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    Modelling mining

    Open pit copper mining in BritishColumbia is modelled to examinehow different price levels affectrates of output and stocks ofreservas and to consider theexistence and distribution ofeconomic rents. The model depictsthe influence of different economiccircumstances on production plans,for example, with respect to scaleof operations and cutoff grade. Toachieve this it represents a balancebetween the high level ofabstraction which hascharacterized the writing of mosteconomists and the degree of detailand complexity required of modelsused by operating mines. Themodel provides a basis for thestudy of the consequences ofalternative forms of taxation, asubject to be pursued insubsequent articles.

    The author is with the Department ofEconomics, University of BritishColumbia, Vancouver, BC V6T lW5,Canada.This article is an edited version ofUniversity of British Columbia,Department of Economics Programme inNatural Resource Economics ResourcesPaper No 31, January 1979. Theprogramme is financed by the SocialSciences and Humanities ResearchCouncil of Canada.

    Open pit copper production inBritish Columbia

    Paul G. Bradley

    The really interesting question is always the particular one; but its onlythe general one that its possible to discuss.

    Lytton StracheyStrachey was concerned with human relationships and addressedquestions at once more subtle and profound than those that willoccupy our attention. However, the practitioner of economicmodelling of resource use is caught in a dilemma akin to his.Frustrated by the barrenness of abstract models of resource use whenit comes to explaining real industry behaviour, one turns towardsthose of the engineers. However, in that body of literature detailoverwhelms the larger issues. Between the two extremes there isperhaps an optimal scale of model, and how it is specified may proveto be, for one with an economists turn of mind, the really interestingquestion.This article describes a model constructed for the purpose ofexamining by means of simulations various economic issues that areencountered in the mining industry. Discussion is restricted to coppermining in British Columbia, Canada. Variations of the model havebeen applied to other branches of the Canadian mining industry, butit was originally developed to describe large-scale open pitexploitation of porphyry orebodies. Concern with attaining anoptimal scale of model has haunted this work, but the ghost is not laidto rest and no prescriptions are offered.We proceed by first mentioning some of the topics that motivateconstruction of a model of open pit mining. More complete discussionof most of these questions and insights provided by the model is, orwill be, provided in other papers.2 We focus instead on the nature ofthe model and the economic features which it depicts of copperproduction in British Columbia. We then explain the structure of themodel, while in the following section we describe some of itsparticular mechanisms. In the concluding portion of this article wepresent simulation results. Operating conditions, reserves, and mineoutputs corresponding to different anticipated price levels arecompared for a sample of five orebodies.

    44 0301-4207/80/060044-16 $02.00 0 1980 IPC Business Press

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    The research reported here incorporateswork that has been carried out over aperiod of several years. My colleagueJohn Helliwell has been closely involvedat various points. Computer programminghas been ably and innovatively done byFrank Flynn. Robin Gregory providedresearch assistance at an early stage ofthe work; more recently John Livernoishas ably contributed to both research onthe industry and computer programming.A number of persons closely associatedwith the mining industry have generouslyoffered encouragement and advice, mostnotably A.J. Sinclair, J.B. Evans, and H.K.Taylor. They have amply warned me ofthe pitfalls of modelling miningoperations, and I must take soleresponsibility for the assumptions andconclusions presented here. Financialsupport for this work was received alsofrom the BC Institute for Economic PolicyAnalysis.

    The quotation is taken from the essay byL. Strachey. The really interestingquestion, in Paul Levy, ed. The ReallyInteresting Question, Coward, McCannand Geoghegan, New York, 1973.*The consequences of alternative taxregimes were examined by J.F. Helliwell,Effects of taxes and royalties on coppermining investment in British Columbia,Resources Policy, Vol 4, No 1, March1978, pp 35-44. This employed aprecursor of the model described here. Arevised version of that paper is beingprepared by Bradley, Livernois andHelliwell.This can be illustrated by perusal of theessays in M. Crommelin and A.R.Thompson, eds, Mineral Leasing as aninstrument of Public Policy. University ofBritish Columbia Press, Vancouver, 1977;and A. Scott, ed, Natural ResourceRevenues, A Test of Federalism,University of British Columbia Press,Vancouver, 1976.The definition is that presented in JoanRobinson, The Economics of imperfectCompetition, Macmillan, London, 1954.Crommelin and Thompson, eds, up cit.Fief 3. p 277.B Helliwell, up cit. Ref 2.

