Introduction

Roof fall accidents currently account for about 40 % of all coal mine accidents (shown in Table 1), ranking first among those in China. In recent years, roof fall accidents at the coal mine workface have significantly declined, due to improvements in the fully mechanized coal-mining equipment, the development of supporting technology, and especially the promotion of a comprehensive workface support quality-monitoring system. Consequently, the number of deaths caused by roof fall accidents has been effectively controlled and reduced lately. However, roof fall accidents continue to significantly contribute to a growing annual number of coal mine accidents. According to incomplete statistics, the number of deaths from roof fall accidents accounts for 50 % of the deaths from all accidents in small coal mines in a number of counties and townships in China. Therefore, a further study of technologies aiming at prevention of roof fall accidents must be undertaken to assure coal mine safety in the future.

Table 1 The statistical analysis of coal mine accidents in China
Full size table

Many scholars had done a lot of researches in the analysis of roof separation for roof control. Ling et al. (2003) comprehensively investigated the condition of the rock medium, environmental occurrence conditions, and engineering factors. In addition, they established a dynamic model for predicting the roof stability of mining fields using artificial neural network tool. Tan et al. (2006) proposed that the occurrence of peaks in the roof separation velocity was applicable for forecasting the falling of inferior roofs. In addition, they developed a roof separation telemetering system for measuring and forecasting roof falls. Zhang et al. (2010) gave the necessity conditions of load, span, and deflection, which were proposed for roof separation in the immediate roof. Tang et al. (2010) analyzed the relationship between the roof separation and the roof deformation under the support method of combined support with anchor. Bertoncini and Hinders (2010) made fuzzy classification of roof fall predictors in microseismic monitoring. Saeedi et al. (2013) presented the effect of exposed area geometry on potential roof falls using the 2D numerical modeling program. A novel fuzzy inference system has been made for predicting roof fall rate in underground coal mines (Razani et al. 2013; Farid et al. 2013; Ghasemi and Ataei 2013). Yan et al. (2014) studied the influencing factors of the roof separation of the roadway with extra-thick coal seam. The underground roadway roof separation was monitored in time by fiber Bragg grating sensors (Zhao et al. 2015). Salient findings of field instrumentation data analysis such as mining-induced vertical stress development over the pillars, load over the support, roof separation at different horizons, and convergence of roof in rooms with face advance were presented (Satyanarayana et al. 2015). Khan et al. (2015) use wireless sensor networks to determine cluster breakage by sudden disruptions caused by roof falls. Xie and Xu (2015) took ground-penetrating radar (GPR) as a means for roof separation detection and analyzed its feasibility for use in detecting roof separation both theoretically and experimentally.

In the performance evaluation of roadway support, roof separation has always been a major index. When roof separation exceeds a certain critical value, the roof is regarded as unstable. Under these conditions, unless timely support measures are taken, a roof fall accident is likely to occur. At present, understanding the real-time changes of the roof separation is mainly achieved by reading the indicator data, but the change trend cannot be predicted. Finding a way to predict the change trend of roof separation will play a very important role in roof control.

Roof separation is induced by multiple geological factors and complex mechanical behaviors that are not yet fully understood. To effectively predict and control roof fall accidents, a clear understanding of the characteristic variations of roof separation is necessary. However, it is not yet clear how to use mathematical models to predict roof separation. Moreover, predicting this complex process is very difficult, due to the many parameters involved. According to the gray system theory proposed by Deng (1987), an irregular information sequence in an arbitrary stochastic process may be transformed into a regular information sequence by generating the transformation. When using this theory, there is no need to know the distribution characteristics of the original data in the mathematical model. Moreover, for a small number of discrete data sequences, the modeling requirements can be met simply by generating the transformation several times. Furthermore, the prediction results from the application of this model are more accurate and more consistent with the real conditions, compared with other prediction models.

The gray model has not been used before to predict the change trend of roof separation, so that this article is the first to attempt such a prediction. When using a gray theoretical analysis, only a small amount of existing roof separation data is required to predict the development trend of roof separation. The results obtained can then provide auxiliary decision-making information for predicting roof fall accidents.

