Breaking law of overlying strata in shallow coal seam fire area under thermal–mechanical coupling effect

The cracks formed by the breaking and collapse of the overlying strata are the oxygen supply channels for the continuous combustion of coal fires. The development of cracks is of great significance for determining the scope of the fire area and controlling the fire efficiently and accurately. To study the development law of cracks, a breaking model of the overlying strata in the shallow coal seam fire area is established on the basis of beam theory considering the coupling effect of gravity load and temperature field. The physical model of the coal fire is established based on the similarity principle. The temperature data from rock strata in the model are monitored by the infrared thermal imager. The dynamic distribution function of the roof temperature field is obtained by fitting. According to the breaking model theory, the position of the maximum tensile stress on the roof of the shallow fire area is the middle of the lower surface. The theoretical calculation value of the first breaking distance of the roof is 0.575 m, which is consistent with the experimental results of the physical model. The maximum tensile stress of the roof under the thermal‐mechanical coupling is slightly less than that under the load only. For shallow coal seams, the thermal stress generated by the coal fire increases the first breaking distance of the roof.


| INTRODUCTION
Coal fires are a global catastrophe. 1 The combustion of coal results in the loss of a lot of coal resources and the release of numerous harmful gases that pollute the environment. 2 The heat from the combustion bakes the ground surface and destroys vegetation and arable land. 3 Coal fires mainly occur in areas with arid climates and shallow buried thick coal seams. 4 At present, the research on coal fires mainly focuses on the field of detection and control. [5][6][7][8] The occurrence mechanism and development law of coal fire have not been accurately mastered. 9 Coal mining activities change the stress balance of coal seams, 10 forming a large number of cracks that provide a good oxygen supply channel and heat storage environment for coal spontaneous combustion. 11 After the fire, the overlying strata are bent and collapsed under the influence of gravity and thermal stress. 12 And new cracks are created to provide new oxygen channels for the continuous combustion of coal fires. 13 Mastering the development law of cracks and the characteristics of underground space is of great significance to determining the combustion state of coal fires and explaining the occurrence and development mechanism of coal fires. The state of crack distribution is also a theoretical guide for fire prevention and control. The researchers studied the permeability and aerodynamic effects of crack zones in the rock formations, established a fire zone collapse model, and delineated the fire zone. 14,15 Then, fire suppression was carried out according to the distribution of surface cracks and indicator gas. 16 The breaking law of the overlying strata in the fire area is the key to explaining the causes of cracks formation and the development process of underground space.
Studies have shown that the underground space in the shallow coal seam fire area has obvious zoning features. [17][18][19][20] In the vertical direction, caved zone, fractured zone, and sagging zone are formed from the bottom up. Along the strike of the coal seam, it is divided into a combustion cavity zone, a combustion zone, and an unburned zone. The schematic diagram of the structure is shown in Figure 1. The movement law of the overlying strata determines the development of cracks in the caved zone and fractured zone. 21 In mining engineering, researchers have introduced the elastic foundation beam theory to study the movement law of the overlying strata and the causes of crack formation. 22 The masonry beam theory and key strata theory based on the beam model can better reveal the mine pressure law and calculate the breaking distance of the roof. 23,24 Based on the beam theory, Tan et al. 25 studied failure evolution and influencing factors of the overlying strata induced by mining multiple coal seams. Gao et al. 26 took the immediate roof as the beam structure and proposed a new numerical simulation method for the continuous collapse of rock strata in the mining area. Different from the coal mining process, coal fire is a complex combustion system with stress and temperature fields in the underground space. 27 The movement and breaking of the overlying strata need to take into account the coupling effect of gravity and thermal stress.
Numerous scholars have conducted a large number of theoretical studies on beams under the thermalmechanical coupling effect and applied them to different materials and engineering backgrounds. Gao et al. 28 carried out a thermal stress analysis on a bimodulus beam to obtain analytical expressions for the stress, bending moment, and displacement of the beam. Xu and Zhou 29 derived the displacement and stress distribution formulas of the simply supported beam with variable thickness subjected to thermal-mechanical loads. When studying the movement and deformation of the overlying strata during underground coal gasification, Xin et al. 30 established a mechanics model of the multilayer thermoelastic foundation beam considering temperature effects and thermal damage factors. The initial spread law of coal fires is to develop from the burning point to the burning line along the fissure or goaf. 31 At this stage, rock strata affected by coal fires can be regarded as beam structures. Therefore, the elastic foundation beam model is suitable for the study of the breaking law of the overlying strata in the shallow coal seam fire area. In the underground space of coal fires, the temperature field is dynamic and irregular. The thermal stress on the overlying strata is also more complex.
The underground space structure of the shallow coal seam fire area. ZHU ET AL.

