# Incorporation of Geometallurgical Attributes and Geological Uncertainty into Long-Term Open-Pit Mine Planning

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## Abstract

**:**

## 1. Introduction

#### 1.1. Mine Planning

#### 1.2. Geometallurgy

#### 1.2.1. Geometallurgical Variables

#### 1.2.2. Metallurgical Recovery

#### 1.2.3. Comminution Performance

#### 1.3. Modeling the Uncertainty

#### 1.4. Background on Direct Block Scheduling

#### 1.5. Contribution of the Research

## 2. Materials and Methods

#### 2.1. Notation

**%**), and (iv) comminution performance $\mathit{T}\mathit{P}{\mathit{H}}_{\mathit{b}\mathit{r}}$ from realization $r$ (in tonnes/hour). The copper-molybdenum concentrate tonnage is denoted as $\mathit{T}\mathit{O}{\mathit{N}}_{\mathit{b}}^{\mathit{c}\mathit{c}}$ (in tonnes). Also, a comminution performance of copper-molybdenum concentrate $\mathit{T}\mathit{P}{\mathit{H}}_{\mathit{b}}^{\mathit{c}\mathit{c}}$ (in tonnes/hour) is given.

Scheme 1 | ${v}_{br1}=\{\begin{array}{c}[\left({P}^{cu}-S{C}^{cu}\right)\cdot f\cdot \overline{{\mathit{G}}_{\mathit{b}}^{\mathit{c}\mathit{u}}}\cdot \overline{{\mathit{R}}_{\mathit{b}}^{\mathit{c}\mathit{u}}}+\\ \left({P}^{mo}-S{C}^{mo}\right)\cdot f\cdot \overline{{\mathit{G}}_{\mathit{b}}^{\mathit{m}\mathit{o}}}\cdot {R}^{mo}-\\ MC-PC]\cdot TO{N}_{b}-P{C}^{mo}\cdot TO{N}_{b}^{cc}\\ \\ -MC\cdot TO{N}_{b}\end{array}$ | $\begin{array}{}\\ \\ ,bisore\\ \\ ,biswaste\end{array}$ | (1) |

Scheme 2 | ${v}_{br2}=\{\begin{array}{c}[\left({P}^{cu}-S{C}^{cu}\right)\cdot f\cdot {\mathit{G}}_{\mathit{b}\mathit{r}}^{\mathit{c}\mathit{u}}\cdot \overline{{\mathit{R}}_{\mathit{b}}^{\mathit{c}\mathit{u}}}+\\ \left({P}^{mo}-S{C}^{mo}\right)\cdot f\cdot {\mathit{G}}_{\mathit{b}\mathit{r}}^{\mathit{m}\mathit{o}}\cdot {R}^{mo}-\\ MC-PC]\cdot TO{N}_{b}-P{C}^{mo}\cdot TO{N}_{b}^{cc}\\ \\ -MC\cdot TO{N}_{b}\end{array}$ | $\begin{array}{}\\ \\ ,bisore\\ \\ ,biswaste\end{array}$ | (2) |

Scheme 3 | ${v}_{br3}=\{\begin{array}{c}\left({P}^{cu}-S{C}^{cu}\right)\cdot f\cdot {\mathit{G}}_{\mathit{b}\mathit{r}}^{\mathit{c}\mathit{u}}\cdot {\mathit{R}}_{\mathit{b}\mathit{r}}^{\mathit{c}\mathit{u}}+\\ \left({P}^{mo}-S{C}^{mo}\right)\cdot f\cdot {\mathit{G}}_{\mathit{b}\mathit{r}}^{\mathit{m}\mathit{o}}\cdot {R}^{mo}-\\ (MC+PC)\cdot TO{N}_{b}-P{C}^{mo}\cdot TO{N}_{b}^{cc}\\ \\ -MC\cdot TO{N}_{b}\end{array}$ | $\begin{array}{}\\ \\ ,bisore\\ \\ ,biswaste\end{array}$ | (3) |

Scheme 4 | ${v}_{br5}=\{\begin{array}{c}\left({P}^{cu}-S{C}^{cu}\right)\cdot f\cdot {\mathit{G}}_{\mathit{b}\mathit{r}}^{\mathit{c}\mathit{u}}\cdot {\mathit{R}}_{\mathit{b}\mathit{r}}^{\mathit{c}\mathit{u}}\cdot \mathit{T}\mathit{P}{\mathit{H}}_{\mathit{b}\mathit{r}}\cdot H+\\ \left({P}^{mo}-S{C}^{mo}\right)\cdot f\cdot {\mathit{G}}_{\mathit{b}\mathit{r}}^{\mathit{m}\mathit{o}}\cdot {R}^{mo}\cdot \mathit{T}\mathit{P}{\mathit{H}}_{\mathit{b}\mathit{r}}\cdot H-\\ MC\cdot TO{N}_{b}-PC\cdot \mathit{T}\mathit{P}{\mathit{H}}_{\mathit{b}\mathit{r}}\cdot H-P{C}^{mo}\cdot TP{H}_{b}^{cc}\cdot H\\ \\ -MC\cdot TO{N}_{b}\end{array}$ | $\begin{array}{}\\ \\ ,bisore\\ \\ ,biswaste\end{array}$ | (4) |

