Prediction of progressive surface subsidence above longwall coal mining using a time function

Transcription

1 International Journal of Rock Mechanics & Mining Sciences 38 (2001) Technical Note Prediction of progressive surface subsidence above longwall coal mining using a time function Ximin Cui a, *, Jiachen Wang a, Yisheng Liu b a Department of Resource Exploitation Engineering, China University of Mining and Technology, Beijing Campus, D11 Xue Yuan Lu, Beijing , People s Republic of China b Qianjiaying Coal Mine, Kai Luan Group Corporation, Tangshan, , People s Republic of China Accepted 2 August Introduction The ground subsidence process induced by underground longwall coal mining is a complicated process, as it is a process of subsidence-induced damage to the surface and sub-surface structures. In order to assess the potential damage, the surface subsidence characteristics should be known. Traditionally, the assessment is made based on the final subsidence. However, this is somewhat misleading [1] because * For super-critical conditions, the bottom of the final subsidence basin is flat and there the surface slope and curvature induced by ground subsidence are zero. This means that there is no permanent differential surface deformation, and thus no damages to those structures located in the subsidence basin bottomfwere the ground is to be simply lowered. However, during the deformation of the subsidence basin, the damage potential of progressive and differential deformation must not be overlooked: proper precautions should be considered accordingly. * Surface movement and deformation during the formation of the subsidence basin are time dependent; therefore, protective measures for structures should be designed with the knowledge of the timedependent behavior. It is well known that the progressive surface movements include progressive vertical subsidence and progressive horizontal displacement. All of the differential subsidence can be derived from knowledge of the vertical subsidence and horizontal displacements. In order to improve the analysis and structural protection in practice, accurate prediction methods of progressive *Corresponding author. Tel: address: cxm@cumtb.edu.cn (X. Cui). subsidence were established and these have attracted considerable attention all over the world [2 6]. Unfortunately, there is no adequate method that could be adopted in practice for the prediction of progressive subsidence in China. So, in this paper a practical method to predict surface progressive subsidence based on the Knothe time function and probabilistic integral method is presented and verified through practice of the Qianjiaying Coal Mine, Kai Luan Group Corporation in northern China. 2. Selection of the time function In 1988, a mathematical model for predicting progressive subsidence over an operating longwall panel was presented by Peng and Luo and was further refined by Luo in 1989 [7]. In this model, it is assumed that the subsidence velocity distribution along the mining direction of the longwall face is a normal distribution with respect to the location of the on-going longwall face. This model is capable of predicting progressive subsidence, slope and curvature at any point in the subsidence basin. However, it cannot deal with the subsidence-developing process of initiation and weakening. In 1952, Knothe proposed a generalized differential equation for the mathematical description of subsidence at a given point in time, the solution being known as Mitscherlich s growth law. This law describes some natural processes such as, for example, the rate of selfcooling of a body warmer than its surroundings, dissolution of a substance in a liquid, a change of an electric current intensity after closure of constant electromotive force, and consideration of the delaying influence of self-induction on radioactive disintegration [5]. In the description of the deformation process, /01/$ - see front matter r 2001 Elsevier Science Ltd. All rights reserved. PII: S ( 0 1 ) 转载

