Introduction

Mine water disasters are among the main factors affecting coal mine safety. With increasing mining depth, the threat of deep Ordovician floor water inrush is becoming increasingly serious in North China1,2,3. Chinese scholars have extensively researched the mechanisms and prevention of floor water inrush4,5. The nature of the rock, geological structures, and mining activities are generally considered the main factors leading to floor water inrush6,7. Water inrush from fault floors is commonly caused by geological structures and mining activities8. Faults provide water inrush channels for aquifers, and mining activities lead to floor failure, which connects the goaf with fault water inrush channels9. A common and effective method to prevent floor water inrush is to reserve waterproof coal (rock) pillars10. Therefore, determining the failure characteristics of the floor is crucial for preventing and controlling floor water inrush and ensuring the safety of coal mine production11.

Researchers have made significant progress in studying fault and floor water inrush, utilizing theoretical calculations12, numerical simulations13, and in-situ measurements14 to investigate floor stress distributions and failure patterns. Theoretical calculations study the distribution laws and failure states of rock strata under fault conditions by establishing mechanical models of coal seam mining. For example, Ma et al.15 proposed a theoretical mechanical model for the Dongjiahe Coal Mine that considers the overall stress and failure analysis of floor confined aquifers, and determines the floor failure morphology above the confined aquifers. Numerical simulation software can reflect stress and strain conditions and rock strata failure during coal mining, setting the physical and mechanical properties of faults and rock strata as needed, which is a powerful tool to address floor water inrush16,17,18. Meng et al.19 used Flac3D software to conduct fluid‒structure coupling numerical simulations on the Zhaolou Coal Mine to study the influence of the mining sequence on floor failure characteristics, showing that floor water inrush occurs when the failure depth exceeds the extension depth of the floor rock mass and when the hydrostatic seepage pressure exceeds the stress state of the fracturing member. Xin et al.20 used microseismic monitoring and numerical simulation to study the plastic failure zone of surrounding rock under different mining steps in the Zhaoku No. 1 Mine, Henan Province, showing that the plastic failure range of the fault zone expands as the distance between the working face and the fault decreases, revealing the activation characteristics of the fault. Ling et al.21 used Flac3D software to study floor failure characteristics under different fault dips, revealing that the maximum failure depth occurs in the rock layer below the coal wall, allowing pressurized water to rise along the fault and seep into the goaf through fissure zones.

However, comprehensively revealing floor failure characteristics and water inrush mechanisms during mining under complex fault conditions is still difficult. Research on floor water inrush under complex fault structures is limited because the stress distribution changes caused by mining in the context of complex fault structures are far more intricate than those in single fault conditions, requiring consideration of the effects of multiple faults on the rock layers below the floor. The existing research and prevention methods for fault floor water inrush are referenced. To investigate the floor failure characteristics and water inrush mechanism under coal seam mining conditions influenced by complex fault structures, theoretical analysis and numerical simulation were carried out. This study proposes an improved theoretical model, including complex fault structures such as horst, graben, and step fault structures. MATLAB is used to theoretically analyze the stress and failure of the coal floor. The calculation formula for waterproof coal pillars under complex fault conditions is established based on the improved theoretical model. Flac3D is adopted to simulate the coal mining process under complex fault conditions, and the failure characteristics and displacement characteristics of the floor are analyzed. Finally, the case of water inrush in the Liangzhuang Coal Mine is used to verify the theoretical model, which is applied to the analysis of water inrush in the 51,101 W working face, aiming to provide references for studying fault floor water inrush under complex fault conditions.

General conditions of the study area

We used ArcGIS 10.2 software (https://www.arcgis.com) to complete the drawing and editing of the map to explain the geographical conditions of the study area (Fig. 1). The Liangzhuang Coal Mine is located in Xintai County, Tai ‘an city, Shandong Province, China, between the two watersheds of Lianhua Mountain and Mengshan Mountain. The strata strike is 290° ~ 310°, inclination is 20° ~ 40°, and dip is 12° ~ 24° (Fig. 1). The coal-bearing strata belong to the North China-type coal-bearing strata, which are composed of the Benxi Formation, Taiyuan Formation, Shanxi Formation, and Shihezi Formation. The overlying strata are Tertiary and Quaternary. The direct water-filled aquifer of the Liangzhuang Coal Mine has sandstone from the Shanxi Formation and limestone from the first and fourth layers of the Taiyuan Formation. Indirect water-filled aquifers include Quaternary water-bearing gravel layers, Tertiary conglomerates, Benxi Formation Xujiazhuang, Caobugou limestone, and Ordovician limestone. The main aquicludes are Tertiary red clayey siltstone, various clay rocks of the Shihezi Formation, siltstone, and argillaceous rock between aquifers and coal seams in coal-bearing strata.

The Liangzhuang Minefield is dominated by fault structures; the structural form is the monoclinic structure. There is only a short axial syncline in the deep part. Most of these faults are high-angle normal faults, large and medium-sized faults are high-angle normal faults, and few reverse faults are present. The main fault combination form is the step fault structure.

The 51,101 W working face is on the west side of the − 580 level in the Liangzhuang Coal Mine’s fifth mining area. The working face is 870 m long in the strike direction and 180 ~ 190 m long in the dip direction. The floor elevation is in the range of -403 m~ -459 m, with the coal seam buried at a depth of approximately 584 ~ 640 m. The thickness of the coal seam in the working face is 1.5 m, and the average mining height is approximately 1.6 m (including stone). The coal seam dip angle is approximately 9° (Fig. 2a). Fault F8−2 and fault FX are located on the west and east sides of the working face, respectively. The drop of fault F8−2 is 15 m, the dip angle of fault F8−2 is 70°, the strike of fault F8−2 is the same as the mining direction of the working face, and the waterproof coal pillar is 30 m. The drop of the FX fault is 7 m, the dip angle of FX fault is 80°, and the strike of the FX fault is 40° with respect to the mining direction of the working face, as shown in Fig. 2b and c.

In June 2000, floor water inrush occurred in the 51,101 W working face of the Liangzhuang Coal Mine. The maximum water inflow was 768m3/h and was stable at 600m3/h, and the cumulative water inflow reached 6 × 106 m3. As shown in Fig. 2c, according to the water quality test data, the water inrush source is the Ordovician limestone aquifer, and the water inrush channel is the FX fault. The distance between water inrush and FX fault is 38 m. The projection distance of the FX fault perpendicular to the mining direction of the working face is 50 m22. According to the requirements of fault waterproof coal pillars in the Coal Mine Water Control Rules, the fault waterproof coal pillar of the Liangzhuang Coal Mine 51,101 W working face needs to be set to 20 m to ensure safety., but water inrush accidents still occur. Therefore, it is necessary to study the failure characteristics of coal mining floors under complex fault structure conditions, which is the key to ensuring the safe mining of coal mines under similar conditions. The coal seam is regarded as a horizontal coal seam for theoretical analysis, as shown in Fig. 2d. After simplification, the angle between fault F8−2 and the coal seam is 79°, and the angle between fault FX and the coal seam is 89°. The two fault structures of the 51,101 W working face can be considered step fault structures, and the working face is located in the middle of the step fault structure.

The water inrush event of the 51,101 W working face shows that the failure of the coal mine floor under complex fault conditions is different from the failure of the coal mine floor under single-fault conditions. The failure characteristics of a coal seam floor under complex fault conditions are the keys to understanding the water inrush mechanism during mining. Various complex fault structures, such as horst structures, graben structures, and step fault structures, are encountered during mining23. Therefore, it is necessary to study and analyze the failure characteristics of coal mine floors under complex fault conditions. Based on the geological conditions and fault structure of the 51,101 W working face, this paper performs theoretical and numerical simulation research on the floor failure of coal seam mining under complex fault conditions, such as the horst structure, graben structure, and step fault structure. The mining direction of the working face is the same as the strike of the fault.

Fig. 1
figure 1

Location and structural geology of Liangzhuang Coal Mine. (a) Location of Liangzhuang Coal Mine in China (The map created by ArcGIS 10.2 software (https://www.arcgis.com)). (b) The geographical ___location and topographic map of Liangzhuang Coal Mine in Shandong Province (The map created by ArcGIS 10.2 software (https://www.arcgis.com)). (c) Geological structure of Liangzhuang coal mine.

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Fig. 2
figure 2

Overview of 51,101 W working face and schematic diagram of fault ___location and water inrush. (a) The coal seam dip angle is approximately 9°. (b) The 51,101 W working face and FX fault relationship plan, the strike of the FX fault is 40° with respect to the mining direction of the working face. (c) The actual fault waterproof coal pillar of 51,101 W working face are 30 m and 50 m wide (d) The angle between fault F8−2 and the coal seam is 79 °, and the angle between fault FX and the coal seam is 89 °.

