Measuring and modelling of dry bulk rock density for mineral resource estimation

Competent non-porous rocks

Dry bulk density (g cm^–3)of a good quality drill core can be determined by dividing its dry mass (g), determined by weighing the dried sample in air, by the volume of the drill core (cm³) (equation (1)) [²V = πR²L, where π = 3.14, R is radius of a circular base of a cylindrical drill core, L is length of core cylinder.]. This approach, is commonly referred to as the Caliper method (Lipton, 2001), requires cutting the core at right angles to the core length and estimating the average length and diameter of the core from several measurements. The method is simple and does not require special equipment, and it can be applied to competent rocks which have natural voids and porosity. However, its application is limited to samples having geometrically accurate shapes. It is not applicable to irregularly shaped samples or samples that are represented by several rock chips, therefore this technique is rarely applied (1)

The bulk volume of solid non-porous rocks is more commonly measured using a water displacement technique, which is based on Archimedes principle (Lipton, 2001). The method requires drying sample and weighing it in air, and then in water. Dry bulk density is estimated using equation (2) (2) where M_S is dry mass of a sample in air (after sample was dried for approximately 24 h at 110°C); M_SW is mass of a sample in water; and ρ_WATER is water density, approximately 1 g/cm³.

The mass of a sample in water (M_SW) is determined by placing the rock chips in a wire basket suspended from a balance and immersing it in water (Fig. 3a). Measurement is made in two steps, first by weighing the empty basket immersed in water (M₁) and then by weighing the basket with the rock specimens immersed in water (M₂). The final mass (M_SW) is determined by subtracting the weight of the empty basket immersed in water (M₁) from the weight of the basket holding the rock chips in water (M₂).

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Figure 3

Different approaches to weighing samples when the water displacement method is applied: a weighing the basket holding the studied rock chips suspended from a balance and immersed in water; b weighing the entire container filled with water. The basket containing the rock specimen is attached to the container

A variation of this method involves weighing an entire container of water with a sample holding basket firmly attached to the container (Fig. 3b). The water container must be weighed twice: first when the wire basket is empty, and then again using the same container and basket with the rock sample placed in it (Fig. 3b). Dry bulk density is estimated using equation (3) (3) where M_S is dry mass of a sample in air (after sample was dried for approximately 24 h at 110°C); M_C+B+SW is mass of a water container with an immersed basket holding rock specimen; M_C+B is mass of a water container with an immersed empty basket; and ρ_WATER is water density of approximately 1 g cm^–3.

Both approaches (Fig. 3) are reliable and give an accurate estimate of the DBD of the sample. However, weighing samples in a basket suspended from electronic balance (Fig. 3a) is used more often. The method is simple, has better ergonomics, does not require special equipment and can be easily implemented at mine sites and exploration camps. DBD measurement facilities are usually installed at core farms and the density is measured at the same time as core is cut for assaying. The procedure of determining the dry bulk rock density is as follows:

The simplicity of the method does not eliminate the possibility for error. The most common sources of the errors are:

The quality of density measurements is usually controlled by check measurements of selected samples in the reputable laboratory, preferably in the same lab where samples are assayed for metal contents. Check measurements should be performed for every sample batch and confirmed by external laboratory tests.

Porous and weathered rocks

The water displacement method described in the previous section is not suitable for porous, clay-rich, or intensely weathered rocks that tend to disintegrate when immersed in water. Application of the water displacement method to porous or soft friable materials requires an additional step in the preparation of the samples. Such samples need to be coated with wax or polymer bags after drying and before immersing in water.

Water displacement methods applied to coated samples

The procedure for measuring the DBD of the porous or soft friable rocks is explained in AS2891.9.1-2005 and briefly summarised below:

•

samples are dried at a temperature that is not destructive to the samples. If necessary, it can be at room temperature. Drying at low temperatures is facilitated by placing samples under a current of air

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dry mass (M_S) is determined by weighing samples in air

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the density of the wax is determined

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the sample is completely sealed by coating it with hot paraffin wax, and then cooled to room temperature

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the sample is weighed again in air to determine the mass of the sample coated by wax (M_S+WAX)

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the sample is transferred to a wire basket suspended from a balance and immersed in water (Fig. 3a). The weight of the coated sample in water (M_{S+WAX IN WATER}) is determined by weighing the sample and subtracting the weight of empty basket in water. Sample mass is recorded to the nearest 0·1 g (AS2891.9.1-2005). Air bubbles adhering to the coated sample shell be removed before sample is weighed

•

DBD of the sample is calculated using equation (4) (4) where M_S is dry mass of a sample in air; M_S+WAX is mass of sample coated with wax and weighed in air; M_{S+WAX IN WATER} is mass of sample coated in wax and immersed in water; ρ_WAX is density of wax; andρ_WATER is water density, approximately 1 g cm^–3.

