Introduction

Coal seam mining causes redistribution of underground stress which is uniformly distributed in surrounding rock before coal seam excavation. Some area in stope become destressed (e.g., the strata above goaf), while the adjacent neighboring solid sections become stress-concentrated (e.g., the coal seam ahead of working face). Accordingly, it could be considered that there is in situ and mining-induced stress in surrounding rock after coal seam excavation (Singh and Singh 2010). Moreover, with working face advancing, the mining-induced stress dynamically changes both in space and in time. The distribution of mining-induced stress, especially the location and amount of maximum, is important for supportive design and safety production. But, it is not easy to make a clear understanding of it. The geological condition of strata, the value of in situ stress, and the mechanical properties of surrounding rock all have a significant influence on distribution mining-induced stress. Besides, the adopted mining technique, the dimension of mining panel, the advancing speed of working face, etc. are other important factors (Feng 2015).

Generally, theoretical analysis, in situ test, physical modeling, and numerical simulation are utilized by researchers to study the development of mining-induced stress (Salamon 1984; Ju et al. 2017; Wang et al. 2016; Toraño et al. 2002). But, in situ test is more time-consuming and expensive. The empirical rules base on past experiences do provide a general guide; however, the heterogeneous nature of mine geology may result in errors. On the contrast, the physical experiment and theoretical analysis are more adoptable.

In this research, theoretical analysis was carried out to study the height of the destressed zone and stress concentration factor in the context of Weijiagou Coal Mine in Southwest China. Meanwhile, physical modeling experiment was also used to study the deformation of strata and the development of mining-induced crack and stress.

Stress distribution around longwall mining panel

Many scholars carried out experiments and analysis to study mining-induced stress in the situation of longwall mining. By supposing the coal seam was infinite, elastic, isotropic, and homogenous, Salamon (1984) developed an analytical equation to evaluate the stress distribution at the edge of longwall panel. Ju et al. (2017) and Wang et al. (2016) studied the characteristics of overlying strata collapse and mining-induced pressure by employing the physical model test. By numerical modeling, Toraño (2002) proved that the concentration factor of mining-induced stress around roadway is approximately 2.3 times of in situ stress. Yavuz (2004) demonstrated that mining height had a positive correlation with the distance of working face to abutment stress. Based on stress evolution both in time and space during longwall mining, Yang et al. (2011) found that the abutment stress changed rapidly within 10 m in front of working face, and the maximum value was approximately twice the in situ stress when the mining size reached 20–30 m. The finite element modeling performed by Khanal et al. (2011); Khanal and Adhikary (2012) demonstrated that the maximum induced vertical stress is approximately four times of the in situ stress. According to the above researches, it can be found that many methods are available to study the mining-induced stress. However, the results obtained from these methods are discrete. A major task is to select a more reliable and intended way to perform a realistic assessment. Additionally, the geological condition of a coal mine is non-uniform and complex, which makes it a challenge to study the distribution and variation of mining-induced stress.

Time-dependent energy model

The transfer of mining-induced stress in strata

The coal seam excavation causes stress redistribution and results in degeneration of integrity of surrounding rock. Generally, according to the deformation and damage state, the overburden strata of goaf could be classified into three regions: caved region, fractured region, and bending region from bottom to top. Because stress in the former two regions has almost totally stress released, this area can be named “destressed zone,” as shown in Fig. 1. The weight of strata above the destressed zone transfers to adjacent coal seam and rock mass. Meanwhile, the abutment strata on the lateral of the destressed zone are the media for stress transfer. As the height of the destressed zone (H d ) increases, the volume of overlying strata and the mining-induced stress reduces. So, H d is the main influence factor to calculate mining-induced stress.

Fig. 1
figure 1

Schematic of stress transfer in longwall mining

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Energy conservation in stope

The combination of coal seam and surrounding rock can be seemed as an enclosing system. Coal seam excavation undermines the balance within the system. The strain energy stored in excavated coal seam shifts, enriches, and rebalances again in other area. Actually, the concentration of stress in coal and rock mass can also be considered as energy accumulation. Based on the law of energy conservation, the total strain energy stored in the mined material is released and consumed in overlying and underlying strata deforming, fracturing, and caving. Therefore, the stored strain energy in the mined material (U m ) should be equal to the stored strain energy of caved material within the destressed zone (U d ) (Rezaei et al. 2015a).

$$ {U}_d={U}_m $$
(1)

U m can be calculated by the following equation:

$$ {U}_m=\frac{\left(1+v\right)\left(1-2v\right)}{2\left(1-v\right)E}\gamma {A}_m{\sigma}_v\left(\frac{h_s^2}{3}+{H}^2+{Hh}_s\right) $$
(2)

where v is Poisson’s ratio of rock mass, E is the elastic modulus of excavated material, γ is the volume weight of rock mass, A m is the cross-section area of the mined panel, σ v is the vertical-induced stress, h s is the mining height, and H is the depth of cover.