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Theoretical analysis of coal seam mining under complex fault structure conditions

Mechanical model of the coal floor

This paper considers the rock masses between two faults as the research objects. The floor rock masses are regarded as a plane strain problem in elasticity and are assumed to satisfy the linear elastic constitutive model. The underground pressure theory is combined with the distribution law of the lateral support pressure on both sides of the working face24,25. The floor forces perpendicular to the mining direction of the working face during mining are shown in Fig. 3. According to elastic mechanics, for a semi-infinite solid subject to a distributed force on the boundary (Fig. 4), the stress components at point M (x,y) are26:

$$\left\{ \begin{gathered} {\sigma _x}= - \frac{2}{\pi }\mathop \smallint \nolimits_{{ - b}}^{a} \frac{{q{x^3}{\text{d}}\xi }}{{{{\left[ {{x^2}+\left( {y - {\xi ^2}} \right)} \right]}^2}}} \hfill \\ {\sigma _y}= - \frac{2}{\pi }\mathop \smallint \nolimits_{{ - b}}^{a} \frac{{qx{{\left( {y - \xi } \right)}^2}{\text{d}}\xi }}{{{{\left[ {{x^2}+\left( {y - {\xi ^2}} \right)} \right]}^2}}} \hfill \\ {\tau _{xy}}= - \frac{2}{\pi }\mathop \smallint \nolimits_{{ - b}}^{a} \frac{{q{x^2}\left( {y - \xi } \right){\text{d}}\xi }}{{{{\left[ {{x^2}+\left( {y - {\xi ^2}} \right)} \right]}^2}}} \hfill \\ \end{gathered} \right.$$
(1)

where \({\sigma _x}\), \({\sigma _y}\) and \({\tau _{xy}}\) are the vertical stress, horizontal stress, and shear stress at any point M (x, y), respectively, which are caused by the point set q between point A (a,0) and point B (-b,0).

Take a tiny length \(d\xi\) at the point where the origin of the distance coordinate O is \(\xi\), and regard the force \(dF=qd\xi\) acting on it as a tiny concentrated force. When Eq. (1) is applied, the stress set q is regarded as a function of \(\xi\).

Fig. 3
figure 3

Schematic diagram of the coal mining load in the direction perpendicular to the mining direction of the working face. The floor forces perpendicular to the mining direction of the working face during mining.

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Fig. 4
figure 4

Schematic of a semi-infinite solid subjected to distributed forces on the boundary. According to elastic mechanics, for a semi-infinite solid subject to a distributed force on the boundary, the stress components at point M can be obtained.

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When mining a coal seam, the stress of the floor is composed of the support pressure and overlying self-weight stress. The support pressure on both sides is simplified as a symmetrically distributed triangular load, as shown in Fig. 5. The load of the floor in the goaf is regarded as a free surface to release the in-situ stress, which is considered to be a uniformly distributed rectangular load15. The overlying self-weight stress is calculated via Eq. (2)27.

$$\left\{ \begin{gathered} {\sigma _{Rx}}=\gamma (h+x) \hfill \\ {\sigma _{Ry}}=\lambda \gamma (h+x) \hfill \\ {\tau _{Rxy}}=0 \hfill \\ \end{gathered} \right.$$
(2)

where \({\sigma _{Rx}}\), \({\sigma _{Ry}}\) and \({\tau _{Rxy}}\) are floor stress components caused by the overlying rock masses, λ is the lateral pressure coefficient, γ is the average bulk density of the overlying rock and soil mass, h is the depth of the coal seam, and x is the vertical distance from point M to the working surface.

Fig. 5
figure 5

Mechanical model of supporting pressure in the direction perpendicular to the mining direction of the working face. When mining a coal seam, the stress of the floor is composed of the support pressure and overlying self-weight stress. The support pressure on both sides is simplified as a symmetrically distributed triangular load.

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Each load q can be expressed via Eq. (3):

$$\left. \begin{gathered} {q_1}= - \frac{{k\gamma h}}{{{y_1}}}(y - {y_2} - \frac{{{y_3}}}{2}) \hfill \\ {q_2}=\frac{{k\gamma h}}{{{y_2}}}(y - \frac{{{y_3}}}{2}) \hfill \\ {q_3}= - \gamma h \hfill \\ {q}^{\prime}_{2}= - \frac{{k\gamma h}}{{{y}^{\prime}_{2}}}(y - \frac{{{y_3}}}{2}) \hfill \\ {q}^{\prime}_{1}=\frac{{k\gamma h}}{{{y}^{\prime}_{1}}}(y - {y}^{\prime}_{2}- \frac{{{y_3}}}{2}) \hfill \\ \end{gathered} \right\}$$
(3)

where K is the support pressure concentration coefficient, generally (1 ~ 3)28,29,30; the y-axis is parallel to the floor and points horizontally to the left wall of the coal rock; and the x-axis is perpendicular to the floor and points downward to the floor. In these equations, \({q_1}\) and \({q}^{\prime}_{1}\) are support pressure loads in the elastic zone, \({y_1}\) and \({y}^{\prime}_{1}\) are the lengths of the elastic zone, \({q_2}\) and \({q}^{\prime}_{2}\) are the support pressure distribution loads in the plastic zone, \({y_2}\) and \({y}^{\prime}_{2}\) are the lengths of the plastic zone, \({q_3}\) is the stress distribution load of the floor in the goaf, and \({y_3}\) is the width of the working face in the goaf. For ease of calculation, the mechanical model is symmetrical.

The stress components caused by the rectangular load \({q_3}\) on the floor in the goaf are expressed via Eq. (4).

$$\left. \begin{gathered} {\sigma _{{q_{3x}}}}= - \frac{2}{\pi }\int_{{ - \frac{{{y_3}}}{2}}}^{{\frac{{{y_3}}}{2}}} {\frac{{q{x^3}d\xi }}{{{{\left[ {{x^2}+\left( {y - {\xi ^2}} \right)} \right]}^2}}}} = - \frac{q}{\pi }\left[ {\arctan \frac{{y+\frac{{{y_3}}}{2}}}{x} - \arctan \frac{{y - \frac{{{y_3}}}{2}}}{x}+\frac{{x(y+\frac{{{y_3}}}{2})}}{{{x^2}+{{(y+b)}^2}}} - \frac{{x(y - \frac{{{y_3}}}{2})}}{{{x^2}+{{(y - \frac{{{y_3}}}{2})}^2}}}} \right] \hfill \\ {\sigma _{{q_{3y}}}}= - \frac{2}{\pi }\int_{{ - \frac{{{y_3}}}{2}}}^{{\frac{{{y_3}}}{2}}} {\frac{{qx{{\left( {y - \xi } \right)}^2}d\xi }}{{{{\left[ {{x^2}+\left( {y - {\xi ^2}} \right)} \right]}^2}}}= - \frac{q}{\pi }\left[ {\arctan \frac{{y+\frac{{{y_3}}}{2}}}{x} - \arctan \frac{{y - \frac{{{y_3}}}{2}}}{x} - \frac{{x(y+\frac{{{y_3}}}{2})}}{{{x^2}+{{(y+\frac{{{y_3}}}{2})}^2}}}+\frac{{x(y - \frac{{{y_3}}}{2})}}{{{x^2}+{{(y - \frac{{{y_3}}}{2})}^2}}}} \right]} \hfill \\ {\tau _{{q_{3xy}}}}= - \frac{2}{\pi }\int_{{ - \frac{{{y_3}}}{2}}}^{{\frac{{{y_3}}}{2}}} {\frac{{q{x^2}\left( {y - \xi } \right)d\xi }}{{{{\left[ {{x^2}+\left( {y - {\xi ^2}} \right)} \right]}^2}}}= - \frac{q}{\pi }\left[ {\frac{{{x^2}}}{{{x^2}+{{(y+\frac{{{y_3}}}{2})}^2}}} - \frac{{{x^2}}}{{{x^2}+{{(y - \frac{{{y_3}}}{2})}^2}}}} \right]} \hfill \\ \end{gathered} \right\}$$
(4)

where \({q_2}=\frac{{K\gamma h}}{{{y_2}}}\left( {y - {y_3}} \right)=\frac{{K\gamma h}}{{{y_2}}}y - \frac{{K\gamma h}}{{{y_2}}}{y_3}\) makes \(k=\frac{{K\gamma h}}{{{y_2}}}\) and \(c= - \frac{{K\gamma h}}{{{y_2}}}{y_3}\), then \({q_2}=ky+c\) makes \(u=\left( {{y_2}+{y_3}} \right)\) and \(v={y_3}\).