    Economic issues in miningModelling m ining

    The question of whether economic rents accrue to the owners oforebodies - and, if so, the magnitude of these rents - has occupiedeconomists since Adam Smith, and arises persistently in discussionregarding public policy towards the mining industry.3 There isagreement on the concept of rent as a surplus earned by a particularfactor of production over and above the minimum earning necessaryto induce it to do its work.4 Where the factor is a natural resource,natures failure to exact payment when the factor is induced to do itswork points to the existence of a surplus to be claimed by whoeverhas gained title to that resource. However, resource allocationproblems must be solved within the context of a given region andtimespan. Thus Crommelin5 in summarizing a volume devoted tomineral leasing as an instrument of public policy, remarks that theessential problem associated with rent collection as a policy objectivein mineral leasing concerns the identification of rent. That is, if policyproblems such as leasing and taxation are to be analysed in thistraditional way, the easy-to-discuss concept of rent is only the startingpoint: circumstances and objectives must be stated and a meansfound for estimating the appropriate magnitudes. In this way theparticular can be rendered capable of discussion.Another avenue of economic inquiry relating to mining has to dowith price determination. Quality differences among orebodiesprovide prima facie evidence for upward sloping mineral supplycurves, but whether or how fast price in a particular market will risewhen demand expands depends on how steeply the correspondingsupply curve slopes upwards. Where a small industry sells in a worldmarket - British Columbias position in copper - the supply curveprovides information about how the level and composition of industryactivity will vary when world price changes. In either case, estimatesof price elasticity of supply are needed; the general proposition aloneis not very useful.An important aspect of the supply-response question is the effectof taxation. Economists have developed some familiar and usefulgeneral propositions, eg the distorting effect of royalty taxation.However, in actual experience a variety of types of taxation areutilized to achieve a blend of objectives. Comparison of tax regimesaccording to an efficiency criterion becomes a question of relatives.Such comparisons, derived using an earlier version of the modeldescribed in this paper, are reported by Helliwell.6The question of mineral scarcity has received considerableattention in the wake of the radical changes experienced in fuelresource markets. Availability is gauged, usually misguidedly, bymeasures of mineral reserves. There is agreement that, in concept,proved reserves, the most widely used measure, refer to stocks whichare known with essential certainty to exist and for which productioncosts are covered at the existing price. Industry statistics for crude oiland natural gas have developed to the point of applying these criteriaquite rigorously. However, the non-homogeneity of orebodies and theeconomic significance of their shapes, not just their sizes, make itdifficult to estimate the amounts which are economic to produce atdifferent price levels. The agreement about the defining characteristicsof mineral reserves has not yet been followed by consistentapplication in practice, as is apparent, in the context of this article,

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    C. Carlisle, The economics of a fundresource with particular reference tomining, The American Economic Review,Vol XLIV, 1954, pp 595-616.* H.K. Taylor, General background theoryof cutoff grades, Transactions of theInstitute of Mining and Metallurgy, Vol81,1972, pp Al 60-A179. H. Hotelling, The economics ofexhaustible resources, Journal of PoliticalEconomy.Vol39, 1931, pp 137-175.lo W.D. Schulze, The optimal use of non-renewable resources: the theory ofextraction, Journal of EnvironmentalEconomics and Management. Vol 1.1974, pp 53-73. T. Puu, On the profitability ofexhausting natural resources, Journal ofEvironmental Economics andManagement, Vol4, 1977. pp 185-l 99.