In general, U d is composed of elastic strain energy and viscoplastic strain energy:

$$ {U}_d=\frac{A_d{H}_d{\sigma}_c^2}{2\left(E+\frac{b_1{\sigma}_c}{b_1-1}\right)}\left(2{Ee}^{- bt}+\frac{\sigma_s^{\mu }}{Bt^{\omega }}-\lambda \right) $$
(3)

Where A d is the unit surface of goaf area, σ c is uniaxial compressive strength of caved material, b1 is the bulking factor, σ s is the threshold value of stress, B is the material constant related to viscosity and temperature, t is the pressure time of caved material, λ is the slope of material hardening stage, and b, ω, and μ are material constants.

Accordingly, the substitution of eq. (2) and (3) into (1) yield the equation for calculating H d in long-term or time-dependent condition (Rezaei et al. 2015b).

$$ {H}_d=\frac{\gamma {A}_m{\sigma}_V\left(1+v\right)\left(1-2v\right){\left(E+\frac{b_1{\sigma}_c}{b_1-1}\right)}^2\left(\frac{h_s^2}{3}+{H}^2+{Hh}_s\right)}{A_dE{\sigma}_c^2\left(1-v\right)\left(2{Ee}^{- bt}+\frac{\sigma_s^{\mu }}{Bt^{\omega }}-\lambda \right)} $$
(4)

where σ V is the vertical mining-induced stress and b1 is the bulking factor of rock.

Beyond the height of the destressed zone, the ground stress transfers to the front abutment strata and adjacent coal seam. In vertical direction, the mining-induced stress applied by abutment rock mass σ av can be obtained by the following equation (Hou and Ma 1989).

$$ {\sigma}_{av}=\left(\gamma H-\frac{\gamma {A}_m{\sigma}_V\left(1+v\right)\left(1-2v\right){\left(E+\frac{b_1{\sigma}_c}{b_1-1}\right)}^2\left(\frac{h_s^2}{3}+{H}^2+{Hh}_s\right)}{A_dE{\sigma}_c^2\left(1-v\right)\left(2{Ee}^{- bt}+\frac{\sigma_s^{\mu }}{Bt^{\omega }}-\lambda \right)}\right)\sqrt{\frac{H_d^2}{H_d^2+{\mathrm{X}}^2}} $$
(5)

where x is the horizontal distance of the stress effect point from the panel edge.

Accordingly, the stress concentration factor k can be determined as follows (Rezaei et al. 2015b, c):

$$ k=\frac{\sigma_v+\left({\sigma}_v-\gamma {H}_d\right)}{\sigma_v}\sqrt{\frac{H_d^2}{H_d^2+{\mathrm{X}}^2}} $$
(6)

Physical modeling of longwall mining

Physical modeling allows researchers to observe development of ground movement, mining-induced stress, and fracture during excavation. Since it has a short test cycle and reliable result (Meguid et al. 2008; Jaiswal and Shrivastva 2004), this research method has been widely used in analyzing mining-related issues.

Similarity theory

In order to capture realistic behavior of strata during mining via physical modeling test, the model is required to be built in accordance with the principle of similarity theory. Moreover, the selection of raw materials and compounding ratios are the key matters in terms of mechanical properties of the modeling material.

The geometry ratio of the model to real stope should be firstly determined. Specifically, the dimension of the model, strength, and density of the modeling material need to match the following criterions (Yang et al. 2011; Meguid et al. 2008; Li 2016).

$$ \frac{C_{\sigma }}{C_{\rho}\times {C}_L}=1 $$
(7)
$$ {C}_L={L}_m/{L}_p,\kern0.5em {C}_{\sigma }={\sigma}_m/{\sigma}_p,\kern0.5em {C}_{\rho }={\rho}_m/{\rho}_p $$
(8)

where C L is the geometry similarity constant, C σ is the strength similarity constant, C ρ is the density similarity constant, L is the geometric dimension, σ is the compressive strength, ρis the density, the subscript P stands for prototype, and m stands for model.

