Empirical methods of subsidence prediction—a case study from Illinois

Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. Vol. 20, No. 4, pp. 153-170, 1983

Printed in Great Britain. All rights reserved

0148-9062:83 $3.00+0.00 Copyright ~ 1983 Pergamon Press Ltd

Empirical Methods of Subsidence Prediction A Case Study From Illinois M. HOOD* R. T. EWY* L. R. RIDDLE* Subsidence profiles above two adjacent panels in Illinois are compared with profiles predicting subsidence behaviour obtained using the (i) National Coal Board (NCB) [1] method, (ii) the profile function method and (iii) the influence function method. The NCB method predicts the maximum subsidence values at the centre of the troughs accurately but the overall shapes of these predicted profiles do not match the profiles from the measured data well. Consistent values for the angles of draw were measured but these angles were different in the transverse (average values 43 °) and in the longitudinal (average value 17.5 °) directions. Time dependent subsidence effects are shown to be small but measurable. These displacements continue at a linear rate for at least a 12 month period. A comparison of measured horizontal distances, interpreted as horizontal strain, and the NCB predictions for strain shows that the peak measured strains are greater by a factor of about four than the predicted strains. The relationship between surface curvature and strain is investigated. Problems associated with calculation of surface curvatures from vertical displacement data are highlighted and a recommendation is made for future studies to consider direct measurement of this parameter. Surface curvatures above a moving face are found to be about three times less than the curvatures at the stationary end of the panel. On the other hand, a hyperbolic tangent profile function is shown to serve as an accurate predictive tool for subsidence behaviour in two adjacent longwall panels at Old Ben Number 24 mine in Illinois. This function predicts not only the vertical displacements but also the surface curvatures above both panels. Influence functions are shown to be more problematic, although potentially more flexible, in their application than the profile functions.

INTRODUCTION Subsidence prediction methods fall into one of two categories: empirical methods and mechanistic methods. Several methods in the former category have been applied successfully in a number of countries, mainly in Europe where this problem has received the most attention. Methods in the latter category have not found widespread application. There are many reasons why this is the case including: the difficulty in determining the behaviour of the overburden rocks; the difficulty in determining the material properties of these rocks; and the mathematical complexity associated with this approach. This is not to say that this more fundamental approach should not be pursued but rather that, with the present state of the art, empirical methods have proved more practical and convenient. *Department of Materials Science and Mineral Engineering, University of California, Berkeley, CA 94720, U.S.A. .... 2o/4-~

Of the empirial techniques three methods: the empirical data method; the profile function method; and the influence function method, have been used successfully for subsidence prediction above coal mines. An example of the first of these methods is the National Coal Board [1] approach. The profile function method employs a mathematical function to match the observed subsidence profile. This technique has been applied successfully in Poland, Hungary and the Soviet Union. Subsidence prediction using influence functions is based on the extraction of infinitesimal elements of area. The extraction of an infinitesimal area, dA, causes infinitesimal subsidence at the surface. This subsidence is described by the influence function p (r) where r is the radial distance from dA. For many influence functions, the function p (r) has a maximum value at r = 0 and diminishes as r increases. The subsidence at a point on the surface is thus determined by its radial distance in plan view from the

153

HOOD et al.: EMPIRICAL METHODS OF SUBSIDENCE PREDICTION

154

extracted element. When more than one element is extracted, the total subsidence at a point is due to the sum of the influence of each element extracted. This sum can be described by mathematical integration. Thus the subsidence profile is:

s(r)= f p ( r ) d A

(1)

When polar coordinates are used, dA = rdrdO equation (1) then becomes

(r)rdOdr

s (r) =

(2)

I

From this equation it can be seen that the subsidence at a point is the volume under the curve p (r) contained in the area defined by r~, r2, Oj, 02 (Fig. 1). One half of the maximum subsidence occurs when r~ = 0. This is the case when the volume is considered over one half of the circular region defined by the radius r2 (Fig. 1). It is worthwhile emphasizing the consequence of this result, namely that by the nature of this influence

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function method the point of half maximum subsidence. that is the inflection point of the subsidence profile. occurs immediately above the ribside. Since it is known that this does not occur all of the time. or even very frequently, in practice, special measures have to be followed to adjust the position of this profile inflection point relative to the ribsides [2]. It is interesting to note that a fundamental assumption made when the influence function method is used is that each elemental extracted area has the same subsidence effect at the surface as any other element since superposition is used to calculate subsidence at a point. This assumption is equivalent to saying that the overburden behaves in a homogeneous, isotropic manner. Also. it should be noted that integration of all of the elemental extracted areas across the mined area gives the profile function. Thus at least in principle, a profile function can be derived from an influence function, and vice-versa. Further details concerning the behavior of profile and influence functions are given in Brauner [2]: Munson and Eichfeld [3]; and Hood et aL [4].

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Subsidence measurements were made above two adjacent longwall retreat panels at Old Ben Coal Company's No. 24 mine, in Illinois. These panels together with the positions of survey monuments are illustrated in Fig. 2 [5]. Two rows of chain pillars were developed along the sides of the panels. One of these rows of pillars was mined as the panel was retreated. Thus a single row of chain pillars 25m wide separated panel l from the adjacent room and pillar panel and a similar pillar row separated panel 2 from panel l after the latter panel was completed. Panel 1 was 538 m long by 148 m wide, and panel 2 was 529 m tong by 147 m wide. Both produced coal from the Herrin No. 6 seam at a depth of 189m. The overburden geology is illustrated in Fig. 3 [6]. The average extracted thickness was reported as 2.13 m. However, in panel 2, where detailed records of this parameter were kept, variations in extracted thickness from 2.07 to 2.68 m with an average value of 2.33 m, were recorded, (Bauer, private communication). Mining of panel 1 commenced on September 3, 1976 and was completed on May 11, 1977. Panel 2 was mined from August 1977 to December 24, 1978. more complete descriptions of the mining and measurement systems are given in references [5] and [7]. DATA ANALYSIS METHODOLOGY

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Fig. 1. The a-eometric relationship between 0 and • for influence functions. The s e q u e n c e (a), (b) and (c) shows the effect on the influence functions of moving the circle increasingly over the mined out area.

