| Citation: | Tetsuya Shoji. Grade-Tonnage, Ore Value-Tonnage, and Enrichment Ratio-Tonnage Models for Resource Assessment. Journal of Earth Science, 2001, 12(1): 45-53.
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According to grade-tonnage diagrams of nickel and zinc deposits, their critical grades are 0.4% and 3.4%, respectively, and hence the former resources can be considered optimistic and the latter pessimistic. The grade-tonnage diagram of gold deposits is convex downwards suggesting that the critical grade is 1×10-6 in the low-grade part. The ore value (
Grade-tonnage diagrams are one of the most useful tools in the resource assessment (e. g. Singer, 1993). One grade-tonnage diagram is well known as Lasky's (1950) equation, which demonstrates the linear relation between average grade (x) and logarithms of Cumulative ore tonnage (T) above a given grade x(0≤x < 1) in the form
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(1) |
where a and b are positive constants. In equation (1), cumulative ore tonnage and average grade are given as follows
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(2) |
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(3) |
where t(x)dx and m(x)dx = x·t(x)dx a are the ore tonnage and the metal amount between grades x and x + dx, respectively, and M (x) is cumulative metal tonnage above a given grade x. If equation (1) is assumed, then m(x) is negative in the region of x < b (DeYoung, 1981) (see Appendix 1).
Shoji (1989) showed that logT(x) given by equation (2) also linearly decreases with increasing x. If the relation,
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(4) |
is assumed,
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(5) |
Equation (5) means that m(x) is always positive. Accordingly, equation (4) does not cause the contradiction pointed out by DeYoung (1981).
The independent variable of a grade-tonnage diagram is the grade of an element. For this reason, a polymetallic deposit is treated as a monometallic deposit enriched in only the specified metal. This treatment implies that many polymetallic deposits seem to be low in ore value. In actual cases, however, ores of polymetallic deposits are frequently valuable, because they consist of many kinds of metallic minerals. To avoid this absurdity, Shoji and Kaneda (1998) introduced the ore value-tonnage relation. Since metal prices vary in time, however, ore values are not universal. For solving this problem, Shoji (submitted a) introduced the enrichment ratio. This paper introduces briefly grade-tonnage diagrams of nickel, zinc (Shoji, 1989) and gold deposits (Shoji, 1993), the ore value-tonnage diagram (Shoji and Kaneda, 1998) and the enrichment ratio-tonnage diagram (Shoji, submitted a) of all deposits in the world, and the correlation between the ore values and the enrichment ratios (Shoji, submitted b).
The data used here are from the database of the Mining Information System (MIS) of Sumitomo Mining Co Ltd. The MIS includes 4 748 deposits. About 40 keywords indicate types of 3 236 of these deposits. Only a a single metal is given for 3 142 of the deposits. The other 1 606 deposits are polymetallic, where grades of multiple metals are described. In the following statements, let us call the former monometallic deposits, the later polymetallic deposits, and the sum of them total deposits.
Figure 1 is a grade-tonnage diagram of nickel deposits (Shoji, 1989). The cumulative log tonnage increases linearly with decreasing grade in the range where the grade is higher than 1%. The flattening of the diagram in the low-grade part is due to economics. Even when low-grade deposits are discovered, many are not reported in public forms. For this reason, some grade-tonnage diagrams show a flat part in the low-grade side. If the flat part is excluded, the relation is well approximated by the straight line (units of x are%)
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(6) |
Figure 2 is a grade-tonnage diagram of zinc deposits. The relation is also approximated by the straight line
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(7) |
Equation (5) means that m(x) has a maximum at x = xc(see Appendix 2). If all operating mines have ore grades higher than xc, m(x)increases with decreasing grade. Resources having such a character can be considered optimistic, because more metal is expected by decreasing grade. In contrast, if some mines are working at grades lower than xc, we cannot expect more metal, because m(x) decreases with decreasing grade in the region of x < xc. Consequently, such mineral resources are classified as pessimistic. Let us say xc is a "critical grade" (Shoji, 1989), because it gives the boundary between the optimistic and pessimistic resources. The critical grade of nickel resources is 0.4%. This value is lower than the grade of presently operating mines. Accordingly, nickel resources are optimistic. In contrast, the critical grade of zinc deposits is 3.4%. Since the value is higher than the lowest grade of operating mines, the zinc resources are pessimistic. According to this definition, most base metals such as copper, lead, zinc, tungsten and molybdenum are pessimistic (Shoji, 1993).