Engineering background

In this research, Weijiagou Coal Mine was selected as prototype for physical modeling. This mine is located in the southwest China. The average slope angle and thickness of coal seam are 15° and 1.2 m which are adopted in the modeling design. The burial depth of coal seam is 530 m; the length of working face is 100 m. According to the synthetic stratum histogram, the overlying and underlying strata with similar lithology can be integrated into the following types: argillaceous sandstone, siltstone, mudstone, medium granular sandstone, sandy mudstone, and coal. The distribution and thickness of each stratum are shown in Fig. 2.

Fig. 2
figure 2

The distribution of lithology and thickness of each strata

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These similarity ratios of the scaled physical model are calculated and assigned according to the uniaxial compressive strength (UCS), physical characteristics, and dimensions of the target prototype case. The final similarity ratios used in this experiment are noted in Table 1. q m is the pressure of overlying strata beyond modeling scale applied on the top of the model.

Table 1 The similarity ratios for the physical models
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Construction of physical modeling

In this research, sand-plaster-lime-water mixtures were utilized to simulate rock layers, and borax was proportionately added to delay the coagulation of plaster. By carrying out orthogonal experiment, the modeling material with different proportions of sand, lime, plaster, and water was tested. A part of the specimens and typical damage state are shown in Fig. 3. According to the similarity ratios, the compounding ratios for every lithology can be acquired (Table 2). Due to restrict of space, the details are not listed here; readers can refer to the thesis of Li (2016) for more information.

Fig. 3
figure 3

The orthogonal experiment on compounding ratios: a Modeling material specimen. b–d The damage state of the specimen

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Table 2 The compounding ratio and quality of the modeling materials for each rock layer
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In accord with the compounding ratios, the raw materials were evenly mixed and casted layer by layer in the self-developed rotatable physical similar test system (Fig. 4a). During the laying process, pressure cells (type SDTY-2) were buried in coal seam floor along dip direction and connected with strainmeter (type WS-3811, Fig. 4b) to collect strain data during excavation. The distance between each pressure cells was 0.07 m. After stacking the modeling material to the top of the frame, the model was fully air-dried. Before excavation, the displacement monitoring points were installed on the surface of the physical model to observe the deformation of strata during mining. The weight of overburden strata beyond the model scale was applied by hydraulic jacks. Digital photographs were used for measurement of the strata movement and crack propagation pattern by means of photogrammetry and digital image correlation (DIC) techniques. The layout of physical model is shown in Fig. 4a. Coal seam in the model was excavated via electric drill. Also, sufficient time was given between each step of excavation to allow the deformations to be fully developed.

Fig. 4
figure 4

The test system and instrument used in physical modeling experiment: a The self-developed rotatable physical similar test system. b The strainmeter used for stress monitoring

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Physical modeling results

Strata deformation

Since the deformation of strata significantly changed during coal seam excavation, displacement monitoring points can quantitatively display the deformation of strata. By taking pictures of the model in different mining stage, excavating the coordinates of monitoring points, calculating the increment of displacement, and importing the data into Sigmaplot, the deformation contour plot could be obtained (Fig. 5).

Fig. 5
figure 5

Deformation contour plot of strata acquired by DIC technique at different mining width: a 36 m. b 60 m. c 72 m. d 90 m

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At the early stage, the excavation only caused deformation in adjacent strata around mining panel (Fig. 5a). As mining width increased, the deformation and movement of strata gradually developed from bottom to top of the overlying strata, while the increasing rate of the height of the destressed zone decreased (Fig. 5b, c).When the mining width exceeded 72 m, the excavation caused large-scale deformation of strata (Fig. 5d), and it developed to the top of the model.

Mining-induced crack propagation

Coal seam mining results in crack propagation both in overlying and underlying strata. In order to acquire clear development of mining-induced crack, binarization method was used for manipulating photographs. Based on the modified photographs, the crack in different stages could be drawn as shown in Fig. 6. According to Fig. 6, it could be found that the connection of cracks resulted in separation and fracture of rock layers. At the early mining stage, only adjacent strata above stope caved (Fig. 6a). As mining width increased, the fracture propagated to the upper seam and generated destressed zone. Above this area, the intact rock strata only bended toward goaf (Fig. 6b). Judging from the spatial relationship between growth direction of crack and layout of strata, there were horizontal and perpendicular cracks in surrounding rock. The horizontal crack was generated by the separation of strata, and the perpendicular crack was produced by the fracture of layers. The height of caved zone firstly increased and become relatively unchanged when the crack propagated to the key stratum (Fig. 6c), similar to the variation of deformation. Generally, because of inclined occurrence of rock layers, the shape of the destressed zone was isosceles trapezoid when the stope reached balance. The next step of excavation re-actived the caved zone and generates new bedding separations and areas of crack closure and opening. Moreover, as the burial depth of the deeper end of working face was greater, the crack in this area appeared more congregated (Fig. 6d, section I and II). Finally, H d was about 29.87 m.

