Dimitrakopoulos (2008) Modelling Geological Uncertainty With Geostatistical Simulations

2008 PROFESSIONAL DEVELOPMENT SEMINAR SERIES STRATEGICRISK QUANTIFICATION ANDMANAGEMENT FOR ORE RESERVES ANDMINE PLANNING

Modelling geological uncertainty with geostatistical simulations: Models, methods, applications and software

February 21, 2008 University of Ottawa, Ontario

© Roussos Dimitrakopoulos McGill University, Canada COSMO – Stochastic Mine Planning Laboratory Department of Mining, Metals and Materials Engineering McGill University, Canada

Table of contents

TABLE OF CONTENTS

CHAPTER 1

1.1 1.2 1.3

1

1.3.1 1.3.2 1.3.3

5 7 9

3.2

3.3

4.2 4.3

4.4

A GENERAL CONDITIONAL SIMULATION FRAMEWORK12

Linking Orebody Uncertainty to Mining, Decision Making and Profitability Modelling the Uncertainty Associated with an Orebody Transfer Functions and the Modelling of a Mining Process Modelling Uncertainty about the Response Two Important Points Some Examples 2.6.1 Uncertainty in the description of an orebody: orebody simulations 2.6.2 Spaces of uncertainty, their mapping, and decision making

12 13 14 14 15 16 16 18

AN INTRODUCTION TO MONTE CARLO SIMULATIONS 21

A House Builder's Anxiety or Will the Big Man Go through the Floor? (A Heavy Friend's example for his Heavier Friend) 21 3.1.1 Drawing values from the histogram a variable (shoe sizes) 22 3.1.2 Drawing values for the correlated variable (weights) 23 Generating Correlated Variables 25 3.2.1 Simulating bivariate normal data 26 3.2.2 Simulating bivariate lognormal data 27 3.2.3 Simulating multivariate normal data 28 What have we seen so far? 28

CHAPTER 4

4.1

2 5

An example from stockwork gold deposit Comparison to thea“real” Comments

CHAPTER 3

3.1

1

Geostatistical Textbooks I’d Rather be Approximately Right than Precisely Wrong: Key Reasons for the Quantitative Modelling of Geological Uncertainty and Risk Traditional Orebody Modelling: Some Limits and Shortcomings

CHAPTER 2

2.1 2.2 2.3 2.4 2.5 2.6

INTRODUCTION

DATA DESCRIPTION

30

The Importance of Outliers 4.1.1 Some observations 4.1.2 The traditional approach in dealing with outliers Estimating a Representative Data Histogram

30 30 31 32

4.2.1 Declustering techniques Univariate and Bivariate Data Description 4.3.1 Univariate description 4.3.2 Summary statistics 4.3.3 Distribution models 4.3.4 Scatterplots 4.3.5 Correlation coefficient 4.3.6 Spatial description Indicator Variables

33 36 36 37 38 40 41 41 45

Applied Risk Assessment for Ore Reserves and Mine Planning: Conditional simulation for the mining industry

i

4.5

4.4.1 Interpretations 4.4.2 Spatial description of indicator variables Conditional Probability Distributions

CHAPTER 5

5.1 5.2

5.3

CHAPTER 6

6.1 6.2 6.3 6.4

6.5

6.6

6.7

ii

CONDITIONAL SIMULATION

Introduction The Turning Bands Method 5.2.1 The estimation plus simulated error algorithm 5.2.2 Comments Generating Sequences of Random Numbers SEQUENTIAL CONDITIONAL SIMULATION METHODS

46 46 47 49

49 50 53 57 57 60

Some Considerations Explaining the Idea with an Example What have we seen so far? A Few Lines of Theory: The Concept of Sequential Simulation 6.4.1 Sequential simulation and related generalisations 6.4.2 Some implementation related comments Sequential Gaussian Simulation (SGS) 6.5.1 Some observations 6.5.2 The SGS simulation algorithm 6.5.2.1 SGS: an implementation of the LU simulation 6.5.2.2 The LU simulation algorithm 6.5.2.3 The SGS algorithm in summary 6.5.2.4 From LU to the SGS algorithm 6.5.3 Monte Carlo drawing from a conditional distribution 6.5.4 Normal score transformations (forward and backward) 6.5.5 A fast sequential Gaussian simulation: Sequential Group Gaussian Simulation 6.5.5.1 The implementation of the SGGS Sequential Indicator Simulation (SIS) 6.6.1 Introduction 6.6.2 Grade indicators and a block of a gold deposit 6.6.3 Exercise 6.6.4 Indicator kriging summary and comments 6.6.5 Median IK 6.6.6 The steps in SIS 6.6.7 Comments on indicator kriging in SIS 6.6.8 Negative weights: a detailed IK example based on median IK 6.6.9 Some comments on IK as used in SIS

60 60 65 65 65 66 67 67 70 71 71 72 73 74 74

6.6.10 usecategorical of IK typevariables approaches in sequential simulation 6.6.11 The SIS of Implementation and Related Intricacies of IK in the SIS of Mineral Deposits: A Shear Hosted Gold Deposit 6.7.1 Shear hosted gold deposit drill hole composites 6.7.2 Economic wireframing options 6.7.3 Interpolation based on economic wireframe envelope 6.7.4 Full indicator kriging constrained by economic factors 6.7.5 Geological wireframing options

92 92

75 77 78 78 78 81 81 82 82 83 84 92

94 94 96 96 97 98


Table of contents

6.8

6.7.6

How to use indicator kriging in practice 6.7.6.1 Kriging: order relation corrections 6.7.6.2 Order relation corrections 6.7.6.3 Median indicator kriging 6.7.6.4 Full indicator kriging 6.7.8 Top cut-off related issues 6.7.8.1 Calculation of block grade 6.7.8.2 Value for top cut-off 6.7.8.3 Influence of number of samples 6.7.8.4 A summary of IK derived block grades

98 100 101 103 104 105 106 107 108 109

6.7.8.5 Order relation corrections Sequential Simulations: Implementation Issues 6.8.1 Generating the path for a sequential simulation 6.8.2 Simple or ordinary kriging 6.8.3 Resolution and variogram reproduction 6.8.4 Sequential simulations: Indicator or Gaussian implementation? 6.8.5 Sequential simulations: some comparisons

110 113 113 113 114 115 116

CHAPTER 7

7.1

7.2 7.3

CONDITIONAL SIMULATIONS IN RESOURCES/RESERVE ESTIMATION 121

Risk Quantification and Limits in Resource Estimates: Example from a Nickel Laterite Deposit 121 7.1.1 Introduction 121 7.1.2 Information available 121 7.1.3 Conditional simulation study: methodology and results 122 7.1.4 Resource variability 123 7.1.5 SMU size sensitivity analysis 125 7.1.6 SMU with varying horizontal block size 125 7.1.7 15 m x 15 m SMU with varying vertical dimension 128 7.1.8 Conditional simulation and ordinary kriged models in 250 m x 250 m mining blocks 130 7.1.9 Conclusions 132 Resources/Reserves Classification and Direct Block Simulation 133 Assessing Risk in Grade-Tonnage Curves in a Complex Copper Deposit, Northern Brazil, based on an Efficient Joint Simulation of Multiple Correlated Variables 153 7.3.1 Abstract 153 7.3.2 Introduction 153 7.3.3 Joint Simulation of correlated variables using minimum/maximum autocorrelation factors 154 7.3.4 The deposit and data available 155 7.3.5 7.3.5.1 Joint simulation of copper, iron and potassium Normal-score transformation

7.3.5.2 MAF transformation 7.3.5.3 Variography of MAF 7.3.5.4 Conditional simulation of MAF 7.3.5.5 Back transformation of MAF 7.3.5.6 Validation of the joint Cu-Fe-K simulation results 7.3.6 Risk assessment results and discussion 7.3.7 Conclusions


158 158 158 158 159 160 161 161 167

iii

7.4 7.5

7.6

7.7

7.8

Optimal Drillhole Spacing and the Role of Stockpile Recoverable Reserves and Conditional Simulation 7.5.1 Introduction 7.5.2 The 'F' factors: Evaluating modelling and mining performance 7.5.3 The idea in brief 7.5.4 A case study at the Fort Knox Gold Project, Alaska (Guadino, et all 1997) Assessment of Dilution in an OP Operation with Conditional Simulation 7.6.1 Introduction 7.6.2 Dilution from support effect

169 179 179 179 180

7.6.3 Dilution from information effect 7.6.4 Combining dilution from support and information effect: the F factors Simulation of Shape, Tonnage and Grade of a Uranium Deposit (David, 1988) 7.7.1 Summary 7.7.2 Deposit characteristics 7.7.3 Modelling the mineralised contour 7.7.4 Estimating the tonnage 7.7.5 Simulating and As U3O8 198 7.7.6 Conclusions Economic Reserves in an Underground Mine 7.8.1 General 7.8.2 Economic reserves assessment in a uranium mine 7.8.3 Discussion

192 195

CHAPTER 8

8.1

8.2

8.3

SIMULATION IN OPTIMISING MINING PARAMETERS

Optimising Mining Parameters: An Example from Ore Control in an Open Pit Epithermal Gold Deposits 8.1.1 Introduction An Application at an Open Pit Gold Mine 8.2.1 Problem statement 8.2.2 Study area: selection and characteristics 8.2.3 Some alternatives 8.2.4 Simulation and related intricacies 8.2.5 Specifying the transfer function 8.2.6 Misclassification and relate effects 8.2.7 Economic considerations and response definition: the cost of dilution 8.2.8 Results, evaluation and decision making Evaluation and Limits 8.3.1 Simulations and transfer functions 8.3.2 Decision making

CHAPTER 9

SIMULATIONS IN GRADE CONTROL

9.1 Grade Control and Objective Functions 9.1.1 Problem definition 9.1.2 The traditional approach 9.1.3 Uncertainty assessment 9.1.4 Objective functions for grade control 9.1.5 Minimising $ loss due to misclassification

iv

181 189 189 189

197 197 197 197 198 199 200 200 200 201 201

202 202 203 203 203 205 206 208 208 212 212 216 216 216 217

217 217 217 218 218 219

Table of contents

9.2

9.1.6 The profit/loss approach 9.1.7 Risk coefficients Grade Control Comparisons in a Fully Known Environment 9.2.1 The Walker Lake gold deposit 9.2.2 Point grade conditional simulations 9.2.3 Change of support 9.2.4 Optimum classification

CHAPTER 10

SIMULATIONS IN MINE PLANNING

222 225 226 226 228 232 233 238

10.1 Uncertainty in Pit Optimation and Simulated Orebodies 238 10.1.1 Open pit optimisation 238 10.1.2 An 'real' deposit, exploration drilling, orebody models and statistical characteristics 239 10.1.3 Open pit design and long term planning: an example of avoiding single and possibly precisely wrong options 241 10.1.4 The Echo Bay, Nevada, example (Rossi & Van Brunt, 1997) 246 10.1.5 Issues for discussion 250 10.2 Minimum Down-side Risk & Maximum Up-side Potential in Open Pit Mine Design 251 10.3 Managing Risk and Waste Mining in Long-Term Production Scheduling of Open Pit Mines 261 10.3.1 Abstract 261 10.3.2 Introduction 261 10.3.3 A new risk-based approach to production scheduling 264 10.3.4 Application in a large open pit gold mine 270 10.3.5 Conclusions 273 10.4 Choosing a Mining Method (Deraisme, 1977) 277 10.4.1 General 277 10.4.2 Mining methods 278 10.4.3 Simulation of the mining process 279 10.4.4 Assessing mill feed variability for the mining methods 280 10.4.5 Comments 281 10.4.6 Discussion 281 10.5 Uncertainty Based Production Scheduling in Open Pit Mining 282 10.5.1 Abstract 282 10.5.2 Introduction 282 10.5.3 Production scheduling under grade uncertainty 283 10.5.3.1 Objective function 284 10.5.3.2 Model constraints 285 10.5.3.3 Probability constraints 285

10.5.3.4 Grade Constraints for equipment 10.5.3.5 blending constraintsaccess and mobility 10.5.3.6 Reserve constraints 10.5.3.7 Processing capacity constraints 10.5.3.8 Mining capacity constraints 10.5.4 Production scheduling under uncertainty in a Ni-Co laterite deposit 10.5.4.1 Deposit, data, deposit models and constraints 10.5.4.2 Application 10.5.4.3 Comparison of risk-based and traditional optimal scheduling


286 286 287 287 287 287 288 289 291

v

10.5.5 Conclusions 10.6 Modelling Uncertainty in Short-Term Mine Production Scheduling 10.6.1 Introduction 10.6.2 Short-term production scheduling 10.6.3 The estimated and simulated deposit 10.6.4 Scheduling with a smoothed, estimated block model 10.6.5 Scheduling with a simulated orebody model 10.6.6 Accounting for uncertainty 10.6.7 Comments and conclusions 10.7 Risk Analysis in Underground Stope Design – Sublevel Stoping at Kidd

294 297 297 300 300 301 301 304 305

Creek 306 10.8 Geostatistical Modelling of Ore Textures in Enhancing Ore Reserve Estimation and Planning 318 10.8.1 Introduction 318 10.8.2 Collecting and compositing texture data 318 10.8.3 Characterising spatial texture continuity 319 10.8.4 Simulating textures 320 10.8.5 Simulation paths and ‘growth’ of SIS – 1 321 10.8.6 Local probability corrections 322 10.8.7 Upscaling texture simulations 322 10.8.8 Economic implications 324 10.8.9 Conclusions 325

vi

Table of contents

Appendix A A Formal Look into Geostatistical Concepts

327

Appendix B

Joint Integration of Assay and Crosshole Tomographic Data in Orebody Modelling: Joint Geostatistical Simulation and Application at Mount Isa Mine, Queensland 344

Appendix C

A. Quantification of Fault Uncertainty and Risk Management in Longwall Coal mining: Back-Analysis Study at North Goonyella Mine, Queensland 356 B. Conditional Simulation of Faults and Uncertainty Assessment in Longwall Coal Mining 366

Appendix D Additional Simulations Algorithms in Brief Appendix E A. Stochastic Integer Programming – Introduction Appendix F

377

B. Conventional Mine Production Scheduling with MIP IXED ine Supplement - Technical papers on A. New and efficient simulation methods B. Stochastic optimization models for

Production Scheduling Stope Design Production Scheduling with SIPs

References

380

Workshop Notes


vii

Chapter 1

CHAPTER 1

1.1

INTRODUCTION

Geostatistical Textbooks

Several Geostatistical textbooks are available today. Although several general textbooks are widely available, textbooks on mining geostatistics are limited and somewhat out dated. The list below includes textbooks of varying levels of complexity. The last book is a useful dictionary for anyone who may be confused by geostatistical jargon terms. and the human ability to increasingly rename the same concept with different

•

•

•

•

•

•

•

•

•

•

Geostatistical Ore Reserve Estimation, David, M., Elsevier Scientific Publishing Co., Amsterdam, 1977. Handbook of Applied Advanced Geostatistical Reserve Estimation , David, M., Elsevier Scientific Publishing Co., Amsterdam, 1988. An Introduction to Applied Geostatistics, Isaaks, E.H. and R.M Srivastava, Oxford University Press, NY, 1989. Applied Mineral Inventory, Sinclair, A. J., Blackwell, G. H., Cambridge University Press, 2002. Geostatistics for the Next Century, Dimitrakopoulos, R., Kluwer Academic Publishers, Dordrecht, 1994. Fundamentals of Geostatistics in Five Lessons, Journel, A.G. Short Course in Geology, v.8, AGU, 1989. Mining Geostatistics, Journel, A.G. and Huijbregts, Ch.J., Academic Press, London, 1978. GSLIB Geostatistical Software Library and User's Guide, Deutsch, C.V. and Journel, A.G., Second Edition, Oxford University Press, NY, 1997. Geostatistics for Natural Resources Evaluation, Goovaerts, P., Oxford University Press, NY, 1997. Geostatistics – Modelling Spatial Uncertainty , Chiles, J-P. and Delfiner, P., Wiley, 1999 Geostatistical Glossary and Multilingual Dictionary, Olea, R., Oxford University Press, New York, 1991.


1

Chapter 1

1.2

1

I’d Rather be Approximat ely Right than Precisely Wrong : Key Reasons for the Quantitative Modelling of Geological Uncertainty and Risk

The present notes are about the conditional simulation of ore bodies and their use in the mining industry. But why bother with these technologies? A short response is that quantification of geological uncertainty and risk can substantially enhance mining project development and mining operations. One way to perhaps consider supporting this statement without reading to the end of the notes, is to consider modern project valuation frameworks. These frameworks canexample elucidateisthe economicframework effects of the quantification of uncertainty and risk. An thepositive “Real Options” (e.g. Amram and Kulatilaka, 1999; Moel and Tufano, 1998) a key characteristic of which is the ability to integrate and manage uncertainty and risk, thus enabling the sheltering of strategic investments while exposing their upside potential. In simple terms, real options may be described as the ability to assess the value of starting a project that gives the right, but not the obligation, to commence operations at a cost of, say, $7M six months from now; and/or to assess the value of delaying production to obtain additional information to reduce uncertainty; or to quantify the value of building in the flexibility to manage uncertainty and risk at any level or aspect of an exploration or mining venture. Figure 1.1 summarises a comparison between a traditional valuation method that does not account for uncertainty and risk, discounted cash flow (DCF) analysis, and real options in assessing current asset value. The figure shows an increase in asset value from the simple step of explicitly quantifying uncertainty and integrating this uncertainty into financial analysis and decision-making. The need to quantify uncertainty in asset valuation and decision-making translates to the need to quantify uncertainty and risk in any pertinent parameters (components). Project risk may arise from three main sources, namely, technical (geological and mining), financial and environmental. The major source of technical risk is uncertainty in grades, tonnages, geology, and geomechanics. Geological risk is seen as the major contributor to not meeting project expectations. For example, at the early stages of a project when establishing investor confidence and repayment of development capital are vital, Vallee (2000) notes that “…in the first year of operation after start-up, 60% of mines surveyed had an average rate of production less than 70% of designed capacity.” While shortfalls in production are also due to problems in scale-up from pilot plant to commercial plant, the quantity and grade of ore are a major contributor to potential shortfalls. Shortfalls from mine production predictions are also common in later stages of production and are substantially contributed to geological reasons (eg Rossi and Parker, 1994). For any mining aspect that involves the orebody, uncertainty on grades, tonnages, geology can be readily modelled and integrated to the modelling, evaluation, design and planning process so as to provide accurate modelling and quantification of uncertainty and risk, rather than a single estimate assessment, for any pertinent parameter, including the project NPV, expected cash flows, gold ounces, and expected production costs. This provides the ability to develop a different, technically sound, risk based approach to valuing an asset, operation or project as well as quantify, and thus minimise, risk.

1

2

Paraphrased form J. M. Keynes, Economist


Chapter 1

Certainly, as the persistent reader of the notes will be convinced that the same concepts can be critical to a mining operation in a multitude of aspects.

$+

Real Options View: Current view of option to produce

e lu a V t e s s A$0 t n rre u C

$ -

Contingent Decision Pay off Function (future price is known)

No production NPV < 0

Production: NPV > 0

Future Gold Price

Traditional DCF View (now, or never.)

Figure 1.1: Accounting for uncertainty increases “Asset Value”: An example using real options, solid black line, where the possible future changes (uncertainty) are accounted for, versus traditional (deterministic) DCF analysis, shown as a dashed grey line.

1

tiy il b a b o r P

Unknown, true value

Reserves

Accurate Uncertainty Modelling

Single, precise, wrong estimate

Reserves

Figure 1.2: Accounting for uncertainty strives for the accurate quantification of uncertainty for the various components effecting mining decisions. This increases the

chances of including the actual but unknown values in the uncertainty models, thus benefiting asset valuation and subsequent decisions. Figure 1.2 illustrates an assessment of uncertainty for a parameter that may, for example, be the ore reserves in a gold mine. Accounting for uncertainty requires the accurate quantification of uncertainty and risk; thus, technical work and evaluation should stress accurate modelling and quantification of uncertainty and risk, not a single estimate or a qualitative type assessment. The technical ability to quantitatively model uncertainty as accurately as possible with the information available at a given time is a


3

Chapter 1

recognised need of paramount importance. Conditional simulation technologies offer a first key step in modelling geological risk that is subsequently transferred downstream. This downstream interaction is critical to the simulation framework, definitive of its practical use and technically challenging as much as rewarding. The next section introduces the conditional simulation concept through traditional ore body modelling and its shortcomings. Chapter 2 introduces the general simulation framework as it has been developed to date. Chapter 3 provides an intuitive approach to the concept of Monte Carlo simulations in mining. Chapter 4 summarises the statistical description of mining data sets. Chapter 5 reviews briefly conditional simulation concepts and techniques. Chapter 6 provides an extensive description of sequential simulation algorithms in a practical way. Chapter 7 is concerned with applications in ore reserve estimation. Chapter 7 extends to the optimisation of mining parameters. Chapter 9 is allocated to grade control and economic functions. Chapter 10 is dedicated to several applications in mine planning including mine design and production scheduling. The key maths of the conditional simulation framework, and several advanced techniques with applications are included in appendixes.

4


Chapter 1

1. 3

Traditional Orebody Shortcomings

Modelling:

1.3.1

An Example from a Stockwork Gold Deposit

Some

Limits

and

The stockwork gold deposit has most of its mineralisation in a 400 x 500m quartz diorite pluton intruding andesitic volcanics with gold in narrow quartz/calcite/pyrite veins dipping 20oNW. About 400 vertical or inclined holes with approximately 25-30m average spacing are available. The operation is open-pit mining on 5m benches. Reserve estimation is based on the geological model and recognised unit contacts, mostly diorite/volcanics on drill sections and benches. Each rock type is modelled using 10 x 10 x 5m blocks, with the gold grade of each block interpolated from the nearby 5m bench composites in the same envelope. We always show the same test bench (#61) with about 1.15Mt diorite material. Resource numbers in Table 1.1 are just for that bench (resources in all the diorite are a bit more difficult to compare since some sectors are not that well documented). There are 77 bench composites and 848 blocks of diorite in the test bench. Figure 1.3 shows the 77 bench composites as well as various gold grade estimates for the 848 10x10m blocks. All the estimates are derived from the same set of 3190 5m bench composites in diorite. The purpose of this figure is to show the simplistic images of gold variation in the bench that is obtained from standard estimates.


5

Chapter 1

Figure 1.3: Interpolated Au grade of test bench (grey scale limits = 0.7 and 1.3 g/t).

6


Chapter 1

Table 1.1 shows resource estimates for the bench as derived by adding the various block estimates. It can be seen that the estimated average grade at no cut-off is rather insensitive to the block interpolation method used (estimates range from 1.2 to 1.41 g/t) but estimates of tonnes and grades above a cut-off vary widely with the method used (a [46.9-32.9]/46.9 = 30% maximum variation for tonnage above 1.3 g/t and a [2.921.82]/2.92 = 38% maximum variation for average grade above the same cut-off).

Au>=0g/t COMPOS 1.41 POLY2D 1.24 POLYANIS 1.24 ID5 1.25 ID2 1.24 ID1 1.29 OK 1.20 MIK 1.30 Table 1.1: Diorite resources bench composites.

%<0.7g/t 42.8 53.5 48.7 28.5 21.1 17.0 21.5 19.9 of test bench

0.7<%<1.3g/t %>=1.3g/t 18.2 39.0 13.6 32.9 16.5 34.8 31.5 40.0 35.1 43.8 36.4 46.6 40.3 38.2 33.2 46.9 from 10x10x5m blocks estimated

Au>=1.3g/t 2.86 2.92 2.72 2.07 1.83 1.82 1.86 1.88 from 5m

COMPOS = statistics of the 77 composites in the test bench POLY2D = nearest-neighbour from composites in the same benches POLYANIS = nearest-neighbour within 100x100x25m ellipsoid tilted 25 o to NW ID5,ID2,ID1 = inverse distance to power 5, 2 or 1 with maximum of 50 composites in the same search ellipsoid OK = ordinary kriging MIK = median indicator kriging It is important to note that the average gold grade at no cut-off is rather insensitive to the interpolation method ranging from 1.20 to 1.41 g/t. However, predicted recoverable resources vary widely with the interpolation method used, as expected. For the 1.3 g/t cutoff, the tonnage recovery varies from 32.9% to 46.9%, and about 37% variation! The average grade above this cut-off jumps from 1.82 g/t to 2.92 g/t. Lastly, the low tonnages correspond to high grades and are obtained with restrictive interpolation conditions, i.e. polygons or restricted distance-weighting.

1.3.2

Comparison to the “real”

Figure 1.4 shows all the 3886 blast holes ultimately drilled in our test bench and within the srcinal limits of the diorite (those srcinal limits, interpreted from exploration drill holes, includes BHs which are tagged as non diorite and some BHs tagged as diorite are outside the srcinal limits). Nominal BH grid is 4x4m and for a better visualisation of BH grade variation, they are gridded on a perfect 4x4m grid. Averaging BHs in 10x10m blocks ( 6 BHs in a block on average) provides "nearly true" block grades which can be compared to the various estimates derived from exploration drill hole bench composites. Figure 1.4 clearly shows that reality is more complex than what was estimated by standard interpolation methods.


7

Chapter 1

Figure 1.4: BH values and interpolated Au grade from DH composites in the test bench.

Table 1.2 compares resources above cut-off in the pit portion of the test bench as derived from BHs and as estimated from exploration drill hole bench composites and various block grade interpolation methods. Reference values (REF BH) are obtained by adding BH block averages (655 blocks with at least 4 BHs). Again, all methods do an acceptable job in predicting the average grade at no cut-off ("real" 1.42 g/t vs estimates from 1.36 g/t to 1.46 g/t) but polygonal estimates are obviously "under-diluted" (too much waste) and distance-weighting estimates are "over-diluted"(too little waste).

8


Chapter 1

Au>=0g/t

%<0.7g/t

0.7<%<1.3g/t

%>=1.3g/t

Au>=1.3g/t

REF BH 1.42 33.5 26.3 40.2 2.56 POLY2D 1.46 44.2 16.1 39.7 2.92 POLYANIS 1.46 42.6 18.0 39.4 2.78 ID5 1.41 19.6 34.8 45.6 2.10 ID2 1.44 10.2 34.5 55.3 1.85 ID1 1.46 6.8 36.9 56.3 1.84 OK 1.36 12.3 41.8 45.9 1.88 MIK 1.46 10.4 34.3 55.3 1.91 Table 1.2: Diorite resources in 655 reference blocks (with at least 4 BHs) of test bench

REF BH = Average BH in block (minimum 4 BHs) POLY2D = nearest-neighbour from composites in the same benches POLYANIS = nearest-neighbour within 100x100x25m ellipsoid tilted 25o to NW ID5,ID2,ID1 = inverse distance to power 5, 2 or 1 with maximum of 50 composites in the same search ellipsoid OK = ordinary kriging MIK = median indicator kriging

1.3.3

Comments

What if we could somehow “simulate” realistic images of the blasthole data or even the actual orebody grades? Then we could, for example, calibrate the recoverable resource/reserve modelling or perhaps find ways to generalise aiming to assess uncertainty in various aspects of resource/reserve modelling and/or optimise various orebody dependent mine operations or ....... Figure 1.5 compares the actual gold grade variation of BHs with another mapping of gold grade from exploration drill hole bench composites but this time using conditional simulation instead of the standard block interpolation techniques. Note the similitude in the "style" of variation. Obviously with conditional simulation, the zones of high and low grade are not necessarily situated in the correct location. This uncertainty about location is reflected by the shift in position from one simulation to the next.


9

Chapter 1

Figure 1.5: Grade variations in BHs and simulated from DHs on the same 4x4m grid.

Table 1.3 show resources above cut-offs in the reference blocks with at least four BHs after reblocking simulated grades in those blocks. As a general rule, estimated proportions in the various categories are better than those derived from standard block grade interpolation methods. Over the five realisations simulated, there is an apparent overestimation of average grade at no cut-off however the last simulation shows that some simulated average grades can be quite low.

Au>=0g/t

10

%<0.7g/t

0.7<%<1.3g/t

%>=1.3g/t

Au>=1.3g/t


Chapter 1

REF BH 1.42 33.5 26.3 40.2 2.56 SGSIM1 1.68 30.4 25.3 44.3 2.96 SGSIM2 1.47 31.1 30.4 38.5 2.74 SGSIM3 1.57 26.4 27.2 46.4 2.58 SGSIM4 1.51 29.9 29.3 40.8 2.66 SGSIM5 1.28 34.7 31.2 34.1 2.43 Table 1.3: Diorite resources in 655 reference blocks with BHs and with simulation from DHs.

Figure 1.6 shows the results of the traditional conditional simulation technique known as the turning band method, using 20 equally spaced lines in 3D space. Note the artefacts generated by the limited number of lines used. These artefacts are not apparent with the sequential Gaussian method.

Figure 1.6: BH and simulated bench composites from various methods.


11

Chapter 2

CHAPTER 2

2.1

A GENERAL CONDITIONAL SIMULATION FRAMEWORK

Linking Orebody Uncertainty to Mining, Decision Making and Profitability

Considering the shortcomings and limits of the ‘traditional’ orebody modelling approaches, it may be worth rationalising a Monte Carlo simulation based alternative (Halton, 1970). The simulation alternative may: (a) enhance the results of older approaches Correction factors for dilution Selection of estimation algorithm ......... (b) provide better solutions to old problems Recoverable reserves Grade control ........ (c) generate new approaches to mining practices that were not attainable before Modelling uncertainty of geological and mining parameters, e.g. mill feed variability, stockpiling Profitability based optimisation, e.g. grade control, stope design ........... Components of a generalised simulation approach include: (a) A model of the uncertainty associated with the orebody. (b) A model of a mining process or mining transfer function. (c) A model of the uncertainty of expected outcomes and mining possibilities. (d) Optimisation of decision making based on desirable criteria. To elaborate further, this section outlines a general framework proposed to deal with the uncertainty and risk of geological attributes of an orebody and their down stream effects, that is uncertainty effects from ore reserves to production. The framework includes three parts: (i) stochastic conditional simulations; (ii) mining transfer functions; and (iii) uncertainty modelling and risk assessment.

12

Applied Risk Assessment for Ore Reserves and Mine Planning: Conditional simulaton for the mining industry

Chapter 2

2.2

Modelling the Uncertainty Associated with an Orebody

Drilling data represents initial information about a given mineral deposit and are used for deposit block modelling. Block models are subsequently used for resource/reserve calculations, mine design and planning, etc. The actual deposit is partially known and the properties of interest such as grades and ore material types are inferred. Can one generate/simulate several block models (images) to deal with the uncertainties of the unknown deposit and its attributes of interest? Geostatistical or stochastic conditional simulation is the tool that generates block models of the ore deposit based on and conditional to the same data and statistical properties. These models represent the same deposit and are all constrained to (a) reproducing all available information, and (b) being equally probable representations of the actual deposit. A series of simulated models of the deposit can represent or capture the uncertainty about the actual description of the deposit. (Figure 2.1).

Actual but unknown Information about mineral de osit the

Probable models of the deposit Deposit model 1

Mineral

Data/Information

Deposit model 2

Deposit model m

Figure 2.1: Description of the uncertainty about a mineral deposit.

Two questions follow: (a) how to simulate equally probable models of the deposit? and (b) how these models can be used to solve specific mining problems? The first question is addressed in the general context of geostatistical or stochastic simulations (Journel and Huijebregts, 1978; Dagbert, 1978; Alabert, 1987; David, 1988; Riplay, 1989; Isaaks, 1990; Dimitrakopoulos, 1990; Verly, 1992; Dietrich, 1993; Journel, 1994; Deutsch, 1994; and others). The second question is addressed next and in the general context of mining transfer functions and optimal decision making.

2.3

Transfer Functions and the Modelling of a Mining Process


13

Chapter 2

A mining process or a sequence of processes, such as open pit design and production scheduling can be conceptualised as a complex transfer function. The specification of a transfer function depends on the problem under consideration. Examples: • • • • • • • •

the grade control and ore classification in a gold mine, additional drilling programs, recoverable reserves, mine optimisation, design and planning, short term scheduling, stope design, and others, etc mill feed quality and blending ..............

Mining processes or transfer functions have specific parameters of interest for analysis or optimisation, for example, open pit optimisation includes parameters such as the maximum NVP pit shell, discounted cash flow, average mill feed grade, recoverable reserves. The output of a transfer function providing the parameter of interest to be analysed and assessed, and upon which decisions will be made is termed a response parameter . Examples of response parameters may include a series of dollar values of possible ore/waste dig lines for ore control, or, in the case of pit optimisation, series of pit shells, cash flow versus pit shell curves, NVP changes, tonnage to the mill feed, and so on.

2.4

Modelling Uncertainty about the Response

For a given possible description of a mineral deposit, a set of possible values for a parameter of interest may be selected. A computerised mining process of interest (the transfer function) can then be applied for each of the selected values. Depending on the mining process at hand, the selection of parameters of interest can be formulated as an optimisation problem where the objective is to maximise profitability. For a set of possible deposit models and each value(s) for the parameter(s) of interest, the transfer function will generate a sequence of distributions of responses that can be seen as a map of the space of uncertainty of the response/parameter. If this mapping is adequate, then the optimisation function will provide the expected results in terms of possible outcomes, ranges of expected values and optimal choices. Figure 2.2 schematically presents the various parts of the methodology suggested.

14


Chapter 2

Possible Deposit Descriptions

M n ng Process

Response Parameter

Response 1 Deposit model 1

Mining Process

Response 2

Deposit model 2 Deposit model m

Parameters of

Response m

interest

Map of Response Uncertainty

Response Distribution

Figure 2.2: Diagrammatic representation of the proposed simulation framework.

2.5

Two Important Points

As noted earlier, mining transfer functions are generally non-linear. As a consequence, (i) an average type block model may not provide an average map of the space of response uncertainty; and (ii) a criterion for generating deposit descriptions may be defined: the simulation technique selected for modelling must be evaluated in terms of the mapping of the response uncertainty. Point (i) above suggests caution when analysing the results of a study involving complex mining processes and predictions. Although predictions can in practice be reasonable, they do not necessarily capture all aspects of the orebody related uncertainties. Point (ii) indicates that one can study the specific effects that different simulation algorithms may have to the specific parameters of interest for a given mining process. Then one can select the algorithms that provide adequate mapping of the related space of uncertainty, thus generate optimal results.


15

Chapter 2

2.6

Some Examples

2.6.1

Uncertainty i n the descrip tion of an orebod y: orebod y simulatio ns

Figure 2.3 shows an example of four conditional simulations of accumulation (the product of thickness and grade) of an uranium roll front deposit. All four simulated models are based and reproduce the exploration drilling data, and their statistics and spatial continuity and are equally probable models of the actual accumulation of the deposit.

LLow High

Figure 2.3: Four conditional simulations showing uncertainty in orebody description.

16


Chapter 2

Figure 2.4: Conditional simulations of gold deposit in diorite (right side) and comparisons to kriging (top left) and blasthole data (bottom left).

Figure 2.4 shows two realisations of a stockwork gold deposit in diorite based on the exploration data shown in Figure 1.3. The grade simulations are compared to blasthole data and kriging. Note the similarity in variability between the blasthole data and the simulations compared to the smooth kriged model. Grades are not the only attribute to simulate. Figure 2.5 shows simulations of geology and Figure 2.6 simulations of ore textures.


17

Chapter 2

Figure 2.5: Simulations of geology (figures from the Centre de Geostatistique).

Figure 2.6: Conditional simulations of ore textures. 2.6.2

Spaces of uncertainty, their mappin g, and decision mak ing

Figure 2.7 presents the results of a simple transfer function ‘areas for additional drilling’, constructed from the combination of 100 simulations of the uranium roll front deposit discussed in the previous section. The black areas on the block model indicate the parts with a greater than 80% chance that accumulation is above an economic cutoff. Light grey outline areas with an over 80% chance that accumulation is below the economic cut-off. The intermediate grey areas are those where additional drilling would most likely provide useful information. Similarly, a map of the probability of being above the economic cut-off in a deposit may be useful in evaluating or classifying resources/reserves, and so on.

18


Chapter 2

Figure 2.7: Probability map of being above the economic cut-off.

Several mining transfer functions and the mapping of ‘spaces of uncertainty’ are given in Figure 2.8 and Figure 2.9. .. 7.968 8.466

8.0925 8.5075

. . . . Blasthole Spacing and Expected Cost 8.9225 8.881 8.4245 9.047 9.213 9.1881 8.881 9.379

. 9.047 9.545

. 8.549 8.881

10 ) n o t/ $ ( t s o C

Simulation Simulation Simulation Simulation

9

1 2 3 4

median average

8 7 12

16

20

24

28

Blasthole Spacing (ft) Figure 2.8: Evaluating dilution in blasthole spacing, space of uncertainty of mining cost in a gold deposit.


19

Chapter 2

Net Present Value Analysis 30,000,000

25,000,000

$ A20,000,000 w o L F h s a 15,000,000 C d te n u o c is 10,000,000 D

5,000,000

0 0

5

10

15

20

25

30

35

40

Pit Number

Figure 2.9: Traditional smooth grade models (crosses) and conditionally simulated models (lines) in mapping possible NVP in the optimisation study of open pit gold mine.

A Question

Discuss the idea of linear and non-linear mining transfer functions based on the previous example.

20


Chapter 3

CHAPTER 3

3.1

AN INTRODUCTION TO MONTE CARLO SIMULATIONS

A House Builder’s Anxiety or Will the Big Man Go through the Floor? (A Heavy Friend’s example for his Heavier Friend)

Let’s say that a friend is building a house and that he is a great fellow but real cheap and also real smart. Can he assess the bare minimum load the floor could support without a fat man with small feet going through the floor? This sounds simple: •

Consider the threshold load, say 200kg per square foot, that the cheapest floor can handle.

•

Find the population’s statistics in terms of shoe sizes and weight, and construct their histograms. Alternatively, it may be easier to generate histograms somehow, knowing the more general shoe/weight population characteristics, e.g. simulate them.

•

Pick a person (shoe size/weight combination) and see whether when standing on 2 one foot the pressure from his/her body exceeds the threshold of 200kg/ft . If it exceeds the threshold, count it as a floor failure

•

Repeat the experiment countless times, say 100,000 or so and count the number of failures.

To complete the above points, it is important that the experiment is based on sound population statistics and honours: • • •

the histogram of the population’s shoe sizes (Figure 3.1) the histogram of the population’s weights (Figure 3.2) the correlation between the shoe size and weight (Figure 3.3)

Figure 3.1: Histogram of the population’s shoe sizes.


21

Chapter 3

Histogra m of the popula tion's body weights 35 n 30 o i t a l 25 u p o 20 p f o 15 t n e 10 c r e P 5

0

50

75

100 125 Body weight (kilograms)

150

Figure 3.2: Histogram of the population’s body weights.

Figure 3.3: Correlation between the shoe size and weight.

3.1.1

Drawing values from the histogram a variable (shoe sizes)

Drawing from any histogram, is a simple process (Figure 3.4): 1. Transform the histogram into a cumulative distribution. 2. Generate a uniform random number between 0 to 1. 3. Read from the cumulative distribution proportion axis across and down to the variable axis.

22


Chapter 3

Figure 3.4: Drawing values from the histogram a variable (shoe sizes).

A large number of drawings will reproduce the srcinal histogram.

3.1.2

Drawing values for the correlated variable (weights)

When two variables correlate, their drawing is a bit more sophisticated as the second variable must correlate with the first F ( igure 3.5). Obviously, independent drawings from histograms will not work. How can the correlation bereproduced?

1. Sample the scatterplots describing the conditional distributions from which the second variable should be drawn. 2. Conditional distribution of body weight given the shoe size.


23

Chapter 3

Figure 3.5: Drawing values for the correlated variable (weights).

Back to our cheap floor and the fat man with the small feet. It is easy now to see how our smart floor builder can set up his problem, quantify all the related factors and assess the probability of a fat man with small feet going through his floor.

24


Chapter 3

In fact, an answer to the problem is that the probability that the floor will be lost to the fat man is a bit less than 1 in a 10,000. What should our friend do? Some ideas:

define criteria, e.g. the cheapest floor is the best, or safety first cost next or safety at affordable cost or ....... take the optimal decision - decide based on his criteria There can be additional constraints . . . . .

3.2

Generating Correlated Variables

From the last example, one may conclude that to generate sequences of correlated pairs of values one may simply: •

draw a value from the distribution of the first variable global distribution unconditional distribution

•

given the choice of the first variable draw the second one from its conditional distribution

If there is a third variable, then given the first and the second variable choices draw from the conditional distribution of the third, and so on. This all can be based on simply large numbers of data, or the distributions - conditional or unconditional can be simulated. The basic ingredients in simulating distributions comes from: Random numbers, data, and possibly theoretical distribution models. Two of the common and convenient theoretical distribution models are the normal or Gaussian and the lognormal, and are examined next for the bivariate case.


25

Chapter 3

3.2.1

Simulating bivariate normal data

Figure 3.6: An old example of bivariate normal data (Krige, 1951).

For generating bivariate normal data, one needs normal data distributions - both marginal and conditional (Figure 3.6). In the previous example or in a more mining related one with, say, two assay values, g/t Au and %Cu for each assay. If we generalise and consider two variables X and Y where X is normal with mean mX and variance σX2 2 Y is normal with mean m Y and variance σY then the following relations are known from statistics:

Given values of X, Y is normally distributed with conditional mean from the regression: 2 X

Y = μY + σXY.(Y-μX)/σ and conditional variance 2

2

2

2

2

σY - (1- ρ ) or σY -(σXY /σX ) σXY is the covariance of X and Y

26


Chapter 3

ρXY=σXY/(σX. σY) is the correlation coefficient of X and Y The parameters defining the bivariate normal distribution model are mX, m Y, σX, σY and σXY or ρXY. Having a model, makes it easy to generate correlated pairs of values from the related parameters. 3.2.2

Simulating bivariate lognormal data

The bivariate lognormal or bilognormal model is an extension of the Gaussian, where the variables are Ln(X) and Ln(Y) (Figure 3.7). It is also defined by five parameters. To simulate pairs of bilognormal data, one generates bivariate normal data and transforms them to the required bilognormal. The relation between normal and lognormal data is straight forward. To generate lognormal data with mean α variance of 2 2, β , from a normal data set with mean μ and variance σ one uses the formulae: 2

2

2

σ = ln(β /α+1)

and

2,

μ = ln(α) - σ /2

Figure 3.7: Example of simulated bivariate data produced. Ln(X) and Ln(Y) are normal with mean 0 and standard deviation 1. Correlation coefficient is -0.65.


27

Chapter 3

3.2.3

Simulating multivariate normal data

The simulation of series of correlated data is an extension of the bivariate case. Some of the main geostatistical simulation algorithms are based on this extension and are discussed in a following section. When more inter-variable correlations are considered, the Gaussian models are generally more tractable. The multivariate normal case will be revisited in a subsequent section (see sequential Gaussian simulation). A smart alternative in generating series of correlated Gaussian variables is to consider their principle components: they are independent. This is an interesting property we will examine in a subsequent section discussing the simulation of an iron ore deposit.

3.3

What have we seen so far?

We can analyse an important problem and effectively assess possibilities. Mining and orebody related problems may be more serious and elaborate than our friend’s cheap floor, but they can also be formulated as a Monte Carlo type simulation. In a simulation context, the various possibilities or uncertainty can be quantified and effectively assessed. Furthermore, one may add criteria for optimal decisions. This is of particular interest to mining applications; profitability is the reason for mining ventures. The basic idea has been to draw values from the distribution of a pertinent variable. If a second correlated variable is considered, it must be drawn taking into account the value of the first, i.e. drawn from the conditional distribution. This in an orebody related study could be at least three things, for example: •

Au and Cu grades in a porphyry copper deposit.

•

Cu grades of this deposit split into two according to a distance.

•

Possible Cu grades in a mining block given the existence of drilling data around the block.

•

Distributions are based on DATA and their pertinent characteristics. Similarly to fat people weight in the example, grade distributions in a deposit MUST be representative and based on data (declustering is a related topic).

We have seen ways to simulate distributions of variables, variable values and values of correlated variables using a theoretical model, either normal or lognormal. In all this, the ability to put together a simulation study relies on a basic ability: the random drawing from a distribution based on a random number generator. Grades and characteristics of mineral deposits have one major difference from the shoe sizes and weights of our example: grades are tied to spatial locations and are spatially correlated. The spatial dependence of grades of mineral deposits and the related spatial correlations are a major element of geostatistical simulations. Optimality criteria are another key

28


Chapter 3

aspect of decision making in mining related problems. Their definition is problem & operator dependent, however there are classes of approaches that have common characteristics.

In mining, optimality criteria are integral/inseparable part of geostatistical simulations.


29

Chapter 4

CHAPTER 4 4.1

DATA DESCRIPTION

The Importance of Outliers

Outliers or ‘abnormally’ high grades are frequent in mineral deposits. Highly skewed grade distributions as shown in Figure 4.1 are common. Ore resource/reserve and orebody modelling are highly sensitive to the presence of even very few extreme values. 4.1.1

Some observations

Consider a mineralisation (e.g. Au) with

median: mean: variance: CV:

3.16 ppm 10 ppm 900 ppm2 3

1.14% of values >100ppm

F

3.16ppm

10ppm

100ppm

Grade

Figure 4.1: Mineralisation.

Proportion of data > 100ppm: ~ 1.14%, mean grade 196 ppm 19 times the mean grade 62 times the median grade 2

Contribution to variance:

(196 - 10) x 0.0114 = 394 .4 or 394.4/900 = 43.8%

Contribution to quantity of gold:

196 x 0.0114 / 10 = 22.4%

Selective mining with cut-off of 3.16 ppm (median) recover

50% of deposit ore - mean 18.7 50% of deposit waste

When outliers are cut out the ‘new mean’ is (18.7x0.5) - (196x0.0114) /0.5 x 0.0014 = 14.65 or 22% underestimation When outliers are set to 100 ppm the ‘new mean’ is (14.56 x 0.4886) + (100 x 0.0114) / 0.5 = 16.51 or 12% underestimation

30


Chapter 4

4.1.2

The traditional approach in dealing with outliers

There is no unique solution used to deal with outliers, at least not in the traditional geostatistical approaches. Empirical criteria are mostly used to cut high values. An alternative is to correct outliers based on a distribution model. Figure 4.2 below shows how outliers in an uranium deposit are scaled back assuming that they follow a lognormal distribution.

Figure 4.2: Cumulative frequency curve of grades and a correction of outliers based on a log-normal distribution (David, 1987).

The ‘scaling’ approach can be enhanced by using geological zones as shown in Table 4.1 below. Zone Outlier cut off grade (% U 3O 8)

I 0.3

II 8.0

III 1.7

2

2

2

Maximum outliers

2.12

10.335

10.000

Corrected maximum

1.00

9.00

2.00

Reduction of mean grade (%) -44% Table 4.1: Geological zones used in ‘scaling approach’.

-3%

-6%

Number of samples above

As far as estimation goes, several approaches may be used to deal with outliers. Indicator kriging may be one. However, simulations are a different type of an algorithm. Dealing with outliers has to be examined along with the algorithm being used. Like in estimation, the estimation of a ‘declustered’ data histogram is a key aspect in conditional simulations.


31

Chapter 4

4.2

Estimating a Representative Data Histogram

The natural tendency and recommended practice is to drill more in high grade areas in order to prove them better. As a result, exploration data are clustered. In specific cases, like a vertically dipping deposit, it is less expensive to drill close to the surface. In regular underground fan drilling the data are clustered close to the drift. When data are not on a regular grid or follow a totally random pattern, such as the typical exploration drilling data (Figure 4.3), the straight use of the full data set will provide a biased estimate of the population mean, variance and, in fact, histogram.

Figure 4.3: Irregular drilling pattern in an uranium deposit (David, 1984).

Any calculation of data statistics from clustered data endangers the reserve/resource calculations.

Figure 4.4: Longitudinal projection of mineralised drill hole intersections from a gold deposit (Sergeri, 1983).

32


Chapter 4

The intuitive solution is to weigh each datum according to its polyhedron of influence as shown in Figure 4.4. However this is far from being practical. Declustering refers to the process of calculating data statistics by ignoring some of the samples in the overdrilled area. There are several declustering algorithms, the major ones are discussed next. 4.2.1

Declustering techniques

Random pick method: Superimpose a regular grid (usually about the size of the average data spacing) on the study area considered and pick at random within each grid cell a single sample in cases where there are more than one.

Averaging method: The values in the cell are averaged. Either all of them or up to a fixed number. The problem of the technique is that there is no unique solution. Different grid sizes will generate different results. Blind averaging is generally not recommended, as it tends to smooth out the data. Figure 4.5 illustrates an example showing the effects of declustering is based on the use of a simulated copper deposit (Helwick et al 1983), with mean of 0.821%Cu and variance 0.475.

Figure 4.5: True average grades of 11m by 11m blocks of the Cu deposit (left); and clustered drilling grid (right).


33

Chapter 4

Figure 4.6 shows the histograms of: (a) the whole deposit shown above, (b) a regular drilling grid, (c) a clustered grid, and (d) a declustered grid using an equal weighting of the data in the grid. The mean and variances are on the figures.

Figure 4.6: Histograms of data sets from a simulated Cu deposit.

Differences in the uranium deposit shown above was 40%! In this case the solution was to retain the cell size equal to the kriged orebody (see details in a following section).

Average grade versus grid size: When rich areas are over represented , plot the average samples available versus the cell size and select the cell size that provides the minimum average grade.

Example: Figure 4.7 shows the declustered mean in a 2D study as a function of the grid sizes in EW vs NS. A grid size of 20 by 24 generates the minimum average, which is 288 compared to the actual mean of 277.

34


Chapter 4

Figure 4.7: Plot of grid sizes versus mean (from Isaaks and Srivastava, 1992).

Global Kriging method: Uses the declustered grid size which gives the same mean as the global ordinary kriging and, possibly, weighs the samples with the ‘global’ kriging weights. In real case studies, global kriging is not feasible. However, there is a reasonable short cut for this problem is as follows: 1.

Do local block kriging throughout the area.

2.

Retain the weights each sample takes when used in estimating various blocks.

3.

Standardise the accumulated weights by the total number of block estimates.

4.

The weights can be used to calculate a global variance.

This can be a time tedious exercise. It is also dependent on the variogram used.

Nearest Neighbour: Uses the nearest neighbour and the fixed grid size to be used for block modelling. This is a common approach. COMMENT: There is no strict unique solution. Any solution should be based on understanding of the problem and a series of trials. The importance of data declustering in various simulation algorithms, is detailed in a subsequent section.


35

Chapter 4

4.3

Univariate and Bivariate Data Description

Routine grade data analysis is based on the graphical representation of the data set and the calculation of simple summary statistics. This section presents an overview of data description based on both parametric and nonparametric statistical measures and indicates the advantages and disadvantages. Data summaries are important in ore reserve estimation. Identification of sample characteristics is a key factor influencing the estimation results as well as geostatistical approaches and validation of results. The term “univariate” refers to the analysis of a single attribute (Cu grade or Pb grade) without reference to any other attribute of the deposit or the location of the samples. The term “bivariate” refers to the description of the relations and dependencies between variables (e.g. Cu and Au in a polymetallic deposit). There is a clear trade-off in summarising data sets: summary statistics are portable and easy to comprehend, but for the same reason they may be oversimplistic (Figure 4.8). The use of statistical distribution models as shown by the histogram of Figure 4.9 (normal, lognormal) is inherent to summarising data sets with as few parameters as possible.

4.3.1

Univariate description

Summarising Data Sets:

Pb (ppm) 192 88 302

180 measurements of Pb concentration

• •

⇓ Sample Histogram Figure 4.8: Listing of sample values.

36


Chapter 4

Figure 4.9: Counting of samples in various classes.

⇓ Summary Statistics Mean (m) = 430 ppm Standard deviation (σ) = 997 ppm

A reporting of various measures

Median (M) = 158 ppm Skewness (sk) = 6.5

4.3.2

•

Summary statistics

Advantages - very condensed, very portable - certain statistics correspond directly to physically relevant parameters

⇔

mean 1

n

n

∑ i

=1

Vi

⇔

average concentration

1

A

∫ V ( x ) dx A

? Median ⇔ effective permeability -

•

can serve as parameters of distribution models

Disadvantages


37

Chapter 4

4.3.3

-

often too condensed certain statistics strongly influenced by extreme values: 2 m σ σ CS

-

certain statistics affected by gaps in the middle of the distribution: M Q 1 Q 3 IQR

Distribution models

An often-used (over-used?) compromise between the over-abundance of detail in the histogram and the lack of detail in summary statistics is a distribution model (Figure 4.10 and Figure 4.11).

m = 430 ppm σ = 997 ppm M = 158 ppm m 1= 78

s e l m a S f o r e b m u N

Pb(ppm)

⇓ Figure 4.10: Histogram and summary statistics.

38


Chapter 4

y c n e u q re F e v it a l e R

Lognormal M = 95 ppm σ = 67 ppm

Figure 4.11: Probability density function and two parameters.


39

Chapter 4

4.3.4

Scatterplots

Figure 4.12: Scatterplots.

40


Chapter 4

4.3.5

Correlation coefficient

Dividing the covariance by standard deviation on the dependence on the magnitude of the data values: Correlation coefficient = ρ =

χ and γ values removes the

Covariance σ

•

x

σ

y

n

σx = st. dev of x values =

1 n

σy = st. dev of x values =

1

∑ [x i =1

i

− m x ]2

i

− m y ]2

n

∑ [y n i =1

The correlation coefficient ranges from –1 to +1 as shown by Figure 4.13 below.

ρ ≈-0.8

ρ ≈

0

ρ ≈

+ 0.8

Figure 4.13: Correlation coefficient.

4.3.6

Spatial description

Univariate and bivariate data descriptions do not relate to the spatial location of the data. Location within the mineral deposit is a key to assay data and resource estimation. This section presents an intuitive introduction to the notion of spatial description and continuity as a key characteristic of mineral deposit data sets. The key geostatistical function the variogram is introduced. Note that although this section discusses only a single variable, the ideas can easily be extended to two variables. But first, why model and what is difficult in geological data?


41

Chapter 4

A data set

One interpretation: A bouncing ball..... t h ig e H

Distance

Another interpretation: The stock market ... te a R t s e tr n I

(from Isaaks and Srivastava, 1988) Time

Orebodies are a bit more complex! This is why one is interested in underlying patterns in datasets that may assist modelling.

Mean St. Dev. CV Median

Au 60

50

29.5 9.9 0.338 30.5

. .

40 30 10

20

30 Easting (m)

40

Au

50

Mean St. Dev. CV Median .

60 50

29.5 9.9 0.338 30.5

.

40 30 10

42

20 30 Easting (m)

40

50


Chapter 4

Geological data show patterns; basic stats is not sufficient to describe these patterns, as this example shows. How can one describe the two grade data sets while accounting for spatial locations of data thus patterns? Consider a data set, say downhole, like the first dataset above; V(x) is variable at location x, V(x+1) is variable 1 unit to the east of x, and so on. Then we have V(x) V(x+1) V(x+2) 34 27 30 30 34 35 34 35 35

)2

) 1

V(x+2) + x (

V(x+1)+ x (

K

K

Γ=18.1 ρ=0.821

Γ=10.3 ρ=0.896

K(x)

Vx

) 3 ( K

Vx

)

V(x+4) 4 + x

V(x+3) + x

( K Γ=36.9 ρ=0.648

Γ=25.9 ρ=0.780

K(x)

Vx

K(x)

Vx

Then as shown above, plot the various V sub-data sets and calculate correlation coefficients or moments of inertia (variograms). Next, plot these versus distance to describe spatial continuity as shown below. .


43

Chapter 4

1.0

Correlogram function ) (h C0.5

0 1.0

Variogram function

0.5

) h ( g

0.5

Separation Distance (h)

0

Spatial continuity describes spatial patterns

Mean 29.5 Stan. Dev. 9.9

60

) 50 m40 p (p e 30 d a r 20 g d l 10 o G 0

CV Median . .

10

20

30

40

0 .3 3 8 30.5

) h (

50


Easting (m) Mean 29.5 Stan. Dev. 9.9 CV 0.338

60

) m50 p (p 40 e d 30 a r g 20 d l o 10 G

Median . .

30.5

) h (

0 10

20

30

40

50


Easting (m)

44


Chapter 4

The simulation idea:

4.4

Indicator Variables

An indicator variable is a variable which has only two possible values: 0 and 1. These values can be used to designate two different ore types, or ranges of grade content, or lithologies . . . A different way to express the above is to consider a binary variable I(x) satisfying a specified condition z, such as a grade cut-off

I(x) =

⎧1 ⎪ ⎩0

Z(x)

≥ z

Z(x) < z

Different categories eg.

⇒1 ⇒0

In an iron ore mine

Hematite Shale

In a gold mine

Oxide ore ⇒1 Refractory ore ⇒ 0

Different classes Example with 3 cut-offs: 0.2 oz/ton, 0.9 oz/ton, and 2 oz/ton, where Z(x) stands for the measured gold content at location x in a deposit. First indicator variable:

0 for Z(x) < 0.2 1 for Z(x) ≥ 0.2

Second indicator variable:

0 for Z(x) < 0.9 1 for Z(x) ≥ 0.9

Third Indicator variable :

0 for Z(x) < 2.0 1 for Z(x) ≥ 2.0


45

Chapter 4

4.4.1

Interpretations

An indicator can be interpreted as the probability of satisfying a given condition. An example using 1 cut-off 0.2 oz/ton, where Z(x) represents measured gold content at location x in a deposit is illustrated by Figure 4.14 below.

• 0.1

• .19 •

•

0.2

•

•0 •

0.21

•

0.6

•

•

•0.13 0.21

•

0

1

1

•

1.1

•

1

•0 1

•1

Figure 4.14: Original Au oz/ton values (left) and indicator transform for 0.2 oz/ton (right).

Like any variable, indicator variables have a mean and variance mean = number of 1s / total number of samples = proportion of values > 0.2 = probability of gold content > 0.2 oz/ton = 5/8 = 62.5% The mean provides the complete description of the distribution indicator variance = mean * (1-mean) 4.4.2

Spatial description of indicator variables

Indicator variables can be grouped by distance similarly to the continuous variables discussed in an earlier section.Figure 4.15 and Figure 4.16 show examples as follows:

1

p01

p11

p00

p10

I(x+h) 0

0 1 I(x) Figure 4.15: The indicator variogram scatterplot for a distance h.

46


Chapter 4

Each pair {I(x) - I(x+h)} of data in the scatterplot will be atone of four possible locations: (0,0), (0,1), (1,0), (1,1)

Figure 4.16: Indicator variograms from a disseminated gold deposit.

Each location in the scatterplotcan be interpreted as a transition probability (or proposition) from one category to an other, fordistance h.

4.5

Conditional Probability Distributions

A conditional probability distribution function (cdf) may be seen as a tool that provides possible values at an unsampled location. The values are conditional to available data. Each possible value has a probability of occurrence as illustrated below in Figure 4.17.

• 0.1 • •

0.19

•

0.2

•

0.21

0.6

• •

?=

•0.13 1.1

•

0.21

Figure 4.17: Example of Au oz/ton.

•

•0

0

• • •

•

0

•

1

•

• • •1

•

1

1 ? =(.625) •0

•1

0

0

1

• •

0

? = (.25) •0

1

•0

Figure 4.18: Indicator transform for 0.2 oz/ton (left) and indicator transform for 0.5 oz/ton (right).


47

Chapter 4

The probability of a value above 0.2 oz/t is 62.5% and above 0.5 oz/t is 25% as shown in Figure 4.18. If additional cut-offs are used, a complete distribution of possible values can be constructed, conditional to the available data (Figure 4.19). The data comes from the same population. The idea is termed stationarity. All estimation methods somehow link to a ‘average type value’ of a conditional probability distribution. For example, simple and ordinary kriging of normally distributed data provide the mean of a conditional probability distribution. Furthermore, the simple kriging estimation variance is the variance of this distribution.

A gold deposit and a block (marked?) the grade of which is to be assessed Histogram of block ? estimated from grade data

2.8g/t 0.07g/t

?

m: 1.24 sd: 2.3

1.3g/t b

Pro

0

Grade g/t Figure 4.19: Conditional probability distribution.

One line of theory (but not the last)

A conditional distribution cumulative distribution function or ccdfis F(x;zc⏐(N)) = Prob{ Z(x) ≤ zc ⏐(N)} where x indicates a location in the deposit Z(x) is the grade at location x the distribution is conditional to N data available in the are of x zc are cut-off.

48


Chapter 5

CHAPTER 5

CONDITIONAL SIMULATION

5.1 Introduction Conditional simulation of spatially dependent geological data, such as mineral deposit grades, was introduced to mining in the 70s. However, the main idea, and, in fact, many of the basic algorithms were developed earlier in other scientific and engineering fields. Spatial stochastic or geostatistical simulations or imaging (to mention all the jargon used for the same type of techniques) may be classified in various groups and implemented in a variety of ways. Although classifications are non-unique, here is a possible one: Gaussian methods

Turning bands Spectral Frequency domain LU decomposition Karhunen-Loeve expansion Stochastic partial difference equations Sequential Gaussian Non-Gaussian, but based on data transformations: Gaussian transformation Logarithmic Indicator Non-Stationary: Turning Bands with generalised variograms Based on the universal kriging decomposition Non-Parametric: Indicator conditional simulation (truncatedGaussian) Sequential indicator simulation and variations Probability field simulation Others

Boolean and Markov based Annealing, Genetic Algorithms and Neural nets Qualitative, . . . . . . The key to all algorithms is the ability to take a set of randomly generated numbers, process them, and generate a new set of numbers, which have certain properties of interest.


49

Chapter 5

The basic idea of spatial simulations is to generate multiple realisations (images) of a pertinent attribute reproducing ALL data/information available. For geological data, the obvious (perhaps) properties are: (i) univariate (or multivariate) statistics, (ii) spatial correlation, and, (iii) as noted earlier, the reproduction of the srcinal data. All the ‘historical’ techniques first generate grids of numbers reproducing the (i) and (ii) above. This first round of simulation is termed ‘unconditional ’ since the statistics are reproduced but not the actual data. The reproduction of the data in the older simulation algorithms comes from a separate post processing step, which enforces the data reproduction in addition to statistics. It is well worth examining how a set of un-correlated numbers can generate a desired, spatially correlated set of values. At the same time we can have a look at the “historical” geostatistical simulation algorithm termed “turning bands” - for reasons that will be quite obvious in the next pages.

5.2 The Turning Bands Method In the seventies and up to mid-eighties it was one of the most efficient and popular techniques. Idea: •

simulation of values on one line in space, based on moving averages and reproducing a given covariance

•

repeat for a number of lines

•

2D or 3D contribution from each line to each point in space

•

move on to the conditioning step

•

repeat the process for more realisations

The turning bands method is based on a simple idea, explained next: Lets generate a series of random numbers, using a random number generator, at equally spaced points on a line:

50


Chapter 5

a(x) .19 -.26 -.30 -.14 .20 .37 -.38 -.01 .05 x (1) (2) (3) (4) (5) (6) (7) (8) (9) The variogram of the above set of numbers will be a pure nugget effect as shown in Figure 5.1 below.

Figure 5.1: Variogram of un-correlated random numbers.

If one uses a moving average on the previous values, such as the 3 y(x) = Σ f(k) a(x) k=-3 where f(k) = k, - 3 ≤ k ≤to 3 and

f(k) = 0, otherwise

One gets:

y(x) -.08 1.15 1.06 -.89 -2.4 -1.69 x (4) (5) (6) (7) (8) (9) For example, y(5)= (-3) (-0.26) + (-2) (-0.30) + (-1) (-0.14) + (0) (0.20) + (1) (0.37) + (2) (-0.38) + (3) (-0.01)


51

Chapter 5

The variogram of the correlated values isshown inFigure 5.2 below:

Figure 5.2: Variogram of correlated values.

In two or three dimensions, one just adds the “contributions” from a number of lines with simulated values as shown inFigure 5.3below.

Figure 5.3: The turning bands method in two or three dimensions.

The value Z at grid point x will be Z(x1) = { (0.7) + (0.4) + (-0.2) } / 3 = 0.3

52


Chapter 5

Some properties: 1.

The simulated values Z(x) follow a normal distribution with a pre specified mean & variance (standard normal).

2.

The simulated values reproduce a given variogram or covariance.

3.

Any number of independent simulations can be generated (withthe same or different statistics).

The unconditional simulation in this section was produced by first spatially averaging uncorrelated values to produce one dimensional “bands” that radiate from a common srcin. Then, the 1D simulations were averaged again by projecting each point on the simulated grid onto these bands.

5.2.1

The estimation plus simulated error algorithm

Having the ability to generate any number of simulated grids with values, provides the ability to generate realisations of grades that also reproduce the available data. The obtain a conditional simulation, the turning bands follows the steps below:

•

Visit a grid node and krige the grid node

⇒

Z*

•

Produce an unconditional simulation, as we saw earlier, with the correct histogram and variogram ⇒ Zucs.

•

Sample the unconditional simulation at the locations where actual data values exist

•

Krige the unconditional simulation ⇒

•

Z*ucs

Use as the simulated value: Estimate Zsim (x) =

Z*(x)

+

Simulated error [Z*ucs(x) - Zucs(x)]


53

Chapter 5

The four steps of the Turning Bands based simulation are seen in their application atthe stockwork gold deposit discussed in the introduction and illustrated byFigure 5.4 through to Figure 5.9.

Figure 5.4: Normalised exploration drill hole data (black above zero, white below).

Figure 5.5: Kriging, Z*, of the normalised exploration drill hole data (black above zero).

54


Chapter 5

Figure 5.6: Simulated values at the data locations.

Figure 5.7: Kriging, Z* zero).

ucs,

of the simulated data at the drill hole locations(black above


55

Chapter 5

Figure 5.8: Unconditionally simulated values, Zucs at the data locations (black above zero).

Figure 5.9: Conditionally simulated values, Z* zero).

56

cs,

at the data locations (black above


Chapter 5

5.2.2

Comments

We have seen the generation of correlated numbers from un-correlated ones. We have also explored the generation of conditional simulations based on the Turning Bands (TB) and the ‘estimate’ plus correlated error approach. In the seventies and to about the mid eighties, the TB method was a very efficient algorithm used to produce unconditional simulations. Even today, it is reasonably efficient for non-conditional simulations. However, considering the whole process, i.e. including the conditioning, the algorithm becomes relatively slow, tedious inflexible and computationally demanding. The use of TB generates artefacts that are visible in th e results. For example, see the ‘lines’ in the simulations of gold grades in the stockwork deposit shown earlier. The lines correspond to the lines used by theTB (Figure 5.8 and Figure 5.9). Although there are remedies that may alleviate the problem, they have a reasonable computational cost. The algorithm has additional disadvantages. It is not well suited for anisotropies, which are ‘accommodated’ by distorting the grid used. Furthermore, 2D simulations have difficulties converging to the appropriate variogram models. Remedies exist, however, they are tedious and are not readilyavailable. These are some of the reasons that have led to the development and use today of faster and more efficient algorithms, such as the ones discussed inthe following sections. Lastly, the estimate plus error approach may be used with othernon-conditional simulation algorithms such as fractal simulations. Our ability to generate sequences of random numbers is the basis ofconditional simulations and it is examined next.

5.3 Generating Sequences of Random Numbers Random numbers: a set of numbers from an independent sequence of random variables uniformly distributed in [0,1]. The existence of random numbers comes from the axioms ofprobability. However, how do we generate them in practice?

Pseudo-random numbers: a sequence of independent numbers in [0,1] generated from deterministic algorithms, BUT with the same statistical properties as a sequence of random numbers


57

Chapter 5

Random number generators are the foundations of the simu lation edifice: The properties of the pseudo-random number generating algorithms can effect thesimulation results. Consider the algorithm ri = (ri-1 + r1-2) mod 1 Any experiment will show that ri never lies between ri-1 and r1-2. This algorithm is inappropriate as a random number generator. Cryptography is a field with a lot of experience in this area.

Stochastic simulation algorithms derive their randomness from the practically infinite supplies of random numbers Calling standard, build-in random number generators may not be optimal - it could be dangerous

The congruential generator is a very common algorithm Ui = (aUi-1 + c) mod M Where, a is the multiplier, c is the shift and M is the full period; a, c, and modulus M are integers. Uo is the seed. A sequence of pseudo random numbers is calculated from Ri = Ui / M When c = 0 the congruential generator is calledmultiplicative Example: a=5, c=1, M=8, U0=1 U1 = (5 x 1 + 1) mod 8 = 6 , R 1 = .75 Find R2,..., R8 and R9, R10

EXERCISE: Can M+1 values be distinct? M=16, c=1, a=1, and a=5 give sequences ofUi, i=1,..., 17 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15, 0 and 0,1,6,15,12,13,2,11,8,9,14,7,4,5,10,3, 0

58


Chapter 5

The sequence of a set of pseudo random numbers is periodic with period≤ M. 30

A congruential generator should have a period as large as possible (> 2) and is chosen so that it gives period M or M-1. Some good implementations: 32

a= 69069, c=1, M=2

(Marsaglia, 1992)

a= 1313, c=0, M= 259 (NAG Fortran library) Some not so successful: a=75, c=0, M=216 + 1 One should test independence, uniformity, pairs of values.

For other geostatistical simulation tasks, multiplicative congruential generators are usually preferred. Congruential pseudo-random number generators are the preference in generating simulation paths in sequentialsimulations. A good pseudo-random number generator should be: Simple and fast Periodic with a very long period Evenly spread in [0,1] Checked! Suitable for the specific job


59

Chapter 6

CHAPTER 6

6.1

SEQUENTIAL CONDITIONAL SIMULATION METHODS

Some Considerations

For sound practical reasons we look into other frameworks for spatial simulations. Traditional simulation algorithms like the Turning Bands and estimate plus errors solutions never became a standard industry tool for good reasons. They tend to be slow and relatively inefficient, take more time, involve more steps, may have undesirable artefacts, and can be tedious and un-tractable when a large number of simulations are needed or several variables are involved. An additional reason is their typical limit to continuous variables or the use of even more complex processes to overcome the limit. The indicator type approaches, popular for good reasons in mining, pose animplementation problem for the old fashioned algorithms. For reasons of efficiency, simplicity, statistical assumptions, data manipulation, flexibility, information integration, suitability for categorical variables, and so on, one may resort to newer but well established and extensively usedsequentialalgorithms. The basic ideas of spatial simulations remain the same: generate multiple realisations (images) of a pertinent attribute reproducing ALL data/information available and their statistical characteristics. The previously considered conditional simulations started from the random drawing of valuesfrom unrelated distributions. What if one draws randomly from distributions somehow related? An example could be conditional distributions like the ones discussed earlier. Instead of drawing random numbers, use some moving average to generate correlation, what happens if one draws from distributions conditioned to previousdrawings?

6.2

Explaining the Idea with an Example

Consider a part of an epithermal gold deposit with five drill holes, as shown below in Figure 6.1 below. There are two categories recognised, ore (black mark) and waste (white mark) based on a 0.006 ounces/ton cut-off. OBJECTIVE: simulate ore or waste categories in each of the blocks of interest in the deposit. Pick one by one the blocks marked in the figure with a ? and simulate and then evaluate the situation with some probabilistic method.

60


Chapter 6

? ?

grade above cut-off

?

? unknown grade

grade below cut-off

(a)

Figure 6.1: Drill hole locations and mining blocks.Two categories are recognised, ore and waste, using a 0.006 ounces/ton cut-off. Black is 'ore' - White is 'waste'

Let’s pick one of the three mining blocks in theFigure 6.1 and find the probability p* that the block is ore (Figure 6.2).

?

grade above cut-of

? unknown grade

grade below cut-off

(b)

*

? has p1 = 0.46

*

Figure 6.2: Finding the probability p that the block is ore.

Use any process available to estimate theprobability of the block being ore. e.g. code ore to 1 and waste to 0 and find the arithmetic average, or... Let’s say here we get 0.46. Translate the probability into ore or waste: Step 1:

Draw a uniform random number r from the interval [0,1].

Step 2:

Compare the random number r to the probability. *

r< p r

≥

*

p

then quality is 'ore' then quality is 'waste'

In our example, p1* = .46 and r = .12 thus the quality is 'ore'.


61

Chapter 6

?

grade above cut-off grade below cut-off

? unknown grade (c)

Figure 6.3: ? Unknown quality is ore.

Code and include the new simulated ‘ore' block into the data setFigure ( 6.3). Move to the next block and repeat the process. Move to the third block and repeat again F ( igure 6.4).

? ?

grade grade above below cut-off cut-off

?

? unknown grade (d)

Figure 6.4: Include new block and repeat.