    when one attempts to add up the reported reserves of copper inBritish Columbia.The matter of mineral reserves leads to the question of cutoff grade.Carlisle insisted that, when looking at mining, economists shouldconsider how total volume of production changes with price, as wellas the traditional question of how rate of output changes. He relatedvolume, or level of recovery, to variation in both ore grade andworkability, or ease of recovery. If nature had conveniently arrangedthe composition of orebodies so that a miner could begin at the richend and move through successively poorer grades of ore, analysis ofthe optimal cutoff grade would be simple, and the implications of achange in market price or in tax policy for mine development and orereserves would be apparent. In reality, there is no convenient positivecorrelation between ore grade and accessibility; selecting an optimalmining sequence involves balancing both factors. Furthermore, inpractice different cutoff trades are defined for different purposeswithin one operation, so that cutoff grade analysis becomes complexas can be seen in Taylors review of the subject.*For all these topics, the significant questions are posed with the aidof concepts of economic analysis. To achieve an understanding of themining industry - which for the policy maker means the ability topredict the consequences of alternative actions - research guided bythese concepts must be directed to the particulars. Resolution of thequestions will take the form of estimation of actual magnitudes andcomparison of specific situations. We return to the really interestingquestion -the design of a suitable model.Recently the literature of mathematical economics has contained aspate of models depicting the extraction of exhaustible, or non-renewable, natural resources. These continue the venerable traditionof Hotelling, and are pitched at a level of abstraction that soarsbeyond mining industry questions of the sort just mentioned.Schulze,l for example, begins by assuming an industry comprisingfirms with identical U-shaped cost functions and output levels whichexploit a homogeneous resource of known total quantity. He isconcerned with the optimal rate of investment when the conditions ofperfect competition are posited. He also deals with the same problemwhere the resource is assumed to be extracted in strict order ofdiminishing quality. Puu examines a single mining enterprise, and isagain concerned with the optimal rate of investment. He assumes thatthe firm can vary its amount of capital continuously by offsettingexponential depreciation with new investment. Again the resource isassumed to be extracted in strict order of diminishing quality.The optimizing models which find application in the miningindustry lie at the opposite pole of abstraction. In large-scale open pitoperations information about the quality of ore is obtained by drillingon a grid pattern so that blocks of material - 40 ft on a side, forexample - are characterized by grade. These are identified by threecoordinates of spatial location within the deposit. A profit-maximizingprogramme then generates an optimal order for mining these blocks,taking account of the desirability of mining the better grades first, butalso recognizing the added cost of early recovery of material whichmay be more accessible later. For our purposes models of this typehave two disadvantages. First, the level of detail means that theamount of information required is costly. Second, while optimalsequence is important, it is only one of the variables controlled by the

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    I2 Construction and use of block modelsis described in T.B. Johnson and D.G.Mickle, Optimal design of an open pit -an application in uranium, in CanadianInstitute of Mining and Metallurgy,Decision-Making in the Mineral Industry.Special Vol 12, 1971, pp 331-337; andby R.E. Davis and C.E. Williams,Optimization procedures for open pitmining scheduling, in J.R. Sturgul, ed,Eleventh Symposium on ComputerApplications in the Minerals Industry,University of Arizona, Tucson, 1973, ppCl-ClB.I3 B.W. Mackenzie and M.L. Bilodeau,Assessing the direct effects of miningtaxation: the case of base metalinvestment in Canada, paper presentedat Workshop on Rate-of-Return Taxationof Minerals, Queens University,December 1977.

    Modelling miningoperator, and we wish to account jointly for all the fundamentaldecision variables. Accordingly, we turn back from this kind of modelby accepting some simplifying assumptions.

