Mini-Max Joint Simulation

Question 1

Name the (3) impacts of having outliers.

Answer 1

• Increase variance
• Increase mean
• Highly right-skewed grade distribution (high grades with low frequencies)

Question 2

Name the impacts of removing outliers from grade distribution.

Answer 2

We observe a decrease of the mean and a decrease of the variance. This results in the underestimation of the deposit and we have smoothing.

Question 3

Why was MAF method developed.

Answer 3

• Common joint simulation methods are too computationally intensive.
• This is a source of complexity = inference of modelling of cross-variograms, and computational inefficiency due to increasing # of variables being co-simulated.

-A potential solution is the decorrelation of variables using PCA.

-Hoever, PCA ignores cross-correlations at distances other than 0 (a limitation).

Therefore need Minimum/Maximum Autocorrelation Factors (MAF).

Question 4

What does the PCA type factorization method allow?

Answer 4

The preservation of the correlation between elements

Question 5

What does PCA stand for?

Answer 5

Principal Component Analysis

Question 6

What is a limitation of PCA and its solution?

Answer 6

PCA ignores cross-correlations at distances other than 0 (a limitation) therefore it’s a limitation in the presence of spatial cross-correlations. Solution: Min/Max Autocorrelation Factors (MAF) in the context of spatial simulation

Question 7

Describe the MAF method in 3 steps.

Answer 7

It is an approach based of PCA,

(1) spatially decorrelates the variables involved to non-correlated factors

(2) independent factors are then individually simulated

(3) back-transformed to the conditional simulations of the correlated deposit attributes

Question 8

What does MAF stand for.

Answer 8

Min/Max Autocorrelation Factors

Question 9

Describe the MAF algorithmin 8 steps

Answer 9

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

Question 10

What is the following case study about: Assessing risk in Grade-tonnage curves in a complex copper deposit, northern Brazil.

Answer 10

Its a mine with a multi-mineral deposit which includes Cu, Fe, and K. They decided to do joint simulation of Cu, Fe and K to:
1. Determine recoverable Cu: copper solubility is controlled by K and Fe content, in addition to Cu grade (aka you can’t feasibly process ore with too much K and Fe)
2. Quantify Risk: generate grade-tonnage curves based on a range of Cu, Fe, and K grades

Question 11

Why did the Brazil mine decide to complete a Joint Simulation with MAF? State all the reasons

Answer 11

• It was for the pre-feasibility study
  The joint simulation with MAF was used because:
• Allows for Multiple elements (3) within the deposit
• Traditional methods for jointly simulating correlated variables are impractical and heavy (cross-variance, cross-variograms, cross-correlation à gigantic matrices)
• MAF allows decorrelation of elements
• MAF can be simulated independently, and the cross-correlations re-appear when MAFs are back-transformed to elements

Question 12

How many and which units were studied in the Brazil Case study?

Answer 12

2 units: sector 11 and 12

Question 13

Describe the simulation process step-by-step for the Brazil case study.

Answer 13

1. Normal-score transformation: normal score transformation performed on the Cu, Fe, and K composites of sectors 11 and 12
2. MAF transformation: generate the 3 min/max autocorrelation factors (As many MAF as there are elements looked at)
3. Variography of MAF: almost no correlation (line around 0) means that the data is not correlated anymore with the factors (variance at approximately)
4. Conditional simulation of MAF: generalized gaussian simulation method, 20 simulations
5. Back-transformations (rotations) of MAFs
6. Validation of the joint Cu-Fe-K simulation results: histograms, experimental variograms and cross-variograms to ensure reproduction of original data characteristics (reproduction is excellent)
   a. Reproduction of data statistics through histograms
   b. Reproduction of spatial correlation through varigorams
   c. Reproduction of correlation coefficients through scatter plots

Question 14

What are the results from the Brazil case study? More precisely about the grade tonnage curves.

Answer 14

Recoverable copper results and risk quantification were derived from metallurgical tests results and from the simulations. From the grade tonnage curves we have a very high change in tonnages as the Cu content increases, a very low change in tonnages as the K content increases, and a very high change in tonnages as the Fe content increases. There is also low variability lines for Fe graph, the lines are closer together than the lines in the Cu graph for a higher grade content.

Question 15

What are the main takeaways from the Brazil case study?

Answer 15

-Recoverable copper tonnage variability is more clearly related to the in-situ copper and potassium content than to iron content. That means for Fe rich ores, no wide variability in copper output is apparent in the grade-tonnage curves.
-Variability in recoverable copper grade is, however, mainly dependent on the in-situ copper content.