2 1058 X. Cui, J. Wang / International Journal of Rock Mechanics & Mining Sciences 38 (2001) Knothe made an assumption that instantaneous subsidence rate is proportional to a difference between the final subsidence, developed as a result of extraction up to a moment t in an area influenced at this point by the extraction, and the quantity of subsidence at the point at moment t: dwðtþ ¼ cðw 0 WðtÞÞ; ð1þ dt where t is the time, WðtÞ is the subsidence at a moment t; W 0 is the final subsidence at a given point and c is the factor of proportionality which is related to the physical and mechanical properties of the overburdened rock mass. Eq. (1) is a linear differential equation with separated variables. We shall solve this equation with the assumption that we execute the extraction of an elementary volume in an immediate manner. Considering W 0 as a constant and the initial condition for W ðt¼0þ as zero, we have WðtÞ ¼W 0 ð1 e ct Þ: ð2þ For purposes of further analysis, Eq. (2) will be described as a function j of the independent variable t; thus, we obtain the following formula: WðtÞ ¼W 0 jðtþ; ð3þ where jðtþ ¼1 e ct : According to Knothe, the factor of proportionality, c; is a parameter describing the influence of geological and mining conditions on the deformation process in time. This parameter is expressed in unit year 1. In China, combining mechanics of the media with rheological mechanics, the progressive subsidence was studied in 1996[8]. Based on the alluvial soil, regarded as a random medium, and the bedrock, looked upon as a viscoelastic beam on a viscoelastic foundation, a predictive equation was established and a similar time function was obtained in 1999 [9]. The Knothe time function can describe the total subsidence process at any surface point and has been accepted by engineers and researchers in China. 3. Determination of the parameter c When we use the Knothe time function to predict progressive subsidence, the key process is to determine the time parameter, c: If surface observations are taken, we can determine it easily. However, how can this parameter be determined when there are no in situ observations? The influence of mining on the rock media, considered as a process operating in time and space, depends on various mining and geological conditions, together with their interactions, which determine the scope and damage potential. According to British and Chinese examinations, the most significant factors influencing the quantity and variability of deformation factors, as well as the duration of deformation movements, are * distance from the working face, * the area already worked out in the seam, * the thickness of overburden, * the rate of advance of the working face, * the method of coal working and * the composition of the rock and the presence of certain geological features. However, it is difficult to determine the above influencing quantities separately. So, how can we compute the influencing factors using a synthetic index? It is fortunate that many practical observations and theoretical studies in China demonstrate that subsidence reaches the surface when the face has moved a distance of 1=421=2H 0 ; where H 0 is the average mining depth, from the set-up entry. As the face continues to advance, the subsidence continues to increase. When the working face has advanced a sufficient distance, e.g., 1:221:4H 0 ; away from the set-up entry, the subsidence reaches the maximum possible value. According to the probabilistic integral method, the maximum subsidence at this time approaches 0.98W 0 : If we assume that the advance rate of a longwall face can be expressed as V; the time when the face point reaches the critical mining condition will be between 1:2H 0 =Vand 1:4H 0 =V: Thus, we have W 0 ð1 e c ð1:2h 0=VÞ ÞX0:98W 0 ; W 0 ð1 e c ð1:4h 0=VÞ Þp0:98W 0 : That is to say the time parameter c satisfies the following condition generally: V ln 0:02 V ln 0:02 pcp : ð6þ 1:4H 0 1:2H 0 Table 1 shows the comparisons of measured c and its estimated values in some coal mines. It can be seen that the measured values are reasonably coincident with the estimated values. If the critical mining dimension L 1 and the advancing rate V of the working face are known, the parameter c of the time function can be determined from the following: c ¼ V L 1 ln 0:02: 4. Computational method for progressive surface subsidence Since the deformation process takes place in a definite time interval, determining instantaneous values of deformation factors, as well as their increase during ð4þ ð5þ ð7þ

