}} \left[{\displaystyle \frac{\mathrm{mm}}{\mathrm{m}}}\right].$$

(7)

As mentioned earlier, to calculate the expected values of the land inclinations using Bialek’s model and Formula (7), it is necessary to know the values of the parameters of this model (the $a$ exploitation coefficient, the tangent of the $\beta$ angle of a range of the main influences ($tan\beta$), and the $A_1$ parameter of the mining margin). The values of these parameters can be obtained in the following three ways:

• Based on the results of the geodetic measurements (of the $W_{\mathrm{i}}$ terrain subsidence) carried out after the exploitation ended at points along the observation line running longitudinally or transversely through the center of the exploitation field. The terrain subsidence represents the profile of a static or quasistatic subsidence trough, in which a maximum value of subsidence ($W_{\max }$) was observed. This solution is the most advantageous from the point of view of the small values of the errors in the determined model parameters, which should then translate into its high efficiency. The limitations are as follows: the possibility of physically stabilizing the observation points at a land surface exactly above the center of the exploitation field due to the complicated landform and its intensive development, as well as the time that the appropriate parameters were obtained (after the exploitation ended), which allows for making backward forecasts (i.e., reforecasts). In this article, this method was used to obtain the values of the model parameters, because they reflect the actual state of the rock mass after the exploitation had ended in the 1/II longwall of the 405/1 hard coal seam;
• As already known, the values of the model parameters from a neighboring mining area or an area where the exploitation was carried out under similar geological and mining conditions;
• Can be assumed as the standard values of the parameters of this model ($a=0.80$, $tan\beta =$ 2.0, and $A_1=0.15$); however, this reduces its effectiveness.

The values of these parameters were determined from geodetic measurements of the land subsidence observed between 4 March 2019 and 17 August 2019 and made at 25 ground points along the observation line stabilized in the southeastern part of the exploitation area, along a dirt road (Figure 8). Because of the complicated landform (escarpments and their steep slopes) and the development of the terrain surface (allotment gardens and fields where agriculture is carried out) above the longwalls, it was not possible to establish a measuring line directly above the exploitation. Because of the unchanged values of the subsidence that occurred after the exploitation ended in the 1/II longwall of the 405/1 hard coal seam, it was concluded that the subsidence trough formed as a result of its exploitation is static and full, because at point no. 4 a subsidence value close to $W_{max}=1.66 \mathrm{m}$ was measured.

The parameter values for Bialek’s model were determined using the TGB1 computer program included in the EDN-OPN program’s package (v.2016-10-06). This program, based on the entered data on the geometry of the exploitation fields (any shape of mine workings), location of the observation points, and values of subsidence measured at these points, calculates the values of the $a$, $tan\beta ,$ and $A_1$ parameters. The criterion for selecting the appropriate values for these parameters was the minimum residual variance between the measured and theoretical model values of the subsidence, as determined by Formula (8), as follows:

$${\displaystyle \sum }_{i=1}^n{\left[W_{Bi theo}\left(a,tan\beta ,A_1\right)-W_{i meas}\right]}^2=\min .\left[{\mathrm{mm}}^2\right],$$

(8)

where the following hold:

• $i$—The number of point;
• $W_{Bi theo}$—The theoretical value of the subsidence calculated for the $i$-th point using Bialek’s model [mm];
• $W_{i meas}$—The practical value of the subsidence measured at the $i$-th point [mm].

The determined values for the parameters in Bialek’s model are presented in

Table 3. Parameter values for Bialek’s model (source: own study).

Parameter	$\mathit{a}$	$\mathit{t}\mathit{a}\mathit{n}\mathit{\beta }$	${\mathit{A}}_1$	${\mathit{R}}_{\mathit{B}}$
Value	0.976 ± 0.0032	2.167 ± 0.0072	0.117 ± 0.0004	0.9967

. The value of the $R_B$ correlation coefficient between the measured and model values of the subsidence was very high ($R_B=0.9967$), which proves that Bialek’s model’s parameter values were correctly determined.