#### 2.2. Ultimate Pit Limit

#### 2.3. LOM Production Scheduling

#### 2.3.1. Deterministic Direct Block Scheduling

**Scheduling Scheme 1:**the economic block model ${v}_{br1}$ is constructed according Equation (1), that is, just considering deterministic variables (E-Type models). The minimum and maximum processing capacities are assumed fixed.

Scheduling Scheme 1 | $\mathit{a}\mathit{t}\mathit{t}\mathit{r}\mathit{i}\mathit{b}\mathit{u}\mathit{t}{\mathit{e}}_{\mathit{b}}=OTO{N}_{b}=\{\begin{array}{c}TO{N}_{b},ifbisore\\ 0,ifbiswaste\end{array}$ | $\forall b\in B$ | (14) |

#### 2.3.2. Stochastic Direct Block Scheduling

**Scheduling Scheme 2**: the economic block model ${v}_{br2}$ is constructed according (2), therefore two simulated variables (Cu grade and Mo grade) are considered. Minimum/maximum processing capacities are fixed.**Scheduling Scheme 3**: the economic block model ${v}_{br3}$ is constructed according (3), therefore three simulated variables (Cu grade, Mo grade and Cu recovery) are considered. As before, minimum/maximum processing capacities are fixed.**Scheduling Scheme 4**: the economic block model ${v}_{br4}$ is constructed according (4), but processing capacity constraint per period changes. While in the previous cases the maximum processing capacities per period are considered in terms of tonnages, in this case the total available times at the milling plant is considered at a given period as constraint. For this purpose, the milling hours for each block are calculated based on TPH model. Therefore, in this case, the four simulated variables are considered (Cu grade, Mo grade, Cu recovery and TPH).

Scheduling Scheme 2,3 | $\mathit{a}\mathit{t}\mathit{t}\mathit{r}\mathit{i}\mathit{b}\mathit{u}\mathit{t}{\mathit{e}}_{\mathit{b}\mathit{r}}=OTO{N}_{br}=\{\begin{array}{c}TO{N}_{b},ifbisoreinscenarior\\ 0,ifbiswasteinscenarior\end{array}$ | $\begin{array}{c}\forall b\in B,\\ r\in R\end{array}$ | (23) |

Scheduling Scheme 4 | $\mathit{a}\mathit{t}\mathit{t}\mathit{r}\mathit{i}\mathit{b}\mathit{u}\mathit{t}{\mathit{e}}_{\mathit{b}\mathit{r}}=\frac{OTO{N}_{br}}{TP{H}_{br}}$ | $\forall b\in B,r\in R$ | (24) |

## 3. Case Study and Results

#### 3.1. Case Study

#### 3.1.1. Block Model

#### 3.1.2. Simulated Geometallurgical Variables

- Grades: Cu (%) and Mo (gr/ton or ppm),
- Cu metallurgical recovery,
- Throughput rate (TPH): this variable predicts the tons per hour that can be processed in the milling circuit and it based on grindability test data (see Section 1.2.3) and the current operational configuration.

#### 3.1.3. Economic Parameters

#### 3.1.4. Technical Parameters

#### 3.2. Results

#### 3.2.1. Ultimate Pit Limit: Key Indicators Results and Risk Analysis

#### Probability Model Results

#### 3.2.2. LOM Production Scheduling

## 4. Discussion

## 5. Conclusions

- Geological and geometallurgical scenarios are considered in one-run as input to the optimization process,
- the extraction period for each mining block is determined,
- the solution achieves the
**maximum discounted economic benefit**of the mining business, - the solution achieves the
**minimum risk of losses**due to potential deviations from the production plan, - the solution satisfies operational constraints, such as slope angles in pit walls and mining capacities.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Traditional methodology for long-term open-pit mine planning (adapted with permission from [5]).

**Figure 2.**Traditional and risk-based approaches applied in mine planning (adapted with permission from [5]).

**Figure 3.**Deterministic Direct Block Scheduling approach, where production scheduling comes out right from solving an optimization model.

**Figure 4.**Two approximations of the extraction in an open-pit mine. (

**a**) In order to extract block 6 first must be extracted blocks 1 to 5, therefore there are five precedence arcs from block 6. (

**b**) In order to extract block 10 first must be extracted block 1 to 9: there are nine precedence arcs from block 10 [43].