Subsidence data from all of the monuments where measurements were taken, for all 29 measurement dates, were read into a computer with an interactive graphics capability. A program was written for this computer that enabled these data to be retrieved and plotted along any selected profile on any chosen measurement date. Furthermore the program was designed to compute curvature from these data, by differentiating the vertical

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The profile and influence function equations ~ere plotted on the computer in a similar manner.

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displacements twice with respect to horizontal distance, and to plot this parameter, again along a selected profile on a predetermined measurement date. The Subsidence Engineers' Handbook [1] employs a suite of charts and tables to predict subsidence. Values from this empirical data set, appropriate to the Old Ben mine geometry were read into the computer and a program similar to that used for the measured data was written to generate profile plots of displacement, curvature and strain for this predicted data set. These plots could be superimposed on the measured data profiles.

This parameter is defined as the angle measured from the edge of the workings to a point at the surface beyond which the effects of subsidence are negligible. Usually this angle is measured from the vertical. The use of angle of draw as a descriptive parameter can be criticized from two viewpoints [2]. First, the definition requires that a judgement be made to establish the point along a subsidence profile that is asymptotic to the original ground surface, beyond which subsidence effects are negligible. This results in an arbitrary definition of this parameter. At the Old Ben mine site the results of relative displacements between monuments are given to an estimated accuracy of 9.1mm [5, 7]. Thus, in this analysis, the surface point considered in calculating angle of draw was taken as that point where at least 9.1 mm of displacement occurred. Second, small errors in the measured subsidence displacements result in large errors in the calculated angle of draw. A sensitivity analysis showed that at the Old Ben site a measurement error of only 3 m m could result in a 5 error in the calculated angle of draw. Despite these difficulties associated with the use of the angle of draw it is an important parameter since knowledge of this parameter is assumed as the starting point for three of the most widely used empirical techniques for subsidence prediction. In one of these methods the National Coal Board [1] assumed an angle of draw of 35 ° for all subsidence profiles. Significant deviations from this 35 ~ value probably will be reflected as differences in the overall shapes of the measured and the predicted profiles. In the other methods, profile and

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157

EMPIRICAL METHODS OF SUBSIDENCE PREDICTION Table 1. Summary of measured angles of draw

Position Transverse profile-over virgin coal Longitudinal profile--stationary end Longitudinal profile--over moving face

Panel

No. of measurements

Mean value (°)

Standard deviation

1 2 1 2 1 2

5 6 8 10 6 9

43.6 42.8 17.8 17.3 17.4 23.9

6.7 6.0 2.3 2.1 10.7 13.3

influence functions require that a parameter termed the critical radius, B, be determined. This parameter is related to the angle of draw, c(, by the expression: B = h tan c( where h is the depth of the workings. The angle of draw was calculated over the west side of the transverse profiles for panels 1 and 2 on all measurement dates after the faces had passed under the transverse monument line. This parameter was not calculated for the east side of these panels since the angle in these regions was affected by previous mining activity. The angle of draw was calculated at both ends of the longitudinal profiles in both panels on all of the measurement dates. Figure 4 shows a measured profile with the calculated angle of draw superimposed on the diagram. A summary of all of the calculated angles of draw is given in Table 1. Three points of interest emerge from a study of this table. First, consistent values of angle of draw are found for the transverse profiles and for the longitudinal profiles at the stationary end of the panel. At each of these positions the values are consistent both within each panel, that is the standard deviations of the mean are relatively low, and also between panels. Second, although the values are consistent, the transverse profiles with mean values of 43.6 ° for panel 1 and 42.8 ° for panel 2 are different from the longitudinal profiles with mean values of 17.8° for panel 1 and 17.3° for panel 2. Third, although the mean values for the angle of draw

above the face, termed travelling angle of draw, are not inconsistent with those from the stationary end in the longitudinal profiles, considerably more scatter, reflected by the high values for the standard deviation, is found in these measurements. The fact that consistent values were recorded for this parameter implies that use of angle of draw in these empirical prediction methods is acceptable. The reason for the large difference in the values of angle of draw between the transverse and the longitudinal directions is not clear at the present time although probably it reflects anisotropic behavior in the superincumbent strata. The large scatter in the results for the travelling angle of draw probably is related to the dynamic conditions that exist around a moving face although no correlation with the angle of draw and the rate of face advance was found.

Comparison of measured and NCB predicted profiles Figure 5 is a transverse section through panels 1 and 2 illustrating both the measured and the predicted subsidence profiles above these panels. This figure shows that the maximum subsidence value is predicted very accurately for panel 1 and is about 150 mm less than the measured value for panel 2. In general however, the overall shape of the predicted profiles does not accurately reflect that of the measured profiles. The trough sides of the former are less steep and the curvatures at both the top and bottom sections of these profiles are less than is the case for the measured profiles. It is for

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Fig. 5. Transverse profile through both panels (monument line C) on Dec. 4, 1978 comparing measured data (diamonds) with NCB predictions (solid line). Summation of the two predicted profiles (dashed line).