3. Results

In this chapter, the results of the geodetic measurements carried out on 51 ground points located around the B1–B8 buildings and the G garage are presented. The observed values of the inclinations were calculated for the sections based on the altitude differences and distances between the neighboring points measured in the initial and final cycles. Because the lengths of the segments between the initial and final observation cycles changed by only a few millimeters, which constituted the accuracy of the distance measurement, the inclination values were not calculated separately for each cycle.

In addition, the results of the calculations of the inclination values for the middle points of the sections using Bialek’s model with the values of its parameters determined in situ and using the EDN-OPN computer program are presented.

3.1. Values of the Inclinations Observed on the Land Surface

The inclination values of the land surface determined using the geodetic measurements were calculated for the middles of the sections located between neighboring points. The value of the inclination ($T$) for the $i,i+1$ section was calculated with Formula (9) as a quotient of the change in the altitude differences ($\mathsf{\Delta }h$) of the $i,i+1$ neighboring ground points between the final ($f$) ($\mathsf{\Delta }{h}_{\left(i,i+1\right)f}$—17 August 2019), and the initial ($s$) ($\mathsf{\Delta }{h}_{\left(i,i+1\right)s}$—4 March 2019) measuring cycles, as well as the length ($l$) between the $i,i+1$ points measured in the final cycle ($f$) ($l_{\left(i,i+1\right)f}$), as follows:

$$T_{\left(i,i+1\right)}={\displaystyle \frac{\mathsf{\Delta }{h}_{\left(i,i+1\right)f}-\mathsf{\Delta }{h}_{\left(i,i+1\right)s}}{l_{\left(i,i+1\right)f}}}\left[{\displaystyle \frac{\mathrm{mm}}{\mathrm{m}}}\right],$$

(9)

where the following holds:

• $f$—The final measuring cycle;
• $i$—The number of measuring points;
• $l$—The length of the section [m];
• $s$—The initial measuring cycle;
• $T_{\left(i,i+1\right)}$—The measured value of the inclination [${\displaystyle \frac{\mathrm{mm}}{\mathrm{m}}}$].

Table 4. The values of the $T_{i,i+1}$ inclinations observed in the sections from March 2019 to August 2019 (source: own study).