Unlike competent non-porous rocks, density of porous samples is usually determined on single hand specimens as it is not practical to coat a sample represented by several rock chips. Specimens used for DBD measurement usually range from 10 to 15 cm in size, which is relatively small, and therefore it is important to assure that the specimen is representative of the studied rock type. Accuracy and reproducibility of the measured density should be controlled by testing duplicate samples. The method is labour intensive and can be very time consuming mainly because the drying process can be slow at low temperatures. It may also be difficult to effectively seal highly porous rocks with wax. Furthermore, the wax coating also make the specimen unsuitable for further analysis or repeat measurements of density for quality control purpose.

An alternative approach to wax coating for sealing porous and friable materials is based on the vacuum-sealing of samples in polymer bags (Fig. 4). The procedure was developed for testing asphalt mixes (ASTM D6752-09) and is known as CoreLok. It has been applied for measuring DBD of friable materials at numerous different mining projects, including the Simandou iron-ore project in Guinea.

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Figure 4

CoreLok device for vacuum-sealing samples in polymer bags

The CoreLok DBD measurement procedure is as follows:

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the sample is dried and its dry mass in air is measured (M_S)

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the mass (M_BAG) and density (ρ_BAG) of the polymer bag is measured

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sample is vacuum-sealed in the polymer bag (Fig. 4)

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the sealed sample is weighed again in air (M_S+BAG). This is a control measurement as the weight of the sealed sample should be equal to the sum of its dry mass and the polymer bag (M_S+BAG = M_S+M_BAG)

•

the sealed sample is placed in a wire basket suspended from a balance, and immersed in water (Fig. 3a). The mass of the immersed sample is measured (M_{S+BAG IN WATER})

•

DBD of the sample is calculated using equation (5) (5) where M_S is dry mass of sample in air; M_BAG is mass of polymer bag in air; M_S+BAG is mass of sample sealed in bag and weighed in air; M_{S+BAG IN WATER} is mass of sealed sample in water; ρ_BAG is density of the polymer bag; and ρ_WATER is water density, approximately 1 g cm^–3.

The CoreLok methodology has the following advantages over conventional coating with wax:

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CoreLok measurements are not dependent on mix specific calibrations

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different sample sizes and shapes can be used, including rock chips

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the method is practically suitable for all types of geological materials, including ones that can be destroyed by immersion in water

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samples remain dry and uncontaminated, and suitable for further testing purposes and control measurements of DBD

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the sealing process is fast, usually taking 2–3 min.

The method can also be used for the determination of rock porosity estimated from the difference between the DBD of the sample and its maximum density. The maximum density is inferred from the mass of a water saturated sample (M_SAT), which is obtained by opening the bag under water after recording the mass of the sealed sample. The samples are usually opened by cutting the sealed bags, or piercing them with a knife. Porosity is estimated using equation (6). The variables are the same as in equation (5), except (M_SAT), which is mass of the submerged sample saturated with water in the polymer bag (6)

The Por(%) is a measure of open porosity as isolated voids can not be saturated by water, therefore it tends to underestimate the actual porosity and should be treated as an indicative measure.

Sand replacement method

Dry bulk density of weathered rocks can also be determined in the field using the Sand replacement method (AS1289.5.3.1-2004). This is a field measurement technique, not a laboratory measurement, and is designed to measure in situ density. It is the main method for determining density at the Weipa bauxite operations in Australia (Abzalov and Bower, 2009). The method involves digging a small cylindrical hole, approximately 300–400 cm³ in volume and 150 to 200 mm in diameter depending on size of equipment used (Fig. 5). The hole is filled with sand of a uniform grain size and known density. Dry bulk density is estimated by measuring the dry weight of material excavated from the hole and the volume of the hole, which is deduced from the volume (amount) of sand that was poured to fill the hole.

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Figure 5

Sand replacement method showing the equipment used for pouring sand

Equipment for the sand replacement method includes a plastic container (bottle) with a funnel shaped bottom which is attached to a plastic cone (Fig. 5). The device is also equipped with a valve located above the cone (Fig. 5) which is used to control sand flow when it is running into the test hole. The assembled apparatus is placed on a flat metal tray which is fixed into the ground through four small corner holes. The central hole in the tray, approximately 150–200 mm in diameter (Fig. 5), is used as the pattern for digging the test hole and also for pouring sand into the hole.

The procedure for measuring in situ DBD using sand replacement method is as follows:

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the density of the sand (ρ_SAND) that is used in the experiment should be thoroughly calibrated using the same sand pouring device (Fig. 5). It is important that each new batch of sand should be calibrated

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at the test site, usually a mine bench, a flat place should be located and cleaned from compacted or disturbed material

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the tray is fixed into place using four corner holes (Fig. 5)

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a cylindrical hole is dug using the opening in the middle of the tray as a template to constrain the diameter of the hole. The test hole should not extend sideways underneath the template rim. The depth of the hole should be approximately the same as the width of the tray

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material dug from the hole is collected and placed in a plastic bag for preventing moisture loss. The filled bag is weighed to determine the raw (wet) mass of the sample (M_SW) subtracting the weight of the bag. It is good practice to prepare and weigh the sample bags in the office before field tests. Allocated sample numbers and weights of the bags should be written on the front sides of the bags and recorded in sample log books

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the container is filled with sand of uniform grain size and known (calibrated) density (ρ_SAND). The bottle filled with sand is weighed (M₁)

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the valve and cone are attached to the container, inverted and placed on top of the template. The valve is opened and the hole slowly filled with sand

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the hole is filled when the sand has stopped running. At this moment the valve is closed and the device removed from the template. It is important that sand is not compacted into the test hole

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the container with remaining sand is weighed again (M₂)

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wet bulk density is calculated using equation (7) (7) where M_SW is wet mass of the sample in air; M₁ is the mass of sand and container before filling the hole; M₂ is the mass of remaining sand and container after filling the hole; M₃ is the mass of sand that filled the pouring cone on the metal tray. This is effectively a constant usually established before digging the hole; and ρ_SAND is calibrated density of the sand used for measurement.

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In order to estimate the DBD all the recovered material is taken to a laboratory and dried at approximately 100°C to determine the dry mass of sample in air (M_S). After that, the DBD is calculated using equation (8). Moisture content can be calculated as the ratio between dry and wet masses (8)

This method is robust and produces accurate density estimates of soft and porous materials, but errors can be introduced if operational procedures are not carefully followed, in particular:

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compaction of the sand in the test hole

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failure to keep the sand dry

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inaccurate calibration of the sand density

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uneven surface where the template was installed

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losses of dug sample material when it is transferred from the hole to the plastic bag or to the laboratory for determining dry weight. Additional errors can be introduced by improper subsampling of material extracted from the test hole, in particular when the sample is comprised of mixed coarse fragments of different sizes and fines. The preferred approach is therefore to dry and weigh whole original sample extracted from the test hole (equation (8))

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the method can be applied only to horizontal surfaces exposed in outcrops or mine workings and is not practical for application to thick or steeply dipping ore bodies.

Non-consolidated sediments

Measuring DBD of soft non-consolidated sands is particularly challenging because of the difficulties in obtaining undisturbed representative samples and assuring that sampled sediments are not accidentally compacted during sampling. The sand replacement method may not be applicable because the dug voids tend to collapse when the technique is applied to free flowing sands. This problem can be partially overcome by using a metal cylinder with both ends open, which is pressed into the sands and used as a casing frame. Sediments inside of the cylinder are collected by a scoop, weighed, and then dried to determine a dry mass (M_S). The volume of the hole is calculated from the known internal diameter of the cylinder and the measured depth of the dug hole. Alternatively, when the bottom of the hole is uneven, the volume can be determined by the sand replacement method (Fig. 5). Limitations of this technique are similar to those of the sand replacement method. The method is not applicable when thickness of the studied mineralisation is greater than several metres and deep parts of the deposit can not be accessed by exploration pits or winzes.

Personal experience of the author in heavy mineral sand deposits (Abzalov, 2009; Abzalov et al., 2011) is that ‘Sonic’ drilling offers the most practical approach for obtaining undisturbed samples of sands located deeply below surface. The samples obtained by ‘Sonic’ drilling can be used for density measurements, in particular if drilling equipment includes stainless steel split liners placed inside the core barrel. Procedure of estimating the density of the sands recovered by the sonic drilling is the same as the Caliper technique explained earlier (equation (1)). This technique proved to be particularly effective for definition of resources of heavy minerals sands at the Richards Bay deposit, where the thickness of mineralised dunes in places reaches 170 m (Abzalov et al., 2011).

Geostatistical classification of the mineral resources

Methodology, proposed in this paper, was developed as mathematically supported approach for choosing optimal distribution of the DBD samples for estimation mineral resources and for the grade control purpose. The classification approach used in the current study links geostatistical estimation errors to the mine production rates (Table 2). It was proposed by Northern American consultants, H. Parker and M. Belanger in the late 90th century (pers. comm.), and at present is broadly used in the mining industry. Their approach has been adopted with minor changes by the current author for classification of resources at some Australian projects (Abzalov and Bower, 2009; Abzalov et al., 2010). A difference was made to the threshold values. The threshold of ±15% (with 90% confidence) suggested by H. Parker and M. Belanger, (pers. comm.) was replaced by ±10% (with 95% confidence) because the latter values are better aligned to actual resource definition practices at the studied Australian mines (Abzalov and Bower, 2009; Abzalov et al., 2010). The same criteria (Table 2) were applied in this paper to studied deposits (Figs. 1 and 2) for optimisation their DBD sampling patterns.

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Table 2

Geostatistical classification of the mineral resources

Resource category	Volume of mineralisation	Estimation error	Confidence level
Measured	3 months of production	±10%	∼95% (2 standard deviations)*
Indicated	12 months of production	±10%	∼95% (2 standard deviations)*

*

2 standard deviation corresponds to 95·45% confidence level or ∼95% if rounded for simplicity.

Exceptions were made for the iron ore mines (Fig. 1) where the historical practice for resource and reserve definition were set to the shorter terms of mine production, requiring accurate planning of monthly (Measured resources) and quarterly (Indicated resources) productions. Some of the nickel sulphide deposit considered in this study (Fig. 1) represents a mining project at an early stage of evaluation. Reserves and a production plan of the project are yet to be defined, therefore the geostatistical estimates are based on the technical and economic assumptions made during the project scoping study.

Grade control

Quantification of grade control errors is another area which historically has largely been based on subjective decisions and usually lacks quantitative estimates of the risks involved. At production stage the rock density estimates are usually based on the data collected during resource definition drilling, therefore, the exploration team needs to assure that DBD samples collected during the drilling campaigns are not only sufficient for definition of resources and reserves, but are also closely enough spaced to allow accurate grade control.

Selectivity of the short term mining plans varies between different operations depending on their mining methods and operational logistics. For example, many sea-borne bauxite mines require accurate planning of the ore parcels loaded to a single ship (e.g. Weipa bauxite operation, Australia). Conversely, many iron ore operations in the Pilbara region of Western Australia constrain their short term mine production plans to the size of the trains transporting ore to the sea port or blending facilities. A common approach is to constrain the short term production plans to the daily production rates at the mines (Abzalov and Bower, 2009; Abzalov et al., 2010; Abzalov, 2011; Abzalov, 2012). This criterion [volume of ore, representing daily production of a given mine, estimated with an error of ±10% (at 2 standard deviations).] is better suited to the objectives of the current study and was therefore used for estimating optimal DBD sampling grids for the studied deposits (Fig. 1).

Geostatistical estimation errors

Estimation error in a formal geostatistical analysis can be directly estimated from stochastic deposit models generated using a variable conditional simulation technique (Goovaerts, 1997; Chiles and Delfiner, 1999; Abzalov and Bower, 2009). The approach is methodologically robust and provides accurate results; however, it can be excessively time consuming and requires full access to the original data which often is not available at the early stages of the mining project evaluation.

Alternatively, estimation errors can be deduced from auxiliary geostatistical functions calculated from variogram models of the studied variables (Annels, 1991). This approach was used in this paper and the estimation errors of the metal grades and rock densities have been deduced from their Gamma-bar parameters (Journel and Huijbregts, 1978; Annels, 1991). This parameter, also known in geostatistics as auxiliary F-function (Annels, 1991; Olea, 1991), represents the mean variogram averaged over all possible vectors contained within a rectangular block (V) (equation (9)) (9)

An important property of the F-function is that it is equal to dispersion variance (D) of a point support (o) within a rectangular block (V), equation (10) (Olea, 1991). In other words, mean variogram over a rectangular block (V) is equal to the extension variance that results from forecasting the average property in the block (V) using the value defined over point (o) which can take any position within the block (V) (10) where F(V) is the value of F-function estimated for the rectangular block of size (V); is mean of the variograms estimated over all possible vectors within a rectangular block (V); and D(o|V) is dispersion variance of a point support (o) within a rectangular block of size (V).

This property of equation (10) links the variogram model with the estimation error and allows the use of the F-function as a measure of resource estimation uncertainty. The F-function is applicable when the drilling grid is quasi-regular, implying that every studied rectangular block contains a sample; however, a sample can take any position within the block (i.e. ‘floating’ at random). Such distribution of data points is usually referred to as a random stratified grid (Journel and Huijbregts, 1978; Annels, 1991) and is one of the most common types of drilling patterns used for estimating mineral resources. The random stratified grid was also used for estimation resources of the deposits studied in the current paper.

The calculation procedure includes the construction of experimental variograms of the studied variable, fitting of a variogram model, and then estimation of the average of the variogram over the rectangular block of interest. The size of the rectangular block chosen for calculation of the F-functions should match the dimensions of the sampling grids proposed for the definition of mineral resources.

In practice, the F-function values corresponding to the blocks of interest are usually deduced from special geostatistical diagrams, which have calibrated the F-function values depending on the variogram model and ratios of the variogram ranges to the dimensions of the investigated drill grids. The diagrams of the geostatistical auxiliary functions can be found in many geostatistical textbooks (e.g. Fig. A4.7 in Annels, 1991). An alternative approach for estimating the F-function value is by using a computer programs. In that case, all possible vectors in the studied rectangular block are approximated by discretising it and presenting as a matrix of data points. Usually, discretisation to a matrix of 10(X)×10(Y)×10(Z) points produces an accurate result. The variogram model, which was deduced from the samples, is applied for calculation of the variograms over all possible vectors between discretisation points contained within a studied rectangular block. The estimated variograms are then averaged to obtain a mean variogram value (i.e. F-function).

Procedure of estimation error calculation and optimisation of sampling grids

The methodology described in this paper has been conducted as follows.

Current practices for determining rock densities of mining projects

The total number of samples collected for chemical assays and rock density measurements for the definition of mineral resources at the several studied mines are summarised in Table 3. The ratio of assay samples to DBD measurements varies at the studied deposits from approximately 1∶1 to 20∶1 (Table 3). The table also shows the average grades and the coefficients of variation of economically valuable metals, deleterious components, and DBD values.

At some of the studied operations, rock densities are measured using the same samples which are collected for estimating the metal grades (e.g. Perseverance nickel sulphide mine, Australia). At such projects, the number of DBD measurements can be several hundreds of thousands (Table 3). However, a more common approach is to collect DBD samples separately from the assay samples using the second half of the drill core. In this case, DBD samples can be taken less frequently than assay samples, usually at regular intervals varying from several metres to several tens of metres. It is noted that different mines, even for the same commodity, can significantly differ in the number of DBD measurements collected. For example, at a CID type iron-ore mine located in the Pilbara region of Western Australia (Table 3), 4984 DBD samples were collected. These represent 24% of the total number of samples assayed for the estimation of metal grades at the deposit. Another case, at a BIF derived type iron-ore mine, also located in the Pilbara region, 103 740 DBD samples were collected, which is 62% of the assayed samples at the deposit (Table 3).

The metal grades in all studied deposits exhibit significantly larger variability than the variability of the DBD values. Coefficients of variations of the DBD values vary from 0·02 to 0·17 whereas coefficients of variations of the metal grades are significantly larger, varying between 0·25 and 6·44 (Table 3).

Geostatistical estimation of variability of the metal grades and the rock densities

Continuity of metal grades and rock densities has been quantified by construction of their variograms (Table 4). Variograms of the metals usually reach a sill (plateau) at a shorter distance than variograms of rock densities, and frequently also exhibit a larger nugget effect (Fig. 6). In other words, the variogram models clearly show that, in all of the studied cases, the metal grades are more spatially variable than rock densities.

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Figure 6

Experimental variograms (shown in red colour) and fitted variogram models (shown in black). N0 – estimated in the direction of Azimuth 0 degrees, D-90 – estimated in the vertically down direction. Cu-porphyry deposit, Australia: a Cu grade, b rock densities

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Table 4

Variogram models of rock densities (DBD) and main metals

Deposit type	Variable	Structure	Sill	Range by ellipsoid radii
				(Azi 000)	(Azi 090)	(Vertical D90)
		Nugget	1·20
Ni-sulphide	Ni	Spherical-1	0·75	30	27	15
(High-grade)		Spherical-2	0·40	200	145	135
		Nugget	0·09
	DBD	Spherical-1	0·05	30	30	20
		Spherical-2	0·073	440	300	620
		Nugget	0·0080
Ni-sulphide	Ni	Spherical-1	0·0092	150	160	16
(Low-grade)		Spherical-2	0·0103	320	200	80
		Nugget	0·0015
	DBD	Spherical-1	0·0061	475	275	23
		Nugget	0·50
IOCG	Cu	Spherical-1	2·20	25	21	25
		Spherical-2	0·52	90	50	60
		Spherical-3	0·10	100	Infinite	Infinite
		Nugget	0·020
	DBD	Spherical-1	0·100	25	20	20
		Spherical-2	0·055	90	55	75
		Spherical-3	0·107	300	310	300
		Nugget	0·0140
U unconformity	U3O8	Spherical-1	0·0210	30	30	40
		Spherical-2	0·0150	215	215	85
		Nugget	0·0022
	DBD	Spherical-1	0·0020	175	200	150
		Spherical-2	0·0010	200	200	450
		Spherical-3	0·0030	Infinite	300	450
		Nugget	0·050
	Cu	Spherical-1	0·225	30	30	50
Cu porphyry		Spherical-2	0·170	130	100	200
		Nugget	0·0010
	DBD	Spherical-1	0·0017	45	20	10
		Spherical-2	0·0015	400	400	275
		Nugget	2·7
	Fe	Spherical-1	0·7	114	313	6
Fe-deposit		Spherical-2	2·4	91	468	11
(CID type)		Nugget	1·00
	SiO2	Spherical-1	0·6	227	101	15
		Spherical-2	1·5	243	101	31
		Nugget	0·8
	Al2O3	Spherical-1	0·6	439	104	6
		Spherical-2	0·7	491	109	15
		Nugget	0·02
	DBD	Spherical-1	0·067	65	65	15
		Spherical-2	0·055	700	300	22
		Nugget	1·30
	Fe	Spherical-1	1·30	80	80	6
Fe-deposit		Spherical-2	0·87	300	190	9
(BIF derived)		Nugget	0·80
	SiO2	Spherical-1	1·0	130	140	6
		Spherical-2	1·5	550	180	10
		Nugget	0·2
	Al2O3	Spherical-1	0·3	90	90	6
		Spherical-2	0·1	175	110	10
		Nugget	0·03
	DBD	Exponential-1	0·105	175	147	33
		Nugget	65·0
Au orogenic	Au	Spherical-1	140·0	100	40	8
		Nugget	0·0025
	DBD	Spherical-1	0·0029	20	30	8
		Spherical-2	0·0014	100	100	90

These results concur with the spatial distributions of metal grades observed on the representative cross-sections (Fig. 2), where metal content can significantly differ even at short distances. The largest spatial variations are characteristic for nickel (Fig. 2a), uranium (Fig. 2b), copper (Fig. 2c) and gold (Fig. 2e). Contrary to this, rock densities exhibit good spatial continuity and change very slowly in the down-hole direction and between drill holes (Fig. 2).

Based on the variogram models the resource uncertainties (estimation errors) have been quantified for rock densities and metal contents (Table 5). Results show that estimation errors of rock densities are consistently lower than those for metal contents. In the studied deposits the errors in estimated average densities vary from 0·6 to 3% (at 2 standard deviation tolerance interval), whereas metal grades are estimated with errors varying from 3·1 to 11·8% (at 2 standard deviation tolerance interval). These results clearly show that accurate estimation of rock densities requires considerably fewer samples than are needed for the modelling of metal grades in the same deposits (Table 5). This finding accords well with current practices in which DBD samples are taken less frequently than assay samples.