Question 16

What is a drilling program?

Answer 16

infill drilling + step-out drilling

Question 17

What is the goal of step out drilling

Answer 17

aim to expand the mineralization zone

Question 18

What is the goal of infill drilling

Answer 18

confirm the presence of mineralization between step-out drill holes

Question 19

What is the concept of Infill drilling

Answer 19

Critical information collection process: ability to assess the performance of potential drilling schemes, prior to drilling is important

Question 20

What are the realted: ‘‘enjeux’’ xd

Answer 20

reducing drilling can enhance the profitability of an operation if misclassification cost does not exceed the saving in drilling.

Question 21

How can we increase the profitability of drilling

Answer 21

Simulations using MAF

Question 22

Describe the whole method to increase profitability of drilling with MAF in 6 points

Answer 22

1. From the exploration drilling data available within the pit, jointly simulate a representation of the deposit using MAFs for the attributes under study —> Generate the “actual” deposit.
2. Virtually sample the above “actual” deposit with the different infill drilling schemes of interest.
3. For each drilling scheme, jointly simulate using MAF the elements of interest conditional to the data from the “actual” deposit in step 2, to obtain several joint realizations.
   a. Re-block the realization for the attributes simulated to produce models of mining blocks to be assessed.
4. Do grade control and classify the blocks (e.g. from their average grades) for each sampling scheme (e.g. milling ore, stockpile, waste). Compare the classification to the “actual” classification using, as economic indicators, the profit per tonne mined and profit per tonne milled.
5. Graph and assess results with respect to the point of diminishing returns.
6. Repeat to assess sensitivity of the results.

Question 23

How can the value of a block in the stockpile be assessed?

Answer 23

Using a discount rate

Question 24

The cost of misclassification of an ore block in the stockpile is a function of what?

Answer 24

a function of the future use of the stockpile.

Question 25

What is the stockpile value?

Answer 25

Its neither ore nor waste. It can be seen as money in the bank without interest = opportunity cost

Question 26

When low-grade ore blocks are stockpiles, the performance of an in-fill drilling scheme is ….

Answer 26

a function of how, when, and if the stockpile would be processed in the future. There is therefore uncertainty linked to the stockpile strategy.

Question 27

How can the uncertainty linked to the stockpile be included into the selection of a drilling scheme?

Answer 27

By depreciating the stockpile value with a discount rate. The higher the rate, the higher the uncertainty.

Question 28

Describe/introduce the Murrin-Murrin deposit case study

Answer 28

• Its a Nickel-cobalt deposit
• There’s 3 attributes of interest: Ni, Co, Mg (magnesium)
• There are milling concerns: given the Mg content in the ore, response of the mill feed to pressure-acid leaching and the cost of acid consumption is a metallurgical issue
  -Exploration drillholes gridded 50x50 m are available: 263 1-m composites.
  -Grade control in-fill drilling typically gridded 12.5x12.5 m with block sizes 15x15 m

Question 29

What is being assessed with the Murrin-Murrin deposit case study

Answer 29

A reduction in drilling would lead to savings in pre-mining costs VS. additional information could improve the quality of the mill feed thus reducing contaminant penalties and improving ore selection in addition to improving short term scheduling performance of the mine. There is also the choice of bench heights looking at 2 m vs. 3 m bench height.
- Can we reduce the drilling budget without endangering the profitability of the operation?
- Can the profitability of be enhanced by increasing the bench height from 2m to 3m?

Question 30

How did the Murrin-Murrin case study simulate Ni, Co, Mg? Also describe the process

Answer 30

Simulated Ni, Co, and Mg with MAF from the exploration drillholes, 4 actual deposits are simulated on point support: 2 with 2m bench height and 2 with 3m bench height
Process:
1. Normal score-transform
2. Ni, Co, and Mg attributes rotated into MAFs.
3. Each factor is then simulated independently one from another.
4. Back-rotation to the Gaussian space.
5. Back-transformation to the original space.
6. Data validation:
a. Cross-variograms: well-reproduced, thus preserving the important spatial relationships between the attributes

Question 31

How did they proceed for the Murrin Murrin case study? Name the 3 main lines

Answer 31

-Simulating Ni, Co, and Mg with MAF
-Simulate the drilling and classification process
-Discounting the stockpile and effects of drilling

Question 32

How did the Murrin Murrin study simulate the drilling and classification process (5)? What are the takeaways?

