Sensitivity of the Plan to Geological Uncertainty
We now address the questions that motivate this chapter: what impact can geological uncertainty have on the production schedule? Is it sure that the production promises and the NPV of the longterm plan will be fulfilled? Is this plan robust against uncertainty in the true grades contained in the deposit?
Table 4.4 Technical and economic parameters for the determination of the final pit of the mine
Parameter 
Value 
Unit 
Rock density 
2.6 
(ton/m^{3}) 
Maximum slope angle 
50.0 
(degrees) 
Copper price 
1.3 
(US$/lb) 
Silver price 
7.0 
(US$/oz) 
Molybdenum price 
15.0 
(US$/lb) 
Copper metallurgical recovery 
85.0 
(%) 
Silver metallurgical recovery 
70.0 
(%) 
Molybdenum metallurgical recovery 
50.0 
(%) 
Mining costs 
l.l 
(US$/ton) 
Processing costs 
6.0 
(US$/ton) 
Copper smelting and refinery cost 
0.4 
(US$/lb) 
Silver smelting and refinery cost 
0.4 
(US$/oz) 
Molybdenum smelting and refinery cost 
2.0 
(US$/lb) 
Mine investment 
1,500 
(US$/extracted tpd) 
Processing plant investment 
4,500 
(US$/ treated tpd) 
Discount rate 
10.0 
(%) 
The values of these parameters have been adjusted to preserve the confidentiality of the real project, without undermining its plausibility.
Figure 4.9 Production schedule defined from the reference block model. The gray bars indicate the tonnages of ore and waste extracted during the ten years of production, whereas the broken lines represent the average annual grades of copper (a) and other elements (b). (Credit: Carlos Montoya.)
Uncertainty in the Grades of the Production Schedule
A sensitivity analysis is carried out by applying the production schedule to each of the previously constructed grade simulations. In each case, the tonnages of ore and waste are the same, but the grade changes depending on the simulation under consideration. These changes are summarized in the following curves, obtained for 10 out of the 40 initially produced simulations (Figure 4.10). Although the reference block model on the basis of which the longterm plan is defined smoothes the grades, it is found that it accurately predicts (without bias) the grades that will be extracted. This property is explained by the absence of conditional bias of the reference block model (Montoya et al., 2012).
In addition, the simulations allow quantifying, for each annual period, the potential deviations between the grades of the ore to be extracted and the grades provided by the
Figure 4.10 Application of the production schedule to ten simulations interpreted as as many geological scenarios. The average grade of each simulation (black dotted curve, average of the ten continuous gray curves) coincides with the grade obtained from the reference block model (gray dotted curve). The plan constructed from this block model therefore provides unbiased predictions of the grades that will be effectively extracted.
production schedule, as well as estimating the probability that the actually extracted grades are greater or less than the planned grades in a certain period. For example, for the first period (year 1), three of the ten simulations exceed the expected value of the arsenic and antimony grades, indicating a probability of about 30% of not achieving the proposed objective for these two grades.
Too high arsenic or antimony grades have an impact on the quality of the concentrate obtained by flotation of the ore and in the smelter, where a fraction of the arsenic and antimony is emitted into the atmosphere. This is the reason why most copper smelters apply severe restrictions to the arsenic grades of the concentrates that they accept to process. The previous sensitivity analysis is then useful for planning engineers: it is not enough for them that the restrictions in the contaminant grades (As and Sb) are fulfilled by the reference block model; it is also necessary that they be fulfilled by most of the simulations in order to minimize risks.
Uncertainty in the Net Present Value of the Mining Plan
The simulations also allow determining the financial risk of the production schedule. The NPV calculated with ten simulations coincides, on average, with the value calculated using the reference block model, which is again explained by the absence of conditional bias of this model and by the fact that, here, the NPV is a piecewise linear function of the grades (Montoya et al., 2012). Variations, positive or negative, of the NPV obtained in the simulations can reach up to 20% of the value predicted by the reference block model (Table 4.5). This difference, 20%, corresponds to the order of magnitude of the expected errors made during the prefeasibility and feasibility studies, which are aimed at demonstrating the technical and economic feasibility of a mining project, while inaccuracies of the order of 30%50% may occur at the prior conceptual study stage that aims to provide a preliminary description of the project.
