Ore-Waste Discrimination with Adaptive Sampling Strategy

Abstract

Grade control and short-term planning determine the performance of a mining project. Improving this decision, by collecting the most informative samples (data) may have significant financial impact on the project. In this paper, a method to select sampling locations is proposed in an advanced drilling grid for short-term planning and grade control in order to improve the correct assessment (ore-waste discrimination) of blocks. The sampling strategy is based on a regularized maximization of the conditional entropy of the field, functional that formally combines global characterization of the field with the principle of maximizing information extraction for ore-waste discrimination. This sampling strategy is applied to three real cases, where dense blast-hole data is available for validation from several benches. Remarkably, results show relevant and systematic improvement with respect to the standard regular grid strategy, where for deeper benches in the field the gains in ore-waste discrimination are more prominent.

Appendices

Appendix A: Pseudo-code

The implemented pseudo-code that summarizes the proposed framework is shown here.

Appendix B: Additional Experimental Results

Case study 1: Cut-off grade $1.241\%$

With Cut-off grade $1.241\%$, Figures 10, 11,  12, and 13 describe the achieved outcome. Figure 13 provides a summary of the confusion matrices for case study 1 considering a Cut-off grade of 1.241%.

Figure 10

Samples for Case Study 1. From left to right: Benches 2–6. Top: Samples from structured sampling. Down: Samples from adaptive sampling using Cut-off grade $1.241\%$

Figure 11

Estimated grade for Case Study 1. From left to right: Benches 2–6. Top: Kriging from structured sampling. Down: Kriging from adaptive sampling using Cut-off grade $1.241\%$

Figure 12

Estimated grade control for Case Study 1. From left to right: Benches 2–6. From top to bottom: Ground truth, structured sampling, adaptive sampling using Cut-off grade $1.241\%$

Figure 13

Confusion Matrix for Case Study 1. From left to right: Benches 2–6. Top: Structured sampling. Down: Adaptive sampling using Cut-off grade $1.241\%$

Case study 1: Cut-off grade $1.518\%$

Considering the Cut-off grade $1.518\%$, the outcome for the benches 2, 3, 4, 5 and 6 is described in Figures 14, 15, 16, and, 17. The Figure 17 provide the performance summary in terms of the confussion matrices for case study 1 and Cut-off grade 1.518%.

Figure 14

Samples for Case Study 1. From left to right: Benches 2–6. Top: Samples from structured sampling. Down: Samples from adaptive sampling using Cut-off grade $1.518\%$

Figure 15

Estimated grade for Case Study 1. From left to right: Benches 2–6. Top: Kriging from structured sampling. Down: Kriging from adaptive sampling using Cut-off grade $1.518\%$

Figure 16

Estimated grade control for Case Study 1. From left to right: Benches 2–6. From top to bottom: Ground truth, structured sampling, adaptive sampling using Cut-off grade $1.518\%$

Figure 17

Confusion Matrix for Case Study 1. From left to right: Benches 2–6. Top: Structured sampling. Down: Adaptive sampling using Cut-off grade $1.518\%$

Appendix C: Economic Evaluation

In order to perform a realistic evaluation of cost and profit, several considerations must be defined for the mining and waste processing. In particular, in this work the variables summarized in Table 10 have been considered, where a range of realistic values for these variables is proposed. This analysis takes into account the block size (fixed to 1000 $\text {m}^\text {3}$, the same block size used in the experimental section), the mining cost by mined ton, the metallurgical recovery, and the stripping ratio (the proportion of tons of expected waste material and ore material). For the purpose of the present analysis, the economic costs considered were the processing cost by processed ton, the price and selling cost per pound of copper.

Table 10 Variables considered in the profit analysis

In practice, the Cut-off grade, Cg, can be defined by,

$$\begin{aligned} Cg = \frac{10000 \cdot (\mathrm{Cos}t^m*(1+S_r) + \mathrm{Cost}^p )}{ (\mathrm{Price}_\mathrm{Cu} - \mathrm{Cost}^s) \cdot R_m \cdot 2204.6}, \end{aligned}$$

(11)

where the value 2204.6 corresponds to the conversion factor from pounds to tons.

Given the set of values for the considered parameters, it is possible to estimate the profit of a block. For a block with an estimated grade under the Cut-off grade value, for simplification it has been defined that the cost to process the waste block in the dump facilities is considered equal for each dumped block without taking into account its actual mineral grade or another variables. For this brief analysis, the revenue from dumping the block is zero (in practice if the block has a zero profit, since it has already been mined then it could be considered as stock pile). Thus:

$$\begin{aligned} \mathrm{Profit}_{d} = - \mathrm{Cost}^m \end{aligned}$$

(12)

In the case of a block estimated as ore (estimated block grade is above the Cut-off grade), it is processed by the mine and its benefit is calculated considering the content of metal in percentage of tons of copper as,

$$\begin{aligned} \mathrm{Cont}_{m} = \frac{\mathrm{Grade}_{b}}{100.0} \cdot size_b \cdot d_b \end{aligned}$$

(13)

The content of recovered metal in tons of copper is estimated as

$$\begin{aligned} \mathrm{Cont}_{m}^r = \frac{\mathrm{Cont}_{m} \cdot R_{m}}{100.0} \end{aligned}$$

(14)

Then, the revenue from mining the block is estimated by,

$$\begin{aligned} \mathrm{Rev}_{b}^{m} = \mathrm{Cont}_{m}^{r} \cdot 2204.6 \cdot (\mathrm{Price}_\mathrm{Cu}-\mathrm{Cost}^s), \end{aligned}$$

(15)

while the processing cost of the block is estimated as

$$\begin{aligned} \mathrm{Cost}_b^p = \mathrm{size}_b \cdot d_b \cdot ({\mathrm{Cost}}^m + {\mathrm{Cost}}^p). \end{aligned}$$

(16)

Finally, the profit of the processed block is defined by,

$$\begin{aligned} \mathrm{Profit}_{b}^p = \mathrm{Rev}_{b}^m - \mathrm{Cost}_{b}^p. \end{aligned}$$

(17)

Therefore, from Eqs. 12 and 17 and the cost of processing ore and waste blocks, it is possible to estimate the profit or loss of any block.

Considering the Cut-off grade of the experimental analysis and Eq. 11, the appropriate economic parameters have been estimated. The initial values shown in Table 10 are provided as an example to obtain the Cut-off grade as the first quartile in Case Study 1. For each case study and for every empirical Cut-off grade, the best set of parameters has been estimated in order to perform the economic analysis.

Then, from the experimental data the profit of every block has been evaluated in the proposed scenarios for the sampling strategies under analysis. Given an estimated bench block model from a specific sampling strategy, the economic profit can be calculated as the sum of the profit for each block conforming this bench.