Generalization of the influence function method in

Generalization of the influence function method in mining subsidence

A. Bello García, A. Menéndez Díaz Department of Construction and Manufacturing Engineering. University of Oviedo J.B. Ordieres Meré, C. González Nicieza Department of Mining Exploitation and Research. University of Oviedo ABSTRACT: A generic approach to subsidence prediction based on the influence function method is presented. The changes proposed to the classical approach are the result of a previous analysis stage where a generalization to the 3D problem was made. In addition other hypothesis in order to relax the structural principles of the classical model are suggested. The quantitative results of this process and a brief discussion of its method of employment is presented at the end of this paper.

1. INTRODUCTION From the study of the available bibliography, subsidence prediction methods can be divided into three categories: - Empirical Techniques: Based on the experience gained from a large number of actual field measurements, the best known example of these methods is that developed by the National Coal Board (NCB 1966), where subsidence values have been related graphically to variable parameters, such as: depth, tilt, thickness, surface topography and seam geometry, etc. - Theoretical Modeling: These models are analytical or mechanistic in nature and are based on the rheology of subsiding materials and their reaction to changing mining geometry. Computer based techniques, such as the Finite Element (FEM), Boundary Element (BEM) and Distinct Element (DEM) methods of modeling of overburden rock mass and simulation of mine geometry have been used recently for the prediction of subsidence over mine panels (Jones 1985).

A. Bello García School of Mines Independencia, 13 - 33004 Oviedo (Spain)

- Influence Functions: These functions are used to describe the amount of influence exerted at the surface by infinitesimal elements of an extraction area. At present, they are found to be the most reliable and practicable. Functional methods have the advantage over the other methods that they can be used for complex mine geometry. Moreover, they allow the application of the time factor (Srivastava 1991). In Spain, the most complete study on mining subsidence has been published by the Instituto Tecnológico Geominero (Ramírez 1986). Here, a generic expression of the influence functions found in the bibliography is proposed:  ρ  −π  ρ  −π   C  C2   ∆w = C1 ⋅ e + n⋅e  3   2

2

  ⋅ ∆A  

(1)

where, w is the elementary subsidence at a point of the surface caused by the extraction of an infinitesimal area, ρ is the horizontal distance, C1 is a constant defined by the geometry of the panel, and C2 and C3 are related to the depth of the seam and reflect the decrease of subsidence 1

as ρ increases. These can be seen as the summary of the mechanical properties of the overlying strata. It seems that linear relationships like C2=kh and C3=2kh are good enough in most influence functions, although there is no information to support them. Thus, the last expression becomes: ρ   −π  ρ  − π    2 kh  kh  ∆w = C1 ⋅  e + n⋅e  2

2

  ⋅ ∆A  

surface and the seam are flat, varying the angle formed by these planes with the horizontal one. The thickness of the seam is 1.5 m and the width of the panel is 100 m.

(2)

where k and n are independent parameters. Furthermore, C1 depends on the lateral extension of the trough, which leads to the limit angle problem. Thus, the influence functions, with n ≥ 0 and assuming that w = 0 when ρ > R , can be expressed as: ρ   −π  ρ  − π    2 kh  kh  ⋅e + n⋅e  2

wmax ∆w = (1 + 4n)( kh) 2

2

  ⋅ ∆A  

(3)

where the parameters k and n reflect the overburden strata properties. In order to consider the thickness m of the panel, the maximum subsidence is expressed as a function of the former, multiplied by a corrector factor a that depends on the postmining treatment of the seam. In this way, the known final expression is: a⋅m ∆w = (1 + 4n)( kh) 2

ρ   −π  ρ  − π    2 kh  kh  ⋅e + n⋅e  2

2

  ⋅ ∆A  

(4)

2. SENSITIVITY ANALYSIS In order to find out the importance and influence of each parameter involved in this formula, three simple exploitation models have been studied, and these are presented below. In every case, the A. Bello, A. Menéndez, J. Ordieres, C. González

Figure 1.- Theoretical models use to evaluate the parameters a, k and n For each model the troughs corresponding to the following range of the parameters have been obtained: a ∈[ −2, 2] ; k ∈[ −2, 2] ; n ∈[ −2, 2]

