A Theoretical Model of Roof Self-Stability in Solid Backfilling Mining and Its Engineering Verification

Abstract

Roof self-stability in backfilling mining was proposed to explore its connotation and characteristics after a comparative analysis of roof structures under long-wall caving and backfilling mining. The mechanical analysis models of roof self-stability along strike and dip were established. After that, the mechanical equations for cooperative roof control were constructed to analyze the elastic foundation coefficients of the backfill, support peak load, unsupported-roof distance, and drilling effect of the working face along strike, the size of the working face, and the section pillar effect along dip. Research showed that the roof self-stability was greatly impacted by the elastic foundation coefficient of backfill, and it was less impacted by the support peak load along strike. The unsupported-roof distance had no obvious effect on roof self-stability. Roof self-stability was significantly affected by the working face and coal-pillar length along the dip. Therefore, the engineering control method of roof self-stability was proposed. The backfilling engineering practice in Xinjulong Coal Mine showed that the maximum roof subsidence was 438 mm, and the backfill ratio was 86.3% when the supporting intensity of backfilling hydraulic support was 0.94 MPa; the advanced distance of the working face was greater than 638 m; the foundation coefficient of backfilling material was 4.16 × 10⁸ Nm⁻³. The roof formed the self-stability structure, which satisfied safe coal mining under buildings, water bodies, and railways.

1. Introduction

Under the background of the construction of ecological civilization, green development and the realization of “double carbon” target in China, filling mining has been gradually developed and applied [1,2]. Backfilling mining is one of the key techniques of the green mining system, and its development has evolved into various forms, including waste filling, water-gravel-mixed filling, cemented filling, solid filling, paste backfilling, and high-water (super high-water) filling [3,4]. Among them, the solid filling mining mainly uses the gangue of particle size below 50 mm, through the porous bottom unloading scraper conveyor, transported to the goaf for filling; the filling body partially or completely replaces the caving gangue to support the overlying strata [5,6]. Compared with the traditional caving method, the filling body plays a supporting role in the roof. When the supporting capacity meets the requirements of roof stability, it can weaken the mine pressure behavior of the working face, control the damage of key rock strata, reduce the surface subsidence, and protect the ecological environment and water resources. Backfilling mining is widely used in coal areas under buildings, water bodies, and railways [7,8].

Massive studies have been conducted by relevant scholars on roof stability control in backfilling mining. The supporting force provided by the backfilling body is not enough to bear the weight of overlying rocks at the initial filling stage, which causes synchronous subsidence of the backfilling body and roof. Strength increases with the continuous compression of the backfilling body and reaches equilibrium with the roof pressure, which stabilizes the roof [9,10]. Liu et al. established a theoretical model of continuous curved beams of overlying strata in backfilling mining [11,12]. The quantitative relationship between continuous curved beams and key strata to study the spatiotemporal characteristics of continuous curved beams. Zuo et al. established a curvature model of backfilling mining strata to analyze the relationship between the backfill ratio and strata curvature [13]. Zhang et al. defined the critical backfilling ratio to solve the critical backfilling ratio for layered fracture of four-layer overlying strata based on the strata movement and control characteristics of backfilling mining [14]. Huang et al. established a coupled roof-controlled mechanical model to analyze the effect law of different factors on the support working resistance under high and low backfilling ratios, as well as the viscoelastic and time effects of the backfilling body [15,16]. Jiang et al. analyzed the self-stability effect of roof arches in goafs with different roof-contacted filling ratios [17]. Sun et al. established a continuous curved mechanical model of backfilling roof strata under non-uniform load to analyze the movement relationship of the overlying strata with different backfilling ratios [18].

The above research positively promotes the development of backfilling mining technology and surrounding rock control theory. The roof-stability control theory is studied from different angles. Roof stability is affected by multiple factors. Therefore, the roof control degrees along strike and dip are different based on different factors, and the quantitative effect research of roof control degrees is not perfect. In this paper, the connotation and characteristics of roof self-stability in solid filling mining are expounded, and the mechanical model of roof along strike and dip direction is constructed. The influence of elastic foundation coefficient of filling body, peak load of support, empty roof distance and footage of working face on roof self-stability along strike direction and the influence of working face length and coal pillar length on roof self-stability along dip direction are studied. Finally, the reliability of the model is verified by the engineering case of Xinjulong Coal Mine. The research content is a further deepening of the research on roof stability control on the basis of the existing research, which enriches the theory of strata control in coal mine filling mining to a certain extent.

2. Roof Structure and Roof Self-Stability Concept in Backfilling Mining

2.1. Roof Structure Characteristic

Backfilling mining is the mining technology where coal resources are replaced with backfill materials. It is different from the traditional roof support structure of long-wall caving based on different space angles. The strata support structure of collapse mining consists of the coal body and support in front of the working face and the rear caving gangue along the strike of the working face. The overlying strata of backfilling mining are synergistically supported by the coal body, backfilling hydraulic support, and backfilling body in front of the working face [19]. The caving and backfilling-mining support structures in front of the stope are both composed of solid coal along the dip of the working face. The caving support structure consists of a caving gangue and coal body, and the backfilling-mining support structure consists of the backfilling body and coal body at the rear of a stope [20].