Gray algebraic curve model-based prediction method

The gray system primarily includes system modeling, model testing, correlation analysis, clustering evaluation, decision making, and planning. The gray system theory is used to reduce the uncertainty resulting from random factors, thus increasing the objective regularity of a system through the mathematical processing of the original system information sequences. The system is then modeled using corresponding mathematical differential equations (Liu 1999). The discrete time series generated by the original sequence accumulation is transformed into a differential equation that is continuous in time. The resulting function model is known as the gray model (GM).

The exponential solutions of the GM model differential equation have been used to determine that the GM prediction method is only applicable to conditions, in which the original sequence shows exponential variations. The gray prediction refers to the generation of cumulative sequence by using an exponential function to simulate the original random sequence. However, the cumulative generation sequence may not vary exponentially. Therefore, the GM model is limited in its capability to analyze complex and nonlinear problems and thus cannot be used to effectively simulate and predict a nonlinear system. For an arbitrary deterministic system, prediction can be approached by using a polynomial function. Thus, the cumulative generation sequence reduces the degree of randomness and enhances the regularity of the original sequence. Meanwhile, by combining this sequence with the polynomial function, it functions intensively in the approach of discrete data and in turn develops into a gray algebraic curve model (GAM) that can more effectively simulate and predict the solutions of nonlinear problems (Qi et al. 2004; Ou et al. 2005; Wang 2012).

We assume here that the original data sequence observed is X (0) = (X (0)(1), X (0)(2),…, X (0)(n)), which develops into X (1) = (X (1)(1), X (1)(2),…, X (1)(n)) after one accumulation.

$$ {X}^{(1)}(i)={\displaystyle \sum_{j=1}^i{X}^{(0)}(j)},i=1,2\cdots n $$
(1)

We then hypothesize that X (1) is the algebraic curve on sequence k. The GAM of X (1) is then

$$ \begin{array}{c}\hfill {\widehat{X}}^{(1)}(k)={a}_0+{a}_1k+{a}_2{k}^2+\cdots +{a}_m{k}^m\hfill \\ {}\hfill ={\displaystyle \sum_{i=0}^m{a}_i{k}^i}\hfill \end{array} $$
(2)

In Eq. (2), the coefficients a 0 , a 1 ,  ⋯  , a m are undetermined parameters, and k = 1, 2,…, n.

Let

$$ A={\left[{a}_0,{a}_1,\cdots, {a}_m\right]}^T $$
(3)
$$ Y=\left[\begin{array}{l}{X}^{(1)}(1)\\ {}{X}^{(1)}(2)\\ {}\kern0.5em \begin{array}{cc}\hfill \vdots \hfill & \hfill \hfill \end{array}\\ {}{X}^{(1)}(n)\end{array}\right] $$
(4)
$$ B=\left[\begin{array}{l}1\begin{array}{cc}\hfill \hfill & \hfill 1\begin{array}{cc}\hfill \hfill & \hfill \begin{array}{cc}\hfill \hfill & \hfill \cdots \begin{array}{cc}\hfill \hfill & \hfill {1}^m\hfill \end{array}\hfill \end{array}\hfill \end{array}\hfill \end{array}\\ {}1\begin{array}{cc}\hfill \hfill & \hfill 2\begin{array}{cc}\hfill \hfill & \hfill \begin{array}{cc}\hfill \hfill & \hfill \cdots \begin{array}{cc}\hfill \hfill & \hfill {2}^m\hfill \end{array}\hfill \end{array}\hfill \end{array}\hfill \end{array}\\ {}1\begin{array}{cc}\hfill \hfill & \hfill 3\begin{array}{cc}\hfill \hfill & \hfill \begin{array}{cc}\hfill \hfill & \hfill \cdots \begin{array}{cc}\hfill \hfill & \hfill {3}^m\hfill \end{array}\hfill \end{array}\hfill \end{array}\hfill \end{array}\\ {}\vdots \\ {}1\begin{array}{cc}\hfill \hfill & \hfill n\begin{array}{cc}\hfill \hfill & \hfill \begin{array}{cc}\hfill \hfill & \hfill \cdots \begin{array}{cc}\hfill \hfill & \hfill {n}^m\hfill \end{array}\hfill \end{array}\hfill \end{array}\hfill \end{array}\end{array}\right] $$
(5)

