strategic design

Strategic Design of an Underground Mine

under Conditions of Metal Price Uncertainty

by

George McIsaac

A thesis submitted to the

Department of Mining Engineering

in conformity with the requirements for

the degree of Doctor of Philosophy

Queen’s University

Kingston, Ontario, Canada

April 2008

Copyright c© George McIsaac, 2008

Abstract

Long-term mine plans are based on forecast future metal prices. By the time the

development is put in place, the forecasts may have been proved wrong and the

production plan might not meet the company’s financial objectives. At that point,

the common reaction to this situation is to create a new revised long-term plan and

spend more capital, only to find out at a later time that the metal prices have changed

again. This results in an inefficient use of capital with low returns to the investors.

The objective of this thesis is to develop a methodology to determine the cut-off

grade and production rate of a narrow-vein underground mine such that the long-term

strategic plan is robust. As a requirement to do so, it is necessary to have a good

understanding of the resources, revenues, capital and operating costs as a function of

the design parameters. Also, the operational limits of the mine must be determined so

that the solution is practical. Afterwards, annual metal prices are randomly generated

with a Monte Carlo process on stochastic metal price model, and the combination of

production rate and cut-off grade yielding the highest net present value is identified

and recorded. This process is repeated many times, and the probabilities of the

solutions occurring at any given design combination are calculated. The results are

plotted on a bubble graph, where the size of a bubble is directly proportional to the

probability a solution occurs at that point. Finally, the combination with the largest

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bubble is the solution, as this point has the highest probability of yielding the highest

net present value in most circumstances.

The model was first tested on an actual gold-copper orebody where very detailed

resource and cost information was available. The methodology was applied with

success and the solution reflected the important impact of the copper milling and

roasting process on revenues. Other tests were then done on an hypothetical gold

orebody and the results showed a great degree of sensitivity to the average grade of

the deposit, and less to the discount rate and to the gold price-reversion factor.

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Acknowledgments

The author would like to dedicate this work to his father’s memory.

The author would like to thank his wife and children. Going back to school at this

later stage of life puts enormous levels of strain on a family. The author wishes to tell

his wife all the appreciation and love he has for her as now comes the time to take

the next step forward.

A very kind and loving word is sent to the author’s mother in recognition for her

unconditional support. This also extends to everyone in the author’s father’s and

mother’s families.

Special appreciation and thanks to Dr. Charles Pelley for his guidance and support,

the long discussions on the mining industry, and for sharing his experience. For their

invaluable support and advice, the author also extends his thanks to all the professors

and staff of the Department of Mining Engineering with whom he had the pleasure of

sharing during the many years spent working there. The author also thanks Professor

Leo B. Jonker of the Department of Mathematics and Statistics for his help with the

calculus in Chapter 3.

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Table of Contents

Abstract i

Acknowledgments iii

Table of Contents iv

List of Tables vii

List of Figures ix

Chapter 1:Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Risk Aversion and Robustness . . . . . . . . . . . . . . . . . . . . . . 121.4 Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Chapter 2:Literature Review . . . . . . . . . . . . . . . . . . . . . . 15

2.1 Production Rate and Cut-Off Grade Selection . . . . . . . . . . . . . 162.2 Metal Price Forecasting Models . . . . . . . . . . . . . . . . . . . . . 472.3 Treatment of Inflation in Discounted Cash Flow Calculations and Dis-

count Rate Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

Chapter 3:The Mathematical Model . . . . . . . . . . . . . . . . . . 60

3.1 Construction of the Model . . . . . . . . . . . . . . . . . . . . . . . . 603.2 Methodology to Determine the Production rate and Cut-Off Grade

under Conditions of Metal Price Uncertainty . . . . . . . . . . . . . . 743.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

iv

Chapter 4:Test on an Actual Mine . . . . . . . . . . . . . . . . . . . 82

4.1 Minable resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 874.2 Production and cash flow model . . . . . . . . . . . . . . . . . . . . . 984.3 Metal prices model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1244.4 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1264.5 Results and analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 1304.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

Chapter 5:Hypothetical Gold Mine . . . . . . . . . . . . . . . . . . . 154

5.1 Hypothetical gold mine model . . . . . . . . . . . . . . . . . . . . . . 1555.2 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1595.3 Base case results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1605.4 Sensitivity to the gold price model mean reversion factor . . . . . . . 1625.5 Sensitivity to the discount rate . . . . . . . . . . . . . . . . . . . . . 1715.6 Sensitivity to the average grade of the deposit . . . . . . . . . . . . . 1735.7 Design analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1795.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

Chapter 6:Conclusions and Recommendations . . . . . . . . . . . . 182

6.1 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1836.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1846.3 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1876.4 Closing remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

Appendix A:Modeling Procedures . . . . . . . . . . . . . . . . . . . . 201

A.1 Model construction procedures . . . . . . . . . . . . . . . . . . . . . . 201A.2 Simulation procedures . . . . . . . . . . . . . . . . . . . . . . . . . . 202

Appendix B:Capital Development . . . . . . . . . . . . . . . . . . . . 203

B.1 Mine A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204B.2 Mine B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213B.3 Mine C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

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Appendix C:Production & Development Indicators . . . . . . . . . 231

C.1 Metres of development per tonne of ore . . . . . . . . . . . . . . . . . 233C.2 Percentage of ore coming coming from development . . . . . . . . . . 236C.3 Tonnes of ore per metre of ore development . . . . . . . . . . . . . . 238C.4 Compilation of data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

Appendix D:Metal Price Model Validation . . . . . . . . . . . . . . 242

Appendix E:Development Constraint . . . . . . . . . . . . . . . . . . 245

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List of Tables

2.1 Optimization Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.2 Rate multipliers, from Tatman [65] . . . . . . . . . . . . . . . . . . . 18

3.1 Comparison of major design features of cost models . . . . . . . . . . 68

4.1 Dilution factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 894.2 Summary of minable resources and grades by vein . . . . . . . . . . . 904.3 Resources by vein and category . . . . . . . . . . . . . . . . . . . . . 914.4 Resources by vein and mining method . . . . . . . . . . . . . . . . . 924.5 Gold cut-off grades and levels of production . . . . . . . . . . . . . . 1004.6 Production and development output . . . . . . . . . . . . . . . . . . . 1014.7 Cash flow and indicators output . . . . . . . . . . . . . . . . . . . . . 1024.8 Smelter contracts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1054.9 Capital development summary . . . . . . . . . . . . . . . . . . . . . . 1094.10 Variable mining unit costs . . . . . . . . . . . . . . . . . . . . . . . . 1144.11 Metal Price Equation Factors . . . . . . . . . . . . . . . . . . . . . . 1254.12 Example of the production profile . . . . . . . . . . . . . . . . . . . . 1314.13 Example of the financial profile . . . . . . . . . . . . . . . . . . . . . 1324.14 Risk analysis example . . . . . . . . . . . . . . . . . . . . . . . . . . 1354.15 Coordinates of hot spots . . . . . . . . . . . . . . . . . . . . . . . . . 1384.16 Occurrences as a function of hot spot and metal price quadrant . . . 1414.17 Solutions for set metal prices . . . . . . . . . . . . . . . . . . . . . . . 142

5.1 Production profile for the gold-copper model . . . . . . . . . . . . . . 1665.2 Production profile for the gold equivalent model . . . . . . . . . . . . 1675.3 Financial analysis of the gold-copper model . . . . . . . . . . . . . . . 1685.4 Financial analysis of the gold equivalent model . . . . . . . . . . . . . 1695.5 Design parameters for the cases studied . . . . . . . . . . . . . . . . . 179

C.1 Calculations of development per tonne indicators . . . . . . . . . . . 234C.2 Development per tonne indicators . . . . . . . . . . . . . . . . . . . . 235C.3 Calculations of percentage production coming from ore development . 237C.4 Calculations of percentage production coming from ore development . 238

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C.5 Tonnes of ore per metre development data . . . . . . . . . . . . . . . 239C.6 Final table of tonne of ore per metre development . . . . . . . . . . . 241

E.1 Annual development . . . . . . . . . . . . . . . . . . . . . . . . . . . 246E.2 Maximum production rate . . . . . . . . . . . . . . . . . . . . . . . . 248

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List of Figures

1.1 Metal Prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1 Operating and capital cost as a function of production rate, fromSmith [62] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2 Maximum Net Present Value as a function of production rate, fromSmith [62] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3 Unit revenue and costs as a function of production rate, adapted fromGray [28] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.4 Total revenue and costs as a function of production rate, adapted fromGray [28] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.5 Effect of discount rates on dynamic optimization of production rates,from Park [50] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.6 Unit revenue and costs as a function of production rate, adapted fromHartwick and Olewiler [29] . . . . . . . . . . . . . . . . . . . . . . . . 29

2.7 Effects on cash flows of operating options of a mine, from Palm, Pear-son, and Read [49] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.8 Levels of recovery unit functions, from Carlisle [12] . . . . . . . . . . 332.9 Levels of recovery total functions, adapted from Carlisle [12] . . . . . 342.10 Effect of discount rates on dynamic optimization of cut-off grades, from

Park [50] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.11 Relationship between rate and level of recovery, from Carlisle [12] . . 402.12 Contour map of PVR in relation to cut-off grade and production rate,

from Wells [73] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.13 Relationship between rate and level of recovery, from Ding [19] . . . . 432.14 Cut-off grade unit functions, from Park [50] . . . . . . . . . . . . . . 442.15 Production rate unit functions, from Park [50] . . . . . . . . . . . . . 442.16 Low-grade zone development and project abandonment boundary, adapted

from Samis et al. [54] . . . . . . . . . . . . . . . . . . . . . . . . . . . 462.17 Average annual gold prices from 1980 to 2005 . . . . . . . . . . . . . 482.18 Average annual copper prices from 1975 to 2005 . . . . . . . . . . . . 492.19 Average annual copper prices in 2001 dollars, from Tilton [69] . . . . 492.20 Falling prices due to a shift in the supply curve, from Tilton [69] . . . 50

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2.21 Copper mine costs (cents/lb, real 2001 dollars), from Tilton [69] . . . 512.22 Causes of market volatility, from Tilton [69] . . . . . . . . . . . . . . 522.23 Example of mean reversion . . . . . . . . . . . . . . . . . . . . . . . . 542.24 Components of real discount rates at different stages of project devel-

opment, from Smith [61] . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.1 Example of lognormal grade distribution . . . . . . . . . . . . . . . . 703.2 Example of Grade-Tonnage curves . . . . . . . . . . . . . . . . . . . . 713.3 Example of NSR-Tonnage curves . . . . . . . . . . . . . . . . . . . . 723.4 Unit revenue curves as a function of the level of production . . . . . . 743.5 Flow sheet of program PeaRL . . . . . . . . . . . . . . . . . . . . . . 783.6 Flow sheet of subroutine Search . . . . . . . . . . . . . . . . . . . . . 793.7 Flow sheet of subroutine NPV . . . . . . . . . . . . . . . . . . . . . . 80

4.1 Actual and budgeted production rates . . . . . . . . . . . . . . . . . 864.2 Grade-tonnage curves based on gold grades . . . . . . . . . . . . . . . 944.3 Production level grades as a function of gold . . . . . . . . . . . . . . 954.4 Grade-tonnage curves based on copper grades . . . . . . . . . . . . . 954.5 Production level grades as a function of copper . . . . . . . . . . . . . 964.6 Metal grade scatter plot for low copper grades . . . . . . . . . . . . . 974.7 Metal grade scatter plot for low gold grades . . . . . . . . . . . . . . 974.8 High-grade copper resources associated with low-grade gold blocks . . 994.9 High-grade gold resources associated with low-grade copper blocks . . 994.10 Mineral processing general flowchart . . . . . . . . . . . . . . . . . . 1034.11 Value-added of roasting copper concentrate . . . . . . . . . . . . . . . 1074.12 Capital development as a function of level of production . . . . . . . 1084.13 Proportion of tonnes mined per mining methods as a function of the

level of production . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1134.14 Proportion of tonnes mined from all sources as a function of the level

of production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1164.15 Regular and capital development as a function of the level of production1174.16 Variable mining cost as a function of the level of production . . . . . 1184.17 Mine services unit cost as a function of the production rate . . . . . . 1194.18 Mill variable unit cost as a function of the production rate . . . . . . 1204.19 Mine monthly fixed cost as a function of the production rate . . . . . 1214.20 Mill and On-Site monthly fixed cost as a function of the production rate1224.21 Total costs over the life of the project for combinations of rates and

levels of production . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1234.22 Unit costs over the life of the project for combinations of rates and

levels of production . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1234.23 Gold price model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

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4.24 Copper price model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1264.25 Feasible area of the possible solutions for the NPV function . . . . . . 1284.26 Example of the output generated . . . . . . . . . . . . . . . . . . . . 1294.27 Average net present value @ 5%, in million dollars . . . . . . . . . . . 1334.28 Standard deviation of the net present value distributions, in million

dollars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1344.29 Probability of positive net present value, in percentage . . . . . . . . 1344.30 Bubble chart of optimal net present value combinations . . . . . . . . 1374.31 Metal prices yielding a solution occurring at hot spot A . . . . . . . . 1394.32 Metal prices yielding a solution occurring at hot spot B . . . . . . . . 1394.33 Metal prices yielding a solution occurring at hot spot C . . . . . . . . 1404.34 Metal prices yielding a solution occurring at other locations . . . . . 1404.35 Solutions for set metal prices . . . . . . . . . . . . . . . . . . . . . . . 1434.36 Copper grade as a function of the level of production . . . . . . . . . 1444.37 Copper NSR factors . . . . . . . . . . . . . . . . . . . . . . . . . . . 1454.38 Value of copper contained in ore . . . . . . . . . . . . . . . . . . . . . 1464.39 Unit revenues and costs for copper price set at 10% confidence interval 1474.40 Unit revenues and costs for copper price set at expectation . . . . . . 1474.41 Unit revenues and costs for copper price set at 90% confidence interval 1484.42 Profit as a function of copper prices . . . . . . . . . . . . . . . . . . . 1484.43 Tactical plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

5.1 Gold equivalent tonnes histogram . . . . . . . . . . . . . . . . . . . . 1575.2 Comparison of original and hypothetical models . . . . . . . . . . . . 1575.3 Bubble graph for gold equivalent grade equal to 8.52 grams per tonne 1615.4 Clustered bubble graph for gold equivalent grade equal to 8.52 grams

per tonne . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1625.5 Base case combinatory table for NPV @ 5%, in million dollars . . . . 1635.6 Comparison of the confidence intervals sensitivity . . . . . . . . . . . 1645.7 Combinatory table of NPV @5% for mean-reversion factor = 0.20, in

MM$ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1645.8 Bubble graph for mean-reversion factor = 0.20 . . . . . . . . . . . . . 1705.9 Clustered bubble graph for mean-reversion factor = 0.20 . . . . . . . 1705.10 Combinatory table of NPV @8%, in MM$ . . . . . . . . . . . . . . . 1715.11 Bubble graph for discount rate = 8% . . . . . . . . . . . . . . . . . . 1725.12 Clustered bubble graph for discount rate = 8% . . . . . . . . . . . . . 1725.13 Combinatory table of NPV @5% for gold equivalent grade equal to

6.50 grams per tonne, in MM$ . . . . . . . . . . . . . . . . . . . . . . 1735.14 Combinatory table of NPV @5% for gold equivalent grade equal to

10.50 grams per tonne, in MM$ . . . . . . . . . . . . . . . . . . . . . 174

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5.15 Bubble graph for gold equivalent grade equal to 6.50 grams per tonne 1745.16 Bubble graph for gold equivalent grade equal to 10.50 grams per tonne 1755.17 Clustered bubble graph for gold equivalent grade equal to 6.50 grams

per tonne . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1755.18 Clustered bubble graph for gold equivalent grade equal to 10.50 grams

per tonne . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1765.19 Average gold grade as a function of the level of production for the three

cases used in the sensitivity analysis . . . . . . . . . . . . . . . . . . . 1775.20 Average revenues and costs as a function of the level and rate of pro-

duction for the three cases used in the sensitivity analysis . . . . . . . 1785.21 Marginal revenues and costs as a function of the level and rate of

production for the three cases used in the sensitivity analysis . . . . . 178

B.1 Mine A, General outline of lateral infrastructure . . . . . . . . . . . . 204B.2 Mine A, Level of production = 1 million tonnes . . . . . . . . . . . . 205B.3 Mine A, Level of production = 2 million tonnes . . . . . . . . . . . . 205B.4 Mine A, Level of production = 3 million tonnes . . . . . . . . . . . . 206B.5 Mine A, Level of production = 4 million tonnes . . . . . . . . . . . . 206B.6 Mine A, Level of production = 5 million tonnes . . . . . . . . . . . . 207B.7 Mine A, Level of production = 6 million tonnes . . . . . . . . . . . . 207B.8 Mine A, Level of production = 7 million tonnes . . . . . . . . . . . . 208B.9 Mine A, Level of production = 8 million tonnes . . . . . . . . . . . . 208B.10 Mine A, Level of production = 9 million tonnes . . . . . . . . . . . . 209B.11 Mine A, Level of production = 10 million tonnes . . . . . . . . . . . . 209B.12 Mine A, Level of production = 11 million tonnes . . . . . . . . . . . . 210B.13 Mine A, Level of production = 12 million tonnes . . . . . . . . . . . . 210B.14 Mine A, Level of production = 13 million tonnes . . . . . . . . . . . . 211B.15 Mine A, Level of production = 14 million tonnes . . . . . . . . . . . . 211B.16 Mine A, Level of production = 15 million tonnes . . . . . . . . . . . . 212B.17 Mine A, Level of production = 16 million tonnes . . . . . . . . . . . . 212B.18 Mine B, General outline of lateral infrastructure . . . . . . . . . . . . 213B.19 Mine B, Level of production = 1 million tonnes . . . . . . . . . . . . 214B.20 Mine B, Level of production = 2 million tonnes . . . . . . . . . . . . 214B.21 Mine B, Level of production = 3 million tonnes . . . . . . . . . . . . 215B.22 Mine B, Level of production = 4 million tonnes . . . . . . . . . . . . 215B.23 Mine B, Level of production = 5 million tonnes . . . . . . . . . . . . 216B.24 Mine B, Level of production = 6 million tonnes . . . . . . . . . . . . 216B.25 Mine B, Level of production = 7 million tonnes . . . . . . . . . . . . 217B.26 Mine B, Level of production = 8 million tonnes . . . . . . . . . . . . 217B.27 Mine B, Level of production = 9 million tonnes . . . . . . . . . . . . 218

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B.28 Mine B, Level of production = 10 million tonnes . . . . . . . . . . . . 218B.29 Mine B, Level of production = 11 million tonnes . . . . . . . . . . . . 219B.30 Mine B, Level of production = 12 million tonnes . . . . . . . . . . . . 219B.31 Mine B, Level of production = 13 million tonnes . . . . . . . . . . . . 220B.32 Mine B, Level of production = 14 million tonnes . . . . . . . . . . . . 220B.33 Mine B, Level of production = 15 million tonnes . . . . . . . . . . . . 221B.34 Mine B, Level of production = 16 million tonnes . . . . . . . . . . . . 221B.35 Mine C, General outline of lateral infrastructure . . . . . . . . . . . . 222B.36 Mine C, Level of production = 1 million tonnes . . . . . . . . . . . . 223B.37 Mine C, Level of production = 2 million tonnes . . . . . . . . . . . . 223B.38 Mine C, Level of production = 3 million tonnes . . . . . . . . . . . . 224B.39 Mine C, Level of production = 4 million tonnes . . . . . . . . . . . . 224B.40 Mine C, Level of production = 5 million tonnes . . . . . . . . . . . . 225B.41 Mine C, Level of production = 6 million tonnes . . . . . . . . . . . . 225B.42 Mine C, Level of production = 7 million tonnes . . . . . . . . . . . . 226B.43 Mine C, Level of production = 8 million tonnes . . . . . . . . . . . . 226B.44 Mine C, Level of production = 9 million tonnes . . . . . . . . . . . . 227B.45 Mine C, Level of production = 10 million tonnes . . . . . . . . . . . . 227B.46 Mine C, Level of production = 11 million tonnes . . . . . . . . . . . . 228B.47 Mine C, Level of production = 12 million tonnes . . . . . . . . . . . . 228B.48 Mine C, Level of production = 13 million tonnes . . . . . . . . . . . . 229B.49 Mine C, Level of production = 14 million tonnes . . . . . . . . . . . . 229B.50 Mine C, Level of production = 15 million tonnes . . . . . . . . . . . . 230B.51 Mine C, Level of production = 16 million tonnes . . . . . . . . . . . . 230

D.1 Gold price model starting in 1980 . . . . . . . . . . . . . . . . . . . . 243D.2 Gold price model starting in 1994 . . . . . . . . . . . . . . . . . . . . 243D.3 Copper price model starting in 1975 . . . . . . . . . . . . . . . . . . . 244D.4 Copper price model starting in 1994 . . . . . . . . . . . . . . . . . . . 244

E.1 Production rates vs. reserves . . . . . . . . . . . . . . . . . . . . . . . 247E.2 Development rates vs. production rates . . . . . . . . . . . . . . . . . 247E.3 Development rates vs. production levels . . . . . . . . . . . . . . . . 248

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Chapter 1

Introduction

1.1 Context

The author worked extensively in the preparation of strategic and tactical long-term

plans for underground mining operations for many years. It is considered normal that

these plans have to be redone on a regular basis as new ore zones are added and older

reserves are depleted. However, this work is subject to rapid obsolescence as metal

prices change. If prices go up, there is a sudden urgency to mine more and faster,

producing a strain on the existing infrastructure. The operation may run out of ore

supply if the development cannot keep up with the extra demand. If prices drop,

the economically-extractable orebody shrinks and the existing development does not

suffice to meet production targets. Extra development must be put in place rapidly

to access whichever ore is available, and the operation may not build enough capacity

for long-term survival.

Over the years, the author has pondered the problem of how a plan can be devised

to incorporate defences against metal price variations. It is obvious that such a mine

1

CHAPTER 1. INTRODUCTION 2

plan cannot be perfect. The concept of economical optimization would be replaced by

that of robustness, presenting the best possibility of surviving future variations and

providing a basis from which to adapt without compromising the deposit. It is hard

to compare the value of a robust plan with that of an optimal one; the optimal plan

is the best possible plan under specific conditions, whereas the robust plan is the best

possible plan under conditions of uncertainty. In the case of the optimal plan, it may

have a very high net present value at the time of conception, but changing orebody

and market conditions may reduce its value very abruptly; on the other hand, the

robust plan’s value may be relatively lower for the same basic conditions, but its value

will not be affected as much by varying factors.

With this in mind, this thesis proposes an approach incorporating reserve evalu-

ation, cost accounting, micro-economics and risk analysis with the objective of iden-

tifying the cut-off grade, the production rate and the level of investment to be used

in the development of long-term production plans.

1.2 Problem Statement

1.2.1 The Problem

In the time spent in mining operations, the author has observed that long-term mine

plans are based on constant metal prices. Cairns [9] reported the same practice, and

Whittle [76] stated that the metal price forecasts are usually provided by the finance

or business evaluation departments at the corporate office. Deterministic models are

used, even though it is recognized that the metal prices will vary. Lane [38] suggested

to do sensitivity analysis to take this into consideration.


Based on accepted future metal prices, the value of each mineralized resource

block is calculated, a plan is built, capital and operating costs are estimated, and the

operation proceeds with the plan if it is accepted. However, whether this is a new

project with many years of preproduction development or an ongoing mine, metal

prices can vary significantly in the short term, thus affecting the initial reserve base

used for planning. This may result in last minute modifications to the development

program and in added strain on the operation.

If prices increase, the production plan of an underground mine can change in

different ways. Taylor [66] argues that the proper course of action is to concentrate

on mining higher grade material only, thus increasing the net present value (NPV)

of the mine even if that leads to condemning the future recovery of lower-grade ore.

However, most operations tend to lower the cut-off grade and increase the production

rate, hoping to maximize the extraction of the orebody. No matter which course of

action is taken, the development in place will probably not be adequate to sustain the

new strategy for a very long time. If Taylor’s approach is chosen, the production rate

will not be sustainable unless the development rate increases. If the option of lowering

the cut-off grade and increasing the production rate is accepted, the development rate

must again be increased. Both cases will also lead to congestion, inefficiencies and

higher costs.

On the other hand, if metal prices decrease, reserves are lost, and the size and

shape of the orebody can be affected. Taylor classified underground orebodies into

two categories, named type A and B. In the case of vein-like deposits of type A, a price

decrease will result in smaller more isolated pods of economical mineralization. In the

case of gradational-grade massive orebodies of type B, the strike length, height and


width will decrease. In both cases, the ratio of tonnes of ore per longitudinal surface

area decreases, and in order to maintain the planned production rate, the development

rate must increase. Even if the operators decide to reduce the production rate, the

ratio of development per tonne of ore mined has to increase. This leads to an increase

in operating and/or capital costs.

In the development of a long-term plan, a base-case scenario is conceived. It is

designed for a constant production rate inspired from the reserves tonnage and its

physical distribution on a longitudinal projection. To limit the capital outlay, the

development of a given sector is done no more than six months ahead of its start of

production.1 However, this limits the time available to do major modifications to the

plan. First, the need to change the plan must be recognized, and that usually takes

a significant time as people tend to look at variations as short-term events and fail to

recognize that a new long-term trend is being established. Once there is consensus to

change, a new plan must be developed and accepted before it can be implemented. By

the time the new plan is in place, metal prices have changed again, making the latest

plan obsolete before it can reap significant benefits. The capital that was spent as

part of the initial plan will probably have a low return or may even not be recovered.

Since this latest plan is based on metal price forecasts following the same approach

as in the previous plan, it may expose the mine to the same problem at some point

in the future.

The risks of a plan can be evaluated prior to its implementation. The two most

common methods are sensitivity and risk analysis. Both quantify how the economic

indicators can be affected if one or more revenue and cost parameters change. The

1There is no hard rule justifying this, it is more an accepted industry rule-of-thumb that takesinto consideration the degradation of the rock mass of a drift that remains unused for too long atime.


drawback of this approach is that the original plan is still used for comparison pur-

poses. Of course, a mine will make less money if revenues decrease while still mining

the same reserves at the same production rate. Risk analysis does not identify what

production rate and cut-off grade would yield better results if revenues decrease. The

same is true if metal prices increase.

Furthermore, prior to performing the risk analysis, there is no guaranty that the

plan is optimum at the outset. Smith [62] mentioned how the production rate is often

decided very early in the development of a project and is not adjusted later on if the

design meets the investor’s economic criteria. From this, it can be deduced that the

initial parameters used for the conception of the plan are not necessarily optimum

and the plan may not be very robust when operating conditions change.

Park [50] and Ding [19] recently developed methods to find the optimal cut-off

grades and production rate yielding the maximum net present value for an operation.

In all cases, metal prices were fixed for the life of the deposit and there was little

discussion on how robust their systems were in case of metal price variations.

Metal price fluctuations are addressed to a certain point in the realm of Real

Options. However, the discussion limits itself to deciding if a mine, as designed,

should operate or not given fluctuating prices. Again, there is no consideration as

to whether the design is optimal to start with. Also, the analysis is limited to yes

or no, operate or do not operate under the current design parameters, and it does

not extend to deciding if changing the production rate and/or the cut-off grade could

yield a better result.

These considerations support the requirement to develop a tool that can estab-

lish the cut-off grade and production rate of an operation under conditions of metal


price uncertainty. The plan produced would propose a constant production rate and

constant cut-off grade over the life of the plan. The plan thus obtained may not

be optimum for the metal prices at the time at which the plan is developed; i.e. it

may not yield the highest possible net present value for those metal prices. It should

however be seen as a defensive plan, one with the minimum risk of failure no matter

what the future may hold, without having to change the development and production

profile of the plan as conditions change. This plan would be designed to best survive

all possible outcomes. Having said that, the plan may not necessarily meet the mini-

mum economic criteria set forth by the operator, but then again, that guaranty never

exists when preparing a plan. However, the probability of meeting the indicators

yielded by this analysis should be very high.

1.2.2 Example

The problem presented can be complex, but it is one that commonly affects mines.

The following provides an example of this problem for an operation over a six-year

period during the last decade. A large underground operation had to be redesigned

yearly as metal prices decreased steadily over many years. To examine this problem,

a brief description of the mine is given; the decrease of gold and copper prices between

the years 1994 and 2000 is reviewed; the actions taken by the operation to counteract

this decrease are discussed; and the implications to the operations, revenues, costs

and reserves are analyzed.


The Mine

The mine contained various deposits consisting of many vein structures with strike

lengths reaching up to 2,000 metres and vertical extent of up to 600 metres. Each

deposit contained many anastomosing veins separated by faults. The widths of both

veins and faults could vary from less than one metre to more than twenty and their

dips ranged from thirty five (35) to ninety (90) degrees. The dips and width of any

given vein or fault could vary significantly within a distance of one hundred metres.

Four metals were mined, but gold and copper accounted for ninety percent (90%)

of the revenues. The metal distribution was uneven throughout the deposit. Some

veins were gold bearing, associated with pyrite in quartz. The background gold grade

was in the order of four grams per tonne, but locally, the grades could surpass 100

grams per tonne. The copper was found mostly in enargite (Cu3AsS4). Copper grades

averaged between four and eight percent but could run as high as fifteen percent. Low

gold grades were also recorded in the copper veins. Silver and arsenic were associated

with the copper and they complemented the revenues.

Given the varying widths and dips of the veins, the proximity of faults, and the

presence of pockets of very high gold grades, many mining methods were employed.