The evolution law of overlying strata structure is determined by multiple factors in backfilling mining. The deformation resistance/backfill ratio is the most significant factor. A smaller deformation resistance/backfill ratio leads to more broken overlying strata and a smaller probability of complete strata. Greater deformation resistance/backfill ratio leads to fewer broken overlying strata and a greater probability of complete strata. It determines whether each stratum is broken. When the backfill ratio reaches a certain degree, the immediate roof only shows the state of bending and mining. The overall roof structure after mining is in the form of a continuous curved beam. In other words, it is a beam although looking like a beam. The movement position is determined by the bending self-stability of the continuous curved beam structure.

Table 1. Comparison of rock-beam structure characteristics.

Item	Precise Backfilling	Long-Wall Caving
Strata control diagram
Goaf	Dense backfilling body	Collapsed bulk gangue
Immediate roof	Mining and bending (collapses)	Mining and collapses
Basic roof	Breakage may not occur (depending on the filling density)	First and periodic breakage
Rock beam structure after mining
	Continuous bending beam: it is a beam although looking like a beam	Masonry beam: it is an arch although looking like a beam
Reconstruction of equilibrium states	Curved arch	Three-hinged arch
Determination of motion states	Bending self-stability	S (slip)-R (roll) instability

shows the comparative roof structure characteristics.

2.2. Roof Self-Stability Concept and Characteristic

There are three types of structural forms of overlying rock beams in backfilling mining. The first type has a low backfill ratio in the goaf and a small elastic foundation coefficient of backfilling body, which causes the failure of immediate and basic roofs, and obvious ground behaviors. It has an insignificant difference from collapse mining. The second type has a high backfill ratio in the goaf and a large elastic foundation coefficient of backfilling body, which causes the failure of the immediate roof, the bending of the basic roof, and relieved ground behaviors. The third type has a particularly high backfill ratio in the goaf, a high elastic foundation coefficient, and a large resistance on the roof. The immediate and basic roofs bend and sink, with no obvious ground behaviors [21].

The concept of roof self-stability in backfilling mining is proposed on this basis. The roof beams are supported by the backfilling body to form a “continuous curved beam” structure, which maintains integrity and stability by its strength. The structure is manifested as the morphological characteristics of unbroken and continuous deformation and is described as a roof self-stability state (corresponding to the third type). If the support degree of the backfilling body is weak, the roof beams cannot form a “continuous curved beam” structure. It cannot maintain integrity and stability by its strength. The structure is manifested as the morphological characteristics of instability and fracture and described as a roof instability state (corresponding to the first and second types). The roof beams reconstruct the stable overlying strata structure to form curved beams in the form of curved arches rather than three-hinged arches. The relevant theory of beams can be used to solve the bending moments, stress, and deflections of continuous bending beams. Stope stress is redistributed to form a stress-relaxed area, a stress-concentrated area, and an initial rock stress area on the roof after collapse mining. The stress-concentrated area has a large stress concentration coefficient, which is not conducive to coal-body stability and safe mining. Roof self-stability inhibits its movement during dense filling. A small stress concentration coefficient and unobvious strata behaviors contribute to ensuring the safety of underground workers. The immediate roof is bent out of shape under the support of the backfilling body according to the equivalent mining-height theory after the self-stabilization of the dense-packed roof. The surface sinks slightly to liberate the coal resources under buildings, water bodies, and railways. Besides, the backfilling body further absorbs the released elastic energy of overlying strata during the self-stabilization process, which reduces the risk of rock bursts.

Roof self-stability is relevant to strike factors (the backfill compactness, support working resistance, unsupported-roof distance, and drilling depth), and the dip factors (the layout of the backfill mining system, coal-pillar width, and working face length). A theoretical model needs to be constructed for specific quantitative analysis. The mechanical models of roof self-stability are established along strike and dip to analyze the roof self-stabilizing strike effects of elastic foundation coefficients, support peak loads, unsupported-roof distances, advancing distances, and the roof self-stabilizing dip effects of working-face lengths and coal-pillar widths.

3. Construction of Roof Self-Stability Model in Backfilling Mining

3.1. Self-Stability Model of Strike Roof

The working face is advanced along strike. The strike roof support system consists of the coal body, backfilling body, and hydraulic support, which cooperate to control the roof. A mechanical model is established to analyze the effect laws of elastic foundation coefficients, support peak loads, and unsupported-roof distances on roof self-stability. It is simplified as a beam model composed of the coal body, backfilling body supported by the elastic foundation, backfilling hydraulic support in the form of linear loads, and unsupported-roof areas (see Figure 1).

Figure 1 shows that q₀ is the initial rock stress; k₁ and k₂ are the stress concentration coefficients of the front and rear coal bodies; q_c is the assumed uniform load on the immediate roof above the backfilling body (the load concentration caused by the potential caving height of overlying strata); k_c is the elastic foundation coefficient of coal body; k_g the elastic foundation coefficient of backfilling body; q₃(x) and q₄(x) are the simplified linear functions of loads above coal body, respectively; q₁(x) and q₂(x) are loads of the top beam before and after backfilling hydraulic support; q is the support load concentration; L₁ and L₆ are the coal body lengths; L₂ is the distance from the support top to the rear column; L₃ the distance from the rear column to the end of the support; L₄ the unsupported-roof distance; L₅ the backfilling body length.