Since the grade-tonnage diagram of gold deposits is convex downwards as shown in Fig. 3, the critical grade seems to be not definable. For this reason, Shoji (1993) concludes that gold resources are. sufficiently optimistic. However, this conclusion is not correct. The relation is approximated by combination of three exponential functions as follows (units of x are 10-6).
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(8) |
The line approximating the low-grade part suggests that the critical grade is 1×10-6.
Since the independent variable of the grade-tonnage diagram is grade of an element, a polymetallic deposit is treated as a monometallic one enriched in only the specified metal. This treatment implies that n many polymetallic deposits seem to be low in ore value. In actual cases, however, ores of polymetallic deposits are frequently of high value, because they consist of many kinds of metallic minerals. For instance, many ores of the massive sulfide type consist of chalcopyrite, sphalerite, galena, and other minerals. Even ores of the porphyry copper type include a remarkable amount t of molybdenite in North and South America, and native gold in the southwestern Pacific for example. Therefore, they must be treated by ways which take account of polymetallic deposits. To solve this problem, an ore valuetonnage diagram has been proposed by Shoji and Kaneda (1998).
The independent variable of the ore value-tonnage diagram is an ore value of each deposit. The ore value is given as a sum of products of grades and metal prices as follows
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(9) |
where v is ore value, and xj and pj are grade and metal price of element j, respectively. If equation (4), where x is replaced by v, is assumed, vc gives the critical ore value.
Figure 4 is the ore value-tonnage diagram of all the deposits which are stored in the MIS (Shoji, submitted b). In this figure, three approximation lines are drawn for the parts more than 450 
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(10) |
the correlation coefficient between the actual (cumulative tonnage and the cumulative tonnage estimated by equation (10) is 0.997. The critical ore value is 21
Ore value can evaluate polymetallic deposits more precisely. Since metal prices vary in time, however, the ore values are not universal. For solving this problem, an enrichment ratio has been proposed by Shoji (submitted a).
The enrichment ratio of a monometallic deposit is defined as the ratio of the grade of the object metal in the deposit to the crustal abundance of the metal, and is expressed by the following equation
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where e, x, and A are enrichment ratio of the metal element, grade of the metal in a deposit, and crustal abundance of the metal, respectively. If a deposit consists of two metals, say metal 1 and metal 2, each enrichment ratio is given as follows
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(11) |
If the ratio of the crustal abundance of metal 2 to that of metal 1 is written as α2/1 = A2/A1, equation (11) gives the following equation
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(12) |
Equation (12) means that x2/α2/1 is the grade of metal 2 norma alized by metal 1. Therefore, the grade summarizing metals 1 and 2 is defined as (x1+x2/α2/1) when metal 1 is the norm.
In this case, the enrichment ratio (e1+2) is given by the following equation
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(13) |
Equation (13) means that the addition theorem is satisfied for the enrichment ratio. The enrichment ratio (eΣ) of a deposit consisting of m metals, therefore, is given as follows
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If equation (4), where x is replaced by e, is assumed, ec gives the critical enrichment ratio.
Figure 5 is the enrichment ratio-tonnage diagram of all deposits (Shoji, submitted a). In this figure, three approximation lines are drawn for the parts more than 20×103, between 15×103 and 1 000, and less than 300. If these exponential functions are combined as follows
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(14) |
the correlation coefficient between the actual cumulative tonnage and the cumulative tonnage estimated by equation (14) is 0.998. The critical enrichment ratio is 250 in the low ratio part. Three approximation lines cross at enrichment ratio values of 16 400 and 621. Let us define, accordingly, that the high, middle and low enrichment ratios are more than 16 000, between 16 000 and 600 and less than 600, respectively.