The competence of the hanging wall and the continuity of the ore were the most

significant factors in the choice of a method. Whenever possible, longhole stopes

were developed, provided that adequate ground support could be installed. In narrow

and relatively vertical veins, Avoca and Eureka were used, while primary-secondary

transverse sequences were established in wider areas of varying dips. If conditions

were such that only small areas of the hanging wall could be stabilized, cut and fill

was used. Many variations were employed, including mechanized, ramp-in-vein, drift


220

240

260

280

300

320

340

360

380

400

420

Jan-94 Jan-95 Jan-96 Jan-97 Jan-98 Jan-99 Jan-00

Date

Go

ld p

rice

($/

oz)

0.50

0.60

0.70

0.80

0.90

1.00

1.10

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pp

er p

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GoldCopper

Figure 1.1: Metal Prices

and fill combined with floor and back slashes, and underhand cut and fill in heavily

fractured ground.

Metal Prices

Between the years 1995 and 1999, gold and copper prices declined substantially (fig-

ure 1.1). During 1994 and 1995, the price of gold had remained steady between $380

and $400 per ounce. However, over the next three years, the price of gold decreased

steadily to $258 per ounce, a decrease of 35%.

As for copper, 1994 was a good year during which its price increased almost 70%

from $0.80 to $1.35 per pound. This price level was maintained throughout most of

1995, but over the next three years, the price plunged by more than 50% down to

$0.62 per pound, followed by a slight rebound back to the $0.80 level.2

2Though there is no reason to believe there is a relation for the simultaneous drop in prices of


Effects of Metal Prices Variations on the Mine Operations

By 1995, the mine had been in operation for more than fifteen years. Over that time,

it had grown from a small bonanza-grade operation, into a large scale mine with

production coming from many deposits. In late 1994, the mine started preparing the

1995 budget as well as the five-year plan. It was estimated that the metal prices

of the time were to remain constant for the following five years. Based on this, the

reserves were updated and a plan was constructed with a production rate of 3,300

tonnes per day.

During 1995, metal prices stayed constant, and production equalled the planned

rate. At that time, the deposit was still open along strike and at depth, and there

was a lot of optimism about both exploration potential and future metal prices. The

talks naturally drifted toward investigating the potential of increasing the production

rate. As different scenarios were drafted to study the possibilities, some development

was put in place in provision for a possible future expansion.

By 1996, metal prices had started drifting lower, but the decrease was not signif-

icant. No adjustments were made to the long-term plan. However, some weaknesses

in the plan were starting to become apparent. The expectations for the discovery

of much more bonanza-grade ore were not met, and the prices of gold and copper

started decreasing. The combination of lower average grade and lower metal prices

led to smaller operating margins. Still, no action was taken during that year.

In 1997, the situation remained the same and by March the need for change was

recognized. The mining reserves were revised and a new long-term plan was completed

both metals, it reminds one of a similar happening in 1982 when gold fell to $325 per ounce andcopper to $0.58 per pound. Many companies had to suspend production at that time and lay-offmany workers. Blais, Poulin, and Samis [5] calculated a correlation coefficient of 0.51 between copperand gold prices during the years 1998 and 2004.


at a rate of 2,200 tonnes per day, with an emphasis on mining copper since its price

was still relatively high at $1.10 per pound. That decision meant developing new

zones that had not been previously considered. By the end of the year, gold had

further dropped to less than $300 per ounce and copper to less than $0.80 per pound

and the plan was successively lowered to 1,800 and then 1,500 tonnes per day.

The mine was able to sustain that production rate throughout 1998, but by the

start of 1999, the prices had fallen to their lowest level in many years. A long-term

plan, based on extracting high-grade material, was established at 1,000 tonnes per

day. The actual production levels stabilized around 800 tonnes per day for the next

two years. High-grade reserves depleted rapidly and the mine eventually closed its

doors.

A mine with a large and valuable resource, dreaming of expansion just a few years

before, had been forced to shut down for lack of reserves. Had it be been able to

survive a few years more, it would have been in excellent position to take advantage

of today’s very high metal prices.

Effects of Metal Prices Variations on the Planning Process

In general, in any change of plan leading to a production rate reduction, each sector

in development or production is re-evaluated. Based on this, each is either left in the

production plan, phased out, or abandoned. To complete the plan, new sectors must

be developed. In a typical vein-type deposit, the stopes in production are usually the

closest to existing infrastructure. The new sectors might actually be located further

than the ones developed ahead of time under the old plan. Furthermore, the new

sectors might be smaller because of the lower value of the ore. All of this implies


that a high quantity of new development must be put in place rapidly in order to

reduce the transition time to the new plan. More crews might be necessary, either

transferred from production, or employed as new employees or contractors. The cost

may appear either as new capital expenditure or as a higher operating cost.

The re-evaluation of the mining sectors first starts through the revision of the

reserves and its value. Lower metal prices lead to smaller net smelter return (NSR)

factors and a decrease in the value of the reserve blocks, to the point that some reserve

blocks are no longer economical. The reserve base decreases and the expected life of

the deposit may be reduced.

Costs are also affected. Variable mining costs usually increase. Even if the mining

method does not change, the stope sizes decrease and there is proportionally more

stope development to do per tonne. Also, in vein-type deposits, there often exists

a correlation between the ore grade and the ground quality: the higher the grade,

the worse the ground. Therefore, in order to mine more high-grade material, more

expensive mining methods must be used. As discussed previously, more development

is required to bring new sectors into production. The fixed cost spent every month

decreases but when that cost is distributed to the tonnes mined, it corresponds to

a higher unit cost. And finally, in parallel to this, there is a renewed interest in

exploration to replace the lost reserves.

The increase in unit costs affects the reserves as more blocks now might have

to be dropped, blocks which values are higher than the previous operating cost but

lower than the new one. If some of these blocks are included in the new plan, they

must now be replaced. New costs are calculated and the process is repeated until the

plan is satisfactory. This is an expanded case of what Ding [19] suggested with the


redistribution of development costs in the determination of a reserve base.

In summary, a reduction in metal prices leads to a lower production rate with a

higher grade and higher unit operating costs. The reserve base is smaller and the

mine life is shorter. The value of the long-term plan is thus much less than it was

previously, and the probability of an economical return is smaller.

1.3 Risk Aversion and Robustness

With all information available a priori, the mine manager might have designed the

mine such that these many variations in metal prices would have had less impact.

Walls and Eggert [72] have shown that by nature, most mine managers take action to

reduce the financial risk associated with their investment decisions. Smith [62] and

Whittle [75] also discuss the fact that most decision-takers in the mineral industry

choose to operate mines at lower production rates, or to mine less reserves, than would

otherwise maximize the NPV in order to lower operating risks and give themselves

time to correct any mistakes made during the initial stages.

The proposed methodology to design a mine plan less sensitive to metal price vari-

ations would certainly allow mine managers to manage their risks without foregoing

too much value. Monkhouse and Yeates [43] refers to this as a robust plan, one where

the mine plan continues to give high values over a wide range of input assumptions.

1.4 Thesis Organization

The following is a summary of the work as it will be presented in the following pages.


Chapter 2 - Literature Review The existing models aiming to define production

rates and cut-off grades will be examined, and it will be shown that very few of the

them can define operating parameters under conditions of metal price uncertainty.

Following that, metal price simulation models will be looked at in order to determine

how metal prices will be generated in this thesis. Finally, a short discussion on

discount rates will be done to decide the rate that will be used.

Chapter 3 - The Mathematical Model Based on the models previously re-

viewed, modifications and adaptations will be done in order to reflect the information

commonly available to a mine planner, and a new cost classification system will be

proposed. Then, the revenue model will be developed and it will be shown that there

exists a direct correlation between the grade-tonnage curves used in resource classifi-

cation and the revenue model. Finally, the structure of the PeaRL (Production Rate

and Level) methodology to solve for robustness against varying metal prices will be

described.

Chapter 4 - Test on Actual Mine A test of the methodology will be conducted

on data obtained from a gold-copper mine, which will be described first. The mineral

resources will then be analyzed and a method will be proposed on how to classify

polymetallic ore when metal prices are ignored. The details of the production, rev-

enues, operating and capital costs functions will be developed, and the metal price

stochastic models will be constructed. The feasible area of the solution will then be

established and the simulation will be run. The chapter will end with a discussion of

the results and the analysis of the main factors influencing them.


Chapter 5 - Hypothetical Gold Mine In this chapter, the procedures used for

building the gold mine model, from the construction of the grade distribution to the

revenue and cost functions will be discussed first, and the simulation will be run

and its results analyzed. Afterwards, certain parameters of the price and cash flow

models will be changed in order to gauge the sensitivity of the methodology. Then,

the sensitivity of the model to the mine average grade will be tested and reviewed.

Finally, a discussion on how a manager may use the results of the simulation to design

a mine closes the chapter.

Chapter 6 - Conclusions and Recommendation This final chapter will give a

summary of the work done, the original techniques developed, the main conclusions

to be drawn, as well as a number of suggested modifications to the technique to adapt

it to other types of deposits, mines, and business opportunities.

Chapter 2

Literature Review

This chapter deals with the review of the work done in three general areas: the

optimization of production rates and cut-off grades under conditions of metal price

certainty and uncertainty, metal price modeling, and the use of inflation rates and

the selection of discount rates in financial evaluation.

Economic techniques were first applied to finding optimal production rates by

Gray [28], and the idea of optimizing cut-off grades was introduced much later by

Carlisle [12]. Solutions to both problems were proposed, separately and jointly, using

a variety of methods.

It is believed that proper metal price models must be constructed in order to have

reasonable answers with the proposed methodology. The model has to go further

than a simple distribution within which prices are randomly generated. There are

some trends in metal price behaviour in both the short and the long term and they

must be taken into consideration.

The treatment of inflation and the selection of a discount rate for the economical

evaluation of an orebody can influence the value of any given plan. The effect of both

15

CHAPTER 2. LITERATURE REVIEW 16

factors are examined in order to guide the selection of the rates to use in this thesis.

2.1 Production Rate and Cut-Off Grade Selection

The available literature can be subdivided into a simple matrix (table 2.1) where the

type of work presented by different authors can be quickly compared. The matrix is

divided into nine sections, three each for work done in relation with the determination

of optimal production rates, cut-off grades and combined production rate and cut-

off grade. The work done in these areas is further split into three, related to static

solutions under conditions of metal price certainty, dynamic solutions under the same

conditions, and solutions under conditions of metal price uncertainty. These nine

sections are further split into subsections based on the approaches taken to obtain a

solution.

Table 2.1: Optimization Matrix

Conditions of Metal Price Uncertainty

Static solution Dynamic solution

Production Rate Physical characteristics Micro-economics Linear programmingTaylor 1986 Gray 1914 Mueller 1994

Tatman 2001 Hotelling 1931 Real optionsEconomic optimality Campbell 1981 Brennan & Schwartz 1985

Glanvi lle 1985 Hartwick & Olewiler 1986Bradley 1985 Cairns 1998Smith 1997

Differential equationsCorbyn 1985

Abdel Sabour 2002

Cut-Off Grade Carlisle 1954 Lane 1991 Dowd 1976Taylor 1972 Fuentes 2003Lane 1991

Combined Carlisle 1954 Park 1992 Samis 2000, 2002Wells 1978Ding 2001

Conditions of Metal Price Certainty


In the coming pages, the various solutions developed for each problem will be

reviewed and some models of interest will be identified for further development in

this thesis.

One thing that becomes obvious in the review of this matrix is that there is a lack

of work done in the area of optimization of both production rates and cut-off grades

under conditions of metal price uncertainty. This thesis proposes a solution to fill

that quadrant of the matrix

2.1.1 Production Rate Optimization

The selection of the production rate has a strong influence on optimizing the value

of the orebody. Bradley [6] summarized the problem by noting that the capital

investment is directly proportional to the production rate, and that the level of capital

investment is significant and is concentrated in an initial dose at the start of the

project. The level of capacity depends on the expectations about the future. If the

level is too big or too small, the value of the resource owner’s asset is reduced. The

expectations about the future are related to metal prices and the size and quality of

the deposit.

Tatman [65] also remarked that there is seldom an engineering-based reason for

selecting the production rate. He noted three methods of selecting a production rate:

general input requirement, economic optimization, and empirical formula.

Empirical static solution based on physical characteristics

The simplest way of choosing a production rate for a deposit is based on the size of

the reserves. The best known approach is that of Taylor [67]. Based on a review of


thirty mining operations, he produced a simple equation (2.1) that is still used today.

However, advances in computer applications and rock mechanics make it such that

better mine models and plans can now yield higher production rates.

Tonnes per year = 5.0× Reserves0.75 (2.1)

Tatman [65] also developed an empirical equation for steeply dipping underground

deposits based on annual vertical depletion rates varying with the horizontal thick-

ness. He developed his formula after studying sixty deposits and noting the risk

associated with their mine plans. His formula expands a general rule of thumb that

is used in narrow vein deposits that states that the mine can produce at a rate equal

to 15 vertical centimetres per day multiplied by the tonnes of reserves per vertical

centimetre. His equation is noted as

Annual production rate = Rate factor× Rate multiplier (2.2)

where the rate factor is the tonnes per vertical metre and the rate multiplier is the

multiple to calculate the annual production rate. The recommended rate multipliers

are listed in table 2.2. Based on this table, 15 centimetres per day would correspond

to a rate multiplier of 37.5, ranking as moderate risk for a narrow vein deposit.

Table 2.2: Rate multipliers, from Tatman [65]

Deposit thickness Rate multiplier(metres) Low risk Moderate risk High risk

< 5 < 20 20 to 50 > 505 to 10 < 50 50 to 70 > 70> 10 < 30 30 to 70 > 70


Figure 2.1: Operating and capital cost as a function of production rate, fromSmith [62]

Static solution based on economic optimality

Some authors recognize that the net present value of a deposit varies with the pro-

duction rate and that there has to be a rate for which this value is maximized. The

basic assumptions behind this work is that the capital cost is proportional to the

production rate and that the operating cost is inversely proportional as economies of

scale can be realized. Glanville [26] developed such functions for a high grade gold

deposits and added a function relating the preproduction time and the capital cost.

Bradley [6] and Smith [62] also developed the concept illustrated in figure 2.1.

The Net Present Value (NPV) and the Internal Rate of Return (IRR) of the

project can be calculated for various production rates, and the results can be drawn,

as shown in figure 2.2. The production rate corresponding to the maximum NPV is

the optimum production rate.


Figure 2.2: Maximum Net Present Value as a function of production rate, fromSmith [62]

One interesting fact developed by all the authors is that the net present value is

fairly insensitive to reductions in production rates around the optimal point, with

reductions between 5 and 10 percent for production rates set at half the optimal rate.

Risk aversion

There is an interesting dialogue in the literature concerning which approach to use to

define the production rate. The proponents of empirical formulas say that, in order

to define the maximum net present value, one must spend too much time and money

conducting feasibility studies at progressively higher production rates, without ever

being certain that the optimal solution can be found. Meanwhile, the proponents

of the maximum net present value find that the empirical solutions are always sub-

optimal and that higher production rates can yield higher net present values.


Smith [62] compared the relationship between the NPV and production rate to the

stress-strain curve obtained by testing the strength of a rock sample. The maximum

point on this curve represents the failure point of the sample, where it is destroyed.

Smith judged that the maximum NPV is a failure point of the project, and if it is

designed at a higher production rate, the chance of failure of the project will be very

high. Given that the NPV is not much lower for production rates smaller than the

optimum, it seems reasonable that a mine be designed at lower rates, thus avoiding

the possible ”failure” of the project without significantly reducing its value.

Static solution mathematically derived

Corbyn [14] used differential equations to describe the financial performance of a

capital expenditure. This approach does not require the calculation of annual cash

flows. The basic parameters used in his equations are the discount rate, the unit

operating profit per tonne of ore, and the incremental capital cost of producing one

more tonne of ore annually. The equations are:

• the marginal payback period as a function of the incremental capital cost;

• the marginal increase in present value as a function of the payback period and

the project life, referred to as the wealth increase function. For any payback

period, there is a project life for which the wealth increase is maximized; and

• the project life as a function of the production period and the discount rate.

Therefore, for any given discount rate, there is a payback period for which the

wealth creation is maximal, to which corresponds project life, and thus production

rate. Overall, this is a good approach, but with one significant simplification: the


operating cost and margin remain constant, independent of the capital spent, or, in

other words, there are no economies of scale as represented in figure 2.1.

Abdel Sabour [1] based his work on that of Wells [73] (reviewed later in this chap-

ter), using Present Value Ratio to determine the optimal production rate. However,

instead of trying to find the maximum of the PVR function, he developed a marginal

analysis where the production rate is optimal when the present value of marginal

revenues is equal to the present value of marginal costs. His equations are developed

from the prefeasibility cost models built by Camm [10], where the operating and cap-

ital costs of every mining method are represented by an equation of the form aQb,

where Q is the production rate.

There are a few problems identified in this paper:

• The optimal production rate tended to be equal to that where the marginal rev-

enue is equal to the marginal cost, a solution that is valid for general industries,

but too high for mines;

• The author used a shrinkage mine as an example and comes to the conclusion

that it should operate at 3,662 tonnes per day, an unreasonable rate from an

operating point of view;

• Abdel Sabour assumed that as the ore grade increases, the optimum production

rate increases also. Again, this makes no sense from an operating point of

view. Since reserves are constant, independent of the grade, the capital cost did

not change with grade. A higher present value of revenues resulted from this,

without a corresponding increase in costs.


Overall, the two approaches do have some merits, but the equations are too simple

and do not take important factors in consideration.

Dynamic solutions

Gray [28] introduced the concept of diminishing productivity. In a producing mine,

as the production rate increases, increasing operating costs yield lower returns until

eventually the marginal revenue is less than the marginal cost. This is illustrated in

figure 2.3. A land owner would operate at the production rate for which both are

equal in order to maximize the NPV (Qm), but the mine owner, faced with limited

resources would tend to operate at a rate for which the average profit is maximum,

or for which the average cost is minimal (Ql). If the discount rate is considered, the

present value of future returns is lessened, and the present value of a tonne mined

now at greater expense might be greater than that of a tonne mined in the future at

lesser expense.

Using a numerical example, Gray indicated that the mine would maximize its

profit during the last year of production, thus working at Ql, and that for each of the

prior years of operation, the production rate is equal to that for which the marginal

profit is equal to the present value of the marginal profit of the last year. This results

in a mine plan with declining production rates annually, where the rate in year 1 is

less than Qm and is equal to Ql in the last year of operation.

It is worth noting that, in this example, Gray worked with a coal mine. With

time, this model was adapted to other commodities, and now the general accepted

model is based on units of metal mined (e.g. pounds of copper), where production

rates are equal to units of metal mined per year and costs and revenues are expressed


Production Rate (units per year)

Rev

enue

s an

d C

osts

($

/ uni

t)

Price = Average Revenue = Marginal Revenue

Marginal Cost

Average Cost

Ql Qm

Figure 2.3: Unit revenue and costs as a function of production rate, adapted fromGray [28]

in dollars per unit of metal.

Hotelling [34] was concerned with the calculus of determining the adequate pro-

duction rate of a deposit such that the NPV is maximized over its life. As had Gray,

he recognized that the economic theory of the firm, in which the production rate

should be static, does not apply in a mine where the supply is finite, and that the

discount rate plays an important role in determining the optimal sequence of produc-

tion rate to maximize the NPV of the deposit. This principle is referred to as the r%

rule. However, the difference between Gray and Hotelling is that Gray concentrated

on one mine while Hotelling was describing the behaviour of the whole industry, and

that the r% rule applies only in cases of scarcity of the metal supply. Nonetheless,

equation 2.3 describes both cases well.


max NPV (q) =

∫ T

0

e−rt[πq(t)]dt (2.3)

where

NPV is the net present value

q is the production rate

T is the life of the mine

r is the discount rate

t is the time

πq(t) is the profit associated to the production rate at time t

Crabbe [15] compared the work of both authors, and listed their major assump-

tions. In the case of Gray:

1. Perfect competition and constant price: The price of the metal remains constant

through time;

2. Grade homogeneity: The grade throughout the deposit is constant;

3. Independence of cost functions from cumulative production: The cost functions

do not change as the operation progresses from mining of principle stopes to

pillar recovery;

4. Time autonomy of cost functions: The cost functions remain the same through-

out the life of the mine;

5. U-shaped curves: The concept of diminishing productivity discussed previously;


6. Free horizon: The life of the mine is not fixed a-priori;

7. Fixed stock:The in-situ resources are fixed;

8. Perfect malleability and shiftability of capital: There is no fixed capital, and

capital can be increased and decreased instantaneously;

9. No storage: the ore is sold immediately upon extraction.

Hotelling’s assumptions are similar, but include:

• #1, Perfect competition and rising prices: The price of the commodity increases

at the same rate as the rate of interest;

• #5, Long-run equilibrium in the mining industry: All firms have identical costs.

As a consequence, the individual mines cannot adjust their production rate

marginally and must either operate or close. The marginal changes in produc-

tion rate are achieved through the opening or closing of mines.

Since this thesis is interested in the design of individual mines, Gray’s model will

be used from now on in this chapter and will be revisited later in chapter 3.

Another aspect of Gray’s model that is of interest is that the fixed cost of operating

a mine is constant for all production rates. This can be better understood by looking

at the total functions corresponding to the unit functions in figure 2.3, illustrated in

figure 2.4.

Park [50] examined the effect of discount rates on the distribution of production

rates in time. Using an hypothetical mine as an example, he produced figure 2.5, in

which he showed that high discount rates force operations to work at high production


Production Rate (units per year)

Rev

enue

s &

Cos

ts (

$)

Revenues

Costs

Fixed

Figure 2.4: Total revenue and costs as a function of production rate, adapted fromGray [28]

Figure 2.5: Effect of discount rates on dynamic optimization of production rates, fromPark [50]


rates in order to minimize the effect of discounting, whereas low discount rates allow

the operation to work at rates where the annual profit can be increased.

Gray recognized that there were some problems with his model:

• The capital outlay required to produce at high production rates early in the

mine life may be wasted as production rates decrease with time. He suggested

it may be better to have uniform production rate to smooth out the capital

outlay. This rate would be between the two extremes defined as the maximum

rate occurring in the first year and the minimum in the last year of production;

• A mine may increase or decrease productivity from one year to another based

on the work done in the mine;

• Metal prices variations would influence the extraction rate.

Campbell [11], Hartwick and Olewiler [29], and Cairns [9] wrote about the rela-

tionship between capital outlay and production rates. The problem lies in the fact

that equation 2.3 does not include the capital outlay. The equation can now be

rewritten as follows:

max NPV (q) =

∫ T

0

e−rt[πq(t)]dt−Kq (2.4)

where Kq is the capital outlay as a function of the production level. This equation

yields a solution where the production rate is constant for most of the life of the mine,

as illustrated by Hartwick and Olewiler in figure 2.6.

Based on this, although dynamic models have higher net present values than static

models, they are not implemented in operations because of the constraint associated

to the initial capital expenditure.


Figure 2.6: Unit revenue and costs as a function of production rate, adapted fromHartwick and Olewiler [29]


Solutions under conditions of metal price uncertainty

Under conditions of metal price uncertainty, the literature is limited to modeling the

behaviour of firms when metal prices change. There is no attempt to develop a mine

plan taking metal price fluctuations into consideration. Instead, the objective is to

maximize the value of a plan that is already in place by modifying the behaviour of

the firm when the prices change.

The first model of interest is that of Mueller[44] in which he built a model relating

mine development (referred to as Effort in the paper), reserves, and the production

rate. He states that reserves change annually through production, discoveries, and

adjustments based on information gathered during the year (geological and price

uncertainty). After statistically testing a few models with data from the oil industry,

he came to the conclusion that firms consider that both geology and metal prices

stay constant through time and try to maximize the NPV using a static model.

Eventually, when the conditions change significantly, a new plan is built with the

objective of maximizing the NPV based on the latest values.

The second paper is from Brennan and Schwartz [7], in which the concept of real

options is introduced as a way to evaluate the true worth of a mine under conditions of

metal price uncertainty. In this model, the mine manager has the option to temporar-

ily suspend the operations of the mine if the metal prices are too low and the cash

flows are negative. The manager can decide to incur the one-time cost of temporarily

shutting down the mine and that of maintaining it ready to reopen, thus losing less

money than if the decision was made to keep it open. The overall effect is that the

total cash flow over the life of the mine will be greater in this case as lower loses are

incurred and as the mine operates only during years of higher metal prices. Figure 2.7


Figure 2.7: Effects on cash flows of operating options of a mine, from Palm, Pearson,and Read [49]

illustrates this concept. Slade [59] and Moel and Tufano [42] demonstrated that mine

managers of copper and gold mines in North America were actually following this

approach during the 1980s and 1990s.

As mentioned earlier, these two papers show that it is possible to maximize the

value of a mine plan under conditions of metal price uncertainty, but there is no

demonstrated way to choose the production rate of a mine under those conditions.

2.1.2 Cut-Off Grade Optimization

Pasieka and Sotirow [51] defined two general types of cut-off grades:

• For strategic planning purposes, the cut-off grade achieves a positive net present

value. It includes geological and planning cut-off grades, used to define mine


reserves and for general design purposes; and

• For operational and corporate planning, the cut-off grade achieves positive net

cash flows. It includes average annual budget cut-off grades for operational

planning, and break-even cut-off grades for ore-waste separation along stope

boundaries.

Taylor [66] offered a detailed breakdown of the points of planning and operational

cut-off grade estimation for Type A and B underground mines and for open-pit mines,

based on the amount of geological information and the costs to incur.

This thesis deals with the definition of the cut-off grade for strategic purposes,

with the objective of defining the reserves and determining the size and capital cost

of the mine. Because of this, the review of cut-off grade optimization deals only with

the first type.

Static solutions

Carlisle [12] introduced the idea that orebodies are not homogeneous and that it

might be more profitable to extract only a fraction with higher average grade. Apart

from the economic aspects, he recognized that the complete extraction of ore has

some practical limitations with respect to the mining method and recovery of pillars

and the mill recovery of low grade ore. In an approach similar to Gray [28], he

talked about economies and diseconomies of scale and related them to the level of

recovery of metal resource, best described as a portion mined from the total metal

resource. Implicitly, it is assumed that, for a given level of recovery, the portion

defined constitutes of resource ore tonnes with the highest available grade, such that

there exists a direct correlation between the level of recovery and the cut-off grade in


Figure 2.8: Levels of recovery unit functions, from Carlisle [12]

which the cut-off grade is inversely proportional to the level.

As seen in figure 2.8, the mine is a price-taker and the price it receives is constant

for all levels. The average total cost curve is u-shaped (ATUC) as is the marginal

cost curve (MCL). For a discount rate equal to 0%, the NPV, i.e. the cash flow, is

maximized if the mine operates at a level for which MCL = price, and the annual

profit is maximized at a level for which the average cost ATUC is at its minimum.

If the discount rate is greater than 0%, the optimum NPV occurs at a level located

between the two end points just mentioned.

In this thesis, Carlisle’s work will serve as a basis for the development of the

mathematical model. For this reason, it is worth examining more the construction


Level of recovery (units of metal)

Rev

enue

s &

Cos

ts (

$)

Revenues

Costs

Capital

Figure 2.9: Levels of recovery total functions, adapted from Carlisle [12]

of the graph. The best approach to this is to build the total function graph as a

complement to the unit graph discussed above. This graph is presented in figure 2.9.

Based on Carlisle’s paper, it can be deduced that the variable cost V CL is the sum of

the fixed and variable mine operating costs. What he refered to as the Total Fixed Cost

is actually the capital cost of mobile equipment and mine development, regardless of

the tonnage recovered, and is represented by the intercept of the cost curve with the

abscissa. The elements of this construction will be revisited in chapter 3.

Whereas Carlisle’s model is conceptual, Lane [37] derived mathematical equations

in order to determine the cut-off grade with the objective to maximize the cash flow

within the operating constraints of the mine. His equation is listed in 2.5. Taylor [66]

also described the same.

c = (p− k)xyg − xh−m− (f + F )τ (2.5)


where

c is cash flow

p is price per unit of mineral

k is marketing cost

x is the proportion of mineralized material classified as ore

y is mill recovery

g is average grade of ore

h is treating cost

m is mining cost

f is time (or fixed) cost

F is the opportunity cost

τ is the time to progress through a unit

In other words, the cash flow is equal to the revenues (as a function of production,

grade, metal price, and smelter contracts) minus mining, treating, and fixed costs.

Within the context of an underground mine, Lane considered that the mining activ-

ities include all lateral and vertical development, and the treatment activities range

from the stoping, tramming and hoisting to surface to crushing, grinding, and sepa-

rating at the mill. Smelting, refining, and selling are considered within the marketing

costs.


The time to progress through a unit is limited by the slowest of the mining,

treating, or smelting, and Lane derived equations to determine the optimal cut-off

grade maximizing the cash flow while remaining within the operating constraints.

Recent examples of the application of this technique in underground mines are made

in Tatiya [64], Wheeler and Rodriguez [74], Poniewierski, MacSporran, and Shep-

pard [53], and Henry [33].

The basic equation presented by Lane is of interest for the development of the

mathematical model in this thesis and will be revisited in chapter 3.

Dynamic solutions

Lane also developed a dynamic solution for optimizing the NPV of a mine. Intuitively,

his solution is easy to understand, with higher cut-off grades employed earlier in

the mine life in order to have higher cash flows and thus higher NPVs. Park [50]

showed that the solution was sensitive to discount rates, as shown in figure 2.10. It

is recognized in the industry that Lane’s solution is easier to apply to open pits than

to underground mines, as it is possible to create lower-grade surface stockpiles that

can be later sent to the mill, whereas low-grade material from underground must

be discarded as waste or blended into the mill feed. The geometry of underground

orebodies and the required sequence of extraction makes it very difficult to establish

strict cut-off grade policies as dictated by Lane. In future work, Lane [38] suggested

that a cut-off grade policy may be a poor one if the early metal prices are low.