    A recent study by Mackenzie and Bilodeau13 warrants mentionbecause it addresses a number of the economic issues in mining thatwere listed above. These authors evaluated mining operations inCanada, considering a sample of 124 deposits discovered in theperiod 1951-54. Mine values at each deposit were computed usingactual observed conditions, such as mine and mill capacity, capitaland operating costs, and recoverable ore reserves. This studytherefore does not contemplate how different values might have beenachieved for any deposit had those making the development andproduction plans entertained a different set of expectations abouteconomic conditions. In that event they would probably have optedfor different capacities, specifications for ore reserves etc. Since weare primarily concerned with examining responses to alternative priceexpectations and tax regimes we require a model that takes account ofalternative strategies for exploiting a deposit.Structure of the model of open pit miningWe describe the nature of the model under several headings: (1)general postulates, (2) circumstances of the industry, and (3)development strategy. This scheme is rounded out in the concludingsection of the article where (4) certain relationships or comparisons ofinterest are specified, and (5) results generated by the model areexamined.General postulatesAs a standard for comparison we are interested in the social value ofporphyry orebodies (denoted in the model VRES), defined as thepresent value of revenues attributable to the mineral less the presentvalue of all costs which must be incurred in obtaining these revenues.Cost is used in the economists sense to refer to the cost of real inputs,exclusive of any transfers from the producer to other claimantsagainst net revenue. Under a given set of physical and economiccircumstances, there exists a social value of a resource correspondingto each alternative plan for its production. The model is designed tofind the highest value - the one which defines optimal production, thebest society can do with what nature has given it.In Canada the actual production plan for the orebody is usuallydevised by a private operator. More precisely, in British Columbia,where mineral rights are vested in the Crown, the operator is thecapitalist who has leased the property. His goal is not to maximizesocial return, but rather to maximize the return to private capital, thatis, the return he receives net of all payments made to any level ofgovernment -taxes, rentals, or whatever. Accordingly, the model alsois designed to find the production plan which maximizes the presentvalue of this private return (designated KRPP$ or KRPPNC%). Thisprovides the basis for predicting what the actual long-term productionplan for the mine will be in particular circumstances.Circumstances of the indust ryVirtually all copper produced in British Columbia is taken fromporphyry orebodies using open pit mining. These orebodies are large,

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    . S.G. Lasky, How tonnage and graderelations help predict ore reserves,Engineering and Mining Journal, No 15 1,April 1950, pp 81-85; G. Matheron.Etude: remarque sur la loi de Lasky. LaChronique des Mines dOutre-Mer et de laRecherche Mini&e, 27 An&e, No 282,December 1959, pp 463-465.$D.A. Singer, D.P. Cox and L.J. Drew,Grade and tonnage relationships amongcopper deposits, Geological SurveyProfessional Paper 907A, USGPO,Washington, DC, 1975.@ Paul G. Bradley, Appendix A of theoriginal version of this paper, University ofBritish Columbia Resources Paper No 31,Vancouver, 1979.

    having a horizontal expanse of as much as several thousand feetbefore mineralization tails off to background level. The ore is low-grade; in fact, British Columbia mines generally yield the lowest-grade copper ore in production anywhere. Mining is commerciallysuccessful because the deposits can be worked by open pitting, andthey are large enough to permit modern removal and recoverytechniques that exploit economies of scale.One of the striking features of porphyry copper deposits is that therichness of mineralization shows certain statistical regularities. Thesehave been examined both with regard to both the distribution ofgrades within a single orebody14 and the distribution of mean gradesacross deposits. l5 For the present, it is the former property that is ofinterest because it affords a basis for generalizing one of the keyphysical parameters in open pit copper production. The propositionthat the grade distribution in porphyry orebodies can be described bythe lognormal probability function commands enough empiricalsupport to justify its use to characterize deposits in British Columbia.Elsewhere,16 some sample data are plotted to illustrate this point.However, grade distribution alone is only part of the story, because