When −L₁ < x < 0, the deflection differential equation of the roof is expressed as

$$EI\frac{d^4{w}_1(x)}{d{x}^4}+k_c{w}_1(x)=q_3(x)$$

(1)

where E is the elastic modulus of the roof, GPa; I the inertia moment of the beam; q₃(x) the load above the front coal body.

$$q_3(x)=\frac{(k_1-1)q_0}{L_1}x+k_1{q}_0$$

(2)

The deflection equation of the roof in segment −L₁ < x < 0 can be obtained.

$$\begin{array}{l}{w}_1(x)=\frac{\frac{(k_1-1)q_0}{L_1}x+k_1{q}_0}{k_c}+e^{-\alpha x}(A_1\cos (\alpha x)+B_1\sin (\alpha x))\\ +e^{\alpha x}(C_1\cos (\alpha x)+D_1\sin (\alpha x))\end{array}$$

(3)

where characteristic coefficient $\alpha =\sqrt[4]{\frac{k_c}{4EI}}$.

When 0 < x < L₂, the deflection differential equation of the roof is expressed as follows.

$$EI\frac{d^4{w}_2(x)}{d{x}^4}+q_1(x)=q_c$$

(4)

where q₁(x) is the front top beam load of the support.

$$q_1(x)=\frac{qx}{L_2}$$

(5)

The deflection equation of the roof in segment 0 < x < L₂ can be obtained.

$$w_2(x)=\frac{-q{x}^5}{120EI{L}_2}+\frac{q_c{x}^4}{24EI}+\frac{A_2{x}^3}{6}+\frac{B_2{x}^2}{2}+C_{2}x+D_2$$

(6)

When L₂ < x < L₂ + L₃, the deflection differential equation of the roof is expressed as follows.

$$EI\frac{d^4{w}_3(x)}{d{x}^4}+q_2(x)=q_c$$

(7)

where q₂(x) is the front top beam load of the support.

$$q_2(x)=\frac{-q}{L_3}x+\frac{L_2+L_3}{L_3}q$$

(8)

The deflection equation of the roof in the segment L₂ < x < L₂ + L₃ can be obtained.

$$\begin{array}{l}{w}_3(x)=\frac{\frac{q{x}^5}{120}-\frac{q{x}^4{L}_2}{24}-\frac{q{L}_3{x}^4}{24}+\frac{q_c{L}_3{x}^4}{24}}{EI{L}_3}\\ +\frac{A_3{x}^3}{6}+\frac{B_3{x}^2}{2}+C_{3}x+D_3\end{array}$$

(9)

When L₂ + L₃ < x < L₂ + L₃ + L₄, the deflection differential equation of the roof is expressed as follows.

$$EI\frac{d^4{w}_4(x)}{d{x}^4}=q_c$$

(10)

The deflection equation of the roof in segment L₂ + L₃ < x < L₂ + L₃ + L₄ can be obtained.

$$w_4(x)=\frac{q_c{x}^4}{24EI}+\frac{A_4{x}^3}{6}+\frac{B_4{x}^2}{2}+C_{4}x+D_4$$

(11)

When L₂ + L₃ + L₄ < x < L₂ + L₃ + L₄ + L₅, the deflection differential equation of the roof is expressed as follows.

$$EI\frac{d^4{w}_5(x)}{d{x}^4}+k_g{w}_5(x)=q_c$$

(12)

The deflection equation of the roof in segment L₂ + L₃ + L₄ < x < L₂ + L₃ + L₄ + L₅ can be obtained.

$$\begin{array}{l}{w}_5(x)=\frac{q_c}{k_g}+e^{-\beta x}(A_5\cos (\beta x)+B_5\sin (\beta x))\\ +e^{\beta x}(C_5\cos (\beta x)+D_5\sin (\beta x))\end{array}$$

(13)

where the characteristic coefficient $\beta =\sqrt[4]{\frac{k_g}{4EI}}$.

When L₂ + L₃ + L₄ + L₅ < x < L₂ + L₃ + L₄ + L₅ + L₆, the deflection differential equation of the roof is expressed as follows.

$$EI\frac{d^4{w}_6(x)}{d{x}^4}+k_c{w}_6(x)=q_4(x)$$

(14)

where q₃(x) is the load above the rear coal body.

$$\begin{array}{l}{q}_4(x)=\frac{-(k_2-1)q_0}{L_6}x+q_0\\ +\frac{q_0(k_2-1)(L_2+L_3+L_4+L_5+L_6)}{L_6}\end{array}$$

(15)

The deflection equation of the roof in segment L₂ + L₃ + L₄ + L₅ < x < L₂ + L₃ + L₄ + L₅ + L₆ can be obtained.

$$\begin{array}{l}{w}_6(x)=\frac{\frac{-(k_2-1)q_0}{L_6}x+q_0+\frac{q_0(k_2-1)(L_2+L_3+L_4+L_5+L_6)}{L_6}}{k_c}\\ +e^{-\alpha x}(A_6\cos (\alpha x)+B_6\sin (\alpha x))+e^{\alpha x}(C_6\cos (\alpha x)+D_6\sin (\alpha x))\end{array}$$

(16)

Rotation angle $\theta (x)$, bending moment $M\left(x\right)$, and shear force $Q\left(x\right)$ of any section of the beam can be calculated by deflection $w\left(x\right)$, respectively.

$$\left\{\begin{array}{l}\theta (x)=\frac{dw(x)}{dx}\\ M(x)=-EI\frac{d{w}^2(x)}{d{x}^2}\\ Q(x)=-EI\frac{d{w}^3(x)}{d{x}^3}\end{array}\right.$$

(17)

The two sides of the immediate roof are taken as semi-infinite rock beams. When $x\to -\infty$ and $x\to +\infty$, the roof subsidence is a constant. In other words, A1 = 0; B1 = 0; C6 = 0; D6 = 0. Equation (18) is obtained according to the boundary and continuity conditions between the top-beam sections.