Building	Section $\mathit{i}\mathbf{,}\mathit{i}\mathbf{+}\mathbf{1}$	4 March 2019	17 August 2019			Category of Mining Area
		$\mathbf{\Delta }{\mathit{h}}_{\mathbf{\left(}\mathit{i}\mathbf{,}\mathit{i}\mathbf{+}\mathbf{1}\mathbf{\right)}\mathit{s}}$ [mm]	$\mathbf{\Delta }{\mathit{h}}_{\mathbf{\left(}\mathit{i}\mathbf{,}\mathit{i}\mathbf{+}\mathbf{1}\mathbf{\right)}\mathit{f}}$ [mm]	${\mathit{l}}_{\mathbf{\left(}\mathit{i}\mathbf{,}\mathit{i}\mathbf{+}\mathbf{1}\mathbf{\right)}\mathit{f}}$ [m]	${\mathit{T}}_{\mathbf{\left(}\mathit{i}\mathbf{,}\mathit{i}\mathbf{+}\mathbf{1}\mathbf{\right)}}$$\mathbf{\left[}\frac{\mathbf{m}\mathbf{m}}{\mathbf{m}}\mathbf{\right]}$
B1	z01p-z02p	−278.1	−305.2	5.091	−5.32	3rd
	z02p-z03p	−136.3	−119.4	4.632	3.65	2nd
	z34d-z35d	151.5	130.7	7.044	−2.95	2nd
	z35d-z36d	−40.8	−41.7	5.764	−0.16	0
B2	z04g-z05g	−212.6	−221.5	6.273	−1.42	1st
	z06g-z07g	−188.8	−194.6	3.399	−1.71	1st
	z07g-z08g	−217.4	−267.2	3.601	−13.83	4th
B3	z09p-z10d	−160.2	−167.3	3.678	−1.93	1st
	z10d-z11d	−221.1	−299.2	5.502	−14.19	4th
	z56d-z57d	130.5	130.7	6.221	0.03	0
	z57d-z58g	62.6	74.4	6.030	1.96	1st
G	z12p-z13p	−37.8	−37.9	4.202	−0.02	0
	z13p-z14p	−32.6	−28.7	5.268	0.74	1st
B4	z15p-z16p	−141.3	−146.1	5.061	−0.95	1st
	z16p-z17p	−98.3	−102.2	3.493	−1.12	1st
	z18g-z19p	−9.7	−10.6	4.567	−0.20	0
	z19p-z20g	13.5	9.4	6.151	−0.67	1st
	z37d-z38d	30.0	41.2	4.001	2.80	2nd
	z38d-z39d	24.1	22.0	4.944	−0.42	0
B5	z21p-z22p	−118.5	−117.6	3.251	0.28	0
	z22p-z23p	−154.6	−157.4	4.742	−0.59	1st
	z53d-z54d	−26.0	−24.1	6.722	0.28	0
	z54d-z55d	−68.2	−67.3	5.583	0.16	0
B6	z24g-z25g	−185.2	−184.0	4.930	0.24	0
	z26g-z27p	−7.3	−3.4	4.379	0.89	1st
B7	z28d-z29d	−39.9	−36.7	4.084	0.78	1st
	z29d-z30d	−21.2	−27.0	4.511	−1.29	1st
	z40d-z41d	35.6	38.5	4.118	0.70	1st
	z41d-z42d	122.7	144.8	4.257	5.19	3rd
B8	z31d-z32d	−74.4	−77.3	6.461	−0.45	0
	z32d-z33d	62.0	61.2	5.544	−0.14	0
	z50d-z51d	−16.2	−14.0	5.492	0.40	0
	z51d-z52d	−6.6	74.5	4.356	18.62	5th
				$S=$	$5.13$
				$\widehat{\sigma }=$	$5.17$

presents the values of the altitude differences ($\mathsf{\Delta }h$) measured at the $i,i+1$ sections in the initial ($s$) and the final ($f$) measuring cycles, the lengths ($l$) of the sections measured in the final ($f$) cycle, the inclinations (T) measured after the end operations in the 1/II longwall of the 405/1 hard coal seam, the $S$ biased and $\widehat{\sigma }$ unbiased estimators of the standard deviation, calculated according to Formulas (10) and (11), as follows:

$$S=\sqrt{{\displaystyle \frac{{{\displaystyle \sum }}_{i=1}^{N_T}{\left(T_i-\overline{T}\right)}^2}{N_T-1}}}\left[{\displaystyle \frac{\mathrm{mm}}{\mathrm{m}}}\right],$$

(10)

$$\widehat{\sigma }={\displaystyle \frac{S}{c_{33}}}={\displaystyle \frac{S}{0.99222}}\left[{\displaystyle \frac{\mathrm{mm}}{\mathrm{m}}}\right].$$

(11)

The results of the geodetic surveys carried out on the ground points indicate that the maximum values of the inclinations of the land surface occurred along the z51d-z52d section ($T_{z51d-z52d}=18.6 {\displaystyle \frac{\mathrm{mm}}{\mathrm{m}}}$, north from the B8 building), the z10d-z11d section ($T_{z10d-z11d}=-14.2 {\displaystyle \frac{\mathrm{mm}}{\mathrm{m}}}$, northeast from the B3 building), and the z07g-z08g section ($T_{z07g-z08g}=-13.8 {\displaystyle \frac{\mathrm{mm}}{\mathrm{m}}}$, northeast from the B2 building). It should be emphasized that to the north of the B2 and B3 buildings, ground steps occurred with heights of 15 cm and 20 cm, respectively, so they may be the cause of the increase in the inclination values in this area. The $\widehat{\sigma }$ unbiased estimator of the standard deviation was equal 5.17 ${\displaystyle \frac{\mathrm{mm}}{\mathrm{m}}}$.

