# Resource and Reserve Calculation in Seam-Shaped Mineral Deposits; A New Approach: “The Pentahedral Method”

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## Abstract

**:**

## 1. Introduction

## 2. Research Objectives

- Three-dimensional (3D) modeling is mandatory;
- It must allow different interpolation algorithms, including geostatistics;
- The calculation detail has to be user-defined;
- Dilution (material below the cutoff) should be considered, if necessary;
- It must allow estimates with a categorization using distances, number of intercepts, or a combination of both parameters;
- It must allow interpolation, taking into account boreholes filtered by distance: neighborhood influence;
- The results will be stored in a Structured Query Language (SQL) database.

## 3. Geological Characteristics and Constraints

## 4. Methodology

- Determination of the seam localization from the geological information and survey data.
- Construction of a 3D-modeled surface at the center of the seam.
- Generation of an equally-spaced cloud of points in the seam centered surface.
- Calculation of the intercept values of the surveys crossing the seam.
- Execution of a geostatistical study of the intercept values.
- Interpolation of the grade and the thickness of each point of the cloud of points, using the intercept information.
- Definition of the estimation categories and generation of the calculation units.
- Database generation and representation.

#### 4.1. Determination of the Seam Localization from the Geological Information and Surveys

#### 4.2. Construction of a 3D Modeled Surface at the Center of the Seam

#### 4.3. Generation of an Equally Spaced Cloud of Points in the Seam Centered Surface

#### 4.4. Calculation of the Intercept Values of the Surveys Crossing the Seam

**Cutoff grade**. This is defined as the minimum grade that allows economical exploitation of the ore body, see [19]. In the example that is being used, the “Zona M” ore body of the Carlés deposit hosts three recoverable elements: Au, Cu, and Ag. A new element called “Au_e” element or “Au grade equivalent” can be defined, where an accumulated value of the three elements is stored according to a price-weighting index scheme. In order to simplify the calculation, an average metal recovery value was assumed, understood as the final percentage of each metal that gives a monetary income, deducting the concentration plant recovery, selling costs, concentration transport, penalties, besides other factors.

**Minimum thickness**. A minimum value of seam thickness must be considered in order to reflect the economic feasibility of the exploitation. It depends on the rock type, the seam dip and the mining method (room and pillar, sublevel stopping, cut and fill). A thickness of 2 m has been considered in this example, according to the mining machinery size that can be used in this deposit mining.

**Mining overbreak**. A lateral dilution has been defined due to unwanted results of blasting and hole deviations. From our experience with this deposit, 50 cm of overbreak (overexcavation, 25 cm each side) is defined, a value that can be user-selectable

**Maximum thickness of waste rock**. A usual situation that is found when the intercept of the borehole and the ore body is calculated is the appearance of inclusions, intervals of waste rock inside the ore body and, more commonly, mineralization below the cutoff grade value. The treatment for this type of situation is achieved by defining a maximum waste thickness value whereby, if the waste thickness value exceeds the maximum, it will not be mined (it will remain as a pillar). If the waste thickness value is calculated to be below the maximum and the average grade is higher than the cutoff grade, it will be considered as ore; however, if the average grade is below the cutoff grade, it will be considered as waste rock and will not be included in the intercept. This maximum waste rock thickness will be referred to as the real thickness of the seam, measured perpendicular to the seam.

**A: Geological Intercept**. This includes only those samples with grades higher than the cutoff grade. If there are no samples higher than that value, the highest-grade sample will be defined as the geological intercept. For instance, borehole S2 shows six samples 1 m in size, but only five will be used when considering a cutoff grade of 2 g Au/t, as can be seen in Figure 4 as per the coding colors.

**B: Minimum Thickness Intercept**. This will be defined as the geological intercept widened as necessary to complete a previously defined minimum real thickness. If the real thickness of the geological intercept is equal to or greater than the minimum thickness value, both intercepts will be the same.

**C: Mining Overbreak Intercept.**This will be defined as the minimum real thickness intercept widened at both sides by a distance defined as over excavation. In this example 50 cm will be used (25 cm on each side). This is an important parameter to consider, as it will simulate the mineral dilution that always occurs in common mining procedures that involve excavation.

