Study guide
7.1 Ore Body Block Modeling & Spatial Variography
PRC Mining Engineer Licensure Examination Philippines
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1 Chapter 1: Surface Mining Methods & Pit Optimization
1.1 Surface Mining Unit Operations & Method Selection 1.2 Bench Geometry, Slope Stability & Geotechnical Pit Design 1.3 Cut-Off Grade Optimization & Lerchs-Grossmann Pit Limits
2 Chapter 2: Underground Mining Methods & Strata Control
2.1 Underground Mining Method Classification & Selection 2.2 Rock Mechanics, Stress Analysis & Strata Control 2.3 Underground Ground Support Systems & Excavation Design
3 Chapter 3: Materials Handling, Equipment & Mine Economics
3.1 Drilling, Blasting & Explosives Engineering 3.2 Mine Loading, Haulage & Materials Handling Systems 3.3 Mine Capital (CAPEX), Operating Costs (OPEX) & Financial Evaluation
4 Chapter 4: Mine Safety, Ventilation & Philippine Mining Laws
4.1 Mine Ventilation Network Analysis & Environmental Quality 4.2 Philippine Mine Safety Rules, Health & Disaster Prevention 4.3 Philippine Mining Act of 1995 (RA 7942), DAO 2010-21 & Mining Ethics
5 Chapter 5: Physical Geology, Mineralogy & Philippine Deposit Types
5.1 Structural Geology, Tectonics & Ore Controls 5.2 Igneous, Metamorphic & Sedimentary Ore Mineralogy 5.3 Major Philippine Ore Deposit Models
6 Chapter 6: Mineral Exploration, Geophysics & Core Logging
6.1 Geochemical & Remote Sensing Exploration Techniques 6.2 Geophysical Exploration Methods in Mining Geology 6.3 Diamond Drilling, Core Logging & QA/QC Protocols
7 Chapter 7: Geostatistics, Resource Estimation & PMRC Code
7.1 Ore Body Block Modeling & Spatial Variography 7.2 Inverse Distance & Ordinary Kriging Grade Estimation 7.3 Philippine Mineral Reporting Code (PMRC 2020) & Resource Classification
8 Chapter 8: Comminution, Sizing & Physical Mineral Separation
8.1 Crushing, Grinding & Bond Work Index Calculations 8.2 Particle Screening, Hydrocyclone Classification & Sizing 8.3 Gravity Concentration & Magnetic/Electrostatic Separation
9 Chapter 9: Froth Flotation & Extractive Hydrometallurgy
9.1 Froth Flotation Chemistry, Reagents & Cell Kinetics 9.2 Leaching Technologies (Heap, Agitation & Tank) 9.3 Precious Metal Recovery: CIP, CIL, Merrill-Crowe & Electrowinning
10 Chapter 10: Process Plant Flowsheets & Environmental Tailings Management
10.1 Metallurgical Plant Flowsheets & Pulp Mass Balances 10.2 Tailings Storage Facility (TSF) Design & Dam Safety 10.3 Acid Mine Drainage (AMD) Mitigation & Environmental Protection (EPEP/FMR/DP)
11 Chapter 11: 2025 PRC Blueprint, Prospecting Tools & Exploration Decisions
11.1 Official 2025 TOS, Three-Paper Structure & Exam Strategy 11.2 Prospecting Instruments, Base Maps & Surface Geological Mapping 11.3 Integrating Direct and Indirect Evidence for Exploration Decisions
12 Chapter 12: Mine Infrastructure, Sustainable Development & Climate
12.1 Power, Water, Drainage & Support-Facility Selection 12.2 Sustainable Development, Progressive Rehabilitation & Community Impact 12.3 Mining, Climate, Reforestation, Coal & Energy-Transition Minerals
13 Chapter 13: Mine Valuation, Financial Analysis & Due Diligence
13.1 Time Value of Money, Future Value & Present Value 13.2 Financial Statements, Mine Economics & External Price Drivers 13.3 Exploration Provenance, Technical Review & Legal Due Diligence