Figure 6 shows the relation between enrichment ratio and ore value based on cut-off grade of each metal (Shoji and Kaneda, 1998). The correlation coefficient between them is quite low (0.28). Figure 7 shows the relation between enrichment ratios and ore values of all deposits (Shoji, submitted a). The correlation l coefficient in this diagram is 0.63 (r=0.76 in linear scale).
Several straight lines of dots are recognized in Fig. 7 (Shoji, submitted b). Each of the lines shows a relation between enrichment ratios and ore values of monometallic deposits of a specified metal. The ore value of deposit i producing only metal j is given by the following equation
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Accordingly, Ajpj gives the intercept of metal j in Fig. 7 (the slope if plotted on a linear scale diagram). Let the intercept be the crustal value of metal j, because the value represents the ore value(
All deposits are classified into three classes of high, middle and low ratios based on their enrichment ratios, and also into three clas lsses of high, middle and low values based on their ore values (Shoji, submitted b). Accordingly, the classifications give nine quality categories of deposits: HH (high ER-high OV), HM (high ER-middle OV), HL (high ER-low OV), MH, MM, ML, LH, LM and LL. Figure 8 shows the relation between the deposits defined by both criterla.
In order to know the relation of deposit types indicated by a keyword (k) and a quality category (q), let us define the frequency proportion (pkq) of a keyword as follows
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where Dkq, DkA, DKq and DKT are the number of deposits having keyword k in category q, the total number of deposits having keyword k, the total number of deposits belonging to category q, and the total number of deposits having any keyword, respectively. If the frequency proportion of keyword k in category q is sufficiently larger than 1 (pkq > > 1), then it is concluded that the keyword characterizes the category. Table 1 lists the keywords whose frequency proportions are larger than 2 in each category. According to Table 1, we can expect the deposits characterized by "unconformity" and "Mississippi Valley" to have high quality in both enrichment ratio and ore value. In contrast, deposits characterized by "porphyry" and "dssemination" are always poor in any of the qualities. Most deposits of the laterite type and many deposits of the orthomagmatic type will be in the middle ore value, but low in enrichment ratio. In contrast, many deposits characterized by "sandstone" or "stockwork" are suggested to be in the middle enrichment ratio, but low in ore value.
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In order to clarify the commodity of each deposit, Shoji and Kaneda (1998) introduced two terms such as "silver-bearing deposit" and "silver deposit". The former term is used for a deposit containing the specified metal (silver here), while the latter term for a deposit in which the value of the specified metal (or metals) exceeds a half the total ore value. The terminology is also applicable to the enrichment ratio (Shoj, submitted a). Accordingly, we have two definitions for "the deposit of the specified metal". If a deposit is monometallic, the deposit is naturally classified as the same commodity by both definitions. In contrast, if a deposit is polymetallic, then the deposit may be classified as different commodities by the definitions. The result is listed in Table 2.
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Table 2 shows the relation between commodities determined by two definitions. If a deposit is classified as a lead deposit on the basis of the enrichment ratio and as a silver deposit on the basis of the ore value, let us say that the deposit belongs to commodity category PbAg (or simply category PbAg). According to the definition, the number of the deposits belonging to category PbAg is4 as shown in cell (Pb, Ag) of Table 2. In Table 2, each diagonal cell shows the case where deposit is classified as the same commodity by the enrichment ratio and the ore value Let us call this the coincident case. On the other hand, every cell other than diagonal cells shows the case of a deposit that is defined as different commodities by the definitions.
The proportion of coincident classification is 40% (= 621/1 606) as a whole. Let us show the coincidence degree in each commodity between the definitions based on the enrichment ratio and the ore value by the following equation
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where superscripts "ER" and "OV" mean enrichment ratio and ore value, respectively, and Djj and DAj are the number of deposits in the coincident case of commodity j, and the total number of the deposits of commodity j, respectively. Large coincidence degrees are seen for nickel (0.76) and gold (0.72). In contrast, the degrees of coincidence of molybdenum (0.21) and lead (0.31) are smal
The other remarkable point indicated in Table 2, where the com-modities are arranged in ascending order of the crustal values, is that figures are written in the cells on the diagonal line or above. This means that any commodity shown by the cells have no symmetric case. For example, category PbAu has 2 deposits, but category AuPb has none. This means that the crustal value of the commodity defined by enrichment ratio is always equal to or lower than that of the commodity defined by ore value. This rule is not satisfied for commodities consisting of multiple metals. For example, category (Au+ Ag)(Pb+ Zn) has 51 deposits, whereas category (Pb + Zn) (Au+ Ag) has 8 deposits.
In order to view the relation between monometallic deposits and commodities, let us define the monometallic proportion of deposits belonging to commodity j (Pjmono as follows
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where Djmono and Bj are the numbers of deposits of commodity j and those of deposits containing commodity j, respectively (the symbols are used for the column titles in Table 2). The percentage is 66% as a whole (Table 2). Chromium and uranium have the largest percentage (ca. 95%). Manganese and tungsten follow them and show 90% and 85%. In contrast, lead, silver and zinc are very low (< 10%). Copper and molybdenum are also low (ca. 40%).
In the case of polymetallic deposits, the number of deposits belonging to each commodity depends on the definitions as shown in Table 2, where row DKjER and column DKjOV represent the total numbers of the deposits of commodity j defined by the enrichment ratio and the ore value, respectively. The difference between the numbers of deposits based on two definitions is small for chromium (6 by ER and 1 by OV), manganese (2 and 3), tungsten (16 and 10) and uranium (4 and 3). All are large in monometallic proportion. In contrast, copper (168 and 454), molybdenum (87 and 4), lead (403 and 39), silver (202 and 65) and zinc (116 and 432) show large differences, and are small in monometallic proportion. Among the latter commodities, if lead and zinc deposits are grouped into a single commodity such as Pb + Zn, the numbers of the deposits are equal between the two definitions (592).
Finally let us compare monometallic deposits and polymetallic one which belong to the same commodity. Table 3 shows log2rkjpoly/mono say rkjpoly/mono frequency ratio of keyword k between polymetallic and monometallic deposits) defined as follows
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where Dkj is the number of the deposits having keyword k and belonging to commodity j, and DKT is the total number of deposits (1 211 for polymetallic deposits, and 2 025 for monometallic ones). The bottom row shows log2 Rjpoly/mono = log2 {(DKjpoly/DKTpoly)/(DKjmono/DKTmono)} (say Rjpoly/mono deposit ratio of commodity j), where DKj is the total number of the deposits having any of keywords in commodity j. If the frequency ratio and also deposit ratio : are larger than 1 (= 20: positive values in Table 3), it means that the deposits belonging to the category defined by keyword k and/or commodity j are relatively dominant in the polymetallic type compared with the monometallic type.
If a deposit ratio of a commodity is smaller than 1, most frequency ratios of the commodity are mall(< 2-1). On the other hand, if a deposit ratio is larg ger that 1, frequency ratios are always large (> 21). Gold, chromium, copper, manganese, molybdenum, tungsten and uranium are examples of the former case, and lead and silver are examples of the latter case. The frequency ratio of negative infinity, which means no example in polymetallic deposits, is found in the combinations of "placer"-Au, " orthomagmatic"-Gr, "placer"-Cr, " chemical"-Cu, "stockwork"-Mo, "vein"-Mo, "garmierte"-Ni, "pegmatite"-U, "sandstone"-U, "unconformity"-U and "vein"-U, where the number of monometallic deposits in each combination is more than 10. The deposits of these combinations scarcely accompany accessory metals. In contrast, the frequency ratio of positive infinity is seen in the combination of "kuroko"-Zn. The frequency ratios of large positive values are found in the combinations of "epithermal" Ag (24), "massive sulfide"-Zn (25) and"volcanogenic"-Zn (24). The deposits characterized by such kinds of keywords are polymetallic in many cases, and hence probably accompany byproduct metals.