**Interpolation parameters**. In this study the Inverse Distance Weighted (IDW) interpolation method will be used; other methods could have been considered: nearest neighbor or kriging, among others. In order to use IDW, a maximum influence distance must be used, calculated through a geostatistical study, independent of the interpolation method shown in this paper and detailed below, as well as a power value. In this case the distance is fixed at 50 m and the power is 3 (Figure 7 shows the input window for these parameters). These values can be different for each intercept, if necessary.

#### 4.5. Geostatistical Study of the Intercept Values

#### 4.6. Interpolation of the Grade and the Thickness at Each Point of the Cloud of Points (NPS)

**W**of the triangle

**a**will be:

_{i}**L**: Metal grade, in this case: Au_eq.**P**: Real thickness of the intercept.**d**: Distance between the NPS point to be interpolated and the center of the intercept.**x**: Inverse distance exponent (in the example, x = 3).**V**: Pentahedron volume.**n**: Borehole identification number.**a**: Seam triangle surface.

**NPS**: Grade multiplied by thickness and thickness for each type of intercept. In this case, the four borehole data (

**S**,

_{1}**S**,

_{2}**S**, and

_{3}**S**) will be used as they are considered to be inside its area of influence.

_{4}**NPS**point network over the seam. Then the three average grades of the

**a**triangle will be:

_{i}_{i}, y

_{i}, z

_{i}are the vertex coordinates:

#### 4.7. Interpolation of the Positions of the Seam Surface Center

#### 4.8. Definition of the Estimation Categories and Generation of the Calculation Units

- Category 1: calculation units with an intercept at less than 10 m
- Category 2: calculation units with two intercepts at less than 20 m
- Category 3: calculation units with two intercepts at less than 30 m
- Category 4: calculation units with one intercept at less than 40 m

#### 4.9. Calculation Units Generation and Categories Definition

## 5. Method Suitability

- Any change in the calculation parameters does not need the calculation units to be redefined.
- When defining the calculation geometry units, these can be separately prepared for subsequent exploitation and scheduling, even exporting them to other software.
- It provides a fairly accurate representation of the thickness of the seam-shaped structure even in the case of low thickness seams, which is extremely difficult to achieve with traditional block models (Figure 19).
- It interpolates intersections, not samples taken from boreholes.
- Geometrical results closely follow the deposit limits, as opposed to block models that create a stair-shaped profile (see Figure 19).
- It includes the possible internal dilution, as well as the lateral dilution due to mining overbreak.
- A minimum thickness can be defined according to mining technical conditions.

## 6. Discussion and Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**(

**A**) Geological map of the Carlés gold-bearing skarn (data provided by Rio Narcea Gold Mines); (

**B**) Section of the Carlés deposit (West Carlés); (

**C**) Section of the Carlés deposit (North Carlés).

**Figure 2.**(

**A**) View of the stratigraphic control of the skarn development at the Carlés skarn. The monzogranite intrusion is also controlled by the sedimentary layers; (

**B**) Close-up view of stratigraphically controlled Carlés mineralized banded skarn. Bands are dominated by pyroxene-magnetite alternate with garnet-amphibole-rich bands.

**Figure 4.**T1 surface generation. (

**A**) A cross section of boreholes is depicted in white and the intercepts with the body are depicted in red. The green line connects the centers of each red line section; (

**B**) Sections and their corresponding centerlines, which are triangulated to generate a surface. (

**C**) Calculated surface (“T1”).

**Figure 5.**Example of base surface T1 (left (

**A**)) converted to calculation Surface T2 (right (

**B**)). Detail of meshing points in sections and in crossing diagonals of any generated square (

**C**).

**Figure 6.**Intercept calculation of four boreholes (S1 to S4) through a seam-shaped mineral ore body. A (geology intercept), B (minimum thickness intercept), C (mining overbreak intercept); U, V, and W are points of the NPS; i1–i4 are the lithology intercept interval; d1–d4 distances between the intercepts center and W point.

**Figure 7.**RecMin Iinput window for intercept parameters. Holes selection (all, in this case), cutoff, minimum thickness, overbreak (one side), and maximum waste thickness.

**Figure 8.**RecMin Output window, intercept graph bars for C1163 borehole: red (geological intercept), yellow (minimum thickness intercept), and white (mining overbreak intercept).