You may repeat the whole process from thebeginning and combine the results (see real life applications in subsequent sections). NOTE: that as expected in a geostatistical type simulation algorithm, one may generate another value for the same block by drawing a new random value and repeating the same steps. Consider now a more complex variation on the same problem, and ADD a second higher cut-off, say 0.04 ounces/ton, for the 'waste’ - 'ore’ classific ation to produce now ‘waste plus low grade ore’ and ‘high grade ore’ categories or in short ‘WLGO’ and ‘HGO’ respectively. OBJECTIVE: Generate a series of possible ore categories for amining block in the deposit. Then, use the results to get one of several grade categories for the block. Drill hole locations and mining blocks as before. This time the categories are two different ones WLGO and HGO discriminated by a cut-off of 0.04 ounces/ton. Black is WLGO White is HGO as shown inFigure 6.5 below.

62


Chapter 6

? ?

?

rade above cut-off grade below cut-

? (a’)

Figure 6.5: Two categories - Waste plus Low Grade Ore- black; High Grade Ore- white.

One repeats the previous process for the new set of 'WLGO' and 'HGO' classification and the given mining blocks. The result is p2* = .58 and r = .82 thus the quality is 'WLGO'

The process can continue as before to generate grade classesFigure ( 6.6).

? ?


?

? (b’)

Figure 6.6: Continuing the process to generate grade classes.

If we consider several grade cut-offs and combine the results in a cumulative histogram, what would we get? In fact: For any block, the probability p2* derived from this second step can be combined with the one derived in the previous one in the form of a probability distribution indicating the grade class assigned to the given block. This is only a distribution of grades and specifically: The conditional cumulative distributions of the block grades, used to simulate grade classes or values.


63

Chapter 6

first mining block

second mining block

1

1 block 1

F

block 2 F

0

0 W

LGO HGO Category

W

LGO HGO (b) Category

(a)

W: waste; LGO: low grade ore; HGO: high grade Figure 6.7: First and second mining block.

In a more typical form the individual block grade conditional cumulative distribution functions (ccdf) would be:

? ?

100

?

Cu m


m: 1.24 sd: 2.3

?

0 Grade g/t Figure 6.8: Cumulative probabilities.

64


Chapter 6

6.3

What have we seen so far?

Starting with a grade data set, we can somehow generate conditional probability distributions for any chosen block based on the data. In a general way, it seems that all we have to do is: 1.

Start with a data set.

2.

Somehow generate the grade conditional distribution of the block.

3.

Generate random numbers, and use them to

4.

Sample the conditional probability distributions of block grades.

5.

Each random number & sampling will generate a different equally possible grade realisation at the unsampled locations.

Interestingly, at a data location there is only one option for the data. As a result, grade data locations in a deposit will be honoured. The addition of a simulated value in the data set before we move on to a block is a distinct characteristic of the sequential approach to simulation. There are theoretical reasons for it, and froma practical viewpoint, it should be easy to see that the use of simulated points ensures that spatial continuity is well considered. The most interesting question is how can we find the conditional distribution (ccdf’s) for each block so that we can sample it. Similarly to the examples, we have to take advantage of convenient theoretical distribution models, like the Gaussian, or consider more ‘distribution free’ alternatives. The main two approaches of estimating ccdf’s are examined in detail in the next sections. However, before considering the estimation of conditional distributions, a few short lines of theory are quite informative.

6.4

A Few Lines of Theory: The Concept of Sequential Simulation

6.4.1

Sequential simulation and related generalisations

Following the standard geostatistical terminology, the grade of a deposit is conceptualised as a random function (RF), Z(x), of spatial coordinates with a multivariate probability density function (pdf) f(x1,...,xN; z1,...,zN). Sequential simulation is based on thedecompositionof the multivariate probability density function, f(x1,...,xN; z1,...,zN), of the RF Z(x) into a product of univariate conditional distributions (Johnson, 1987).


65

Chapter 6

f(x1,...,xN; z1,...,zN) = f(x1;z1) x f(x2;z2⏐Z(x1)=z1) x ... x f(xN;zN⏐Z(xa)=za, a=1,...,N-1) where N is the number of locations in the deposit, and za is a value at location xa { a=1,...,N-1}. If ALL the conditional distributions are known, realisations of the grade Z(x) can be generated by random sequential drawings from each of the N univariate conditional distributions. The result of each drawing is used to condition the next univariate distribution before the next drawing. The simulation and drawings stop when the last univariate distribution is conditioned and a realisation is drawn randomly. When an initial data set, {z(xa), a=1,...,M < N}, is available, then the sequential drawing begins at the M+1 step and the first value is drawn from the univariate conditional distribution, f(xM+1;zM+1/Z(xa)=za, a=1,...,M)

6.4.2

Some implementation related comments

In practice, one needs to know the complete sequence of univariate conditional probability distributions: A variety of geostatistical techniques can be used to build these univariate distributions EXERCISE:

Discuss techniques that could be used to estimate conditional distributions

A practical problem that may arise is the number N of points at which realisations need to be drawn compared to the available M data < < N EXERCISE:

An iron ore deposit is represented by a 500x500x10 grid and about 10 composites in each of the available 50 drill holes. How large is the kriging matrix at (i) the first grid node to be simulated, and (ii) at the last one?

In practice one uses only datalocally, i.e. within a given distance from the simulated node ("screening" of distant data). The effect of the screeningis deemed insignificant, although long distance variability can also beaccommodated

66


Chapter 6

EXERCISE:

• • • • •

What could be the effect of screening on the reproduction of second order statistics such as variograms?

Sequential simulations are easy to understand Can be implemented in a variety ofways / algorithms All one needs to do isestimate conditional distributions Are sensitive to various implementation related factors The two main implementations are discussed next

6.5

Sequential Gaussian Simulation (SGS)

6.5.1

Some observations

The basic idea of drawing from a distribution can be based on a Gaussian framework. Surprisingly, simple kriging is a main factor in this Gaussian framework. SGS comes from the following: The decomposition of the probability distribution function presented in the previous section. The case of Gaussian distributions, the univariate conditional distributions in the sequential decomposition have conditional means and variances that are given by the corresponding simple kriging (SK) mean and variance respectively.

Msk=1.24 sdsk: 2.3 2.8g/t

Prob

Randomly draw a sample

0.07g/t

?

1.3g/t

0 Figure 6.9: Sampling an estimated Gaussian distribution in SGS.

Recall that SK is the estimate of the grade of a block like the one marked with ? in the ? above figure, say z , and it is ?

z = (w1 * 1.3 + w 2 * 0.07 + w2 * 2.8 ) + (1 - w1 - w2 - w2) *1.39

where


67

Chapter 6

the weights wi come from the solution of the SK system discussed below. In general, the SK formulation is asfollows

Σi=1n wi zi

*

zsk =

+ [1 -

Σi=1n wi ] m

where zi are the available data, m the data mean, and wi the simple kriging weights derived from the solution ofthe following system

.

C11 . . . . . . . . Cn1 . . .

.

C

w1

1n

. . . C

X nn

C01 .

. . wn

=

. . C0n

Cij are the covariance values between the available data points and C0i the covariances between the estimated point (block) and the data. Note that the covariances can be replaced by the corresponding variogram or correlogram values. The SK estimation variance is

σsk2

=

σdata2

-

Σi=1n wi C0i

where

σdata2 is the data variance The property of SK to provide the mean and variance of a conditional distribution is quite convenient. Once there is a conditional distribution, one can randomly sample it to generate simulated grades forthe corresponding block. IMPORTANT: If one sequentially draws from a conditional distribution the corresponding covariance or variogram will be reproduced. It is also important to remember that the above approach assumes normally distributed data. This is rare in real data sets and one resorts to normalising the data set, and back transforming the data at the end (this is discussed in a subsequent section).

68


Chapter 6

SK as seen above implies that the global mean is known, which is rarely the case. In addition, it implies that the actual variation of the grade mean locally is relatively small Another option: Both these reasons lead to the frequent replacement of SK with ordinary kriging (OK), which reasonably approximates the conditional mean andvariance. Recall the last example with SK, if one used ordinary kriging we would have ?

z = w1 * 1.3 + w 2 * 0.07 + w2 * 2.8 where the weights wi come from the solution of the OK system discussed below. In general, the OK formulation is asfollows zok* =

Σi=1n wi zi

where zi are the available data, and wi the ordinary kriging weights derived from the solution of thefollowing system C11 . . . .

.

1

C

1n

. . . . . . . . Cn1 . . . C nn . . . 1 0

1 . . 1

w1 .

X

. . wn

C01 . =

μ

. . C0n 1

Similarly to SK, Cij are the covariance values between the available data points, C0i the covariances between the estimated point (block) and the data, and

μ is a constant which comes from the derivation of the equations. The covariances can be replacedby the corresponding variogram or correlogram values. The OK estimation variance is

σok2

=

σdata2

-

Σi=1n wi C0i + μ

where

σdata2 is the data variance


69

Chapter 6

The major difference from before is that OK does not require the knowledge of the global mean. In practice, OK implicitly uses the mean of the local data used in kriging. One option is to combine the OK mean and the SK variance. The practical differences between SK and OK in simulating grades isdiscussed later on. EXERCISE: Consider the situation below. If the SK estimate is 0.0023 and the corresponding variance is 0.49, how could you generate a few simulated values?

?

high grade

? unknown grade

low grade Figure 6.10: Simulating grades.

6.5.2

The SGS simulation algorithm

Define a path to be followed to visit each location x (or grid node) to be simulated. There are N grid nodes to be visited and M data ({z(x a), a=1,...M
Zsk*(x1) = E{Z(x1)/M)}

3. variance:

σsk2(x1)

= Var{ Z(x1)/M)}

4. Draw a value from the conditional distribution at the first location 1xand add the corresponding results in the data set. The new data set is now {z(xa,), a=1,...,M+1}. 5. Move to the second location in the path, say x 2, and estimate the local conditional distribution as above but with the new dataset where mean:

Zsk*(x2) = E{Z(x2)/M+1)}

variance:

σsk2(x2)

= Var{ Z(x2)/M+1)}

6. Draw from the estimated conditional distribution, add the value in the data set, move to x3, and repeat the process until a value is drawn from the last distribution at location xN.

70


Chapter 6

More realisations? Set a new path throughthe grid nodes and repeat the process. What do we do if grades are not normally distributed? Normal score transforms are discussed next. Are there issues related to the implementation fo the algorithm? YES, and they are examined in a subsequent section.

6.5.2.1

SGS: an implementation of the LU simulation

The SGS algorithm is an ‘iterative’ implementation of the simulation method termed “LU”. This method is based on the decomposition of the covariance matrix to lowerL() and upper (U), and it is outlined in this section.

6.5.2.2

The LU simulation algorithm 1

Given a grade data set , say, {z i, i = 1, ..., n} with mean m, and let C be the covariance matrix, the simulation zcs of a grid with grade values Z and covariance C is as follows: Decompose the covariance matrix into lower L and upper U

C = LU

where

T L =U Now consider the vector z= Lw + m

where T z = (z'1, ..., z'n, z1, ..., zN) , and {z'i, i = 1, ..., n} and {zi, i = 1, ..., N} are the data and required conditional simulation respectively, and w is a vector of random numbers from N(0, 1) Partition C as ⎛⎜CDD CTGD⎞⎟ ⎜⎝ CGD CGG ⎟⎠ where

CDD is the covariance matrix between data locations,


71

Chapter 6

CGD is the covariance matrix between grid locations and data locations, and CGG is the covariance matrix between grid locations

This gives:

⎛ LDD 0 ⎞ ⎛UDD UGD⎞ ⎟⎜ ⎟ ⎝LGD LGG⎠ ⎝ 0 UGG ⎠

C=⎜

Partition w as T w = (w D

T T w ) G

where wG is a vector of independently standard normally distributed random numbers, and wD is the conditioning vector LDDwD = zD- mD

where zD and mD are vectors of the data and means, respectively.

The required simulation zG at the G grid nodes is then -1 zG = [LGDL (z - mD) + mG] + LGGwG DD D -1 Note that [LGDL (z -m )+mG] is the simple kriging vector derived from the DD D D conditioning grades. If N=1, we are back to the expressions of the simple kriging estimate skz* and simple 2 kriging variance σsk . A major limitation to the method is the size [(N+n)x(N+n)] of the covariance matrix C which practically cannot exceed 1000x1000 to be decomposed by numerical methods.

UNLESS LU is implemented in an ‘iterative’ way, which is theSGS algorithm.

6.5.2.3

The SGS algorithm in summary

1. Normalise and standardise existing sample data. 2. Compute and model the variogram, covariance, or correlogram of the normalised data. 3. Define a random path that goes through each node of the grid representing the deposit. 4. Do simple (or ordinary) kriging of the normalised value at the selected node using both actual and simulated data to estimate the normal local conditional distribution.

72


Chapter 6

5. Simulate the value by randomly sampling the estimated normal local conditional distribution having the SK (or OK) estimate and its variance as mean and variance respectively. 6. Add the new simulated value to the conditioning data set and move to the next grid node. 7. Repeat the process until all nodes are simulated. 8. "Denormalise" the simulated values and validate the results.

6.5.2.4

From LU to the SGS algorithm

It is easy to see why LU implemented in an ‘iterative’ way is the same as SGS. Rewrite the LU using L = (lij) z'1 - m'1 .

. . . . z'n - m'n z1 - m1

.

L11

0

w1

.

=

. . . . Ln1 . . . Lnn

.

Ln+11. . .

.. zN - mN

.. Ln+N 1 . .

. . wn

. Ln+1n+1

wn+1 . . .

.

.

.

Ln+Nn+N

.. wn+N

The conditional simulation is zi = Ln+i 1.w1 + ... + Ln+i n+i .wn+i+mi, i = 1, ..., N If we let vector LiIT = (ln+i 1, ..., ln+i n+i-1) and wiT = (w1, ...,wn+i-1), the above formula can then be rewritten as follows: zi = LiITwi+mi + Ln+i n+i .wn+i, i = 1, ..., N which is the equivalent of the iterative implementation of the LU algorithm and completely equivalent and identical to the sequential Gaussian simulation algorithm.


73

Chapter 6

6.5.3

Monte Carlo drawing from a conditional distribution

To generate a realisation (simulated grade value) csz at any location (x), one needs to draw a value (class) from the estimated conditional distribution. The drawing has two steps: Step 1: draw r, a uniform random number from [0,1] Step 2: retrieve the R-quantile of the estimated co nditional distribution zcs(x) = F*-1(x;R/(M)) such that F*(x;zcs/(M)) = R

6.5.4

Normal score transformations (forward and backward)

Normal score or Gaussian data transformations are used to ‘normalise’ data sets prior to the application of techniques that assume that the data distribution is normal (forward transform). Several ways are available for normal score transformations: graphical, based on hermitian polynomials, etc. The simplest and most efficient is the graphically obtained Gaussian transformation, φ. It is generated from a one-to-one correspondence of the standard normal distribution and the distribution of the data asshown inFigure 6.11.

Figure 6.11: Distributions of the data.

Transformation φ is devided by a series of NB bounds: {zk, yk ; yk = G [F (Zk) / (1 + ε)]}, k = 1, ..., NB -1

*

where, *

F is the estimated cdf of the data, G the standard normal cdf, and

ε a very small positive number (<<< 1 -to avoid G-1(1)= ∞ ) The function ϕ(y) is defined in the interval ]-∞, + ∞[, but in practice it is sufficient to use the interval [-7.0, +7.0]

74


Chapter 6

The corresponding value to the above zmin bound is either 0 or the minimum value in the data set. The zmax bound is either equal or higher than the highest value in the data set. NOTE that an artificially high value for z max will create artificially high values when back transforming values. The number of bounds should be sufficiently large to well define the transformationφ. Usually, 300 to 500 bounds aresufficient. The spacing of the bounds can be defined in different ways. However, regularly spaced bounds for y are by far the most efficient. In most applications, the final results, y*, are back-transformed to follow their srcinal characteristics (backward transformation): z* = φ(y*) The values z* are obtained directly by interpolation: z* = zk + (zk+1 -zk) [(y* - yk)/Δy] where,

Δy is the y bound spacing, and k = int[(y*-y min)/Δy] + 1, while yk < y* < yk+1. Normal score transformations are sensitive to a variety of factors:

•

The transform is based on the estimated data distribution, which should be derived from a declustered data set.

• •

The back transform is sensitive tothe selection of the upper bound.

6.5.5

A fast sequential Gaussian simulation: sequential group Gaussian simulation (SGGS)

The transformation is sensitive to spikes inthe data distribution.

The equivalence of the SGS and LU gives the opportunity to implement a newer SGS algorithm. The sequential process is performed from a group of nodes closest to the next, which generates a very fast sequential algorithm.


75

Chapter 6

1

2

3

4

data location 1

node to be simulated

Figure 6.12: Four overlapping neighbourhoods of four closest nodes.

For reasons of convenience, the number of nodes in most groups can be identically designed, the algorithm can then be defined by an order or neighbourhood size κ. Note that when κ = 1, the algorithm is identical to SGS; and when κ equal to the number of grid points the algorithm is identical to the LU implementation. A convenient name is sequential group Gaussian simulation (SGGS). SGGS is a series of sequential Gaussian simulation algorithms associated with different local neighbourhoods (groups) (Figure 6.12) and in a way a generalisation of the LU/SGS implementations. The major advantage of SGGS is SPEED and STORAGE requirements. Program FASTSIM in the workshop implements SGGS. The details of the mathematics are beyond the scope of the present document. The basic idea comes from sequential decomposition of a conditional distribution and its estimation through a ‘screen effect’ approximation. Similarly SGS, the algorithm on the decomposition conditional probabilitytofunction to the productisofbased the conditional distributions of of the k groups of N nodes. SGGS uses two paths to visit all grid nodes: an internal path and an external path. Why bother? Time improvement is substantial, and can be as low as 10% of a SGS run; the memory requirements are also quite lower.

76


Chapter 6

Studies show that the best algorithm in terms of computing time is the one whose group size is around 80% of the bound of the data neighbourhood used.

6.5.5.1

The implementation of the SGGS

Define an external path that visits each group in the grid, and an internal path that visits all nodes in each group. 1. Find a neighbourhood for the current group to be simulated. 2. Calculate the conditional mean vector and covariance matrix of the current group. 3. Generate the simulated values of the current group. 4. Add the simulated values of the current group into the data. 5. Proceed to the next group until all groups are simulated.


77

Chapter 6

6.6

Sequential Indicator Simulation (SIS)

6.6.1

Introduction

We mentioned that the sequential approach requires the estimation of the local conditional distributions. SGS does this using the convenient properties of the Gaussian models. What other options are there if one does not find the Gaussian options suitable? This section introduces the alternative to the Gaussian sequential simulations. This alternative is based on a ‘non-parametric’ indicatorapproach.

6.6.2

Grade indicators and a block of a gold deposit

Consider the Figure 6.13 on the following page. A block with a sample of 0.008 oz/t is inside the block and is surrounded by sample grades0.002, 0.012, 0.054 and 0.250 (oz/t). Lets use three cut-offs of for the grade z: 0. 005 oz/t, 0.015 oz/t, and 0.10 oz/t and the transform

i(x;z) =

⎧1 ⎨ ⎩0

Z(x;z)

≥ z

Z(x;z) < z

The first cut-off is used to differentiate mineralised and barren rock, and it leaves one sample indicator at 0 outside the block. A likely estimate fromISD or OK could be 90%. This means that 90% of that block is mineralised. The second cut-off (0.015 oz/t) isolates very low grade material aroun d 0.01 oz/t. We now have three zeros, one of them in the middle of the block. A likely estimate for that indicator is 0.30 ( about 15% weight for each of the outside samples and 40% for the inside). The last cut-off (0.10 oz/t) leaves only one outside sample coded as 1. A reasonable block estimate for that cut-off is 0.10.

78


Chapter 6

Figure 6.13: Example of the IK grade estimation approach in a block within a gold deposit (values in oz/t).


79

Chapter 6

For this block we have 1 - 0.9 = 10% of the block within the non-mineralised material of zerograde 0.9 - 0.3 = 60% of the block with low grade material of 0.009 oz/t 0.3 - 0.1 = 20% of the block with medium grade material of 0.039 oz/t 0.1 = 10% of the block with high grade material of grade 0.332 oz/t The fixed grade in each category is the average of the samples in the category. An estimate for the entire block is: 0.1x0 + 0.60x0.009 + 0.20x0.039 + 0.10x0.332 = 0.046 oz/t The above block estimate compares with the direct estimate which can be calculated from assigning a 40% weight to the sample at the centre of the block and a 15% weight to each of the surrounding samples. 0.15x0.002 + 0.15x0.012 + 0.15x0.25 + 0.15x0.054 + 0.40x0.008 = 0.051 oz/t NOTE: that if our high grade was an outlier of 2 oz/t instead of 0.25 oz/t the indicator estimate would be the same butthe direct would be 0.313 oz/t. We have just seen an example of the technique termed Indicator Kriging (IK). Advantage of the IK is the ability to generate proportions of grade classes and estimates of block recoveries - the block’s internal waste. The above example suggests 30% material is above 0.015 oz/t and grade 0.137 oz/t. Returning to the comments in the start of the section, let’s see take a step further. Continuing the example with the block of the gold deposit, we can use the various proportions to construct the conditional grade distribution of the block. For the given configuration and cut-offs, the estimated conditional distributions for IK are constructed (Figure 6.14).

80


Chapter 6

Figure 6.14: Estimated conditional distributions with IK.

Values can now be drawn from thisdistribution using a random number generator.

6.6.3

Exercise 3

Generate a uniform random number R=U1/2 using 6 as seed in the congruential generator 3 U1 = (5 x Uo + 1) mod 2 R=? Generate realisations from the IK ccdf ics-IK = Result: R: U=7 and R=0.875

6.6.4

Indicator kriging summary and comments

At the unsampled location (block) xo, a grade value may be estimated as the weighted average of surrounding data *

n

i(x;z) = Σi=1 wi i(xi) using, Ordinary Kriging or Simple Kriging If a series of cut-off values is used to create different sets of binary data, one may repeat the estimation at the same point with the different binary data sets. The process generates a series of probabilities. An important point is that using kriging requires a series of indicator variograms. Indicator variograms are average probabilities of encountering one ore grade range at a distance from another:


81

Chapter 6

We use average probabilities in theform of variograms to estimate probabilities. Here we use not only one variogram but a series ofthem: we use more information.

6.6.5

Median IK

Median IK is based on the premise that indicator variograms at different cut-offs are proportional to each other. This means that the indicator correlograms for all cut-offs are the same. A general interesting and practical relation is that the median indicator correlogram is the same as the correlogram of the continuous grade.

ρI (h;z)

=

ρz(h)

= 1 - γR(h) / σR2

where

ρI (h;z) = C I (h;z)/ σI,z2 is the median indicator correlogram, ρz(h) is the correlogram of the actual grades, γR(h) is the relative variogram, σR2 is the relative variance, CI (h;z) is the indicator covariance, and

σI,z2 the indicator variance at 50% 6.6.6

The steps in SIS

Having looked at IK, it is easy to see now that the conditional distributions generated can be used in a sequential simulation algorithm. Consider the grade Z(x) of a deposit discretised into K mutually exclusive classes, using a series of cut-offs. The SIS primary objective is the simulation of the spatialdistribution of the K class indicators. Again, SIS is the sequential simulation implementation based on the estimation of univariate conditional distributions with an indicator typekriging approach. The SIS algorithm is as follows:

•

Define a path to be followed in visiting each location x (or grid node) to be simulated. There are N grid nodes to be visited.

•

Estimate at the first location, say x1, the whole ccdf of Z(x) using IK for k classes zk, k=1,..., K. F(x1;zk/(M))* = Prob *{Z(x 1) ≥ z/(M)}

82


Chapter 6

= [I(x1;zk)/Z(xa)=za, a=1,...,n]* M

= Σa=1 wa i(xa;zk)

•

Draw a value (class) from the conditional distribution at the first location 1xand add the corresponding results to the data set. The new data set is now {za, a=1,...,M+1}.

•

Move to the second location in thepath, say x2, and estimate the local ccdf for the k classes. *

*

F(x2;zk/(M+1)) = Prob {Z(x 2) ≥ z/(M+1)} *

=[I(x2;zk)/Z(xa)=za, a=1,...,M,M+1] M+1

= Σa=1

•

wa i(xa;zk)

Draw from the estimated ccdf, add the value(s) to the data set, move to 3x, and repeat the process until a value is drawn from the last ccdf atlocation xN.

To generate additional realisations, a new path is set and the process is repeated. Shortcuts are possible but not recommended. To generate continuous variables, one may draw values from within each class.

•

Sequential Indicator Simulation is a ‘ distribution free’sequential algorithm.

•

SIS reproduces the indicator data and their spatial indicator statistics.

6.6.7

Comments on indicator kriging in SIS

IK is a ‘distribution free’ approach in estimating conditional distributions, and the key mechanism in SIS. The characteristics of IKthat affect the implementation and results of SIS, are pointed out here. Note that the estimation of a grade cumulative conditional distribution function has the two requirements. 1.

Estimated probabilities are in the interval[0,1].

2.

For a sequence of increasing cut-offs,each estimated cumulative probability is greater or equal to the previous one.

The OK used in IK does NOT ensure the above conditions and generates “order relation problems”.

•

The kriging weights are not necessarily positive.


83

Chapter 6

• 6.6.8

Different weights are used for eachcut-off at the same location.

Negative Weights: a detailed IK example based on median IK

Eight gold samples with the spatial configuration shown below in Figure 6.15 are used to estimate the grade distribution (conditional probability distribution function or cdf) of a block. The IK approach used is median IK.

Figure 6.15: Spatial configuration of 8 gold samples.

The indicator variogram model is isotropic, spherical with the following characteristics: Nugget effect C0 = 0.0 Sill C1 = 1.0 Range a = 20m1

84


Chapter 6

The cut-off grades we wish to consider are 0.2, 0.8, 1.5, 3.0, 7.5, and 15.0 g/t The kriging weights are (obviously the same for each cut-off) as shown below in Figure 6.16.

Figure 6.16: Kriging weights for the 8 gold samples.

The conditional distribution at the unknown node is determined by estimating its probability using the six cut-offs indicated above. Figures 6.16 through to 6.28 illustrate this process.


85

Chapter 6

Figure 6.17: Indicator coding for a 0.2 g/t cut-off. Sample

Grade

1 0.01 2 0.10 3 0.25 4 0.50 5 1.0 6 2.0 7 5.0 8 10.0 Estimated Prob.

Weight

0.2

0.1667 0.1087 -0.0916 0.1934 0.3588 0.0667 -0.1604 0.3577

1 1 0 0 0 0 0 0

0.8

1.5

3.0

7.5

15

0.1667 0.1087 0 0 0 0 0 0 0.275

Table 6.1: Samples, weights and estimated probabilities. 1

y ti 0.8 li b a b 0.6 ro P d 0.4 te a m it 0.2 s E 0 0

0.2

0.8

1.5

3

7.5

15

Gold Grade (ppm)

Figure 6.18: Estimated conditional distribution at the 0.2 g/t cut-off.

86


Chapter 6

Figure 6.19: Indicator coding for a 0.8 g/t cut-off.

Sample

Grade

1 0.01 2 0.10 3 0.25 4 0.50 5 1.0 6 2.0 7 5.0 8 10.0 Estimated Prob.

Weight

0.2

0.1667 0.1087 -0.0919 0.1934 0.3588 0.0667 -0.1604 0.3577

1 1 0 0 0 0 0 0

0.8 0.1667 0.1087 0 0 0 0 0 0 0.275

1.5

1 1 1 1 0 0 0 0

3.0

7.5

15

0.1667 0.1087 -0.0919 0.1934 0 0 0 0 0.377

Table 6.2: Samples, weights and estimated probabilities.

1

ty 0.8 lii b a b 0.6 ro P d 0.4 e t a m ti 0.2 s E 0 0

0.2

0.8

1.5

3

7.5

15

Gold Grade (ppm)



87

Chapter 6


Grade

1 0.01 2 0.10 3 0.25 4 0.50 5 1.0 6 2.0 7 5.0 8 10.0 Estimated Prob.

Weight

0.2

0.1667 0.1087 -0.0919 0.1934 0.3588 0.0667 -0.1604 0.3577

1 1 0 0 0 0 0 0

0.8 0.1667 0.1087 0 0 0 0 0 0 0.275

1.5 1 1 1 1 0 0 0 0

0.1667 0.1087 -0.0919 0.1934 0 0 0 0 0.377

3.0 1 1 1 1 1 0 0 0

7.5

15

0.1667 0.1087 -0.0919 0.1934 0.3588 0 0 0 0.736


1


0.2

0.8

1.5

3

7.5

15

Gold Grade (ppm)


88


Chapter 6


Grade

Weight 0.2

0.8

1

0.01

0.1667

1

0.1667

1

0.1667

1

0.1667

1

2

0.10

0.1087

1

0.1087

1

0.1087

1

0.1087

1

0.1087

3

0.25

-0.0919

0

0

1

-0.0919

1

-0.0919

1

-0.0919

4

0.50

0.1934

0

0

1

0.1934

1

0.1934

1

0.1934

5 6

1.0 2.0

0.3588 0.0667

0 0

0 0

0 0

0 0

1 0

0.3588 0

1 1

0.3588 0.0667

7

5.0

-0.1604

0

0

0

0

0

0

0

0

8

10.0

0.3577

0

0

0

0

0

0

0

Estimated Prob.

1.5

0.275

3.0

0.377

7.5

0.736

15

0.1667

0 0.802


y itl 0.8 i b a b 0.6 o r P d 0.4 e t a itm 0.2 s E 0 0

0.2

0.8

1.5

3

7.5

15

Gold Grade (ppm)



89

Chapter 6


Grade

Weight 0.2

1

0.01

0.1667

1

0.1667

1

0.1667

1

0.1667

1

0.1667

1

0.1667

2

0.10

0.1087

1

0.1087

1

0.1087

1

0.1087

1

0.1087

1

0.1087

3

0.25

-0.0919

0

0

1

-0.0919

1

-0.0919

1

-0.0919

1

-0.0919

4

0.50

0.1934

0

0

1

0.1934

1

0.1934

1

0.1934

1

0.1934

5 6

1.0 2.0

0.3588 0.0667

0 0

0 0

0 0

0 0

1 0

0.3588 0

1 1

0.3588 0.0667

1 1

0.3588 0.0667

7

5.0

-0.1604

0

0

0

0

0

0

0

0

1

-0.1604

8

10.0

0.3577

0

0

0

0

0

0

0

0

0

Estimated Prob.

0.8

1.5

0.275

3.0

0.377

7.5

0.736

0.802

0 0.642


y itl 0.8 i b a b 0.6 ro P d 0.4 e t a m ti 0.2 s E 0 0

0.2

0.8

1.5

3

7.5

15

Gold Grade (ppm)


90


Chapter 6


Grade

1 0.01 2 0.10 3 0.25 4 0.50 5 1.0 6 2.0 7 5.0 8 10.0 Estimated Prob.

Weight 0.1667 0.1087 -0.0919 0.1934 0.3588 0.0667 -0.1604 0.3577

0.2

0.8

1 1 0 0 0 0 0 0

0.1667 0.1087 0 0 0 0 0 0 0.275

1.5 1 1 1 1 0 0 0 0

0.1667 0.1087 -0.0919 0.1934 0 0 0 0 0.377

3.0 1 1 1 1 1 0 0 0

0.1667 0.1087 -0.0919 0.1934 0.3588 0 0 0 0.736

7.5 1 1 1 1 1 1 0 0

15

0.1667 0.1087 -0.0919 0.1934 0.3588 0.0667 0 0 0.802

1 1 1 1 1 1 1 0

0.1667 0.1087 -0.0919 0.1934 0.3588 0.0667 -0.1604 0 0.642


1


0.2

0.8

1.5

3

7.5

15

Gold Grade (ppm)



91

1 1 1 1 1 1 1 1

0.1667 0.1087 -0.0919 0.1934 0.3588 0.0667 -0.1604 0.3577 1.000

Chapter 6

6.6.9

Some comments on IK as used in SIS

There are other kriging techniques that canbe used to estimate conditional grade distributions, e.g. probability, multi-Gaussian, disjunctive kriging. What makes IK and SIS interesting?

•

IK is a 'non-parametric' technique and resistant to outliers (recall non-parametric statistics in first part of thiscourse).

•

IK uses more information (e.g. a seriesof variograms).

•

Since one deals with indicator data that can be interpreted as probabilities, one can use ‘soft’ data in the estimation process.

•

It has order relation problems.

•

There are several corrections and these are discussed in a subsequent section. There is difficulty in interpreting variograms at high cut-offs.

•

It relies on strict stationarity assumptions, like SGS.

6.6.10 The use of IK type approaches in sequential simulation

•

Adds more structural information to the simulation: direct inference of the covariances of different classes or categories or types of data.

•

Integrates different types ofdata/information.

•

Does not need assumptions or parameter inference for adistribution model.

•

IK related assumptions should not be totally ignored.

6.6.11 SIS of categorical variables

SIS for categorical variables (e.g. geological units, ore textures, etc) proceeds in a slightly different way. For k mutually exclusive categories, the indicator I(x) is set to 1 if the location x belongs to category k, and 0otherwise Prob{I(x)=1/(M)} = E{I(x)/(M)} For SIS an ordering of the categories is needed. This is a cdf type ordering of the K categories in the probability interval [0,1], to accommodate the application of the technique. The ordering of K intervals (categories) looks like:

92


Chapter 6

*

*

*

*

K-1

[0,p1 ], (p1 ,p1 +p2 ],...,(1-Σk=1

*

pk ,1]

The last interval constrains the sum of probabilities to 1. The ordering is a matter of judgement. The generation of a category at any location is based on the drawing of a uniform random number in [0,1] as before. This reduces the importance of the ordering.


93

Chapter 6

6.7

Implementation and Related Intricacies of IK in the SIS of Mineral Deposits: A Shear Hosted Gold Deposit

What is discussed in this section?

• • • •

Deposit and Available Data Wireframing and Interpolation Problems and Options Recipes for Indicator Kriging

••

Revisiting Wireframing IK Implementation Guidelines

6.7.1

Shear hosted gold deposit drill hole composites

Figure 6.29: Shear hosted gold deposit drill hole composites.

Gold associated with quartz veining hosted by metamorphically altered granite. Some isolated high grade lodes outside main shear zone.

94


Chapter 6

Figure 6.30: Grade histogram.

Figure 6.31: Log-grade histogram.

Figure 6.32: Grade probability plot.

95

Risk Analysis for Ore Reserves and Strategic Mine Planning: Stochastic models and applications

Chapter 6

6.7.2

Economic wireframing options

Figure 6.33: Economic wireframing options.

6.7.3

Interpolation based on economic wireframe envelope

Figure 6.34: Interpolation based on economic wireframe envelope

96


Chapter 6

6.7.4

Full indicator kriging constrained by economic factors

Figure 6.35: Full indicator kriging constrained by economic factors. Threshold (%) Cum. Prob.1 Cum. Prob.2

10

20

0.000 0.000

0.000 0.000

30 0.000 0.000

40 0.000 0.000

50 0.000 0.000

60

70

80

90

95

97.5

99

0.000 0.000

0.067 0.067

0.139 0.139

0.314 0.314

0.500 0.435

0.718 0.544

0.370 0.544

Table 6.7: 1 – before order relation corrections : 2 – after order relation corrections

1 order relation corrections Total magnitude of order relation corrections = 34.8%

97


Chapter 6

6.7.5

Geological wireframing options

Figure 6.36: Geological wireframing options.

6.7.6

How to use indicator kriging in practice

1. Estimate the cdf by selecting cut-offs. 2. Define an indicator function for cut-off z k (k=1,....) a. ⎧ i(x;zk)

0, if z(x)

⎨ ⎩

= b.

>zk

≤ 1, if z(x)

zk

3. Develop variograms for each cut-off. *

4. Use kriging to obtain i(x;z k) for any block. 5. Generate the conditional distribution and sample it. 6. Move to the next block.

98


Chapter 6

Selecting Cut-offs: Important Considerations

Adequate discretisation of the ccdf • from 6 to 15 cf’s (or 9-12) • no need for cf’s in uninteresting parts Number of samples per class • statistical reasons, e.g. 5 samples? • deciles of conditional distribution (ccdf)

•

miss the important part of ccdf

Quantity of metal per class • poor for low grade part of ccdf IMPORTANT

Geologic considerations • distinct changes in variograms

Figure 6.37: Probability plot showing significant changes in slope and discontinuities.

99


Chapter 6

Features of probability plots

• •

changes in slope discontinuities

Economic cut-offs may not always be important

• •

after support correction grade cf’s change no need if discretisation is adequate

Statistical considerations

•

variograms below the 10th and above the 90th percentiles are meaningless

The blind man’s approach

•

9-12 cf’s

•

half from splitting # of data reasonably

•

half from splitting quantity of metal

And the last class?

6.7.6.1

Kriging: order relation corrections

Kriging does not ensure that

•

F(u;zk/(n)) ≥ F(u ;zj /(n)), for any z k > zj

•

F(u;zk/(n)) is in [0 ,1]

Main sources of problem

•

negative weights

•

number of samples in a class too low

Large differences suggest implementation problems Order relations corrections

• • •

Upwards - systematic bias (underestimates) Downwards - systematic bias (overestimates) AVERAGE

Search radius must be the same from cut-off to cut-off. The local ccdf must be estimated from the same volume of interest.

6.7.6.2

100

Order relation corrections


Chapter 6

Figure 6.38: Order relation corrections.

Alternate order relation corrections if no data in local neighbourhood belongs to the z6 –z7 class

Figure 6.39: An alternate order relation correction.

If no data in the local neighbourhood is in the z6 – z7 class then the indicators used for interpolation must be the same for thresholds z6 and z7. Thus the significant order relation problem must be due to kriging parameters changing between thresholds z6 and z7.

101


Chapter 6

Quantifying Order Relation Problems Number of order relation 300 Median IK 250

Full IK - 1

r 200 e b m 150 u N 100

Full IK - 2 Full IK - 3

50 0 1 +2

2+3

3+4

4+5

5 +6

6+7

7+8

8 +9

9+

10 + 10

11

Cut-offs

violations age magnitude of error

Aver

10 9

Median IK

) 8 7 % ( 6 e d5 tu i 4 n3 g a2 M1

Full IK - 1 Full IK - 2 Full IK - 3

0 1+2

2+3

3+4

4+5

5+6

6+7

7+8

Cut-offs

8+9

9+10

10+ 11

Maximum magnitude of errors 20

) % ( 15 e d u ti 10 n g 5 a M

Median IK Full IK - 1 Full IK - 2 Full IK - 3

0 1+2

2+3

3+4

4+5

5+6

6+7

7+8

8+9

9+10

10+ 11

Cut-offs

6.7.6.3

102

Median indicator kriging


Chapter 6

Figure 6.40: Median indicator kriging.

Threshold (%) Cum. Prob.1 Cum. Prob.2

10 0.000 0.000

20 0.056 0.056

30 0.195 0.195

40 0.275 0.275

50 0.275 0.275

60 0.275 0.275

70 0.332 0.332

80 0.332 0.332

90 0.450 0.450

95 0.626 0.626

97.5 99 0.806 0.868 0.806 0.868


No order relation corrections made.

103


Chapter 6

6.7.6.4

Full indicator kriging

Figure 6.41: Full indicator kriging.

Threshold (%) Cum. Prob.1 Cum. Prob.2

10 0.000 0.000

20 0.055 0.055

30 0.189 0.189

40 0.270 0.268

50 0.275 0.271

60 0.267 0.271

70 0.307 0.307

80 0.332 0.332

90 0.468 0.468

95 0.619 0.619

97.5 99 0.802 0.677 0.740 0.740


2 ORDER RELATION CORRECTIONS Total magnitude of order relation corrections = 13.3%

104


Chapter 6

Full indicator kriging with rotating anisotropy

Figure 6.42: Full indicator kriging with rotating anisotropy.

6.7.8

Top cut-off related issues

Choosing the highest cut-off

•

Quantity of metal is attractive

Choosing the mean of the last class

•

Mean of samples

•

Median

•

Top cutoff

Need to check!

105

•

Model

•

Mean grade


Chapter 6

6.7.8.1

Calculation of block grade

Threshold

IK Cdf

Class Prob.

Class Mean

Grade Component

0.00

0.000

-

-

-

0.07

0.000

0.000

0.01

0.000

0.14

0.055

0.055

0.11

0.006

0.22

0.189

0.134

0.18

0.024

0.33

0.268

0.079

0.27

0.021

0.47

0.271

0.003

0.39

0.001

0.69

0.271

0.000

0.58

0.000

1.06

0.307

0.036

0.86

0.031

1.70 3.10

0.332 0.468

0.025 0.136

1.34 2.28

0.034 0.310

5.03

0.619

0.151

3.88

0.586

8.24

0.740

0.121

6.25

0.756

16.12

0.740

0.000

10.85

0.000

-

1.000

0.260

27.50

7.150

Block grade:

8.93

Table 6.10: Detailed calculation of IK estimated grade in a block.

106


Chapter 6

6.7.8.2

Value for top cut-off

CLASS CLASS GRADE PROB. MEAN COMPONENT

THRESHOLD

IK CDF

0.00

0.000

-

-

-

0.07

0.000

0.000

0.01

0.000

0.14

0.055

0.055

0.11

0.006

0.22

0.189

0.134

0.18

0.024

0.33

0.268

0.079

0.27

0.021

0.47

0.271

0.003

0.39

0.001

0.69

0.271

0.000

0.58

0.000

1.06

0.307

0.036

0.86

0.031

1.70

0.332

0.025

1.34

0.034

3.10

0.468

0.136

2.28

0.310

5.03

0.619

0.151

3.88

0.586

8.24

0.740

0.121

6.25

0.756

16.12

0.740

0.000

10.85

0.000

-

1.000

0.260

?

?

Table 6.11: IK classes and class contributions to estimated grade.

107

•

mean = 27.5

grade comp. = 7.15

grade = 8.93

•

median = 25.0

grade comp. = 6.50

grade = 8.28

•

threshold =16.12

grade comp. = 4.19

grade = 5.97


Chapter 6

6.7.8.3

Influence of number of samples

THRESHOLD IK CDF 10 IK CDF 15 IK CDF 20 SAMPLE SAMPLE SAMPLES S S 0.00

0.000

0.000

0.000

0.07

0.000

0.000

0.000

0.14

0.000

0.055

0.088

0.22

0.085

0.189

0.185

0.33

0.085

0.268

0.321

0.47

0.085

0.271

0.378

0.69

0.085

0.271

0.378

1.06

0.199

0.307

0.423

1.70

0.199

0.332

0.423

3.10

0.409

0.468

0.545

5.03

0.415

0.619

0.664

8.24

0.711

0.740

0.743

16.12

0.711

0.740

0.743

-

1.000

1.000

1.000

Number order

4

2

3

relation Total magnitude

7.5%

13.3%

15.1%

Block grade

10.41

8.93

8.42

Table 6.12: Effect of sample numbers on IK estimate.

108


Chapter 6

6.7.8.4

A summary of IK derived block grades

Economic wireframe

Indicator kriging

14.24 *

8.93

Upwards order relation correction Downwards order relation correction

10.27

“

Median for last class

8.28

“

Threshold for last class

“

10 samples

10.41

“

20 samples

8.42

“ “

7.60

5.97

Table 6.13: Effect of order relation corrections on grade estimate.

* Averaging order relation correction MEAN FOR TOP CLASS

15 SAMPLES

How We Implement IK in SIS Does Matter

109


Chapter 6

6.7.8.5

Order relation corrections

GRADE

IK CCDF

UPWARDS

DOWNWARD S

AVERAGING

0.00

0.000

-

-

-

0.07

0.000

0.000

0.000

0.000

0.14

0.055

0.055

0.055

0.055

0.22

0.189

0.189

0.189

0.189

0.33

0.270

0.270

0.267

0.268

0.47

0.275

0.275

0.267

0.271

0.69

0.267

0.275

0.267

0.271

1.06

0.307

0.307

0.307

0.307

1.70

0.332

0.332

0.332

0.332

3.10

0.468

0.468

0.468

0.468

5.03

0.619

0.619

0.619

0.619

8.24

0.802

0.802

0.677

0.740

16.12

0.677

0.802

0.677

0.740

-

1.000

1.000

1.000

1.000

Block Grade

-

7.60

10.27

8.93

Table 6.14: Order relation corrections in IK.

110


Chapter 6

Can we validate? Median IK section represents 9.2 Mt @ 2.7 g/t Au while the Full IK section 8.3 Mt @ 3.1 g/t Au

Figure 6.43: Section model from Median IK and Full IK of the same section.

Visual inspection is not sufficient even for block estimates. How about distributions of IK blocks (see Figure 6.44)?

Problem Good

Figure 6.44: Average block grade distribution (dots) and declustered histogram of the data (crosses).

111


Chapter 6

A validated IK model should show that the histograms of the declustered data set and the average histogram of IK blocks should match as the Figure 6.45 below.

Figure 6.45: Histogram of declustered data and average of IK blocks.

112


Chapter 6

6.8

Sequential Simulations: Implementation Issues

6.8.1

Generating the path for a sequential simulation

The path followed by the simulation algorithm may be either sequential, i.e. from node to node by row or column, or defined randomly. A sequential path will generate artefacts on the simulation output (i.e. effects on higher order moments). An example of the artefacts is shown in the exhaustive variograms from SGS conditional simulations (Isaaks, 1991), in Figure 6.46.

Random path

Sequential path (row by row)

Figure 6.46: Simulation algorithms and effect of random and sequential paths.

The discrepancy in the Y direction is due to the row by row sequential path. Artefacts will appear stronger in SIS.

6.8.2

Simple or ordinary kriging

Does the use of simple or ordinary kriging influence the sequential simulation results? In theory, SK provides the exact conditional means and variances. In practice, conditional means from SK are influenced by the global mean, NOT the actual data values.


113

Chapter 6

EXERCISE: For any configuration of four data points with the same value (say, 2.5g/t Au) calculate the value at a fifth nearby location using OK and discuss the differences from SK. OK is more sensitive to actual data values, and it may be better to estimate conditional means from OK. SK provides the exact conditional variance. The above comments suggest that in SGS, OK is probably better in calculating the mean and SK the variance of the local distributions. In SIS, OK is commonly more preferable to SK.

•

The use of OK in sequential simulations stresses the characteristics of the local conditioning data.

•

The use of SK in sequential simulations generates more ‘mixing’ of values.

6.8.3

Resolution and variogram reproduction

Due to the increasing number of conditioning points at each step of sequential simulations, one needs to use local neighbourhoods. The use of neighbourhoods influences the reproduction of the variogram, which is reproduced up to the size of the neighbourhood. The multiple grid solution is used to reproduce variograms at any distance: Step 1: Use a coarse grid as shown in Figure 6.47. The large neighbourhood allows the reproduction of data at large distances.

Figure 6.47: Simulation on a coarse grid.

Step 2: Use a dense grid and smaller neighbourhoods for the remaining grid nodes as shown in Figure 6.48. This will ensure the reproduction of variograms at short distances.

114


Chapter 6

Figure 6.48: Simulation on a dense grid after the coarse grid. 6.8.4

Sequential simulations: indicator or Gaussian implementation?

SIS is not directly or indirectly related to Gaussian RF models. There is no need for prior estimation of distribution parameters. One can easily include soft data. These properties are particularly attractive when Gaussian implementations are inappropriate: Spatial correlation between extreme values need to be maximised. SIS may be more appropriate when indicator data reveal high correlations at high cut-offs. This is less tractable and computationally efficient. The problem of resolution: SIS may be affected by ‘class resolution.’ Consider the following case as illustrated by Figure 6.49 (Isaaks, 1991):

Figure 6.49: Definition of the last class in IK maybe difficult.

What happens if the value to be simulated belongs to the last class? In the present example the last class extends from 784 to 9,499! The choices are: to choose uniformly (equal probability assigned to each value) or assume some distribution within the last class. The distribution used to draw values from the last class is based on the global data distribution: i.e. the value drawn is not conditional to the local data. The above is also


115

Chapter 6

valid for any other class. The lack of local conditioning may be a weakness of indicator simulations. One solution is to use additional cut-offs for the last class and the variogram/covariance defined using the 9th decile. SGS is faster, less demanding in memory, and tractable. SGS like all Gaussian models tends to minimise the correlation between the extreme values. And, it is preferable when one is interested in ‘spatial averages’ of attributes rather than correlation of extreme values.

•

The choice of an indicator over a Gaussian based simulation technique depends on:

Data types and their characteristics. The objectives of the study and theproblem at hand. Constraints in: time, hardware and software, expertise. •

Generally, indicator simulations are preferable when outliers area serious issue and one is justifiably interested inthe correlation/connectivity of extreme values.

•

Generally, Gaussian simulations are usuallybetter when dealing with continuous variables, and one is interested in spatial averages.

•

The group based Gaussian algorithm is by far themost efficient and the preferred Gaussian choice.

•

Combination of indicator and Gaussian techniques may oftenbe more effective, e.g. combination of rock types withgrades.

6.8.5

Sequential simulations: some comparisons

Simulating Geological Environment for comparing SIS and SGS (Deutsch, 1992). The environment is an eolian sandstone shown in Figure 6.50 and Figure 6.51.

116


Chapter 6

Figure 6.50: Sandstone cut (20 cm by 10 cm represented by 328 by 171 square pixels). The figure represents an 20 cm by 10 cm sandstone cut represented by 328 by 171 square pixels. White areas on the figure represent well sorted sands with high values (permeability) and black are low values in fine grained sandstone.

Figure 6.51: Upscaled version of the sandstone cut. The figure represents the ‘upscaled’ version of the above image (164 by 85 pixels) and is used as a reference. The curvilinear features of this environment are a challenge for most simulation techniques.

The idea here is to compare (non-conditional) SIS and SGS with the reference image.


117

Chapter 6

h

h Figure 6.52: The variogram of the reference image (normalised values).

Several nested structures are used to model the variogram in Figure 6.52 above. Note the anisotropy (6 to 1) in the two principal directions. Note the periodicity.

Figure 6.53: Two realisations from SGS results, note the ‘salt and pepper’ like texture.

In Figure 6.53, the ‘continuity’ of the features is not reproduced well. The curvilinear features of the srcinal image are not reproduced (refer to Figure 6.51).

118


Chapter 6

Results from SIS using indicator variograms with the same form as before: two realisations. In Figure 6.54, the ‘continuity’ of the features is much improved compared to the reference image. The curvilinear features of the srcinal image are still not reproduced.

Figure 6.54: Using indicator variograms with the same form as before, note the absence of the ‘salt and pepper’ like texture.

Figure 6.55 shows an additional example reproducing cross-indicator covariances as well.

Figure 6.55: Example reproducing cross-indicator covariances.


119

Chapter 6

•

Note that this was a comparison among non-conditional simulations.

•

Simulations reproducing two-point statistics may be limited in reproducing elaborate patterns such as the curvilinear features in this example.

120


Chapter 7

CHAPTER 7

7. 1

7.1.1

CONDITIONAL SIMULATIONS IN RESOURCES/RESERVE ESTIMATION

Risk Quantification and Limits in Resource Estimates: Example from a Nickel Laterite Deposit Introduction

Risk and uncertainty in resource/reserve estimation are key concerns in both feasibility and production. This section presents an example showing a conditional simulation (CS) approach as used for the explicit quantification of reserve risk/uncertainty quantification. The study assesses the sensitivity of resource/reserve variability and mining selectivity based on a conditional simulation study of nickel grades in a nickel laterite deposit. More specifically, the aim of the CS study was two fold: •

Assess the resource/reserve variability in terms of nickel grade based on multiple CS runs and compare the results with the srcinal ordinary kriged (OK) resource estimate at the 25 by 25 by 1 metre block size. •

7.1.2

Assess the impact of mining selectivity on ore tonnage, nickel head grade and nickel metal quantity by considering various selective mining unit (SMU) sizes for several CS models of the nickel grade. Information available

The information available for the study consisted of the srcinal 1m composited drill hole data as well as the 25 m x 25 m x 1 m resource block model estimated by ordinary kriging. Both datasets were restricted to nickel grades. The statistical characteristics of nickel mineralisation are summarised in Table 7.1.

SUMMARY STATISTICS OF NICKEL MINERALISATION Mean (% Ni) Variance CV All Samples 0.94 0.094 0.33 Declustered Samples 0.91 0.090 0.33

# of Samples 15 574 9 756

Table 7.1: Summary statistics of nickel mineralisation.


121

Chapter 7

NICKEL VARIOGRAM PARAMETERS Sills and Ranges Nugget, 1st Sill, 2nd Sill 1st Ranges C0, C1, C2 a1 0.1, 0.46, 0.44 70 m x 80 m x 6.5 m Directions Major Axis Intermediate Axis azimuth 130o 40azimuth o

2nd Ranges a2 400 m x 370 m x 7 m Minor Axis Vertical

Table 7.2: Nickel variograms (spherical models)

Variography is summarised in Table 7.2. Nickel mineralisation is quasi-isotropic in the horizontal plane with short and long ranges respectively in the order of 80 m and 400 m. The maximum vertical range was set at 7 m. 7.1.3

Conditional simulation study: methodology and results

The sequential Gaussian simulation algorithm (Journel, 1994; Johnson, 1987) was use to generates equally probable models of the deposit, which reproduce the in-situ grade variability. All CS models reproduce the available data and their statistical characteristics (grade histogram and variogram). The methodology includes: •

•

•

•

•

The SGS process assumes that the distribution of the dataset is Gaussian, thus the first step in the process consists of transforming the srcinal dataset to a dataset which follows a normal distribution (termed “N-scored” data). Variograms are completed on the N-score data to define the CS parameters relative to the spatial continuity of the mineralisation. Multiple CS runs are completed using a fine grid block resolution. Each CS run block grade is then back-transformed to the srcinal data distribution. Simulated models are validated by comparing the statistical characteristics of the simulated grades to the srcinal data, as well as by visual checking of the simulated values against the input data. A change of CS support from small blocks to the block sizes required for the study is completed.

Results may be summarised as follows: Ten CS models of the deposit grade were generated using 5 m x 5 m x 1 m (“5x5x1”) blocks. Examples of Ni realisations are shown in Figure 7.1. The 5x5x1 simulations were reblocked into models of various block size to complete the resource variability and SMU size sensitivity analysis. The statistical characteristics of the CS models are presented in Table 7.3 for comparison with the srcinal input data. The CS models reproduce the statistical characteristics of the srcinal declustered data.

122


Chapter 7

Figure 7.1: Ni data and three conditional simulations

SUMMARY STATISTICS OF CS RUNS Mean (% Ni) Declustered Samples 0.91 CS model 2 0.91 CS model 3 0.92 CS model 4 0.91 CS CS CS CS CS CS

model 5 model 6 model 7 model 8 model 9 model 10

0.91 0.91 0.92 0.92 0.91 0.91

Variance 0.090 0.092 0.091 0.093

Coefficient of Variation 0.33 0.33 0.33 0.34

0.092 0.094 0.091 0.093 0.091 0.092

0.33 0.34 0.33 0.33 0.33 0.33

Table 7.3: Summary statistics of CS runs.

7.1.4

Resource variability

To assess the resource variability, nine CS models completed using 5 m x 5 m x 1 m resolution were reblocked into 25 m x 25 m x 1 m (“25x25x1”) models for comparison with the srcinal 25x25x1 OK model (Figure 7.3). Table 7.4 summarises the grade, tonnage and quantity of metal of the OK model and CS results for nickel cut-off grades close to the likely mining cut-off grade. CS results are compared with the OK results in terms of metal quantity, ie minimum and maximum metal quantities obtained from the nine CS runs are reported with the corresponding tonnage and grade. A grade-tonnage diagram of the OK model and CS model 3 is presented in Figure 7.2:.


123

Chapter 7

Figure 7.2: Grade-tonnage curve. One conditional simulation for various selectivities and kriging.

15m x 15m 25m x 25m

Simulation

Figure 7.3: The fine-scaled simulations can be re-blocked to different SMU’s to provide grade-tonnage information at various levels of selectivity The small differences observed between the minimum and maximum metal quantities obtained from the various CS models as reported in Table 7.4, indicates that globally the resource is well defined and presents a low variability. This supports the Measured status the resource reported in past studies, in relation to the current drilling density of 50 m x 50 m.

124


Chapter 7

ORDINARY KRIGED MODEL AND MINIMUM-MAXIMUM METAL QUANTITIES FROM CONDITIONAL SIMULATION MODELS Ordinary Kriged Model Minimum and Maximum Metal Quantities with corresponding Tonnage and Grade Minimum Metal Quantity Maximum Metal Quantity Cut-off Tonnage Grade Metal Metal Tonnage Grade Metal Tonnage Grade Grade Quantity Quantity Quantity (% Ni) (kt) (% Ni) (t) (t) (kt) (% Ni) (t) (kt) (% Ni) 0.8 22 915 1.04 238 316 227 693 21 685 1.05 233 550 22 033 1.06 0.9 17 165 1.10 188 815 183 266 16 363 1.12 186 872 16 685 1.12 1.0 1.1

11 608 6 970

1.17 1.25

135 814 87 125

137 436 94 490

11 453 7 382

1.20 1.28

140 496 97 203

11 708 7 594

1.20 1.28

Table 7.4: Ordinary kriged model and minimum-maximum metal quantities from conditional simulation models.

The differences observed between the OK model and the CS models (Figure 7.2:), arise from the absence of smoothing in the CS models in comparison with the OK model. The CS models reproduce the variability of the statistical characteristics of the input data, which leads to a different grade-tonnage relationship for the CS models by comparison with the OK model. The CS models define less tonnes at higher grades at the lower end of the curve up to a cut-off grade close to 1.0% Ni; above this cut-off grade the tendency is reversed with the CS models defining slightly more tonnes at a higher grade than the OK model. Practically, close to the likely mining cut-off grade (0.9% - 1.0% Ni), the CS models report higher grades than the OK model for small positive or negative tonnage to quantities the absence of smoothing. Furthermore, the CS models report highervariations, minimum due metal than the OK model at cut-off grades greater than 1.0% Ni.

7.1.5

SMU size sensitivity analysis

The variability of the resource as observed in the previous section does not account for mining selectivity. The grade-tonnage-metal variations observed in Section 5 between the CS and OK models are emphasised when mining selectivity is accounted for, ie when the resource/reserve is simulated on blocks smaller than 25x25x1. The second aspect of the CS study considers the effects of improved mining selectivity, as defined by the SMU size, on potential ore tonnage, nickel head grade and nickel metal. An assessment of the recoverable resource can be made for decreasing SMU sizes.

7.1.6

SMU with varying horizontal block size

The minimum SMU size considered is a 5 m x 5 m x 1 m block, which is the size on which the CS models were completed. The models were re-blocked considering blocks sizes of 10 m x 10 m, 15 m x 15 m and 20 m x 20 m which represent realistic mining unit sizes on which to assess the impact of decreasing selectivity from the extreme 5 m x 5 m case.


125

Chapter 7

TONNAGE vs NICKEL CUT-OFF GRADE Minimum and Maximum Values of Conditional Simulation Runs for Various Block Sizes OK Model 25x25x1 Min 25x25x1 Max 25x25x1

25 000

Min 15x15x1 Max 15x15x1 Min 10x10x5

20 000

Max 10x10x1

)t (k 15 000 e ag n n o 10 000 T

Min 5x5x1 Max 5x5x1

5 000

0 0.8

0.9

1.0

1.1

1.2

1.3

1.4

Cut-off Grade (% Ni)

Figure 7.4: Tonnage versus cut-off grade.

METAL QUANTITY vs NICKEL CUT-OFF GRADE Minimum and Maximum Values of Conditional Simulation Runs for Various Block Sizes OK Model 25x25x1 Min 25x25x1

250 000

Max 25x25x1 Min 15x15x1

200 000

Max 15x15x1 Min 10x10x5

) 150 000 (t la te M100 000

Max 10x10x1 Min 5x5x1 Max 5x5x1

50 000

0 0.8

0.9

1.0

1.1

1.2

1.3

1.4


hb Figure 7.5: Metal versus cut-off grade.

The grade-tonnage diagram presented in Figure 7.2: illustrates the impact of increasing mining selectivity on potential ore tonnage and head grade for block sizes 25x25x1,

126


Chapter 7

15x15x1, 10x10x1 and 5x5x1 in the case of the CS model 3. Figure 7.4 and Figure 7.5 illustrate tonnage and metal quantity variations (minimum and maximum values obtained for the 9 simulation models) for nickel cut-off grades close to the likely mining cut-off grade. Table 7.5 summarises the grade, tonnage and quantity of metal of the OK model and CS model 3 for nickel cut-off grades close to the likely mining cut-off grade. •

•

•

The CS results indicate that, as expected, the 5x5x1 case allows recovery of significantly higher tonnages of material at a higher grade than the other options, for example at a 1.0 % Ni cut-off grade, the 5x5x1 case defines 155 000 t of metal against 136 000 t for the OK model, a variation of 14 %. The 5x5x1 CS option can be considered as an ideal-best case scenario. It is should be noted that globally the results for 15x15x1 and 10x10x1 are very close, which indicates that mining using SMUs of 10x10x1 rather than 15x15x1 would not benefit greatly the operation in terms of head grade. Conversely, the difference in outcome between 25x25x1 and 15x15x1 is significant. The CS test indicates that to achieve the best head grade outcome with practical selective mining practices, the preferred mining option to consider would be to use 15x15x1 SMUs, as a reduction of the block size to 10x10x1 would bring little overall improvement. The tonnage and metal quantity variations vs nickel cut-off grade diagrams in Figure 7.4and Figure 7.5 and Table 7.5 show that the impact of selectivity on recovered metal and ore tonnage increases significantly when the mining cut-off grade departs from 0.9% - 1.0% Ni. As an example, at a 1.1% Ni cut-off grade, the metal quantity increase over the OK model figures is 21% for a 10x10x1 SMU compared with 12% for a 25x25x1 SMU. By contrast, at a 0.9% Ni cut-off grade, metal quantities are similar for both SMU sizes. This confirms the intuitive idea that the higher the effective mining cut-off grade is set from a 0.9% - 1.0% Ni cutoff grade (close to the deposit average grade), the more attention must be given to the choice of mining selectivity to achieve targeted ore tonnage and head grade. The CS test quantifies this effect and allows measurements of the consequences of a given mining selectivity on production figures. Similarly, these results quantify the different production outcomes that can be achieved depending on the grade control procedure applied for a given mining selectivity, i.e. it is frequently observed that when a cut-off grade is applied on raw grade control assays rather than SMUs, the effective mining cut-off will be higher than the planned cut-off grade with the related consequences on tonnage and metal quantity.

•

The changes in grade-tonnage relationship observed for the nickel mineralisation between the OK and CS results may be emphasised in other zones of lateritic nickel mineralisation.


127

Chapter 7

ORDINARY KRIGED MODEL AND CS MODEL 3 RESOURCES REPORTED FOR VARIOUS BLOCK SIZES OK Model 25x25x1 Cut-off Grade (% Ni)

Tonnage

0.8

22 915

0.9

17 165

1.0

11 608

1.1

6 970

(kt)

CS Model 3 25x25x1 Grade

Metal Quantity (t)

Tonnage

1.04

238 316

22 033

1.06

1.10

188 815

16 685

1.12

(% Ni)

1.17

1.25

135 814

(kt)

11 708

87 125

7 594

Metal Quantity (t)

Tonnage

OK Model 25x25x1

Metal Quantity (% Ni) (t)

1.28

0.8

22 915

1.04

238 316

21 395

1.07

0.9

17 165

1.10

188 815

16 369

1.14

1.0

11 608

1.1

6 970

Grade (% Ni)

1.17

1.25

135 814

(kt)

11 756

87 125

7 852

Metal Quantity (t)

Tonnage

OK Model 25x25x1

Metal Quantity

233 550

-4%

2%

-2%

186 872

-3%

2%

140 496

97 203


9%

Grade

0.8

22 915

1.04

238 316

20 992

1.08

0.9

17 165

1.10

188 815

16 211

1.15

1.0

11 608

1.1

6 970

(% Ni)

1.17

1.25

135 814

(kt)

11 780

87 125

8 057

228 927

-7%

3%

-4%

186 607

-5%

4%

142 248

102 076

Metal Quantity (t)

Tonnage

OK Model 25x25x1

13%

Tonnage

Grade

0.8

22 915

1.04

238 316

21 044

1.1

0.9

17 165

1.10

188 815

16 648

1.17

1.0

11 608

1.1

6 970

(% Ni)

1.17

1.25

135 814

87 125

(kt)

12 381

8 789

5%

17% Metal Quantity

226 714

-8%

4%

-5%

186 427

-6%

5%

143 716

105 547

1%

16%

-1% 4%

5%

6%

21%

Variation from OK Model


1.33

4% Grade

Grade

1.25

-1% 3%

Tonnage

CS Model 3 5x5x1


1%



1.31

3%

12% Metal Quantity

Grade

1.22

2% Grade

CS Model 3 10x10x1

Tonnage

-1% 3%

Tonnage

1.21

1.30

1%


Grade


(kt)

Grade

CS Model 3 15x15x1

Tonnage

(kt)

Tonnage

1.20


(kt)


Grade

Tonnage

Grade

Metal Quantity

231 484

-8%

6%

-3%

194 782

-3%

6%

154 763

116 894

7%

26%

3% 7%

6%

14%

34%

Table 7.5: Ordinary kriged model and CS model 3 resources reported for various block sizes.

7.1.7

15 m x 15 m SMU with varying vertical dimension

The results in the previous section suggest that the preferred mining option should consider a 15x15 m horizontal mining block size. This section quantifies the dilution for a varying bench height, specifically it compares grade-tonnage curves based on 3 SMU sizes: 15x15x1, 15x15x2 and 15x15x4. The grade-metal diagram presented in Figure 7.6 illustrates the impact of changing the bench height from 1 m to 2 m and 4 m on the reserve potential for CS model 3. The relationship between the various curves is similar as observed in Figure 7.5 and demonstrates support/selectivity effects on the expected grade-tonnage curves. The negative effects of a less selective operation using 4 m bench height are apparent from Figure 7.6.

128


Chapter 7

Table 7.6 summarises the sensitivity of recoverable metal quantities to vertical selectivity. It should be noted that Table 7.6 suggests that changing “vertical selectivity” from 15x15x1 to 15x15x2 has relatively little effect on recovered metal quantities at the economically interesting cut-off grades, suggesting that the 2 m bench height, which is a more practical option, could be chosen without detrimental effects on expected production numbers. The expected differences from a 4 m bench height seem large as one would anticipate. METAL QUANTITY vs NICKEL CUT-OFF GRADE Bench Heigth Sensitivity - Conditional Simulation Run 3 block size 15x15x1, 15x15x2 and 15x15x4 OK Model 25x25x1 15x15x4

250 000

15x15x2 15x15x1

200 000

150 000 ) (t la et M 100 000

50 000

0 0.8

0.9

1.0

1.1

1.2

1.3

1.4


Figure 7.6: Metal quantity variations for nickel cut-off grades ranging from 0.8% Ni to 1.4% Ni.

ORDINARY KRIGED MODEL AND MINIMUM-MAXIMUM METAL QUANTITIES FROM CONDITIONAL SIMULATION MODELS FOR VARIOUS BENCH HEIGHTS Ordinary Kriged Model Minimum and Maximum Metal Quantities Cut-off Tonnage Grade Metal Minimum Metal Quantity Maximum Metal Quantity Grade Quantity (t) (t) (% Ni) (kt) (% Ni) (t) 15x15x1 15x15x2 15x15x4 15x15x1 15x15x2 15x15x4 0.8 22 915 1.04 238 316 226 241 229 458 232 138 228 927 232 267 236 257 0.9 17 165 1.10 188 815 183 791 184 484 185 648 186 607 186 981 188 245 1.0 11 608 1.17 135 814 139 247 137 700 135 086 142 248 140 988 138 402 1.1 6 970 1.25 87 125 99 111 95 859 90 430 102 076 98 278 92 698

Table 7.6: Ordinary kriged model and minimum-maximum metal quantities from conditional simulation models for various bench heights.


129

Chapter 7

7.1.8

Conditional simulation and ordinary kriged models in 250 m x 250 m mining blocks

To elucidate on the simulation versus estimation approach, the 25x25x1 and 15x15x1 results for the CS model 3 were regularised in the 250 m x 250 m mining blocks and compared with the equivalent results obtained from the OK model. Despite the smoothing effect of the regularisation process, the scatter plots and quantile-quantile (Q-Q) plot presented in Figure 7.7 illustrate, as expected, the higher variability of the CS model in comparison with the OK model. The CS model tends to define lower mining block grades at the low cut-off grades, approximately below 0.7 % Ni, and slightly higher block grades above. Because the 250 m x 250 m mining blocks have been designed according to the OK model, it is expected that this trend would be emphasised should the mining blocks be based on CS models. However, the correlation between the 250 m x 250 m mining block grades derived from OK and CS is high confirming the low variability of the resource globally. The Q-Q plot of 25x25x1 CS and OK results presented in Figure 7.8 illustrates more clearly the difference between CS and OK results. At this scale, the absence of smoothing of the CS results is clearly shown with the CS model defining less low grade blocks (points below the bisecting line), and more higher grade blocks above a 0.9 % Ni cut-off grade (points above the bisecting line) as was reported globally on the gradetonnage diagram presented in Figure 7.2. The higher variability of grades in the CS results and the absence of smoothing imply that locally, mining selection would be in some instances different from the outcome obtained from the smoother OK model.

130


Chapter 7

SCATTER PLOT

SCATTER PLOT Ordinary Kriged Model and Conditional Simulation Run 3 25x25x1 regularised in 250 m x 250 m Mining Blocks

Ordinary Kriged Model and Conditional Simulation Run 3 15x15x1 regularised in 250 m x 250 m Mining Blocks 1.40

1.40

1.20

1.20

)i1.00 N % ( 3 n u r S C0.80

)i1.00 N (% 3 n ru S C0.80

0.60

0.60

0.40

0.40 0.40

0.60

0.80 1.00 OK model (% Ni)

1.20

0.40

1.40

0.60

0.80 1.00 OK model (% Ni)

1.20

1.40

Q-Q PLOT Ordinary Kriged Model and Conditional Simulation Run 3 15x15x1 regularised in 250 m x 250 m Mining Blocks 1.40

1.20

i)1.00 N % ( 3 n u r S C0.80

0.60

0.40 0.40

0.60

0.80 1.00 OK model (% Ni)

1.20

1.40

Figure 7.7: Scatter plots and quantile-quantile (Q-Q) plot illustrating the higher variability of the CS model in comparison with the OK model.

Q-Q PLOT Ordinary Kr iged Model and Conditional Si mulation Run 3 on 25 m x 25 m x 1 m blocks, based on 3206 blocks from bench 405.5 mRL 1.70

1.50

1.30

i) N1.10 (% 3 n u r S0.90 C

0.70

0.50

0.30 0.30

0.50

0.70

0.90 1.10 OK model (% Ni)

1.30

1.50

1.70

Figure 7.8: Q-Q plot of 25x25x1 CS and OK results.


131

Chapter 7

7.1.9

Conclusions

Typically, a kriged model, because of inherent smoothing, will have a tendency to generate a larger amount of low grade material and a reduced amount of high grade blocks than will be mined, i.e. the variability between block grades is reduced when compared with the actual grade variability. CS models do not incorporate smoothing, but can replicate actual high and low grades within the deposit at the same frequencies as the srcinal data. Because CS reproduces actual in-situ grade variability together with the spatial grade distribution and continuity, the CS results associated with a given SMU size represent a potentially ‘mineable’ resource. As far as the resource variability goes, the conditional simulation study indicates that globally the nickel limonite resource is well defined and has low variability. The SMU size sensitivity analysis suggests that the quantitative information obtained from the CS study provides a base from which to choose a practical mining method and realistic mining selectivity which will satisfy a given production target in terms of nickel head grade and metal recovered. The study also indicates that increasing mining selectivity from a 15x15x1 SMU to a 10x10x1 SMU would bring little overall improvement on ore tonnage and head grade, but that the difference between 15x15x1 and 25x25x1 SMU is significant. The impact of selectivity on recovered metal and ore tonnage increases significantly when the mining cut-off grade departs from 0.9% - 1.0% Ni. Once the economic cut-off grade has been ascertained, this is an issue to keep in mind when choosing the grade control procedure. If mining blocks are delineated on raw blast hole assays (mining polygons drawn from the blast hole assay bench maps), rather than SMU size blocks (blast hole assays interpolated into SMU block size by kriging or CS, which are then used to define the mining blocks), it is highly likely that a slightly higher effective cut-off grade than planned will be used. The CS study allows quantification of this effect in terms of production results. When considering a 15x15x1 SMU, the CS results indicate that mining can expect to achieve an improved nickel head grade from 1.17 % Ni from the OK model to 1.21 % Ni for a minor tonnage increase at a 1.0 % Ni cut-off grade. At a 0.9 % Ni cut-off grade, the head grade increases from 1.10 % Ni (OK model) to 1.14 % Ni (CS model) for a minor tonnage decrease.