14 Chapter 14: Mine Organization, Communication & Accountable Leadership
14.1 Line and Staff Functions in a Mining Organization 14.2 Communication With Superiors, Peers & Subordinates 14.3 Authority, Responsibility, Delegation & Safety Accountability
15 Chapter 15: Philippine Mining Law, Indigenous Rights & Professional Ethics
15.1 Legal Hierarchy, Mineral Tenure & Compliance Analysis 15.2 Constitutional Participation, IPRA, FPIC & Ancestral Domains 15.3 RA 4274, Professional Scope, Ethics & Document Integrity
16 Chapter 16: Applied Mathematics & Mine Surveying
16.1 Engineering Mathematics for Safe, Efficient Mine Decisions 16.2 Survey Control, Instruments, Traverses & Error Adjustment 16.3 Topographic Maps, Underground Set-Out & Survey Timeliness
17 Chapter 17: Feasibility Studies, Data Quality & Project Decisions
17.1 Scoping, Pre-Feasibility & Final Feasibility Study Maturity 17.2 Data Accuracy, Assumption Control & Go/No-Go Decisions 17.3 Multidisciplinary Integration, Interfaces & Independent Review
18 Chapter 18: Mining Computer Applications, Digital Control & Global Developments
18.1 Computer Functions, Office Applications & Data Governance 18.2 Geological Modeling, Mine Planning, Scheduling & Valuation Software 18.3 Real-Time Control, Critical Minerals & Regional Mining Developments
19 Chapter 19: Earth Systems, Geologic Time & Philippine Hazards
19.1 Solar-System Origin, Earth Interior & Atmospheric Layers 19.2 Rock Cycle, Landforms & the Geologic Time Scale 19.3 Philippine Tectonics, Geologic Hazards & Mitigation
20 Chapter 20: Metallurgy, Foundry Practice & Assaying
20.1 Extractive, Physical & Adaptive Metallurgy Principles 20.2 Foundry Materials, Casting, Solidification & Defects 20.3 Sampling, Fire Assay, Cupellation, Parting & Assay Calculations
21 Chapter 21: Solid–Liquid Separation, Environmental Law & Mine Ecology
21.1 Thickening, Filtration & Solid–Liquid Separation 21.2 Philippine Air, Water, Waste & Hazardous-Substance Controls 21.3 Mine Ecology, Closure, Tailings, Waste & Community Livelihood
7.1 Ore Body Block Modeling & Spatial Variography
Key Takeaways Explain This Topic AI
3D geological block modeling partitions a deposit into parent blocks for estimation and may use sub-blocking to improve geometric-volume approximation at wireframe boundaries.
Parent-block dimensions are selected from data spacing, geology, variogram support, estimation strategy, change of support, and Selective Mining Unit; fractions of drill spacing are screening heuristics, not a universal rule.
Spatial continuity is quantified by the experimental semi-variogram γ ( h ) = 1 2 N ( h ) ∑ [ Z ( x i ) − Z ( x i + h ) ] 2 \gamma(h) = \frac{1}{2N(h)} \sum [Z(x_i) - Z(x_i+h)]^2 γ ( h ) = 2 N ( h ) 1 ∑ [ Z ( x i ) − Z ( x i + h ) ] 2 , evaluating average squared grade differences between sample pairs.
Key variogram parameters include the nugget effect (C_0), structural variance (C), total sill plateau (C_0 + C), and spatial correlation range (a).
Spatial anisotropy is classified as geometric (range varies with direction while sill remains constant) or zonal (sill plateau varies across directional domains).
Last updated: August 2026
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.