$$\left\{\begin{array}{l}{w}_1(0)=w_2(0);{\theta }_1(0)={\theta }_2(0)\\ M_1(0)=M_2(0);Q_1(0)=Q_2(0)\\ w_2(L_2)=w_3(L_2);{\theta }_2(L_2)={\theta }_3(L_2)\\ M_2(L_2)=M_3(L_2);Q_2(L_2)=Q_3(L_2)\\ w_3(L_2+L_3)=w_4(L_2+L_3)\\ {\theta }_3(L_2+L_3)={\theta }_4(L_2+L_3)\\ M_3(L_2+L_3)=M_4(L_2+L_3)\\ Q_3(L_2+L_3)=Q_4(L_2+L_3)\\ w_4(L_2+L_3+L_4)=w_5(L_2+L_3+L_4)\\ {\theta }_4(L_2+L_3+L_4)={\theta }_5(L_2+L_3+L_4)\\ M_4(L_2+L_3+L_4)=M_5(L_2+L_3+L_4)\\ Q_4(L_2+L_3+L_4)=Q_5(L_2+L_3+L_4)\\ w_4(L_2+L_3+L_4+L_5)=w_5(L_2+L_3+L_4+L_5)\\ {\theta }_4(L_2+L_3+L_4+L_5)={\theta }_5(L_2+L_3+L_4+L_5)\\ M_4(L_2+L_3+L_4+L_5)=M_5(L_2+L_3+L_4+L_5)\\ Q_4(L_2+L_3+L_4+L_5)=Q_5(L_2+L_3+L_4+L_5)\end{array}\right.$$

(18)

The above deflection equations are substituted in Equation (18) to solve C₁, D₁, A₂, B₂, C₂, D₂, A₃, B₃, C₃, D₃, A₄, B₄, C₄, D₄, A₅, B₅, C₅, D₅, A₆, and B₆ combined with engineering parameters of the specific mine. Therefore, the deflection and stress of the immediate roof along strike can be obtained.

3.2. Dip Roof Self-Stability Model

The working face is advanced along strike, with retained section pillars between working faces. The inclined roof consists of the backfilling body, coal body, and section pillars, which jointly support the overlying strata. The section pillars are affected by the mining of the backfilling working face on both sides, with superimposed abutment pressure. The distribution law and size of coal-pillar abutment pressure are relevant to its length. When the coal pillar is short, the abutment pressure is presented in the form of a parabola. When the coal pillar is long, the abutment pressure takes the form of a saddle-shaped curve. A mechanical model of dip roof self-stability is established to explore the effects of the working face and coal-pillar lengths on roof self-stability. The coal pillars are simplified into the intersected linear loads on both ends. The remaining simplifications are consistent with strike. Figure 2 shows the mechanical model of dip roof self-stability.

Q₁(x), Q₂(x), Q₃(x), and Q₄(x) are the immediate roof loads above the coal body at different positions, respectively, which are all set as linear loads; K₁, K₂, K₃, K₄, and K₅ are the stress concentration coefficients, respectively; k_g1 and k_g2 the elastic foundation coefficients of backfill; l₁ and l₆ the coal-body lengths on two sides; l₂ and l₅ the lengths of the sectional working faces; l₃ and l₄ the sectional pillar lengths.

The deflection differential equations of roof segments from left to right are expressed as Equation (19) According to Figure 2.

$$\left\{\begin{array}{l}EI\frac{d^4{w}_1(x)}{d{x}^4}+k_c{w}_1(x)=Q_1(x)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} -L_1<x<0\\ EI\frac{d^4{w}_2(x)}{d{x}^4}+k_{g1}{w}_2(x)=q_c\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} 0<x<L_2 \\ EI\frac{d^4{w}_3(x)}{d{x}^4}+k_c{w}_3(x)=Q_2(x)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} L_2<x<L_2+L_3 \\ EI\frac{d^4{w}_4(x)}{d{x}^4}+k_c{w}_4(x)=Q_3(x)\\ \hspace{1em}\hspace{1em} L_2+L_3<x<L_2+L_3+L_4\\ EI\frac{d^4{w}_5(x)}{d{x}^4}+k_{g2}{w}_5(x)=q_c\\ \hspace{1em}\hspace{1em} L_2+L_3+L_4<x<L_2+L_3+L_4+L_5\\ EI\frac{d^4{w}_6(x)}{d{x}^4}+k_c{w}_6(x)=Q_4(x)\\ \hspace{1em} L_2+L_3+L_4+L_5<x<L_2+L_3+L_4+L_5+L_6\end{array}\right.$$

(19)

where

$$\left\{\begin{array}{l}{Q}_1(x)=\frac{(K_1-1)q_{0}x}{l_1}+K_1{q}_0\\ Q_2(x)=\frac{(K_3-K_2)q_{0}x}{l_3}-\frac{q_0{l}_2(K_3-K_2)}{l_3}+K_2{q}_0\\ Q_3(x)=\frac{(K_4-K_3)q_{0}x}{l_4}-\frac{(K_4-K_3)q_0(l_2+l_3)}{l_4}+K_3{q}_0\\ Q_4(x)=\frac{(1-K_5)q_0}{l_6}x-\frac{(1-K_5)q_0(l_2+l_3+l_4+l_5)}{l_6}+K_5{q}_0\end{array}\right.$$

(20)

The deflection equations of segments are calculated by Equations (19) and (20).