Acknowledgement: Thanks are in order to Annick Manfrino, Minproc, for her contribution to this study.

132


Chapter 7

7. 2

Resources/Reserves Simulation

Classification

and

Direct

Block

Resources/Reserves Classification and Direct Block Simulation

Some Common Concepts in Resources/Reserves Classification and Drilling Targeting Variogram Range Number of Samples . . .

18.2

16.5 low variability

?

17.1

17.9

0.03

0.25

Estimation Variance

Estimation variances reflect the geometry of the drilling, NOT in-situ orebody variability

high variability

11.54

? 6.03


133

Chapter 7

Some measures measures of (local) local) uncertainty revisited A cumulative conditional distribution function (ccdf)

Measures of location 

1.0



0.75

a b o r P

y ti il b

0.50

Measures of spread 0.25

Conditional variance Coefficient of Variation(CV  Interquartile range (IQR)  

0 1

234

56

Probability

Grade (g/t)



E-type estimate conditional mean or Median

E-type estimate or conditional mean:

m(x) = m1*p1 + m2*p2 + …. + mk*pk where mk and pk are the mean and probabilities of class k



Median:



Conditional variance:

q0.5 σ2(x) = (m- m 1)2*p1 + (m- m 2)2*p2 +

… + (m- m k)2*pk Coefficient of Variation: CV(x) = σ(x) / m(x)  Interquartile range: IQR(x) = q 0.75 - q0.25 

134


Chapter 7

Some measures measures of (local) local) uncertainty revisited

it n ag M

d u

e

n g a M

u it

e d

Samples

Kriging variance

Interquartile range

Conditional variance

Eastin (x)

Eastin (x)

Alternative Approaches in an Example

o A direct block simulation o Example in a gold deposit o Conclusions


135

Chapter 7

Uncertainty Assessment -> Stochastic Simulation Sequential Algorithms

Point support

Block support

Some disadvantages of the traditional techniques

o Size of the simulation domain o Ability to handle data with different support

136


Chapter 7

DBSIM – Direct block simulation The method is a further development of the Generalized Sequential Gaussian simulation algorithm The simulation is carried out simultaneously for a group of nodes which coincide with the internal points of each block The simulated nodes are averaged out into a block value and the internal points are discarded The simulation is sequential: Simulated block values are put in the data set

Conditioning to the data ⎡ [C ] nn [ ] C12 C = ⎢ 11 ⎣[C21] Nn[ ] C22

⎤ ⎥ NN ⎦ nN

u8 u2 u5

u3

u6

⎡ [y1 ]n ⎤ ⎡ L11 ⎢[y ] ⎥ = ⎢ L ⎣ 2 ⎦ ⎣ 21 N

0 ⎤ ⎡ w1 ⎤ ⋅ L22 ⎥⎦ ⎢⎣ w2 ⎥⎦

u3

u4 u1

u7

y2 = L21 L11-1 y1 + L22 w2

N

nodes

n cond. data


137

Chapter 7

138


Chapter 7

DATA WITH DIVERSE SUPPORT SIZE Drill holes

Mined-out stopes


139

Chapter 7

Validation Study

]281;561]

[0; 281]

]561;842]

]842;1123]

250

200

th r o 150 N

1000

750

500

250

0

100

50

50

100

150

200

250

East

140


Chapter 7

Block Variograms


141

Chapter 7

SGSIM

Real

DBSIM

1000

750

500

250

0

Example

o 60,010 blocks -> 2,561,345 nodes o 40 Realisations -> ~ 5h o 3 CPUs R10000/175MHz

142


Chapter 7

m 0 1

4m 10m

Model characteristics: o Large number of blocks o Multiple domains o Resource classes with specific sample selection criteria

A gold load


143

Chapter 7

Simulation run Model size: 27 888 147 blocks Number of realisations: 20 Total number of simulated nodes: 557 762 940 Hardware: 3 CPUs with 175Mhz (RS10000) Processing time: 46 hours* * time includes data compositing, variogram and histogram calculations

Variogram reproduction

144


Chapter 7

Histogram reproduction P-P Plot

1

0.9

0.8 0.7

0.6

DATA S3 S6 S9 S12 S15 S18 S21 S24 S27 S30 S33 S36 S39 S42 S45 S48

n io t 0.5 a l u m i S . 0.4 b ro P

0.3

0.2

0.1

S1 S4 S7 S10 S13 S16 S19 S22 S25 S28 S31 S34 S37 S40 S43 S46 S49

S2 S5 S8 S11 S14 S17 S20 S23 S26 S29 S32 S35 S38 S41 S44 S47 S50

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Prob. Reference

Examples in



Aid to resources resources classification



Targeting of areas for additional drilling


145

Chapter 7

Current resources resources classification



to Measures, Inferred and Indicated is based on number of samples within given distances

How about local variability/uncertainty?

Interquartile range

146


Chapter 7

E-type

Resource classification


147

Chapter 7

Probability

148


Chapter 7

Classified as measured but low probability

Targeting of areas for additional infill drilling



The mine uses fixed drilling patterns, usually 18 x 16 RC drilling


149

Chapter 7

Probability E-type estimate

Before and after RC drilling Probability

RCno

150

RC

Level -210

Gradeestimates

RCno

RC


Chapter 7

Identify zones of high grade and high variability

Interquartile range

E-Type estimate

Classified as waste but high probability


151

Chapter 7

Classified as waste but high probability

Comments Comments Resource classification could be based on E-type estimates Probability of being ore Iterquartile range Infill drill based on chances to increase resources, could use similar indicators as above

152


Chapter 7

7.2.1

Resource/Reserve Uncertainty

Classification

and

Measures

of

Local

Conditional simulation enables a resource/reserve classification methodology that simultaneously considers data density and orebody continuity as well as in situ variability. In this section, key aspects of the methodology are described including appropriate measures of uncertainty to be used for resource/reserve classification.

Classification Methodology. A resource classification methodology is outlined here and consists of five steps: (1) generation of suitable realizations of the deposit at the blocksupport scale considered appropriate; (2) derivation of conditional distribution functions (cdf) of block grades or attribute selected; (3) calculation of uncertainty measure(s) to be used for resource classification based on the cdf of each block in the orebody model; (4) classification of resources/reserves; and (5) evaluation and production of risk profiles for estimates. Measures of Local Spatial Uncertainty. A key part of the classification methodology is the use of summary statistics and measures of uncertainty which are derived from the conditional distribution function (cdf) of the grade Z(u) of a block at location u within the deposit (Figure 1). Basic measures include the conditional variance, conditional coefficient of variation, inter percentile ranges and probability interval. The conditional variance (CV) measures the spread of the cdf around its mean value and is given by K +1

CV (u) =

∑ [z

− z E* (u ) ] [F (u; z k ) − F (u; z k −1 )] 2

k

(7.2.1)

k =1

where, zk k=1,…K, are K threshold values discretizing the range of variation of z values, z k is the mean of the class zk-1,(zk-1,zk] which in case of a within-class linear interpolation model corresponds to zk = (zk-1 + zk)/2, and

z E* (u ) =

K +1

∑ z [F (u; z k

k

) − F (u; z k −1 )]

is the discrete sum

k =1

approximation to the expected value of the cdf. The conditional coefficient of variation (CCV) corresponds to the conditional standard deviation divided by the mean or relative standard deviation. It expresses variability as a percentage of the mean, and is calculated as 2

K +1

CCV(u) =

∑ [z

k

− z E* (u )] [F (u; z k ) − F (u; z k −1 )]

k =1

z E* (u )

(7.2.2)

The CCV better discriminates zones in terms of sample density (or availability of conditioning information). An advantage of the CCV over the CV is that it is expressed directly as a percentage of the mean and therefore filters out proportional effects. The inter quartile range (IQR) is defined as the difference between the upper (q0.75) and the lower (q0.25) quartiles of the distribution

IQR(u) = q0.75(u) – q0.25(u) = F -1(u;0.75) – F –1(u;0.25)


(7.2.3)

153

Chapter 7

Similarly, the generalization to any inter percentile range (IPR), such as a 90% confidence intervals, is defined as

IPR(u) = q0.95(u)– q0.05(u) = F -1(u;0.95) – F –1(u;0.05)

(7.2.4)

Unlike the CV and CCV, IPR and its variations are not affected by the magnitude of grades in the cdf and are thus not sensitive to outliers. The IPR may also be scaled by the mean or median of the corresponding cdf. Related measures of uncertainty are the probability that a block grade, Z(u) is above, below or within a probability interval. Figure 1 shows related 2 summary statistics and uncertainty measures for the cdf of thickness of a 100 x 100 m block. The cdf itself provides the model of uncertainty for the value of the thickness attribute for a given block at location u within the deposit. Each discrete point in the cdf corresponds to a simulated value. The cdf is a cumulative histogram of all simulated values assigned to the simulation realizations of the deposit. A continuous function is interpolated between the discrete points to enable the assessment of probabilities for any cdf value.

Fig. 1: A conditional distribution function (cdf) showing summary statistics and uncertainty measures for the thickness of a 100 x 100 m 2 block in a diamond deposit. The cdf is generated by calculating the associated probabilities for a given block from multiple simulated realizations.

Resource/Reserve Classification Measures. The uncertainty measures in the previous section can be used to define resource/reserve classification criteria. These criteria require the use of threshold values, similarly to past efforts, where thresholds were applied to estimation variances (e.g. review in Sinclair and Blackwell, 2000). Threshold values may reflect the error tolerance that is acceptable locally for the block estimates as well as globally for the resource estimate. The block estimate herein corresponds to the mean value (or e-type estimate). It is possible to categorize resources with block precision and using resource models derived from multiple simulation realizations to obtain the uncertainty in global resources in each category. A first confidence interval based on the CCV involves fixed threshold CCVs that are compared to the the CCV. This is equivalent to acomparing confidence interval aboutasthea mean of +/standard deviation against threshold the confidence interval defined proportion of the mean. The CCV classification criterion are given by a

Measured if CCV < Threshold_ 2 CCV { Threshold_ 2 ≤ Indicated if CCV
154


Chapter 7

As an example, consider a CCV classification with a threshold CCV of 0.5. In this case, all blocks for which the confidence interval bounded by +/- the standard deviation is over 50% of the value of the mean will be classified as inferred. The remaining blocks are classified as Indicated or Measured. The selection of threshold values is deferred until a latter section. A second confidence interval based on 90% probability (a relative percentile range or RPR) is a confidence interval about the mean that is compared with a threshold confidence interval defined as a proportion of this mean. The CCV classification is defined based on the standard deviation, here the confidence interval is defined considering the relative th th position of the mean to the 5 and 95 percentiles, termed negative and positive differences. The symmetric confidence interval used in this criterion is

⎡ z * (u ) − F −1 (u;0.05) z *E (u ) − F −1 (u ;0.95) + ⎤ Confidence Interval= ± ⎢ E w− + w ⎥ (7.2.6) * z E (u ) z E* (u ) ⎢⎣ ⎥⎦ -

+

th

where w and w are weights for the relative differences to the 5 and 95 respectively. The weights are determined based on the probability intervals w− =

[0.05 − F (u; z 0.9

* E

(u ))]

w+ =

[0.95 − F (u; z

* E

th

percentiles,

(u ))]

0.9

Note that setting w- and w+ to 0.5 corresponds to defining the confidence interval as +/- the average between the negative and positive differences. The classification criterion is

Measured if Threshold
Confidence Interval_2 ≤

{

Indicated if Threshold
≥

(2.1.7)

Confidence Interval_1

A Simple Example in a Diamond Deposit: See Dugan abd Dimitrakopoulos, 2006 (Banff, Geostats Conference 2004).


155

Chapter 7

7.2.2 Selecting Thresholds and using Probabilities at a Gold Mine The example of an open pit gold mine in Western Australia is used to further explore CCV based classification. Issues addressed include the selection of threshold values for reserve classification and the extension of the previous concepts to further integrate an assessment of risk by using the probability that a block is ore. The example comprises a set of epithermal gold lodes in a zone of the deposit which are to be classified into Measured, Indicated or Inferred ore reserves. Figure 2 shows tonnage of ore above cut-off (1g/t) and average grade above cut-off plotted against the CCV for 25 realizations of the orebody. The figure shows a general and distinctly increasing spread (risk) in ore tonnage and average grade as the CCV increases, which is particularly strong in the ore tonnages. This trend is also seen in Table 1 detailing CCV and summary statistics for metal content in the gold lode. The characteristics of the spread and slope of the curves in Figure 2 and values reported in Table 5.5 can be used to guide selection of thresholds for classification. For example, Figure 2 suggests that there is little risk for a CCV up to around 0.65 as the related tonnage above the cut-off curves are mostly overlapping and the average grade above cut-off appears to have stabilized. In addition, the figure shows that after a CCV of about 1.1 the spread of tonnages and grades is at its greatest and remains stable for higher CCV values. Hence, the higher CCVs can be selected for the classification of the high risk resource. In an alternative analysis, Table 1 shows a CV or relative standard deviation with a minimum of 4.2% at the 0.65 CCV threshold, suggesting that this threshold corresponds to the most certain measure of metal quantity and would be a reasonable threshold to differentiate between Measured and Indicated reserves. The differentiation between Indicated and Inferredend reserves may a CV ofonly abouta 6.7% CCV of which identifies the highest of risk. Atbe a based higheron threshold small or amount of 1.1, additional metal would be delivered. For a CCV value between 0.65 and 1.1 both metal quantity and uncertainty increase substantially. Hence, there is a trade off between gains in metal quantity and increase in risk when differentiating between Indicated and Inferred reserves. Note that Table 5.5 allows straightforward reporting of average metal for each category as well as a confidence interval, minimum, maximum and proportion of range.

Risk Profile - Tonnage, Zonecode 2020 (25 Realsiations AU > 1g/t)

Average Grade above cut off garde of 1 g/t

3,000,000

AU1

3.500

AU2

T ON1

AU3

T ON2 T ON3 T ON4

2,500,000

AU4

3.000

AU5 AU6

T ON5 T ON6 T ON7

t/ g 2,000,000 1 e v o b a 1,500,000

T ON8 T ON9 T ON10 T ON11 T ON12 T ON13

e g a n n to 1,000,000

T ON14 T ON15 T ON16 T ON17 T ON18 T ON19 T ON20

AU7

ff o 2.500 t u c e v o b 2.000 a e d a r g 1.500 e g ra e v 1.000 a

AU8 AU9 AU10 AU11 AU12 AU13 AU14 AU15 AU16 AU17 AU18 AU19 AU20 AU21 AU22

T ON21

500,000

AU23

T ON22 T ON23

0.500

AU24 AU25

T ON24 T ON25

0 0.350

0.450

0.550

0.650

0.750 CCV

0.900

1.100

1.300

1.500

0.000 0.350

0.450

0.550

0.650

0.750

0.900

1.100

1.300

1.500

CCV

Fig. 2: Average grade above the 1g/t cut-off (left) and tonnage of ore above cut-off (right) plotted versus the CCV for 25 realizations of a gold load.

156


Chapter 7

CCV Min Metal (oz) 0.35 576 0.40 2,925 0.45 6,721 0.50 14,245 0.55 29,146 0.60 48,050 0.65 72,354 0.70 97,457 0.75 127,570 0.80 151,644 0.90 191,436 1.00 212,766 1.10 223,778 1.20 228,104 1.30 229,903 1.40 230,305 1.50 230,549

Max Metal (oz) 1,151 4,645 9,834 20,000 36,105 58,724 83,978 116,560 155,776 187,932 238,876 270,189 286,719 295,563 298,081 299,453 299,779

Range

Average CV Metal (oz) (%)

575 1,720 3,113 5,755 6,959 10,674 11,624 19,102 28,206 36,288 47,440 57,423 62,941 67,459 68,178 69,148 69,230

812 3,849 8,840 18,120 33,353 53,916 77,909 107,747 140,369 168,126 212,314 238,691 251,968 258,527 261,089 262,317 262,641

17.0 10.3 7.8 6.3 4.4 4.3 4.2 4.7 5.3 5.4 6.0 6.4 6.7 6.9 6.9 7.0 7.0

Increase in Metal quality

3,037 4,991 9,280 15,233 20,563 23,993 29,838 32,622 27,757 44,188 26,377 13,277 6,559 2,562 1,227 324

Table 1: CCV and summary statistics for metal content in a gold load.

1.0 0.9 0.8 0.7 0.6 B O 0.5 R P

0.4 0.3 0.2 0.1 0.0 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

CCV

Fig. 3: Probability of blocks to be ore (gold grade above 1 g/t) versus the CCV for 25 realizations of a gold load and ore classification based on both.


157

Chapter 7

Fig. 4: Blocks in a section of a gold load classified using the CCV classification (left) and using available drilling (right).

Category CCV

Tonnage Average (t) (g/t) Measured < 0.65 790 ± 23 2.80 0.13 ± Indicated 0.65 - 1.1 1,734 ± 8 7 2.84 0.17 ± Inferred > 1.1 103 ±9.4 2.920.16 ±

grade Metal (oz)

77.9 174.0 10.5

± 2.3 ± 14.0 ± 1.9

Table 2: Reporting of classification and evaluation of ore reserves for metal, tonnes and average grade.

Incorporating Probabilities. It is reasonable to consider that a probability value can be used block to further ascertain the classification resources and of reserves. With itthe each within the deposit available fromofthe realisations the orebody, is ccdfs simplefor to plot CCV values against probability of ore, as shown in Figure 3 where ore in this example is defined as grades above 1 g/t. The presence of two decision characteristics provides additional information to the classification procedure. For example, here Measured is considered the reserve where a block has a CCV less than 0.65 and probability of over 80%. Indicated are blocks where the CCV is 0.65 < CCV < 1.1 with probability > 60% and CCV < 0.65 with probability 60% < probability < 80%. Inferred are blocks with CCV > 1.1 with probability probability > 40%, and blocks of CCV < 1.1 with probability 40% < probability < 60%. Any block with a probability less than or equal to 40% of being ore is classified as waste. Table 2 provides the detailed reporting of the reserve classification. An important difference from traditional classification methods in this example is that the classification is independent of their average grade of the blocks.

Comments. It is important to note that the thresholds used in the above example will change from load to load, different geological zones, deposits and commodities. However, the process to generate CV or CCV values will remain the same. “Probability of ore” thresholds may not change and depend on predefined decision criteria. In addition, it is important to reiterate that unlike traditional approaches, classification of the blocks using any of the proposed measures generated through use of simulations and reflect both data density and local grade variation. Figure 4 shows the CCV classification of blocks in a section of the gold load discussed above and the available drilling data. Although drilling density varies, the classification does not always show a strong correlation with drilling density. For example, a densely drilled area may not always be classified to Measured as traditional classification approaches may imply.

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Blocks reported in resource classification should be of the same support size. If this is not the case, support-independent measures may be derived. A known approach is to standardize the CCV of a block of a given support size using the square root of the corresponding dispersion variance (e.g. Saburin, 1982). Another practical comment is that the number of simulations required in this approach depends on the sensitivity of the results. If additional simulations do not effect the overall classification, they are not required. All classification approaches may be improved by choosing a rational spatial grouping of classified blocks into a category to avoid their unrealistic dispersion. Last but not least, the ability to efficiently simulate large deposits and manage the data generated is of critical importance to implement the suggested approaches. The methods discussed do not consider production scheduling in reserve classification. It can be argued that sequencing and scheduling of mining adds an additional dimension to decisions about reserve classification. This is because the criteria for deciding what is acceptable risk in one year will not necessarily be the same as the criteria for decisions 5 years later. An approach which integrates geological uncertainty more fully into scheduling is to apply the concept of “geological risk discounting” within the scheduling formulations. This discounting concept is developed in an other section of the notes, and leads to a planned blending of materials that provides the closest approximation to the desired risk profile in each scheduling period.

Other Comments. A key aspect of the approach in this section is the construction of simulated orebody models that reflect the spatial variability at the block support size considered. If they are constructed by reblocking “point” support realisations, these realisations must be generated on a sufficiently dense grid of nodes to ensure that the reblocked values do not overestimate spatial variability. To illustrate this point, consider the example of a small geological zone of a nickel laterite deposit. The zone has 28,766 blocks of size 25 x 25 x 1 m3. Block grades are generated for three cases: simulating the deposit on a relatively sparse grid where there are 4 points (nodes) in a block, simulating on a grid where there are 25 points in a block, and simulating on a dense grid with 100 points in a block. The three cases correspond to simulating 115,064, 718,825 and 2,875,300 nodes, respectively. Table 4 shows the effect of using different simulation grid densities. There is a large variance for blocks from the 2 x 2 discretisation (4 points in a block), and a variance reduction from 11% to 18% when 25 points per block are used. Variance reduction is less, when a 100 point discretization is used and approaches the theoretical dispersion variance for the corresponding block size. The block discretisation does not need to be on a square grid in order to reflect spatial continuity.

Ni

Nodes: 2 x 2 Nodes: 5 x 5 Nodes: 10 x 10 Variance / CV Variance / CV Variance / CV 0.09 / 0.35 0.08 / 0.34 0.073 / 0.32

MgO 2.66 Fe2O3 317.2 / SiO3 420.6 / Al2O3 1.82

/ 1.44 2.19 / 1.31 1.99 0.30 280.23 / 0.28 272.3 / 0.84 370.5 / 0.83 351.2 / / 0.53 1.56 / 0.49 1.46

/ 1.28 0.27 0.83 / 0.45

Variance Reduction % 2x2 to 5x5 5x5 to 10x10 11% 8%

18% 12% 12% 15%

10% 3% 5% 6% 3

Table 4: Global variance and coefficient of variation of 25 x 25 x 1 m block simulations from different block discretization in a zone of a Ni laterite deposit.


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7.2.3 Estimating Recoverable Reserves though the Chain of Mining and Future Data A aspect to note in the previous sections that, similarly to grade-tonnage curves at the beginning of this chapter, if a geostatistical simulations can be used to assess modelling, mining performance and related risk, they can also be used directly calculate recoverable reserves. This can be achieved through directly simulating the orebody, and, subsequently, the grade control process using “future data”, that is, simulated blasthole or RC data. In addition, other pertinent aspect of the “chain of mining” from drilling to production can be integrated. This approach differs from the known in the past estimation approaches for recoverable reserve assessment (e.g. David, 1988; Rossi and Parker, 1994) in that it accounts for mining selectivity and support effects as all existing methods do and, in addition, incorporates the expected mining dilution and ore loss while mining as well as operational constraints. The term “chain of mining” is used for this reason, to stress the calculation of recoverable reserves that accounts for imperfections in both modelling and mining practices.

Method. An approach that utilises all the above concepts in calculating recoverable reserves is outlined below. Journel and Kyriakides (2004) provide details on the basic concepts and initial studies in controlled environments, showing superior performance when compared to common recoverable reserve estimation approaches, including simulations, all of which assume perfect selectivity. The method may be describes as follows. 1. Simulate a realisation of the “actual” deposit using exploration data, at the scale of mining (selectivity) considered. 2. Generate a realisation of future grade control data from the simulation and available information on sampling and assaying practices and errors. 3. Simulate the grade control classification for a given mining selectivity and grade control method. 4. Account for blast movements as lateral displacement and vertical heave. 5. Account for any other mining consideration such as mine production sequencing. 6. Define a series of cut-offs. 7. Calculate quantities of interest, including tonnage, grade, metal and profit indicators from the smu selected in Step 3 to be above the chosen cut-offs, using the actual deposit grade for the smu known from Step 1. 8. Repeat with several simulations, possibly different smu sizes of interest, or mining options, and tabulate the expected recoverable reserve estimates and risk profiles per production period or pushback. 9. If production data available, test models and forecasts against production records to calibrate models. 10. If of interest, classify recoverable reserves based on their risk profiles and mining sequence considered. The steps in the above method are graphically shown in Fig. 5.37, as implementated at the Escondida Copper mine, Chile, discussed in a subsequent parahraph. In Step 7, tonnage, grades above cut-off, metal quantity and profit indicators are estimated as follows. The ore tonnage mined is:

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N t v (z)c = ∑ i** v (u;j z)c j=1

⎧ 1 Z** v (u)>Z c where i** is an indicator of where Z** v (u; z c )= ⎨ 0 else v (u) is the grade of an smu ⎩ estimated using the simulated grade control data and Z c is a selected cut-off grade. Quantity of metal actually mined: N

**

q v (zc ) = ∑ i v (u; z c )Z v (u ) j j j=1

Average grade above cut-off: m v (zc ) =

q v (zc ) t v (z c )

An undiscounted economic value or profit indicator may also be considered as an increasing function of quantity of metal and a decreasing function of ore tonnage: p v (z c ) = C(z c )[q v (z c ) - t v (z c ).zc ] - C 0

where C(z) is related to the cost of mining and processing, and C0, an initial capital investment.

Case Study:

Richard Peatty, AngloGold Ashanti And Case study as Escondida Cu Mine, Chile


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7. 3

7.3.1

Assessing Risk in Grade-Tonnage Curves in a Complex Copper Deposit, Northern Brazil, based on an Efficient 1 Joint Simulation of Multiple Correlated Variables Abstract

Risk quantification in grade-tonnage curves is critical for capital investment in mining projects and can be obtained through geostatistical simulations of orebodies. A practical difficulty may arise in multi-element deposits, as the joint modelling of the related attributes using the traditional co-simulation approaches is computationally intensive and may be impractical for use in the industrial environment. This paper presents the construction of risk-integrating grade-tonnage curves for a complex copper deposit in northern Brazil, by jointly simulating its key geochemical attributes of interest: Cu, Fe and K. The joint conditional simulation of these elements is based on Minimum/Maximum Autocorrelation Factors (MAF). MAF is an approach, based on principal components, that spatially decorrelates the variables involved to noncorrelated factors. MAF’s spatial decorrelation at any lag distance is the main and critical difference of this approach from the principal component approach attempted in the past. In the MAF approach, the independent factors are individually simulated and back-transformed to the conditional simulations of the correlated deposit attributes that reproduce the cross-correlations of the srcinal variables. Keywords: Joint simulation; mini/max autocorrelation factors; grade-tonnage curves. 7.3.2

Introduction

Capital investment in mining projects requires the quantification, understanding and assessment of risk in grade-tonnage curves. Geostatistical simulation technologies 1 provide an increasingly recognized tool to model geological uncertainty and quantify geological risk associated with grade-tonnage curves. Frequently, mineral deposits and their geological characteristics are described by a multitude of geochemically interrelated attributes. The joint modelling of these attributes assists the geological plausibility of complex orebody models as well as the modelling of individual attributes. A key bottleneck, however, is that the common joint simulation methods 2,3 are too computationally intensive to be of practical use in the industrial environment, particularly when more than two attributes are considered. Contributors to complexity include the tedious inference and modelling of cross-variograms, and computational inefficiency, substantially increasing with the number of variables being co-simulated. A practical alternative to the ‘direct’ co-simulation of variables is the decorrelation of variables introduced by David 4. His approach, demonstrated in the joint simulation of a uranium5,deposit, is based on the decorrelation of variables using principal component analysis 6, 7 (PCA). The effectiveness of this approach, in the presence of spatial crosscorrelations inherent in mineral deposits, is limited because PCA ignores crosscorrelations at distances other than zero. To overcome the limitations of the ‘direct’ cosimulation methods and PCA, Desbarats and Dimitrakopoulos8 have suggested the use of the so-termed Minimum/Maximum Autocorrelation Factors, or MAF, in the context of spatial simulation. The MAF approach may be described as an approach, based on 1

Dimitrakopoulos and Fonseca, 2003


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principal components, that spatially decorrelates the variables involved to noncorrelated factors. The independent MAF are individually simulated and backtransformed to the conditional simulations of the correlated deposit attributes, that reproduce the cross-correlations of the srcinal variables. This paper presents the joint simulation of copper, iron and potassium in an oxide copper deposit located in northern Brazil (Figure 7.9) and the resulting assessment of risk in grade-tonnage curves for copper. The geological environment of the deposit is complex. It occurs within a faulted sequence of Archean felsic-basic metavolcanics hosting hydrothermal breccias and alteration haloes. Copper is dispersed throughout the weathering profile but is also disseminated and enriched in sheet-like saprolitic units outside of the weathered mineralized breccia bodies. The ability of the deposit to supply ore for a SX-EW metallurgical process plant is a key point assessed in a pre-feasibility study, and the joint simulation is particularly important in assessing copper solubility controlled by iron and potassium content. The following sections include; a summary of the method of joint simulation of multiple correlated variables based on MAF; a description of the deposit and the data available; the results of the joint simulation; the presentation of grade-tonnage curves and risk analysis; and conclusions.

7.3.3

Joint simulation of correlated variables using minimum/maximum autocorrelation factors

In geostatistical terminology, the attributes of a multi-element mineral deposit are represented by a multivariate stationary and ergodic random function. Consider a multivariate, l dimensional, Gaussian, stationary and ergodic spatial random function T Z ( x ) = [ Z1 ( x),..., Z l (x)] . The Minimum/Maximum Autocorrelations Factors are defined as the l orthogonal linear combinations Yi ( x ) = aiT Z(x), i = 1,..., l of the srcinal multivariate vector Z ( x ) . MAF are derived assuming that Z ( x ) is represented by a two-structure linear model of coregionalisation6. The MAF transformation can be rewritten as Y(x) = AMAF Z(x)

(1)

and the MAF factors are derived from AMAF = Q 2 Λ1−1Q1

(2)

where the eigenvectors Q1 and eigenvalues Λ1 are obtained from the spectral decomposition of the multivariate covariance matrix B of Z ( x ) at zero lag distance. More specifically, T

Q1 BQ1 = Λ1

(3)

and Q2 is the matrix of eigenvectors from the spectral decomposition T

⎛1 ⎝2

Q 2 M( Δ)Q 2 = Q 2 ⎜

154

[[

] [ T + ]ΓY (Δ) ]⎞⎟QT2

Γ Y ( Δ)

⎠

(4)


Chapter 7

where matrix ΓY (Δ ) is an asymmetric matrix variogram at lag distance Δ for the regular PCA factors Y(x)= Z(x) A, where A = QΛ -1/2. In practice, several Δ lag distances may be used for values lower than the range and the resulting eigenvectors averaged.

Given the MAF transformation above, the joint simulation of multiple correlated variables using the MAF approach proceeds as follows:

• • • • • • • •

Normalize the variables to be simulated. Use MAF to generate the MAF non-correlated factors. Produce variograms for each MAF. Conditionally simulate each MAF using any Gaussian simulation method. Validate the simulation of factors. Back-transform simulated MAF to variables and denormalize. Validate the final results. Generate additional simulations, as needed.

In most cases the reblocking of the generated realizations to a block support model is required and may be seen as an additional step in the above algorithm. 7.3.4

The deposit and data available

The copper oxide deposit considered in this study is located in northern Brazil (Figure 7.9, top). It is approximately 2 km long and 500m wide and appears as a prominent NW-SE aligned hill. The copper mineralization is hosted by copper oxide minerals, mainly malachite, in a weathered hydrothermal breccia and in a large weathering cap where copper is disseminated and enriched in sheet-like mineral-rich saprolitic units, as shown schematically in Figure 7.9 (bottom). Metallurgical studies indicate that iron and potassium are key elements for predicting copper recovery, indicating the need to evaluate all these three elements throughout the deposit. The orebody mineralization model supplied by project staff is a weathering envelope that hosts oxide ore, Cu-oxide rich ore and Cu enriched in saprolite ore, with saprolitic ore superimposed over mainly basic volcanics. The deposit in this study is divided into two units: Sector 11 and Sector 12. The available data include 654 RC and 310 DDH drillholes with samples analyzed by ICP-Plasma (Cu-K) and X-Ray. There are 1136 5-metre composites available in Sector 11 and 866 in Sector 12. The descriptive summary statistics for Cu, Fe and K composites for Sectors 11 and 12 are given in Table 7.7. Figure 7.10 shows the corresponding histograms for Sectors 11 and 12, respectively.


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Figure 7.9: Location map of the project and study area (top); and a schematic vertical section of the geological model of the deposit (bottom).

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Cu % - Sector 11 Mean 0.41 Median 0.24 Std Deviation 0.59 Kurtosis 75.83 Asymmetry 6.81 Minimum 0.01 Maximum 9.03

Fe % - Sector 11 Mean 10.93 Median 10.86 Std Deviation 3.77 Kurtosis -0.28 Asymmetry 0.29 Minimum 2.68 Maximum 25.04

K ppm – Sector 11 Mean 14341.70 Median 11111.60 Std Deviation 14038.68 Kurtosis -0.52 Asymmetry 0.71 Minimum 123.00 Maximum 60300

Cu % - Sector 120.62 Fe % - Sector 1213.21 K ppm % - Sector 12 Mean Mean Mean 13715.20 Median 0.42 Median 13.70 Median 7679.90 Std Deviation 0.59 Std Deviation 3.65 Std Deviation 15334.67 Kurtosis 20.82 Kurtosis 0.11 Kurtosis 2.32 Asymmetry 2.98 Asymmetry -0.37 Asymmetry 1.44 Minimum 0.02 Minimum 0.01 Minimum 42.00 Maximum 7.46 Maximum 28.38 Maximum 93528 Table 7.7: Descriptive statistics of 5-metre composites for Sectors 11 and 12.

Figure 7.10: Data histograms of Cu, K and Fe in Sector 11 (left) and Sector 12 (right).


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7.3.5

Joint simulation of copper, iron and potassium

7.3.5.1

Normal-score transformation

Following the simulation steps using MAF described earlier, a normal-score transformation is performed on the Cu, Fe and K composites available in Sector 11 and Sector 12. Normal score transformations are based on rank ordering of the data and decrease the influence of outliers. This, in turn, assists the inference of the variogram and estimation of covariance matrixes in the simulation process that follows.

7.3.5.2

MAF transformation

The transformation matrix AMAF (Eq. 1) used to generate the three min/max autocorrelation factors in Sectors 11 and 12 is shown in Table 7.8. MAF are calculated by multiplying the vector of elements Cu, Fe and K by a vector of loadings from the rows of the transformation matrix. It should be noted that the MAF loadings are quite different from the ones derived by PCA 8. The lag Δ in Eq. 4 used in this example is 20 metres and was derived experimentally by testing several lag distances to assure a suitable decorrelation and stable MAF decomposition. Figure 7.11 shows examples of cross-variograms between MAF from the present study that demonstrate variable decorrelation. Experimental variograms and cross-variograms for Cu, Fe and K are shown and discussed in more detail in a subsequent section.

Sector 11

-0.930 0 .007 -0.126 Sector 0.704 -0.633 0.495 0.363 0.971 -0.400 12 0.697 0.773 0.008 0.057 0.238 0.908 -0.135 0.051 0.869 Table 7.8: Transformation matrix AMAF for Sectors 11 and 12. 0.4

0.4

0.2

0.2

) 0 h (

) 0 h (

γ -0.2

γ

Cross-Variogram MAF Y1 - Y2

-0.2

-0.4

Cross Variogram MAF Y2 - Y3

-0.4

0

40

80

120

0

Distance (m) - h

40 80 Distance (m) - h

120

Figure 7.11: Cross-variograms of MAF showing decorrelation.

7.3.5.3

Variography of MAF

Variography on each MAF is performed. Figure 7.12 shows the experimental and model variograms fitted to the three MAF in Sector 11 and Sector 12. Note that all variogram models are spherical. MAF variograms are subsequently used in the simulation of each factor and the validation of the MAF simulation results. MAF variograms show clear

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spatial patterns, as expected. It should be noted that MAF variograms are linear combinations of the variograms of the srcinal (normal score) variables.

7.3.5.4

Conditional simulation of MAF

Conditional simulation is performed independently on the three MAF using a sequential algorithm 9 based on the generalized sequential Gaussian simulation method 10. The simulations are performed on a grid of 320,075 nodes within the geological limits of Sector 11 and 239,375 nodes within Sector 12. Twenty simulations are generated in this study for each of the two sectors and are validated in detail for reproduction of data, histograms and variograms. The validation of the MAF simulations is not presented here as a subsequent section presents the validation of realizations on the data space.

Figure 7.12: Experimental and model variograms of the three MAF in Sector 11 (left column) and Sector 12 (right column).


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7.3.5.5

Back transformations of MAF

The realizations of MAF were transformed back to simulated normal score variables by multiplying a column vector of simulated MAF in each grid node with the corresponding inverse matrix of the MAF loadings in Table 7.8. Subsequently, the normal score Cu, Fe and K realizations are back transformed to the data space, and point support realizations are reblocked to 10 x 10 x 8 metre blocks. Figure 7.13 shows three realizations of Cu through horizontal sections and a vertical section of the deposit.

Figure 7.13: Selected conditional simulations of Cu for Sectors 11 and 12 for level #325 (left column) and cross-section 250NW (right column).

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7.3.5.6

Validation of the joint Cu-Fe-K simulation results

Several validation checks are performed to assess the results of the joint simulations of Cu, Fe and K using the MAF transformations in both Sectors 11 and 12. Validation involves calculation of histograms, experimental variograms and cross-variograms of the simulated realizations to ensure reproduction of srcinal data characteristics. Figure 7.14 shows histograms from simulated realizations of each variable in both sectors, and these can be compared to the histograms of srcinal data in Figure 7.10. Figure 7.15 shows plots of variograms and cross-variograms for data and simulations in Sector 11. All results suggest that the reproduction of the srcinal data spatial characteristics by simulation is excellent. The reproduction of data variograms and cross-variograms for Sector 12 is also excellent. Recall that the variograms and crossvariograms of srcinal variables are not directly used in the joint simulation based on MAF, which used the variograms of the MAF independent factors.

7.3.6

Risk assessment results and discussion

The mining project considered in this study represents a US$120 million capital investment and aims to operate an SX-EW copper recovery plant using traditional sulphuric heap leach technology. One of the most important factors in the economics of this project is the variability of recoverable copper and the associated risk. Preliminary metallurgical tests, summarized in Table 7.9, indicate that recoverable copper is controlled by the content of iron and potassium, in addition to the srcinal copper grade. This highlights the importance of integrating the geological model of the deposit with jointly simulated orebody scenarios and metallurgical results so as to better identify the possible variability of copper ore output (Cu tonnes) from the SX-EW recovery plant. ORETYPE

NUMBER

AVERAGE GRADE %

OF TESTS Cu (Head grade) Al SAPROLITE A SAPROLITE B SEMIWHEATHERED SEMIWHEATHERED A

K Mg Fe Fe2O3C u% (Rec) Rec%

55 14

0.83 0.28

7.3 0.9 1.3 1.0 7.8 1.2 1.0 0.6

14.1 16.8

0.57 0.09

70.02 35.49

9

1.2

5.1 0.4 2.3 4.0

6.43

1.01

84.65

6

0.45

6.3 1.1 1.7 1.5

12.9

0.28

62.53

Table 7.9: Preliminary metallurgical Cu recovery tests for the deposit under study .

SAPROLITE A SAPROLITE B SEMI-WHEATHERED

Cu Rec (%) = 0.05+0.93*Cu-0.018*Fe+0.007*K Cu Rec (%) = 0.04+0.62*Cu-0.009*Fe+0.02*K Cu Rec (%) = 0.04+0.69*Cu-0.030*Fe-0.04*K

Table 7.10: Recoverable Cu equations from metallurgical tests and for different rock types.


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Figure 7.14: Histograms for joint simulation of Cu, K, Fe using MAF. Sector 11 is shown on the left column and Sector 12 in the right.

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0.4

Cu

1.2

Cu-K

0.2

0.8

0

)

)

h ( 0.4

h ( -0.2

Data Simulation

γ

Data Simulation

γ

0

-0.4

0

40

80

120

160

200

240

0

40

Distance (m) - h

K 0.8

γ

0.4

160

200

240

Fe-K

0

1.2 )

120

0.2

1.6

h (

80

Distance (m) - h

-0.2 )

(h-0.4

Data Simulation

Data Simulation

γ

0

-0.6

0

40

80 120 160 Distance (m) - h

200

240

Fe

1.2 0.8 ) h ( 0.4

40

80 120 160 Distance (m) - h

) 0.2 h (

240

Data Simulation

γ

0

200

Cu-Fe

0.4

Data Simulation

γ

0

0 0

40

80 120 160 Distance (m) - h

200

240

0

40

80 120 160 Distance (m) - h

200

240

Figure 7.15: Reproduction of data variograms and cross-variograms of Cu, K, Fe, CuK, Fe-K and Cu-Fe in a simulation of Sector 11. Note that MAF uses only the variograms of the MAF independent factors.

Given the above reasoning, the assessment of Cu variability and risk in grade-tonnage curves for the deposit is based on recoverable Cu. Recoverable Cu is calculated as a function of the jointly simulated Cu, Fe and K contents for each block of the orebody models and the relationships derived from metallurgical tests, shown in Table 7.10. The joint simulations described in the previous section provide the information needed to generate realizations of recoverable Cu at the support scale of 10 x 10 x 8 metre blocks, and to subsequently generate grade-tonnage curves for the deposit based on a range of Cu, Fe and K grades. The resulting grade-tonnage curves are then suitable for risk analysis as well as for ‘average type’ assessments for project economics.


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Figure 7.16, Figure 7.17 and Figure 7.18 present the grade-tonnage curves for recoverable Cu as a function of Cu, Fe and K cutoffs. All figures plot the results from individual simulations and the average (e-type) from the simulations. It is apparent from all grade-tonnage graphs that recoverable copper tonnage variability is more clearly related to the in-situ copper and potassium content than to iron content. Although there is no restriction to a specific cut-off grade for Fe and K in the metallurgical process, it is clear that for Fe rich ores no wide variability in copper output is apparent in the gradetonnage curves. This is supported by the geological model, where Fe-rich grade zones are normally rich in malachite, a mineral with better kinetics in the metallurgical process, compared with iron for acid consumption, and therefore not reducing copper recovery. Variability in recoverable copper grade is, however, mainly dependent on the in-situ copper content. These results reflect the related metallurgical tests and the recovery functions developed, where the in-situ copper content is the predominant control on recoverable copper. The grade-tonnage curves in Figure 7.16, Figure 7.17 and Figure 7.18, show that the assessment of variability of recovered copper can be investigated by integrating all three elements with the geology of the deposit. This assessment of variability provides a clear description of the inherent uncertainty of potential copper outputs at the plant when analyzing cash flows and other financial aspects in the evaluation of the present project. For example, consider a preliminary operational Cu cut-off grade for the project of 0.6% Cu. Figure 8 suggests that the output from the SX-EW recovery process could range from 145000 to 210000 tonnes of copper. With a planned annual Cu production of 45000 tonnes, the results suggest that the potential exists for the project life to be substantially expanded. At present, no restrictions to any specific cut-off grades for Fe and K are considered in the evaluation of the deposit. However, it is clear from Figure 7.17 that for the Fe rich ores, which are mainly composed of hematite and goethite oxidized breccia zones, there does not appear to be wide variability in recoverable copper outputs in the gradetonnage curves. This may result from the regular occurrence of malachite with Fe rich ores, which has better kinetics in the metallurgical process, compared with iron, for acid consumption.

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400000.00

1.60 1.40

)t ( u C e g a n o n T

300000.00

1.20

d e l a 200000.00 c S

1.00

100000.00

0.40

0.80 0.60

0.20 0.00

) d re e v o c e R ( u C g v A

% e d a r G

0.00 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

1

Cutoff Cu% Cus1 (t) Cus5 (t) ETYPECus(t) Cus12 (t) Cus16(t)

Cus2 (t) Cus6 (t) Cus9(t) Cus13 (t) ETYPECus%

Cus3 (t) Cus7 (t) Cus10(t) Cus14 (t)


Figure 7.16: Grade-tonnage curves for recoverable Cu as a function of Cu cutoffs. Thin lines represent individual simulations and thick dashed lines represent the average (etype) from the simulations.

400000.00

t)( u C e g a n n o T

0.70 0.60

300000.00

0.50

d e l a c 200000.00 S

0.40 0.30 0.20

100000.00

0.10 0.00

) d e r e v o c e R ( u C g v A

% e d a r G

0.00 0.00

2.00

4.00

6.00

8.00

10.00

12.00

14.00

16.00

Cutoff Fe% Cus1(t) Cus6(t)

Cus2(t) Cus7(t)

Cus3(t) Cus8(t)

Cus10(t) Cus15(t)

Cus11(t) Cus16(t)

Cus12(t) ETYPECus%

Cus4(t) ETYPECus(t) Cus13(t)

Cus5(t) Cus9(t) Cus14(t)

Figure 7.17: Recoverable copper tonnage as a function of iron content; results from individual simulations and average (e-type) from the simulations.


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0.50

400000.00

0.40

)t 300000.00 ( u C e 200000.00 g a n n o T

0.30

0.20 100000.00

0.10

% e d a r G ) d e r e v o c e R ( u C g v A

0.00

0.00 0

400

800

1200

1600

2000

2400

2800

3600

4000

Cutoff K (ppm) Cus1 (t) Cus5 (t) ETYPECus(t) Cus12 (t) Cus16(t)

Cus2 (t) Cus6 (t) Cus9(t) Cus13 (t) ETYPECus%



Figure 7.18: Recoverable copper tonnage as a function of potassium content; results from individual simulations and average (e-type) from the simulations.

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7.3.7

Conclusions

Risk analysis in grade-tonnage curves for a complex multi-element copper deposit in northern Brazil, or for any multi-element deposit, being considered in a feasibility study is critical for capital investment. This study presented the use of an efficient joint simulation approach based on min/max autocorrelation factors. The approach simplifies joint simulations in mineral deposits by decorrelating variables to independent factors that are then simulated independently. The case study presented herein shows the key practical aspects of the MAF approach and the excellent validation of the results it generates. The case study has presented three developments: (i) a methodology for investigating uncertainty via jointly simulating Cu, Fe and K; (ii) the construction of recoverable Cu models as a function of solubility, given the Fe and K content and the simulated realizations of the deposit; and (iii) the quantification of related risk and range of outcomes in Cu grade-tonnage curves. As a result, valuable information is generated for the further analysis of Cu grade and tonnage, solubility and recoverability effects on the appraisal of the project.

Acknowledgements

This study had the support of Companhia Vale do Rio Doce. Data of the project presented in this paper are confidential. The results have therefore been selectively scaled to avoid disclosure of information critical to the project, but without affecting the approach developed in the paper. References

Dimitrakopoulos, R., 2002, Conditional simulations for the mining industry: Orebody uncertainty, risk assessment and profitability in recoverable reserves, ore selection, and mine planning. BRC, Brisbane, 355p. Chiles, J.P. and Delfiner, P., 1999, Geostatistics: Modelling spatial uncertainty. John Wiley and Sons, New York, 695p. Gutjahr, A., Bullard, B. and Hatch, S., 1997, General joint conditional simulations using a fast Fourier transform method. Mathematical Geology, v. 29, no. 3, pp. 361-389. David, M., 1988, Handbook of applied advanced geostatistical ore reserve estimation. Elsevier, Amsterdam, 216p. Davis, J.C., 1986, Statistics and data analysis in geology: John Wiley, New York, 646p. Wackernagel, H.J., 1995, Multivariate geostatistics, Springer, Berlin, 256p. Goovaerts, P., 1993, Spatial orthogonality of the principal components computed from coregionalized variables. Mathematical Geology, v.25, no.3, p. 281-302. Desbarats, A.J. and Dimitrakopoulos, R., 2000, Geostatisical simulation of regionalized pore-size distributions using min/max autocorrelation factors. Mathematical Geology, v.32, no.8, p. 919-942. Dimitrakopoulos, R., 2002, Joint simulation using min/max autocorrelation factors (MAF Factors), Docegeo/CVRD internal report, 18p.


167

Chapter 7

Dimitrakopoulos, R. and Luo, X., 2002. Generalized sequential Gaussian simulation on group size ν and screen-effect approximations for large field simulations. Mathematical Geology. Deustch, C.V. and Journel, A.G., 1997, Gslib: Geostatisitcal software library and user´s guide (2nd Edition). Oxford University Press, New York, 369p.

168


Chapter 7

7. 4

Optimal Drillhole Spacing and the Role of Stockpile

Optimal Drillhole Spacing and the Role of the Stockpile

Outline • • •

Multivariable simulations with mini/max The use of joint simulation for infill drilling performance Case study at Murrin Murrin

• • •

Evaluation infill-drilling schemes and bench heights Relation between the use of the stockpile and a drilling scheme performance

Conclusions and future work


169

Chapter 7

Joint simulation with mini/max autocorrelation factors

• With the possible exception of gold deposits, most deposits are multivariable

• The mini/max autocorrelation factors, is a PCA type factorisation method that allows the preservation of the correlation between the elements

• The factors can be then simulated independently, the cross-correlations re-appear when back rotated

Multivariate simulation with mini/max autocorrelation factors The PCA only decorrelate at lag 0 F1

F2

Collocated factors F2

F3

F3 F1

The MAF is two consecutive PCA, one at lag 0 followed by one at lag h

170


Chapter 7

Multivariate simulation with mini/max autocorrelation factors PCA

MAF

F2(x+h)

F2(x+h)

F1(x) F1(x)

Non-proportional structures in the multivariable covariance

For up to two nested structures and a nugget effect

Multivariate simulation with mini/max autocorrelation factors MAF factors can be expressed as a linear combination ⎡ ⎡Fu ⎤1 ( ) ⎢ ⎢F⎥ ( u ) = ⎢ ⎢ ⎥2 ⎣⎢ ⎣⎢Fu ⎦⎥3 ( )

x1 N i +⎤ y1 C o + z 1 M g

⎥ ⎥ x 3 N i +⎦⎥ y 3 C o + z 3 M g

x2 N i + y2C o + z2 M g

(2)

Ni Co Mg

(1) MAF rotation

MAF1 MAF2

Ni

SGS

(3)

SGS

Back rotation

SGS

Co Mg

MAF3


171

Chapter 7

Method to assess future information Partially known deposit Actual #1 Exhaustively known deposit

Actual #2

Distribution of indicators

Case study at Murrin Murrin deposit

172

•

A laterite Ni-Co deposit, located in the Eastern Goldfield province of the Yilgarn Block, WA

• •

Three elements of interest Ni, Co and Mg

•

Study done on a part of the deposit moderately drilled

Mining of high grade ore only at this point


Chapter 7

Challenges at Murrin Murrin Questions

•

Can we reduce the drilling budget without endangering the profitability of the operation?

•

Can the profitability be enhanced by increasing the bench height from 2m to 3m? 0.6

“Point of diminishing returns” is a function of the increased drilling cost with drilling density

Drilling cost per tonne

e n0.4 n o t r e0.2 p $ 0 x 50

50

25

5 8 8 2 0 x2 8x1 2x1 2x1 0x1 1 1 1 1

8 8x

5x

5

Parameters of Murrin Murrin •

In the test area, the selection of waste, stockpile or ore for the 338 15x15m blocks is based on Ni %

• • •

•

Ni<0.8: waste 0.81.2: ore (ROM)

Four drilling schemes are assessed for 2 and 3m benches 12m x 12m (512 holes), 18m x 12m (320 holes) 18m x 18m (210 holes),

• •

25m x 25m (105 holes)

Two actual deposits 30 joint simulations of Ni,Co, and Mg per drilling scheme


173

Chapter 7

Simulation of Murrin Murrin

Ni

Co

Mg

MAF yields simulations honoring the histograms and variograms Ni

2.00

Co

2.00

m1.00 a r g o i r a V

m 1.00 a r g io r a V

0.00

m a r g o i r a V

0.00 0

100

200 h

300

1.00

0.00 0

400

Mg

2.00

100

200 h

300

400

0

100

200 h

300

400

Simulation of Murrin Murrin

Ni

Co

Mg

MAF yields simulations honoring the histograms and variograms Ni-Co

1.00

Ni-Mg

1.00

0.50

Co-Mg 0.00

-1.00

174

0.00

0.00

0

100

200

300

0

-0.50

100

200

300

0

100

200

300

-1.00


Chapter 7

Calculating economics • •

Revenue is a function of Ni and Co

• •

Revenue = Ni grade x $Ni + Co grade x $Co

Mg is a penalty that incr eases the cost of processing

Cost = (mining + milling cost) + Mg grade x penalty + drilling cost

Block classification (based on average grade)

True classification

2m bench Black: Waste Blue: Stockpile Red: ROM

12x12

18x12

18x18

25x25


175

Chapter 7

Histograms of profit of ore tonnes mined 10

h c n e5 b

12x12

18x12 m=5.40

m=6.05 σ

2

σ

=0.17

2

18x18 m=4.78

=0.22

σ

2

25x25

m=4.51

=0.19

σ

2

=0.60

m 2 10 m=6.71 h 2 c σ =0.20 n e5 b m 3 0

2

m=6.14 σ

4

6

8

2

2

m=4.63

m=5.06

=0.33

4

σ

6

8

2

=0.28

2

σ

4

6

8

2

4

2

=0.63

6

8

• The mean decreases with a sparser drilling pattern • The variation of outcomes (risk) is significantly reduced with denser drilling

Economic profiles of drilling scheme e0 n n o t r e-1 p ti f o r-2 p . c Stockpileasore n I -3 12x12 18x12 18x18

0

-20

-40

Act 2; 3m e c n e r fe fi D %

Act 1; 3m Act 2; 2m Act 1; 2m

Stockpileaswaste 25x25 12x12

18x12

18x18

25x25

Drilling scheme

•

An increase of $0.08 per tonne in drilling increases profit by - $2 if stockpile is waste - $0.5 if stockpile is ore

•

176

The results are insensitive to the bench height


Chapter 7

The effect of stockpiling •

The cost of misclassification of an ore block in the stockpile is a function of the future use of the stockpile

•

The value of a block in the stockpile can be assessed using a discount rate

•

All other discount rates are intermediate scenarios between the two “end-point” cases: The stockpile is considered as ore or the stockpile will never be processed

•

The formulation used here is simplistic and aims to show the effect of mine planning on drilling decisions

Effect of discount rate on profit -0.5

e n -1.5 n to Stockpile as ore r -2.5 e p itf -0.5 o r p-1.5 . c Discount rate n I -2.5 =0.10

• •

Act 2; 3m Act 1; 3m Discount rate =0.06

Discount rate =0.08

Act 2; 2m Act 1; 2m

Discount rate

Stockpile as

=0.20

waste

12x12 18x12 18x18 25x25

The decrease of profit per tonne is due to the increased discou nt rate The importance of the scheme de pends on what is intended to be done with the stockpile


177

Chapter 7

Conclusions

178

•

The MAF based joint-simulation is an efficient method to simulate correlated variables

•

In this case study, the 12x12m drilling scheme appears to be the

•

most efficient with the 3m bench height The method developed provides the means to test and assess the effectiveness of drilling schemes

•

Once the simulations of the deposit are available, it is straightforward to compare cut-offs

•

Mine planning aspects are shown to be a key factor in deciding drilling densities


Chapter 7

7. 5 7.5.1

Recoverable Reserves and Conditional Simulation Introduction

Recoverable reserve estimation may be seen as an exercise in which one attempts to assess the efficiency (ability) to both estimate and mine in-situ reserves. The effectiveness of this exercise is based on the validation of the production schedule from an orebody model. The key element in assessing the validity of a production schedule and recoverable reserves ahead of production is the ability to simulate the orebody as well as the grade control process prior to actual mining and production. This can only be done using a conditional simulation approach. 7.5.2

The ‘F’ factors: evaluating modelling and mining performance

Three ‘F’ factors may be used to evaluate the efficiency in estimating in-situ reserves and ability to mine and produce from them. These are: I. The efficiency in estimating in-situ reserves or factor F1 for tonnes, grades, and quantity of metal over a period of time (eg annually): F1 = Actual In-situ Quantities / Estimated In-situ Quantities Obviously, accurate assessment of the in-situ reserves with respect to tonnage, grade, and metal means that F1 should be equal to unity i.e. the estimated reserves are unbiased. However, many reasons can cause these factors to deviate from unity including biased or poor data, ore methods reserve estimation approach, over smoothing of grades, poorquality change of poor support in predicting recoverable reserves. F1 is larger than one when the in-situ reserves are under-estimated and less than one when over-estimated. II. The efficiency in mining in-situ reserves or factor F2 for tonnes, grades, and quantity of metal over a period of time (eg annually): F2 = Run-of-Mine Quantities / Actual In-situ Quantities Similarly to F1, if the in-situ reserves are perfectly identified with respect to tonnage, grade, and metal AND mined, then F2 is equal to one. F2 is larger than one when waste is being mined as ore. F2 is less than one when ore is being sent to the waste. Poor sampling of blast holes or RC, high nugget effect, and operational constraints generating ore dilution can cause F2 to deviate from unity. III. The combined impact of reserve estimation (F1) and ore selection errors (F2) in predicting recoverable reserves or factor F3 is: F3 = F1 x F2 = Run-of-Mine Quantities / Estimated In-situ Quantities The F3 factors for tonnage, grade, and metal indicate if the production schedule and goals will be met. F3 value close to unity indicates that the production schedule is being met but it may also indicate two errors. For example, 1.2 times 0.84 is 1.004, if F1 is 1.2 then ore reserves are underestimated by 20% and if F2 is 0.84 only 84% if the


179

Chapter 7

reserves are mined as ore then F3 is 1.004 thus the production schedule is being met very well. However, the operation can really perform much better. It should be noted that all three factors can and should be analysed by mining period or bench as to monitor performance and minimise adverse financial effects. 7.5.3

The idea in brief

A conditional simulation approach allows the generation of a ‘perfectly known’ equally probable orebody and the calculation of the actual in-situ quantities that is factors F1 and F2. F3 can be calculated as well as monitored if mining is carried out. F1 and F2 are calculated for drill hole assay or blast hole grades and selective mining unit (SMU) grades. As the approach does not generate a unique model, the calculation of factors is repeated for several orebody realisations and several also equally probable factors whose expected value provide an assessment of recoverable reserves and production as well as uncertainties involved in meeting expectations.

180


Chapter 7

7.5.4

A case study at the Fort Knox Gold Project, Alaska (Guadino, et al 1997)

Production Schedule 

A detailed Production Schedule for the mine

Pe r i o d

t o ns

1 2 3 4 5 6 7 8 9 10 11 12

0.911 6.520 6.060 4.595 4.377 2.629 1.413 1.027 0.000 0.000 0.000 0.000

Total





Summary of Production Schedule by period as provided by FGMI St o c k pi l e M il l mi l+ ls t o c k pi l e w a st e g r a d Ounce t o ns G r a de o u n c e s t o ns g ra de o unc es t o ns e s 0.0154 14.0 1.595 0.0306 48.8 2.506 0.0250 62.8 9.578 0.0153 100.1 13.510 0.0323 437.0 20.030 0.0268 537.0 15.264 0.0151 91.5 12.872 0.0320 412.3 18.932 0.0266 503.8 13.002 0.0145 66.6 11.898 0.0299 355.4 16.493 0.0256 422.0 13.791 0.0146 63.9 14.153 0.0309 437.6 18.530 0.0271 501.5 19.934 0.0140 36.8 14.145 0.0269 380.3 16.774 0.0249 417.1 22.654 0.0140 19.8 11.980 0.0251 300.6 13.393 0.0239 320.4 17.199 0.0140 14.4 14.467 0.0232 335.8 15.494 0.0226 350.2 16.662 0.0000 0.0 11.287 0.0235 265.6 11.287 0.0235 265.6 12.420 0.0000 0.0 13.464 0.0243 327.5 13.464 0.0243 327.5 9.824 0.0000 0.0 13.818 0.0256 353.8 13.818 0.0256 353.8 8.140 0.0000 0.0 0.974 0.0258 25.1 0.974 0.0258 25.1 0.050

2 7. 53 2

0.0 1 48

4 07. 0

1 34. 16 2

0 .0 274

36 79 .9

1 6 1. 69 5

0. 02 53

4 08 6 .9

1 5 8.51 8

The Production Schedule is detailed by period (year), by phases (I,II,III), and by benches It predicts recoverable reserves for mill ore and stockpile ore with respect to tonnage, grade, and metal

Block Model 

Block model which was estimated using Multiple (Full) Indicator Kriging MIK

Characteristics: Block size: 50 by 50 by 30 feet Mill ore:

Stockpile ore:

- Phase 1: cut off 0.0185 oz/t

- Phase 1: cut off 0.0125 oz/t

- Phase II: cut off 0.0155 oz/t - Phase III: cut off 0.0125 oz/t

- Phase II: cut off 0.0125 oz/t - Phase III: cut off 0.0125 oz/t

Benches: 910 through 1900 totalling 34 Mine life: 12 years


181

Chapter 7

Simulations to Calculate F1











Simulation technique: SGS Simulation grid: 10 by 10 by 10 feet Number of nodes simulated: 4,039,200 Phases simulated: I, II, III Number of simulations: 10








182

SMU size: 50 by 50 by 30 feet SMU simulated grade is the average of 25 simulated nodes that fall within the block Number of SMUs simulated: 161,568


Chapter 7

Summary of F1 Factors Tonnage

grade

Metal

Estimated

Simulated

Estimated

Mill

134,156,862

147,062,118

0.0274

0.0272

Stockpile

27,532,170

23,609,478

0.0148

0.0148

mill+stockpile

161,689,032

170,671,596

0.0253

simulated

estimated 3,679,890 407,017

0.0255

4,086,907

Simulated 4,004,536 348,436 4,352,973

f1 Tons

grade

metal

Mill

1.096

0.993

1.088

Stockpile

0.858

0.998

0.856

mill+stockpile

1.056

1.009

1.065

F1 = Actual In Situ Quantities / Estimated In Situ Quantities

F1 Factors by Period: Mill Ore

F1 Factor:Tonnage

1.8 1.6

e g 1.4 a n n 1.2 o T 1.0 1 F 0.8

Average Minimum Maximum

0.6 1

2

3

4

5

6

7

8

91 0 11 12

Period

F1 Factor: Grade

1.8 1.6 e d 1.4 ra 1.2 G 1 F 1.0


0.8 0.6 1

2

3

4

5

6

7

8

91 0 11 12

Period

If F1 > 1.0 in situ reserves are underestimated

F1 Factor: Metal



If F1 < 1.0 in situ reserves are overestimated 

1.8 1.6


l ta 1.4 e M1.2 1 1.0 F

0.8 0.6 1

2

3

4

5

6

7

8

91 0 11 12

Period


183

Chapter 7

F1 Factors by Period: Stockpile Ore

1.4 e 1.2 g a n n 1.0 o T 1 0.8 F


2.0 1.8 1.6 1.4 e1.2 g a n n1.0 o T 10.8 F

0.6 1

2

0.6

3

0.4

Average

F1 Factor: Tonnage

Minimum Maximum

4

5

6

7

8

91 0 11 12

0.2 0.0 1

2

3

4

Period 5

6

7

8

9

101

11

2

Period

1.6 1.4

e d 1.2 ra G1.0 1 F


F1 Factor: Grade

0.8 0.6 1

2

3

4

5

6

7

8

91 0 11 12

Period

If F1 > 1.0 in situ reserves are underestimated 

If F1 < 1.0 in situ reserves are overestimated 

1.6 1.4

l ta 1.2 e M 1 1.0 F


0.8

F1 Factor: Metal

0.6 1

2

3

4

5

6

7

8

91

0 11 12

Period




Simulation technique: SGS Phases simulated: I Grid: 5 by 5 by 30 feet Number of simulations: 1 Number of nodes simulated: 5,997,600 





184




Chapter 7








Blast hole spacing: 15 by 15 by 30 feet Blast hole simulated grade corresponds to the grade of the simulated node plus a random error Number of blast holes simulated: 114,889

Grade Control on Simulated Blast Holes





Grade control is done manually Panels have to be at least 45 by 30 feet in size in order to allow for operational constraints


185

Chapter 7

Summary of F2 Factors Factors t o n n a ge s i m u l at ed

g ra de

RO M

Metal

s i m u l a t ed

ROM

S i m u l a t ed

RO M

m il l

41 ,58 6,00 0

3 8 ,0 3 9 , 0 4 0

0 .0 3 1 9

0 .0 3 2 8

1 , 3 2 8 ,4 4 3

1,24 6,29 0

s to c kpi l e

14 ,3 34,00 0

1 2 ,4 4 6 , 4 6 0

0 .0 1 5 6

0 .0 1 7 3

2 23 ,4 26

2 1 4 ,9 6 9

m il l + s t o c k p i l e

5 5,9 20,0 00

5 0,48 5,5 00

0 .0 2 7 8

0 .0 2 8 9

1 ,55 1,86 9

1 ,46 1,25 9

f2 ton s

g rad e

m etal

mill

0 .9 1 5

1 .0 2 5

0 .93 8

stockpile

0 .8 6 8

1 .1 0 8

0 .96 2

mill+stockpile

0 .9 0 3

1 .0 4 3

0 .94 2

F2 = Run-of-Mine Quantities / Actual In Situ Quantities

F2 Factors by Period: Mill Ore

e g1.4 a1.2 n n1.0 o T0.8 20.6 F

F2 Factor: Tonnage 1

4

7

0 1

Period

1.4

e d1.2 a r1.0 G 20.8 F0.6

F2 Factor: Grade 1

4

7

0 1

Period

If F2 > 1.0 for tonnage, waste is mined as ore 

If F2 < 1.0 for tonnage, ore is sent to waste 

186

l 1.4 a t1.2 e 1.0 M 20.8 F

F2 Factor: Metal

0.6 1

4

7

0 1

Period


Chapter 7

F2 Factors by Period: Stockpile Ore

e1.4 g a1.2 n n1.0 o T0.8 20.6 F

F2 Factor: Tonnage 1

4

7

0 1

Period 1.4

e d1.2 ra1.0 G 20.8 F0.6

F2 Factor: Grade 1

4

7

0 1

Period If F2 > 1.0 for tonnage, waste is mined as ore 

If F2 < 1.0 for tonnage, ore is sent to waste 

l 1.4 ta1.2 e 1.0 M 20.8 F0.6

F2 Factor: Metal 1

4

7

0 1

Period

F2 Factors by Bench Mill And Stockpile Ore MillMill OreOre 1900

Stockpile Ore

1900

1900 1870 1870 1840 1840 1810 1810 1780 1780 1750 1750 1720 1720 16901690 16601660 1630 h h c 1630 c n n1600 e e1600 B B1570 1570 15401540 15101510 14801480

1900 1870 1870 1840 1840 1810 1810 1780 1780 1750 1750 1720 1720 1690 Tonnage 1690 1660 Metal h Grade c1630 n1600 e B 1570 1540 1540 1510 1510 1480 1480

14501450 14201420

1450 1450 1420 1420 1390 1390 1360 1360 1330 1330 1300 1300

13901390 13601360 13301330 13001300 0.70 0.70 0.80

0.90 1.00 1.10 0.80 0.90 1.00 1.20 1.10 1.20 F2 Factor

F2 Factor

0.80

Tonnage Metal Grade

Tonnage Metal Grade

0.70 1.00 0.801.10 0.901.20 1.00 1.10 1.20 F2 Factor F2 Factor

0.90


187

Chapter 7

Summary of F3 Factors Factors

Mill

Tonnage Grade Metal Tonnage

Stockpile

Grade Metal Tonnage

Mill + Stockpile Grade Metal

Factors F1 F2 F3 1 .0 9 6 0 . 91 5 1.003 0 .9 9 3

1 . 02 5

1.018

1 .0 8 8

0 .9 3 8

1.021

0 .8 5 8

0 . 86 8

0.745

0 .9 9 8

1 . 10 8

1.106

0 .8 5 6

0 .9 6 2

0.823

1 .0 5 6

0 . 90 3

0.954

1 .0 0 9

1 . 04 3

1.052

1 .0 6 5

0 .9 4 2

1.003

Mill Ore: Stockpile Ore: Production schedule is Production schedule is not met for tonnage, met for tonnage and metal grade, and metal 



Conclusions 

F1, F2, and F3 factors are useful tools to identity and quantify problems in mining operations



F1 verifies the accuracy of the block model



F2 quantifies ore losses and dilution at the time of mining



F3 measures the net effect of F1 and F2



Conditional simulation techniques can spatial be used to reproduce realistic scenarios of grade variability

188


Chapter 7

7.6

Assessment of Dilution Conditional Simulation

7.6.1

Introduction

in

an

OP

Operation

with

We consider a 10m bench in an open-pit operation mining a low grade gold deposit. Figure 7.19 shows drill hole bench intercepts as well as the polygons of influence of those above the cut-off of 0.7 g/t. Those sample data (and the others in the surrounding benches) are used to simulate the gold grade of 10m bench intercepts on a 2x2m grid filling the designed pit limit in the bench (Figure 7.19). All together, we get 32,285 bench intercepts with gold grade ranging from 0 to 41.8 g/t and averaging 0.64 g/t. 9876 i.e. 30.6% of the intercepts are above the cut-off of 0.7 g/t and they average 1.49 g/t. 7.6.2

Dilution from support effect

To simulate the grade of 5x5x10m SMUs, we just have to average the simulated bench intercepts in cells of a 5x5m grid (we get from 4 to 9 intercepts in a cell with an average of 6 intercepts). Map of the 5135 simulated SMUs is shown in Figure 7.19. SMU grades range from 0.01 to 10.9 g/t with the same mean of 0.64g/t. 1637 i.e. 31.9% of the SMUs are above the 0.7 g/t cut-off and they average 1.39 g/t. Based on the elongated N-S shape of the mineralised structures, we might be able to keep the 5m selectivity across zones (hence E-W) but increase it to 10m along faces (hence N-S). In that case, SMUs are 5x10x10m. To simulate their grades, we just need to average the simulated bench intercepts in cells of a 5x10m grid. We now get 2533 SMUs at least 10 824 intercepts and calculated grades to 7.5 0.64 g/twith (Figure 7.20). i.e. 32.5% of the SMUs are from above0.01 0.7g/t withg/tanaveraging average grade of 1.37 g/t. What would happen if the SMU can not be smaller than 10x10x10m? Let's average the simulated bench intercepts in 10x10m cells. We get 1228 SMUs (Figure 7.20) with 25 simulated intercepts and calculated grade ranging from 0.01 to 5.7 g/t and averaging 0.65 g/t. 410 i.e. 33.4% of the SMUs are above 0.7 g/t with an average grade of 1.35 g/t.


189

Chapter 7

Figure 7.19: Test 10m bench of an OP gold mine with real and individual + block average simulated DH intercepts.

190


Chapter 7

Figure 7.20: Test 10m bench of an OP gold mine with individual + block average simulated DH intercepts.


191

Chapter 7

Recoverable reserves as a function of SMU size are summarised in Table 7.11. At all cut-offs, tonnage is increasing and grade is decreasing as we increase the SMU size. Using numbers corresponding to point recovery (0x0x10m SMU) as reference, we get about the same percent tonnage increase at all cut-offs (about 4% for 5x5m, 6% for 5x10m and 8% for 10x10m) while the percent decrease of grade is increasing with the cut-off (maximum 8% at 0.5 g/t cut-off to 10% at 1 g/t cut-off for 10x10m SMU). Cut-off

SMU size

0x0x10m

5x5x10m

5x10x10m

10x10x10m

0 g/t

Grade

0.64

0.64

0.64

0.65

0.5 g/t

Tonnage

39.8%

41.8% (+4.8%)

42.4% (+6.1%)

43.2% (+7.9%)

Grade (g/t)

1.28

1.20 (-6.2%)

1.19 (-7.0%)

1.18 (-7.8%)

Tonnage

30.6%

31.9% (+4.1%)

32.5% (+5.9%)

33.4% (+8.4%)

Grade (g/t)

1.49

1.39 (-6.7%)

1.37 (-8.0%)

1.35 (-9.4%)

Tonnage

19.7%

20.3% (+3.0%)

20.8% (+5.3%)

21.1% (+6.6%)

Grade (g/t)

1.84

1.70 (-7.6%)

1.66 (-9.8%)

1.65 (-10.3%)

0.7 g/t

1.0 g/t

Table 7.11: Assessing the support effect dilution in a low grade gold open-pit at various cut-offs. 7.6.3

Dilution from information effect

In selective open-pit mining operations, the "support effect" (i.e. SMU size) is not the only source of dilution. One also needs to recognise the actual grade of each SMU. SMU grades are actually "estimated" from grade control samples in blast holes. If estimates are not good enough because blast hole sampling error is high or blast hole spacing is large, there will be some misclassification of the SMUs, i.e. SMUs actually above cut-off are estimated below (ore mined as waste) and SMUs actually below cut-off are estimated above (waste mined as ore). The misclassification of SMUs above and below cut-off also generates dilution but this time the dilution is a result of "information effect" i.e. limited knowledge of grade variations from blast hole samples. The magnitude of the dilution from the information effect can also be assessed through simulation. To continue the previous story, we assume that at the time of mining, we will have a gold assay in each blast hole on an 8x8m grid. To simulate blast hole values, we need to pick simulated bench intercepts on such a grid (i.e. one every four in both X and Y direction). Polygonal map of simulated blast holes is on Figure 7.21. We have 2021 blast holes with values from 0 to 9.03 g/t and mean of 0.65 g/t. 625 (i.e. 31%) of the blast holes are above 0.7 g/t with a mean of 1.51 g/t. Assuming a SMU size of 5x5x10m, we can simulate the estimation of the grade of each SMU from the grades of nearby blast holes. Simulation is done by nearest-neighbour (Figure 7.21) or ordinary kriging (Figure 7.21) In both cases, we estimate recoverable reserves by adding all the SMUs estimated above cut-off but we mine the real grade of those SMUs. Comparison between the average grade of SMUs mined and average grade of all SMUs actually above the cut-off provides the dilution srcinating from the information effect. Results are summarised in Table 7.12. With blast hole kriging, the tonnage increase varies from 2% to 12% while the grade decrease varies from 3% to 9%. Like dilution from support effect, worst results occur at high cut-offs. 192


Chapter 7

If we compare predicted grade and real grade of SMUs predicted above the cut-off, we can see the advantage of kriged estimates over nearest-neighbour estimates (or polygons) i.e. we mine what we predict and not less than what we predict. For example, with kriging, we predict that the average grade of the 1672 SMUs kriged above 0.7 g/t is 1.37 g/t and in reality it is 1.34 g/t. With nearest-neighbour, we predict an average grade of 1.51 g/t for the 1583 SMUs estimated above 0.7 g/t but in reality it is only 1.31 g/t.