- Modeling Parameter Parent Block Sub-Block (Child Cell)
- Primary Purpose Grade estimation & kriging matrix calculation Geological wireframe volume representation
Dimension Criteria Selected and validated for data, geology, estimation support, and mining use Refined enough to meet a stated geometric-volume tolerance
Mining Significance Represents estimation support and may relate to mining selectivity Approximates contact geometry and volume to a validated tolerance
- Estimation Handling Direct neighborhood search & weight assignment Inherits parent grade or regularized grade
- 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 γ ( h ) \gamma(h) γ ( h ) for a distance lag vector h h h is defined as one-half the average squared difference between sample pairs separated by distance h h h : γ ( h ) = 1 2 N ( h ) ∑ i = 1 N ( h ) [ Z ( x i ) − Z ( x i + h ) ] 2 \gamma(h) = \frac{1}{2 N(h)} \sum_{i=1}^{N(h)} [Z(x_i) - Z(x_i + h)]^2 γ ( h ) = 2 N ( h ) 1 i = 1 ∑ N ( h ) [ Z ( x i ) − Z ( x i + h ) ] 2
Where:
γ ( h ) \gamma(h) γ ( h ) = Semi-variogram value at lag distance h h h .
N ( h ) N(h) N ( h ) = Total number of sample pairs separated by lag vector h h h .
Z ( x i ) Z(x_i) Z ( x i ) = Assay grade value at spatial location x i x_i x i .
Z ( x i + h ) Z(x_i + h) Z ( x i + h ) = Assay grade value at spatial location x i + h x_i + h x i + h .
To calculate the experimental variogram from irregularly spaced drillhole assays, directional parameters are applied:
Lag Distance ( h h h ) and Lag Tolerance ( Δ h \Delta h Δ 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 h h ) versus semi-variance ( γ ( h ) \gamma(h) γ ( h ) ). A theoretical mathematical model must be fitted to these experimental points for use in kriging algorithms. Key variogram parameters include:
Nugget Effect ( C 0 C_0 C 0 ) : The discontinuity between γ ( 0 ) = 0 \gamma(0)=0 γ ( 0 ) = 0 and the modeled limit as h → 0 + h \to 0^+ h → 0 + . It can reflect micro-scale variability occurring at distances smaller than the sample spacing, as well as sampling, sub-sampling, and assaying errors.
Sill / Total Variance ( C 0 + C C_0 + C C 0 + C ) : The plateau where the variogram flattens out, representing the overall sample population variance. C C C represents the structural variance component.
Range ( a a 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 C_0 + C 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 a a : γ ( h ) = { C 0 + C [ 3 2 ( h a ) − 1 2 ( h a ) 3 ] , for h ≤ a C 0 + C , for h > 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} γ ( h ) = { C 0 + C [ 2 3 ( a h ) − 2 1 ( a h ) 3 ] , C 0 + C , for h ≤ a for h > a
Exponential Model : Reaches the sill asymptotically. With scale parameter a e a_e a e in the equation below, the common 95% practical range is approximately 3 a e 3a_e 3 a e :
γ ( h ) = C 0 + C [ 1 − exp ( − h a e ) ] \gamma(h) = C_0 + C \left[ 1 - \exp\left(-\frac{h}{a_e}\right) \right] γ ( h ) = C 0 + C [ 1 − exp ( − a e h ) ]
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): γ ( h ) = C 0 + C [ 1 − exp ( − h 2 a 2 ) ] \gamma(h) = C_0 + C \left[ 1 - \exp\left(-\frac{h^2}{a^2}\right) \right] γ ( h ) = C 0 + C [ 1 − exp ( − a 2 h 2 ) ]
Spatial Anisotropy: Geometric vs. Zonal
Mineral deposits rarely exhibit identical spatial continuity in all 3D directions. Spatial anisotropy describes directional variations in variogram properties.
Geometric Anisotropy : The variogram range a a a changes with direction (e.g., longer range along strike, shorter range cross-dip), but the total sill ( C 0 + C C_0 + C 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.
Zonal Anisotropy : The total sill ( C 0 + C C_0 + C 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.