Internal and external dilution are central to converting geological resources into realistic mineable reserves, because they directly control plant feed grade, tonnage, costs, and ultimately project value. Modelling these effects explicitly in block models rather than applying generic factors gives a more reliable basis for mine design and long‑term scheduling.
Core Concepts and Types of Dilution
Dilution is the proportion of waste mass in the total extracted material (ore + waste), often expressed as a percentage. One formal definition is the ratio between the tonnage of waste and the total tonnage of ore and waste in the mined unit (Masoumi, Kamali and Asghari, 2019). When densities differ, the same definition can be written in terms of volumes and specific gravities (Masoumi, Kamali and Asghari, 2019).
For block‑model‑based planning, it is useful to distinguish:
- Internal dilution
Waste or low‑grade material inside the orebody, such as gangue inclusions, barren lenses, or internal layers, that is unavoidably mined together with ore within a given block or selective mining unit (SMU). It also reflects the effect of averaging small‑scale variability when grades are reported at block scale. Internal dilution is fundamentally a geological and support issue. - Geologic contact dilution
Mixing at ore–waste boundaries because block geometry, minimum mining width, or fixed block size forces inclusion of some external material near contacts. In a fixed‑dimension block model, boundary blocks partially outside the true orebody are specifically highlighted as locations of such geologic contact dilution (Masoumi, Kamali and Asghari, 2019). - External (operational) dilution
Additional waste incorporated from outside the designed ore volume through drilling, blasting, loading, excavation overbreak, blast movement, poor boundary control, or equipment constraints. Many analytical, empirical and numerical models are tailored to operational parameters such as blast‑hole pattern and equipment type (Masoumi, Kamali and Asghari, 2019).
This article focuses on how to determine internal vs. external dilution quantitatively and how to model their effects within block models, with emphasis on methods that use geological and grade data.
Economic Motivation for Dilution Modelling
Dilution reduces ore grade, increases extracted tonnage, lengthens mine life, and lowers NPV and ROI. Along ore–waste boundaries, dilution may depress grade below the economic cut‑off so that material becomes uneconomic for processing (Masoumi, Kamali and Asghari, 2019). Higher waste proportion increases extraction and processing costs, directly eroding project value and complicating dump and stockpile management (Masoumi, Kamali and Asghari, 2019).
Because decisions about cut‑off grade, pit limit, and scheduling depend on block grades and tonnages, a block model that already accounts for dilution provides more realistic long‑term production forecasts and feed‑grade variability control (Masoumi, Kamali and Asghari, 2019).
Mathematical Description of Dilution and Diluted Grades
Dilution as Mass or Volume Fraction
If (mW) is waste tonnage and (mO) is ore tonnage, dilution (D) is:
D = mW / (mW + mO)
This mass‑based definition can be converted to a volume‑based expression by including specific gravities of ore and waste, (ρO) and (ρW), and corresponding volumes (vO) and (vW), which is useful when block models track volumes and densities explicitly (Masoumi, Kamali and Asghari, 2019).
Grade of Diluted Material
For a single element, with ore grade (gO) and waste grade (gW), and tonnages (mO) and (mW), the final diluted grade (gT) of the material mined and potentially sent to plant is:
gT = (gO x mO + gW x mW) /(mO + mW)
This expression is used block‑by‑block to compute the ultimate grade of extracted material after accounting for internal dilution (Masoumi, Kamali and Asghari, 2019). If a block contains only ore (no waste), the diluted and undiluted grades coincide (Masoumi, Kamali and Asghari, 2019).
In practice, one may assume representative average waste grades and densities to compute (gT) and (D) for each block. For example, in one iron‑ore case study, Fe and FeO waste grades were taken as 5% and 2% with densities of 2.65 g/cm³ for waste and 4 g/cm³ for ore (Masoumi, Kamali and Asghari, 2019).