    spatial arrangement must also be considered. Here too there is ageneral pattern. In British Columbia orebodies, the richest materialoccurs in vertical pipes nearest a central axis; ore grade diminishesgoing outwards.Value maximization requires taking the highest-grade ore first,other things being equal, and this is a principle which is wellestablished in the mining industry. However, all ore is not equallyaccessible, nor, as noted above, is there a positive correlation betweenrichness and accessibility. While there is a typical pattern to thearrangement of ore by grade in a porphyry deposit, the productionplan is constrained by the shape which the pit can take. The walls ofthe pit cannot exceed a critical angle, so one can visualize thepossibilities as variations on the shape of an inverted cone. It isevident that the sequence in which ore is removed will reflect thespatial relation betwen the contours defining ore grade within theorebody and the feasible pit shapes. The actual variation in grade ofore produced will thus differ markedly from the sequence whichwould be chosen if ordering were costless.On the strength of this cursory description of open pit coppermining, one can immediately identify some of the economicallysignificant physical parameters. They include the size and shape ofthe orebody and the grades of ore which it contains. Grades will haveto be specified as a frequency distribution, with attention paid to howdifferent ore grades are arranged within the deposit. The mainconstraint on pit design is the maximum pitch the wall can safelyassume; pit angles typically vary from 30-45 measured from thehorizontal. Another important physical parameter is overburdendepth, which indicates the amount of extraneous material that mustbe stripped away before mining of the orebody can proceed.Turning to the technology, open pit mining entails breaking therock, loading it on trucks, and delivering the ore to be processed tothe mill. There it is ground and then flotation is used to separateparticles rich in copper from the others, the tailings. Mines in BritishColumbia produce a concentrate containing 2530% copper which isshipped to Japan for smelting.Some investigation was undertaken to determine cost equations for

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    Modelling miningeach of four categories: operating costs, both for mining andprocessing, and capacity costs, again for both mining and processing.Information from mining engineers indicates scale economies incapacity which are particularly pronounced for the concentratingoperation. The latter were detectable by regression analysis with thefragmentary investment data available. Cost equations for the fourcategories are presented elsewhere. I7 The exponents showing scaleeconomies were chosen after inquiries about industry experience; theremaining coefficients were estimated using available cost data.Thus far, the concern has been long-term decision making, that is,choice of scale of operations and time pattern of copper output. Priceexpectations are foremost among the parameters that characterize theeconomic circumstances of the industry. A limitless number of futureprice patterns could be hypothesized; consideration has beenrestricted to different uniform expected price levels. Other marketparameters, which are not elaborated, include opportunity cost ofcapital to the industry, the rate of social time preference, and the rateof inflation.Controversies in recent years have highlighted the circumstances ofthe industry with regard to taxation. Specification of the taxes leviedby federal and provincial governments is routine, but complicated.One feature that can be noted is the importance of distinguishing thesituation where a mine represents the only activity of a firm from thatwhere a firm has several mines or other business ventures. In the lattercase, various advantageous deductions given to mining can becharged against combined income flow, with the result that they canoften be used sooner and therefore have a higher present value. Thisdistinction is the reason why two alternative maximands were notedearlier for the case of private returns (KRPP$ and KRPPNC$).Development strategyThe mine operator, in a particular set of circumstances, must studyengineering possibilities and economic projections and decide oninvestment and production plans. For modelling purposes we need toselect the key decision variables which affect the value of the orebody.The model will be designed to optimize over these variables.Scale of operation, or rate of output, will certainly be important.When the ultimate quantity of a resource to be produced is fixed,long-term unit capacity cost will tend to rise with higher rates ofoutput because the larger investment required is borne by the sametotal output. However, in the present case, and typically, there aresignificant economies of scale in capacity investment. Both tendenciesmust be taken into account.With rate of output fixed, rising operating costs per unit of mineraleventually dictate the end of production. This can occur for ahomogeneous mineral where cost per unit of ore is rising, as wouldoccur, for example, with increasing depth of pit. Were cost per unit ofore constant, steadily diminishing ore grades would cause the cost perunit of mineral to rise, again eventually signalling an end toproduction. Here both changing grade of ore and rising cost of oreremoval must be taken into account in determining the optimalvolume of production.

