1. Introduction
Underground mining operations lead to a disturbance of the initial structure of the strata. This is related to the disturbance of stresses in the rock mass, which may lead to an increase in seismic activity or the occurrence of rock outbursts in the area of exploitation [
1]. Additionally, as a result of underground mining operation, deformations are created on the ground surface, which can lead to so-called mining damage. This problem has been the subject of numerous scientific studies conducted around the world since the 19th century. However, the greatest development of modelling and computational methods and their modifications occurred in 1950–1990. The methods of that time focused on analytical solutions: geometric-integral and stochastic. Their advantages and disadvantages were described in many works. An example of such analyses conducted for the Knothe method [
2,
3], which is one of the most popular geometric-integral methods used in science and mining industry, are the experiments described by [
4,
5,
6,
7,
8,
9,
10], among others. As can be noticed, the topic of using analytical methods to predict ground deformation is still popular despite the passage of years, and research is being conducted on adapting methods and parameters to mining and geological conditions [
11]. These methods owe their popularity to the ease of conducting analyses for a large number of exploitation fields and obtaining quick forecast results. However, the effectiveness of forecasts based on these methods depends on the complexity of the geological structure of the region, including tectonic disturbances of the rock mass and the occurrence of layer packages with different mechanical properties.
Meanwhile, in parallel, since the 1990s, there has been a significant increase in interest in using solutions based on numerical methods [
12,
13,
14] to determine changes in the state of stress and strain of the rock mass caused by underground mining operations. The authors of these solutions used a variety of numerical methods and physical and mathematical models [
15,
16,
17]. These methods allow taking into account many additional factors affecting the nature and development of surface deformation [
12,
18,
19,
20,
21,
22,
23], which is largely an advantage over classical computational methods. However, attention should also be paid to the problems that occur during rock mass modelling. One of the main problems, in addition to the selection of the material law and the size of the model, the density of the elements, and the compensation of numerical disturbances, is the proper calibration of the mechanical parameters of the rock layers. The first work on this topic was based on mapping the rock mass with isotropic elastic models [
24,
25,
26], which, however, due to their nature, showed the correctness of the subsidence trough calculations only in terms of the vertical displacement index. A further attempt to use elastic models modified in terms of not transferring tensile stresses by the modeled elements to predict land sur-face deformations in the mining area was conducted by Chrzanowska-Szostak [
27]. She used these elements for the weakening zone whose geometry was determined a priori. Some scientists used elastic–plastic models to describe the impact of exploitation on the surface. Adopting this model in a physical sense has greater understanding because rocks exhibit elastic properties only to a very small extent. However, studies conducted by Siriwardane [
28], Whittaker and Reddish [
29], Najjar and Zaman [
13], and Derbin [
30] showed that the modeled subsidence trough has a much larger extent than indicated by surface observations. In parallel, work was carried out on the use of transversally isotropic models to analyze the impact of exploitation on the land surface. In 1964, Berry [
31] was the first to notice the advantages of using this type of models to limit the width of the modeled trough. Further research presented by Hazine [
25] and McNabb [
32] confirmed the validity of the adopted model. In 2019, Derbin [
33] conducted analyses for several constitutive models: Mohr–Coulomb and modified Hoek–Brown. They confirmed the research results of their predecessors that elastic–plastic models overestimate the width of the subsidence trough. A similar conclusion was presented after using the Clay and Sand Model or CASM models [
30]. Further work is mainly related to combining continuous and discontinuous numerical methods in order to obtain satisfactory results of deformation indicators, including Vyazmensky [
34], Zhang [
35], or the work of Liu [
36], and Li [
37]. Another important problem concerns the selection of parameter values for the adopted model. This issue was the subject of separate studies, and their results have been presented in various articles, among others [
16,
22,
38,
39,
40]. Further development of numerical methods on the issue of land surface deformations resulting from underground mining works made it possible to use them to analyze this phenomenon regardless of the nature and cause of the deformation, including strip pillar exploitation [
41], underground coal gasification [
42], land uplift when flooding underground mines [
43,
44,
45,
46], determining their impact on surface objects [
47].