Figure 2.10: Effect of discount rates on dynamic optimization of cut-off grades, fromPark [50]


As in the case of production rates, there is a fair amount of literature on how the

cut-off grade of an operating mine should be adjusted when metal prices fluctuate.

Taylor [66] is the best-known author, and he concluded that the cut-off grade of a

mine should increase with metal prices in order to increase the cash flow and thus

increase the net present value of the mine. He noted that it contradicted the observed

behaviour in most mines, where the cut-off grade is lowered in hope of increasing the

reserves. He argued that current practices lead to mining more tonnes for a greater

total cost while producing the same amount of metal. The revenues from production

would be higher however, and might profitably offset the increase in costs.

Dowd [21] used dynamic programming to determine the sequence of annual cut-off


grades with the objective to maximize the net present value. Even though the title

of his paper mentioned production rate optimization, his examples are constrained

by the mill which remains full in all the cases presented, and his work showed only

cut-off grade optimization. He developed a stochastic distribution of metal prices,

with Bayesian probabilities of getting a metal price j in year n + 1 given a metal

price i in year n. He developed and solved two examples, one for a deposit with an

homogeneous grade-tonnage distribution throughout, and one where five production

levels are defined, each with its own grade-tonnage relationship. He recognized that

his model was oversimplified, but that it served the purpose of introducing a general

model to generate a solution.

Fuentes [24] looked at the effect of metal price uncertainty on the planning of

block caving operations. He wanted to develop a methodology to determine which

mineralized columns to mine and the order in which to mine them to increase the

economic robustness. He introduced the concept of the Price Certainty Parameter

(PCP), the probability that a column can participate in a profitable design, ie. be

considered in the mining plan.

Each ore column must be evaluated individually. Therefore, for each column, one

must:

• Randomly generate a metal price from a predetermined distribution;

• Evaluate the operating benefit of each resource block in the column;

• Calculate the cumulative block benefits from the bottom up, including the cost

of development and mine preparation to the bottom block;

• Identify the maximum cumulative benefit and its corresponding column height;


and

• The PCP is equal to the percentage of times a simulation generated a positive

maximum cumulative benefit.

For mine design purposes, a given column is included if it adds to the overall

profit or if its inclusion permits the addition of columns that increase profit. Once

the boundaries of the area to mine are determined, the sequence of extraction is set

by starting in the highest PCP sector and progressively adding lower PCP sectors

over time.

In a case study, the stochastic model was compared to plans built with various

metal prices. In general, the stochastic plan was very similar to that generated with

a metal price equal to the average of the metal price distribution. However, upon

doing a risk analysis for each plan, it was shown that the stochastic plan was more

robust, with a higher probability of achieving a positive net present value.

2.1.3 Combined Models

Static solutions

Carlisle [12] combined his recovery level model with Gray’s production rate model

and suggested that, for a mine trying to optimize its annual cash flow, the cut-off

grade should be increased such that the total average cost is at a minimum and the

production rate should be increased such that the marginal cost is equal to the metal

price. In the case of a mine trying to maximize its total profit over the life of the

mine, the level of recovery should be such that the marginal cost is equal to the metal

price and the production rate is such that the average total cost is at a minimum. He


Figure 2.11: Relationship between rate and level of recovery, from Carlisle [12]

produced a graph (figure 2.11) showing the relationship between levels of recovery,

production rates, and average and marginal cost curves, with the objective to identify

operating conditions.

Wells [73] used the Present Value Ratio 1 to determine the best combination of

production rate and cut-off grade. In the paper, the PVR is equal to the ratio of

the present value of the positive cash flows (PVOUT) divided by the present value

of the negative cash flows (PVIN), and the mine is economical if the PVR is greater

than 1. He developed the relationship of capital cost as a function of production

rate and assumed that the preproduction time is equal to four years, independently

of the production rate, and that production starts after that time. Positive cash

1Gentry and O’Neil [25] refers to this as the Benefit Cost Ratio


flows assume a constant operating cost independent of the production rate, thus no

economies of scale. As for the cut-off grade, he used an approach similar to Lane’s,

where each tonne mined must pay for its variable operating cost. The last step in his

optimization process was to calculate the PVR function for many other cut-off grades

and plot a contour map of the PVR as a function of the production rate and cut-off

grade, as shown in figure 2.12. An island of PVRs greater than 1 can be defined and

the economical engineering designs can thus be constrained to the ones within the

island.

Ding [19] developed an iterative process to define the optimal cut-off grade and

production for an underground mine. An initial production rate is selected and the

cost and revenue curves common to Gray and Carlisle are constructed, and from these

curves, the optimum production level is defined as the one for which the marginal cost

is equal to marginal revenues. Using the grade-tonnage curves of the deposit, a new

reserve base is calculated. Accordingly, the cost and revenue curves are adjusted to

reflect this new information, and the production rate for which the NPV is maximized

is chosen as the optimal. This production rate is now the basis for the next iteration of

the optimization process, and the process ends when there is little difference between

two consecutive optimal production rate determinations. The process is presented in

figure 2.13.

Dynamic solutions

Park [50] proposed an iterative process to find a dynamic solution to find the optimal

set of varying annual production rates and cut-off grades. The process is very similar

to Ding’s static solution, with the difference that another level of iteration is added


Figure 2.12: Contour map of PVR in relation to cut-off grade and production rate,from Wells [73]


Figure 2.13: Relationship between rate and level of recovery, from Ding [19]

by optimizing each year of production. The process starts by assuming that the

deposit is in its last year of production. In this case, the objective is to maximize the

total profit, and this is achieved by operating at a cut-off grade for which the unit

profit is maximum, or point CB in figure 2.14, and at a production rate for which

the maximum unit profit is achieved, rate XA in figure 2.15. For each consecutive

step of the iteration, as time proceeds back towards the start of the mine life, the

cut-off grade and the production rate increase as a function of the discount rate.

For each year of production, an iterative process is necessary to find the equilibrium

between the optimum cut-off grade and production rate as each function influences

the other. The production rate is first calculated and is integrated into the cut-off

grade functions which are then optimized. The latest solution is fed back into the

production rate functions which are then optimized again. The process is carried on

until the solution becomes stable.


Figure 2.14: Cut-off grade unit functions, from Park [50]

Figure 2.15: Production rate unit functions, from Park [50]



Samis [55] and Samis, Laughton, and Poulin [54] looked at the case of a mine exploit-

ing the high-grade core of an orebody and studying whether to mine the low-grade

extension. Three options are open:

• Mine the high-grade until depletion and close the mine;

• While mining the high-grade, develop the low-grade extension, increase the mill

capacity, and mine both sectors simultaneously; or

• Deplete the high-grade and put the low grade in production without having to

increase the mill capacity.

Samis developed fixed production plans (FPP) for many variations on the three

options, including different timing for the introduction of the low-grade expansion

during the mining of the high-grade. He combined all these variations into a flexible

discrete mine production (XDFP) project structure. This structure was used to de-

velop a tool to be used by the mine manager to decide on which previously-mentioned

option to incorporate at any point in the mine life based on the current metal price.

This tool is illustrated in figure 2.16. For example, if the metal price is equal to

$0.60 per unit of mineral with 5 years left in the mine life, the high-grade zone alone

should be mined; if the price is greater than $0.70, both zones should be operated. In

essence, this tool analyzes all the options available under many metal price conditions

and later allows the mine manager to chose both the production rate and the cut-off

grade of the operation based on the current metal prices.


Figure 2.16: Low-grade zone development and project abandonment boundary,adapted from Samis et al. [54]

2.1.4 Summary of reviewed literature

In general, the existing literature concentrates mostly on two aspects:

• The optimization of the Net Present Value through the variation of the produc-

tion and/or the cut-off grade and examining the effect of the marginal profit

of the total value. Production rate variation equations were shown to be in-

complete as they did not consider the impact of capital costs and also to be

impractical at the operations themselves. Cut-off grades variations were shown

to be impractical underground because of the impossibility to create low-grade

stockpiles on surface.

• The optimization of the Net present Value through the option to suspend a

given mine plan when metal prices are low and projected losses are too high;

the mine can be put back in operation when metal prices increase again. Mine

managers were shown to exercise this option when times were bad.


In other words, almost every paper talks about developing an optimal mine plan

under conditions of metal price certainty or optimizing an existing plan under condi-

tions of metal price uncertainty. Only Samis proposed a method to choose between

operating options such that the choice has the highest probability of success, or the

highest degree of robustness.

2.2 Metal Price Forecasting Models

This section discusses a metal price generation model. One must first take a look

at long-term trends in gold and copper prices, in terms of both nominal and real

dollars and explain why long-term tendencies are towards increasing production and

lower prices. The next step is to look at factors that drive short-term price volatility.

Finally, this section reviews stochastic model for generating metal prices for use in

Monte Carlo application.

2.2.1 Long-Term Metal Price Tendencies

The author began investigating average annual gold and copper prices in nominal

terms, starting in 1980 for gold (figure 2.17) and 1975 for copper (figure 2.18). In

real dollars (figure 2.19), the obvious long-term trend in copper prices is downward.

Tilton [69] and [70] said that this tendency is due to a downward shift in the supply

curve, represented by the marginal production cost curve. Such a shift not only

results in a decrease in metal prices but also in an increase in production, as seen in

figure 2.20. Tilton continued by showing that the marginal copper production cost

curves have gone down over the years, mostly as a result of technological advances


0

100

200

300

400

500

600

700

800

900

1000

1980 1985 1990 1995 2000 2005

Year

Ave

rage

ann

ual g

old

pric

e ($

/oz)

Figure 2.17: Average annual gold prices from 1980 to 2005

and scale efficiencies (figure 2.21).

2.2.2 Short-term price volatility

Whereas long term variations in metal prices are related to supply conditions, Tilton

went on to describe how short-term volatility is driven by demand and the slow

reaction time of the industry to add productive capacity. Figure 2.22 illustrates a

condition where the demand for a metal increases rapidly, shown as a shift of the

demand curve to the right. As the demand increases, mines increase their production

as best they can, but after relatively small short-term increases, the operating mines

simply do not have the capacity to develop new reserves, either by expanding current

operations or by developing new deposits. This inability to react in the short term

is illustrated as a capacity constraint, forcing the supply curve upwards and driving


0

20

40

60

80

100

120

140

160

180

1975 1980 1985 1990 1995 2000 2005

Year

Ave

rage

ann

ual c

oppe

r pr

ice

(cen

ts /

lb)

Figure 2.18: Average annual copper prices from 1975 to 2005

Figure 2.19: Average annual copper prices in 2001 dollars, from Tilton [69]


Figure 2.20: Falling prices due to a shift in the supply curve, from Tilton [69]


Figure 2.21: Copper mine costs (cents/lb, real 2001 dollars), from Tilton [69]

prices to vary significantly with small changes in demand.

Over the medium and the long term, new productive capacity will be added, thus

pushing the capacity constraint line to the right and bringing prices down.

2.2.3 Stochastic metal price models

In the realm of Real Options, metal price models are used to study the effect of

metal prices on the value of production plans. Dixit and Pindyck [20] went through

the more common ones in their book. However, one particular model seems to be

preferred by most authors. This model, which takes into consideration both the

short-term volatility of prices and the general long-term tendencies, is referred to as

a mean-reverting process. Amongst others, Brennan and Schwartz [7] used the model

with real options to evaluate the practice of temporary mine closure when prices are


Figure 2.22: Causes of market volatility, from Tilton [69]


low, Davis [16] used the same process to evaluate mines using various metal price

models, and Samis et al. [54] adapted it to take production decisions in a multi-zone

orebody. Schwartz [56] analyzed fifteen (15) years of gold and copper price series to

establish the main characteristics of each metal’s model.

The model is described as a Wiener process or a Brownian motion with a mean-

reversion process. The Wiener process is a continuous-time process with three charac-

teristics. First, it is a Markov process, meaning that all future values can be calculated

from the current one. Second, it has independent increments, meaning that the prob-

ability distribution for the change in the process is independent from one time period

to another. And third, changes in the process are distributed normally and with the

variance increasing over time. This model however tends to wander far from its start-

ing point. As presented by Tilton, metal prices tend to have long-term average prices

related to the marginal production cost. Therefore, the Weiner process is modified to

include mean-reversion, thus forcing the model to tend towards a long-term average

price, as shown in equation 2.6.

xt = E(xt) + dx (2.6)

Where

xt is the metal price at time t,

E(xt) is the expectation of the price at time t, and

dx is the short-term metal price variation

The first argument of the equation is the expectation of the price in the future,

and is a function of the tendency for the price to revert to a long-term average, also


50

60

70

80

90

100

110

120

130

140

150

0 2 4 6 8 10 12 14 16 18 20

Time (years)

Met

al p

rice

(ce

nts/

poun

d)

100

25.0

=

=

x

η

Figure 2.23: Example of mean reversion

referred to as the degree of mean-reversion. It is expressed by equation 2.7, and

figure 2.23 shows an example of how prices would revert to a long-term average for

various initial prices. Schwartz [56] noted that both copper and gold tended to revert

towards long-term averages, though gold’s degree of reversion is very weak, ie. it

takes much more time for gold to revert.

E(xt) = x + (x0 − x) exp(−ηt) (2.7)

Where

x is the long-term average of the metal price,

x0 is the price at time zero, and

η is the degree of mean reversion.


The second argument is the short-term shock resulting in sharp increases or de-

creases in price over short time intervals. This variation is a random function known

as a Wiener process.

dx = σdz (2.8)

Where

σ is the variance of the metal price variation, and

dz is an increment of a Wiener process. dz = ε√

t where ε is a normally distributed

random variable with a mean of zero and a standard deviation of 1.

2.3 Treatment of Inflation in Discounted Cash Flow

Calculations and Discount Rate Selection

2.3.1 Treatment of Inflation

It is widely recognized that cost inflation has an impact on cash flow, and much

literature shows the effect of different types of inflation

Heath, Kalcov, and Inns [32] stated that inflation rates can be reasonably predicted

for a period of three to five years, but are hard to project afterwards.

Gentry and O’Neil [25] showed that inflation negatively affects after-tax cash flows

when considering the effect of rising costs on depreciation, working capital and capital

gains taxes. In all their examples, revenues and costs increase at fixed or variable

rates, and the resulting after-tax financial indicators are considerably lower than the


cases where inflation is not considered. Smith [60] proceeded to the same type of

exercise with the same conclusions.

Gentry and O’Neil then discussed how projects at the evaluation stage should

be studied without adjustments for inflation to be able to identify variations due to

technical factors without being obscured by economic effects.

Smith [63] considered that economic analysis without considering inflation, the

”bare bone” case, is the indicated way to compare many projects by using a common

reference point. He added that, if the bare bone scenario is attractive, scenarios with

inflation and other assumptions will be attractive also.

For the project considered in this thesis, inflation is not included for a number of

reasons.

• inflation has a greater impact on after-tax analysis. Since this thesis deals with

pre-tax cash flows, the impact of inflation is not as important,

• all studies on inflation assume that both costs and revenues increase at the same

rate. In this thesis, the annual price of metals varies as per a stochastic process,

and the model does not consider that long-term metal prices increase with time.

It could be argued that revenue fluctuates according to a variable inflation (and

deflation) rate, thus concentrating any effect of inflation on revenues only.

• In order to identify the elements driving the methodology developed in this

thesis, cost inflation is assumed to be equal to zero to truly concentrate on

technical aspects of the problem.


2.3.2 Discount Rate Selection

There are two major lines of thoughts in literature aimed at selecting a discount

rate. The first is based on the CAPM model and the second tends to assign risk in a

project according to the available information for that project. The two approaches

are summarized here.

Gentry and O’Neil [25] considered that the discount rate must be sufficient to cover

base opportunity costs, the transaction cost, the increment for risk varying with the

nature of the project, and an increment for inflation. This rate is calculated as the

weighted average cost of acquiring capital for a given company and is a function of

the cost of debt and the cost of equity calculated as a function of the correlation of

the risk of the company’s equity to that of the market in general. This correlation

factor is referred to as the Beta factor.

Smith [61], [63] discussed the same concept. He then pointed out that the Beta

factor measures the variability of an entire company, and not of individual projects.

As practiced in the industry, the discount rate varies from project to project, based

on its location in the world and the type of metal mined. The total discount rate

is equal to the sum of an risk-free rate, the project-related risks, and the country

risk. The project risks include the mine life, the revenue factors (reserves and metal

prices), and the operating and capital costs. He then showed how the discount rate

diminishes throughout the life of the project as more information is available, as

shown in figure 2.24.

In this thesis, the mine on which the proposed methodology is tested is in ad-

vanced stages of production, at a point somewhat similar to mid-life as referred to

by Smith [61]. At this stage, he considered that the discount rate should be equal to


Figure 2.24: Components of real discount rates at different stages of project develop-ment, from Smith [61]


5% including revenue risks equal to 1.5 or 2%. Based on this, the discount rate to be

used in this project is 5%.

Chapter 3

The Mathematical Model

The objective of this chapter is to build the mathematical tools required by the

engineer to solve the problem of preparing a strategic mine plan under conditions of

metal price uncertainty.

Two principle points are covered in this chapter:

• The construction of a micro-economic model defining revenues and costs as a

function of production rates and cut-off grades, and

• the development of the methodology to select the production rate and cut-off

grade of a mine under conditions of metal price uncertainty using this micro-

economic model.

3.1 Construction of the Model

The purpose of the model that is developed in this chapter is to permit the mine

planning engineer to have a tool that he or she can manage easily. For this reason, the

60

CHAPTER 3. THE MATHEMATICAL MODEL 61

theoretical models that were of interest in Chapter 2 will be briefly presented again,

with the emphasis on identifying their limitations in the eyes of the engineer. The

new model will then be built on the foundations of previous work while incorporating

elements of revenues and costs managed on a daily basis by the mine planner. It

will then be shown that there is a direct relationship established between the Grade-

Tonnage curves of an orebody and this model.

3.1.1 Characteristics of Existing Models

Three models are of interest for the construction of the new one: Gray’s production

rate model, Carlisle’s level of recovery model, and Lane’s cut-off grade model. Each

will be briefly discussed in the light of identifying the elements that the mine planner

can and cannot manage easily.

Gray

The main characteristics of Gray’s model are the following:

• All functions are related to units of metal produced, pounds of copper for ex-

ample;

• The production rate is expressed as units of metal produced per time period;

• Revenues are equal to the sale price of the metal;

• Costs are equal to the variable cost of extracting and treating a unit of metal;

• Fixed costs are constant for all production rates; and

• Capital costs are a function of production rates, as per Campbell and others.


Carlisle

Carlisle’s level of recovery model has the following characteristics:

• As in Gray’s model, all functions are related to units of metal produced;

• The level of recovery is inversely proportional to the cut-off grade of the deposit.

A low level of recovery corresponds to a high cut-off grade and only the portion

of the deposit with a grade equal or greater than the cut-off is mined;

• The capital cost of the mine is constant for all levels of recovery.

Lane

Lane’s general equation (2.5) has the following characteristics:

• Revenues are a function of the cut-off grade and are expressed in terms of tonnes

of ore mined and its average grade;

• Mining and treatment costs per tonne of ore mined are constant for all produc-

tion rates and cut-off grades;

• Mine development costs are constant for all cut-off grades and production rates;

• Fixed costs are constant for all production rates and cut-off grades.

3.1.2 The Cost Model

The models reviewed above are found to be limited and lacking in the representation

of the activities found in an underground narrow-vein mine. In this part, these models

will be expanded upon by integrating many aspects of costing and revenue generation


that are drawn from the experience of the author. The purpose is to create a more

realistic model taking into account the many activities found in a mine.

At this time, the model is built for a mine already in production, with the mill

commissioned, the ore handling system in place, and with some of the vertical and

lateral development infrastructure in place. Some aspects that are developed are

limited to this condition, akin to that of a mine in the first third of its life. A project

at the feasibility stage would carry all the elements of the present with additional

considerations. That case will be discussed in Chapter 6.

Observations of a Mining Engineer

In this section, general background information will be compiled from professional

experience.

Working units In financial reporting, it is very common to explain to all stake-

holders the profitability of an operation by comparing the realized price of sale of the

metal produced with the cost of producing one unit of metal. These numbers are

calculated by dividing the total operating cost at the mine by the number of metal

units produced, and are very easy to produce once all the cost accounting information

has been compiled. However, during mine planning, costs are measured exclusively

as a function of the quantity of tonnes of ore to move, with no direct reference to the

quantity of metal that can be extracted from the rock. Given this information, the

main working reference should therefore be the tonnes of ore sent to the mill.

Based on this: the production rate axis is scaled in tonnes of ore sent to the mill

per day, month, or year; the level of recovery axis is scaled in tonnes of ore; and

costs and revenues are expressed in dollars per tonnes or ore. For the purpose of this


model, the term level of production will be used as a replacement for level of recovery

to describe the tonnes of ore with an average grade higher than the cut-off grade.

This relationship will be expanded upon later in this chapter.

Capital costs For a mine in production, there are three circumstances in which

capital can be spent. The first case is for the preparation of development for the

purpose of accessing new ore zones, the second is for the replacement of mobile and

fixed equipment in the mine, and the third is for the replacement of equipment in the

mill and other surface installations. Each case can be better described as a function

of either the level of production or the rate. Each case will be studied individually.

Capital development in underground mines consists of shafts, ramps, raises, and

lateral transport drifts required to access ore zones with expected utility greater

than one year. In the typical Taylor [66] Type A deposit under study in this thesis,

mineralization is erratically distributed along large horizontal and vertical extensions.

Given this nature, it is reasonable to assume that the amount of capital development

required would vary with the level of production. As the best million tonnes is

developed, a certain amount of development is required. If the next best million

tonnes is developed, part of the increment would already be included within the first

million tonne envelope, but extra development would be required to encompass the

rest. This incremental relationship would continue until all resources are included,

and would result in a exponential mathematical relationship of the style aLb where

L is the total number of tonnes mined in the orebody, and b would be less than zero.

Sustaining capital for the mine, mill and on-site would consist mostly of replace-

ment for fixed and mobile equipment necessary for operational purposes. In this case,

it is logical to assume that this cost is related to the quantity of equipment required


to produce and thus to the production rate. A mine producing at twice the produc-

tion rate would need twice as many drills, shovels, and trucks and twice the number

of processing circuits at the mill. The relationship is obviously not so linear, and

the function would be modeled by the formula aQb, where Q is the production rate,

as discussed by O’Hara [48]. However, one must also consider the resources mined.

A deposit exploited at a given rate will obviously need more sustaining capital if

more resources are mined and the life is much longer. Therefore, one can expect the

relationship to be of the form aQbLc.

Variable costs Variable costs are incurred directly in the production process and

are constant independent of the quantity mined and processed. This cost includes the

ore and waste development of individual stopes, the actual stoping activities, the mine

services providing logistical support to the miners, and the milling and processing of

the ore at the plant.

In a Type A mine, if the whole resource is mined, large contiguous mining areas

are formed, where low-cost bulk-mining methods are used. As the level of production

decreases, these areas become smaller and smaller, requiring the adoption of expensive

selective mining methods in increasing proportions. Furthermore, if the grade across

the width of the vein varies, the stope widths become narrower, again forcing the use

of selective mining methods. Therefore, one can expect the variable cost of mining

to be inversely proportional to the level of production.

Associated to the mining method, the ore and waste development of individual

stopes is also affected. In general, more selective mining methods require more de-

velopment per tonne of ore mined. Because of this, the development unit cost is also

inversely proportional to the level of production.


Mine services include ancillary services (ventilation, compressed air, industrial

and potable water, waste water disposal, etc.), definition diamond drilling, mobile

equipment maintenance, and ore handling from the orepass to the mill (including

ramp and hoist). In this case, it would be reasonable to assume that the variable cost

is as a function of the quantity of service provided, and that if more ore is produced

on a daily basis, more equipment is required and more services are necessary. Most

ancillary services are provided by a number of large fixed pieces of equipment (com-

pressors, fans, pumps) and handle fixed quantity of air or water. For high production

rates, every piece of equipment would be functioning, and as the production rate

decreases, the individual pieces would be taken off-line one by one as their capacity is

no longer required. Therefore, the total cost of each service would be a step function

consisting of constant cost within the equipment operational range, and increasing to

the next step of the function at production rates where more pieces of equipment are

required. However, when all services are considered together, the production rates at

which the steps occur are different for each service, and the total cost curve can now

be approximated as a continuous function of the form aQb.

Similarly, in the mill, processes are carried out in a number of rod and ball mills,

cyclones, and flotation cells that are put in or taken out of operation as the production

rate varies. Individually each process is a step function but, combined together,

they can be considered as continuous. It could be argued also that there is some

relationship between the cost of processing and the level of production as consumption

of reagents can change significantly if the average feed grade is much higher or much

lower that for which the plant was designed. This aspect is not considered in this

thesis.


Fixed costs Fixed costs are incurred by the operation on a time basis, at a steady

rate every month or year, independently of the number of tonnes mined or units of

metal produced. These costs usually include the cost of general and area management,

engineering, geology, accounting, surface maintenance, human resources, logistics,

sales, and many more. Generally speaking, small variations in planned production

do not affect the fixed costs, but significant changes force management to review the

number of people required to service the operation. As in the case of mine services

variable costs, each cost center would be represented as a step function, but the sum

of all centers would give a continuous curve.

For the purpose of this model, fixed costs can be split into two groups, the mine

fixed costs and the mill and on-site costs, where the items directly related to mine

operations are regrouped under that banner, and all the rest, including general man-

agement, are regrouped in the mill and on-site category.

The Resulting Cost Model

The resulting model is quite different from those currently available in literature. It

is based on tonnes of ore mined and milled, and it takes into consideration the actual

subtleties related to the various cost centers considered in a mine budget. Table 3.1

briefly summarizes the differences between the new model and the classic ones.

3.1.3 The Revenue Model, Relationship with the Grade-Tonnage

Relationship

Revenues are also expressed as a function of tonnes of ore, and thus directly as a

function of the average grade of the ore. This grade is independent of the production


Table 3.1: Comparison of major design features of cost models

McIsaac Gray et al. Carlisle Lane

Units Tonnes ore Metal units Metal units Tonnes ore

Operating CostVariable Costs

Mining LevelRegular Development Level

Mine Services RateMilling Rate

Subtotal Rate Rate LevelFixed Costs

Mine RateMill & On-Site Rate

Subtotal Constant Constant ConstantCapital Costs

Capital Development LevelOther Mine Capital Rate & LevelMill & On-Site Rate & LevelTotal Capital Cost Rate Constant Constant

rate but is related to the level of production. The revenue component of Lane’s

equation (2.5), presented here as equation 3.1, serves as the starting point for the

development of the model derived in this thesis.

R = (p− k)xyg (3.1)

where

R is the revenues,

p is the price per unit of mineral,

k is the marketing cost,

x is the proportion of mineralized material classified as ore,


y is the mill recovery, and

g is the average grade of ore

Let’s declare φ as a constant representing the net smelter return factor, equal to

revenues generated by selling one unit of mineral from the mine to a smelter.

φ = (p− k)y (3.2)

and the revenues are now equal to

R = xgφ (3.3)

Based on this, the objective now is to develop this relationship as a function of

the level of production to then be able to estimate the average and marginal revenue

functions. Thus equation 3.3 becomes:

Rl = xglφ (3.4)

where

Rl is the revenues as a function of the level of production, and

gl is the average grade as a function of the level of production

The grade-tonnage curves

The relationship between the cut-off grade, tonnage above cut-off, and average grade

of the tonnes above cut-off are commonly represented by the grade-tonnage curves.


Grade

Ton

nes

Figure 3.1: Example of lognormal grade distribution

The construction is derived from the frequency distribution of grades within the

deposit, as shown in figure 3.1.

The level of production is the total tonnes of ore mined given a particular cut-off

grade given by the formula as:

Tgc = T0

∫ ∞

gc

f(x)dx (3.5)

where

gc is the cut-off grade,

Tgc is the tonnes of ore with grade greater than gc,

T0 is the total tonnes of resources in the deposit of production, and

f(x) is the frequency distribution function of rock of grade x in the deposit


Cut-Off Grade

Ton

nes

abov

e cu

t-of

f gr

ade

Ave

rage

gra

de o

f to

nnes

abo

ve c

ut-o

ff

Tonnes

Average Grade

Figure 3.2: Example of Grade-Tonnage curves

The metal contained in tonnes of ore Tgc is

Mgc = T0

∫ ∞

gc

xf(x)dx (3.6)

and the average grade of tonnes of ore Tgc is

g =Mgc

Tgc

(3.7)

and these relationships are usually presented in graphical form as in figure 3.2.

The grade-tonnage curves can easily be transformed in revenue-tonnage curves by

multiplying all grades by φ, the NSR factor, as seen in figure 3.3.

Formally, the total revenue function is equal to the metal contained multiplied by

the NSR factor.


Cut-Off NSR ($/tonne)

Res

ourc

es A

bove

Cut

-Off

NSR

(to

nnes

)

Ave

rage

Val

ue o

f R

esou

rces

Abo

ve C

ut-O

ff N

SR

($/t

onne

)

TonnesAverage NSR

Figure 3.3: Example of NSR-Tonnage curves

Rgc = φT0

∫ ∞

gc

xf(x)dx (3.8)

A direct relationship can be established between the NSR-tonnage curves and the

unit revenues as a function of the level of production.

