Planning, Designing and Optimising Production Using Geostatistical Simulation

Abstract

The full potential of geostatistical simulation as a tool for planning, designing and optimising production is only realised when it is integrated within the entire design and production cycle. In the planning and design stages this involves the simulation of components of the production cycle that depend on (simulated) grades and geology. In the production stage it involves integration with the mining method and the type and use of equipment. This paper explores the general concepts of integrated geostatistical simulation and illustrates them with particular reference to blast design, equipment selection and the associated quantification of ore loss, ore dilution and the ability to select ore on various scales. The critical component of most metalliferous open pit mining operations is ore selection, i.e. the minimisation of ore loss and ore dilution during extraction. In general, extraction comprises drilling, blasting and loading, all of which are planned and designed on the basis of uncertain models of geology and grade. The application describes the integration of geostatistically simulated grade, geological and geomechanical models with blast modelling to provide a link between the estimated in situ characteristics of the orebody and the locations of the same (displaced) characteristics following the blast. This approach provides a means of evaluating different types of selection and thereby enables planners to optimise the selection process in terms of blast design, type and size of loading equipment, maximisation of ore recovery and minimisation of ore loss and dilution. This conversion of the in situ/block model resource to a realistically recoverable reserve may, in many instances, be the most significant source of uncertainty in reserve estimation.

Appendices

Appendix A: Blast Modelling

The adapted version of the SCRAMBLE/SABREX blast modelling code used in this study is an energy-based approach comprising two separate models: heave mechanics and fragmentation. The heave mechanics are based on the energy released from the adiabatic expansion of the explosive gases following detonation. Fragmentation is based on the powder factor (ratio of charge weight in kilograms to mass in tonnes of rock broken by the charge) converted to an energy equivalent via the Bond Index.

The velocity of detonation for each blasthole is taken as infinite and the wall is allowed to expand until it reaches a state of equilibrium determined by the isotropic expansion characteristics of the quasi-static gas pressure and the elastic resistance of the rock. The expanded blasthole sets up hoop stresses in the surrounding rock, creating a system of radial cracks that, because of tensile failure, spread away from the hole. The radial fractures, together with any pre-existing geological discontinuities, define the damage created in the rock mass by the blast.

The gaseous detonation products flow into the fractured rock mass at the local speed of sound until the gas vents through a free face; at this stage a rarefaction wave travels back toward each blasthole decompressing the cracks. As the rarefaction wave travels through the rock, the pressurised crack system imparts an impulse, which heaves the broken rock mass out from the bench.

In generating the muck pile, empirical routines are used to limit the angle of repose whilst producing a smooth surface and adding swell factors.

Equation of State for Explosive Gases

The equation of state for the gaseous products of detonation is:

$${p} = \frac{{\alpha E\rho (1 + \beta \rho )^{3} }}{{100\left( {1 + 2\beta \rho } \right)}}$$

(A1)

where:

p :

is the gas pressure in kbars

ρ :

is the gas density in g.cm⁻³

E :

is the available energy in J.g⁻¹

α and β :

are dimensionless constants.

The available energy E is the work done by the explosive gases in expanding adiabatically from the density ρ to ambient conditions, and is obtained from:

$${ \ln }\left( {\frac{E}{{E_{0} }}} \right) = \alpha \left( {\frac{{(\beta \rho )^{2} + 5\beta \rho }}{4} - \frac{{(\beta \rho_{0} )^{2} + 5\beta \rho_{0} }}{4} + \frac{1}{8} { \ln }\left[ {\frac{{\rho \left( {1 + 2\beta \rho_{0} } \right.}}{{\rho_{0} \left( {1 + 2\beta \rho } \right)}}} \right]} \right)$$

(A2)

where:

ρ ₀ :

is the initial gas density after detonation (equal to the explosive density)

E ₀ :

is the initial available energy

The values for E ₀, α and β can be obtained from an ideal or non-ideal detonation model. An ideal detonation model is adequate for the large diameter holes used in this study; more accurate data could be obtained from non-ideal models such as CpeX (Leiper and Plessis 1987).

Equation A1 reduces to the ideal gas law for small gas densities and, together with Eq. A2, allows available energy and pressure to be generated as a function of their density during the expansion process.