In this article, the authors initially used the methodology for estimating the geomechanical parameters of rock layers developed by Tajduś [
39], which was optimized for the advancing mining front, based on the author’s Python script-based program. In addition, a modification related to the evaluation of the error of estimated parameter values described in this article was introduced in the program. Numerical analyses were carried out for a selected region of the Piast–Ziemowit mine (Poland) using both mining, geological and geodetic data.
3. Results of Conducted FEM Simulations for LW501 and LW502 Longwall Panels Exploitation
The development of the subsidence trough, observable on the observation line during the operation of the LW501 and LW502 longwalls, is presented in
Figure 5. Due to the significant distance of the observation line points from the operation carried out from June to December 2021, i.e., the LW501 longwall, the recorded subsidence does not exceed 4 cm at the end of November 2021. Only the exploitation of the LW502 longwall carried out directly under the observation line leads to larger displacements of the terrain surface. Exploitation was carried out in direction from point 40 to point 1, which shows the development of the subsidence trough in
Figure 5. The operation was completed at the end of May 2022, and a slight increase in subsidence between the last two observation series (12 May 2022 and 4 August 2022) indicates the presence of residual subsidence on the land surface. The maximum observed subsidence on the observation line equals
at point 23.
The second series of land surface subsidence measurements were carried out at the end of November 2021. As of this day, the face advance of longwall panel 501 was determined at a distance of 500 m from the beginning of the longwall. The maximum measured subsidence was 36 mm and it was located in point no. 23. Having this information, the parameters of the rock layers were calibrated using ABAQUS 2022 hf4 software and author’s Python 3.9 script in order to adjust the simulated subsidence to the real values. The estimated values of mechanical parameters are listed in
Table 5 and the plot of fit is shown in
Figure 6.
In
Figure 7, the distribution of land surface subsidence based on FEM analysis is shown. For the analyzed mining situation, the maximum subsidence of 433 mm was obtained.
The fifth series of land surface subsidence measurements were carried out at the beginning of August 2022. As of this day, the mining of the longwall panel 501 and 502 have been completed. The maximum measured subsidence was 637 mm and it was located in point no. 23. Having this information, the parameters of the rock layers were calibrated using Abaqus software and Python script in order to adjust the simulated subsidence to the real values. The estimated values of mechanical parameters are listed in
Table 6 and the plot of fit is shown in
Figure 8.
In
Figure 9, the distribution of land surface subsidence based on FEM analysis is shown. For the analyzed mining situation the maximum subsidence of 917 mm was obtained.
The obtained error values (see chapter 2.4) of numerical modelling (
Table 7) indicate the correct determination of the deformation model parameters. Positive
values mean an average underestimation of the subsidence, while negative values mean higher modelled values in relation to the observed subsidence. The increase in the absolute error values (
,
) from 1 to 35 mm for successive states of exploitation is related to the development of the trough and the occurrence of subsidence in the range of exploitation impact. The percent error for maximum subsidence
describes the accuracy of determining the maximum subsidence. Its value does not exceed
.
4. Discussion
As part of the conducted analyses, numerical calculations were performed to assess the impact of the ongoing mining exploration of the KWK Piast–Ziemowit mine in panels LW501 and LW502 on surface deformation. The calculations were carried out using a custom script based on Python, which allows for the automatic estimation of the values of parameters of the modelled rock layers located above the formed caving zone. To ensure the clarity of the results, the parameter values in the caving zone were assumed to be constant for all numerical models. This assumption does not align with observations of the behavior of caving zones resulting from mining activities. Previous studies unequivocally indicate that as mining progresses, the caving zone increases in height, causing various rock strata to collapse. Unfortunately, this assumption was necessary to ensure the clarity of the solution based on a program that automatically determines the mechanical parameters of individual rock layers. The program performed calculations according to the specified conditions and then compared the estimated deformation indicators with the measured values. The results are presented for a selected region of the underground coal mine. However, the methodology used allows for the determination of deformation indicators and rock mass parameters also for other regions of the mine. It is limited only by knowledge of mining conditions and having geodetic measurements and geological structure with preliminary parameters. Based on these data, the program will adjust the values of “real” parameters to the measurement results using the method of similarity of geode measurement results described in this article. According to the procedure, calculations were conducted until the consistency of the selected deformation indicators was achieved, i.e., obtaining the global minimum of the specified objective function, which was the RMSE error. Subsequently, calculations were carried out for the next specified mining advance.
