Excessive strain energy accumulation in the coal seam is one of the essential conditions for rockbursts to occur. It is imperative to understand the parameters that affect the strain energy accumulation in retreating longwall mining to optimise the design and minimise rockbursts occurrences. In this paper, new analytical models were developed to calculate the strain energy accumulation considering the current state-of-the-art longwall machinery being used in the industry. Seven parameters, i.e., mining depth, length of the cantilever roof, coal seam thickness, thickness of the immediate roof, Young’s modulus of coal and the roof, and Poisson’s ratio of the coal seam, were identified as the parameters affecting the strain energy accumulation. A detailed statistical analysis of the parameters was conducted using the orthogonal testing method, which revealed that mining depth, Young's modulus of coal and the coal seam thickness significantly influence the strain energy accumulation within a 99% confidence interval.
Notes
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Abbreviations
\(A\)
Area of cross-section (m2)
\({A}_{1}\), \({A}_{2}\)
Intermediate calculation variables (–)
\({b}_{{\text{r}}}\)
Width of the roof (m)
\(C\)
Elastic foundation modulus of the coal seam (GPa/m)
\({df}_{e}\)
Degree of freedom of the error term (–)
\({df}_{j}\)
Degree of freedom of each parameter (–)
\(e\)
Error term (–)
\({E}_{{\text{coal}}}\)
Young’s modulus of the coal seam (GPa)
\({E}_{{\text{roof}}}\)
Young’s modulus of the cantilevered roof (GPa)
\({F}_{j}\)
F-Value of parameter (–)
\({F}_{\delta }\)
F-Value of the confidence interval (–)
\(g\)
Acceleration due to gravity (m/s2)
\({h}_{{\text{coal}}}\)
Coal seam thickness (m)
\({h}_{{\text{roof}}}\)
Thickness of the immediate roof (m)
\(H\)
Mining depth (m)
\(i\)
Variation levels (–)
\(I\)
The second moment of inertia (m4)
\(j\)
Parameters (–)
\({k}_{i}\)
Average of parametric interaction at a level (J/m3)
\({K}_{{\text{beam}}}\)
Flexural rigidity of the roof (Pa m4)
\({K}_{{\text{i}}}\)
Sum of all parametric interactions in a level (J/m3)
\(l\)
Abutment stress influence zone in the solid coal pillar (m)
\(L\)
Length of the cantilevering roof (m)
\(m\)
Number of unique scenarios (–)
\({M}_{0}\)
Initial bending moment (N m)
\(M\left(x\right)\)
Bending moment of the cantilevered roof (N m)
\(p\)
Stress acting on the cantilevered roof (MPa)
\({P}_{{\text{o}}}\)
Peak abutment stress acting in the solid coal pillar (MPa)
\(P\)
In situ Stress acting due to mining depth (MPa)
\(P\left(x\right)\)
Stress per unit length acting on different sections (MPa)
\(q\)
Reaction force acting against the abutment stress (MPa)
\({R}_{j}\)
Range of a parameter \(j\) (–)
\({{\text{SS}}}_{{\text{e}}}\)
Sum of the square of experimental error (J2/m6)
\({{\text{SS}}}_{j}\)
Sum of the square of each parameter’s average deviation (J2/m6)
\(V\)
The volume of the roof rock (m3)
\({V}_{e}\)
The variance of the error term (–)
\({V}_{j}\)
The variance of each parameter (–)
\({V}_{0}\)
Initial shear force (MPa)
\(V\left(x\right)\)
Shear force developed in the clamped end (MPa)
\({W}_{{\text{cant}}}\)
Strain energy density in the cantilever roof (J/m3)
\({W}_{{\text{coal}}}\)
Strain energy density in the coal seam (J/m3)
\({W}_{i}\)
Strain energy density (J/m3)
\({W}_{{i}^{2}}\)(\({i}^{j}\))
Orthogonal testing matrix (–)
\({W}_{{\text{roof}}}\)
Strain energy density in the supported roof (J/m3)
\({W}_{{\text{total}}}\)
Total strain energy density accumulated in the longwall mining (J/m3)
It is suggested that coal mining will contribute around 22% to the world’s electricity until 2040 (World Coal Association 2020). The future of coal mining lies deep underground as shallow-depth deposits are fast depleting (Zuo et al. 2020; Ghosh et al. 2020). Longwall coal mining is a proven technology with high production rates achieved in deep underground mines around the world. Around 90% of underground coal production in Australia comes from longwall mining panels (Ralston et al. 2017). With the increase in mining depth, a high in situ stress state combined with the mining-induced stresses presents severe ground control challenges, increasing the risk of rockbursts (Heib 2018; Si et al. 2018; Agrawal et al. 2019).
Rockbursts, an energy phenomenon, are characterised by a sudden release of stored strain energy leading to the violent ejection of failed coal/rock at a very high speed into the mining workings (Singh 1988; Cai et al. 2016; Mark 2016; Zhou et al. 2018; Canbulat et al. 2019). The mining-induced stresses acting ahead of the longwall face develop strain energy in the coal and the surrounding rock mass. The stored energy may release suddenly due to the cantilevering roof breakage or coal failure in retreating longwall mining, triggering rockbursts (Cai et al. 2016, 2019; Mark 2016; Fedotova et al. 2017; Zhou et al. 2018; Agrawal et al. 2019; Canbulat et al. 2019; Huang et al. 2020).
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Rockbursts may impede coal production, cause injuries to mining workers, and destroy underground structures (Miao et al. 2016; Agrawal et al. 2021). It may affect ventilation and trigger gas outbursts enhancing the severity of the hazard (Cai et al. 2016; Dou et al. 2018). Since the first occurrence of rockbursts in 1738 in a British coal mine (Bukowska 2006; Zhang et al. 2017), over 40,000 rockbursts have been reported around the world (Xue et al. 2021). The frequent occurrence of rockbursts highlights it as the most complex, difficult to predict, and longstanding underground mining problem that remains to be solved (Mark 2016; Zhang et al. 2017; Sabapathy et al. 2019; Agrawal et al. 2022).