The average revenue curve is thus

ARTgc=

Rgc

Tgc

(3.9)

As for the marginal revenue curve, note that

MRgc =dRgc

dgc

=d

dgc

(φT0

∫ ∞

gc

xf(x)dx) = −φT0gcf(gc) (3.10)

and that


dTgc

dgc

=d

dgc

(T0

∫ ∞

gc

f(x)dx) = −T0f(gc) (3.11)

Using the chain rule to combine these last two equations:

MR(Tgc) =dRgc

dTgc

=dRgc

dgc

× 1dTgc

dgc

=−φT0gcf(gc)

−T0f(gc)= φgc (3.12)

Therefore the marginal revenue as a function of the level of production is equal to

the cut-off grade multiplied the NSR factor. Based on this, the construction of the

unit revenue curves as a function of the level of production becomes very simple:

1. Build the Grade-Tonnage curves of the deposit (figure 3.2);

2. Convert to NSR-Tonnage curves by multiplying all grades by the NSR factor

constant (figure 3.3);

3. Shift the Tonnes above cut-off NSR from the y to the x-axis and shift the cut-off

NSR from the x to the y-axis; and

4. Positioning the average grade of tonnes above cut-off grade on the y-axis allows

calculation of the average grade as a function of the tonnes.

Figure 3.4 presents the resulting graph.


Level of Production (tonnes)

Uni

t re

venu

e ($

/ to

nne)

Marginal Revenue

Average Revenue

Figure 3.4: Unit revenue curves as a function of the level of production

3.2 Methodology to Determine the Production rate

and Cut-Off Grade under Conditions of Metal

Price Uncertainty

Now that the cost and revenue models are set up, the next step consists in developing

the procedure to solve under conditions of metal price uncertainty.

The general approach is inspired by the Brennan and Schwartz [7] model in which

the cash flows of a given mine plan are repeatedly evaluated under many metal price

conditions generated from equation 2.6 and a Monte Carlo simulation process. Monte

Carlo simulations are typically used in economic analysis to calculate the risks associ-

ated with a particular project; by combining the possible outcomes of many uncertain

variables, the probabilities of success can be estimated. It is not used for this purpose


in this model, as the generation of many possible outcomes is the feature of interest.

The purpose is to generate many combinations of possible yearly metal prices over

the life of a project, based on the same metal price stochastic models.

For each simulation, metal prices are forecast and the net present value function

is calculated as a function of production rates and production levels. This function

is convex and has an optimal solution corresponding to its apex. After each run, the

combination of rate and level of production with the maximum NPV is recorded, and

the process is repeated. This data is then compiled and analyzed. The coordinates

of the maxima found during each run are plotted in a scatter diagram with levels of

production on one axis and production rates on the other, and the density of plotted

points is calculated. This information is transformed into a bivariate probability

function, which is analyzed to determine its mode. The production rate and level

corresponding to this mode are chosen as the design parameters. In other words,

given the many possible outcomes of future metal prices, it is most probable that the

project NPV will be maximized if the mine is designed along these parameters.

3.2.1 Structure of the Computer Program PeaRL

In this section, the flow sheets of the program to solve the decision model are pre-

sented. The process is fairly simple, and consists mostly of a number of repetitive

subroutines. A general description of the process is provided in the following para-

graphs.

The general process consists of calculating the combination of production level

and production rate for which the net present value is maximized for a randomly-

generated set of metal prices, and this process is repeated two thousand times to


generate a graph where all combinations are plotted1.

In particular, for each generated set of metal prices, a three-dimensional function is

drawn showing the net present value as a function of production rate and production

level. The objective is to identify the maximum NPV combination. The process used

to identify this point is called the ”Method of Steepest Ascent”, using a modification

of the ”Golden Section Search” approach to solve it. See Winston [77] for the details

of the process. Generally, this approach is a systematic search process that allows

one to zoom in on the maximum point of a function by comparing the values of a

number of close points on the function, determining the highest gradient vector, and

repeating the process in an area further along that vector. This process is repeated

as often as necessary until the maximum point is found. The final steps of the search

are refined by reducing the distance intervals between the points.

The program is called PeaRL (Production Rate and Level) and the general flow

sheet is presented in figure 3.5; the two main subroutines Search and NPV are illus-

trated in figures 3.6 and 3.7.

List of symbols

The symbols that are used in the flow sheets are described below.

q is the production rate (tonnes per day),

Q is the production rate for which the Net Present Value is maximum for a given set

of metal prices,

l is the production level (tonnes),

1Trials were conducted for various numbers of repetition and 2,000 seemed like a reasonablecompromise between replication of results and process time


L is the production level for which the Net Present Value is maximum for a given set

of metal prices,

MP is the set of metal prices over the life of the mine ($/oz or $/lb),

t is time (years),

T is the mine life (years),

Delta is the degree of refinement of the search pattern in the Golden Search subrou-

tine. Delta increases from 1 to 3, and the size of the search pattern decreases

with each increase in Delta,

dl is the size of the search interval of the level of production as a function of Delta,

dq is the size of the search interval of the production rate as a function of Delta,

P(m,n) is the relative coordinate of production level and production rate during a

search step,

s is the search step counter used to track the progress of a search in the Golden

Search subroutine.

n is a counter

3.3 Summary

It was seen in this chapter how the general aspects of Gray’s, Carlisle’s, and Lane’s

models are adapted to conditions seen in the industry to formulate a new cost model

relating operating and capital costs to production rates and levels based on tonnes of


ProgramPeaRL

Do for n = 1 to 2000

Generate a set of metal prices MP

Run subroutine Search to find production level L and rate Q

yielding maximum NPV

Store L, Q, production and financial summaries

Next n

Plot graph of all L and Qcombinations

End

Figure 3.5: Flow sheet of program PeaRL


Subroutine Search

Calculate P(m, n) for m = {-1, 0, 1} and n = {-1, 0, 1}

where P(m, n) = (l+m*dl, q+n*dq)

Run Subroutine NPV for all P(m,n)Let (L,Q) be P(m,n) with Max NPV

l = 5,000,000 tonnesq = 2,000 tonnes per day

P(0,0) = (l, q)

Delta dl dq(tonnes) (tpd)

1 1,000,000 5002 250,000 2503 100,000 100

Transform coordinates of (L,Q)sinto starting coordinates for the

next search step s+1P(mL,nQ)s = P(-m, -n)s+1

(L,Q) = P(0,0)?

Delta = 3?

Do for Delta = 1 to 3

Set search step s = 1

If s>1,(L,Q)s = (L,Q)s-1?

Y

N

Y

N

Next Delta

Set dl, dq

Y

s = s+1

Maximum NPVis found at

(L,Q)

Return

N

Figure 3.6: Flow sheet of subroutine Search


Subroutine NPV

Calculate mine life T

Calculate annual revenues

Do for t = 1 to T

Calculate annual cash flow

Calculate net present value

Return

Calculate annual operatingand capital costs

Next t

Figure 3.7: Flow sheet of subroutine NPV


ore moved. This provides a tool to the mining engineer working at the mine site to

generate cost equations in a system related to the mine accounting structure.

It was then demonstrated how the grade-tonnage relationship is directly related

to the average and marginal revenue as a function of the level of production. This

tool will be very useful in the analysis of polymetallic ore distribution in chapter 4

and in the understanding of the solutions in chapter 5.

The procedure for determining the production rate and cut-off grade under con-

ditions of metal price uncertainty was then presented along with the structure of

the program PeaRL designed to carry out the procedure. A summary of the proce-

dures for the construction of the model and the execution of the model is listed in

appendix A.

With the tools in place to solve the problem, the process will be used in the next

chapter to determine the strategic parameters of an actual mining operation.

Chapter 4

Test on an Actual Mine

In this chapter, the model is tested on data obtained from an actual mine. A practical

example is based on the data accumulated at the mine described earlier in Chapter 1.

That mine is chosen because detailed cost accounting data was available. There are

six years of information from which to determine fixed and variable operating costs,

along with capital expenditures for production rates varying between 800 and 3 300

tonnes per day and various levels of production, from one hundred percent of the

deposit to almost nothing.

The practical example allows performing certain verifications that cannot be done

on a theoretical model. First, the construction of true micro-economic curves will help

identify the limitations of this exercise. The second advantage of solving a practical

example is that the methodology developed can be immediately validated.

In this chapter, the objective is to determine the design parameters for a life-of-

mine plan starting in 1996. As mentioned before, the mine at that time was scheduled

to work at 3,300 tonnes per day based on historically-high gold and copper price

forecasts. As the prices decreased in the following years, the plan had to be revised

82

CHAPTER 4. TEST ON AN ACTUAL MINE 83

downward as the reserves were shrinking, prompting many new plans with smaller

and smaller production rates. By conducting the validation work on that mine, it

will be possible to compare the results of the model with the actual outcomes.

The mine that so graciously provides the information for this work has asked

for anonymity. This thesis must provide sufficient information to give a sense to

the numbers being presented, while holding back details that could help identify the

operation. Also, information of confidential nature cannot be revealed. This includes,

but is not limited to, the details on capital expenditures, the smelting contracts, and

the financial statements. That information is necessary for the completion of this

work, but is presented in a very general way.

4.0.1 Description of the operation

The orebody consists of a large number of parallel echelon veins, varying in horizontal

thickness from one to 8 metres, and dipping between forty-five and ninety degrees.

These veins are spatially distributed over a volume of approximately two kilometres

along strike, by 600 metres in height, and 500 metres in thickness. As such, this

volume is divided into three operational areas, each with its own lateral and vertical

mine infrastructure, but all connected to a central ore haulage system. In these three

mines, a total of 15 veins are extracted economically, and are the only ones considered

in this thesis.

The mine economically exploits three metals: gold, copper, and silver, with the

first two generating more than ninety percent of all revenues. Copper is usually found

in enargite (copper arsenic sulphide), with grades reaching as high as 15%. Gold can

be found with enargite or in quartz associated with pyrite. The grade in enargite is


usually around 4 grams per tonne, but can average 20 grams per tonne in quartz,

with extremes of more than 3,500 grams per tonne in some veins.

The quality of the rock mass varies considerably throughout the deposit. Faults

are present between the veins as well as in the footwalls and hanging walls, with

thicknesses comparable to those of the veins, and many of them are directly on the

hanging wall of some veins. Their quality can be described as ranging from very

bad to bad, and most will readily cave into open stopes when exposed. The veins

themselves have rock masses varying from very bad in the upper part of the mine to

very good and excellent in the lower sectors. As a result of these conditions, stopes

in the upper sectors are developed in very bad ground and require very selective and

expensive mining methods, stopes in the central elevations require cut and fill mining,

and lower sectors are mined by long-hole.

The orebody is accessed by adits at many elevations. Inside, two main ramps

connect all the levels, and a winze allows hoisting the ore from the lower sectors to

the elevation of the mill. Many ore passes are laid out throughout the mine, reaching

a lower main transport level where the ore is trucked to the hoisting facilities. At the

mill elevation, the ore is transported by train through an adit. Although most of the

ore reaches the mill through the shaft, a certain proportion is trucked from isolated

stopes.

The mill is designed for copper sulphide flotation and roasting of the concentrate.

The concentrate resulting from flotation (green concentrate) contains high levels of

arsenic. The roasting process allows the separation of the arsenic from the copper,

as well as further concentration of the copper, thus adding significant value to the


product. The mill has a capacity of 3,500 tonnes per day and the roaster has an equiv-

alent capacity of 2,200 tonnes per day. Four concentrates are sold: copper calcine,

commercial-quality arsenic trioxide (As2O3), electrostatic precipitation (ESP) dust

with high levels of contaminants, and green concentrate when the roasters’ capacity

is exceeded.

4.0.2 Production profile

In order to relate costs and production rates, it is important to take a close look

at the production profile over the time period for which accounting data exists, and

establish domains over which costs can be grouped and analyzed separately.

Figure 4.1 shows the average daily actual and budgeted production rates on a

monthly basis from January 1996 to December 2000. The year 1996 was budgeted at

a rate of 2,500 tonnes per day (tpd), but the actual rate was lower in the first half

of the year and tended to be higher for the second half. In 1997, budget was set at

3,200 tonnes per day, but the first four months averaged 500 tonnes per day less, and

the production rate afterwards was ranging between 1,400 and 2,000 tpd as a result

of an important reevaluation of the mine plan. Afterwards, production went from

1,500 tpd in 1998 to 1,000 in 1999 and 2000. Over that three-year period, actual

rates closely followed the budget. One notable exception occurred in the second half

of 1999, when production rates could be maintained at 1,000 tpd even though the

budget was set at a much lower rate.


0

500

1,000

1,500

2,000

2,500

3,000

3,500

Jan-96 Jul-96 Jan-97 Jul-97 Jan-98 Jul-98 Jan-99 Jul-99 Jan-00 Jul-00 Jan-01

Month

Pro

duct

ion

rate

(to

nnes

per

day

)

Budget

Actual

Figure 4.1: Actual and budgeted production rates

4.0.3 Available information

The mathematical model can only be as good as the information used to construct

it. A great number of reports were made available to the author and include the

following:

• Long-term mine plans;

• Annual budgets;

• Production forecasts produced on a regular basis as updates to the budgets;

• Monthly accounting summaries providing detailed budgeted and actual spend-

ing by cost centre and account number;

• Monthly actual and budgeted development and production details broken down


by work place;

• Detailed listing of reserves and resources blocks;

• Mine plans and sections for each vein indicating existing development; and

• Mill and roaster flowsheet with capacities and metal recovery equations.

In brief, monthly and annual information was collected for a five-year period ranging

from 1996 to 2000, and even though some months are missing, more than 85% of all

data was recovered, leaving the author satisfied that enough information was gathered

to make the analysis significant.

4.1 Minable resources

The resource estimation is based on that compiled for the January 1996 official re-

serves. The database contains information for a great number of veins that may or

may not be included in the mine plans. As such, the following analysis of the resources

will limit itself to the 15 economic veins. For each block, the following information is

known:

• Vein;

• Mine in which it is located;

• Block reference number;

• Bottom elevation of the block;

• Section of the block corresponding to the centre;


• Category1 (proven, probable, possible)

• In-situ horizontal width in metres;

• In-situ tonnes;

• In-situ gold grade in grams per tonne;

• In-situ silver grade in grams per tonne; and

• In-situ copper grade in percent.

For sake of anonymity, the real names are not used here. However, since the mines

and veins are often referred to in this document, a nomenclature must be agreed upon.

Therefore, the three mines are called A, B, and C. Mine A contains six veins labeled

A1 to A6, Mine B also has six veins, labeled B1 to B6, and Mine C has three veins

labeled C1 to C3. Veins A1, A6, B4, B5, B6 and C3 are sulphide veins, generally

steeply dipping with horizontal widths greater than 3 metres. Veins A3, A4, A5, B1,

B2, B3, C1, and C2 are quartz veins, generally dipping around 60 degrees and width

between 1 and 4 metres. Vein A2 starts off near the top of the mine as a flat-dipping

quartz vein and changes into steeply dipping sulfide vein at around elevation 270.

At this stage, it is important to select the blocks with a sufficiently-high level of

confidence, i.e. the proven and probable blocks (2P reserves). For each of the veins

previously enumerated, the proven and probable blocks are drawn on a longitudinal

projection. With the idea that a certain level of continuity must be present for

mining to be practical, isolated blocks are identified and eliminated from the database.

1These reserve categories were determined before the current resource classification standardswere established.


Conversely, some 2P blocks can be put in production if some possible blocks are

located between them and are added to the planning reserves.

Next, the in-situ resource blocks are transformed into minable blocks by adjusting

each one by the expected dilution as a function of the mining method required for each

block. Originally, each block is adjusted to a minimum working width of 3 metres.

All blocks measuring less than that have internal dilution added to them. Then, each

block is located on its vein longitudinal projection and is assigned a mining method

and expected external dilution as per the available information in the mine plans. The

internal and external dilution criteria are listed in table 4.1. The resulting database

on which the model is built is now composed of minable resource blocks.

Table 4.1: Dilution factors

Mine method dilution minimum width (m.) external dilution rateA Long Hole low 3 15%

Long Hole high 3 25%Cut & Fill low 3 15%

Pillar recovery high 3 25%B Long Hole regular 3 20%

Cut & Fill low 3 15%Pillar recovery high 3 25%

C Long Hole low 3 12%Cut & Fill low 3 12%

As a result of these operations, minable resources total 15,700,000 tonnes aver-

aging 2.9 grams per tonne of gold and 3.5% copper. Summaries are presented in

tables 4.2, 4.3, and 4.4.


Table 4.2: Summary of minable resources and grades by vein

Mine Vein Tonnes Gold (g/t) Silver (g/t) Copper (%)

A A1 235,810 4.0 36 2.9A2 252,194 7.1 18 1.8A3 35,742 4.1 29 1.5A4 440,597 5.1 39 3.7A5 1,990,146 1.5 40 4.0A6 774,894 2.0 47 5.9

A Total 3,729,383 2.6 39 4.1B B1 1,351,186 1.6 45 4.5

B2 2,804,945 2.6 27 2.1B3 435,647 8.2 11 1.2B4 564,682 7.5 26 2.6B5 425,926 3.0 11 0.3B6 883,398 3.1 46 4.4

B Total 6,465,783 3.3 31 2.8C C1 470,601 3.9 15 0.7

C2 271,738 5.5 44 1.8C3 4,761,038 2.4 54 4.4

C Total 5,503,378 2.7 50 4.0Grand Total 15,698,543 2.9 40 3.5


Table 4.3: Resources by vein and category

categoryMine Vein Proven Probable Possible TotalA A1 178,284 1,026,788 146,114 1,351,186

A2 739,744 1,515,434 549,766 2,804,945A3 249,516 186,131 435,647A4 266,252 298,429 564,682A5 73,436 352,489 425,926A6 124,493 758,905 883,398

A Total 1,631,726 4,138,176 695,880 6,465,783B B1 69,933 89,312 76,565 235,810

B2 75,282 170,482 6,430 252,194B3 7,414 28,328 35,742B4 328,063 112,534 440,597B5 237,426 1,729,306 23,413 1,990,146B6 373,053 280,685 121,156 774,894

B Total 1,083,758 2,389,734 255,891 3,729,383C C1 326,901 143,701 470,601

C2 109,723 162,016 271,738C3 994,587 2,737,475 1,028,976 4,761,038

C Total 1,104,310 3,226,391 1,172,677 5,503,378Grand Total 3,819,794 9,754,300 2,124,449 15,698,543

(tonnes)


Table 4.4: Resources by vein and mining method

Mining MethodMine Vein Cut & Fill Longhole Pillar Recovery TotalA A1 55,307 1,235,542 60,337 1,351,186

A2 736,803 1,832,143 235,998 2,804,945A3 380,488 55,159 435,647A4 489,682 74,999 564,682A5 425,926 425,926A6 75,343 635,466 172,589 883,398

A Total 2,163,548 3,703,150 599,084 6,465,783B B1 230,501 5,309 235,810

B2 217,010 35,184 252,194B3 35,742 35,742B4 153,770 185,973 100,855 440,597B5 20,549 1,936,299 33,297 1,990,146B6 102,132 602,254 70,508 774,894

B Total 759,703 2,724,526 245,153 3,729,383C C1 470,601 470,601

C2 271,738 271,738C3 4,406,901 354,138 4,761,038

C Total 5,149,240 354,138 5,503,378Grand Total 2,923,252 11,576,917 1,198,375 15,698,543

(tonnes)


4.1.1 Determination of the primary metal

The next step consists of constructing the grade-tonnage curves of the deposit in order

to determine the relationship between cut-off grades and levels of production. In a

deposit producing one metal or in a polymetallic orebody with one main economic

metal, the problem is fairly simple. However in this deposit, two metals, gold and

copper, contribute almost equally to the revenues of the mine2. In a case where metal

prices are given, this problem would be resolved by calculating the net smelter return

of each resource block and then working with NSR-tonnage curves, and an example

of this can be found in Baird and Satchwell [3]. However, in this thesis, the main

assumption is that metal prices are stochastic, and thus the value of every block

changes with each simulation as well as the relative ranking of each block. Therefore,

it is difficult to define which metal should be used to construct the curves.

As a main consideration, the use of one metal as primary reference must not

compromise the recovery of the secondary metal. In other words, high-grade gold

blocks must not be left behind when the copper cut-off is low, and vice-versa.

Starting with gold as the primary metal, a standard grade-tonnage curve is built

and presented in figure 4.2. Based on the work in chapter 3 for the development of

the average and marginal revenues as a function of the level of production, the same

development used for calculating the marginal revenue function in equation 3.12 can

be used for calculating the marginal metal grades, as seen in figure 4.3. The grade-

tonnage curves show that for low gold cut-off grade, the average copper grade is

between 3.5 and 4 percent, but it is impossible to deduce more information. Using

figure 4.3, it becomes evident that the marginal copper grade decreases to zero as

2The revenue model will demonstrate that in an upcoming section of this chapter.


0

2

4

6

8

10

12

14

16

0 1 2 3 4 5 6 7 8

Gold cut-off grade (g/t)

Res

ourc

e ab

ove

cut-

off

grad

e(m

illio

n to

nnes

)

0

3

6

9

12

15

18

21

24

Average gold grade (g/t)

Average copper grade (%)

Resources above cut-off

Ave

rage

gra

de o

f to

nnes

abo

ve c

ut-o

ff g

rade

(g/

t A

u, %

Cu)

Figure 4.2: Grade-tonnage curves based on gold grades

the level of production tends towards 16 million tonnes and the gold cut-off grade

reaches zero. From this, it can be deduced that low copper grades are associated with

resource blocks with low gold grades.

The same comparison can be made by setting copper as the primary metal. The

resulting graphs are in figures 4.4 and 4.5. In this case, the marginal gold grade has

a tendency to increase with high levels of production. This indicates that gold could

be left behind if the copper cut-off grade is low, thus making copper a bad candidate

as the primary metal.

Further analysis is required to confirm these observations and to quantify the

possible loss of metal production in each case. The first step is based on the ob-

servation of the scatter diagrams showing the pairings of gold and copper grades of

resource blocks. In figure 4.6, pairings are shown such that the analysis can be concen-

trated on resource blocks with low copper grades. The figure clearly shows that there


0

2

4

6

8

10

0 2 4 6 8 10 12 14 16

Gra

des

(g/t

Au,

% C

u)

AG

MG

AC

MC

Level of production = tonnes above cut-off grade(million tonnes)

AG = average gold grade (g/t)MG = marginal gold grade (g/t) = gold cut-off grade (g/t)AC = average copper grade (%)MC = marginal copper grade (%)

AG

MG

AC

MC

Figure 4.3: Production level grades as a function of gold

0

2

4

6

8

10

12

14

16

0 1 2 3 4 5 6 7 8 9 10

Cut-Off Copper Grade (%)

Res

ourc

e ab

ove

cut-

off

grad

e (m

illio

n to

nnes

)

0

3

6

9

12

Average gold grade (g/t)

Average copper grade (%)

Resources above cut-off

Ave

rage

gra

de o

f to

nnes

abo

ve c

ut-o

ff g

rade

(g/

t A

u, %

Cu)

Figure 4.4: Grade-tonnage curves based on copper grades


0

2

4

6

8

10

12

0 2 4 6 8 10 12 14 16

Level of production (million tonnes)

Gra

de (

g/t

Au,

% C

u)

AG

MG

AC

MC

AC = average copper grade (%)MC = marginal copper grade (%)AG = average gold grade (g/t)MG = marginal gold grade (g/t)

AG

MG

AC

MC

Figure 4.5: Production level grades as a function of copper

are resource blocks with very high gold grades independently of the blocks’ copper

grades, confirming the previous observation. The corresponding graph concentrating

on blocks with low gold grades (figure 4.7) also supports the observations made in

figure 4.3, as the number of high-grade copper blocks decreases along with the gold

grade.

The final step is to quantify the amount of high-grade secondary metal tied into

low-grade primary blocks. Defining what constitutes as a high grade is somewhat

arbitrary. For the purpose of this exercise, high-grade is defined as the minimum grade

required for a resource block to be economical when only one metal is considered.

Then again, this definition implies that the revenues from the metal sales are known,

implying that the metal price is known. Since that is not the case in this problem,

arbitrary numbers based on the experience of the mine operator are chosen. Therefore,

the gold high-grade is defined as any grade greater or equal to 10 grams per tonne,


0

5

10

15

20

25

30

35

40

45

50

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

Copper Grade (%)

Gol

d G

rade

(%

)

Figure 4.6: Metal grade scatter plot for low copper grades

0

3

6

9

12

15

0 1 2 3 4 5 6

Gold Grade (g/t)

Cop

per

Gra

de (

%)

Figure 4.7: Metal grade scatter plot for low gold grades


and copper high-grade is equal to 8.5%3. Based on this, two graphs are compiled and

presented in figures 4.8 and 4.9.

Figure 4.8 demonstrates well that there are no high-grade copper resources for

a gold cut-off grade less than 2 grams per tonne. Within the following gold grade

interval, from 2 to 3 g/t, some high-grade copper starts to appear. These resource

blocks correspond to 1.5% of the total resource tonnage, and contain 1.3% of all the

gold and 4.6% of all the copper. These percentages decrease as the the gold cut-off

increases. In figure 4.9, the picture is completely different. Within the copper grade

cut-off interval ranging from 0 to 1%, high-grade gold resource blocks count for 1.3%

of the tonnes but almost 8% of all gold contained in the orebody. More than 6% of

all the gold is found in the next interval.

The consequences of this are significant. If copper were to be chosen as the primary

metal, large quantities of gold contained in economic resource blocks would be lost.

The situation is the opposite if gold were to be the primary metal. Based on these

observations, gold is chosen as the primary metal. The relationship between cut-off

grades and levels of production is represented by the marginal gold grade curve in

figure 4.3, and, as a reference, in table 4.5.

4.2 Production and cash flow model

In this section, the objective is to build the equations that will be used in the simu-

lation. These equations will be based on actual and budgeted production, revenues,

and costs gathered at the mine site over a period of many years, during which time,

3This number seems ridiculously high, but it is the result on a combination of factors includinglow smelter payback for concentrates containing arsenic as a deleterious element and high operatingcosts


0%

1%

2%

3%

4%

5%

0 to 1 1 to 2 2 to 3 3 to 4 4 to 5

Gold cut-off grade intervals (g/t)

Per

cent

age

cont

aine

d in

inte

rval

% resource tonnes

% total gold resources

% total copper resources

Figure 4.8: High-grade copper resources associated with low-grade gold blocks

0%

1%

2%

3%

4%

5%

6%

7%

8%

0 to 1 1 to 2 2 to 3 3 to 4 4 to 5 5 to 6

Copper cut-off grade intervals (%)

Per

cent

age

cont

aine

d in

inte

rval

% resource tonnes

% total gold resources

% total copper resources

Figure 4.9: High-grade gold resources associated with low-grade copper blocks


Table 4.5: Gold cut-off grades and levels of production

Gold Level Referredcut-off (tonnes) to as5.59 1,000,000 1 mt4.05 2,000,000 2 mt3.32 3,000,000 3 mt2.84 4,000,000 4 mt2.58 5,000,000 5 mt2.32 6,000,000 6 mt2.17 7,000,000 7 mt2.07 8,000,000 8 mt1.91 9,000,000 9 mt1.78 10,000,000 10 mt1.57 11,000,000 11 mt1.33 12,000,000 12 mt1.08 13,000,000 13 mt0.72 14,000,000 14 mt0.40 15,000,000 15 mt0.00 15,698,543 16 mt

the mine experienced many variations in production rates. It is believed that this

wide range of data can give a good estimate of the actual situation.

All the production and cash flow information that is required to run the model

and that is generated by it, is summarized in tables 4.6 and 4.7. To aid the reader,

the table also indicates the section of this chapter in which each item is discussed.

4.2.1 Revenue model

This model mimics the actual flowsheet of the mill and roaster complex, presented

in figure 4.10. The mill has a capacity of 3,500 tonnes of ore per day, where the ore

is crushed and ground and floated to produce a green concentrate containing copper,

gold, silver, and arsenic. The green concentrate is sent to a roaster with the equivalent


Table 4.6: Production and development output

Item Section where item is discussed

Production and DevelopmentResource (tonnes) Simulation input variableAverage grades Gold (g/t) see Section 4.1.1

Copper (%)Production Rate (tonnes per day) Simulation input variableDevelopment (metres)

Regular see Section 4.2.5Deferred see Section 4.2.2Total Development

Production (tonnes) see Section 4.2.5Cut and FillLongholePillar RecoveryOre DevelopmentTotal production

Mill recovery (%) see Section 4.2.1GoldCopper

Metal Recovered Gold (’000 oz) see Section 4.2.1Copper (’000,000 lbs)


Table 4.7: Cash flow and indicators output

Item Section where item is discussed

Cash flow (MM$)Revenues Gold Sales

By-product creditsTotal revenues see Section 4.2.1

Operating costs Variable mining see Section 4.2.5Regular development see Section 4.2.5Mine services see Section 4.2.6Variable milling see Section 4.2.7Subtotal variable

Fixed mine see Section 4.2.8Fixed mill see Section 4.2.8Subtotal fixed

Total Operating

Capital costs Deferred Development see Section 4.2.2Mine sustaining see Section 4.2.3Mill sustaining see Section 4.2.4Total Capital

Cash FlowNPV @5%

Indicators Net Smelter Return ($/t)Operating cost ($/t)Operating margin ($/t)

Operating cost ($/oz)


Mine Mill RoasterOre

Tailings

Green Concentrate

Sales

Calcine

ESP Dust

As2O3

Figure 4.10: Mineral processing general flowchart

capacity of 2,200 tonnes of ore per day. All unprocessed green concentrate is sold to

smelters. The roasting process generates three salable products, a copper calcine

containing very little arsenic, a high-quality arsenic trioxide concentrate, and small

quantities of electrostatic precipitation dust containing very high levels of arsenic

contaminants.

