Determination of the Caving Zone and Fracture Zone Heights by Using the LK-Means Algorithm

. The caving zone and fracture zone frequently appear after coal excavation. However, the caving zone and fracture zone can hardly be obtained with an empirical equation. Therefore, identifying a proper method to determine the caving zone and fracture zone is critically important to engineering practice. In this paper, Universal Distinct Element Code numerical simulations were conducted. Furthermore, the LK-means algorithm was used to determine the caving zone and fracture zone heights. To verify the validity of the proposed method, two engineering case studies were used. It could be found that the height calculated by the proposed method agrees well with that determined in engineering tests. Hence, the proposed method can be used in engineering practice.


Introduction
After coal is excavated, the rock mass above the goaf can generally be classified into three zones [1,2], which is the basis of coal excavation design [3,4].Based on the ground pressure and in situ tests, the upper rock mass of the goaf can be divided into three zones [5,6]: the caving zone, fracture zone, and curve subsidence zone.
Due to the importance of these three zones in coal excavation design, gas drainage and ground building structures, the study of the three zones has remained a hot topic.Fu et al. [7] studied the caving zone height in the Shangwan Coal Mine by using ANSYS software.Moreover, the caving zone height in the Shangwan Coal Mine was analyzed.Through the analysis of numerical simulation results and in situ test data, it could be found that the caving zone height increases with increasing length of the working face and working face height, while the influence of the working face was less notable, and an empirical equation for calculating the caving zone in the Shangwan Coal Mine was proposed.Song et al. [8] found that the pressure on the support remarkably increases with increasing working face length and concluded that the caving zone increases with increasing working face length; however, in their study, no method for calculating the caving zone height was proposed.Teng et al. [9] found that the coal location greatly influences the fracture zone height and caving zone height.This conclusion is important for coal excavation design.Therefore, coal excavation location selection is critical.Xu et al. [10], Yi et al. [11], and Xu et al. [12] conducted engineering experiments and concluded that the primary key stratum imposes a notable influence on the caving zone height; hence, the primary key stratum location should be considered in estimating the caving zone height, and these results were also verified in other studies.Many other researchers have measured the three-zone height [13][14][15][16][17][18][19][20][21]; however, they mainly focused on the measurement of the caving zone height and fracture zone height in the field, which is a time-consuming and noneconomical process.Until now, no method has been determined for estimating the height of the three zones.Therefore, proposing a method for estimating the caving zone height and fracture zone height is critical for engineering practice.
To resolve this problem, the Universal Distinct Element Code (UDEC) numerical simulation method and the LK-means algorithm were combined to determine the caving zone height and fracture zone height.UDEC numerical simulations were conducted first.Subsequently, the displacement magnitude was exported by using FISH language.Via the use of the LK-means algorithm, the caving zone and fracture zone were determined, and the caving zone height and fracture zone height were then determined.To verify the validity of the proposed method, two numerical simulation examples were given, and the numerical simulation results were compared to in situ test data.The difference between the calculation results and the in situ test data is small.The proposed method is valid and can be used to calculate the heights of the caving zone and fracture zone.

Determination of the Caving Zone and
Fracture Zone Heights In this paper, the upper rock mass zone of the goaf was classified into a caving zone and a fracture zone based on the displacement of the upper rock mass.To implement the process, the LK-means algorithm was used, and the classification process is explained below.