193

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Figure 7.21: Same test bench with simulated BHs, block estimates from BHs and real block grade.

194


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Cut-off applied to

Number of SMUs Real grade above Predicted grade above above cut-off cut-off (g/t Au) cut-off (g/t Au) 0.5 g/t cut-off

Real grades

2150

1.20

Polygon grades

2046 (-4.8%)

1.17 (-2.5%)

1.30

Kriged grades

2192 (+1.9%)

1.16 (-3.3%)

1.18

0.7 g/t cut-off Real grades

1637

1.39

Polygon grades

1583 (-3.3%)

1.31 (-5.8%)

1.51

Kriged grades

1672 (+2.1%)

1.34 (-3.6%)

1.37

1.0 g/t cut-off Real grades

1040

1.70

Polygon grades

1058 (+1.7%)

1.51 (-11.2%)

1.84

Kriged grades

1162 (+11.7%)

1.55 (-8.8%)

1.60

Table 7.12: Assessing the information effect dilution in a low grade gold open-pit at various cut-offs.

7.6.4

Combining dilution from support and information effect: the F factors

F factors allow us to evaluate the performance of any step in the resource/reserve prediction, mining and recovery process. At the reserve estimation stage, the F1 factor compares the actual in-situ quantities (tonnage, grade and metal above a given cut-off in SMUs of any given size) to estimated in-situ quantities from drill hole information. In our test bench of the gold open-pit, actual in-situ quantities for SMU of various sizes and cut-offs of 0.5, 0.7 and 1.0g/t cut-offs are derived from the simulation and shown in Table 7.11. Predicted quantities depend of the interpolation method from drill hole data. We have considered two methods: ordinary kriging and nearest neighbour. Both apply to 5x5m blocks but for both methods, block size does not significantly influence the results of the long term prediction. Once we have set the long term prediction method, we can easily calculate the F1 factors corresponding to that method. For example, at the 0.7g/t cut-off, we have 34.2% of 5x5m blocks with a kriged grade from DHs above that limit with an average estimated grade of 1.25 g/t. Since simulation tells us that 31.9% of 5x5m blocks are above 0.7 g/t with an average of 1.39 g/t (see Table 7.11), the F1 of tonnage is 31.9/34.2 = 0.93, that of grade is 1.39/1.25 = 1.11 and that of metal is the product i.e. 1.04 (see Table 7.13). If long term interpolation is done by simple nearestneighbour with no dilution factor, 29.4% of blocks have a polygon grade above 0.7g/t with an average of 1.54 g/t. In that case, F1 for tonnage is 31.9/29.4 = 1.09, that of grade is: 1.39/1.54 = 0.90 and that of metal is 0.98. F1 factors for other cut-offs are listed in Table 7.13.


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At the grade control stage, the F2 factor compares the ROM corresponding to any given ore delineation from blast holes to actual in-situ quantities for the same cut-off. They are readily obtained from numbers in Table 7.12. For example, at the 0.7 g/t cut-off, if SMU selection is done by BH kriging, we would get 1672 5x5m SMUs estimated above cutoff with an average real grade of 1.34 g/t whereas only 1637 SMUs are really above cutoff with an average grade of 1.39 g/t. Hence the F2 factor for tonnage is 1672/1637 = 1.02 whereas that of grade is 1.34/1.39 = 0.96. If dig lines are based on polygonal estimates of 5x5m SMUs from BHS, then we would mine 1583 SMUs averaging 1.31 g/t hence, in that case, F2 for tonnage is 1583/1637 = 0.97 and F2 for grade is 1.31/ 1.39 = 0.94. F2 factors for other cut-offs are listed in Table 7.13. The impact of both estimation (F1) and selection (F2) errors is measured by factor F3 which is the product of F1 and F2. As illustrated in Table 7.13, this factor depends on the way the estimation from DHS and the selection from BHs are conducted. If we concentrate on results at the 0.7 g/t cut-off, we can see that we have the best performance in terms of metal (F3 for metal closer to 1) when both interpolation from DHs and selection from BHs are done by kriging. Any procedure that uses nearestneighbour either at the interpolation or selection stages is likely to generate less metal than anticipated. This is not necessarily true at all cut-offs: at the high 1.0 g/t cut-off, we have a better prediction of metal when selection is based on nearest neighbour rather than kriging. F1 F2 F3 F1 F2 F3 F1 F2 F3 Long term = OK 5x5m from DHs- Short term = OK 5x5m from BHS Cut-off 0.50 0.50 0.50 0.70 0.70 0.70 1.00 1.00 1.00 Tonnage Grade Metal

0.94 1.02 0.96 0.93 1.02 0.95 1.07 1.12 1.09 0.97 1.05 1.11 0.96 1.07 1.07 0.91 1.03 0.99 1.01 1.04 0.98 1.02 1.14 1.02 Long term = NN 5x5m from DHs- Short term = OK 5x5m from BHS Cut-off 0.50 0.50 0.50 0.70 0.70 0.70 1.00 1.00 Tonnage 1.06 1.02 1.08 1.09 1.02 1.11 0.99 1.12 Grade 0.93 0.97 0.90 0.90 0.96 0.87 0.91 0.91 Metal 0.98 0.99 0.97 0.98 0.98 0.96 0.91 1.02 Long term = OK 5x5m from DHs- Short term = NN 5x5m from BHS Cut-off 0.50 0.50 0.50 0.70 0.70 0.70 1.00 1.00 Tonnage 0.94 0.95 0.90 0.93 0.97 0.90 1.07 1.02 Grade 1.09 0.98 1.06 1.11 0.94 1.05 1.07 0.89 Metal 1.03 0.93 0.95 1.04 0.91 0.95 1.14 0.90 Long term = NN 5x5m from DHs- Short term = NN 5x5m from BHS Cut-off 0.50 0.50 0.50 0.70 0.70 0.70 1.00 1.00

1.19 0.97 1.16 1.00 1.11 0.83 0.92 1.00 1.09 0.95 1.03 1.00

Tonnage 1.06 0.95 1.00 1.09 0.97 1.05 0.99 1.02 1.01 Grade 0.93 0.98 0.91 0.90 0.94 0.85 0.91 0.89 0.81 Metal 0.98 0.93 0.91 0.98 0.91 0.89 0.91 0.90 0.82 Table 7.13: F factors assuming final selection on interpolated grade of 5x5m SMU from BHs.

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7. 7

Simulation of Shape, Tonnage and Grade of a Uranium Deposit (David, 1988)

7.7.1

Summary

This case study will demonstrate how conditional simulations can be used to model the tonnage, mineralisation envelope and grades associated with a uranium deposit.

7.7.2

Deposit characteristics

Gaertner is an 'unconformity'-type uranium deposit occurring in proterozoic sandstone underlying metasediments. The deposit exceeds 1400m in length with an average width of 15-50m and extends 5080m below the surface. Significant U308 and as concentrations are present with U 308 varying from traces to 50%. Drilling consists of 477 drill holes on a 25 x 10m (or smaller) grid with drilling density being greater in high grade areas. Why Simulations? The narrow shape of the deposit, as well as with mine planning and safety issues require estimating U308 and As grades, and tonnages on a very fine grid (2 x 2 x lm).

7.7.3

Modelling the mineralised contour

Two thirds of the U3O8 grades in the drill hole composites are barren and will cause problems. First, because these samples will dominate the U308 distribution and so the economically important high grade values will play a minor role in the simulation process. Secondly, the preponderance of low grade samples will make normalisation of the distribution difficult. As a result the data was treated as two populations, mineralised (U3O8, > 0.05) and barren (U 3O8, ≥ 0.05). The mineralised contour was then defined by estimating the probability that a point was mineralised on a 2 x 2 x lm grid. This probability was obtained by kriging an indicator variable imin(X) where:

0⎧ imin(X) = ⎨ 1⎩

if

U3O8 < 0.05

if

U3O8, ≥ 0.05

and x is the point's location


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A point is within the mineralised contour if it's indicator value is greater than a random number drawn from [O..1]. An example of a mineralised contour simulation is shown in Figure 7.22 below.

Figure 7.22: Bench map with simulated mineralisation contour. 7.7.4

Estimating the tonnage

Due to extreme grade fluctuations, the density is also highly variable, hence the grade is not sufficient to estimate a tonnage and so a density is required at each point. In the Gaertner deposit the majority of the available samples do not have a dry density (DD) measurement, however some have a wet density (WD). A regression study revealed that: DD = 0.983 5 * WD - 0.06

p = 0.9835

and in areas with no wet density measurements: DD = 0.0278 * U3 08 - 2.1545

7.7.5

p = 0.6412

Simulating U3O8 and As

Random Pick Declustering was performed to compensate for the clustering of samples in the high grade zones. Prior to performing the simulations, a principal component analysis of collocated U308 and As values was performed. This resulted in components Cl and C2 where:

where:

198

As = normalised As grades U308 = normalisedU308grades P = the correlation coefficient


Chapter 7

Gaussian simulation was used to generate the simulations and the simulated components were then rotated back to yield U308 and As values.

7.7.6

Conclusions

This case study has shown that conditional simulation can be used to create an orebody model that is useable in mine planning. A comparison with block models derived from kriging show that both methods produced comparable results. The main difference is due to the declustering method used. This indicates that the declustering method may have a significant effect on any model.


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7. 8 7.8.1

Economic Reserves in an Underground Mine General

The discovery of an orebody is followed with more drilling to outline the extent of a mineralisation and provide a first evaluation of the economic reserves. Economic reserves are defined here as the part of the geological reserves that are to be mined given specific economic conditions and technical mining constraints. Economic conditions are summarised by a cut-off grade. Then, the economic reserves are linked to the proportion of the orebody blocks estimated to be above the cut-off. The above is a reasonable approach for open pit mining. However, in underground operations technical constraints add additional constraints. Generally, these constraints make the appraisal of economic reserves somewhat complex. Typically, economic blocks isolated from the main orebody are left aside. To assess the effects of mining constraints on economic reserves in an underground mine, conditional simulations may provide a solution. An example follows.

7.8.2

Economic reserve assessment in a uranium mine

The deposit to be evaluated is a uranium deposit. It occurs in a deep, nearly horizontal layer of about 5m thickness, with high uranium concentrations in places. The deposit has a rectangular shape and is drilled on a 100m by 50m grid. Uranium grades are lognormal, with a mean of 0.02% U 3O8 and a coefficient of variation of 1.4. Variography shows a 30% nugget effect and a geometric anisotropy with a 400m range in the E-NE and a 200m range in the N-NW. The cut-off grade considered here is 0.03% U 3O8 applied on small blocks in a roomand-pillar type structure. The initial calculation of economic reserves based on the cut-off grade, is 17% of the mineralised intersections, the data distribution and an average specific gravity of 2.6 gives reserves Ore tonnage (tons)

Ore grades (% U3O8)

2,272,000

0.066

This is, at best, an upper limit since it implies that all the economic intersections which can be encountered will be mined. This may be the case in an open pit, but it is not the case in an underground operation. For instance, one needs to consider the economics of extending galleries to mine additional ore, which depends on the geometry and size of the isolated pods of ore.

200


Chapter 7

Conditional simulations assist at first with the visualisation of various spatial arrangements of ore. One can then consider mining constraints and outline the ore to be mined. An example is shown in Figure 7.23 below.

Figure 7.23: Two realisations of the simulated level.

Consideration of mining constraints leads to the delineation of selected ore zones to be mined and a new evaluation of the economic reserves. Based on the simulation we now have values pertaining to ore and waste as illustrated by Table 7.14 below.

Zone A Zone B

Ore and Waste Tonnage Grade (tons) (% U3O8) 1,830,400 0.052 1,034,800 0.036

Ore Tonnage (tons) 1,21,200 525,200

Total 2,865,200 0.046 1,726,400 Table 7.14: Ore and waste estimates.

Grade (% U3O8) 0.070 0.054

Waste Tonnage (tons) 629,200 509,600

Grade (% U3O8) 0.017 0.017

0.065

1,138,800

0.017

The reserves in the above table indicate that the average ore grade remains the same, however, the estimated tonnage is substantially less. 7.8.3

Discussion

Can we generate confidence intervals for the economic reserves?


201

Chapter 8

CHAPTER 8

SIMULATION IN OPTIMISING MINING PARAMETERS

8.1

Optimising Mining Parameters: An Example from Ore Control in an Open Pit Epithermal Gold Deposits

8.1.1

Introduction

Effective selective mining is critical in the successful exploitation of epithermal gold deposits. The degree of selectivity is controlled by both metallurgical and mining constraints. Most often, the later constraints dictate whether the desired selectivity can be achieved. Mining constraints may be seen to consist of two broad categories. The first category includes ore control related constraints, specifically, bench height and blast hole spacing, blast hole sampling and assaying, and computer based modelling of the blast hole data and ore classification. The second category includes operations related constraints such as the mining equipment used, procedures followed in the pit, and truck dispatching. Bench height and blast hole spacing are critical factors controlling selectivity. For example, when the bench height is increased, selectivity is reduced. This is because the selection of the material mined is based on higher columns of muck where the entire column is put into one material class and processed accordingly. When the blast hole spacing is increased, is decreased as well, control / material classification uses lessselectivity information and produces less because accurateore classification of the material mined. Starting with pragmatic needs at an open pit mine, this case study revisits the question of selecting bench height and blast hole spacing. A new general methodology is developed here based on maximum profitability criteria. The methodology is based on the framework presented in Chapter 2 and includes three ingredients: (a) geostatistical simulations, used to generate representations of the deposit being studied in terms of grade and material types; (b) the concept of a transfer function, used here to account for the ore control processes and characterise the misclassification of material mined; and (c) decision making using unifying economic criteria, based upon the mapping of the so called ‘space of uncertainty’ of the economic parameter of interest. In the following sections, an application at a typical open pit mine is developed in detail.

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8.2

An Application at an Open Pit Gold Mine

8.2.1

Problem statement

The pit of a disseminated gold deposit is mined using two different bench heights. On the west, representing almost 50% of production, the bench height is 25ft. In the remaining of the pit, 20ft benches are used. Converting from 20ft to 25ft in the parts of the pit generates substantial operating cost savings, estimated at $0.04/ton. The areas of savings include drilling, blasting, sampling and assaying, support equipment and loading. As indicated in the above report, increased bench heights will result in increased dilution, thus the effects of dilution on expected savings needs to be studied. Three considerations may be pointed out. Firstly, decision making can be based on the quantification of dilution cost in terms of dollar loss per ton of material mined. The dilution dollar value is directly comparable to the operational savings discussed above. Secondly, flexibility in decision making may be added if the study includes various blast hole spacings. For instance, increased bench height combined with a denser blast hole spacing may show the same dilution effects as the current bench height with a sparse blast hole spacing. Lastly, the evaluation of dilution effects must be based on jointly considering grade content and material type. This is particularly important given changes in economics based on both. Oxide and refractory material types are considered here, while mined materials are classified in four categories, oxide leach, oxide mill, refractory mill and waste. For reasons of completeness, the quantities of interest considered here are bench heights of 10, 15, 20 and 25ft and blast hole spacings of 14, 16 and 18ft.

8.2.2

Study area: sele ction and characteristics

A suitable study area is selected within the pit. To account for the expected dilution, the material in the study area must be as similar as possible to the material planned to be mined in the next months. For this reason, the selected study area is in the middle of the pit and in very recently mined benches, as shown in Figure 8.1. Most of the study area belongs to a single geological zone. The stripping ratio in this area is 4/1.


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Figure 8.1: Location of the study area within the open pit.

The statistics of the blast hole data (9953 blast holes from 20ft benches; see Figure 8.2) in the study area are given in Table 8.1. Data include fire assays (FA) and values of fire assays to atomic absorption (AA). The values are used to disseminate two material types, specifically oxide from refractory ore. The variogram models for the relevant geologic zones are reported in Table 8.2. Note that these are the same variogram models used for grade control. Variogram models and geologic units are discussed in a subsequent section.

Statistic Mean Median Variance Relative Variance

FA 0.021 0.006 0.001 2.260

AA/FA ratios 0.58 0.70 0.23 0.68

Table 8.1: Statistics of blasthole data in study area.

Zone

Strike

Dip

Pitch

Vario type

Sill

MIMU

-20

0

0

Nug Sph 1

0.20 0.37

Sph 2 0.62 Table 8.2: Relative variogram models used in this study.

204

_|_ pitch

|| pitch

_|_ plane

17

20

15

120

175

85


Chapter 8

High Oxide Refractory Low Figure 8.2: Blast hole data; FA’s on the left and AA/FA ratios.

8.2.3

Some alternatives

There are two other alternatives in the selection of the study area. First, the study area could be an area within the part of the pit that will be mined in the next months. In this case, exploration data instead of blast hole data would be used. The disadvantage of this approach is that the local information from the exploration data is limited. The second option is to select a study area extending from a recently mined part of the pit to the part to be mined. In this case, both exploration and blast hole data would be integrated. Nevertheless, the present choice of study area is sufficient. Should there be drastic changes in the material mined during the year, the study should be repeated.


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8.2.4

Simulation and related intricacies

Using the sequential indicator simulation (SIS) algorithm in Chapter 6, nine images of the grade and material type (oxide and refractory) have been generated. All images are generated on a dense grid with a 5 by 5 by 5 ft resolution and 345,600 grid nodes. Practical aspects in setting up the simulations are addressed in this section. The available blast hole data represent samples of a 20 ft length. To use them as conditioning data in the simulations they need to be ‘de-regularised’ to represent 5ft composites. ‘De-regularisation’ is the reverse of the typical procedure of compositing from exploration data and amounts to splitting each 20ft length to four 5ft samples with the same mean as the srcinal sample and a variance equal to the variance difference between 20ft and 5ft composites. The latter variance is derived from geostatistical charts (David, 1977). The implementation of SIS and, specifically, the implementation of the SIS algorithm is based on median IK and the ‘mosaic’ model (Journel, 1983). Accordingly, the indicator correlogram (or variogram) at the median cut-off is used, assuming that indicator correlograms at different cut-offs have the same type and anisotropy. The median indicator correlogram used in this study is generated from ρI (h;z )

where

γR(h)

samples,

σR 2

=

ρz (h)

= 1 - γR(h)/σR2 with F(z) = 0.50

(8.1)

is the global relative variogram (e.g. Table 8.2) de-regularised for 5ft is a relative variance, and

ρz (h)

is the correlogram of the actual grades.

FA grades and AA/FA ratios are simulated independently, using the same input variograms. AA images are generated from the AA/FA and FA simulations. The reasoning behind the simulation of AA/FA ratios instead of directly simulating AA grades is that simulating AA/FA and then generating AA images experimentally generates a reasonable correlation between FA’s and AA’s. Furthermore, it generates appropriate classifications of the required oxide and refractory material types. Alternative approaches could be used to directly generate correlated FA and AA grades. Different variograms are used to address the question of uncertainty of the so called ‘model parameters’ as well as concerns about geological zone uncertainties. Model parameters refer to the variograms inferred from the data. Being inferences, variogram models include a degree of uncertainty in ranges, anisotropy directions and type. Figure 8.3 material and Figure 8.4respectively. shows different images of simulated FA grades andsimilar, oxiderefractory types, General patterns in the figure are quite as expected from the exceptionally large number of conditioning data. The reproduction of different proportions in some of the simulations are shown in Table 8.3 and Table 8.4, all simulations show excellent reproductions. Considering the large number of conditioning data, the use of global instead of local variograms does not affect the results. For the same reason, explicit comparisons of input variograms and variograms from simulations was not pursued.

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High

Low

Figure 8.3: Images of simulated grades.

Oxide Refractory

Figure 8.4: Images of simulated oxide and refractory material types. Cut off

Ccdf FA data

Ccdf FA sim #3

ccfd FA sim#6

Ccdf FA sim #8

0.004

0.40

0.42

0.41

0.43

0.006

0.46

0.51

0.50

0.50

0.010

0.59

0.61

0.60

0.62

0.040

0.86

0.85

0.85

0.84

0.100

0.92

0.96

0.96

0.95

Table 8.3: Global cumulative distributions of FA data and simulations in the study area.


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ccdf AA/FA ccdf AA/FA ccfd AA/FA ccdf AA/FA data sim#3 sim#6 sim#8 0.28 0.30 0.31 0.31 0.10 0.45 0.49 0.48 0.49 0.60 0.50 0.55 0.56 0.56 0.70 0.75 0.79 0.78 0.79 0.90 0.96 0.95 0.95 0.94 1.50 Table 8.4: Global cumulative distributions of FA/AA data and simulations in the study area. Cut off

8.2.5

Specifying the transfer function

The transfer function specific to the present study is defined in this section. It is implemented as a computational analogue of the ore control process and the related economics. A simulated image of the deposit in the study area is treated as the real and fully known deposit. Since, the deposit is fully known, a variety of ore control alternatives can be implemented. Following actual practices, the blasting-sampling process is first implemented. More specifically, the 5 by 5 by 5 ft deposit is sampled using, say, a 20ft bench height and 14ft blast hole spacing. Similarly, the deposit is resampled according to the desired bench heights and blast hole spacings. Having generated the blast hole data for each set of the quantities of interest, the next step in ore control follows. Using the grade control parameters used by the operation, a kriged block model with blocks of 20 by 20 ft is produced. It is assumed here that the 20ft cubes represent average SMU’s. The effects of the SMU size used here is examined later on. Based on grade and material type, as well as the currently used cut-off grades each 20 by 20 by 20 ft block is assigned to a category, such as oxide leach, oxide mill, refractory mill, and waste. When kriging blastholes, modifications are made in search distances so that the same number of samples is used in all cases. The processing part and related economics are considered next. Each of the above ore control blocks is evaluated based on (a) the category it has been classified, and (b) its actual grade and material type. Subsequently, the average dollar value per ton is calculated, using current economics. The process is repeated for all the combinations of the bench heights and blast hole spacings.

8.2.6

Misclassificat ion and related effects

Prior to examining the actual optimisation of the quantities of interest, it is worth focusing directly on the ore classification. Table 8.5, Table 8.6 and Table 8.7, show the actual versus ore control classification of the 345,600 blocks of size 5 by 5 by 5 ft in one of the simulated deposits and for different bench heights and blast hole spacings. Table 8.8 shows proportional differences between classifications in terms of ore. As expected, when going from 20ft to 25ft benches, misclassification increases in ore categories ( Table 8.5 and Table 8.6, respectively). Similarly, when blast hole spacing is increased from 14ft to 18ft (Table 8.5 and Table 8.7, respectively), dilution and misclassification are increased as well. The discrepancies from the actual categories are

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attributed to two factors. Firstly, when the SMU support increases small pockets of one category are included in a dominating category. Secondly, as bench heights and blast hole spacings increase, the information used decreases and the errors in ore control increase. Misclassification in one of the simulations can be observed in Figure 8.5. The larger blocks in the figure correspond to blocks with FA values generated from ore control, while the small ones represent the actual deposit grade. The colour classification is the same as in Figure 8.3 (yellow is below 0.006 ounces/ton). Dominant factors in classification are the spatial variability and connectivity of both the orebody grade as well as material types. When, for example, orebody variability and connectivity better horizontally, theblast bench willand be soa on. more important contributor toare misclassification than the holeheight spacing This is also reflected in the proportion of ore that can be classified correctly when bench heights and blast hole spacings increase. The effects of misclassification are reflected in the quantification of the cost of dilution in the following section. Note that the problem considered here, i.e. classification based on both grade and material type, generates a non linear system. Consequently, the effects of support and information on the quantities of interest are generally non linear, i.e. more information does not necessarily generate better classification and a tighter mapping of the space of uncertainty. A similar behavior has been observed in other applications (e.g. Deutsch, 1994). A factor found to effect results is the correlation of simulated FAs and simulated AAs. The first are used for grade related classification, while the second for oxide and refractory delineation. When the reproduction of the correlation of FA’s vs AA’s is poor, the results may also be poor. This observation generates a ‘problem specific’ criterion in implementing a simulation technique in similar problems.


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Figure 8.5: Actual grades and grades from ore control.

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Classification from Ore Control Oxide Mill Oxide Waste Refrac Mill 15379 2067 335 152 957 29066 2280 45 15324 2594 43 341 5849 16142 434 55168 33728 2112

Refr Waste 20544 3820 28496 2172 118344 173376

Total 71589 41084 62101 6106 164720 345600

Table 8.5: Misclassification of blocks; bench height is 20ft and blast hole spacing 14ft. Classification from Ore Control Oxide Leach Oxide Mill Oxide Waste Refrac Mill 15820 2024 405 A O xide Leach 31752 7404 114 638 C Oxide Mill 28654 15515 2264 43 T Oxide Waste 14271 1013 2655 56 U Refr Mill 207 24216 6832 17635 457 A Refr Waste 79900 56225 34100 1750 L TOTAL

Refr Waste 21588 4274 30008 2175 115580 173625

Total 71589 41084 62101 6106 164720 345600

Refr Waste 20696 4378 29306 2132 118208 174720

Total 71589 41084 62101 6106 164720 345600

A O xide Leach C Oxide Mill T Oxide Waste U Refr Mill A Refr Waste L TOTAL

Oxide Leach 33264 7089 15956 956 23951 81216

Table 8.6: Misclassification of blocks; bench height is 25ft and blast hole spacing 14ft.

A O xide Leach C Oxide Mill T Oxide Waste U Refr Mill A Refr Waste L TOTAL

Oxide Leach 33221 7407 16793 1062 25229 83712

Classification from Ore Control Oxide Mill Oxide Waste Refrac Mill 15147 1820 705 117 966 28216 1887 75 14040 2511 35 366 5615 14964 704 53376 30976 2816

Table 8.7: Misclassification of blocks; bench height is 20ft and blast hole spacing 18ft.


211

Chapter 8

Bench/Blast sizes 20ft / 14ft 25ft / 14ft 20ft / 18ft

Total Ore Blocks 118779 118779 118779

Correctly Classified % 52.76 51.03 52.03

Table 8.8: Proportion of correctly classified blocks of ore in tables 8.5 to 8.7.

8.2.7

Economic considerations and response definition: the cost of dilution

Cost of dilution is the response parameter of the transfer function and is defined in this section. Considering that the deposit is known (simulated) at any specified level (resolution) and assuming perfect selectivity at this level, the calculation of the dollar value for each block or the average dollar value per ton of material mined, V ps, is straight forward. Similarly, for each of the specific combinations of the quantities of interest, the recovered dollar value per ton of material mined, Voc, can be calculated as well, using the classifications from ore control and the actual grades and material types of the known deposit. The dilution cost Cd is then defined as the difference

Cd = Vps - Voc

(8.2)

Note that Cd is a dilution cost only under the constraint that mining costs are kept constant across the board, so that the differences account for dilution only. Alternatively, one may consider total costs, i.e. include drilling, blasting, assaying, as well as different operating costs for each set of quantities of interest. The present study uses a resolution of 5 by 5 by 5 ft to represent the actual deposit. The resolution is sufficient for the calculation of the cost in equation 8.2, particularly when the bench height and blast hole spacing combinations define blocks larger than 10 by 10 by 10 ft. The dilution costs are calculated based on the mining costs corresponding to the 20ft bench and 16ft blast hole spacing. Having defined the case specific response parameter, the application of the methodology generates the mapping the dilution cost for the different combinations of the quantities of interest.

8.2.8

Results, evaluation and decision making

The dilution costs for each of the nine simulations are tabulated in Table 8.9, together with their corresponding bench heights and blast hole spacings. The table is said to represent the mapping of uncertainty associated with the cost of dilution, for the various bench heights and blast hole spacings considered in this study. Table 8.10 is a summary of Table 8.9, in terms of the mean, median and variance. The statistic used for decision making depends on the problem and approach taken. Here, the mean is considered adequate, particularly since there are no substantial differences from the median. Figure 8.6 plots the average or expected cost of dilution for the different benches and blast hole spacings. Note that the zero dilution cost in the figure corresponds to perfect selectivity

212


Chapter 8

at the 5 by 5 by 5ft orebody resolution. The figure also represents e.g. 5 and generates an optimal selection, given the question posed. For the current problem of 20ft against 25ft bench heights, Figure 8.6 suggests the expected dilution cost to be in the order of 0.07$/ton. Considering that the operational savings expected to be on the order of 0.04$/ton, the economically optimal decision is to not switch to 25ft benches. In the calculation of the above costs, it is assumed that the mining width is about 20ft. To test the sensitivity of the results to this assumption, a 30ft width was used as well and the results were compared. Table 8.11 shows the difference in dilution cost for 20ft and 30ft mining widths. Although there are differences due to dilution, the order of differences remains quite similar from bench height to bench height. As noted earlier, this study includes blast hole spacings in addition to bench heights in order to add flexibility in decision making. A possible example of alternative choices could be illustrated in Figure 8.6, where the cost of dilution between 25ft with a 14ft blast hole spacing is comparable with the dilution cost of 20ft benches and a 18ft blast hole spacing. The magnitude of the dilution cost depends on the characteristics of the ore body considered. For example, the dilution cost in a nearby pit may vary from 0.20$/ton to 0.40$/ton depending on the variability and connectivity of the orebody. The differences in $/ton also reflect the higher proportions of ore misclassification. This observation, points out the significance of the adequate simulation of orebody variability and connectivity, both in terms of grade and material type.


213

Chapter 8

Dilution Cost ($/ton) Simulation Run Number cs01 cs02 cs03 cs04 cs05 cs06 cs07 cs08 cs09 0.57 0.48 0.54 0.60 0.59 0.62 0.65 0.60 0.65 0.57 0.55 0.55 0.61 0.63 0.66 0.68 0.61 0.69 0.60 0.55 0.57 0.63 0.66 0.69 0.70 0.65 0.70

Bench Height (ft) 25

Blast hole spacing (ft) 14 16 18

20

14 16 18

0.45 0.47 0.50

0.46 0.49 0.54

0.46 0.49 0.53

0.55 0.58 0.60

0.54 0.53 0.57

0.57 0.60 0.62

0.57 0.55 0.61

0.51 0.57 0.58

0.62 0.63 0.64

15

14 16 18

0.46 0.51 0.52

0.42 0.46 0.47

0.43 0.45 0.47

0.45 0.50 0.50

0.48 0.48 0.51

0.55 0.56 0.61

0.52 0.55 0.57

0.46 0.49 0.49

0.50 0.50 0.60

10

14 16 18

0.40 0.40 0.44

0.37 0.39 0.40

0.32 0.38 0.40

0.39 0.42 0.42

0.36 0.38 0.42

0.48 0.50 0.53

0.44 0.47 0.51

0.39 0.43 0.43

0.52 0.56 0.54

Table 8.9: Dilution cost for different simulations. Expected Dilution Cost ($/ton) Bench Height (ft)

Blasthole spacing (ft)

Average

Median

Variance

25

14 16 18

0.59 0.62 0.64

0.60 0.61 0.65

0.0026 0.0025 0.0027

20

14 16 18

0.53 0.55 0.58

0.54 0.55 0.58

0.0031 0.0026 0.0019

15

14 16 18

0.47 0.50 0.53

0.46 0.50 0.51

0.0016 0.0012 0.0025

10

14 16 18

0.41 0.44 0.45

0.39 0.42 0.43

0.0034 0.0034 0.0028

Table 8.10: Expected dilution cost.

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Chapter 8

Figure 8.6: Effect of bench height on dilution.

Dilution Cost ($/ton) – Sim. #3 Bench Blasthole Difference ($/ton) height (ft) spacing (ft) Width 20ft – 30ft 0.039 25 14 0.049 16 0.065 18 20

14 16 18

0.032 0.057 0.078

Table 8.11: Dilution cost difference for different SMU’s (see text for discussion).


215

Chapter 8

8.3

Evaluation and Limits

8.3.1

Simulations and transfer functions

A specific criterion that was discussed in a previous section is the adequate mapping of the space of response uncertainty. Prior to fully implementing a given algorithm, the effects of the algorithm on the space of uncertainty map should be examined. This is important as well as quite straight forward in terms of implementation. That is, the SIS algorithm can be replaced with the equivalent annealing one, then one of the previous studies repeated and the results compared. This should be done prior to implementing joint simulations with annealing. Note that although the SIS algorithm may not be the best solution in the type of problems discussed here, it is preferable for other applications such as deciding on additional drilling.

8.3.2

Decision making

The main strength of the proposed methodology is that decision making is based on the economic parameters of interest, namely cost and profit. In the context of a global mine environment, profitability is considered to be maximised by locally minimising opportunity type costs. The idea may be examined in a variety of (local) processes of a mining venture, such as ore control, truck dispatching, ore processing, marketing and sales, etc. The present study is an example of minimising local costs (or dollar losses), where the cost of switching to 25ft benches is expected to generate a loss of 0.04$/ton. If the thousands of tons of material mined from the pit are considered, the effects and the dollar value of a wrong decision can be readily calculated.

216


Chapter 9

CHAPTER 9

SIMULATIONS IN GRADE CONTROL

9.1

Grade Control and Objective Functions

9.1.1

Problem definition

Grade control is the operation where the mine geologist must flag truckloads as ore or waste based on blast hole, drill hole and geologic information. In grade control, classification errors generate expenses. If a block of ore is misclassified, a net loss ($) occurs. For example, a truckload of Such milling ore wrongly classified as never waste result represents an actual loss in potential revenue. misclassification errors can in a net $ gain. Figure 9.1 below is a plot of true grades versus estimated grades for a certain number of blocks. The application of the cut-off grade (Zc) divides the plot into four quadrants.

Figure 9.1: Plot of true grades versus estimated grades for a certain number of blocks. Question: What is the $ loss associated with misclassification errors and how can this loss be minimised? 9.1.2

The traditional approach

The block selection is typically carried out by comparing a deterministic cut-off grade (Zc) to an estimated block grade. This estimated block grade is often calculated by using traditional estimators, such as ISD, polygons, kriging. One key point to remember is that the relationship between dollar losses and the classification error are not symmetric.


217

Chapter 9

9.1.3

Uncertainty assessment

We have seen the major difference between estimation and conditional simulation; the former returns a single estimate, whereas the latter returns a range of possible values from a conditional cumulative distribution function ( cdf) of the type:

F(x;z|(n)) = Prob{Z(x) ≤ z|(n)} where Z(x) represents a way to look at modelling uncertainty about the unknown true value, (n) represents local conditioning data within the specific neighbourhood of location (x). The objective here is to assess the conditional distribution function of block grades. The steps needed to obtain a sample of the cdf for point or selective mining unit (SMU) grade consist of: •

Generating a number of equiprobable conditional realisations of point support grades.

•

Changing the support if required i.e., obtain the average of point grades over the volume of any SMU.

•

Obtain the conditional distribution of the point or SMU grades Z(x).

Note: In practice one does not need to use SMU’s, but draw dig lines based on quasipoint values (grade, probability, loss, profit). SMUs are used in the following examples as a way to ‘emulate’ mining and study effects from different classification methods.

9.1.4

Objective functions for grade control

Material that was once mined as simply ore or waste may now be classified into one of several possible categories. Some objective functions are examined in this section. Once the conditional distribution function (cdf) has been established, we should retain a classification for the SMU, which optimises an economics based objective function. Different objective functions can produce different “best” classifications for the same cdf. The obvious goal of any decision is to minimise the expected loss ($) that can occur as a result of the decision. The loss associated with any estimate can be viewed as a consequence of the uncertainty introduced by the lack of perfect knowledge. The loss associated with each type of misclassification error can be expressed as an objective function L of the actual, but unknown, grade z(x) (Isaaks, 1990). Another objective function was proposed by Glacken (1996) and involves specifying the profit or loss associated with each possible scenario of ore classification. Variations of these two approaches may also be developed.

218


Chapter 9

9.1.5

Minimising $ loss due to misclassificati on

The principle here is that any value selected other than the (unknown) true value will incur a loss. Example: At the time of mining, each SMU is classified as one of four types; waste, ROM, agglomeration, or milling ore. The costs, associated recovery factors, and the price are those shown in Table 9.1.

Min. + Procs. Costs ($/ton) Recovery

Waste

ROM

Agg.

Milling

$1.50

$2.94

$5.14

$10.14

0%

45%

70%

95%

0.008 - 0.022

0.022 – 0.05]

0.05 +

0.0 - 0.008] Grade range (oz/ton) Table 9.1: Classification of ore types.

Let’s define marginal cut-off grade ( Zc), separating each ore type

Zc (oz/ton)* 400($/oz) * 45% - 2.94 $/ton = -1.5 $/ton where Z = 0.008 oz/ton c

Similarly, the cut-off grades separating ROM from agglomeration ore and agglomeration ore from milling are calculated to be 0.022 $/ton and 0.5 oz/ton. The actual loss ($) associated with each type of misclassification error is simply the potential value of the SMU less the actual value recovered:

loss($) = potential value - recovered value Returning to our example, suppose the true grade of an SMU is z(x) ∈ ]0.022,0.05]. Then the potential value of the SMU is given by:

potential value = Z(x) * 0.70 * 400 - 5.14 If the same SMU is misclassified as milling ore, its recovered value is given by

recovered value = Z(x) * 0.95 * 400 - 10.14 The actual loss ($) is given by

loss($) = 5.00 -100 * Z(x), Z(x) ∈ ]0.022,0.05]


219

Chapter 9

The loss ($) can be calculated similarly for each type of classification error. For our example, there are 12 possible misclassification errors. These have been calculated and are given in Table 9.2.

Loss for each type of misclassification True grade z(x)(oz/ton)

[0.0, Waste

0.008] [0.008, 0.022] [0.022, ROM ore Agg. ore

>0.05 Mill ore

380 z(x)- 8.64 200 z(x)– 7.20

– 2.20 [0.022, 0.05] 280 z(x) - 3.64 100 z(x) Agg. ore

100 z(x) - 5.00 0

- 1.44 0 [0.008, 0.022] 180 z(x) ROM ore

0.008] Waste

0

0.05] > 0.05 Mill ore

5.00 – 100

z(x)

z(x)

7.20 – 200

z(x)

3.64 – 280 z(x)

8.64 – 380

z(x)

2.20 – 100 1.44 – 180 z(x)

0

Table 9.2: Actual dollars lost due to misclassification of ore type at the time of mining.

Thus, given the previous expressions of loss as a function of Z(x), the expected conditional loss can be calculated for any SMU, given:

• the cdf of the SMU, FZ(x)(z|(n)), • an estimate Z(x) = z(x) of the SMU, • and a particular loss function L(Z(x)|Z(x) = z(x)) the expected conditional loss ($) is given by

E{ L(Z(x)|Z(x) = z(x))} = ∫ L(Z(x)|Z(x) = z(x)) dFZ(x)(z|(n)) = ψ(z(x),(n)) The minimum expected loss can be found by calculating the conditional expected loss for all possible values of the estimates Z(x). The optimum classification Lopt(x) for the SMU in question is the one providing the smallest expected loss, i.e.,

ψmin(z(x),(n)) = min [ψ(z(x),(n))] Lets suppose a sample of a cdf with 100 realisations for the gold grade in a hypothetical SMU as shown in Figure 9.2.

220


Chapter 9

1.00

0.80

Number of Data 100

0.60

mean 0.0252 std. dev. 0.0196 coef. of v ar 0.7778 maximum upper quartile median lower quartile minimum

0.40

0.20

0.0946 0.0359 0.0197 0.0101 0.0019

0.00 0.000

0.020

0.040

0.060

0.080

0.100

value

Figure 9.2: Sample of a cdf with 100 realisations for the gold grade in a hypothetical SMU.

If we know the distribution of grade Z(x), we can calculate the probability P(a < Z(x) ≤ b) corresponding to any cut-off range. We can now go through each cut-off range defined in Table 9.1 and compute the probabilities. We have:

P(0.0 0.05)

= = = =

0.12 0.44 0.36 0.08

We can now calculate the loss for each type of misclassification by finding the expected value within each range

E{ Z(x)| 0.00.05)}

= = =

0.00554 0.01356 0.03478 = 0.07516

We can now define an array of the dollars lost due to the misclassification errors Lk,j (Table 9.3 below). Each column is related to a different possible classification k of the SMU, where k=1,..,4. The rows represent the unknown true classification j, where j= 4,..,1.


221

Chapter 9

Estimated SMU grades (oz/ton) True grade [0.0, 0.008] [0.008, 0.022] [0.022, 0.05] > z(x) (oz/ton) Waste k=1 ROM ore k=2 Agg. Ore k=3 Mill >=0.05 Mill 19.9218 ore j=4

7.83250

2.51625

0.00000

[0.022,

0.05] 6.09856

1.27806

0.00000

1.52195

j=3 Agg. ore [0.008, 0.022] 1.00104 ROM ore j=2

0.00000

0.84386

4.48773

0.44250

2.08833

6.53417

<= Waste

0.008 0.00000 j=1

0.05 k=4

Table 9.3: Array of the dollars lost due the misclassification errors.

The expected loss ($) for each type of classification k is 4

E{L(Z(x) | a
∑ Lk , j ∗ Pj j =1

where a and b are the limits defining the class k. Pj is the probability for the class j. We then have:

E{L(Z(x)| 0.00.05))}

= = =

4.22968 1.13980 0.82320 = 3.30660

The optimum classification Lopt for the SMU in question is the one providing the smallest loss ($), i.e.,

ψmin(z(x),(n)) = E{L(Z(x)| 0.022
9.1.6

The profit/loss approach

This approach involves the incorporation of all possible classification scenarios, and instead of basing our decision only on expected loss, we will select the classification that maximises the expected profit or cash flow. There are now 16 scenarios of possible classifications for the SMU. Following the classification of ore types given in Table 9.1 we can define a profit function 222


Chapter 9

Tk,j(Z(x)|Z(x) = z(x)) where k=1,..,4 is related to the different possible classifications of the SMU; j=4,..,1 represents the unknown true classifications. Then, we have two cases defining our objective function Tk,j such as: • when k ≥ j

This expression covers overestimation and the correct classification scenarios, being the difference between revenue and cost:

Tk,j(Z(x)|Z(x) = z(x)) = recovered value • when k < j

This expression covers underestimation scenarios and reflects the opportunity cost of losing the difference between the potential and recovered value

Tk,j(Z(x)|Z(x) = z(x)) = - (potential value - recovered value) The actual profit ($) function associated with each type of the 16 possible scenarios has been calculated and is given in Table 9.4.

Profit/loss for each possible scenario

Zv(x) (oz/ton) >0.05 ore [0.022, Agg. Ore

0.008] [0.008, 0.022] [0.022, ROM ore Agg. ore

0.05] > 0.05 ore

Mill -(380 z(x) – 8.64) -(200 z(x) – 7. 20) -(100 z(x)- 5.00) – 5.14 0.05] -(280 z(x) – 3.64) -(100 z(x) – 2.20) 280 z(x)

[0.008,0.022] ROM ore 0.008]

[0.0, Waste

Mill

380 z(x) – 10.14 380

z(x) – 10.14

-(180 z(x) – 1.44) 180 z(x)– 2.94

280 z(x) – 5.14

380

z(x) – 10.14

180 z(x)– 2.94

280 z(x) – 5.14

380

z(x) – 10.14

Waste -1.5

Table 9.4: Actual profit equation for each possible classification scenario.

Using the same hypothetical SMU cdf showed in Figure 9.2, we have the following class means


223

Chapter 9

E{ Z(x)| 0.00.05)}

= =

0.00554 = 0.01356 0.03478 = 0.07516

or expected value of Z(x) inside each cut-off class. The corresponding probabilities of grade values being in the class is, also obtained from the cdf in Figure 9.2, are:

P(0.0
=

0.12

P(0.008 0.05)

= = =

0.44 0.36 0.08

We can now calculate the profit for each scenario and define the array of profits Tk,j (Table 9.5). Like in the previous section, each column is related to a different possible classification k of the SMU, where k=1,..,4 and the rows represent the unknown true classification j, where j= 4,..,1.

Estimated SMU grade (oz/ton) True grade [0.0, 0.008] [0.008, 0.022] [0.022, 0.05] > 0.05 z(x) (oz/ton) Waste k=1 ROM ore k=2 Agg. ore k=3 Mill k=4 -7.8320

-2.5160

18.4208

-1.2780

4.5984

3.0764

[0.008, 0.022] -1.0008 ROM ore j=2

0.4992

-1.3432

-4.9872

<= Waste

-1.9428

-3.5888

-8.0348

>=0.05 Mill =4 [0.022, Agg. ore

-19.9208

ore 0.05] -6.0984 j=3

0.008 -1.5000 j=1

Table 9.5: Array of the profits (Tk,j).

The expected profit ($) for each kind of classification k is given by 4

T ∗ Pj E{T(Z(x) | a
where a and b are the limits defining the class k. Pj is the probability for the class j. Then, we have:

E{T(Z(x)| 0.0
224

= =

-4.40944 -1.10013


Chapter 9

E{T(Z(x)| 0.0220.05))}

= =

0.43248 -0.57738

The optimum classification for the SMU in question is the one that provides the greatest profit ($), i.e.,

ψmax(z(x),(n)) = E{T(Z(x)| 0.022
9.1.7

Risk coefficients

The loss is typically a function of the error, i.e. the difference between the estimated classification and the true classification. In any mine operation, the penalty attached to overestimation is not the same as that of underestimation. We should then define an array of coefficients ϖkj to quantify the impact of loss. These coefficients can be divided in two categories, the ones that quantify the magnitude of the loss due to under-estimation and those related to the opportunity cost of overestimation . Thus, we have two cases defining ϖkj as: • if k

′ j

kj is a positive coefficient of loss; the magnitude of related to the value ϖ ϖkj is of company places on losing resources and the consequence producing belowwhich target;the if we do not wish to reject blocks wrongly, then ϖkj will be high

• if k

> j

ϖkj is positive too and it is related to the degree of risk-aversion of the company; if the company is especially risk-averse, then ϖkj will be high. Overestimation may result in a sub-grade block being treated, with a relatively small loss. However, underestimation may cause rejection of an ore block and subsequent loss of the profitable contained metal. The coefficients ϖkj are linear multipliers of the objective function, being minimised. An example of the use of risk coefficients combined with the objective function is presented in Table 9.6.


225

Chapter 9

Profit for each possible scenario Zv(x) (oz/ton)

[0.0, Waste

0.008] [0.008, ROM ore

>0.05 Mill ore

-(380 z(x) 8.64)∗ϖ14

- -(200 z(x) 7.20)∗ϖ24

– -(100 z(x) 5.00)∗ϖ34

z(x) [0.022, 0.05] -(280 Agg. Ore 3.64)∗ϖ13

- -(100 2.20)∗ϖ z(x)

– 280 z(x) – 5.14

[0.008, 0.022] ROM ore 0.008] Waste

- 180 z(x)– 2.94

(280 z(x) 5.14)∗ϖ32

– (380 z(x) – 10.14) ∗ϖ4

(180 z(x) 2.94)∗ϖ21

– (280 z(x) 5.14)∗ϖ31

– (380 z(x) – 10.14) ∗ϖ4

-(180 z(x) 1.44)∗ϖ12 -1.5

2j

0.022] [0.022, Agg. ore

0.05] > Mill ore - 380 z(x) – 10.14

(380 z(x) – 10.14) ∗ϖ4

Table 9.6: Actual profit equation for each possible classification scenario.

Comment on risk aversion coefficients

The objective function approach derived in the previous sections shows alternatives through the use the coefficients ϖkj . However, their choice is arbitrary and changes locally within a pit. The same comment applies to the use of risk coefficients for the minimum loss objective function. In practice, the plotting of the actual probabilities above the cut-off is a good practical alternative.

9.2

Grade Control Comparisons in a Fully Known Environment

9.2.1

The Walker Lake gold deposit

The Walker Lake data set consists of V, U and T measurements at each of 78,000 points on a 1 x 1 m 2 grid ( exhaustive data set, presented in Figure 9.3). From this dense data set, a subset of 195 samples (Figure 9.4) has been adapted to represent a typical gold mine data set. Each of the 78,000 values has been divided by 10,000 and is interpreted as oz/ton. Descriptive statistics are given in Figure 9.5. Let’s define four ore types k, where k=1,..,4. The processing and mining costs, the recovery factors, and the grade ranges based on $400/oz are shown in Table 9.7.

226

0.05


Chapter 9

Costs

($/ton)

k=1

K=2

k=3

k=4

Waste

ROM

Agg.

Milling

$2.94

$5.14

$10.14

45%

70%

95%

0.008 – 0.022

0.022 - 0.05

0.05 +

$1.50

Recovery

0%

Grade

range 0.0 - 0.008

(oz/ton) Table 9.7: Classification of ore types.

Figure 9.3: A categorical posting of the 78,000 Walker Lake point gold grades. 300

250

Ore type

200

=< 0.008 ]0.008,0.022] 150

]0.022,0.05] > 0.05

North 100

50

0 0

50

1 00

1 50

2 00

250

East

Figure 9.4: A categorical posting of the 195 conditioning data on a regular grid.


227

Chapter 9

Figure 9.5: Histograms for the exhaustive data set and sample conditioning data.

9 2.2 Point grade conditional simulations .

1.

Perform the normal score transform of z-values, i.e. define y-values such that, by construction, they are normally distributed. Normal score transformation consists of transforming, mathematically or graphically, (see section 6.5.4) the srcinal data histogram ( z) into a standard normal distribution with mean zero and variance one ( y) (Figure 9.6).

The Gaussian program used inisthe first workshop) workssequential with normal scoressimulation of the srcinal data.(SGSIM, The simulation performed in the normal space, and the simulated values are back transformed.

Figure 9.6: Histogram of the y-values. 2.

228

Calculate the sample variogram. The models used for this case study are given in Figure 9.7.


Chapter 9

(a) Cov(h) = 0.35 + 0.55 Sph30 (h)

(b)

Cov(h) = 0.35 + 0.55 Sph

45

(h)

Figure 9.7: Experimental and the modelled variograms fitted to y-values. The Gaussian sample covariance is shown by • symbol while its model is shown by the solid line (models extracted from Isaaks, 1990; pg. 106). 3.

Select the simulation parameters and proceed with sequential Gaussian simulation. The simulation parameters used here are shown in Figure 9.8. Parameters for SGSIM walker.dat \file with data 1 2 0 3 0 0 \ columns for X,Y,Z,vr,wt,sec.var. -1.0e21 1.0e21 \ trimming limits 1 \transform the data (0=no, 1=yes) walker.trn \ file for output trans table 0 histsmth.out 1 0 0.0 0.4077 1 0.0 4 2.0 2 walker.dbg walker.out 100 260 1.0 1.0 300 1.0 1.0 1 0.0 0.0 140371 2 16 12 1 0 0 2 45.0 45.0 0.0 0.0 0.0 0.0 0 0 ../data/ydata.dat 0 1 0.35 1 0.65 90.0 0.0 0.0 45.0 30.0 0.0

\ consider ref. dist (0=no, 1=yes) \ file with ref. dist distribution \ columns for vr and wt \ zmin, zmax(tail extrapolation) \ lower tail option, parameter \ upper tail option, parameter \debugging level: 0,1,2,3 \file for debugging output \file for simulation output \number of realizations to generate \nx, xmn, xsiz \ny, ymn, ysiz \nz, zmn, zsiz \random number seed \min and max srcinal data for sim \number of simulated nodes to use \assign data to nodes (0=no, 1=yes) \multiple grid search (0=no, 1=yes),num \maximum data per octant (0=not used) \maximum search radii (hmax, hmin, vert) \angles for search ellipsoid \ktype: 0=SK,1=OK,2=LVM,3=EXDR,4=COLC \file with LVM, EXDR, or COLC variable \column for secondary variable \nst, nugget effect \it,cc,ang1,ang2,ang3 \a_hmax, a_hmin, a_vert

Figure 9.8: Parameter file for sequential Gaussian simulation. 4.

Check the simulation results in order to confirm if the main statistical features, one point and two point statistics, were properly replicated. Four simulated


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images are randomly selected and are shown in Figure 9.8, their histograms are given in Figure 9.9.

Figure 9.9: Maps for simulated point grades (realisations #81, #100, #60 and #1).

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Figure 9.10: Histograms of four realisations.

Figure 9.11: Check on the reproduction of the covariance function. Variograms for three randomly-chosen realisations (#1, #60, #81: dotted line) are compared with the sample covariance values (blue line).


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Chapter 9

9.2.3

Change of support

For the sake of this example, point grades are to be re-blocked to SMUs as the goal is to determine the conditional probability distribution of ore blocks that are classified one by one. Thus, the 78,000 points of each simulated image were averaged into 780 blocks or SMUs, i.e., each SMU grade is the arithmetic average of 100 contained point grades (Figure 9.12). Figure 9.13 shows a map and histogram simulated of the trueSMU 780 SMU gold grades. Figure 9.14 shows two examples of the conditionally gold grades. Figure 9.15 shows the histograms relative to the maps presented in Figure 9.14.

Figure 9.12: Change of support from point to block.

Figure 9.13: Histogram and map of the 780 Walker Lake SMU gold grades.

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Figure 9.14: Maps of SMU grades for realisations #1 and #60.

Figure 9.15: Histograms of SMU grades for realisations #1 and #60.

9.2.4

Optimum classification

A computer program, LOPT, has been developed to address the problem of SMU classification (see workshop 3). The program implements block selection for two objective functions that optimise the classification of SMU’s, given some economic constraints. The program reads a set of simulated realisations and uses the conditional probability distribution of the SMU grades combined with a particular objective function to classify blocks within the given marginal cut-offs. After applying the L-optimum objective function in a set of 100 realisations of the 780 SMU gold grades we have the classification scheme showed in Figure 9.16 with the associated values found in Table 9.8. The classification scheme presented in Figure 9.17 is the actual classification of SMUs in Walker Lake. Table 9.9 corresponds to the actual recoveries for each class given perfect classification.


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Figure 9.16: Map of the 780 L-optimum classified SMUs.

Figure 9.17: Map of the true 780 Walker Lake SMU classification.

ORE GRADE TYPE Waste ROM ore Agg. ore Mill ore

TONS

TOTAL VALUE

AVERAGE

(%) 13.97 48.08 31.79 6.15

($) -163.50 48.76 1256.98 1198.73

(oz/ton) 0.00805 0.01706 0.03646 0.09240

Combined 100.0 2340.98 Table 9.8: Actual recoveries given the Lopt. classification.

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ORE TONS GRADE TYPE (%) Waste 34.10 ROM ore 26.03 Agg. ore 23.46 Mill ore 16.41 Combined 100.0 Table 9.9: Actual recoveries for each classification.

TOTAL VALUE

AVERAGE

($) (oz/ton) -399.00 0.00257 - 60.56 0.01468 783.48 0.03365 2855.90 0.08540 3179.81 ore type of Walker Lake given perfect

ORE TONS TOTAL VALUE AVERAGE GRADE TYPE (%) ($) (oz/ton) Waste 19.74 -231.00 0.01091 ROM ore 41.54 143.39 0.01879 Agg. ore 29.23 782.37 0.03061 Mill ore 9.49 1530.21 0.08110 Combined 100.0 2224.97 Table 9.10: Actual recoveries given the OK classification.

Figure 9.18: Map showing the ore classification based on ordinary kriged estimates.


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Figure 9.19: Map showing the ore classification based on profit/loss classification. ORE TYPE Waste ROM ore Agg. ore Mill ore Combined

TONS TOTAL VALUE AVERAGE GRADE (%) ($) (oz/ton) 16.93 -198.50 0.00927 21.15 -107.60 0.01271 37.95 513.15 0.02455 23.97 1965.87 0.05435 100.0 2173.42

Table 9.11: Actual recoveries given the profit/loss classification.

The classification scheme shown in Figure 9.18 is based on ordinary kriging. From this the recoveries were calculated using actual SMU grades. Table 9.10 corresponds to the actual recoveries for each class given the kriging classification. The classification scheme shown in Figure 9.19 is based on the profit/loss approach described earlier. Table 9.11 corresponds to the actual recoveries for each class given profit/loss classification. In terms of efficiency, the OK estimates recovered (2225/3180)*100 or approximately 70% of the total dollar value, i.e., 30% of the actual total dollar value is lost because of misclassification of ore types using OK. The LOPT classification recovered (2341/3180)*100 or approximately 73.6% or 3.6% more then by using OK. The profit/loss classification recovered (2173/3180)*100 or approximately 68.3%. We observe from the map given in Figure 9.19 that the profit/loss approach was very aggressive in classifying high-grade blocks. This criterion resulted in a loss of accuracy and a subsequent lost of efficiency. The classification scheme shown in Figure 9.20 is based on the profit/loss approach. Here, it used a risk coefficient for underestimation scenarios. We made ϖkj=3 when k
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underestimation. Table 9.12 corresponds to the actual recoveries for each class given in this classification.

Figure 9.20: Map showing the ore classification based on profit/loss classification combined with risk coefficients. ORE

TONS TOTAL VALUE

AVERAGE GRADE

TYPE (%) ($) (oz/ton) Waste 8.08 -94.50 0.00675 ROM ore 22.56 -148.85 0.01163 Agg. ore 43.85 409.15 0.02263 Mill ore 25.51 1986.50 0.05295 Combined 100.0 2152.30 Table 9.12: Actual recoveries given the profit/loss combined to risk coefficients classification.


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CHAPTER 10 10.1 10.1.1

SIMULATIONS IN MINE PLANNING

Uncertainty in Pit Optimisation and Simulated Orebodies Open pit optimisation

Pit design and scheduling is based on a single orebody block model used as an input to an optimiser. There are several pit optimisers, such as floating cone, Lerchs-Grossmann, etc. These optimisers provide an optimal pit and a mining schedule (a mining scenario) given (i) technical mining specifications as slope considerations, excavation capacities, plant performance, etc., and (ii) such economic considerations including capital and operating costs, and commodity prices. The optimisation results, or mining scenarios, are sensitive to the uncertainties of the above mentioned specifications. A lot of effort has been put into developing techniques dealing with the related uncertainties. The effects of pit design, scheduling and related predictions have major consequences to management of cash flows, which are typically in the order of millions of dollars. The optimiser widely used in the industry is Whittle 4D (and 3D). It implements the so called nested Lerchs-Grossmann algorithm. An additional issue of critical importance to optimisation results, is the orebody and the related in-situ grade variability and material type distribution. Not surprisingly, optimisation results and mine schedules are particularly sensitive to the orebody model used. The wide discrepancies between predictions and realisations are a well known phenomenon. Orebody models and their geological characteristics are widely acknowledged as a major source of uncertainty and risk. The use of widely spaced exploration data, sampling practices, reserve calculation methods, poor recoverable reserve calculations, are some of the known sources of problems. No matter what the reason for discrepancies between predictions and realisations, it is widely recognised that one needs to assess the various possibilities. Whittle 4D, for example, allows sensitivity testing of the optimisation results to grade variability or metal values based on global changes (Whittle, 1993) which, however, can not account for the critical local block grade variability. Other options explored in the past were the efforts to address the issue of block grade uncertainty in terms of the estimation variance or an estimate of the expected distribution of block grades. Unfortunately, estimation variances are a dead end..... The option of using block distributions as input to Whittle 4D/3D is feasible, however, we have seen in this course that, say IK, will smooth the block grade distribution. Finally, it is crucial to understand that the whole optimisation, pit design and production scheduling process, is a non-linear process, thus any confidence intervals directly from block models are not the best option. A first example of a possible orebody model may be a good start in understanding the problem and a solution. At this point the reader should probably expect the results of the example that follows.

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10.1.2

A ‘real’ deposit, exploration drilling, orebody models and statistical characteristics

(a)

(b)

(c)

Figure 10.1: (a) Exhaustive data; (b) sample data locations with clusters in the areas of high values; (c) histogram of declustered sample data

(a)

(b)

(c) Figure 10.2: (a) An ordinary kriging block model of the data; (b) histogram of the kriged grades in (a); and (c) experimental variograms of grades in (a) and model used in kriging

Some questions: Can the geological uncertainty be assessed? Can it be translated to mineable/recoverable reserves? Can we asses the response to questions like: are the few high grade composites towards the deeper part of the orebody driving the pit downwards without much certainty? Will the stockpiled grades be there during the stripping periods? Will the grades be there after the completion of a push back? Is there more waste and where?


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Figure 10.3: Two conditionally simulated models using the data in the previous figures and the SIS algorithm.

Figure 10.4: Sample histograms and variograms from different simulated realisations Simulations are a tool to address all the questions in the previous section. Examples from Dimitrakopoulos, Farrelly and Godoy (2001) and Rossi & Van Brunt (Optimising with Whittle ‘97) provide an interesting start.

10.1.3

1

Open pit design and long term plann ing: an example of avoiding single and possibly precisely wrong options 1

Dimitrakopoulos, Farrelly and Godoy, 2001

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Chapter 10

The availability of conditional simulation technologies allows the integration of geological uncertainty into optimisation studies and related decision making. This differs from the traditional grade estimation model used in pit optimisation studies. The example in this section elucidates the differences. The examples shown in this section are from the optimisation study of a typical disseminated low-grade epithermal quartz breccia-type gold deposit, hosted in intermediate-felsic volcanic rocks and sediments. Free milling and refractory ores are to be mined by open pit. Ore is to be processed via a CIL processing plant, with a floatation circuit added for the refractory ore. The question of geological uncertainty and risk in the design, planning and production expectations is accentuated by the generally low ore reserve grade, and a variable, depressed metal price. The example starts with the traditional way of pit optimisation in which an estimated orebody model of the deposit is used. Subsequently, fifty realisations of the deposit are developed to quantify geological risk for a given mine design and long-term mine plan. This is implemented by replacing the estimated orebody model with each one of the fifty simulations and re-running the optimisation, while all other mining and economic parameters are kept the same in all runs. Figure 10.5 shows the analysis of net present value. Orebody simulations have produced a range of financial outcomes, which in this example, is in contrast to the single estimate expected from the traditional approach. The project NPV is shown to vary drastically, with about 80% of the outcomes covering a range of A$5 million. The NPV outcome for the traditional approach is shown to be higher than the 95 th quantile of the distribution, that is, there is a 95% probability of the project returning a lower NPV than predicted by the estimated orebody model. The median NPV from the conditional simulations is A$16.0 million, approximately 25% lower than the estimated model indicates; and the worst case scenario from the simulations has an NPV 45% lower than the estimated orebody. Figure 10.6 shows the distribution of project NPVs more clearly and summarises the distribution for possible project NPV for the pit design considered. Operating cost of production may be cited as a benchmark when comparing gold mining projects. Figure 10.7 shows the geological uncertainty and risk integrated to the expected production cost per ounce of gold produced for the deposit considered here and the given mine layout. The analysis of Figure 10.7 shows the cost of production per ounce of gold is most likely to be underestimated from the kriged grade model. However, the small range of outcomes would provide confidence that the cost of production for the project is not likely to exceed A$463/oz gold. This may be very useful quantitative information to have if the company involved in the project is sensitive to production cost variation. The cost per ounce is shown in Figure 10.7 in terms of nested pit shells for comparison with the NPV chart. Note that the range, or spread, of cost per ounce outcomes is fairly constant across all the pit optimisation shells. This shows the cost of production is insensitive to the size of the open pit, and is not significantly improved by increasing the scale of the mining operation. The cost per ounce on a period or annual basis could also be calculated to investigate the cost profile over time.


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Project Net Present Value 25,000,000 Probability The NPV determined from OK has only a 2-4% probability of occuring.

20,000,000

The most likely NPV for this optimum pit is m$16.5, 25% less than the kriged model.

15,000,000 $ A V P

) % 8 = i N (

10,000,000 Pit design and production schedules might typically be founded on pit shell 41, with the highest NPV.

5,000,000 CS Realisations Ordinary Kriging 0 0

5

10

15

20

25

30

35

40

45

50

Optimised Pit Shells

Figure 10.5: Geological uncertainty and risk in NPV of a disseminated gold deposit. Distribution of NPV

+/- 1 std dev.

y t lii b a b ro P

lie t n cer e P th 0 1 S C

11,000,000

13,000,000

V P N e ag er v A S C

15,000,000

el ti n e rce P h t 0 9 S C

1 7 ,8 5 5 ,5 6 1 $ A

17,000,000

19,000,000

0 6 ,2 7 2 ,1 2 2 $ A . K . O

0 0 ,0 7 5 ,6 9 1 $ A IK

21,000,000

23,000,000

25,000,000

NPV A$

Figure 10.6: Distribution of NPV from conditionally simulated realisations and from kriged orebody estimates.

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Chapter 10

Cost er Ounce 600 Cost per Ounce of Gold Produced Kriged Model

500

$440.00

CS Realisations Mean Standard Deviation Minimum Maximum

400 u A z O / $ 300

$436.60 $ 14.50 $397.80 $463.30

ts A o C

200

100

CS Realisations Ordinary Kriging

0 0

5

10

15

20

25

30

35

40

45

Optimised Pit Shell

Figure 10.7: Geological uncertainty in cost of production per ounce of metal produced.

The physical parameters of ore tonnes mined and milled are also major sources of risk, particularly in the early years of a project. The risk of designing and constructing a plant unsuited to the available ore feed will be better understood when the uncertainty in the feed quantity and grade is known. As an example, uncertainty in the mill feed grade to a gold plant was analysed. The analysis shown in Figure 10.8 is the average mill feed grade for the life of the project. Figure 10.8 shows a large range of possible average feed grades for the project, information useful to have when designing a processing plant. The conjugate partner to mill feed grade is the ore tonnage, shown in Figure 10.9. The simulated realisations show consistently lower total ore tonnage, down by an average of 12.5% compared with the kriged model. The variable grade and lower tonnage of mill feed indicated by the simulated realisations suggest that this project will have difficulty in achieving scheduled mill throughput and feed grade for the life of the mine, if based on the kriging model. The lower ore tonnage indicated from the simulations may suggest a design change for the processing plant. In addition to sensitivity analysis on key ‘life of mine’ project parameters, geological uncertainty in the mine production schedule can also be quantified. Consider, for example, the discounted cash flow (DCF) for a mine calculated for production periods of 3 months. What is the variation expected to arise from a production schedule DCF due to geological uncertainty in the ore reserve estimation model? Are there periods of greater risk, and when do such periods occur? Figure 10.10 demonstrates the uncertainty in a quarterly discounted cash flow, in comparison with the single estimate used in Figure 10.8 and Figure 10.9.


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Chapter 10

Mean Feed Grade 2.10

Single kriging estimate is shown in solid black, with 50 grade realisations are shown in white.

2.05 2.00 1.95 u A1.90 t/ g d 1.85 ee F lli M1.80

1.75 1.70 1.65 1.60

Possible Average Mill Feed Grade

Figure 10.8: The average grade of mill feed ore over the life of mine is shown for a series orebody realisations, and for a single estimate .

Mean Feed Tonnage 5,000,000 4,500,000 4,000,000

Single kriging estimate is shown in solid black, with 50 alternative CS realisations are shown in white.

3,500,000 e g a n 3,000,000 n o T d 2,500,000 ee F lli 2,000,000 M

1,500,000 1,000,000 500,000 0

Possible Average Ore Tonnes

Figure 10.9: Average ore tonnages, available to the mill over the life of mine, are shown for a series of CS realisations, and for a single kriging estimate.

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Chapter 10

Discounted Cash Flow (by 3 month periods)

9,000,000 First 2 years of production 7,000,000

A P $ A w o l F sah C

5,000,000 ) % 8 d te 3,000,000 n u isco (d1,000,000

Last year of production : Cash flow highly likely to be negative

0 -1,000,000

Cash Flow: Variation ranges from $m 1.7 to $m 3.7 for this Period

-3,000,000

CS Realisations

{Refer Distribution curve for this period}

OK

-5,000,000 0

2

4

6

8

10

12

14

Production Period (1/4 Year)

Figure 10.10: Distribution of outcomes for discounted cash flow by 3 month production periods. Distribution of Discounted Cash Flows From CS for Period 10 Kriging model : single, precise estimte

CS Realisations : asymmetrical distribution of cash flow outcomes. High probability of negative cash flow

y it il ab b o r P

$(2,000,000)

$(1,500,000)

$(1,000,000)

$(500,000)

$-

$500,000

Discounted Cash Flow

Distributiontoofsingle DCF estimate. outcomes in period 10 of the mine production Figure schedule10.11 and :comparison Figure 10.10 shows that the anticipated cash flow for this minable reserve has a reasonable probability of materialising for accounting periods during the first two years of this project. It is more likely that cash flows will be less than forecasted, but there is a small probability that cash flows would actually exceed expectations during the first two years. The probability of the last year of production achieving the forecasted cash flow is very low; it is much more likely that the discounted cash flow during the last


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year (periods 8 to 12) will in fact be negative. The cash flow distribution in period 10 is shown as a probability density function in Figure 10.11. The variation in DCF outcomes, shown in Figure 10.10, highlights high-risk periods during the life of the project. The distribution of DCF’s for a single period, shown in Figure 10.11, quantifies the risk of negative cash flows in the last stages of this production schedule. Having access to such information prior to mining is a valuable asset when determining such key parameters as the ultimate size of the project, and the risk profile for that life.

10.1.4

The Echo Ba y, Nevada, Exam ple 2

The deposit is a volcanic hosted gold, located in the margin of a collapsed caldera. The mineralisation is fracture controlled. Oxide material is being mined, refractory gold is at depth. The open pit gold deposit is mined at 200,000 tpd on 35ft benches. Mining equipment includes 28 yd3 electric shovels and CAT 992 & 994 loaders, CAT 785 haulage units and CAT 789 trucks. The operation feeds three plants using two separate crushing streams and a run of mine feed. These include an 8,000 tpd mill, an asphalt based, crushed ore, off loadable leach pad. All processes operate with fixed cut-off grades. Typically, statistics and variography is done by rock type and used to simulate the deposit with SGS. Five realisations are run and used as input to Whittle 4D. An additional run is made with the kriged orebody model. Note that Whittle 4D provides some facilities to test for sensitivity (in addition to global grade changes). This may be done by varying the mining dilution factor and the mining recovery factor Example: 110% metal = 0.91 Mining dilution factor * 1.1 Mining recovery factor =100% Tons The mining recovery factor is used to control the quantity of metal in all blocks, and the mining dilution factor corrects tonnages (the total tonnage does not vary from the srcinal value): the percentage adjustment is arbitrary. The study is based on 35 pit shells for the kriged model, the 90% and 110% metal content adjustment of the kriged model, and the five SGS models. Figure 10.12 and Figure 10.13 below are typical Whittle 4D results of discounted cash flow analysis. Here the initial pits have a cash flow variation, BUT ultimately the difference in discounted cash flow is NOT affected by grade variation.

2

Rossi and Van Brunt, 1997

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Chapter 10

Discounted Cash Flow Analysis 450

400

w o l 350 F h s a C d et n u o cs i D

s n io l il

M

300

250

Adjusted Metal 90% Kriged Metal Adjusted Metal 110%

200 0

5

10

15

20

25

30

35

40

Pit Shell Number

Figure 10.12: Whittle 4D results for discounted cash flow analysis Is Figure 10.12 showing an accurate description of the situation? Figure 10.13 shows the results from the kriged model and the five SGS models. Yes, the initial discounted cash flows from the pits will most likely be variable.

Discount ed Cash Flow A nalysis 45 0

40 0

w lo F h s a C d et n u o cs i D

35 0

s n io lli M 30 0

Kriged Metal Sim_1

25 0

Sim_2 Sim_3 Sim_4 Sim_5

20 0 0

5

10

15

20

25

30

35

40

Pit Shell Number

Figure 10.13: Results from the kriged model and the five SGS models.