Heave Mechanics

All regions within the gas envelope have a common gas density and pressure. The leading edge of the envelope is regarded as the gas front, which is assumed to move at the local speed of sound (m.s⁻¹) given by:

$$c = \left( {\frac{100 000\gamma p}{\rho }} \right)^{{\frac{1}{2}}}$$

(A3)

where $\gamma$ is the adiabatic exponent for the gases at pressure p (kbar) and the density ρ(g.cm⁻³), $\gamma$ is given by:

$$\gamma = \left( {\frac{{1 + \alpha (1 + \beta \rho )^{3} }}{1 + 2\beta \rho }} \right) + \left( {\frac{3\beta \rho }{1 + \beta \rho }} \right)$$

(A4)

and is derived from the equation of state given in Eq. A1.

To calculate the necessary density and pressure of the gas within the envelope the volume of rock within the envelope is assumed to be in a state of hydrostatic compression at pressure p. The resultant reduction in the volume of rock is given by:

$$\Delta V + \frac{Vp}{G}$$

(A5)

V :

is the initial volume (m³)

G :

is the bulk modulus

∆V :

is the volume increase in the envelope contributing to the reduction in gas density and pressure.

Another small increase in volume is associated with the gas pressure compressing the rock below and behind the blasthole.

As the gas expands with the moving gas front, the local speed of sound in Eq. A3 falls and a time-stepping loop is used to track the expansion of the gas. The time steps used are defined by:

$$\Delta t = \frac{{b +\Delta b}}{c}$$

(A6)

where:radius b +  Δb is the equilibrium radius blasthole.

Equation A6 shows that, although the time steps can vary, the corresponding spatial steps are constant and equal to the equilibrium borehole radius.

The time-stepping procedure is:

• 1. calculate the initial local speed of sound from Eqs. A3 and A4 prior to the expansion of gas into the rock mass;

• 2. calculate the appropriate time step from Eq. A6 and generate the appropriate gas front profile;

• 3. calculate the increase in volume from Eq. A5 and then calculate the new gas pressure and density using Eqs. A1 and A2;

• 4. recalculate the local speed of sound using Eqs. A3 and A4; and

• 5. repeat the steps while keeping track of the total elapsed time.

Venting of the explosive gas begins when the gas front meets a free face. At that time the gas fronts retrace their original paths and, during this period of contraction, the gas density, pressure and speed of sound are assumed to be constant within the volume of the gas envelope. The respective constant values are those that were calculated at the time of venting, while the pressure beyond the gas fronts is assumed to be insignificant.

At the time of venting, the rock mass is assumed free to move, reacting to a momentum impulse that is imparted on the rock mass. The calculated impulse is based on the assumption that the rock mass does not start to move until the gas fronts have (kg.ms⁻¹) is given by:

$$10^{8} \mathop \smallint \limits_{{t_{v} }}^{{t_{0} }} p\left( {t_{v} } \right)A\left( t \right)dt - M.v$$

(A7)

where:

t _v :

is the time (s) at which gas venting takes place

t ₀ :

is the time (s) at which the contracting gas fronts reach their blastholes

p(t _v ):

is the gas pressure (kbar) in the gas envelope

A(t):

is the area (m²) over which the pressure is applied

M :

is the mass (kg) associated with each blasthole

v :

is the velocity (ms⁻¹) with which the rock mass is heaved

t :

is the time (s).

To derive heave velocities from Eq. A7 an expression for M can be applied for a vertical free face to calculate the mass of rock associated with each blasthole using:

$$M = B.S.H\rho_{R} .1000$$

(A8)

where:

B :

is the burden (m)

S :

is the hole spacing (m)

H :

is the bench height (m)

ρ _R :

is the rock density (g.cm⁻³).

In practical situations the highwall of a bench is not vertical and the program has an input variable for face angle to calculate the true mass of rock associated with the first row of holes.

The momentum impulse for each blasthole is resolved into the vertical and horizontal directions on the basis of the areas defined by the gas envelope. For the vertical impulse the area at the base of the envelope is used in Eq. A7. However, due to the angled highwall, the front row has an inconsistent burden and the area is taken as an average of the areas at the top and bottom of the explosive column length.

Two impulses are computed in the horizontal direction. The first is the section of rock between the toe of the bench and the top of the explosive column, and the second impulse is the region at the top of the bench where the blasthole is filled with stemming material.

A similar averaging process is used to account for the effect of the front row of holes in the calculation of the horizontal impulse, which results in three horizontal heave velocities defining the heave velocity profile. On subsequent rows the effective free face is assumed to be vertical.