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this reason that the prediction of the subsidence above the chain pillars between panels 1 and 2 is more than the measured value. Longitudinal profiles for panel 1 showing both the measured and the predicted subsidence curves are given in Figs 6 and 7. Figure 6 shows the panel immediately after mining was completed. Figure 7 shows the same section some 18 months later, after panel 2 was mined. The predicted profiles shown in these figures are similar to the transverse predicted profiles in terms of both giving an accurate prediction for the maximum value for subsidence, and having slope angles for the trough sides that are less than the measured slope angles. This behaviour results in an inaccurate representation of the measured subsidence profile and was a common feature in all of the plots of transverse and longitudinal profiles that were analyzed in this study• This difference in the

observed profile shape when compared with NCB predictions could be explained by either the overburden geology or the depth of the workings. O'Rourke and Turner [6] have noted that the overburden in the Illinois Basin has a geologic and tectonic structure similar to that of the Midlands and Yorkshire Coalfields in the U.K. Additionally, they note that the NCB predictions for virgin workings are all based on deep mines (often greater than 365 m with small width to depth ratios). These conditions have been shown to promote gentle curvatures in the subsidence profiles. Old Ben contrasts those conditions with a depth of only 190 m which could explain the difference in profile shapes. A striking feature of these curves is the irregular nature of the profile along the bottom of the trough. This behaviour was observed also in panel 2. Munson and Eichfeld [3] correlated the irregularities in the profile

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curve for panel 2 with variations in the extracted thickness of the seam. They claim, and our analysis supports this claim [4], that these perturbations in the subsidence profile are caused by variations in the extracted thickness. Another feature illustrated in these figures is the downward movement of the measured profile with time. part of this movement undoubtedly is the result of mining the adjacent panel 2. However, when panel 1 was mined the maximum vertical displacement observed over the centre of panel 2 was 30 ram. Inspection of these figures shows that the average increase in vertical displacement in the middle of the trough was almost 100 mm. The difference between these two values must be explained by time-dependent behaviour, termed residual subsidence in the Subsidence Engineers' Handbook, [1].

linear rate indefinitely but this behaviour was observed to occur over at least a 12 month period, (Fig. 8). A detailed examination of subsidence behaviour above a stopped face was made for panel 2 where mining ceased for 4 months, after only about one third of the panel had been extracted, during the miners' strike. Figure 9b plots the movements of monuments in panel

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Time-dependent subsidence behaviour The behaviour of individual monuments located along the panel 1 centreline before, during and after they were undermined is illustrated for three representative monuments in Fig. 8. The gradient of the steep portion of these curves probably is controlled by the rate of face advance. The tail ends of these curves represents the residual subsidence. Interestingly the rate of increase of this residual subsidence is found to be about the same for all of the monuments in the centre of a panel. Also, this subsidence increase is linear with time. The measured average rates of residual subsidence were 3.35ram/month for panel 1 and 4.88mm/month for panel 2. The higher subsidence rate measured in panel 2 may have been influenced by the local geology. It is known that a fault ran longitudinally through the panel and that the overburden was less competent in panel 2 than in panel 1. Dames and Moore [5] give R Q D values for overburden as 93~ for panel I and 57~ for panel 2. Obviously the residual subsidence cannot increase at a

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(b) Fig. 9 (a). Portion of longitudinal profile through panel 2 (monument line W) on Jan. 3 1978, almost I month after strike began. Face position is immediately to the left of monument 38. (b). Subsidence of monuments in Fig. 9(a) at three different time periods during the strike, showing a relationship between the subsidence profile and the residual subsidence. Some data is missing due to ice coverase of the monuments, (Note dates are given month/day/yr).

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2 above the face during this stoppage. These results, and the results from analysis in panel 1, shows that when a face stops moving the subsidence immediately above the face continues at a faster rate than elsewhere along the profile. This causes the point of half maximum subsidence, the inflection point of the profile curve, to move closer to the face. Also this curve, which has a more gentle slope than the profile curve at the stationary end of the panel, steepens after the face stops, although it never becomes as steep as the stationary end profile. This behaviour can be seen in Figs 6 and 7 and is similar to British experience of time-dependent subsidence behaviour [1]. Curvature and strain Differential movements of the ground surface above a mining panel produces horizontal tensile and compressive ground strains. Damage to surface structures has been correlated directly with the magnitude of these strains [1]. Thus, in many cases the strains induced by the mining activity are of at least as much interest as the displacements within the subsidence trough. Strain is a function of surface curvature and curvature can be approximated by: d2x

lllp ~ dy2

where x is the vertical displacement (subsidence), y is the horizontal distance, p is the radius of curvature, and thus lip is the curvature. Since the slope of the ground surface G, is given by: G=--

dx

dy

curvature can be found directly from the vertical displacement measurements used to obtain the subsidence

trough. First the slope between monuments is calculated and then curvature is obtained by computing change of slope for unit horizontal distance. This procedure, which is described in the Subsidence Engineers' Handbook [i], was followed to compute curvatures above both transverse and longitudinal profiles at the Old Ben site. A sample of one of these profiles is given in Fig. 10. The general form of this curvature profile above panel 2 is as expected with positive values for curvature (equivalent to tensile strains) measured close to the panel sides and negative values for curvature (equivalent to compressive strains) measured in the panel centre. The profile above panel 1 however, is not as easily interpreted. The same general comments regarding the location of zones of positive and negative curvatures could be made if it were not for the perturbations in the curves marked J and K in this figure. An investigation to determine the reason for these perturbations showed that in the equivalent subsidence profile (Fig. 4) minor slope changes occured at locations J and K. Reasons for these changes in slope are not clear but possibly they are related to either variations in the extracted :thickness or to geological features in the overburden. In any event, because the curvature profile is obtained by twice differentiating the subsidence profile these minor slope changes are reflected as major perturbations. The fact that these perturbations are found in panel 1 rather than in panel 2 is related directly to the closer monument spacing of 4.57 m that was employed on the west side of this panel. On the east side of panel 1 and across panel 2 where these major perturbations are not as evident, a monument spacing of 9.1 m was employed (Fig. 2). Direct measurements of the changes in horizontal distance were made along the transverse monument line across the west half of panel 1. Two independent sets of measurements, using a tape extensometer and using manual taping, were made and the results were used to compute horizontal strain. These strain profiles are illustrated in Fig. 11. Both sets of measurements show

