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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:Title of paper:
Authors:Authors:
Kazem Oraee; University of Stirling, UK
Ezzeddin Bakhtavar; Urmia University of Technology
29th International Conference on Ground Control in Mining: 29th International Conference on Ground Control in Mining: Morgantown, WV; 2010
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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.
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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
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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.
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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
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Open-pit limit
Transition depth
Underground layout
Crown pillar
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.
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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.
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29th International Conference on Ground Control in Mining: 29th International Conference on Ground Control in Mining: Morgantown, WV; 2010
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29th International Conference on Ground Control in Mining: 29th International Conference on Ground Control in Mining: Morgantown, WV; 2010
Dimensional 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.
There are two main systems:
- Mass system
- Force system
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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.
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29th International Conference on Ground Control in Mining: 29th International Conference on Ground Control in Mining: Morgantown, WV; 2010
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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.
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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.
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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.
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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
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Effective parameters
Variable
Crown pillar thickness t
Block span s
Block height h
Rock mass rating RMR
Cohesion strength C
Specific weight of rock γr
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:
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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.
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Table 2- Dimensional values
Variable (unit) Dimensiont (m) [L]s (m) [L]h (m) [L]RMR [1]C (ton/m2) [FL-2]γr (ton/m3) [FL-3]
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.
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Table 3- Dimensional matrix
Dimension Quantity t s h RMR C γr
F 0 0 0 0 1 1L 1 1 1 0 -2 -3T 0 0 0 0 0 0
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:
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Dimension
Quantity
t s h RMR C γr
F 0 0 0 0 1 1L 1 1 1 0 -2 -3T 0 0 0 0 0 0
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
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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.
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Dimension
Quantity
t s h RMR C γr
F 0 0 0 0 1 1L 1 1 1 0 -2 -3T 0 0 0 0 0 0
k = 2
n = 6
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.
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Dimension
t s h RMR C γr
K1 K2 K3 K4 K5 K6
F 0 0 0 0 1 1L 1 1 1 0 -2 -3T 0 0 0 0 0 0
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 K1, K2, K3 and K4 and hence
K5 and K6 are calculated. In this way, matrix of responses
can be made as in Table 4.
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Table 4- Matrix of responses
K1 K2 K3 K4 K5 K6
t s hRMR
C γr
π1 1 0 0 0 -1 1π2 0 1 0 0 -1 1π3 0 0 1 0 -1 1π4 0 0 0 1 0 0
29th International Conference on Ground Control in Mining: 29th International Conference on Ground Control in Mining: Morgantown, WV; 2010
Thus, four (π1-π4) independent dimensionless products are:
Then the independent dimensionless products can be written as:
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K1 K2 K3 K4 K5 K6
t s hRMR
C γr
π1 1 0 0 0 -1 1π2 0 1 0 0 -1 1π3 0 0 1 0 -1 1π4 0 0 0 1 0 0
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:
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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:
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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:
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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.
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Table 5- Data of the real case studies
Case studies
Valuest s h RMR C γr
1 200 180 400 62.5 0.75 2.72 200 220 400 75 2.9 3.13 180 190 230 48 1 2.754 230 250 460 70 0.82 2.81
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:
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29th International Conference on Ground Control in Mining: 29th International Conference on Ground Control in Mining: Morgantown, WV; 2010
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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.
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Table 6- An hypothetical case example
t s h RMR C γr
? 200 300 45 0.9 3
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.
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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.
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29th International Conference on Ground Control in Mining: 29th International Conference on Ground Control in Mining: Morgantown, WV; 2010
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