The residual vector E = (Y − BA). If E 2 = (Y − BA)2 → min, which yields Eq. (6) by the least squares method,

$$ A={\left({B}^TB\right)}^{-1}{B}^TY $$
(6)

Substituting the coefficients in Eq. (6) in the GAM equation, we obtain the cumulative generating sequence. After a one-time regressive reduction, the prediction data are obtained as

$$ {\widehat{X}}^{(0)}(k)={\widehat{X}}^{(1)}(k)-{\widehat{X}}^{(1)}\left(k-1\right) $$
(7)

To summarize, the GAM proceeds as follows:

  1. (1)

    Calculate the one-time cumulative generating sequence X (1) according to the original data sequence X (0);

  2. (2)

    Determine the maximum times m of the optimal curve;

  3. (3)

    Establish the B and Y matrices;

  4. (4)

    Calculate the uncertainty coefficients a i according toA = (B T B)−1 B T Y;

  5. (5)

    Establish the prediction model;

  6. (6)

    Obtain the prediction sequence \( {\widehat{X}}^{(0)} \); and

  7. (7)

    Determine the accuracy of the model.

The accuracy of the simulated data obtained from above should be determined to investigate the goodness of fit of the prediction model. The original sequence is assumed to be X (0) = (X (0)(1), X (0)(2),…, X (0)(n)), and the corresponding simulated sequence is \( {\widehat{X}}^{(0)}=\left({\widehat{X}}^{(0)}(1),{\widehat{X}}^{(0)}(2),\cdots, {\widehat{X}}^{(0)}(n)\right) \).

The mean of the original sequence X (0) is

$$ \overline{X}=\frac{1}{n}{\displaystyle \sum_{k=1}^n{X}^{(0)}(k)} $$
(8)

The variance of the original sequence X (0) is

$$ {s}_1^2=\frac{1}{n}{\displaystyle \sum_{k=1}^n\Big({X}^{(0)}(k)}-\overline{X}\Big){}^2 $$
(9)

The residual sequence from the original and simulated sequences is

$$ \begin{array}{c}\hfill {\varepsilon}^{(0)}=\left\{\varepsilon (1),\varepsilon (2),\cdots \varepsilon (n)\right\}\hfill \\ {}\hfill =\left\{{X}^{(0)}(1)-{\widehat{X}}^{(0)}(1),{X}^{(0)}(2)-{\widehat{X}}^{(0)}(2),\cdots, {X}^{(0)}(n)-{\widehat{X}}^{(0)}(n)\right\}\hfill \end{array} $$
(10)

The mean of the residual is

$$ \overline{\varepsilon}=\frac{1}{n}{\displaystyle \sum_{k=1}^n\varepsilon (k)} $$
(11)

The variance of the residual is

$$ {s}_2^2=\frac{1}{n}{\displaystyle \sum_{k=1}^n{\left(\varepsilon (k)-\overline{\varepsilon}\right)}^2} $$
(12)

The relative error sequence from the original and simulation sequences is

$$ \varDelta =\left\{\left|\frac{\varepsilon (1)}{X^{(0)}(1)}\right|,\left|\frac{\varepsilon (2)}{X^{(0)}(2)}\right|,\cdots, \left|\frac{\varepsilon (n)}{X^{(0)}(n)}\right|\right\}=\left\{{\left.{\varDelta}_k\right|}_1^n\right\} $$
(13)

The ratio of the mean square error is

$$ C=\frac{s_2}{s_1} $$
(14)

The probability of a small error is

$$ P=P\left(\left|\varepsilon (k)-\overline{\varepsilon}\right|<0.6745{s}_1\right) $$
(15)

Table 2 shows the precision inspection grade of the prediction model.