Nickel resources show a good linear relation between average grade and logarithms of cumulative ore tonnage above a given grade except deposits whose grades are less than 1%. The relation suggests that the critical grade is 0.4%, and hence the nickel resources are classified as optimistic. The grade-tonnage relation of zinc resources is also quite linear. The critical grade (3.4%) indicates that the resources are pessimistic. The grade-tonnage diagram of gold deposits is convex downwards. The critical grade obtained from the approximation line in the lowgrade part is 1×10-6.
The ore value (OV)-tonnage diagram of all deposits in the world consists of three parts: high-, middle- and low-value classes. The boundaries among the three classes are 400 and 40
If the commodity of a deposit belonging to the polymetallic type is defined by both of the enichment ratio and the ore value, the proportion of the coincident classification is 40% as a whole. The relatively high coincident classification is realized for nickel and gold. The classifications for copper, silver and zinc are moderately coineident, but those for molybdenum and lead are largely different. If the commodity of Pb+ Zn is defined, the classification is highly coincident. The complimentary character of lead and zinc is also seen in the relation between the classifications and the keywords characterizing the commodities.
The comparison between monometallic and polymetallic deposits suggests that accessory metals are not commonly expected for the deposits of orthomagmatic chromium, chemically precipitated copper and sandstone-type uranium. In contrast, deposits of kuroko-type zinc, epithermal silver, massive sulfide-type zinc and volcanogenic zinc accompany by-product metals with high probability.
If M(x) and T(x) are assumed to be negligible small in the range of x > 1, equations (1) and (3) gives
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The differentiation of the above equation gives the following equatlon
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(A1) |
Equation (A1) means that m(x) is negative if x < b, because t(x) is always positive.
The differentiation of equation (5) gives the following equation
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(A2) |
Equation (A2) means that dm(x)/dx > 0 in the region of x < xc and dm(x)/dx < 0 in the 1 region of x > xc, and that therefore m(x) has a maximum at x=xc.
ACKNOWI EDGMENTS: I would lke to thank Sumitomo Metal Mining Co. Ltd., which kindly allowed using the data of their Mining Information System (MIS). We are grateful to Dr. D. A. Singer of U. S. Geological Survey who gave many important comments. The work is partially supported by the Grand-in-Aid for Scientific Research (No.10041136) from Ministry of Education of Japan.| DeYoung J H, 1981. The Lasky Cumulative Tonnage-Grade Relation-A Reexamination. Economic Geology, 76: 1067-1080 doi: 10.2113/gsecongeo.76.5.1067 |
| Lasky S G, 1950. How Tonnage and Grade Relations Help Predict Ore Reserves. Engineering Mining Jour, 151(4): 81-85 |
| Shoji T, 1989. Resources and the Environment: Which Does Limit Economic Growth? Kyoto, Japan: MMIJ/IMM Joint Sym. 109-114 |
| Shoji T, 1993. Optimistic and Pessimistic Resource Estimates. In: Henein H, Oki T, ed. 1st Intern Conf Processing Materials for Properties. The Minerals Metals & Materials Soc, 463-466 |
| Shoji T, Kaneda H, 1998. Ore Value-Tonnage Diagrams for Resource Assessment. Nonrenewable Resources, 7(1): 25-36 doi: 10.1007/BF02782506 |
| Shoji T, submitted a. Enrichment Ratio-Tonnage Diagrams for Resource Assessment. Natural Resources Research |
| Shoji T, submitted b. Correlation between Enrichment Ratio and Ore Value Defined for Assessing Metallic Mineral Resources. Natural Resources Research |
| Singer D, 1993. Basic Concept in Three Part Quantitative Assessments of Undiscovered Mineral Resource. Nonrenewable Resources, 2(2): 69-81 doi: 10.1007/BF02272804 |
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Tetsuya Shoji. Grade-Tonnage, Ore Value-Tonnage, and Enrichment Ratio-Tonnage Models for Resource Assessment. Journal of Earth Science, 2001, 12(1): 45-53.
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