The calculations showed a very high consistency of deformation indicators (see
Table 7), meaning that for the assumed physical model, the actual values of the mechanical parameters of the rock layers were determined with high probability. Another way to assess accuracy of numerical modelling result is another deformation indicator. Based on the field observation data the tilts of subsidence trough profile were compared. The fitting errors of the modelled inclination
were determined in relation to the measured values
similarly to those for subsidence, in accordance with Formulas (6)–(9). The results for individual operating states are presented in
Table 8.
Similarly to the subsidence,
Figure 10 shows the tilts profiles of the subsidence trough after the end of operation of the LW 501 and LW502 walls.
The obtained results indicate a good and very good match of the tilts of the subsidence trough. The increased fluctuations in relation to the subsidence result from the nature of the phenomenon, which was confirmed on the basis of research on the variability of deformation indicators and forecasting accuracy in numerous publications (e.g., [
71]).
The last period (4 August 2022) is characterized by increased values of error characteristics compared to the previous ones (see
Table 8). The reasons for this state of affairs should be sought primarily in the measurement technology used to determine the subsidence in observation points. For this purpose, the results of RTN-GNSS measurements were used (see
Section 2.2), which are characterized by a much larger error in determining the height (m
H < ± 0.05) compared to the heights determined based on geometric levelling (m
H ≤ 0.015 m).
The mechanical parameters, particularly the ratio E1/E3, played a crucial role in the accuracy of the numerical models. This ratio, which compares the horizontal Young’s modulus (E1) to the vertical Young’s modulus (E3), is essential in determining the anisotropic behavior of rock layers. Anisotropy in rock mechanics refers to the directional dependence of rock properties, and the E1/E3 ratio is a key indicator of this behavior. It is also responsible for the change in the slope within the subsidence trough, while Young’s modulus (E3) influences the value of land surface subsidence. Poisson’s number causing a slight change in subsidence trough course [
72]. Recent studies have highlighted the impact of anisotropy on rock deformation and subsidence predictions. For example, Lekhnitskii’s theory on anisotropic elasticity provides a foundation for understanding how the different directional moduli influence stress distribution and deformation in rock masses. In particular, research has shown that the elastic stiffness parameters, including Young’s modulus, exhibit strong correlations that significantly affect the rock’s response to loading conditions [
73]. Additionally, the variation in the E1/E3 ratio influences the deformation characteristics during mining activities. By incorporating the E1/E3 ratio into our numerical models, we achieved high consistency in deformation indicators, which aligns with empirical observations and enhances the reliability of our predictions. This adjustment is crucial for accurately modelling the complex behavior of rock layers under mining-induced stress.
It is noteworthy that high accuracy was also achieved in the case where a full subsidence trough did not form, i.e., for the LW501 panel. As the surface measurements conducted from June 2021 to November 2021 showed, the first panel led to slight surface deformation along the observation line, with a maximum subsidence indicator of 38 mm. The maximum estimated subsidence value was 433 mm in the central part of panel LW501. This most likely means that the extraction of this panel did not lead to the fracture (or breaking) of the thick and strong sandstone layer located directly above the mined seam, which has a thickness of up to 100 m. This assumption is supported by reports on the registration of mining tremors recorded on the surface in the area of Lędziny city (
https://grss.gig.eu/en/seismic-map/, accessed on 9 July 2024). From June 2021 to the end of 2022, monitoring stations recorded tremors mainly in the energy magnitude range of
J according to the GSIS-2017 scale by PGVH
max [
74], with the intensity of these tremors classified as the lowest category, i.e., 0.