$$\left\{\begin{array}{l}{w}_1(x)=\frac{\frac{(K_1-1)q_{0}x}{l_1}+K_1{q}_0}{k_c}+e^{-\alpha x}(a_1\cos (\alpha x)+b_1\sin (\alpha x))\\ +e^{\alpha x}(c_1\cos (\alpha x)+d_1\sin (\alpha x)) -l_1<x<0\\ w_2(x)=\frac{q_c}{k_{g1}}+e^{-{\beta }_{1}x}(a_2\cos ({\beta }_{1}x)+b_2\sin ({\beta }_{1}x))\\ +e^{{\beta }_{1}x}(c_2\cos ({\beta }_{1}x)+d_2\sin ({\beta }_{1}x)) 0<x<l_2\\ w_3(x)=\frac{\frac{(K_3-K_2)q_{0}x}{l_3}-\frac{q_0{l}_2(K_3-K_2)}{l_3}+K_2{q}_0}{k_c}\\ +e^{-\alpha x}(a_3\cos (\alpha x)+b_3\sin (\alpha x))+e^{\alpha x}(c_3\cos (\alpha x)+d_3\sin (\alpha x))\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{l}_2<x<l_2+l_3\\ w_4(x)=\frac{\frac{(K_4-K_3)q_{0}x}{l_4}-\frac{(K_4-K_3)q_0(l_2+l_3)}{l_4}+K_3{q}_0}{k_c}\\ +e^{-\alpha x}(a_4\cos (\alpha x)+b_4\sin (\alpha x))+e^{\alpha x}(c_4\cos (\alpha x)+d_4\sin (\alpha x))\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} l_2+l_3<x<l_2+l_3+l_4\\ w_5(x)=\frac{q_c}{k_{g2}}+e^{-{\beta }_{2}x}(a_5\cos ({\beta }_{2}x)+b_5\sin ({\beta }_{2}x))\\ +e^{{\beta }_{2}x}(c_5\cos ({\beta }_{2}x)+d_5\sin ({\beta }_{2}x))\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} l_2+l_3+l_4<x<L_2+l_3+l_4+l_5\\ w_6(x)=\frac{\frac{(1-K_5)q_0}{l_6}x-\frac{(1-K_5)q_0(l_2+l_3+l_4+l_5)}{l_6}+K_5{q}_0}{k_c}\\ +e^{-\alpha x}(a_6\cos (\alpha x)+b_6\sin (\alpha x)) +e^{\alpha x}(c_6\cos (\alpha x)+d_6\sin (\alpha x)) \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} l_2+l_3+l_4+l_5<x<l_2+l_3+l_4+l_5+l_6\end{array}\right.$$

(21)

where characteristic coefficients ${\beta }_1=\sqrt[4]{\frac{k_{g1}}{4EI}}$ and ${\beta }_2=\sqrt[4]{\frac{k_{g2}}{4EI}}$.

The elastic foundation coefficient is defined as follows [15].

$$k_g=\frac{E_g}{h_g}$$

(22)

where $E_g$ is the elastic modulus of stratum, GPa; $h_g$ the thickness of stratum, m.

The deflection equation and corresponding engineering parameters are substituted into Equation (18) to obtain the sectional deflection and stress of the immediate roof along the dip according to the boundary conditions and continuity.

4. Effect Analysis of Roof Self-Stability Factors

4.1. Effect Analysis of Roof Self-Stability Factors along Strike

(1)

Preliminary setting of parameters

The parameters are preliminarily selected in different directions to study the key factor effects of roof self-stability based on the engineering geological conditions of a certain mine [22].

The buried depth of the coal seam is denoted as 700 m along strike; initial rock stress q₀ = 17.5 × 10⁶ N/m; load concentration of overlying roof in goaf q_c = 2 × 10⁶ N/m; immediate roof thickness h = 3.7 m; elastic modulus of roof E = 18 GPa; elastic foundation coefficient of coal body k_c = 3 × 10⁸ N/m³; elastic foundation coefficient of backfilling body k_g = 9 × 10⁶ N/m³; support peak load q = 1.62 × 10⁶ N/m; stress concentration coefficient of roof k₁ = k₂ = 2.3; solid-coal length L₁ = L₆ = 50 m; front top beam length L₂ = 3.56 m; rear top beam length L₃ = 2.81 m, unsupported-roof distance L₄ = 1 m; backfilling body L₅ = 60 m. The above parameters are used to analyze the effect law of the elastic foundation coefficient of backfill, support peak load, and unsupported-roof distance on roof stability along strike.

(2)

Effect analysis of elastic foundation coefficient of backfill

The elastic foundation coefficients of backfill are set to be 8, 9, and 10 MN/m³, respectively. Figure 3 shows the correlation curves among the elastic foundation coefficient, roof deflection, and bending moment through mechanical calculations.

The immediate roof subsidence presents a parabolic curve under the elastic-foundation coefficient of backfill. The roof subsidence decreases with the increased elastic-foundation coefficient. Besides, the position of maximum roof subsidence moves close to the stope coal wall. When the elastic foundation coefficient rises from 8 to 10 MN/m³, the maximum roof subsidence is decreased by 20.23% (from 0.262 to 0.209 m). A negative correlation exists between the roof bending moment and the elastic-foundation coefficient of backfill. The maximum roof bending moment appears near the stope coal wall. However, the maximum roof bending moment shifts to the direction of the goaf with the increased elastic-foundation coefficient. When the elastic-foundation coefficient rises from 8 to 10 MN/m³, the maximum roof bending moment is reduced from 40.9 to 26.06 MN·m, with a change rate of 36.28%. Therefore, the elastic-foundation coefficient of backfill significantly impacts roof self-stability. The elastic foundation coefficient of filling directly affects the strength of filling body and changes the supporting ability of filling body to overlying rock. In order to make the roof reach the self-stable state, the filling elastic foundation coefficient can be improved by increasing the filling rate, optimizing the filling material ratio and optimizing the filling material.