Determining Internal Dilution: Geological and Geostatistical Approaches
Conceptual Basis
Internal dilution arises when a block contains a mixture of ore and waste, or ore at varying grades. This can be due to:
- Thin internal waste layers or gangue bands embedded within ore
- Barren dykes or lenses
- Geological variability over distances smaller than block size
- Fixed, relatively large block dimensions that cross true geological boundaries
A fixed‑dimension geological block model can cause blocks near the orebody edge to extend outside the ore domain, automatically increasing dilution in these “border blocks” (Masoumi, Kamali and Asghari, 2019).
Workflow Using Sequential Indicator Simulation (SIS) and Multivariate Simulation
A comprehensive method has been demonstrated on the Gohar Zamin iron ore deposit, where internal dilution is quantified per block and grades corrected accordingly (Masoumi, Kamali and Asghari, 2019). The workflow is as follows:
- Data preparation and coding
- Drill‑core intervals are composited (e.g., 2 m) and coded as ore or waste based on detailed lithological logs (Masoumi, Kamali and Asghari, 2019).
- Assay data for key elements (e.g., Fe, FeO, P, S) are composited at a suitable length (e.g., 6 m) for grade simulation (Masoumi, Kamali and Asghari, 2019).
- Geological domain and block model definition
- A primary geological model is constructed based on the first and last occurrence of ore in each drill hole, defining the simulated internal‑dilution space (Masoumi, Kamali and Asghari, 2019).
- A regular block model is built with fixed cell dimensions (e.g., 10 × 10 × 15 m) covering this space (Masoumi, Kamali and Asghari, 2019).
- Rock‑type simulation using SIS
- Rock types (ore vs. waste) are treated as categorical variables and simulated with SIS using indicator variograms that capture anisotropy and spatial structure (Masoumi, Kamali and Asghari, 2019).
- A large number of realizations (e.g., 100) are generated; the E‑type (average) probability of ore occurrence per block is then computed from these realizations (Masoumi, Kamali and Asghari, 2019).
- Resulting E‑type maps show spatial patterns, such as lower ore probabilities at deposit margins (Masoumi, Kamali and Asghari, 2019).
- Internal dilution calculation per block
- For each block, the proportion of ore and waste volumes is derived from ore occurrence probabilities, densities, and total block volume, and internal dilution (D) is evaluated using the volume‑density formulation (Masoumi, Kamali and Asghari, 2019).
- High‑dilution blocks correspond to zones with thick waste layers or blocks partly outside the orebody. These high‑dilution blocks can be mapped to identify problematic areas (e.g., blocks with dilution greater than 50% highlighted in blue) (Masoumi, Kamali and Asghari, 2019).
- Joint simulation of grade variables with MAF
- Continuous grade variables (Fe, FeO, P, S) are jointly simulated using the minimum/maximum autocorrelation factor (MAF) approach. First, variables are transformed to Gaussian scores and a covariance matrix is computed; eigen‑decomposition yields principal components, which are then rotated to MAF factors that are spatially decorrelated (Masoumi, Kamali and Asghari, 2019).
- Each MAF factor has approximately zero mean and unit variance, and cross‑variograms confirm spatial decorrelation (Masoumi, Kamali and Asghari, 2019).
- Factors are simulated independently (e.g., by sequential Gaussian simulation) and back‑transformed to recover correlated grade variables that reproduce original correlations satisfactorily (Masoumi, Kamali and Asghari, 2019).
- Computation of ultimate (diluted) block grades
- For each block, the internal dilution proportion and simulated ore grades are combined (with assumed waste grades) to calculate the ultimate Fe and FeO grades using the mixing equations described above (Masoumi, Kamali and Asghari, 2019).
- In areas with very low ore occurrence probability (e.g., 10%), ultimate Fe and FeO grades can drop to the point where the block should be treated as waste rather than ore. Conversely, modest internal dilution may reduce Fe grade from, for example, 62.2% to 58.95% where a thin internal waste layer is present (Masoumi, Kamali and Asghari, 2019).