For the heave action, the blocks comprising the block model are treated sequentially within a time-stepping loop using a raster pattern starting at the toe of the bench with priority given in order to z, x and then y. For each run through the time-stepping loop all block positions and velocities are recalculated from ballistic trajectory equations and the revised values are stored in three-dimensional arrays; in-flight interactions with other blocks are not modelled. Each block remains in the time-stepping loop until it travels to a point in space at which, ahead or below it, another three-dimensional array describing the mine floor has a positive value, defining that volume of space as containing a block.

When a block drops out of the time-stepping loop to form part of the muck pile it immediately comes to rest on the ground and becomes part of the array that defines the floor and the developing muck pile. The input value for maximum angle of repose ensures that if the defined angle is exceeded in the generation of the muck pile then the block is moved down the surface of the muck pile until it reaches a point of stability.

When all blocks have come to rest, swell is applied to the muck pile by raising each block by a pre-defined factor proportional to the change in vertical height the block underwent in moving from the bench to the muck pile.

Fragmentation

The Bond Index equation from comminution theory is used to assess the effect of different blasting practices on the degree of fragmentation resulting from a blast (Van Zeggeren and Chung 1975; Nielsen 1983). The equation relating energy input to degree of comminution is:

$$W = K_{B} \left[ {\frac{1}{{P^{1/2} }} - \frac{1}{{F^{1/2} }}} \right]$$

(A9)

where:

W :

is the energy input to a machine reducing particle size (kWh.t⁻¹)

F :

is the feed size, measured in microns (10⁻⁶ m), and defined as the mesh size of a screen that allows 80% of the material to pass

P :

is the product size in microns also at 80% passing

K _B :

is a constant determined for a specific feed material

The constant K _B is determined by rearranging Eq. A10 to give:

$$K_{B} = W\left[ {\frac{{P^{1/2} F^{1/2} }}{{F^{1/2} - P^{1/2} }}} \right]$$

(A10)

and the amount of energy required to reduce a known feed size to a given product size is measured. For MRT the amount of energy needed to reduce the secondary crushed product from −19 mm to a final product size of −210 microns was, on average over a two-month period, 16.10 kWh.tonne⁻¹. As the Bond Index works on 80% passing size, the feed and product sizes are taken as 16,300 microns (16.3 mm) and 180 microns respectively. Substituting these values into Eq. A10 gives:

$$K_{B} = 16.10\times\left[ {\frac{{180^{1/2}\times16300^{1/2} }}{{16300^{1/2} - 180^{1/2} }}} \right]=\text{241.4}kWh - {micron^{1/2}.tonne^{-1}}$$

Equation A9 can also be rearranged to calculate the energy required to reduce an infinite feed size (F = ∞) down to any product size P. This is referred to as the total energy (W _t ) and is given by:

$$W_{t} = K_{B} \left[ {\frac{1}{{P^{1/2} }} - \frac{1}{{\infty^{1/2} }}} \right]= \frac{{K_{B} }}{{P^{1/2} }}$$

(A11)

Based on Eq. A11, the Bond Work Index (W _i ) is the amount of energy required to reduce an infinite feed size down to an 80% passing size of 100 microns. This is used as a common basis of comparison across different materials and processes and is given by:

$$W_{i} = K_{B} \left[ {\frac{1}{{100^{1/2} }} - \frac{1}{{\infty^{1/2} }}} \right] = \frac{{K_{B} }}{{100^{1/2} }}$$

(A12)

Substituting the calculated K _B in Eq. A12 gives:

$$W_{i} = \frac{241.4}{{100^{1/2} }} = 24.1kWh.tonne^{ - 1}$$

From Eqs. A11 and A12 it is possible to calculate the energy required to reduce material from an infinite size down to the desired 80% passing size as:

$$W_{t} = W_{i} \left[ {\frac{100}{P}} \right]^{1/2}$$

(A13)

If it is assumed that the only factor that influences the degree of fragmentation in blasting is the amount of energy imparted to the rock mass, and that the energy distribution and initiation variables can be ignored, then Eq. A13 should give a good representation of the energy input from the explosive in a blast, based on the resulting fragmentation.