(3)

Effect analysis of supporting-peak loads

The peak loads of backfilling hydraulic support are set to be 0.62, 2.62, and 4.62 MPa, respectively. Figure 4 shows the correlation curves among the support peak load, roof deflection, and bending moment through mechanical calculations.

A parabolic curve exists between the immediate roof subsidence and the advancing distance of the working face along the strike of coal seams. The roof subsidence reaches the maximum in the middle of the goaf, and it decreases with increased supporting-peak loads. When the supporting-peak load rises from 0.62 to 4.62 MPa, the maximum roof subsidence is decreased by 1.3% (from 0.233 to 0.230 m). The roof subsidence is insignificantly impacted by the support peak load. The roof bending moment above the goaf presents asymmetric M-shaped curves due to the existence of the backfilling hydraulic support and unsupported-roof distance in the support system. The maximum roof bending moment continuously decreases with increased supporting-peak loads. When the support peak load rises from 0.62 to 4.62 MPa, the maximum roof bending moment is decreased by 33.34% (from 37.37 to 24.91 MN·m). The peak load reaches a maximum near the front coal wall, where the roof is damaged at first. Therefore, the support force of hydraulic support can be appropriately increased to ensure roof stability.

(4)

Effect analysis of unsupported-roof distances

The unsupported-roof distance is not more than 1.5 m according to the actual situation of the stope. It is set to be 0.5, 1.0, and 1.5 m, respectively. Figure 5 shows the correlation curves among the unsupported-roof distance, roof deflection, and bending moment through mechanical calculations.

The effect of the unsupported-roof distance on roof subsidence can be ignored. A positive correlation exists between the unsupported-roof distance and roof bending moment. When the unsupported-roof distance increases from 0.5 to 1.5 m, the maximum roof bending moment is increased by 9.9% (from 31.42 to 34.52 MN·m). The overall roof stability is insignificantly impacted by the unsupported-roof distance. However, in the poor condition of the roof, the empty top distance is large, and the supported area of the roof is reduced, which will also cause various disaster problems.

(5)

Effect analysis of advancing distance of working face

The advancing distance of the working face is set to be 47.37, 57.37, and 67.37 m, respectively. Figure 6 shows the correlation curves among the advancing distance of the working face, roof deflection, and bending moment through mechanical calculations.

The maximum roof subsidence and maximum bending moment increase with the increased advancing distance of the working face. When the advancing distance increases from 47.37 to 67.37 m, the roof maximum subsidence is increased by 4.00% (from 0.225 to 0.234 m); the maximum roof bending moment is increased by 22.20% (from 34.41 to 42.05 MN·m). The advancing distance of the working face slightly impacts roof self-stability under dense backfilling. The backfilling body reaches a certain strength to support the roof after dense tamping, which suppresses the roof movement. The stress concentration coefficient is small in the initial and subsequent mining process of the working face. Besides, the support stress area does not increase significantly with the advancement of the working face.

4.2. Effect Analysis of Roof Self-Stability Factors along Dip

(1)

Preliminary setting of parameters

It is denoted that coal-body length l₁ = 50 m and l₆ = 50 m at the left and right ends along dip; stress concentration coefficient K₃ = 1 in the middle of the coal pillar.

The stress concentration coefficient of the coal pillar is affected by the working-face length, coal-pillar length, and elastic-foundation coefficient of the backfilling body.

The functional relationship between the stress concentration factor and backfilling working face is expressed as Equation (23) when the coal-pillar length is 100 m, and the working-face length is within a certain range [23].

$$K=a{e}^{bl}$$

(23)

where K is the stress concentration coefficient of the coal pillar; l the length of the backfilling working face, m; a and b are coefficients.

When lengths of the backfilling working face l₂ = 87 m and l₅ = 87 m, the coal-pillar length and stress concentration coefficient are fitted [24] (see Figure 7).

Other parameters along the dip are consistent with those along the strike. The effect law of the coal-pillar length and working face length on roof stability is analyzed along dip.

(2)

Effect analysis of working face size

The working face length is set to be 50, 60, and 70 m, respectively. Figure 8 shows the correlation curves among the working face length, roof deflection, and bending moment through mechanical calculations.

An M-shaped curve relationship exists between the working face length and roof subsidence along the dip of coal seams. Roof subsidence increases with the increased working-face length, and the increase rate becomes smaller. When the working-face length increases from 50 to 70 m, maximum roof subsidence is increased by 6.82% (from 0.22 to 0.235 m). Roof subsidence is restrained by adopting dense filling and retaining a long coal pillar between backfilling working faces. The roof bending moment curve above the goaf is transformed from M-shaped to parabolic with increased working face length, and the maximum bending moment gradually increases. When the working-face length increases from 50 to 70 m, the maximum roof bending moment increases by 41.86% (from 29.93 to 42.46 MN·m). The working face length impacts roof failure. In the process of working face mining, the increase of face length causes the increase of roof exposed in the goaf area, which is not conducive to the formation of self-stabilizing structure of roof. In order to improve the stability of roof, the length of working face can be appropriately reduced.

(3)

Effect analysis of section coal-pillar size

The section coal-pillar length is set to be 60, 70, and 80 m, respectively. Figure 9 shows the correlation curves among the coal-pillar length, roof deflection, and bending moment through mechanical calculations.

The coal-pillar length is negatively correlated with roof subsidence. The coal pillar has no obvious effect on roof subsidence under dense filling. The overall roof stability depends on the elastic-foundation coefficient and roof load concentration. The possible failure position changes with increased coal-pillar length, and the maximum roof bending moment gradually decreases. Therefore, the coal-pillar length significantly impacts roof failure. When the coal-pillar length increases from 60 to 80 m, the roof maximum bending moment is decreased by 32.05% (from 39.91 to 27.12 MN·m). In the process of passive support roof, the support capacity provided by coal pillar is far greater than the filling body; therefore, in order to maintain the stability of the roof and maximize the exploitation of coal resources, it is recommended to optimize the length of coal pillar to reach the best state.

5. Engineering Control Method for Roof Self-Stability

The section discusses two aspects of engineering design and construction stages.

5.1. Engineering Design Stage

Roof self-stability control is realized by material selection, ratio optimization, reasonable designs of working face size, and coal-pillar width before mining the working face.

(1)

The above mechanical model analysis shows that the roof self-stability is restrained by the working face length and promoted by the coal-pillar length. Reasonable designs of working-face and coal-pillar sizes are relevant to the geological conditions of mines. When the overlying strata are stable, the working face length can be appropriately increased to decrease the coal-pillar length, and vice versa.

(2)

Material selection, size grading, and material ratio play an important role in roof self-stability control. The backfill materials include gangue, aeolian sand, river sand, and fly ash. Different materials have different physical and mechanical properties. The materials with higher strength are more easily to achieve roof self-stability. Gangue is used as a bulk filling material. Different size grading leads to different compression deformation properties. Therefore, the experimental methods are used to select the size grading with optimal deformation resistance at the engineering design stage. The material ratio plays an important role in roof self-stability as one of the key factors affecting the deformation resistance of the backfilling body. Therefore, the material ratio needs to be optimized.

5.2. Engineering Construction Stage

The roof is ensured to form a self-stable structure through optimization designs of backfill technology, support structure, and backfill effect monitoring in the mining process of the working face.

(1)

The backfill technology is accurately implemented by reducing the unsupported-roof distance and increasing the backfilling ratio. The unsupported-roof distance has little influence on roof self-stability. However, the backfilling ratio is one of the main factors for roof self-stability. A larger backfilling ratio leads to smaller roof subsidence and easier achievement of roof self-stability.

(2)

Roof stability is promoted by the top beam of the backfilling hydraulic support. The initial support force and working resistance of hydraulic support are increased to reduce the advanced roof subsidence, which ensures the filling space.

(3)

Filling effect monitoring is an effective method of determining whether the roof forms a self-stable structure. The roof state is reflected by monitoring the support working resistance, advanced abutment pressure, roof subsidence in the goaf, and backfill stress.

Figure 10 shows the control method for roof self-stability.

6. Engineering Verification

6.1. Mining Geological Conditions

First-mining working face 1303N-1# has a face length of 115.66 m and an advanced length of 911.6 m. The coal seam is No. 3 upper coal, with an elastic modulus of 9.6 GPa, a thickness of 2.2–3.63 m, an average thickness of 3.19 m, a coal-seam inclination angle of 9–13°, an average inclination angle of 11°, and a buried depth of 760–780 m. Figure 11 shows the working face layout. The immediate roof is siltstone with a thickness of 19.87 m, a hardness coefficient of 5.0, an elastic modulus of 15.6G Pa, and a tensile strength of 1.21 MPa. The direct floor is mudstone with a thickness of 1.45 m and a hardness coefficient of 4.0. The basic floor is siltstone with a thickness of 3.40 m and a hardness coefficient of 5.

6.2. Parameter Calculation for Roof Self-Stability Control

Xinjulong Coal Mine requires waste filling of 1.5 million t/a. The backfilling working face is arranged by gob-side entry retaining, and the surface structures exist above the working face. The core parameters such as the support working resistance and backfill ratio are designed to realize roof self-stability, which satisfies protection level I of structures. Roof self-stability characteristic refers to bending deformation rather than broken collapse, and the roof has a certain bearing capacity. Roof self-stability conditions are proposed on basis of this: (1) The tensile strength of the roof is not greater than the allowable tensile strength; (2) the roof subsidence is not greater than the allowable subsidence.

The strength and backfill ratio of the backfilling hydraulic support are designed to keep Xinjulong Coal Mine in line with the roof self-stability conditions. The specific process is as follows.

The support strength is calculated to be 0.71–0.93 MPa according to the overburden failure caused by the mining disturbance and overlying strata characteristics in Xinjulong Coal Mine. It is suggested that the support strength is greater than 0.93 MPa by considering roof lithology and working face safety.

The top of the backfilling working face is mainly villages where the horizontal deformation of surface structures is generally no more than 2 mm/m. The maximum horizontal deformation is set to 1 mm/m to increase the safety factor. The probability integration method is used to predict the mining subsidence of waste filling based on the equivalent mining-height model. After that, a reasonable backfill ratio is obtained through inversion. The specific formula is as follows.

$$\left\{\begin{array}{l}\epsilon =1.52\frac{W_0}{r}\\ W_0=H_{\mathrm{z}}q\cos \alpha \\ H_{\mathrm{z}}=(1-\eta )M\\ r=\frac{H}{\tan \beta }\\ \epsilon \le \left[\epsilon \right]\end{array}\right.$$

(24)

where ε is the maximum horizontal deformation; b the horizontal movement coefficient; W₀ the maximum subsidence of the surface, m; r the main influencing radius, m; H_z the equivalent mining height, m; q the subsidence coefficient; α the coal-seam inclination; H the buried depth, m; tan β the main influencing tangent angle; M the mining height, m; η the backfill ratio.

When the backfill ratio is greater than 84.6%, the roof can form a self-stable structure according to the geological and mining conditions of the Xinjulong Coal Mine.

6.3. Engineering Verification of Roof Self-Stability

(1)

Backfilling hydraulic support strength

The strength of backfilling hydraulic support needed to be greater than 0.93 MPa to meet the basic requirements for roof self-stability. Xinjulong Coal Mine adopted the C7000/20/38 four-column backfilling hydraulic support with a supporting intensity of 0.94 MPa.

(2)

Backfill ratio

①

Measuring-point layout

The roof is supported by the backfilling body after the goaf is filled in coal mining. Roof displacement sensors are arranged inside the backfilling body to monitor the self-stable state of the roof after mining.

There are 25 roof displacement sensors installed in the backfilling body. The first row of sensors is installed at the points that are 350 m from the open-off cut of the working face. When the working face is advanced to 450, 550, 700, and 850 m, the second, third, fourth, and fifth rows are installed, respectively. Figure 12 shows the layout of monitoring points.

The data transmission lines of roof displacement sensors are easily damaged in the goaf. Therefore, the data lines are placed in a special pipeline.

 ② Result analysis

A roof subsidence monitor is used to monitor the roof subsidence of the goaf after backfilling. The monitor is installed in the goaf during the mining period to measure and record the roof subsidence of the goaf. The monitoring data are transmitted through a wired connection. When the working face is advanced to 450 m, measuring point 1^# in the middle of the second row is selected for the monitoring analysis of dynamic roof subsidence. Figure 13 shows the monitoring results of dynamic roof subsidence.

(a)

Dynamic roof subsidence above the goaf is divided into three stages with the advancement of the working face: the accelerated subsidence stage (the distance between the working face and measuring point is 0–88 m); the slowly-increased subsidence stage (the distance is 88–188 m); the creep deformation stage (the distance > 188 m).

(b)

The backfilling body is not compacted in time with the increased advancing distance at the accelerated subsidence stage. The overlying strata cannot be supported by the backfilling body, which accelerates roof subsidence. The dynamic backfill ratio of the roof is 89.78% when the working face is 88 m away from the measuring point (the working face is advanced by 538 m). A stable roof structure leads to its bending subsidence.

(c)

The backfilling body is gradually compacted with the increased advancing distance at the slowly-increased subsidence stage. The compaction rate continuously decreases. The roof subsidence rate is lower than the compaction rate at the accelerated deformation stage of surrounding rocks. The roof bending amplitude increases, but no caving failure occurs.

(d)

When the distance between the working face and measuring point is larger than 188 m (the advanced distance is larger than 638 m), the roof subsidence enters the creep deformation stage of surrounding rocks. The backfilling body is gradually compacted to bear a load of overlying strata with the advanced working face, which stabilizes the roof subsidence. When roof subsidence is stable, the foundation coefficient of the backfill material is 4.16 × 10⁸ Nm⁻³; the maximum subsidence at measuring point 1^# is 438 mm. The backfill ratio at measuring point 1^# is calculated to be 86.3%, which is greater than the roof self-stability condition. Therefore, the roof forms a self-stable structure.

7. Conclusions

(1)

The concept and connotation of roof self-stability were proposed by analyzing the roof structure types along the strike and dip of coal seams and under different backfill degrees. After that, the structural characteristics of self-stable rock beams were expounded from the perspectives of roof stress distribution, overburden subsidence, and energy transfer.

(2)

A mechanical model of roof self-stability along strike was established based on the elastic-foundation beam theory to analyze the effect laws of backfilling elastic-foundation coefficients, supporting-peak loads, and unsupported-roof and advancing distances on roof self-stability along strike. Roof stability was greatly impacted by backfilling elastic-foundation coefficient and slightly impacted by supporting-peak load and advancing distance. Unsupported-roof distance had no obvious effect on roof stability. A mechanical model of roof self-stability along dip was established to analyze the effect laws of the working face and coal-pillar lengths on roof stability. The working face and coal-pillar lengths significantly affected roof breakage. The reserved coal-pillar length was appropriately reduced within a certain range.

(3)

The control method of roof self-stability engineering was put forward at the engineering design and construction stages according to the analysis of the roof self-stability effect.

(4)

The control parameters for roof self-stability were calculated combined with the geological and mining conditions of the Xinjulong Coal Mine. Roof self-stability conditions were satisfied when the supporting intensity of backfilling hydraulic support was greater than 0.93 MPa; the backfill ratio was greater than 84.6%. Engineering practice showed that the maximum roof subsidence was 438 mm, and the backfill ratio was 86.3% when the supporting intensity of backfilling hydraulic support was 0.94 MPa; the advancing distance of the working face was greater than 638 m; the foundation coefficient of backfill material was 4.16 × 10⁸ N·m⁻³. The roof formed a self-stabile structure, which met the requirements of coal mining under buildings, water bodies, and railways.