Table 4.5 Uncertainty in the net present value of the production schedule
Net present value (MUS$) 
Variation with respect to the reference block model (%) 

Reference block model 
1,204.66 
0.00 
Simulation n°l 
1.III.80 
8.35 
Simulation n°9 
1,173.17 
2.68 
Simulation n°IO 
1,247.11 
3.40 
Simulation n°l9 
1,186.56 
1.53 
Simulation n°27 
1,028.60 
17.12 
Simulation n°3l 
1,507.65 
20.10 
Simulation n°33 
1,326.84 
9.21 
Simulation n°35 
1,269.28 
5.09 
Simulation n°37 
1,340.93 
10.16 
Simulation n°39 
1,034.75 
16.42 
Average 
1,222.67 
1.47 
The value calculated in a simulation may differ from the value calculated in the reference block model by up to MUS$300, representing 20% of the reference value. However, on average over ten simulations, the calculated value is very close to the value of the reference model, which therefore delivers an unbiased prediction of the true value
Discussion
The previous results highlight the importance of the absence of conditional bias of the block model used in longterm planning, a property that can be used to predict without bias the grades to be extracted and the NPV of the mining project. In general, the absence of conditional bias is emphasized for shortterm planning, when deciding the destination of the extracted blocks, but it is rarely mentioned as a significant factor for longterm planning, which, however, is an important property. It should be remembered that, often, conditional bias is caused by a poor design of the moving neighborhood used in kriging or cokriging (see Section A.3 in Appendix).
Looking for the Optimal Final Contour
In the previous section, a mining plan was sensitized to different geological scenarios to quantify the changes in tonnages, average grades and NPVs that can occur between one scenario and another. But the longterm plan has been defined from a smoothed block model and has no reason to be the optimal plan for each scenario.
The search for an ‘optimal’ plan that incorporates geological uncertainty in open pit mining is not trivial, since the algorithms for determining the final pit (Eq. 4.6) and the successive intermediate pits are formulated for a unique block model containing the information of the block grades.
Of course, it is always possible to apply these algorithms to several geological scenarios obtained by conditional simulation. The final pits and production schedules are compared based on waste tonnages, ore tonnages, ore grades and NPVs. An example for the Ministro Hales deposit is shown in Table 4.6. It is seen that the ore grades in the final pit are higher and the ore tonnage is smaller when the final pit is optimized in each simulation instead of the reference block model that corresponds to the simulation average. These differences, which have a significant impact on the NPV of the mining project, are explained by the smoothing effect caused by averaging simulations: the occurrence of intermediate grades increases (high and low grades become scarcer), which causes an increase in the tonnage above the cutoff grade that defines the ore and a decrease in the average grade of this ore.
Hence, the final pit that the planning engineer usually calculates from a smoothed block model is far from being optimal. In the previous example, the N PV obtained with the reference block model (Million US$ 1,204.66) can overestimate or underestimate
Table 4.6 Results of the optimization of the final pit, for the reference block model (average of 40 simulations) and for two specific simulations
Reference block model 
Simulation n°3l 
Simulation n°39 

Total extracted tonnage (Mton) 
547.16 
562.82 
358.16 
Ore tonnage (Mton) 
181.68 
157.84 
121.30 
Copper grade (%) 
l.ll 
1.42 
1.25 
Silver grade (g/t) 
22.66 
30.47 
26.19 
Molybdenum grade (g/t) 
71.73 
74.40 
70.41 
Arsenic grade (g/t) 
973.37 
1,167.51 
1,191.14 
Antimony grade (g/t) 
83.62 
102.81 
100.87 
Net present value (MUS$) 
1,204.66 
1,507.65 
1,034.75 
by up to 20% the value of a production schedule developed on the basis of a single simulation. This percentage represents the order of magnitude of the financial uncertainty of the mining project due to uncertainty in the grades.
An adaptation of a wellknown algorithm to determine the final mining pit, the so called ‘floating cone’ method (Carlson et al., 1966), which allows the optimization of the pit over a set of geological scenarios, is presented in the following. This adaptation was originally proposed by Reyes et al. (2012) and Reyes (2017).
The floating cone is an iterative algorithm based on the geometric shape of the extraction cone of a block (Figure 4.7). At each iteration, a new cone is placed in a block chosen at random in the block model: if this cone improves the economic value of the pit, it is accepted and the pit is increased with this cone; in the opposite case, the cone is rejected and the pit is not modified. A known drawback of the algorithm is that it does not necessarily converge toward the optimal pit, a drawback that can be eliminated using the socalled ‘simulated annealing’ technique. Let us consider the following ingredients:
 • an initial state: a pit equal to the empty set;
 • a transition kernel, which allows proposing a new pit from the current pit, consisting in choosing at random between removing one of the cones of the pit or adding a new cone positioned randomly in the block model;
 • an objective function О that, for any possible pit, measures its ‘value’;
 • a positive scalar /(0) known as the initial ‘temperature’;
 • a ‘cooling schedule’, that is, a decreasing function that indicates, for iteration k, the temperature t(k) used for this iteration, for example, t(k) = t(0) e^{k} with e< 1;
 • a stopping criterion that can be, for instance, the maximum number of iterations.
At iteration k, the transition that improves the objective function is always accepted, as in the original floating cone algorithm. However, a transition that deteriorates the objective function is not always rejected: it is accepted with probability ехр(ДOlt(k)) where AO represents the deterioration of the objective function (loss of value of the pit).
