29th International Conference on Ground Control in Mining: 29th International Conference on Ground...

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


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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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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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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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A crown pillar between open pit and underground mines

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Open-pit limit

Transition depth

Underground layout

Crown pillar


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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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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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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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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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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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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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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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


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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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]


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


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


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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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


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


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


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


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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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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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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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


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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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


Assigning different values to each variable, the results of

sensitivity analysis are shown in Figure 2.

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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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