By substituting Eq. (3) into Eq. (1), the floor stress components caused by the support pressure and the floor pressure (Eq. (4), Eq. (5)) can beobtained. Using Eq. (4) and Eq. (5), the mechanical model of support pressure in the coal seam is established as Eq. (5) based on the principle of stress superposition.

$$\left. \begin{gathered} {\sigma _x}={\sigma _{{q_{1x}}}}+{\sigma _{{q_{2x}}}}+{\sigma _{{q_{3x}}}}+{\sigma _{{{q}^{\prime}_{1x}}}}+{\sigma _{{q}^{\prime}_{2x}}} \hfill \\ {\sigma _y}={\sigma _{{q_{1y}}}}+{\sigma _{{q_{2y}}}}+{\sigma _{{q_{3y}}}}+{\sigma _{{q}^{\prime}_{1y}}}+{\sigma _{{q}^{\prime}_{2y}}} \hfill \\ {\tau _{xy}}={\tau _{{q_{1xy}}}}+{\tau _{{q_{2xy}}}}+{\tau _{{q_{3xy}}}}+{\tau _{{{q}^{\prime}_{1xy}}}}+{\tau _{{q}^{\prime}_{2{xy}}}} \hfill \\ \end{gathered} \right\}$$
(5)

The mechanical model of the floor in coal seam mining is obtained by substituting Eq. (2) into Eq. (5), expressed as Eq. (6) :

$$\left. \begin{gathered} {\sigma _x}={\sigma _{{q_{1x}}}}+{\sigma _{{q_{2x}}}}+{\sigma _{{q_{3x}}}}+{\sigma _{{q}^{\prime}_{1x}}}+{\sigma _{{q}^{\prime}_{2x}}}+{\sigma _{Rx}} \hfill \\ {\sigma _y}={\sigma _{{q_{1y}}}}+{\sigma _{{q_{2y}}}}+{\sigma _{{q_{3y}}}}+{\sigma _{{q}^{\prime}_{1y}}}+{\sigma _{{q}^{\prime}_{2y}}}+{\sigma _{Ry}} \hfill \\ {\tau _{xy}}={\tau _{{{q}^{\prime}_{1xy}}}}+{\tau _{{q_{2xy}}}}+{\tau _{{q_{3xy}}}}+{\tau _{{q}^{\prime}_{1{xy}}}}+{\tau _{{{q}^{\prime}_{2xy}}}}+{\tau _{Rxy}} \hfill \\ \end{gathered} \right\}$$
(6)

Floor stress components caused by complex fault structures

Because graben, horst and step faults have different tectonic fault tendencies and strata movement directions on both sides of faults, the magnitudes and directions of the forces are different. Therefore, it is necessary to analyze the stresses in horst, graben and step fault structures, respectively. To analyze the floor stress components caused by complex fault structures, the rock masses below the floor are taken as the research objects. Rock masses with a height of ∆x are selected for force analysis to obtain the horizontal stress components21,31. The vertical and shear stress components are derived with reference to the supporting beam under uniform load32.

Components of floor stress caused by the horst structure

The horst structure exhibits a grooved fault block configuration. The middle fault block B rises relatively, bounded by the falling fault blocks A and C on both sides with opposite tendencies (Fig. 6a). The rock masses below the floor of the working face of intermediate fault block B are taken as the mechanical analysis objects to study the floor stress components caused by the horst structure. The force of the rock masses below the floor is composed of the support pressure and overlying self-weight stress, the positive pressure exerted by blocks A and C, and the friction between the two faults. These combined forces determine the horizontal stress components of the floor caused by the horst structure, as shown in Fig. 6b. The vertical and shear stress components of the floor caused by the horst structure are analyzed, as shown in Fig. 6c.

In accordance with underground pressure theory, Saint Venant’s principle in elastic theory (Eq. (7)) and equilibrium conditions \(\sum {{F_{\text{x}}}=0}\) and \(\sum {{F_{\text{y}}}} =0\) are used to obtain Eq. (8) as follows:

Fig. 6
figure 6

Schematic diagram of the components of floor stress caused by the horst structure. (a) The horst structure exhibits a grooved fault block configuration. The middle fault block B rises relatively, bounded by the falling fault blocks A and C on both sides with opposite tendencies. (b) The force of the rock masses below the floor is composed of the support pressure and overlying self-weight stress, the positive pressure exerted by block blocks A and block C, and the friction between the two faults. These combined forces determine the horizontal stress components of the floor caused by the horst structure. (c) The vertical and shear stress components of the floor caused by the horst structure are analyzed.

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$$\left\{ \begin{gathered} {F_T}_{{_{1}}}=\int_{0}^{{\Delta x}} {\frac{{{\sigma _{yT}}_{{_{1}}}}}{{\sin \theta }}dx} \hfill \\ {F_{{T_2}}}=\int_{0}^{{\Delta x}} {\frac{{{\sigma _{y{T_2}}}}}{{\sin \alpha }}dx} \hfill \\ {F_s}=\int_{0}^{{\Delta x}} {{\tau _{xy}}dx} \hfill \\ \end{gathered} \right.$$
(7)
$$\left\{ \begin{gathered} {q_{Fr}}{l_1} - {q_{Fr}}{l_2}+{F_{{T_1}}}\cos \theta +{F_{{T_2}}}\cos \alpha +{F_{{f_1}}}\sin \theta +{F_{{f_2}}}\sin \alpha \hfill \\ = - \Delta Q+{F_{{T_1}}}\cos \theta +{F_{{T_2}}}\cos \alpha +{F_{{f_1}}}\sin \theta +{F_{{f_2}}}\sin \alpha =0 \hfill \\ {F_{{T_1}}}\sin \theta +{F_{{f_2}}}\cos \alpha ={F_{{T_2}}}\sin \alpha +{F_{{f_1}}}\cos \theta \hfill \\ {F_f}=\mu {F_T} \hfill \\ \Delta Q={q_{Fr}}\Delta l=\frac{{\tan \alpha +\tan \theta }}{{(\tan \alpha \times \tan \theta )}}\Delta x \hfill \\ \theta <\alpha \hfill \\ \end{gathered} \right.$$
(8)

where qFr is the uniform load of floor rock masses during mining, N/m; FTi is the supporting force applied by blocks A and C, N; Ffi is the friction force between blocks A and B, C and B determined by FTi, N; FS is the shear force of the supporting beam, N; µ is the friction factor; α and θ are fault dip angles; L1 and L2 are the lengths of q, m; and ∆x is the unit height, m.

Consider blocks A and C to be infinite and without boundary displacement. The friction between the rock masses, the dead weight of the rock masses on both sides and the influence of the bending moment of the rock masses are ignored.

Equation (7) is substituted into Eq. (8) to obtain the floor stress components of the horst structure (Eq. (9)).

$$\left\{ \begin{gathered} {\sigma _{{H_x}}}=\frac{{\Delta Q}}{{{L_1}}} \hfill \\ {\sigma _{{H_y}}}={\sigma _{yT}}_{i} - {\sigma _{y{f_i}}}=\Delta Q\left( {\frac{{\left( {\sin \theta - \mu \cos \theta } \right)\left( {\sin \alpha - \mu \cos \alpha } \right)}}{{\left( {\left( {1 - {\mu ^2}} \right)\sin (\theta +\alpha ) - 2\mu \cos (\theta +\alpha )} \right)}}} \right) \hfill \\ {\tau _{{H_x}_{y}}}=\frac{{\Delta Q}}{2} - \frac{{\Delta Q}}{{{L_1}}}y \hfill \\ \end{gathered} \right.$$
(9)

where \({\sigma _{{H_x}}}\), \({\sigma _{{H_y}}}\) and \({\tau _{{H_x}_{y}}}\) are the floor stress components caused by the horst structure.

Components of floor stress caused by the graben structure

The graben structure exhibits a grooved fault block configuration. Middle block B is relatively small and is bounded by the relatively rising fault blocks A and C on both sides with opposite tendencies (Fig. 7a). The rock masses below the floor of the working face of intermediate fault block B are taken as the mechanical analysis objects to study the floor stress components caused by the graben structure. The force of the rock masses below the floor is composed of the support pressure and overlying self-weight stress, the positive pressure exerted by blocks A and C, and the friction between the two faults. These combined forces determine the horizontal stress components of the floor caused by the graben structure, as shown in Fig. 7b. The vertical and shear stress components of the floor caused by the graben structure are analyzed, as shown in Fig. 7c.