**Figure 9.**Table of calculated intercepts calculated in RecMin. Results for several boreholes. First column is the name of the borehole (down hole). Each hole has three data rows showing the intercept data.

**Figure 10.**Definition of the guide line of the interpolation ellipse drawn over T2. A theoretical borehole crosses the seam and defines several possible categories of resources, according to the distance between the NPS points and the location of intersection center point.

**Figure 11.**Calculation unit as used in the pentahedral method (

**A**). Volume calculation (

**B**) and calculation graphic exploded (

**C**).

**Figure 12.**Mineral seam filled with calculation units, as automatically obtained by the software. Boreholes are colored by lithology. The legend refers to grade over seam (ppb).

**Figure 15.**Numerical results of the “Pentahedral Method,” classified by intercept type and category.

**Figure 16.**Assistant module used to open the pentahedral files and obtain a 3D visualization, as per SQL database conditions restrictions.

**Figure 17.**3D view, pentahedral filled seam representing orebody “Zona M” of the Carlés deposit (

**A**). Detail of the variable width of the calculated seam (

**B**). The legend shows the Au grade, expressed in ppb.

**Figure 19.**Graphic comparison between the block model (conventional method) and the “Pentahedral Method”. Left (

**A**): plane section of the blocks that define the seam colored according to grade and the same calculated seam; Right (

**B**): sections of the three different types of calculation done in the same seam.

**Figure 20.**3D model of the calculated seam colored by grade, including in the user selectable layers the topography and boreholes colored according to geology and sample grading.

Borehole S_{1} | Borehole S_{2} | Borehole S_{n} |
---|---|---|

A_{1}: Geological Intercept | A_{2}: Geological Intercept | A_{n}: Geological Intercept |

(L.P): Grade multiplied by thickness _{A1} | (L.P) Grade multiplied by thickness_{A2}: | (L.P) Grade multiplied by thickness_{An}: |

P thickness_{A1}: | P thickness_{A2}: | P thickness_{An}: |

B_{1}: Minimum Thickness Intercept | B_{2}: Minimum Thickness Intercept | B_{n}: Minimum Thickness Intercept |

(L.P) Grade multiplied by thickness_{B1}: | (L.P) Grade multiplied by thickness_{B2}: | (L.P) Grade multiplied by thickness_{Bn}: |

P thickness_{B1}: | P thickness_{B2}: | P thickness_{Bn}: |

C_{1}: Mining Intercept | C_{2}: Mining Intercept | C_{n}: Mining Intercept |

(L.P) Grade multiplied by thickness_{C1}: | (L.P) Grade multiplied by thickness_{C2}: | (L.P) Grade multiplied by thickness_{Cn}: |

P thickness_{C1}: | P thickness_{C2}: | P thickness_{Cn}: |

NPSID | Dist | % | D.Hole | P_R | <Au> | <Ag> | <Cu> | <As> | <Au_e> |
---|---|---|---|---|---|---|---|---|---|

30038000070 | 23.9 | 36.36% | C1043 | 1.06 | 1975 | 2.4 | 1200 | 253 | 1975.18 |

30038000070 | 26.9 | 25.50% | C1041 | 0.98 | 800 | 0.1 | 0 | 3900 | 800.00 |

30038000070 | 30.1 | 18.20% | C1042 | 3.03 | 9992 | 6.3 | 5433 | 15114 | 9992.76 |

30038000070 | 32.4 | 14.59% | C1012 | 0.59 | 2050 | 0.5 | 570 | 1387 | 2050.08 |

30038000070 | 45.3 | 5.34% | C1089 | 0.71 | 4850 | 8.4 | 4000 | 4400 | 4850.60 |

30038000070 | - | 100.00% | - | 1.31 | 3299.1 | 2.6 | 1722.1 | 4275.1 | 3299.36 |

