Introduction

With the continuous growth in energy demand and the increase in mining intensity and depth, the geological conditions of mines are becoming increasingly complicated1. Disturbances induced by geological conditions in stress fields can lead to many engineering disasters such as rock bursts, roadway collapses, and coal and gas outbursts. The evolutionary characteristics of coal seam stress affected by geological conditions are of great significance for the prevention of engineering disasters2,3,4.

Changes in coal seam stress caused by geological anomalies such as faults, folds, coal seam phase transformation and thick hard roof are the main influencing factors of rock bursts, coal and gas outburst accidents5,6,7.

The rockburst mines belonging to Wushen Banner in Inner Mongolia are all mines with complex hydrogeological conditions, and mining activities are affected by roof water8. The mining activities of 2–2 and 3 − 1 coal seams in the main mining are affected by the rich water in the overlying Zhiluo formation and Yan’an formation, and the strata water is locally irregular9,10,11. The roof water found in Yingpan trench Coal Mine and Shilawusu Coal Mine is extremely irregular, as shown in Fig. 1.

After the confined water in the roof is released, the lithology of the drainage area in the water-rich layer changes, and the water pressure decreases, leading to the movement of the overlying strata and affecting the geostress distribution12,13. Many scholars have studied the influence of roof water drainage of coal seam on the stress distribution of rock strata14,15,16,17,18. The influence of the boundary angle of water-rich abnormal area on the stress distribution in coal seam has not been reported. However, predicting disasters and implementing disaster prevention rely on controlling local pressure in coal seams19,20,21. Therefore, it is necessary to study the influence of the boundary angle of the water-rich abnormal area on the stress distribution. This has theoretical and engineering significance for preventing and controlling dynamic disasters such as rock bursts, coal and gas outbursts in water-rich Kuangqu areas.

Fig. 1
figure 1

Distribution of irregular water-rich areas in the Shaanxi-Inner Mongolia mining area.

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Theoretical analysis of coal seam stress after drainage

The theoretical basis of the calculation of coal seam stress is rock formation movement and stress transfer12,13,18. The research is divided into two parts. One is to calculate the stress of drainage rock strata based on existing research14,15,16,17 and theory of rock formation movement. The second is to calculate the coal seam stress based on the stress transfer theory.

Stress calculation of water-rich strata after drainage

Calculating the stresses of water-rich strata forms the basis of coal seam stress calculation. In the process of calculating the stress of rock strata after drainage, it is assumed that the water pressure is uniformly distributed and the damage of rock strata is uniform. The stress calculation model of coal seam after drainage in water-rich abnormal area is shown in Fig. 2.

Fig. 2
figure 2

Stress model after drainage.

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Figure 1 illustrates the coordinates system, in which ‘o’ is the origin located at the top center of the water-rich abnormal area. H1 is the distance from the top of the water-rich area to the coal seam roof. H2 is the drainage influence height of the water-rich area and H3 the thickness of the static load strata. d represents the width of the water-rich area, and α is the movement angle of the overlying strata within the water-rich area. A and B represent the affected strata area on the section.

Following the drainage, if the shape of the drainage area is not taken into consideration, the stress distribution of the rock strata on both sides remains identical, and calculations focus only on the stress on the far-right side of the origin.

Post drainage, the elastic modulus of the rock layer within the drainage area transitions from K1 to K2, accompanied by a shift in strain ε1 to ε2.

$${\sigma _1}={k_1}{\varepsilon _1}=\gamma \left( {{h_2}+{h_3}} \right)$$
(1)

where γ is the bulk density of rock strata.

After dewatering, the stress in the drainage area remains σ2, as shown in Eq. 2.

$${\sigma _2}={k_2}{\varepsilon _2}=n\gamma \left( {{h_2}+{h_3}} \right)$$
(2)

The weight transferred by the rock mass in area B on the right is QB, as shown in Eq. 3.

$${Q_B}=\frac{d}{2}\left( {1 - n} \right)\gamma \left( {{h_2}+{h_3}} \right)$$
(3)

In Eq. 3, n represents the coefficient of drainage pressure relief, where the value of 0 < n < 1 correlates to the degree of drainage.