In accordance with underground pressure theory, Saint Venant’s principle in elastic theory (Eq. (10)) and equilibrium conditions \(\sum {{F_{\text{x}}}=0}\) and \(\sum {{F_{\text{y}}}} =0\) are used to obtain Eq. (11) as follows:

Fig. 7
figure 7

Schematic diagram of the components of floor stress caused by the graben structure. (a) The graben structure exhibits a grooved fault block configuration. Middle block B is relatively small and is bounded by the relatively rising fault blocks A and C on both sides with opposite tendencies. (b) The force of the rock masses below the floor is composed of the support pressure and overlying self-weight stress, the positive pressure exerted by blocks A and C, and the friction between the two faults. These combined forces determine the horizontal stress components of the floor caused by the graben structure. (c) The vertical and shear stress components of the floor caused by the graben structure are analyzed.

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$$\left\{ \begin{gathered} {F_T}_{{_{1}}}=\int_{0}^{{\Delta x}} {\frac{{{\sigma _{yT}}_{{_{1}}}}}{{\sin \theta }}dx} \hfill \\ {F_{{T_2}}}=\int_{0}^{{\Delta x}} {\frac{{{\sigma _{y{T_2}}}}}{{\sin \alpha }}dx} \hfill \\ {F_s}=\int_{0}^{{\Delta x}} {{\tau _{xy}}dx} \hfill \\ \end{gathered} \right.$$
(10)
$$\left\{ \begin{gathered} q{l_1} - q{l_2} - {F_{{T_1}}}\cos \theta - {F_{{T_2}}}\cos \alpha - {F_{{f_1}}}\sin \theta - {F_{{f_2}}}\sin \alpha \hfill \\ =\Delta Q - {F_{{T_1}}}\cos \theta - {F_{{T_2}}}\cos \alpha - {F_{{f_1}}}\sin \theta - {F_{{f_2}}}\sin \alpha =0 \hfill \\ {F_{{f_2}}}\cos \alpha +{F_{{T_1}}}\sin \theta ={F_{{T_2}}}\sin \alpha +{F_{{f_1}}}\cos \theta \hfill \\ {F_f}=\mu {F_T} \hfill \\ \Delta Q=q\Delta l=\frac{{\tan \alpha +\tan \theta }}{{(\tan \alpha \times \tan \theta )}}\Delta x \hfill \\ \theta <\alpha \hfill \\ \end{gathered} \right.$$
(11)

Equation (10) is substituted into Eq. (11) to obtain the floor stress components of the graben structure (Eq. (12)):

$$\left\{ \begin{gathered} {\sigma _{{G_x}}}= - \frac{{\Delta Q}}{{{L_2}}} \hfill \\ {\sigma _{{G_y}}}={\sigma _{yT}}_{i} - {\sigma _{y{f_i}}}=\Delta Q\left( {\frac{{\left( {\sin \theta - \mu \cos \theta } \right)\left( {\sin \alpha - \mu \cos \alpha } \right)}}{{\left( {\left( {1 - {\mu ^2}} \right)\sin (\theta +\alpha ) - 2\mu \cos (\theta +\alpha )} \right)}}} \right) \hfill \\ {\tau _{{G_x}_{y}}}= - \left( {\frac{{\Delta Q}}{2} - \frac{{\Delta Q}}{{{L_2}}}y} \right) \hfill \\ \end{gathered} \right.$$
(12)

where \({\sigma _{{G_x}}}\), \({\sigma _{{G_y}}}\) and \({\tau _{{G_x}_{y}}}\) are the floor stress components caused by the graben structure.

Components of floor stress caused by the step fault structure

The step fault structure exhibits a grooved fault block configuration. The same inclined fault bounds middle block B and blocks A and C on both sides (Fig. 8a). The rock masses below the floor of the working face of intermediate fault block B are taken as the mechanical analysis object to study the floor stress components caused by the step fault structure. The force of the rock mass below the floor is composed of the support pressure and overlying self-weight stress, the positive pressure exerted by blocks A and C, and the friction between the two faults. These combined forces determine the horizontal stress components of the floor caused by the step fault structure, as shown in Fig. 8b. The vertical and shear stress components of the floor caused by the step fault structure are analyzed, as shown in Fig. 8c.

In accordance with underground pressure theory, Saint Venant’s principle in elastic theory (Eq. (13)), and equilibrium conditions \(\sum {{F_{\text{x}}}=0}\) and \(\sum {{F_{\text{y}}}} =0\) are used to obtain Eq. (14) as follows:

Fig. 8
figure 8

Schematic diagram of the components of floor stress caused by the step fault structure. (a) The step fault structure exhibits a grooved fault block configuration. The same inclined fault bounds middle block B and blocks A and C on both sides. (b) The force of the rock mass below the floor is composed of the support pressure and overlying self-weight stress, the positive pressure exerted by blocks A and C, and the friction between the two faults. These combined forces determine the horizontal stress components of the floor caused by the step fault structure. (c) The vertical and shear stress components of the floor caused by the step fault structure are analyzed.

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$$\left\{ \begin{gathered} {F_T}_{{_{1}}}=\int_{0}^{{\Delta x}} {\frac{{{\sigma _{yT}}_{{_{1}}}}}{{\sin \theta }}dx} \hfill \\ {F_{{T_2}}}=\int_{0}^{{\Delta x}} {\frac{{{\sigma _{y{T_2}}}}}{{\sin \alpha }}dx} \hfill \\ {F_s}=\int_{0}^{{\Delta x}} {{\tau _{xy}}dx} \hfill \\ \end{gathered} \right.$$
(13)
$$\left\{ \begin{gathered} {q_{Fr}}{l_1} - {q_{Fr}}{l_2}+{F_{{T_1}}}\cos \theta - {F_{{T_2}}}\cos \alpha - {F_{{f_1}}}\sin \theta +{F_{{f_2}}}\sin \alpha \hfill \\ = - \Delta Q+{F_{{T_1}}}\cos \theta - {F_{{T_2}}}\cos \alpha - {F_{{f_1}}}\sin \theta +{F_{{f_2}}}\sin \alpha =0 \hfill \\ {F_{{T_1}}}\sin \theta +{F_{{f_1}}}\cos \theta ={F_{{T_2}}}\sin \alpha +{F_{{f_2}}}\cos \alpha \hfill \\ {F_f}=\mu {F_T} \hfill \\ \Delta Q=q\Delta l=\frac{{\tan \alpha - \tan \theta }}{{(\tan \alpha \times \tan \theta )}}\Delta x \hfill \\ \theta <\alpha \hfill \\ \end{gathered} \right.$$
(14)

Equation (13) is substituted into Eq. (14) to obtain the floor stress components of the step fault structure (Eq. (14)).

$$\left\{ \begin{gathered} {\sigma _{{s_x}}}=\frac{{\Delta Q}}{{{L_1}}} \hfill \\ {\sigma _{{s_y}}}={\sigma _{yT}}_{i}+{\sigma _{y{f_i}}}=\Delta Q\left( {\frac{{\left( {\sin \theta +\mu \cos \theta } \right)\left( {\sin \alpha +\mu \cos \alpha } \right)}}{{\left( {(1 - {\mu ^2})\sin (\alpha - \theta )} \right)}}} \right) \hfill \\ {\tau _{{s_x}_{y}}}=\frac{{\Delta Q}}{2} - \frac{{\Delta Q}}{{{L_1}}}y \hfill \\ \end{gathered} \right.$$
(15)

where \({\sigma _{{S_x}}}\), \({\sigma _{{S_y}}}\) and \({\tau _{{S_x}_{y}}}\) are the floor stress components caused by the step fault structure.

Model of a coal floor under complex fault conditions

The total stress of the floor is composed of the floor stress components caused by the mining and the floor stress components caused by the complex fault structure. The mechanical model of the coal floor is used to derive the floor mechanics model under such fault structure conditions, in which the stress components caused by complex fault structures are introduced. These mechanical models provide the horizontal, vertical, and shear stresses at any point M (x, y) below the floor of the working face during mining under complex fault conditions.