Several hypotheses leading to different empirical prediction indices were developed to understand rockburst occurrences over the last few decades. Most of these indices were developed based on laboratory-scale analysis of coal/rock, however, very few researchers have investigated the strain energy accumulation in a retreating longwall coal mining panel.
Hetenyi (1946) pioneered the differential equation analysis of the roof to evaluate the effect of elastic abutments on beam deflection resting on elastic supports. Holland (1955) analysed the strain energy stored in the cantilevering roof and coal seam using cantilever beam models but ignored the energy stored in the supported roof. Cook (1965) presented a theoretical analysis of energy accumulation and release during mining leading to rockbursts. Stephansson (1971) developed mathematical solutions assuming roof beds as horizontal beams to emphasise the effect of abutment stresses on the deflection of the roof. He estimated deflection, stress and bending moment for different roof configurations occurring in underground coal mining on elastic supports.
Haramy and McDonnell (1988) used Holland (1955)’s equations to analyse the performance of strong roof beds as a cantilever beam in response to longwall coal mining on an elastic foundation and evaluated strain energy accumulation under uniformly distributed overburden stress. They assumed constant applied stress and thus ignored the mining-induced stresses that concentrate ahead of the active face. This limits the applicability of the equations to accurately evaluate the total amount of strain energy stored in the roof and coal (Wu 1995).
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Xu (2009) extended the analysis by considering the longwall roof as a cantilever beam during different weighting stages of longwall coal mining, modelling the mining-induced abutment stresses as an exponentially decreasing function ahead of the active face. He considered overburden weight to be acting on the cantilever roof, which leads to an error in the deflection calculation as the Euler–Bernoulli theory assumes the cantilever beam to be stressed under its own weight. He also allowed significant roof deflection at the active face to represent the time-intensive manual installation of chock shield supports at the goaf edge. The state-of-the-art mining technology now available for longwall coal mining, using hydraulic-powered supports, advances soon after the longwall shearer cuts the coal face to support the immediate roof and prevents excessive deflection (Rezaei et al. 2015; Agrawal et al. 2019). The loading conditions considered in the coal seam are too simplistic with only the reaction force considered to be acting on the coal seam. In practice, the mining-induced stresses also affect the coal seam. This limits the application of previously developed energy equations to evaluate strain energy accumulation in state-of-the-art longwall mining panels.
To overcome the limitations of earlier energy equations and to reflect the use of current state-of-the-art support systems now available in longwall mining, a new analytical model was developed to calculate strain energy accumulation as presented in this paper. The analytical model was developed considering (a) new boundary conditions, (b) appropriate loading conditions to incorporate the effect of abutment stresses and (c) the presence of hydraulic-powered supports to prevent excessive roof deflection. The model was calibrated against a numerical model to evaluate its suitability. The identified parameters were tested using an orthogonal testing matrix to determine the dominance of each parameter in strain energy accumulation.
2 The Analytical Model
A vertical cross-section of a retreating longwall coal mining panel assuming full extraction of the coal seam is shown in Fig. 1. The coal seam is sandwiched between a roof and floor layer with three different loading sections,
(a)
Cantilever beam section (\(-L<x\le 0)\)
In this section, the lower portion of the immediate roof in the area where coal is extracted gradually separates along the bedding plane to form a gravity-loaded rock layer. The rock layer is clamped at the excavation boundary (\(x=0\)) by the overburden stress and hydraulic-powered support (HPS) to form a cantilever beam. A uniformly distributed stress \(p=\rho g{h}_{{\text{roof}}}\) (where \(\rho\) is the density of rock (kg/m3), \(g\) is the acceleration due to gravity (m/s2), \({h}_{{\text{roof}}}\) is the thickness of the immediate roof (m)) acts on the cantilever beam (Galvin 2016). The beam hangs freely in the goaf (\(x=-L\)) to represent the periodic and main weighting cycle (Huang et al. 2020). \(L\) is a direct manifestation of the weighting cycles which has not been incorporated in any previous research.
(b)
Abutment stress section (\(0\le x<l)\)
In this section, an exponentially decreasing abutment stress function, \({P}_{o}{e}^{-\alpha x}\) is considered with its peak at \(x=0\) (\({P}_{o}= \sim 2P\)) to act on the roof and the coal seam. The stress decreases to \(P=\rho gH\), at \(x=l\) (\(l=0.12H)\), inside the solid coal seam (Sheorey 1993). Using the abutment stress value at \(x=l\), one gets,
The elastic coal foundation exerts a reaction force \((q=Cy)\) to the abutment stress inside the solid coal seam having a foundation modulus (Stephansson 1971),
where \(C\) is the foundation modulus (GPa/m), \({E}_{{\text{coal}}}\) is Young’s modulus of the coal seam (GPa), \({h}_{coal}\) is the coal seam thickness (m) and \({\upsilon }_{{\text{coal}}}\) is the Poisson’s ratio of the coal seam.
(c)
Normal stress section (\(x\ge l)\)
In this section, the position is far inside the solid coal seam, so the effect of mining-induced stresses is negligible, and the coal seam exhibits in situ stress conditions. The deflection, stress and strain energy accumulated in this region will not affect the rockburst occurrence at the excavation boundary (Agrawal et al. 2019).
Fig. 1
Schematic vertical cross-section layout of a retreating longwall coal mining panel, where \(L\) is the length of the cantilever roof, \(l\) is the length of abutment zone, \({P}_{o}\) is the abutment stress, \(p\) is the weight of the cantilever roof, \({P}_{o}{e}^{-\alpha x}\) is the exponentially decreasing abutment stress function, \(P\) is the in situ stress, \({h}_{{\text{roof}}}\) is the thickness of the immediate roof, \({h}_{{\text{coal}}}\) is the coal seam thickness, \(C\) is the foundation modulus,\(q\) is the reaction force, \(M(x)\) is the bending moment, \(V(x)\) is the shear force, \(HPS\) is the hydraulic-powered support, \({W}_{{\text{cant}}}\) is the energy stored in the cantilever roof, \({W}_{{\text{roof}}}\) is the energy stored in the supported roof, and \({W}_{{\text{coal}}}\) is the energy stored in the coal seam (Modified after Haramy and McDonnell 1988)
×
Horizontally bedded mining roofs separated by bedding planes are often considered as simply supported rock beams (Caudle and Clark 1955; Obert et al. 1960; Wu 1995). Stephansson (1971) emphasised the applicability of the Euler–Bernoulli beam theory to analyse horizontally bedded roofs with no slip along the boundaries of the rock layers in a long underground opening. He observed that either uniform loading on the top surface of the layer or loading under its weight does not induce any measurable error in the normal stress distribution. The critical stress values remain the same irrespective of the two loading conditions. Thus, the weight of the cantilevered roof beam was approximated by applying a uniformly distributed stress per unit width to its top surface (Galvin 2016).
