369 Minmg Science and Technology, 13 (1991) 3 6 9 - 3 8 2 Elsevier Science Publishers B.V., A m s t e r d a m A critical review of mine subsidence prediction methods P.P. Bahuguna a, A.M.C. Srivastava a and N.C. Saxena b a Department of Civll Engineering, Umversity ofRoorkee, Roorkee 247667, Indm b Substdenee Engineenng Group, Central Mining Research Statwn, Dhanbad 826001, India (Received October 1, 1990; accepted January 4, 1991) ABSTRACT Bahuguna, P.P., Srivastava, A.M.C. and Saxena, N.C., 1991. A critical review of mine subsidence prediction methods. Min. Sci. Technol., 13: 369-382. A review of the application of existing subsidence prediction methods available is presented. A brief description of all the three methods--empirical, functional and mechanistic--is given and merits and demerits of each method are discussed. A review of the applications of each method m different parts of the world and recommendations for research on further improvements in each method are given for the state of the art of subsidence prediction. Methods using the functional approach, especially those of influence function, are found, at present, to be the most reliable and practicable, however there is still scope for improvements m this approach, to incorporate the effects of various subsidence-inducing parameters by the use of complementary functions. Introduction The void created by large amount of extraction of coal or minerals from underground mines causes the overlying strata to collapse, resulting in subsidence on the surface, which may be an unwanted, if not incovenient, phenomenon. It causes damage to surface features; both m a n m a d e and natural. Increasing demands on energy and mineral resources have resulted in the mechanisation and rapid expansion of mining activities all over the world. With the increase in mining activities there will be a further corresponding increase in mine subsidence problems and the possibility of more damage unless proper subsidence control measures are taken. Subsidence control measures can be taken in the following three stages: (1) prediction, (2) prevention, (3) protection. The effectiveness of preventative and protective measures greatly depend upon the accuracy of prediction of subsidence and associated parameters, such as subsidence, horizontal displacement, slope and curvature of the subsidence trough and associated tensile and compressive strains needed to asses the possible damage to surface features. Once the maxi m u m subsidence and the profile of the subsidence curve is known, other parameters can be calculated. Subsidence prediction methods Subsidence precalculation methods can be roughly divided into three categories, based on: (i) empirical techniques, (ii) influence functions, and (iii) theoretical modelling [1]. Empirical techniques are based on the experience gained from large number of actual field measurements. The influence function methods are based on model propositions and 0 1 6 7 - 9 0 3 1 / 9 1 / $ 0 3 . 5 0 © 1991 - Elsevier Science Publishers B.V. All rights reserved 370 P.P. BAHUGUNA ET AL. mathematical assumptions to a greater or lesser degree; whereas theoretical model methods are analytical or mechanistic in nature and are based on the rheology of subsiding materials and their reaction to changing mining geometries. tional Coal Board (NCB) in the United Kingdom [2]. In their handbook on subsidence engineering, subsidence values have been related graphically (Fig. 1) to variable parameters, such as: thickness, depth, dip, panel geometry of the seam and the surface topography which influence subsidence. The magnitude and profile of the subsidence can be predicted with the help of graphical charts given in the handbook. Other related paramters, such as horizontal displacements, slope, curvature and associated horizontal strains can be calculated from the subsidence profile data. The empirical methods are quick, simple to use and yield fairly satisfactory results. The main disadavantage is that they are sitespecific; their use is restricted to areas having identical geological and mining conditions. Methods using the empirical approach Empirical methods are further divided into graphical and profile function techniques. Graphical methods rely on compilation and a summary of case histories in graphical form, from which a prediction of subsidence can be made. Graphical methods The best known example of graphical empirical methods is that developed by the Na~E I, 0 (a) Coving u~ JE O~ 0.8 0.6 = o / u S 0.4 u 0,2 or strip packing ./ / / / / / f Solid / stowing // \\ // x\. ,k/ M ,/ .0 0 0,2 04 0,6 0.8 i i I 1 0 1,2 I.~, width/depth ~, u~ 1.6 1.8 h ._L 2.0 2.2 (w/h) 1.0 (b) -~ ~0.8 o 0.6 .c ~: 0 . 2 ~ / ~3 0 50 I00 150 200 250 300 Extractton 350 400 width 450 500 550 600 650 (w).m Fig. 1. Nomograms developedby NCB. (a) To show the influence of width/depth on subsidence, (b) To show the influenceof extraction width. A CRITICALREVIEWOF MINE SUBSIDENCEPREDICTIONMETHODS To establish an empirical method for a particular coalfield a large number of field observations must be carried out first. Profile function methods The empirical profile function methods are basically curve fitting techniques for matching the predicted profiles with measured profiles to obtain a mathematical formula for the profile curve. Further predictions can then be made on the basis of the derived formula. A few profile functions formulae, developed in various parts of the world, are given below: (1) Donetz Trigonometrical Function. In the USSR, the following profile function was developed by VNIMI (General Institute of Mine Surveying, Leningrad) (in 1958) to predict subsidence in Donetz basin (described by Branner [3]): s=S [1 - ~ +x sin 2~r (1) and peak value of subsidence, S, is calculated by: S = a M cos a n~n% (2) 371 This method makes use of a formula derived from an empirically obtained ratio between subcritical and critical extraction areas Asub and Acrit. The equation gives fairly close agreement with measured subsidence values in Donetz and some other European coalfields. (2) Polish Profile Function. The profile function developed by Kowalczyk in Poland on the basis of numerous data in upper Silesian coalfields is given by [4]: S = Smax e x p ( - n x 2 ) (4) where: Smax n = R2~ (5) Where R is the radius of critical area. Here, account is taken of incomplete settlement at the edge of panel and ?, the average roof settlement, is not equal to aM. (3) Hungarian Profile Function. In Hungary the profile function developed by Martos [1] is represented by: and S / Asub Smax = V Aerit where: x and L (3) = distances of calculation point and trough margin from the centre of the subsidence trough (m), a = subsidence factor; M = thickness of seam (m); o/ = dip of seam; = m a x i m u m possible subsidence S~x occuring at critical width (m); S = m a x i m u m subsidence at the centre (m); s = subsidence at any point P along the profile (m); Asub, Acrit = subcritical and critical areas of extraction (m2); = constants for the particular mine n 1, n 2 geometry. for critical and supercritical widths, and ( --X2) s = S exp\ ~ 7 - (7) for subcritical widths where: W S = Smax2R (8) where x and I are the distances of calculation point and transition point from the centre of the panel and w is the subcritical width of the panel. This function produces a relatively flatter and wider subsidence trough because the observations in Hungarian Coalfields indicate the transition point to be not over the face edge but over the margin zone of the stowed goaf. 372 P.P.BAHUGUNAETAL. (4) Niederhofer's Profile Function. This method is based on mathematically obtained formulae in which some empirically determined factors are used. It is specially useful for calculation of subsidence profiles for inclined seams and complex geometry with the help of computers. It is represented by [11: s ,9) w Px,y x mflucnce fur~tK)n ex - AA ~ i" _ _ x;y' ~1 -I Fig. 2. Superimposition of elementary troughs (after [11). (10) (5) Indian Profile Function: The profile function used in Indian coalmines makes use of a constant n [5] and is given by, for subcritic~ widths: nx 2 results, but they can only applied to simple two-dimensional problems of rectangular extraction. Methods using influence functions s= S e (11) for critical widths. s=S e r I, in its simplest form, where: p = half width of subsidence profile, i.e.: p = ~- + R r [ p4--X4 1 (12) This method gives broader subsidence trough than observed in the field. (6) Hyperbolic Function. This formula developed by King and Whetton [6] is given below and gives fairly satisfactory results for British coalfields: s = S [1 - tan h(~--~x)l (13) (7) Trigonemetrical Profile Function. The trigonometrical profile function derived by Hoffman [7] only gives satisfactory results for some of the European coalfields " 2[qT/X s = S sin [ ~-~ -~ -- 1 )] These methods, based on influence functions, are used to describe the amount of influence exerted by infinitesimal elements of an extraction area. The extraction of infinitesimal area d A causes infinitesimal subsidence at the surface. The elementary subsidence of point P moving radially within an elementary trough can be described as influence function kz(r ), where r is the radial distance of P from dA. The function kz(r ) generally has a maximum value at r = 0 and diminishes as r increases. For more than one extracted element, the total subsidence at a point P is due to