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Chapter 10

Also, there is a two out of five or 40% chance that the discounted cash flow for the ultimate pit may be significantly less than predicted from in the previous more conventional formulation. What happens to the mill feed release (see Figure 10.14 and Figure 10.15)? Mill Feed Release 9500

9000

y a D 8500 re p d es a el e R l ta o 8000 T

7500


7000 0

5

10

15

20

25

30

35

40

Pit Shell Number

: The Figure short fall10.14 under 8000traditional tpd for theorebody middle model to late predicts years. that a possible mill production

M il l Feed R eleas e

11500

10500

y a D er p d es ea le R l a t o T

9500

8500 Kriged Metal

Sim_1 Sim_2

7500

Sim_3 Sim_4 Sim_5

6500 0

5

10

15

20

25

30

35

40

Pit Shell Number

Figure 10.15: The simulations suggest that stockpiling in the early years however mill feed will very likely be greater than predicted by the traditional approach.

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Chapter 10

What happens to the mill feed grade? The mill feed grade from the kriged model . . . . is marginally conservative as shown by comparing Figure 10.16 with Figure 10.17.

Average Grade of Mill Feed 0.015

0.010

e d a r G d ee F l il

M 0.005


0.000 0

5

10

15

20

25

30

35

40

Pit Shell Number

Figure 10.16: Mill feed grade from kriging. Ave rage Grade of Mil

l Fe ed

0.015

0.010 e d a r G d ee F lli M

0.005 Kriged Metal

Sim_1 Sim_2 Sim_3 Sim_4 Sim_5

0.000 0

5

10

15

20

25

30

35

40

Pit Shell Number

Figure 10.17: Mill feed grade from five SGS models.


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10.1.5

Issues for discussion

Is Whittle 4D a linear transfer function? Is simulation a tool for pit optimisation and scheduling? Do we need to look into more integrated approaches?

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Chapter 10

10.2

Minimum Down-side Risk & Maximum Up-side Potential in Open Pit Mine Design

Minimum Down-side Risk & Maximum Up-side Potential in Open Pit Mine Design

Outline • The case study • Open pit design based on quantified risk • Key project indicators • Decision making criteria

• Effect of the gold price • Conclusions


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Chapter 10

A Low Grade Epithermal Gold Deposit

Grade legend >10.0 >5.00 >1.50 >0.60 >0.35 >0.01

Process • Design a pit for each simulated orebody Procedure

• Quantify risk in each pit design Procedure • Decide based on risk • Meeting mill requirements - ore tonnes • Minimum acceptable return • Other

• Select a pit design Next

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Chapter 10

Simulated ore body mode ls

Project indicators

Pit designs Optimisation

DCF Ore tonnage

1

1

Gold

DCF Ore tonnage

2

Gold

2

DCF Ore tonnage Gold

13 13

Risk analysis


DCF (m$)

24 16 8

1

0

Pit design

CB-1

CB-2

CB-3

Ore (Mt)

3 2

22

1 0 CB-1

CB-2

CB-3

Gold (Mgr)

4 3 2 1

13

0 CB-1

CB-2


CB-3

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Chapter 10


Risk analysis ) $ M ( F C D

1

Pit design

CB-1

CB-2

CB-3

CB-1

CB-2

CB-3

CB-1

CB-2

CB-3

M er O

2

7 )r M( l G

13

Pit Designs in Case Study 1

2

3

4

5

6

7

8

9

10

11

12

13

CB-1 CB-2 CB-3

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Chapter 10

Ore Tonnes 2.5

O

t)

2

(M re

1.5 1 0.5 0 CB-1

CB-2

CB-3

Key Project Indicator: Ore Tonnes Cutback - 1

1.4

1.2 M

1

0.8

0

2

4

6

Pit design

8

10

12

14

Pit designs with probabilities greater than 70% in achieving mill target are considered for the analysis


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Chapter 10

Upside Potential / Downside Risk Uncertainty Risk

Reward

MAR

Average

MAR: Minimum Acceptable Return…

Up-side Potential / Down-side Risk

F C D

Upside Average

Downside Value 1

2

Pit design

Avg ] * probability ) ∑ ([Value − MAR Downside = ∑ ([Value − MAR ] * probability )

Upside = Downside

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Chapter 10

Key Project Indicator: DCF 21

Cutback - 1

) $ 18 m ( F C15 D

MAR

12 9 2

4

12

6

Pit design

MAR = m$12

Pit design

Upside (m$)

Downside (m$)

2

2.3

0.0

4

1.3

-0.079

6

2.4

0.0

12

2.9

0.0


Cutback - 2

) $ m (18.5 F C D 15.0

MAR 11.5

2

4

6

Pit design

MAR = m$14

12 Pit

Upside

Downside

design 2

(m$) 2.4

(m$) -0.08

4

2.1

-0.2

6

2.4

-0.02

12

2.4

-0.2


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Chapter 10


Cutback - 3

) $ m ( 19 F C D14

MAR

9 2

4

6

Pit design

MAR = m$15

12 Pit design

Upside (m$)

Downside (m$)

2

1.8

-.0.20

4

1.6

-0.51

6

1.9

-0.28

12

1.2

-0.96

Decision Making Upside Potential (m$) Downside Potential (m$)

Pit Design

CB-1

CB-2

CB-3

2.3

2.41

1.8

0.0

1.3

2.1

1.6

2.4

2.43

2.9

2.40

CB-1

CB-2

CB-3

-0.079

-0.20

-0.78

-0.15

-0.51

1.9

0.0

-0.022

-0.28

1.2

0.0

-0.16

2

4

6

-0.96

12

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Decision Making Upside Potential (m$) Downside Potential (m$)

Pit Design

CB-1

CB-2

CB-3

2.3

2.41

1.8

0.0

1.3

2.1

1.6

2.4

2.43

2.9

2.40

CB-1

CB-2

CB-3

-0.079

-0.20

-0.78

-0.15

-0.51

1.9

0.0

-0.022

-0.28

1.2

0.0

-0.16

2

4

6

-0.96

12

Outline • Case study • Traditional pit design and risk assessment • Open pit design based on quantified risk • Key project indicators • Decision making criteria

• Effect of the gold price


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Gold Price Effect

$600/oz

$650/oz

$700/oz

Gold Price Effect

32

25 m F D

18

11

MAR

550

600

650

700

750

Gold price ($/Oz) Minimum Acceptable Return: m$15

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Gold Price Effect Metal Price

Pit Design

Upside Potential (m$) $600/oz

2

$650/oz

$700/oz

1.8

0.7

5.4

1.9

4.0

9.7

6

Z

North

Section 10000-E

Conclusions • Geology (grade) uncertainty is critical in open pit design

• Quantification of uncertainty can be used to enhance open pit design

• Decision making criteria are inherently linked to risk quantification

• Quantifying upside potential and downside risk are important for project evaluation


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10.3 10.3.1

Managing Risk and Waste Mining in Lon Production Scheduling of Open Pit Mines

g -Term 3

Abstract

Open pit mine design and production scheduling deals with the quest for the most profitable mining sequence over the life of a mine. The dynamics of mining ore and waste, and spatial grade uncertainty make predictions of the optimal mining sequence a challenging task. A new optimization approach to production scheduling, based on effective management of waste mining and orebody grade uncertainty, is presented in this paper. The approach considers an economic model, mining specifics including production equipment, and the integration of multiple equally possible representations of an orebody. The utilisation of grade uncertainty and optimal mining rates leads to production schedules that meet targets whilst being risk resilient and generating substantial improvements in project net present value. A case study from a large gold mine demonstrates the approach.

10.3.2

Introduction

Valuation and related decision-making in surface mining projects require the assessment and management of orebody risk in the generation of a pit design and longterm production schedule. As the most profitable mining sequence over the life of a mine determines both the economic outcome of a project and the technical plan to be followed from mine development to mine closure, the effect of orebody risk on performance is critical (Rendu, 2002; Dowd, 1994; Ravenscroft, 1992). Geological risk is a major contributor in not meeting expectations in the early stages of a project (Vallee, 2000), when repayment of development capital is vital, as well as to production shortfalls in later years of operation (Rossi and Parker, 1994). The adverse effects of orebody uncertainty on the traditional optimisation of pit designs and corresponding key project performance indicators are documented in various studies (e.g. Dimitrakopoulos et al, 2002; Farelly, 2002; Dowd, 1997). These past efforts deal with the use of stochastic simulation methods in assessing project risk for a given mine design and mining sequence. They do not, however, address the generation of optimal conditions under uncertainty, long-term production schedules, or operational issues and interactions of ore and waste within the orebody space over the life of mine. New integrated approaches can be developed to effectively deal with orebody uncertainty in production scheduling while maximising cash flows, and may be based on two elements. The first element is the ability to represent orebody uncertainty through the stochastic simulation of multiple, equally probable deposit models. Although the technologies available (e.g. Dimitrakopoulos, 2002), the isuse multiple orebody models for are production scheduling, instead of a single model, notofa trivial exercise. Generally, traditional optimisation formulations are not compatible with stochastic modelling approaches. The second element in dealing with risk is a modified optimisation framework that, while compatible with orebody uncertainty, integrates a variety of mining issues, particularly management of waste, equipment utilisation, mill demand, and technological, financial and environmental constraints. 3

Godoy and Dimitrakopoulos, 2003

262


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This paper presents a novel optimisation approach that is shown to effectively integrate grade uncertainty into the optimisation of long-term production scheduling in open pit mines. The approach is founded upon two key elements: (i) a framework for long-term production scheduling based on the concept of a “stable solution domain”; and (ii) a new scheduling algorithm based on simulated annealing. The approach generates “100% confidence” in the contained ore reserves, given the understanding of the orebody, and minimises deviations from production targets to acceptable ranges. Related to the approach presented herein are concepts in Tan and Ramani (1992) and Rzhenevisky (1968), where open pit production scheduling is seen as the determination of a sequence of depletion schedules in which at least two types of products, ore and waste, are removed to meet the mine’s demand. The optimal schedule maximises the net present value (NPV) of the project subject to constraints, including: (i) feasible combinations of ore and waste production (stripping ratio); and (ii) ore production rates that meet mill feed requirements. At the same time, an optimal schedule defers waste mining as long as possible and, in doing so, considers the mining equipment and capacity available. This approach is limited in that no physical mining schedule is produced and issues of uncertainty are not addressed, as they are in the approach presented herein. Godoy (2003) provides a detailed review of past work and new applications in the context of the nested Lerchs-Grossman algorithm and nested pits that can be mined independently (Whittle and Rozman, 1991; Hustrulid and Kuchta, 1995). It should be noted that an optimal long-term mine production schedule can be found within a “domain of feasible solutions”, that is, within combinations of ore and waste that can be produced from a specific orebody. The nested pit optimisation framework, mentioned above, establishes this domain, based on two extreme cases of mining waste deferment. The worst mining case (Figure 10.18(a)) where a bench is mined out before starting the next, is producing the maximum quantity of waste from the pit needed to recover a certain amount of ore (highest stripping ratio). This schedule shows a poor NPV as the expense for mining waste at the periphery of the pit is incurred early, and thus discounted little, whereas the income from mining ore at the bottom of the pit is delayed for later periods, and thus is heavily discounted. The opposite happens in the best mining case (Figure 10.18(b)), corresponding to the sequential mining of the independent nested pits, where mining occurs in each successive bench of the smallest pit and then each successive bench of the next pit and so on. This schedule has the lowest stripping ratio and highest NPV, whilst providing the necessary working room and safety conditions for mining operations. An intermediate mining schedule in Figure 10.18(c) shows mining of the first bench leading to the commencement of mining in the next cutback. In searching for an optimal ore production and waste removal schedule, a feasible solution domain can be represented in a cumulative graph, bounded by the curves of the best and worst mining cases. The solution domain accounts for all physically possible combinations of stripping ratios. Figure 10.19 shows the solution domain of the gold deposit discussed in a subsequent section. Any non-decreasing curve within the solution domain characterises a production schedule having different combinations of stripping ratios over the life of the mine, and reflects possible spatial arrangements for working zones. There are many feasible schedules of waste removal given a single ore demand scenario from the mill. An optimal schedule in terms of NPV will tend to follow the Applied Risk Assessment for Ore Reserves and Mine Planning: Conditional simulation for the mining industry

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curve representing the minimal quantity of waste (Tan and Ramani, 1992; Godoy, 2003), that is, where mining waste is deferred as long as possible. In the following sections, a risk-based approach to life-of-mine production scheduling is presented. It includes (i) the determination of optimum mining rates for the life of mine, whilst considering ore production, stripping ratios, investment in equipment purchase, and operational costs; and (ii) the generation of a detailed mining sequence from the previously determined mining rates, focusing on spatial evolution of mining sequences and equipment utilisation. The approach is then demonstrated through an application at the Fimiston Gold Mine (Superpit), Western Australia. The results of the new approach are compared with traditional production scheduling. Finally, the benefits of the approach are presented in the conclusions.

(a)

(b)

(c) Figure 10.18: Schematic representation of three mining schedule configurations: (a) worst case mining schedule; (b) best case mining schedule; and (c) intermediate mining schedule.

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Feasible Dom ain of Ore Production and Waste rem oval Mt

e st a W fo y itt n a u Q e ivt la u m u C

Best Case Schedule Mt

Worst Case Schedule

Mt

Mt

Mt

Mt

Mt Mt

Mt

Mt

Mt

Mt

Mt

Mt

Mt

Mt

Mt

Mt

Cumulative Quantity of Ore

Figure 10.19: Solution domain of ore production and waste removal.

10.3.3

A new risk-based approach to production scheduling

The risk-based approach presented in this section differs conceptually from traditional approaches in many aspects. For a start, all traditional approaches use a single estimated orebody model to produce a mining schedule. Such an estimated orebody model is based on imperfect geological knowledge, so estimation errors are propagated to the various mining processes involved in the optimisation, and related geological uncertainty is not included or assessed. The approach presented here quantifies geological uncertainty using a series of stochastically simulated, equally probable models of the orebody. Subsequently, a multi-stage optimisation process utilises these models to produce a risk-resilient, long-term mining schedule. The multi-stage process starts by generating a series of mining schedules, each corresponding to one of the simulated spatial distributions of orebody grades representing the possible orebody. These mining sequences are optimised within their common feasible solution domain, termed “stable solution domain” (SSD), and post-processed to provide a single mining sequence. This optimisation process has four stages, as shown in Figure 10.20 and presented below: 1.

Derive a solution domain of ore production and waste removal “stable” to all simulated models of the distribution of the grades of the deposit.

2.

Determine the optimal production schedule of waste removal and formation of mining capacity within the stable solution domain from Stage 1. This generates optimal mining rates for the life of mine, given the equipment considered.

3.

For each one of the available simulated orebody models, generate a physical mining sequence constrained to the mining rates from, and equipment

4.

selection in, Stage 2. Combine the mining sequences generated in Stage 3 to produce a single mining sequence that minimises the chances of deviating from production targets.

These four stages are discussed in detail below.


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Stage 1: Derivation of the stable solution domain (SSD) The derivation of the stable solution domain starts from a design with ultimate pit limits, a sequence of cutbacks and a set of stochastically simulated orebody models. The SSD is generated from the cumulative graphs of ore production and waste removal from each one of the simulated orebody models and the ultimate pit limits and cutbacks available. Figure 10.21 presents cumulative graphs and solution domains for a series of simulated orebody models and grade distributions in an open pit gold mine discussed in a subsequent section. The common part of all the cumulative ore and waste graphs forms the SSD. This new domain represents a solution domain that, according to the orebody grade uncertainty quantification from the set of the available stochastic simulations, provides 100% confidence in the contained reserves. Note that this procedure is general and independent of the objectives driving the optimization of production scheduling.

MULTI-STAGE OPTIMISATION ALGORITM

S1 S2 SN

S1 S2

CALCULATE STABLE SOLUTION DOMAIN (SSD)

SCHEDULE OPTIMISATION (LP MODEL)

SSD

SCHEDULER TO PRODUCE MINING SEQUENCE

SN MINING RATES

S1 S2 SN Seq1 Seq2

COMBINATORIAL OPTIMISATION

E L U D E H C S

SeqN

Figure 10.20: Schematic representation of the process developed for optimising longterm production scheduling. (S stands for simulated orebody model and Seq. for mining sequence).

Stage 2: Schedule optimisation Given the SSD from the previous stage, a linear programming (LP) optimisation formulation results in a schedule for ore production and waste removal, and the formation of optimal mining capacities within the SSD. The LP formulation discussed next, is based on the following objective function: n

∑ d (1 − R ) ⎡⎢⎣(−S

MAX

i

ma C− +(i i )γ

i

i =1

n

+ Cmp C pp)( )Ct i

i

−1 α i

pi

⎤ M − p ⎥⎦ i

n

− ∑ di Cms (α s )−1 M s − ∑ d iC w Wi − i

i =1 K

Z

n

i

i

i

i =1 K

Z

n

kzi di h kzi NC d iu kzi DC kzi − ∑∑∑ === − k == 1= z 1 i 1 k 1 z 1 i 1 ∑∑∑

(1)

where, i=1,...,n denotes time periods considered. Definitions of constants and variables in the objective function and constraints are given in Table 10.1 and Table 10.2. The objective function (1) corresponds to the schedule’s economic outcome on the basis of discounted cash flow analysis, before taxation and without treatment of related depreciation and depletion allowances. The objective function represents a mining

266


Chapter 10

operation where the secondary ore is only stockpiled. The main variables of the optimization model are the time-related primary ore metal, secondary ore metal and waste. While the variables corresponding to the waste quantities allow for the ore-waste relation to be optimised over time, the metal variables allow for the metal quantities to be optimized. The metal optimisation accounts for the ore quality at different parts of the orebody. The remaining variables of the optimisation model are the added capacity and decreased capacity of each type and model of the mine equipment, which deals with the stabilisation of the mining rate over time as a function of capacity. Mining rates are also stabilised through the economic parameters of unit purchase and ownership costs of each type and model of equipment. The unit purchase cost is determined by the value of the equipment divided by its production capacity. The unit ownership cost is determined by the ownership cost of the equipment divided by the production capacity. Thus, the penalty for decreased capacity is defined as being equivalent to the ownership cost, which reflects a penalty for having idle equipment. In this context, the stabilization of the mining rate over time is determined as a search for the balance between the purchase and ownership costs of the production capacity, and represents a direct incorporation of the capital investments in the optimization. As noted earlier, and although developed in a different context, the LP formulation relates conceptually to that in Tan and Ramani (1992). It is also analyzed in detail in Godoy (2003). Figure 10.22 displays the SSD and a typical solution produced by the LP model. This optimum solution corresponds to a production schedule that maximises the NPV within the SSD. This is unique in the sense that the geological uncertainty has been effectively integrated into the optimisation process.

Stable S olution Domain of Ore Production and Waste Removal Mt

a tes a W f o y ti t n a u Q e v it la u m u C

Mt Mt Mt

SSD

Mt Mt Mt Mt Mt

Mt

Mt

Mt

Mt

Mt

Mt

Mt

Mt

Mt


Figure 10.21: A stable solution domain (SSD) derived from six simulated orebody models.


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Chapter 10

Variab le

Definition

M pi

Primary and secondary ore metal

M si

Secondary ore metal

Wi

Waste quantity to be removed

NCkzi

Added capacity for k-th type, z-th model of production equipment

Decreased capacity for k-th type, zth model of production equipment Table 10.1: LP model variables in objective function (1). DCkzi

Consta nt

Definition Number of time periods to be considered Number of types of mining equipments Number of models of production equipment Discount factor di=(1+r)-i, with r = interest rate

n

Z J di

Si Cmpi , Cmsi C ppi , Cspi Cmwi

Selling price of metal Unit mining costs of primary and secondary ore Unit processing costs; primary & secondary ore Unit mining cost of waste removal

C

Marketing cost per unit payable metal

R

Royalty as % of the net revenue

ma i

α

pi

γi

, αs

i

Primary and secondary ore metal grade Total recovery of the payable metal

Time costs for operating support services Capacity limit of k-th type and j-th max Ckj model of production equipment Unit purchase cost of k-th type, z-th h kzi model of mine equipment Unit ownership cost of k-th type, zu kzi th model of mine equipment Table 10.2: LP model constants in objective function (1). Cti

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Chapter 10

Optimimun Solution Constrained to the SSD Mt Optimum Schedule

0 tes Mt a W f Mt o y itt n Mt a u Q e Mt v ti la u Mt m u C

Best Case Schedule Worst Case Schedule

SSD

Mt

Mt Mt

Mt

Mt

Mt

Mt

Mt

Mt

Mt


Mt

Mt

Figure 10.22: Optimal solution (small dark dots) obtained inside the SSD, derived from a series of simulated resource models.

Stage 3: Mining sequencing The LP in Stage 2 generates a set of optimal mining rates. The third stage uses these mining rates to produce a series of physical production schedules that describe the detailed spatial evolution of the working zones in the pit over the life of the mine. The sequencing needs to obey slope constraints, consider equipment utilisation and meet mill requirements, while matching the mining rates previously derived. Any scheduling algorithm that accommodates these criteria may be used. This stage generates multiple mining sequences, one for each simulated grade model representing the orebody. The alternative mining sequences present two characteristics that allow the derivation of a single mining sequence. These characteristics are that all schedules are (i) technically feasible solutions that maximise the project’s NPV within a common solution domain; and (ii) based on distinct but equally probable models of the spatial distribution of grades within the deposit.

Stage 4: Combinatorial optimisation The fourth stage considers the production schedules generated in Stage 3 and derives a single mining sequence. A combinatorial optimisation algorithm based on simulated annealing has been developed and is outlined here. The basic idea in simulated annealing is to continuously perturb a sub-optimal configuration until it matches some pre-specified characteristics coded into an objective function (Kirkpatrick et al., 1983). Each perturbation is accepted or not depending on whether it carries the objective function value towards a predefined minimum. To avoid local minima, some unfavourable perturbations maybe accepted based on a probability distribution (Metropolis et al., 1953). The annealing formulation first selects an initial mining sequence, where blocks with maximum probability (e.g. 95%) of belonging to a given mining period are frozen to that period and not considered further in the combinatorial optimization process. Block probabilities are calculated from the results of Stage 3. The initial sequence is perturbed by random swapping of (non-frozen) blocks between the candidate mining periods. Favourable perturbations lower the objective function and are accepted; unfavourable


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perturbations are accepted using an exponential probability distribution. Annealing stops when perturbations no longer lower the objective function or when a specified minimum objective function value is reached. The objective function is a measure of the difference between the desired characteristics and those of a candidate mining sequence. Consider, for example, the objective of meeting a series of optimal mining rates derived in Stage 2, i.e., the prescription of ore production and waste removal for the life of the mine. If a mining sequence achieves that objective for all the equally probable simulated orebody models, there is 100% chance that the production targets will be met, given the knowledge of the orebody as represented in the simulations. An objective function is built to measure the average deviation from the production targets for a given mining sequence over a series of simulated orebody grade models. The objective function is defined as the sum of components representing mining periods, O=

N

∑O

(2)

n

n =1

where On, n=1,…,N are component objective functions and N is the total number of production schedule periods. For each n component (period), the objective function measures the average deviation of ore and waste production θn ( s) and ωn ( s) of the *

*

perturbed mining sequence from the target productions

θ n ( s)

and

ωn ( s)

over the S

simulated grade models, with s=1,…,S: On

=

1

S

∑ θ−( sθ+) *

n

S

n

1 S ( s )−ω ω

∑

S

s =1

*

n

(s)

n

(s )

(3)

s =1

The decision to accept or reject a perturbation is based on the change to the objective function, N

ΔO = ∑ ΔOn n =1

(4)

The resulting sequence meets the production target for each period with minimum chance of deviation. That is, this mining sequence will achieve the production targets, within the prescribed mining rates, given any of the simulated orebodies. None of the individual mining sequences from Step 3 will meet these requirements. Note that the objective function can be modified to include other production targets, such as head grade, metal quantities and blending requirements. An important aspect of the procedure is the mechanism of swapping blocks. To ensure the final solution avoids physically inaccessible blocks in any period, the perturbation mechanism must be set to recognise the spatial evolution of the mining sequence. To achieve this, the perturbation mechanism is defined so as to restrict the candidate periods, of any given block, to only those having physical access to the block without violating slope constraints (Godoy, 2003).

10.3.4

270

Application in a large open pit gold mine


Chapter 10

The practical aspects of the proposed method were tested in a case study using data from the Fimiston open pit (Superpit) in Western Australia. Fimiston is operated by Kalgoorlie Consolidated Gold Mines. The gold deposit is an intensely mineralised shear system developed largely within the so-called Golden Mile dolerite. The mineralisation is localised in mainly steeply dipping, NNW to NW striking lodes, consisting of a highgrade lode shear zone and a lower grade alteration halo. Gold lodes can be up to 1,800 meters long, have vertical extents of 1,200 metres and be up to 10 metres wide. The Fimiston pit is a conventional open pit, truck-and-loader, operation. It has a mining rate of approximately 85 million tonnes per year, making it the single largest open pit operation in Australia, on a tonnes per year basis. Of this, some 12 million tonnes of ore are produced and milled through the Fimiston mill. The mill currently consists of a grind-float circuit for processing refractory sulphide ore, electrowinning, smelting and then pouring of gold bullion. The orebody block model used in this application included 648 individual mineralised lodes discretised into 321,937 ore blocks. Block grades were simulated 20 times using the direct block sequential simulation method (Godoy, 2003). For scheduling, all models were reblocked to 20x20x20 metre blocks. It is important to note that the determination of the ultimate pit limits and cutbacks is outside the scope of this application. The risk-based schedule developed was based on predefined ultimate pit limits and sequence of cutbacks, which were derived using the traditional block model of the deposit and the nested pit implementation of the LerchsGrossman pit optimization algorithm (Lerchs and Grossman, 1965; Whittle, 1999). The application of the proposed method started with Stage 1 and the derivation of the SSD (Figure 10.21), followed by the optimisation of the production schedule in terms of ore production and waste removal (Figure 10.22). The schedule of ore production was identified with the mill demand over 15 production periods and is shown by the dotted line in Figure 10.23. It is important to note that the fluctuations in ore production do not indicate a variable mill production rate. The mill production rate is constant over the life of the mine. Periods characterised by a reduction in ore demand only indicate input of ore from other sources, such as underground operations. The LP optimisation model in Stage 2 produced the optimal formation of mining capacity as a combination of Komatsu PC8000 face shovels, CAT994 loaders, CAT793C trucks and nine pieces of support equipment including dozers, graders and water carts. It has been assumed that the purchase of the starting fleet was carried out at the first production period and, therefore, the respective capital costs were charged to the first year. The replacement costs were charged to the first year after the end of the equipment life. The ore production target and the optimal formation of mining capacity, as produced by the LP model, are presented in Figure 10.23. The increased mining capacity in periods 9 to 14 clearly shows the deferment of waste mining, which, as noted earlier, is a characteristic of the optimisation model. Stage 3 of the proposed approach is the mining sequencing, where the Stage 2 prescription of ore and waste production and equipment selection forming the mining capacity are used to generate the mining sequences. The Milawa algorithm (Whittle, 1999) was used in this case study to generate one sequence for each of the 20 simulated grade models of the Fimiston loads. In the final stage, the combinatorial optimisation Applied Risk Assessment for Ore Reserves and Mine Planning: Conditional simulation for the mining industry

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algorithm was used to combine the multiple mining sequences. One of the mining sequences was randomly selected as the starting mining sequence for annealing. Figure 10.24 shows the ore and waste component objective functions versus the number of accepted perturbations. The optimisation stopped after 202,669 perturbations, with 8716 being accepted, when reached the maximum and no change in the objective function. Figure 10.25 and Figure 10.26 illustrate the evolution of the component objective function for periods 4 and 7, respectively. The figures show the percentage deviation from target ore tonnages plotted against the number of accepted perturbations for a set of simulated models. Figure 10.25 shows that the swapping of blocks between different periods causes an increase in the values of the component objective functions related to period 4 for up to the first 1400 perturbations. For the same perturbations, the component objective functions related to period 7 (Figure 10.26) show the opposite behaviour. This is because the decision rule of whether to accept or reject the perturbations is based on a global average over all production periods. In fact, as shown by the global component objective function of ore production (bottom curve in Figure 10.24), the region of up to 1400 perturbations presents the steepest descent of the optimisation process. What is achieved here is a swapping of volumes between different production periods, to distribute regions of high-grade uncertainty among production periods where their negative impact to ore production is minimised. Note that none of the individual mining sequences from the 20 simulated orebodies minimises the effect of grade uncertainty, meets production requirements or maximizes NPV. The effectiveness of the proposed method is demonstrated in Figure 10.27, which shows the risk profile in ore production for the final mining sequence produced by the riskbased approach in this study. The bars indicate average expected percent deviation from the target ore production. The largest deviations are in periods 2, 5 and 8, and are respectively -3%, +3.5% and +2.7%. The magnitude of these deviations is considered very small and is easily managed by re-handling ore from alternative sources for the periods presenting a shortfall. Risk profiles for any schedule are generated from the comparison of the schedule with each of the simulated orebodies. Figure 10.28 compares the proposed approach to the conventional approach, using NPV. Grade risk analysis on the base-case life-of-mine schedule from the traditional scheduling approach, where grade uncertainty is not taken into account, shows that forecast NPV will not be reached. Similarly, Godoy (2003) shows deviations for various production periods in the order of 13%. Compared to the base case schedule, the approach in this study shows an NPV increase of about 28%, illustrating the benefits of both managing waste mining and integrating orebody uncertainty in production scheduling.

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Formation of Mining Capacity and Ore Production Mt

Mt

Mt

Mt

Mt

Mt

Mt

Mt

yt cia Mt p a C Mt gn i Mt in

Mt Mt Mt

MMt

n o tic u od r P er O

Mt

Mt

Mt

Mt

PC8000

FEL994

Mt

OreProduction

Mt

Mt 1

2

3

4

5

6

7

8

9

10 1 1 1 2 1 3 1 4 1 5

Producti on Pe riod

Formation capacity, produced by theperiods. LP optimisation Figure 10.23 constrained to:the SSD, andof oremining production target as over 15 production Com ponent Objective Functions versus Attem pted Swaps 0.9 Obj_Waste

0.8

e 0.7 u al V 0.6 n oi tc 0.5 n u F e 0.4 ivt ecj 0.3 b O 0.2

Obj_Ore

0.1 0 10

1010

2010

3010

4010

5010

6010

7010

8010

Number of Attempted Swaps

Figure 10.24: Evolution of component objective functions: deviations in total ore (top line) and waste (bottom line) quantities versus attempted number of swaps. Com ponent Objective Function - Ore Production in Period 4 8% S1 S2 S3 S4 S5 S6 S7

6%

a ) 4% % ( te gr 2% a T m 0% or f n -2% iot iav -4% e D

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

-6% -8%

Number of Pe rturbations

Figure 10.25: Evolution of component objective functions: deviations of ore production in period 4 for seven simulated models. Com ponent Objective Function - Ore Production in Period 7 10% S1 S2 S3 S4 S5 S6 S7

8%

a ) 6% (% etg 4% r aT m or f n oi t ia ev D

2% 0% 0

1000

2000

3000

4000

5000

6000

7000

8000

9000

-2% -4% -6% -8%

Number of Perturbations

Figure 10.26: Evolution of component objective functions: deviations of ore production in period 7 for seven simulated models.


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10.3.5

Conclusions

A new risk-based, multi-stage optimisation process for long-term production scheduling has been presented in this paper. The process integrates orebody uncertainty, waste management, economic and mining considerations to generate a prescription of optimal mining rates, aiming to maximise a project’s NPV. The subsequent utilisation of grade uncertainty and optimal mining rates leads, through combinatorial optimisation, to lifeof-mine production schedules that meet required targets, whilst being risk resilient and substantially improving project NPV. A case study at the Fimiston open pit, Western Australia, shows how the approach capitalizes on mining waste deferment and quantified grade uncertainty to provide a risk-resilient, life-of-mine schedule, and simultaneously increase asset value. Key elements of the approach are the assessment of the inherent source of orebody uncertainty and the ability to drive the mining sequence through zones where the risk of not achieving the target ore production is minimised. Comparison of results with those of the traditional scheduling practices shows the potential to considerably improve the valuation and forecasts for life-of-mine schedules.

Acknowledgements The work in this paper was funded from the Australian Research Council grant #C89804477 to R. Dimitrakopoulos, “General optimization and uncertainty assessment of open pit design and production scheduling”. Support from Kalgoorlie Consolidated Gold Mines Pty Ltd, WMC Resources Ltd and Whittle Programming Pty Ltd is gratefully acknowledged. Thanks to P. de Vries, K. Karunaratna, W. Li and C. Reardon, KCGM, for facilitating research, providing data and comments. Uncertainty in Ore Production - Final Schedule Mt

70%

Mt

60%

Mt

50%

n iot c Mt u d ro P Mt re O

40% Avrg. Deviation Target Ore Production Maximum Ore Expected Ore Minimum Ore

Mt

3.0%

1.2% 0.7%

3.5%

20% 10%

Mt

0.5%

30%

) % ( n oi t iav e D eg rae v A

1.9%

0.2%

2.7% 1.6%

0.0% 0.6% 0.4% 0.1% 0.0% 0.7%

0%

Mt 1

2

3

4

5

6

7

8

9

10 1 1 1 2 1 3 1 4 1 5

Period

Figure 10.27: Risk profile for ore production in the final schedule.

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Chapter 10

Uncertainty in NPV - Base Case x Risk-Based Approach $M

28% + Risk-based approach

$M

Traditional approach

$M

V P N

$M $M $M

Risk analysis on traditional approach

$M $M 1

2

3

4

5

6

7

8

9

10 1 1 1 2 1 3 1 4 1 5 1 6 1 7

Period

Figure 10.28: Risk profile on cumulative NPV for the schedule produced by the riskbased approach (top three lines), cumulative NPV as forecast by the base-case schedule (single middle line) and risk profile on cumulative NPV on the schedule developed using the traditional approach (bottom three lines). References Dimitrakopoulos, R. 1998. Conditional simulation algorithms for modellingorebody uncertainty in open pit optimization, International Journal of Surface Mining, Reclamation and Environment, 12: 173-179. Dimitrakopoulos, R., Farrelly, C. and Godoy, M.C., 2002. Moving forward from traditional optimization: Grade uncertainty and risk effects in open pit mine design. Transactions of the IMM, Section A Mining Industry, 111: A82-A89. Dimitrakopoulos, R., 2002. Orebody uncertainty, risk assessment and profitability in recoverable reserves, ore selection, and mine planning. Short course notes, SME Annual Meeting & Exhibit, Phoenix, Az, February, 22-24,355p. Dowd, P.A. 1994. Risk assessment in reserve estimation andopen pit planning, Transcript of the IMM, Section A: Minerals Industry,103: A148-A154. Dowd, PA, 1997. Risk in minerals projects: Analysis, perception and management. Transcript of the IMM, Section A: MineralsIndustry, 106: A9-18 Farrelly, C.T. 2002. Risk Quantification in Ore ReserveEstimation and Open Pit Mine Planning. MSc thesis, The University of Queensland, Brisbane, 510p. Godoy, M.C. 2003. The Effective Management of Geological Riskin Long-term Production Scheduling of Open Pit Mines. PhD thesis, The University of Queensland, Brisbane, 256p. Hustrulid, W. and Kuchta, M. 1995. Open Pit MiningPlanning and Design. A.A. Balkema, Rotterdam, 636p. Kirkpatrick, S., Gelatt, C.D. and Vecchi, M.P., 1983. Optimization by simulated annealing. Science, 220: 671-680. Lerchs, H. and Grossman, L., 1965. Optimum design of open-pit mines. Trans. CIM, LXVII:17-24.


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Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H. andTeller, E. 1953. Equations of state calculations by fast computing machines, Journal of Chemical Physics, 21:1087-1092. Ravenscroft , P.J., 1992. Risk analysis for mine scheduling by conditional simulation. Transcript of the IMM, Section A: MineralsIndustry, A104-A108. Rendu, J-M., 2002. Geostatistical simulations for riskassessment and decision making: The mining industry perspective. International Journal of Surface Mining Reclamation and Environment, 16:122-133. Rossi, M.E. and Parker, H.M. 1994. Estimating recoverable reserves:Is it hopeless? in Dimitrakopoulos, R. (Ed.) Geostatistics for the Next Century,Kluwer Academic Publishers, 259-276. Rzhenevisky, V.V. 1968. Open Pit Mining. Nedra Publications, Leningrad, USSR (in Russian), 312p. Tan, S. and Ramani, R.V. 1992. Optimization models for scheduling oreand waste production in open pit mines. 23rd APCOM Symposium, SME-AIME, Littleton, 781791. Vallee, M., 2000. Mineral resource + engineering, economic and legal feasibility = ore reserve. CIM Bulletin, 90:53-61. Whittle, J. and Rozman, L. 1991. Open pit designin 90's. Proceedings Mining Industry Optimization Conference, AusIMM, Sydney. Whittle, J. 1999. A decade of open pit mine planning and optimization – The craft of turning algorithms into packages. 28th APCOM Symposium, SME-AIME, Golden, 15-24.

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10.4 10.4.1

Cho osi ng a Mi nin g Met hod

4

General

The deposit used here represents the sulphide zone of a porphyry-copper deposit, and it is a single, homogenous zone. The dimensions of the zone are 450m by 450m and consist of four levels of blocks of size 18x18m horizontally and 5m height. The mean grade is 1% Cu, dispersion variances of true block grades of 0.235%2 or a 48% relative standard deviation and a histogram shown in the next figure, and an isotropic variogram with a 70m range. Histogram of true (simulated) and kriged block grades are shown in Figure 10.29; means are at 1% Cu, the dispersion variance from kriging is 0.185% 2

Figure 10.29: Histogram of true (simulated-dotted) and kriged block grades (solid).

The minimum mining units are blocks of 18m by 18m by 5m, containing 4730 tons of ore each. Mining rate is 35,000 tons/day or eight blocks. The production year is 280 days. A mill is planned to treat the daily ore production; the metal recovery is a function of the mill’s capacity to absorb variations in the mill feed grade. Automatic control of the mill’s operation is expensive and although responds to hourly variability it can not easily respond to daily or bi-weekly fluctuations. The regulation of the overall mill feed grade period is two weeks in this example. We predict the quality of the ore to be sent to the mill during the two week period, and we need to ensure that actual variations do not drastically deviate from predictions. To make things simpler we will assume that the grade is the major factor affecting the mill. We will next test the effectiveness of three mining methods. 4

Deraisme, 1997


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10.4.2

Mining methods

Three non-selective methods are considered to meet the required degree of homogenisation of daily grades over the two week period.

Method 1a:

•

The deposit is mined with two 10m high benches;

•

athe shovel extracts two continuous blocks 18x18x10 blocks perall day; shovels can only retract to the nextoffront beforemextracting the ore of the current front - length of 18x25 =450m; all ore is sent directly to the mill - no homogenisation

•

Figure 10.30: Mining fronts for mining methods 1a and 1b.

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Method 1b: Same as method 1a but some ore can be stockpiled and homogenised. The stockpile consists of two piles: rich and poor the total capacity is two days production (70,000 tons or eight large blocks); when a block is sent to the pile there is a possibility it will drastically effect the daily mill feed grade; when a block is put on the rich pile, a block is removed from poor one to meet the 35,000 tons daily tonnage. There is also some flexibility on how may blocks of ore will be extracted and from where, but generally the norm is followed, i.e. two blocks per bench per day

Method 2: The zone is mined by four 5m high benches as shown in Figure 10.31. • A mobile shovel extracts two 18x18x10 m blocks per day; • production is 35,000 tons and it is sent straight to the mill; • the shovel’s mobility allows the selection of blocks to be used to stabilise the daily mean grades over the two weeks period; • like before, all ore in a front must be mined before we can start at the next front; • the total moving distance of the shovel is to be kept at a minimal; • the homogenisation comes from blending the ore coming from four working levels (obviously at a higher cost).

Figure 10.31: Mining fronts for method 2 (5m benches).

10.4.3 Simulation of the mining process 10.8.1 Method 1 is easy to simulate as the path of the big shovels is fixed. Methods 1a and 2 are based on decisions of whether a block should be mined, when, and if a block is to be sent to the mill or to the stockpile. The latter cases require the actual ‘simulation’ of the process by simply visualising the fronts and deciding by considering the past and future mill feed grades. Discussions are based on a kriged model of the blocks of the Cu deposit. Note that the true grades are delivered to the mill not the kriged estimates.


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The blending performed in Method 1b in the sub-piles assumes that the true grade in the pile has mean equal to the true mean of the sub-pile at the time of extraction (1.6% for the rich pile and 0.6% for the poor); the relative standard deviation is fixed to 0.25%2 for the rich and 0.3%2 for the poor. In general, the model of blending considered here is simplistic but sufficient to show the order of magnetite of the influence of the stockpile on the homogenisation of the mill feed grades

10.4.4

Assessing mill feed variability for the mining methods

Figure 10.32 shows the fluctuation of daily mean grades for two months of continuous production.

Fluctuations of the daily mean feed grades; true grades; - - - - estimated grades

______

Figure 10.32: Fluctuation of daily mean grades for two months of continuous production.

Dispersion variances for the three mining methods are shown in Table 10.3. Note that when there are two values, the upper value is the true grade and the lower the estimated.

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Dispersion Variance 1 in 7 days 1 in 15 days 1 in 30 days 1 in 1 year

Method 1a

Method 1b

Method 2

0.0452 0.0362 0.0501 0.0422 0.0525 0.0474 0.0816

0.0011 0.0040 0.0136 0.0052 0.0156 0.0070 0.0405

0.0074 0.0018 0.0082 0.0016 0.0106 0.0038 0.0310

0.0260 0.0249 15 days in 1 year Table 10.3: Dispersion variances for the three mining methods.

10.4.5

0.0227

Comments

There is a considerable variation in daily fluctuations of grades depending on the mining method. The dispersion variance of daily grades in 15 days for method 1a is 6 times greater (!) than method 2. These orders of magnitude of mill feed grades can only be predicted with simulations. Kriging or any estimation method will be irrelevant to the prediction of their variability. The above techniques don’t account for mining, stockpiling, trucking, and so on. The deviation between the variances increases as the time interval increases: The amount and location of the ore extracted during a two week period is almost constant (last line in Table 10.3) over one year; however, methods 1b and 2 are very efficient in stabilising the daily grades within any one or two weeks. The dispersion of the estimates is, as expected, always less than that of the actual grades. For method 2, although the homogenisation is nearly perfect, there is still fluctuation of true grades, because blending is based on smooth estimates. The experimental variances of the daily grades for the three methods are 0.0089%2; 0.0131%2; 0.0061 %2. Method 1b produces dispersions of the same order as those of method 2 at a lower mining cost (10m benches instead of 5m) and would probably be chosen.

10.4.6

Discussion

Could this be done in a different way?


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10.5 10.5.1

Uncert a inty Based Production Scheduling In Open Pit Mining 5 Abstract

Optimisation of long-term production scheduling is important for managing the substantial cash flows inherent in open pit mining ventures. Discrepancies between actual production and planning expectations arise through uncertainty about the orebody, in terms of ore grade, tons and quality. These aspects of uncertainty are integrated in a new optimisation formulation for multi-element production scheduling, which also takes into account risk quantification, equipment access and mobility, and other operational requirements such as blending, mill capacity and mine production capacity. Furthermore, the approach introduces the concept of orebody risk discounting. In a case study of an Australasian nickel-cobalt laterite orebody, this new risk-based approach produced better results than traditional approaches.

10.5.2

Introduction

Production scheduling is a critical mechanism in the planning of surface mining ventures. It deals with the effective management of a mine’s production and cash flows in the order of millions of dollars). Long-term production scheduling is used to maximise the net present value (NPV) of the project and focuses on the sequencing of materials to be mined in space over time, under technical, financial and environmental constraints. The importance of incorporating uncertainty and risk from technical, geological and mining, sources in mine production schedules, particularly the possible in-situ variability of pertinent orebody grade and ore quality characteristics, is well appreciated. Discrepancies from planning expectations to actual production may occur at any stage of mining. For example, Vallee (2000) reports that 60% of surveyed mines had an average rate of production less than 70% of designed capacity in the early years. Others (eg. Rossi and Parker, 1994) report shortfalls against predictions of mine production in later stages of production, mostly attributed to orebody uncertainty. The detrimental effects of grade uncertainty and in-situ variability in optimising open pit mine design are shown in recent studies. Dimitrakopoulos et al. (2002) show the substantial conceptual and economic differences of risk-based frameworks. Dowd (1997) proposes a framework for risk integration in surface mining projects. Godoy and Dimitrakopoulos (2003) present a new approach for risk-inclusive cutback designs, which yield substantial NPV increases. Ravenscroft (1992) discusses risk analysis in mine production scheduling, where the use of stochastically simulated orebodies shows the impact of grade uncertainty on production scheduling. He concludes that conventional mathematical programming models cannot accommodate quantified risk, thus there is a need for a new generation of scheduling formulations to overcome infeasible or unrealistic scheduling and account for production risk. Smith and Dimitrakopoulos (1999) show additional examples using mixed integer programming to verify the above conclusion in the context of short-term planning. Kumral and Dowd (2001) use stochastic simulations and optimisation in short-term planning. 5

Dimitrakopoulos and Ramazan, 2003

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Past efforts to deal with uncertainty attempt to sequentially link stochastic orebody models with conventional optimisation formulations, with the exception of Godoy and Dimitrakopoulos (2003). This sequential process is inefficient and, although it assesses risk in a schedule, it does not produce optimal scheduling solutions in the presence of uncertainty. In addition, these efforts do not consider multi-element deposits with complex ore quality constraints, such as nickel laterites, iron ore or magnesium deposits. Furthermore, dealing with orebody uncertainty and in-situ variability accentuates the need to consider issues of equipment access and mobility on the related ‘stochastic’ optimisation formulations. In the above context, this paper presents a new, risk-based production scheduling formulation for complex, multi-element deposits. The formulation is based on expected block grades and probabilities of grades being above required cut-offs, both values being derived from jointly simulated ore deposit models (Dimitrakopoulos, 2002). Expected block grades and probabilities are integrated with equipment constraints and the practical feasibility of mining sequencing in a linear programming model. The later model typically considers homogenisation and blending, mill and mining capacities and performs multi-period optimisation. A key effect of such a probabilistic approach is that the more certain areas of the deposit are mined in earlier production periods, leaving uncertain areas for later periods, when additional information usually becomes available. The probabilistic approach followed in this paper introduces a technical link to the new concept of “risk discounting” that explicitly integrates orebody uncertainty to production scheduling and, inevitably, project valuation. In the following sections, the new production scheduling formulation under conditions of orebody uncertainty is first presented and combined with equipment constraints, to generate practical scheduling patterns. Subsequently, the formulation is applied to a nickel-cobalt laterite deposit, elucidating the practical aspects of the formulation. Next, the practical differences between this approach and the traditional scheduling approach are discussed. Lastly, the conclusions of this study follow.

10.5.3

Production scheduling under grade uncertainty

The mathematical programming model developed in this section is based on linear programming (LP) and takes into account the geological uncertainty and equipment mobility and access required for scheduling and excavating mining blocks. In this scheduling approach, a probability is assigned to each block to represent the “desirability” of that block being mined in a given period. This probability represents the chances that a block will contain the desired grade and ore quality and quantity, including ore grades above given cut-offs, deleterious elements within required ranges, recovery characteristics, and processibility indexes. The probability is calculated from simulated orebody models representing the mineral deposit (eg. Dimitrakopoulos, 2002). This model can be easily extended to a mixed integer programming (MIP) model (Ramazan, 2001) simply by defining the variables as binary instead of linear, as needed. The model contains an objective function and a set of constraints as follows.


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10.5.3.1 Objective function The objective function formulation is Minimize Z =

pmax ⎡ t

⎛ nblock ⎜ ∑ ⎜ ⎝ i =1

∑ ⎢C1t * Y1t + = 1 ⎢⎣ ⎞⎤

C2 * (Y2it ) + C3 * (Y3it ) ⎟⎟⎥

(1)

⎠⎦⎥

where, pmax is the total time period for scheduling; nblock is total number of blocks in the model; Y1t is the percent deviation from having 100% probability that the material mined in period t would have the desired properties; C 1t is the cost coefficient for the 3 probability deviation in period t, such that C 11>C 12>C 1>…..> C 1pmax . Discussion of C1t t and Y1 is deferred for a subsequent paragraph. In the presence of orebody uncertainty, a number of blocks will have a probability of less than 100%, and thus the schedule will have a deviation from this target probability. This deviation can be seen as the risk of not meeting production targets for the related parameters and has a cost for the objective function. Costs for the objective function are set so that a unit of deviation is more costly in the first period than in the second period, which is higher than the third period, and so on. Thus, the objective function will find the blocks with the highest probabilities for the first period, lower probability blocks for the second period and so on. Coefficient variables C 2, and C 3 are cost coefficient variables for Y2 and Y3 percent deviations from mining targets, relating to the smoothness of the mining operation. More specifically, Figure 10.33 shows mining blocks and two concentric windows that move as the central block i moves. The optimisation model is set to mine block i together with the blocks within the inner (smaller) window. If all the blocks within the inner window can not be mined out, the tonnage of the blocks that can not be mined is a “deviation” referred to as Y2 in percentage, and each percentage costs C 2 for the objective function. The mining blocks within the outer (large) window will be mined, if possible, and Y3 and C3 correspond to the related deviation and cost. The smoothing formulation can ensure minimum mining width for the available equipment access and mobility. If C 2 is set equal to C 3, the objective function will be penalised twice for the deviation of the inner window compared with once for the outer window. This set-up means that when block i is mined it is more desirable to mine it together with the neighbouring blocks (in the inner window) than the blocks farther from it (outer window). But it is even better for smoothness of mining, to also mine the farther blocks too with block i, if feasible.

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i

Figure 10.33: Inner and outer windows around block i are set up to establish a smooth schedule forshow equipment access and a required mining width. (Solid lines show windows; dotted lines blocks). The model in Equation (1) requires suitable cost coefficients (C 1t) for deviations from 100% probability and a smooth schedule, and these are derived through a trial and error approach, as follows. At the start, a low cost penalises the deviation from the smoothness in Eq. (1), to result in a widely spread mining pattern for various periods, while the probability will be expected to be relatively high in the first period. Incremental cost increases over time will lead to a required mining width and suitable equipment access in the schedule. This schedule is considered as optimum for maximising probabilities, given the degree of the smoothness obtained. The objective function in Eq. (1) does not directly maximise net present value (NPV). Rather, it opts to provide a feasible scheduling pattern and ensure a desired grade and quality of the ore produced. The reason is that feasible scheduling patterns and the amount of ore having the desired quality to be sent to the mill need to be priorities, indirectly leading to a practically maximum NPV that is realistic. Otherwise, the generated NPV would only be optimal in the mathematical sense, and not in mining practice. Furthermore, the risk of producing adequate ore having the desired properties is integrated in the process, to maximise the chances of delivering to the mill the amount and quality of ore required during mining operation. Risk minimisation and feasible patterns result in practically maximum NPV.

10.5.3.2

Model constraints

The proposed scheduling optimisation model in Eq. (1) contains a series of constraints. These include probability targets and equipment accessibility and mobility, as well as the more traditional constraints of grade blending mill requirements, mill capacity, upper and lower bounds for ore quality parameters, mining capacity and others that depend on the conditions of the given mine, such as stripping ratio and wall slope. The constraints considered here and used in the subsequent case study are as follows.

10.5.3.3

Probability constraints

nblock

1 t t ( P i − 100.0) * OT i + Y1 * ( )=0 TO

∑

i =1

(2)

where, P i is the probability of block n having a grade within a desired interval, P i≤100. The constant 100.0 is the target probability for the schedule; OTit is the ore tonnage scheduled from block i to be mined in period t; TO is a constant number representing


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total ore tonnage to be scheduled in period t; Y1t is the percent deviation from the probability target at period t. Note that Y1t is penalised at a rate of C 1t in the objective function. In the first period, each unit of Y11 will cost C11 unit, a unit Y12 will cost C12 and a unit Y13 will cost C13 for the objective function. Since the objective function is a minimisation, the blocks with the highest probabilities will be scheduled in the first period to have the minimum possible C 11*Y11 due to the fact that C 11 is larger than C 12 and C 13.

10.5.3.4

Constraints for equipment access and mobility

The two windows discussed in the previous section (Figure 10.33) are used to set up the objective function and constraint formulations. Constraints for the inner window and for mining block i at period t are: nb1

t

t

− ∑ K1 j * OT j + K2 i * OTi − Y2 it ≤ 0 j=1

(3)

where, K1 j = 1/TO j and K2 i = nb1/TO i are the coefficients to convert ore tons to percentage; TOj is the total ore tonnage available in mining block j; nb1 is the total number of blocks within the inner window excluding the central block, which is 8 in the Figure 10.33; Y parameters are deviations from smoothness for this inner window. The constraint formulation for the outer window is similar to the inner window nb2

t

t

− ∑ K1 j * OT j + K2i * OTi − Y3it ≤ 0 j=1

(4)

where, Y3 parameters are the equivalent to Y2 parameters the outer window; nb2 is the total number of blocks within the outer window. In this case, K2 i= nb2/TO i.

10.5.3.5

Grade blending constraints

Upper bound constraints require the average grade of the material sent to the mill to be less than or equal to a certain value, Gr max (

nblock

∑ (Gri - Gr max ) * OTit ) ≤ 0

(5)

i =1

where, Gri is the average gra de of block i. Lower bound constraints require the average grade of the material sent to the mill to be greater than or equal to a certain value, Gr min (

nblock

∑ (Gri - Gr min) * OTit ) ≥ 0

(6)

i =1

where t=1, 2, ……, pmax.

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10.5.3.6

Reserve constraints

The set of reserve constraints is used to require all the available ore tons in a block to be mined. The following formulations are written for each block. pmax

∑ OT = (Tot.Ore) t i

i

(7)

t =1

where i =1, 2, ……, nblock.

10.5.3.7

Processing capacity constraints

Processing is constrained by the maximum production capacity of the plant (PCap max) and the minimum production requirement (PCap min). These upper and lower bounds are necessary to ensure a smooth feed of ore to mill. Upper bound constraints for each period: nblock

∑ OTit ≤ PCapmax

(8)

i =1

Lower bound constraints for each period: nblock

∑ OTit ≥ PCapmin

(9)

i =1

10.5.3.8

Mining capacity constraints

Mining capacity constraints represent the actual available equipment capacity (MCapmax) during each production period, and are nblock

∑ (OTit + WTit ) ≤ MCapmax

(10)

i =1

where WT it is the waste tonnage scheduled from block i to be m ined in period t.

10.5.4

Production scheduling under uncertainty in a Ni-Co laterite deposit

The case study considers a part of a typical laterite nickel deposit in Australasia. The deposit is expected to produce around 30,000 tonnes of nickel metal and 3000 tonnes of cobalt metal per year, with a mine life estimated to exceed 20 years. The operation is expected to recover high purity nickel and cobalt by electrowinning. Important metallurgical issues are the response of the ore to pressure-acid leaching, given the magnesium and aluminium content in the ore being processed, and the forecasting of acid consumption in the mill due to this content. Orebody variability and uncertainty are considered critical in achieving “multi-variable” mine optimisation and production scheduling.


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10.5.4.1 De po si t, da ta , de po si t mo de ls an d co ns tr ai nt s The geology of the orebody shows a layer of waste material on top of limonite and saprolite layers, here combined to a zone (LS), with rocky saprolite (RS) below. Both LS and RS may contain high-grade nickel. For classification of ore and waste, the cut-off grade is set at 0.5%Ni. The deposit is characterised by seven attributes: Ni, Co, Mg and Al grades, volume of percent rock (Vol%R), thickness of LS and thickness of RS. The main mineral considered for profit in this project is nickel. Cobalt is a by product with limited contribution to overall mine cash flows. Magnesium and aluminum are relevant to the acid consumption at the processing plant and have a major influence on processing costs. For the purpose of scheduling, the deposit is represented by 2030 blocks, each 40m x 40m. An orebody model is generated using the technique of joint conditional simulation, as detailed below. The model comprises total tonnage, economic value, % tons of the LS layer, % tons of the RS layer at –2mm, Ni, Co, Mg, Al, volume % rock and % total ore tons (% tons of LS + % tons of RS at –2mm) within eachblock The seven deposit attributes mentioned above are simulated, jointly and conditionally, using a 5m x 5m grid for 35 realisations. The 5mx5m grids are then reblocked to the 40m x 40m block size. Each set of joint simulation models generated is equally likely to be the real deposit, given the available information. The joint conditional simulation of these attributes is based on the so-called simulation with minimum/maximum autocorrelation factors. This is an approach that spatially decorrelates the variables involved to non-correlated factors. The independent factors are individually simulated and back-transformed to the conditional simulations of the correlated deposit attributes that reproduce the cross-correlations and individual correlations of the srcinal variables (Desbarats and Dimitrakopoulos, 2000). Simulated representations of the orebody are used to generate average block grades and probabilities of values of different attributes to be within given ranges of interest and as needed for the scheduling optimisation formulation in the previous section. This study considers that ore material sent to the processing plant during each production period should have an average Ni grade in the range of 1.3±0.1%. The probability of each block having the Ni grade in the desired feed range for the mill is used to minimise the geological risk in mine optimisation, as discussed earlier. The total ore tonnage, total tonnage, total undiscounted economic value, average Ni, Co, Mg and Al grades in the simulation based model (SM) and the probabilities are shown in Table 10.5, together with the corresponding values for a traditional model (TM) discussed in a subsequent section. The lower and upper bound constraints on Ni grade for each period are 1.2% and 1.4%, respectively. Ore production is limited to between 9.5 million tons and 10 million tons per period, because the scheduling model is designed over a period of three yearsand average periodical ore tons is around 9.64 million tons. Overall average Mg and Al are around 4.5% and 0.6%, respectively. Minimum and maximum periodical ranges are selected as 4.0% 5.0% for Mg and 0.6% - 0.7% for Al. Following established practices, the economic value of each block is calculated to include clearance, mining, processing and administration costs, recovery and price for Ni and Co, overburden and suitable densities. The steps followed in this project can be summarised as:

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1. Provide jointly simulated models of the deposit attributes of interest: Ni, Co, Mg, Al, Vol%R, thickness of LS, and thickness ofRS. 2. Assign probabilities to each block for aNi grade between the desired bounds (1.2% and 1.4%) from the jointly simulated models of the deposit in Step 1. 3. Generate the orebody SMby averaging the joint simulations in each scheduling block. 4. Schedule the orebody SMusing the formulations inEq (1) through (10). 5. Quantify the risk in theoptimal production schedule using the jointly sim ulated, equally probable deposit models of pertinent attributes in Step 1.

10.5.4.2 Ap pl ic at io n The production scheduling results obtained by applying the optimisation formulation in Eq. (1) and constraints in Eqs. (2) to (9) to the orebody SM of the Ni laterite deposit (Table 10.5) are shown in Figure 10.34 and summarised in Table 10.6. The table includes ore and total tonnages mined, undiscounted economic value (UEV) and NPV, and average grades per scheduling period. In the schedule, UEV is higher for the first period than the second period ($543 million compared with $536.5 million), and is highest in the last period ($561 million). The total UEV is estimated to be around $1640 million. Although the economic value is high in the last year, the probability of meeting the required average grade is low, reflecting high risk in achieving the planned metal production in the last period. At about $503 million, NPV is the highest in the first period, and decreases to about $445 million in the last period. Total project NPV is about $1408 million, which is less than 2% different from the optimisation using an objective function directly maximising NPV.

Ore (106 tons) SM 28.91 TM 28.83

Tonnage (106 tons) 47.45 48.32

UEV ($106)

Ni (%)

1640.56 1.29 1655.20 1.30

Co (%)

Mg (%)

Al (%)

0.090 0.088

4.50 4.70

0.58 0.67

Table 10.5: Average values in the simulation-based model (SM) and traditional model (TM) (Ore represents ore tonnage, Tonnage is total tonnage, UEV is total undiscounted economic value).


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Periods

(years)

Tonnage Ore (106tons) (106tons)

UEV ($106)

NPV ($106)

Ni (%)

Co (%)

Mg (%)

Al (%)

Prob. (%) 88.29

1 2 3

9.53

14.98

543.03

502.81

1.28

0.091

4.23

0.62

9.58

16.25

536.53

459.99

1.30

0.090

4.65

0.53

84.28

9.81

16.22

561.00

445.34

1.31

0.090

4.60

0.59

78.83

Tot/Avg

28.91

47.45

1640.56

1408.14

1.30

0.090

4.50

0.58

83.75

Table 10.6: Summary results of production scheduling using the simulation-based model (SM) (UEV is undiscounted economic value; Prob is probability; Tot/Avg are total of columns for tonnage and economic values, and average of the remaining columns). As shown in Table 10.6, the schedule has the highest probability to achieve the desired properties of the ore produced in the first year (88.3%), a lower probability in the second year (84.3%) and the lowest in the last year (78.83%), exactly as intended in the scheduling optimisation model. This shows that the available risk of not achieving the production targets could be distributed over the different time periods by controlling their costs in the objective function. In the present case study, an 8% “risk discount rate” is used to discriminate the costing between time periods. This risk discount rate can be viewed as a parameter controlling the ‘orebody/grade’ risk distribution over time, which is distinctly different from the discount rate conventionally applied to economic values. If a higher rate is used, the differences in the probabilities between different periods are expected to be higher. Table 10.6 shows that the LP model is scheduling a little over 9.5 million tons of ore in the first period, barely satisfying the processing minimum capacity constraint. This low rate in the ore production leads to lower NPV at the end of first year. The reason for this is that the objective function is refusing to mine the ore tonnage with lower probabilities to meet the grade requirements; this refusal is because of the relatively high cost assigned for probability deviation factors in the objective function. The reduction of the cost of probability deviations will result in the LP model producing more ore tonnage with higher NPV at the end of first production period. However, the probability of meeting production targets would be lower. Thus, if the decision maker is willing to tolerate additional risk, the cost on the deviations of the probabilities can be reduced. This example suggests a trade off between risk and targets as well as between NPV and the utility of risk quantification. Additional characteristics of the mine production schedule include the effect on the project economics of small variations in the average Ni grade, as well as the average grade of Co, Mg and Al, in the three production periods. The LP model did not produce significant partial block mining, because the costs of deviations are different between periods. Almost 8% of the blocks were partially mined, with most of the 8% scheduled in a single period. Figure 10.34 shows the scheduling patterns generated from the proposed LP model. They suggest that the scheduled blocks can be mined at two faces. The production schedule allows equipment access and mining in a continuous manner once mining is started from a certain location. For example, during the first year, some equipment may

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start mining downwards from the top of the deposit, while other equipment may start mining the bottom part of the first year’s scheduling pattern and mining upwards before moving to the small patch on the left side of the deposit. The second and third years’ scheduling patterns are also easy to mine continuously.

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Figure 10.34: Production scheduling results incorporating orebody risk and equipment access (from SM). 10.5.4.3 Comparison of risk-based and traditional optimal scheduling In this section, the production schedule generated in the previous section is compared with the schedule generated by a traditional approach (TM). The TM uses an estimated model of the deposit (as summarized in Table 10.5), commonly generated through an approach such as kriging (David, 1988). The scheduling optimisation does not include probabilities and the corresponding penalties for related deviations. Figure 10.35 shows the optimal production schedule for TM and Table 10.7 summarises the results.


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Periods (years) 1 2 3 Tot/Avg Table 10.7:

Ore (106tons) 9.78 9.66 9.44 28.83 Summary

Tonnage UEV NPV Ni Co Mg Al Prob. (10 6tons) ($106) ($106) (%) (%) (%) (%) (%) 14.98 660.89 611.94 1.394 0.104 4.327 0.666 82.19 15.77 555.51 476.26 1.292 0.088 4.812 0.741 82.91 17.56 438.85 348.37 1.210 0.072 5.023 0.621 85.59 48.32 1655.2 1436.57 1.300 0.088 4.718 0.676 83.55 results of production scheduling using the traditional model

(TM) (UEV is undiscounted economic value; is probability; Tot/Avg are total of columns for tonnage and economic values, and Prob average of the remaining columns). In Table 10.7, the probabilities of meeting the schedule are calculated by comparing the TM schedule with each of the equally possible SM representations of the deposit, in a way similar to Step 5 in the previous section. Evidently, the effect of not factoring risk in the scheduling optimisation formulation generates lower probabilities for meeting production targets. Furthermore, the non-use of orebody “risk discounting” leads to the ordering of probabilities being the reverse of that of the SM schedule. In the TM schedule, the probability of achieving the desired properties of the ore produced is lowest in the first year (82.2%), higher in the second year (82.9%) and highest in the last year (85.6%). This trend is usually not desirable, as it is expected that more information will be available as a result of experience gained with the deposit as the mining operation proceeds. Uncertainty in the riskier areas would therefore be expected to decrease, enabling decision maker to improve decisions on short-term production scheduling and blending processes in the future. In addition,sothe objective is usually to secure productionascharacteristics early stages of a project, as to secure cash flows and loan repayment, well as improveatoverall financial aspects of a project. In comparing the scheduling patterns of TM in Figure 10.35 and SM in Figure 10.34, the SM scheduling pattern appears practical for mining in two phases, whereas the TM pattern is spread over the deposit and does not appear feasible in practice. This is a common concern with traditional MIP/LP scheduling models. The spread of scheduling patterns in Figure 10.35 means that mining equipment would need to be moved often in a given period. In addition, mining blocks may not provide access to equipment, as may be the case for the blocks on the top and centre part of the deposit scheduled for the second period. Either their excavation will have to be in the third period or other blocks scheduled for later periods will have to be mined first in order to reach them. These issues are not considered in traditional optimisation. The changes that are not considered by the MIP optimiser in an operation may cause infeasibilities in the model constraints andnot in materialise. terms of ore tonnage, grade and quality and sub-optimal NPV or in a NPV that will Figure 10.36 summarises the comparison of the risk-based (SM) and traditional (TM) formulations, showing the average deviations per mining period from expected ‘optimal’ production targets, and the probability of deviations in ore production per mining period occurring. The values plotted in the figure are generated by calculating the deviations of each schedule with respect to the 35 jointly simulated orebody models. Figure 10.36(a) shows the average of these deviations. Figure 10.36(b) is obtained by

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finding the ratio of the number of models in which the ore tonnage constraints are violated to the total number of simulated orebody models. There is littledifference in the first year in the probability of deviation occurring. However, during the second year of production, the risk-based SM schedule has about 28% (100,000 tons) less deviation in expected ore production compared with the traditional schedule. Furthermore, the probability of deviation in ore production occurring is around 10% less than in the traditional schedule (The ore tonnage is directly related to Ni grades, and increasing the probabilities to meet Ni grade constraints increases the chance of producing the required ore tons.). There are no significant deviations in grades, which means that grade constraints are not as tight as processing capacity constraints. The proposed risk-based LP schedule performs substantially better than the traditional schedule when comparing the overall deviations in ore production during the first two periods that the LP model considers.

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Figure 10.35: Production scheduling results from the traditional scheduling approach (TM).


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Conclusions

This paper has presented a new, risk-based optimisation formulation for long-term production scheduling in open pit mines. It is particularly suitable for complex, multielement orebodies such as Ni laterites, iron ore, or magnesium mines. The mathematical programming formulation integrates orebody uncertainty in respect of grade, ore quality and quantity, and risk quantification as well as equipment access and mobility and other typical operational requirements. A key part of the formulation is that it is based on the probabilities of grades of different elements to be above relevant cut-offs or within a given range. This provides the opportunity to: Generate schedules that aim to reduce risk at early production stages when secure cash flows are most critical. And later production periods will benefit from additional information that becomes available as mining operations proceed. Explicitly set up the approach to reduce risk in meeting production expectations. Introduce the concept of an “orebody risk discount rate” that can account for orebody uncertainty and be used in combination with the common approach of employing discount rates in dealing with mining project uncertainty. To generate production schedules with feasible miningwhich patterns, formulation is coupled with equipment accessibility and mobility constraints, aimthe to minimise inefficiencies in the utilisation of mining equipment. The practical aspects of the risk-based approach were shown in an application at a Ni laterite deposit. Relevant attributes and input to the scheduling formulation (Ni, Co, Mg and Al grades, volume of percent rock, thickness of layer LS and thickness of layers) were jointly simulated, conditional to all available drilling information. Thirty-five equally

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possible simulated representations of the deposit were used to generate probabilities and averages for the optimisation, as well as quantify the risk in meeting production schedules. The comparisons of the results with traditional long-term production scheduling based on NPV optimisation verifies the expectations for the new risk-based formulation that risk in meeting production targets is minimised and risk is lower in the first period, than the second and so forth. In addition, the scheduling patterns generated from the proposed approach are feasible and superior to those from the traditional optimisation. Although the proposed approach is not set up to explicitly maximise NPV, it generates a realistic NPV, which is the best under the scheduling considerations. It is obvious that NPV can be increased by forcing the probabilistic LP model to mine high-grade blocks in the early periods through the use of high and tight grade constraints in those periods. However, increasing NPV will generally increase the risk of not meeting production targets. The traditional model shown in this study produced 2% higher total NPV due to the high cash flows in the first scheduling period of the model. However the risk of not meeting production targets in the first period was about 6% higher than for the proposed risk-based LP model. There are also the practical mining issues mentioned above to take into consideration. Future work could consider additional testing as well as a more direct integration of orebody uncertainty in production scheduling formulations.

10.8.2

References

David, M., 1988. Handbook of applied advanced geostatistical ore reserve estimation. Elsevier, The Netherlands, 217p. Desbarats, A. and Dimitrakopoulos, R., 2000. Geostatistical simulation ofregionalized pore size distributions using min/max autocorrelation factors. Math Geology, v.30, pp.919-942. Dimitrakopoulos, R., Farrelly, C. and Godoy, M.C., 2002. Moving forward from traditional optimization : Grade uncertainty and risk effects in open pit mine design. Transactions of the IMM, Section A MiningIndustry, v. 111, pp. A82-A89. Dimitrakopoulos, R., 2002. Orebody uncertainty, risk assessment and profitability in recoverable reserves, ore selection, and mine planning. BRC Notes, SME Annual Meeting & Exhibit, Phoenix, Az, February, 22-24, p.355. Dowd, PA, 1997. Risk in minerals projects: Analysis, perc eption and management. IMM (Sect. A: Min. Ind.), 106: A9-18 Godoy, M.C. and Dimitrakopoulos, R., 2003. risk and waste mining in longterm production scheduling. 2003 SME Managing Annual General Meeting, Cincinnati. Kumral, M. and Dowd, P.A., 2001. Short-term scheduling for industrial mineralsusing multi-objective simulated annealing, APCOM 2001, Phoenix, Az, pp. Ramazan, S., 2001. Open pit mine scheduling basedon fundamental tree algorithm. PhD Thesis, Colorado School of Mines, Golden, Co. Ravenscroft , P.J., 1992. Risk analysis for mine scheduling by conditional simulation. Trans. Instn Min. Metall. (Sect. A: Min. industry),v.101, pp.A104-A108.


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Rossi, M. and Parker, H.M., 1994. Estimating recoverable reserves- Is it Hopeless? In, Dimitrakopoulos, R. (ed), Geostatistics for the NextCentury, Kluwer Academic, 259-276. Smith, M.L. and Dimitrakopoulos, R., 1999. The influence of deposituncertainty on mine production scheduling. Int. J. Surface Mining. Vallee, M., 2000. Mineral resource + engineering, economic and legal feasibility = ore reserve. CIM Bulletin, 90:53-61.

