In view of the three-dimensional dynamic abutment pressure, the influence of the far-field hard stratum (FHS) in deep, thick coal seams is indeterminant. Based on elastic foundation theory, a three-dimensional dynamic prediction model of the abutment pressure was established. Using this model, the dynamic change in the coal seam abutment pressure caused by the movement of the FHS was studied, and a method for determining the dynamic change range of the abutment pressure was developed. The results of the new prediction model of the abutment pressure are slightly higher than the measured values, with an error of 0.51%, which avoids the shortcomings of the results because the Winkler foundation model results are lower than the measured values and have an error of 9.98%. As time progresses, the abutment pressure and its distribution range are affected by the FHS movement, which has the characteristics of gradually increasing dynamic change until the FHS fractures. The peak value of the abutment pressure increases linearly with time, and the influence range increases with time following a power function with an exponent of less than 1. The influence range of the FHS movement on the abutment pressure ahead of the working face, behind the working face, and along the working face is 10 times, 25 times, and 17 times the mining thickness, respectively. According to the actual geological parameters, the dynamic change range of the coal seam abutment pressure was determined by drawing an additional stress curve and by determining the threshold value. These research results are of great significance to the partition optimization of the roadway support design of deep, thick coal seams.

Abutment pressure is the main reason for the compression and shear failure of the rock mass around the goaf. After the coal seam is mined, the abutment pressure acts on the rock body, which causes the cracks in the rock body to expand significantly. This causes the rock surrounding the adjacent working face roadway to be greatly deformed [

Based on this, many scholars have carried out fruitful research. The most common method used is to establish a mechanical model based on the roof collapse angle [

In the above studies, in the study based on the roof collapse angle, the influence of the mechanical properties of the interlayer between the HS and the coal seam was not considered, but in the study based on the Winkler foundation model, it was considered. However, there are some disadvantages to using the Winkler foundation model to predict the abutment pressure. One of which is the assumption that the roof collapse angle is 90°, which contradicts the actual situation, and the roof collapse angle has an important influence on the abutment pressure distribution [

Far-field hard stratum (FHS) is a kind of HS that is located far away from the coal seam under certain conditions, so it is more difficult for it to collapse in a short time period after mining [

The goal of this study is to establish a three-dimensional dynamic abutment pressure prediction model to solve the problem of the influence of the FHS movement on the abutment pressure distribution being indeterminant. Subsequently, the dynamic evolution law of the coal seam abutment pressure is clarified, and a method to determining the distribution range of the abutment pressure is proposed. The research results clarify the abutment pressure distribution of the rocks surrounding the goaf in deep, thick coal seams, which is conducive to the reasonable selection of a support form. In the subsequent sections, first, the prediction model is introduced and validated using field data. Then, the validity of the model is validated, and the dynamic evolution law and distribution range of the abutment pressure are studied according to the model.

(1) Thin plate hypothesis

The thickness of the thin plate to the minimum side length ratio is less than 1/5–1/8 [

(2) Assumption of an elastic foundation

The FHS is clamped between the overlying strata and underlying strata. After the coal seam is mined, the FHS can only move downward due to the

The main mining method used for thick coal seams is the longwall top coal caving (LTCC) method [

The FHS mechanical model was established, as shown in

In addition to the FHS, there are many layers of HS in the strata overlying the coal seam. The HS load transfer is shown in _{m+1} transfers the load to HS_{m}, causing HS_{m} to generate additional stress. Then, HS_{m} transfers the load to HS_{m–1}, causing HS_{m–1} to generate additional stress, that is, HS_{m–1} generates additional stress due to the HS_{m+1} load and the HS_{m} load transfer. The HS mechanical model is similar to the FHS mechanical model.

As shown in the FHS mechanical model (

After the coal seam is mined out, the overlying stratum subsides until it is stable, and its movement time is from the beginning of the movement until stability. However, it is very difficult and unnecessary to obtain the complete movement track of a certain rock boundary. Therefore, selecting the spatial location of the key time nodes of the rock movement is a feasible method. The key time nodes of the FHS movement can be divided into the start of the movement, touching the gangue, before fracturing, and after fracturing.

(1) Displacement of edge CD at the beginning of the FHS movement

The movement time of the FHS lags behind the movement time of the direct roof, which also causes the change in the coal seam abutment pressure to lag. When the underlying rock layer collapses and falls behind, a separation layer appears between the FHS and the underlying rock layer, and the stress balance is broken. Therefore, it can be assumed that the time when the FHS separation layer first appears is the starting time. At this time, the FHS has just separated from the underlying rock layer, and it has not had enough time to subside, so the displacement of edge CD is 0:

(2) Displacement of edge CD when the FHS touches the rock block

After the FHS starts to move, edge CD gradually subsides. When the edge touches the goaf rock block, edge CD of the FHS changes from the friction Q of the rock block after the fracturing of the FHS to the resultant force of friction Q and the supporting force F_{Z} of the goaf rock block, as shown in

At this time, the allowable movement distance of edge CD is the space between it and the goaf rock block, which is due to the compression effect of the gravity of the rock block between the caving zone and the FHS, resulting in the reduction of the bulking factor of the gangue in the caving zone. The length of

The deflection _{Z} is as follows:

where

Considering the quasi consolidation characteristics of the gangue [

where

Combining

The displacement of edge CD touching the goaf rock block can be obtained by solving

(3) Displacement of edge CD after the FHS touches the goaf rock block and before it fractures

After the FHS touches the goaf rock block, the length

where

(4) Displacement of edge CD after the FHS fractures

After the FHS fractures, the simply supported edge changes from edge CD before fracturing to edge AB after fracturing. At this time, edge AB is only affected by the friction Q, as shown in

The abutment pressure of the coal seam was obtained by calculating the force that each HS exerts on the foundation using the HS mechanical model and the stress at the different positions of the coal seam according to the load transfer model.

Taking the FHS as an example, the force the FHS exerts on the foundation was obtained by solving for its deflection and foundation coefficients. The deflection was obtained using the finite difference method (FDM) to construct algebraic equations according to the deflection differential equation and the boundary conditions of the thin plate. The calculation process of any HS force acting on the underlying rock is similar to that for the FHS force acting on the underlying rock.

(1) Determination of the foundation coefficients

According to the simplified elastic space method [

As can be seen from

where

The elastic modulus

where

(2) Deflection differential equation of the FHS

When the Winkler elastic foundation model is used, the deflection differential equations of the FHS in S1, S2, and S3 can be summarized as follows:

where

When the Kerr elastic foundation model is used, the deflection differential equations of the FHS in areas S1, S2, and S3 can be summarized as follows:

where

(3) Inner boundary conditions

Before the failure of the FHS, at the boundary between S1 and S2, the deflection and rotation angles are continuous. The S2 and S3 boundary conditions are similar to the S1 and S2 boundary conditions:

After the FHS breaks, the AB, BC, and AD boundaries are the existing deflections of the FHS:

(4) Outer boundary conditions

A_{2}B_{2}, B_{2}C_{2}, and A_{2}D_{2} are fixed support boundaries; the rotation angle and deflection are 0; C_{2}D_{2} is a simply supported boundary; the bending moment of C_{2}D_{2} is 0; and the boundary displacement at different times is obtained from

(5) The force that the FHS exerts on the foundation

Based on the Winkler foundation model, the Kerr foundation model [

where

Because the force of the HS acting on the underlying strata is not evenly distributed, the equivalent-load method was used to divide the force into several small areas and to synthesize the force on them into a small concentrated force. When several concentrated forces act on the foundation, by applying the superposition principle, the additional stress at any position is

When the coal seam is not excavated, it is already in the

The field measured abutment pressure data are from the 14201 working face and the 14202 tailgate of Majialiang mine [

HS | D/(GPa*h^{3}) |
k_{w1}/(GPa*m^{–1}) |
k_{w2}/(GPa*m^{–1}) |
q/(GPa) | η |
---|---|---|---|---|---|

HS1 | 1640 | 1.049 | 1.049 | 0.014 | |

HS2 | 8022 | 0.003 | 0.43 | 0.01 | 1.20 |

HS3 | 30016 | 0.003 | 0.07 | 0.026 | 1.15 |

HS4 | 30016 | 0.003 | 0.07 | 0.026 | 1.15 |

HS5 | 30016 | 0.003 | 0.07 | 0.026 | 1.15 |

FHS | 20067 | 0.008 | 0.06 | 0.01 | 1.15 |

According to the abutment pressure prediction model, the influences of the distance between each HS and the coal seam and the roof collapse angle were fully considered, and the forces of each HS on the underlying strata were calculated when they were fracturing. The curve of the HS1 abutment pressure and the comparison between the new prediction model, the traditional prediction model, and the field measurements of the abutment pressure is shown in

As can be seen from

As can be seen from

As can be seen from

Based on the above-described evidence, the prediction model proposed in this paper can effectively predict the abutment pressure on the goaf rock surrounding deep, thick coal seams, and the predicted abutment pressure meets the safety requirements of the support design.

It is necessary to define the movement process of the FHS to determine how the dynamic evolution of coal seam abutment pressure is affected by FHS. The movement process of the FHS is the change in the foundation’s reaction at different times when the FHS is clamped between the overlying strata and the underlying strata. Using

As can be seen from

The contour map of the foundation reaction at different times during the FHS movement shows that the following. As time progresses, the simple boundary of the FHS gradually subsides, which causes the stress balance between the FHS and the underlying strata to be broken, and the range of the unbalanced force (i.e., the force on the underlying strata) increases rapidly, which remains unchanged when the FHS is touching the rock block. However, the unbalanced force increases slowly, resulting in the stress on the underlying strata to be in a state of dynamic change until the FHS fractures. The response of the underlying strata to the unbalanced force is to continually adjust its internal elastic-plastic area to adjust the stress balance. In particular, the strength of the coal body in the underlying strata is low. When the coal body changes from elastic deformation to plastic deformation, plastic expansion will occur, resulting in the displacement of the coal wall.

In summary, the FHS force on the underlying stratum increases with the advancement of the working face until it reaches the maximum value when the FHS breaks. According to the load transfer model, the FHS movement continuously affects the abutment pressure of the coal seam until the FHS fractures. However, it is not clear how the FHS affects the dynamic change of the coal seam’s abutment pressure. Therefore, at a distance of 0.5 m from the coal wall of the goaf and 141 m behind the working face, the additional stress on the coal seam along the advancement direction of the working face and along the working face from the beginning of the movement to its fracturing was calculated and plotted in

As can be seen from

In summary, over time, the FHS continues to subside until it breaks, which causes the coal seam abutment pressure and its influence range to increase. The influence range of the FHS movement of the coal seam’s abutment pressure ahead of the working face is more than 10 times that of the thickness of the coal seam, the influence range of that behind the working face is more than 25 times, and the influence range of the lateral abutment pressure is more than 17 times. Thus, the influence of the FHS movement on the abutment pressure distribution along the advancement direction of the working face is greater than that along the working face. Both along the advancement direction of the working face and along the working face, the peak value of the stress increases linearly with time, and the influence range increases with time following a power function with an exponent of less than 1. The predicted results of the traditional model for the influence range and maximum stress of the abutment pressure are lower than those of the proposed model by about 5%–7% and 13%, respectively.

As can be seen from the above section, the FHS movement plays a decisive role in the dynamic change characteristics of the coal seam abutment pressure, and this change is mainly manifested as the change in the additional stress. Regardless of how the coal seam abutment pressure changes, the distribution of the additional stress can be summarized as a curve that increases initially and then decreases. When the additional stress is 0, the corresponding distance from the working face is the influence range of the FHS movement on the abutment pressure along the advancement direction of the working face, and the corresponding distance from the coal wall is the influence range along the working face. Under actual geological and mining conditions, the abutment pressure prediction model can be used to draw the additional stress curve of the coal seam at different times during the FHS movement. Given the threshold value of the additional stress judgment, the influenced range of the FHS movement on the coal seam abutment pressure at different times can be determined based on the corresponding range of the additional stress curve.

In this paper, we proposed a three-dimensional dynamic abutment pressure prediction model, which clarifies the dynamic evolution of the abutment pressure affected by the FHS movement. It can also be used to determine the influences range of the FHS movement. The calculation results of the Kerr elastic foundation were used to modify the Winkler elastic foundation results, and the abutment pressure prediction model was established by combining the FHS structure and the bearing characteristics. Using the model, the dynamic evolution of the abutment pressure and the determination method of its distribution range were obtained. The results of this study are as follows.

The prediction results of the model were obtained and compared with the field data. The results show that the error in the peak abutment pressure predicted by the new model is 0.51%, and the predicted value is slightly larger than the measured value, so it meets the safety requirements of the working face and the roadway support design based on the abutment pressure.

The FHS movement causes the peak value and the distribution range of the abutment pressure to increase linearly and as a power function with an exponent of less than 1 with time until the FHS fractures.

The influence range of the FHS movement on the abutment pressure ahead of the working face, behind the working face, and along the working face is 10 times, 25 times, and 17 times the mining thickness, respectively.

Under actual geological and mining conditions, the abutment pressure distribution range was obtained by drawing the additional stress curve of the coal seam affected by the FHS movement and combining it with the threshold value.

In the future, we intend to apply the prediction model results to the optimization of roadway support designs in deep, thick coal seam mining. For example, according to the variation in the abutment pressure’s distribution range with time, the roadway support’s form can be designed in different sections.

We would like to give our sincere thanks to the editors and anonymous reviewers for their constructive opinions and comments. This work was funded by the National Natural Science Foundation of China [U1810102]. We thank LetPub (