| 1771
This paper attempts to apply the elastic foundation beam theory to solve the problem of breaking the overlying strata in the shallow coal seam fire area. Furthermore, the development law of coal fire underground space and the reasons for the formation of combustion cracks are explained. First, according to the stress distribution characteristics, the overlying strata of shallow coal seams can be regarded as a simply supported beam model. The breaking mechanics model of overlying strata under the action of the thermal-mechanical coupling is derived. According to the similarity principle, a similar physical model of the coal fire is established, and the dynamic distribution function of the roof temperature field is fitted. Comparing the theoretical calculation value of the first breaking distance of the roof with the experimental results of the physical model, the rationality of the overlying stratabreaking model in the shallow coal seam fire area is verified.

| Models and assumptions
To study the breaking problem of the overlying strata in the coal fire area, the elastic foundation beam mechanical model as shown in Figure 2 is established. The assumptions and model parameters symbols in this paper are as follows: (1) The overlying strata in the fire area are regarded as an ideal elastic material with a certain elastic coefficient. Because the concentrated stress of the coal bodies at both ends of the roof of the shallow coal seam (buried depth less than 150 m) is small, the overlying strata are regarded as elastic simply supported beams. 32,33 (2) Coal seams and rock strata in the fire area are considered to be horizontal. Coal fires mostly occur in the mined-out area, and the combustion direction points to unmined coal seams. A coordinate system is established as shown in Figure 2. The x-axis is horizontal to the right, located on the roof neutral plane, and the y-axis is vertically upward at the starting point of combustion. The x-axis represents the strike of the coal seam, and the y-axis represents the height.

| Effect of gravity load on the beam
The effect of the gravity load formed by the selfweight of the rock formation on the roof is considered. It is assumed that the coal fire only spread to the right. The left end of the combustion zone is supported by the goaf coal pillar, and the right end is supported by the unburned coal seam. The immediate roof of the combustion zone is used as F I G U R E 2 Mechanical model of overlying strata in the fire area. the beam structure for research. The x-axis is located at the middle plane of the beam model, and the y-axis is located at the leftmost end of the beam model. In the model, the x-axis represents the strike of the coal seam, the z-axis represents the dip direction, and the y-axis represents the height. The beam model for the shallow coal seam is simply a supported beam force model. It is assumed that the overlying strata are isotropic and have a uniform mass distribution. Therefore, the simply supported beam is subjected to a uniform load q i formed by the self-weight of the rock formation. The same support forces R exists at both ends of the simply supported beam. The simply supported beam stress model of the overlying strata in the shallow coal seam is shown in Figure 3. The restrained extension in the axial direction and the restrained bending do not exist in the beam. The uniform load is the load generated by i layers of rock on the first layer of rock. 34 (1) The exposed length of the beam is L, the height is h 1 , and the width is b. The bending moment formula of any section (A-A) of the simply supported beam can be obtained through the force analysis, as shown in Equation (2). According to the bending moment formula, the stress formula at any point of the simply supported beam is obtained, as shown in Equation (3).
In Equation (3), I Z is the moment of inertia of section (A-A) with respect to the z-axis, and the value is bh 12 1 3 . It is specified that the bending moment in tension on the lower surface of the beam is positive and that on the upper surface is negative. For the stresses on the beam, the positive value is compressive stress, and the negative value is tensile stress. According to Equation (2), the maximum bending moment of the simply supported beam is located in the middle of the beam, which is a positive bending moment. And the value is . The tensile strength of rock is generally less than the compressive strength. Therefore, the main basis for judging whether a rock stratum is fractured is the maximum tensile stress value. According to Equation (3)

| Effect of temperature field on beam
According to the model assumptions, the temperature field T x y ( , ) only has temperature changes in the x-axis and y-axis directions, as shown in Figure 4. The temperature change produces thermal stress acting on the beam structure. Under free boundary conditions, the beam is not constrained by bending, and axial extension and the thermal stress on the beam consists of three parts. The first part is the compressive thermal stress caused by thermal expansion. The second part is the tensile stress due to axial strain in the x-axis direction. The third part is the stress caused by the bending moment resulting from bending. Therefore, the thermal stress formula of the beam under unconstrained conditions is shown in Equation (4). 35 Simply supported beam force model.
When the rock strata are deformed, Poisson's ratio μ of the material is considered. The simply supported beam ZHU ET AL. | 1773 has no restrained extension and restrained bending. All three parts of the thermal stress are present, as shown in Equation (5).