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10.6

Modelling U ncertainty in Short-Term Mine Production Scheduling 6

10.6.1

Introduction

Traditional orebody modelling techniques (eg David, 1997; 1988) generate smooth representations of the deposit under study. Although these methods are appropriate for ore reserve estimation, recent developments (eg Journel, 1992) suggest that a different class of techniques, namely conditional simulations, are appropriate for orebody modelling. An alternative approach is particularly needed when the in-situ variability of the deposit affects the mining process under study and evaluation. Several concerns arise when mine production scheduling is considered in combination with traditional orebody models, including: (a) the effect of in-situ grade variability on the scheduling algorithm, (b) the effects of grade uncertainty on the pattern of mining, and (c) the effect on a non-linear process such as scheduling when using ‘expected’ or ‘average’ type orebody models. Conditional simulations (Journel, 1994; Dimitrakopoulos, 1990; 1994; Armstrong and Dowd, 1994; and others) are stochastic simulation techniques that are used to generate equally probable models of orebodies, all reproducing the available data, both in statistics as well as spatial continuity. A conditional simulation utilising optimisation of mining processes and quantification of related uncertainty is detailed in Dimitrakopoulos (1998); this framework is highly applicable to mine production scheduling. Short and long-term scheduling routines are becoming more commonplace in mine software packages, and a number of production scheduling algorithms have been reported in the literature (Chanda, 1990, Trout, 1995, Winkler, 1996, Smith, 1998). All of these algorithms use a block model for input data, which are processed as a series of records with each record corresponding to a block. The fields in the record include the volume of ore and waste and the quality characteristics; each of these fields must be estimated as being a continuous random function, yet the data used for estimation are discrete. Thus, uncertainty is associated with the fields in a block model, which represents one of many possible realisations of the deposit. Typically, a short-term schedule attempts to find a combination of blocks which satisfies ore demand over a timeframe of one to several weeks. Ore demand may be quantified in terms of one or many variables such as tonnage of ore and waste, grades and contaminant levels. The scheduler seeks operationally feasible combinations of blocks, which satisfy production criteria. The feasibility of any combination of blocks is defined by constraints, which limit the solution to allowable mining practice. All of these scheduling algorithms assume that the input from grades the block is deterministic, i.e., there will be no significant fluctuation of block and model tonnage, or if there are, they all average out and won’t strongly influence either the mining sequence or the actual production. In fact, many deposits are characterised by high spatial variability in grade and tonnage, so much so that production targets can be widely divergent from off their planned values even over relatively short scheduling periods (Ravenscroft, 1992). Estimation methods that result in smoothed deposit 6

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models such as kriging will intensify this problem, especially when used in combination with a production scheduling algorithm which aims at minimising the deviation from production targets. While the true deposit variability may reduce the validity of, say, a kriging-based production schedule optimisation, this does not invalidate an optimisation methodology. As will be shown, schedules that are not based on the minimisation of production target deviation suffer to at least the same extent when based on a smoothed deposit model. The underlying solution to the production scheduling and mill reconciliation problem is to adopt a stochastic approach to production schedule optimisation.

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10.6.2

Short-term production scheduling

In the simplest case, a production engineer only seeks to select a series of blocks satisfying the demand for production tonnage in the forthcoming planning period. While seeking a set of blocks that meet the demand for ore, operational constraints must also be satisfied such as finding a contiguous set of blocks for each production unit (e.g., shovel) that can be mined in a fashion that will provide sufficient working room for the equipment. Even satisfying this one production criterion can be a difficult task: there may have to be multiple production faces (e.g. one per shovel) with sufficient accessible material to maintain high equipment productivity. When dealing with a relatively thin deposit, locating readily accessible faces with good digging conditions and sufficient ore can be difficult, requiring significant time using a trial-and-error approach. Production goals can also be represented as constraints in situations where tonnage and quality can be within specific limits. In this case, the objective might be to minimise operational costs while producing a certain tonnage whose quality characteristics have minimum and maximum allowable levels. In other cases, it may be difficult to estimate operational costs which might not vary significantly over the scope of the exposed benches. The objective in this case might be to minimise costs in the mill by providing a blend of ore that has the least deviation from an ideal mill feed that would maximise recovery and production while minimising the power and reagent consumption. When an objective such as cost or grade deviation minimisation drives production scheduling, mathematical programming methods are used. This study is based on the application of Mixed Integer Programming (MIP) for optimising short-term production schedules. Additional information on operations research aspects of the study may be found in Smith and Dimitrakopoulos (1999). Note that the LP approach is not used as it may not be entirely suitable to short-term scheduling problems in which a specific set of blocks has to be identified for each mining period, although heuristic solution strategies based on LP are available (Dagdelen, 1992) but risk missing the optimal solution and may be no faster than an MIP for complex blending problems. Identification of the set of blocks to be mined in a specific production period requires the introduction into the model of a binary variable, one for each block to indicate its mined status (1 = mined, 0 = not mined). These binary variables are used in precedence constraints that ensure adequate working slopes are maintained and blocks are accessible for mining. Formulation details for an MIP similar to the one used in this study are given in Smith (1999).

10.6.3

The estimated and simulated deposit

The present study is based on a roll-front uranium deposit and available exploration diamond drillholes, the locations of which are shown in Figure 10.37. The deposit is thin and flat, thickness and accumulation are use for its modelling (David, 1988). Figure 10.38 shows the estimated grade and thickness orebody models generated using ordinary kriging. Figure 10.39 shows one realisation of grade and thickness for the deposit. The realisations were generated using SIS.

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10.6.4

Scheduling with a smoothed, estimated block model

Short-term scheduling of the roll-front uranium deposit in Figure 10.40 is based here on a Goal Programming formulation with the objective of minimising grade and tonnage target deviations was used to find a series of single period optimal production schedules based on the estimated deposit model shown in Figure 10.40. The estimated deposit consists of 7,654,670 bcm of ore averaging (by volume) 0.064% U2O3. The deposit was scheduled over seven intervals with goals of 1,000,0000 bcm and the average grade. The deviation variables (defining the divergence from the production goals) were bound by upper limits of 0.0005% for grade and 10,000 bcm for volume. For the following simulations, greater upper bounds on grade were occasionally necessary. Penalty weights associated with the deviation variables in the objective function were selected to give positive and negative deviations of both grade and volume equal importance. Constraints consisted of volume and grade deviationvariable definitions (goal constraints) and block precedence relationships (e.g. the covering blocks must be mined first). The mining patterns for a seven period schedule based on the estimated deposit and one simulated deposit is given in Figure 10.40. Note the significant difference in the pattern of mining that occurs as a result of using a simulated deposit model. The optimisation was run using Cplex’s Mixed Integer Solver. Grade and volume deviations are given in Figure 10.41 and Figure 10.42. In comparing Figure 10.39 and Figure 10.40, it is interesting to note the strong influence that the spatial distribution of grade and thickness can have on the pattern of mined blocks in any one period. In this deposit, grade is positively correlated with thickness. The deposit thickens toward the middle with a low grade, thin ore zone between block columns 60 and 80. Otherwise, grade and thickness is reasonably uniform along the strike of the deposit. These same general deposit characteristics were preserved in the simulations. The production schedule optimisation must find a set of high and low grade blocks which is as close to the deposit’s average grade as possible. This objective is difficult to achieve in the initial production periods due to the dominance of thin, low grade blocks around the deposit’s edge. Thus, the model drives the production face as deep into the centre of the deposit as is feasible given the 3-to-1 precedence constraints, resulting in a pyramidal pattern of mined blocks in each period as the scheduler cuts across the strike of the deposit in order to blend both low and high grade material.

10.6.5

Scheduling with a simulated orebody model

Eight simulated realisations of the deposit generated with SIS were used to test the influence of deposit variability on production. The optimal schedule using the estimated block model was used as the base case. For each of the simulations, the volume and average grade was found for each period in the base case’s production schedule by selecting the same set of blocks for each of the simulations as was found to be the optimal solution based on an estimated deposit. Figure 10.41 and Figure 10.42 show how the grade and volume deviate from the production targets for estimation and the simulations. Again, note that the apparent “optimal” solution based on kriging has virtually no deviation from the production targets: grade deviations are limited to thousandths of a percentile and the volume is off by only single digits. While this is an excellent demonstration of the effectiveness of the MIP Solver used, the deviations Applied Risk Assessment for Ore Reserves and Mine Planning: Conditional simulation for the mining industry

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given by the simulations are on orders of magnitude greater. The same pattern of deviations is seen as in the kriging base case with low volumes and grades in the initial periods and positive grade and volume deviations in the later periods. Grade can vary from the base case by as much as 9% while the volume can be off target by over 200,000 bcm, 20% under or over the required amount.

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“optimal” solution as long as the production targets can be met with a given deposit model. However, “optimality” is limited to the given deposit model used not the actual deposit schedule. Figure 10.41 (labelled “priority”) shows the scheduling of the estimated, “base-case” orebody model using a scheduler where the objective was to maximize the accessibility of the blocks. In this approach, each block is assigned a priority level that is a function of the blocks that are “covered” by that block, i.e. the underlying blocks that are dependent on the mining of an overlying block. The summed value of all covered blocks is then used to assign a priority level to the mining of the covering block. In this approach, blocks near the exposed face that are in advance of high grade zones will be assigned a high priority (Gershon, 1987). The pattern of blocks mined in each period is much more linear and continuous than was the case for the goal programming formulation and appears much more similar to a conventional production pattern in terms of simplicity and continuity. The grade deviations for this schedule labelled ‘priority’ are shown in Figure 10.41 along with the deviations resulting from the goal programming formulation. In this example, the grade deviations are orders of magnitude greater than was the case when the goal programming formulation was used on the same data set. Additionally, even though the smoothed deposit model is being used, these deviations exceed those seen for the simulations in every period except the last. Even in the last period, the relatively poor performance of the goal programming formulation results from the exhaustion of the deposit and the resulting loss of flexibility when forming an “optimal” solution. The significance of these results is that a production schedule based on any one deposit model particularly if the deposit is estimated i.e. smoothed out may vary significantly from the actual schedule. Any production schedule shouldinaccount for uncertainty those deposit characteristics that influence the optimisation, this example grade andinthickness. Period 1 Block Probability (100% certain blocks shown)

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Accounting for uncertainty


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Past optimisation applications using a LP formulation account for uncertainty using sensitivity analysis (Bazaara, et al., 1990): shadow prices and reduced costs are examined to identify constraints that are binding and parameters wherein a change in value will lead to a change in the optimal basis. Unfortunately, this standard approach to uncertainty is not practical and may also be misleading, as it is ‘global’ in any production scheduling formulation.

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Additionally, when using an MIP formulation a sensitivity analysis can only be applied to the LP relaxation of the optimal solution, and this is likely to be far from integer feasible. The approach in the current approach accounts for uncertainty in the optimisation process suggests that an optimal schedule is one that gives the greatest certainty of meeting the production goals given the likely distribution of the deposit’s key characteristics. An example of this is shown in Figure 10.43 showing the probability that a block will be mined in scheduling periods 1 or 2. A series of simulations were run, for each of these realisations, the optimal production schedule was determined. If a block was mined in the same period for all simulations, then the likelihood that that block would be mined in that period is 100%. Thus, a block probability level can be calculated that indicates patterns of mining that will reduce the impact of deposit uncertainty on production. As shown in Figure 10.44, the desired certainty of obtaining a target tonnage can be compared against the tonnage that will most likely be produced. In this case, there is only a 50% chance that the period 1 goal of 2,000,000 bcm will be achieved! Given these results, what strategy of risk reduction should be used? Some blocks are clearly of low risk, especially in the first period when a significant portion of the demand can be found in 100% certain blocks. Conversely, large portions of the deposit never fall within the optimal schedule for any realisation and should be ignored as candidate blocks. The blocks whose probability level is somewhere between the categories of always mined or never mined in a given period require closer attention.

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The pattern of mined blocks also indicates an overall strategy for scheduling the deposit. In this case, production should start on the western half of the deposit with a deep penetration around column 22. In the second period, production will be concentrated in the middle of the deposit. By the third period, production will have shifted to the east with additional blocks being mined across the length of the deposit in advance of periods 1 and 2. The fourth period simply cleans up the remaining blocks in this small example.

10.6.7

Comments and conclusions

Mine/mill ore reconciliation will continue to be a major obstacle as long as the stochastic nature of deposit models is ignored during scheduling. While significant improvements in blending can be achieved by using optimisation algorithms, major deviations in production goals will continue as long as a single expected value block model is used as input. If improvements in the performance of production scheduling algorithms are to be realised, the uncertainty of the deposit model must be quantified and integrated into the scheduling process. A methodology for dealing with uncertainty is suggested herein: (1) quantify uncertainty by simulating a series of realisations of the deposit which are representative of the range of block values likely to be encountered during mining, (2) apply the scheduling algorithm to each realisation generating a response distribution in the form of the pattern of mining blocks over time, (3) determine the probability of a block being mined in a given period, and (4) use the patterns of probability as a guide in setting a risk-based schedule. While time-consuming, this approach is no more tedious than the traditional trial-and-error methods, will minimise production target deviation, and, most importantly, will bring deposit model uncertainty into the scheduling process. Additional work is needed to enhance the interface between simulation and optimisation. The optimisation process reported herein is limited, thus additional formulations should be considered (e.g. stochastic programming).


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10.7

Risk Analysis in Underground Stope Design – Sublevel Stoping at Kidd Creek

Risk Analysis in Under ground Stope Design Sublevel Stoping at Kidd Creek

Outline 1. Kidd Creek Mine and Sublevel Open Stoping 2. Traditional Stope Design 3. Grade Simulations in Evaluating Risk 4. Optimising Stope Design Based on Risk 5. Conclusions

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Kidd Creek No. 3 Mine • Source data • Stringer ore • No. 3 Mine

• Sublevel stopes: • 15m wide x 30m long x 30m high • Selective rings 3-4 m in width (minimum of 2 per stope) Study Area

• Cutoff grade 3% Cu • Mill Recovery 95% • Mining and Milling Cost $55/tonne

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Steps Involved in the Evaluation of Risk for Traditional Stope Design 1. 2. 3. 4.

5.

Simulate 40 realisations of the deposit. Reblock each realisation to selective ring sizes. Filter out the rings below the cutoff grade of 3% for each simulated realisation. Calculate the tonnes of ore, tonnes of Cu, grade and dollar value for each realisation as if it was the real deposit mined to the specifications of the traditional stope design. Summarize the results quantifying the risk associated with the ‘traditional stope design’.

Example: Realisation No. 17 Realisation 20

Realisation 35

Estimated Deposit

Deposit cells resized to selective ring sizes (3m x 15m x 30m).

310


Chapter 10

Example Cont.

Deposit cells of Realisation #17 below Cutoff Grade (3%) are filtered out of model.

Example Cont.

Bottom Level

Top Level

Plan View of Realisation # 17 after being filtered. The stope outline within which all calculations are made is represented in white.


311

Chapter 10

Example Cont. Results of Realisation # 17 Simulated Realisation … 13 14 15 16 17 … E s tima te d

Volume ……… 58,050 58,050 66,825 64,800 6 2 ,7 7 5 ……… 7 5 ,6 0 0

Tonnes of Ore

Tonnes of Waste

156,735 156,735 180,428 174,960 169,493

47,385 47,385 23,693 29,160 34,628

167,670

36,450

Cu Grade … 4.75 5.10 4.82 5.12 4.89 … 4.87

Tonnes of Cu … 7,445 7,993 8,697 8,958 8,288 … 8,166

Dollar Value … $ (660,073) $ 48,683 $ 483,264 $ 930,274 $ 174,283 … $ 52,263

Tonnes of Ore Within Stope Outline (all 40 simulations) 13 12 11 10 9 y8 c n7 e u q6 re F5 4

30 % Frequency Probability 20 %

Estimated Deposit 15 % 13%

13%

3 2 1 0

8% 3%

156735

161899

167063

172226 Tonnes of Ore

177390

182554

187718

Traditional Stope Design Expects = 167670 tonnes

312


Chapter 10

Tonnes of Copper Within Stope Outline (all 40 simulations) 13 12 11 10 9 y 8 c n e 7 u eq r 6 F 5 4 3 2 1 0

35%

Simulation Probability

30%

30% 25%

Estimated Deposit

25% 20%

18% 15%

10%

10% 10% 5%

5% 3%

0% 7445

7697

7949

8201

8454

8706

8958

Tonnes of Metal

Traditional Stope Design Expects = 8166 tonnes

Dollar Value Within Stope Outline (all 40 simulations) 11 10 9 8

40%

Simulations

35%

Probability Estimated Deposit

7 y c n e 6 u eq r 5 F 4

30%

25%

25%

23% 20%

20%

15%

15%

3

8%

2 1

8%

10% 5%

3%

0

0% -660073

-395015

-129957

135100

400158

665216

930274

Present Value

Traditional Stope Design Expects = $52 263


313

Chapter 10

Optimising Stope Design based on Grade Risk 1. “Optimisation” using a Linear Programming –

formulation to account for: Number of stopes

–

Size of stopes

–

Location of stopes

2. Quantification of risk using a similar method to the one presented for ‘traditional stope design’ 3. Rerun at a number of simulations and choose a stope design that performs better based to selected criteria

Linear Programming Formulation Objective Function: MAX

Σ(gradei • probabilityi • xi ) where: xi = {0,1} 1≤ i ≥ # of blocks

Subject to the following constraints:

Σ Probability • x i

Σ Probabilityi

i

≥ 60%

Minimum and maximum size constraints Minimum number of rings ≥ 2 Maximum number of rings ≤ 7

314


Chapter 10

Design – 3 - LP

Traditional Design

Design – 13 - LP

Design – 35 - LP Design – 17 - LP

Design – 20 - LP

Comparison of Dollar Values Comparison of Dollar Value s 2000000

1500000 e1000000 lu a V r a ll o 500000 D

1,049,359.78

766203.6767 649,273.61

643742.3565

656,455.79

mean median min max 10th per 90th per Expected

0

-500000 Simulation 3 Simulation 13 Simulation 17 Simulation 20 Simulation 35

Design


315

Chapter 10

Comparison of Tonnes of Ore 200000


180000

re160000 O f o s140000 e n n o120000 T 100000

80000 Simulation 3

Simulation 13

Simulation 17

Simulation 20

Simulation 35

Design

Comparison of Tonnes of Metal 10000

9000


l ta 8000 e M f o 7000 s e n n o 6000 T 5000

4000 Simulation 3

Simulation 13

Simulation 17

Simulation 20

Simulation 35

Design

316


Chapter 10

Conclusions • Simulations provide a means of evaluating risk related to traditional sublevel stope design. • Simulations provide a means of generating risk robust designs. • This was the beginning of the story.


317

Chapter 10

10.8

10.8.3

Geostatistical Modelling of Ore Textures in Enhancing Ore Reserve Estimation and Planning Introduction

Ore textures are an important factor in the liberation of the economic components of ore. If the grade varies throughout an orebody then it is possible that the ore texture, and hence the metallurgical performance and metal recovery are heterogeneous. Constructing models of ore textures involves: 1. 2. 3. 4. 5.

10.8.4

the unbiased collection of texture data at the core scale; compositing texture data to equal sample support; characterising the spatial texture continuity; generating multiple simulations of textures; upscaling the texture simulations to generate the expected mining block/stope texture distribution.

Collecting and compositing texture data

Textures occur at irregular sample sizes, geostatistical modelling requires compositing to equal sample support. How do you composite texture (categorical) data? Figure 10.45 shows two options: (a) assign the dominant texture (mode) to the composite, which intrigues bias in the samples; and (b) use small size composites to generate statistically representative samples.

Logged Textures

e it s o p m o C

Composite Composite Method 1 Method 2 (Mode) (Small intervals)

l a rv e t n I

Local and global bias Texture 1

Texture 2

Statistically representative Texture 3

Figure 10.45: Approaches to compositing textures.

318


Chapter 10

10.8.5

Characterising spatial texture continuity

Each texture has its own spatial continuity. Some concerns include: Do we use individual measures of spatial continuity or a single ‘representative’ correlogram? Or the multi-texture indicator correlogram is an average type indicator correlogram as shown in Figure 10.46 below.

0.8 Multi

0.7

Cat 1 Cat 2

0.6

Cat 3 Cat 4

0.5

Cat 5

m ra g o l 0.4 e rr o C0.3

Cat 6 Cat 7 Cat 8 Cat 9 Cat 10

0.2 0.1 0.0 0

2

4

6

8

10 Dist ance (m)

12

14

16

18

20

Figure 10.46: Indicator and multi-texture indicator correlograms for 10 texture categories.


319

Chapter 10

10.8.6

Simulating textures

Traditional implementation of categorical simulation algorithms like SIS performed poorly, failing to honour spatial structures and global proportions as shown in Figure 10.47 below.

Figure 10.47: Texture simulation produced using traditional SIS algorithm.

Two modifications to the traditional SIS algorithm produced significantly better texture simulations (e.g. Figure 10.48).

Figure 10.48: Texture simulation produced using modified SIS algorithm.

320


Chapter 10

10.8.7

Simulation paths and ‘growth’ of SIS - 1

Figure 10.49: The growth algorithm in operation


321

Chapter 10

10.8.8

Local probability corrections

If, in the early stages of a simulation, the simulated points tend to be located close to the experimental data of the one category its proportion tends to increase rapidly above the global value and rarely returns to the objective of reproducing the global proportion of each category (see Figure 10.50).

25 20 )% (15 n 10 o ti a i 5 v e0 D-5 0 e -10 iv t -15 a l e -20 R

70000

140000

210000

-25

Iteration Num ber

Figure 10.50: The proportion of eight textures relative to the experimental data as a simulation progresses along 208,000 iterations.

A correction to the local estimated category probabilities, based on the current global category deviations, can be implemented

25 ) 20 % ( 15 n 10 o ti ia 5 v 0 e D -5 0 e v -10 ti a l -15 e R-20

-25

70000

140000

210000

Iteration N um ber

Figure 10.51: The global proportion of 8 mesotextures relative to the experimental data as a simulation using local probability corrections progresses along 208,000 iterations.

10.8.9

Upscaling texture simulations

Each of the 10 texture categories perform, metallurgically, in one of three ways, thus, the textures can now be grouped. At each node the simulations results are combined to provide an expected histogram of texture groups (see Figure 10.52).

322


Chapter 10

Figure 10.52: Histograms of mesotextures at “point” locations

Using the point distributions, such the ones in Figure 10.52, one can build the distribution of textures for any block size or shape as seen in Figure 10.53.

Figure 10.53: Generating block support textures. WARNING: taking the average or mode of block estimates is incorrect and may generate misleading results.


323

Chapter 10

Based on metallurgical studies of the metal recoveries from the three texture groups the metal recovery for each mining block can be quantified.

Figure 10.54: Calculating recoveries without discriminating textures.

10.8.10

Economic implications

Figure 10.55: Net value of blocks if mined and processed incorporating only grade information.

324


Chapter 10

Figure 10.56: Net value of blocks if mined and processed incorporating both grade and texture information.

10.8.11

Conclusions

Texture modelling is based on core descriptions. The economic benefit from modelling textures is, in part, related to the integrity of the texture model at a mining scale. The question of scale and when to upscale during data collection, compositing, and texture modelling is critical in honouring the global texture statistics and spatial structures. The presence of different metal recoveries is the justification for modelling the ore textures. Accounting for the relationship between ore texture and metallurgical performance provides several benefits, including: 1.

enhanced assessment of resources and reserves;

2.

better ore/waste delineation;

3.

reduced block misclassification;

4.

blending of ore textures to maintain a constant metallurgical performance.


325

Chapter 10

THIS IS THE END OF THE NOTES BUT NOT THE END OF THE STORY

326


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Rossi, M., and Parker, H. (1994). Recoverable Reserves - Is there Light at the End of the Tunnel? . In, R. Dimitrakopoulos, ed., Geostatistics for the Next Century, Kluwer Academic Publishers, Dordtrecht, The Netherlands, pp. 259-276. Rossi, M. and Van Brunt, B. (1997). Optimizing Conditionally Simulated Orebodies with Whittle 4D. Optimising with Whittle, Perth 1997. Rubinstein, 1981. Simulation and Monte Carlo method. Wiley, New York, 278p. Sichel, H.S., 1973, Statistical valuation of diamondiferous deposits, Jour. S-Afr. Inst. Min. and Metall., v.73, p.235-243. Sims D. A. and Bartrop S. B. Exploration in amature mine - The Mount Isa Minestrategy. International Mining Geology Conference, AusIMM, 1993, 25 – 29 Sinclair, A. J., Blackwell, G. H., 2002, Applied Mineral Inventory Estimation. Cambridge University Press Smith, M. (1993). “The use of fractals in quantifying pyrite textures to determine ore liberation in base metal ores.” GSA North-Central Section 27th annual meeting, Abstracts with programs - Geol. Soc. of America v.25, no.3: 82. (Appendix D) Smith, M.L. and Dimitrakopoulos, R. (1999). Influence of deposit modelling on mine production scheduling. Int. J. of Surface Mining, v. 13, pp. 173-178. Srivastava, R.M. (1992). Reservoir characterization with probability field simulation. SPE paper No. 24753. (Appendix C) Srivastava, R.M. and Dimitrakopoulos,R. (1992). Characterisation and Management. AAPC, Calgary.

Geostatistics

for

Reservoir

Srivastava, R. M., Hartzell D. & Davis B. (1994). Enhanced metal recovery through improved grade control. In Proceeddings of the 23th APCOM Symposium, Kim Y. Ed., AIME, New York : 243-249. Stolarczyk L. G. Definition imaging of an orebody with the radio imaging method (RIM). IEEE Transactions Industy Applications, 28, 1992, 1141-1147. Stoyan, D. and Stoyan, H. (1994). Fractals, random shapes and point fields - Methods of geometrical statistics, John Wiley& Sons, Chichester, UK, 389p. (Appendix C) Suro-Perez, V. and A.G. Journel, 1991, Indicator Principal Component Kriging, Math. Geol., v.23, no.5, p.759-792. Switzer, P. and A.A. Green (1984). Min/Max autocorrelation factors for multivariate spatial imaging, Tech. Rep. no. 6,Dept. of Statistics, Stanford University, 14p. Tercan, A.E., 1999, Importance of orthogonalization algorithm in modeling conditional distributions by orthogonal transformed indicator methods, Math.Geol., v.31, p.155-173. Vallee, M., 2000. Mineral resource + engineering, economic and legal feasibility = ore reserve. CIM Bulletin, vol.90, pp. 53-61. Verly, G. (1992). Sequential Gaussian Co-Simulation: A Simulation Method Integrating Several Types of Information. In, A. Soares, ed.,Geostatistics Troia’92, Kluwer Academic, Dordrecht: 543-554. Wackernagel, H., 1995, Multivariate Geostatistics, Springer, Berlin, 256p.


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388


Appendix A

APPENDIX A

A.1

A FORMAL LOOK INTO GEOSTATISTICAL CONCEPTS

Random Functions and Related Notations

A.1.1 Random variable

Random Variable (RV) is a variable, say Z, that takes values according to a probability distribution. Cumulative distribution function (cdf), of a continuous RV at location x, Z(x), is denoted: F(x;z) = Prob{Z(x) ≤ z} Conditional cumulative distribution function (ccdf), of a continuous RV Z(x), is the cdf given specific n pieces of related data/information and is denoted: F(x;z/(n)) = Prob{Z(x) ≤ z/(n)}

The distribution of a categorical RV Z(x) which can take any of L values l=1,...,L is denoted: f(x;l/(n)) = Prob{Z(x) belonging to l/(n)}

NOTE: Categorical variables do not always have a predefined ordering (e.g. lithologies). Thus, the above may not be a ccdf but a conditional probability density function (cpdf).

A cdf (or pdf) of a RV characterizes the uncertainty about a possible value at location x. A ccdf updates a cdf to available information.

A.1.2 Random function Random Function (RF), {Z(xi), i=1,...,N, xi ∈ A} or Z(x) for simplicity, is a set of random variables defined over an area of interest, e.g. a reservoir.

The N-variate or multivariate cdf of the RF Z( x) is denoted: F(x1,...,xN;z1,...,zN) = Prob{Z(x1) ≤ z1,..., Z(xN) ≤ zN}

The above cdf is characterised by its statistical properties below.


327

Appendix A

Mathematical expectation, E, of a RF Z(x) is: E{Z(x)} = m(x) 2

Variance:

Var{Z(x)} = E{{Z(x)-m(x)] }

Covariance:

C(x1,x2) = E{[Z(x1) - m(x1)][Z(x2) - m(x2)]}

Variogram:

γ (x1,x2) = Var{Z(x1) - Z(x2)}

The Random Function model, explicitly describes/models thedependence of RV's that can be subsequently used to update priorcdf's at unsampled locations.

A.1.3 The bivariate (or two-point) case

When N=2, the cdf of the corresponding random function {Z(x1),Z(x2)}, is characterised by the bivariate distribution: F(x1,x2;z1,z2) = P`rob{Z(x1) ≤ z1,Z(x2) ≤ z2}

The covariance of the above is (if it exists, see following pages) C(x1,x2) = E{Z(x1)Z(x2)} - E{Z(x1)}E{Z(x2)} An interesting summary of a bivariate cdf is denoted as: F(x1,x2;z1,z2) = E{I(x1;z1)I(x2;z2)} I(x,z) is the indicator transform of the bivariate RF Z(x):

I(x;z) =

328

⎧ 1 if Z( x) ≤ z ⎨ ⎩ 0 otherwise


Appendix A

NOTE: The bivariate cdf above is termed the 'non-centred covariance' of the indicator RF I(x,z), and is, in practice, inferred from the data. For two categorical variables, the corresponding bivariate pdf or two-point histogram is denoted: f(x1,x2;l1,l2) = Prob{Z(x1) belonging to category l1, Z(x2) belonging to category l2}

A.2

Comments

The two RV's Z(x1) and Z(x2) can be seen to represent: (i) different attributes at the same location, e.g. the porosity and permeability of a core plug; (ii) the same attribute at different locations, e.g. porosity at locations x1 and x2 (= x1+h); or (iii) two attributes at different locations.

Conventional geostatistics is based on two point-statistics: (i) The covariance or variogram of the bivariate cdf. OR (ii) A series of indicator covariances or variograms.

A.3

More on the Multivariate (or Multiple-Point) Case

Similarly to the two-point case, a multivariate cdf can be the product of a series of indicator functions: F(x1,...,xN;z1,...,zN) = E{I(x1;z1)I(x2;z2) ... I(xN;zN)} For a series of categorical variables, the two-point histogram of the bivarate case becomes multiple-point histogram: f(x1,...,xN;l1,...,lN) = Prob{Z(x1) belonging to category l1,..., Z(xN) belonging to category lN} l1,...,lN = 1,...,L


329

Appendix A

A.4

Inference and Assumptions

In practice, any random function model requires inference of its characteristics from a set of data and it requires some assumptions: A RF Z(x) is said to be stationary, if E{Z(x)} = m = constant and the covariance exists, C(h) = E{[Z(x+h)Z(x)] - m} for every x, x+h

NOTE: A RF is said to be strictly stationary, if the whole corresponding multivariate cdf remains unchanged under rotation of the N coordinate vectors xk. A RF Z(x) is said to be intrinsic, if E{Z(x)} = m = constant and the variogram exists, 2 } γ (h) = Var{Z(x+h) - Z(x)} = E{[Z(x+h) - Z(x)]

NOTE: This hypothesis extends the idea of stationarity to increments of the RF. In the case ofnon-stationary RF's one may consider decomposing the RF in to a trend function and a stationary residual RF, OR consider RF's withstationary increments of order k.

In practice, one mostly needs to assumelocal stationarity. Stationarity of any form is only a property of the corresponding RF model not a property of nature. When one is considering models the main concerns focus on: relevance, trade-off's and common sense. Choosing a model is not the only question: the practical estimation of the parameters of the modelis an additional concern.

330


Appendix A

A.5

Expanding on Geostatistical Estimation

A.5.1 Ordinary kriging (OK)

Recall the estimation problem previously discussed. One may reconsider the formulation of a linear combination in the context of RF's. Specifically, a stationary RF Z(x) to be estimated at location x o, from the values of n RV's at neighbouring locations is a linear combination of the RV's: Z(xo)* = Σi=1n wi Z(xi)

where wi are weights derived (recall previous section) from -1

w = D C

One good reason to consider an RF model in the above estimation is the properties of unbiasness and minimum estimation variance, as previously discussed. For example, the average estimation error avg r = 1/n

Σi =1n (zi* -zireal)

as such is of no use, since it involves unknown actual values However, if one considers the estimation error as a RF, R(x), and the above properties the OK system of equations is derived. The OK system is valid for stationary or intrinsic RF's. The OK system requires that the mean of the corresponding RF is a constant, BUT it does not require its inference. The OK system rewritten in the form of a set of equations:

Σi=1n wi C(xi-xj ) + μ = C(x-xj ), j=1,...,n Σi =1n wi = 1

A.5.2 Simple kriging (SK)

If the mean, m, of a stationary RF, Z(x), is known, then the OK system is simplified to

Σi=1n wi C(xi-xj ) = C(x-xj ), j=1,...,n and the kriging estimator is now: Z(xo)* = Σi=1n wi Z(xi) + [1 - Σi=1n vi] m


331

Appendix A

OK filters the mean from the SK estimator and is more general SK and OK weights for the same configuration are not identical What if the mean of the RF is not aconstant?

A.5.3 Universal kriging (UK)

UK considers the decomposition of a RF Z(x): Z(x) = m(x) + R(x)

where, R(x) is a stationary RF and m(x) a trend (drift): m(x) = E{Z(x)} = Σl=1 al fl(x) L

in 2D, m(x) = a1 + a2x + a3y + a4x2 + a5y2 + a6xy + ...

NOTE: al are unknown coefficients; m(x) is in practise a polynomial of order up to 2. The estimation formulation remains the same, however, the weights are now derived from:

Σi=1n wi C(xi-xj ) + Σl=1L ai fl(xj ) = C(x-xj ), Σi=1n wi fl(xi) = fl(u),

j=1,...,n

l=1,...,L

where C(h) is the covariance of R(x)

332


Appendix A

In matrix notation, the UK system of equations is -1

w = D C

where

C11 . . . C . . . . . . . . . . . . . . . Cn1 . . . C C =

1

. . . f2(x1) . . . f2(xn)

. . . . .

1n

nn

1 . . . . .

. fL(x1) . . . . . . . fL(xn)

1 f2(x1) . .

. . . fL(x1) . . . .

. . . . 1 f2(xn)

. . . . . . . . . . . fL(xn)

0

0 0 . . . 0

. . . 0 . . . 0

w1 . . . wn w

=

μ1 μ2 . . .

μl

. . . . .

. . . . .

0 . . . . .

0 . . . 0

C01 . . . Cn D

=

1 f1(x0) . . . fL(x0)


333

Appendix A

•

The UK system of equations is a generalisation of OK

•

decomposition of a RF resembles to: Egg The UK or chicken first?

•

In practice one needs to decide on the trend and find the residual.

•

The trend component is usually considered to vary smoothly.

•

The UK system is identical to kriging with Intrinsic Random Functions of order k, with two main differences: i) One does not need to use the UK decomposition. ii) One needs to substitute the covariance of the residual with the covariance of certain linear combinations of the data. This covariance is termed generalised covariance.

•

UK may be quite flexible for extrapolation

•

The UK system is identical to Generalised Least Squares.

•

The UK system can be further expanded to consider a supplementary variable.

334


Appendix A

A.5.4 UK with a supplementary variable

Consider a RF with a very smooth trend m( x) = a1 + a2f(x), and a second relevant variable, say y(x). Then, one may rewrite: m(x) = E{Z(x)} = a1 + a2y(x)

The UK system and kriging with the above alternative or external drift, is identical.

•

Using the second variable must make physical sense; the final maps will reflect the trends of the second variable.

•

The second variable must vary smoothly.

•

The supplementary variable must be known at the same locations as the primary variable.

•

The covariance used is a residual covariance.

•

The correlation of the two variables is irrelevant to the UK system.

A.5.5 Co-kriging or vector kriging

If data on two or more attributes are available, it seems reasonable to suggest that the one may combine all data, if possible, to estimate one or all attributes simultaneously. The idea seems particularly attractive if the attributes are somehow physically and/or statistically related. One may also consider the case of estimating one under-sampled variable from others more systematically sampled. Examples of related variables may be: seismic transit time and porosity or logpermeability and porosity, volume of sand and porosity, etc.


335

Appendix A

Figure A.1: Related variables: seismic transit time and porosity.

In an alternative situation one may be faced with data measurements of the same attribute from different sources and different reliability. Examples are porosity and fluid saturations derived from core and well logs.

Figure A.2: Related variables: porosity (%)-core data and porosity (%)-well log data.

336


Appendix A

A.5.5.1

Co-kriging with two variables

Use data from two variables to estimate values at unknown locations

Figure A.3: Co-kriging with two variables.

Consider Z(x) to be the variable of primary interest and Y( x) the supplementary. The co-kriging estimate at location xo, from the neighbouring data is a linear combination: Z(xo)* = Σi=1n wi Z(xi) + Σi=1m vi Y(xi)

where wi and vi are weights. Similarly to the development of the OK equations, one considers an the properties of unbiasness and minimum estimation variance, and the RF estimation error R(x): *

R(x) = Z(xo) - Z(xo)

A.5.5.2

Formulation

The conditions of unbiasness and minimisation of the estimation variance, provide the following co-kriging system of equations:

Σi=1n wi CZ(xi-xj ) + Σi=1m vi CZY(xi-xj ) + μ1 = CZ(x-xj ),

j=1,...,n

Σi=1n wi CYZ(xi-xj ) + Σi=1m vi CZ(xi-xj ) + μ2 = CZY(x-xj ),

j=1,...,n

Σi=1n wi = 1 Σi=1n vi = 0


337

Appendix A

In matrix notation, the co-kriging system of equations is -1

w = D C

where

CZ11 . . . C . . . . . .. .. .. .. .. CZn1 . . . C

C =

.

CYZ11. . . . . . . . . . . . . . . CYZm1 . . .

1 0

338

C

Z1n

Znn

YZ1n

. . . CYZmn

. . . 1 . . . 0

CZY11 . . . CZY1m . . . . .

1 0 . .

.. .. .. .. .. CZYn1 . . . CZYnm

.. .. 1 0

CY11 . . . CY1m . . . . . . . . . . . . . . CYm1 . . . CYmm

0 1

. . . 0 . . . 1

0 1 . . . . . . 0 1

0 0 0 0


Appendix A

w1 . . . wn

w

=

v .1 . . vm

CZ01 . . . CZ0n

D

=

C . ZY01 . . CZY0m

μ1

1

μ2

0

The above kriging system appears as a general form of OK, and the basic properties of the system are the same (see discussion on OK) One important difference of from previous formulations is the use of both the crosscovariance and covariance the second variable. The cross-covariance CZY is generally not symmetric, i.e. different from CYZ, however, in practise is most often modelled as symmetric. In order to infer, model and use variograms and cross-variograms, the cross-covariance must be symmetric. In the case of block estimation the only changes are the covariance and crosscovariance terms, in a similar fashion as the OK block estimation system. The set of conditions on the supplementary variable, i.e. weights sum up to 0, limits the influence of the supplementary variable. Question: What are the weights for the supplementary data if they are only two?


339

Appendix A

A.5.5.3

Exercise

Set up the co-kriging matrix C for the following data configuration

Figure A.4: Estimating with co-kriging.

An alternative formulation of co-kriging with two variables, gives a more equal weighting to the primary and secondary variables:

Σi=1n wi + Σi=1n vi = 1 and the estimation is reformulated as: Z(xo) = Σi=1 wi Z(xi) + Σi=1 vi [ Y(xi) - mY + mZ ] *

n

m

The mean values mZ and mY are used to 'scale' the second variable. In practice, mZ and mY are taken as the arithmetic averages of the data. This formulation may be physically more meaningful (e.g. only two measurements of the supplementary variable are available at equal distances from the estimated point). Appears to be less prone to generating negative estimates. Is based on the estimated means of the two variables.

340


Appendix A

If the means are to be estimated, then one may consider the co-kriging extension of simple kriging, i.e. without constraints on the weights and by using variables whose means are standardized to 0.

A.5.5.4

Coregionalisation

A phenomenon may be seen as represented by several inter correlated variables that are examined simultaneously, as shown previously. This amounts to working with a set of inter-correlated {Z1(x),...,Zk(x),...,Zk'(x),...,ZK(x)}. Assuming stationarity one defines:

RF's

• the mean E{Zk(x)} = mk = constant

• cross-covariance between any of the RF's, Zk(x) and Zk'(x), Ck'k(h) = E{[Zk'(x+h) Zk(x)] - mk'mk } for every x,x+h

• cross-variogram

γk'k(h) = E{[Zk'(x+h) - Zk'(x)] [Zk(x+h) - Zk(x)]} For k'=k one derives the definitions of the variogram and covariance The cross-variogram may take negative values (e.g. a negative correlation of the two variables), while the regular variogram is always positive The existence of cross-covariances leads to the cross-variogram: 2γk'k(h) = 2Ck'k(0) - Ck'k(h) - Ckk'(h)

The cross-variogram is symmetric. The cross-covariance is NOT. The linear model of coregionalisation is used to model experimental variograms and cross-variograms, in most practical cases and is:

γk'k(h) = Σi=1n bkk'i γi(h),

with bkk' = bk'k

where γi(h) is the basic the basic variogram, and bkk' are a matrix of coefficients. For the case of two RF's (K=2) the above coefficients must be : l

b11

≥ 0 and ⏐b11l⏐ = ⏐b12l⏐ ≤ √

l

l

b11 b22

The above restrictions ensure that the co-kriging system has a unique solution.


341

Appendix A

The restriction imposed by the above relations may, in some cases, make the modelling of variograms and cross-variograms difficult. Note that the basic model included in a variogram does not have to be included in a cross-variogram. Also, the basic model that appears in a cross-variogram does not have to be included in the variogram model. Example: Modelling variograms and cross-variograms (from Dowd, 1971).

k

k’

Figure A.5: Experimental variograms and models (variables k and k’)

kk’

Figure A.6: Experimental cross-variogram and model.

342


Appendix A

The indexes k and k' are (k, k' = 1 to 2) Basic model: a nugget effect (i=1) and a spherical with range of 60 (i=2). The coefficient matrixes are (no nugget in x-vario = 0 diagonals). i=1 k k' k 11 0 k' 0 9

i=2 k k' k 39.0 14.5 k' 14.5 15.0

A.5.6 Calculating experimental cross-variogra ms

Recall the basic idea of an h-scatterplot using one variable. The same idea can be used on two different variables. The scatterplots of variable z(xa) at sampled locations xa against the variable Y(xa+h), i.e. at distance h, are called cross h-scatterplots. The experimental cross-variogram of variables, say z and y, is calculated from:

γ(h) = 1/n(h) Σi=1n(h) [z(xi)-z(xi+h)][y(xi)-y(xi+h)] Compare the above equation with that used to estimate a variogram.

• • • • • • •

Co-kriging integrates different data inestimating. Co-kriging is computationallyintensive. In many cases where both variables are well sampled at the same locations and the variogram and cross-variogram are more or less proportional to the same model: the improvement in the results is not substantial. There is no reason to co-krige uncorrelated variables. Co-kriging in its general form is the vector form of kriging; in most practical cases two variables are used. Co-kriging can be expanded to include a trend Before considering co-kriging the specific characteristics of the variables involved should be considered.


343

Appendix B

APPENDIX B

JOINT INTEGRATION OF ASSAY AND CROSSHOLE TOMOGRAPHIC DATA IN OREBODY MODELLING: JOINT GEOSTATISTICAL SIMULATION AND APPLICATION AT MOUNT ISA MINE, QUEENSLAND

(R. Dimitrakopoulos and K. Kaklis, 2000 –IMM)

B.1

Introduction

Traditional orebody modelling1,2 is based on borehole assay data. Recent developments in geophysical imaging in metalliferous mining environments,3,4 particularly cross-hole tomography, have generated a substantial amount of information which can be incorporated into the orebody modelling process. However, the ‘fuzzy’ nature of the cross-hole tomographic data requires technologies that can integrate diverse data. This issue of data integration or fusion is not unique to orebody modelling and mining. Integrating diverse information such as borehole and geophysical data or, more generally, integrating ‘hard’ and ‘soft’ spatial data, is well known in the petroleum industry.5-8 However, despite developments in other fields, orebody modelling based on diverse data integration is in its early stages due to the fairly recent development of inmine geophysical methods as well as related problems unique to the hard rock mining 3

environment. Geostatistical simulations9-12 are increasingly used for orebody modelling and mine planning in both open pit and underground mining ventures.13-18 Conditional simulation techniques provide the means to quantify and assess the geological uncertainty of orebody attributes, as well as link this uncertainty to engineering processes, profitability criteria and decision-making. This study presents an extended conditional indicator simulation method for the integration of cross-hole tomographic and borehole assay data, as well as its application at Mt Isa Mines, Queensland, Australia. In the next section, the conditional simulation method is first outlined. Subsequently, the application at Mt Isa Mines demonstrates the use of the algorithm in the integration of cross-hole RIM (Radio Imaging Method) conductivity data with geochemical copper diamond drill hole (DDH) assay data. Conclusions and recommendations follow.

B.2

Sequential Indicator Simulation with Soft Data

This section presents an extended sequential simulation algorithm that can be used to generate equally probable models of orebody attributes while integrating diverse data.

344


Appendix B

B.2.1 The concept of sequential simulation: A recall

Consider a stationary and ergodic RF Z(x) with a multivariate probability density function f(x 1,...,xM; z 1,...,z M). Z(x) could, in geostatistical terminology, represent the grade of a mineral deposit. Sequential simulation is based on the decomposition of the multivariate probability density function (pdf) of the RF Z(x) into a product of univariate conditional distributions,20 f(x1,...,xM; z 1,...,z M) = f(x 1;z 1) • f (x2 ;z2⏐ Z(x11)=z ) • ... • f(xM;zM⏐Z(xa)=za, a=1,...,M-1)

(1)

where M is the number of locations within the orebody and z a is a value at location x a {a=1,...,M-1}. The decomposition in eq. 1 shows that when generating a realisation of Z(x), the first draw ing comes from the marginal distribution f(x 1;z 1). The second drawing comes from the distribution f(x 2;z 2⏐ Z(x11)=z ), which is conditional to the value z1 drawn from f(x 1;z 1), and so on. If all the univariate conditional distributions functions (cdf) in eq. 1 are known, images of Z(x) can be simulated by sequentially drawing from each of the M conditional distributions. In practice, an initial data set, {z(x a), a=1,...,N < M}, is available. The sequential drawing begins at the N+1 step and the first value comes from the univariate conditional distribution f(x N+1;z N+1⏐Z(xa)=za, a=1,...,N). Subsequently, the remaining pdf’s are estimated one by one, each time conditional to the previously drawn values. The simulation and drawings stop when the last conditional distribution is estimated and a value is picked from it. B.2.2 Joint sequential indicator simulation for hard and soft data

Sequential indicator simulation or SIS is the sequential simulation implementation based on the estimation of univariate conditional distributions using an indicator kriging approach suggested by Journel and Alabert.6 The method can be extended to account for both ‘soft’ and ‘hard’ data. Frequently, additional deposit attributes may be available as secondary data sets which can be integrated into the modelling process. For example, in a copper deposit, a primary or ‘hard’ data set may represent copper drillhole assays, while a secondary or ‘soft’ data set may consist of conductivity derived from RIM tomographic surveys in the mine. Consider a RF Z(x), as described above, discretised in to K mutually exclusive classes, I(x;zk) using a series of k cut-offs. The SIS objective is the simulation of the spatial distribution of the K class indicators. A secondary data set {y(x), a=1,...,M} may be integrated into the estimation of the local cumulative cdf (ccdf) of the primary attribute Z(x) by first transforming it to an indicator RF Y(x;z k) similarly to the indicator transform I(x;z k) of Z(x). The related alg orithm is as follo ws:

•

Define a random path to be followed by visiting each location x (or grid node) to be simulated. There are L grid nodes to be visited.


345

Appendix B

•

Estimate at the first location, say x 1, the whole ccdf of Z(x), using indicator kriging and for K classes zk, k=1,..., K F(x1;z|(N))*=Prob*{Z(x1)≥z|(N)} = [I(x ;z )1 k | Z(xa)≥za, a=1,...,N]* =∑a=1NwaI(xa;zk) +

∑b=1Mvb Y(xb;zk)

(2a)

•

Draw a value from the ccdf at the first location x 1 and add the corresponding results in the data set. The new data set is now {z(xa), a=1,...,N+1}

•

Move to the second location in the path, say x 2, and estimate the local ccdf for the K classes F(x2;zk| (N+1))*=Prob*{Z(x2)≥z|(N+1)} = [I(x 2;zk)| Z(xa)≥za, a=1,...,N, N+1]* = ∑a=1N+1waI(xa;zk) + ∑b=1Mvb Y(xb;zk)

•

(2b)

Draw a value from the estimated ccdf, add the value in the data set, move to x3, and repeat the process until a value is drawn from the last ccdf at location x L.

The weights wa and vb are weights for the hard and soft data respectively. Derivation of the data weights required for the above are derived from the solution of the following system of equations, termed indicator cokriging,

∑a=1NwaCI(xa-xa’) +∑b=1MvC b IY (xb- xa’)) + μ1 = CI(xo- xa), a’=1,...,N ∑a=1NwaCYI(xa-xb’) +∑b=1MvbCY(xb- xb’)) + μ2 =C IY(x o- xb’), b’=1,...,M

(3)

∑i=1Nwa = 1 , ∑i=1Mvb = 0 where CI (h;z k), C Y(h;zk), C IY(h;z k ) are the indicator covariances for cut-off kz of the primary data set, secondary data set and their cross-covariance, respectively. To reduce the need of inference and modelling of each of the above covariances (eq. 3), a Markov-Bayes formalism may be used, as suggested by Zhu, 8 to allow the calculation of CY(h;zk), C IY(h;z k) directly from C I (h;zk ), CIY(h;zk) = B(z k) C I(h;zk) CY(h;zk) = B(z k) 2C I(h;zk), for h > 0 and CY(h;zk) = |B(zk)| CI(h;zk), for h = 0

(4)

where B(zk) is generated from the calibration of the primary and secondary data. B(z) is an accuracy index, which is equal to one when the secondary data is fully equivalent to the primary.

346


Appendix B

B.3

Case Study at Mt Isa Copper Mine

B.3.1 General geology and RIM tomography

The simulation technique outlined in the previous section is used in this section to demonstrate the practical aspects of the technique in an application at the Mt Isa Copper Mine in Queensland, Australia. The case study uses conductivity data in a cross-hole region derived from a RIM tomographic survey in part of a copper orebody at Mt Isa Mines as well as DDH Cu assay data to simulate Cu grades in the cross-hole region. Mt Isa Mines are located in north-western Queensland and exploit complex copper and silver-lead-zinc orebodies. Orebodies occur within the upper part of the Urquhart Shale, a dolomitic, calcareous and pyritic unit within the Western Fold Belt of the Proterozoic Mt Isa Inlier. The formation forms part of the Proterozoic siltstone shale sequence known as the Mt. Isa Group.21 At Mt Isa, there are 35 orebodies that can be mined individually with mineralisations grouped into either stratabound silver-lead-zinc, occurring as massive sulphide beds, or discordant copper, primarily chalcopyrite occurring as irregular bodies surrounded by a silica dolomite alteration halo. 22 The so called Radio Imaging Method or RIM tomographic survey described by Stolarczyk23 is a geophysical imaging method that can be used to delineate conductive mineralization, such as copper orebodies at Mt Isa. The method was used at Mt Isa Copper Mine in 19953 at a site was located at Level 19, almost 1 km below surface. The orebody at that level is predominantly chalcopyrite with associated pyrite and pyrotite. Fig. 1 shows the location of cross-holes 951106 and 951107 used for the RIM survey. Fig. 2 shows the resulting cross-hole RIM conductivity tomogram. In addition, Fig. 2 shows the %Cu assay data at the drill hole locations. The RIM survey and conductivity tomogram at Mt Isa are detailed in Fullagar et al3 and Zhou et al,24 and are beyond the scope of this paper. Fig. 2 suggests that the overall correspondence between the RIM conductivity and the Cu assay data is reasonable. However, it is also evident that at various parts of the drillholes, there are areas of better correspondence as much as worst. Although the RIM tomogram provides valuable information, orebody modelling requires much more accurate copper grade data. Thus, there is a need to develop methods that can use the secondary ‘soft’ RIM data jointly with primary or ‘hard’ geochemical Cu assay data, when and where appropriate as to ensure the use of all available pertinent information as well as the reliability of the orebody models produced. The availability of secondary information from the RIM survey generates a new technical challenge that the technique described previous sectionto addresses. accommodates integration of secondary in datathe whose correlation the primaryIt data changes for the various ranges of values as well as spatially. This integration is obtained with the general indicator simulation approach used here (including the use of indicator cross-covariances) and the calibration through the B coefficients in eqs. 4. B.3.2 Data characteristics and spatial statistics


347

Appendix B

Characteristics of the %Cu assays and the ‘co-located’ conductivity data are summarised in Figs. 3 and 4 respectively. Data statistics are based on one-meter composites available at the cross-holes. The correlation of the co-located %Cu and conductivity data are shown in Fig. 5. The figure shows that, generally, the correlation of the co-located data is relatively weak and the correlation of the data for various Cu grade cases and conductivity vary. The implementation of the extended SIS technique is based on the selection of a series of Cu grade cut-offs. Six cut-offs are used here: 0.5, 1.0, 1.7, 2.1 and 5.2 %Cu. The 1.0 % Cu corresponds to the median grade of the drillholes available at the location of the RIM tomogram. The selection of cut-offs is combined with cut-offs on the secondary conductivity data such that the combination maximises the correlations for the two series of cut-offs. The implementation of eqs. 3 is based on the ‘mosaic’ indicator kriging model. 25 Accordingly, the indicator correlogram (or covariance) at the median cut-off is used. The use of this model is sufficient to demonstrate the technique, although further work on site could enhance future implementation. The median indicator correlogram used in this study is generated from

ρI (h;z ) = ρz (h) = 1 -γR(h)/σR2

with F(z) = 0.50

where γR(h) is the global relative variogram of the corresponding ore zone, σR2 is a relative variance, and ρz (h) is the correlogram of the actual grades. The variogram of the orebody zone the data corresponds to is given in Table 1 and is used here in the context of the above equation. The application of the simulation algorithm described earlier requires the selection of cut-offs for the secondary variable. The selected cut-offs on conductivity are 1.50, 2.00, 2.25, 2.50, 3.00 and 3.5 mV. The corresponding calibration coefficients B(z) are calculated as described by Deutsch and Journel 25 and are 0.13, 0.21, 0.09, 0.11, and 0.08 for the % Cu cut-offs 0.25, 0.50, 1.0, 2.1 and 5.2, respectively. Note that the generally low B(z) coefficients reflect the relatively low correlation of the %Cu and conductivity. This is expected as the cross-hole locations the conductivity signals are weaker in their correlations with copper grades. Note, however, that the correlation is different among cut-off combinations and that these differences are reflected in the following results. B.3.3 Simulation results and discussion

Results of the joint geostatistical simulation of %Cu and conductivity data are shown in Fig. Fig. 6The depicts two equally probable realisationsmeter of copper grades in Typically, the crosshole 6. region. simulations were run on a one-by-one grid resolution. orebody grades reflect the actual in-situ grade variability and suggest possible variations of the orebody. For instance, Fig. 6(b) indicates the possibility that there may a copper string in the top of the section that is not apparent in the first simulation in Fig. 6(a). Fig. 7 shows the copper grades in the cross-hole region derived from the averaging of 50 simulations (e-type estimate). The figure shows that there is a discontinuous copper string in the upper part of the section between the cross-holes as well as various small copper pockets above the main orebody in the lower part of the section. It is interesting 348


Appendix B

to compare this last figure to the corresponding one derived using only the locally available assay drillhole data. Fig. 8 shows the expected copper grade (e-type estimate) in the cross-hole region, derived with the same copper assay data, method and parameters as the ones used for the copper grade modelling in Fig. 7, but without the RIM conductivity data. There are several interesting points to be made. The orebody model based exclusively on the drillhole data shows a general tendency to link the grades from one drillhole to another. This appears in Fig. 8, for example, to generate a continuous string of copper in the upper part of the section as well as a more extensive orebody in the lower part of the section. With the use of background conductivity data, Fig. 7 suggests a less continuous copper string at the top of the section, a more confined orebody delineation in the bottom part of the cross-hole region and shows a more realistic representation of the orebody. As expected, the use of additional, reliable and relevant information can lead to both the better delineation of the orebody and the accurate representation of the in-situ copper grades and their variability.

B.4

Conclusions

An extended joint geostatistical simulation technique can be used to generate orebody models of ore grades using both primary (‘hard’) and secondary (‘soft’) data. This is important considering the ever-increasing generation of diverse, indirect grade data in the mine environment, including in-mine geophysical data that can further assist ore delineation, grade estimation and orebody modelling. The application of the method is demonstrated in the modelling of Cu grades in the cross-hole region in an copper orebody at Mt Isa Copper Mine, Queensland. Copper grade modelling is based on the joint use of Cu grade assays and the cross-hole RIM conductivity data. The method presented is general and can be used to integrate most types of geophysical imaging data with the orebody modelling and subsequent mine planning processes. Further improvements of the method and its application could include (i) the enhancement of results from the use of co-located drillholes and tomogram other than the cross-holes; (ii) the further application of the techniques to additional case studies and in-mine geophysical data types, as to generate more experience in the application of the technologies; and (iii) the investigation of possible support-effects that may arise from the nature of the secondary data. Acknowledgements

Thanks are in order to Mount Isa Mines and particularly Garry Fallon, Principal Geophysicist, MIM Exploration, and Peter Forrestal, General Manager - Technical Development, MIM Holdings for providing data and technical collaboration.

Strike

Dip

Pitch

Variogram Sill _|_ type pitch 160 25 0 Nugget 0.27 Spherical 0.35 115 Table 1: Relative variogram model used in this study.

|| pitch

_|_ plane

90

45


349

Appendix B

Figure 1. Schematic geological section and drillholes for the RIM conductivity

tomogram at Mt Isa Mine, Queensland.

Figure 2: RIM tomogram (conductivity, mV) and borehole grades.

Note that the conductivity and Cu grades have a different range of values; however, the sequencing represented by the colour scheme is the same in both cases. ADD SCALE

350


Appendix B

Samples 59 Mean 1.73 SD 2.14 CV 1.24 Min 0.38 25th% 0.42 Median 0.97 75th% 2.28 Max 8.11

Figure 3: Histogram of the diamond drillhole derived %Cu assays.

Samples Mean CV Min 25th % 2.09 Median 2.69 75th % 3.22 Max

59 2.72 0.29 1.07

5.06

Figure 4: Histogram of the co-located conductivity (mV) data from the RIM survey

Figure 5: Scatter plot of the %Cu or primary versus conductivity (mV) or secondary

data.


351

Appendix B

(a)

(b) Figure 6: Two realisations of %Cu grade in the cross-hole region generated from both

Cu DDH assay and RIM derived conductivity (mV) data.

352


Appendix B

Figure 7: An average (e-type) model of %Cu grade in the cross-hole region generated

from 50 simulations based on both DDH Cu assay and RIM derived conductivity (mV) data.

Figure 8: An average (e-type) model of %Cu grade in the cross-hole region generated

from 50 simulations and based exclusively on DDH Cu assay data.


353

Appendix B

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

12. 13. 14. 15. 16. 17. 18. 19. 20. 21.

354

David M. Geostatistical ore reserve estimation, Elsevier Scientific Publishing, Amsterdam, 1977, 364p. David M. Handbook of applied advanced geostatistical ore reserve estimation. Elsevier Scientific Publishing, Amsterdam, 1988, 216p. Fullagar P. K. et al. Implementation of geophysics at metalliferous mines. Report No MM 1-9611 1, AMIRA Project P436, 1996. Fallon G. N., Fullagar P. K. and Sheard S. N. Application of geophysics in metalliferous mines. Australian J. of Earth Sciences, 44, no. 4, 1997, 391-409. Yarus J. M. and Chambers R. L. Stochastic modeling and geostatistics. AAPG Computer Applications in Geology, no.3, Tulsa, Ok, 1994, 380p. Journel A. G. and Alabert F. New method for reservoir mapping. Journal of Petroleum Technology, February 1990, 212-218. Verly G. Sequential Gaussian co-simulation: A simulation method integrating several types of information. In Soaers A. ed. Geostatistics Troia'92, 1992 (Dordtrecht: Kluwer Academic, 1992), 543-554. Zhu H. Modeling mixtures of spatial distributions with integration of soft data. Stanford: PhD thesis, Stanford University, 1991, 180p. Journel A. G. Geostatistics for the conditional simulation of orebodies. Economic Geology, 69, 1974, 673-680. Dimitrakopoulos R. Conditional simulation of intrinsic random functions of order k. Mathematical Geology, 22, 1990, 361-380. Journel A. G. Modelling uncertainly: Some conceptual thoughts . In Dimitrakopoulos R. ed. Geostatistics for the Next Century, 1994 (Dordrecht: Kluwer Academic Publishers, 1994), 30-43. Armstrong, M. and Dowd P. A. Geostatistical simulations. Kluwer Academic Publishers, Dordtrecht, 1994, 182p. Dowd P. A. Risk assessment in reserve estimation and open pit planning. Trans. IMM (Sect. A: Min. Industry), 103, 1994, A148-A154. Dowd P. A. Risk in mineral’s projects: analysis, prediction and management. Trans. IMM (Sect. A: Min. Industry), 106, 1997, A9-A17. Ravenscroft P. J. Risk analysis for mine scheduling by conditional simulation. Trans. Instn Min. Metall. (Sect. A: Min. Industry), 101, 1992, A104-A108. Dimitrakopoulos R. Conditional simulation algorithms for modelling orebody uncertainty in open pit optimisation. Int. J. of Surface Mining, 12, no. 4, 1998, 173-179. Dimitrakopoulos R. Geostatistical simulations for the mining industry. WH Bryan Mining Geology Research Centre, Brisbane, Australia, 1999. Glacken I. Change of support and use of economic parameters for block selection. In Baafi E. and Schofield N. eds. Geostatistics Woolongong '96, 1997 . (Dordtrecht: Academic, 723- 234 Halton J. H. Kluwer A retrospective and1997), prospective survey of the Monte Carlo method. In Zemanian A. H. ed. S.I.A.M. Review, Society for Industrial and Applied Mathematics, 1970, 1-63 Johnson M. Multivariate statistical simulation. John Wisley & Sons, New York, NY, 1987. Perkins W.G. Mount Isa silica-dolomite and copper orebodies; the result of a syntectonic hydrothermal alteration system. Economic Geology, 79, 1984, 601 – 637. Applied Risk Assessment for Ore Reserves and Mine Planning: Conditional simulation for the mining industry

Appendix B

22. 23. 24. 25. 26.

Sims D. A. and Bartrop S. B. Exploration in a mature mine - The Mount Isa Mine strategy. International Mining Geology Conference, AusIMM, 1993, 25 - 29 Stolarczyk L. G. Definition imaging of an orebody with the radio imaging method (RIM). IEEE Transactions Industy Applications, 28, 1992, 1141-1147. Zhou B., Fullagar P. K. and Fallon G. N. Radio frequency tomography trial at Mt Isa Mine. Exploration Geophysics, 29, 1998, 675-679. Journel A. G. Non-parametric estimation of spatial distributions. Mathematical Geology, 15, no. 3, 1983, 445-468. Deutsch C. V. and Journel A. G. GSLIB: Geostatistical software library and user’s guide. Oxford University Press, New York, NY, 1998, 369p.


355

Appendix C

APPENDIX C.A

C.A.1

QUANTIFICATION OF FAULT UNCERTAINTY AND RISK MANAGEMENT IN LONGWALL COAL MINING: BACKANALYSIS STUDY AT NORTH GOONYELLA MINE, QUEENSLAND

Abstract

A method foris fault and risk assessment based the concept of stochastic simulations used uncertainty to back-analyse data from mined outonlongwall panels at North Goonyella mine, Queensland. The results from back-analysis show that (i) fault risk can be quantified; (ii) quantified fault risk can be integrated into longwall design and assist decision making; and (iii) if the simulation technologies were available earlier, geological risk at North Goonyella would have been substantially better understood, therefore could have had a major positive economic impact. Lastly, a comparison with a specific part of the Goonyella-Riverside area suggests that the latter is less risky for a comparable longwall design. The study shows the contribution of the quantified risk approach to reducing coal mining investment risks and facilitating more informed decisions. C.A.2

Introduction

Fault uncertainty and risk have widely recognised adverse impacts on the exploration and mining of underground coal deposits, especially longwall mining. Geological uncertainties may cause significant delays in production schedules, impose substantial changes to mine plans, reduce expected recoverable coal quantities, adversely affect safety, and heavily influence the financial viability of a mine. As Australia’s coal mining industry is becoming increasingly reliant upon longwall mining, there is a need to implement a more effective, quantitative and practical approach to geological risk modelling, uncertainty assessment and integration of risk management. This will enable mining companies to better plan underground exploration activities and longwall operations. The ACARP project C7025 “Quantification of fault uncertainty and risk management in underground coal mining” which aimed to contribute to meeting the above needs was recently completed. The methods developed, case studies and tests are detailed in Dimitrakopoulos et al (2001), and the tools assisting the implementation of the methods are presented in Li et al (2001). A key aspect of the above project is the back-analysis at the North Goonyella mine. Back-analysis aims to (a) assess the effectiveness and validity of methods for fault uncertainty and risk quantification developed by the project in a mined out part of the mine, where all faults have been mapped in detail; and (b) show that if the technologies from the above project were available earlier, the fault related risk at North Goonyella would have been more accurately understood, therefore improving decision-making. Complementary to the above is the (c) comparison of quantified risk at North Goonyella to that of Goonyella-Riverside. These three aspects of the ACARP project are presented in this paper.

356


Appendix C

The results reported here are based on a new method developed for modelling fault uncertainty and quantifying risk. The method and the back-analysis are based on the concept of spatial stochastic simulations. The key idea in stochastic fault simulation is that from an initial set of fault data, one can generate multiple equally probable models (realisations) of the faults within a study area. The combination of these equally probable models provides the means to quantify fault risk. Characteristics of fault simulations include: (i) fault realisations are based on and reproduce all the available data and geological interpretations available; (ii) realisations reproduce the statistical characteristics of the fault data including the key “power-law” relationships of fault size distributions and length versus maximum throw of the fault data. For further information on the method, the reader is referred to the ACARP C7025 project report available from ACARP. C.A.3

Back-analysis: steps and assumptions

North Goonyella mine is located in the Bowen Basin, Central Queensland. Backanalysis uses mined out longwalls of the mine where faults have been mapped in detail. Geological interpretations or other information are not available. Figure CA.1(a) shows the available and completely known dataset. The steps involved in back-analysis are as follows. (i) The fully known fault data are sampled to generate a sample (sub-) fault dataset as shown in Figure CA.1(b); (ii) 50 fault simulations are run based on the sample fault data set and its statistical characteristics; (iii) fault probability maps derived from the simulations are compared to the complete fault dataset; (iv) a longwall design is used to quantify fault risk and is compared to the known risk of longwall panels in the study area. The back-analysis study assumes that there would be a reasonable level of exploration carried out in the parts of the area to be considered, and that the proportion of known smaller faults reflects the levels to which such faults are known after typical exploration activities. C.A.4

Fault simulation, uncertainty assessment and risk mapping

Two simulated realisations of the fault populations in the study area are shown in Figure CA.2. The comparison to the complete dataset in Figure CA.1(a) provides an insight to the concept of fault simulations. The specifics of the fault simulation within the study area are beyond the scope of this paper and are given in the C7025 project report. Figure CA.3(a) shows the fault probability map generated from the 50 fault realisations used here. The fault probability map can be assessed and compared against the fully known fault dataset.


357

Appendix C

Figure CA.1: (a) Mapped

fault dataset from mined

(a)

(b)

out part at North Goonyella mine; (b) sample fault dataset (faults shown have a throw ≥ 1m).

Figure CA.2: Two fault

realisations using the sample fault dataset at North Goonyella mine (faults shown have a throw ≥ 1m).

358


Appendix C

2 1 2 1 3 1

Figure CA.3: Probabilities from: (A)

50 fault realisations and the fault simulation approach in this study; (B) the sample fault dataset alone, without the use of fault simulations; and (C) from the fully known fault dataset and the real fault probability map in the study area. Figure CA.3 compares various fault probability maps, showing that the technologies used inthe thisprobability study can map predict risk upon particularly well. specifically, CA.3 (B) shows based the faults in More the sample dataset Figure alone (70 faults with throw ≥1m). As discussed earlier, the quantification of risk based on the sample data seems severely reduced in comparison with that seen in Figure CA.3 (A), which is the probability map resulting from 50 fault realisations (about 207 faults with throw ≥1m). Figure CA.3 (C) presents the probability map from the fully known fault dataset (231 faults with throw ≥ 1m). The fault simulation technologies are able to use an exploration-like level of information to generate a more reasonable assessment of fault risk throughout the study area than the spatially limited and incomplete sample dataset alone that could be any “exploration data”. The probability map based upon 50 fault realisations, shown in Figure CA.3 (A) can be compared to the probability map of the complete data set in Figure CA.3 C. In Figure CA.3, location numbered with 1 highlights an actual high-risk part of the study area that is predicted by the fault simulation methods developed in this project. Number 2 highlights a part of the study area in which risk is overestimated by the fault simulation methods. Number 3 highlights a part of the study area in which a high-risk area in the probability map based upon the realisations is slightly shifted with respect to its position in the fully known fault dataset. It should be noted that the risk assessments based strictly on the sample dataset underestimate risk within the mining area, as is expected. The probability map resulting from the fault realisations provides a substantially closer estimate to the actual risk scenario than that based upon the sample dataset.


359

Appendix C

Data collection expenditure (drilling or geophysical surveying) is one additional example where quantification of geological risk has the potential to improve current practices. For instance, if one considers Figure CA.3, the expenditure for further seismic surveys may be prioritised and there may be two or three approaches to the task. The first involves the targeting of low risk panels to verify the simulation predictions. Secondly, if an ambitious approach is decided upon, the risky parts of the area may be investigated to determine if indeed there are significant faulting concerns or not. Thirdly, as the middle values of fault probability reflect those areas about which the least is known, they can be explored (their exploration may be prioritised) and then classified with greater certainty.

C.A.5

Integration of quantified risk in longwall decisions making at north Goonyella

The levels of risk associated with a longwall layout used at North Goonyella mine are assessed in this section. In all cases presented here, risk is calculated and reported for an equivalent longwall panel size of 200m x 2000m. Risk is calculated and reported for a given (constant) longwall panel size in order to be physically and statistically meaningful and can be calculated for any longwall panel sizes as needed. Figure CA.4 shows the probability and risk associated with the mine and the longwall panels when all faults within the fully known dataset with throw greater than or equal to 1 m are considered. This is the ‘true risk’ scenario. Figure CA.5 shows the fault probability and risk associated with the mine and the longwall panels based upon the probability map obtained from fault simulation. Spatial distribution of risk, the histogram of the risk distribution and the related descriptive statistics are shown. The assessment of risk shown in Figure CA.5 is very close to that of the actual scenario shown in Figure CA.4. Figure CA.6 shows the fault probabilities and risk associated with mine longwall panels based upon the sample fault dataset only. The spatial distribution of risk, the histogram of the risk distribution and the related descriptive statistics are shown. The figure shows a fault risk assessment that could be anticipated at a relatively early stage of exploration without the use of computerised fault modelling technologies. By comparing Figure CA.6 with the risk distribution of the fully known fault dataset in Figure CA.4, it can be seen that the use of exploration data alone results in severe underestimation of geological risk. Comparison of Figure CA.4 and Figure CA.5 suggests that the use of fault simulation technologies provide an excellent assessment of the actual fault risk levels in the North Goonyella example. Comparison to fault CA.6dataset showsalone. that the estimated fault risk is far closer than that obtained using the sample

360


Appendix C

0.6 0.9 - 1.0 0.8 - 0.9 0.7 - 0.8 0.6 - 0.7 0.5 - 0.6 0.4 - 0.5 0.3 - 0.4 0.2 - 0.3 0.1 - 0.2

0.5

0.4 y c n e u0.3 q e r F 0.2

0.1

0 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fault probability

Descriptive Mean S ta n d a rd Er r o r

Figure CA.4: Spatial

stat istics

Mo ed de ian M S a mp l e V a r i a n c e M inim um M a ixm um Num ber of equ iv al ent pan el s

(a) (a)

distribution of fault

0 .61 0 .04 .63 00.69 0 .01 0 .49 0 .72 7

(b)

Fault probability in mined longwall panels a) spatial distribution; b) histogram distribution (fault throw >= 1m, panel size 0.2x2.0km)

probability and ‘true’ risk in a longwall layout at North Goonyella in (a); histogram and descriptive statistics of same in (b) using the fully known dataset.

0.6 0.5

0.9 - 1.0 0.8 - 0.9 0.7 - 0.8 0.6 - 0.7 0.5 - 0.6 0.4 - 0.5 0.3 - 0.4 0.2 - 0.3 0.1 - 0.2

0.4 y c n e u 0.3 q e r F 0.2


0.1

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0 .2

0.3

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0 .5

0 .6

0.7

0 .8

0.9

1

Fault probability

D e s c rip tiv e s ta ti s tic s

M ean

0 .5 9

S t a n d a r dE r r o r M e d ia n M ode S a m p l eV a r i a n c e M in im u m M a x im u m Nu mbe r of eq ui val en t pa ne l s

0 .0 4 0 .5 6 n /a 0 .0 1 0 .4 5 0 .7 6 7

(a)

(b)

Fault probability and risk in North Goonyella mine based on the simulated fault probability using a sample dataset a) spatial distribution; b) histogram distribution (fault throw >= 1m, panel size equivalent 0.2 x 2.0 km)

0.9 - 1.0 0.8 - 0.9 0.7 - 0.8 0.6 - 0.7 0.5 - 0.6 0.4 - 0.5 0.3 - 0.4 0.2 - 0.3 0.1 - 0.2

distribution of fault probability and risk in a longwall layout at North Goonyella mine in (a); histogram and descriptive statistics of same in (b) as calculated using the fault probability map generated from realisations based on the sample fault dataset.