For the 6.5 m × 8 m MRT blast designs, the material in the resulting muck piles had an 80% passing size of approximately 0.5 m. From Eq. A13 the energy imparted by the explosive is:

$$W_{t} = \left[ {\frac{100}{{5 \times 10^{5} }}} \right]^{1/2} = 0.34 \,kWh.tonne^{ - 1} = 1.23 \,MJtonne^{ - 1}$$

The energy supplied by the explosive acting on the rock mass can be derived from the known powder factor (PF) at 0.31 kg.tonne⁻¹ for the blasts and the energy contained in the explosive used. The energy for the heavy ANFO used, with specific density 1.2 g.cm⁻³, is 4.5 MJ.kg⁻¹. The explosive energy per tonne is therefore:

$${\text{PF}} \times {\text{Explosive}}\,{\text{Energy}} = 0.31 \times 4.5 = 1.40\,\,{\text{MJ}}.{\text{tonne}} ^{- 1}$$

This value of 1.40 MJ.tonne⁻¹ compares favourably with the value of 1.23 MJ.tonne⁻¹ derived using the Bond Index for comminution (Hustrulid 1999). If it is assumed that the difference in values is due to slight differences in the efficiencies of the two processes then it is reasonable to reconcile the two values by applying a factor (α) that is appropriate over a range of energies.

By rearranging Eq. A13 and applying the correction factor α (Van Zeggeren and Chung 1975) the equation for product size from the powder factor used in the blast design is:

$$P = \left( {\left[ {\frac{{W_{i} }}{{W_{t} }}} \right] \times \alpha } \right)^{2}$$

(A14)

where:

W _t :

is the energy equivalent of the powder factor.

Appendix B: Costing the Blasting and Selection Processes

To simplify calculations, all process costs are calculated as cost per tonne worked.

Drilling Costs

Drilling costs are expressed as a cost per metre drilled (DC _m ) for the 250 mm hole diameter used in this study. The tonnage of rock associated with each blasthole, taken as a standard for a specific hole pattern, is given by Eq. A8, divided by 1000 to give tonnes. Cost per tonne (DC _t ) is then:

$$DC_{t} = \frac{{DC_{m} \times HL}}{M}$$

(B1)

where:

HL :

is the hole length (m), including subdrill.

Blasting Costs

The initial costs are calculated for a single hole and are divided into fixed costs per hole—booster, detonator, surface connection and manpower costs—and the variable cost of the main charge placed in the hole. The main charge costs (EX _m ) in dollars are calculated using:

$$EX_{m} = EX \times (A_{h } \times EC_{l} ) \times \rho_{e}$$

(B2)

where:

EX :

is the cost of the explosive ($.kg⁻¹)

A _h :

is the cross-sectional area of the hole (m²)

EC _l :

is the charge length of the explosive in the hole (m)

ρ _e :

is the density of the explosive used (g.cm⁻³).

Loading Costs

The loading costs for the original blast design are taken as $0.14/tonne for a muck pile with 80% passing size of 0.5 m. Within reasonable limits, as passing size decreases loading costs decrease, due mainly to an increase in the ease of digging, which leads to faster loading rates and reduced maintenance costs. MacKenzie (1966) reports a linear relationship between cost per unit loading and degree of fragmentation for Quebec Cartier’s 16-D iron ore mine. Van Zeggeren and Chung (1975) found that their data followed a square root relationship and Nielsen (1983) used a variable exponent selected by the user. For this application, with too few operational data to derive an appropriate relationship, the linear equation is:

$$C_{l } = (D80 \times \alpha ) + (SC_{l } - \beta )$$

(B3)

where:

Cl :

is the adjusted loading cost ($.tonne⁻¹)

D80:

is the calculated 80% passing size (m) using Eq. A14

SC _l :

is the standard loading cost (0.14 $.tonne⁻¹)

α, β :

are constants.

The incorporation of the standard loading cost (SC _l ) in Eq. B3 allows the loading cost relationship to be adjusted for different loaders with different attributes.

Haulage Costs

Haulage costs also decrease with muck pile particle size because the truck is more completely filled, providing the ore density allows it. The relationship used for haulage costs, (Ch), in $/tonne is:

$$C_{h} = \chi .e^{D80}$$

(B4)

where:

$\chi$ :

is a constant.

Primary Crushing Costs

Because variations in feed size to the primary crusher affect power costs much more than general maintenance and plate replacement costs, the Bond Index Eq. A9 was used to calculate crushing costs:

$$C_{cr} = \delta \times 241.4 \times \left( {\frac{1}{{16 300^{1/2} }} - \frac{1}{{D80^{1/2} }}} \right)$$

(B5)

where:

C _cr :

is the adjusted crushing cost ($.tonne^-1)

$\delta$ :

is a constant.

Costs Unaffected by Blasting Practices

Costs incurred in producing a saleable product that are not affected by blasting practices include mining services and the entire mineral processing operation downstream of the primary crushing. These values, also expressed as $/tonne, are assumed to remain constant.