Table 2 Precision inspection grading table
Full size table

Numerical simulations

To simulate the separation development in a composite roof, we use a discrete element numerical simulation software, Universal Distinct Element Code (UDEC). Figure 1 shows the two-dimensional numerical calculation model of case 1. This model is 150 m long and 55 m high. The horizontal coal seam therein is 3 m thick. The immediate roof is 1 m thick, and the blocks of the immediate roof are further divided into blocks of 0.5 × 0.5-m2 size. The main roof is 3 m thick, and the main roof is divided into blocks of 10 × 3 m2. The boundary of the model is fixed by displacement, and the model obeys the Mohr–Coulomb constitutive relationship. Table 3 shows the lithology, thickness, and mechanical parameters of the main rock strata in the model.

Fig. 1
figure 1

Simulation model

Full size image
Table 3 Parameters of each rock stratum in the model
Full size table

To explore the influence of different values of the immediate roof thicknesses on roof separation, three additional model cases are added to the model above. The immediate roof thickness in cases 2 to 4 are set at 3, 5, and 7 m, respectively, while the other parameters are the same as those in case 1.

The model coal mine was excavated in the following two steps in the simulation:

Step 1: Initial balancing of the model; and

Step 2: Excavation of the coal mine at a pace of 1.0 m at each step of the simulation, from left to right, while the roof is supported by pillars.

The three additional model cases were simulated following the two steps listed above. Figure 2 shows the roof separation produced by the simulations for each case.

Fig. 2
figure 2figure 2

State of roof separation in the four simulation models. a State of roof separation in the simulation model of case 1. b State of roof separation in the simulation model of case 2. c State of roof separation in the simulation model of case 3. d State of roof separation in the simulation model of case 4

Full size image

Table 4 lists the roof separation data obtained by roof separation observations at the workface of the four model cases in the numerical simulations and by recording the variations in the roof separation from generation to gradual expansion.

Table 4 Roof separation data of the four simulation cases
Full size table

GAM prediction results and discussion

Table 5 presents Lagrange interpolation results of the experimental data from Table 4.

Table 5 Lagrange interpolation results from the experimental data of roof separation and the results from the four simulation cases
Full size table

The cumulative sequence of the first seven data in Table 4 was calculated using Eq. (1). According to Eq. (4), we determined that

$$ \begin{array}{l}{Y}_1={\left[5.20\ 19.25\ 51.27\ 108.47\ 198.96\ 329.46\ 511.26\right]}^{\mathrm{T}}\hfill \\ {}{Y}_2={\left[2.18\ 9.96\ 27.97\ 61.95\ 118.64\ 205.72\ 331.71\right]}^{\mathrm{T}}\hfill \\ {}{Y}_3={\left[2.04\ 5.54\ 14.49\ 34.74\ 73.98\ 141.77\ 249.51\right]}^{\mathrm{T}}\hfill \\ {}{Y}_4={\left[1.00\ 3.45\ 8.20\ 16.29\ 28.94\ 47.56\ 73.74\right]}^{\mathrm{T}}\hfill \end{array} $$

By comparison, we determined that the fitting is optimal for a maximum time of m = 3. Using Eqs. (3)–(6), we find that