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§ -,.o Fig. 11. Strain over the west half of panel 1 (monumentline C) just after the panel was completed. The NCB predicted strain profile (solid line) is compared with strain computed from horizontal monument positions determined by manual taping (triangles) and strain computed from tape extensometermeasurements (squares). the same general trend with a zone of maximum tension inside the panel rib and a zone of maximum compression closer to the panel centre. The strain at the panel centre is approaching, but has not yet quite reached, zero, indicating that the panel width is just subcritical [2]. The magnitudes of the strains are approximately the same for these two measurement sets at the point of peak tension. However, the tape extensometer measurements, which are claimed by O'Rourke and Turner [6] to be about one order of magnitude more accurate than the manual tape readings, indicate higher values for peak compressive strain. Also shown in this figure is the predicted strain profile using the National Coal Board method. It is evident that the measured peak strains are about four times greater than the strains predicted by this method. This result is not surprising in view of the more gentle shapes of the predicted subsidence profiles (Figs 4 to 6), It is perhaps interesting to note that the position of the zone of measured maximum compression is the same as that for the predicted curve and that the predicted curve also indicates that this panel is barely subcritical. However, this curve predicts the zone of maximum tension to be immediately above the ribside whereas the measured position of this zone is 30 m in from the rib. It is worthwhile noting that although some fluctuations in the tape extensometer data plot are evident, the major perturbations indicated in the curvature plot (Fig. 10) are not reflected in this direct strain plot. This would suggest that curvature fluctuations while they reflect real features at the surface are not indicative of potential damage to surface structures. The National Coal Board [1] claims the following relationship for curvature and strain: l i p = kE 2

where

The nature of the relationship between these two parameters was investigated using the data from the Old Ben mine. The best fit to these data was provided, again by a parabolic relationship of the same form: but where k = 5.25 m - I This fit to the data is illustrated graphically in Fig. 12. The curvature profile obtained from the vertical displacement data is smoothed in this plot in the sense that displacement readings from the two monuments that produced major fluctuations in the curvature profile (Fig. 10) were ignored. From this figure it is evident that good agreement is obtained in the overall shape of these two profiles. Also, the magnitudes of the peak positive curvature are almost the same. The magnitude of the peak negative curvature is somewhat greater for the curvature obtained from direct horizontal measurement. Figure !3 compares the curvatures calculated from vertical subsidence measurements above a travelling face with those at the stationary end of the panel. In all cases the curvatures, and thus the strains, above the moving face are about one third of those at the stationary end. This result, which is similar to findings reported by the National Coal Board [1], reflects the more gentle slopes of the profiles over the travelling face, (Fig. 6).

OBJECTIVE OF PROFILE FUNCTION ANALYSIS A number of different types of functions have been used as subsidence profile functions. Examples of five of these function types are given in Table 2. These functions are all of the form: s = Sf(B,

y, c)

where

k = 4 1 . 7 ( m ) -1

S is the subsidence at the panel centre,

E is the strain.

B is a parameter that controls the range of the function,

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EMPIRICAL M E T H O D S OF SUBSIDENCE PREDICTION

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for non-asymptotic functions B is the critical radius and is given by

above two adjacent longwall panels at the Old Ben No. 24 mine. The third function, the exponential, was found not to fit the measured data very well, (Eichfeld, personal communication). The study described in this paper set out to extend the work of Munson and Eichfeld by investigating whether, in addition to using a profile function to fit a few selected subsidence profiles, the profile function method could be used as an accurate predictive tool. The same field results from the case study at the Old Ben No. 24 mine site in Illinois were used for this study. Because some of the functions listed in Table 2 had been examined already to some extent by Munson and Eichfeld, a different function, the hyperbolic tangent, was selected for a detailed comparison with the field data for this investigation.

B = h tan ~, where h = depth and ct = angle of draw, y is the horizontal distance, c is either a function or a constant. Three of the functions given in Table 2, the error, the exponential and the trigonometric were examined to a limited extent by Munson and Eichfeld [3] to determine whether they could be used to describe measured subsidence profiles in the Illinois coal basin. These workers found that by suitable adjustment of the constants used in the equations, two of these functions, the error and the trigonometric, could be made to fit measured profiles

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Comparison of profile function with measured data A program was written for a computer with an

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H O O D et al.:

163

EMPIRICAL METHODS OF SUBSIDENCE PREDICTION Table 2. Profile functions ~sted by Brauner [2]

Profile function Hyperbolic

s (y) = 0.5 S[I - tanh(by/B)]

Error

s(y)=0.5 S[I

Exponential Donets coal field Trigonometric

s(y) = s e x p [ - ( 1 / 2 ) [ ( y + B)/B] 2] s (y) = 0.5 S[I - (y/B) - ( l / n ) sin n (y/B)] s (y) = S sin"[(n/4)(y/B - 1)1

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Area of application

Equation number

References

U.K.

(3)

[8-10]

Upper Silesia, Poland

(4)

[11]

(5) (6) (7)

[12, 13] [14] [151

Hungary USSR

b = an arbitrary constant, B = radius of a critical area of excavation, u = an integration variable.

and

interactive capability to plot the profile function on the terminal screen. The profile function constants then were adjusted until a best fit image with the measured profile was obtained. The ground surface was taken as the y-axis. The negative x-axis was a vertical line through monument 1 for the three survey lines used at the Old Ben mine. For this geometry the profile function equation becomes: s (y) = 0.5 S [1 - tan h(b (y - I)/h)]

s(y)=0.5S[-1 + tan h(br(y - Ir)/h)]

the values b,, I,, br, It, obviously being the respective parameters for the left and fight portions of the profile. These parameters could be varied independently for each portion of the curve. While the inflection points of the curve measured from a reference point at the surface are used as input for the purposes of the calculations, it is of interest to express these inflection points in terms of their relationship to the panel edges. The distance from the left hand curve inflection point to the left hand panel edge is termed a~ and the equivalent distance on the right hand side of the panel is termed dr. The definitions of all of these terms are illustrated in Fig. 14. The best fit b and d values for both the left and right sections of the profiles were recorded for all 29 measurement dates along all three monument lines. Representative examples of both transverse and longitudinal plots are given in Figs 15 and 16. These figures show that the predicted profiles match the measured profiles quite closely over the majority of the transverse and longitudinal profiles for panel 2. The panel 1 predicted transverse profile given in Fig. 15 was a good fit with the measured data prior to mining panel 2. From this figure it can be seen that extraction of this adjacent panel has:

(8)

The parameters S, b, h, and I are required input into the program. S = subsidence at the centre of the trough, b = constant controlling the slope at the inflection point of the curves, h = mining depth, I = horizontal distance to the inflection point from monument 1, y = horizontal distance from monument 1. The program was written in a flexible manner to allow for different slopes for each of the trough sides. In order to achieve this flexibility the complete subsidence profile was calculated in two parts with the relevant equation for the left and right portions of the curve respectively, being: s (y) = 0.5 S [ - 1 - tan h (b~(y - I~)/h)]

(9)

14k-III[3 F I ~ II

IN

m

1401~tl14E]~ I 7M

m

II

WmI ImJ U

-11.6

Ir " ~ "

If ~x

\

J

1 I

-! .6

--2 8°

t

m

I_

-I .I

(10)

I I I I

I

t I I I

Illlll

Fig. 14. Profile function illustrating parameters Sm~ I,, If, ~ and d r.

g~

164

HOOD et al.:

E M P I R I C A L M E T H O D S O F SUBSIDENCE P R E D I C T I O N Jqb-TRES FI~M IIONUI41E~T I

8

P,

tim

288

~

408

~

~

780

808

080

-e.6

ILl

t~

- I .8

Id ,'5 M e) e)

"-;:.8 --

IIII

--

IIII I J

[11

Fig. 15. Comparison of hyperbolic tangent profile function (solid line) with measured data (diamonds) for transverse profiles through panels 1 and 2 on Oct. 5 1978, showing the poor prediction of subsidence values between panels above the chain pillar.

increased the subsidence in the centre of panel 1; reduced the slope of the west trough side; and increased the subsidence in the region between the two panels. None of these features is taken into account by the present profile function prediction technique. However, as noted previously, a study of the measured subsidence data reveals that time-dependent, or residual, subsidence accounted for most of the increased movement in the centre of panel 1. Also, crushing of the chain pillar between the two panels accounted for about 180 mm of the movement in this region [4]. If the profile function technique was modified to take these factors into account then this method would provide an accurate fit to the measured data almost everywhere along the p/'ofiles. The one region where the measured data does not fit the predicted profile well is at the upper corner of the trough sides, (Figs 15 and 16). The fundamental problem

here is that the predicted profile is antisymmetric whereas the measured profile is not. Therefore, although the fit might be improved by using a different function [3], it will never be matched exactly unless a nonantisymmetric function is used. On the other hand, from these figures it is evident that the curvatures of the predicted profiles are steeper in the these upper corner regions than the profiles for the measured data. Curvature often is used as a parameter to indicate damage to surface structures [1]. Thus this function will tend to overestimate curvature and consequently it will act as a conservative prediction technique. A summary of all of the best fit b and d values is given in Table 3. This table highlights a number of interesting points. First, the right hand sides of the transverse profiles, where the panel adjacent to the panel that is being mined is virgin coal, show remarkable consistency

NE'TRCS ~

B

J 08

2~

44m

~

r~O

I

680

7M

8gg

g~

-11.6 hJ j.hJ -I

.1~

Z h.j

- ! .5

-2.8

mm mm

m

IllllllII

Ill. . . .

Fig. 16. Comparison of hyperbolic tangent profile function (solid line) with measured data (diamonds) for a longitudinal profile through panel 2 on June 1, 1978. Values obtained for the equation parameters in this best fit curve were:/~ = 8, b r = 14, ~ = 43,

dr = 40.

H O O D et al.:

EMPIRICAL METHODS OF SUBSIDENCE PREDICTION

165

Table 3. A s u m m a r y o f the calculated slope and inflection point parameters Left side of Panel Mean value SD

Right side o f Panel Mean value SD

Remarks

Transverse profiles b~ = 11.5

0

br = 11.5

0

d t = 37.7

0

dr = 41.1

0.6

b~ = 10.6

1

b r = 11.6

1.5

d~ = 31.4

3.1

d r = 45.1

1.25

bI = dl= b~ = dl=

2.2 7.2 2.5 7.6

br = dr = br = dr =

0.7 1.3 1.8 2.3

Panel No. 1

Panel No. 2

Panel adjacent to left side is room and pillar Panel adjacent to right side is virgin coal Panel adjacent to left side is longwall Panel adjacent to right side is virgin coal

Longitudinal profiles Panel No. 1 Panel No. 2

8.8 48.2 7.7 45.5

both in the slope values and in the location of the inflection points of the curves relative to the ribsides. The consistency of these results is indicated by the low values for the standard deviations of the mean. Significantly, this consistency holds not only for different profiles within a single panel as the face advances, but also between panels. A similar result is obtained comparing these parameters for the longitudinal profiles above the stationary (right hand) end of the panels. The importance of this result is that it indicates that, where boundary conditions are known and when dynamic effects do not play a significant role, average values for the parameters used in this profile function can be used to predict subsidence behaviour along a panel that is being mined. Also these same parameter values can be used to make a priori predictions of subsidence behaviour for panels that are to be mined in the same local area using the same longwall mining method. Second, it is interesting to note that, comparing the transverse profiles on the left hand sides of panels 1 and 2 (Table 3), the slope parameter, b, for panel 1 adjacent to the room-and-pillar panel is the same value, 11.5, as for the condition when the adjacent panel is virgin coal. The inflection point of this curve is somewhat closer to the ribsides than is the case when the adjacent panel is unmined. This small difference in parameters on either side of the transverse profile is surprising in that the pillars were partially extracted in the room and pillar area and for this reason it might be expected that the slope over this side would be less than that over virgin coal. This expectation is met in panel 2 where the slope parameter of the transverse profile over the ribside next to the longwall is 10.6 (Table 3), substantially less than the 11.6 value obtained over the virgin coal. Also, the inflection point of this profile is substantially closer to the ribside where the adjacent panel has been mined. These results show that more subsidence is experienced over the longwall panel than over the room and pillar panel, reflecting the greater extraction from the former panel.