(5)

as from an analytical point of view the negative values of these parameters seem appropriate, even though they can not be easily explained from a physical point. Nine examples of each model were selected to carry out a comparative study. In each one, the variation of the trough is related graphically with one of the parameters, the other two being constant. In the light of the results, it can be concluded that a has a multiplying effect, without a qualitative influence over the trough, which keeps its shape. On the other hand, k has a direct influence over the trough's shape. The trough becomes 2

wider but shallower when k increases. On the contrary, for decreasing values of k the trough gets narrower and deeper, disappearing for k=0. Finally, the parameter n has a local influence over the shape, having a discontinuity for n = −0.25, when the trough drastically changes its shape. For increasing values of n, not only positive but also for negatives ones, the trough keeps its traditional shape. However, for n ≈ −2 the central part of the trough changes its behavior and its convexity increases as n tends to -0.25. Apart from the specific study of each parameter, in this analysis the effect of the steepness, not only that from the seam but also that from the surface, on the trough's shape can be seen. Obviously, the first of these effects is more important for negative values of n, whereas the effect of surface steepness leads to asymmetric troughs, even for general values of a and k. This deformation increases as the depth of the seam decreases. In practice, the lack of surface flatness seems to be the most important cause of the asymmetry of the troughs. It is necessary to emphasize that usually, when using the NCB's Handbook or any other techniques, the ranges of the parameters involved in the influence function are limited, and vary from one expression to another. Generally, the commonest values used are (Ramírez 1986): a ∈[ 0,1] ; k ≥ 0 ; n ≥ 0

that implies the relaxation of the principle of rotational symmetry. Secondly, the term 1 + 4n is removed and this supposes a relaxation of the volume constancy, including the possible variation in the parameter a. This way, the first expression used in this project is:  ρ   ρ −π   m ⋅ aθ ⋅ ∆A  − π  kθ h  2k h ∆w( ρ, θ , h) = e + nθ ⋅ e  θ  2 ( kθ h)  2

∆v = ∆w

ρ

2

   

(6)

h

where m, ∆A are respectively the thickness and area of the extraction element, aθ is the subsidence

coefficient

and

kθ ,

nθ

are

independent parameters. The value these parameters depends on the direction θ from the point P at the surface, where the subsidence is evaluated, towards the centroide of the analyzed element, belonging to the exploitation. This way, the affected rock mass is supposed to behave orthotropically, improving the characterization of its properties, since it will not generally behave isotropically as is proposed in the principle of rotational symmetry. Somehow, the rock mass properties are included in the parameters a, k and n, and thus we try to give them some variability in order to improve the prediction, allowing a local orthotropical behavior.

3. GENERALIZATION OF THE INFLUENCE FUNCTION From a formal point of view, here some modifications of the final expression of the influence function are presented in order to relax some of the previous conditions imposed. First, a generalization of the horizontal problem is proposed, becoming a three dimensional one, A. Bello, A. Menéndez, J. Ordieres, C. González

3

yp

P ayp

aθ θ

axp

xp

Figure 2.- Characteristic values of the parameter a

program, based on the finite element method (SRAC 1994), was used to analysis the subsidence phenomenon. This way the results of each model can be compared in order to characterize the properties of the rock mass affected by the exploitation. In the following figures, the geometry of each one of the models evaluated is presented. Each one includes a coal seam with a thickness of 2 meters, and the width of the panel varies from model to another. Furthermore, the steepness of the surface and the seam varies in each model.

The value of each parameter can be expressed the following way, as a function of its characteristic values, presented in figure 2 for the parameter a ( axp , ayp ): aθ =

(a

xp

cos θ

) + (a 2

yp

sin θ

)

2

(7)

where the origin of the coordinate system is the centroide of the extraction element considered in each moment. From a physical point of view the proposed model assumes the study of subsidence along two orthogonal vertical planes passing through the centroide of each extraction element. The continuity of the phenomenon in all other planes is obtained by a smooth variation of the parameters included in the influence function. Emphasis must be placed on the relaxation of the constancy of each parameter in all the exploitation panels (principle of equivalence), since the model proposed assumes that this constancy takes place only in each individual panel, changing all of them from one panel to another. This way the model can be adapted to the singularities of the rock mass like fractures, faults, etc., and not simply to its orthotropical behavior that is a global property. In order to evaluate the influence function model proposed in equation 6 a commercial A. Bello, A. Menéndez, J. Ordieres, C. González

4

200

150

100

50

0

-50

-100 -150 200

-100

-50

0

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-100

-50

0

50

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-100

-50

0

50

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150

100

50

0

-50

-100 -150 200

150

100

50

0

-50

-100 -150

Figure 4.- FEM meshes of each model

Figure 3.- Models I, II and III evaluated using FEM Next, the FEM meshes obtained in each case are presented using the same coordinate system in order to ease the comparison between them.

A. Bello, A. Menéndez, J. Ordieres, C. González

The vertical displacement of the surface was obtained in two stages: first, an analysis was carried out assuming no exploitation at all of the coal seam. This way, the displacement due to soil compactation can be subtracted from the one obtained in a second analysis where the panel has been fully extracted. The results are presented in the following graphs, where the subsidence is related to the horizontal position of each point of the surface mesh. 5

0.02

0

subee cosmos 0.018

0.02 compactacion subsidencia sin compac. subsidencia con compac.

0.04

0.016

0.014

0.06 0.012

0.08 0.01

0.1 0.008

0.12 0.006 -60 0.24

0.14 -60 0

-40

-20

0

20

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-40

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40

60 subee cosmos

60 0.22

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0.2 compactacion subsidencia sin compac. subsidencia con compac.

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1

1.2 -100 0

0.08 -100 0.016

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0.015

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subee cosmos

100 0.014

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0.012

compactacion subsidencia sin compac. subsidencia con compac.

0.011

0.15 0.01

0.2 0.009

0.25

0.008

0.3

0.007

0.35

0.006 0.005 -100

0.4 0.45 0.5 -100

-80

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0

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Figure 5.- Surface subsidence of each model Then, the parameters of the influence function were estimated for each model. However, since these three models were bi-dimensional, the orthotropical behavior of the rock mass was discarded assuming that the characteristic values of each parameter are the same: a x = ay ; k x = k y ; nx = ny

The results obtained with the influence function model and COSMOS/M are compared in the following figures: A. Bello, A. Menéndez, J. Ordieres, C. González

-80

-60

-40

-20

0

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100

Figure 6.- Comparison of the results for each model with 3 parameters The results achieved with the influence function model were not satisfactory enough, since the trough's shape obtained with this method was quite different from the one obtained with FEM, especially in those cases where the surface and the seam are inclined. Therefore, a new parameter was introduced in order to improve the adaptation of the second term of the function to the variations observed in the models. This way, the new expression of the influence function is

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− πb   m ⋅ a ⋅ ∆A  − π  kh  e ∆w( ρ , h) = + n ⋅ e  kh  2 ( kh)   ρ

 ρ

2

2

  

(8)

where b is the new parameter. The results achieved with this new formula were:

error (the sum of the differences in every control point) and the maximum error (the biggest difference in one control point) obtained with the last two expressions of the influence function are presented.

0.019

Table 1. Total errors using the influence functions with 3 and 4 parameters Number of Parameters Model 3 4 (eq 8) I 0,017785 0,001210 II 0,724150 0,685652 III 0,055471 0,053379

subee cosmos 0.018 0.017 0.016 0.015 0.014 0.013 0.012 0.011 0.01 0.009 -60 0.24

-40

-20

0

20

40

60 subee cosmos

0.22

0.2

0.18

Table 2.- Maximum errors using the influence functions with 3 and 4 parameters Number of Parameters Model 3 4 (eq 8) I 0,002193 0,000118 II 0,007092 0,084789 III 0,005254 0,005305

0.16

0.14

0.12

0.1

0.08 -100 0.016

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-60

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subee cosmos

0.015 0.014 0.013 0.012

Bearing in mind that the lack of fit is due to the surface steepness, the function was modified for including a new linear term, reflecting the vertical distance between the surface and the extraction elements. This way, the varying depth along the model is considered in the influence function. The new expression becomes:

0.011

ρ  ρ  − πb   m ⋅ a ⋅ ∆A  −π  kh   kh    (9) ∆w( ρ , h) = e + n ⋅ e + c ⋅ h ( kh) 2   2

0.01 0.009 0.008

2

0.007 0.006 0.005 -100

-80

-60

-40

-20

0

20

40

60

80

100

Figure 7.- Comparison of the results for each model with 4 parameters In the light of these results, it can be concluded that model I, where the surface and the seam are horizontal, is perfectly estimated by the function reflected in equation 8. However, in the other two models only a slight improvement is observed. In the following tables the total A. Bello, A. Menéndez, J. Ordieres, C. González

where c is the fifth parameter included in the influence function. Table 3.- Total errors using the influence functions with 5 parameters Number of Parameters Model 5 (eq 9) II 0,118869 III 0,001840

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Table 4.- Maximum errors using the influence functions with 5 parameters Number of Parameters Model 5 (eq 9) II 0,011760 III 0,000280 The comparison between both methods in models II and III are presented in the following figures, since the inclusion of the parameter c does not suppose a noticeable improvement in model I. 0.23 subee cosmos 0.22

0.21

0.2

0.19

0.18

0.17

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0.15 -100 0.015

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subee cosmos 0.014

0.013

0.012

0.011

0.01

0.009

0.008

0.007 -100

-80

-60

-40

-20

0

20

40

60

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100

Figure 8.- Comparison of the results for model II and III with 5 parameters

A. Bello, A. Menéndez, J. Ordieres, C. González

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Table 5.- Improvements obtained related to the number of parameters Model No of % of fitting Parameters 3 100,00 I 4 6,80 5 3 100,00 II 4 94,68 5 16,41 3 100,00 III 4 96,23 5 3,32 As the reference value for the creation of this table, the total errors obtained for each model when applying the function with 3 parameters is taken. The errors achieved with the other expression of the function are reflected in relation to them.

4. CONCLUSIONS As a result of this comparative analysis the following conclusion can be drawn: In order to improve the fitting of the exponential terms of the influence function, a new parameter is included in the exponent of the second term, which causes a great improvement of the model. Also, the influence function must be modified to take into account the varying distance between the surface and the exploitation. This supposes the introduction of a new linear term. Furthermore, if the anisotropical behavior of the rock mass is taken into account in a generic three-dimensional model, some variability of the parameters must be introduced. Then, the characteristic values of each parameter are defined. Besides, the set of all these values has a local meaning, varying from one panel to another. The only negative aspect is the proper selection technique of each of these parameters. A. Bello, A. Menéndez, J. Ordieres, C. González

This way, the final expression of the influence function is: m ⋅ a θ ⋅ ∆A  −π  kθ h  e ∆w( ρ , θ , h) = + 2 (kθ h)   ρ 

+ nθ ⋅ e

 ρ   − πbθ   kθ h 

2

2

 + cθ ⋅ h   

(10)

This formula leads to a considerable improvement of subsidence prediction as the analysis presented in this paper shows.

5. REFERENCES Bello García, A.; Morís M., G.; Menéndez Díaz, A.; González Nicieza, C, Rodríguez Díaz, M.A. 1993. Aspectos gráficos en la predicción de la subsidencia minera. V Congreso Internacional de Expresión Gráfica en la Ingeniería, Vol 2, pp 9-18. Asturias, España. Bello García, A.; González Nicieza, C.; Ordieres Meré, J.B.; Ariznavarreta Fernández, F. 1994. Predicción de la subsidencia minera. XI Congreso Nacional de Ingeniería Mecánica. Valencia, España. Bello, A.; Rusev, P. 1994. Caracterización gráfica de la zonas de riesgo producidas por el fenómeno de subsidencia minera. VI Congreso Internacional de Expresión Gráfica en la Ingeniería, Vol 2, pp 217-227. Toledo, España. Gómez de las Heras, J.; Ochoa Bretón, A.; González Nicieza, C.; Bello García, A. 1994. El programa itgesub como aplicación a los estudios de subsidencia minera. IX Congreso Internacional de Minería y Metalurgia, pp 435-450. León, España. González Nicieza, C.; Toraño Alvarez, J.; Bello García, A.; Fernández Fernández, V. 1992. Efectos de la subsidencia sobre las construcciones en superficie. Anales de Ingeniería Mecánica. Año 9, nº5, pp 201-205. Madrid, España.

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Jones, T.Z.; Kohli, K.K. 1985. Subsidence over room and pillar mine in Appalachian coal province and the of subsidence predictive methods. Proc. 26th U.S. Symp. on Rock Mechanics: 179-187. USA. Karmis, M.; Agioutantis,Z.; Jarosz A. 1990. Subsidence prediction techniques in U.S.A.: state of the art review. Min. Res. End., v. 3, No. 3, pp 197-210. USA N.C.B., Production Department. 1966. Subsidence engineer’s handbook. London. Pariseau, W.G.; Duan F. 1989. Finite element analyses of the Moetake mine study slope: an update. Proc. 3rd Int. Conf. on Numerical Model in Geomech: 566-576. NY, USA. Ramírez Oyanguren, P.; Rambaud Pérez, C. et al. 1986. Hundimientos mineros: métodos de cálculo. Instituto Tecnológico Geominero de España. Madrid, ESPAÑA. Salamon, M.D.G. 1989. Subsidence prediction using a laminated linear model. Symposium on Rock Mechanics: Rock Mechanics as a Guide for Efficient Utilization of Natural Resources: 503-510. U.S.A. Srivastava, A.M.C.; Bahuguna, P.P. 1991. A critical review of mine subsidence prediction methods. Min. Sci. & Technol., v. 13, No. 3, pp 369-382. Saxena, N.C., USA SRAC, Structural Research & Analysis Corporation. 1994. Cosmos/m: finite element analysis system user guide. USA.

A. Bello, A. Menéndez, J. Ordieres, C. González

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