In this study, the deflection, stress, and strain energy were calculated using the Euler–Bernoulli beam theory for the cantilevered rock beam. Several assumptions were made to apply the Euler–Bernoulli beam theory to determine the effect of overburden and abutment stress on the deflection of the roof. It is assumed that the beam is isotropic, homogeneous, free of any defects/discontinuities, linearly elastic, perfectly straight along its axes, initially stress-free, loaded only normal to its faces, of uniform flexural rigidity and symmetric about an axis in the plane of bending (Galvin 2016). The thickness of the immediate roof is assumed to be less than one-fifth of the length of the roof so the influence of shear stresses can be neglected (Wu 1995). The deflection of the immediate roof during the period of HPS advancement compared to the thickness of the immediate roof is assumed to be negligible for calculation purposes.
2.1 Deflection of the Roof
The roof will deflect due to its weight or due to the mining-induced stresses acting on them. As per the Euler–Bernoulli beam theory, the bending moment for a deflection is given as (Young et al. 2012; Galvin 2016),
where \(M(x)\) is the bending moment of the cantilevered roof (N m), \(y(x)\) is the deflection of the roof (m), \({K}_{{\text{beam}}}\) = \({E}_{{\text{roof}}}I\) is the flexural rigidity of the roof (Pa m4), \({E}_{{\text{roof}}}\) is Young’s modulus of the cantilevered roof \((\)GPa\()\), and \(I={b}_{{\text{r}}}{h}_{{\text{roof}}}^{3}/12\) is the second moment of inertia of the roof (m4).
The shear force developed across the clamped end is given as, \({M}^{\mathrm{^{\prime}}}\left(x\right)=V\left(x\right)\), and the stress per unit length \(P(x)={M}^{{\prime}{\prime}}\left(x\right).\) It is apparent that,
Due to the influence of HPS, roof deflection at the longwall face was prevented resulting in changed boundary conditions (zero deflection and zero slope at the longwall face). The bending moment (Equ. (3)) and roof deflection (Equ. (4)) for different loading conditions were calculated by applying suitable loading and boundary conditions and integrating the equation in the appropriate range (Table 1). The loading conditions considered in different sections of the roof by Xu (2009) are high in the cantilever section and oversimplistic in the abutment stress section. Correspondingly, the roof deflection values are significantly different from those previously developed due to the change in the boundary conditions as Xu (2009) allowed excessive roof deflection at the excavation boundary. The bending moment developed by Xu (2009) does not consider the influence of \(L\), which has been incorporated into the new model.
Table 1
Different scenarios in longwall mining and corresponding stress, boundary conditions, deflection, and bending moment
The external work done on the rock by overburden stress is stored in the elastically stressed rock as strain energy (Young et al. 2012; Agrawal et al. 2019). The strain energy stored in the roof and the coal seam can be calculated assuming that the stress–strain relationship from Hooke’s law follows (Wu 1995; Young et al. 2012; Fedotova et al. 2017; Dong et al. 2018; Xue et al. 2021),
where \({W}_{i}\) is the strain energy density, \({\sigma }_{x}, {\sigma }_{y}{, \sigma }_{z}, {\tau }_{xy}, {\tau }_{yz}, {\tau }_{xz}\) are stress components and \({\varepsilon }_{x}, {\varepsilon }_{y}, {\varepsilon }_{z}, {\gamma }_{xy}, {\gamma }_{yz}, {\gamma }_{xz}\) are strain components.
For a bending beam, the non-zero stress components are flexural stress and shear stress. The flexural stress and strain can be calculated as (Young et al. 2012),
The horizontal beam deflects to produce a continuously distributed reaction force \(q=Cy(x)\) in the coal seam, which acts vertically and opposes the deflection of the beam. As it is assumed that the beam is loaded normal to its faces, the loaded beam may deflect beside the vertical reaction, and there may be some horizontal forces originating along the surface between the beam and the elastic supports, but the horizontal forces would be low in magnitude and thus neglected in the current analysis (Stephansson 1971). The stresses due to the Poisson’s effect make the stress in the coal seam biaxial,
The total strain energy (\({W}_{{\text{total}}})\) stored in the system comprises energy stored in the cantilevered roof \({(W}_{{\text{cant}}})\), supported roof \(({W}_{{\text{roof}}})\), and in the coal seam \({(W}_{{\text{coal}}})\),
The strain energy accumulated in the cantilevering roof and the supported roof was calculated using Equ. (7) and for the coal seam using Equ. (13) by integrating the bending moment and roof deflection in the suitable range (Table 2). The strain energy accumulated in the cantilever roof is the same for the two approaches except for the fact that Xu (2009) considered abutment stress \(P\) while the correct stress acting should be \(p\). This suggests that the modified boundary conditions considered in this paper are correct. The strain energy accumulated in the supported roof and the coal seam varies due to the changed boundary conditions. The main parameters involved in the strain energy equation are mining depth (\(H\)), length of the cantilever roof (\(L\)), coal seam thickness (\({h}_{{\text{coal}}}\)), thickness of the immediate roof (\({h}_{{\text{roof}}}\)), Young’s modulus of coal (\({E}_{{\text{coal}}}\)), Young’s modulus of roof (\({E}_{{\text{roof}}}\)), and Poisson’s ratio of the coal seam (\({\upsilon }_{{\text{coal}}})\).