**Figure 7.**Plan view of geometallurgical models (top) with the associated histogram (bottom) for realization #1. Variables: (

**a**) Cu grade (%) and (

**b**) Mo grade (gr/ton).

**Figure 8.**Plan view of geometallurgical models (top) with the associated histogram (bottom) for realization 1. Variables: (

**a**) Cu metallurgical recovery (%) and (

**b**) throughput rate (ton/h).

**Figure 9.**Grade-tonnage curves, with cutoff grade (% Cu). (

**a**) Average grade above cutoff (% Cu), and (

**b**) Average grade above cutoff (ppm Mo). Results from simulations and E-Type models.

**Figure 10.**Grade-tonnage curve for estimated (E-Type) and simulated models, when considering (

**a**) total resources, and (

**b**) economic reserves within optimal ultimate pit limits defined for Scheme 2.

**Figure 11.**S-N sections, E views for probability models (FP prob) and a specific probabilistic ultimate pit limit for 90% reliability (FP 90%). (

**a**) Scheme 2—probability final pit; (

**b**) Scheme 2—90% probability final pit; (

**c**) Scheme 3—probability final pit; (

**d**) Scheme 3—90% probability final pit; (

**e**) Scheme 4—probability final pit; (

**f**) Scheme 4—90% probability final pit.

**Figure 12.**Ore and waste tonnages and net present value from production schedule obtained according (SCHED) by using scheduling Scheme 1. Results are assessed along all realizations into (STO) by using scheduling Scheme 3 for fair comparison.

**Figure 13.**Ore and waste tonnages and net present value from production schedules obtained according (STO) by using scheduling Scheme 2. Results are re-evaluated by using scheduling Scheme 3 for fair comparison.

**Figure 14.**Ore and waste tonnages and net present value from production schedules obtained according (STO) by using scheduling Scheme 3.

**Figure 15.**Ore and waste tonnages and net present value from production schedules obtained according (STO) by using scheduling Scheme 4. Results are re-evaluated by using Scheme 3 for fair comparison and a tonnage per block as TPH times H.

**Figure 16.**S-N sections, E views of the production schedules obtained by each scheduling scheme on 90% probability ultimate pit obtained by using Scheme 3. (

**a**) Scheduling Scheme 1; (

**b**) Scheduling Scheme 2; (

**c**) Scheduling Scheme 3; (

**d**) Scheduling Scheme 4.

Symbol | Unit | Parameter |
---|---|---|

${P}^{cu}$ | USD/lb | Cu price |

${P}^{mo}$ | USD/lb | Mo price |

$S{C}^{cu}$ | USD/lb | Cu selling cost |

$S{C}^{mo}$ | USD/lb | Mo selling cost |

$MC$ | USD/ton | Mining cost |

$PC$ | USD/ton | Processing cost (main) |

$P{C}^{mo}$ | USD/ton | Mo processing cost |

Scheme | Cu Grade | Mo Grade | Cu Recovery | TPH |
---|---|---|---|---|

1 | ✕ | ✕ | ✕ | ✕ |

2 | ✓ | ✓ | ✕ | ✕ |

3 | ✓ | ✓ | ✓ | ✕ |

4 | ✓ | ✓ | ✓ | ✓ |

**Table 3.**Summary of economic parameters for block valuation applied in case study. Unitary costs to be considered: (*) scheduling Schemes 2 and 3; and (**) scheduling Scheme 4.

Symbol | Unit | Parameter | Value |
---|---|---|---|

${P}^{cu}$ | USD/lb | Cu price | 1.80 |

${P}^{mo}$ | USD/lb | Mo price | 6.00 |

$S{C}^{cu}$ | USD/lb | Cu selling cost | 0.40 |

$S{C}^{mo}$ | USD/lb | Mo selling cost | 1.72 |

$MC$ | USD/ton | Mining cost | 3.79 |

$PC$ | USD/ton | Processing cost (main) | 11.35 |

$P{C}^{mo}$ | USD/ton | Mo processing cost | 15.58 |

$c{p}^{-}$ (*) | USD/ton | Unitary shortage cost | 30.00 |

$c{p}^{+}$ (*) | USD/ton | Unitary surplus cost | 30.00 |

$c{p}^{-}$ (**) | USD/h | Unitary shortage cost | 70,000.00 |

$c{p}^{+}$ (**) | USD/h | Unitary surplus cost | 70,000.00 |

**Table 4.**Summary of technical parameters for case study. Bounds on processing capacity to be considered: (*) scheduling Schemes 2 and 3; and (**) scheduling Scheme 4.