By substituting Eq. (9) into Eq. (6) based on the principle of stress superposition, the mechanical model of the coal floor under the horst structure condition (Eq. (16)) can be obtained.

$$\left. \begin{gathered} {\sigma _x}={\sigma _{{q_{1x}}}}+{\sigma _{{q_{2x}}}}+{\sigma _{{q_{3x}}}}+{\sigma _{{q}^{\prime}_{1x}}}+{\sigma _{{q}^{\prime}_{2x}}}+{\sigma _{Rx}}+{\sigma _{{H_x}}} \hfill \\ {\sigma _y}={\sigma _{{q_{1y}}}}+{\sigma _{{q_{2y}}}}+{\sigma _{{q_{3y}}}}+{\sigma _{{q}^{\prime}_{1y}}}+{\sigma _{{q}^{\prime}_{2y}}}+{\sigma _{Ry}}+{\sigma _{{H_y}}} \hfill \\ {\tau _{xy}}={\tau _{{q_{1xy}}}}+{\tau _{{q_{2xy}}}}+{\tau _{{q_{3xy}}}}+{\tau _{{{q}^{\prime}_{1xy}}}}+{\tau _{{{q}^{\prime}_{2xy}}}}+{\tau _{Rxy}}+{\tau _H}_{{_{{xy}}}} \hfill \\ \end{gathered} \right\}$$
(16)

By substituting Eq. (12) into Eq. (6) based on the principle of stress superposition, the mechanical model of the coal floor under the graben structure condition (Eq. (17)) can be obtained.

$$\left. \begin{gathered} {\sigma _x}={\sigma _{{q_{1x}}}}+{\sigma _{{q_{2x}}}}+{\sigma _{{q_{3x}}}}+{\sigma _{{q}^{\prime}_{1x}}}+{\sigma _{{q}^{\prime}_{2x}}}+{\sigma _{Rx}}+{\sigma _{{G_x}}} \hfill \\ {\sigma _y}={\sigma _{{q_{1y}}}}+{\sigma _{{q_{2y}}}}+{\sigma _{{q_{3y}}}}+{\sigma _{{q}^{\prime}_{1y}}}+{\sigma _{{q}^{\prime}_{2y}}}+{\sigma _{Ry}}+{\sigma _{{G_y}}} \hfill \\ {\tau _{xy}}={\tau _{{q_{1xy}}}}+{\tau _{{{q}^{\prime}_{2xy}}}}+{\tau _{{q_{3xy}}}}+{\tau _{{q}^{\prime}_{1{xy}}}}+{\tau _{{q}^{\prime}_{{2xy}}}}+{\tau _{Rxy}}+{\tau _{{G_{xy}}}} \hfill \\ \end{gathered} \right\}$$
(17)

By substituting Eq. (15) into Eq. (6) based on the principle of stress superposition, the mechanical model of the coal floor under the step fault structure condition (Eq. (18)) can be obtained.

$$\left. \begin{gathered} {\sigma _x}={\sigma _{{q_{1x}}}}+{\sigma _{{q_{2x}}}}+{\sigma _{{q_{3x}}}}+{\sigma _{{q}^{\prime}_{1x}}}+{\sigma _{{q}^{\prime}_{2x}}}+{\sigma _{Rx}}+{\sigma _{{s_x}}} \hfill \\ {\sigma _y}={\sigma _{{q_{1y}}}}+{\sigma _{{q_{2y}}}}+{\sigma _{{q_{3y}}}}+{\sigma _{{q}^{\prime}_{1y}}}+{\sigma _{{q}^{\prime}_{2y}}}+{\sigma _{Ry}}+{\sigma _{{S_y}}} \hfill \\ {\tau _{xy}}={\tau _{{q_{1xy}}}}+{\tau _{{q_{2xy}}}}+{\tau _{{q_{3xy}}}}+{\tau _{{q}^{\prime}_{1{xy}}}}+{\tau _{{{q}^{\prime}_{2xy}}}}+{\tau _{Rxy}}+{\tau _{{s_x}_{y}}} \hfill \\ \end{gathered} \right\}$$
(18)

Distribution law of floor stress under complex fault conditions

The mechanical models of the coal floor under complex fault conditions are used to analyze the stress distribution during mining. Based on geological data and relevant information from the 51,101 W working face in the Liangzhuang Coal Mine, the parameters are determined for the floor stress components. The depth of the design coal seam is 610 m, the width of the working face is 166 m, the length of the elastic zone is 30 m, the length of the plastic zone is 3.2 m, the average bulk density of the overlying rock and soil is 26 kN/m3, the fault is 50 m from the working face, and the pressure is positive. Take the stress concentration coefficient K is 1.5, and the side pressure coefficient λ is 0.3. The left fault dip \(\theta\) is 79°, which is named the small-dip fault. The right fault \(\alpha\) is 89°, which is referred to as the large-dip fault. The stress curves at various depths below the floor (Fig. 9) and contour plots of the floor stress (Figs. 10, 11, 12 and 13) are obtained viaMATLAB.

As shown in Figs. 9a and 10, the vertical, horizontal, and shear stresses of the floor decrease with increasing depth under normal mining conditions in the vertical direction. As the distance from the working face increases, tensile stress caused by the goaf load transitions into compressive stress because of the support pressure in the horizontal direction. The vertical stress in the coal seam near the floor on both sides is greater due to the support pressure, which diffuses outward and downward, forming a “pressure bulb” with a downward stress direction. The vertical stress direction in the goaf is upward due to mining. Horizontal stress produces compressive stress near the floor on both sides and tensile stress near the goaf. Under the combined action of floor pressure and goaf pressure, the magnitude and influence range of compressive stress are significant, whereas the influence range of tensile stress is limited. Owing to the symmetrical distribution of floor pressure on both sides, two shear stress bulbs of equal size and opposite directions form in the pressure zones on either side and spread outward. The force direction in the goaf is opposite to the pressure zones on both sides, creating two shear stress bulbs of equal size and opposite directions in the goaf area that spread inward under the combined effect of pressure from both sides and the goaf.

Fig. 9
figure 9

Stress diagram under normal mining and complex fault structure conditions. (a) The vertical, horizontal, and shear stresses of the floor decrease with increasing depth under normal mining conditions in the vertical direction. (b) Under the horst structure condition, the vertical, horizontal, and shear stresses of the floor decrease with increasing depth in the vertical direction. (c) Under the graben structure condition, the vertical, horizontal, and shear stresses of the floor decrease with increasing depth in the vertical direction. (d) Under the step fault structure condition, the vertical, horizontal, and shear stresses of the floor decrease with increasing depth in the vertical direction.

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Fig. 10
figure 10

Stress distribution of floor under normal mining conditions. As the distance from the working face increases, tensile stress caused by the goaf load transitions into compressive stress because of the support pressure in the horizontal direction. The vertical stress in the coal seam near the floor on both sides is greater due to the support pressure, which diffuses outward and downward, forming a “pressure bulb” with a downward stress direction. Horizontal stress produces compressive stress near the floor on both sides and tensile stress near the goaf. Two shear stress bulbs of equal size and opposite directions form in the pressure zones on either side and spread outward. The force direction in the goaf is opposite to the pressure zones on both sides, creating two shear stress bulbs of equal size and opposite directions in the goaf area that spread inward under the combined effect of pressure from both sides and the goaf.

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As shown in Figs. 9b and 11, under the horst structure condition, the vertical, horizontal, and shear stresses of the floor decrease with increasing depth in the vertical direction. The tensile stress caused by the goaf load transitions into compressive stress because of the support pressure in the horizontal direction.

Compared with that under normal mining conditions, the change in vertical stress under the horst structure condition is not obvious in the vertical direction. In the horizontal direction, the horizontal stress increases by approximately 20%, the tensile stress area of the goaf decreases, and the pressure ball caused by the floor pressure presents a symmetrical distribution. The shear stress on the small-dip fault side decreases, whereas the shear stress on the cell side of the large-dip fault increases under the horst structure condition. Therefore, the horst structure affects mainly the horizontal stress and shear stress.

Fig. 11
figure 11

Stress distribution of floor under the horst structure condition. Compared with that under normal mining conditions, in the horizontal direction, the horizontal stress increases by approximately 20%, the tensile stress area of the goaf decreases, and the pressure ball caused by the floor pressure presents a symmetrical distribution. The shear stress on the small-dip fault side decreases, whereas the shear stress on the cell side of the large-dip fault increases under the horst structure condition.

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As shown in Figs. 9c and 12, under the graben structure condition, the vertical, horizontal, and shear stresses of the floor decrease with increasing depth in the vertical direction. The tensile stress caused by the goaf load transitions into compressive stress because of the support pressure in the horizontal direction.

Compared with that under normal mining conditions, the change in vertical stress under the graben structure condition is not obvious in the vertical direction. In the horizontal direction, the horizontal stress increases by approximately 20%, the tensile stress area of the goaf decreases, and the pressure ball caused by the floor pressure presents a symmetrical distribution. The shear stress on the side of the small-dip fault increases under the graben structure, whereas the shear stress on the side of the large-dip fault decreases. Therefore, the graben structure mainly affects the horizontal stress and shear stress of the floor.

Fig. 12
figure 12

Stress distribution of floor under the graben structure condition. Compared with that under normal mining conditions, in the horizontal direction, the horizontal stress increases by approximately 20%, the tensile stress area of the goaf decreases, and the pressure ball caused by the floor pressure presents a symmetrical distribution. The shear stress on the side of the small-dip fault increases under the graben structure, whereas the shear stress on the side of the large-dip fault decreases.

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As shown in Figs. 9d and 13, under the step fault structure condition, the vertical, horizontal, and shear stresses of the floor decrease with increasing depth in the vertical direction. The tensile stress caused by the goaf load transitions into compressive stress because of the support pressure in the horizontal direction.