$$ \begin{array}{l}B=\left[\begin{array}{llll}1\hfill & 1\hfill & 1\hfill & 1\hfill \\ {}1\hfill & 2\hfill & 4\hfill & 8\hfill \\ {}1\hfill & 3\hfill & 9\hfill & 27\hfill \\ {}1\hfill & 4\hfill & 16\hfill & 64\hfill \\ {}1\hfill & 5\hfill & 25\hfill & 125\hfill \\ {}1\hfill & 6\hfill & 36\hfill & 216\hfill \\ {}1\hfill & 7\hfill & 49\hfill & 343\hfill \end{array}\right]\hfill \\ {}{A}_1=\left[\begin{array}{l}-0.2586\hfill \\ {}3.3850\hfill \\ {}0.5863\hfill \\ {}1.3378\hfill \end{array}\right]\hfill \\ {}{A}_2=\left[\begin{array}{l}-4.6471\hfill \\ {}8.1073\hfill \\ {}-2.6783\hfill \\ {}1.1972\hfill \end{array}\right]\hfill \\ {}{A}_3=\left[\begin{array}{l}-8.7371\hfill \\ {}16.6332\hfill \\ {}-7.0907\hfill \\ {}1.4374\hfill \end{array}\right]\hfill \\ {}{A}_4=\left[\begin{array}{l}-0.8728\hfill \\ {}1.8974\hfill \\ {}-0.2833\hfill \\ {}0.2191\hfill \end{array}\right]\hfill \end{array} $$

Using Eq. (2), we obtain the following corresponding gray curve model equations:

$$ \begin{array}{l}{{\widehat{X}}_1}^{(1)}(k)=-0.2586+3.385k+0.5863{k}^2+1.3378{k}^3\hfill \\ {}{{\widehat{X}}_2}^{(1)}(k)=-4.6471+8.1073k-2.6783{k}^2+1.1972{k}^3\hfill \\ {}{{\widehat{X}}_3}^{(1)}(k)=-8.7371+16.6332k-7.0907{k}^2+1.4374{k}^3\hfill \\ {}{{\widehat{X}}_4}^{(1)}(k)=-0.8728+1.8974k-0.2833{k}^2+0.2191{k}^3\hfill \end{array} $$

Substituting for the number k (from 1 to 7) into the formulas above, the k represents the time point which is equivalent to the time as kΔt, according to the actual situation; here, Δt can be several days or hours, etc. The predicted sequence, the residual, and the relative error can be solved by Eqs. (7), (10), and (13), respectively, as shown in Tables 5, 6, 7, and 8.

Table 6 GAM-predicted data, residuals, and relative errors of the roof separation in case 1
Full size table
Table 7 GAM-predicted data, residuals, and relative errors of the roof separation in case 2
Full size table
Table 8 GAM-predicted data, residuals, and relative errors of the roof separation in case 3
Full size table

The values for k = 8 and k = 9 are predicted to be 238.269 and 303.656 mm, respectively, in case 1. In Table 4, the two time points of case 1 are experimentally set at 236.75 and 300.60 mm, respectively. The residuals of the predicted data compared with the experimental data of the two time points are −1.52 and −3.01 mm, while the relative errors are −0.64 and −1.02 %, respectively. Error detection in the predicted data in Table 6 shows that the mean square deviation ratio solved by Eq. (14) is C = 0.01 < 0.35, while the small error probability solved by Eq. (15) is P = 1.

The values for k = 8 and k = 9 are predicted to be 170.260 and 222.369 mm, respectively, in case 2. In Table 4, the two time points of case 2 are experimentally set at 174.22 and 232.48 mm, respectively. The residuals of the predicted data from the experimental data of the two time points are 3.96 and 10.11 mm, while the relative errors are 2.3 and 4.3 %, respectively. Error detection in the predicted data in Table 7 shows that the mean square deviation ratio solved by Eq. (14) is C = 0.03 < 0.35, while the small error probability solved by Eq. (15) is P = 1.

The values for k = 8 and k = 9 are predicted to be 153.193 and 208.007 mm, respectively, in case 3. In Table 4, the two time points of case 3 are experimentally set as 160.96 and 229.29 mm, respectively. The residuals of the predicted data with the experimental data of the two time points are 7.77 and 21.28 mm, while the relative errors are 4.8 and 9.3 %, respectively. Error detection in the predicted data in Table 8 shows that the mean square deviation ratio solved by Eq. (14) is C = 0.05 < 0.35, while the small error possibility solved by Eq. (15) is P = 1.

