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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:—

1 The profit that may be won from ore exposed;

2 The prospective profit to be derived from extension of the ore beyond exposures;

3 The effect of a higher or lower price of metal (except in gold mines);

4 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, W₁, V, V₁ etc., represent different samples, we have:—

V(WLD/C) + V1(W1 L1 D1/C) + V2(W2 L2 D2/C)	= average value.
WLD/C + W1L1D1/C + W2L2D2/C

This may be reduced to:—

(VWLD) + (V1 W1 L1 D1) + (V2 W2 L2 D2,), etc.

(WLD) + (W1L1D1) + (W2L2D2), 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, D₁, D₂ may be considered as equal and the equation becomes:—

(VWL) + (V1W1L1) + (V2W2L2), etc.

(WL) + (W1L1) + (W2L2), 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, L₁, L₂ becoming thus equal can in such case be eliminated, and the equation becomes:—

(VW) + (V1W1) + (V2W2), etc.

W + W1 + W2, 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 régime 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%.