LK-Means
Algorithm.The LK-means algorithm is a combination of the XK-means clustering algorithm, Levy flying algorithm, and K-means algorithm [22].The K-means clustering algorithm attempts to divide n groups of data into k classes, and each data point belongs to the nearest cluster of center points.For example, for a set of n groups of data ðx 1 ; x 2 ; x 3 ; ::::::; x n Þ, this set of data is ddimensional.With the use of the K-means clustering algorithm, these data are divided into k (k ≤ n) sets (S ¼ S 1 ; S 2 ; S 3 ; ::::::S k ) to minimize the variance, which can be described as follows: where u i is the mean value of the points in S i .In the clustering process, the above is equivalent to minimizing the squared deviation of points in the same cluster. argmin It can be deduced as follows: However, the K-means algorithm can easily converge at local optima.To solve this problem, the XK-means clustering algorithm was proposed.In this algorithm, a random vector θ j is added to the clustering centroid of the K-means algorithm.
where θ j is a random search vector, which can be determined as follows: The relationship between a i and b i can be expressed as follows: where β is a coefficient with a range of [0, 1].Before the next iteration, the value of b i can be expressed as follows: where α is a constant value, which occurs within the range of [0, 1).However, based on previous studies, the XK-means algorithm cannot prevent convergence at local optimal values.Hence, a new clustering algorithm, namely, the LK-means algorithm, was proposed by Dasgupta et al. [23].This algorithm is inspired by the XK-means algorithm, Levy flying algorithm, and K-means algorithm.
In the LK-means algorithm, the Levy flight path increases the diversity of random search vectors, and Levy flight strategies can be used to generate the new search vector, which can be expressed as follows: where μ is a random number within the range of [0, 1], and sign½rand − 1 2 is the direction of the vector, which has three values: 0, 1, and −1.
In the exploration process, the number of iteration steps increases, which can be expressed as follows [24]: The Levy flight strategy exhibits both large and small steps in the search process.To imitate a lambda stable distribution by a random behavior similar to that of Levy flights, Mantegna's algorithm was proposed, which can be expressed as follows: where s is the random step number, and γ ¼ 1:5; λ ¼ 1 þ γ; ρ, and v obey a normal distribution: 2 Advances in Civil Engineering The Levy flight strategy can be generated based on the above equations.
A flowchart of the process of the LK-means algorithm is shown in Figure 1.

Determination of the Caving Zone and Fracture Zone
Heights by Using the LK-Means Algorithm.To determine the caving zone and fracture zone heights, UDEC numerical simulations were conducted, and the block displacement was then obtained by using FISH language (secondary embedded language in the UDEC).Then, the LK-means algorithm was used to determine the fracture zone height and caving zone height.In our study, based on the displacement magnitude, the displacement data can be divided into two clusters, namely, a small displacement zone and a fracture zone, and the fracture zone can be further divided into two parts: a caving zone and another part of the fracture zone.It should be noted that the caving zone exhibits a larger displacement than the other part of the fracture zone.The determination process can be described as follows.
Step 1: The parameters of the LK-means algorithm are initialized.In this study, the upper rock mass of the goaf can be divided into two parts; hence, the K value is 2.
Step 2: Based on engineering practice, the UDEC numerical simulation model was constructed, and the block displacement was obtained by using the embedded FISH language in the UDEC.
Step 3: By using the LK-means algorithm, the upper rock mass of the goaf can be divided into two parts: a small displacement zone and a fracture zone.
Based on the geometric parameters, the height of the fracture zone can be determined.
Step 4: Based on the above results, the fracture zone can be further divided into a caving zone and another part of the fracture zone by using the LK-means algorithm, and the height of the caving zone can then be determined.
Step 5: The process is terminated.
The flowchart of the process for determining the caving zone height and fracture zone height is shown in Figure 2.

Example Verification of the Proposed Method.
To verify the validity of the proposed method, an engineering practice example is given.A goaf exists in the Hongqing Coal Mine, which is located in Inner Mongolia in China, and the burial depth is 1,100 m.As shown in Figure 3, the size of the numerical simulation model is 196 m × 240 m, the width of the goaf is 100 m, the height of the goaf is 6.14 m, and the height of the model is 196 m.However, the buried depth is 1,100 m.Therefore, extra stress was loaded onto the top of the model, and the extra stress reached 22 MPa.In the numerical simulations, the Mohr-Coulomb model was used, and the corresponding mechanical parameters of the numerical simulations are listed in Table 1.
After the numerical simulations, a contour map of the displacement of the upper rock mass of the goaf can be obtained, as shown in Figure 4.
To determine the caving zone height and fracture zone height, the LK-means algorithm was utilized.To reduce the calculation time, the displacement at the center of the goaf was used.Based on the LK-means algorithm, the fracture zone height was determined, as shown in Figure 5.