| Breaking model of overlying strata under thermal-mechanical coupling
In Equation (3), the negative value of stress generated by the load is tensile stress, and the positive value is compressive stress. In Equation (5), the thermal stress formed by the temperature field has a positive value of tensile stress and a negative value of compressive stress. Therefore, when gravity and the temperature field are coupled, the load stress minus the thermal stress. The negative value is tensile stress, and the positive value is compressive stress. The stress formula at any point of the overlying strata in the shallow coal seam fire area is Equation (6).
When the stress value is greater than the tensile strength of the rock, the rock stratum breaks. L in Equation (6) represents the first breaking distance of the overlying strata. The coal fires continued to burn, and the roof periodically collapses. And the regional underground space in the fire area is formed.

| Building a similar physical model
In this paper, a similar physical model of the coal fire was established based on the geological conditions of the Wuda mining area in Inner Mongolia, China. 36 The length, width, and height of the model experimental platform were 2, 0.2, and 1 m, respectively. The length and width represent the strike and dip direction of the model, respectively. Coal seams and rock strata were horizontal. The buried depth of the coal seam was 25.9 m, which belonged to the shallow deep coal seam. The immediate roof was 10.4 m thick siltstone. The experimental procedure of the model was recorded using a digital camera. The temperature data of coal seams and rock strata in the model were recorded using the infrared thermal imager. A similar physical model of shallow coal seam fire was shown in Figure 5. The properties and model parameters of each rock layer in the model were shown in Table 1.
The coal fire model needs to follow the similarity principle to ensure the accuracy of the experimental results. 37 In this experiment, the model and the prototype are required to achieve similar in stress, geometry, volumetric weight, elastic modulus, and thermal stress. Equation (7) shows the relationship between the similarity constants 38 : In Equation (7), C σ , C E , C γ , C L , and C σT denote stress, elastic modulus, volumetric weight, and geometric and thermal stress similarity constants, respectively. The specific calculation method of each similarity constant is shown in Equation (8). In Equation (8), the subscript p represents the parameter of the prototype, and the subscript m represents the parameter of the model. According to the data in Table 1, γ p (the average volumetric weight of all rock strata of the prototype) is 2475 kg/m 3 . The similar materials of the physical model are mainly sand, lime, gypsum, mica, and water. γ m (the average volumetric weight of similar materials) is 1600 kg/m 3 . According to Equation (8), the calculated value of C γ (volumetric weight similarity constant) is 1.55. In the prototype, the total thickness of the rock strata is 47.8 m, and C L (geometric similarity constant) is 50 based on the height of the test platform. According to Equation (7), the values of C σ , C E , and C σT are all 77.5. The conclusion that can be obtained in Equation (8). Based on the mechanical parameters of the rock strata in the prototype, the parameters of the model are calculated, as shown in Table 2.
Similar materials with different composition ratios were made into cylindrical test pieces for uniaxial compression experiments and splitting experiments.  The ratios in line with the mechanical properties of the rock were obtained, as shown in Table 1. Dividing each rock layer into several layers equally was not only convenient for laying similar materials, but also could simulate the bedding of rock. The baffles were fixed on both sides of the experimental platform, and each layer of similar materials was laid in turn. All the ingredients for each layer were quickly mixed uniformly, spread out in the experimental platform, and tamped. Each layer took no more than 20 min to make. A suitable amount of mica flakes were sprinkled between each layer to simulate a rock layer. After the physical model is built, it will be naturally dried for a week. After removing the baffles, allowed to dry for another week. Observation points were set every 0.1 m in the overlying strata of the coal seam. A 0.1 m cavity is excavated in the coal seam to simulate the goaf. Ignition of the coal seam to simulate the goaf fire. The coal seam burnt and spread to the right. Physical model combustion images and infrared images were recorded every 12 h.

| Overlying strata-breaking process
By observing the experimental process of the coal fire model, it was found that the movement law of the overlying strata had the characteristic of stages. The movement process of the overlying strata could be divided into four stages according to the movement scale and breaking degree of the overlying strata. The overlying strata bending deformation stage, the immediate roof first breaking stage, the roof periodically collapses stage, and the coal fire underground space formation stage. In the bending deformation stage, the overlying strata were affected by gravity and thermal stress, and the bending phenomenon occurred, as shown in Figure 6A. As the coal fire continued to burn, the exposed length of the immediate roof increased, and the deflection also increased. When the coal fire burnt for 144 h, horizontal fissures appeared between the rock strata. The immediate roof broke when the tensile stress on the immediate roof exceeded the tensile strength of the material. The exposed length of the roof at the time of breaking was the first breaking distance. It could be seen from Figure 6B that the first breaking distance of the immediate roof was 0.59 m after burning for 168 h.
The fire continued to advance into the unburned coal seam, and the roof collapsed periodically. When the coal fire burnt for 216 h, longitudinal cracks appeared in the rock strata, and the caved zone was formed in the underground space of coal fires, as shown in Figure 6C. At this time, the collapse angle formed by the longitudinal crack near the goaf was 23 continued to collapse as coal fires spread. The rock strata above the immediate roof were wider and more complete when they collapsed, forming a fractured zone. The sagging zone was formed when the uncollapsed overlying strata bent, and the surface subsided after continuous burning for 264 h, as shown in Figure 6D. Longitudinal cracks further increased. The collapse angle formed by the longitudinal crack near the goaf was 30.7°. The collapse angle formed by the longitudinal crack near the combustion zone was 43.9°. These cracks were possible to develop into cracks through the ground surface. The cracks increased the oxygen supply for coal combustion, and thermal anomalies appeared on the ground surface of the fire area. According to the surface cracks and the sagging zone, the position of the combustion zone of the coal fire can be calculated, and then the fire extinguishing work could be guided.

| Overlying strata temperature field
The temperature distribution data of the shallow coal fire model were recorded by the thermal imager. The body of the model's similar material was sand, therefore the emissivity parameter of the thermal imager was set to 0.9. 39 Atmospheric temperature, and relative humidity were measured by corresponding equipment. The model width is only 0.2 m, ignoring the loss of energy conduction in the width direction. Therefore, the model surface temperature can be approximately equal to the temperature inside the model. The thermal images of the four stages of the coal fire strike model are shown in Figure 7. The center of combustion translates uniformly with time. The isotherm range continues to expand, reaching the maximum at the stage of the roof periodically collapsing and then decreasing gradually. The red, green, and blue isotherms are 20°C, 60°C, and 100°C, respectively. The combustion front gradually tilts to the upper right, resulting in isotherms that are flat on the upper left and steep on the upper right, as shown in Figure 7A-D. In Figure 7D, the overlying strata collapse on a large scale, forming cracks through the strata. The heat generated by coal combustion is transferred into the crack, and there is a temperature gradient. A temperature difference is formed between the cracks and the outside world, resulting in an air pressure difference, which further intensifies the flow of air. This phenomenon proves that cracks are important oxygen supply channels for underground coal fires.
According to the breaking theory of the overlying strata in the second part of this paper, it is necessary to obtain the temperature distribution function of the immediate roof before breaking. The immediate roof of the coal seam in the coal fire physical model is the No. 1 rock stratum. The length of the burned coal seam is 1.4 m, so the immediate roof dimensions are 1.4, 0.2, and 0.208 m in length, width, and height, respectively, as shown in Figure 7A,B. According to the model assumptions, the immediate roof is at the same temperature in the width direction. According to the coordinate system of Figure 4, the plane image and threedimensional (3D) image of the immediate roof temperature distribution are drawn, which are shown in the first and the third layer images in Figure 8, respectively. Based on the six groups of temperature field data before the first breaking of the immediate roof, the temperature field image and distribution function are fitted. The Lorentz 2D surface function is selected for fitting, 40 and its expression is Equation (9). A represents the highest temperature of the fitted temperature field, and a represents the lowest temperature of the fitted temperature field. x c and y c are the abscissa and ordinate of the highest temperature point of the immediate roof. Comparing Figures 6 and 7, it is found that the highest temperature point occurs at the position where the combustion front contacts the immediate roof. There is a gap between the ash on the left side of the highest temperature point and the immediate roof, which cannot support the roof. Therefore, the abscissa of the highest temperature point is considered to be the length L of the beam model.
The fitting results are shown in Figure 8. The second and the fourth layer images are the immediate roof temperature field fitting plan and 3D images, respectively. In the 3D graph, the x-axis and y-axis represent the x-direction and y-direction of the immediate roof in the breaking model, and the z-axis represents the temperature. Among them, the temperature value of the third layer image directly corresponds to the z-axis coordinate. The image temperature value of the fourth layer is less than the z-axis coordinate of 30. The specific fitting parameters are shown in Table 3.
The mean value of the fitted correlation coefficient R 2 of the six groups of data is 88.73%. x c is represented by the beam length L, and other parameters are averaged. The immediate roof temperature distribution function T x y ( , ) of the coal fire model is obtained, as shown in Equation (10):