    Ibid, Appendix 0.Cutoff grade decisions must be made with reference to the cost ofremoving particular units of ore. An especially rich lode 300 ft downin a deposit may look less attractive than relatively low-grade ore near

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    I8 Ibid, Appendix C.

    50

    the top. We have already noted that grade-cost possibilities in openpit mining depend on the relation between the location of contoursdepicting ore grade in the deposit and the shape of the pit. Acceptingthis relationship as determining the grade of ore coming from the pit,there are at least two cutoff grades which may be specified. The firstpertains to pit design. With ore grade tailing off as the pit expands,there will be a cutoff grade beyond which it will not pay to expand aparticular bench. It should be noted, however, that the bench may beexpanded anyway to gain access to the bench below it. This leads to asecond cutoff grade specification, which pertains to the grade of orefed to the concentrating process. For any pit design, the decision mustbe made whether to send broken rock coming to the surface to themill or to the dump. Because the cost of milling is less than thecombined cost of mining and milling, we may expect this cutoff gradeto be lower than the one used in pit design.So far we have specified four possible design variables. The valuesof these which yield a maximum value for the mine are jointlydetermined. If many values for each are tested, the number of possiblecombinations quickly mounts, and with it the cost of computer time.As will be described when dealing with some of the workings of themodel, we accepted certain simplifications. In the model a singlecutoff grade is used for pit design and processing. Suppressing for themoment the question of when to shut down, two optimizing variablesremain, cutoff grade and scale, or rate of production. That theseparameters do indeed have a significant effect on the value oforebodies can be observed in Figure 1, which shows the net presentvalue of the orebody when different operating conditions are specified.For example, at a throughput of 50 000 tons of ore/day, lowering thecutoff grade from 0.30% to 0.25% reduces the value of this particularorebody by roughly 15%. The maximum social value is obtained at athroughput of 40 000 tons/day and a cutoff grade of 0.30%.Mechanisms of the modelTo summarize, the model determines the value of a given orebody fora specified set of economic conditions. This value is posited to be thehighest of the values that can be attained with possible combinationsof operating conditions. We now consider more detailed features ofthe model. The concern throughout has been to strike a balancebetween maintaining sufficient generality to be able to deal with alarge variety of mining conditions while not neglecting therelationships known by those experienced in the mining industry tohave economic significance.A flow chart depicting the complete model is presented elsewhere.*Although this will be of assistance to one actually working with thecomputer program, it only serves to illustrate here that even the mostbasic circumstances and operating variables which we have taken intoaccount are enough to generate a formidable model. In describingsome of its workings, we wish primarily to draw attantion to a few ofthe assumptions which have been made in an effort to generalizeabout an industry which inevitably displays a great deal of variety.Discussion here is related to the schematic flow diagram shown inFigure 2.The three diamond-shaped boxes at the top of Figure 2 show themajor control variables: operating rate (or capacity), cutoff grade,

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    Figure 1. Mine value v cutoff grade atvarious daily rates of mill throughput.1 = 30 000 tons/day2 = 40 000 tons/day3 = 50 000 tons/day4 = 60 000 tons/dayNote: Mine value refers to social value(VRES) rather than private value net oftaxes and other payments to governments(KRPP$ or KRPPNC$).

    @Production in all runs reported hereproceeded with an initial 6 year pitfollowed by 3 year pushbacks. Theeconomic significance of the timesequence can be seen by consideringextremes. If the mine were developedwithout a series of pushbacks - ie if theentire uppermost wafer were exploitedbefore going down to the ore below it -all investment in overburden removalwould be incurred initially and lower-grade ore from outer edges of upperwafers would be removed before high-grade ore within reach from lower wafers.Frequent, short pushbacks, on the otherhand - amounting to continual sharingof all benches - would necessitate costlyrelocation of roadways.