Taking into account the categories of the mining area for the values of the inclinations presented in

Table 5. Categories of the mining area based on the inclination values (source: [3,5]).

Category of Mining Area	$\mathbf{Inclination}$ $\mathit{T} \left[\frac{\mathbf{m}\mathbf{m}}{\mathbf{m}}\right]$
0	$\left|T\right|\le 0.5$
1st	$0.5<\left|T\right|\le 2.5$
2nd	$2.5<\left|T\right|\le 5.0$
3rd	$5.0<\left|T\right|\le 10.0$
4th	$10.0<\left|T\right|\le 15.0$
5th	$\left|T\right|>15.0$

, the investigated mining area can be appropriately categorized.

Each observed value of the $T_{i,i+1}$ inclination was assigned an appropriate mining area category (Table 4), and then quantitative descriptions of the determined categories were made, as shown in

Table 6. Determined categories of a mining area due to a number of $T_{i,i+1}$ observed inclinations (source: own study).

Category of Mining Area	Number of Measured Inclinations ${\mathit{N}}_{{\mathit{T}}_{\left(\mathit{i},\mathit{i}+1\right)}}$	Percent of Total Number Measured Inclinations ${\mathit{P}}_{\mathit{\Sigma }{\mathit{N}}_{\mathit{T}\left(\mathit{i},\mathit{i}+1\right)}}$ [%]
0	12	36.36
1st	13	39.39
2nd	3	9.10
3rd	2	6.06
4th	2	6.06
5th	1	3.03
Total	33	100.00

.

From the data presented in Table 6, the results show that the high values for the observed inclinations (4th and the 5th categories of mining area) represent 9.1% of all observed values for the inclinations. The largest share of all observed values had the inclinations of a terrain, which classifies it in the 1st category of mining area (39.4%).

The repeatability of the presented results is possible under the conditions of researching an observation network of a similar character, located near buildings with similar geometries, construction features, methods of use, and above all, an exploitation area with similar geological and mining conditions and rock mass structure.

3.2. Values of Inclinations Calculated Using Bialek’s Model

The theoretical values of the inclinations were calculated using Bialek’s model which was implemented in the EDN-OPN computer program.

The EDN-OPN program includes a package of programs enabling the calculation of the values of any mining area deformation indicators at any number of points, located at any distance from the edges of any number of mining fields, and at any time of exploitation. The results were obtained in tabular form using the EDBJ1 program and as surface maps of the distribution of the deformation indicator values using the EDBJ2 program. Moreover, the TGB1 program also made it possible to determine the parameter values for the selected forecasting model based on the subsidence values measured at points representing the profile of a static, full subsidence trough formed after the end of a given stage of exploitation.

For proper operation, the EDN-OPN program requires four types of files with data on the following:

• The geometry and location of the mining fields (a file with the .eks extension);
• The location of the calculation points where the values of the deformation indicators are calculated (a file with the .pkt extension);
• The geological structure of the rock mass and the thickness of the overburden over the hard coal seams (a file with the .ndk extension);
• Connecting the abovementioned files into one file and controlling the EDBJ1 and EDBJ2 programs (files with the .st1 and .st2 extensions, respectively).