Revenues, or Net Smelter Return (NSR), correspond to the net value of the ore

once physical losses in the mill and smelting charges are taken into consideration.

Goldie and Tredger [27] provided examples of typical calculations for various types

of metal concentrates. As part of the calculations, NSR factors for each metal are

also estimated. These factors constitute very simple indicators of the relative value of

each metal, allowing for a rapid comparison as feed grades and metal prices change.

See Lafleur [36] for sample calculations.

Physical losses occur in the flotation of the ore and in the roasting of the green

concentrate, where typical recoveries are in the order of 95% for copper and 90% for

gold, with actual recoveries fluctuating as a function of the feed grades.

The general contract parameters of each concentrate sale is presented in table 4.8.

As a general rule for the design of this model, the characteristics of the mine’s actual

smelter contracts are respected, but the real numbers used in the contracts are not


quoted. Instead, the numbers used are derived from these contracts and from spot

contract numbers published regularly in the ”Mining Journal” magazine, taking into

consideration the variations to key elements as the prices of copper and gold changed

over the years. The factors included in the Net Smelter Return calculations include:

• Metal deductions (D);

• Treatment charges (TC);

• Price participation (PP);

• Transport, loading and representation (TLR);

• Penalties for content of contaminants (PC); and

• Refining charges (RC).

Concentrate charges Treatment charges, Loading, and penalties are charged ex-

clusively to the copper, and not to gold and silver. The reasoning behind this is that

the mill produces and sells copper concentrates, and the aforementioned charges are

applied to the concentrates independently of the quantity of precious metal contained

in the concentrates. For this reason, only metal deductions and refining charges are

applied to gold and silver.

Price participation Price participation is the mechanism through which the treat-

ment charge is adjusted for copper price variations. In this exercise, the base metal

copper price is $1.00 per pound, and any variation is charged at a rate equivalent

to 10% of the difference between the current and the base prices. That way, copper

prices lower than $1.00 result in a decrease in total charges and vice-versa.


Table 4.8: Smelter contracts

Green Concentrate Copper Gold Silver

Metal deduction 1.15 unit 4% 10%Treatment Charge $150 per tonne of concentrate nil nilPrice Participation Payable pounds of copper per tonne of

concentrate multiplied by 10% of the difference between current price and $1.00

nil nil

Transport, Loading,& Representation

$45 / tonne concentrate nil nil

Penalties $60 / tonne concentrate nil nilRefining Charge $0.10 per pound for base copper price of

$1.00 per pound. Charge varies proportionally with the price of copper.

$7.50 per ounce of gold

$0.37 per ounce of silver

Calcine Copper Gold Silver



nil nil







ESP Dust Copper Gold Silver



nil nil








Silver sales Since silver revenues count for a very small proportion of total revenues,

the silver price is kept constant at $5.00 per ounce.

Arsenic trioxide Arsenic trioxide sales calculations are simplified in this model,

assuming the sales price per tonne of concentrate includes all transport and transfor-

mation costs, and this price is kept constant independently of all other metal prices.

For calculation purposes, all As2O3 credits are applied to copper since the arsenic is

mineralogically associated with it.

Value-added of roasting The sales of green concentrate sent to the roaster and

afterwards sold as three separate concentrates is treated as one revenue stream, re-

ferred to as calcine. This simplification makes it easier to compare the value added

of roasting. In figure 4.11, one can readily compare the NSR factors of copper sold

as a green concentrate or as a calcine at different copper prices. In this example, the

copper grade of the ore fed to the mill is equal to 3.50%, and the roasting adds $6.70

per percent of copper or approximately $0.30 per pound of copper. The effect on gold

price is negligible as roasting does not affect the quality of the gold sold.

4.2.2 Capital development

Capital (deferred) development is the development required to extend the mine infras-

tructure in order to put new sectors in production and to haul the ore to surface. This

includes extensions to the main ramp, development of transport drifts and ventilation

raises to bring fresh air to the new work levels.

The procedure to determine the relationship between the metres to develop and the

level of production is the following: first, for each mine, draw a longitudinal projection


-5.00

0.00

5.00

10.00

15.00

20.00

25.00

30.00

35.00

0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25

Copper price ($/lb)

NS

R F

acto

r ($

/ %

)

Calcine

Green concentrate

Figure 4.11: Value-added of roasting copper concentrate

showing the existing infrastructure; then draw the position of every resource block

included in the level of production equal 1 million tonnes, corresponding to blocks

with a gold grade equal to or greater than 5.59 grams per tonne, and determine

the metres of ramp, lateral development and ventilation raise required to put all the

blocks in production. Repeat the process for increments of levels of production of 1

million tonnes until the whole resource is examined. The figures used to calculate

these numbers are found in Appendix B.

The unit cost per metre of development is calculated from information gathered in

the production and accounting records dating from 1997 to 1999. Over that period,

7,068 metres of capital development was done for a total cost of $8,466,331, equivalent

to an average cost of $1,198 per metre. Over that period, capital development was


0

5

10

15

20

25

30

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17


Cap

ital

Dev

elop

men

t co

st (

mill

ion

$)

Data Regression

CD = -0.0015 l4 + 0.0526 l

3 -0.647 l

2 +4.7781 l

R2 = 0.9974

Figure 4.12: Capital development as a function of level of production

done by mine contractors; the cost included direct mining costs as well as supervi-

sion, mobile equipment maintenance and provision for replacement, fixed costs and

overhead.

A table summarizing the information is in table 4.9 and in figure 4.12.The equation

used for modeling capital development costs as a function of the level of production

is:

CD = −0.0015× l4 + 0.0526× l3 − 0.647× l2 + 4.7781× l (4.1)

where

CD is the capital development cost in million dollars; and

l is the level of production in million tonnes.


Tab

le4.

9:C

apit

aldev

elop

men

tsu

mm

ary

Leve

l of p

rodu

ctio

n (m

illio

n to

nnes

)1

23

45

67

89

1011

1213

1415

16G

old

cut-o

ff (g

/t)5.

604.

053.

332.

842.

582.

322.

172.

071.

901.

781.

581.

331.

08

0.72

0.40

0.00

Late

ral D

evel

opm

ent

Min

e A

1,09

01,

958

2,47

33,

232

3,40

33,

438

3,87

53,

875

3,90

04,

076

4,12

64,

541

4,75

35,

831

6,45

76,

595

Min

e B

8590

115

270

365

375

450

550

1,09

52,

045

2,94

53,

170

3,5

203,

845

4,02

04,

095

Min

e C

1,98

63,

272

3,80

04,

102

4,57

25,

247

6,38

76,

787

7,34

07,

465

7,84

78,

063

8,32

18,

696

8,77

68,

776

subt

otal

3,16

15,

320

6,38

77,

604

8,34

09,

060

10,7

1211

,212

12,

335

13,5

8514

,917

15,7

7416

,594

18,3

7119

,253

19,4

65

Ram

p D

evel

opm

ent

Min

es A

& B

020

080

01,

200

1,20

01,

200

1,20

01,

200

1,93

32,

267

2,26

72,

267

2,26

72,

267

2,26

72,

267

Min

e C

080

080

080

01,

200

1,40

01,

400

1,40

01,

400

1,40

01,

4001

,400

1,40

01,

400

1,40

01,

400

subt

otal

01,

000

1,60

02,

000

2,40

02,

600

2,60

02,

600

3,33

33,

667

3,66

73,

667

3,66

73,

667

3,66

73,

667

Ven

tilat

ion

Min

es A

& B

030

120

180

180

180

180

180

290

340

340

340

340

340

340

340

Min

e C

012

012

012

018

021

021

021

021

021

021

021

021

021

021

021

0su

btot

al0

150

240

300

360

390

390

390

500

550

550

550

550

550

550

550

Tota

l (m

etre

s)3,

161

6,47

08,

227

9,90

411

,100

12,0

5013

,70

214

,202

16,1

6817

,802

19,1

3419

,990

20,8

1022

,588

23,4

6923

,682

Uni

t Cos

t ($/

m)

1,19

81,

198

1,19

81,

198

1,19

81,

198

1,19

81

,198

1,19

81,

198

1,19

81,

198

1,19

81,

198

1,19

81,

198

Cos

t ('0

00 $

)3,

786

7,75

09,

856

11,8

6413

,298

14,4

3516

,41

517

,014

19,3

6921

,326

22,9

2223

,948

24,9

3127

,060

28,1

1628

,371

Ave

rage

cos

t per

tonn

e ($

/t)3.

823.

943.

302.

972.

642.

392.

342.

132.

132.

142.

092.

001.

92

1.93

1.87

1.81

(met

res)


4.2.3 Mine sustaining capital costs

The mine sustaining capital is based on life-of-mine plans developed yearly between

the years 1994 and 2000. These plans cover levels of production between 1,026,000 and

14,224,000 tonnes and production rates between 937 and 3,897 tonnes per day. Life-of-

mine plans are used because they provide an estimate of all foreseeable expenditures

required over the remaining life of the project. Actual numbers (as compared to

forecasted) are never really available because conditions change constantly over time,

especially when the production rate of the mine changes constantly.

Mine sustaining capital includes mobile equipment, fans, pumps, electrical and

communications systems, equipment for the underground maintenance shops, and

auxiliary underground development.

As discussed in chapter 3, a power function is calculated as a function of the rate

and level of production and the resulting equation is

MSC = 7.09× 10−6 × q1.634 × l0.886 (4.2)

where

MSC is the mine sustaining capital cost in million dollars;

q is the production rate in tonnes per day; and


with a correlation coefficient of 0.88.

It is surprising to notice that the production rate power factor is greater than 1.

This implies that the cost increases exponentially as the production rate increases,


which would not make sense under normal circumstances. This high factor is ex-

plained in this case by the fact that the life-of-mine plans were constructed for a mine

already in production. Every time a new plan was built, economic conditions had

worsened, and the production rate was smaller. A mine reducing its production rate

winds up with some spare capacity and under-utilized equipment. This results in

lower replacement costs than if the mine were built from scratch and did not possess

any equipment to start with.

4.2.4 Mill and On-site sustaining capital costs

The same data is used for the mill and sustaining capital cost. It includes mechanical

and electrical supplies for the mill, roaster, assay laboratory, central maintenance and

other surface installations, mobile equipment for surface work, tailings dam expan-

sions, and other purchases for management, accounting and human resources.

The resulting equation is as follows:

OSC = 9.19× 10−10 × q3.04 × l−0.04 (4.3)

where

OSC is the mill and on-site sustaining capital cost in million dollars;

q is the production rate in tonnes per day; and


with a correlation coefficient of 0.92.

As in the case of the mine sustaining capital, the production rate power factor

is very high, and can be explained by the same reasoning. The level of production


factor is very close to zero, and the analysis of the regression reveals that there is a

very high probability that that factor is equal to zero. That would imply that the

mill sustaining cost is really only a function of the rate of production and that the

life of the mine has little influence on the cost. That can be understood by the fact

that the very high cost items in a mill have very long lives and might actually have

useful life left over once the mine is closed.

4.2.5 Variable mining and regular development costs

The cost of mining varies as a function of the level of production. As a starting point

for this exercise, each resource block in the database is assigned a mining method

based on the continuity of the ore and the quality of the rock mass of the ore, footwall

and hanging wall where it is located. The blocks located in the older sectors of the

mine generally have lower dips, narrower horizontal widths, and bad rock mass quality

and are assigned with cut and fill or pillar recovery, based on historical considerations.

At depth, the dip tends to be steeper, the width increases, and the rock mass quality

tends to increase. In these sectors, long hole mining is assigned.

This relationship between the mining methods and the levels and production can

be clearly understood by looking at figure 4.13. For low levels of production, the

high-grade ore is primarily located in the upper portions of the orebody where min-

ing conditions are more complicated, resulting in a higher proportion of ore coming

from pillar recovery and cut and fill. As the cut-off grade decreases and the level of

production increases, more ore is located in better ground and long hole becomes the

principal method of extraction.

Variable mining costs include the direct costs of ore and waste development of the


0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16


Per

cent

age

of t

onne

s m

ined

per

min

ing

met

hod Pillar Recovery

Long Hole

Cut & Fill

Figure 4.13: Proportion of tonnes mined per mining methods as a function of thelevel of production

drifts around and in the stopes, and drilling, blasting, and mucking the ore out of

it. The costs also include those of all the activities indirectly related to production

such as ground support, stope ventilation, and other services. Regular development

includes the costs of developing through waste from the capital infrastructure to the

stopes and the ore development required to put the stope in production. All costs

include labour, materials, contracts, and others.

In order to calculate the variable mining cost function, data had to be extracted

from different sources. The cost centers were set up such that development and

stoping expenditures could be tracked for each mining method in each mine. The

actual development metres and production tonnes were compiled on a monthly basis,

with each heading and stope clearly identified. From this, the overall cost per metre

development and the cost per tonne of ore can be calculated for each mining method.


Table 4.10: Variable mining unit costs

Activity Cost (MM$)

Development 23,888 metres 18.4 772 $/metre

Cut & fill 130,040 tonnes 2.6 19.96 $/tonne

Longhole 1,209,251 tonnes 9.9 8.22 $/tonne

Pillar recovery 256,496 tonnes 5.6 21.66 $/tonne

Actual Unit Cost

These costs are summarized in table 4.10. This cost was based on information covering

45 months.

However, this information is not enough to determine the variable mining cost as

a a function of the production level. This function must take into consideration the

following points:

Metres of regular development per tonne of ore: It is important to know how

many metres of development are required to prepare one tonne of ore. It follows

from this that each mining method has a different proportion, and that further

variations occur between the veins as the quality of the rock mass and the

widths of the veins have an influence on the size of the stopes.

Percentage of ore coming from development: For any given stope, part of the

ore sent to the mill comes from the development done in the ore. Again, this

factor varies with the mining method and the vein.

Tonnes per metre of ore development: For each metre of development driven in

ore, a certain number of tonnes of ore are generated, and this quantity varies

with the method and vein.

For each point, constants are calculated for all combinations of mining method

and vein. From them, it now becomes possible to calculate for each resource block


in the database the expected metres of ore and waste regular development required

to develop it, the tonnes of ore expected to be generated by the ore development,

and the tonnes of ore coming from stoping activities. When looking at each resource

block individually, these numbers are not truly representative, but their sums over

large numbers of resource blocks will tend to give a good picture of what can be

expected.

Once combined with the information presented in figure 4.13, it becomes possible

to calculate a number of level of production functions:

1. A breakdown of production tonnes, split into the three mining methods and the

ore development;

2. The metres of regular development;

3. The variable cost of stoping activities (variable mining cost); and

4. The variable cost of regular development.

The development of the indicators is given in appendix C and the results and

functions are summarized here.

Regarding sources of production as a function of the level of production, figure 4.13

must be modified to include the breakdown of development and stoping. The result

is shown in figure 4.14, and the equations are listed below.

CFT = −0.00015× L4 + 0.00563× L3 − 0.0731× L2 + 0.490× L (4.4)


0%

20%

40%

60%

80%

100%

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Level of Production (million tonnes)

Rel

ativ

e so

urc

e o

f p

rod

uct

ion

Pillar Recovery

Long Hole

Cut & Fill

Ore Development

Figure 4.14: Proportion of tonnes mined from all sources as a function of the level ofproduction

LHT = −0.0018× L3 + 0.050× L2 + 0.218× L (4.5)

PRT = −8.285×10−6×L5 +0.000311×L4−0.00343×L3 +0.000588×L2 +0.207×L

(4.6)

ODT = 4.59× 10−5 × L3 + 0.0016× L2 + 0.162× L (4.7)

where

CFT are cut and fill tonnes;

LHT are longhole tonnes;


0

20

40

60

80

100

120

140

160

180

0 2 4 6 8 10 12 14 16


Dev

elo

pm

ent

(th

ou

san

d m

etre

s)

Operating Development

Capital Development

Figure 4.15: Regular and capital development as a function of the level of production

PRT are pillar recovery tonnes;

ODT are ore development tonnes; and

all tonnages are in million tonnes.

The metres of regular development as a function of the level of production are

graphed in figure 4.15 alongside the capital development requirements for sake of

comparison. The regular development function is as follows:

RD = 0.0195× L3 − 0.582× L2 + 14.714× L (4.8)

where RD is regular development, expressed in thousand metres.

The variable mining operating and regular development costs as a function of

the level of production can now be calculated based on the information in the new


0

20

40

60

80

100

120

140

160

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Production Level (million tonnes)

Var

iabl

e M

inin

g C

ost

(mill

ion

$)

Figure 4.16: Variable mining cost as a function of the level of production

database. The variable mining cost equation is graphed in figure 4.16.

V MO = 0.0227× L3 − 0.696× L2 + 15.031× L (4.9)

RDC = RD × 772÷ 1000 (4.10)

where V MO is the variable mining operating and RDC is the regular development

cost in million dollars.

4.2.6 Mine services costs

Mine services include ancillary (ventilation, drainage, etc.), diamond drilling, main-

tenance, and transport. Actual monthly costs for the period from 1996 to 2000 is

divided by the actual production to get average monthly costs on which a regression


0.00

5.00

10.00

15.00

20.00

25.00

30.00

35.00

40.00

45.00

50.00

0 500 1,000 1,500 2,000 2,500 3,000 3,500

Production Rate (tonnes per day)

Cos

t ($

per

ton

ne)

Figure 4.17: Mine services unit cost as a function of the production rate

is done, which can be seen in figure 4.17.

MSUC = 2104×Q−0.6199 (4.11)

where MSUC is the mine services unit cost in dollars per tonne.

4.2.7 Variable mill costs

The mill variable costs include direct milling and roasting costs, as well as services.

Costs are calculated from the same information as for the mine services. The data is

graphed in figure 4.18.

MUC = 328.23×Q−0.3356 (4.12)

where MUC is the mill unit cost per tonne.


0.00

5.00

10.00

15.00

20.00

25.00

30.00

35.00

40.00

45.00

50.00

0 500 1,000 1,500 2,000 2,500 3,000 3,500 4,000

Production rate (tonnes per day)

Mill

uni

t co

st (

dolla

rs p

er t

onne

)

Figure 4.18: Mill variable unit cost as a function of the production rate

4.2.8 Fixed costs

Fixed costs are split into two, the mine fixed and the mill and on-site fixed costs. The

mine costs include the mine administration and planning of services. The mill costs

include the administration of the mill and roaster. On-site costs are more general and

include management, accounting, human resources, engineering, geology, purchasing,

legal, etc., as well as surface maintenance.

The data used to determine the relationship between costs and the production

rate was based on actual costs over periods of six months. The reasons for this were:

• Certain costs, such as labour, occur on a monthly basis. However, some equip-

ment purchases and contracted work occur on a less regular basis. It was judged

that monthly data would show too much variance and that six-month averages

would tend to include all relevant costs; and


y = 204.32x + 21403

R2 = 0.8218

0

100

200

300

400

500

600

700

0 500 1,000 1,500 2,000 2,500 3,000 3,500


Ave

rage

cos

t ('0

00 d

olla

rs p

er m

onth

)

Figure 4.19: Mine monthly fixed cost as a function of the production rate

• As metal prices changed, budgets had been invalidated and new forecasts were

built. Those were usually put into action in the second half of the calendar

year, leading to changes in production and costs.

Therefore, the average monthly fixed costs are calculated for data ranging from

1996 to 2000, split in the two halfs of the calendar year, with the data and regressions

shown in figures 4.19 and 4.20. To calculate the cost over the life of the mine, these

costs are multiplied by the mine life expressed in months.

4.2.9 Cost curves analysis

Figures 4.21 and 4.22 show the total costs over the mine life and the average unit

cost per tonne mined for every combination of production rates and levels. The first

figure gives a good impression of the levels of expenditures required, and shows how

costs escalate rapidly at low production rates, a result of increasing fixed costs.


y = 687.2x + 314507

R2 = 0.824

0

500

1,000

1,500

2,000

2,500

3,000

0 500 1000 1500 2000 2500 3000 3500


Ave

rage

cos

t ('0

00 d

olla

rs p

er m

onth

)

Figure 4.20: Mill and On-Site monthly fixed cost as a function of the production rate

The unit costs curves are particularly interesting. It would be most improbable

that the mine would function at a rate of 500 tonnes per day as the unit cost would

be greater than $155 per tonne, requiring either very high metal prices for very long

periods, or more practically the extraction of the very highest grades available in

the deposit. Options at 1,000 tonnes per day seem more possible as costs of $120

per tonne can more easily be met by high-grading the deposit, and this condition

would be best if metal prices are low. Between production rates of 2,000 and 3,500

tonnes per day, costs do not vary by more than 10% and it could be conceivable that

optimum operating conditions might exist within that range under other metal price

conditions.


0

500

1,000

1,500

2,000

2,500

0 500 1000 1500 2000 2500 3000 3500 4000


Co

sts

(MM

$)

1 Mt2 Mt

3 Mt4 Mt

5 Mt6 Mt

8 Mt

10 Mt

12 Mt

15 Mt

Level of Production

Figure 4.21: Total costs over the life of the project for combinations of rates and levelsof production

80

90

100

110

120

130

140

150

160

170

180

0 2 4 6 8 10 12 14 16


To

tal u

nit

co

st (

$/to

nn

e)

500 tpd

1000 tpd

1500 tpd

2000 tpd

3500 tpd

Figure 4.22: Unit costs over the life of the project for combinations of rates and levelsof production


4.3 Metal prices model

In this section, the equations 2.6 to 2.8, introduced in chapter 2, are developed to

mimic gold and copper price behaviour so that they can be used in the simulation.

Both have different characteristics with significantly different factors. Some insight

can be found in literature, but a subjective approach is relied upon to choose the

model factors.

Dixit and Pindyck [20] presented some numerical examples of their equations and

used a long-term average of $1.00 per pound of copper with a variation of 0.20 and

various reversion factors. Schwartz [56] studied the data of the previous ten years

to estimate the factors for copper, and gold. He calculated that copper’s long-term

average was equal to $1.28 per pound, with a variance of 0.233 and a mean-reversion

factor of 0.369. As for gold, Schwartz did not notice any mean reversion over a ten-year

period. Davis [16] looked at the effect of the factors on the evaluation of gold mines.

He considered a long-term average price of $326.26, with reversion rates ranging from

0.2 to 0.4, and price variations ranging from 5% to 15%. In a subsequent article,

Davis [17] used a gold price variation of 20%. Samis [55] studied the impact of price-

reversion on deciding wether or not to operate sectors of a multi-zoned copper mine.

In his study, he used a long-term average copper price of $1.00, a mean-reversion

factor of 0.223, and a price variation of 25%.

The author combined his observations and understanding of the actual metal

price variations over the last twenty-five years with the information discussed above

to define the factors to be used in the simulations. These factors are summarized in

table 4.11. Furthermore, in addition to the models used in the literature, the gold and

copper models used in this thesis have floor prices established, such that the prices


Table 4.11: Metal Price Equation Factors

Metal Long-Term Average Variation Mean Reversion Floor PriceGold $400 / oz 0.15 0.03 $250 / oz

Copper $1.00 / lb 0.25 0.25 $0.50 / lb

200

300

400

500

600

700

800

1995 2000 2005 2010 2015

Year

Go

ld p

rice

($/

oz)

10% confidence interval


expectation

Figure 4.23: Gold price model

can never go below these numbers.

Based on these factors, the 10% and 90% confidence intervals can be drawn for

both metals. Gold and copper are presented in figures 4.23 and 4.24. The 10%

confidence interval and the floor price lines are merged into one. As can be seen

in the figures, the top confidence interval for gold diverges with time because the

factor of mean reversion is very small, whereas that of copper becomes constant after

seven years because of a higher degree of mean reversion. A discussion covering the

validation of the models to be used in the simulation is given is appendix D.


40

60

80

100

120

140

160

180

1995 2000 2005 2010 2015

Year

Co

pp

er p

rice

(ce

nts

/lb)



expectation

Figure 4.24: Copper price model

4.4 Simulation

At this time, all the elements are in place to do the simulation. In this section, the

details of this particular simulation are discussed, with particular attention to the

definition of the feasible area, and to the general modifications brought in as a result

of the fact that the NPV function can be bimodal in certain cases.

The life-of-mine plan starts in 1997. The standard planning process, as done at

the mine site, is followed here where the plan is built approximately six to eight

months before the start of the year. The annual metal prices are known up to the

year 1995, meaning that for the purpose of this exercise, the metal price simulation

starts with the year 1996.


4.4.1 Feasible area

The limits of the feasible area are established. The first obvious one is the total

resource, such that the total production over the life of the mine can not exceed the

available resource. The second one is set by the mill capacity, such that the daily

production rate can not exceed 3,500 tonnes per day.

The third one is the development constraint and is related to the rate of develop-

ment that is possible in the mine as a function of the rate and level of production.

For low levels of production, there are a limited number of available stopes and de-

velopment headings thus restricting the daily development capacity, and reducing

the potential to produce many tonnes on a sustainable basis, thus leading to low

production rates. It can be expected that as the level of production increases, the

sustainable production rate also increases. The calculation of this constraint is based

on the analysis of historical data and is presented in appendix E.

All three constraints are drawn in figure 4.25, and the feasible area is identified,

area within which solutions are possible.

4.4.2 Review of the unimodal assumption for the NPV func-

tion

Before running the simulations, it is now important to verify the assumption of uni-

modality of the Net Present Value function, discussed in chapter 3, as this assumption

is necessary to be able to get a solution using the Method of Steepest Ascent and the

Golden Search. The way to check the hypothesis is to calculate the NPV function for

many points within the feasible area and look at the results. The test is run for levels

of production between 1 and 16 million tonnes in increments of one million tonnes,


0

2

4

6

8

10

12

14

16

18

0 500 1,000 1,500 2,000 2,500 3,000 3,500 4,000


Pro

duct

ion

leve

l (m

illio

n to

nnes

)

Feasible Area

Resource constraint

Development constraint

Milling constraint

Figure 4.25: Feasible area of the possible solutions for the NPV function

and for production rates between 500 and 3,500 tonnes per day in increments of 500

tonnes per day.

As it turns out, the NPV function is sometimes bimodal, and the process suggested

earlier is not valid. However, the process can easily be adapted by calculating the

NPV for all the incremental points within the feasible area, picking the maximum,

and then follow up with the the Method of Steepest Ascent, starting from that point

in order to determine the true maximum combination for much finer increments.

4.4.3 Procedures

The procedure for one run of the simulation is better presented with an example.

First, gold and copper annual prices are generated. The long-term metal price aver-

ages used are $400 per ounce for gold and $1.00 per pound for copper, with copper


15 -212 -147 -77 -93 -139 -159 -18114 -196 -112 -46 -71 -106 -123 -15013 -180 -85 -28 -57 -76 -89 -12612 -163 -59 -19 -34 -49 -67 -8811 -146 -34 -6 -16 -23 -54 -5010 -136 -22 -6 -2 -17 -31 -289 -125 -21 0 9 -12 -4 -198 -114 -34 5 12 7 6 -137 -106 -42 6 11 20 5 -186 -97 -38 9 29 22 4 -275 -80 -21 20 38 23 04 -77 -12 29 33 123 -63 0 21 202 -46 -8 -11 -12 6

500 1,000 1,500 2,000 2,500 3,000 3,500

Leve

l of p

rodu

ctio

n (m

illio

n to

nnes

)Net Present Value @ 5%


Figure 4.26: Example of the output generated

showing a high degree of mean-reversion and gold showing very weak mean-reversion.

The seed prices at the start of the simulations are the average metal prices for 1995,

set at $384 per ounce of gold and $1.33 per pound of copper. Second, the net present

values @ 5% for many combinations of levels and rates of production are calculated

and presented in table form, referred to as a combinatory table as seen in figure 4.26.

The values shaded in gray are negative, and the value in red is the highest, with

coordinates of 5,000,000 tonnes and 2,000 tonnes per day and a value of 38 million

dollars in this example.

The next step consists of conducting a detailed search around the coordinates

of the highest value until the maximum NPV is found, using the process discussed

previously. For the purposes of this exercise, it is judged that a resolution to the

nearest 100,000 tonnes and 50 tonnes per day is sufficient. In this example, the

maximum is found at 4,750,000 tonnes and 1,750 tonnes per day with a value of 40.9

MM$. The production profile of this combination is shown in table 4.12 and the


financial profile is in table 4.13. The first numerical column of these two profiles is

the total over the mine life and this column is saved for further analysis.

Therefore, for each metal price generation, the combinatory table of NPVs and

the production and financial profiles of the combination with the maximum NPV are

saved. This procedure is repeated 2,000 times.

4.5 Results and analysis

4.5.1 Combinatory tables

The data of all 2,000 combinatory tables is compiled to calculate the average net

present value and the standard deviation of every combination, as well as the proba-

bility of yielding a positive NPV. See figures 4.27 to 4.29. The information in these

three figures is read as in the following example: for a production rate of 2,500 tonnes

per day and a level of production of 7 million tonnes, the distribution of the 2,000

NPV calculations has an average of 47 MM$ with a standard deviation of 123 MM$,

and 61% of the NPVs are positive.

There is a general northeast-southwest trend in this table, showing a small gradi-

ent when both the production levels and rates vary proportionally and a very steep

gradient when they do not. In other words, if the mine wants to change its operating

point without varying too much the expected NPV, it would increase or decrease

both the production rate and the production level. Any other action would result in

a large variation of the expected NPV.