12.3 41.9 13.3 42.0

It is probably worth commenting on the fact that ideally the final transverse profiles across a number of adjacent panels would be flat in the centre or, if ridges do occur between panels, they should have gentle slopes. This type of behaviour would help to ensure that any damage that was caused to surface structures by the mining activity would be minimized or that agriculture at the surface was affected a minimal amount. In terms of this objective longwall mining appears to offer more desirable subsidence effects than retreat room and pillar mining, although even with adjacent longwall panels a substantial ridge remains between the panels. The impediment to eliminating this ridge completely is the chain pillar that is left between panels. Mining this pillar, which would have major beneficial results in terms of ameliorating the effects of subsidence at the surface and in the subsurface, could complicate the coal extraction procedure if a standard triple entry system is employed. Third, the profiles above the advancing faces on panels 1 and 2, the so-called development curves, show different values for, and wide scatter in, both the slope and inflection point parameters. This scatter is illustrated by the high standard deviations of the mean for these parameter values (Table 3). These variations must be related in some way to the dynamic conditions that exist above the advancing face and to the time dependent behaviour of the superincumbent strata. An attempt was made to correlate the results obtained for these parameters with the rate of face advance but no direct correlation was evident from this analysis. On the other hand, if these different parameter values and scatter are due to time-dependent effects only, then after the panel is completed it would be expected that the scatter would be reduced and that the parameter values would change and become closer to the values at the stationary end of the panel. The former expectation was met but the latter was not, indicating perhaps the effects of local geology on slope at the different ends of the panel. Fourth, it is very interesting to note that one set of

166

HOOD et al.:

E M P I R I C A L M E T H O D S OF SUBSIDENCE PREDICTION

~ I

!oo

zoo

aoo

F'I~EqFg~INENT

4co

r,m

I

eeo

7o0

mo

mo

.~4.S

7.~ *I.O IlL.

I

.

-'.,.~~ I

~lk..

-! .I 20 -S.O'

"'"

..4.I Fig. 17. Comparison of curvatures calculated from measured data (diamonds) for a longitudinal profile through panel 1 on March 4, 1977.

slope and inflection point parameters are self-consistent between panels for the transverse profiles and another set of parameters are self-consistent between panels for the stationary end of the longitudinal profiles. The variation in these parameters between the transverse and the longitudinal panels must be due to either the different mining geometry, that is in transverse section the panel is narrow but long, whereas in longitudinal section the panel is wide but short, or because of the anisotropic behaviour of the superincumbent strata. Since this phenomenon is not commented upon in the literature perhaps the latter explanation is the most likely. The hyperbolic tangent function was used also to compare the predicted values for curvature to the measure.d curvature data. The function with the appropriate best fit values of the b and I parameters was differentiated twice with respect to the horizontal distance to obtain the predicted curvatures. A numerical differentiation procedure using the measured vertical displacements was employed to obtain the 'measured curvature' values. A sample curvature plot is given in Fig. 17. This figure shows that curvature is predicted very accurately along most of the length of these profiles although some slight overprediction is observed. Perturbations in the measured profile were discussed pre-

viously and are caused by the technique used to calculate curvature.

INFLUENCE

FUNCTION

BEHAVIOUR

Several influence functions have been developed and a selection is given in Brauner [2]. Eight of the functions listed by Brauner and given in Table 4 were compared with the Illinois field data. Brauner [2] also describes the hand calculation methods that can be used to generate subsidence profiles using the influence function method. Other influence functions described by Bals [16], Pottgens [20] and Marr [21] utilize hand calculation methods. These methods are time consuming and prone to errors. For continuous influence functions (Table 4), they are an approximation. In this study, these continuous functions were computerized to speed up the analytic process, to reduce errors and to aid in comparison with measured data. Again the program was written for an interactive graphics machine. The first parameter that must be determined prior to conducting calculations using influence functions is the critical radius, B. This is the distance from the panel centre to the edge of the excavation beyond which, if the excavation is enlarged, no additional subsidence is observed at the surface above the panel centre. At the Old

Table 4. Influence functions listed by Brauner [2] and applied in this investigation Influence function p ( r ) = (Smax) B 3 tan3)'/nr (sin), cos)' + ~/2 - )')(r 2 + B 2 tan2?) 2 p(r) = 3(S~)[1 - r/B)"]2/gB 2 p ( r ) = 2(Sam) e x p [ - 4 ( r / B ) ~ ] / ( n x//nBr) n = 1 p (r) = 0.216(Sm~) e x p [ - 4 ( r / B ) 6 ] / B r n = 3 p (r) = ( S ~ ) exp [ -- n ( r / B )2]/B 2 n = 1 p (r) =-4.6(Stud e x p [ - 4 . 6 ( r / B ) " ] / n B : p (r) = 2 ( S = . ) e x p [ - 2 ~ ( r / B ) z ] / a 2 n = 2 p (r) = 7(Smx) e x p [ - 6 . 6 5 ( r / B ) ] / B 2 B = 6.65 r0, n = 1

Where n is a parameter characterizing strata conditions

Equation number

References

(11) 02) (13) (14) 05) (16) (17) (18)

[16] [17] 12] [2] [18] [19] [2] 12]

HOOD et al.: 120 -

EMPIRICAL METHODS OF SUBSIDENCE PREDICTION Table 5. Percent Smax at B

/

I00

8C

1113

-

/

~

~60

trl

14

#

167

Equation

Percent Sm~. at B

11 12 13 14 15 16 17 18

99.76 99.99 99.07 99.57 95.73 99.01 99.02 98.48

40

20

0

t 0

0.5

1.0

I

I

1.5

2.0

RADIUSOF INTEGRATION//CRITICALRADIUS(R/R)

Fig. 18. Predictedpercentageof S~x as a functionof R/B for equations (IlL (12), (13), (14) and (15).

