Principles of Mining

Valuation of Copper, Gold, Lead, Silver, Tin, and Zinc Lode Mines.

DETERMINATION OF AVERAGE METAL CONTENT; SAMPLING, ASSAY PLANS,
CALCULATIONS OF AVERAGES, PERCENTAGE OF ERRORS IN ESTIMATE FROM
SAMPLING.

The following discussion is limited to _in situ_ deposits of copper,
gold, lead, silver, tin, and zinc. The valuation of alluvial deposits,
iron, coal, and other mines is each a special science to itself and
cannot be adequately discussed in common with the type of deposits
mentioned above.

The value of a metal mine of the order under discussion depends
upon:--

_a_. The profit that may be won from ore exposed;
_b_. The prospective profit to be derived from extension of the
ore beyond exposures;
_c_. The effect of a higher or lower price of metal (except in
gold mines);
_d_. The efficiency of the management during realization.

The first may be termed the positive value, and can be approximately
determined by sampling or test-treatment runs. The second and the
third may be termed the speculative values, and are largely a matter
of judgment based on geological evidence and the industrial outlook.
The fourth is a question of development, equipment, and engineering
method adapted to the prospects of the enterprise, together with
capable executive control of these works.

It should be stated at the outset that it is utterly impossible to
accurately value any mine, owing to the many speculative factors
involved. The best that can be done is to state that the value
lies between certain limits, and that various stages above the
minimum given represent various degrees of risk. Further, it would
be but stating truisms to those engaged in valuing mines to repeat
that, because of the limited life of every mine, valuation of such
investments cannot be based upon the principle of simple interest;
nor that any investment is justified without a consideration of
the management to ensue. Yet the ignorance of these essentials
is so prevalent among the public that they warrant repetition on
every available occasion.

To such an extent is the realization of profits indicated from
the other factors dependent upon the subsequent management of the
enterprise that the author considers a review of underground engineering
and administration from an economic point of view an essential to
any essay upon the subject. While the metallurgical treatment of
ores is an essential factor in mine economics, it is considered that
a detailed discussion of the myriad of processes under hypothetic
conditions would lead too far afield. Therefore the discussion is
largely limited to underground and administrative matters.

The valuation of mines arises not only from their change of ownership,
but from the necessity in sound administration for a knowledge
of some of the fundamentals of valuation, such as ore reserves
and average values, that managerial and financial policy may be
guided aright. Also with the growth of corporate ownership there
is a demand from owners and stockholders for periodic information
as to the intrinsic condition of their properties.

The growth of a body of speculators and investors in mining stocks
and securities who desire professional guidance which cannot be based
upon first-hand data is creating further demand on the engineer.
Opinions in these cases must be formed on casual visits or second-hand
information, and a knowledge of men and things generally. Despite
the feeling of some engineers that the latter employment is not
properly based professionally, it is an expanding phase of engineers'
work, and must be taken seriously. Although it lacks satisfactory
foundation for accurate judgment, yet the engineer can, and should,
give his experience to it when the call comes, out of interest
to the industry as a whole. Not only can he in a measure protect
the lamb, by insistence on no investment without the provision of
properly organized data and sound administration for his client, but
he can do much to direct the industry from gambling into industrial
lines.

An examination of the factors which arise on the valuation of mines
involves a wide range of subjects. For purposes of this discussion
they may be divided into the following heads:--

1. _Determination of Average Metal Contents of the Ore._
2. _Determination of Quantities of Ore._
3. _Prospective Value._
4. _Recoverable Percentage of Gross Value._
5. _Price of Metals._
6. _Cost of Production._
7. _Redemption or Amortization of Capital and Interest._
8. _Valuation of Mines without Ore in Sight._
9. _General Conduct of Examination and Reports._

DETERMINATION OF AVERAGE METAL CONTENTS OF THE ORE.

Three means of determination of the average metal content of standing
ore are in use--Previous Yield, Test-treatment Runs, and Sampling.

PREVIOUS YIELD.--There are certain types of ore where the previous
yield from known space becomes the essential basis of determination
of quantity and metal contents of ore standing and of the future
probabilities. Where metals occur like plums in a pudding, sampling
becomes difficult and unreliable, and where experience has proved
a sort of regularity of recurrence of these plums, dependence must
necessarily be placed on past records, for if their reliability is
to be questioned, resort must be had to extensive test-treatment
runs. The Lake Superior copper mines and the Missouri lead and zinc
mines are of this type of deposit. On the other sorts of deposits
the previous yield is often put forward as of important bearing
on the value of the ore standing, but such yield, unless it can
be _authentically_ connected with blocks of ore remaining, is not
necessarily a criterion of their contents. Except in the cases
mentioned, and as a check on other methods of determination, it
has little place in final conclusions.

TEST PARCELS.--Treatment on a considerable scale of sufficiently
regulated parcels, although theoretically the ideal method, is,
however, not often within the realm of things practical. In examination
on behalf of intending purchasers, the time, expense, or opportunity
to fraud are usually prohibitive, even where the plant and facilities
for such work exist. Even in cases where the engineer in management
of producing mines is desirous of determining the value of standing
ore, with the exception of deposits of the type mentioned above,
it is ordinarily done by actual sampling, because separate mining
and treatment of test lots is generally inconvenient and expensive.
As a result, the determination of the value of standing ore is,
in the great majority of cases, done by sampling and assaying.

SAMPLING.--The whole theory of sampling is based on the distribution
of metals through the ore-body with more or less regularity, so
that if small portions, that is samples, be taken from a sufficient
number of points, their average will represent fairly closely the
unit value of the ore. If the ore is of the extreme type of irregular
metal distribution mentioned under "previous yield," then sampling
has no place.

How frequently samples must be taken, the manner of taking them,
and the quantity that constitutes a fair sample, are matters that
vary with each mine. So much depends upon the proper performance
of this task that it is in fact the most critical feature of mine
examination. Ten samples properly taken are more valuable than
five hundred slovenly ones, like grab samples, for such a number
of bad ones would of a surety lead to wholly wrong conclusions.
Given a good sampling and a proper assay plan, the valuation of a
mine is two-thirds accomplished. It should be an inflexible principle
in examinations for purchase that every sample must be taken under
the personal supervision of the examining engineer or his trusted
assistants. Aside from throwing open the doors to fraud, the average
workman will not carry out the work in a proper manner, unless
under constant supervision, because of his lack of appreciation of
the issues involved. Sampling is hard, uncongenial, manual labor.
It requires a deal of conscientiousness to take enough samples and
to take them thoroughly. The engineer does not exist who, upon
completion of this task, considers that he has got too many, and
most wish that they had taken more.

The accuracy of sampling as a method of determining the value of
standing ore is a factor of the number of samples taken. The average,
for example, of separate samples from each square inch would be
more accurate than those from each alternate square inch. However,
the accumulated knowledge and experience as to the distribution
of metals through ore has determined approximately the manner of
taking such samples, and the least number which will still by the
law of averages secure a degree of accuracy commensurate with the
other factors of estimation.

As metals are distributed through ore-bodies of fissure origin
with most regularity on lines parallel to the strike and dip, an
equal portion of ore from every point along cross-sections at right
angles to the strike will represent fairly well the average values
for a certain distance along the strike either side of these
cross-sections. In massive deposits, sample sections are taken
in all directions. The intervals at which sample sections must
be cut is obviously dependent upon the general character of the
deposit. If the values are well distributed, a longer interval
may be employed than in one subject to marked fluctuations. As
a general rule, five feet is the distance most accepted. This,
in cases of regular distribution of values, may be stretched to
ten feet, or in reverse may be diminished to two or three feet.

The width of ore which may be included for one sample is dependent
not only upon the width of the deposit, but also upon its character.
Where the ore is wider than the necessary stoping width, the sample
should be regulated so as to show the possible locus of values.
The metal contents may be, and often are, particularly in deposits
of the impregnation or replacement type, greater along some streak
in the ore-body, and this difference may be such as to make it
desirable to stope only a portion of the total thickness. For deposits
narrower than the necessary stoping width the full breadth of ore
should be included in one sample, because usually the whole of
the deposit will require to be broken.

In order that a payable section may not possibly be diluted with
material unnecessary to mine, if the deposit is over four feet and
under eight feet, the distance across the vein or lode is usually
divided into two samples. If still wider, each is confined to a
span of about four feet, not only for the reason given above, but
because the more numerous the samples, the greater the accuracy.
Thus, in a deposit twenty feet wide it may be taken as a good guide
that a test section across the ore-body should be divided into
five parts.

As to the physical details of sample taking, every engineer has
his own methods and safeguards against fraud and error. In a large
organization of which the writer had for some years the direction,
and where sampling of mines was constantly in progress on an extensive
scale, not only in contemplation of purchase, but where it was also
systematically conducted in operating mines for working data, he
adopted the above general lines and required the following details.

A fresh face of ore is first broken and then a trench cut about
five inches wide and two inches deep. This trench is cut with a
hammer and moil, or, where compressed air is available and the
rock hard, a small air-drill of the hammer type is used. The spoil
from the trench forms the sample, and it is broken down upon a
large canvas cloth. Afterwards it is crushed so that all pieces
will pass a half-inch screen, mixed and quartered, thus reducing the
weight to half. Whether it is again crushed and quartered depends
upon what the conditions are as to assaying. If convenient to assay
office, as on a going mine, the whole of the crushing and quartering
work can be done at that office, where there are usually suitable
mechanical appliances. If the samples must be taken a long distance,
the bulk for transport can be reduced by finer breaking and repeated
quartering, until there remain only a few ounces.

PRECAUTIONS AGAINST FRAUD.--Much has been written about the precautions
to be taken against fraud in cases of valuations for purchase. The
best safeguards are an alert eye and a strong right arm. However,
certain small details help. A large leather bag, arranged to lock
after the order of a mail sack, into which samples can be put
underground and which is never unfastened except by responsible
men, not only aids security but relieves the mind. A few samples
of country rock form a good check, and notes as to the probable
value of the ore, from inspection when sampling, are useful. A
great help in examination is to have the assays or analyses done
coincidentally with the sampling. A doubt can then always be settled
by resampling at once, and much knowledge can be gained which may
relieve so exhaustive a program as might be necessary were results
not known until after leaving the mine.

ASSAY OF SAMPLES.--Two assays, or as the case may be, analyses,
are usually made of every sample and their average taken. In the
case of erratic differences a third determination is necessary.

ASSAY PLANS.--An assay plan is a plan of the workings, with the
location, assay value, and width of the sample entered upon it. In
a mine with a narrow vein or ore-body, a longitudinal section is
sufficient base for such entries, but with a greater width than one
sample span it is desirable to make preliminary plans of separate
levels, winzes, etc., and to average the value of the whole payable
widths on such plans before entry upon a longitudinal section. Such
a longitudinal section will, through the indicated distribution
of values, show the shape of the ore-body--a step necessary in
estimating quantities and of the most fundamental importance in
estimating the probabilities of ore extension beyond the range of
the openings. The final assay plan should show the average value
of the several blocks of ore, and it is from these averages that
estimates of quantities must be made up.

CALCULATIONS OF AVERAGES.--The first step in arriving at average
values is to reduce erratic high assays to the general tenor of
other adjacent samples. This point has been disputed at some length,
more often by promoters than by engineers, but the custom is very
generally and rightly adopted. Erratically high samples may indicate
presence of undue metal in the assay attributable to unconscious
salting, for if the value be confined to a few large particles
they may find their way through all the quartering into the assay.
Or the sample may actually indicate rich spots of ore; but in any
event experience teaches that no dependence can be put upon regular
recurrence of such abnormally rich spots. As will be discussed
under percentage of error in sampling, samples usually indicate
higher than the true value, even where erratic assays have been
eliminated. There are cases of profitable mines where the values
were all in spots, and an assay plan would show 80% of the assays
_nil_, yet these pockets were so rich as to give value to the whole.
Pocket mines, as stated before, are beyond valuation by sampling,
and aside from the previous yield recourse must be had to actual
treatment runs on every block of ore separately.

After reduction of erratic assays, a preliminary study of the runs of
value or shapes of the ore-bodies is necessary before any calculation
of averages. A preliminary delineation of the boundaries of the
payable areas on the assay plan will indicate the sections of the
mine which are unpayable, and from which therefore samples can
be rightly excluded in arriving at an average of the payable ore
(Fig. 1). In a general way, only the ore which must be mined need
be included in averaging.

The calculation of the average assay value of standing ore from
samples is one which seems to require some statement of elementals.
Although it may seem primitive, it can do no harm to recall that if
a dump of two tons of ore assaying twenty ounces per ton be added
to a dump of five tons averaging one ounce per ton, the result has
not an average assay of twenty-one ounces divided by the number of
dumps. Likewise one sample over a width of two feet, assaying twenty
ounces per ton, if averaged with another sample over a width of five
feet, assaying one ounce, is no more twenty-one ounces divided by
two samples than in the case of the two dumps. If common sense were
not sufficient demonstration of this, it can be shown algebraically.
Were samples equidistant from each other, and were they of equal
width, the average value would be the simple arithmetical mean of
the assays. But this is seldom the case. The number of instances,
not only in practice but also in technical literature, where the
fundamental distinction between an arithmetical and a geometrical
mean is lost sight of is amazing.

To arrive at the average value of samples, it is necessary, in
effect, to reduce them to the actual quantity of the metal and volume
of ore represented by each. The method of calculation therefore
is one which gives every sample an importance depending upon the
metal content of the volume of ore it represents.

The volume of ore appertaining to any given sample can be considered
as a prismoid, the dimensions of which may be stated as follows:--

_W_ = Width in feet of ore sampled.
_L_ = Length in feet of ore represented by the sample.
_D_ = Depth into the block to which values are assumed to penetrate.

We may also let:--

_C_ = The number of cubic feet per ton of ore.
_V_ = Assay value of the sample.

Then _WLD_/C_ = tonnage of the prismoid.*
_V WLD_/C_ = total metal contents.

[Footnote *: Strictly, the prismoidal formula should be used, but
it complicates the study unduly, and for practical purposes the
above may be taken as the volume.]

The average value of a number of samples is the total metal contents
of their respective prismoids, divided by the total tonnage of
these prismoids. If we let _W_, _W_1, _V_, _V_1 etc., represent
different samples, we have:--

_V(_WLD_/_C_) + _V_1 (_W_1 _L_1 _D_1/_C_) + _V_2 (_W_2 _L_2 _D_2/_C_)
---------------------------------------------------------------------
_WLD_/_C_ + _W_1 _L_1 _D_1/_C_ + _W_2 _L_2 _D_2/_C_
= average value.

This may be reduced to:--

(_VWLD_) + (_V_1 _W_1 _L_1 _D_1) + (_V_2 _W_2 _L_2 _D_2,), etc.
---------------------------------------------------------------
(_WLD_) + (_W_1 _L_1 _D_1) + (_W_2 _L_2 _D_2), etc.

As a matter of fact, samples actually represent the value of
the outer shell of the block of ore only, and the continuity of
the same values through the block is a geological assumption.
From the outer shell, all the values can be taken to penetrate
equal distances into the block, and therefore _D_, _D_1, _D_2
may be considered as equal and the equation becomes:--

(_VWL_) + (_V_1 _W_1 _L_1) + (_V_2 _W_2 _L_2), etc.
---------------------------------------------------
(_WL_) + (_W_1 _L_1) + (_W_2 _L_2), etc.

The length of the prismoid base _L_ for any given sample will be
a distance equal to one-half the sum of the distances to the two
adjacent samples. As a matter of practice, samples are usually taken
at regular intervals, and the lengths _L_, _L_1, _L_2 becoming thus
equal can in such case be eliminated, and the equation becomes:--

(_VW_) + (_V_1 _W_1) + (_V_2 _W_2), etc.
----------------------------------------
_W_ + _W_1 + _W_2 , etc.

The name "assay foot" or "foot value" has been given to the relation
_VW_, that is, the assay value multiplied by the width sampled.[*]
It is by this method that all samples must be averaged. The same
relation obviously can be evolved by using an inch instead of a
foot, and in narrow veins the assay inch is generally used.

[Footnote *: An error will be found in this method unless the two
end samples be halved, but in a long run of samples this may be
disregarded.]

Where the payable cross-section is divided into more than one sample,
the different samples in the section must be averaged by the above
formula, before being combined with the adjacent section. Where
the width sampled is narrower than the necessary stoping width,
and where the waste cannot be broken separately, the sample value
must be diluted to a stoping width. To dilute narrow samples to
a stoping width, a blank value over the extra width which it is
necessary to include must be averaged with the sample from the
ore on the above formula. Cases arise where, although a certain
width of waste must be broken with the ore, it subsequently can
be partially sorted out. Practically nothing but experience on
the deposit itself will determine how far this will restore the
value of the ore to the average of the payable seam. In any event,
no sorting can eliminate all such waste; and it is necessary to
calculate the value on the breaking width, and then deduct from
the gross tonnage to be broken a percentage from sorting. There
is always an allowance to be made in sorting for a loss of good
ore with the discards.

PERCENTAGE OF ERROR IN ESTIMATES FROM SAMPLING.--It must be remembered
that the whole theory of estimation by sampling is founded upon
certain assumptions as to evenness of continuity and transition
in value and volume. It is but a basis for an estimate, and an
estimate is not a statement of fact. It cannot therefore be too
forcibly repeated that an estimate is inherently but an approximation,
take what care one may in its founding. While it is possible to
refine mathematical calculation of averages to almost any nicety,
beyond certain essentials it adds nothing to accuracy and is often
misleading.

It is desirable to consider where errors are most likely to creep
in, assuming that all fundamental data are both accurately taken
and considered. Sampling of ore _in situ_ in general has a tendency
to give higher average value than the actual reduction of the ore
will show. On three West Australian gold mines, in records covering
a period of over two years, where sampling was most exhaustive as
a daily r�gime of the mines, the values indicated by sampling were
12% higher than the mill yield plus the contents of the residues.
On the Witwatersrand gold mines, the actual extractable value is
generally considered to be about 78 to 80% of the average shown
by sampling, while the mill extractions are on average about 90
to 92% of the head value coming to the mill. In other words, there
is a constant discrepancy of about 10 to 12% between the estimated
value as indicated by mine samples, and the actual value as shown
by yield plus the residues. At Broken Hill, on three lead mines,
the yield is about 12% less than sampling would indicate. This
constancy of error in one direction has not been so generally
acknowledged as would be desirable, and it must be allowed for
in calculating final results. The causes of the exaggeration seem
to be:--

_First_, inability to stope a mine to such fine limitations of
width, or exclusion of unpayable patches, as would appear practicable
when sampling, that is by the inclusion when mining of a certain
amount of barren rock. Even in deposits of about normal stoping
width, it is impossible to prevent the breaking of a certain amount
of waste, even if the ore occurrence is regularly confined by walls.

If the mine be of the impregnation type, such as those at Goldfield,
or Kalgoorlie, with values like plums in a pudding, and the stopes
themselves directed more by assays than by any physical differences
in the ore, the discrepancy becomes very much increased. In mines
where the range of values is narrower than the normal stoping width,
some wall rock must be broken. Although it is customary to allow for
this in calculating the average value from samples, the allowance
seldom seems enough. In mines where the ore is broken on to the
top of stopes filled with waste, there is some loss underground
through mixture with the filling.

_Second_, the metal content of ores, especially when in the form of
sulphides, is usually more friable than the matrix, and in actual
breaking of samples an undue proportion of friable material usually
creeps in. This is true more in lead, copper, and zinc, than in
gold ores. On several gold mines, however, tests on accumulated
samples for their sulphide percentage showed a distinctly greater
ratio than the tenor of the ore itself in the mill. As the gold is
usually associated with the sulphides, the samples showed higher
values than the mill.

In general, some considerable factor of safety must be allowed
after arriving at calculated average of samples,--how much it is
difficult to say, but, in any event, not less than 10%.