the sum of the influence of each element extracted (Fig. 2). Thus, the subsidence is given by: S(r)= f k z ( r ) . d A (15) (14) The profile function methods are simple to use and need little input data for application. The profile functions are easy to calibrate with field data and also yield satisfactory When polar coordinates are used, dA = r- d r .dO S(r)=Li~foi2kz(r).r.dO.dr (16) 373 A CRITICAL REVIEW OF MINE SUBSIDENCE PREDICTION METHODS and the equation for the normal subsidence profile (in polar form) is: P SURFACE / I\\ / / / \ h / / / i \ / s=S e(-~) \ \ / \ \ \ \ / \ \ I I I" ~ 2 3 i t 2R ' 1 . - e( - - ~ (2) Keinhorst's method. This method makes use of a formula which gives a simplified subsidence profile. The profile function [9] is: 77- max. tan2/3 S k z = 3~r(tan2/3_ tan27) " R--5 Fig. 3. Calculation of subsidence by the integration grid method. The value of influence function kz can be determined from measured values of subsidence S due to an area of extraction A. The influence of the mined area can also be shown in a graphical form by employing an integration grid or grid zones. A critical circular extraction area with critical width as diameter is drawn on tracing paper and divided into annular zones of equal subsidenceinduced influence. The required subsidence influence of the given extraction at a surface point on the grid is obtained by placing the centre of the grid on that point and adding up the number of grid meshes and their parts covering the workings (Fig. 3). Some of the selected influence functions are given below: (1) Knothe's method. The formula derived by Knothe [8] is based on a Gaussian distribution of probabilities: k z = N1e - ~ (18) (17) (19) where: 7 -- angle of influence of the outer zone (angle of draw); /3 = angle of break of the inner zone; R = h cot 7; h = depth of extraction. (3) Bals' method. Bal's formula is based on Newtonian gravitational law, that is, the influence on the surface being inversely proportional to the square of distance of the particular element. The function is expressed by [1,4]: C k z - R2 + h2 d a (20) and in usable form: kz=C ¼(sin2am + 2 a m ) (21) where: C = constant, a m = angle of influence measured to the vertical. (4) Beyer's method. This influence function for calculating subsidence [1,4] is: kz-~rR23S [ 1 - ( R ) 2 1 2 (22) A table for k z is prepared semigraphically for various stages of r/R to calculate the subsidence values. 374 P.P. B A H U G U N A ET AL (5) Sann's method. Sann's formula for calculating subsidence profile is [1,4]: k z = 2 . 2 5 6 1 e -4rz r (23) This method predicts a trough with a deeper central area and, therefore, higher values are obtained for partial extractions. (6) Litwiniszyn's method. Based on probability considerations, and well supported by field and experimental observations, this method has also been verified with the theory of stochastic rock movements. The formula used is [4]: kz =nS [ [r ~-~exp[-n~r[~)2] @ Keinhorst Bells Sann Bev~r (24) where n is a constant usually equal to 1. This method has further been modified by Kochmanski. Figure 4 shows a comparison of various influence function zones in graphical form. The merits of the influence function are that: (1) they are also applicable to complex mine geometry, (2) they can be mathematically validated, (3) they are applicable in various types of mining situations, (4) some factors other than mine geometry can be used in the form of complementory functions. For these reasons, influence function methods are widely used with considerable success in most of the mining areas in the world. The demerits of this method are: (1) They become more complicated than profile functions when an extensive area of irregular configuration is encountered. (2) They predict symmetrical subsidence profiles about the trough centre which is not always the case. (3) The inflection point in the influence function is located just above the ribside, which is also not always the case. Disadvantages (2) and (3) can, however, be over- Ehrnard~¢ a n d $auer Knothe Fig. 4. Various influence function integration zones (after [4]). come by modifying the influence function accordingly. Methods using theoretical models These methods are based on statistical or mechanistic laws considering the material of the overlying strata as a model of either a cohesionless stochastic or elastic or even plastic, isotropic or anisotropic medium. Computer-based techniques, such as the Finite Element (FEM), Boundary Element (BEM) and Distinct Element (DEM) methods of modelling of overburden rockmass and simulation of mine geometry have been used recently for the prediction of subsidence over mine panels. In FEM the structural analysis of the overburden and