The values for k = 8 and k = 9 are predicted to be 34.675 and 44.626 mm, respectively, in case 4. In Table 4, the two time points of case 4 are experimentally set as 35.53 and 46.83 mm, respectively. The residuals of the predicted data with the experimental data of the two time points are 0.86 and 2.20 mm, while the relative errors are 2.4 and 4.7 %, respectively. Error detection in the predicted data in Table 9 shows that the mean square deviation ratio solved by Eq. (14) is C = 0.09 < 0.35, while the small error possibility solved by Eq. (7) is P = 1.

Table 9 GAM-predicted data, residuals, and relative errors of the roof separation in case 4
Full size table

By observing the errors in Tables 5, 6, 7, and 8 and referring to the precision detection grade control in Table 1, we find that most of the GAM prediction results are within precision grade 1. Figure 3 shows the curves from the GAM predictions and the experimental data of roof separation in Tables 5, 6, 7, and 8. As shown in Fig. 3, it is clear that the GAM prediction data show little error with respect to the experimental data. Moreover, the prediction trend reflects the experimental results. For comparison, when the traditional gray model GM(1,1) is used to forecast the experimental data, most of the predicted results do not meet the necessary requirements (Fig. 4). Thus, we prove that the GAM approach is more effective in predicting roof separation trends than GM(1,1).

Fig. 3
figure 3

GAM-predicted and original curves for the experimental data

Full size image
Fig. 4
figure 4

GM(1,1)-predicted and original curves for the experimental data

Full size image

A case study

As an example, the Qidong coal mine in the Anhui Province of China makes use of a mechanical displacement meter to monitor the roadway roof separation. The shallow and deep basis points of the displacement meter were set at 3 and 6 m, respectively, above the roof interface (Fig. 5).

Fig. 5
figure 5

Roof separation field-monitoring diagram

Full size image

By selecting the first seven data from field measurement, and using the GAM method, we obtain the corresponding gray curve model equation as follows:

$$ {\widehat{X}}^{(1)}(k)=4.5000-6.3492k+4.1786{k}^2+0.2778{k}^3 $$
(16)

Substituting k with 1 to 7 into the formulas above, we can solve the predicted sequence, residual, and relative error with Eqs. (7), (10), and (13), as shown in Table 9.

The values for k = 8 and k = 9 are predicted to be 103.278 and 124.970 mm, respectively. The two time points of field measurement data were 101 and 123 mm, respectively. The residuals of the predicted data compared with the experimental data of the two time points are −2.28 and −1.97 mm, while the relative errors are −2.3 and −1.6 %, respectively. Error detection in the predicted data in Table 10 shows that the mean square deviation ratio solved by Eq. (14) is C = 0.03 < 0.35, while the small error probability solved by Eq. (7) is found to be P = 1, and the prediction level is grade 1. Figure 6 shows the curves corresponding to the GAM and GM(1,1) predictions and the field measurements.

Table 10 GAM-predicted data, residuals, and relative errors of the roof separation from the field measurements
Full size table
Fig. 6
figure 6

GAM- and GM(1,1)-predicted and original curves for the field measurements

Full size image

Conclusions

  1. (1)

    We introduce the GAM modeling approach and apply it for the first time to forecast the change trend of roof separation.

  2. (2)

    To build our prediction model based on the GAM method, we use the roof separation data from a UDEC numerical simulation as the original observation data. We then apply the prediction model to predict the development trends of roof separation. The prediction data show little error with respect to the experimental data, and the trend is consistent with the experimental observations. For the sake of comparison, we also apply the traditional GM(1,1) method and find that the predicted error was larger than that in the GAM application. Thus, we prove that GAM is the more effective method for predicting roof separation trends.

  3. (3)

    In the case study making use of field-monitoring roof separation data from a mine, the GAM method leads to good prediction results. When we get the warning criterion of roof separation of a roadway according to empirical data statistical analysis, and set roof separation indicators to get a small amount data, then, we can predict the development trend of roof separation using GAM method based on the existing data. The GAM method provides the desired auxiliary decision-making information that can be used to issue warnings about roof fall accidents and thus has the greatest potential to help reduce and avoid these accidents.