0. 6

0. 5

0. 4 y c n e u 0. 3 q e r F 0. 2

0. 1


0 0 .1

0 .2

0 .3

0 .4

0 .5

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0 .8

0 .9

1

Fault Probability Descriptive Statistics

n a eS Mt andar d E r r or

0 . 2 80 . 0 4

indae M e d oM Sampl e Var i ance M i inm u ixM am m u N u m b e r o f e q u iv a le n t p a n e ls

0.25 0.30 0.01 0.18 0.48

(a)

7

(b) Fault probability for a longwall layout at North Goonyella mine based on the sample dataset a) spatial distribution; b) histogram distribution (fault throw >= 1m, panel size equivalent 0.2 x 2.0 km)

distribution of fault probability and risk in a longwall layout at North Goonyella mine in (a); histogram and descriptive statistics of same in (b) as calculated based on the sample dataset only.


361

Appendix C

Exploration provides fault data is commonly used ‘as is’ which implicitly corresponds to the level of risk in the faults identified in the dataset. If technologies such as seismic methods are used, faults below resolution are not detected and as a result any ‘risk’ assessment based on the dataset ‘as is’, will underestimate the true risk. The opposite is also possible, for example, other geophysical methods such as aeromagnetics are highly interpretative and may result in the over representation of interpreted faults which if used ‘as is’ will produce an overestimated risk assessment. Qualitative risk assessments are, in addition to the above, difficult to use because they implicitly reflect fixed information similarly to the use of the ‘dataset as is’. Furthermore, quantitative assessments are highly subjective and a relative measure, not a physically meaningful probability. The geological interpretations that are used in the coding of soft data do not necessarily reflect the geological risk assessment provided by the simulations nor the true risk scenario. This is perhaps obvious, but also useful to recall in uncertainty modelling and risk assessment. To enhance the understanding of an area, further exploration could be carried out which may be costly and would not guarantee uncertainty reduction or the optimisation of data collection. Using fault simulation technologies provides a means to utilise exploration data to gain a more complete and quantitative understanding of a relatively unknown area than otherwise achievable (from qualitative investigations). Simulation technologies such as those developed in this project could be utilised in combination with the information acquired from remote sensing surveys. Although the example presented here is simple, it shows that probabilistic models, in combination with current exploration, offer a way to obtain a more complete picture. C.A.6

Is a longwall mine at Goonyella-riverside as risky as north Goonyella mine?

A comparison can be made between the levels of risk at North Goonyella mine and a potentially mineable part of Goonyella-Riverside. The histograms of risk associated with a longwall panel layout can be seen in Figure CA.7. The longwall layout shown for North Goonyella Mine is the same as that used above in the quantification of fault risk. The layout shown for the part of Goonyella-Riverside has been designed without consideration of quantified geological risk - while the “exploration data” is available. Using the basic map of fault locations, it is apparent that risk is fairly high and may be classified as such using a qualitative approach. It is only through quantifying risk using true probabilities that the difference in geological risk between the two areas can be grasped. Using the longwall layouts presented, the average fault probability at North Goonyella is 61%, and for the study area in Goonyella-Riverside it is 39%. The comparison shows that the longwall layout designed for part of Goonyella-Riverside has, on average, a substantially reduced level of risk compared to the mined-out parts of North Goonyella. Usingit the quantified risk assessment provided the technologies developed in this study, is possible to redesign the longwall layoutby of both locations to reduce risk. The quantitative approach is the only method that can determine which parts of the mineable area have lower risk and lead to an optimal redesign of the layout. C.A.7

Fault uncertainty and mineable coal reserve risk

To link the quantification of fault risk to mine economics, the risk associated with the mineable coal reserve can be calculated. In all cases presented here, calculations

362


Appendix C

assume an equivalent longwall panel size of 200m x 2000m, an average coal seam thickness at North Goonyella Mine of 4.5m, and a coal relative density of 1.6. Figure CA.8(a) shows a histogram of the classification of quantities of mineable coal reserve at various risk levels as calculated using the fully known fault dataset. This is the true risk scenario. Figure CA.8(b) shows a histogram of the classification of quantities of mineable coal reserves and their fault risk levels based upon the fault simulation technologies in this study. Figure CA.8(c) shows a histogram of the classification of quantities of mineable reserve using only the sample dataset. Figure CA.8(a) can be compared to Figures CA.8(b) and CA.8(c) to evaluate the impact of considering quantified geological risk on mineable coal reserves. The estimate generated from the fault simulation technologies (Figure CA.8(b)) is very close to that of the “true” scenario (CA.8(a)). Risk is underestimated when only the available data is used (Figure CA.8c). When mine design and planning is conducted without the explicit consideration of risk, it is in fact conducted as if the risk levels were those shown in Figure CA.8(c) (i.e. largely underestimated). A comparison with Figure CA.8(a) shows that the real risk situation is not reflected by a sample fault dataset. The results in Figure CA.5 and Figure CA.8(b) provide information that could have been used in the decision-making processes at the time of longwall layout design. Histogram of fault probability distribution for 0.2 by 2.0km longwall panels

0.9- 1.0 0.8- 0.9 0.7- 0.8 0.6- 0.7 0.5- 0.6 0.4- 0.5 0.3- 0.4 0.1- 0.2 0.1- 0.2

0.6

0.9-1.0 0.8-0.9 0.7-0.8 0.6-0.7 0.5-0.6 0.4-0.5 0.3-0.4 0.2-0.3 0.1-0.2

0.5

0.4

0.5

0.4

y c n e u

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q re F

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0.4

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0.9

1

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Fault probability

statis tics M nae 0 .61 Sta ndard Er ro r 0. 04 Medi an 0 .63 M edo 0 .69 S a m p le V a ria n c e 0.01 Mi ni mum 0 .4 9 Maxi mum 0 .72 Num be r o f p ane ls 7

0.2 0. 3

0.4 0. 5

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1

Fault probability

Descriptive

(a)

Histogram of fault probability distribution for 0.2 by 2.0km longwall panels 0.6

(a)

(b)

Fault probability mined in longwall panels a) spatial distribution; b) histogramdistribution (fault throw >= 1m, panel size 0.2x2.0km)

Descriptive statistics

Mean

0.39

Fault probability in StandardError 0.02 proposed longwall panelsMedian 0.40 Mode 0.40 Layout without considering SampleVariance 0.01 quantified fault probability 0.20 a) spatial distribution; Minimum Maximum 0.70 b) histogramdist ribution Total area of panels 19200000 (fault thro w 1m, panelsize Number of equivalent longwall panels 48 0.2x2.0km) (b)

Figure CA.7: Spatial distribution of fault probability and histograms of risk for North

Goonyella mine based on mined out dataset (left) and for part of Goonyella-Riverside (right) based upon simulations using the available sample dataset and calculated for given layouts and a panel size equivalent of 200m x 2000m.


363

Appendix C

10

10 8.64

9 t) 8 (M e7 v r e s6 e R l 5 a o C4 le b3 a in M2

8 t) 7 M ( e

5.76

R l 5 a o

2.88

C le 4 b a n i M 3

2.88

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0.3

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Risk

(b) Figure CA.8: (a) ‘True’ classification of

10

9

8

a in M

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0.1

C e l b

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0

R l a o

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1

t) (M e rv e s e

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9

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5

4 2.88

3

2

1

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Risk

C.A.8

0 .8

0 .9

1

(c)

mineable coal reserves based on quantified fault risk, calculated from the fully known fault dataset; (b) estimated classification of mineable coal reserves based upon quantified fault risk, calculated following use of the computerised fault simulation technologies developed for this study and; (c) estimated classification of mineable coal reserves based upon quantified fault risk, calculated from the sample fault dataset only.

Conclusions

The back-analysis study at North Goonyella mine shows that the quantification of fault uncertainty is feasible and contributes to the implementation of risk management strategies, both in terms of exploration and mine planning. In addition, it shows that the technologies generated in ACARP project C7025 work very well. The real risk of faults in longwall panels in North Goonyella is particularly well predicted from a subset of faults within the same area and the fault simulation method in the ACARP project C7025. More specifically, this means that if the technologies from this project were available at the time North Goonyella mine was design, subsequent substantial fault related problems could have been minimized. The comparison of mined out longwalls at North Goonyella to a potential longwall mine located in the Goonyella-Riverside area provides an eloquent example of the use of quantified uncertainty and risk for investment decision-making. The ability to compare risk in a longwall layout at Goonyella-Riverside to that in North Goonyella is uniquely based on the quantification of geological risk. The back-analysis study also shows that the traditional use of geological data from exploration programs ‘as is’ may severely underestimate geological risk. Qualitative risk assessments, although technically simpler, may be misleading and difficult to use because they implicitly reflect information similar to that from exploration programs taken ‘as is’. Furthermore, qualitative assessments are highly subjective and do not provide an objective “number” (probability) as needed for risk comparison. 364


Appendix C

Qualitative risk assessments are generally difficult to link meaningfully to mine design and planning, whereas quantitative risk assessments that incorporate local geological understanding offer accurate risk assessments and can be used directly and routinely in longwall mine design and planning. Coal reserve risk is a specific topic that the methods and examples presented in this paper can be applied to. Reserves can be classified based on their fault risk level and the possibility of not being recovered from mining. Cost effective data collection, be it additional drilling or geophysical surveys, can be supported and strategies for prioritised data gathering developed based on fault risk quantification. Lastly, technologies developed in this project can serve as a starting point for further development of risk quantifying technologies that can offer an inexpensive method to acquire information to substantially assist decision-making in both longwall exploration, longwall mine design, mine development and operations.

C.A.9

Acknowledgement

Acknowledgements are in order to the Australian Coal Research Association for selecting and funding this research project, Anglo Coal, Newlands Coal Mine for additional funding and collaboration; BHP Coal, CSIRO, North Goonyella Mine and MIM Holdings for support and collaboration. Thanks are in order to the project monitors: A. Wilson, P. Forrestal, S. Peau, D. Mathew, P. Caddihy and J. Sleeman. We appreciated their dedication and pragmatic views. We thank D. Dunn and B. Coutts, (BHP Coal), A. Laws (Anglo Coal), M. Barker (Newlands) and T. Britton (North Goonyella Coal Mines), as well as CSIRO’s J. Esterle, G. LeBlanc Smith and R. Sliwa for technical collaboration and data provision. The support and contribution of P. Hatherly (CSIRO/CMTE), particularly in the conception of this project, was greatly appreciated. Last but not least, we thank Bruce Robertson (Anglo Coal) and Alan Davis (BHP Coal) for supporting and encouraging the undertaking of ACARP C7025.

C.A.10

References

Dimitrakopoulos, R., Li, S., Scott, J. and Mackie, S., 2001, Quantification of Fault Uncertainty and Risk Management in Underground Longwall Coal Mining , ACARP Project C7025 Report, Volume I, W H Bryan Mining Geology Research Centre, The University of Queensland, 215p. Li, S., Dimitrakopoulos, R., Scott, J. and Mackie, S., 2001, Quantification of Fault Uncertainty and Risk Management in Underground Longwall Coal Mining , ACARP Project C7025 Report, Volume II, W H Bryan Mining Geology Research Centre, The University of Queensland, 88p.


365

Appendix C

APPENDIX C.B

C.B.1

CONDITIONAL SIMULATION OF FAULTS AND UNCERTAINTY ASSESSMENT IN LONGWALL COAL MINING

Introduction

Longwall mining sustains heavy financial losses by the presence of faults that generate delays in production schedules, changes in mine plans, and loss of coal reserves, making the assessment and quantification of fault uncertainty of major concern in mine planning and uncertainty operations. of Mining will improve if methodsand arequantitatively developed to integrate quantify the faults,operations assess geological interpretations such assessments in mining decisions and longwall mining practices. A key requirement is the ability to quantitatively model the uncertainties associated with faults and quantify the associated risk. To date, research has been limited in the areas of understanding fault uncertainty and simulating fault systems in longwall coal mining. Hatherly et al. (1993) present a study on fault prediction based on structural and geophysical methods. Fault simulation in longwall mining is shown in Li et al. (1999) and is founded on the inhomogeneous Poisson process, an approach limited in accounting for spatial correlations in fault locations and soft information. Theoretical developments and applications of stochastic simulation of faults are available in the petroleum literature, where interest is focused on the simulation of sub-seismic faults. Recent developments include marked point processes and factorial kriging (Wen and Sinding-Larsen, 1997), Gibbs fields (Omre et al, 1994; Munthe et al., 1993) and hierarchical modelling (Ivanova, 1997). Fault systems are simulated with a new approach as to quantify uncertainty in longwall mining. The approach is based on the use of correlated probability values for the thinning of a Poisson process used to generate the locations of the centres of the fault traces. Probability fields are simulated in an indicator framework and accommodate soft data coded as prior distributions. In the following paragraphs, the proposed stochastic fault conditional simulation algorithm is described. The application of the algorithm in a coal field follows. Subsequently, the quantification of fault uncertainty is presented and ramifications for longwall mining are discussed. C.B.2

Statistical description of fault populations

Fault populations are characterised by statistically describing fault parameters, specifically fault strike, dip, length, and throw, as well as the spatial arrangement of the centres of the fault traces. The relationship between fault parameters such as length versus maximum throw are also of critical importance in characterising populations. Conditional fault simulations are based on the statistical description of fault populations. Fault populations may be geometrically subdivided using differences in strike or dip (Fossen and Rornes, 1996). Fault strikes may reflect different structural events and are observed in various data sets when plotted as a histogram or rose diagram. In addition to strike, fault size (length and throw) distributions are key parameters; fault size distributions seem to follow a power-law (fractal) model over a wide range of fault size (Childs et al., 1990; Heffer and Bevan, 1990; Walsh et al., 1994; Marrett and Allmendinger, 1991, 1992; Fossen and Rornes, 1996) such that 366


Appendix C

logNS = α - β logS

(1)

where NS is the cumulative number of faults with either length or throw greater than size S; S is the fault length (L) or throw (T); α is a scaling factor function of fault density in given area, when α is high the fault density is high; β is the fractal dimension of the fault population that defines the relative number of large and small faults, when β is high the number of small faults is high relative to the number of large faults. The interrelationship between fault length and maximum throw is widely accepted to follow a power-law (Watterson, 1986; Walsh and Watterson, 1988; Marrett and Allmendinger, 1991; Cowie and Scholz, 1992; Clark and Cox, 1996) written as Tmax = c Ln

(2)

where Tmax is the maximum fault throw; L is the fault length; c is a constant reflecting rock properties and n a constant. Both c and n are obtained from data and are subject to ambiguities in fault interpretations. C.B.3

Fault simulation based on probability fields

The algorithm proposed herein is based on the statistical description of a fault population as described in the previous section and the thinning of a Poisson process (Stoyan and Stoyan, 1994) using a correlated probability field similar to the probability fields proposed by Srivastava (1992). The proposed conditional simulation of faults proceeds as follows: 1. Define a random path to be followed in visiting locations x to be considered as centers of fault traces. There are N locations or grid nodes {xi, i=1,…N} to be potentially visited. The N locations exclude the known fault centers. 2. Generate a realization of an autocorrelated probability field {p(xi), i=1,…,N} reproducing the uniform marginal cdf and the covariance C p(h) corresponding to the covariance Cx(h) of the uniform transform of the fault densities in the study area. 3. Estimate at the first location xi the intensity function of an inhomogeneous Poisson process λ(xi) using a planar Epanecnikov kernel (see Appendix 1). 4. Use the probability value p(xi) at location xi to thin a Poisson process from 1 - p(xi) < λ(xi) / λ*

(3)

where λ* is the intensity of a corresponding homogenous Poisson process and λ(x) ≤ λ*. If the above constraint is met, a fault center exists at xi. If not, the next node on the random path is visited until the constraint is met. 5. Select a maximum fault throw from the power-law model of the fault size distribution in Eq. 1. 6. Grow the fault from the center of a fault trace in opposite directions from the following steps: Risk Analysis for Ore Reserves and Strategic Mine Planning: Stochastic models and applications

367

Appendix C

6.1 6.2 6.3 6.4

Define the fault strike by randomly drawing from the fault strike distribution. Define the fault length from the power-law model between fault length and maximum throw in Eq. 2. Use a distance step and a directional tolerance at each step to grow the fault. Stop the growth when the fault length has been reached.

7. Repeat points 3 to 6 until the number of total faults satisfies the fault size distribution in Eq. 1. 8. Repeat the process to generate additional realisations. The fault simulations described above reproduce the faults identified within a study area while existing and simulated faults reproduce the desired statistical fault characteristics. The simulation of the probability field in step 2 of the above algorithm requires further consideration. The probability field p(xi) can be generated by either a Gaussian or indicator non-conditional simulation algorithm. An indicator sequential simulation algorithm (Alabert, 1987) facilitates the use of multiple secondary information and it is implemented for this study. C.B.4

Simulating fault systems in a coal field

C.B.4.1

Data description

The study area shown in Figure CB.1 covers approximately 130 km 2 in an area where the of potential underground mines3D is being investigated. Faults in the areadevelopment are interpreted from drilling, surface coal mapping, seismic surveys and studies of the regional structural geology. Faults are mapped out on a coal seam horizon and are spatially distributed as shown in Figure CB.1. One population of normal (dip-slip) faults oriented NE-SW is present in the study area. The Rose diagram of fault strikes is also shown in Figure CB.1 and Figure CB.2 shows four zones in the central and southern part of the study area with different ground conditions, these conditions are defined by sedimentological studies and are linked to their susceptibility in structural deformations and presence of faults. Zones A, B and D are expected to have more faults than zone C due to weak ground conditions. The corresponding fault size (throw) power-law distribution is modelled using Eq. 1 and is shown in Figure CB.3 (a). The specific models inferred for the first and second population are log(NT)=2.02-1.36*log(T). It is interesting to note that the data points at the top and bottom end of each curve in Figure CB.3 (a) deviate from the model fitted. The top end deviation is due to small faults not identified by drilling, mapping, 3D seismic and geological interpretations. The bottom end deviation is due to the limited area covered during the geological study upon which the analysis is based. The relationship between fault length and throw is shown in Figure CB.3 (b). A powerlaw model is fitted according to Eq. 2 and it is Tmax=1.69L0.15 . The variogram of the uniform transform of fault density in he area is shown in Figure CB.4.

368


Appendix C

C.B.4.2

Conditional simulation

The fault simulation algorithm described in Section 3 is used here to simulate the fault population in the study area. The simulation is based on the population statistics discussed in Section 4.1. To illustrate the simulation process, Figure CB.5 shows three realisations of the probability field {p( xi), i=1,…,N} generated in Step 2 of the simulation algorithm based on the variogram in Figure CB.4. Figure CB.6 shows the fault density field, {λ(xi)/λ*, i=1,…,N} as modelled using the planar Epanecnikov kernel and a ‘smoothing factor’ of 3.5 km. The smoothing factor is selected so that all nodes in the study area use at least one existing fault center when local densities are estimated. Distance step and direction tolerance used in step 6.3 of the simulation are 200m and 50 respectively. Figure CB.7 shows 3 realisations of fault simulation in the study area based on the three probability fields in Figure CB.5. The realisations reproduce the srcinal faults and their statistical characteristics. Figure CB.7 also shows the reproduction of the strike for the fault population in the area and the reproduction of the power-law models of the throw in the same realisation; the example suggests that the reproduction of data characteristics is excellent. To utilise the geological interpretations on ‘weak ground condition’ as presented in Figure CB.2, interpretations are coded as prior probabilities. Zones A, B and D have on average high probabilities on the order of 80%, 75% and 70% respectively, while zone C has an average probability for fault occurrence set at 30%. Sequential indicator simulation is then based on updated conditional probability density functions, and is used to generate a second set of probability fields. Figure CB.8 shows three realisations of the probability field generated using the soft information. Note the similarities of the realisations in Figs. 2 and 8 with the realisations in Figure CB.5 and in the northern parts of the maps as there is no soft information in the northern part of the study area. Figure CB.9 shows the corresponding fault realisations and fault statistics. In comparison to Figure CB.7, the new set of fault simulations reflects the soft information and shows a higher fault concentration in the central part of the study area which corresponds to geological zone A and prior probabilities at about 80%. C.B.4.3

Uncertainty assessment

A large number of fault realisations are combined to generate probability maps over the study area, showing the probability of having faults of given specifications of interest. Figure CB.10 and Figure CB.11 show probability maps having faults with throws over 1 meter both without and with prior information respectively; both cases are based on 100 fault realisations. The key difference between Figs. 10 and 11 is that the high risk (probability over 40%) area is extended when soft information is used. This is expected as three of the four geological zones have a high likelihood of having faults. The probability maps quantify the uncertainty of having faults with throws larger than 1 meter that can be used to select low risk areas for mine development such as the area in the northeast part of Figure CB.11 or the southeast and southwestern part of the study area. Furthermore, the probabilities of faults may also be locally used to optimise the layout of longwall panels so as to minimise the fault occurrence within them.


369

Appendix C

Additional scenarios may be generated based on different geological interpretations, coding of prior information or the use of additional sources of soft data. Finally, the uncertainty of the fault statistics used may be assessed. C.B.5

Summary and conclusions

A method to simulate fault systems in two dimensions is founded upon the statistical characterisation of fault populations and the use of this characterisation to simulate equally probable, geologically meaningful fault populations. A new algorithm is based on the thinning of a Poisson process used to place the centres of fault traces and on generally accepted power-law models that define interrelations of fault attributes. The thinning of a Poisson process is based on a simulated probability fields reflecting the spatial continuity of fault densities and integrates soft data, including geological and geo-mechanical interpretations. The algorithm also honours the available faults at their locations and reproduces their statistics. The combination of large numbers of realisations of fault systems in an area of interest for longwall coal mining provides probability maps that quantify the uncertainty of having a fault in a given location. Probability maps are tools that can enhance longwall design, as well as plan and manage risk. However, the probability maps and related use of these maps for longwall design are currently limited in the sense that the proposed simulation algorithm does not account for the uncertainties of the statistical characterisation of fault populations; further, it is limited to fractal models which may be less flexible than other models such as multi-fractals. As the available data include geological fault interpretations, surface and underground mapping, and 3D seismic interpretations, data ‘reliability’ discrimination is an additional concern and should be further assessed. Conditional simulation of fault systems provides a new tool in assessing the spatial distribution, strike, length and throw of fault populations, as well as in quantifying fault uncertainties and risk for longwall coal mine design, planning and production management.

Strikes

Figure CB.1: Fault location and strikes distribution.

370


Appendix C

A

B C

D

Figure CB.2: Geological units with broadly similar ground conditions based on degrees

of variability in sedimentology and structure. Distr ibutio n o fFa ult Thr ow Po pula tio n 1

Fault Thr ow vs Length Population 1

100

= > w o r h T th i w s lt u a F f o . o N

10

) m ( w o r h T tl u a F

10

1

1 1

10 Fault Throw(m)

log10(N)=2.02-1.36log10(T)

1

10

100

1000

10000

Fault Length (m)

T max=1.69L 0.15

Figure CB.3: (a) Fault throws distribution for population one in the study area; (b)

relationship between fault length and throw for population one in the study area.


371

Appendix C

0.4

0.35

0.3

0.25 m a r g io r a V

0.2

0.15

North East

0.1

North-- Type=Sph Nug=0.01 C=0.255 a=900

0.05 East-- Type=Sph Nug=0.01 C=0.255 a=600

0 0

200

400

600

800

1000

1200

1400

1600

1800

2000

Distance (m)

Figure CB.4: Experimental and model variograms of fault density.

Figure CB.5: Probability field simulated without using soft data.

372


Appendix C

(a)

(b)

Figure CB.6: Fault density (a) and density ratio (b) with smoothing factor 3.5 km.

Strikes

Number vs Throw

Figure CB.7: Three realisations of faults generated without soft information.


373

Appendix C

Figure CB.8: Simulated probability fields based on soft information (Figure CB.2).

Strikes

Number vs Throw

Figure CB.9: Three realisations of faults using soft information.

374


Appendix C

Figure CB.10: Probability of having faults generated without soft information.

Figure CB.11: Probability of having faults generated with soft information.


375

Appendix C

APPENDIX I

The planar Epanecnikov kernel used to estimate the density function of an inhomogeneous Poisson point process is (Stoyan and Stoyan, 1994):

λ ( x) =

8 n ∑ eh (| x − xi |) 3πh i=1

⎧3 (| x − x |)2 i ) eh (| x − xi |) = ⎪ h2 ⎨ 4h (1 − ⎪⎩0

for...(| x − xi |) < h otherwise

where λ(x) is the density function, x is the location where the density is estimated, xi is a location within distance h from x, and h is a distance termed ‘smoothing factor’.

376


Appendix D

APPENDIX D

D.1

ADDITIONAL SIMULATIONS ALGORITHMS IN BRIEF

Probability Field Simulation (PFS)

In the previous section, the sequential algorithm used every simulated value together with the actual data set to estimate the lcpd at each grid node. In order to save computational time from re-estimating lcpd’s, Probability Field Simulation (PFS) can be used. (Figure D.1). PFS consists of the following steps:

2.8g/t

0.85 g/t

0.07g/t grade below new cut-off

?

1.3g/t 100%

(a)

Cum Prob

m: 1.24 sd: 2.3

(b)

P-field 1

0.36

0.28

0.21

0

grade below new cut-off

Grade g/t

(c) 0.85 g/t

P-field 2

.09

(e) 0.1

grade below new cut-off

0.12

0.85 g/t

0.95g/t

1.3g/t

(d)

Simulation 2

1.07g/t

1.3g/t Simulation 1

(f)

Figure D.1: Probability field simulation of gold grades in a deposit: (a) deposit; (b) estimated conditional distribution at a grid node; (c) and (d) probability fields; (e) and (f) simulations at the grid node from the sampling of conditional distributions in (b) using (e) and (f).


377

Appendix D

1. Use srcinal data to estimate the local conditional distribution at all grid nodes 2. Create a non-conditional simulation (probability field) in [0,1] using a scaled variogram of the srcinal data 3. Retain the percentile corresponding to the probability field at each the grid node 4. Repeat point 2 and retrieve percentiles in point 3 to generate additional realisations The major difference between PFS and the sequential algorithms, as well as the reason for increased speed, is in estimation of the local conditional distributions at the grid nodes. Conditional distributions are estimated in a single step and are conditional only to the srcinal data. The control over the final results and the reproduction of variograms comes from step 2, as neighbouring locations will tend to have similar probability values. This is in contrast to the sequential methods where continuity is ensured by adding the simulated data to the actual data set. For step 2 of the PFS algorithm, any simulation algorithm can be used (Srivastava, 1992), however, as speed is a concern SGS is a common choice.

D.2

Simulated Annealing (SA)

Unlike sequential and probability field simulations, simulated annealing (SA) is a combinatorial optimisation algorithm based on an analogy to the physical process of annealing as known, for example, in metallurgy. SA consists of the following steps: 1. Establish a grid of blocks with values honouring the available data and their distribution 2. Define a global ‘goodness’ criterion (objective or ‘energy' function) that minimises the miss-match of desired properties and the grid values 3. Swap randomly several non-data grid values 4. Retain the swap if the ‘energy’ function lowered, otherwise reject the swap and try another 5. Reduce the number of swaps as the ‘energy’ function gets closer to the target 6. If objective function close enough to desired value, stop the process SA is commonly implemented on a simple variogram-based objective function, where the desired objective function is the difference between the model and observed variogram. However, the objective function can be generalised as a weighted average of several components such as variograms, indicator variograms, conditional distributions, and so on. SA is not as computationally intensive as it may appear (Deutsch, 1994). For instance, annealing implementations allow several swaps and then update the ‘energy’ function instead of recalculating it after each swap. An additional aspect of annealing is that in step 4, energy or the value of the objective function is allowed to increase based on criteria such as the Boltzman distribution, thus the tolerance differences and number of swaps are larger at early stages of annealing process and decrease as time passes. The major advantage of annealing is flexibility in the reproduction of characteristics of interest. In this regard, mining applications may be improved by enhancing the spatial relations of grades when simulating a deposit. For instance, in addition to the reproduction of variograms (two point statistics), the annealing process can be formulated to impose the reproduction of multi-point statistics, which describe the

378


Appendix D

connectivity of various ore grade categories. An additional advantage of SA is that it can use experimental statistics, such as smoothed experimental variograms that may be more suitable in some applications. Some comparisons

Figure D.2: Two conditionally simulated models using the sample data and sequential Gaussian simulation algorithm.

Figure D.3: Two conditionally simulated models using the Walker Lake data and the sequential indicator simulation algorithm.

Figure D.4: Two conditionally simulated models using the Walker Lake data and the probability field simulation algorithm.


379

Stochastic Programming • Stochastic programming is an optimization technique that is used for optimizing the problems that involve uncertainty. •The goal is to a find solution that is feasible for all the possible data instances that optimize the problem

Example;(from (Birge, 1999) Intro. Stochastic Programming • • • • • •

Consider farmer Tom can plant his farm with grain, corn, and sugar beets on his 500 acres of land. Tom knows that at least 200 ton of wheat and 240 ton of corn are needed to feed his cattle. Any production in excess of these amount can be sold for $170/ton for wheat and $150/ton for corn. Any shortfall must be purchased for $238/ton for wheat and $210/ton for corn. He can also produce sugar beets and sell it for $36/ton for the first 6000 tons. However, there is an economic quota for the excess of 6000 tons sugar beets production and can be sold only at $10/ton. Based on his experience the mean yield on his land is roughly 2.5 tons/acre for wheat, 3 ton/acre for corn, 20 ton/acre sugar beats.

1

Wheat

Corn

Sugar Beets

Yield (ton/acre) Planting co st ($/acre)

2.5 150

3 230

20 260

Selling Price ($/ton)

170

150

36 (=<6000

Purchase price ($/ton) Minimum requireme nt (ton)

238 200

210 240

tons) 10 (>6000 tons ) -

500 acres available for plantin g

Linear programming formualation (LP) • Define your variables: • x1,2,3: acres of wheat, corn, sugar beets planted.(x1: wheat, x2: corn, x3: sugar beets) • w1,2,3: tons of wheat, corn, sugar beets at the favorable price. • w4: tons of sugar beets at the lower price. • y1,2: tons of wheat, corn purchased.

2

LP formulation Obj. Function: min 150x1+230x2+260x3-170w1-150w2-36w310w4+238y1+210y2 Subject to: • x1+ x2+ x3<=500 • 2.5x1+ y1- w1>=200 • 3 x2+ y2- w2>=240 • w3<=6000 • 20 x 3- w3- w4>=0 • x1,x2,x3,w1,w2,w3,w4,y1,y2 >=0

LP solution

Culture

Wheat

Corn

Sugar Beans

Plant area (acres) Production (ton) Sales (ton) Purchase (ton)

120 300 100 0

80 240 0 0

300 6000 6000 0

Profit:$118,600

3

Become friendly with the weather

• Good weather: 1.2Yield, • Bad weather: 0.8Yield,

The formulation for the 1.2Yields: Obj. Function: min 150x1+230x2+260x3-170w1-150w2-36w310w4+238y1+210y2 Subject to: • x1+ x2+ x3<=500 • 3x1+ y1- w1>=200 • 3.6x2+ y2- w2>=240 • w3<=6000 • 24x3- w3- w4>=0 • x1,x2,x3,w1,w2,w3,w4,y1,y2 >=0

4

Solution with the good weather Culture

Wheat

Corn

Sugar Beans

Plant area (acres)

183.33

66.67

250

550 350 0

240 0 0

6000 0

Production Sales (ton) (ton) Purchase (ton) Profit:$167,667

The formulation for the 0.8yields: Obj. Function: min 150x1+230x2+260x3-170w1-150w2-36w310w4+238y1+210y2 Subject to: • x1+ x2+ x3<=500 • 2x1+ y1- w1>=200 • 2.4x2+ y2- w2>=240 • w3<=6000 • 16x3- w3- w4>=0 • x1,x2,x3,w1,w2,w3,w4,y1,y2 >=0

5

Solution with the bad weather Culture Plant area (acres) Production (ton) Sales (ton)(ton) Purchase

Wheat

Corn

Sugar Beans

100 200

25 60

375 6000

0

0 180

6000 0

Profit:$59,950

What will he do? • The optimal solution is very sensitive to change on yields. The overall profit ranges from $59,950 to $167,667 • Long term weather forecasts would be very important here • The most important variable is sugar bean production • He realizes that it is impossible to make perfect decision, because the planting decision must be made now, purchase and sale decision can be made later

6

Maximization of expected profit • Assume three scenarios occur with equally probability s=1,2,3 for above average, average, below average respectively and add to purchase and sales variables. We will have for example, w32 : the amount of sugar beet sold @ favorable price if yields is average. w21 : the amount of corn sold @ favorable price if yields is above average. w13 : the amount of weat sold @ favorable price if yields is below average.

LP formulation Obj. Function: min 150x1+230x2+260x3+1/3(-170w11-150w21-36w31 -10w41 +238y11+210y21)+ 1/3(-170w12-150w22 -36w3210w42+238y12+210y22)+ 1/3(-170w13-150w23-36w33 -10w43+238y13+210y23 )

Subject to: •

• • • • • • • • • • • • •

x1+ x2+ x3<=500 3x1+ y11- w11>=200 2.5x1+ y12- w12>=200 2x1+ y13- w13>=200 3.6x2+ y21- w21>=240 3x2+ y22- w22>=240 2.4x23+ y23- w23>=240 w31 <=6000 w32 <=6000 w33 <=6000 24x31 - w3- w4>=0 20x32 - w3- w4>=0 16x33 - w3- w4>=0 All variables >=0

How do we solve this problem? SIMPLEX METHOD

7

Solution of the model Wheat

Corn

Sugar Beans

First Stage

Area (Acres)

170

80

250

S=1 Abo ve

Production (ton) Sales(ton) Purchase

510 310 0

288 48 0

375 6000(Favor.price) 0

S=2 Aver age


425 225 0

240 0


S=3 Below


340 140 0

192 48


Profit = $108,390

What is this solution telling us; • The most profitable decision for sugar beet land allocation is always avoid to sell sugar beans at unfavorable price. • Plant the corn so that it meets the production requirement. • The remaining land is devoted for the wheat. This area large enough to cover minimum requirement and sales always occur.

8

Expected value of perfect information (EVPI ) • • • • • •

Now assume yields vary over the years, but on a random basis. If the farmer gets the information on the yields before planting it he will choose one of the following solutions. Good yields: (183.33, 66,67,250), Profit: $167,667 Average yields: (120, 80,67,300), Profit: $118,600 Bad yields: (100,25,375), Profit: $59,950, In the long run, if each yield is realized one third of the years, the farmer will get expected profit of $115,406. As we all know, the farmer doesn’t get prior information on the yields. The best he can do in the long run is take the solution as given last table. In this case farmer gets expected profit of $108,390. The differences between $115,406-$108,390=$7,016 is called expected value of perfect information (EVPI). The farmer is willing to pay $7,016 for the perfect information.

Value of a stochastic solution • Another approach farmer may have is to assume expected yields and allocate the optimum planting surface according to this yields. How: • Solve the “mean value” to get first stage solution. • Mean yields: (2.5,3,20) • We have solution for this problem which is: x1:120 x2:80 x3:300. • Fix the first stage solution at that value of x, and then solve all the scenarios to see farmer’s profit of each

9

If we solve these three problem we will get: Yield (acres)

Profit($)

Good Averag e Bad

148,000 118,600 55,120

If outcomes have the same probability, in the long run he will expect to make profit of: 1/3(148,000)+1/3*(118,600)+1/3*(55,120)=$107,240 If the farmer implement policy based on stochastic programming problem x1=170 x2= 80 x3=250 He would expect to make $108,390. The difference of the values $108,390-$107,240=$1,150 is the value of the stochastic solution. “What it means is that if farmer use stochastic solution every season he would get $1,150 more than using mean value solution”.

10

APPENDIX D: Classical Surface Mine Production Scheduling with Mixed Integer Programming The purpose of long term scheduling in a given mine is to maximize the overall discounted net profit, while adhering set of operational constraints such as mining slope, grade blending, ore production and mining capacity. Mixed integer programming (MIP) is considered to be powerful tool in optimizing mine scheduling. The following model is considered to be basic optimization mathematical model for optimizing surface mine production scheduling. The objective function The main objective of the long term production scheduling is to maximize net present value (NPV) of the mine. In order to achieve this purpose the following formula is used. p

n

Maximize ∑ ∑ C it * X it t =1 i =1

here p is the maximum number of scheduling periods, n is the total number of blocks to be scheduled, Cit is the NPV to be generated by mining block i in period t and X it is a binary variable, equal to 1 if the block i is to be mined in period t, 0 otherwise.

Grade blending constraints The ore delivered from the mine to the mill or to the customer has to be in certain grade ranges. The generated schedule has to satisfy these upper and lower bound grade rage requirements. The following two constraints are generated to hold this requirement. a. Upper bound constraints: The average grade of the material sent to the mill has to be less than or equal to a certain grade value, Gmax, for each period, t n

t

∑ ( g i - G max) * O i * X i ≤ 0

i =1

where gi is the average grade of block i and Oi is the ore tonnage in block i. b. Lower bound constraints: The average grade of the material sent to the mill has to be greater than or equal to a certain value, Gmin, for each period, t n i∑ =1

t

( g i - G min ) * O i * X i ≥ 0

1

Reserve constraints In general reserve constraint holds for not mining more than what is available from the mine and a block cannot be mined more than once. By using inequalities to state that all the blocks in the orebody model considered can be mined only once the following formula can be used. p

t

∑ Xi ≤ 1

t =1

Processing capacity constraints The amount of produced ore from the mine can be milled depending on the mill capacity. The total amount of mined ore at a given time has to be in a certain range in order to feed the mill in constant rate. The following constraints hold upper and lower bound of the milling capacity.

a. Upper bound: The total tonnage of ore processed cannot be more than the processing capacity (PCmax) in any period, t n

∑ (O i

* X it ) ≤ PC max

i =1

b. Lower bound: The total tonnage of ore processed cannot be less than a certain amount (PCmin) in any period, t n

∑ (O i

* X it ) ≥ PC min

i =1

Mining capacity The total amount of material (waste and ore) to be mined cannot be more than the total available equipment capacity (MCmax) for each period, t n

∑ (O i

t

+ Wi ) * X i ≤ MC max

i =1

where Wi is the tonnage of waste material in block i.

2

To force the MIP model to produce balanced waste production throughout the periods, a lower bound (MCmin) may need to be implemented as follows: n

∑ (O i

t

+ Wi ) * X i ≥ MC min

i =1

Slope constraints All the overlying blocks that must be mined before mining a given block. This can be implemented through one or more cone templates representing the required wall slopes of the open pit mine. There are two ways of implementing these constraints:

a. The classical application of the slope constraint is generated one constraint for each block per period: t

x tk − ∑ X lr ≤ 0 r =1

This constraint can be implemented with the following way with less constraint.

b. Using one constraint for each block per period: t

mx kt − ∑ ∑ X lr ≤ 0 l

r =1

where k is the index of a block considered for excavation in period t, m is the total number of blocks overlying block k, l is the counter for the m- overlying blocks.

Example; The example of surface mine scheduling is generated for copper mine. The formulation of the model is driven for three time periods (p=3) and 9 production blocks (n=9). The grade and tonnage of the each block is presented in the following figure. Economic block value of the each block is assumed to be calculated and represented as V1, V2 etc. The objective function of the model is to maximize the net present value (NPV) of the mine and the restrictions of the model are grade blending, reserve, mining capacity, processing capacity, and block sequencing. Figure

3

1 represents ore grade, ore tonnage and waste tonnage of the block for example, ore grade, ore tonnage and waste tonnage of the first block is g1=%0.50Cu, O1=50 tons, W1=0 respectively. g1=%0.50 Cu O1=50 tons W1=0 g4=%0.52 Cu O4=55 tons W4=0

g2=%0.45 Cu O2=0 tons W2=55 g5=%0.50 Cu O5=70 tons W5=0

g3=%0.47 Cu O3=0 tons W3=60 g6=%0.48 Cu O1=0 tons W6=70

g7=%0.9 Cu O8=50 tons W8=0

g8=%0.85 Cu g9=%0.80 Cu O9=70 tons O10=60 tons W9=0 W10=0

p=3, Max # of scheduling period n=9, total # of blocks to be scheduled Cutoff=%0.49 Cu Economic value of each block; V1=100$,V2= -30$, V3=-30$, V4=$120, V5=$110, V6=-35$, V7=150$, V8=155$,V9=$130. Obj. Function Max p

Maximize

n

t

t

∑ ∑ Ci * X i

t =1 i =1

∗

1

(100 X1

−

∗

1

30 X 2

+−

∗

1

+

∗

1

30 X 3 120 X 4

+

∗

1

110 X 5

+−

∗

1

35 X 6

+

∗

1

150 X 7

+

∗

1

+

∗

1

∗

1

+

155 X 8

130 X 9 ) ( (1 + 0.10)1 ) 1 2 2 2 2 2 2 2 2 100 ∗ X1 − 30 ∗ X 2 + −30 ∗ X 3 + 120 ∗ X 4 + 110 ∗ X 5 + −35 ∗ X 6 + 150 ∗ X 7 + 155 ∗ X 8 + 130 ∗ X 92 ∗ ( )+ (1 + 0.10) 2 100 ∗ X13 − 30 ∗ X 23 + −30 ∗ X 33 + 120 ∗ X 34 + 110 ∗ X 53 + −35 ∗ X 63 + 150 ∗ X 37 + 155 ∗ X 83 + 130 ∗ X 93 ∗ (

1 ) (1 + 0.10) 3

Grade Blending Constraints: Upper bound constraints: n

t

∑ ( g i - G max) * Oi * X i ≤ 0

i =1

Lower bound constraints: n

t

∑ ( g i - G min ) * O i * x i ≥ 0

i =1

Gmax=%0.95 Cu, Gmin=%0.5 Cu Upper bound constraint: Time period 1 (((0.5 - 0.95) ∗ 50 ∗ X 11 ) + ((0.5 − 0.95) ∗ 55 ∗ X 14 ) + ((0.5 − 0.95) ∗ 70 ∗ X 51 ) + ((0.9 − 0.95) ∗ 50 ∗ X 17 ) + ((0.85 − 0.95) ∗ 70 ∗ X 81 ) + ((0.80 − 0.95) ∗ 60 ∗ X 91 )) ≤ 0

Lower bound constraint:

4

(((0.5 - 0.5) ∗ 50 ∗ X 11 ) + ((0.5 − 0.5) ∗ 55 ∗ X 14 ) + ((0.5 − 0.5) ∗ 70 ∗ X 51 ) + ((0.9 − 0.5) ∗ 50 ∗ X 17 ) + ((0.85 − 0.5) ∗ 70 ∗ X 81 ) 1

+ ((0.80 − 0.5) ∗ 60 ∗ X 9 )) ≥ 0

Upper bound constraint: Time period 2 (((0.5 - 0.95) ∗ 50 ∗ X 12 ) + ((0.5 − 0.95) ∗ 55 ∗ X 24 ) + ((0.5 − 0.95) ∗ 70 ∗ X 52 ) + ((0.9 − 0.95) ∗ 50 ∗ X 72 ) + ((0.85 − 0.95) ∗ 70 ∗ X 82 ) 2

+ ((0.80 − 0.95) ∗ 60 ∗ X 9 )) ≤ 0

Lower bound constraint: (((0.5 - 0.5) ∗ 50 ∗ X 12 ) + ((0.5 − 0.5) ∗ 55 ∗ X 24 ) + ((0.5 − 0.5) ∗ 70 ∗ X 52 ) + ((0.9 − 0.5) ∗ 50 ∗ X 72 ) + ((0.85 − 0.5) ∗ 70 ∗ X 82 ) 2

+ ((0.80 − 0.5) ∗ 60 ∗ X 9 )) ≥ 0

Upper bound constraint: Time period 3 (((0.5 - 0.95) ∗ 50 ∗ X 13 ) + ((0.5 − 0.95) ∗ 55 ∗ X 34 ) + ((0.5 − 0.95) ∗ 70 ∗ X 53 ) + ((0.9 − 0.95) ∗ 50 ∗ X 37 ) + ((0.85 − 0.95) ∗ 70 ∗ X 83 ) 3

+ ((0.80 − 0.95) ∗ 60 ∗ X 9 )) ≤ 0

Lower bound constraint: (((0.5 - 0.5) ∗ 50 ∗ X 13 ) + ((0.5 − 0.5) ∗ 55 ∗ X 34 ) + ((0.5 − 0.5) ∗ 70 ∗ X 53 ) + ((0.9 − 0.5) ∗ 50 ∗ X 37 ) + ((0.85 − 0.5) ∗ 70 ∗ X 83 ) 3

+ ((0.80 − 0.5) ∗ 60 ∗ X 9 )) ≥ 0

Reserve constraints: p

t

∑ Xi ≤ 1

t =1

X 11 + X 12 + X 13 ≤ 1 , X 12 + X 22 + X 32 ≤ 1 , X 13 + X 32 + X 33 ≤ 1 1

2

3

1

2

3

1

2

3

X 4 + X 4 + X 4 ≤ 1, X 5 + X 5 + X 5 ≤ 1 , X 6 + X 6 + X 6 ≤ 1 X 17 + X 72 + X 37 ≤ 1 , X 18 + X 82 + X 83 ≤ 1 , X 19 + X 92 + X 39 ≤ 1

Processing capacity constraints: Upper bound constraints: n

∑ (O i

* X it ) ≤ PC max

i =1

Lower bound constraints: n

∑ (O i

* X it ) ≥ PC min

i =1

PCmax=60tons/period, PCmin=30tons/period Upper bound constraint: Time period 1 ((50 ∗ X 11 ) + (55 ∗ X 14 ) + ((70 ∗ X 51 ) + (∗50 ∗ X 17 ) + (70 ∗ X 81 ) + (60 ∗ X 91 ) ≤ 60

Lower bound constraints: ((50 ∗ X 1 ) + (55 ∗ X 1 ) + ((70 ∗ X 1 ) + (∗50 ∗ X 1 ) + (70 ∗ X 1 ) + (60 ∗ X 1 ) ≤ 30 1

4

5

7

8

9

Upper bound constraint: Time period 2

5

((50 ∗ X 12 ) + (55 ∗ X 24 ) + ((70 ∗ X 52 ) + (∗50 ∗ X 72 ) + (70 ∗ X 82 ) + (60 ∗ X 92 ) ≤ 60

Lower bound constraint: ((50 ∗ X 12 ) + (55 ∗ X 24 ) + ((70 ∗ X 52 ) + (∗50 ∗ X 72 ) + (70 ∗ X 82 ) + (60 ∗ X 92 ) ≤ 30

Upper bound constraint: Time period 3 ((50 ∗ X 13 ) + (55 ∗ X 34 ) + ((70 ∗ X 53 ) + (∗50 ∗ X 37 ) + (70 ∗ X 83 ) + (60 ∗ X 93 ) ≤ 60

Lower bound constraint: ((50 ∗ X 13 ) + (55 ∗ X 34 ) + ((70 ∗ X 53 ) + (∗50 ∗ X 37 ) + (70 ∗ X 83 ) + (60 ∗ X 93 ) ≤ 30

Mining Capacity Constarints MCmax120tons/period, MCmin=70tons/period Upper bound constraint: n

∑ (O i

t

+ Wi ) * X i ≤ MC max

i =1

Lower bound constraint: n

∑ (Oi

t

+ Wi ) * X i ≥ MC min

i =1

Upper bound constraint: Time period 1 ((50 + 0) ∗ X 11 ) + ((0 + 55) ∗ X 12 ) + ((0 + 60) ∗ X 13 )((55 + 0) ∗ X 14 ) + ((70 + 0) ∗ X 51 ) + ((0 + 70) ∗ X 16 ) + ((50 + 0) ∗ X 17 ) + ((70 + 0) ∗ X 81 ) + ((60 + 0) ∗ X 91 ) ≤ 120

Lower bound constraint: ((50 + 0) ∗ X 11 ) + ((0 + 55) ∗ X 12 ) + ((0 + 60) ∗ X 13 )((55 + 0) ∗ X 14 ) + ((70 + 0) ∗ X 51 ) + ((0 + 70) ∗ X 16 ) + ((50 + 0) ∗ X 17 ) + ((70 + 0) ∗ X 81 ) + ((60 + 0) ∗ X 91 ) ≥ 70

Upper bound constraint: Time period 2 ((50 + 0) ∗ X 12 ) + ((0 + 55) ∗ X 22 ) + ((0 + 60) ∗ X 32 )((55 + 0) ∗ X 24 ) + ((70 + 0) ∗ X 52 ) + ((0 + 70) ∗ X 62 ) + ((50 + 0) ∗ X 72 ) + ((70 + 0) ∗ X 82 ) + ((60 + 0) ∗ X 92 ) ≤ 120

Lower bound constraint: ((50 + 0) ∗ X 12 ) + ((0 + 55) ∗ X 22 ) + ((0 + 60) ∗ X 32 )((55 + 0) ∗ X 24 ) + ((70 + 0) ∗ X 52 ) + ((0 + 70) ∗ X 62 ) + ((50 + 0) ∗ X 72 ) + ((70 + 0) ∗ X 82 ) + ((60 + 0) ∗ X 92 ) ≥ 70

Upper bound constraint: Time period 3 ((50 + 0) ∗ X 13 ) + ((0 + 55) ∗ X 32 ) + ((0 + 60) ∗ X 33 )((55 + 0) ∗ X 34 ) + ((70 + 0) ∗ X 53 ) + ((0 + 70) ∗ X 36 ) + ((50 + 0) ∗ X 37 ) + +

∗

3

+

+

∗

3

≤

((70 0) X 8 ) ((60 0) X 9 ) 120

Lower bound constraint:

6

((50 + 0) ∗ X 13 ) + ((0 + 55) ∗ X 32 ) + ((0 + 60) ∗ X 33 )((55 + 0) ∗ X 34 ) + ((70 + 0) ∗ X 53 ) + ((0 + 70) ∗ X 36 ) + ((50 + 0) ∗ X 37 ) + ((70 + 0) ∗ X 83 ) + ((60 + 0) ∗ X 93 ) ≥ 70

Slope Constraints:

Using one constraint for each block per period t

X tk − ∑ X lr ≤ 0 r =1

Block 5 can be mined if the block 1,2,3 are being mined (45o slope angle considered). The sequencing constraint for the block 5 ; Time period 1 X15 − X11 ≤ 0,

X15 − X12 ≤ 0,

X15 − X13 ≤ 0

Time period 2 X 52 − (X11 + X12 ) ≤ 0,

X15 − (X12 + X 22 ) ≤ 0,

X15 − (X13 + X 32 ) ≤ 0

Time period 3 X 35 − (X11 + X12 + X13 ) ≤ 0,

X15 − (X12 + X 22 + X 32 ) ≤ 0,

X15 − ( X13 + X 32 + X 32 ) ≤ 0

Slope Constraints: Using m- constraints for each block per period: t

mX k − ∑ ∑ X l ≤ 0 t

r

l

r =1

Block 5 can be mined if the block 1,2,3 (m=3) are being mined (45o slope angle considered). The sequencing constraint for the block 5 ; Time period 1 3X15 − X11 - X12 − X13 ≤ 0

Time period 2 3X 52 − X11 - X12 − X13 − X12 - X 22 − X 32 ≤ 0 Time period 3 3X 35 − X11 - X12 − X13 − X12 - X 22 − X 32 − X13 - X 32 − X 33 ≤ 0 As it can be seen from the example, the number of required constraint is a lot less in the second method compare to first one. In other words, three constraints have to be setted in order to hold sequencing requirement for the first time period with the first method, this number reduced to one for the second method.

7

Stochastic Optimization Models for Open Pit Mine Planning: Application and risk analysis at a copper deposit

COSMO – Stochastic Mine Planning Laboratory Department of Mining, Metals and Materials Engineering

Outline 

Introduction



Stochastic optimization with simulated annealing



Case study at a disseminated copper deposit



Conclusion



Future research

1

Introduction A stochastic mine production scheduling technique based on simulated annealing 

Past work tested the method in a highly variable gold deposit 

This work studies an application of the method in a deposit with relative low grade variability 

Stochastic optimization Simulated annealing as optimization tool to obtain a minimum risk life-of-mine (LOM) schedule an initial state is gradually perturbed until a final acceptable state is obtained 



incorporates geological grade uncertainty

different targets can be included into the objective function 

2

Stages involv ed in t he process

Pit limit and mining rates definition

S1 S2 … Sn

Seq1 Scheduler to produce mining sequences

Seq2 … Seqn

S c h e d u le

Simulated annealing algorithm

A m in i m u m ri s k

Pit limits and mini ng rates


S1 S2 … Sn

Scheduler to produce mining sequences

Seq1 Seq2 … Seqn


S c h e d u l e

3

Pit limits and mini ng rates Whittle nested pits used to define a final pit

Conditional simulations


S1 S2 … Sn


Seq1 Seq2 … Seqn


S c h e d u le

4

Conditional simulations Generates equal probable scenarios of the deposit

1

2

… n

Mine produc tion sch edule A simulation can be seen as the actual deposit n

schedules produced, one for each realization 1

2

…

n

5

Simulated annealing technique


S1 S2 … Sn


Seq1 Seq2 … Seqn


S c h e d le u

Simulated annealing technique

Min Oi

1

2

3

if O ( i − 1) > Oi ⎧1 ⎪ Prob{accept ith pert} = ⎨ [O ( i − 1) − Oi ] )otherwise ⎪⎩exp( ti

6

Stochastic optimization

Obje ctive func tion

Min O =

⎛

N

S

∑ ⎜⎝ ∑sθ(−s) θ+ n =1

* n

( s)− ω sn

n

s =1

S

∑ω (

*

)

n s =1

( )

N

is the number of mining periods

S

is the number of simulation

θ n ( s ), ωn( s) * n

* n

θ (s), ω ( s)

⎞ ⎟ ⎠

ore and waste of the perturbed sequence and targets at period n on simulation s


Starting Mine Sequence

• Freeze blocks if prob=1

Remaining blocks are assigned to a period according to one of the mining sequences

1

2

3

7

Simulated annealing graphically

no

Randomly select a block

Does the block have connectivity?

no

STOP

Acc ept/rej ect using a cdf

Acc ept s wap

yes yes

Swap the block to a possible period

yes

Are s to pp in g criteria met?

no

Is objective function decreased?

Compone nt ob jective function vs. Swa ps

n io t c n fu j. b O

Att empt ed sw aps

8

Case study    

Disseminated low-grade copper deposit Orebody dips mainly N180/60S 185 DH in a pseudo-regular grid of 50x50m2 Mineralized envelop defined using the drill core logs

Dire ct block simulation 20 simulations, directly generated in a 20x20x10m3 mining block size 

Pit opti mization Conventional model with a fixed cut-off of 0.3% Cu Final pit defined by Whittle implementation of LerchsGrossman algorithm  

9

Stochastic LOM schedule









A common final pit limit 7.5 Mt of ore and 20.5 Mt of waste 20 LOM schedules using Whittle Millawa algorithm 1 minimum risk schedule as the final result

Stocha stic s chedule : Risk profi le of ore production 9

8

7

) s e n n6 to (M e r5 O

Maximum Minimum Stochastic Schedule

4

Target 3

2 01234

5678 Period (years)

10

Stocha stic s chedule : Risk profi le of wa ste productio n 25

20

) s e n n 15 to (M e t s a 10 W

5

Maximum Minimum Stochastic schedule Target

0 01234567

8 Period (years)

Compa rison with a conventional LOM sche dule 

Standard ordinary Kriging



Smoothes the distribution of grades

Schedule obtained using Whittle Millawa algorithm with same parameters used before 

11

Compa rison – physica l schedule s Stochastic schedule

Conventional schedule

Compa rison - ore prod uction Stochastic schedule 9

8

7

n n6 o t x M e r5 O 4

Maximum Minimum StochasticSchedule Target

3

2 012345678 Period (years)

9 8 7 s e n n6 to x M5 re O 4

Maximum Minimum Expected

3


2 0

2

4

6

8

10

Period

12

Compa rison - ore prod uction 9

8

7

n n6 to x M re5 O

Maxim um Minim um Stochastic Schedule Target

4

3

2


012345678 Period (years)

9 8 7

s e n n6 to x M5 e r O

Maximum Minimum

4

Expected 3


2 0

2

4

6

8

10

Period

Compa rison - waste production Stochastic schedule 25

20

n n to

15

x M te s a 10 W

5

Maximum Minimum Stochastic schedule Target

0 012345678 Period (years) 25

20 s e n n15 to x M te10 s a W 5

Maximum Minimum Expected Conventional schedule

0 0123456789 Period

13

Compa rison - waste production 25

20

n o t 15 x M te s a 10 W

Maxim um Minim um Stochastic schedule

5

Target

0


012345678 Period (years)

25

20 s e n n15 to x M e t s 10 a W 5

Maximum Minimum Expected Conventional schedule

0 0123456789 Period

Compa rison – ore body cumula tive di scounte d cash flows 300

250

26%

200

) $ (M150 V P N Conventional Sched.

100

Max Stochastic Sched. Min. Stochastic Sched.

50

Stochastic Sched. 0 0123456789

Period (years)

*10% discount rate

14

Compa rison – ore body cumula tive di scounte d cash flows 300

250

15%

200

) $ M ( 150 V P N Max . Conventional Sched. Conventional Sched. Min. Conventional Sched. E-Type Conventional Sched. Max Stochastic Sched. Conventional Sched. Min. Stochastic Sched. Max Stochastic Sched. Min. Stochastic Schedule Stochastic Sched. Stochastic Sched.

100

50

0 0123456789 Period (years)

*10% discount rate

Conclusions Mine of high-probable ore blocks in early periods, once the objective of minimizing target deviations over all periods considering all simulations 

Conventional approach @ 0.3% cut-off in this case overestimates ore tonnes although underestimates the value of the mine 

Method proved to obtain positive results even when applied to a low variability deposit 

15

EXTRA SLIDES

Conventional mineralized envelop definition

16

Cu(%) histogram

Geostatistical modelling Variogram data space 0.3 0.25 0.2

Sph (0.086,[100,100,2 0],[ 0,6 0,0 ]) + γ ( h ) = 0.019 +

) h ( 0.15 Y

Strike Dip Plunge Model Plunge Model Strike/Dip

0.1 0.05

Sph (0.086,[250,250,45],[0, 60,0])

0 0

50

100

150

200

250

300

350

400

Lag

Variogram - Normal space 1.4 1.2 1 ) h ( Y

γ ( h ) = 0.01 +Sph (0.045,[100,100,2 0],[ 0,6 0,0 ]) +

0.8 0.6

Main Intermediate Minor Model Strike/Dip Model Plunge

0.4 0.2

Sph (0.045,[30 0,300, 60],[0,60, 0])

0 0

50

100

150

200

250

300

350

400

450

Lag

17

Risk ana lysis on pit s hells NV per pit

Ore tonnage per pit

@ 0.3% Cu cut-off

@ cut-off of 0.3 Cu(%)

450

60

400 50

350 s e 40 n n to x 30 M e r O 20

300 $ M250 V N200 150 100

10

50 -

1

2

3

4

5

6

7

8

0

9 1 0 1 1 1 2 1 3 1 4 1 5 1 6 1 7

-

1

2

3

4

5

6

7

8

9 1 0 11 12 13 14 15 16 17

Pit #

Pit #

Head grade per pit

Striping ratio per pit



0.008 2.50

0.007

2.00

e d a r g d a 0.006 e h u

o ti ra g in p i rt S

C

0.005

1.50

1.00

0.50

0.004 -

1

2

3

4

5

6

7

8

-

9 1 0 1 1 1 2 1 3 1 4 1 5 1 6 1 7

-

Pit #

1

2

3

4

5

6

7

8

9

10 1

1 1

2 1

3 1

4 1

5 1

6 1

7

Pit #

estimated model

Whittle Milla wa scheduler 1st pushback

2nd pushback

Final pit

Push-backs selected among Whittle nested pits obtained by incremental changes in the price/cost factor, corresponding to ~ 3 years of ore production

18


Connectivity and slope constraint checks

1st test 2nd test 4

3 4

3

3

3 4

4

5

Case study

Grade ton nage curves 20x20x10m 3 blocks

ff o -t u c e v o b a e g a n n o T

120

1.4

100

1.2 1

) 80 s e n n to 60 M (

0.8 Estimation

0.6

Simulations

40

0.4 20

0.2

% u C ff -o t u c e v o b a e d a r g . rg e v A

0

0 0

0.2

0.4 0.6 Cut-off Cu (%)

0.8

1

19

Stochastic Optimisation of Long-term Production Scheduling for Open Pit Mines with a New Integer Programming Formulation

Contents • Introduction • Stochastic integer programming (SIP) model • Risk management using the SIP model • Case study • Conclusions

1

Introduction • The traditional “optimum scheduling methods” are based on mathematical models with inputs of 100% certainty. • Uncertainty may exist from technical, environmental and market sources. Grade variability is examined in this presentation. • A recently developed Stochastic Integer Programming (SIP) model uses multiple simulated orebody models in optimising long-term production schedules in open pit mines

Models for Optimisation Orebody model

Integer Programming

c1

An objective function Maximise (c1x1

1+c

2x 2

c2

c3

c4 1+….

)… c = constant X11 = binary variable

Subject to c1x11+c2x21+…. = b 1

Period 1

c1x1p+c2x2p+…. = b p

Period p

2

Stochastic Integer Programming The objective function now is …..

s 11 s 21 s 31

Maximise (s11x11+s21x21+…. s12x11+s22x21+….) …

1

s11s4 s21 s31 s11 s1 21 s31 s1ns4s21n s3n s4

Subject to

1

s11x11+s21x21+…. = b1

Period 1 Simulated model 1 Simulated model 2

s4

Simulated model r s11x1p+s21x2p+…. = b1 s12x1p+s22x2p+…. = b1 s1rx1p+s2rx2p+…. = b1

Period p

Stochastic Integer Programming (SIP) 





Account for uncertain inputs Consider simulated grade realizations in the optimisation process; and Minimise the risk of not meeting production targets caused by geological variability

3

SIP - Production Scheduling Model Objective function P

N

t =1

i =1

Max ∑ [ ∑ E{(NPV) it} b it

Part 1

Mill & dump

U - ∑ E{(NPV) it + MC}it *s it i =1

Part 2

Stockpile input

M

t

+ ∑ (SV) st (P) q s

Part 3

Stockpile output

s =1 M

ty

ty ty ty - ∑ (c u d su + c l d sl )] s =1

Part 4

Risk management

Objective function - Deviations

R Minimise........

−

ty

∑ (c u d ruty

+

ty

ty c l d rl )].....

r =1

Part 2

Deviation from production targets c tyu and ctyl penalized by dtyru and dtyrl for each simulation r

4

Stochastic Integer Programming - SIP Deviation 1 Orebody Model 1 A production schedule

Ore Grade 1 Metal …

- TARGET [ ] Deviation 2

1 3

4

2

Orebody Model 2

Ore Grade 2 Metal …

… …

Orebody Model R

- TARGET [ ]

Deviation R Ore Grade R Metal …

- TARGET [ ]

Managing Risk within a Given Period Ore Producti on 1.6 ) n o il li m ( s e n n o T

1.3

1.0

0.7 1

2

3

Schedules Cl3 t > (Cl1 t = Cu1 t ) > (Cl2 t = Cu2 t ) > Cu3

5

Managing Risk Between Periods Deviations from metal production target 33 y itt n a u

) g K 0 lq 0 0 ta 1 e ( M

2.5

22 1.5

11 0.5

00 01234

2

1

3

Periods

Ct=Ct-1 * RDFt-1 RDF – risk discounting factor

RDFt=1/(1+r)t r – orebody risk discount rate

Case Study on a Large Gold Mine General information Total b locks

22,296

Block dimensions (m)

20 x 20 x 20

Processing i nput c apacity (P C)

18 Mtpa

Metal prod ucti on capacity (MC )

28,000 Kg p a

Total minin g capacity (TC)

85 Mtpa

Stockpil e capacity (SC )

5 Mt

Stockpile re- handling cost

0.6 $/t

Discount rate

10 %

Mine Life

6 yrs

6

Case Study on a Large Gold Mine The SIP specific information

Orebody risk discounting rate

20 %

Cost of shortage in ore production

10,000 /t

Cost of exce ss ore productio n

1,000 /t

Cost of shortage in meta l produ ction

20 /gr

Cost of excess meta l pro duction

20 /gr

Numbe r of simulate d o rebody models

15

The SIP Model Information Periods

1-4

4-6

Tota l bl ocks

11,301

10,995

Constraints

33,273

21,363

Total variables Binary:

53,301 18,540

37,286 9,580

<04:49:55

<37:15:33

T Time (hr:min :sec)

Supercomputing system is used (parallel processors≤ 8)

7

Cross-Sectional Views of the Schedules SIP

Whittle Four-X

Periods 1 2 3 4 5 6

Deviations from Production Targets Metal Production - 20 ty ti n a u q l a t e M

- 16

) g K 0 - 12 0 0 1 ( -8

SIP model WFX

-4 0 123456

Periods

8

6 ) n io lli m ( s e n n o T

Stockpile’s Profile Available ore at the end of each period

4 2

0

2

1

3

4

5

SIP model

6

WFX

5 ) n o lil i (m s e n n o T

4 3

Ore taken out from the stockpile

2 1 0 1

23456

Periods

Uncertainty is Good: Traditional vs Risk-Based Stochastic Integer Programming

$723 M Risk Based

1000 800 ) n io lil m ( $

$609 M Traditional

600 400

Difference = 17%

200 0 1

Cumulative NPV values SIP model

WFX

2

3

4

5

6 Periods

Average NPV values SIP model

WFX

Geological Risk Discounting= 20%

9

Some Key Comments The new SIP production scheduling model: • Uses individual realisations, thus explicitly accouns for geological risk • Allows the risk management at three levels: 1. manage the magnitude of risk within a period 2. manage the variability of risk 3. control the risk distribution between time periods • Maximises NPV for a desired risk profile • The SIP is efficient: Contains less binary variables than traditional MIP models

10

Grade Uncertainty in Stope Design: Improving the the Optimisation Process

Outline •

Introduction

•

Kidd Creek and study area

•

Development of a conventional stope design

•

Assessing the risk in the ‘conventional stope’ design

•

Development of risk based stope designs

•

Conclusions and recommendations

1

Introduction •

Decisions in the mining industry are made in the presence of uncertainty

•

Uncertainty is present no matter what is mined or how it is mined

•

Geological risk associated with open pit optimisation and scheduling What about underground mining applications?

•

Apply similar concepts to those for open pit operations, underground

Kidd Creek

Kidd Creek

•

27km North of Timmins, ONT, Canada

•

Discovered by Texas Gulf Sulphur Co (1963)

•

Division of Falconbridge Ltd. (1986)

•

Volcanic massive sulphide deposit containing 2 orebodies consisting of 3 ore types • stringer ore • massive banded and bedded sulphides • sulphide breccia Major products include – Cu, Zn, Ag

• •

Ore reserves as at 2001 25M @ 2.20% Cu

2

Study Area • • •

No. 3 Mine – 4700L Stringer ore Open stopes • 15m wide x 20m long x

Study Area

40m high • rings spaced every 3m (min 2, max 7 per stope)

• • • •

Cutoff grade 3% Cu Mill recovery 95% Mining cost $35/tonne Milling cost $20/tonne

Some Terminology

Orebody model

Layer

Panels

Rings

3

Study Area 2 4 0

2 8 0

3 2 0

3 6 0

4 0 0

1920

1920

1880

1880

Cu (%)

0 [ 0.80 , 1.40 ]

[ 0.00 , 0.80 ] [ 1.40 , 2.00 ] [ 2.00 , 2.50 ] [ 2.50 , 3.00 ] [ 3.00 , 4.00 ]

W 1840

[ 4.00 , 5.00 ] [ 5.00 , 8.00 ] [ 8.00 , 14.00 ]

0 4 2

0 8 2

0 2 3

0 6 3

[ 14.00 , 20.00 ]

27.59

[ 20.00 , 27.59 ]

0 0 4

1840

Quantifying Stope Risk - Methodology Conve ntional Stope Design

• •

Estimate zone of interest Establish economic ore zone using Floating Stope

Risk Quantification

• •

Simulate  40 orebodies  ring dimensions

•

Quantify risk – risk profiles for key project indicators

Assess risk – running ‘conventional design’ through each simulated orebody

4

Conventional Stope Design Estimated Zone

Cu% 0

>14

Conventional Stope Design

0 Cu%

Upper Level Outline (plan) Vertical Section

>14

Lower Level Outline (plan)

West

5

Conventional Stope Design Conventional Stope Outline

Estimated Zone

Cu% 0

>14

Risk Quantification Estimated Zone

Simulated Orebodies Ring sizes – 15m x 3m x 40m

Cu% 0

>14

6



Cu% 0

>14



Cu% 0

>14

7

Risk Quantification As ses sin g Potenti al Risk

Upper Level

Simulated Orebody 14

Lower Level rings – 15m x 3m x 40m

Conventional design outline (white line).

Risk Quantification Model

Ore

Metal

Cu

(tonnes)

(tonnes)

(%)

Economic Econ.Pot. Potential ($) %difference

3

191,909

8,769

4.57

2,285,625

- 33

18

167,306

8,228

4.92

1,858,484

- 46

31

216,513

11,187

5.17

5,905,110

+ 73

35

211,592

10,492

4.96

4,820,407

+ 41

Est.

196,830

9,490

4.82

3,412,999

8

Risk Quantification Tonnes of Ore 300,000 250,000 ) s e 200,000 n n o t( 150,000

Est. Min 25th 75th Max Avg

re 100,000 O 50,000

0 Lower

Upper

Both

Level

Risk Quantification Stope Design ’ s Economic Potential ) M $ ( l a it n te o p c i m o n o c E

7 6 5

Est.

4

Min

3

25th 75th

2

Max

1

Avg

0 -1

Lower

Upper

Both

Level

9

Risk Based Designs

How does one ca ptu re the upsi de potential of the ore de posit in an optimal stope

des ign?

A probabilistic linear programming formulation that seeks to locate the highest grade stope in the presence of grade uncertainty.

Risk Based Designs - Methodology Formulating Linear Program

•

Objective function – incorporating average grades and associated probabilities

•

Constraints – geometric, geotechnical, probabilities indicating acceptable risk

Risk Based De sig ns

•

Produce designs based on varying levels of acceptable level of risk

•

Evaluate noting design the by passing through each simulated orebody, effect onitkey project indicators

10

Risk Based Designs - LP Objective Function

#

panels # rings

∑ ∑

Maximise

j =1

gij ⋅ pij ⋅ xij

i =1

pij g ij

- probability of a ring being above cut-off grade

xij

- ring (binary variable = 0 or 1)

- average grade of a ring

Orebody model

Layer

Panels

Rings

Risk Based Designs - LP Constraints # rings

∑

1. Level of risk, PL

pij ⋅ xij ≥ PL

i =1

Rings, i 1 2 :

p51 + p61+ p71+ p81+ p91

5 rings

≥

:

PL

: : : 15

Panels, j =

1

2

11

Risk Based Designs - LP Constraints 2. Stope size

# rings

∑

# rings

∑

xij ≥ MIN

i =1

xij ≤ MAX

i =1

Rings, i 1

MIN = 2

2 :

MAX = 7

: : : : 15

Panels, j =

1

2

3

Risk Based Designs - LP Constraints 3. Pillar size – related to stope sizes i=15

STP = 6 rings PIL = 5 rings

i=1

12

Risk Based Designs - LP

60%

Probability of rings being above 3% Cu

100%

Conventional Design

80%

Risk Based Designs - LP Tonnes of Ore 200,000 160,000

) s e 120,000 n n o t( e r O

Min 25th 75th Max Avg

80,000 40,000

0

60%Prob

80%Prob

90%Prob

100%Prob

Design

13

Risk Based Designs - LP Stope Design ’ s Economic Potential ) M 8.0 $ ( l a i 6.0

Min

tn e t o 4.0 p c i m 2.0 o n o c 0.0 E

25th 75th Max Avg

60%Prob 80%Prob 90%Prob 100%Prob

Design

Conclusions & Recommendations •

Quantification of uncertainty provides a means of evaluating the geological risk related to conventional stope design

•

Quantification of uncertainty provides a means of generating risk based designs

•

It is possible to design stopes that capture the upside potential of the deposit while ensuring a minimum acceptable level of production performance

Future Work

• •

Stochastic approach to generating risk based designs Integrate additional geotechnical considerations

14

Affects of the Simulation Method What if we used sequential indicator simulation (SIS) ?