For any given column (where the production rate is held constant), the highest

NPVs are always associated with the lowest feasible production level. This seems


Yea

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9719

9819

9920

0020

0120

0220

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Rev

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97.6

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13.1

13.1

13.1

13.1

13.1

13.1

5.7

Mill

ing

127.

217

.117

.117

.117

.117

.117

.117

.17.

5S

ubto

tal

328.

344

.444

.444

.444

.444

.444

.444

.417

.8F

ixed

Cos

tsM

ine

33.8

4.5

4.5

4.5

4.5

4.5

4.5

4.5

2.0

Mill

& O

n-S

ite13

4.7

18.1

18.1

18.1

18.1

18.1

18.1

18.1

7.9

Sub

tota

l16

8.5

22.7

22.7

22.7

22.7

22.7

22.7

22.7

9.9

To

tal O

per

atin

g C

ost

s49

6.9

67.0

67.0

67.0

67.0

67.0

67.0

67.0

27.7

Cap

ital C

ost

Cap

ital D

evel

opm

ent

13.0

1.8

1.8

1.8

1.8

1.8

1.8

1.8

0.3

Oth

er m

ine

5.6

0.8

0.8

0.8

0.8

0.8

0.8

0.8

0.3

Mill

& O

n-S

ite6.

30.

90.

90.

90.

90.

90.

90.

90.

4T

ota

l Cap

ital

Co

sts

24.9

3.4

3.4

3.4

3.4

3.4

3.4

3.4

1.0

Cas

h F

low

55.8

0.1

-8.7

-1.7

14.0

17.9

15.2

11.5

7.4

Net

Pre

sen

t V

alu

e @

5%

40.9

Indi

cato

rsN

et S

mel

ter

Ret

urn

($/t)

121.

6010

9.02

95.4

210

6.24

130.

3913

6.55

132.

3712

6.69

127.

86O

pera

ting

Cos

t ($/

t)10

4.61

Ope

ratin

g M

argi

n ($

/t)16

.99

Ope

ratin

g C

ost (

$/oz

Au)

106

Tab

le4.

12:

Exam

ple

ofth

epro

duct

ion

pro

file


Yea

r19

9719

9819

9920

0020

0120

0220

0320

04T

otal

12

34

56

78

Fin

anci

als

('000

,000

$)

Rev

enue

s Gol

d S

ales

220.

125

.225

.225

.227

.930

.331

.236

.718

.5B

y-P

rodu

ct C

redi

ts35

7.5

45.4

36.6

43.6

56.5

58.1

54.5

45.3

17.6

To

tal R

even

ues

577.

670

.661

.768

.884

.488

.485

.782

.036

.1O

pera

ting

Cos

tV

aria

ble

Cos

tsM

inin

g58

.17.

87.

87.

87.

87.

87.

87.

83.

4R

egul

ar D

evel

opm

ent

45.4

6.3

6.3

6.3

6.3

6.3

6.3

6.3

1.2

Min

e S

ervi

ces

97.6

13.1

13.1

13.1

13.1

13.1

13.1

13.1

5.7

Mill

ing

127.

217

.117

.117

.117

.117

.117

.117

.17.

5S

ubto

tal

328.

344

.444

.444

.444

.444

.444

.444

.417

.8F

ixed

Cos

tsM

ine

33.8

4.5

4.5

4.5

4.5

4.5

4.5

4.5

2.0

Mill

& O

n-S

ite13

4.7

18.1

18.1

18.1

18.1

18.1

18.1

18.1

7.9

Sub

tota

l16

8.5

22.7

22.7

22.7

22.7

22.7

22.7

22.7

9.9

To

tal O

per

atin

g C

ost

s49

6.9

67.0

67.0

67.0

67.0

67.0

67.0

67.0

27.7

Cap

ital C

ost

Cap

ital D

evel

opm

ent

13.0

1.8

1.8

1.8

1.8

1.8

1.8

1.8

0.3

Oth

er m

ine

5.6

0.8

0.8

0.8

0.8

0.8

0.8

0.8

0.3

Mill

& O

n-S

ite6.

30.

90.

90.

90.

90.

90.

90.

90.

4T

ota

l Cap

ital

Co

sts

24.9

3.4

3.4

3.4

3.4

3.4

3.4

3.4

1.0

Cas

h F

low

55.8

0.1

-8.7

-1.7

14.0

17.9

15.2

11.5

7.4

Net

Pre

sen

t V

alu

e @

5%

40.9

Indi

cato

rsN

et S

mel

ter

Ret

urn

($/t)

121.

6010

9.02

95.4

210

6.24

130.

3913

6.55

132.

3712

6.69

127.

86O

pera

ting

Cos

t ($/

t)10

4.61

Ope

ratin

g M

argi

n ($

/t)16

.99

Ope

ratin

g C

ost (

$/oz

Au)

106

Tab

le4.

13:

Exam

ple

ofth

efinan

cial

pro

file


Net Present Value @ 5% (M$)15 -243 -213 -164 -131 -146 -160 -17814 -226 -181 -125 -96 -107 -119 -13713 -210 -152 -90 -65 -74 -84 -9812 -193 -123 -56 -36 -42 -51 -6111 -177 -97 -27 -11 -15 -22 -3110 -165 -79 -10 8 4 -3 -159 -152 -62 7 23 22 15 18 -140 -48 19 36 36 29 167 -129 -36 26 47 47 39 286 -112 -20 39 59 58 51 365 -89 2 55 74 72 654 -65 20 66 83 803 -38 36 71 892 -14 42 651 30 62

500 1,000 1,500 2,000 2,500 3,000 3,500

Lev

el o

f pr

oduc

tion

(mill

ion

tonn

es)


Figure 4.27: Average net present value @ 5%, in million dollars

very reasonable, as the mine is being high-graded. For a given production rate, the

highest-grade ore is mined, resulting in higher revenues per tonne and a shorter mine

life, decreasing the effect of discounting on the cash flows.

For any given row (where the production level is kept constant), the highest NPV

is associated to the highest production rate, again resulting from the fact that the

mine is delineated in less time, thus reducing the effect of time discounting. So, in

general, it is not surprising that the better values are located along the limits set by

the development and milling constraints, as it is always more valuable to mine the

ore as fast as possible .

The combinatory tables can be looked at as having done metal price risk analysis

for a great number of projects corresponding to each combination reported in the

tables. For example, three mine plans are analyzed in order to take a production

decision. In this example, based on certain considerations, management decides that

three options should be studied : 1500 tpd - 2 Mtonne; 2500 tpd - 4 Mtonnes; and


Standard Deviation (M$)15 47 88 118 142 160 177 19114 50 91 121 145 163 180 19313 52 94 124 146 165 181 19312 55 97 126 147 166 180 19211 57 98 127 148 165 178 18910 59 99 125 144 160 171 1819 61 98 122 140 153 163 1728 62 97 118 133 144 153 1597 62 93 111 123 132 139 1446 63 89 104 113 120 125 1295 63 85 96 103 108 1114 62 78 85 89 923 58 66 70 732 49 51 521 36 35

500 1,000 1,500 2,000 2,500 3,000 3,500

Lev

el o

f pr

oduc

tion

(mill

ion

tonn

es)


Figure 4.28: Standard deviation of the net present value distributions, in million dol-lars

Probability of Positive NPV (%)15 0 2 9 17 17 17 1714 0 4 15 23 23 22 2113 0 7 21 29 29 29 2712 1 10 29 36 36 34 3311 1 16 36 42 41 40 3810 2 19 42 46 45 44 419 2 23 47 51 50 49 468 3 28 50 56 56 53 497 3 31 53 61 59 56 536 5 36 59 67 65 62 565 8 44 69 75 73 704 14 54 77 83 803 22 66 86 902 33 78 911 78 99

500 1,000 1,500 2,000 2,500 3,000 3,500Production rate (tonnes per day)

Lev

el o

f pr

oduc

tion

(mill

ion

tonn

es)

Figure 4.29: Probability of positive net present value, in percentage


Table 4.14: Risk analysis example

Case Production rate Level of production Average NPV @ 3% Standard deviation(tonnes per day) (tonnes) (million $) (million $)

1 1,500 2,000,000 65 52 91%2 2,500 4,000,000 80 92 80%3 3,500 6,000,000 36 129 56%

Probability of positive NPV

3500 tpd - 6 Mtonnes. A standard risk analysis would consist of collecting exactly

the information that is seen in the combinatory tables. A risk-neutral mine manager

would thus look at table 4.14 and probably declare that case 2 is the best because of it

having the highest NPV, and even though the standard deviation might be high, the

scenario still has eighty percent probability of being positive. A risk-adverse manager

might choose case 1 because of the highest probability of success.

Up to now, there is nothing new under the sun. A mean-variance risk analysis

(limited to metal prices only) has been conducted on many mine plans. The risk

analysis technique is well known, though probably not used often at mine sites. What

may be considered as a contribution here is the fact that the risk analysis is done on

so many mine plans at once. The only reason why this is possible is because there

is a very detailed knowledge of all cost functions. This ties in with Smith [62] who

said that, too often, a production decision is taken after having considered only one

option because calculating others would take too much time and effort.

As a tool for the mine manager, the combinatory tables add more information

that what is commonly available at the time of making a decision - a manager may

only have two options to choose from. As such, it is nice to have this information,

but the decision will still be influenced by the manager’s level of risk adversity.


4.5.2 Bubble graph

Now the mine manager is also shown the bubble graph (figure 4.30). The bubble

graph adds other dimensions to the analysis, dimensions that will be revealed by

analyzing the graph and its components.

General observations The graph is built such that its presentation has the same

orientation as the combinatory tables in order to refer between them rapidly. The

constraints and feasible area are easily identifiable. Each circle represents the coor-

dinates for which the maximum NPV is realized at least once during the simulation.

The size of the circles is proportional to the number of times the maximum NPV oc-

curred at those coordinates. For example, the addition of all bubbles at and around

2000 tpd and 3 mtonnes is equal to 33%, meaning that in thirty three percent of all

simulations, the optimal NPV occurred at or near that combination.

Approximately two thirds of all results are located on the perimeter set by the

development and milling constraints, with the majority occurring along the devel-

opment constraint. The rest of the answers are distributed within the feasible area.

Most of the bubbles are concentrated in specific points, where high concentration

areas are formed, surrounded by areas of nil probability. This is a surprising result,

as it was expected that high concentrations would occur with probabilities decreasing

concentrically around them. This aspect will be revisited in the upcoming pages.

Difference between being on periphery or within the feasible area Any

result located within the feasible area and not on the periphery indicates that the

string of metal prices generated consists of low prices at the start and higher prices

later in time. The mine would probably generate negative cash flows in the early


0

2

4

6

8

10

12

14

16

0 500 1,000 1,500 2,000 2,500 3,000 3,500 4,000


Lev

el o

f pr

oduc

tion

(m

illio

n to

nnes

)

33%

28%

6%

Figure 4.30: Bubble chart of optimal net present value combinations

years followed by large positive cash flows offsetting the early years. The proportion

of results located in this area should be proportional to the discount rate. If a higher

discount rate were to be used in this exercise, the discounted value of future cash

flows would be much smaller, eventually forcing these combinations to yield negative

NPVs. As a consequence, more results would be found along the periphery and less

in the north-west quadrant of the graph.

The fact that most results are located on periphery indicates that, in general, the

short term behaviour of metal prices is more influential than the long-term, again

for reasons of time discounting. If a higher discount rate were to be used, near-term

metal prices should be more influential.


4.5.3 Hot spots

The hot spots are combinations of production rates and levels around which the

results accumulate and form high concentrations. As mentioned earlier, there should

be only one hot spot, and not a number of isolated ones like seen in the bubble graph.

Three main hot spots are identified and are listed in table 4.15.

Table 4.15: Coordinates of hot spots

Hot Spot Production level Production rate(tonnes) (tonnes per day)

A 1,000,000 1,000B 3,000,000 2,000C 4,750,000 2,850

Many factors may influence the formation of these hot spots, and they will be

reviewed individually in this section with the objective of identifying the main one.

Under review will be metal prices, the cost and the revenue functions.

Metal prices Metal prices play a role in the formation of the hot spots. To illustrate

this, the combinations of gold and copper prices are plotted for each hot spot, where

the metal prices are equal to the average price sold over the life of the mine in each

case. In each figure, the chart area is divided into four quadrants4, separating sectors

above and below the expected long term prices of $400 per ounce gold and $1.00

per pound copper. Four graphs are plotted, one for each of the three hot spots and

one for the rest of the data. See figures 4.31 to 4.34. Also, table 4.16 indicates the

distribution of occurrences as a function of the hot spot and the quadrant.

Hot spot A corresponds to mining the very best fraction of the orebody at the

4The quadrant are numbered as in classical mathematics, with Quadrant 1 being the top right-hand, and the others following clockwise.


0

20

40

60

80

100

120

140

0 100 200 300 400 500 600 700

Gold price ($ / oz)

Cop

per

pric

e (c

ents

/ lb

)2%

21%

60%

17%

Figure 4.31: Metal prices yielding a solution occurring at hot spot A

0

20

40

60

80

100

120

140

160

0 100 200 300 400 500 600 700 800 900

Gold price ($ / oz)

Cop

per

pric

e (c

ents

/ lb

)

26%

23%

3%

48%

Figure 4.32: Metal prices yielding a solution occurring at hot spot B


0

20

40

60

80

100

120

140

160

180

200

0 100 200 300 400 500 600 700

Gold price ($ / oz)

Cop

per

pric

e (c

ents

/ lb

)51%

0%0%

49%

Figure 4.33: Metal prices yielding a solution occurring at hot spot C

0

50

100

150

200

250

300

0 200 400 600 800 1,000 1,200 1,400 1,600 1,800 2,000

Gold price ($ / oz)

Cop

per

pric

e (c

ents

/ lb

)

56%

6%0%

37%

Figure 4.34: Metal prices yielding a solution occurring at other locations


Table 4.16: Occurrences as a function of hot spot and metal price quadrant

Hot Spot Metal Prices Quadrant

1 2 3 4 Total

A 0% 6% 16% 5% 28%

B 8% 8% 1% 16% 33%

C 3% 0% 0% 3% 6%

others 19% 2% 0% 13% 34%

Total 31% 16% 17% 36% 100%

fastest rate possible, an approach commonly referred to as high-grading the deposit.

Figure 4.31 shows clearly that this solution occurs when both gold and copper price

averages are below the long-term expected prices (third quadrant), and furthermore,

based on the information in table 4.16, lower metal prices almost always result in the

high-grading solution.

Hot spot B does not correspond to any particular operating situation, and yet

one third of all cases fall on that spot. Looking at the metal prices distribution in

figure 4.32 does not yield any real insight, though it seems that there is a better than

fifty percent chance that hot spot B will be the preferred operating design if the gold

price is high and the copper price is low (second quadrant). However, this situation

occurs only eight percent of the time and does not really explain the formation of a

hot spot at this location. Furthermore, most of the hot spot B solutions are obtained

when copper prices are higher than one dollar, conditions that are equal to those at

hot spot C and all other cases.

Based on this, metal prices seem to have a definite influence on the high-grading

solution but not the others, and therefore do not explain the hot spots.


Cost functions The cost functions seem to have no influence on the the formation

of hot spots. Figure 4.21 indicates that the total cost function seems very smooth,

without discontinuities that could favor the accumulation of results in a particular

point.

Revenue functions The easiest way to see how revenues can influence the forma-

tion of hot spots is by analyzing said functions with set metal prices. Nine combi-

nations of prices are tested, combining the 10% confidence interval, the expectation,

and the 90% confidence interval of gold and copper, as traced in figures 4.23 and 4.24.

The results of these tests are in table 4.17 and plotted in figure 4.35. From this table,

it seems evident that the results are very sensitive to the price of copper: the high-

grading solution occurs when copper prices are low; the solutions occur mostly around

hot spot B when the copper prices are equal to the expectation, and the solutions are

associated to high levels of production when copper prices are high. Meanwhile, the

price of gold does not seem to influence the results. Therefore, the analysis must be

concentrated on the copper revenue functions.

Table 4.17: Solutions for set metal prices

Copper10% confidence interval expectation 90% confidence interval

Gold (tonnes-tpd) (tonnes-tpd) (tonnes-tpd)10% c.i. 1,000,000 - 1000 1,100,000 - 1050 9,950,000 - 2850

expectation 1,000,000 - 1000 3,100,000 - 1900 10,950,000 - 315090% c.i. 1,250,000 - 1125 4,750,000 - 1750 11,250,000 - 3250

Two factors affect copper revenues: the annual capacity of the roasters and the

average grade of the mill feed. The roasters have the capacity to receive 114,000

tonnes of green concentrate per year. All excess production from the mill is sold


0

2

4

6

8

10

12

14

16

18

0 500 1,000 1,500 2,000 2,500 3,000 3,500


Lev

el o

f P

rod

uct

ion

(m

illio

n t

on

nes

)

10 - 1010 - ExpExp - 1090 - 10

Legend1st number is "Gold price"2nd number is "Copper price"

where"10" = 10% confidence interval"Exp" = expextation"90" = 90% confidence interval

Exp - Exp

90 - Exp

90 - 90

10 - 90

Exp - 90

Figure 4.35: Solutions for set metal prices

directly to the smelters at a lower value. Depending on the copper grade of the mill

feed, the roaster capacity is reached when the mine works at close to 2000 tonnes per

day.

The copper grade varies with the level of production, as discussed in section 4.1.1.

Figure 4.36 shows that it increases as the level of production increases from 1 to 6

million tonnes where it reaches a maximum.

For the remainder of this section, the numbers and relationships that are analyzed

are all confined to the axis represented by the development constraint. This axis is

chosen because the three major hot spots are located on it, as well as most of the

solutions obtained for the set metal prices presented in table 4.17.

By combining the average feed grade to the mill as well as the production rate at

which it is fed, it is possible to calculate the Net Smelter Return factor of one percent

copper at different copper prices. Figure 4.37 shows how the factors change as values


2.00

2.50

3.00

3.50

4.00

4.50

0 1 2 3 4 5 6


Co

pp

er a

vera

ge

gra

de

(%)

Figure 4.36: Copper grade as a function of the level of production

increase along the development constraint axis. As expected, the factors are higher

for higher copper prices, and it is clear that at a rate of 2,000 tonnes per day and

a level of 3 million tonnes, the roaster capacity is reached, resulting in progressively

lower NSR factors beyond that point.

By multiplying the copper grade by the copper NSR factors, figure 4.38 is gener-

ated. Of interest here is the fact that the curves have different shapes for given copper

prices. This is due to the fact that, proportionally, the variations of the copper NSR

factors for low copper prices are much greater than for high prices. As a result, for

low copper prices, the value of copper in the ore sent to the mill reaches a maximum

around 2000 tpd - 3 mt, and decreases afterwards. As the copper price increases, this

rate of decrease is less significant until eventually the value of copper in the ore stays


0

5

10

15

20

25

30

1000 tpd - 1mt 1500 tpd - 2mt 2000 tpd - 3mt 2500 tpd - 4mt 3000 tpd - 5mt 3500 tpd - 6mt

Production rate - level (tonnes/day - million tonnes)

Co

pp

er N

et S

mel

ter

Ret

urn

Fac

tor

($ p

er p

erce

nt

cop

per

)

Cu price = 10% c.i.

Cu price = 90% c.i.

Cu price = expectation

Figure 4.37: Copper NSR factors

constant when copper prices are high. This would tend to indicate that under condi-

tions of high copper prices, there are more opportunities for solutions to be obtained

at high production rates and levels, whereas for low copper prices, solutions would

tend to happen around 2000 tpd - 3 mt.

The shape of the copper unit revenue curve thus has an influence on the total

unit revenue curve. Three figures, 4.39 to 4.41 illustrate this point. Using figure 4.39

as an example, the copper unit revenue curve for the copper price equal to the 10%

confidence interval is reproduced. The total value of the ore is equal to the value of

the copper plus that of gold for gold prices equal to expectations. And finally, the

unit cost curve is included on the graph to get an understanding of the relationship

between revenues and costs. By comparing the three figures, the slope of the total

value of the ore increases as a function of the copper price, reaching a point where


0

10

20

30

40

50

60

70

80

90

100



Co

pp

er V

alu

e ($

per

to

nn

e)

Cu price = 10% c.i.

Cu price = 90% c.i.


Figure 4.38: Value of copper contained in ore

the costs and revenue curves are almost parallel in the case of high metal prices.

The effect of this can clearly be seen by comparing the total profit curves for the

three copper price conditions. Figure 4.42 gives a very clear indication that profits are

maximized at 1,000 tonnes per day when the copper prices are low, at 2,000 tonnes

per day when copper prices are at expectation, and above 3,000 tonnes per day when

copper prices are high, corresponding well to the solutions obtained under known

metal prices conditions.

Based on this information, it is easier to understand the formation of the hot spots

resulting from the simulation. For low copper prices, the solutions would naturally

tend to 1,000 tonnes per day and 1 million tonnes. Again, for most cases averaging

close to the expectation, the solution would tend to converge towards the 2,000 tonne

per day solution. In the case of high metal prices, the total profit curve does not


0

40

80

120

160

200



Val

ue

or

Co

st (

$ p

er t

on

ne

of

ore

)

Copper value for price = 10% c.i.

Total cost

Total value of ore

Figure 4.39: Unit revenues and costs for copper price set at 10% confidence interval

0

50

100

150

200



Val

ue

or

Co

st (

$ p

er t

on

ne

of

ore

)

Total cost

Total value of ore

Copper value for price = expectation

Figure 4.40: Unit revenues and costs for copper price set at expectation


0

50

100

150

200

250



Val

ue

or

Co

st (

$ p

er t

on

ne

of

ore

)

Total cost

Total value of ore

Copper value for price = 90% c.i.

Figure 4.41: Unit revenues and costs for copper price set at 90% confidence interval

-200

-150

-100

-50

0

50

100

150

200

250



To

tal p

rofi

t (m

illio

n $

)

Cu price = 10% c.i.

Cu price = 90% c.i.


Figure 4.42: Profit as a function of copper prices


show a unique maximum and solutions would be more dispersed.

It is thus very clear that, although this mine is considered as a gold mine and

that all relationships should be developed with that in consideration, the mine design

parameters are most sensitive to copper prices.

4.5.4 Mine design

Using the information compiled in the bubble graph, the following design is suggested

for the mine:

Based on the highest concentration of solutions, design the mine for a production

rate of 2,000 tonnes per day and a level of production of 3,000,000 tonnes. The

important aspect of this first decision is the establishment of the level of capital

investment that is to be done. This decision establishes the extents of the lateral

development to be done, thus fixing the final envelop of the mineralization to consider

in the plan. Another aspect is that the upper limit for the production rate is fixed.

This means that parts of the mill can be moth-balled and put on care and maintenance

in case new nearby deposits can be brought in production in the future.

This solution can be considered as the final step of the strategic design. In sum-

mary, this plan has an expected value of 89 million dollars with a standard deviation

of 73 million dollars and a 90% probability of being positive. However, more impor-

tantly, this design has a 33% probability of being the best design given uncertain

metal prices, the best probability available.

This level of probability is very low and a manager might be hesitant to use this

design based on these criteria. However, the manager might also want to consider that

other available designs would not be as robust as this one, and that choosing to go


with a design with lower probability would certainly mean that any variation in metal

prices would lead to a plan that is far from optimum, and would force management

to redesign the mine in very short order.

A note on tactical planning Up to now, the objective of this work has been to

develop a static mine plan, one where the production rate and cut-off grade remain

fixed throughout the life of the mine. However, once in place, tactical planning can

start in order to refine the extraction sequence and also to conduct marginal analysis

on sectors not included in the strategic plan.

The design, as it stands, considers only the best 3 million tonnes of the deposit,

but the envelope of the reserve contains much more mineralized material that could

be considered for extraction. As such, all the capital lateral development is considered

covered by the reserves. This development goes by mineralized areas contained within

the reserves envelope, and only regular development would be required to bring them

in production. Graphically, this could be shown as in figure 4.43. The new feasible

area would be limited to the right by the strategic production rate set at 2,000 tonnes

per day. Also, the grade-tonnage curves of the mineralization contained within the

reserves envelope can be estimated, meaning that the feasible area now finds itself

shrunken in height and in width. The overall effect on the tactical plan would be a

shift upwards from the strategic design.

Once this plan is in place, metal prices will fluctuate, and adjustments might

be made on a temporary basis to take advantage of higher prices or to mitigate the

effects of lower prices. The practice to adopt in these cases is to limit modifications to

movements along the new tactical constraint lines, either up or down the production

constraint to include or exclude new ore from within the reserves envelop, or sideways


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Strategic Design

Tactical Design

Less Resources

Constrained Production Rate

Figure 4.43: Tactical plan

along the development constraint. The advantages are that no new capital investment

would be required as the mineralized envelop remains fixed and no equipment is

purchased as the production rate does not increase. This approach would lead to a

robust plan able to weather any situations.

More work would be required to expand on this subject, and it is suggested that

the topic of tactical planning be studied at some further time as part of another study.

4.6 Summary

In this chapter, a detailed model was constructed and solved for a polymetallic mine.

This mine was chosen because of the large quantity of available information concerning

its resources and costs.

The fact that two metals contribute significantly to revenues complicated the


task of determining the main metal on which to base the model. An innovative

approach was developed to make that determination, where the relationship between

the resource base and revenue functions developed in section 4.2.1 was combined

with an analysis of the quantity of high-grade secondary metal contained in low-

grade primary blocks. It became obvious that gold was omnipresent in various grades

independently of the copper grade whereas copper was not present when the gold

grades were low. For this reason, gold was chosen as the primary metal for this

deposit.

Revenues functions were developed, incorporating the various streams of products

coming from the mill and roaster, and taking into consideration the effect of metal

prices on the treatment and refining charges for each product.

The relative spatial position of the resource blocks was reviewed and this informa-

tion was combined to the available operating and capital cost information to generate

the various cost functions classified as per the proposal made in table 3.1.

Gold and copper metal price models were prepared, for the purpose of running

the Monte Carlo simulations.

The feasible area within which solutions are possible was shown to be constrained

by the available resources, the milling capacity, and the physical limitation of devel-

oping sufficient work places to sustain given production rates as a function of the

level of production.

The simulation was run and the results were presented in the form of a bubble

graph, showing a few design combinations with the higher probabilities of occurrence.

This result was counter-intuitive as it was thought that there should be only one

solution to the problem. The analysis showed that the multiple solutions were due to


the influence of the copper feed grade and the roaster capacity.

Chapter 5

Hypothetical Gold Mine

Chapter 4 showed that the methodology produced viable results, but that they were

grouped into hot spots. As discussed, these hot spots were not expected as it was

thought that the solution would consist of a single higher-probability combination

surrounded by concentrically decreasing probabilities. The objectives of this chapter

are to determine if this expected solution would be generated if the effect of the copper

revenue function were eliminated, and to evaluate the sensitivity of the model to its

composing factors.

This chapter is divided into seven parts:

• The creation of a new mine model will be described. The cost functions remain

the same as in the previous chapter, but the revenues are based exclusively

on gold. In order to compare the results between the actual mine and this

hypothetical model, copper grades are converted into gold grades;

• The assumptions used in the simulation will be discussed along with the pa-

rameters on which sensitivities will be run;

154

CHAPTER 5. HYPOTHETICAL GOLD MINE 155

• The results of the simulation on the base case will be reviewed and compared

to those of the gold and copper model from the previous chapter;

• The sensitivity of the model to the discount rate will be presented;

• The sensitivity of the model to the gold price mean-reversion factor will be

examined;

• The results of a sensitivity analysis based on the deposit average grade will be

presented; and

• Finally, the chapter will close with a discussion on how a mine manager might

use the results to decide on the mine design.

5.1 Hypothetical gold mine model

In creating a gold mine model, the objectives are: to have an orebody with similar

grade and spatial distribution as the original one in order to use the same costs as

a function of the level of production; and to simplify the milling process such that

revenues as a function of the production rate are smoothed out with the elimination of

the roasting process. In this section, the assumptions used to create the hypothetical

gold mine are discussed, the PeaRL simulation is run on the new model, and the

results are discussed.

5.1.1 Grade distribution of the mineralization

The first step is to create a gold orebody with similar grade-tonnage functions as the

original polymetallic orebody. The approach to do so is the following:


• Using the long-term metal price expectations, calculate the NSR factors for the

average metal grades of the total resource;

• Using these NSR factors, calculate the value in dollars per tonne of each resource

block;

• Calculate the average NSR of the deposit and the standard deviation;

• Divide the average and standard deviation by the gold NSR factor to get the

average gold equivalent grade and standard deviation.

The deposit has an average value of $89.17 per tonne and the gold equivalent

grade is equal to 8.52 grams per tonne with a standard deviation of 6.53 grams per

tonne. The histogram of deposit resources expressed as gold equivalent (figure 5.1)

seems to have two modes as would be expected from a polymetallic orebody with two

dominant metals.

The orebody model is simplified by transforming it into a unimodal log-normal

distribution with average grade and standard deviation equal to those calculated.

The grade-tonnage curves of the original and hypothetical models are compared in

figure 5.2. The tonnage curves are almost equal, but the average grade curves diverge

for cut-off grades higher than 11 grams per tonne, corresponding to levels of produc-

tion less than 4 million tonnes, with the real model having much higher grades. The

fact that the real model is bimodal plays a role in this as the highest mode tends to

be truncated and the high-grade values present in the deposit are under-evaluated.

However, since the purpose of this section is to test the shape of the results of the

simulation, the hypothetical model is accepted.