Ben site the panel widths were such that this critical radius was never, or was only just reached in the transverse profiles, that is, in transverse section these panels remained subcritical or barely critical. Therefore the longitudinal profiles were used to determine this critical radius parameter. Longitudinal profiles of the measured subsidence data from both panel 1 and panel 2 were plotted for every measurement data. These profiles were inspected and the critical radius was determined to lie in the range 73-110 m. The impracticality of determining this parameter with more precision prompted an analysis of the error that is introduced in the calculated subsidence values as a function of the change in critical radius. Figures 18 and 19 plot the percentage of maximum subsidence as a function of R / B for all of the influence functions that are considered in this study. These graphs reveal a number of interesting points. First, while most of the equations behave as

120

expected and asymptote to 100~o of the maximum subsidence at R = B, three of the equations (15), (11) and (12) do not. These last two equations indicate that if integration is carried on beyond the critical radius then subsidence values in excess of 100% of maximum subsidence are calculated. For these equations then it is important that the upper limit of B be know accurately. Equation (15), on the other hand, shows that at R = B only 96~o of the maximum subsidence is calculated and integration must be carried on beyond the critical radius to approach 100Y/o of S=ax. This behaviour makes a nonsense of the definition of critical radius. A summary of the values of percent S=ax at R - - B is given in Table 5. From Figs 18 and 19 it can be seen that the error that would be introduced if the actual critical width is ! 10 m but integration is performed only to a 73 m radius depends on the function that is being considered. At first glance it may seem that this error can be quite high, as much as 20%. However consideration of the actual mine geometry fortunately shows that this is not the case. Only those portions of the annulus between the 73 and l l 0 m radii that are extracted need to be considered. Thus the error is reduced and in fact the error that would be introduced is less than one half of the values indicated by these plots. For the equations considered in this paper (Table 4) this error is no more than 6%. This maximum error occurs at the panel centre; towards the ends where the fraction of the annulus that is extracted is reduced, the error is less. This maximum error of 6~o was considered acceptable for the purposes of this study. Examination of the function listed in Table 4 shows that with the exception of equation (11) they can be written in the form p ( r ) = klSm~,(B , r, k2)

8C

where x

o

6o

4o i 2c o

Smax is the maximum possible subsidence, B is the critical radius, r is the radial distance from the point under consideration, k; and k: are constants; equation (11) has an extra input variable.

7t6 l

0...5

I.o

1.5

I

z.o

RADIUSOF INTEGRATION//CRITICAL RADIUS(R,~) Fig. 19. Predicted percentage of Smx as a function of R / B for equations (16), (17) and (18).

The first two of these, S~x and B, are input parameters. The other parameters control the slope of the profiles but they are fixed quantities. These equation constants are interdependent; if one of them is changed without the other then errors in the form of wild variations in S, the

168

HOOD et al.:

EMPIRICAL METHODS OF SUBSIDENCE PREDICTION NE'I"I~S FROM ~

lea

28Q

3BQ

~

.S~

I 6~

eQ~

aSO

F

-11.6

W r, O 2: b,I H e)

7M

- ! .8

- ! .6

-2.8

III

I

i

H

I

If

Fig. 20. Comparison of influence function, equation (11), (solid line) with measured data (diamonds) for a longitudinal profile through panel l on March 4, 1977.

subsidence at the panel centre, occur [4]. Thus there is no simple means for adjusting the slope of the profiles when influence functions are used. Also, a special technique, either a compensation zone method [2] or simply a curve translation method, [4] must be applied to move the location of the predicted profiles whenever the inflection point of the measured profiles does not overlay the panel edge. Application of these methods can lead to the introduction of considerable errors in the value of S is the distance through which it is necessary to move the predicted curve is large.

Comparison of influence functions with measured data For the purposes of this analysis B was held constant at 73 m. The value of Sn~x was taken as the maximum value of the measured data for the particular profile that was being plotted. Suites of curves, similar to those described for profile functions, were generated for the

il

IN

~

various influence functions along both longitudinal and transverse profiles for a number of measurement dates. Two plots (Figs 20 and 21) serve to illustrate the results from one of these suites. Inspection of these graphs reveals that none of the influence functions fits the complete subsidence profile in a satisfactory manner. Some equations fit the stationary end of the profile well and some others a fit the different slope of the profile over the advancing face well. Some reflection about the nature of influence functions reveals that, in a situation such as that illustrated in these figures where the measured data is not antisymmetric about the inflection points on the trough sides but is antisymmetric about the centreline of the extracted region, a single influence function can never be made to fit the entire profile in a satisfactory manner. The fundamental assumption that is made by the integration process used in the influence function method is that superposition can

m

m

•

,

M

" f

-8.5

hi

kd - ! .ll O H 44 44

- 2 .ll

'

I

1. . . .

Fig. 21. Comparison of influence function, ~uation (12), (solid line) with measured data (diamonds) for a longitudinal profile through panel 1 on March 4, 1977.

HOOD et al.:

EMPIRICAL METHODS OF SUBSIDENCE PREDICTION

169

caused to a 23 m structure if it is subjected to 0.5°.:o strain. The measured peak strains in this study were in excess of 2.0~. This major difference in strain behaviour means that particular attention should be paid to the prediction of strain in the United States. A potential problem was shown to exist when vertical displacement data is used to compute curvatures. If the measurements are taken too close together then major perturbations, reflecting real but small slope changes in the subsidence profile, are introduced into the curvature plot. On the other hand if measurements are taken too far apart then the magnitudes of the peak curvatures will be less than is actually the case. The National Coal Board recommends a monument spacing of 0.05-0.1 times the seam depth for this computation, and from the Old Ben mine experience this seems a sensible choice. A better technique would be to make independent strain and/or curvature measurements. These latter measurements could be made using inclinometers. The relationship between curvature and strain was investigated and it was found lip = 5.25E 2 provided the best fit to the data. Also it was found that the curvatures DISCUSSION AND C O N C L U S I O N S over the travelling face were much less than, usually less Reproducible measurements for the angle of draw than one third, the curvatures over the stationary end of were obtained in this case study when conditions outside the panel. This result has implications for mine layouts the panel boundary were known and where dynamic when it is desired to minimize damage to surface struceffects caused by a moving face were not involved. When tures. In summary, when taken out of the context for which this latter condition did apply the mean value for this parameter was similar to the value obtained at the it was developed, namely British coal mines, the stationary end of the panel but the scatter in the results National Coal Board method does not appear useful as about this mean was greater. A considerable difference a predictive tool for subsidence. This finding is supwas found between the angles of draw calculated for the ported by analysis from other subsidence studies [22-24]. transverse profiles (about 43 °) and for the longitudinal Although this method predicts the magnitudes of the profiles (about 17.5°). No explanations for this difference maximum subsidence accurately it fails to predict the are available at the present time but one suggestion that overall shape of the trough well. As a consequence the could be investigated is anistropic behaviour in the forecasts made for curvature and strain using this techoverburden. Both of these values for the angle of draw nique are less, in this case study, than the measured are very different from the single 35° angle reported for values by a factor of at least three. The hyperbolic tangent profile function is shown to be all subsidence profiles by the National Coal Board [1]. A comparison of the measured and NCB predicted capable of providing an accurate description of measubsidence profiles showed that although the maximum sured subsidence data at the Old Ben No. 24 mine in values for subsidence were predicted fairly accurately by Illinois. In addition, and more significantly, this function this method, the shapes of the predicted profiles did not is shown to predict accurately the shape of both the match the measured profiles well. The predicted curves subsidence trough and the curvature profile. This latter were consistently less steep along the trough sides. A findings is particularly important since curvature is direct correlation between subsidence measurements and known to be related directly to damage of surface structures. Other work [3] has shown that different extracted seam thickness was noted. Time dependent subsidence effects were found to be profile functions can be used to describe the measured small but measurable. It was found that the displace- subsidence profiles at this mine site. Thus it is concluded ment at a monument would continue to change in a that, provided the input parameters are known, the linear fashion for at least 12 months after the face had profile function method offers an accurate and easy-toadvanced well beyond the monument location. Also it use technique for subsidence prediction at least when was found that when a face is stopped the profile slope mining conditions are similar to those at this site in above the face steepens to approximate more closely the Illinois. The limitations of the profile function approach are profile slope at the stationary end of the panel. Measured strains were shown to have the same general described by [2]. The major limitation is that the use of trend predicted by the National Coal Board method. profile functions essentially is restricted to mine geomeHowever the magnitude of the peak strains was about tries that are simple. When longwall mining is practiced four times that of the predicted strains. Also, the then, almost by definition, the panel geometry is simple; National Coal Board reports that severe damage will be usually it is rectangular. However, since currently less

be applied. Thus if the mine geometry is symmetrical the final predicted curve must be antisymmetric about the inflection point and symmetric about the panel centre line. This result illustrates a striking difference between the use of profile functions and influence functions. With the former it is possible to use the same function to describe both ends of a longitudinal profile such as those illustrated in these figures, where the slopes at either end are changed simply by changing a parameter in the profile equation. The parameters in influence functions, on the other hand, have a complex interrelationship and it may prove to be easier, or even necessary to use two functions to describe complete antisymmetrical profiles of this type. Predicted values for curvature were obtained by twice differentiating the subsidence profiles calculated by the influence functions. Comparisons of these predicted and the measured curvature profiles showed that these two sets of curves matched well where the influence function subsidence profile was a good fit to the measured data.

R~MS 20/4~

R

170

HOOD et al.: EMPIRICAL METHODS OF SUBSIDENCE PREDICTION

t h a n 10% o f the coal m i n e d b y u n d e r g r o u n d m e t h o d s in the U n i t e d States is e x t r a c t e d using the longwail technique a n d r o o m a n d pillar panels are n o t necessarily o f simple g e o m e t r y , then a t t e n t i o n s h o u l d be d i r e c t e d tow a r d s investigating m e t h o d s which p e r m i t p r e d i c t i o n o f s u b s i d e n c e b e h a v i o u r a b o v e i r r e g u l a r l y s h a p e d panels. T h e influence function a p p r o a c h is such a m e t h o d . A c o m p a r i s o n o f v a r i o u s influence functions with m e a s u r e d d a t a f r o m this field s t u d y revealed two imp o r t a n t p r o b l e m s t h a t are a s s o c i a t e d with this m e t h o d . One, the subsidence profile o b t a i n e d is a n t i s y m m e t r i c a b o u t the inflection p o i n t s o f the t r o u g h sides a n d s y m m e t r i c a b o u t the p a n e l centre, w h e r e a s often, if n o t usually, this is n o t the case for the o b s e r v e d subsidence profile. It s h o u l d be n o t e d t h a t these p r o b l e m s can be h a n d l e d by profile functions simply by selecting n o n a n t i s y m m e t r i c functions [2]. A l s o , lack o f s y m m e t r y a b o u t the panel centreline is easily d e a l t with w h e n profile functions are used b y assigning different c o n stants to the profiles on either side o f the subsidence t r o u g h . W h e n a n influence f u n c t i o n a p p r o a c h is used however, it m a y require t h a t two different f u n c t i o n s be e m p l o y e d in o r d e r to o b t a i n a n a c c u r a t e predictive tool for the c o m p l e t e t r o u g h . T w o , this m e t h o d a s s u m e s t h a t the profile inflection p o i n t is l o c a t e d i m m e d i a t e l y a b o v e the p a n e l edge. W h e n this is n o t the case special m e a sures have to be e m p l o y e d to shift the l o c a t i o n o f the p r e d i c t e d curve. I f the d i s t a n c e t h r o u g h which this curve is shifted is large then i n a c c u r a c i e s in the p r e d i c t e d subsidence can result. These p r o b l e m s need to be addressed. Acknowledgements~This work was conducted as part of the DOE

Fossil Energy subsidence program under subcontract number 62-0200 through Sandia Laboratories. The results express the opinion of the authors only. Received 26 January 1982; revised 20 January 1983. REFERENCES

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