The theoretical inclination values were calculated for the middle points of the measurement sections numbered from 1 to 33, after the end of the exploitation in the 1/II longwall of the 405/1 hard coal seam (inclination values for the period from 4 March 2019 to 17 August 2019) using Bialek’s model with the following values for its parameters:

• The $a$ exploitation coefficient (roof rocks’ deflection coefficient): $a=0.976$;
• a tangent of the $\beta$ angle of a range of main influences: $tan\beta =2.167$;
• The $A_1$ parameter of the mining margin: $A_1=0.117$;
• The $B$ Awierszyn’s proportionality coefficient: $B=0.32r$ [m];
• The $k$ coefficient of the impact deviation (due to the slope of the hard coal seam): $k=0.7$;
• The $A_{rel}$ relaxion coefficient: $A_{rel}=0.4$;
• The $T_{rel}$ relaxion time: $T_{rel}=1.0$;
• The $C_1$ coefficient of the subsidence velocity, calculated with Formula (12), as follows:

  $$C_1={\displaystyle \frac{140}{r\left(0.175-\frac{A_1}{tan\beta }\right)}}={\displaystyle \frac{140}{240.886 \left[\mathrm{m}\right]\left(0.175-\frac{0.117}{2.167}\right)}}=4.803\left[{\displaystyle \frac{1}{\mathrm{m}}}\right],$$

  (12)
• The $C_2$ coefficient of the subsidence velocity, calculated by Formula (13), as follows:

  $$C_2={\displaystyle \frac{0.6}{0.175-\frac{A_1}{tan\beta }}}={\displaystyle \frac{0.6}{0.175-\frac{0.117}{2.167}}}=4.958.$$

  (13)

Table 7. The values of the $T_j$ inclinations calculated for the middles of the sections by Bialek’s model and the EDBJ1 program in the period of March–August 2019 (source: own study).

Building	Point $\mathit{j}$	17 August 2019	Category of Mining Area
		${\mathit{T}}_{\mathit{j}}$ $\left[\frac{\mathbf{m}\mathbf{m}}{\mathbf{m}}\right]$
B1	3	1.8	1st
	4	1.8	1st
	2	1.9	1st
	1	1.8	1st
B2	5	1.3	1st
	6	1.2	1st
	7	1.2	1st
B3	8	0.9	1st
	9	0.9	1st
	11	0.8	1st
	10	0.8	1st
G	12	0.7	1st
	13	0.7	1st
B4	16	1.3	1st
	17	1.2	1st
	18	1.2	1st
	19	1.1	1st
	15	1.4	1st
	14	1.4	1st
B5	20	0.8	1st
	21	0.7	1st
	23	0.7	1st
	22	0.7	1st
B6	24	0.6	1st
	25	0.5	0
B7	28	0.9	1st
	29	0.8	1st
	27	0.9	1st
	26	1.0	1st
B8	30	0.6	1st
	31	0.5	0
	32	0.5	0
	33	0.5	0
	$S=$	$0.41$
	$\widehat{\sigma }=$	$0.41$

juxtaposes the values of the $T_j$ inclinations calculated by Bialek’s model and the EDBJ1 program at the 1–33 points constituting the middles of the measurement sections, and the $S$ biased and $\widehat{\sigma }$ unbiased estimators of the standard deviation, calculated according to Formulas (10) and (11).

The model inclinations did not have large values, and their maximum value did not exceed 2 ${\displaystyle \frac{\mathrm{mm}}{\mathrm{m}}}$ (points 1, 2, 3, and 4 near the B1 building). The highest inclinations values occurred in the area located closest to the edge of the exploitation (Figure 9). The $\widehat{\sigma }$ unbiased estimator of the standard deviation was 0.41 ${\displaystyle \frac{\mathrm{mm}}{\mathrm{m}}}$.

Each calculated value of the $T_j$ inclination (Table 7) was assigned an appropriate category of the mining area presented in Table 5, and then quantitative descriptions of the obtained categories were created, as shown in

Table 8. Obtained categories of the mining area due to the number of $T_j$ model inclinations (source: own study).

Category of Mining Area	Number of Model Inclinations ${\mathit{N}}_{{\mathit{T}}_{\mathit{j}}}$	Percent of Total Number of Model Inclinations ${\mathit{P}}_{\mathit{\Sigma }{\mathit{N}}_{{\mathit{T}}_{\mathit{j}}}}\mathbf{[}\mathbf{\%}\mathbf{]}$
0	4	12.12
1st	29	87.88
2nd	0	0
3rd	0	0
4th	0	0
5th	0	0
Total	33	100.00

.