Answer 32

1. Virtually sample the actual deposit with specific drilling schemes
   a.12x12 m
   b.18x12 m
   c.18x18 m
   d.25x25 m
2. For each drilling drilling scheme, 30 simulations are performed conditional to the prior exploration holes and posterior virtual infill-sampling. Overall, 480 simulations.
3. 480-point simulations transformed into blocks 15x15 m.
4. Block selection for each sampling scheme based on the E-type mimicking the actual selection process for a mine operation.
   a. Ni < 0.8 = waste
   b. 0.8 < Ni < 1.2 = stockpile
   c. Ni > 1.2 = ore (ROM)
5. Calculate economic indicators aka the profit per tonne mined and profit per tonne milled.

Takeaways:
i. Mean decreases with a sparser drilling pattern.
ii. The variation of outcomes (risk) is significantly reduced with denser drilling.
iii. The 12x12 scheme is the most advantageous when considering both the efficiency and the uncertainty.

Question 33

How did the Murrin Murrin study go on about discounting the stockpile and effects of drilling? What were the results

Answer 33

6 discount rates on a period of 10 years assessed based on the 12x12 m grid:
-Discount rate = 0 –> no cost of opportunity –> the stockpile is considered as ore
-Discount rate = 1 –> 100 % opportunity cost –> the stockpile is considered as waste
- 0 < discount rate < 1 –> intermediate scenarios

Results:
a. The decrease of profit per tonne is due to the increased discount rate.
b. The importance of the drilling scheme depends on what is intended to be done with the stockpile.

Question 34

What are all the results from the Murrin Murrin case?

Answer 34

• Mean decreases with a sparser drilling pattern.
• The variation of outcomes (risk) is significantly reduced with denser drilling.
• The 12x12 scheme is the most advantageous when considering both the efficiency and the uncertainty, even at a low cut-off (stockpile as ore).
• Increase of 0.08$/tonne drilling cost between 12x12 m and 25x25 m.
• Profit/tonne mined improves up to 2$ when considering stockpile as waste (high cut-off)
• Profit/tonne mined improves up to 0.50$ when considering stockpile as pre (low cut-off)
• 12x18 m is also appropriate as it does not decrease the profit too much.
• Results insensitive to bench height.

Question 35

What is the conclusion from the Murrin Murrin case

Answer 35

• MAF based joint simulation is an efficient method to simulate correlated variables.
• 12x12 m bench height is optimal (with the 3-m bench height)
• The method developed provides the means to test and assess the effectiveness of drilling schemes.
• Mine planning (stockpiling) aspects are shown to be a key factor in deciding drilling densities.

Question 36

What does this variography of MAF show

Answer 36

almost no correlation (line around 0)–>data not correlated anymore with the factors

Question 37

Here is the Cu grade tonnage curve from the Brazil Case. Comment

Answer 37

Very high change in tonnages as the Cu content increases.

Question 38

Here is the Fe grade tonnage curve from the Brazil Case. Comment

Answer 38

• Very high change in tonnages as the Fe content increases.
• No wide variability lines for Fe graph are closer together than lines in Cu graph for higher grade content.

Question 39

What is the Mount Keith case about? What is the situation?

Answer 39

It’s a Ni and Co deposit with multiple lithologies. We want to assess uncertainty in mine planning and scheduling

Question 40

Describe the process taken at the Mount Keith case?

Answer 40

1. Determine domains according to lithologies
2. Create composites.
3. Transform attributes to normal scores.
4. De-correlate attributes using the MAF method.
5. MAF factors are uncorrelated (correlation coeff = 0)
6. Variography of factors.
7. Simulate: for each domain, visit wach location a grid and simulate using Gaussian simulation. Combine all domains for complete deposit realization.
8. Validate the conditional simulations. Compare each realization for each variable and domain, to the data:
   a. Histograms and quantile-quantile plots
   b. Directional variograms and variograms from composite data

Question 41

Present the results of Mount Keith case

Answer 41

• Simulations are validated.
• Simulated data for all domains are combined into a realization for the deposit.
• Each realization is presented for a cross section and long section through the middle of the deposit.
• Each realization also is reblocked to orebody block size, for use in assessing uncertainty in mine planning and scheduling.

Question 42

What is rebooking and why is it useful?

Answer 42

-Reblocking = data is averaged within the desired orebody block volume
-Why is it reblocked? –> large number of simulated values on a dense grid pose problems for mine planning.

