What do you understand by 'tonnage factor'? Discuss the geometric and graphic methods used in reserve calculation.

What is 'Tonnage Factor'?

The 'tonnage factor' is a critical parameter in mineral reserve estimation, representing the relationship between the volume of an ore body and its mass. Essentially, it is the inverse of the bulk density of the ore material. It allows geologists and mining engineers to convert the calculated volume of a mineral deposit (e.g., in cubic meters or cubic feet) into an estimated tonnage (e.g., metric tons or short tons).

Mathematically, the tonnage factor is expressed as:

Tonnage Factor (metric) = Density (tonnes/cubic meter)

Tonnage Factor (English) = Volume (cubic feet) / Weight (short ton)

For example, if an ore has a specific gravity of 2.8, meaning it weighs 2.8 times more than an equal volume of water, its density would be 2.8 tonnes/cubic meter. In the English system, this might translate to approximately 11-15 cubic feet per ton, varying with the ore type and deposit characteristics. Accurate determination of the tonnage factor is vital, as errors here can significantly impact the overall reserve estimate, affecting financial viability and mine planning.

Geometric Methods in Reserve Calculation

Geometric methods are traditional techniques that rely on the geometric interpretation of drill hole data or mine workings to delineate areas of influence and calculate volumes. These methods assume a certain spatial continuity of mineralization.

1. Polygon Method (or Area of Influence Method)

The polygon method is a widely used geometric technique, particularly for tabular or broadly disseminated deposits, and is based on defining an area of influence around each drill hole.

• Principle: Each drill hole's assay values (grade and thickness) are assumed to be representative of a polygonal area around it.
• Construction:
  • Lines are drawn between adjacent drill holes.
  • Perpendicular bisectors are then constructed for each of these lines.
  • The intersection points of these bisectors define the vertices of polygons around each drill hole.
• Calculation:
  • The area of each polygon is measured (manually or digitally).
  • The volume of ore within each polygon is calculated by multiplying its area by the average thickness of the ore intersected in the central drill hole.
  • Tonnage is then derived by multiplying the volume by the tonnage factor (or specific gravity).
• Advantages: Relatively simple to apply, especially with computer-aided design (CAD) software.
• Limitations: It assumes the grade of the central drill hole is uniform throughout its polygon, which may not always be accurate, especially in heterogeneous deposits. Extreme high or low values in a single drill hole can disproportionately influence a large area.

2. Triangle Method

The triangle method addresses some limitations of the polygon method by averaging values over a smaller, more localized area.

• Principle: The area of influence is defined by triangles formed by connecting three adjacent drill holes.
• Construction:
  • Drill holes are connected to form a network of non-overlapping triangles across the ore body.
• Calculation:
  • The area of each triangle is calculated.
  • The volume of ore within each triangular prism is determined by multiplying the triangle's area by the average thickness of the ore encountered in the three corner drill holes.
  • The average grade for each triangle is typically a weighted average of the grades from the three corner drill holes, often weighted by their respective thicknesses.
  • Tonnage is computed by multiplying the volume by the tonnage factor.
• Advantages: Provides a smoother estimate of grade and thickness variations compared to the polygon method as it considers three data points.
• Limitations: Can be more computationally intensive than the polygon method for large datasets. The accuracy still depends on the assumption of linear grade variation within the triangle.

Graphic Methods in Reserve Calculation

Graphic methods typically involve creating maps or cross-sections to visually represent the ore body and then calculating volumes based on these representations.

1. Cross-Sectional Method (or Sectional Method)

This method is commonly used for steeply dipping, irregular, or lenticular ore bodies where geological continuity can be best visualized in cross-sections.

• Principle: The ore body is divided into a series of parallel cross-sections, usually perpendicular to the strike of the deposit.
• Construction:
  • Geological data (drill hole intercepts, underground mapping) are plotted on a series of vertical cross-sections.
  • The ore boundaries are interpreted and drawn on each section.
  • The area of ore on each section is then measured.
• Calculation:
  • The area of ore on each section is determined (e.g., using a planimeter or CAD software).
  • The volume of ore between two adjacent sections is calculated by averaging their areas and multiplying by the distance between them (prismoidal formula approximation for greater accuracy).
  • Total volume is the sum of volumes of all blocks.
  • Tonnage is obtained by multiplying the total volume by the tonnage factor.
• Advantages: Provides a good visual representation of the ore body's geometry and allows for detailed geological interpretation. Effective for complex structures.
• Limitations: The accuracy heavily depends on the spacing and orientation of the cross-sections. Closely spaced sections are time-consuming to prepare. Assumptions are made about the continuity of the ore body between sections.

2. Isochore/Isopach Method (or Contour Method)

This method is particularly suitable for stratiform or tabular deposits with varying thickness and/or grade, where contours can be effectively drawn.

• Principle: Lines of equal thickness (isopachs) or equal grade (isochores/isograde) are drawn on a plan map of the deposit.
• Construction:
  • Drill hole data (thickness, grade) are plotted on a plan map.
  • Contour lines are interpolated between data points, connecting points of equal thickness or grade.
  • These contours define areas with specific thickness or grade ranges.
• Calculation:
  • The area enclosed by each contour interval is measured.
  • The volume for each contoured layer is calculated by multiplying the average thickness of that layer by its area. Alternatively, for grade, the average grade within a contour interval is multiplied by the corresponding volume to get metal content.
  • Summation of these volumes or metal contents yields the total reserve.
• Advantages: Excellent for visualizing grade and thickness distribution and identifying trends. Can provide more refined estimates for irregularly shaped and varying thickness deposits.
• Limitations: Labor-intensive for manual contouring, especially for multi-element deposits. Requires sufficient and well-distributed data points for accurate contouring.

These methods, whether geometric or graphic, form the bedrock of conventional ore reserve estimation. While more advanced geostatistical methods like kriging offer probabilistic analyses and account for spatial variability, the fundamental understanding and application of geometric and graphic techniques remain essential for initial assessments and validating complex models. The choice of method largely depends on the ore body's geology, data density, stage of exploration, and required accuracy.