Practical Introduction to Sampling Theory
The subject of sampling is almost entirely omitted from the syllabus taught to geologists and mining engineers in Australia and we suspect elsewhere. This is unfortunate, given the fact that most of the important economic and technical decisions made by geologists and mining engineers in an operating mine are based on data collected by a sampling process of some type!
An understanding of the importance of sampling practice and sampling theory is critical in the design and implementation of correct grade control.
During the 1950’s and 1960’s Dr. Pierre Gy formulated a generalised Theory for the Sampling of Particulate Materials. This theory developed in parallel with Prof. George Matheron’s development of the Theory of Regionalised Variables (which is the basis of modern geostatistics), and not without some cross-fertilisation. Gy’s Sampling Theory is the only worked-out theory for particulate sampling, and it is general enough to be applied to most of the sampling problems seen in a grade control context.
However, the application of this theory has been limited in the mining industry, largely because of the technicality and mathematical prose of Gy’s book (Gy, 1982). The publications of Dominique François-Bongarçon and Francis Pitard (see
references) have provided more accessible presentations of Gy’s theory. The paper of Dominique François-Bongarçon and Pierre Gy (2001) is a particularly good, and up-to-date, starting point.
A second problem with the acceptance of Gy’s theory has been that, to quote Assibey-Bonsu (1996):
“…in spite of its renowned theoretical validity, Gy’s theory has been found to have some limitations in its implementation, which are
mainly due to the misapplication of the model”
This course cannot substitute for a sampling course12, but sampling is such a critical
activity (and source of serious error) that we present a digested summary here. Components of the Total Sampling Error
Components of the Total Sampling Error
In general, the sample grade the lab returns to us is not, in fact, the true grade of the material delivered to the lab. There is always a sampling error, and this error can be viewed as being made up of several components:
1. Fundamental Sampling Error (FSE): due to the irregular
distribution of the economic mineral in the lot to be sampled. The FSE is the smallest achievable residual average error, i.e. the component of total sampling error that cannever be totally eliminated.
12 The authors regard attendance at a specialised sampling course as a fundamental step for mine
geologists dealing with exploration, grade controlor resource/reserve issues. Sampling Theory
Sampling Theory
One might say that particulate sampling theory is about the ‘internal architecture’ of the nugget effect.
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2. Segregation and Grouping Error (SGE): due to lack of
homogeneity of the lot and grouping of the fragments by increments in the sample. To quote Dominique François-Bongarçon“…no matter how counter intuitive (the) idea may appear, it is close to impossible to successfully homogenise a lot of broken ore solely by mixing”. The usual case is that attempts to homogenise a lot by mixing result in the opposite effect: we segregate the lot because of a variety of physical processes, chief
among them gravimetric separation of grains and grain-sorting by size (granulometry). With very fine materials, static electric effects may play a role, with some clayey materials the relative adhesion (“stickiness”) of particles is important. Note that the SGE is difficult to quantify and its magnitude may exceed that of the FSE.
3. Analytical Error: The variance of differences between duplicate analyses is equal to twice the variance of the analytical error plus the FSE for sampling an additional aliquot from the pulp.
Note also that operator error—that is, using the correct sampling device improperly, or failing to follow a prescribed sampling procedure—is a serious issue in sampling, because correctly designed sampling devices that are not used in the proper manner can result in serious biases or large degrees of imprecision. Gy’s Theory of Fundamental Sampling Error
Gy’s Theory of Fundamental Sampling Error
The following presentation is based in part on that of Winfred Assibey-Bonsu’s excellent 1996 summary paper and on the various papers of Dr. Dominique François-Bongarçon (see references).
Gy’s srcinal model for the fundamental sampling error (FSE) can be written:
σ R s l n M M f g c l d 2
= ⎛
1−
1 3⎝
⎜
⎞
⎠⎟ ⋅ ⋅ ⋅ ⋅ ⋅
where: σ R 2is the relative variance of the FSE.
d n is the nominal top-size of the fragments in the sample. This is the maximum
particle size in the lot to be sampled. In practice,d n is taken as the mesh size that
retains 5% of the lot being sampled (i.e. the 95% pass diameter). The formula for FSE presumes thatd n is expressed in centimetres (cm).
f is the particle shape factor . This is an index which varies between 0 and 1 in most cases. In practice, most values range between 0.2 and 0.5, depending on the shape of the particles. For most ores, f is assigned the value 0.5.
g is the granulometric factor . Granulometry is a term that describes grain-size distribution. The granulometric factor,g, can, like f , assume values between 0
Segregation Segregation
This error is serious in many practical sampling systems and may in fact be the largest type of error.
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and 1. Low values ofg indicate a wide range of grain sizes within the lot, whereas high values ofg denote relative uniformity of grain-size. A value ofg equal to 1.0 indicates that all the particles in the lot are of identical size ; for most practical situations, a value forg of 0.25 is realistic.
c is the mineralogical composition factor , and can be expressed:
(
)
[
]
c a a a m a g=
1−
1− ⋅ ρ + ⋅
ρ
where:a is the decimal proportion of the mineral.
ρ m is the density of the valuable constituent.
ρ g is the density of the gangue.
a refers to the decimal proportion of the ore mineral. Note that densities are specified in grams per cubic centimetre (g/cm3 ).
For example, for zinc, occurring in pure sphalerite (ZnS), an assay of 5% is equivalent to a decimal proportion of 0.075:
a
=
64 32+
×
=
64 5 100 0075.
Note also that:
c t
m
=
ρ wheret is the grade of low-grade ores, for example, gold. It is important to note thatt is expressed as a proportion, i.e. grams/gram;notgrams/tonne. Hence a grade of 10 g/t is expressed as:
t g
g
=
10=
1 000 000, , 000001.
M s is the mass of the sample, measured in grams.
M l is the mass of the lot, measured in grams.
l is the liberation factor , which is a number (once again) varying between 0 and 1.0. Gy assumes that the liberation factor, for unliberated particles, is:
l d
d n
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where d 0 is the liberation size, i.e. the maximum particle size that ensures effectively complete liberation of the mineral. Note thatd 0 is measured in centimetres (cm).
The result we calculate for the FSE isvery sensitive to the value assumed for this liberation factor (we discuss this more, below).
A Simplification A Simplification
In almost all grade control situations, M s is much smaller than M l. Because the
term
1
M l in this case approximates zero, the condition that M s is much smaller
than M l logically leads to a simplification of the formula for FSE:
σ R n s f g c l d M 2 3
= ⋅ ⋅ ⋅ ⋅
More About the Liberation Factor More About the Liberation Factor
Gy proposed an approximation to obtain a value for the liberation factor,l . As stated above, the calculation of FSE is highly sensitive to the value ofl we employ. The empirical estimate proposed by Gy is, as previously noted:
l d
d n
=
0It is important to note that this empirical result, which has been applied indiscriminately to all types of ores, was obtained by experiment from specific ores which were high-grade. By ‘high grade’ we mean that average grades exceed several
percent.
Low grade ores (gold, PGE, uranium, diamonds and some copper and nickel mineralisation) are typical in large-scale, modern mining operations. The general use of Gy’s approximation forl can, especially in the case of low grade ores, produce results that are meaningless and this has led to many practitioners abandoning Gy’s model (François-Bongarçon, 1996). Winfred Assibey-Bonsu (1996, p290) gives the following example:
“Take, for instance, the use of Gy’s model (based on Gy’s empirical liberation factor) in the calculation of a minimum sample size for a typical South African gold mine in production. Assume a top particle size of 13cm. With a gold grade of 5 g/t and a gold density of 19 g/cm 3 , the mineralogical factor, c, is:
c density grade
=
=
×
+ 38 10 6 . g / cm 3 Liberation factor Liberation factorThe liberation factor is critical is correct application of Gy’s formula.
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With grade expressed as a proportion, as previously explained. For a gold grain top size of 75 μ m (i.e. 7.5 x 103
cm), Gy’s empirical liberation factor is:
l d d cm cm n
=
0=
×
=
×
− 3 2 7 5 10 13 2 4 10 . .At a relative precision of 10%, i.e. a variance of 0.01 g/t
2
, the use of the above computed parameters gives a minimum sample mass of 2,507 tonnes. For a typical production rate of between 1,000t and 10,000t per shift, this minimum sample mass is practically unacceptable”.
Assibey-Bonsu is not the first to report this kind of result, and those interested are referred to the papers of François-Bongarçon for further details. François- Bongarçon and Gy (2001) give an equally ludicrous result when a particle size for gold is calculated to be smaller than an atom of gold!
At this point, many geologists or mining engineers would abandon the theory, but perhaps we should carefully consider the reasons for such unacceptable results.