Question 43

Introduce the Yandi iron case? What’s the goal?

Answer 43

Evaluate critical geochemical parameters that influence the physical and chemical properties of the product and the performance of the beneficiation process
- Iron content
- Silica content (SiO2)
- Alumina content (Al2O3)
- Phosphorus content
- Water and organic content measured as loss on ignition (LOI)

Question 44

1.Why do we want to study the Yandi Iron case? 2.What do we want to adapt?

Answer 44

1. In-situ variability and incomplete spatial distribution of the elements in the orebody are most critical for client. –> use of methods that incorporate in-situ variability and geological uncertainty. –> Joint direct block simulation approach
2. We want to adapt the generated schedule to actual life aka generate a practical mining schedule –> refine stochastics results using manual mine design and haul road construction –> some kind of smoothing

Question 45

What are the parameters for the Yandi Iron case? (# of simulations, Operational parameters, Economical and risk controlling parameters)

Answer 45

20 simulations of the main ore zone considering the 5 elements –> guarantee of the local reproduction of cross-correlation between elements
- Fe is strongly correlated with the elements SiO2 and Al2O3

Operational parameters:
- Maximum mining capacity: 20,000,000 t/period
- Slope angle is 45°
- Cut-off Fe grade of 0.56 %

Economical and risk-controlling parameters (risk of not meeting production targets of produced element-grades):
- Price
- Mining cost: blasting, extraction, transportation
- Processing cost: crushing, conveying, stockpiling
- Discount rates: economical discount rate (discount cash flows over periods), geological discount rate (controls the risk of producing grades that fall outside the limits over the period)
- Recovery is 100 %

Question 46

In the Yandi Iron case, the results are evaluated in terms of what?

Answer 46

• ore and waste tonnages
• risk profiles of produced grades per period: the spread of the different realizations provide an indication about uncertainty.

Question 47

What are the results based on the analysis of risk profiles for the Yandi case?

Answer 47

• Fe, P, and LOI results: no risk of deviating from production targets
• SiO2 and Al2O3: critical in meeting targets–> to mitigate risk additional schedules were generated applying penalties.
  o Low penalty: 1 $/unit deviation per tonne
  o Medium penalty: 10 $/unit deviation per tonne
  o High penalty: 100 $/unit deviation per tonne

Question 48

What are the results based on the schedules for the Yandi case?

Answer 48

Dispersion of schedules increases with the magnitude of the penalties.
o Low penalties = the extraction sequence is smooth.
o Medium penalties = extraction sequence is more dispersed.
o High schedule = very dispersed schedule = higher selectivity

Question 49

What are the results based on the analysis of risk profiles for SiO2 and Al2O3 for the Yandi case?

Answer 49

SiO2: the effect of increasing penalties becomes obvious in the case of medium penalties –> compared to low penalties, the fluctuation of grades between periods decreases significantly and there exists only a slight probability of deviating from targets in period 2, 3, and 4. Higher penalty only improves marginally the results.

Al2O3: decrease in probability of deviating from targets is recognizable with higher penalties but there is still a certain amount of risk. A potential solution to decrease the risk: blend the ore from different mines

Generally: less risk of deviation comes with a cost of higher selectivity which is caused by  minimizing risk of deviating from production targets and generating a smooth schedule.

Question 50

Comparison to traditional production scheduling approaches for the Yandi case.

Answer 50

• Both stochastic and deterministic schedules have a smooth sequence (Figure 7)
• Risk profiles for the critical elements Sio2 and Al2O3:
  o E-type is not able to account for geological uncertainty.
  o Mean values of the element grades produced fall within the production targets but deviations from upper and lower production limits for SiO2 and Al2O3 are visible. (Figure 8)
   Stochastic: 2 periods deviate with a probability of 5 % and 25 % each.
   Deterministic: all periods deviate from the target production with an average probability of 30 %.
   Therefore, deterministic has almost twice as high probabilities than stochastic to deviate from upper and lower limits.

Question 51

Conclusion from Yandi Case

Answer 51

• Efficient simulation with MAF and DBSIM facilitates mine production scheduling that manages risk in meeting production targets.
• Stochastic scheduling allows controlling geological multi-element uncertainty of generated extraction schedules.
• Stochastic scheduling is efficient, considering the size of the problem.
• Stochastic schedule shows a higher probability in meeting production targets, which decreases overall project risk and can increase project value.