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Figure 5.1: Gold equivalent tonnes histogram

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Tonnes

Grade

Lines in red represent the original modelLines in black the hypothetical model

Figure 5.2: Comparison of original and hypothetical models


5.1.2 Spatial distribution of the mineralization

In order to keep the same cost functions as developed in chapter 4, it is assumed that

the resources of the hypothetical model have the same spatial distribution as that

of the polymetallic model. In other words, the resources corresponding to a level of

production of 2 million tonnes are at the same location in the mine for both models.

Furthermore, the mining methods and relationships between sources of production

and levels of production do not change.

5.1.3 Milling process

The roasting process is eliminated and all the gold is recovered through flotation

using the same metal recovery equations. The green concentrate NSR factor for gold

is used for revenue calculations, thus eliminating the effect caused by green and calcine

concentrate differentials.

5.1.4 Operating and capital costs

All cost functions remain exactly the same. In the case of milling, although only

flotation is considered, the full costs of the roaster are kept in order to produce a fair

comparison. Since there are no changes to the spatial distribution of the resources,

all development and mining costs as a function of the level of production stay the

same.

5.1.5 Solution feasible area

The constraints remain identical, resulting in the same feasible area.


5.1.6 Numerical modeling procedures

The modeling procedures remain the same except for two small modifications:

• Only the price of gold is simulated; and

• the annual gold NSR factors are based on the sales of green concentrate. It

must be noted though that the difference with the sale of calcine is one cent or

approximately one tenth of one percent.

5.2 Assumptions

The assumptions used for the simulations are the following:

Average grade of the deposit As discussed previously, the average grade of the

deposit is equal to 8.52 grams per tonne of gold. To gauge the sensitivity of the model

to the grade, this base case will be compared to a low-grade deposit averaging 6.5 g/t

Au and to a high-grade deposit averaging 10.5 g/t Au.

Discount rate The discount rate to be used in the base case will be 5% since the

mine is already in production and the capital and operating costs and the reserves

are well understood. However, it could be assumed that the mine recently went into

production and that some of these revenue and cost functions are still ill-defined.

Therefore, the model will be run with a discount rate of 8% to gauge the effect.

Gold price model parameters As in the previous chapter, the long-term average

price is equal to $400 per ounce with a floor price of $250, the price variation is equal

to 0.15, and the mean-reversion factor is equal to 0.03. In the sensitivity analysis, the


mean-reversion factor will be increased to 0.20 to estimate the effect on the solution

of reducing the possibility of obtaining very high gold prices on the solution.

5.3 Base case results

The base case results are presented in figure 5.3. The differences are quite marked

from the original model in figure 4.30:

• The hot spots have disappeared. The elimination of the revenue function dis-

continuities has resulted in the smoothing out of the results along the constraint

lines, thus proving the initial assumption.

• There are no results associated with the high-grading option at 1,000 tonnes

per day, and the lowest production rate along the development constraint line

is located at a rate of 1,575 tonnes per day. This tends to confirm the points

made in the previous chapter about the influence of copper prices in the final

solution; when copper is taken out of consideration, the bias towards high-

grading disappears and the unique solutions related to various copper prices

are also eliminated;

• The proportion of results located on the constraint lines has decreased from two

thirds to one half. Given the fact that the gold price model variance increases

with time, there are more opportunities to have very high gold prices in the

future, thus leading to more situations where the production rate should be

reduced in order to be in production when prices are high. This in turn leads

to solutions located to the left and above of the constraint lines.


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Base CaseAverage grade = 8.52 g/t AuDiscount rate = 5%Mean reversion factor = 0.03

Figure 5.3: Bubble graph for gold equivalent grade equal to 8.52 grams per tonne

It is very hard to determine the final design from this graph as there is too much

information on it. The figure is therefore simplified by grouping all the individual

results into clusters measuring 250 tonnes per day and 500,000 tonnes. The resulting

graph is shown in figure 5.4. There is a shift of the bubbles towards the right,

indicating that higher production rates and levels might be indicated. The highest

concentration of results is located between 1,750 and 2,500 tonnes per day along the

development constraint line, but there is no clear-cut design to be chosen from this

graph. The mine design is analyzed later in this chapter.

The values in combinatory tables (figure 5.5) are remarkably higher, with the

NPVs twice as high as than those observed in the gold-copper case. The variation

might come from a two factors:

• The copper concentrates have higher treatment and refining charges than the


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4%8%

9%9%

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6%

6%


Figure 5.4: Clustered bubble graph for gold equivalent grade equal to 8.52 grams pertonne

gold’s. By converting the copper into a gold equivalent, the associated charges

are eliminated, yielding higher revenues and thus higher NPVs.

• The gold price model has a low mean-reversion factor than that of the copper.

It is therefore possible that higher revenues can be generated from gold alone.

5.4 Sensitivity to the gold price model mean re-

version factor

To test the influence of the mean-reversion factor on the results of the base case, this

factor will be increased in order to reduce the probability of achieving very high gold

prices in the future. The results should be closer to those obtained in the previous


Net Present Value @ 5% (M$)15 -213 -167 -115 -75 -49 -34 -3014 -194 -133 -75 -32 -4 12 1713 -174 -99 -34 11 40 57 6312 -152 -65 4 51 81 98 10411 -131 -33 39 86 116 133 13810 -109 -3 70 117 147 163 1679 -87 26 98 143 172 186 1898 -64 52 122 166 192 204 2047 -40 77 144 185 208 218 2166 -14 101 164 200 220 227 2235 13 123 180 211 226 2314 41 140 189 214 2263 64 149 187 2052 77 139 1651 66 99


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Figure 5.5: Base case combinatory table for NPV @ 5%, in million dollars

chapter.

For this exercise, the mean-reversion factor was increased to 0.20, compared to

0.03 in the base case. This results is a narrower band of probable answers, as shown

in figure 5.6.

The results of the simulation are presented in the combinatory table in figure 5.7.

5.4.1 Effect of treatment charges

The values read in the combinatory table are very similar to those in the base case,

which seems to show that the mean reversion factor plays a very small role in the

increase of the NPV. Therefore, one must assume that the role of the copper treatment

and refining charges is very important. It is possible to get a feel of their impact by

comparing the gold-copper case to the gold equivalent case for a given combination

of production rate and level under known metal price conditions. For the sake of this

exercise, a production rate of 2,500 tonnes per day and a level of production of 4


200

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1995 2000 2005 2010 2015

Year

Go

ld p

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($/

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10% confidence interval, both cases

90% confidence intervalreversion factor = 0.03

expectation

90% confidence intervalreversion factor = 0.20

Figure 5.6: Comparison of the confidence intervals sensitivity

Net Present Value @ 5% (M$)15 -235 -197 -144 -101 -69 -50 -4314 -217 -164 -103 -56 -22 -2 513 -197 -128 -61 -10 24 44 5312 -176 -94 -21 32 67 87 9611 -155 -60 16 70 104 124 13210 -134 -28 50 103 136 156 1629 -111 2 81 132 164 181 1858 -88 31 109 157 186 201 2037 -62 60 133 178 204 216 2166 -34 88 156 196 218 226 2245 -4 113 175 209 226 2324 27 134 187 214 2263 56 146 186 2062 74 139 1661 66 100


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Figure 5.7: Combinatory table of NPV @5% for mean-reversion factor = 0.20, inMM$


million tonnes was evaluated at metal prices equal to their respective expectations.

The production profiles are presented in tables 5.1 and 5.2 and the financial analyzes

are in tables 5.3 and 5.4. It is very evident from these tables that, for equal conditions,

the revenues and the operating margin are much greater, thus showing that most of

the variation seen in the values in the combinatory table in the last section is due to

the copper concentrate charges.

5.4.2 Effect of the mean-reversion factor

The simulation bubble graphs are presented in figures 5.8 and 5.9. Their analysis

shows that the mean-reversion factor plays an important role in the distribution of

the results:

• The number of results located in the top left corner of the graphs have decreased

noticeably. This is due to the lower probability of having very high gold prices

in the far future, where the solution would be to mine the deposit very slowly

to take advantage of the future high prices;

• There is a shift of the solutions towards the milling and development constraint

lines, resulting in a tighter distribution. This can also be explained by the

elimination of the probability of very high gold prices in the future.

• The higher concentration of bubbles has moved slightly towards the right along

the development constraint line. This can be explained by the fact that, as

the probability of very high future gold prices decreases, the time value of cash

flows favours the rapid mining of the orebody, thus pushing the results to the

right.


Table 5.1: Production profile for the gold-copper model

Year 1997 1998 1999 2000 2001Metal Prices Average

Au ($/oz) 386 385 385 386 386 387Cu ($/lb) 112 118 113 110 107 105

NSR FactorsAu ($/g) 10.35 10.33 10.34 10.35 10.36 10.38Cu ($/%) 13.78 14.71 13.84 13.40 12.93 14.42

Production TotalTonnes 4,000,000 912,500 912,500 912,500 912,500 350,000Grade

Au (g/t) 6.61 6.61 6.61 6.61 6.61 6.61Cu (%) 3.83 3.83 3.83 3.83 3.83 3.83

Tonnes per day 2,500Mine life (years) 4.38

Development (metres)Regular 50,795Deferred 9,803Total 60,598

Mining (tonnes)Cut & Fill 1,112,952Long Hole 1,563,600Pillar Recovery 688,404Ore Development 635,044

Mill RecoveryGold 88.6%Copper 96.6%

Metal recoveredGold ('000 oz) 1,257 172 172 172 172 66Copper ('000,000 lbs) 785 74 74 74 74 29


Table 5.2: Production profile for the gold equivalent model

Year 1997 1998 1999 2000 2001Metal Prices Average

Au ($/oz) 386 385 385 386 386 387

NSR FactorsAu ($/g) 10.84 10.82 10.84 10.85 10.86 10.87

Production TotalTonnes 4,000,000 912,500 912,500 912,500 912,500 350,000Grade

Au (g/t) 15.51 15.51 15.51 15.51 15.51 15.51

Tonnes per day 2,500Mine life (years) 4.38

Development (metres)Regular 50,795Deferred 9,803Total 60,598

Mining (tonnes)Cut & Fill 1,112,952Long Hole 1,563,600Pillar Recovery 688,404Ore Development 635,044

Mill RecoveryGold 92.8%

Metal recoveredGold ('000 oz) 3,359 422 422 422 422 162


Table 5.3: Financial analysis of the gold-copper model

Year 1997 1998 1999 2000 2001Financials ('000,000 $) TotalRevenues

Gold Sales 273.6 62.3 62.4 62.4 62.5 24.0By-Product Credits 235.1 56.9 53.8 52.3 50.7 21.4Total Revenues 508.7 119.2 116.2 114.7 113.1 45.4


Mining 50.4 11.5 11.5 11.5 11.5 4.4Regular Development 39.2 9.5 9.5 9.5 9.5 1.3

Mine Services 65.9 15.0 15.0 15.0 15.0 5.8Milling 95.0 21.7 21.7 21.7 21.7 8.3

Subtotal 250.6 57.7 57.7 57.7 57.7 19.8Fixed Costs

Mine 28.0 6.4 6.4 6.4 6.4 2.4Mill & On-Site 106.6 24.3 24.3 24.3 24.3 9.3

Subtotal 134.6 30.7 30.7 30.7 30.7 11.8Total Operating Costs 385.1 88.4 88.4 88.4 88.4 31.5

Capital CostCapital Development 11.7 2.8 2.8 2.8 2.8 0.4Other mine 8.6 2.0 2.0 2.0 2.0 0.8Mill & On-Site 18.9 4.3 4.3 4.3 4.3 1.7Total Capital Costs 39.3 9.1 9.1 9.1 9.1 2.8

Cash Flow 84.3 21.6 18.7 17.2 15.6 11.1Net Present Value @ 5% 74.0

IndicatorsNet Smelter Return ($/t) 127.17 130.58 127.34 125.74 124.00 129.84Operating Cost ($/t) 96.28Operating Margin ($/t) 30.89

Operating Cost ($/oz Au) 119


Table 5.4: Financial analysis of the gold equivalent model

Year 1997 1998 1999 2000 2001Financials ('000,000 $) TotalRevenues

Gold Sales 672.8 153.2 153.4 153.5 153.7 59.0By-Product CreditsTotal Revenues 672.8 153.2 153.4 153.5 153.7 59.0


Mining 50.4 11.5 11.5 11.5 11.5 4.4Regular Development 39.2 9.5 9.5 9.5 9.5 1.3

Mine Services 65.9 15.0 15.0 15.0 15.0 5.8Milling 95.0 21.7 21.7 21.7 21.7 8.3

Subtotal 250.6 57.7 57.7 57.7 57.7 19.8Fixed Costs

Mine 28.0 6.4 6.4 6.4 6.4 2.4Mill & On-Site 106.6 24.3 24.3 24.3 24.3 9.3

Subtotal 134.6 30.7 30.7 30.7 30.7 11.8Total Operating Costs 385.1 88.4 88.4 88.4 88.4 31.5

Capital CostCapital Development 11.7 2.8 2.8 2.8 2.8 0.4Other mine 8.6 2.0 2.0 2.0 2.0 0.8Mill & On-Site 18.9 4.3 4.3 4.3 4.3 1.7Total Capital Costs 39.3 9.1 9.1 9.1 9.1 2.8

Cash Flow 248.4 55.7 55.8 56.0 56.2 24.7Net Present Value @ 5% 217.6

IndicatorsNet Smelter Return ($/t) 168.20 167.87 168.07 168.26 168.44 168.66Operating Cost ($/t) 96.28Operating Margin ($/t) 71.92

Operating Cost ($/oz Au) 115


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Figure 5.8: Bubble graph for mean-reversion factor = 0.20

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7%9% 11%

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Figure 5.9: Clustered bubble graph for mean-reversion factor = 0.20


Net Present Value @ 8% (M$)15 -145 -129 -101 -74 -54 -40 -3614 -133 -108 -72 -40 -16 -1 513 -121 -84 -41 -5 21 39 4512 -109 -61 -11 28 57 75 8211 -96 -38 17 59 89 107 11410 -83 -15 43 87 117 134 1419 -69 7 68 112 141 157 1638 -54 28 91 135 162 177 1817 -38 50 113 154 179 192 1946 -20 73 134 172 194 204 2045 1 95 152 186 205 2124 24 116 165 194 2083 47 128 169 1902 63 126 1541 60 94


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Figure 5.10: Combinatory table of NPV @8%, in MM$

5.5 Sensitivity to the discount rate

The choice of the original discount rate was a function of the state of production of

the mine. It was judged that the mine had been in production long enough to have a

good control of its costs and a good definition of its resources, and for those reasons,

the discount rate was selected to be equal to 5%. It could be argued that the mine

had not been in production long enough to reduce the risk associated to these factors,

and that the discount rate should have been set higher, equal to 8%. The results of

this simulation are presented in figures 5.10 to 5.12.

The combinatory table shows that the NPV values have decreased by a small

factor as it would be expected when a higher discount rate is used. The bubble graphs

show fewer results in the top left corner of the graphs, meaning that the higher rate

overcomes the effect of very high gold price in the very far future. However, the

overall result is unchanged, and it could be said that the outcome is not sensitive to

the discount rate.


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Figure 5.11: Bubble graph for discount rate = 8%

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9% 10% 11%8% 9%

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Figure 5.12: Clustered bubble graph for discount rate = 8%


Net Present Value @ 5% (M$)15 -307 -329 -319 -306 -296 -294 -29914 -290 -297 -278 -260 -247 -243 -24713 -271 -262 -234 -211 -196 -190 -19212 -251 -226 -191 -164 -146 -139 -14011 -230 -191 -149 -119 -101 -92 -9310 -209 -157 -110 -79 -59 -50 -519 -186 -123 -74 -41 -22 -13 -148 -162 -90 -39 -7 13 21 187 -136 -56 -5 27 45 52 486 -105 -20 29 60 76 80 755 -71 16 64 91 104 1064 -33 51 94 117 1263 6 81 116 1332 37 95 1191 47 79


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Figure 5.13: Combinatory table of NPV @5% for gold equivalent grade equal to 6.50grams per tonne, in MM$

5.6 Sensitivity to the average grade of the deposit

The objective of this section is to test how the simulation results are affected by the

average grade of the deposit. In order to do this, the base hypothetical model is

modified by changing its average grade while keeping the same standard deviation.

All other assumptions discussed previously remain. To test the sensitivity, two cases

are analyzed, a low-grade deposit averaging 6.5 grams per tonne and a high-grade

deposit averaging 10.5 grams per tonne.

5.6.1 Results

The variation of the average grade has a marked effect on the results of the simulation.

The low grade case (figure 5.15) shows a strong shift towards high-grading whereas

the high grade case (figure 5.16) shows a shift towards the maximal utilization of the

mill. The clustered data is presented in figures 5.17 and 5.18.


Net Present Value @ 5% (M$)15 -128 -20 70 133 176 203 21614 -107 13 108 172 216 243 25613 -86 46 144 209 252 279 29112 -65 77 176 240 283 309 32011 -44 105 203 267 307 332 34210 -23 130 226 287 326 348 3569 -2 152 244 302 338 357 3648 18 171 258 311 344 361 3647 39 187 268 316 345 358 3596 60 200 273 316 339 349 3485 81 209 273 309 327 3344 98 211 264 293 3063 109 201 242 2632 107 173 2001 80 114


Lev

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Figure 5.14: Combinatory table of NPV @5% for gold equivalent grade equal to 10.50grams per tonne, in MM$

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Low-grade CaseAverage grade = 6.5 g/t AuDiscount rate = 5%Mean reversion factor = 0.03



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14

16

0 500 1,000 1,500 2,000 2,500 3,000 3,500 4,000


Lev

el o

f pr

oduc

tion

(m

illio

n to

nnes

)

High-grade CaseAverage grade = 10.5 g/t AuDiscount rate = 5%Mean reversion factor = 0.03


0

2

4

6

8

10

12

14

16

0 500 1,000 1,500 2,000 2,500 3,000 3,500 4,000


Lev

el o

f pr

oduc

tion

(m

illio

n to

nnes

)

7%11%

14%17%

14%9% Low-grade Case

Average grade = 6.5 g/t AuDiscount rate = 5%Mean reversion factor = 0.03

Figure 5.17: Clustered bubble graph for gold equivalent grade equal to 6.50 gramsper tonne


0

2

4

6

8

10

12

14

16

0 500 1,000 1,500 2,000 2,500 3,000 3,500 4,000


Lev

el o

f pr

oduc

tion

(m

illio

n to

nnes

)

5%5%

5%4%

14%

6%

High-grade CaseAverage grade = 10.5 g/t AuDiscount rate = 5%Mean reversion factor = 0.03

Figure 5.18: Clustered bubble graph for gold equivalent grade equal to 10.50 gramsper tonne

5.6.2 Analysis

These results can be directly related to the available quantity of high-grade ore in

each model. This is illustrated in figure 5.19 in which the average grades of the three

models are plotted as a function of the level of production.

The grades can be transformed into NSRs by multiplying them by a factor of

$9.60 per gram corresponding to the long-term expected gold price of $400 per ounce,

thus resulting in a graph similar to the average revenue curve presented in figure 3.4.

These unit revenues can now be combined to the average cost curves of figure 4.22

to create the average revenue and cost curves in figure 5.20. This figure shows that,

for a gold price of $400 per ounce and a production rate of 1,500 tonnes per day, the

low-grade case has five million tonnes of ore with a value higher than the average


5

10

15

20

25

0 2 4 6 8 10 12 14 16


Ave

rag

e g

rad

e (g

ram

s p

er t

on

ne)

Low-grade case

High-grade case

Base case

Figure 5.19: Average gold grade as a function of the level of production for the threecases used in the sensitivity analysis

cost. For the base and high-grade cases, the reserves are equal to 8 and 13 million

tonnes. Thus the greater the reserves, the greater the tendency to operate the mill

at full capacity.

The marginal revenue and cost curves are drawn in figure 5.21, and this graph

might bring some insight into the results of the simulations. As discussed by Gray [28],

Hotelling [34] and others, an operation will maximize its value when the marginal

revenue is equal to the marginal cost when the discount rate is nil or very small, as

is the case under study. For the low-grade case, the intercept between the revenue

and cost curves is at two million tonnes and 1,500 tonnes per day. For the base case,

the lowest intercept occurs at 4 Mt and 2,500 tpd, and for the high-grade case, at 7

Mt and 3,500 tpd. These intercepts correspond closely to the coordinates with the

highest concentrations of results in each simulation.


80

100

120

140

160

180

0 2 4 6 8 10 12 14 16


Ave

rag

e re

ven

ues

& c

ost

($/

ton

ne)

500 tpd

1000 tpd

1500 tpd

2000 tpd

3500 tpd

Low-grade

High-grade

Figure 5.20: Average revenues and costs as a function of the level and rate of pro-duction for the three cases used in the sensitivity analysis

80

100

120

140

160

180

0 2 4 6 8 10 12 14 16


Mar

gin

al r

even

ues

& c

ost

($/

ton

ne) 500 tpd

1000 tpd

1500 tpd

2000 tpd

3500 tpd

Figure 5.21: Marginal revenues and costs as a function of the level and rate of pro-duction for the three cases used in the sensitivity analysis


One can now imagine how the marginal revenue curves move laterally as gold

prices change in the simulations, yielding new solutions each time. The compilation

of all solutions obtained is comparable to the bubble graphs presented. Based on these

observations, there seems to be a correlation between the results of the simulations

and the marginal analysis.

5.7 Design analysis

Table 5.5 summarizes the design parameters for the three cases under study. In

the first instance, the highest percentage combinations are reported, and secondly,

high-probability operating ranges are indicated.

Table 5.5: Design parameters for the cases studied

Average gold equivalent grade

Best case combination

Percentage Operating range

Percentage

(g/t) (tpd - mt) (%) (tpd - mt) (%)

1250 - 1.502000 - 3.001750 - 2.502500 - 4.002500 - 4.003500 - 7.50

10.5 3500 - 6.00 14% 42%

8.52 2250 - 3.50 9% 35%

6.5 1750 - 2.50 17% 56%

It is apparent that no single design is robust. It would be difficult to convince a

mine manager that a design with 15% probability is the one to use, even though it

is the highest percentage available. It would be necessary to look at the proposed

operating ranges where the combined probabilities are more significant.

For the base case, the mine could be designed at 2,500 tonnes per day and 4,000,000

tonnes. This is based on the general consideration related to the possible flexibility


that could be built into the tactical plan. By limiting the mine to 2,500 tpd, and by

applying the concepts discussed at the end of the previous chapter, the operations

would have flexibility to modify its budgeting parameters while still keeping all the

robustness of the plan. A more conservative evaluator may decide that the design

should be made at 2,000 tonnes per day and 3,000,000 tonnes with similar reasoning.

Nonetheless, it is clear that the possible solutions are located within a very narrow

range between 1,750 and 2,500 tpd. This model and simulations have provided this

very useful information.

Applying the same logic, the low-grade case design would be set at 1,750 tpd

and 2,500,000 tonnes. The high-grade case would be set at 3,500 tpd and 6,000,000

tonnes.

5.8 Summary

It was shown that the simulation run on the gold-equivalent model resulted in a

continuous distribution of solutions as originally thought, and that the hot spots

generated in the work of the previous chapter were due to discontinuities in the

revenue functions.

The transformation of the copper to a gold equivalent resulted in much higher

net present values as the copper concentrate treatment and refining charges were

eliminated. This shows that one must be aware of all related charges when one

decides to work with metal equivalents in the evaluation of a deposit.

A higher mean-reversion factor had little influence on the net present value. How-

ever, it had the effect of eliminating the combinations related to high gold prices far

in the future. It also resulted in shifting the high concentrations of results to the right


of the bubble graph.

A higher discount rate had the expected effect on the net present values, resulting

in slightly lower values. In the bubble graphs, the higher discounting had a similar

effect to that of the mean-reversion factor with the elimination of the high future gold

price effect.

It was also shown that the solutions are sensitive to the average grade of the

deposit, and that the concentrations represented on the bubble graphs tend to shift

position as the average grade is varied.

None of the solutions are particularly robust however, and it is much easier to

identify an operating range than a single design parameter. The level of risk-adversity

of the mine manager would play a great role in finalizing the choice.

Chapter 6

Conclusions and Recommendations

The objective of this thesis was to develop a methodology to select the cut-off grade

and production rate of a mine under conditions of metal price uncertainty. It was

demonstrated that the approach and procedures developed in this thesis produce

results that can be used for selecting these strategic parameters and come up with a

final design. This process is very important as a design tool as it provides information

to the mine manager that is not currently available through standard sensitivity

and risk analysis. The method pinpoints the design criteria that will increase the

robustness of the mine, by increasing its ability to survive under conditions of various

possible metal prices.

This chapter reviews the original elements that were incorporated in order to

make this work possible, discusses the conclusions that can be drawn from this work,

and proposes several lines of research to expand on the possible applications of this

research.

182

CHAPTER 6. CONCLUSIONS AND RECOMMENDATIONS 183

6.1 Contributions

Some existing concepts were reviewed and expanded upon, other concepts were born

of necessity in order to solve specific problems encountered in this work, and the

final improvements were developed with the objective of deriving a general solution

to the problem set in this thesis. This section summarizes the new elements that are

introduced.

Addition to the existing models For the past one hundred years, most of the

models have been concentrating on selecting production rates such that the net

present value of the mine is maximized for specified metal prices. More recently,

models were introduced where the objective was to solve for the cut-off grade or for

the combination of both parameters. There has been less literature dealing with

conditions of metal price uncertainty, and only recently has Samis [55] dealt with

robustness by developing a methodology to establish the production rate and cut-off

grade of a deposit under these conditions. This present work adds to the existing

models by going beyond the study of fixed mine plans and by determining at the

design stage which strategy would be the most robust.

Development of a novel analysis of polymetallic grade-tonnage relation-

ships In general in the industry, and as discussed by Baird and Satchwell [3], metal

grades are transformed into dollar values and resource blocks are sorted on this mea-

sure. Obviously, this system works only if metal prices are assumed. Since this is not

the case in this work, a new method was developed to analyze metal distributions

and to determine the main metal on which to sort resources.


Establishment of the relationship between the grade-tonnage relationship

and the revenue function It was demonstrated that there is a direct correlation

between the grade-tonnage curves representing the distribution of resource blocks

and the average and marginal revenues as a function of the level of production. This

relationship is useful to understand metal distributions in polymetallic deposits and

to do marginal profit analysis.

Generation of a new underground capital and operating cost classifica-

tion scheme The schemes presented in literature are derived from general micro-

economics principles where fixed costs are not as sensitive to production rates vari-

ations and where capital development is not considered. The proposed scheme de-

scribes much more accurately the existing relationships.

Creation of a methodology to choose the production rate and the cut-

off grade of an underground mine under conditions of metal price uncer-

tainty A new methodology was developed permitting determination of the most

robust combination of production rate and cut-off grade such that the design has the

best possibility of success independently of future metal prices.

6.2 Conclusions

As this work was progressing, many problems were encountered and their solutions

brought some insight on the general process. This section discusses these aspects.

More comprehensive classification of costs in a narrow-vein underground

mine The major cost classification systems found in literature tend to regroup costs


into major components (variable and fixed operating, and capital) and very often

concentrate in solving problems by varying only the variable operating costs. The

cost classification discussed here is much closer to the actual cost accounting found

in operating mines, and the relationship between these costs and the main design

parameters is important to solve the relationship between capital and operating costs

and production rates and cut-off grades.

Problems related to the grade-tonnage analysis of a polymetallic deposit

Polymetallic deposits are usually analyzed by converting the various metal grades

into revenues, and classifying the resource blocks by value. However, this requires the

engineer to make assumptions about metal prices, and this assumption is voluntarily

not made in this study. Thus another approach must be developed to classify resources

as a function of the level of production.The solution provided in this thesis is based

on the study of the distribution of each metal contained within low-grade resource

blocks, and it ensures that few high-value blocks are left behind in the mine. The

method is not perfect and there is room for improvement when detailed analysis of

sectors is conducted during the tactical planning phase.

Limits of the feasible area In this study, the feasible area was defined by three

constraints, the available resources, the mill capacity, and the ability to have enough

development fronts to sustain a given production rate. Apart from the available

resources, these constraints are not universal and would change from case to case. For

example, this exercise may be done on a grass-roots project where no infrastructure

is present and therefore no milling capacity is preset. Would it be reasonable to

assume that the milling restriction could be replaced by a sustainable production


rate restriction, something akin to Taylor’s rule where the production rate would

increase as a function of the level of production? As for deposits that require a very

strict control on the sequence of extraction, could there also be a ”rock mechanics”

or ”mine sequence” constraint that would reduce the sustainable production rates as

a function of the level of production?

Robust solutions The PeaRL procedure yielded solutions in every case tested,

with actual and hypothetical mine data. The results obtained during the sensitivity

analysis of the hypothetical mine mimic the behaviour of Taylor Type A mines as is

observed broadly in the industry. In general, low-grade deposits tend to be mined by

extracting the high-grade ore using high operating costs and low production rates,

whereas high-grade deposits are mined at higher production rates while trying to

recover as much of the resource as economically possible by using lower-cost bulk

mining methods. However, one would contend that is common sense mining and not

necessarily robustness. It is obvious that an operator with many years experience will

have a good idea of how a mine should be designed, and this design might include,

purposefully or otherwise, some degree of robustness. However, PeaRL will define the

operating parameters more objectively than reliance on instinct.

Low probability of one robust design The results of the simulations done on

the hypothetical mine showed that it is very complicated to come up with one specific

design and that the level of risk adversity of the manager may play an important role

in the final selection within a reasonable range of possible solutions. A risk-adverse

manager might want to expand the mine at a later date if the operating results are

good, while this work proposes that further capital spending be avoided.