Table 2
Strain energy accumulation in different sections of the longwall panel
where \(\eta =[\left(1+{\upsilon }_{{\text{coal}}}\right)\left(1-2{\upsilon }_{{\text{coal}}}\right)]/[2{E}_{{\text{coal}}}\left(1-{\upsilon }_{{\text{coal}}}\right)]\)
3 Methodology
3.1 Numerical Simulation
A 3-dimensional numerical model was developed in FLAC3D with the dimensions 300 m × 50 m × 120 m having a grid size of 1 m × 1 m × 1 m along X-, Y-, and Z-directions respectively. A coal seam of 2 m thickness was simulated with a modelled roof of 70 m and a floor of 48 m in the numerical model. The model was constrained with stress boundaries on both sides (X- and Y-axes) and the bottom (Z-axis) was fixed. A uniform density of 2360 kg/m3 was initialised in the model and the model was loaded under gravity. The top surface of the model was free to deform. A vertical stress was applied at the top surface to simulate the overburden stress. The roof, coal and floor were assigned elastic constitutive material properties (Table 3).
Table 3
Rock properties used in numerical simulations
Rock layer
Roof
Coal
Floor
Bulk modulus, GPa
6.67
2.45
6.67
Shear modulus, GPa
3.07
0.94
3.07
After the basic model was developed, boundary conditions were applied to the model (Fig. 2a) and the model was loaded to initialise in situ stress conditions (Fig. 2b). The main gate and return gate of 5 m width were developed to form a 180 m wide longwall panel and 55 m wide rib pillars (Fig. 2c). The coal was extracted sequentially from the longwall panel of dimensions 180 m × 1 m × 2 m along X-, Y- and Z-directions, respectively, in each excavation step by using the null constitutive model in FLAC3D. To represent the cantilevering roof conditions hydraulic-powered supports were incorporated in the model as an equivalent force applied on each grid point, acting on the 5 m wide immediate roof, while the unsupported roof behind the powered supports contribute towards the cantilever roof. The length of the cantilever roof section increases as mining progresses, as the hydraulic-powered supports also move along with the mining face to support the immediate roof. The hydraulic-powered supports’ movement is achieved by removing the equivalent force from the previous grid points and initiating the equivalent force at the freshly cut longwall face (Fig. 2d). A flow chart of the numerical simulation procedure is presented in Fig. 3.
Fig. 2
a Vertical cross-section of a representative FLAC3D model grid with applied boundary conditions, b the model with initialised stresses, c development of the main gate and return gate in the equilibrated model, and d hydraulic-powered supports installed along the longwall face
Fig. 3
A flow chart for the numerical simulation procedure
×
×
3.2 Sensitivity Analysis
Analysis of a full factorial combination of parameters affecting strain energy accumulation will be time-consuming and computationally expensive. To identify the influence of each parameter in the multi-parameter equations, an orthogonal testing method developed on the probability theory and mathematical statistics was used (Wu and Leung 2011; Yuan et al. 2018; Wang et al. 2019). The orthogonal testing method is designed based on the work of Taguchi where each parameter is listed in a different column and has the same variation level (Yuan et al. 2018). Several random unique scenarios can be obtained by juxtaposing any two columns (Bai et al. 2010; Wu and Leung 2011). These scenarios are free of experimental or personal bias (Bai et al. 2010).
3.2.1 Identification of Variation Levels
(i)
Mining Depth \((H)\)
The severity and frequency of rockbursts increase with mining depth as the in situ stresses acting on the coal seam increase (Zhang et al. 2017). Iannacchione and Zelanko (1995) and Mark (2016) observed that no rockbursts occurred up to 300 m mining depth and most were deeper than 400 m depth in the US. Therefore, \(H\) was varied from 400 to 1200 m to encompass shallow-depth to deep-seated coal deposits.
(ii)
Length of the Cantilever Roof/Periodic or Main Weighting (\(L\))
\(L\) and \({h}_{{\text{roof}}}\) is correlated to obey the elastic theory (\(L>2{h}_{{\text{roof}}}\)). \(L\) was used in the orthogonal testing method, which is an important parameter representing roof weighting that influences the roof rock breakage while \({h}_{{\text{roof}}}\) was not considered for statistical analysis. The length of the cantilevering roof is dependent on the mining height and the thickness of the immediate roof strata (Huang et al. 2020). Changing \(L\) represents different weighting stages before the roof falls. Lower values of \(L\) represent easy-to-cave strata while higher values of \(L\) represent massive, strong, and competent strata that are generally difficult to cave.\(L\) was varied from 5 to 30 m.
(iii)
Coal Seam Thickness \(({h}_{{\text{coal}}})\)
The coal seam thickness can influence the magnitude of rockburst damage (Brauner 1994). Undulation in the coal seam thickness may lead to an area of high-stress concentration that may be burst-prone (Agapito and Goodrich 1999; Petros and Premysl 2000; Furniss 2009). \({h}_{{\text{coal}}}\) was varied from 2 to 6 m to incorporate a regular thickness coal seam to thicker coal seams.
(iv)
Young’s Modulus of the Coal Seam \({(E}_{{\text{coal}}})\)
The ability of the coal seam to store strain energy has a significant effect on the rockburst liability of the coal seam (Zhang et al. 2017). As coal is a compressible material, a large amount of strain energy can be stored in the coal seam at low stress levels (Holland and Thomas 1954). \({E}_{{\text{coal}}}\) was varied from 1.50 to 3.50 GPa to represent soft coal to relatively stiff coal.
(xxii)
Young’s Modulus of the Roof \({(E}_{{\text{roof}}})\)
The stiffness of the roof and strength of the roof rock play a major role in the strain energy accumulation in the cantilever and supported roof. A massive, strong, and competent roof will cause more mining-induced stress in the solid coal pillar ahead of the excavation boundary due to a long hanging cantilever roof. It has been observed in several mines that the rockbursts are violent in the case of stiff strata (Brauner 1994; Huang et al. 2020). \({E}_{{\text{roof}}}\) was varied from 4 to 20 GPa to encompass easily caving to massive roof strata.
(vi)
Poisson’s Ratio of the Coal Seam \(\left({\upsilon }_{{\text{coal}}}\right)\)
Coal at the excavation boundary will experience lower confining stress as compared to that present deep inside the solid coal pillar, which will affect Poisson’s ratio of the coal seam. \({\upsilon }_{coal}\) was varied from 0.20 to 0.50 to represent different confining stress scenarios that the coal seam may experience.