The data contained in Table 8 show that the mining area under consideration, according to the results of the theoretical calculations carried out using Bialek’s model, should be included in the 1st category of mining area (87.9% of all calculated values of inclinations are values included in the 1st category). Bialek’s model did not indicate increased inclination values in any region, in particular in the region of the discontinuous linear deformations, where, according to the geodetic measurements, the 4th category of mining area occurred.

Figure 9 presents a map of the distribution of inclination values on the land surface and the categories of mining area (Category 0—white in color, 1st category—yellow in color, 2nd category—orange in color), created in the EDBJ2 program. The 1/II longwall of the 405/1 hard coal seam and the 1–33 calculation points to the north of the longwall are marked on the map.

4. Discussion

This section presents a comparison of the results of the observations of the practical inclination values of the land surface using the appropriate geodetic measurements and the calculations of the inclinations’ theoretical values for the mining area made using Bialek’s model implemented in the EDN-OPN computer program. Inclination values were obtained for 33 points constituting the centers of measurement sections located next to 9 objects and were caused by the exploitation of the 1/II longwall in the 405/1 hard coal seam. Discontinuous linear deformations (the ground steps) also appeared near the B1, B2, B3, and G objects.

The results of the comparison are shown in

Table 9. Comparison of the $T_{\left(i,i+1\right)}$ practical and $T_j$ theoretical values of the inclinations of the land surface (source: own study).

Building	Section $\mathit{i},\mathit{i}+1$	Absolute Value of Practical Inclination $\mathbf{\left|}{\mathit{T}}_{\mathbf{\left(}\mathit{i}\mathbf{,}\mathit{i}\mathbf{+}\mathbf{1}\mathbf{\right)}}\mathbf{\right|} [\frac{\mathbf{m}\mathbf{m}}{\mathbf{m}}]$	Point $\mathit{j}$	Value of Theoretical Inclination ${\mathit{T}}_{\mathit{j}} \mathbf{\left[}\frac{\mathbf{m}\mathbf{m}}{\mathbf{m}}\mathbf{\right]}$	Difference between the Theoretical and Practical Values of Inclinations $\mathit{∆}{\mathit{T}}_{\mathit{j}-\mathbf{\left(}\mathit{i}\mathbf{,}\mathit{i}\mathbf{+}\mathbf{1}\mathbf{\right)} }[\frac{\mathbf{m}\mathbf{m}}{\mathbf{m}}]$	Deviations of the Model from the Practical Values of Inclinations ${\mathit{D}}_{\mathit{j}}$[%]
B1	z01p-z02p	5.32	3	1.8	−3.52	−66.19
	z02p-z03p	3.65	4	1.8	−1.85	−50.67
	z34d-z35d	2.95	2	1.9	−1.05	−35.66
	z35d-z36d	0.16	1	1.8	1.64	1052.80
B2	z04g-z05g	1.42	5	1.3	−0.12	−8.37
	z06g-z07g	1.71	6	1.2	−0.51	−29.68
	z07g-z08g	13.83	7	1.2	−12.63	−91.32
B3	z09p-z10d	1.93	8	0.9	−1.03	−53.38
	z10d-z11d	14.19	9	0.9	−13.29	−93.66
	z56d-z57d	0.03	11	0.8	0.77	2388.40
	z57d-z58g	1.96	10	0.8	−1.16	−59.12
G	z12p-z13p	0.02	12	0.7	0.68	2841.40
	z13p-z14p	0.74	13	0.7	−0.04	−5.45
B4	z15p-z16p	0.95	16	1.3	0.35	37.07
	z16p-z17p	1.12	17	1.2	0.08	7.48
	z18g-z19p	0.20	18	1.2	1.00	508.93
	z19p-z20g	0.67	19	1.1	0.43	65.03
	z37d-z38d	2.80	15	1.4	−1.40	−49.99
	z38d-z39d	0.42	14	1.4	0.98	229.60
B5	z21p-z22p	0.28	20	0.8	0.52	188.98
	z22p-z23p	0.59	21	0.7	0.11	18.55
	z53d-z54d	0.28	23	0.7	0.42	147.65
	z54d-z55d	0.16	22	0.7	0.54	334.23
B6	z24g-z25g	0.24	24	0.6	0.36	146.50
	z26g-z27p	0.89	25	0.5	−0.39	−43.86
B7	z28d-z29d	0.78	28	0.9	0.12	14.86
	z29d-z30d	1.29	29	0.8	−0.49	−37.78
	z40d-z41d	0.70	27	0.9	0.20	27.80
	z41d-z42d	5.19	26	1.0	−4.19	−80.74
B8	z31d-z32d	0.45	30	0.6	0.15	33.68
	z32d-z33d	0.14	31	0.5	0.36	246.50
	z50d-z51d	0.40	32	0.5	0.10	24.82
	z51d-z52d	18.62	33	0.5	−18.12	−97.31