Advances in Civil Engineering
Because the fracture zone includes the caving zone, the fracture zone can be further divided into the caving zone and another part of the fracture zone based on the displacement magnitude, and the corresponding caving zone height can be obtained.The caving zone height is 48.07 m, as shown in Figure 6.
To validate the rationality of the proposed method, the caving zone height and fracture zone height were obtained.The caving zone height is 44.50 m, and the fracture zone height is 110.30m, which are quite close to the numerical simulation results (the caving zone height is 48.07 m, and the fracture zone height is 108.13 m).The difference between the numerical simulation results and engineering test data is    Advances in Civil Engineering small, and the error is acceptable, indicating that the proposed method is reliable.

Another Example.
To further verify the validity of the proposed method, another example is given.In the Dongqu Coal Mine, the buried depth is 200 m, and the height of the working face is 3.19 m.The geometry of the UDEC numerical simulation model is shown in Figure 7.
The mechanical parameters of the numerical simulation model are listed in Table 2.
After the numerical simulation model was constructed, a contour map of the displacement magnitude was obtained, as shown in Figure 8, and the displacement magnitude was exported by using FISH language embedded in the UDEC.The exported data were used for determining the caving zone height and fracture zone height.
Based on the displacement magnitude and the LK-means algorithm, the fracture height and caving zone height were determined, as shown in Figure 9.
By using the LK-means algorithm, the fracture zone height and caving zone height were determined: the caving zone height was 13.17 m, and the fracture zone height was 43.31 m.In the field, the caving zone height and fracture zone height were measured: the caving zone height was 11.1 m, and the fracture zone height was 42.20 m.The fracture zone height and caving zone height calculated by using the LK-means algorithm are quite close to the measured values, indicating that the proposed method is reliable and can be applied in engineering practice.

Discussion
Caving zones and fracture zones frequently emerge when coal is excavated, and the caving zone height and fracture zone height are critically important for coal excavation design and gas drainage design.However, until now, there has been no empirical equation or method for estimating the caving zone height and fracture zone height.

Advances in Civil Engineering
To resolve this problem, the UDEC numerical simulation method and LK-means algorithm were combined to determine the caving zone height and fracture zone height.The determination process is as follows: UDEC numerical simulations were first conducted.Thereafter, the simulated displacement magnitude was exported by using FISH language (embedded secondary language in the UDEC).The upper rock mass of the goaf was first divided into a fracture zone and a small displacement zone, and the fracture zone height was determined.Then, the fracture zone was divided into a caving zone and another part of the fracture zone, and finally, the caving zone height was determined.The classification method is based on the LK-means algorithm.
To verify the validity of the proposed method, two examples were given, and the corresponding caving zone height and fracture zone height were determined.Furthermore, the caving height and fracture zone height were measured in situ.By comparing the numerical simulation results and the in situ test data, the calculation results were quite close, indicating that the proposed method is reliable and can be applied in engineering practice.
However, many other factors may influence the fracture zone height and caving zone height, such as gas and underground water.These factors were not considered in our manuscript, which will be our next task.

Conclusions
To estimate the fracture zone height and caving zone height, UDEC numerical simulations were conducted.By using the LK-means algorithm, the caving zone height and fracture zone height were calculated, and the main conclusions of this paper can be summarized as follows: (1) Based on engineering practice, the UDEC numerical simulation model was constructed, and numerical simulations were conducted.By using the LK-means algorithm, the upper rock mass of the goaf was divided into two parts: a fracture zone and a small displacement zone.Subsequently, the fracture zone was further divided into a caving zone and another part of the fracture zone.Then, the fracture zone height and caving zone height were determined.(2) To verify the validity of the proposed method, two examples were given.The simulated fracture zone height and caving zone height were quite close to those obtained in engineering tests, which indicates that the proposed method is reliable.The proposed method can be applied to estimate the fracture zone height and caving zone height in engineering practice.Advances in Civil Engineering

FIGURE 2 :
FIGURE 2: Flowchart of the determination of the caving zone height and fracture zone height.

FIGURE 3 :FIGURE 4 :
FIGURE 3: Geometry of the numerical simulation model.

FIGURE 5 :
FIGURE 5: Determination of the fracture zone height.

FIGURE 6 :
FIGURE 6: Determination of the caving zone height.

TABLE 2 :
Mechanical parameters of the numerical simulation model.