3.2.2 Orthogonal Testing Matrix
An orthogonal testing matrix was designed to accommodate the identified parameters and their variation levels. The matrix is defined as \({W}_{{i}^{2}}\)(\({i}^{j}\)), where \(W\) indicates the name of the matrix, \({i}^{2}\) in subscript indicates the number of random unique scenarios generated for the analysis, \(i\) indicates the number of variation levels of each parameter and \(j\) indicates the number of parameters plus an error term \((e)\) in the matrix (He et al. 2018; Yuan et al. 2018). An error term is introduced to obtain the influence of missing parametric interactions that were not considered in the scenarios generated. \(e\) also varies in the same level as the identified parameters and are denoted by arbitrary letters or numbers to distinguish different error term levels. Each \(e\) identifies the error associated in that level and indicates the health of the experimental design (Wu and Leung 2011; Yuan et al. 2018).
3.2.3 Range Analysis
Range analysis is a popular two-step method that is concise, simple and easy to understand (He et al. 2018). In the first step, interactions at every level \((i)\) for each parameter \((j)\) is grouped together to calculate the sum and mean value. \({K}_{i}\) presents the sum of all interactions for a particular level \((i)\) (Wu and Leung 2011; He et al. 2018; Yuan et al. 2018; Wang et al. 2019). \({k}_{i}\) provides the mean of \({K}_{i}\). \({R}_{j}\) provides the range. These parameters can be calculated as (Yuan et al. 2018),
In the second step, the calculated data is used to judge the importance of the parameter \((j)\)(Wu and Leung 2011; He et al. 2018). Larger \({R}_{j}\) means a greater weightage to the parameter (Wu and Leung 2011; Wang et al. 2019). Range analysis is limited to distinguishing data fluctuations occurring due to the influence of \(e\) at each variation level (Wu and Leung 2011; He et al. 2018; Wang et al. 2019). The experimental error cannot be determined in range analysis (Wu and Leung 2011). It cannot evaluate the differences among the mean values and fails to indicate the dominance of the parameters in the system (Wu and Leung 2011; Wang et al. 2019). To overcome the limitations, analysis of variance (ANOVA) is done (He et al. 2018).
3.2.4 Analysis of Variance (ANOVA)
Analysis of variance is a standard statistical technique to determine parametric dominance, estimate the confidence interval of each parameter and evaluate experimental errors (Wu and Leung 2011; He et al. 2018; Yuan et al. 2018; Wang et al. 2019). The dominance of each parameter is evaluated using \(F\)-value which is the ratio of the sum of the square of each parameter’s average deviation (\({{\text{SS}}}_{j}\)) to that of the experimental error (\({{\text{SS}}}_{e})\). It is used to indicate the magnitude of each parameter and the data are analysed using a \(F\)-test. The sum of the square deviation for each parameter can be calculated as (Wu and Leung 2011; Yuan et al. 2018),
The degree of freedom of each parameter \({df}_{j}=i-1\), similarly the degree of freedom of the error term for \(m\) unique scenarios \({df}_{e}=m-1\)(Yuan et al. 2018; Wang et al. 2019). The variance of each parameter \(\left({V}_{j}\right)\) and experimental error \(\left({V}_{e}\right)\) can be calculated as,
$${V}_{j}={{{\text{SS}}}_{j}/df}_{j}$$
(19)
$${V}_{e}={{\text{SS}}}_{e}/{df}_{e}$$
(20)
Thus, the \(F\)-value for each parameter can be calculated as,
$${F}_{j}={V}_{j}/{V}_{e}$$
(21)
Depending on the inspection level for the \({F}_{j}\) value, different critical values \(({F}_{\delta })\) can be determined from the \(F\)-test distribution table, where \(\delta\) is the confidence interval (Wu and Leung 2011). The effect of any parameter is prominent when \({{F}_{j}>F}_{\delta }\) (Wang et al. 2019). The effect of a parameter is very small when \({V}_{j}{\le 2V}_{e}\) (Wu and Leung 2011; Yuan et al. 2018; Wang et al. 2019).
4 Results
4.1 Comparison of Analytical and Numerical Model Findings
A numerical model was run to compare the deflection, stress and total strain energy developed in the numerical model with the analytical model results for a mining depth of 800 m. Other parameters were set as follows: \({h}_{{\text{coal}}}=\) 2 m, \({h}_{{\text{roof}}}=\) 4 m, \({E}_{{\text{coal}}}=\) 2.5 GPa, \({\upsilon }_{{\text{coal}}}=\) 0.33, \({E}_{{\text{roof}}}=\) 8 GPa, and \({\upsilon }_{{\text{roof}}}=\) 0.3. Hydraulic-powered supports applied a vertical reaction force of 10 MPa on the roof and the floor at the working face. The vertical stress acting ahead of the longwall face shows a significant variation within ~ 20 m distance ahead of the active longwall face (Fig. 4a). The stress has a maximum value at 1 m inside the coal face and decreases exponentially as the distance inside the coal seam increases. This can be attributed to the fact that an elastic model has been considered in the model and thus the plastic zones are not formed, else the maximum stress will occur at a farther distance ahead of the active longwall face.
Fig. 4
a Vertical stress redistribution ahead of the coal face and b strain energy comparison between the analytical and numerical model
×
The total strain energy accumulated in the numerical model was compared with that of the analytical model in Sect. 2.2. The peak strain energy accumulated in the numerical model is ~ 20% lower than that calculated from the analytical model at a 1 m distance inside the solid coal seam (Fig. 4b). The total strain energy decreases with the increase in the distance ahead of the longwall face. The trend of the strain energy accumulating inside the solid coal seam obtained from the numerical model agrees well with that calculated using the analytical model. The equations proposed by Xu (2009) also had a similar trend but the magnitude of strain energy was high by an order of magnitude due to the incorrect stress initialisation and excessive deflection at the active longwall face.