gob is made by dividing and A CRITICAL REVIEW OF MINE SUBSIDENCE PREDICTION 375 METHODS subdividing it into individual structural elements (Fig. 5). Because of stresses in the overburden body, the nodes of the mesh experience strains and get displaced. The amount of displacement of each element depends on the level of stress and material properties of each element. In FEM the effect of regular and large numbers of geological discontinuities such as joints, faults, bedding planes, etc., and different types of rock layers in the overburden, can be taken into account as the finite element mesh is spread all over the body of the overburden. At the same time, however, this makes the method more voluminous and time consuming. In the boundary element method of subsidence simulation the element mesh is not spread all over the body of the overburden but only at the boundary, that is, on the ground surface. This method is more suitable for cases where geological discontinuities are comparatively less because the method is simpler than FEM. The distinct element method represents the rockmass as a discontinuous system of interacting blocks. This method is suitable for modelling a jointed rockmass where the deformation mechanism is mainly block separation, rotation, or slip, and there are large relative movements. This method has yet to establish its credit in satisfactory subsidence prediction. State of the art of subsidence prediction The existing methods of subsidence prediction do not have universal application and were generally developed for local considerations. The state of art of various subsidence prediction methods is discussed below. Empirical methods A good number of predictive empirical methods can be considered as the extension of empirical methods developed in the U.K. by the National Coal Board (NCB) [2]. Most of these empirical methods have been developed for European coalfields which have consistant geological conditions. A variety of predictive methods based on British practice have ~ -'~X Surface 1206 Sand stone Shale Sand s~one Silt stone Sand stone Shaley Sand stone c o ": 5 0 6 Sand stone U.l Shale Sand stone Shale San~ stone 30$ 300 Coal Sh~?g&nd ~ond stone 0 615 Distance from panel c e n t r e ~ f e e t Fig. 5. The finite elementmesh, a sectionalview (after [10]). 376 P.P. B A H U G U N A E T A [ 0 Distance(m) I00 I D i s t a n c e (ft) 0 T 200 S t a t i o n no, 8 0~-L. 9 400 11 I0 300 200 I 600 12 . 800 . . . . . . I 1000 13 'A. 15 16 17 18 19 20 21 . 4-00 I . 1200 22 . . 23 ~C-00 24" . / . I 2:5 - 26 ~ hO ~'/ 0.4, E \ \ ~, 0 ' 6 'ID / / \ '~ O.B 2,0 / u "7, .o 3,0 curve Predicted-Hyperbolic Predlcted-NCB Predicted- Finite Element ~ Field 1.0 partial Coal 21m __ F No. 2Panel Mined out ll2rn T Barrier proposed partial Mined out L . .4,8m __ 91 m -I- (20') 4,'0 No. l P a n e l (370') No. 4 Panel Coal I (160 ~) (300'} Fig. 6. Predicted (hyperbolic function, NCB, finite element methods) and measured subsidence profiles (after [10]). yielded satisfactory results only for those coalfields having similar geological and mining conditions and nowhere else. Jones and Kohli [10] found that NCB methods predicted almost 100% higher values of maximum subsidence (Smax) in U.S. coal- 0 JW~, a // , 20~ /. ~o ~\. " "h~. i ~. \ ',%), 4'0 ea R- , ~/ z //o" ,~./ o9 ,2,;/ x / J / //./. ,I ~. \ . . + . j~.>~'" / \ 60 a3 o---.--o ~- . . . . ~ ~-----~ 4- . . . . 4- 8O 100 0 Measured subsidence NCB p r e d i c t e d s u b s i d e n c e ( s t a b l e chain p l l [ a r s ) NCB p r e d i c t e d s u b s i d e n c e (failed p i l l a r s ) p r o f i l e ~ u n c t i o n - - D o n e t z 25° I 1000 I I 2000 3000 D i s t o n c e j ~eet Fig. 7. Measured versus theoretical subsidence profiles (after [11]). 4000 A CRITICAL REVIEW OF MINE SUBSIDENCE PREDICTION 377 METHODS fields (Fig. 6), whereas predictions form the Hyperbolic Function Method and FEM were reported to be matching well with the measured values. Considerably large values of Sm~x (Fig. 7) have also been reported with these methods in the coalfields of the western United States by Feejes [11]. Holt and Mikula [12] found that empirical methods developed elsewhere gave erroneous results when used in Australia. In India, as well, the empirical methods developed for European coalfields have been said to behave unsatisfactory by Kumar et al. [5] as they do not satisfy the boundary conditions. The above discussion implies that most of the empirical methods prove to be good only for the localities for which they were evolved, or for the regions having identical geological and mining conditions. There is scope for further improvements in empirical graphical methods for incorporating site-based parameters. Analysis of the empirical