, in which the $\left|T_{\left(i,i+1\right)}\right|$ absolute values of the practical inclinations, $T_j$ theoretical values of the calculated inclinations, $∆T_{j-\left(i,i+1\right)}$ differences between the theoretical and practical values of the inclinations and the $D_j$ deviations of Bialek’s model from the practical inclination values in percentage terms are juxtaposed. The $D_j$ model deviations were calculated with Formula (14), as follows:

$$D_j={\displaystyle \frac{T_j-\left|T_{\left(i,i+1\right)}\right|}{\left|T_{\left(i,i+1\right)}\right|}}·100\%$$

(14)

Taking into account the values of the $D_j$ deviations of Bialek’s model, it can be seen that in the case of 15 points (45.45% of all points), this model underestimated the theoretical inclination values by an average of −53.54%. The highest underestimated values (−91.32%, −93.66%, and −97.31%) were obtained for the middle points (7, 9, and 33, respectively) of the sections, where the highest values for the inclinations were observed on the land surface (13.83 ${\displaystyle \frac{\mathrm{mm}}{\mathrm{m}}}$, 14.19 ${\displaystyle \frac{\mathrm{mm}}{\mathrm{m}}}$, and 18.62 ${\displaystyle \frac{\mathrm{mm}}{\mathrm{m}}}$, respectively). The average underestimation of the maximum values of the measured inclinations was −94.10%. Such a large underestimation of the value may result from the appearance of discontinuous linear deformations in the area of points 7 and 9 (ground steps with heights of 15 cm and 20 cm, respectively), which run perpendicular to the z07g-z08g and z10d-z11d measurement sections. It can be seen that the values of the $D_j$ deviations ($D_7=-12.63{\displaystyle \frac{\mathrm{mm}}{\mathrm{m}}}$ and $D_9=-13.29{\displaystyle \frac{\mathrm{mm}}{\mathrm{m}}}$) increased with the height of the ground steps. Previous studies have shown [60] that discontinuous linear deformations may also increase the values of mining area deformation indicators observed in situ.

In the case of 18 points (54.55% of all points), Bialek’s model overestimated the theoretical values of the inclinations by as much as 461.90% on average. The highest overestimated values (2841.40%, 2388.40%, and 1052.80%) were obtained for the middle points (12, 11, and 1, respectively) of the sections where, generally, the smallest values for the inclinations were observed (0.02 ${\displaystyle \frac{\mathrm{mm}}{\mathrm{m}}}$, 0.03 ${\displaystyle \frac{\mathrm{mm}}{\mathrm{m}}}$, and 0.16 ${\displaystyle \frac{\mathrm{mm}}{\mathrm{m}}}$, respectively). Such a huge overestimation of these values may result from the fact that near points 1, 11, and 12, there were ground steps with heights of 10 cm and 25 cm, respectively, which run parallel to the z35d-z36d and z56d-z57d measurement sections. Such arrangement of the longitudinal axis of the ground steps may reduce the values of the land surface inclinations observed in situ. A similar phenomenon of reductions in the values of a building’s deflections in a situation where the discontinuous linear deformation runs parallel to the building’s wall was observed in the work [61].