Case Study Outcomes and Interpretation
Applied to the northern part of Gohar Zamin, this integrated SIS–MAF approach produced an average internal dilution of about 10% across all simulated blocks (Masoumi, Kamali and Asghari, 2019). This is slightly higher than experimentally assumed mine‑wide internal dilution (7.5–8%), but provides block‑by‑block values and corrected grades.
On average, considering internal dilution reduced both Fe and FeO grades by around 10% relative to simulations that ignored dilution (Masoumi, Kamali and Asghari, 2019). Many blocks with dilution exceeding 50% were located where thicker waste layers intersect blocks, guiding selective mining or redesign.
Crucially, internal dilution estimates from this method agreed well with measurements from blast‑hole lithological logs. For example, one blasting pattern showed 612 m of ore and 137 m of waste within ore‑containing areas, corresponding to 11% dilution—close to the simulation‑derived ~10% (Masoumi, Kamali and Asghari, 2019). This validation supports using the method in long‑term planning.
Determining External Dilution and Distinguishing It from Internal Dilution
The Gohar Zamin study primarily targets internal, geologically controlled dilution. External dilution involves additional processes:
- Overdigging ore boundaries in open pits due to equipment geometry and operator error
- Inclusion of hangingwall or footwall rock in underground stopes due to overbreak
- Blast‑induced movement and mixing of ore and waste before loading
- Minimum mining width or safety factors that force waste extraction at contacts
Although many prior models focus on these operational aspects, they often do not integrate detailed geological data and rock types into dilution estimation (Masoumi, Kamali and Asghari, 2019).
When building a block model, a practical separation is:
- Internal/geologic dilution: predicted using methods like SIS + MAF, as above, representing the mixture within the geological orebody.
- Contact and operational (external) dilution: estimated through geometric ore–waste adjacency analysis, blast‑movement models, or empirical relationships and applied as additional modifying factors or as separate attributes.
In other words, internal dilution defines the ultimate grade of each geological block, while external dilution modifies the grade and tonnage actually delivered from that block during mining.
Integrating Dilution into Block Models
Representation Choices
A dilution‑aware block model can store:
- Undiluted grade estimates (geological resource)
- Internal dilution percentage (Dint) per block
- Diluted grades (gT,int) after internal mixing
- Optional external dilution attribute (Dext) per mining unit or boundary block
- Fully diluted grades (gT,full) including internal and external dilution
Such a structure allows using undiluted grades for geological reporting, while diluted grades feed pit optimization, reserve declaration, and scheduling. The Gohar Zamin methodology effectively populates (Dint) and (gT,int) at block scale (Masoumi, Kamali and Asghari, 2019).
1. Undiluted Grade (Geological Resource)
gO
2. Internal Dilution Percentage per Block
Dint = mW,int / (mO + mW,int)
3. Diluted Grade after Internal Mixing
gT,int = gO x mO + gW x mW,int / (mO + mW,int)
4. External Dilution Percentage (Optional)
Dext = mW,ext / (mO + mV,int + mW,ext)
5. Fully Diluted Grade (Internal + External)
gT,full = gO x mO + gW x (mW x (mW,int + mW,ext) / ( mO + mW,int + mW,ext)
Block Size and Boundary Effects
The fixed grid size used in Gohar Zamin (10 × 10 × 15 m) means that some blocks near deposit boundaries inevitably span both ore and waste, thereby contributing significantly to internal and contact dilution. Mapping these high‑dilution blocks highlights where reducing block size or introducing sub‑blocking could lower geologic and external dilution in mine design. The study explicitly notes dramatic dilution increases in border blocks due to constant block dimensions (Masoumi, Kamali and Asghari, 2019).
In a broader context, careful selection of SMU size—guided by support‑effect analysis and economic criteria—helps balance grade smoothing (internal dilution) and operational selectivity. Where narrow orebodies or complex boundaries dominate, smaller blocks or variable‑size models can reduce both internal and external dilution, at the cost of computational complexity.