    3001

    150 L0 20

    ;4 I1 I I I I025 0 30 0 35 040

    Cutoff grade (% capper)

    and production sequence. With regard to the latter, the inputspecification consists of the duration of an initial production periodand the durations of succeeding periods, in industry parlancepushbacks. The aggregate production period is the sum of the initialperiod plus all the pushbacks which the model finds to be economic.In practice we have focused on operating rate and cutoff grade as thecrucial value-determining variables, setting the intervals forproduction from the initial pit and for subsequent pushbacks inaccord with prevailing industry practice.igIt is necessary to specify the shape of the orebody. Actual orebodyconfigurations do display some regularities, but do not, of course,conform to convenient geometric shapes. An exact rendering ofnature - ie the specification of mineralization at all points in threedimensional space - would require an enormous input of information,a demand which cannot be denied in actual mining operations. Formodelling purposes, however, we construct various symmetricalshapes by the device of assembling wafers - our designation forcylinders whose heights are very short relative to their diameters -along a central axis. This affords considerable flexibility, because thecomposite shape can be varied between such extremes as pencil-likecylinders, near-spheres, or discs. Computer input thus includes thetotal volume of mineralized rock, together with the number, thickness,and relative diameters of the wafers into which the orebody beingmodelled has been resolved.

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    Choose0utoffgmde ElI I I1 -I-. 1Define Define Go to nextconflgurotlon conflgurotlon l pushbockof orebody of pit (J+l)

    I I~--&~ 3_IJCompute Yes Computemine output, 0rIos0rlos processingmlnlng costs costs

    1constructore outputsequence

    -

    Figure 2. Abbreviated flow chart showing computational sequence.While we mentioned above that two or more cutoff grades may be

    used in practice at different stages of the mining operation, we have sofar employed a single cutoff grade which governs both pit design andmill feedstock. Referring to the box in Figure 2 designated defineconfiguration of pit, the problem posed is to determine, starting froman initial size and shape, the final dimensions of the pit such that overthe prescribed interval the material removed will supply the mill foroperation at design capacity. Broken rock emerging from the pit isdivided: that above cutoff grade, now distinguished as ore in the stricteconomic sense, goes to the mill, while that below cutoff is waste andgoes to the dump. Related to the assumption that the differencesassociated with multiple cutoff grades would not have first ordersignificance is the assumption that there would not be a very largeincrement to value to be gained by processing low-grade materialfrom the dump once the mining part of the operation had ceased.Redesign of the pit occurs before each pushback period. Arelationship of economic importance in open pit mining is the stripratio, defined here as the ratio of the total amount of material that isremoved to the usable portion, that is, the ratio [ (ore + waste)/(ore) 1.If one visualizes a pit in the shape of an inverted cone exploiting aregularly shaped, homogeneous orebody, it is clear that more wastematerial must be removed to get at successively deeper ore. A risingstrip ratio implies increasing cost per unit of ore, and acts inconjunction with trends in grade to determine the optimal ultimatesize of the pit. The economic significance of the increasing strip ratio

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    20 R.H. Seraphim, Some problems met inthe evaluation of porphyries , WesternMiner, April 1973, pp 91-97.2 H.K. Taylor, A mainly-technical reviewof the paper (Helliwell, 1978) andassociated matters, personalcorrespondence, January 1977.Helliwell, 1978. refers to op cit. Ref 2.z Bradley, op cit. Ref 15, Appendix D. 1.23We have assumed that the distributionof mineralized material by grade islognormal. The first partial amount of thisdistribution is also lognormal and isinterpreted as the fraction of coppercontained in material above a specifiedgrade. See J. Aitchison and J.A.C. BrownThe Lognormal Distribution, CambridgeUP, Cambridge, 1963, p 12. Using thisdistribution we can readily calculate theamount of copper contained in the orewithin any range of grades. Bradley, op tit, Ref 16, Appendix D.2.