Figure 10 presents the correlation between the $\left|T_{\left(i,i+1\right)}\right|$ absolute values of practical inclinations observed in the terrain and their $T_j$ theoretical values calculated using Bialek’s model. As can be seen from the graph, their correlations were rather weak, because the value of the $R_T$ correlation coefficient did not exceed 0.24 ($R_T=0.2356$).

From the above considerations, it follows that Bialek’s model generally copes well in assigning mining area categories (the 1st category was determined for both the practical and theoretical inclination values) [50,52], but in the case of small inclination values and their drastically increased values due to the occurrence of discontinuous linear deformations, it becomes defective. This model does not include a module that takes into account the possibility of the occurrence of discontinuous deformations in areas affected by underground mining of hard coal seams, where continuous deformations of the land surface occur. A solution to this problem could be the introduction of a safety factor for the theoretical values calculated by the model if there is a risk of discontinuous linear deformations in a given area. The values of this coefficient may depend on the heights of the ground steps. The above factors are planned to be considered in future research, which will be conducted in various mining areas (combinations of operation depths, thicknesses and declinations of hard coal seams, geological structures of the rock masses, especially a compact rock mass, etc.).

5. Conclusions

Bialek’s model based on Knothe’s geometric integral theory of exploitation influences is the most popular model used in Central–Eastern Europe, especially by the Jastrzebie Coal Company, from the south of Poland, for forecasting the values of continuous deformation indicators of a mining area (e.g., subsidence, inclinations, and curvatures) caused by underground mining exploitation of hard coal deposits. Generally, the effectiveness of this model is rather high when only discontinuous deformations are revealed on a land surface [50,52].

An important problem arises when, in addition to continuous deformations, discontinuous linear deformations (i.e., ground steps) unexpectedly appear on the land surface, as shown in this article, for example, by inclinations in a mining area. The program does not contain a module that allows for taking into account the changes in the values of the indicators of the continuous deformations of a land surface, which were caused by discontinuous deformations. For this reason, the efficiency of the model decreased significantly, because, for approximately 45.5% of the inclination values observed in situ, the model underestimated their values by an average of −53.5% (Table 9). For the remaining part of the examined sample (54.5%), their values were overestimated on average by as much as 461.9% (Table 9). The largest anomalies (underestimations and overestimations) were observed in the case of inclination values coming from the points located near the ground steps. As the analyzed example shows, the location of the longitudinal axis of the discontinuous linear deformation concerning the measurement sections is also significant (Figure 5). In the case in which the axis ran transversely to the measurement sections (the z07g-z08g section and the z10d-z11d section), the inclination values observed in these sections increased. In the situation in which the axis of the ground steps ran along the measurement sections (the z35d-z36d section and the z56d-z57d section), the inclination values of the land surface decreased.

As observed, discontinuous linear deformations may be problematic not only due to the mining damage to the ground surface [60] and the buildings [56,61] but also during forecasting of the impacts of mining exploitation using Bialek’s model. A solution to this problem could be the introduction of a safety factor for the forecasted values calculated by this model if there is a risk of the occurrence of discontinuous linear deformations. The values of this coefficient may depend on the heights of the ground steps, which requires further research. The basis for this statement is that the values of the inclinations observed in the sections located near the ground steps change depending on their height, as presented in Section 4.

Despite these shortcomings, Bialek’s model will still be widely used in forecasting continuous deformations of land surfaces, because it well reflects the categories of mining areas (in the analyzed case, the 1st model mining area category—Table 8 corresponds to the 1st observed mining area category—Table 6) important from the point of view of the design of construction structures in mining areas and building’s reinforcements against the adverse effects of underground mining.