Compared with that under normal mining conditions, the change in vertical stress under the horst structure condition is not obvious in the vertical direction. In the horizontal direction, the horizontal stress is significantly greater than that in normal mining conditions, and the pressure ball caused by floor pressure is symmetrically distributed. Compared with normal mining conditions, the floor shear stress of the step fault structure decreases. The shear stress on the small-dip fault side is lower than that on the side of the large-dip fault under the step fault structure condition. Therefore, the step fault structure affects mainly the horizontal stress and shear stress of the floor.

Fig. 13
figure 13

Stress distribution of floor under the step fault structure condition. Compared with that under normal mining conditions, in the horizontal direction, the horizontal stress is significantly greater than that in normal mining conditions, and the pressure ball caused by floor pressure is symmetrically distributed. The floor shear stress of the step fault structure decreases. The shear stress on the small-dip fault side is lower than that on the side of the large-dip fault under the step fault structure condition.

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In summary, compared with normal mining conditions, horizontal stress increases under the horst structure condition, and the shear stress on the side of the large-dip fault increases. The horizontal stress of the floor increases, and the shear stress of the side of the small-dip fault increases under the graben structure condition. Under the step fault structure condition, the horizontal stress of the floor increases significantly, and the shear stress decreases.

Failure characteristics of the mining floor under complex fault conditions

In Section of distribution law of floor stress under complex fault conditions, there are tensile, compressive, and shear stresses in the rock masses below the floor. The modified Mohr‒Coulomb failure criterion for compressive and tensile failure is used to reflect the failure of the rock under tensile, compressive, and shear stresses, as outlined in Eq. (19).

$$\left\{ \begin{gathered} {\sigma _{1,3}}=\frac{{{\sigma _x}+{\sigma _y}}}{2}+\sqrt {{{\left( {\frac{{{\sigma _x}+{\sigma _y}}}{2}} \right)}^2}+{\tau _{xy}}^{2}} \hfill \\ {\tau _{\hbox{max} }}=\frac{{{\sigma _1} - {\sigma _3}}}{2}=\frac{{\left( {{\sigma _1}+{\sigma _3}} \right)\sin \varphi }}{2}+C\cos \varphi \hfill \\ {\sigma _1}=\sigma {}_{c} \hfill \\ {\sigma _3}={\sigma _t} \hfill \\ \end{gathered} \right.$$
(19)

where \({\sigma _{1,3}}\) are the maximum and minimum principal stresses on the floor; \({\tau _{\hbox{max} }}\) is the maximum shear stress on the floor; \(\varphi\) is the angle of internal friction; C is the cohesion; \(\sigma {}_{c}\) is the compressive strength; and \({\sigma _t}\) is the tensile strength.

There are three types of floor failure resulting from mining: tension failure in the middle of the floor, compression failure on both sides, and external shear failure. The minimum principal stress in the middle of the floor is tensile stress, which exceeds the tensile strength of the rock stratum, leading to tension failure. The maximum principal stress is compressive stress, and when it exceeds the compressive strength of the rock mass, compression failure occurs on both sides of the floor. The support pressure in the goaf floor is directed opposite to the pressure on the coal walls on both sides, resulting in shear failure beneath and on both sides of the goaf. Under normal mining conditions, the floor failure depth is 20 m and the floor failure width is 23 m. (Figures 14a and 15a).

As shown in Fig. 14b, under the horst structure condition, the maximum principal stress is compressive stress. And the minimum principal stress is tensile stress under the goaf and the minimum principal stress is compressive stress on both sides of the working face. As the distance from the working face increases, the maximum principal stress increases first and then decreases, in the horizontal direction. The peak value of the maximum principal stress is located at 6 m away from the working face, on the side of the large-dip. As the distance from the working face increases, the minimum principal stress increases first and then decreases. The peak value of the minimum principal stress is located at 15 m away from the working face, on the side of the large-dip, in the horizontal direction. As the distance from the working face increases, the maximum shear stress increases first and then decreases. The peak value of the maximum shear stress is located at 3 m away from the working face, on the side of the large-dip, in the horizontal direction. The maximum and minimum principal stresses and shear stress of the floor decrease with increasing floor depth in the vertical direction. Compared with the normal mining condition, the maximum and minimum principal and the maximum shear stresses on the side of the large-dip fault increase.

The floor failure depth under horst structure mining is 24 m, which is 20% greater than under normal mining conditions. The failure width on both sides is 24 m, which is 4.3% greater than under normal mining conditions. The area of compressive failure increases, and the appearance of a shear failure region on the large-dip fault side, as shown in Fig. 15b.

As shown in Fig. 14c, under the graben structure condition, the maximum principal stress is compressive stress. And the minimum principal stress is tensile stress under the goaf and the minimum principal stress is compressive stress on both sides of the working face. As the distance from the working face increases, the maximum principal stress increases first and then decreases, in the horizontal direction. The peak value of the maximum principal stress is located at 6 m away from the working face, on the side of the small-dip. As the distance from the working face increases, the minimum principal stress increases first and then decreases, and there is a 3 m wide tensile stress zone on the side of the small-dip. The peak value of the minimum principal stress is located at 14 m away from the working face, on the side of the small-dip, in the horizontal direction. As the distance from the working face increases, the maximum shear stress increases first and then decreases. The peak value of the maximum shear stress is located at 3 m away from the working face, on the side of the small-dip, in the horizontal direction. The maximum and minimum principal stresses and shear stress of the floor decrease with increasing floor depth in the vertical direction. Compared with the normal mining condition, the maximum and minimum principal and the maximum shear stresses on the side of the small-dip fault increase.

The floor failure depth under the graben structure condition is 24 m, which is 20% greater than under normal mining conditions. The failure width on both sides is 23 m, the same as under normal mining conditions. The area of compressive failure increases, and the appearance of a shear failure region on the small-dip fault side, as shown in Fig. 15c.

As shown in Fig. 14d, under the step fault structure condition, the maximum principal stress is compressive stress. And the minimum principal stress is tensile stress under the goaf and the minimum principal stress is compressive stress on both sides of the working face. As the distance from the working face increases, the maximum principal stress increases first and then decreases, in the horizontal direction. The peak value of the maximum principal stress is located at 5 m away from the working face, on the side of the large-dip. As the distance from the working face increases, the minimum principal stress increases first and then decreases, and there is a 3 m wide compressive stress zone on the side of the small-dip. The peak value of the minimum principal stress is located at 8 m away from the working face, on the side of the small-dip, in the horizontal direction. As the distance from the working face increases, the maximum shear stress increases first and then decreases. The peak value of the maximum shear stress is located at 2 m away from the working face, on the side of the large-dip, in the horizontal direction. The maximum and minimum principal stresses and shear stress of the floor decrease with increasing floor depth in the vertical direction. Compared with the normal mining condition, the maximum and minimum principal and the maximum shear stresses on the side of the large-dip fault increase.

The floor failure depth under the step fault structure condition is 24 m, which is 20% greater than under normal mining conditions. The failure width on both sides is 23 m, the same as under normal mining conditions. The area of compressive failure increases, and the appearance of a shear failure region on the large-dip fault side, as shown in Fig. 15d.

Fig. 14
figure 14

Principal stress map below the floor for normal mining and complex fault structure mining. (a) The minimum principal stress in the middle of the floor is tensile stress. The maximum principal stress is compressive stress. The support pressure in the goaf floor is directed opposite to the pressure on the coal walls on both sides. (b) Under the horst structure condition, the maximum principal stress is compressive stress. And the minimum principal stress is tensile stress under the goaf and the minimum principal stress is compressive stress on both sides of the working face. As the distance from the working face increases, the maximum principal stress increases first and then decreases, in the horizontal direction. (c) Under the graben structure condition, the maximum principal stress is compressive stress. And the minimum principal stress is tensile stress under the goaf and the minimum principal stress is compressive stress on both sides of the working face. As the distance from the working face increases, the maximum principal stress increases first and then decreases, in the horizontal direction. (d) Under the step fault structure condition, the maximum principal stress is compressive stress. And the minimum principal stress is tensile stress under the goaf and the minimum principal stress is compressive stress on both sides of the working face. As the distance from the working face increases, the maximum principal stress increases first and then decreases, in the horizontal direction.