As is known from engineering experience and the literature on the subject, longwall panels can be classified as sub-critical, critical, and supercritical [
75,
76]. The critical panel is defined as the width of an extracted panel for which the maximum possible subsidence should occur at one point. The critical width depends upon the geological characteristics of the overburden. In single seam coal mining operations in New South Wales, Australia, the critical value is typically 1–1.6 times the depth of the overburden [
38,
77]. In Europe (including Germany, Poland, and the Czech Republic) [
78] it is estimated that a full subsidence trough will form for an extracted space that meets the condition d ≥ 2R, where R is the radius of main influence range determined by the formula
, and d is the shorter dimension of the coal longwall (width or length). This means that the condition for the occurrence of a full subsidence trough is related to the mining depth (H) and the angle of the main influence range (β), whose values should consider the quality and structure of the rock mass. However, describing the rock mass with a single parameter is a challenging task. Therefore, numerical analyses that consider the structure and tectonics of the rock mass allow for a more accurate description of its behavior. As shown by further geodetic measurements, the increase in the exploited area by panel LW502 led to a significant increase in the subsidence indicator (compared to the results obtained for panel LW501). Research carried out so far indicates that in order to properly model the subsidence basin caused by mining, an anisotropic model should be used. Elastic–plastic models are used in the literature. It should be borne in mind that these models allow for high consistency of the modelled subsidence compared to those observed from measurements. However, these models have a big problem with reflecting the range of the main influences
R. In most cases, this range is obtained as significantly exceeding the values observed in reality (overestimated). The situation is similar in the case of horizontal displacements, which also deviate significantly from the measured values. The use of an anisotropic model allows to limit the value of
R obtained from numerical models. However, it is very difficult to determine all the parameter values describing this model. Therefore, the authors use a simplified transversally isotropic model, which has only 5 parameter values (see
Section 2.3). The problems with this model are:
Proper selection of parameter values, for this reason the authors used a solution used in science, which allows determining the elastic parameters G12 and G13 = G23, which are difficult to determine for mining conditions.
Unfortunately, this model still incorrectly determines the values of horizontal displacements (like other models presented in the literature). Research [
79] suggested that the use of contact between the modelled layers allows solving this problem, but these studies did not provide a clear solution (for which layers contact should be used, what friction coefficients should be used, etc.). For this reason, the authors did not take into account the change in friction values and the use of contact between layers in the described tests. Research on this issue will be the subject of further research by the authors.
Further numerical analyses based on FEM and the custom script continued to demonstrate high matching accuracy and allowed the determination of the parameters of the rock layers.
5. Conclusions
This article presents numerical modelling of the advancing mining front at the Piast–Ziemowit mine using Abaqus software and a custom Python script. The research aimed to determine the impact of mining operations on surface displacements and to estimate the mechanical parameters of rock layers. Numerical computation results were compared with geodetic measurements of the surface, verifying the model, and calibrating the mechanical parameters of the rock mass. The presented solution allows for the automation of the process of optimizing the parameters of the adopted geomechanical model. Determining the optimal set of parameters requires finding the minimum of the objective function defined as the RMSE error value. Iterative calculations end when the criterion of convergence of two consecutive simulations is reached. The case study of the Piast–Ziemowit mine was characterized by the presence of a strong rock layer in the rock mass, which prevented the formation of a complete subsidence trough on the surface. Additionally, the dimensions of operation classify it as sub-critical or critical. This means that the observed reductions are smaller than those resulting from empirical formulas for hard coal exploitation. For this reason, the use of numerical modeling for impact calculations for ongoing exploitation requires taking into account changes in the geomechanical properties of the rock mass. This solution is an innovative approach and requires the development of a method for determining them. Nevertheless, the applied numerical modelling method demonstrated high accuracy in predicting surface displacements. The computational results showed good agreement with field measurements, confirming the validity of the model assumptions and the accuracy of the mechanical parameters of the rock layers. It worth to notice, that was also achieved in the case where a full subsidence trough did not form, i.e., for the LW501 panel. This article detailed the calibration process of the numerical model, including methods for error assessment and adjustment of computation results to real measurements. The custom Python script, integrated with Abaqus, enabled automated computation and efficient analysis of large datasets. This facilitated multiple computation iterations in a short time, significantly speeding up the research process.
In conclusion, this study demonstrated that numerical modelling of the advancing mining front using advanced computational tools such as Abaqus and Python script is an effective method for predicting the impact of mining operations on surface displacements. Despite the specific geological conditions of the Piast–Ziemowit mine, this method provided reliable results, confirming its utility in engineering practice.