Symbol | Unit | Parameter | Value |
---|---|---|---|

${R}^{mo}$ | - | Mo metallurgical recovery | 0.55 |

$H$ | hr/block | Average processing time | 3.18 |

$\alpha $ | degree | Overall slope angle | 42.00 |

$h$ | - | Height (number of upper benches) | 3.00 |

${M}_{t}^{-}$ | Mton | Lower bound mining cap. | 0.00 |

${M}_{t}^{+}$ | Mton | Upper bound mining cap. | 80.00 |

${P}_{t}^{-}$ (*) | Mton | Lower bound process. cap. | 25.00 |

${P}_{t}^{+}$ (*) | Mton | Upper bound process. cap. | 40.00 |

${P}_{t}^{-}$ (**) | hour | Lower bound process. cap. | 10,000.00 |

${P}_{t}^{+}$ (**) | hour | Upper bound process. cap. | 15,710.00 |

|T| | year | Planning horizon | 30.00 |

$d$ | - | Discount rate | 0.10 |

**Table 5.**Key indicators such as value, tonnages of rock, ore, Cu and Mo metal contents (average and coefficient of variation) from ultimate pit limits obtained by different schemes. The economic values for Schemes 1,2 and 4 were re-evaluated by using the economic model of Scheme 3 for comparison.

Scheme | Undiscounted Value (MUSD) | Rock (Mton) | Ore (Mton) | Cu Metal (Mton) | Mo Metal (Kton) | |||||
---|---|---|---|---|---|---|---|---|---|---|

Avg | CV% | Avg | CV% | Avg | CV% | Avg | CV% | Avg | CV% | |

1 | 9453 | - | 3071 | - | 1433 | - | 17.5 | - | 388.1 | - |

2 | 10,528 | 2.5 | 3372 | 4.1 | 1602 | 2.9 | 18.5 | 1.9 | 471.6 | 3.2 |

3 | 10,725 | 3.1 | 3558 | 3.7 | 1682 | 2.4 | 18.9 | 2.5 | 467.2 | 3.8 |

4 | 10,472 | 2.1 | 2972 | 3.2 | 1572 | 3.1 | 17.9 | 2.2 | 417.6 | 3.5 |

**Table 6.**Summary of ultimate pit limits by using hybrid pits methodology [50]. Values were re-evaluated by using the economic model of Scheme 3 for comparison.

Minimum Probability | Scheme 2 | Scheme 3 | Scheme 4 | ||||||
---|---|---|---|---|---|---|---|---|---|

Value | Ore | Rock | Value | Ore | Rock | Value | Ore | Rock | |

(MUSD) | (Mton) | (Mton) | (MUSD) | (Mton) | (Mton) | (MUSD) | (Mton) | (Mton) | |

1.0 | 9062 | 1120 | 2320 | 9310 | 1184 | 2492 | 8633 | 988 | 2125 |

0.9 | 9134 | 1199 | 2417 | 9422 | 1233 | 2509 | 8708 | 1005 | 2298 |

0.8 | 9161 | 1274 | 2721 | 9501 | 1311 | 2718 | 8638 | 1072 | 2407 |

0.7 | 9241 | 1295 | 2882 | 9638 | 1375 | 2925 | 8715 | 1163 | 2539 |

0.6 | 9395 | 1342 | 2955 | 9793 | 1432 | 3046 | 8865 | 1203 | 2626 |

0.5 | 9462 | 1473 | 3086 | 9941 | 1555 | 3173 | 8972 | 1298 | 2710 |

Scheduling Scheme | Expected NPV | Relative Variation | Expected CDDC | Relative Variation |
---|---|---|---|---|

(MUSD) | CDDC % | (MUSD) | CDDC % | |

1 | 2949.3 | - | 891.7 | - |

2 | 3024.4 | +2.5 | 396.5 | −55.5 |

3 | 3162.2 | +7.2 | 389.8 | −56.3 |

4 | 3227.3 | +9.4 | 280.4 | −68.6 |

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**MDPI and ACS Style**

Morales, N.; Seguel, S.; Cáceres, A.; Jélvez, E.; Alarcón, M.
Incorporation of Geometallurgical Attributes and Geological Uncertainty into Long-Term Open-Pit Mine Planning. *Minerals* **2019**, *9*, 108.
https://doi.org/10.3390/min9020108

**AMA Style**

Morales N, Seguel S, Cáceres A, Jélvez E, Alarcón M.
Incorporation of Geometallurgical Attributes and Geological Uncertainty into Long-Term Open-Pit Mine Planning. *Minerals*. 2019; 9(2):108.
https://doi.org/10.3390/min9020108

**Chicago/Turabian Style**

Morales, Nelson, Sebastián Seguel, Alejandro Cáceres, Enrique Jélvez, and Maximiliano Alarcón.
2019. "Incorporation of Geometallurgical Attributes and Geological Uncertainty into Long-Term Open-Pit Mine Planning" *Minerals* 9, no. 2: 108.
https://doi.org/10.3390/min9020108