3 X. Cui, J. Wang / International Journal of Rock Mechanics & Mining Sciences 38 (2001) Table 1 Comparison of the measured c and its estimated value Name of coal mine Mining depth (m) Advance rate (m/yr) Measured c (1/yr) Estimated c (1/yr) Upper Silesia Coal Basin, Poland Lower Rhine Coal Basin Suncun Coal Mine, Xinwen, China Fuli Coal Mine, Hegang, China Ping an Coal Mine, Fuxin, China Lower Silesia Coal Basin, Poland Xiaqiao Coal Mine, Xuzgou, China the deformation movements, is necessary. From the point of view of protection of objects being influenced by the mining works, a unit change of the deformation value is essential. Frequently, even surface structures resistant to deformations are damaged as a result of large increases in the deformation during the subsidence movements. In the light of this, establishing the influence of the working face advance on the quantity and distribution of deformations is essential. In order to fix an optimal extraction rate, for the given mining and geological conditions, we should have an appropriate mathematical technique. We can divide the total working face into certain elements according to the advancing rate or mining duration. For example, assuming that the mining duration of the first element is t 1 ; and its advancing rate is V 1 ; we have the mining length V 1 t 1 : For the ith mining element, its mining duration is t i and the advancing rate is V i ; the ith mining length V i t i can also be determined. The maximum progressive subsidence of mining element from 1 to n can be computed, respectively, for moment t: W 1 ðtþ ¼W 0 jðtþ; W 2 ðt t 1 Þ¼W 0 jðt t 1 Þ; W 3 ðt t 1 t 2 Þ¼W 0 jðt t 1 t 2 Þ; W n ðt t 1 t 2? t n 1 Þ ^ ¼ W 0 jðt t 1 t 2? t n 1 Þ; W 0 ¼ mq cos a; where m is the mining thickness, q is the subsidence coefficient and a is the inclination of the coal seam. According to the superposition principle, the progressive subsidence of any point in the trough at time t can be expressed as follows: Wðx; tþ ¼jðtÞ½WðxÞ Wðx V 1 t 1 ÞŠ þ jðt t 1 Þ½Wðx V 1 t 1 Þ Wðx V 1 t 1 V 2 t 2 ÞŠ þ jðt t 1 t 2 Þ½Wðx V 1 t 1 V 2 t 2 Þ ð8þ Wðx V 1 t 1 V 2 t 2 V 3 t 3 ÞŠ ^ þ jðt t 1 t 2? t n 1 Þ ½Wðx V 1 t 1 V 2 t 2? V n 1 t n 1 Þ Wðx V 1 t 1 V 2 t 2? V n t n ÞŠ: If the advancing rate of the working face is constant, V 1 ¼ V 2 ¼? ¼ V n and the mining durations of the elements are equal, t 1 ¼ t 2 ¼? ¼ t n ; then Eq. (9) can be revised as Wðx; tþ ¼jðtÞ½WðxÞ Wðx V 1 t 1 ÞŠ þ jðt t 1 Þ½Wðx V 1 t 1 Þ Wðx 2V 1 t 1 ÞŠ þ jðt 2t 1 Þ½Wðx 2V 1 t 1 Þ Wðx 3V 1 t 1 ÞŠ ^ þ j½t ðn 1Þt 1 ÞfW½x ðn 1ÞV 1 t 1 Š Wðx nv 1 t 1 Þg: ð10þ In the above analysis, the related formula for the probabilistic integral method is applied directly: " WðxÞ ¼ W pffiffiffi! # 0 p erf 2 r x þ 1 : ð11þ The progressive horizontal displacement, progressive slope, progressive curvature and progressive horizontal deformation can be obtained similarly. On the other hand, the differential movement indices can also be computed using non-linear geometric field theory and the strain rotation (S R) decomposition theorem to improve the prediction accuracy [10]. 5. Acase study of progressive subsidence prediction for the 1176E working face in Qianjiaying Coal Mine, China Qianjiaying Coal Mine is located at Tanshan, Hebei Province of northern China. From March 1, 1999 to March 21, 2000, the longwall face of 1176E was extracted. The surface altitude is 19 m above Huanghai sea level and the working face depth is 445 m below sea level. The average mining depth is 464 m. The mining thickness is 3.0 m. The dip angle of the coal seam is 31 and the average mining rate of the longwall face is ð9þ

4 1060 X. Cui, J. Wang / International Journal of Rock Mechanics & Mining Sciences 38 (2001) m/yr. The strike and dip length of the working face are 912 and 160 m, respectively. In March 1999, the observation station was set up along a small road with 30 measuring positions or monuments. The monuments layout and mining progress are shown in Fig. 1. The first complete measurement set was completed finished in April 1999 and 13 subsidence observations were made from July 22, 1999 to September 14, The total station method was employed and the observation precision depended on the China Coal Mines survey handbook. The time-dependent subsidence curves that were obtained are shown in Fig. 2. It can be seen that the subsidence values at the measuring stations Nos. 9 and 13 are abnormal. This phenomenon A B C D E F G H E Mining durations of A: Mar B: Jul C: Aug D: Oct E: Nov F: Dec G: Jan H: Feb Mar Mining direction Dip direction Fig. 1. Plan layout of measuring stations (monuments) and mining progress (dimensions in metres) Fig. 2. Measured subsidence development curves.

5 X. Cui, J. Wang / International Journal of Rock Mechanics & Mining Sciences 38 (2001) Fig. 3. (a k)comparison of measured and predicted subsidence curves with mining progress.