The strain energy accumulation calculated from the analytical model shows a low value (zero) at the excavation boundary accumulated mostly due to the cantilevering roof. The energy sharply peaks at ~ 1 m inside the solid coal pillar where the maximum stress abutment acts and then decreases gradually as the distance ahead of the longwall face increases, which is similar to the field observations (Cao et al. 2018). The analytical model proposed in this paper provides a more realistic estimate of the strain energy accumulation in the coal seam overcoming the limitations of previous models. Detailed parametric analysis was conducted on the analytical model to identify the dominance of each parameter in the strain energy accumulation.
4.2 Range Analysis
Table 4 lists different parameters and their variation levels. To accommodate six parameters from equations and an error term (\(j=7\)) and five variation levels (\(i=5\)), a \({W}_{25}\)(\({5}^{7}\)) orthogonal testing matrix was constructed by juxtaposing columns to generate \(m=25\) random unique scenarios (Table 5). The total strain energy (\({W}_{{\text{total}}}\)) accumulated in the retreating longwall face was calculated for each unique scenario (Table 6).
Table 4
Different parameters and their variation levels considered for analysis (after Agrawal 2022)
Level \((i)\)
Parameters \((j)\)
\(H\) (m)
\(L\) (m)
\({h}_{{\text{coal}}}\) (m)
\({E}_{{\text{coal}}}\) (GPa)
\({E}_{{\text{roof}}}\) (GPa)
\({\upsilon }_{{\text{coal}}}\)
\(e\)
1
400
5
2
1.5
4
0.20
1
2
600
10
3
2.0
8
0.25
2
3
800
15
4
2.5
12
0.30
3
4
1000
20
5
3.0
16
0.40
4
5
1200
30
6
3.5
20
0.50
5
Table 5
The \({5}^{7}\) orthogonal testing matrix listing 25 random scenarios (modified after He et al. 2018; Yuan et al. 2018)
Test no. \(m\)
Parameters \((j)\)
\(H\)
\(L\)
\({h}_{{\text{coal}}}\)
\({E}_{{\text{coal}}}\)
\({E}_{{\text{roof}}}\)
\({\upsilon }_{{\text{coal}}}\)
\(e\)
1
1
1
1
1
1
1
1
2
1
2
2
2
2
2
2
3
1
3
3
3
3
3
3
4
1
4
4
4
4
4
4
5
1
5
5
5
5
5
5
6
2
1
2
3
4
5
5
7
2
2
3
4
5
1
4
8
2
3
4
5
1
2
3
9
2
4
5
1
2
3
2
10
2
5
1
2
3
4
1
11
3
1
3
5
2
4
4
12
3
2
4
1
3
5
3
13
3
3
5
2
4
1
2
14
3
4
1
3
5
2
1
15
3
5
2
4
1
3
5
16
4
1
4
2
5
3
3
17
4
2
5
3
1
4
2
18
4
3
1
4
2
5
1
19
4
4
2
5
3
1
5
20
4
5
3
1
4
2
4
21
5
1
5
4
3
2
2
22
5
2
1
5
4
3
1
23
5
3
2
1
5
4
5
24
5
4
3
2
1
5
4
25
5
5
4
3
2
1
3
Table 6
\({W}_{25}\)(\({5}^{7}\)) orthogonal testing matrix with corresponding total strain energy values
Test no. \(m\)
Parameters \((j)\)
\(H\)
\(L\)
\({h}_{{\text{coal}}}\)
\({E}_{{\text{coal}}}\)
\({E}_{{\text{roof}}}\)
\({\upsilon }_{{\text{coal}}}\)
\(e\)
\({W}_{m}\)
1
400
5
2
1.50
4
0.20
1
2.08 × 105
2
400
10
3
2.00
8
0.25
2
2.03 × 105
3
400
15
4
2.50
12
0.30
3
1.96 × 105
4
400
20
5
3.00
16
0.40
4
1.75 × 105
5
400
30
6
3.50
20
0.50
5
1.81 × 105
6
600
5
3
2.50
16
0.50
2
2.06 × 105
7
600
10
4
3.00
20
0.20
3
4.21 × 105
8
600
15
5
3.50
4
0.25
4
3.68 × 105
9
600
20
6
1.50
8
0.30
5
1.09 × 106
10
600
30
2
2.00
12
0.40
1
3.56 × 105
11
800
5
4
3.50
8
0.40
3
4.59 × 105
12
800
10
5
1.50
12
0.50
4
1.11 × 106
13
800
15
6
2.00
16
0.20
5
1.63 × 106
14
800
20
2
2.50
20
0.25
1
5.42 × 105
15
800
30
3
3.00
4
0.30
2
7.30 × 105
16
1000
5
5
2.00
20
0.30
5
2.05 × 106
17
1000
10
6
2.50
4
0.40
4
1.43 × 106
18
1000
15
2
3.00
8
0.50
3
2.85 × 105
19
1000
20
3
3.50
12
0.20
2
7.96 × 105
20
1000
30
4
1.50
16
0.25
1
2.46 × 106
21
1200
5
6
3.00
12
0.25
1
2.30 × 106
22
1200
10
2
3.50
16
0.30
2
7.81 × 105
23
1200
15
3
1.50
20
0.40
3
2.22 × 106
24
1200
20
4
2.00
4
0.50
4
1.33 × 106
25
1200
30
5
2.50
8
0.20
5
2.52 × 106
Range analysis was then performed on the completed orthogonal testing matrix. In the range analysis, \({K}_{1}\) for \(H\) represents the sum of all strain energy variations corresponding to \(H\) = 400 m (Table 7). Similarly, \({K}_{1}\) for \(L\) represents the sum of all strain energy variations corresponding to \(L\) = 5 m and so on for all parameters. Table 7 lists the sum of all level-wise parametric interactions \({K}_{i}\), the corresponding average value \({k}_{i}\) and the variation \({R}_{j}\).