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Fig. 15
figure 15

Failure range of floor in normal mining and complex fault structure mining. (a) Under normal mining conditions, the floor failure depth is 20 m and the floor failure width is 23 m. (b) The floor failure depth under horst structure mining is 24 m, which is 20% greater than under normal mining conditions. The failure width on both sides is 24 m, which is 4.3% greater than under normal mining conditions. The area of compressive failure increases, and the appearance of a shear failure region on the large-dip fault side. (c) The floor failure depth under the graben structure condition is 24 m, which is 20% greater than under normal mining conditions. The failure width on both sides is 23 m, the same as under normal mining conditions. The area of compressive failure increases, and the appearance of a shear failure region on the small-dip fault side. (d) The floor failure depth under the step fault structure condition is 24 m, which is 20% greater than under normal mining conditions. The failure width on both sides is 23 m, the same as under normal mining conditions. The area of compressive failure increases, and the appearance of a shear failure region on the large-dip fault side.

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In conclusion, floor failure caused by coal mining comprises tension failure in the middle, compression failure on both sides and external shear failure under normal mining conditions.

Under the horst structure condition, the floor failure characteristics primarily show an increase in the compression failure area on both sides of the floor and shear failure occurs on the side of the large-dip fault. Compared with normal mining conditions, the floor failure depth under the horst structure condition, increased by 20% and the floor failure width increased by 4.3%.

Under the graben structure condition, the floor failure characteristics primarily show an increase in the compression failure area on both sides of the floor and shear failure occurs on the side of the small-dip fault. Compared with the normal mining conditions, the floor failure depth under the graben structure condition increases by 20%, and the floor failure width is the same as that of normal mining conditions.

Under the step fault structure condition, the primary failure is the compression failure on both sides of the working face. The maximum compressive stress on the large-dip fault side is 20% higher than under normal mining conditions, significantly increasing the floor compression failure area on the large-dip fault side. Compared with normal mining conditions, the floor failure depth under the horst structure condition, increased by 45% and the floor failure width increased by 26%.

Calculation formula for the width of waterproof coal pillars under complex fault structure conditions

By determining the failure width on both sides of the coal floor and the depth of floor failure under complex fault structures, a calculation formula for the waterproof coal pillars on the hanging wall and foot wall of the fault was established, as shown in Fig. 16. This formula is based on the position of the mining face relative to the fault, the depth of floor failure, and the failure width on both sides of the floor (Eq. (20)).

$$\left\{ \begin{gathered} {a_h}={w_B}+\frac{Z}{{\tan \delta }} \hfill \\ {a_f}={w_B} \hfill \\ a=\hbox{max} \left\{ {{a_h},{a_f}} \right\} \hfill \\ \delta =\hbox{min} \{ \alpha ,\theta \} \hfill \\ \end{gathered} \right.$$
(20)

where a is the width of the fault waterproof coal pillars, m; ah is the width of the upper wall waterproof coal pillars, m; af is the width of the floor waterproof coal pillars, m; Z is the depth of the floor failure calculated by the mechanical model, m; and wB is the failure range of both sides of the floor, m.

Fig. 16
figure 16

Waterproof coal pillars in complex fault structure. A calculation formula for the waterproof coal pillars on the hanging wall and foot wall of the fault was established, which depends on the position of the working face and the fault.

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The water pressure of the Ordovician tuff water in the 51,101 W working face in the Liangzhuang Coal Mine is 4.0 MPa. The width of the fault coal pillars calculated in accordance with Eq. (21) of the Coal Mine Water Control Rules is 20 m.

$$L=0.5{K_{\text{s}}}M\sqrt {\frac{{3P}}{{{K_p}}}} \geqslant 20{\text{m}}$$
(21)

where L is the width of the coal pillars, m; Ks is the safety coefficient, usually taken as 2 ~ 5; M is the thickness or mining height of the coal, m; P is the actual head value, MPa; and Kp is the tensile strength of the coal, MPa.

Fig. 17
figure 17

Width of waterproof coal pillars. The width of the waterproof coal pillar for the horst structure fault is 24 m, which is 20% greater than the width calculated for the water-conducting fault by the Coal Mine Water Control Rules. The width of the waterproof coal pillar for the graben structure fault is 27.6 m, which is approximately 40% greater than the width calculated for the water-conducting fault by the Coal Mine Water Control Rules. The width of the waterproof coal pillar for the step fault structure is 30 m, which is 50% greater than the width calculated for the water-conducting fault by the Coal Mine Water Control Rules.

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Therefore, excluding aquifer effects, according to Eq. (20), as shown in Fig. 17, the width of the waterproof coal pillar for the horst structure fault is 24 m, which is 20% greater than the width calculated for the water-conducting fault by the Coal Mine Water Control Rules. The width of the waterproof coal pillar for the graben structure fault is 27.6 m, which is approximately 40% greater than the width calculated for the water-conducting fault by the Coal Mine Water Control Rules. The width of the waterproof coal pillar for the step fault structure is 30 m, which is 50% greater than the width calculated for the water-conducting fault by the Coal Mine Water Control Rules.

To obtain the calculation result straightforwardly and rapidly, the width specified for waterproof coal pillars in the Coal Mine Water Control Rules is adopted as the reference standard for establishing such pillars in areas with complex fault conditions. The width under the horst structure condition needs to be increased by 20%. The width under the graben structure condition needs to be increased by 40%. The width under the step fault structure condition needs to be increased by 50%.

Numerical simulation analysis under complex fault conditions during mining

Numerical simulation scheme

Numerical simulation models under complex fault structure conditions are established based on the geological and mining conditions of the 51,101 W working face in the Liangzhuang Coal Mine. These models are built with dimensions of 700 m×500 m×300 m (length×width×height), as shown in Fig. 18. The fault dip angles are 79° and 89°, respectively. The mining direction of the working face is parallel to the fault strike, with the fault located 50 m away from the working face. The working face has a width of 166 m, with 115 m left on each side and 200 m in front and behind. The mining thickness is 2 m, and each span 20 m to maintain balance, with a total mining length of 400 m. The geological data from the Liangzhuang Coal Mine are used to determine the physical and mechanical parameters of the rock in the study area. To better analyze the changes in the properties of the surrounding rock and fault rock during coal seam mining, 50 monitoring points were established at 10 m intervals between the coal floor to monitor fault and floor displacement. The monitoring point is located at the 400 m mining point and records the change of floor displacement after the 400 m mining process. Based on drilling data and ensuring calculation accuracy, adjacent rock masses with similar physical and mechanical properties were simplified.

The surrounding rock weakening method is used to address the problem of numerical simulation faults, which takes the fault zone as a rock mass with a specific thickness. This range is considered an integral fault structure, dividing the model into fault and nonfault rock and weakening the mechanical parameters of the fault rock layers to form a fault-weakening zone with independently defined physical and mechanical properties24. Moreover, the effect of fault drop is neglected. The boundary conditions are set to zero horizontal displacements at the lateral boundaries, a fixed bottom boundary in both directions and a free top boundary. A uniformly distributed vertical load (12.87 MPa) is applied to the top boundary to simulate gravity, excluding aquifer effects. Table 1 shows the physical and mechanical parameters for each rock layer and fault.

Table 1 Physical and mechanical parameters of rock strata (faults) in the numerical model.
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Fig. 18
figure 18

Numerical simulation model of a complex fault structure and the schematic layout of displacement detection points. (a) Numerical simulation model under normal mining is built with dimensions of 700 m×500 m×300 m (length×width×height). (b) Numerical simulation model under the horst structure condition is built with dimensions of 700 m×500 m×300 m (length×width×height), the fault dip angles are 79° and 89°, respectively. (c) Numerical simulation model under the graben structure condition is built with dimensions of 700 m×500 m×300 m (length×width×height), the fault dip angles are 79° and 89°, respectively. (d) Numerical simulation model under the step fault structure condition is built with dimensions of 700 m×500 m×300 m (length×width×height), the fault dip angles are 79° and 89°, respectively. (e) Taking mining under normal mining as an example, to analyze the changes in the properties of the surrounding rock and fault rock during coal seam mining, 50 monitoring points were established at 10 m intervals between the coal floor to monitor fault and floor displacement.

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Analysis of the numerical simulation results

Rock failure occurs 400 m from the working face after completing the 400 m mining model, as illustrated in Fig. 19. Table 2 presents the specific numerical simulation results of floor failure. The simulation results align with the theoretical calculations, with some discrepancies due to rock layer characteristics and other factors. Compared with those under normal mining conditions, the failure characteristics of the floor under complex fault structure conditions are as follows : (1) The failure width on the large-dip fault side under the horst structure condition increases by 25%, and the failure width and depth on the small-dip fault side remain the same as those under normal mining conditions. (2) The failure width on the small-dip fault side under the graben structure condition increases by 50%, with the large-dip fault side showing failure widths similar to those under normal mining conditions. (3) The failure width on the small-dip fault side under the step fault structure condition increases by 15%, the large-dip fault side increases by 50%, and the floor failure depth increases by 20%.

The displacement of the floor monitoring points revealed symmetrical displacement under normal mining (Fig. 20a).