6 1062 X. Cui, J. Wang / International Journal of Rock Mechanics & Mining Sciences 38 (2001) may be caused by damage to the monuments. According to the monitoring results in Qianjiaying, we have the subsidence parameters: subsidence factor q ¼ 0:73; tangents of major influence angle for strike, rise and dip are tg b strike ¼ 1:9; tg b rise ¼ 1:9 and tgb dip ¼ 1:6; respectively. The displacement distance of the inflection point is 0:05H 0 : Substituting the average mining depth and advancing the rate of 1176E working face into Eq. (6), we find that the domain of time parameter c in Qianjiaying is 9:03=yrpcp10:54=yr: Finally, we select the value c ¼ 9:4=yr for the simulation. The comparisons of the timedependent subsidence curves and the predicted subsidence development curves are shown in Figs. 3 and 4, respectively. In order to evaluate the predictions, we assume that there are no monitoring errors in practice and that prediction errors are equal for all points. Thus, we can calculate the mean square errors using the Bessel formula: rffiffiffiffiffiffiffiffiffiffiffi ½vvŠ m ¼ 7 ; ð12þ n 1 where the v s are the differences between measured values and predicted values respectively for the surface points and the [ ] component represents the square sum. The relative errors can be computed by f ¼ jmj ; ð13þ W i 0 where W0 i are the maximum subsidence values in each measurement. The calculated results demonstrate that the average relative error is 8% only. That is to say that the prediction method of progressive subsidence established in this paper is valid and can be used in practice. On the other hand, the comparative analysis of the measured and predicted subsidence in 1117E illustrates that the tangent of the major influence angle is constant in the subsidence developing process. This may indicate that the viewpoint that the tangent of the major influence angle is variable is not correct. 6. Conclusions and discussion Based on the Knothe time function, the prediction of progressive subsidence has been demonstrated, and the factor of proportionality, c; describing the influence of geological and mining conditions on the deformation process in time has been determined when there is no measured material in practice. The prediction of progressive subsidence for the 1176E working face, Qianjiaying Coal Mine, indicates that this method is effective and easy to apply for analysis in practice. The differential subsidence characteristics, such as progressive slope, progressive curvature and horizontal strain can also be calculated similarly. Although we found that the tangent of major influence angle is constant, in fact this is different from the traditional Chinese researchers standpoint. However, determining the real distribution of a time function is not an easy problem in theory and the imperfections of Knothe time function still occur. In order to determine Knothe time function s character, we make an assumption that the variable t has values in the range from zero to þn: Results of the analysis Fig. 4. Predicted subsidence development curves.

7 X. Cui, J. Wang / International Journal of Rock Mechanics & Mining Sciences 38 (2001) demonstrate that in this model the time function increases from a minimal value equal to zero to a maximal value limited by the horizontal asymptote. The first derivative is a decreasing function, and its maximum occurs for a time t ¼ 0: The second derivative is a function increasing from a minimal value to zero. It must be noted that the subsidence rate initially increases from a value equal to zero (for t ¼ 0) to a maximal value, and then it decreases to zero (for t ¼þN) in real conditions. The second derivative, which represents the subsidence acceleration, increases from a value equal to zero (for t ¼ 0) to a maximal value, then it decreases to a minimal value less than zero, and then it increases asymptotically to zero. That is to say, we cannot simulate the subsidence rate and acceleration using the Knothe time function. Thus, further aspects now need to be investigated and remain to be solved in the future. Acknowledgements The authors are grateful to Professor J.A. Hudson for encouraging them to do the further research work and for assistance with the editorial work. The study is financially supported by the National Natural Science Foundations of China under Grant No References [1] Peng SS. Surface subsidence engineering. New York: SME, [2] Liu TQ. Surface movement, over burden failures and its application. China: Coal Industry Press, 1981 (in Chinese). [3] Cui XM, Miao XX, Zhao YL, Jin RP. Discussion on the time function of time dependent surface movement. J China Coal Soc 1999;24:453 6(in Chinese). [4] Cui XM, Miao XX, Jin RP. Time-based computation of surface movement process. China Min Mag 1999;6: [5] Kwinta A, Hejmanowski R, Sroka A. A time function analysis used for the prediction of rock mass subsidence. In: Gui yuguang, Golosinski TS, editors. Mining science and technology. A.A. Balkema: Rotterdam, Brookfield, p [6] Litwiniszyn J. The influence of the rate of mining operations on the development of subsidence troughs. Arch Min Sci 2000;45: [7] Luo Y. An integrated computer model for predicting surface subsidence due to underground coal mining (CISPM). PhD dissertation, Deptartment. of Mining Engineering, West Virginia University, Morgantown, WV, December [8] Ma FH, Wang YJ, Fan XL. Successive medium rheological theory and its application in stratum subsidence progressive course. Chin J Nonferrous Met 1996;6(4):7 12. [9] Zhang XD, Zhao YH, Liu SJ. A new method of calculation surface subsidence and deformations under thick alluvial soil. Chin J Nonferrous Met 1999;9: [10] Cui XM, Miao XX, Wang JA. Improved prediction of differential subsidence caused by underground mining. Int J Rock Mech Min Sci 2000;37: