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29th International Conference on Ground Control in Mining: 29th International Conference on Ground Control in Mining: Morgantown, WV; 2010 Determination of the optimum crown pillar thickness between open-pit and block caving Title of paper: Authors: Kazem Oraee; University of Stirling, UK Ezzeddin Bakhtavar; Urmia University of Technology

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29th International Conference on Ground Control in Mining: 29th International Conference on Ground Control in Mining: Morgantown, WV; 2010 2 Introduction Dimensional analysis Modeling Conclusions

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29th International Conference on Ground Control in Mining: 29th International Conference on Ground Control in Mining: Morgantown, WV; 2010 There are many mines that will have to change from open-pit to underground mining due to increasing depths and environmental requirements. The only underground methods whose costs are comparable with surface mining are caving methods, especially block caving. In these cases, it is often necessary to leave a crown pillar between the open-pit floor and underground workings. 3

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29th International Conference on Ground Control in Mining: 29th International Conference on Ground Control in Mining: Morgantown, WV; 2010 The main duties of such pillars are: To provide ground control for the mines, both surface and underground To minimize interference between the two mines To prevent water from entering underground mine from the surface pit To confine caving forces within the block to encourage caving to begin 4

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29th International Conference on Ground Control in Mining: 29th International Conference on Ground Control in Mining: Morgantown, WV; 2010 There are three possibilities: Surface mining before underground Simultaneous mining in both Underground mining before surface In all cases, provision of a crown pillar is necessary. 5

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29th International Conference on Ground Control in Mining: 29th International Conference on Ground Control in Mining: Morgantown, WV; 2010 A crown pillar between open pit and underground mines 6 Open-pit limit Transition depth Underground layout Crown pillar

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29th International Conference on Ground Control in Mining: 29th International Conference on Ground Control in Mining: Morgantown, WV; 2010 Determination of the optimal thickness of a crown pillar in a combined mining method using open-pit and block caving is an interesting and important decision faced by the mining engineer. Leaving a pillar with optimal thickness will minimize detrimental interference between the two working areas, whilst maximizing ore recovery. 7

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29th International Conference on Ground Control in Mining: 29th International Conference on Ground Control in Mining: Morgantown, WV; 2010 In this paper, a formula is established by using the available methods for surface crown pillars. Consideration of the effective parameters is the basis of determinations for properly dimensioning a crown pillar. This has been done by dimensional analysis procedure. The established formula can be used as a useful tool in all similar mining situations by mining design engineers to calculate the optimal crown pillar thickness. 8

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29th International Conference on Ground Control in Mining: 29th International Conference on Ground Control in Mining: Morgantown, WV; 2010 9 Dimensional analysis Modeling Conclusions Introduction

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29th International Conference on Ground Control in Mining: 29th International Conference on Ground Control in Mining: Morgantown, WV; 2010 DDimensional analysis is a technique for restructuring the original dimensional variables of a problem into a set of dimensionless products using the constraints imposed upon them by their dimensions. TThere are two main systems: - Mass system - Force system 10

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29th International Conference on Ground Control in Mining: 29th International Conference on Ground Control in Mining: Morgantown, WV; 2010 In mass system, three units are regarded as fundamental: mass (M), length (L), and time (T). Force system considers force (F), length (L), and time (T). In this paper, the force system is the basis of modeling. Any other physical unit is regarded as a derived unit, since it can be represented by a combination of these base units. Each base unit represents a dimension. 11

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29th International Conference on Ground Control in Mining: 29th International Conference on Ground Control in Mining: Morgantown, WV; 2010 12 Introduction Dimensional analysis Modeling Conclusions

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29th International Conference on Ground Control in Mining: 29th International Conference on Ground Control in Mining: Morgantown, WV; 2010 First, the most important determinants of pillar thickness are decided. Since both conditions and concept are similar, the methodology of surface crown pillar thickness determination has been used. Then, on the basis of the selected parameters, the main model is established by dimensional analysis. 13

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29th International Conference on Ground Control in Mining: 29th International Conference on Ground Control in Mining: Morgantown, WV; 2010 Considering the most important aspects of “crown pillars between open-pit and block caving” and the available methods in relation to “surface crown pillars”, the most effective parameters (variables) are: Block span and height: geometry of the block RMR: discontinuities and their characteristics, uni-axial compressive strength and groundwater pressure are reflected in geomechanics as RMR classification. 14

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29th International Conference on Ground Control in Mining: 29th International Conference on Ground Control in Mining: Morgantown, WV; 2010 Cohesion strength: an important parameter that determines crown pillar stability. Specific weight of rock mass: another important parameter that affects crown pillar stability. 15

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29th International Conference on Ground Control in Mining: 29th International Conference on Ground Control in Mining: Morgantown, WV; 2010 Table 1- Effective parameters 16 Effective parametersVariable Crown pillar thickness t Block span s Block height h Rock mass rating RMR Cohesion strength C Specific weight of rock γr γr

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29th International Conference on Ground Control in Mining: 29th International Conference on Ground Control in Mining: Morgantown, WV; 2010 The crown pillar thickness (t) is assumed to be a function of these variables: To specify the relationship between the independent and dependant variables of the problem, this is transformed into the Equation: 17

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29th International Conference on Ground Control in Mining: 29th International Conference on Ground Control in Mining: Morgantown, WV; 2010 Adopting the force system for the expression of the dimensions, the dimensional values for each variable are shown in Table 2. 18 Table 2- Dimensional values Variable (unit)Dimension t (m)[L] s (m)[L] h (m)[L] RMR[1] C (ton/m 2 )[FL -2 ] γ r (ton/m 3 )[FL -3 ]

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29th International Conference on Ground Control in Mining: 29th International Conference on Ground Control in Mining: Morgantown, WV; 2010 In order to make a dimensional matrix, the variables should be arranged as in Table 3. 19 Table 3- Dimensional matrix DimensionQuantity tshRMRC γrγr F000011 L1110-2-3 T000000

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29th International Conference on Ground Control in Mining: 29th International Conference on Ground Control in Mining: Morgantown, WV; 2010 The determinant of the right section of the dimensional matrix is calculated as: 20 DimensionQuantity tshRMRC γrγr F000011 L1110-2-3 T000000

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29th International Conference on Ground Control in Mining: 29th International Conference on Ground Control in Mining: Morgantown, WV; 2010 When the determinant of this matrix is zero, on the basis of Buckingham theorem the following Equation can be used: m is the number of dimensionless products n is the number of dimensional variables k is the number of primary quantities 21

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29th International Conference on Ground Control in Mining: 29th International Conference on Ground Control in Mining: Morgantown, WV; 2010 On the basis of k=2 and n=6, there are four dimensionless products. 22 DimensionQuantity tshRMRC γrγr F000011 L1110-2-3 T000000 k = 2 n = 6

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29th International Conference on Ground Control in Mining: 29th International Conference on Ground Control in Mining: Morgantown, WV; 2010 The homogeneous linear algebraic equations (as below) can be derived from the dimensional matrix. 23 Dimension tshRMRC γrγr K1K1 K2K2 K3K3 K4K4 K5K5 K6K6 F 000011 L 1110-2-3 T 000000

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29th International Conference on Ground Control in Mining: 29th International Conference on Ground Control in Mining: Morgantown, WV; 2010 In order to solve the two previous Equations, different values should be allocated to K 1, K 2, K 3 and K 4 and hence K 5 and K 6 are calculated. In this way, matrix of responses can be made as in Table 4. 24 Table 4- Matrix of responses K1K1 K2K2 K3K3 K4K4 K5K5 K6K6 tshRMRC γrγr π1π1 1000 1 π2π2 0100 1 π3π3 0010 1 π4π4 0001 00

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29th International Conference on Ground Control in Mining: 29th International Conference on Ground Control in Mining: Morgantown, WV; 2010 Thus, four (π 1 -π 4 ) independent dimensionless products are: Then the independent dimensionless products can be written as: 25 K1K1 K2K2 K3K3 K4K4 K5K5 K6K6 tshRMRC γrγr π1π1 10001 π2π2 0100 1 π3π3 0010 1 π4π4 000100

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29th International Conference on Ground Control in Mining: 29th International Conference on Ground Control in Mining: Morgantown, WV; 2010 We now have to choose the equation type: either linear or non-linear. Linear and non-linear equations can be written as: Linear: Non-linear: 26

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29th International Conference on Ground Control in Mining: 29th International Conference on Ground Control in Mining: Morgantown, WV; 2010 Experience shows that non-linear Equations are often more suitable. After making slight simplifications it can be transformed into the following Equations: 27

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29th International Conference on Ground Control in Mining: 29th International Conference on Ground Control in Mining: Morgantown, WV; 2010 Hence the following basic formula is derived. This formula determines the optimal thickness of the crown pillar between open-pit and underground mining in the case of block caving: 28

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29th International Conference on Ground Control in Mining: 29th International Conference on Ground Control in Mining: Morgantown, WV; 2010 The case studies coefficients in the basic Equation can be determined on the basis of a data from some real situations as in Table 5. 29 Table 5- Data of the real case studies Case studies Values tshRMRC γrγr 1 20018040062.50.752.7 2 200220400752.93.1 3 1801902304812.75 4 230250460700.822.81

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29th International Conference on Ground Control in Mining: 29th International Conference on Ground Control in Mining: Morgantown, WV; 2010 After assignment of the coefficients using SPSS software (version 14), the final formula for determination of a practical crown pillar thickness between open-pit and underground becomes: 30

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29th International Conference on Ground Control in Mining: 29th International Conference on Ground Control in Mining: Morgantown, WV; 2010 31 Conclusions Introduction Dimensional analysis Modeling

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29th International Conference on Ground Control in Mining: 29th International Conference on Ground Control in Mining: Morgantown, WV; 2010 For sensitivity analysis of the selected variables an hypothetical example is studied as in Table 6. Using the established formula, crown pillar thickness is calculated to be equal to 221 m. 32 Table 6- An hypothetical case example tshRMRC γrγr ?200300450.93

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29th International Conference on Ground Control in Mining: 29th International Conference on Ground Control in Mining: Morgantown, WV; 2010 Assigning different values to each variable, the results of sensitivity analysis are shown in Figure 2. 33

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29th International Conference on Ground Control in Mining: 29th International Conference on Ground Control in Mining: Morgantown, WV; 2010 The established formula can be practicable in all situations where a combined open-pit and block caving method is used. Similar methodology can be used to determine appropriate formulae when other underground methods are used. Sensitivity analysis shows that the crown pillar thickness is most sensitive to the block dimensions and least sensitive to specific weight. 34

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29th International Conference on Ground Control in Mining: 29th International Conference on Ground Control in Mining: Morgantown, WV; 2010 35

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