This algorithm allows two other notable improvements for the definition of the pit. The first one is the incorporation of geological uncertainty through the definition of the objective function. In fact, this function allows not only the valuation of a pit for a single block model but also its valuation for a set of geological scenarios obtained by simulation. This could be, for example, the value of the pit obtained on average in all the scenarios, or the worst value if the planning engineer has a strong aversion to risk. In summary, the choice of the objective function can incorporate the planner’s preferences or the policy of the mining company.
The second improvement refers to the operationalization of the final pit. Defining a pit as the union of extraction cones as shown in Figure 4.7 is only an approximation to the problem of designing an openpit operation. Spiky cones provide a global geomechanical stability of the pit, but not its operational capacity: a minimum basic contour at the bottom of the mine should be considered to operate the excavators and the vehicles that transport the extracted material, as well as a minimum bench height and protection and containment berms to ensure slope stability. Instead of spiky cones, it is possible to apply the floating cone algorithm with more realistic objects (pseudocones) that include these geometric characteristics (Figure 4.11). Likewise, the application of mathematical
Figure 4.11 Pseudocone including a minimum base (pseudoellipse similar to a rectangle with rounded angles), a ramp and benches with walls of fixed height and with safety berms. (Credit: Manuel Reyes.)
morphology tools (dilation, erosion, opening and closing) can be useful to eliminate protuberances or cavities with acute angles and to smooth the contour of the pit obtained (Reyes, 2017; Bai et al., 2018), so that it is not only optimal but also operational.
The previous example is intended to provide an overview of the potential of geosta tistical tools, in particular, of conditional simulation, when combined with optimization techniques to define the final contour of the mine, the extraction sequence and the production schedule.
For more than one decade, an important research effort in ‘stochastic mine planning’ has been carried out, where the longterm plan is no longer defined on the basis of a single smooth block model, but on the basis of many geological scenarios that represent geological uncertainty (Dimitrakopoulos, 2007, 2011, 2018; Dimitrakopou los and Ramazan, 2008; Espinoza et ah, 2013a, b; Marcotte and Caron, 2013; Vargas et ah, 2014; Aguirre et ah, 2015; Moreno et ah, 2017; Maleki et ah, 2020).
Classification of Mineral Resources and Ore Reserves
Planning in the long term also means converting mineral resources into ore reserves. The technical and public reports on the prediction of resources and reserves should consider several categories according to the degree of confidence in the prediction. For the mineral resources, three categories are defined: ^{[1]}
The definition of the categories is a very subjective exercise, with greater reason because the international codes giving the classification guidelines do not provide an exact definition of the concepts of significant or reasonable confidence, leaving the definition in the hands of an expert called a ‘competent’ or a ‘qualified’ person, a statute that is acquired by affiliation with a professional association or the cooptation of a peer, as well as by the justification of an adequate university degree and a regular and consistent industrial mining practice.
It is worth noting that nothing in these codes indicates how to make the predictions, or sometimes in a very relaxed way, which gives a considerable amount of latitude to these calculations and places results obtained in very different ways at the same level. In the absence of being able to propose a universal classification criterion, we here limit ourselves to a few thoughts:
 • Geostatistics provides a set of tools and techniques to measure the uncertainty in the grade of each block of the deposit. Some practitioners use the variance of the kriging error, whereas others prefer the variance obtained on a set of conditional simulations, the conditional coefficient of variation (square root of the conditional variance divided by the average of the simulated grades), the width of a given probability interval or the width divided by the average value, to name a few examples.
 • The classification is sensitive to the chosen uncertainty measure and to the threshold values used to separate the categories (Emery et al., 2006). In particular, there is no onetoone relationship between the above measures.
 • The classification is also sensitive to the support of the mining block (David, 1988). The uncertainty on the grades can be reduced for a large block but becomes high when this block is subdivided into smaller ones, especially if the spatial grade correlation is low. In practice, the uncertainty measure is often established for a block support considered relevant for the technical and economic evaluation, for example, a quarterly, semiannual or annual volume of production.
 • The quality of the sampling, from the geological mapping of the drillhole cores to the chemical analysis of the grades, the degree of knowledge of the geology of the deposit and the quality of the interpreted geological model must be taken into account when classifying the mineral resources. In the case of the ore reserves, one should also consider the degree of knowledge of the modifying factors, including the economic factors and the factors related to the mining, mineral and metallurgical processes (Muller and de Nordenflycht, 2006).
Chapter S
 [1] ‘measured’ resources, predicted with a significant confidence; • ‘indicated’ resources, predicted with a reasonable confidence; • ‘inferred’ resources, predicted with a low confidence. The ore reserves are divided into two classes: • ‘proven’ reserves, originating from measured resources; • ‘probable’ reserves, originating from indicated resources, but sometimes alsofrom measured resources.