Fig. 6
figure 6

The growth of mining-induced crack in surrounding rock

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Mining-induced stress concentration

Based on the monitoring data acquired from pressure cells that were buried in coal seam floor, the variation of stress concentration factor k could be calculated, as shown in Fig. 7.

Fig. 7
figure 7

The variation of stress concentration factor

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Because coal seam mining caused stress relief, k appeared negative above goaf. Since the length of working face had a positive correlation with the volume of overlying strata of the destressed zone, the mining-induced stress became greater with mining width increased. The maximum of k in coal seam is about 1.94 in deep end of working face, and it was about 1.69 in shallow end when the mining width reached 72 m.

Comparative analysis

In order to carry out comparative analysis, the time-dependent energy model was used to calculate H d and k. The utilized constant values in this calculation were referred to Rezaei et al. (2015a, c) and Zhang et al. (2011), as shown in Table 3.

Table 3 The value of experimental parameters used in calculation
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The average bulking factor of overlying rock strata was set to 1.5. Generally, threshold value of stress was the yield point of stress-strain curve which was commonly assumed to be about 80% of the peak compressive stress (Falakian 2012). The pressure time of caved rock mass was set to 1.5 years (Rezaei et al. 2015a). Other geometrical and mechanical parameter values of coal seam and rock strata were listed in Table 4.

Table 4 Average values of geometrical and mechanical parameters
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By taking the values of parameters into eqs. (4) and (6), the output of k was 2.13, and H d was 23.53 m. When comparing with the modeling results, k obtained from the theoretical model was larger, but H d was smaller. Generally, k had inverse relationship with the distance from the panel edge. Meanwhile, the panel edge was related to the length of working face. Because of the sidewall breakage of gate and pillars, which was widely existed in underground mining, actually, the length of working face became greater. It increased the height of the destressed zone which generates a lower stress concentration. As shown in Fig. 6d, the deep end of working face damaged and caused an apparent decrease of stress concentrate (Fig. 7). Hence, the mining panel width should consider the plastic zone of the gate.

Hou and Ma (1989) proposed a method to calculate the width of the limit equilibrium zone of stress (i.e., the plastic zone x0) as shown in the following equation:

$$ {x}_0=\frac{mA}{2\tan {\phi}_0}\ln \left(\frac{k\gamma H+\frac{C_0}{\tan {\phi}_0}}{\frac{C_0}{\tan {\phi}_0}+\frac{P_x}{A}}\right) $$
(9)

where C0 and ϕ0 were the fraction angle and cohesion between coal seam and rock layer (14° and 7.4 MPa), A was the ratio between horizontal and vertical stress (1.5), and p x was supportive pressure (0.7 MPa).

Consequently, the equation of A m could be modified as below:

$$ {A}_m=\left({l}_w+2{x}_0\right){h}_s $$
(10)

Based on the above modification, the values of H d and k were 27.94 m and 1.82 which had a higher consistency with physical modeling. However, it should be noted that the presence of geological features, especially to the joints and faults, in different mining states led to variations in stress transfer and concentration. But, these geological features were not considered in the physical model and theoretical analysis as presented in this study. Actually, because of the initial damage existed in overlying strata, the height of the destressed zone may propagate higher.

Conclusions

In this research, in order to reveal the characteristics of deformation, damage, mining-induced stress, and crack in strata during long wall mining, the time-dependent energy model and physical modeling test were carried out based on the geological condition of Weijiagou Coal Mine. The physical modeling experiment was designed in accordance with the laws of similarity theory and conducted on the self-developed rotatable physical similar test system. Accordingly, the mechanical properties including strength and density of the modeling material were calculated. Orthogonal experiment method and mechanics tests were also performed to obtain the mechanical behavior of specimen with different compounding ratios. Accordingly, the best ratios for every lithology were determined.

Via the physical modeling experiment, strata movement, crack propagation, and mining-induced stress concentration during working face advancing were observed. The result showed that the deformation and crack developed to upper strata with mining width increase. But, the development stoped when the destressed zone grew to the bending strata. The excavation finally caused breakage of key strata, and it led to large-scale deformation of strata.

The structure of caved strata showed that the height of the destressed zone had a significant influence on stress distribution. When compared to the time-dependent energy model, the result obtained from physical modeling test indicated that the height of the destressed zone was greater and the stress concentration factor was smaller. In order to reduce the deviation, the dimension of the plastic zone was introduced into mining panel width. Due to the modification, it drew a higher consistency with physical modeling.