relationship of measured data from various coalfields and mines in the world could be improved by including the local geology and rock conditions as parameters for the study of their effect on the amount and extent of mine subsidence. Development of some simple formulae for Sma~ based on field data for different types of geological and other site parameters, could be explored for use with the profile functions. Mechanistic models Simple analytical models have proven incapable of simulating the complex strata behaviour encountered in the process of subsidence [13]. Analytical or mechanistic methods based on computerised mathematical models using FEM have been employed recently with limited success for subsidence prediction by some researchers world over. Jones and Kohli [10], using FEM, could obtain the predicted surface subsidence profiles matching within 15% of the measured profiles (Fig. 6). 0 ~ m 0 6 ~ 11 -- FIELD ~0 INEAR N o i sb ido HORIZONTAL ~o 260 2~o DISTANCE ) m ~ r ~ s Fig. 8. Subsidence profiles based on two-dimensional finite dement analysis (after [14]). Siriwardane [14] using two-dimensional and one-dimensional finite element idealization in the study of subsidence in longwall mines, concluded that these procedures failed to give satisfactory prediction of maximum subsidence and needed some improvements (Fig. 8). Later Siriwardane and Amanat [15] could get better results by using the displacement discontinuity method for the same mines (Fig. 9). Similar investigations on the use of FEM for subsidence modelling did not yield satisfactory predictions when conducted in China by Sugwara et al. [16]. The discrepancies were, however, attributed to the fact that the rheological behaviour and fractures of the cap rock layers could not be taken into account. Dahl and Choi [13] suggested that the strength and modulus of a large section of the Carboniferous and overlying rocks should be drastically reduced from those obtained from laboratory tests in order to account for the lower strength and elastic modulus of in situ jointed rocks than of intact laboratory samples. Since this reduction is arbitrary, and in a way manipulative in order to obtain the predicted values elose to the measured value, the approach gives solutions of localized nature. Coulthard and Dutton [17] used continum and distinct element stress analysis to study the subsidence and found that this approach gave considerably shallower troughs; the rea- 378 P.P. B A H U G U N A Horizontal (a) dlstance~ E T AL, MeGsured feet 200 4.00 . . . . ~.---- LI n e a r eIa Stl C L i n e a r eIos-~lC w i t h ~nterfQce Linear reduced elastic with modulus 7.':_ ,~ iIJ 6 u -,, F i e l d 1foot = 0.305 m -- "~ - 2 ] ", 15° I_.l ~ Predictions w/crack g, predictions w:lO c r a c k Horizontal (b) • ,o,o , distance, ~oo Fig. 10. Subsidence comparison with prediction with interface and reducedmodulus(after [18]). feet 6~o , s,oo o E u~ 0.2 Srnax ( f e e t ) 0.4 "o 3.28 ~ 0.6 _N son for the discrepancy was again attributed to arbitrary selection of material properties and jointing pattern in the overburden. Hazen and Sargand [18] used in the three predictive methods, namely the NCB graphical, functional and finite element method, and reported excellent agreement of the subsidence values (Fig. 10) predicted by profile functions with the measured values. FEN gave reasonable agreement for strain values only. In India, Shankar and Dhar [19] attempted numerical modelling using /satrap/c, transversely isotropic and multi membrane models. 0.8 E 1.0 # : # // / ~ ~Predicted 3.69 --Cry, ~ z I foot : 0.305m Fig. 9. Comparison of measured and predicted subsidence based on the displacement discountinuity method. (a) subsidence versus distance from the excavation. (b) Normalized subsidence, S/Sma x, against distance from the excavation. lOO lOO ~ ,,,) : 0.5 NCB .. 80 V : 0,25 -e~ ff'~ 4C i _,_r ' 0 J'~'Ji// " 20 i 40 i i 60 80 100 120 140 W/H Fig. 11. The calculated values of Tsur Lavie and Denekamp subsidence and NCB measurements (after [20]). 