It is assumed that the stress increment of rock strata resulted from drainage follows a triangular distribution22, with the rock mass transfer stress in region B denoted as σB.

$${\sigma _B}=\left\{ {\begin{array}{*{20}{l}} {\frac{{2{\sigma _{B\hbox{max} }}\tan \alpha }}{{{h_2}}}\left( {{x_1} - \frac{d}{2}} \right){\text{ }} \quad \left( {\frac{d}{2},\frac{d}{2}+\frac{{{h_2}}}{{2\tan \alpha }}} \right]} \\ {2{\sigma _{B\hbox{max} }}\left[ {1 - \frac{{\tan \alpha }}{{{h_2}}}\left( {{x_1} - \frac{d}{2}} \right)} \right]{\text{ }} \quad \left( {\frac{d}{2}+\frac{{{h_2}}}{{2\tan \alpha }},\frac{d}{2}+\frac{{{h_2}}}{{\tan \alpha }}} \right]} \\ {0{\text{ }} \qquad\qquad \left( {\frac{d}{2}+\frac{{{h_2}}}{{\tan \alpha }},+\infty } \right)} \end{array}} \right.$$
(4)

In Eq. 4, x1 is the distance from the origin and the maximum value of rock mass transfer stress in region B is σBmax.

$${\sigma _{B\hbox{max} }}=\frac{{2{Q_B}\tan \alpha }}{{{h_2}}}=d\left( {1 - n} \right)\gamma \left( {{h_2}+{h_3}} \right)\frac{{\tan \alpha }}{{{h_2}}}$$
(5)

The gravity stress of drainage strata is as follows:

$${\sigma _{{\text{r}}z}}=\left\{ {\begin{array}{*{20}{l}} {{\sigma _2}{\text{ }} \qquad\left( {0,\frac{d}{2}} \right]} \\ {{\sigma _2}{\text{+}}\frac{{\left( {1 - n} \right)\gamma \left( {{h_2}+{h_3}} \right)\tan \alpha }}{{{h_2}}}\left( {{x_1} - \frac{d}{2}} \right){\text{ }} \quad \left( {\frac{d}{2},\frac{d}{2}+\frac{{{h_2}}}{{\tan \alpha }}} \right]} \\ {{\sigma _1}{\text{ }} \qquad \left( {\frac{d}{2}+\frac{{{h_2}}}{{\tan \alpha }},+\infty } \right)} \end{array}} \right.$$
(6)

The stress distribution of drainage strata results from the superposition of dead weight stress and stress increment, as shown in Eq. 7.

$${\sigma _r}=\left\{ {\begin{array}{*{20}{l}} {n\gamma \left( {{h_2}+{h_3}} \right){\text{ }} \quad \left( {0,\frac{d}{2}} \right]} \\ {n\gamma \left( {{h_2}+{h_3}} \right){\text{+}}\left[ {2{\sigma _{B\hbox{max} }}+{w_1}} \right]\frac{{\tan \alpha }}{{{h_2}}}\left( {{x_1} - \frac{d}{2}} \right){\text{ }} \quad \left( {\frac{d}{2},\frac{d}{2}+\frac{{{h_2}}}{{2\tan \alpha }}} \right]} \\ {n\gamma \left( {{h_2}+{h_3}} \right){\text{+}}2{\sigma _{B\hbox{max} }}+\left[ {{w_1} - 2{\sigma _{B\hbox{max} }}} \right]\frac{{\tan \alpha }}{{{h_2}}}\left( {{x_1} - \frac{d}{2}} \right){\text{ }} \quad \left( {\frac{d}{2}+\frac{{{h_2}}}{{2\tan \alpha }},\frac{d}{2}+\frac{{{h_2}}}{{\tan \alpha }}} \right]} \\ {\gamma \left( {{h_2}+{h_3}} \right){\text{ }} \quad \left( {\frac{d}{2}+\frac{{{h_2}}}{{\tan \alpha }},+\infty } \right)} \end{array}} \right.$$
(7)

Stress calculation of coal seam after drainage

The change of stress within coal seam is influenced by the drainage of rock formation. During the process stress transfer caused by the drainage, the total stress remains constant:

$$\int_{0}^{{\frac{{{h_2}}}{{\tan \alpha }}}} {\Delta {\sigma _{\text{d}}}} {{\text{d}}_{\text{x}}}=\int_{0}^{{\frac{{{h_2}}}{{\tan \alpha }}+\frac{{{h_1}}}{{\tan \beta }}}} {\Delta {\sigma _{\text{c}}}} {{\text{d}}_{\text{x}}}$$
(8)

In Eq. 8, Δσd represents the stress increment of rock strata, and Δσc the stress increment of coal seam.