Table 7
Range analysis of the parameters affecting elastic strain energy accumulation
\(H\)
\(L\)
\({h}_{{\text{coal}}}\)
\({E}_{{\text{coal}}}\)
\({E}_{{\text{roof}}}\)
\({\upsilon }_{{\text{coal}}}\)
\({K}_{1}\)
9.62 × 105
5.22 × 106
2.17 × 106
7.08 × 106
4.07 × 106
5.57 × 106
\({K}_{2}\)
2.44 × 106
3.95 × 106
4.15 × 106
5.56 × 106
4.55 × 106
5.87 × 106
\({K}_{3}\)
4.47 × 106
4.69 × 106
4.86 × 106
4.89 × 106
4.75 × 106
4.85 × 106
\({K}_{4}\)
7.02 × 106
3.93 × 106
6.22 × 106
3.91 × 106
5.25 × 106
4.64 × 106
\({K}_{5}\)
9.13 × 106
6.24 × 106
6.63 × 106
2.59 × 106
5.41 × 106
3.11 × 106
\({k}_{1}\)
1.92 × 105
1.04 × 106
4.34 × 105
1.42 × 106
8.13 × 105
1.11 × 106
\({k}_{2}\)
4.88 × 105
7.90 × 105
8.30 × 105
1.11 × 106
9.10 × 105
1.17 × 106
\({k}_{3}\)
8.94 × 105
9.39 × 105
9.72 × 105
9.79 × 105
9.50 × 105
9.69 × 105
\({k}_{4}\)
1.40 × 106
7.85 × 105
1.24 × 106
7.81 × 105
1.05 × 106
9.28 × 105
\({k}_{5}\)
1.83 × 106
1.25 × 106
1.33 × 106
5.17 × 105
1.08 × 106
6.21 × 105
\({R}_{j}\)
1.63 × 106
4.63 × 105
8.91 × 105
8.99 × 105
2.69 × 105
5.52 × 105
The effect of each parameter on strain energy accumulation for selected levels of variation is shown in Fig. 5. The variation in the strain energy accumulation due to \(H\) is directly correlated and shows a linearly rising trend with a correlation coefficient of 0.99 (Fig. 5a), which indicates that more strain energy will be accumulated at higher depths. The variation in the strain energy accumulation due to \(L\) follows a non-linear dependence with a correlation coefficient of 0.81 (Fig. 5b). It can be seen from Fig. 5b that, with the increase in the length of the cantilevered roof to 30 m, the strain energy accumulated increases sharply which suggests that the model is responding to the length of the cantilever roof. The variation in the strain energy accumulation due to \({h}_{{\text{coal}}}\) is directly correlated and exhibits a linearly rising trend with a correlation coefficient of 0.95 (Fig. 5c), which suggests that thicker coal seams will accumulate more strain energy. The variation in the strain energy accumulated due to \({E}_{{\text{coal}}}\) is inversely correlated and shows a declining trend with a correlation coefficient of 0.98, which suggests that hard coal will tend to absorb less energy (Fig. 5d). The variation in the strain energy accumulation due to \({E}_{{\text{roof}}}\) has a direct correlation and follows a rising trend with a gentle slope having a correlation coefficient of 0.97 (Fig. 5e), which suggests that as the roof stiffness increases more strain energy will be accumulated. The variation in the strain energy due to \({\upupsilon }_{{\text{coal}}}\) is inversely correlated with a declining trend having a correlation coefficient of 0.87 (Fig. 5f). The magnitude of strain energy calculated using the analytical model is comparable to those observed in underground mining. Wen et al. (2016) observed that the strain energy accumulated at 5 different longwall faces where rockbursts occurred was in the range of MJ/m3 while Xue et al. (2021) observed that the strain energy was in the range of MJ/m3 for 6 outburst cases in China.
Fig. 5
Variation in total strain energy accumulation due to different influencing parameters a mining depth, b length of the cantilever roof, c coal seam thickness, d Young’s modulus of the coal seam, e Young’s modulus of the roof, and f Poisson’s ratio
×
Based on the \({R}_{j}\) value, the hierarchy of dominance of parameters can be listed as \(H>{E}_{{\text{coal}}}>{h}_{{\text{coal}}}>{\upsilon }_{{\text{coal}}}>L>{E}_{{\text{roof}}}\) (Fig. 6). \(H\) has maximum dominance in the strain energy accumulation which explains why the risk of rockburst increases with depth. The coal seam foundation \(C\) depends on \({E}_{{\text{coal}}}\), \({h}_{{\text{coal}}}\) and \({\upsilon }_{{\text{coal}}}\) that reflects the ability of coal to accumulate strain energy which is necessary for violent ejection and thus influences the strain energy accumulation. \(L\) and \({E}_{{\text{roof}}}\) also influence the strain energy accumulation but in terms of its relative magnitude, the influence is less pronounced.
Fig. 6
Range analysis showing the relative dominance of different parameters affecting strain energy accumulation in a retreating longwall panel
×
4.3 Analysis of Variance (ANOVA)
Analysis of variance conducted for the parameters considered in the orthogonal testing matrix for strain energy accumulation shows that the \({V}_{H}\), \({V}_{{h}_{{\text{coal}}}}\), and \({V}_{{E}_{{\text{coal}}}}\) values are more than twice \({V}_{e}\) thus dominant, while \({V}_{L}\), \({V}_{{E}_{{\text{roof}}}}\), and \({V}_{{\upsilon }_{{\text{coal}}}}\) are less than twice \({V}_{e}\) (Table 8). The \(F\)-value for a confidence interval of 99%, i.e., \({F}_{\delta =0.01}\left(\mathrm{4,24}\right)=4.22\) was determined from the \(F\)-test table (Dinov 2020). The hierarchy obtained from the F-value is \({F}_{H}>{{F}_{{h}_{{\text{coal}}}}>F}_{{E}_{{\text{coal}}}}>{F}_{\delta =0.01}>{{F}_{{\upsilon }_{{\text{coal}}}}>F}_{L}>{F}_{{E}_{{\text{roof}}}}\). This suggests that \(H\), \({h}_{{\text{coal}}}\) and \({E}_{{\text{coal}}}\) will significantly affect the strain energy accumulation in a retreating longwall mining panel within a 99% confidence interval.