379 A CRITICAL REVIEW O F M I N E S U B S I D E N C E P R E D I C T I O N M E T H O D S They reported good agreement among the three models but the results did not match well with the measured values. Two-dimensional and three-dimensional boundary element modelling have also been attempted by some researchers, such as Tsur Lavie and Denekamp [20] and McNabbe [21]. McNabbe's investigations were directed only to the study of the relative importance of some parameters in numerical modelling and no attempt was made to check the results with actual observed values. Tsur Lavie and Denekamp [201 recommended a lower Poisson's ratio (0.25) for higher depths and higher values for shallower depths. Their findings are shown in Fig. 11. This proposition has also been supported by other investigators. The use of equivalent material and other laboratory models has, however, helped in understanding the mechanism of subsidence and they have been found to be satisfactory for qualitative results but not for quantitative predictions. The mechanistic methods have, so far, not found widespread application in predicting subsidence for three major reason: the difficulty in determining the behaviour of overburden strata, difficulty in measuring and estimating correct material properties of massive overburden rock layers, and the necessity of making simplified assumptions to simulate the complex,field problems. As stated earlier, the laboratory values of elastic moduli need to be reduced for use in the calculations in order to account for the lower strength and presence of joints in the in situ rock mass. This reduction, which varies with the mine geometry, depth of the mine and other site factors, needs further study. This will help in choosing the right elastic parameters to yield reliable predictions. Functional approach It has become apparent that functional methods currently represent the most realistic Metres from m o n u m e n t 0 200 400 600 ] BOO I000 0 u,-05 E u g, -lO -15 -20 Fig. 12. Comparison of calculated influence function values with measured data (dots) (after [24]). and reliable approach for subsidence prediction. The functional methods have the advantage over the empirical methods in that they can be used for complex mine geometry. At the same time, these methods allow application of the time factor. Suitable supplementary functions can be evolved to take care of various parameters effecting the mine subsidence. These statements have been confirmed by Karmis et al. [22] and Steed et al. [23] and have further been substantiated by Jones and Kohli [10], who found hyperbolic functions predicting accurate results than FEM. After testing influence functions, Hood et al. [24] reported remarkable agreement between predicted and measured profiles (Fig. 12). Aston et al. [25], comparing the results obtained by six computer programs, three based on empirical and other three on mechanistic models, found that he later models predicted comparatively shallower subsidence troughs because these models were overstiff. The usefulness of functional methods have been further enhanced by incorporating suitable supplementary functions for different parameters. Sutherland and Munsion [26,27] incorporated the effect of mined and unmined zones to predict subsidence, which is specially useful for bord and pillar mining (Fig. 13). They further recommended the inclusion of material response-based formulations as supplementary functions to encom- 380 P.P. B A I - I U G U N A 0 • • - .~ i ET AL p • "-% 't \ ?e, A F I E L Do OA, DATA , \ I, \\ 0.6 \\ \ , 1.0 I 0.5 0 TRANSVERSE 0.5 DISTANCE (V/W) Fig. 13. Field and predicted subsidence (after [26]). pass material properties of the overburden and geological factors. Tandanand and Powell [28] have attempted to consider the effect of the lithological distribution of hard and soft rocks in overburden in the functional methods. There is further scope for work in this direction. Heasley and Saperstein [29] have taken into account the edge effect, on and near the ribside by introducing corrective parameters. Based on NCB experience, similar edge effect adjustments have been introduced into influence functions by Ren, Reddish and Whittaker [30]. Hellwell [31] has given an empirical formula for inclusion of the effect of geological faults and suggested that further research should be carried out on this aspect. Conclusions Of the existing subsidence prediction methods which have been considered, the empirical graphical methods are too localized in nature and need measurements and study of the necessary parameters afresh for each new region. The mechanistic approach, however, is more fundamental, but the real problem re- mains again of determining the parameters afresh in each case and analyzing the physical properties of rock layers on a large scale, which is normally beyond the means of mine operators. The arbitrary reduction or adjustment of laboratory values of modulii for use in mathematical modelling further makes it unreliable. The functional methods, besides being simple in use, are also capable of inchiding the effects of various influencing parameters and complex mine geometry. Research needs to be directed towards the mechanistic approach to make it more meaningful, reliable and practical. Until this has been achieved research on further improvements in the functional methods should continue. 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