The stress increment of coal seam follows Eq. 9:

$$\Delta {\sigma _c}=\left\{ {\begin{array}{*{20}{l}} {2{\text{k}}{\sigma _{B\hbox{max} }}{w_2}\left( {{x_1} - \frac{d}{2}} \right){\text{ }} \quad \left( {\frac{d}{2},\frac{{d+{w_2}}}{2}} \right]} \\ {2k{\sigma _{B\hbox{max} }}\left[ {1 - {w_2}\left( {{x_1} - \frac{d}{2}} \right)} \right]{\text{ }} \quad \left( {\frac{{d+{w_2}}}{2},\frac{d}{2}+{w_2}} \right]} \\ {0{\text{ }} \quad \left( {\frac{d}{2}+{w_2},+\infty } \right)} \end{array}} \right.$$
(9)

where β represents the stress transfer angle, H1 is the distance between the coal seam and the drainage roof, and d is the width of the drainage area.

The self-weight stress of coal seam is the sum of rock layer self-weight stress and interlayer rock weight, indicated by Eq. 10:

$${\sigma _c}=\left\{ {\begin{array}{*{20}{l}} {{\sigma _2}{\text{+}}\gamma {h_1}{\text{ }} \quad \left( {0,\frac{d}{2}} \right]} \\ {{\sigma _2}{\text{+}}{w_3}\left( {{x_1} - \frac{d}{2}} \right){\text{+}}\gamma {h_1}+2k{\sigma _{B\hbox{max} }}{w_2}\left( {{x_1} - \frac{d}{2}} \right){\text{ }} \quad \left( {\frac{d}{2},\frac{{d+{w_2}}}{2}} \right]} \\ {{\sigma _2}{\text{+}}{w_3}\left( {{x_1} - \frac{d}{2}} \right){\text{+}}\gamma {h_1}+2k{\sigma _{B\hbox{max} }}\left[ {1 - {w_2}\left( {{x_1} - \frac{d}{2}} \right)} \right]{\text{ }} \quad \left( {\frac{{d+{w_2}}}{2},\frac{d}{2}+{w_2}} \right]} \\ {{\sigma _1}{\text{+}}\gamma {h_1}{\text{ }} \quad \left( {\frac{d}{2}+{w_2},+\infty } \right)} \end{array}} \right.$$
(10)

The applicability of the geologic conditions

The theoretical model is based on the geological conditions of the Shaanxi-Inner Mongolia mining area in China and the theory of coal mining. However, only irregular water-rich areas are emphasized and the geological conditions are not strictly restricted for the calculation process. Therefore, it is applicable to the condition that the roof contains irregular water-rich areas of coal mining.

Example analysis of coal seam stress after drainage

The theoretical calculation is validated by the examination of the relationship between coal seam stress and drainage area in the mining operations of the 221upper18 face in Shilawusu Coal Mine.

The average mining depth of 221upper18 working face is 684 m, with an average thickness of 5.6 m and an average dip angle of 1°. The relative positions of the water-rich abnormal area and the stress sensors are shown in Fig. 3.

Fig. 3
figure 3

Relative positions of water-rich abnormal area and stress sensors.

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By taking the water-rich abnormal area in 221upper18 working face as an example, where d is 100 m, h1 is 37.8 m, h2 is 50 m, h3 is 596.2 m, α is 81°, β is 45°, γ is 25 kN, n is 0.8, and applying Eq. 10, the theoretical stress of coal seam is obtained as shown in Fig. 4.

Fig. 4
figure 4

Theoretical stress of coal seam.

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As shown in Fig. 4, the calculation shows that the high stress range of coal seam spans 46 m, the peak stress is 23.02 MPa and the baseline stress is 17.10 MPa. Additionally, the stress concentration factor is 1.35 m.

Throughout the mining operations of the 221upper18 working face, stress monitoring data from 10 groups of stress gauges installed in the water-rich abnormal area were collected. The relationship between the drainage area and the stress monitoring results is shown in Fig. 5.

Fig. 5
figure 5

Comparison of calculated stress results with measured stress.

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In Fig. 5, the red square dots represent the peak stress at the deep hole measurement points during mining. Blue dots represent peak stress at shallow hole measurement points during mining. The blue line represents the extent of the drainage area. The black dashed line depicts the theoretical stress distribution curve. It is evident that drainage in the water-rich area impacts the stress distribution of the coal seam. The theoretical results allign with the measured coal stress, validating the accuracy of the theoretical calculation.