Table 8
The analysis of variance of the strain energy accumulation in a retreating longwall panel
Parameter
\(j\)
\({{\text{SS}}}_{j}\)
\({df}_{j}\)
\({V}_{j}\)
\({V}_{j}\le 2{V}_{e}\)
\({F}_{j}\)
\(H\) (m)
1
8.83 × 1012
4
2.21 × 1012
No
18.64
\(L\) (m)
2
7.50 × 1011
4
1.88 × 1011
Yes
1.58
\({h}_{{\text{coal}}}\) (m)
3
2.54 × 1012
4
6.34 × 1011
No
5.36
\({E}_{{\text{coal}}}\) (GPa)
4
2.30 × 1012
4
5.74 × 1011
No
4.85
\({E}_{{\text{roof}}}\) (GPa)
5
2.36 × 1011
4
5.89 × 1010
Yes
0.50
\({\upsilon }_{{\text{coal}}}\)
6
9.25 × 1011
4
2.31 × 1011
Yes
1.95
\(e\)
2.84 × 1012
24
1.18 × 1011
5 Discussion
Several parameters are related in a complex manner to determine the strain energy accumulation in a retreating longwall coal panel explaining the complex nature of rockburst hazards that have been reported by several researchers (Mark 2016; Zhang et al. 2017; Sabapathy et al. 2019). The strain energy accumulated in the cantilevered section of the roof in the analytical model proposed in this paper is the same as that reported by earlier researchers (Stephansson 1971; Xu 2009; Young et al. 2012; Galvin 2016), which suggests that the modified boundary conditions considered are in agreement. However, the strain energy accumulated in the supported roof and the coal seam is significantly large as compared to those proposed in this paper. The strain energy variation calculated from the analytical model matches well with the numerical model with maximum variations occurring within ~ 40 m ahead of the coal face in the solid coal seam. A similar observation was made by Cao et al. (2018) who observed from field monitoring micro-seismicity that the maximum events lie within ~ 40–70 m of the excavation boundary inside the solid coal seam ahead of the coal face.
The parametric analysis of strain energy using the orthogonal testing method provides a logical hierarchy. It is obvious that with an increase in \(H\) the in situ stress increases, which in turn leads to an increased strain energy accumulation. Hence, it is the most important parameter. This observation matches the findings of previous researchers who have suggested that rockbursts are likely to occur only at a mining depth of more than 400 m (Brauner 1994; Iannacchione and Zelanko 1995; Mark 2016; Zhang et al. 2017; Canbulat et al. 2019). The ability of coal to store strain energy is dependent on the foundation modulus of coal \(C\) (Wu 1995; Xu 2009), thus, the next three parameters in the hierarchy \({E}_{{\text{coal}}}, {h}_{{\text{coal}}}\), and \({\upsilon }_{{\text{coal}}}\) are obvious candidates to have dominance.
Wu (1995) observed that increasing the foundation modulus of coal leads to a decrease in the total stored energy. The strain energy increases monotonically with an increase in \({h}_{{\text{coal}}}\) that matches the observation of previous researchers (Campoli et al. 1987; Brauner 1994; Iannacchione and Zelanko 1995; Wu 1995; Iannacchione and Tadolini 2008; Mark and Gauna 2016; Zhang et al. 2017; Agrawal et al. 2019). The strain energy accumulated in the roof is proportional to the length of cantilevering rock (\(L\)) which manifests the effect of periodic/main roof weighting before caving which has serious implications in terms of mining-induced stress and corresponding strain energy accumulation. In the case of a strong, stiff roof with high \({E}_{{\text{roof}}}\), the roof overhangs for a large distance in the goaf before cracks develop (Huang et al. 2020) and thus increases the periodic/main roof weighting distance.
Due to the complex interactions between different parameters of strain energy equations, it is difficult to forecast rockbursts using a single index (Zhou et al. 2018; Agrawal et al. 2022). Researchers in the past have developed over one hundred empirical indices based on strength, strain and strain energy calculated in laboratory-characterised coal and surrounding coal measure rocks to analyse the potential for rockbursts occurrence (Qiu and Feng 2018; Li et al. 2019; Shirani et al. 2020). Most of these indices use mechanical properties like unconfined compressive strength (UCS), maximum and minimum principal stresses, tensile strength, peak strength, Young’s modulus, elastic strain energy, and plastic strain energy to determine the rockbursts liability (Zhou et al. 2018; Bacha et al. 2020).
The mechanical properties indicate that the presence of hard coal and massive strata are conducive to rockbursts. UCS has a strong positive correlation and has been used as a direct indicator of rockburst potential as stronger rocks allow larger strain energy build-up which is a necessary condition for rockbursts occurrence (Yang et al. 2018; Zhou et al. 2018). High-strength brittle rocks, high-stress environments (occurring due to deep mining) and increasing peak strength with confinement allow more strain energy accumulation to meet the necessary conditions for rockbursts to occur (Miao et al. 2016). The strain energy accumulated in the coal seam and surrounding strata is the main source of energy leading to rockbursts (Wu 1995; Zhou et al. 2018). Previous researchers used field microseismic monitoring data to back-calculate the strain energy threshold beyond which rockbursts may be expected (Bukowska 2006; Mark 2016). These field-monitored threshold values can be used as a trigger to forecast the risk of rockbursts in different underground mining scenarios.
6 Conclusion
The analytical model proposed in this paper overcomes the limitations of previous approaches by presenting a realistic representation of the field conditions that exist around the current state-of-the-art longwall faces with hydraulic-powered supports. The orthogonal testing method revealed the hierarchy of parametric dominance as \(H>{E}_{{\text{coal}}}>{h}_{{\text{coal}}}>{\upsilon }_{{\text{coal}}}>L>{E}_{{\text{roof}}}\) which is free from any sampling, personal or experimental bias. \(H, {E}_{{\text{coal}}},\) and \({h}_{{\text{coal}}}\) affect the strain energy accumulation within a 99% confidence interval. The parameters considered in the equations are readily available and can be suitably used to facilitate safe and efficient longwall mining panel design at the planning and designing stage. These equations are easy to use and do not require specific software knowledge which makes them suitable to be widely used in the mining industry.
Acknowledgements
The first author is thankful to the ITASCA Consulting Group for the FLAC3D version 6.0 license under the IEP program.
Ethics
Conflict of interest
The authors have no conflicts of interest to declare. All co-authors have seen and agree with the contents of the manuscript and there is no financial interest to report.
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