•

Generate 40 simulated orebodies using SIS

• •

Reblock orebodies to ring sizes Generate a risk based design with an 80% acceptable level of risk

•

Run the design through all simulated orebodies observing the behaviour of key project indicators

•

Compare results with the risk based design using SGS

Risk Based Designs - LP Stope Design ’ s Economic Potential ) M $ ( l a it n e t o p c i m o n o c E

7.0 6.0 5.0

MIN 25TH 75TH MAX AVG

4.0 3.0 2.0 1.0 0.0 -1.0 60%Prob 80%Prob 90%Prob 100%Prob

Design

15

Risk Based Designs - LP Tonnes of Ore 250,000

) 200,000 s n 150,000 e n o t( e r 100,000 O

MIN 25TH 75TH MAX AVG

50,000 0 60%Prob

80%Prob

90%Prob

100%Prob

Design

16

Grade Control and Simulation Methods

Economic Functions & Comparisons • Comparisons using a fully known deposit: • The Walker Lake data set (Isaacs and Srivastava, 1989)

• Simulation Techniques: • Sequential Gaussian Simulation • Sequential Indicator Simulation • Probability Field Simulation

• Economic Functions: •M inimumLoss

(economicfunctionI)

• Maximum Profit

(economic function II)

• Maximum Profit Exc.Min.

(as above but excluding mining cost)

1

Economic Functions & Comparisons • Comparisons using a fully known deposit: • The Walker Lake data set (Isaacs and Srivastava, 1989) • Grade control bench gold deposit (the “Anglo” deposit)

• Simulation Techniques: • Sequential Gaussian Simulation • Sequential Indicator Simulation • Probability Field Simulation

• Economic Functions: •M inimumLoss

(economicfunctionI)

• Maximum Profit

(economic function II)

• Maximum Profit Exc.Min.

(as above but excluding mining cost)

Case Studies and Procedure 1. Walker Lake is a fully known data set of 78,000 values 2. 195 values are sampled from the Walker Lake ‘deposit’ to simulate a 6 x 6m grade control program 3. Conditionally simulate with three techniques (100 realisations each time) 4. Re-block to smu’s of 10 x 10m, and cl assify using each of the economic functions 5. Compare the $ value and quantity of selected ore with the case for perfect selection from the known data set 6. Repeat steps 2 – 4 using 725 data on a 3 x 3m grid

2

Case Studies and Procedure 1. Walker Lake is a fully known data set of 78,000 values 2. 195 values are sampled from the Walker Lake ‘deposit’ to simulate a 6 x 6m grade control program 3. Conditionally simulate with three techniques (100 realisations each time) 4. Re-block to smu’s of 10 x 10m, and cl assify using each of the economic functions 5. Compare the $ value and quantity of selected ore with the case for perfect selection from the known data set 6. Repeat steps 2 – 4 using 725 data on a 3 x 3m grid 7. Repeat steps 2 – 6 using the “Anglo” gold deposit

Walker Lake Deposit and Data Sets

3

Data Set and Blasthole Patterns Reference (0.3x0.3)

195 Samples (6x6m)

725 Samples (3x3m)

ORE WASTE 0

15

30

45

60

N= Mean Std. Dev. Coef. Var. Maximum 75% Median 25%

195 0.73 1.42 1.94 12.68 0.73 0.16 0.02

(m)

N = 78,000 Mean Std. Dev. Coef. Var. Maximum 75% Median 25%

0.83 1.52 1.84 29.55 0.99 0.18 0.02


725 0.87 1.56 1.80 12.68 0.96 0.17 0.22

Validation of Simulations l m

Gaussian

Indicator

Probability Field

Walker Lake Gold grades (g/t)

l m

[4.0;12.86] ]0.86;4.0[ ]0.17;0.86] [0; 0.17]

4

Validation : Walker Lake SGS 195 Data Variogram of multiple simulations (SGSIM195) NS 1.2 1 0.8 a m 0.6 Histogram reproduction - SGSI M (195) m a

25

0.4 G 0.2

20

0

y15

1

c n e u q e r10 F

3

5

7

9 11 13 15 1 7 19 21 2 3 25 2 7 29 3 1 33 3 5 37 3 9 41 4 3 45 4 7 49

Distance (m)

5

0 0.01 0.04 0.08 0.17 0.32 0.65 0.74 0.86 1.21 1.76 2.32 3.05 4.04 4.53 5.69 6.92

Bin (g/t)

Validation : Walker Lake SIS 195 Data Variogram of m ultiple sim ulations (50 %) NS 0.3

0.25

0.2 a m 0.15 m a G

Histogram reproduction - SISIM (195) 25 0.1

0.05

20

0 1

3

5

7

9

11 13 15 1 7 19 2 1 23 2 5 27 2 9 31 3 3 35 3 7 39 4 1 43 4 5 47 4 9

15

Distance (m)

y c n e u q e r F

10

5

0 0.01

0.04

0.08

0.17

0.32

0.65

0.74

0.86

1.21

1.76

2.32

3.05

4.04

4.53

5.69

6.92

Bin (g/t)

5

Validation : Walker Lake PFS 195 Data Variogram of mu ltiple simul ations (PFSIM19 5) NS 1.4

1.2

1

0.8

a Histogram reproduction - PFSIM (195) m

35

m a G0.6

30

0.4

25

0.2

0 y20 c n e u q re15 F

1

3

5

7

9

11 13 1 5 17 1 9 21 2 3 25 2 7 29 3 1 33 3 5 37 3 9 41 4 3 45 4 7 49

Distance (m)

10

5

0 0.01

0.04

0.08

0.17

0.32

0.65

0.74

0.86 1.21 Bin (g/t)

1.76

2.32

3.05

4.04

4.53

5.69

6.92

Selectivity is Based on 10x10m SMU’s

6

Costs, Recoveries, Gold Prices and Cut-off Grades for three scenarios A

B

C

Processing Costs ($/t)

8.0

8.64

8.85

Mining Costs [1] ($/t)

2.00

2.00

1.50

Recovery (%)

80

95

80

Cut-off grade (g/t)

[ 0.65 +

[ 0.74 +

[ 0.86 +

Gold price ($/oz)

480

380

400

[1] The cost of mining ore and waste is assumed to be the same

Legend : Profit Charts

7

Legend : Tonnage Charts Ore Tonnes Mined Low Cut Off (0.65 g/t) Walker Lake 725 Data 65%

s e 55% n n o T l ta o T45% f o e g a t n 35% e c r e P

n io tc lee S er O tc ef re P

25%

15% True

Kriging

m-type Min Loss Profit +

Profit - Min Loss Profit +


Profit -

Selection Method

Bench Profit Af ter Ore Selection - Walker Lake 72 5 Data Low Cut Off (0.65 g/tAu)

Classification Based on 725 Samples (10x10m SMU) Low cutoff (0.65 g/t)

250,000

Se quen ti al Ga us si an

Sequ en ti al In di ca to r

Pr obab il it y Fie ld

200,000

n o i cte le S er O cte fr e P

150,000

$ A itf ro P

100,000

w o L n ea

50,000

M

0 True

Kriging




Profit -

Ore/ Waste Selection Method Ore TonnesMined Low Cut Off (0.65 g/t) Walker Lake 725 Data 65% s e n n o T l a t o T f o e g a t n e c r e P

55%

45%

n o i cte le S

35%

re O tc fer e P

25%

Legend

15% True

Kriging

m-type

Min Loss

Prof it +

Profit -

Min Loss

Profit +

Profit -

Min Loss

Profit +

Profit -

Selection Method

8

Bench Profit After Ore Selection - Walker Lake 72 5 Data Medium Cut Off (0.74 g/tAu)

Classification Based on 725 Samples (10x10m SMU) Med. cutoff (0.74 g/t) 140,000

Sequ en ti al Ga us si an

Se qu en ti al In di ca to r

Pr ob abil it y Fie ld

120,000

n io tc el e S er O cte fr e P

100,000

$ A t fi ro P

ed

80,000

60,000

n ai

40,000

M n ea M

20,000

0 True

Kriging




Profit -

Ore/ Waste Selection Method OreTonnesMined Medium Cut Off (0.74 g/t) Walker Lake 725 Data 65% 60% 55%

s e n n o T l a t o T f o e g a t n e c r e P

50% 45% n o tic el e S er O cte fr e P

40% 35% 30% 25%

Legend

20% 15%

True

Kriging

m-type

Min Loss

Prof it +

Profit -

Min Loss

Profit +

Profit -

MinL oss

Profit+

Profit-

Selection Method

Classification Based on 725 Samples (10x10m SMU) High cutoff (0.86 g/t)

Bench Profit After Ore Selection - Walker Lake 72 5 Data High Cut Off (0.86 g/tAu)

100,000

Se qu en ti al Ga us si an

Sequ en ti al In di ca to r

Pr obab il it y Fie ld

90,000 80,000

n o tic el e S er O tc ef re P

70,000

$ A itf o r P

H

h gi

60,000 50,000 40,000 30,000

n ea M

20,000 10,000 0 True

Kriging




Profit -

OreTonnesMined High Cut Off (0.86 g/t) Walker Lake 725 Data Ore/ Waste Selection Method

60% 55% s e n n o T l a t o T f o e g a t n

50%

e c r e P

30%

45% 40% 35%

25%

Legend

20%

n o i cte el S er O cte fr e P

15% True

Kriging

m-type

Min Loss

Profit +

Profit -

Min Loss

Profit +

Profit -

Min Loss

Profit +

Profit -

Selection Method

9

Classification Based on 195 250,000 Samples (10x10m 200,000 SMU) Low cutoff (0.65 150,000 g/t)

Bench Profit After Ore Selection - Walker Lake 19 5 Data Low Cut Off (0.65 g/ tAu) Sequentia l Gaussian

ow L n ea M

Probability Field

n io t c e l e S e r O t c e f r e P

$ A t fi o r P

100,000

N= 195 Mean 0.73 Std. Dev. 1.42 Coef. Var. 1.94 Maximum 12.68 75% 0.73` Median 0.16 25% 0.02

Sequentia l Indica tor

50,000

0 True Kri gin gm-ty pe

Min Loss

Prof it +Pro fit - Min Loss

Profi t +Pro fit - Min Loss

Profit +Profit -

Ore/ Waste Selection Method

Ore TonnesMined Low Cut Off (0.65 g/t) Walker Lake 195 Data

60% 55% s e n n o T l a t o T f o e g a t n e c r e P

50% n io cte el S re O cte fr e P

45% 40% 35% 30% 25%

Legend

20% 15% True

Kriging

m-type

Min Loss

Profit +

Profit -

Min Loss

Profit +

Profit -

Min Loss

Profit +

Profit -

Selection Method

Classification Based on 195 Samples (10x10m SMU) Med. cutoff (0.74 g/t) ed M

n ai

n ea M


Bench Profit After Ore Selection - Walker Lake 195 Data Medium Cut Off (0.74 g/tAu)

140,000

Se qu en ti al Ga us si an

Se qu en ti al In di ca tor

Pr ob abil it y Fie ld

120,000

n o i ct el e S e r O cte fr e P

100,000

195 0.73 1.42 1.94 12.68 0.73 0.16 0.02

$ A ti f ro P

80,000

60,000

40,000

20,000

0 True

Kriging

m-type Min Loss Profit + Profit - Min Loss Profit + Ore TonnesMined


Profit -

Medium Cut Off (0.74 g/t) Walker Lake 195 Data

60%

Ore/ Waste Selection Method

55% s e n n o T l a t o T f o e g a t n

50%

e c r e P

30%

45% 40% 35%

25%

Legend

n o tic el e S re O cte fr e P

20% 15% True

Kriging

m-type

Min Loss

Profit +

Profit -

Min Loss

Profit +

Profit -

Min Loss

Profit +

Profit -

Selection Method

10

Classification Based on 195 Samples (10x10m SMU) High cutoff (0.86 g/t)

Bench Profit After Ore Selection -Walker Lake 195 Data High Cut Off (0.86 g/tAu) 100,000

Sequ en ti al Ga us si an

90,000

n ea M

N= Mean

195 0.73

Std. Coef.Dev. Var. Maximum 75% Median 25%

1.42 1.94 12.68 0.73 0.16 0.02

Pr ob ab il it y Fie ld

n io tc el e S er O tc ef re P

70,000 60,000 $ A itf o r P

50,000 40,000

H

h gi

Se qu en ti al In di ca to r

80,000

30,000 20,000 10,000 0 True

Kriging

m-type

Min Loss

Profit +

Profit -

Min Loss

Profit +

Profit -

Min Loss

Profit -

Min Loss

Profit +

Profit -

Ore/ Waste Selection Method Ore TonnesMined High Cut Off (0.86 g/t) Walker Lake 195 Data

60% 55% s e n n o T l a t o T f o e g a t n e c r e P

50% 45% 40% n o i cte le S

35% 30%

er O

25%

Legend

tc fer e P

20% 15% True

Kriging

m-type

Min Loss

Profit +

Profit -

Min Loss

Profit +

Profit +

Profit -

Selection Method

A Summary •

For cutoffs quite above the mean, SIS and minimum loss works well.

•

For cutoffs quite below the mean, the differences are small.

•

Generally, minimum loss performs well.

•

For dense grade control drilling, the grade control method does not matter; but what is ‘dense’ drilling?

•

What is the point of diminishing returns?

•

P-field did not perform well; a similar observation as in the section comparing simulation methods.

11

Current Limitations • No consideration of short term production scheduling • Non-linear metallurgical recoveries • Ore “processability” aspects not included • Multiple ore and waste categories • Complex economic functions for multiple elements, including contaminants • Complex simulations including simulations based on high-order connectivity functions

12

Simulation of Orebody Geology with Multiple-Point Geostatistics — Application at Yandi Channel Iron Ore Deposit, WA and Implications for Resource Uncertainty V Osterholt1 and R Dimitrakopoulos2 ABSTRACT Development of mineral resources is based on a spatial model of the orebody that is only partly known from explor ation drilling and associated geological interpretations. As a result, orebody models generated from the available information are uncertain and require the use of stochastic conditional simulation techniques. Multiple-point methods have been developed for petroleum reservoir modelling enabling reproduction of complex geological geometries for orebodies. This paper considers a multiple-point approach to capture the uncertainty of the lithological model at the Yandi channel iron ore deposit, Western Australia. Performance characteristics of the method for the application are discussed. It is shown that the lithological model uncertainty translates into considerable grade-tonnage uncertainty and variability that is now quantitatively expressed.

INTRODUCTION Geological controls of physical-chemical properties of ore deposits are important, thus, understanding and modelling the spatial distribution of deposit geology is critical to grade estimation, as well as the modelling of any pertinent attributes of orebodies (eg Sinclair and Blackwell, 2002; King et al , 1986). In iron ore deposits, for example, geological domains typically include lithology, weathering, ore and contaminant envelopes. Domains for other physical properties such as density, hardness and lump-fines yield may be required. The traditional approach to model geological domains is theresulting drawinginofanoutlines of the geological units by the geologist, over-smoothed subjective interpreta tion. Automatic interpretations are rare and include solids models that are, however, also inherently smooth. Furthermore, such single ‘best-guess’ interpretations do not account for uncertain ty about the location of boundaries and corresponding volumes, leading to inconsistencies between mine planning and production. Stochastic simulation technique s address the above type of challenges in modelling the geology of, or the uncertainty about, a deposit. Unlike in the petroleum industry, stochastic simulation of geological units of mineral deposits has been limited in the mining industry due to the above-mentioned traditional practices, despite early efforts (David, 1988). The principle behind stochastic simulation is interpreting the occurrence of a geological unit at a location as the outcome of a discrete random variable. This probabilistic approach honours the fact that the geology at any location cannot be known precisely from drilling data. All available information including data, data statistics/geostatistics, and geological interpretations are included in such an approach to yield the most realistic models. Stochastic simulation methods have been developed and tested on geological models of mineral deposits. Methods mainly consist of sequential indicator simulation or SIS (Goovaerts, 1997) type 1.

MAusIMM, Resour ce Geolo gist, BHP Billiton Iron Ore, 225 St Georges Terrace, Perth WA 6000, Australia. Email: [email protected]

2.

MAusIMM, COSMO Laboratory, Department of Mining, Metals and Materials Engineering, McGill University, Frank Dawson Adams Building, Room 107, 3450 University Street, Montreal QC H3A 2A7, Canada. Email: [email protected]

OrebodyModellingandStrategicMinePlanning

SpectrumSeriesVolume14

approaches and the truncated pluri-Gaussian simulation approach or PGS (Le Loc’h and Galli, 1997; Langlais and Doyle, 1993). Various implementations and applications include the modelling of mineralised envelopes with a predecessor to SIS approach (David, 1988), simulating geologic units with nested indicators (Dimitrakopoulos and Dagbert, 1994), generation of ore textures with ‘growth’ (Richmond and Dimitrakopoulos, 2000), simulation of oxidisation fronts with PGS (Betzhold and Roth, 2000), ore lenses in an underground mine (Srivastava, 2005), uranium roll-fronts (Fontaine and Beucher, 2006) and kimberlite pipes (Deraisme and Field, 2006). Alternative approaches include methods based on Markov transition probabilities (Carle and Fogg, 1996; Li, 2007) and object based methods (eg Seifert and Jensen, 2000). The main drawback of the above methods is their inability to capture non-linear geological complexities, and it becomes obvious when curvilinear features such as faults, multiple superimposed geological phases, fluvial channels, or irregular magmatic bodies are simulated. The reason for this limit is that conventional methods represent geological complexity in terms of second order (two-point) statistics. Variograms describe the variability of point-pairs separated by a given distance and, although they capture substantial geological information (David, 1988), there is a limit to the information they can convey (Journel and Alabert, 1988; Journel, 2007, this volume). Figure 1 illustrates limitsFigure of variograms in fully characterising geological the patterns. 1 shows three geological patterns with different spatial characteristics where the variograms of the three patterns cannot differentiate between the three geological patterns.

1

2

1

2

3

3

FIG 1 - Vastly different patterns show same variogram (modified from Journel, 2007, this volume).

1

V OSTERHOLT and R DIMITRAKOPOULOS

In advancing from the above limits, substantial efforts have been made to develop new techniques that account for the so-called high-order spatial statistics . These include the most well established multiple- point (multi-poin t or MP) approach (Strebelle, 2002; Zhang et al , 2006), as well as Markov random field based, high-order statistical approaches (Daly, 2004; Tjelmeland and Eidsvik, 2004) or computer graphic methods that reproduce multiple-point patterns (Arpat and Caers, 2007). These efforts replace the two-point variogram with a training image (or analogue) so as to account for higher order dependencies in geological processes. The training image is a geological analogue of a deposit that describes geometric aspects of rock patterns assumed to be present in the attributes being modelled and reflects the prior geological understanding of a deposit considered. The multiple-point or MP simulation approach examined herein and adopted for the modelling of the geological units of an iron ore deposit is base d on the MP exte nsion of SIS (Guardiano and Srivastava, 1993; Strebelle, 2002; Liu and Journel, 2004), where MP statistics are inferred by scanning a training image (TI). The TI is regarded as a geological analogue, forms part of the geological input, and it should contain the relevant geometric features of the units being simulated. Until recently, the MP simulation approach has mainly been used for modelling of fluvial petroleum reservoirs. It is logical to extend its application to modelling mineral deposits, where the TI can be derived from geological interpretations of the relatively dense exploration or grade control drill hole data, and/or face mappings. This paper revisits multiple-point simulation as an algorithm for the simulation of the geology for mineral deposits. In the next sections, the MP method is first reviewed and outlined. Subsequently, an application at the Yandi channel iron ore deposit is detailed. Implementation issues, the characteristics of the resulting simulated realisations and the resource uncertainty profile are also discussed. Finally, conclusions from this study are presented.

SIMULATION WITH MULTIPLE-POINT STATISTICS REVISITED Definitions Multiple-point or MP statistics consider the joint neighbourhood of any number n of points. As indicated above, the variogram can be seen as a MP statistic consisting of only two points; hence, it cannot capture very complex patterns. Using MP statistics sequentially on difference scales, large and complex patterns can be reproduced with a relatively small neighbourhood size n of about 20 to 30. MP statistics can be formulated using the multiple-point data event D with the central value A. The geometric configuration of D is called the template τ n of size n . Figure 2 shows an example of a data event on a template with n=4.

D

The size n of the template and its shape can be adjusted to capture any data events informing central value A. As MP statistics characteris e spatial relations of closely spaced data, they may not always be calculated directly from drilling data. The method used for this study defines MP statistics on a regular grid, and are inferred from the TI, a regular cell model that serves as a 3D representation of the geological features concerned. The geometries contained in the TI should be consistent with the geological concept and interpretation of the deposit. In practice, this can always be confirmed by a geologist familiar with the deposit.

A conditional simulation algorithm Consider an attribute S taking K possible discrete states {sk, k=1, …, K}, which may code lithological types, metallurgical ore types, grindability units, and so on. Let dn be a multiple-point data event of n points centred at location x. d n is associated with the data geometry (the data template τn) defined by the set of n vectors {hα, α=1, …, n} and consists of the n data values s(x+hα) = s(xα), α =1, …, n. While traditional variogram-based simulation methods estimate the corresponding conditional distribution function (ccdf) by somehow solving a kriging system consisting on the two-point covariances, the MP ccdf is conditioned to single joint MP data events dn :

f(x;sk | dn) = E{I(x;sk) | dn}= Pr{S(x) = sk | dn}, k=1,…,K

Let Ak denote the binary random variable indicating the occurrence of category sk at location x :

Ak

1,i f S( x) = s k = 0, otherwise

(2)

Similarly, let D be a binary random variable indicating the occurrence of data event d n. Then, the conditional probability of node x belonging to state sk is given by the simple indicator kriging (SIK) expression:

f(x;sk | dn) = Pr{Ak=1 | D=1} = E {Ak} + λ[1-E{D}]

f(x;sk | dn) = E{Ak}+

E {AkD }

− {E }A{k }E D

{

; 1{ , , J} i = 1| =Aα =; si α, ,∈ J

2

=

E {D}

Pr{Ak

=,1 D =} 1 (4) = 1}

Pr{D

Therefore, given a single global conditioning data event, this solution is identical to Bayes’ defi nition of the conditional probability. However, one might consider decomposing the global event D J into more simple components whose frequencies are easier to infer. From its definition, it is obvious that DJ can be any one of the 2 J joint outcomes of the J binary data events Aα = A(x+hα), α=1, …, J with Aα ∈ {0,1}. Equivalent to the common SIK estimate, the conditional probability of the event A0 = 1 can be written in a more general form as a function of the J conditioning data (Guardiano and Srivastava, 1993): Pr A0

+

A

(3)

where, E{D} = Pr{D=1} is the probability of the conditioning data event dn occurring, and E {Ak} = Pr{S(x) = sk} is the prior probability for the state at x to be s k. Solving the simple kriging system for the single weightλ leads to the solution of Equation (3):

[ λ(α11){ Aα 1 − E A0 }] + = E{}A0 + ∑ J

J

α1= 1 J

∑ ∑ ∑ )

1 K J

J

∑ ∑

}

λ(α2[1)α 2 Aα 1 Aα{2}]− E A0

α 1= 1 α 2 > α 1

( λ(α31 α 2 α 3 [ Aα 1 A2 α3 Aα −0{) }] E A

+ + λ J

α 1 = 1 α2>α 1 α3 > α 2 J

FIG 2 - Naming conventions to define MP statistics.

(1)

 J  ∏ Aα − E ∏ Aα  α = 1  α = 1



(5)

SIMULATION OF OREBODY GEOLOGY WITH MULTIPLE-POINT GEOSTATISTICS

The 2 J - 1 weights λ (αij) call for an extended system of normal equations similar to a simple kriging system that takes into account the multiple-point covariances between all the possible subsetsD A J' J , ' J⊆{ ,1  , } of the global event DJ. These β

∏

β ∈J '

multiple-point covariances are inferred by scanning the training image for each specific configuration. For the case when all J values a α are equal to 1, Equation (4) is identical to Bayes’ relation for conditional probability. The decomposition of the global event DJ illustrates that the traditionally used two-point statistics lose their exclusive status in an extended simple kriging system. The numerator and denominator of Equation (4) are inferred by scanning a training image and counting both the number of

dn), and replicates of event c(ones, thethe number ck(the dn),conditioning c previous replicates among the data with centralof value S( x) = sk. In the Single Normal Equation Simulation or SNESIM algorithm (Strebelle, 2002), these frequencies are stored in a search tree enabling fast retrieval. The required conditional probability is then approximated by: f(x;sk | dn) = Pr{Ak=1 | D=1}≈

c k ( dn )

(6)

c( dn )

To simulate an unknown location x, the available conditioning data forming the data event d n is retained. The proportions for building the ccdf (Equation 6) are retrieved from the search tree by searching the retained data event and reading the related frequencies. The SESIM algorithm and the options provided in the implementation have been covered elsewhere (Strebelle, 2002; Remy, 2004; Liu, 2006) and will not be repeated here in detail. An overview of the genera l steps of the simula tion is given below: 1.

Scan and occurrences data D. training eventsthe This mayimage be seen asstore building a databaseofofall jigsaw puzzle pieces of different shapes (D) and their centra l values (A) from the TI.

2.

Define a random path and visit nodes one by one.

3.

Simulate each node by:

4.

•

retrieving all data events (ji gsaw puzzle pieces ) fitting the surrounding data and previously simulated nodes,

•

derive the local probabili ty distribution from stored frequencies of central values; the probability of finding a certain lithology at the node given the surrounding data event D is given by Bayes relation for conditional probability, and

•

pick randomly from the distribution and add simulated node to the grid.

Start agai n at Step 1 for th e next re alisation, as may be needed.

CASE STUDY Geology of the Yandi channel iron ore deposit A number of operations in the Pilbara region of Western Australia produce iron ore from clastic channel iron ore deposits (CID) formed in the Tertiary. These deposits contribute a significant portion of the overall production from the region. Their formation in a fluvial environment with variable sources and deposition of the material as well as post-depositional alteration resulted in very large high quality but complex iron orebodies. The CID consists of an incised fluvial channel filled with detrital pisolite ore that is affected by variable clay content. Ore qualities depend on lithological domains that are modelled using sectional interpretat ions and grade cut-offs . Defining and modelling boundaries low-grade overburden and to internal high-aluminous areas to cause problems in the current resource estimation, assessment and modelling practices. Figure 3 shows a schematic cross-section through the CID showing the various lithologies. The erosional surface of the incised channel is covered by the BCC. From bottom to top, LGC, GVL, GVU, WCH and ECC sequentially fill the channel. ALL covers the whole channel sequence including the surrounding WW bedrock. The GVU and the GVL are the only units that currently fall within economic mining parameters. The WCH is a high SiO2 waste unit with a gradational uncertain boundary to the GVU below. These two ore bearing lithologies and the transitional WCH are encapsulated by high Al2O3 waste (WAS), which consists of various clay-rich low-grade strata in both the hanging wall and the footwall (ALL, ECC, LGC, BCC and WW). The study area is located at Junction Central deposit of the Yandi CID (Figure 4) and consists of the so-called Hairpin model area. The existing Hairpin resource orebody model is rotated by 45°. To accommodate for this rotation, this case study was performed in a local grid with north oriented to 285°. All results are presented in this rotated grid. The study area has been drilled out in various campaigns to nominal spacing of 100 m by 50 m. This data and the knowledge of absence of CID outside the drilled area are used to interpret the deposit. To introduce the knowledge about undrilled areas into the simula tions, the areas around the drilled CID was ‘infilled’ using 50 m by 50 m spaced data points with WAS code assigned (Figure 5).

Deriving a training image The training image (TI) has to contain the relevant geological patterns of the simulation domain. In the context of the Yandi CID, this means that the TI has to characterise the shape of the channel and of the internal boundaries within the study area. The geological model of the mined out initial mining area (IMA) is the best available source for this information: 1.

the model is based on relatively dense exploration drilling on a 50 m × 50 m grid, and

2.

it consists of a straight section of the CID thus having a constant channel axis azimuth.

FIG 3 - Schematic cross-section through CID showing the various lithologies. Lithologies: ALL – alluvium, ECC – eastern clay conglomerate, WCH – weathered channel, GVU – goethite-vitreous upper, GVL – goethite-vitreous lower, LGC – limonite-goethite channel, BCC – basal clay conglomerate, WW – Weeli Wolli formation.



3


FIG 4 - Location map of the study area.

FIG 5 - Drill hole data set (left) and infilled grid data (right).

The TI was generated as a regular geological block model of the IMA prospect into 10 m × 10 m × 1.25 m blocks. This resulted in 80 × 80 × 125 or 800 000 blocks in total. Four slices of the training image are depic ted in Figure 6 and show the main direction of the channel (EW) and the slight undulation of the channel axis. The boundaries between the various units are smooth, reflecting the wireframe model upon which the TI was based. As such, ensuring that the training image is consistent with the available data within the simulation domain is a measure needed to assess the validity and limits of the TI. Here, the variograms and cross-variograms of the geological categories are used for this validation. Two data sets will be compared with the TI: 1.

4

the data at IMA that was use d for constructing the geological model; this shows the differences of two-point statistics occurring between exhaustive 3D data and sparse drill hole data, and

FIG 6 - Bench sections (channel bottom to the top) of the lithological interpretation at Yandi IMA used as training image.

2.

the data available in th e simulation domain (HPIN) then serves the validation of the TI for use within that domain.




1

1 WAS

TI

0. 8

0. 8

data IMA-data

0. 6

0. 6

0. 4

0. 4

0. 2

0. 2

0

0 0

200

400

60 0

800

1

WCH

0

200

400

60 0

800

1 GVU

GVL

0. 8

0.8

0. 6

0.6

0. 4

0.4

0. 2

0.2 0

0 0

200

4 00

600

800

0

200

400

600

800

FIG 7 - Variograms of the four categories parallel to the channel axis for the TI, the data (at HPIN) and the data at the IMA, ie the area of the TI. The x-axis shows the lag in metres, the gamma values are given on the ordinate.

Note that this procedure change of(IMA) the two-point statistics between the datachecks in thethe TI-domain and the simulation domain (HPIN) . The statistics of the TI help to evaluate this change: If the differences between the TI-statistics and the HPIN-statistics are grossly larger than those between the TI and the IMA statistics, one will have to consider the reasons for and consequences of these differences. Overall, the variograms of the four categories perform well in this validation (Figure 7). For unit WAS (please refer to the geological unit abbreviations in the previous section), the HPIN variogram coincides more closely with the TI than the IMA. The WCH variogram of the HPIN data shows larger values at lags up to 350 m than both IMA data variograms and TI. However, these differences are relatively small. The GVU variograms follow a very similar structure; only at short lags do the HPIN variograms have slightly larger values. For the GVL, both data variograms have almost the same values but they are smaller than the TI variogram, suggesting stronger continuity.

Sim2

Sim1

Geological units (top figures) P(WAS)

P(WCH)

Wireframe

Probabilities (bottom figures) P(GVU)

P(GVL)

Simulation results To assess the geological uncertainty 20 realisations were generated. Each realisation of the 3.6 m nodes took 7.5 minutes on a 2.4 GHz personal computer, makin g the proces s very practical in terms of computational requirements. Figures 8 and 9 show bench 490RL and a cross-section of the channel, respectively; each figure includes two realisations along with the interpreted deterministic model (wireframe). The bench view shows that the overall shape of the channel has been well reproduced. The incised shape of the channel was generated on a large scale and the stratigrap hic sequence has been reproduced. The continuation of the tributary in the north-east was not generated due to very widely spaced drilling in the area. The



FIG 8 - Two simulations and wireframe interpretation (top); probability maps (bottom) for bench 490mRL – units WAS, WCH, GVU and GVL.

5


P(WAS)

P(WCH) P(GVU)

P(GVL)

Sim1

Sim2

Wireframe FIG 9 - Two simulations and interpretation and indicator probability for an E-W cross-section (see Figure 8 for colour coding).

proportion of GVL in this bench is higher in the simulations than in the interpreted model however, globally, proportions were reproduced. Boundaries in the simulations are less smoothed for both the GVL-GVU and the GVL-WAS contacts. In some areas, channel material was generated in small pods outside the continuous channel. The cross-section view supports these observations. However, on the channel margins, holes and saw-tooth shaped contacts are inconsistent with the depositional environment of the deposit. Bench 490RL (Figure 8) cross-cuts the boundary of GVL and GVU. The boundary is undulating and shows an increased

(dark grey line), and of the 20 simulations (bright grey lines). The consistency of the data and the TI was described earlier. Two interesting aspects are compared here: (a) simulations versus TI; and (b) simulations versus data. The WAS variograms are well reproduced in the main direction (EW), but the experimental data variograms suggest less continuity of lags up to 350 m, although this difference is not excessive. For WCH, the variogram reproduction is mediocre and suggests more continuity of the simulations compared to the data. The simulations deviate for lags larger than 50 m and reach the sill of the TI-variogram only at a lag of about 450 m. GVU

irregularity comparison withof GVU the wireframe model. Furthermore, in the overall proportion in bench 490RL is larger in the realisations than in the HIY model. On average probability for unit GVU (P(GVU)), the locations of the lowermost parts of the GVU are related to the wireframe model. However, there are areas in the northern part of the channel where the realisations contain GVU, while the wireframe model consists mainly of GVL. At the southern end of the channel, the GVU patches in the realisatio ns have an increased extension compared to the wireframe model. The outline of the GVU to the surrounding WAS in the realisations is very fuzzy, overall, compared with the wireframe model. This higher disorder occurs on two scales:

and GVL variograms are well reproduced and correspond to the experimental data variograms. Cross-variogram reproduction for WAS/GVL and GVU/GVL is good regarding the TI, however there is inconsistency with the data.

1.

On a very fine scale of a few blocks, the outline is strongly undulating.

2.

On a larger scale of about 15-25 blocks, the undulations are less extreme. However, they are still present and not consistent with the TI.

In the cross-section in Figure 9, the shapes of the channel margins are not well reproduced. Instead of an expected rather smooth outline as in the wireframe model, the appearance is sharply stepped (left margin of Sim1 and Sim2). The top part of the channel is very fringy. All the sections depicted in Figure 9 show saw-tooth shaped features at the channel margins, indicating slight problems of the algorithm to reproduce the patterns of the channel margins.

Reproduction of two point statistics The validation takes the major direction of continuity, EW or along the channel axis, into account: Figur e 10 shows the experimental variograms of the data (black diamonds), of the TI

6

Volumetric differences with deterministic wireframes and uncertainty in grade tonnage curves The intersection of stochastic realisations and estimated grades allows an assessment of uncertainty due to uncertain geological boundaries. For example, Al2O3 is chosen here to show the differences between simulated geology and conventional wireframing, because Al2O3 is not a well understood variable in the resource model of the deposit. Grade-tonnage curves below Al2O3 cut-offs are generated to reflect ore cut-offs. Blocks were selected only within the limits of the ultimate pit as optimised for the deposit at Harpin and below the WCH/GVU boundary that serves as the hanging wall ore limit. The grades used in the comparisons are estimated conventionally (ordinary kriging) and within each of the 20 simulated lithology models. A two per cent Al2O3 cut-off was applied to the Yandi Hairpin block grades to generate a product of about 1.35 per cent Al 2O3. Figure 11 shows the grade – tonnage curve of Al 2O3 for the resource within the ultimate pit limits and the uncertainty profile for Al 2O3 grade and resource tonnage. The two figures compare results based on the simulated lithology models (solid lines) and the deterministic (wireframe) lithology model (dashed line). The grade uncertainty appears relatively small. However, the resource tonnage indicated by simulations is on average 12 Mt (nine per cent), smaller than the tonnage indicated by the best-guess wireframe model. The simulations allow for estimating a tonnage confidence interval. With 70 per cent confidence, the final pit at




WA S - EW

WCH-

EW

1.2

1.2 Simulations 1

1

TI Data

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0 0

10 0

200

300

400

500

0

10 0

2 00

GV U -EW

3 00

4 00

5 00

4 00

5 00

GVL - EW

1.2

1.2 Simulations

1

TI

1

Data 0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0 0

100

2 00

300

4 00

500

0

0

10 0

2 00

3 00

0

WAS/GVL -0.01

-0.005

-0.02

-0.01

-0.03

-0.015

-0.04

-0.02

-0.05

-0.025

GVU/GVL -0.06

-0.03 0

10 0

20 0

3 00

4 00

5 00

60 0

0

10 0

20 0

3 00

40 0

500

600

FIG 10 - Variogram and cross-variogram reproduction of simulations versus TI and data for various units.

CONCLUSIONS

geologically meaningful. The MP method can incorporate information from dense drill hole data as available in typical mining applications. The visual validation showed inconsistencies of the algorithm, reproducing patterns at the margins of the channel. In bench

Multiple-point simulation provides a practical and powerful option to assess uncert ainty in the geolog ic units of mineral deposits. The application of the MP method at Yandi utilises geometric information from a mined-out area. The generated realisations are easily comparable to the existing geological model and reproduce general channel shapes and the rotation of the channel axis. Geometries borrowed from the mined-out area are, in general, well reproduced. The position of boundaries in between drill holes changes from realisation to realisation, thus reflecting the uncertainty about the boundaries’ ex act shape. On the margins of the channel, the generated patterns are not always

views, the outline ofblocks. the GVL, the GVU, and WCH undulates on a scale of 15-25 Additionally, the the simulations show a strong, short-scale fuzziness for the GVU and the WCH. This visual impress ion is underpinned by the larger perimeter-tovolume ratio of the realis ations compar ed to the TI. In the cross-sections, the major critical observation is that the erosional contact to the Weeli-Wolli formation is not consistent with observations in the pit nor with geological knowledge srcinating from modern geomorphologic analogues. Two sources for these issues with pattern reproduction have to be considered, ie the TI and the algorithm.

the Hairpi n deposit conta ins 95-97 Mt of ore within Al 2O3 specifications. This shows that the contribution of the geological uncertainty to the overall grade uncertainty is considerable.



7


FIG 11 - Al2O3 grade-tonnage curves between WCH and the ultimate pit limits (left) and uncertainty profile at two per cent Al2O3 (right) where bars depict the mean of the simulation ±1σ . Note that the grade variability (<1 per cent) is not significant. (T – tonnage, Al – Al2O3, Sim – simulations; HIY – wireframe).

It was shown that the TI and the data in the simulation domain are not fully consistent with respect to two-point statistic s. The extent to which this influences the quality of reproduced patterns is difficult to assess. Using a set of different training images can provide further insight. Resource grade and tonnage uncertainty due to uncertain lithological boundaries was assessed by combining probabilistic realisations of the geology with a standard grade estimation technique. At an alumina cut-off of two per cent, the ore tonnage based on the simulated geology ranges from 94.5 to 97.5 Mt (wireframe model: 107 Mt) with bulk alumina grades below the cut-off ranging insignificantly between 1.357 per cent and 1.37 per cent (interpreted model: 1.37 per cent). Using grade simulation instead of grade estimatio n techniques would add realistic grade variability to this model and allow the assessment of total grade tonnage uncertainty. Potential areas of application are in areas of little geological understanding or definition of boundaries by drilling. At Yandi, internal clayey high-aluminous waste that cannot be defined with the 50-100 m spaced resource evaluation drilling and simulation could create value by better defining grade tonnage curve with regard to contaminants. Training images could be constructed from geological interpret ation and data gained in previously mined areas of the deposit.

ACKNOWLEDGEMENTS Special thanks to Michael Wlasenko and Jim Farquhar from Rio Tinto Iron Ore for their support of the case study at Yandi.

REFERENCES Arpat, G B and Caers, J, 2007. Conditional simulation with patterns, Mathematical Geology, 39(2). Betzhold, J and Roth, C, 2000. Characterising the mineral variability of a Chilean copper deposit using pluri-Gaussian simulations, Journal of the SAIMM, March/April, pp 111-120. Carle, S F and Fogg, G E, 1996. Transition probability-based indicator geostatistics, Mathematical Geology, 28(4):5453-5476. Daly, C, 2004. High order models using entropy, Markov random fields and sequential simulation, in Geostatistics Banff 2004 (eds: O Leuangthong and C Deutsch) Vol 2, pp 215-224 (Springer: Dordrecht). David, M, 1988. Handbook of Applied Geostatistical Ore Reserve Estimation (Elsevier: The Netherlands). Deraisme, J and Field, M, 2006. Geostatistical simulations of kimberlite orebodies and application to sampling optimisation, in Proceedings Sixth International Mining Geology Conference, pp 193-203 (The Australasian Institute of Mining and Metallurgy: Melbourne).

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Dimitrakopoulos, R and Dagbert, M, 1993. Sequential modeling of relative indicator variables: Dealing with multiple lithological types, in Geostatistics Troia ‘92 (ed: A Soares) Vol 2, pp 413-424 (Kluwer Academic Publishers). Fontaine, L and Beucher, H, 2006. Simulation of the Muyumkum uranium roll front deposit by using truncated plurigaussian method, in Proceedings Sixth International Mining Geology Conference, pp 205-215 (The Australasian Institute of Mining and Metallurgy: Melbourne). Goovaerts, P, 1997. Geostatistics for Natural Resource Evaluation (Oxford: New York). Guardiano, F and Srivastava, R M, 1993. Multivar iate geostatist ics: Beyond bivariate moments, in Geostatistics Troia ’92 (ed: A Soares) Vol 1, pp 133- 144 (Kluwer Academic Publishers: Dordrech t). Journel, A G, 2007. Roadblocks to the evaluation of ore reserves — The Simulation overpass and putting more geology into numerical models of deposits, in Orebody Modelling and Strategic Mine Planning — Uncertainty and Risk Management Models (ed: R Dimitrakopoulos) Second Edition, pp XX-XX (The Australasian Institute of Mining and Metallurgy: Melbourne). Journel, A G and Alabert, F A, 1988. Focusing on spatial connectivity of extreme-valued attributes: Stochastic indicator models of reservoir heterogeneities. SPE paper 18324. King, H F, McMahon, D W, Bujtor G J and Scott, A K, 1986. Geology in the understanding of ore reserve estimation: An Australian viewpoint, in Ore Reserve Estimation, Applied Mining Geology (ed: D E Ranta) Vol 3, pp 55-68 (Society for Mining, Metallurgy, and Exploration: Littleton). Langlais, V and Doyle, J, 1993. Comparison of several methods of lithofacies simulation on the fluvial gypsy sandstone o f Oklahoma, in Geostatistics Troia ’92 (ed: A Soares) Vol 1, pp 299-310 (Kluwer Academic Publishers: Dordrecht). Le Loc’h, G and Galli, A, 1997. Truncated Plurigaussian method: Theoretical and practical points of view, in Geostatistics Wollongong ’96, (eds: E Y Baafi and N A Schofield) Vol 1, pp 211-222 (Kluwer Academic Publishers: Dordrecht). Li, W, 2007. A fixed-path Markov algorithm for cond itional simulation of discrete spatial variables, Mathematical Geology, 39(2). Liu, Y, 2006. Using the Snesim program for multiple-point statistical simulation, Computers and Geosciences, 32(10):1544-1563. Liu, Y and Journel, A, 2004. Improving sequential simulation with a structured path guided by information content, Mathematical Geology, 36(8):945-964. Remy, N, 2004. S-GEMS – Stanford Geostatistical Earth Modeling Software: User’s Manual. Stanford University [online]. Available from: http://sgems.sourceforge.net/doc/sgems_manual.pdf [Accessed: 13 March 2006]. Richmond, A J and Dimitrakopoulos, R, 2000. Evolution of a simulation: Implications for implementation, in Geostatistics Cape Town 2000 (eds: W J Kleingeld and D G Krige) Vol 1, pp 135-144 (Geostatistical Association of South Africa: Johannesburg).




Seifert, D and J L Jensen, 2000: Object and pixel-based reservoir modelling of a braded and fluvial reservoir, Mathematical Geology, Vol 32, p p 581-603. Sinclair, A J and Black well, G H, 2002. Applied Mineral Inventory Estimation, 381 p (Cambridge University Press, Cambridge). Srivastava, R M, 2005. Probabilistic modeling of ore lens geometry: An alternative to deterministic wireframes, Mathematical Geology , 37(5):513-544. Strebelle, S, 2002. Conditional simulation of complex geological structures using multiple-point statistics, Mathematical Geology , 34(1):1-21.



Tjelmeland, H and Eidsvik, J, 2004. Directional Metropolis: Hastings updates for posteriors with non linear likelihood, in Geostatistics Banff 2004 (eds: O Leuangthong and C Deutsch) Vol 1, pp 95-104 (Springer: Dordrecht). Zhang, T, Switzer, P and Journel, A, 2006. Filter-based classification of training image patterns for spatial simulation, Mathematical Geology, 38(1):63-80.

9

10



Grade Uncertainty in Stope Design — Improving the Optimisation Process N Grieco1 and R Dimitrakopoulos2 ABSTRACT Decisions in the mining industry are made in the presence of uncertainty whether it is in the form of technical, financial or environmental risk. In recent years, the main focus of uncertainty has been the mineral resource. Methods for assessi ng and quantifying grade risk in open pit operations has lead to the ability to forecast problems and improve the design and planning process by integrating this risk. This paper successfully implements these risk-based methods in an underground stoping environment using data from Kidd Creek Mine, Ontario, Canada. Risk is quantified in terms of the uncertainty a conventional stope design has in contained ore tonnes, grade and economic potential. A mathematical formulation optimising the size, location and number of stopes in the presence of uncertainty is introduced and applied. The implementation of different geostatistical simulation methods to the optimisation formulation is discussed briefly and observations made.

INTRODUCTION Risk is present in all facets of mining be it technical, financial or environmental (Rendu, 2002). When determining the feasibility of a project the uncertainty associated with all sources must be considered and contingencies made. Geological uncertainty is a major component of technical uncertainty, along with mining, and has been isolated as a primary source of risk affecting the viability of projects. This uncertainty is recognised as the key factor responsible for many mining failures (Baker and Giacomo, 1998; Vallee, 1999). Hence, the necessity to quantify geological risk is well appreciated. Modelling geological uncertainty in a mineral resource can be achieved through conditional simulation technologies. The last few years in open pit mining these technologies have been coupled with mine design optimisation methods to assess risk in conventionally generated mine designs and production schedules. The approach allows planners to anticipate fluctuations in key project parameters that would otherwise be impossible (Ravenscroft, 1992; Dowd, 1997; Dimitrakopoulos, Farrelly and Godoy, 2002). These studies have also documented that conventional methods may be misleading in their forecasts as they assume certainty. Recent developments in open pit mining show that direct integration and management of inherent grade risk in mine design and planning have begun (Dimitrakopoulos and Ramazan, 2004; Ramazan and Dimitrakopoulos, 2007, this volume; Menabde et al , 2007, this volume; Froyland et al , 2007, this volume; Dimitrakopo ulos, in press). The developments provide the opportunity to generate substantially more profitable mine designs; for example, Godoy and Dimitrakopoulos (2004) report a 28 per cent higher NPV from managing geological risk. It is logical to consider how to develop concepts and similar risk-based technologies for underground mining methods. Optimisation in underground mine design has had less routine application than open pit mines, which is attributed to the diversity of underground mining methods that does not allow the production of general optimisation tools. Related in the technical literature is the work of Ovanic (1998) who considers the 1.

AMEC Americas Limited, 2020 Winston Park Drive, Oakville ON L6H 6X7, Canada. Email: [email protected]

2.

MAusIMM, COSMO Laboratory, Department of Mining, Metals and Materials Engineering, McGill University, Frank Dawson Adams Building, Room 107, 3450 University Street, Montreal QC H3A 2A7, Canada. Email: [email protected]



economic optimisation of stope geometry, a topic directly linked to the present study; and work on conventional stope optimisers (Thomas and Earl, 1999; Ataee-pour and Baafi, 1999). None of these approaches consider risk and hence assume the inputs are certain. Limited initial work reported, combines simulated orebodies and grade risk models with conventional optimisers (Standing et al , 2004); these however, are limited in their assessment as optimisation formulations are, in general, a non-linear process. Geological risk-based approaches to stope optimisation that directly integrate risk have been recently introduced (Grieco, 2004) and open the possibility to further develop risk-based underground mine design. Current efforts, however, focus on the issue of grade uncertainty. In the longer run these developments need to be fused with geotechnical issues critical to underground mining (Bawden, 2007, this volume). This paper stems from the need to explore the contribution of geological uncertainty quantification and the direct integration to stope optimisation through a new, risk-based approach to stope design. In the following sections a conventional stope design in a part of Kidd Creek base metal mine, Ontario, Canada, is assessed in terms of copper grade risk, to explore uncertain ty in terms of upside potential as well as downside risk. Subsequently, a probabilistic mathematical programming optimisation formulation is outlined and applied. The question of the sensitivity to the geostatistical simulation method is briefly visited. Finally, summary and conclusions follow.

QUANTIFYING GRADE RISK IN CONVENTIONAL STOPE DESIGNS: AN EXAMPLE Grade risk quantification in a given underground stoping design is similar to that used in the design and production schedule of an open pit mine (Dimitrakopoulos, Farrelly and Godoy, 2002). The quantification process requires two main components: 1.

the design of a stoping outline generated using a conventionally estimated orebody model; and

2.

a series of simulated realisations of the orebody, quantifying the uncertainty and in situ variability.

By putting each realisation through the stoping outline, as if the realisation is the actual orebody being mined, and accounting for potential productio n from the design, distributions or risk profiles for the pertinent project indicators are generated, thus allowing the quantification of geological uncertainty and risk assessment for the design being considered.

The deposit and study area Applying the concepts outlined for quantifying the grade risk in a conventional stope design is presented with a case study involving data from Falconbridge Ltd’s Kidd Creek Mine. Kidd Creek is a volcanic massive sulfide deposit located in Ontario, Canada and produces about 7000 tonne per day (Roos, 2001) from two major orebodies containing silver, copper, zinc and lead, the main commodities. Production began in 1966 via an open pit mine and has extended into three underground mines reaching depths of over 2000 m and employing various mining methods including sublevel caving, open stoping and sublevel stoping.

1

*

N GRIECO and R DIMITRAKOPOULOS

The focus of this study is a densely drilled area located in the copper concentrated stringer ore 1400 m below the surface in Phase I of Mine No 3. The drill hole configuration consists of 37 drill holes with 1.5 m copper composites in nine vertical fans that are spaced approximately four metres apart. The resulting samples show a high-grade zone in the central region. Statistics outlining the distribution of declustered copper samples is given in Table 1. Mining in this region is via open stoping methods with stope sizes typically 15 m wide by 20 m long by 40 m high. Blast rings are spaced generally every three metres and have a copper cut-off of three per cent.

Generating estimated and s imulated orebody models Estimation methods are by construction smoothing operations. Conditional simulatio n methods aim at modelling the in situ spatial variability of a given attribute and, unlike the equivalent estimation approaches, reproduce the data histogram and spatial continuity. At Kidd Creek, the study area is first geostatistically estimated, producing 16 236 blocks within the orebody model. Blocks are estimated with a block size of 3.0 m × 3.0 m × 4.5 m, spanning 123 m in the east, extending 51 m in the north and reaching 99 m in the vertical direction. A horizontal section of this estimated model is shown in Figure 1. The same area of the deposit is then geostatistically simulated using the wellestablished sequential Gaussian simulation method or SGS

TABLE 1 Declustered data statistics of copper. S t a t is t i c Numberofsamples Mean Standarddeviation

D e c l u s t er e d d a t a s e t 2723 2.43% 3.17%

Maximum 75thpercentile

27.59% 3.00%

Median

1.34%

25thpercentile

0.54%

Minimum

0.0%

(Goovaerts, 1997). Forty realisations of the deposit are generated on a 1.5 m × 1.5 m × 1.5 m grid of 19 880 nodes. Figure 2 shows a simulated realisation of copper grades of the same horizontal section as in Figure 1. When comparing the estima ted and simulated models in Figure 1 and Figure 2, both reproduce the regions of high-grade mineralisation in the drill hole configuration. The figures also show the typically smooth representation of reality by the estimated model whilst the simulated realisation reflects the likely in situ copper variability.

Risk quantification In establishing a conventional stope design, a conceptual stoping layout recognising potential development and stoping levels must be first determined. Due to the vertical extent of the orebody models, two potential stoping levels are (Figure configured accounting for required drilling and hauling levels 3). It is assumed that the lower level will be mined and backfilled before the upper level is extracted. Accounting for this stoping layout, a stope outline is produced given the estimated copper grade model using the DATAMINETM floating stope facility, hence providing a conventional design for which a risk quantification and analysis can be performed. Figure 4 shows a three-dimensional view of the conventional outline generated here incorporating both stoping levels. For the quantification of copper grade risk in this conventionally generated stope design, first, the simulated copper realisations are re-blocked into mineable rings by averaging the nodes contained within consecutive ring dimensions (15 m × 3 m × 40 m). Then, the conventional design outline is put through each of the orebody realisations and values pertaining to copper grades are recorded. It is subsequently simple to calculate for a set of realisations, such as the 40 here, the ore tonnage, metal, average grade and revenues or any other project indicator, the corresponding histogram of possible outcomes and from that histogram statistics of interest such as the various percentiles and so on. The following discussion refers to the risk profiles of some project indicators. Figure 5 depicts the risk profiles for the upper and lower stoping outlines providing a means of quantifying copper grade risk in terms of the potential average copper grade the conventional design could contain. The conventional design and approach tend to underestimate the likely contained grade in the lower stoping level, while in the upper level tends to overestimate copper grade.

FIG 1 - Horizontal section of the estimated orebody model.

2



GRADE UNCERTAINTY IN STOPE DESIGN — IMPROVING THE OPTIMISATION PROCESS

FIG 2 - Horizontal section of a simulated orebody model.

FIG 3 - Stoping layout indicating two stoping levels.



3


Uncertain ty in Copper Grades 300,000

6.00

250,000

5.00

s e200,000 n n to d150,000 e in a t n100,000 o C 50,000

4.00 3.00 2.00 1.00

0

0.00 Lower Level

N

) (% u C

ESTIMATE

Upper Level MIN

25TH

Both Levels

75TH

MAX

AVG

15m

FIG 5 - Quantifying the conventional stope envelope’s uncertainty in copper grade. FIG 4 - Conventional stope envelope.

For analysis purposes only, the rings within the design outline that are less than three per cent copper are removed to uncover how the grade uncertainty within the orebody model effects the amount of ore tonnes, metal and economic potential that could, in reality, be realised. Figures 6, 7 and 8 illustrate the resulting risk profiles of these parameters respectively. Figure 6 also highlights the amount of material within the original design outline before any waste rings are removed (black diamonds). This demonstr ates a potential for the conventional outline (both levels) to contain up to 32 per cent waste, significantly affecting the tonnes expected to reach the mill. Both Figures 6 and 7 illustrate a generally small risk the conventional outline presents in the amount of ore and metal tonnes expected from the upper level, as the extreme grade values present a tight distribution in which the expected values fall. Figure 8 shows the results of an economic evaluation of the stoping levels using values representing the present value before tax. The figure illustrates signifi cant risk in the conventional outline’s ability to predict its potential economic value in each level. In addition, the singl e estimate in the lower leve l is 17 per cent less than the average predicted economic potential expected, while the estimate in the upper level is 33 per cent above this equivalent average value. Since each level will likely be mined in separate periods, the profit made in the upper level cannot compensate for the potential loss (seven per cent) in the lower level. This potential to incur monetary losses on production could, for example, affect monthly profits expected from this part of the mine. The conventional stoping design in this specific example is generally straightforward and is found to provide a reasonable assessment of the average economic value of the design. However, several points can be made, including the following: 1.

if the ability to quantify risk was not available, the assessment would not be possible; and most importantly

3.

conventionally, one is unable to fores ee the signi ficant upside potential and/or downside risk the conventional design may actually produce (eg Table 2).

In the example presented here, quantifying the risk in terms of economic potential recognises the potential to earn 62 per cent more and the risk of earning 38 per cent less than expected. In dollar terms, this conventional design could be worth as little as 1.8 million dollars or as much as 5.9 million dollars. The above lead to considerations such as:

Ore tonnes

250,000 s 200,000 e n n to 150,000 re O 100,000

50,000 0 Lower Level ESTIMATE

MIN

Upper Level 25TH

75TH

MAX

Both Levels AVG

ORIGOUTLINE

FIG 6 - Quantifying the conventional stope envelope’s uncertainty in contained ore tonnes.

Quantifying Risk Quantity of metal 12,000 10,000 8,000 6,000 4,000 2,000

the siz e of the s tudy ar ea is sm all and at th e same tim e uncommonly well drilled (nearly three times the density of fans normally expected), thus results are not surprising;

2.

4

QuantifyingRisk 300,000

0 Lower Level ESTIMATE

Upper Level MIN

25TH

75TH

Both Levels MAX

AVG

FIG 7 - Quantifying the conventional stope envelope’s uncertainty in contained metal.

1.

Can grade uncertainty be not only quantified for a design, but also employed during the design process to capture the upside economic potential of the deposit?

2.

Can designs be based on a minimum acceptable risk? And generally, can the design process manage grade risk directly and generate benefits?




up of a series of rings. With multiple simulated orebodies available, each ring can be identified by a probability to be above any cut-off grade and have an average grade, hence introduci ng grade risk into the process. The objective function of the formulation focuses on maximising the grade content within a layout in the presence of grade uncertainty, and is:

Quantifyi ng Risk Economic potential 7 6 5 4 3

m

n

Maximise∑ ∑ g ij pij Bij

2

(1)

j=1 i=1

1

where:

0 Lower Level

-1

ESTIMATE

Upper Level MIN

25TH

75TH

Both Levels MAX

AVG

FIG 8 - Quantifying the conventional stope envelope’s uncertainty in economic potential.

m

is the number of panels within the orebody model

n pij

is the number of rings within a panel is the probability of ring ij being above a specified cut-off

gij

is the expected grade of ring ij above the cut-off

Bij

is a binary varia ble representing every ring within the model and identifies whether it has been selected (Bij =1) or not (Bij =0) in the optimal layout

TABLE 2 Project indicators based on the conventional stope design. M o del

Ore (t)

Metal (t)

Cu (%)

Economic Economic potential potential ($) % difference

Estimate

196 830

9490

4.82

3 412 999

--

Realisation 3

191 909

8769

4.57

2 285 625

- 33

Realisation 18

167 306

8228

4.92

1 858 484

- 46

Realisation 31

216 513

11 187

5.17

5 905 110

+ 73

Realisation 35

211 592

10 492

4.96

4 820 407

+ 41

In the last decades, major improvements have been made to the time-consuming manual approach to stope design. However, these computer-aided tools are limited in their ability to mathematically optimise the location of designs under uncertainty similarly to the optimisation methods in open pit mine design. With a methodology in place for quantifying grade risk in conventional mine design, the limitations of existing computer planning and optimisation tools force the development of a new optimisation approach based on and integrating grade uncertainty directly into the optimisation process, essentiall y creating a more versatile computer-aided tool.

GENERATING RISK-BASED DESIGNS Mathematical programming methods provide a means of optimising an objective function subject to a set of constraints through a mathematical formulation. Such methods allow the development of formulations that integrate grade uncertainty directly into the optimisation process, as well as allow the consideration of a user-selected minimum acceptable risk. In this section, a mathematical programming formulation considering the above to optimise the location of stopes in the presence of grade uncertainty is presented and used at Kidd Creek to produce a risk-based design for comparison and analysis.

The optimisation formulation A mixed integer programming (MIP) formulation with the aim of locating an optimal stope layout is presented here. This optimal layout is defined by the size, location and number of stopes within an orebody model. Such a model is described as consisting of a series of layers, each of which is composed of a number of rows referred to as panels, where the panels are made



Further to the above, the presence of simulated orebody models allows risk-based designs to be generated for a given minimum level of acceptable risk specified by the planner or decision-maker. The following constraint restricts the total average probability of selected rings within a panel to be greater than or equal to an assigned value representing the minimum acceptable level of risk (PL). n

∑ (p

ij

)

− PL B ≥0 ij

(2)

i=1

By changing the value of the minimum acceptable level of risk, PL, a number of different risk-based designs can be generated, compared and assessed. Risk profiles can then be generated for the key project indicators by putting each outline through all simulated realisations, in the same procedure that was used to quantify risk in the conventional design of the previous section. A design that best suits the operational requirements can be selected with the risk being quantifiably assessed (Grieco, 2004). The formulation above is also constrained by limitations on the stope size – both minimum and maximum, which are a direct reflection of the geotechnical restrictions and production requirements of the area. These stope size constraints are based on the number of consecutive rings allowed to form a single stope. The size of the pillars between two primary stopes is also considered. This algorithm determines the minimum number of rings to be left un-mined between stopes and is directly related to the size of the stopes surrounding them. The larger a stope, the larger the pillar is.

Application at Kidd Creek The MIP formulation for optimisi ng a stope as above is applied to the study area at Kidd Creek mine. Geotechnical requirements in the region restrict a given stope to consist of a minimum of two rings and a maximum of seven. Applying a cut-off grade of three per cent, each ring within the re-blocked orebody model (same configuration as the one used in simulation) is represented by the probability of being above three per cent copper and the average copper grade above this cut-off. A risk-based design with a minimum acceptable level of risk at 80 per cent is generated. Figure 9 illustrates a three-dimensional aspect of the resulting design layout using the simulated model, with dark grey rings representing primary stopes and light grey rings the

5


recoverable pillars. In comparing the size and shape of the conventional design outline (Figure 4) with the new, risk-based design, a notable difference in size is recognised. Introducing the minimum acceptable level of risk has limit ed the amount of waste (tonnes) contained within the new design as it forces the stopes within a given panel to have an average probability above 80 per cent. This approach grants the planner control over the level of risk permissible within a given design. The formulation constraints require the stopes and pillars to contain a minimum of two rings and a maximum of seven, providing an optimal combination for obtaining the most metal. The conventional approach produces an envelope of rings for which some combination satisfies the minimum grade and size requirements and further development of a mineable stope layout is needed. The fluctuation in copper grade within the risk-based design can be predicted by putting the outline through all simulated realisations generat ed with the SGS method, similarly to the conventional design in a previous section. Figure 10 illustrates the amount of contained material within the primary stopes and recoverable pillars, and the potential grade variation within each. Although grade uncertainty has been accounted for within these

N

15m

FIG 9 - LP stope design layout based on SGS and 80 per cent acceptable level of risk.

SGS vs SIS Risk Based Designs 80% minimum acceptable level of risk

160,000

8.00

140,000

7.00

se120,000 n n100,000 o t d e 80,000 in a t 60,000 n

6.00 5.00 ) % 4.00 (u C 3.00

o 40,000 C

2.00

20,000

1.00

0

0.00 SGS STOPES

SIS PILLARS

MIN

SGS 25TH

75TH

SIS MAX

AVG

FIG 10 - Primary stoping layout for LP designs based on SGS and SIS orebodies and 80 per cent acceptable level of risk.

6

designs, the simulated realisations reflect the variability in grade within this area. Additional information shown in Figure 10 is discussed in the next section.

Effects of the simulation method Conventional estimation approaches used for orebody modelling differ in their formulations as well as orebody models they generate from the same srcinal dataset. Similarly, different implementations of the same method will result in somewhat different represent ations of the orebody being modelled. The same is also true for simulation methods and the orebody models generated, including the average ring grades and probabilities above the cut-off consider ed in the stope optimisa tion approach used here. Thus, it may be of interest to consider how the stope optimisation results may differ, if the orebody used was simulated independently and with a different simulation method. For this study, an alternative method is the sequential indicator simulation method or SIS (Goovaerts, 1997) and was implemented independently from this study at Kidd Creek by Kay (2001). The latter study provides 40 simulated realisation s of the same broader domain. Figure 10 compares the two designs (both with an 80 per cent acceptable level of risk) in terms of the contained tonnage and grade for both the primary stoping and pillar recovery layouts. As expected, these design layouts contain the same amount of tonnes with only slight variations in potential copper grade. The wider risk profile in the pillar recovery layout is not unexpected due to the limited selection of rings remaining for the second pass of the optimiser. The limited extent of pillar recovery can be explained using the same rationale. From the observations made from Figure 10, the difference in simulation method cannot be said to affect the stoping optimising process. Figures 11 and 12 illustrate, on a given section, the location and size of the relative stoping (dark grey) and pillar (light grey) layouts based on the simulated orebody with the two different methods at an 80 per cent probability above cut-off set as the minimum acceptable risk. The figures reflect how the central high-grade zone evident in the drill holes is consistently reproduced by both simulation techniques, as expected, and hence located by the optimisation process at the specified probability constraint. The lower level stoping layouts are almost identical. In the upper level, the SGS-based layout considers a stope in the north-east part of the study area not included in the layout shown in the figure based on SIS at the same 80 per cent probability. However, if the minimum acceptable level of risk governing these designs is lowered to say, 70 per cent, the same part of the study area is highlighted as the location of a possible stope by the optimisation based on the SIS models. The stoping layout in the upper level based on the SIS orebody model s, recognises a larger stope in the sixth panel whose extent is not considered by the layout based on the SGS models. These minor differences between designs are normal and not significant. Similarly to the various conventionally used estimation methods for orebody modelling leading to variations in stope designs, different simulation methods will perform somewhat differently from each other, as their specific technical specifications and characteristics dictate. For example, SGS is based on one grade variogram whilst SIS requires multiple variograms, each for a series of grade cut-offs (Goovaerts, 1997). The discrepancies arising from different methods are more extensively documented in other areas of application of simulations such as grade control that have long been in practice (Dimitrakopoulos, in press). Independent implementations provide a source of variance for the results, because the detailed specifications of the simulated orebody models and the parameters for their generation are different. These deviations become apparent in the stoping layouts generated.




FIG 11 - Horizontal section of the risk-based stope designs in the upper level from the orebodies generated with SGS (left) and SIS (right), for 80 per cent acceptable level of risk.

FIG 12 - Horizontal section of the risk-based stope designs in the lower level from the orebodies generated with SGS (left) and SIS (right), for 80 per cent acceptable level of risk.

SUMMARY AND CONCLUSION This paper extends concepts and technologies used in managing geological risk in open pit mines to underground mining methods. It shows that geostatistical simulation technologies allow grade risk quantification in a stoping design. The example from the Kidd Creek mine, Ontario, Canada illustrates how conventional technologies cannot quantify risk, thus are unable to foresee a significant upside potential and/or downside risk for the conventionally produced designs. The example shows a conventional design could be valued from as little as 1.8 million dollars to as much as 5.9 million dollars. To provide the means of incorporating risk in stope design, geological uncertainty is integrated into the design process through a new mathematical programming formulation that uses risk grades above a cut-off value for rings within a stope, as well as geometric and other traditional constraints. An additional constraint introduced is the minimum acceptable risk allowed in a design. The application shows that the risk-based approach has the ability to generate different that meet the pre-specified minimum acceptable risk with designs a desired risk profile accommodating the selection of designs with preferred upside/downside profiles. Grade uncertainty quantificat ion may be based on different simulatio n methods. A comparison of orebody models constructed independently with the sequential Gaussian and indicator simulation methods show stope designs with some variation, which is not significant and considered normal when different methods are used. The work presented here could be further developed. Such developments could include:



1.

the formulation of a stop e optimisation formulation tha t replaces the probability of grades above cut-off with the direct use of all available simulated orebodies, and thus integrate more geological information;

2.

consider sequencing and thus accommodate risk management and/or geological risk discounting as part of the stope design process; and

3.

extend to integrate geotechnical uncertainties starting from over-breaking and under-breaking.

ACKNOWLEDGEMENTS Special thanks to Paul Roos and Arie Moerman from Falconbridge, Kidd Creek mine who supplied the data and provided support. Thanks also to Mark Noppe and Jörg Benndorf for their constructive comments.

REFERENCES Ataee-pour, M and Baafi, E Y, 1999. Stope optimisation using the maximum value neighborhood (MVN) concept, in Proceedings 28th International Symposium on the Application of Computers and Operations Research in the Mineral Industry (ed: K Dagdelen), pp 493-501 (Colorado School of Mines: Golden). Baker, C K and Giacomo, S M, 1998. Resource and reserves: their uses and abuses by the equity markets, inOre Reserves and Finance, pp 65-76 (The Australasian Institute of Miningand Metallurgy: Melbourne). Bawden, W F, 2007. Risk assessment in strategic and tactical geomechanical underground mine design, in Orebody Modelling and Strategic Mine Planning, Second Editio n, pp XXX-XXX (The Australasian Institute of Mining and Metallurgy: Melbourne).

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DATAMINETM, 1995. Floating Stope Optimiser User Guide, Edition 1.2, 20 p (Mineral Industries Computing Limited). Dimitrakopoulos, R, in press. Applied risk analysis for ore reserves and strategic mine planning: Stochastic simulation and optimisation, 350 p (Springer – SME: Dordrecht). Dimitrakopoulos, R, Farrelly, C T and Godoy, M, 2002. Moving forward from traditional optimisation: grade uncertainty and risk effects in open-pit design, Trans Inst Min Metall (Section A), 111:A82-A88. Dimitrakopoulos, R and Ramazan, S, 2004. Uncertainty based production scheduling in open pit mining, SME Transactions, 316:106-112. Dowd, P A, 1997. Risk in minerals projects: analysis, perception and management, Trans Inst Min Metall (Section A), 106:A9-A18. Froyland, G, Menabde, M, Stone, P and Hodson, D, 2007. The value of additional drilling to open pit mining projects, in Orebody Modelling and Strategic Mine Planning, Second Edition, pp XXX-XXX (The Australasian Institute of Mining and Metallurgy: Melbourne).

Myers, P, Standing, C, Collier, P and Noppè, M, 2007. Assessing underground mining potential at Ernest Henry Mine using conditional simulation and stope optimisation, in Orebody Modelling and Strategic Mine Planning, second edition, pp XXX-XXX (The Australasian Institute of Mining and Metallurgy: Melbourne). (not mentioned in text) Ovanic, J, 1998. Economic optimization of stope geometry, PhD thesis, Michigan Technological University, Houghton. Ramazan, S and Dimitrakopoulos, R, 2007. Stochastic optimisation of long-term production schedu ling for open pit mines with a new integer programming formulation, in Orebody Modelling and Strategic Mine Planning, Second Edition, pp XXX-XXX (The Australasian Institute of Mining and Metallurgy: Melbourne). Ravenscroft, P J, 1992. Risk analysis for mine scheduling by conditional simulation, Trans Inst Min Metall (Section A), 101:A104-A108. Rendu, J-M, 2002. Geostatistical simulations for risk assessment and

Godoy, M C and Dimitrakopoulos, R, 2004. Managing risk and waste mining in long-term production scheduling, SME Transactions, 316:43-50. Goovaerts, P, 1997. Geostatistics for Natural Resources Evaluation , 483 p (Oxford University Press: New York). Grieco, N J, 2004. Risk analysis of optimal stope design: incorporating grade uncertainty, MPhil thesis, University of Queensland, Brisbane, 204 p. Kay, M H, 2001. Geostatistical integration of conventional and downhole geophysical data in the metalliferous mine environment, MSc thesis, University of Queensland, Brisbane, 204 p. Menabde, M, Froyland, G, Stone, P and Yeates, G A, 2007. Mining schedule optimisation for conditionally simulated orebodies, in Orebody Modelling and Strategic Mine Planning, Second Edition, pp XXX-XXX (The Australasian Institute of Mining and Metallurgy: Melbourne).

decision making: the mining perspective, Journal of Surface Mining,industry Reclamation and International Environment, 16:122-133. Roos, 2001. Underground tour guidebook, Kidd Creek Mine, p 21. Standing et al, 2004. (see page 1) and advise. Thomas, G and Earl, A, 1999. The application of second-generation stope optimisation tools in underground cut-off grade analysis, in Proceedings Strategic Mine Planning, pp 175-180 (Whittle Programming Pty Ltd: Perth). Vallee, M, 1999. Resource/reserve inventories: what are the objectives? CIM Bulletin, 92(1031):151-155.

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2008 PROFESSIONAL DEVELOPMENT SEMINAR SERIES STRATEGICRISK QUANTIFICATION ANDMANAGEMENT FOR ORE RESERVES ANDMINE PLANNING

Modelling geological uncertainty with geostatistical simulations: Models, methods, applications and software

Workshop Notes February 21, 2008 University of Ottawa, Ontario

© ROUSSOS DIMITRAKOPOULOS COSMO – Stochastic Mine Planning Laboratory Department of Mining, Metals and Materials Engineering McGill University, Canada

Dimitrakopoulos (2008) Modelling Geological Uncertainty With Geostatistical Simulations

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