Validation and Reconciliation
Robust dilution modelling must be validated against operational data:
- Comparison with mine‑assumed dilution factors
Average modelled internal dilution (~10%) compared favourably with mine practice (7.5–8%), although slightly higher because it was computed independently per block and propagated through grade calculations (Masoumi, Kamali and Asghari, 2019). - Blast‑hole lithology comparisons
- For each bench pattern, the lengths of ore and waste intervals from blast‑hole logs are used to compute an empirical dilution ratio (e.g., 612 m ore vs. 137 m waste → ~11% dilution) (Masoumi, Kamali and Asghari, 2019).
- These are then compared to model‑derived dilution in the corresponding blocks, showing “a high level of conformity and agreement” (Masoumi, Kamali and Asghari, 2019).
- Lithological pattern consistency
Where the model indicates low ore occurrence probabilities (<25%), blast‑hole lithology usually records waste, demonstrating consistent internal dilution prediction. Small discrepancies are attributed to differences in data resolution (2‑m composites vs. real variation) and spacing between exploration and blast holes (Masoumi, Kamali and Asghari, 2019).
Continual reconciliation between modelled and measured dilution supports iterative refinement of both internal and external dilution models over the mine life.
Practical Guidelines for Determining and Modelling Dilution
Drawing from the case study and broader geostatistical principles, a practical framework for internal vs. external dilution in block models is:
- Define objectives clearly
Decide whether the block model is to represent undiluted geology, in‑pit ore, plant feed, or all three via different attributes. This determines how extensively dilution must be embedded. - Use geological data to model internal dilution
- Encode ore/waste categories from high‑resolution lithological logs.
- Apply SIS to simulate rock types and obtain ore/waste occurrence probabilities per block.
- Compute internal dilution as the waste proportion per block, considering density contrasts.
- Jointly simulate grades with multivariate methods such as MAF to preserve cross‑correlations and obtain ultimate grades.
- Quantify contact and external dilution separately
- Analyse blocks at ore–waste boundaries to see where block geometry causes internal + contact dilution.
- Superimpose operational envelopes (dig lines, stope outlines, safety margins) and quantify extra waste included, distinguishing this external component.
- Capture scale effects
Develop grade‑tonnage and metal‑tonnage curves at different support sizes to understand how internal dilution changes with block or SMU size, and adjust block design accordingly. - Validate against operational data
Compare modelled dilution with blast‑hole lithological logs, grade control, and reconciliation; adjust assumptions on waste grades, densities, and ore probabilities where needed. - Feed into planning and economic evaluation
Use diluted grades and tonnages in pit optimization and scheduling, while retaining undiluted models for resource statements. Quantify the impact of dilution on NPV, cut‑off selection, and plant feed variability.
Concluding Remarks
Internal and external grade dilution are intimately tied to geology, block size, and mining practice. The SIS + MAF‑based workflow implemented at Gohar Zamin shows that internal dilution can be estimated block‑by‑block from exploratory drillhole data and lithological logs, and that grades can be corrected accordingly to produce realistic ultimate grades for Fe and FeO (Masoumi, Kamali and Asghari, 2019).
Average internal dilution of around 10% and corresponding 10% reductions in Fe and FeO grades, confirmed by blast‑hole data, highlight how substantial internal dilution can be, even before external and operational effects are considered. Embedding such dilution modelling into block models enables more reliable long‑term mine planning, improved dump and plant feed management, and better‑quantified project risk.
If desired, this framework can be expanded with explicit external dilution modules—e.g., blast‑movement simulation, ore–waste boundary overdig models, or stope overbreak predictors—to produce fully realistic, dilution‑aware block models that bridge the gap between geological resources and actual plant feed.
References
Masoumi, I., Kamali, G., & Asghari, O., 2019. Assessment of an ore body internal dilution based on multivariate geostatistical simulation using exploratory drill hole data. Journal of Mining and Environment. https://doi.org/10.22044/jme.2019.7622.1618