| Breaking law of overlying strata under thermal-mechanical coupling
Based on Equation (1), the own load q 1 of the first rock layer is calculated: Calculate the load (q 2 ) 1 formed by the two layers of rock on the first layer: According to Equations (11) and (12) L takes any value in the range of 0 to 1.4 m and is substituted into Equation (13). The calculation results show that the location of the maximum tensile stress on the immediate roof is always at x = L/2 and y = −0.104. Taking L = 0.5 as an example and substituting it into Equation (13), a 3D diagram of the immediate roof stress distribution, as shown in Figure 9. The x-axis and y-axis represent the strike and height of the immediate roof, and the z-axis represents the stress, with the negative value being tensile stress. In Figure 9, the red surface represents the stress distribution under thermal-mechanical coupling, and the blue surface represents the stress distribution under the load only. At the lower right of the immediate roof, the red surface is slightly higher than the blue surface. This phenomenon indicates that the high temperature reduces the tensile stress of the simply supported beam. In the coal fire model, the dominant part of the stress on the immediate roof is the gravity load.
Substituting x = L/2 and y = −0.104 into Equation (13), the formula for the maximum stress on the immediate roof under the thermal-mechanical coupling is obtained, as shown in Equation (14). According to Equation (14), the calculation result of the first breaking distance of the immediate roof is 0.575 m. In the coal fire similar physical model, the theoretical calculation results of the first breaking distance are basically consistent with the experimental results. When the immediate roof is only affected by the load, the first breaking distance is calculated to be 0.567 m according to Equation (3). According to the calculation results, the coal fire reduces the tensile stress in the middle of the lower surface of the roof, thereby increasing the first breaking distance of the roof. The fire continues to burn, and the overlying rock of the coal seam periodically collapses. The shallow coal seam fire area is prone to cracks running through the surface. The high-temperature area of the coal seam is determined according to the inclination angle of the longitudinal cracks, and a reasonable fire extinguishing scheme is implemented. The evolutionary law of inclination angles of cracks is summarized in Section 3.2. The collapse angle of the longitudinal cracks near the combustion zone is slightly larger than that of the longitudinal cracks near the combustion cavity zone. At the same time, according to Figure 7, it can be seen that the longitudinal crack temperature is higher in the combustion zone. According to these two characteristics, the combustion direction and combustion center of the coal fire can be deduced. The hightemperature area on the surface corresponds to the combustion center or the longitudinal cracks near the combustion zone.

| CONCLUSION
The breakage of the overlying rock of the shallow coal seam fire area creates longitudinal cracks, which become new oxygen supply channels in the fire area and accelerate coal fire combustion. There are gravity and temperature field in the fire area, and the roof is subjected to the coupling effect of load stress and thermal stress. Based on the elastic foundation beam theory, this paper establishes the breaking model of the overlying strata in the shallow coal seam fire area. The breaking theory of overlying strata is verified by coal fire similar physical experiment. The detailed conclusions are as follows.
(1) Beam theory is used to study the movement of overlying strata in coal seams. The elastic simply supported beam model is used to explain the movement law of the roof based on the stress distribution of the overlying strata in the shallow coal seam. At the same time, the influence of the dynamic temperature field on the overlying strata is considered. A breaking model of overlying strata under thermal-mechanical coupling is established. (2) A similar physical model for the shallow coal fire based on the similarity principle is established. The model temperature field distribution data is recorded using a thermal imager. The dynamic distribution function of the temperature field at the immediate roof of the coal fire is obtained by fitting, and the average fitting correlation coefficient is 88.73%. The highest point of the temperature of the immediate roof occurs at the contact position between the combustion front and the roof, which is also the position where the unburned coal seam effectively supports the roof. (3) According to the proposed breaking theory of the overlying strata, the maximum tensile stress of the immediate roof is located in the middle of the lower surface, and the first breaking distance is 0.575 m. In the coal fire similar physical experiment, the first breaking distance of the roof is 0.59 m, and the break position is in the middle. The theoretical calculation values are in good agreement with the experimental results. The coal fire continues to burn, and the overlying strata continue to collapse. Periodic collapse occurs, forming longitudinal cracks and coal fire underground spaces. (4) The motion law of the roof under the thermalmechanical coupling is compared with that under the load only. Thermal stress will reduce the maximum tensile stress value of the roof in the shallow coal seam fire area and increase the first breaking distance.