Influence of all metals on the solution for a polymetallic orebody It seems

very evident from the work done in this thesis that equivalent metal grades should

not be used for evaluating an orebody. Metal price behaviour, mineral processing

recoveries, and treatment and refining charges are significantly different for each one,

and thus revenue functions should be estimated for each major metal in an orebody.

Baird and Satchwell [3]’s approach might be justified in deposits where one metal

dominates all others but it is necessary to understand the influence of all metals,

not just in the geological context as discussed previously, but also in the economical

aspects.

Sensitivity to the discount rate and the mean-reversion factor Higher

mean-reversion factors and discount rates tighten the distribution of possible combi-

nations in the bubble graph by eliminating the possibility of having high metal prices

in the future or by decreasing their effect. However, their effect on the solution is

very weak.

6.3 Future work

The methodology presented here is definitely not limited to narrow vein mines and

some possible applications are presented here.

Development of the tactical planning methodology based on the results of

the strategic design Once a strategic design has been chosen, the resource base

is reduced to reflect what is readily available within the limits of the envelope of the

mine development. While the mine is in operation, metal price variations will occur


and the mine manager will want to maximize cash flows by varying the cut-off grade

and at times reducing the production rate. The notes on tactical planning that are

presented at the end of chapter 4 should be expanded upon in order to develop a

practical methodology.

Expansion of this methodology to include underground Taylor type B de-

posits Taylor type B deposits are more compact and grade distributions in space

are more continuous, meaning that their sensitivity to capital development and min-

ing operating costs should be lower. The solution to the problem now becomes a

boundary problem where the question could be simplified to determining the lateral

extent of the development and figuring out the rate at which to mine it.

Expansion of this methodology to consider grass-root projects at the fea-

sibility stage As discussed above in the conclusions, a different set of constraints

must be understood to define the feasible area, reflecting the productive limits of

an orebody as a function of its shape and size and the restrictions dictated by the

sequence of extraction.

Expansion of this methodology to include open pit mines The computerized

systems used in open pit optimization can determine production phases and final pit

limits such that the net present value is maximized for a given set of metal prices. In

the absence of this set of prices, the problem becomes the identification of the final

envelope to be mined.

Evaluation of the sensitivity to Real Options In this model, very high metal

prices occurring far into the future result in combinations where the mine operates


a very low production rates for a very long time in order to take advantage of those

prices. In practice, this would never happen as the mine operator would certainly

choose to postpone the operation to a later date rather than operating at a loss

in the present. The work of Brennan and Schwartz could be incorporated into the

methodology developed in this thesis to evaluate the impact of incorporating the

postponement of mining the deposit.

Also, the application of individual discount rates to the various components of

the cash flow equations, reflecting the risk directly related to each component, could

be studied as a comparison with the classic method used in Discounted Cash Flow

analysis.

Evaluation of the sensitivity to Conditional Simulation In this thesis, it is

assumed that the mine operator has perfect information on the resources and the

individual resource blocks, but in practice, the grade of each resource block will

vary from its estimate. Conditional simulation could be incorporated to measure the

sensitivity of the methodology to the uncertainty of the resource estimation.

Inclusion of operating and capital cost variations in the simulation In

this thesis, it is assumed that the operating and capital costs are well known; a more

complete analysis should include the addition of these two factors. It seems reasonable

to assume that the results would show more diffusion.

Comparison of mutually-exclusive scenarios In essence, the model compares

many possible mine operating conditions in order to choose the best alternative.

By extension, this model could be used to compare two or more mutually exclusive


investment alternatives. Some examples:

• Keep operating the mine at current levels or go ahead with a mill expansion to

increase the production rate in the future. The mine would build a life-of-mine

plan for each alternative, and the methodology would determine which of the

two is more robust to metal price variations.

• A large mining company may have many projects under consideration, but

may be constrained by the cash available. Projects could be ranked using this

model, thus providing one more tool to make decisions. Furthermore, projects

with low rankings could be redesigned and tested with this methodology in

order to increase their general robustness before being presented again to the

board.

6.4 Closing remarks

This thesis addressed some problems that had been bothering the author for many

years. Throughout the development of this thesis, he was able to elaborate on con-

cepts in ways that cannot be done while working in the industry, and he hopes that

some of these tools can now be transferred back to the mines and that they can

stimulate the imagination of other engineers.

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February:48–54, 1997.

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[63] L. D. Smith. Discounted cash flow analysis methodology and discount rates. In

Special Session on Valuation of Mineral Properties, Mining Millennium 2000,

Toronto, Canada, page 15 pages, 2000.

[64] R. R. Tatiya. Cutoff-grade decisions in relation to an indian copper-mining

complex. Transactions of the Institute of Mining and Metallurgy (Section A:

Mining Industry), Vol.105, May-August:A127–131, 1996.

[65] C. R. Tatman. Production-rate selection for steeply dipping tabular deposits.

Mining Engineering, Vol. 53, No. 10, October:62–64, 2001.

[66] H.K. Taylor. General background theory of cutoff grades. Transactions of the

Institution of Mining and Metallurgy, Vol. 81, July:A160–A179, 1972.

[67] H.K. Taylor. Rates of working of mines - a simple rule of thumb. Transactions

of the Institute of Mining and Metallurgy (Section A: Mining Industry), Vol.95,

October:A203–204, 1986.

[68] Dino J. Tessaro. Factors affecting the choice of a rate of production in mining.

Transactions of the Canadian Institute of Mining and Metallurgy, Vol. 63:573–

581, 1960.

[69] John E. Tilton. Long-term trends in copper prices. Mining Engineering, Vol. 54,

No. 7, July:25–35, 2002.

[70] John E. Tilton. Outlook for copper prices - up of down? Mining Engineering,

Vol. 58, No. 8, August:16–20, 2006.

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[71] Dana R. Walls, Michael R.; Clyman. Risky choice, risk sharing and decision

analysis: Implications for managers in the resource sector. Resources Policy,

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Belmont, California, third edition edition, 1994.

Appendix A

Modeling Procedures

This appendix provides a summary of the procedures to follow for the construction

of the mine model and for the execution of the PeaRL simulation.

A.1 Model construction procedures

This section deals with the information to gather and the steps to follow to prepare

the mathematical model prior to the execution of the simulation.

1. Estimate the diluted and recovered resources on a block by block basis;

2. Draw the grade-tonnage curves for the orebody. Identify the primary metal in

case of a polymetallic deposit;

3. Spatially locate the position of each resource block within the mine;

4. Estimate the lateral and vertical infrastructure required to put each fraction of

the resources as a function of the level of production and calculate the capital

cost;

201

APPENDIX A. MODELING PROCEDURES 202

5. Determine the mining method required to mine each resource block, the regular

development required, and the production expected to come from ore develop-

ment and from stoping activities. Estimate the corresponding operating costs;

6. Estimate the development and mining operating costs as a function of the level

of production.

7. Estimate the capital and operating costs of mine services, milling, and on-site

costs as a function of production rates and levels.

8. Determine the mill recoveries and smelter contracts.

A.2 Simulation procedures

Once the functions are established, the procedures to run the simulation are:

1. Establish the parameters of the metal price model: long-term average price,

degree of reversion, and variance of the short-term price variations.

2. Determine the number of simulations to run in order to get significant results.

3. For each simulation, generate a series of annual metal prices, and determine

which combination of rate and level of production yields the highest net present

value.

4. Plot the relative frequencies of all answers obtained in the simulations in a

bubble graph format.

5. Identify the highest-frequency combination and analyze the graph to establish

the final solution.

Appendix B

Capital Development

At continuation, the longitudinal projections of each mine show the position of the

resource blocks relative to the existing mine infrastructure. This appendix is divided

into three sections, one for each mine. In each section, a general outline of the

lateral infrastructure shows the elevation and lateral extent of the main levels and

the position of the ramp along which ore and waste are hauled either to surface or

to ore and waste passes. Afterwards, sixteen figures show the position of resource

blocks according to the level of production, starting with the best one million tonnes

and increasing in increments on one million tonnes until all resources are shown. The

blocks of all veins in a given mine are projected on the longitudinals since they all

are dependent on the transport drifts to be put in production.

Each resource block is represented by a single point corresponding to the bottom

elevation of the centre of the block. This point could be considered as the initial

entry point for production purposes, and therefore, it represents the extent to which

a transport drift would have to be developed in order to put that block in production.

The capital development necessary to put the blocks associated with a given level

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APPENDIX B. CAPITAL DEVELOPMENT 204

of production is equal to the metres of ramp to reach that level, the lateral drifting

required to reach the blocks and the raises necessary to supply ventilation to them.

B.1 Mine A

Mine A has lateral development extending from section -250 to section 850 metres

at various elevations between 150 and 560 metres, accessed either through adits or

from the main ramp centered on section 0. Accessing resource blocks located above

elevation 270 would entail extending the existing drifts laterally, and blocks located

below that elevation would require driving the ramp downwards and establishing new

transport drifts. These blocks would be mined by longhole and the transport drifts

would be located every 30 metres.

When all resource blocks are considered, the mine extends between sections -230

and 825 metres and between elevations 30 and 550.

0

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B.2 Mine B

Mine B is located close to Mine A, and shares the same main ramp. The existing

infrastructure extends from section -380 to 180 metres and from elevation 140 to 560.

When all resource blocks are considered, the mine extends from section -450 to section

180 and from elevation -100 to 560 metres. In that mine, the ground in some sectors

is classified as fair and for that reason, only 20 metres separate the levels in those

areas.

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Figure B.18: Mine B, General outline of lateral infrastructure


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Figure B.19: Mine B, Level of production = 1 million tonnes

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B.3 Mine C

Mine C is located at a certain distance from the other two mines. It has its own

independent ramp extending between elevations 290 and 560 metres. The existing

transport drifts extend from section -230 to 775 metres. The resource blocks’ limits are

sections -338 to 930 and elevations 80 to 580. Lower resources are mined by longhole,

with 15-metre level intervals adjacent to the current infrastructure and increasing to

30 metres at lower elevations.

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Figure B.35: Mine C, General outline of lateral infrastructure


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Figure B.36: Mine C, Level of production = 1 million tonnes

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Appendix C

Production & Development

Indicators

Three main indicators are required to determine the relationships between regular

development, production, and costs as a function of the level of production. They

are metres of regular development (ore and waste) required to put a tonne of ore

resource in production, the percentage of this development located in ore, and the

tonnes of ore generated per metre of ore development. These indicators are constants

for each mining method and for each vein. In this appendix, the concepts of ore and

waste regular development are defined for each mining method, then the constants

are calculated and are combined with the resource block database to generate the

production, development and costs functions.

In longhole mining, the typical development of a stope consists of developing two

levels, one at the top elevation and another at the bottom. On both elevations,

development typically includes the access from the main infrastructure to a point

approximately located 10 metres away from the stope. A drift that extends for the

231

APPENDIX C. PRODUCTION & DEVELOPMENT INDICATORS 232

length of the stope is set up parallel to the ore. From this drift, cross-cuts are driven

into the ore at every ten metres, and a drift is driven on the vein for the whole length

of the stope. A slot raise is sunk in the ore at an extremity of the stope as a last step

before the start of production. The vertical separation between the two elevations

can vary between 16 and 20 metres at the mine, according to the quality of the rock

mass and the variation of the method that is used.

Ramp-in-vein cut and fill is commonly used at the mine, with some minor use

of conventional mechanized cut and fill. Ramp-in-vein are usually 20 metres high,

and requires waste access drifts at the bottom and at the top of the stope. The

bottom drift accesses the ore where the whole length of the stope is developed; this

cut is considered as development. All other cuts above this one are considered as

production.

In pillar recovery, there are no hard rules as this is more of a scavenging operation

than a strict mining method. In general, a drift is driven from the infrastructure to

the ore on the bottom elevation. A drift in ore is driven along the length of the stope.

Often, no more development is required, however, sometimes a ramp might be driven

in the waste to gain access of the upper elevation of the stope.

From the descriptions of the mining methods and required development, one can

understand that the metres of development per tonne of ore indicator is a function

of the size of the stopes and the distance separating the individual veins and the

infrastructure. In relative terms, cut and fill stopes are generally the biggest stopes,

followed in size by longhole and then by pillar recovery. The second indicator, per-

centage of development done in ore, would tend to be higher for longhole mining

given the fact that and overcut and undercut are required, compared to the other two


methods where only one ore drift is driven. As for the third indicator, tonnes of ore

per metre of ore development, this would be independent of the mining method, and

directly related to the type of mineralization and the width of the vein.

C.1 Metres of development per tonne of ore

The best way to calculate this indicator would be to take the block plan of each

individual stope or mine sector and average the numbers out over all the projects,

but this information was not available. However, long-term mine plans incorporate

all the information contained in these block plans, and thus became the source of

information. Plans from 1997 to 2001 were used. The compiled data is in table C.1.

It is worth noting that these numbers are compiled for a resource of 2.9 million tonnes,

a small subset of the total resource, but the overall average indicator (expressed as

metres per 1,000 tonnes of ore for sake of simplicity) seem reasonable at ten meters, an

average that the author encounters in many operations. However, closer examination

of the numbers is warranted.

Starting with cut and fill, vein A5 seems high, but one must consider that that

vein is fairly remote and that a lot of waste development is required to reach it.

Meanwhile, vein B1 seems high and veins B4 and B5 seem low. One must consider in

this case that all three veins are closely related and that they share development. It

would be logical to consider all three together. Overall, it seems right that vein A5

averages more than twice the average of the other veins. Therefore, for calculation

purposes, vein A5 is set at 14.8 metres per 1,000 tonnes and all the other veins are set

at 6.3 (see table C.2). These numbers make sense given the nature of ramp-in-vein

cut and fill with short crosscuts to the ore and the stope undercut.


Table C.1: Calculations of development per tonne indicators

Metres development Tonnes mined metres per 1,000 tonnes of oremethod vein ore waste Total Total ore waste Total

Cut & Fill A2 205 65 270 29,215 7.0 2.2 9.2A3 202 89 291 50,611 4.0 1.8 5.7

A4 859 403 1,262 139,058 6.2 2.9 9.1A5 275 275 18,600 0.0 14.8 14.8

A6 100 190 290 58,462 1.7 3.2 5.0

B1 50 133 183 13,525 3.7 9.8 13.5B2 101 101 13,151 0.0 7.7 7.7

B4 110 110 48,205 0.0 2.3 2.3B5 35 295 330 95,260 0.4 3.1 3.5

Cut & Fill Total 1,451 1,661 3,112 466,087 3.1 3.6 6.7

Longhole A1 237 134 371 27,151 8.7 4.9 13.7

A2 1,992 1,449 3,441 373,899 5.3 3.9 9.2A3 41 41 11,184 3.7 0.0 3.7

A4 20 20 9,600 2.1 0.0 2.1A6 1,935 2,303 4,238 259,619 7.5 8.9 16.3

B4 1,185 634 1,819 301,287 3.9 2.1 6.0

B6 1,536 768 2,304 253,365 6.1 3.0 9.1C1 1,300 1,616 2,916 178,120 7.3 9.1 16.4

C2 1,659 2,343 4,002 193,094 8.6 12.1 20.7C3 2,822 2,453 5,275 753,514 3.7 3.3 7.0

Longhole Total 12,727 11,700 24,427 2,360,833 5.4 5.0 10.3

A2 150 858 1,008 31,796 4.7 27.0 31.7

A3 83 101 184 3,600 23.1 28.1 51.1A4 6,450 0.0 0.0 0.0

B1 55 184 239 3,240 17.0 56.8 73.8B2 89 30 119 12,310 7.2 2.4 9.7

pillar Total 377 1,173 1,550 57,396 6.6 20.4 27.0

Grand Total 14,555 14,534 29,089 2,884,316 5.0 5.0 10.1

Pillar Recovery


Table C.2: Development per tonne indicators

Vein Cut & fill Longhole Pillar recovery

A1 6.3 13.7 27.0A2 6.3 9.2 27.0A3 6.3 2.9 27.0A4 6.3 2.9 27.0A5 14.8 2.9 27.0A6 6.3 16.3 27.0B1 6.3 2.9 27.0B2 6.3 2.9 27.0B4 6.3 7.4 27.0B5 6.3 2.9 27.0B6 6.3 7.4 27.0C1 6.3 18.6 27.0C2 6.3 18.6 27.0C3 6.3 7.0 27.0

For longhole stoping, veins A1, A6, C1, and C2 seem high, but they are all reason-

able numbers since the longhole sectors of veins A1 and A6 are located far from the

infrastructure, and veins C1 and C3 are very narrow and have fewer tonnes on which

to average out the development. Veins A3 and A4 are very small, but the longhole

sectors are remnants of cut and fill stopes requiring very little development to put in

production.

Pillar recovery information is very sparse and it was decided that the average of

27 metres per 1,000 tonnes was reasonable and should be applied to all veins.

Table C.2 list the indicators used for further calculations.


C.2 Percentage of ore coming coming from devel-

opment

Production sent to the mill comes from two sources, the development done in ore and

the stoping activities. In general, the ore produced by development costs more that

by stoping, and development needs to be minimized in order to generate more profits.

However, this proportion can vary given the size of the stopes and the spacing between

sublevels. Using the same source of information as previously, the initial estimation

of the proportions is presented in table C.3, and the final indicators are listed in

table C.4.

The numbers show a lot of spread, it would be better to compile the data for

similar circumstances. In a cut and fill stope, only the first cut of a stope is considered

development. For a level spacing of 20 meters, one cut of 4 metres followed by 16

metres stope corresponds to a 4 ÷ 20 or 20% ratio. However, many stopes are not

bounded by levels and can go much higher. Therefore, the average of 17% seems

reasonable. Pillar stopes need very little ore development given the type of operation

they represent. The 7% average seems reasonable. Longhole stopes can be broken

down in sectors. Mines A and B copper veins, Mine C gold veins C1 and C2, and vein

C3. Mines A and B copper veins have 20 metres sublevels, vein C3 has 25 metres,

and veins C1 and C2 are narrow with a lot of internal dilution in development. In

all cases, development consists of an undercut and an overcut, but since the overcut

counts as the next stope’s undercut, and that in general four sublevels are needed per

three stopes, one must assume one and one third development per stope. Using these

numbers, the theoretical numbers are quite similar to the calculated ones.


Table C.3: Calculations of percentage production coming from ore development

Sourcemethod vein Development Production Total % dev/total

Cut & Fill A2 9,350 19,865 29,215 32%A3 8,521 42,090 50,611 17%A4 31,405 107,653 139,058 23%A5 0 18,600 18,600 0%A6 10,262 48,200 58,462 18%B1 0 13,525 13,525 0%B2 0 13,151 13,151 0%B4 4,490 43,715 48,205 9%B5 0 95,260 95,260 0%

Cut & Fill Total 64,028 402,059 466,087 14%

Longhole A1 14,376 12,775 27,151 53%A2 92,471 281,428 373,899 25%A3 1,394 9,790 11,184 12%A4 0 9,600 9,600 0%A6 87,039 172,580 259,619 34%B4 65,597 235,690 301,287 22%B6 65,750 187,615 253,365 26%C1 55,820 122,300 178,120 31%C2 44,149 148,945 193,094 23%C3 120,348 633,166 753,514 16%

longhole Total 546,944 1,813,889 2,360,833 23%

A2 5,201 26,595 31,796 16%A3 0 3,600 3,600 0%A4 0 6,450 6,450 0%B1 0 3,240 3,240 0%B2 0 12,310 12,310 0%

pillar Total 5,201 52,195 57,396 9%

Grand Total 616,173 2,268,143 2,884,316 21%

Pillar Recovery

(Tonnes)


Table C.4: Calculations of percentage production coming from ore development

Vein Cut & Fill Longhole Pillar RecoveryA1 14% 26% 9%A2 14% 26% 9%A3 14% 26% 9%A4 14% 26% 9%A5 14% 26% 9%A6 14% 26% 9%B1 14% 26% 9%B2 14% 26% 9%B4 14% 26% 9%B5 14% 26% 9%B6 14% 26% 9%C1 14% 27% 9%C2 14% 27% 9%C3 14% 16% 9%

C.3 Tonnes of ore per metre of ore development

Since veins have various horizontal width and mineralogy, it is important to know how

many tonnes of ore are generated in one metre of ore development. The horizontal

width of each stope varies, meaning that the proportion of ore to waste also varies for

a constant development cross-section; and the specific density of the mineralization

also changes wether the ore is in quartz or in sulfide. By combining them, the number

of tonnes of ore per metre advance in a stope changes based on the mining method

and the vein.

The procedures to calculate this indicator are as follows. Based on the monthly

data collected from 1997 to 2001, match the number of metres developed and the

tonnes of ore reported per heading. Sum up each heading over its life, and add up

all the headings as a function of the vein and mining method. The tonnes of ore per

metre of ore development is calculated by dividing the tonnes by the metres. The


Table C.5: Tonnes of ore per metre development dataCross sectional area of ore development Specific density Tonnes per meter advance in ore(square metres) (tonnes per cubic metre)

vein cutfill longhole pillar cutfill longhole pillar cutfill longhole pillarA1 10 3.57 36 X XA2 12 14 10 2.77 3.26 2.77 33 46 29A3 11 12 15 2.91 2.84 2.38 33 34 35A4 12 3.00 35A5 XA6 13 14 3.41 3.41 44 47 XB1 X XB2 X XB3 XB4 14 3.86 X 53 XB5 X X XB6 16 3.26 X 52 XC1 15 2.60 40C2 13 2.71 35C3 15 3.62 54 X

data is summarized in table C.5. Unfortunately, not all the combinations of vein and

mining methods are evaluated by this approach, as indicated by the letter ”X” in the

last table.

The only way to fix this is by applying factors based on the experience of the mine

engineers and geologists of the mine.

• Vein A1 is a massive sulphide and its thickness increases with depth. In the

case of longhole, the cross-sectional area of development would be in the order

of 14 square metres giving a factor of 48 tonnes per metre. Pillar recovery would

occur near the top of the vein in narrower ore and a lot of internal dilution would

be taken in development, thus reducing the specific density. It is estimated that

31 t/m would be produced.

• Vein A5 is a narrow quartz vein and the indicator in cut and fill would probably

be very similar to other cut and fill in similar veins. Use 37 t/m.

• Vein A6 is a massive sulfide with narrow width near the top where pillar recovery


would be conducted. Use 31 t/m as in the case of A1.

• Vein B1 is a narrow quartz vein. For cut and fill, use the same factor as for vein

A5, and for pillar recovery, use 31 t/m.

• Veins B2 and B3 are similar to vein B1, use the same factors.

• Veins B4, B5, and B6 are similar to vein A6 but narrower near the top where

selective mining is conducted. Use 37 t/m for cut and fill and 31 t/m for

pillar recovery. In the case of longhole mining in vein B5, the vein is somewhat

narrower than veins B4 and B6, and the factor is reduced marginally to 48 t/m.

• Vein C3, pillar recovery is conducted in the narrower upper portion of the vein

and is assigned the same as the other veins, 31 t/m.

Table C.6 shows the factors that are used for the rest of calculations. The author

judges that, though many numbers seems to be judged arbitrarily, the error is not

more than 15% and does not have a significant impact.

C.4 Compilation of data

With these three indicators in place, it now becomes possible to apply these factors

to each resource block in the database to calculate the following:

• Breakdown of resource tonnes into development and stoping tonnes;

• Metres of regular development associated with each block;

• The variable stoping cost; and


Table C.6: Final table of tonne of ore per metre development

(tonnes per meter advance)vein cutfill longhole pillarA1 36 48 31A2 33 47 35A3 33 34 42A4 35A5 37A6 43 45 31B1 37 31B2 37 31B3 37B4 37 55 31B5 37 48 31B6 37 52 31C1 41C2 35C3 55 31

• The variable regular development cost.

The database can now be queried to determine these four factors as a function of

the level of production.

Appendix D

Metal Price Model Validation

Shannon [58] suggested that a model is valid if it has face validity, does not generate

absurd answers, and generally looks right. The argument presented here is that the

model fits well with the observed prices of the last thirty years. Obviously, the future

cannot be predicted from the past, but a solid base is at least set.

The gold model is compared to actual prices starting in 1980 in figure D.1 and

1994 in figureau1994, 1 and copper is shown starting in 1975 in figure D.3 and in 1994

in figure D.4.

In general, the models seem to fit well with past behaviour, and it could be

conceived that they will represent adequately future price fluctuations.

11994 is chosen a a reference year because it is the start of price modeling for the project; thefirst year of the plan is 1996, and the model should be constructed 6 months earlier in June 1995,so the latest annual average price available is 1994.

242

APPENDIX D. METAL PRICE MODEL VALIDATION 243

0

100

200

300

400

500

600

700

800

900

1000

1980 1984 1988 1992 1996 2000 2004 2008

Year

Ave

rage

ann

ual g

old

pric

e ($

/oz)



minimum price = $250

Expected price

Actual price

Figure D.1: Gold price model starting in 1980

0

100

200

300

400

500

600

700

800

900

1000

1980 1985 1990 1995 2000 2005

Year

Ave

rage

ann

ual g

old

pric

e ($

/oz)



minimum price = 250

Expected price

Actual price

Figure D.2: Gold price model starting in 1994

APPENDIX D. METAL PRICE MODEL VALIDATION 244

0

20

40

60

80

100

120

140

160

180

1975 1980 1985 1990 1995 2000 2005

Year

Ave

rage

ann

ual c

oppe

r pr

ice

(cen

ts /

lb)



minimum price = 50

Expected price

Actual price

Figure D.3: Copper price model starting in 1975

0

20

40

60

80

100

120

140

160

180

1975 1980 1985 1990 1995 2000 2005

Year

Ave

rage

ann

ual c

oppe

r pr

ice

(cen

ts /

poun

d)



minimum price = 50

Expected price

Actual price

Figure D.4: Copper price model starting in 1994

Appendix E

Development Constraint

The objective in this part is to establish the relationship existing between the quantity

of ore to extract in a mine plan and the capacity to develop it such that a given

mining rate can be sustainable over the life of the project. As the level of production

decreases, so does the number of headings available for development, thus limiting

the potential daily production rate. Any attempt to increase the development rate

beyond the sustainable rate only leads to inefficiencies and higher unit costs, without

any benefits to production.

The data available to establish the relationship consists of a series of life-of mine

plans and annual budgets, listed in table E.1.

Figure E.1 shows a correlation between production rate and production level for

production levels up to 6,000,000 tonnes. Above that level, production rates stay

constant at 3,300 tonnes per day as that is the installed capacity of the mine. The

equation of the regression curve is similar to that developed by Taylor [67], with the

power of this equation established at 0.56 versus a power of 0.75 for Taylor. Figure E.2

shows the correlation between annual development capacity and production rates

245

APPENDIX E. DEVELOPMENT CONSTRAINT 246

Table E.1: Annual development

Year Plan Reserves Production Rate Development(tonnes per day) (metres per year)

1999 Annual Budget 750 2,0002000 Life-of-mine 700,000 1,000 4,0002000 Annual Budget 1,000 5,8002001 Life-of-mine 1,000,000 1,200 8,0001998 Annual Budget 1,680 5,7001998 Life-of-mine 2,400,000 2,200 11,0001995 Annual Budget 3,050 14,0001994 Life-of-mine 5,300,000 3,100 12,0001996 Life-of-mine 6,000,000 3,300 19,0001997 Life-of-mine 12,000,000 3,300 16,0001997 Annual Budget 3,300 22,000

based on the data from budgets and life-of-mine plans. The regression shows a factor

of approximately 5 metres of development per daily tonnes of ore.

A relationship between the upper limit of annual development and the production

level is deduced from these two functions, as shown in figure E.3. This function

is valid for production levels less than 6,000,000 tonnes, and for higher levels, the

development rate is equal to 17,300 metres.

Using these development rate limits, the minimum mine life and the maximum

production rate as a function of production levels can be calculated. To simplify

calculations, the production rates are increased to the next higher increment of 500

tonnes per day. These constraints will be used in the simulations. The numbers are

summarized in table E.2.


0

500

1,000

1,500

2,000

2,500

3,000

3,500

4,000

0 2 4 6 8 10 12

Production level (million tonnes)

Pro

duct

ion

rate

(to

nnes

per

day

)

rate = 0.5185 x level0.5628

R2 = 0.9923

Figure E.1: Production rates vs. reserves

0

5

10

15

20

25

0 500 1,000 1,500 2,000 2,500 3,000 3,500


Dev

elop

men

t ('0

00 m

etre

s pe

r ye

ar)

Dev = 5.1 * rate

R2 = 0.84

Figure E.2: Development rates vs. production rates


0

2

4

6

8

10

12

14

16

18

0 1 2 3 4 5 6 7


Dev

elop

men

t rat

e ('0

00 m

etre

s pe

r ye

ar)

Dev = 2.64 * level0.5628

Figure E.3: Development rates vs. production levels

Table E.2: Maximum production rate

Production Total Maximum Minimum Maximumlevel development development mine production

rate life rate(tonnes) (metres) (metres per year) (years) (tonnes per day)1,000,000 17,600 6,300 3.0 1,0002,000,000 33,400 9,300 3.8 1,5003,000,000 47,600 11,700 4.3 2,0004,000,000 60,600 13,700 4.7 2,5005,000,000 72,600 15,600 4.9 3,0006,000,000 83,900 17,300 5.1 3,5007,000,000 94,700 17,300 5.7 3,5008,000,000 105,200 17,300 6.3 3,5009,000,000 115,500 17,300 6.9 3,50010,000,000 125,800 17,300 7.5 3,50011,000,000 136,100 17,300 8.1 3,50012,000,000 146,600 17,300 8.7 3,50013,000,000 157,300 17,300 9.3 3,50014,000,000 168,100 17,300 10.0 3,50015,000,000 179,000 17,300 10.6 3,500

strategic design

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