Numerical simulation analysis of coal seam stress after drainage

Simulation results of stress in coal strata after drainage

To address the issue of drainage stress transfer in water-rich abnormal areas, stress cloud maps of the aquifer and coal seam after drainage are compared (Fig. 6).

The stress cloud map is divided into two sections: the upper part illustrates the stress distribution in the aquifer after drainage, while the lower part shows the stress distribution in coal seam after drainage. Stress monitoring lines are established for both coal seam and rock strata in the unbounded affected area.

Fig. 6
figure 6

Stress distribution of rock strata after drainage in water-rich abnormal area.

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In Fig. 6 it is observed that after drainage, the stress in the drainage area decreases, with a minimum stress of 12.32 MPa and an increased boundary stress reaching a maximum of 23.91 MPa. Upon transfer to the coal seam, the maximum stress values range between 21.00 MPa-22.00 MPa, while the minimum stress values increase, ranging from 13.00 MPa to 14.00 MPa. To note, the degree of stress concentration decreases. The stress of coal seam is compared with that of the rock in Fig. 7.

Fig. 7
figure 7

Stress comparison between stress monitoring line and corresponding coal seam stress monitoring line.

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In Fig. 7, the red line represents the stress in the strata while the black line the coal seam stress.

The stress distribution on both sides of the drainage area is largely uniform. The peak stress of the water-rich strata on either side of the drainage area measures 20.71 MPa and 20.56 MPa, with the corresponding stress offset values of 14.29 MPa and 14.28 MPa, respectively.

Similarly, the peak stresses of the coal seams on both sides of the drainage area are 20.07 MPa and 19.83 MPa, with the corresponding stress offset values of 15.45 MPa and 15.09 MPa, respectively. Following dewatering, the stress of water-rich strata increases by 20% and 19%, respectively. Conversely, after transfer to the coal seam, stress concentration diminishes, resulting in an approximate increase of 13% and 11% in coal seam stress, respectively.

Simulation results that the stress of coal seam after drainage is affected by the boundary of drainage area.

The shape of the water-rich abnormal area significantly affects the stress distribution after drainage. In order to analyze the influence of the shape of the water-rich abnormal area (i.e., the plane angle occupied by the water-rich abnormal area, such as the angle in Fig. 8, the angle of the drainage area is 90°, the angle of the drainage area is 90°) on the drainage of the water-rich abnormal area.

Fig. 8
figure 8

Angular dimension.

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To investigate the correlation between the angle of the water-rich abnormal zone and the stress values after drainage, the maximum and minimum values of each angle are determined. The relationship between the minimum value and the angle and, between the maximum value and the angle are shown in Figs. 9 and 10.

Fig. 9
figure 9

Statistics of angle and minimum stress.

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Fig. 10
figure 10

Statistics of angle and maximum stress.

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The minimum and maximum values of vertical stress at different angles are observed at 36°and 270°, respectively. Based on the linear trend observed in the minimum distribution, a correlation analysis is conducted using linear fitting. The fitting formula is given by Eq. 11, illustrated by the red solid line in Fig. 9. An R2 of 0.98234 indicates a good correlation.

$$y=0.01972x+9.92584$$
(11)

The linear distribution trend of the maximum value and its correlation is analyzed using linear fitting with Eq. 12 (Fig. 10).

$$y=0.02668x+15.70096$$
(12)

The coefficient of determination R2 of 0.98154 indicates a good fit.

Conclusion

The influence of drainage on coal seam stress was studied through theoretical analysis, numerical simulation, and on-site monitoring. The conclusions are following:

(1) This is research conclusions on stress transfer and downward transfer. Local stress within the coal seam increases after drainage. The theoretical stress within the coal seam measures 17.10 MPa.The maximum stress in the stress rising zone is 23.02Mpa, and the stress concentration factor is 1.35. The theoretically calculated stress influence range aligns with the measured results from the mining case of the working face.

(2) This is the influence of the boundary angle of the water-rich abnormal area. The boundary angle of the water-rich abnormal area significantly affects drainage stress transfer. The extreme value of drainage stress increases with the expansion of the boundary angle of the water-rich abnormal area. The extreme value and angle are linearly and positively correlated, With an R2 value exceeding 0.98.