The displacement increases on the large-dip fault side under the horst structure condition (Fig. 20b). As the depth increases, the distance between the floor failure area and the large-dip fault is smaller than that between the floor failure area and the small-dip fault, so the side of the large-dip fault is more susceptible to the joint action of coal seam mining and faults.

The displacement increases on the small-dip fault side under the graben structure condition (Fig. 20c). As the depth increases, the distance between the floor failure area and the small-dip fault is smaller than that between the floor failure area and the large-dip fault, so the side of the small-dip fault is more susceptible to the joint action of coal seam mining and faults.

The displacement under the step fault structure condition is obviously inclined to the side of the large-dip fault (Fig. 20d). As the depth increases, the distance between the floor failure area and the large-dip fault is smaller than that between the floor failure area and the small-dip fault, so the side of the large-dip fault is more susceptible to the joint action of coal seam mining and faults.

In conclusion, the difference in floor displacement under complex fault conditions is caused by different fault tendencies and dips of different fault combinations.

Fig. 19
figure 19

Numerical simulation results of floor rock failure in normal mining and complex fault structure mining. (a) Numerical simulation results under normal mining conditions show that the floor failure depth is25 m and the floor failure width is 20 m. (b) The failure width on the large-dip fault side under the horst structure condition increases by 25%, and the failure width and depth on the small-dip fault side remain the same as those under normal mining conditions. (c) The failure width on the small-dip fault side under the graben structure condition increases by 50%, with the large-dip fault side showing failure widths similar to those under normal mining conditions. (d) The failure width on the small-dip fault side under the step fault structure condition increases by 15%, the large-dip fault side increases by 50%, and the floor failure depth increases by 20%.

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Fig. 20
figure 20

Monitoring results of vertical displacement of floor. (a) The displacement of the floor monitoring points revealed symmetrical displacement under normal mining. (b) The displacement increases on the large-dip fault side under the horst structure condition. (c) The displacement increases on the small-dip fault side under the graben structure condition. (d) The displacement under the step fault structure condition is obviously inclined to the side of the large-dip fault.

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Table 2 Results of numerical simulation of floor failure.
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Comparing the numerical simulation results under normal mining conditions, we find obvious differences in floor failure characteristics under complex fault conditions. Under the horst structure condition, the failure width of the large-dip fault side and the displacement of the horst structure increase significantly, indicating a failure trend toward the large-dip fault side. Under the graben structure condition, the floor failure width and displacement on the small-dip fault side increase significantly, indicating a failure trend toward the small-dip fault side. Under the step fault structure condition, the floor failure width and displacement on both sides increase, with the failure width on the small-dip fault side increasing by 15% and the failure depth on the large-dip fault side exceeding 20% of normal mining conditions.

Application

Background of the case

A numerical simulation model is established, utilizing the observed water inrush data at the 51,101 W working face in the Liangzhuang Coal Mine. This model is built with dimensions of 1000 m×500 m×300 m (length× width× height), as shown in Fig. 21. During mining, we set the working face width to 166 m, leaving 160 m in front and 300 m behind. The mining thickness is 2 m, and each span is 20 m to maintain balance, with a total mining length of 540 m. The length of the working face area near the FX fault is 140 m, where the fault strikes at a 40° angle to the working face mining direction. The experimental scheme and the physical and mechanical properties of the rock layers and faults are set based on the site conditions and numerical simulation scheme described in section of numerical simulation scheme.

Fig. 21
figure 21

Numerical model of No. 51,101 W working face. A numerical simulation model is established, utilizing the observed water inrush data at the 51,101 W working face in the Liangzhuang Coal Mine. This model is built with dimensions of 1000 m×500 m×300 m (length× width× height), with a total mining length of 540 m. The length of the working face area near the FX fault is 140 m, where the fault strikes at a 40° angle to the working face mining direction.

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Treatment of water inrush

As shown in Fig. 22, when mining reaches 60 m from the water inrush point, the rock masses near the FX fault are not damaged (Fig. 22a).When mining reaches 40 m from the water inrush point, the area near the FX fault has already experienced advanced failure due to coal mining (Fig. 22b). At 20 m from the water inrush point, the failure range near the FX fault extends to 20 m (Fig. 22c). When mining reaches the water inrush point, the failure range on both sides of the floor is 30 m. The mechanism of water inrush results from the floor failure area and the advanced fault failure area working together, creating a continuous water inrush channel (Fig. 22d). Observations from the mechanical analysis and numerical simulation based on the step fault structure indicate that, under the step fault structure condition, the floor tends to failure towards the side of the large-dip fault. This results in a significant increase in the failure width and depth on the FX fault side, ultimately leading to the connection of the floor failure area at the 51,101 W working face with the FX fault, consequently causing the water inrush event.

Given the 40° angle between the FX fault strike and the working face mining direction, as seen in Fig. 23, the water inrush channel forms because of the failure zones on both sides of the working face and the advanced failure zone along the working face strike. Therefore, advanced failure conditions should be considered for waterproof coal pillars15,24. Considering these conditions, the width of the waterproof coal pillar is calculated to be 42 m, exceeding the current 38 m pillar. The projection distance of the fault to the working face in the direction perpendicular to the mining area is 54 m, which is greater than the actual distance of 50 m. Therefore, establishing waterproof coal pillars based on theoretical analysis of complex fault structure conditions can effectively prevent floor water inrush events under such conditions.

Fig. 22
figure 22

Water inrush mechanism of No. 51,101 W working face. (a) When mining reaches 60 m from the water inrush point, the rock masses near the FX fault are not damaged. (b) When mining reaches 40 m from the water inrush point, the area near the FX fault has already experienced advanced failure due to coal mining. (c) At 20 m from the water inrush point, the failure range near the FX fault extends to 20 m. (d) When mining reaches the water inrush point, the failure range on both sides of the floor is 30 m. The mechanism of water inrush results from the floor failure area and the advanced fault failure area working together, creating a continuous water inrush channel.

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Fig. 23
figure 23

Top view of No. 51,101 W working face floor failure. (a) There is no water inrush channel in the goaf. (b) A water inrush channel begins to form at 5 m below the goaf, because of the failure zones on both sides of the working face and the advanced failure zone along the working face strike. (c) A water inrush channel is formed 15 m below the goaf, because of the failure zones on both sides of the working face and the advanced failure zone along the working face strike. (d) A water inrush channel is formed 25 m below the goaf, because of the failure zones on both sides of the working face and the advanced failure zone along the working face strike.

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Conclusions

In this study, an improved theoretical model is first proposed, which comprehensively considers the action of complex fault structures and mining activities. The underground pressure theory and elastic theory are used to establish a mechanical model of the coal floor under horst, graben, and step fault conditions based on the typical water inrush case of the 51,101 W working face in the Liangzhuang Coal Mine, and the floor stress and failure characteristics are analyzed. The retaining formula of waterproof coal pillars under complex fault conditions is established based on the improved theoretical model. In combination with MATLAB and Flac3D, the stress, displacement, and failure characteristics of the coal floor under complex fault conditions are investigated. Finally, we explained the mechanism of water inrush under complex fault conditions to verify the theoretical model. The following conclusions can be drawn:

  1. (1)

    Under the horst structure condition, compared with the normal mining condition, the horizontal stress of the floor increases, and the shear stress on the side of the large-dip fault increases. The failure characteristics primarily include an increase in the compression failure area on both sides of the floor, and shear failure occurs on the side of the large-dip fault. Based on the theoretical analysis, the width of waterproof coal pillars needs to be increased by 20%. The numerical simulation results reveal that the failure width on the large-dip fault side increases by 25%. The displacement increases on the large-dip fault side, indicating a failure trend toward the large-dip fault side.

  2. (2)

    Under the graben structure condition, compared with the normal mining condition, the horizontal stress of the floor increases, and the shear stress of the side of the small-dip fault increases. The failure characteristics primarily include an increase in the compression failure area on both sides of the floor, and shear failure occurs on the side of the small-dip fault. Based on the theoretical analysis, the width of waterproof coal pillars needs to be increased by 40%. The numerical simulation results reveal that the failure width on the small-dip fault side increases by 25%. The displacement increases on the small-dip fault side, indicating a failure trend toward the small-dip fault side.

  3. (3)

    Under the step fault structure condition, compared with the normal mining condition, the horizontal stress of the floor increases significantly, and the shear stress decreases. The floor failure characteristics primarily reveal that compression failure on both sides of the working face, significantly increases the floor compression failure area on the large-dip fault side. The width of waterproof coal pillars needs to be increased by 50%, based on the theoretical analysis. The numerical simulation results show that the failure width on the small-dip fault side increases by 15%, the large-dip fault side increases by 50%, and the floor failure depth increases by 20%. The displacement is obviously inclined to the side of the large-dip fault.