7.1 Ore Body Block Modeling & Spatial Variography

Geological block modeling and spatial variography represent the foundation of modern computer-based mineral resource estimation. A three-dimensional (3D) block model provides a digitized spatial framework of an ore body, partitioning the subsurface into discrete volume elements (blocks) to which geological attributes, bulk densities, structural domains, and estimated metal grades are assigned.

3D Geological Block Modeling Framework

The creation of a 3D geological block model begins with explicit or implicit wireframing (3D solid modeling) based on interpreted drillhole intercepts, geological mapping, and structural cross-sections. Wireframes delineate geological boundaries such as lithological units, alteration zones, mineralization envelopes, and structural faults.

Parent Block Selection vs. Sub-blocking

Choosing the optimal block dimensions is a critical decision in resource estimation.

• Parent Block Size: The primary grid resolution is dictated by the drillhole network density, geological continuity, and the Selective Mining Unit (SMU) size—the smallest volume of material that can be classified and mined as ore or waste by open-pit or underground loading equipment. Drill spacing can inform parent-block dimensions, but no fixed fraction applies along every axis. Choose block size from deposit geometry, data spacing and orientation, mining selectivity, compositing, estimation support, computational needs, and validation. A model might test 25 m by 25 m lateral blocks on a 50 m grid, but its vertical dimension and final resolution require separate justification.
• Sub-blocking (Sub-celling): To prevent volume errors where parent blocks intersect irregular wireframe boundaries, sub-blocking divides boundary parent blocks into smaller sub-cells (e.g., down to 1/8th or 1/16th of the parent block size). Sub-blocks improve geometric approximation along domain contacts, with residual volume error controlled and reported through validation. During grade estimation, grades are typically estimated into the parent block and assigned uniformly to child sub-blocks, or sub-blocks are re-aggregated (regularized) back to parent blocks to preserve computational efficiency and avoid over-smoothing.

Spatial Continuity & Experimental Variography

Variography quantifies spatial correlation—the principle that two sample points closer together in space are more likely to have similar mineral grades than points farther apart. Spatial variability is evaluated using the semi-variogram (commonly referred to simply as the variogram).

Experimental Variogram Formula

The experimental semi-variogram $\gamma(h)$ for a distance lag vector $h$ is defined as one-half the average squared difference between sample pairs separated by distance $h$:

$\gamma(h) = \frac{1}{2 N(h)} \sum_{i=1}^{N(h)} [Z(x_i) - Z(x_i + h)]^2$

Where:

• $\gamma(h)$ = Semi-variogram value at lag distance $h$.
• $N(h)$ = Total number of sample pairs separated by lag vector $h$.
• $Z(x_i)$ = Assay grade value at spatial location $x_i$.
• $Z(x_i + h)$ = Assay grade value at spatial location $x_i + h$.

To calculate the experimental variogram from irregularly spaced drillhole assays, directional parameters are applied:

• Lag Distance ($h$) and Lag Tolerance ($\Delta h$): The step interval and allowable distance margin for pairing samples.
• Azimuth / Dip Angle and Angular Tolerance: The directional vector cone specifying the search direction (e.g., along strike, down dip, or cross dip).

Variogram Parameters & Model Fitting

An experimental variogram displays discrete points plotted on a graph of lag distance ($h$) versus semi-variance ($\gamma(h)$). A theoretical mathematical model must be fitted to these experimental points for use in kriging algorithms. Key variogram parameters include:

1. Nugget Effect ($C_0$): The discontinuity between $\gamma(0)=0$ and the modeled limit as $h \to 0^+$. It can reflect micro-scale variability occurring at distances smaller than the sample spacing, as well as sampling, sub-sampling, and assaying errors.
2. Sill / Total Variance ($C_0 + C$): The plateau where the variogram flattens out, representing the overall sample population variance. $C$ represents the structural variance component.
3. Range ($a$): The spatial correlation limit. For a bounded model at distances beyond its modeled range, covariance is effectively zero and semi-variance reaches the total sill; this is modeled lack of correlation, not proof of statistical independence ($C_0 + C$).

Semi-Variance γ(h)
    ^
Sill|                     +--------------------- (C0 + C)
 (C0+C)                 /
    |                 /  <- Structural Variance (C)
    |               /
    |             /
    |   +-------+
    |   | Nugget (C0)
    +---+---------------------------------------> Lag Distance (h)
        0                Range (a)

Standard Theoretical Variogram Models

• Spherical Model: A commonly applied model in mining geostatistics, demonstrating linear behavior near the origin and reaching the sill exactly at range $a$:

$\gamma(h) = \begin{cases} C_0 + C \left[ \frac{3}{2} \left(\frac{h}{a}\right) - \frac{1}{2} \left(\frac{h}{a}\right)^3 \right], & \text{for } h \le a \\ C_0 + C, & \text{for } h > a \end{cases}$

• Exponential Model: Reaches the sill asymptotically. With scale parameter $a_e$ in the equation below, the common 95% practical range is approximately $3a_e$:

$\gamma(h) = C_0 + C \left[ 1 - \exp\left(-\frac{h}{a_e}\right) \right]$

• Gaussian Model: Exhibits parabolic behavior near the origin, indicating extreme spatial continuity (common in smooth physical features such as stratum thickness or top-of-bed elevations):

$\gamma(h) = C_0 + C \left[ 1 - \exp\left(-\frac{h^2}{a^2}\right) \right]$

Spatial Anisotropy: Geometric vs. Zonal

Mineral deposits rarely exhibit identical spatial continuity in all 3D directions. Spatial anisotropy describes directional variations in variogram properties.

1. Geometric Anisotropy: The variogram range $a$ changes with direction (e.g., longer range along strike, shorter range cross-dip), but the total sill ($C_0 + C$) remains constant in all directions. Geometric anisotropy is handled in geostatistics by applying an ellipsoid rotation matrix to transform coordinates into isotropic space.
2. Zonal Anisotropy: The total sill ($C_0 + C$) varies depending on the spatial direction, regardless of lag distance. This frequently occurs in layered sedimentary or vein-hosted deposits where crossing geological domain boundaries causes step changes in overall grade variance. Zonal anisotropy is modeled by combining directional variograms or modeling domains separately.