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Table 5

Sampling grids and estimation errors by the mineral resource categories

Deposit type	Sampling grid by the resource categories and estimation error (at 2 standard deviations)				Reference
	Assay samples		Density samples
	Measured	Indicated	Measured	Indicated
Nickel sulphide, high grade	40×40×1 (9·5%)	80×80×1 (9·9%)	40×40×1·08 (1·3%)	80×80×1·08 (1·4%)	Fig. 2a
Nickel sulphide, low grade	75×75×1 (7·4%)	150×150×1 (8·6%)	75×75×1·16 (0·6%)	150×150×1·16 (0·7%)
Iron oxide copper gold	35×35×1 (9·5%)	70×70×1 (10·3%)	35×35×1·10 (0·9%)	70×70×1·10 (1·0%)
Unconformity uranium	35×35×1 (9·3%)	70×70×1 (10·3%)	35×35×20 (1·4%)	70×70×20 (1·6%)	Fig. 2b
Copper porphyry	40×40×1 (10·5%)	80×80×1 (11·8%)	40×40×15 (1·2%)	80×80×15 (1·3%)	Fig. 2c
Iron-ore, CID type	50×50×1 (4%)	150×150×1 (7·1%)	50×50×4 (1·7%)	150×150×4 (3%)
Iron-ore, BIF derived	50×50×1 (3·1%)	150×150×1 (5·8%)	50×50×1·60 (1·1%)	150×150×1·60 (2%)	Fig. 2d
Orogenic gold	30×30×1 (7·2%)	60×60×1 (8·3%)	30×30×10 (1·1%)	60×60×10 (1·2%)	Fig. 2e
Diamond, kimberlite pipe	15×15×10	30×30×10	15×15×10	30×30×10
Diamond, kimberlite pipe	15×15×10	30×30×10	15×15×20	30×30×20

Geostatistically optimal dry bulk density sampling grids

Mineral resources

Geostatistically deduced estimation errors (Table 5) show that some mining projects have tended to collect an unnecessarily large number of DBD samples. In order to find an optimal ratio between chemical assays and DBD samples, the different data configurations have been analysed and their impacts on resource estimation errors have been geostatistically estimated. Results are presented on diagrams where estimation errors are plotted against drilling grids (Fig. 7).

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Figure 7

Estimation errors at different sampling grids: a high-grade Ni-sulphide deposits (Fig. 2a); b unconformity uranium (Fig. 2b); c Cu porphyry (Fig. 2c); d BIF derived iron ore (Fig. 2d); e orogenic gold (Fig. 2e); f Channel Iron Deposit type iron ore; g low-grade Ni-sulphide; h IOCG type

The diagrams of estimation errors vs sampling distances (Fig. 7) allow us to both optimise drilling grids and to determine the optimal numbers of density measurements for estimation of mineral resources and for grade control purposes. Geostatistically optimal proportions between chemical assays and the DBD measurements deduced from the diagrams (Fig. 7) are summarised in Table 6, which can be used as an approximate guide for choosing a sampling grid at early stages of exploration.

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Table 6

Sampling grids proposed for definition mineral resources and for grade control

	Metal grade		Dry bulk rock density
Deposit type	Measured	Indicated	Measured	Indicated	Grade control	Reference
Ni sulphide (high grade)	40×40	80×80	250×250	500×500	40×40	Fig 2a	Fig 7a
Ni sulphide (low grade)	75×75	150×150	500×500	800×800	200×200		Fig 7g
IOCG	35×35	70×70	300×300	550×550	50×50		Fig 7h
Unconformity Uranium	35×35	70×70	175×175	350×350	50×50	Fig 2b	Fig 7b
Cu-porphyry	40×40	80×80	300×300	550×550	40×40	Fig 2c	Fig 7c
Iron ore (BIF derived)	50×50	150×150	200×200	300×300	70×70	Fig 2d	Fig 7d
Iron ore (CID)	50×50	150×150	150×150	500×500	70×70		Fig 7f
Au orogenic	30×30	60×60	250×250	450×450	60×60	Fig 2e	Fig 7e