30038000060 | 19.5 | 59.73% | C1043 | 1.06 | 1975 | 2.4 | 1200 | 253 | 1975.18 |

30038000060 | 30.5 | 15.61% | C1041 | 0.98 | 800 | 0.1 | 0 | 3900 | 800.00 |

30038000060 | 34.6 | 10.69% | C1042 | 3.03 | 9992 | 6.3 | 5433 | 15114 | 9992.76 |

30038000060 | 36.5 | 9.11% | C1012 | 0.59 | 2050 | 0.5 | 570 | 1387 | 2050.08 |

30038000060 | 45 | 4.86% | C1089 | 0.71 | 4850 | 8.4 | 4000 | 4400 | 4850.60 |

30038000060 | - | 100.00% | - | 1.2 | 2795.3 | 2.6 | 1544 | 2716.1 | 2795.57 |

30038000060 | 17.4 | 66.80% | C1043 | 1.06 | 1975 | 2.4 | 1200 | 253 | 1975.18 |

30038000060 | 29.5 | 13.71% | C1041 | 0.98 | 800 | 0.1 | 0 | 3900 | 800.00 |

30038000060 | 35.9 | 7.61% | C1042 | 0.59 | 2050 | 0.5 | 570 | 1387 | 2050.08 |

30038000060 | 36.5 | 7.24% | C1012 | 3.03 | 9992 | 6.3 | 2533 | 15114 | 9992.76 |

30038000060 | 42.3 | 4.65% | C1089 | 0.71 | 4850 | 8.4 | 4000 | 4400 | 4850.60 |

30038000060 | - | 100% | - | 1.1 | 2562.8 | 2.5 | 1442.1 | 2158.1 | 2563.00 |

TOTAL | - | - | - | 1.22 | 2885.75 | 2.56 | 1569.37 | 3049.37 | 2885.98 |

D.Hole | Tons | Au_e | Cu | Au | Ag |
---|---|---|---|---|---|

CN1014 | 1.5% | 1.90% | 0% | 2% | 1% |

CN1015 | 2.9% | 14.90% | 9.7% | 15.4% | 16.6% |

CN1016 | 1.9% | 2.00% | 1.2% | 2.1% | 1.5% |

CN1017 | 0.4% | 1.40% | 0.8% | 1.5% | 1.2% |

CN1018 | 0.2% | 1.30% | 0.9% | 1.3% | 0.8% |

CN1019 | 0.6% | 0.40% | 0.5% | 0.4% | 1.0% |

CN1020 | 1.1% | 0.60% | 0.7% | 0.6% | 0.9% |

CN1021 | 0.7% | 0.90% | 1.0% | 0.9% | 1.4% |

CN1022 | 0.6% | 0.10% | 0.0% | 0.1% | 0.2% |

CN1023 | 1.5% | 1.50% | 1.6% | 1.4% | 2.1% |

CN1024 | 1.2% | 0.70% | 0.0% | 0.8% | 0.0% |

CN1025 | 0.6% | 0.50% | 0.6% | 0.5% | 0.9% |

CN1026 | 0.6% | 0.50% | 0.7% | 0.4% | 0.9% |

CN1027 | 0.9% | 0.60% | 1.1% | 0.6% | 0.8% |

CN1028 | 0.9% | 0.50% | 0.6% | 0.5% | 0.4% |

CN1029 | 0.6% | 0.70% | 0.0% | 0.8% | 0.0% |

CN1030 | 0.7% | 0.10% | 0.0% | 0.1% | 0.1% |

CN1031 | 0.7% | 0.20% | 0.0% | 0.2% | 0.0% |

CN1032 | 0.7% | 0.60% | 0.8% | 0.6% | 0.5% |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Castañón, C.; Arias, D.; Diego, I.; Martin-Izard, A.; Ruiz, Y.
Resource and Reserve Calculation in Seam-Shaped Mineral Deposits; A New Approach: “The Pentahedral Method”. *Minerals* **2017**, *7*, 72.
https://doi.org/10.3390/min7050072

**AMA Style**

Castañón C, Arias D, Diego I, Martin-Izard A, Ruiz Y.
Resource and Reserve Calculation in Seam-Shaped Mineral Deposits; A New Approach: “The Pentahedral Method”. *Minerals*. 2017; 7(5):72.
https://doi.org/10.3390/min7050072

**Chicago/Turabian Style**

Castañón, César, Daniel Arias, Isidro Diego, Agustin Martin-Izard, and Yhonny Ruiz.
2017. "Resource and Reserve Calculation in Seam-Shaped Mineral Deposits; A New Approach: “The Pentahedral Method”" *Minerals* 7, no. 5: 72.
https://doi.org/10.3390/min7050072