    Modelling miningusually observed over the period of mine operation is emphasized bySeraphim*O and Taylor.*l Elsewhere simulation results are presentedto illustrate the variation in strip ratios which is encountered with ourmodel.**With the configuration of the orebody fixed and a procedure thatestablishes the shape and size of the pit, the mining sequence isdetermined. The quantity of ore produced over any interval hasalready been specified, but in addition we are able to compute thegrade of ore going to the mill. This computation depends onassumptions mentioned above, namely that grade diminishesregularly in each wafer along a horizontal ray from the central axis,and that within any wafer the grade distribution of ore conforms tothe lognormal probability function. Provision is made for thepossibility of dilution, defined as the presence within the orebody ofvarying percentages of non-mineralized rock. The actualcomputational procedure, which is not detailed here, relies for itssimplicity on properties of the distribution of the first partial momentof the lognormal variate.23The product of the mill, as the concentrating operation is labelled inFigure 2, is copper concentrate which is sold at a price referred to asthe net minesite realization. Revenue is thus generated in each period;corresponding operating costs are also computed. Thus companyoperating earnings are determined, and tax calculations can be made,to be added to other transfers that may be required between theprivate operator and the federal or provincial government. We do notdiscuss specific tax regimes here. The general point is that for eachperiod through which the model iterates the incomes accruing to theprivate operator, the provincial government and the federalgovernment are calculated. These increments, with appropriatediscounting, are accumulated, so that at the end of the mines life wecan observe how the value of the orebody was divided amongclaimants, here numbering three.

    The final feature to which attention is drawn is determination of thelife of the mine. If ore could be removed in strict order of diminishinggrade, specification of a cutoff grade would determine the volume ofore to be produced, so that for a particular operating rate the mineslife would be determined in advance. Here the grade sequencedepends on the orebody configuration and the pit design. If the gradeof ore being removed did not vary, the tendency for an increasingstrip ratio previously noted would induce a monotonic decline inprofitability. To date we have assumed that the grade distribution ineach constituent wafer reflects the overall distribution, so that there isno systematic quality variation vertically in the deposit. Fluctuationsin grade of ore are observed, however, with markedly higher gradessent to the mill when production advances to the richer centralportion of a deeper wafer. This is seen in the figures elsewhere,24giving average grade of ore by year for several simulation runs.Variation in profitability attributable to changing ore grade will besmoothed out when annual values are averaged over a pushbackperiod. We have assumed that profitability measured over pushbacksdoes decline monotonically, and, on this assumption, mining iscontinued until a pushback is encountered which would not add to thevalue of the mine. Referring to Figure 2, in the final period of apushback the profitability of the next possible pushback is computed.If it is zero or negative, operations cease without its being undertaken.

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    Modelling miningIf it is positive, operations continue and the next pushback isevaluated. The operating profit figures given elsewherez5 support themonotonic decline assumption for orebodies of the sort consideredhere. There would be situations, however, where the presence of arelatively large amount of richer ore in the lower part of a depositmay cause the assumption to be violated, so in general one shouldverify that mine value cannot be increased by a programme whichincludes several more pushbacks.Simulation resultsTo apply the model which has now been described we must specifyrelationships which are of interest, simulate mine development andproduction, and analyse the results. Furthermore, if the simulationresults are to be of use, we must establish the models validity, ie inferfrom the reasonableness of both the assumptions which have beenmade and the resulting observations that the model does in fact depictrelationships which are true for the mining industry.We have constructed a sample of five orebodies in which we seek torepresent the primal features of porphyry copper deposits in BritishColumbia - those features taken into account in the model. Theparameter values assigned to each orebody in the sample reflect thecharacteristics of an actual orebody, but it is important to statecarefully the degree of correspondence between the sample depositand the real one. Clearly the models symmetry assumptions limit thefaithfulness with which the sample orebody can reproduce theoriginal. This applies to the configuration of the deposit and to thepattern of distribution of mineral values by grade. Furthermore, withrespect to the grade distribution, the assumption of a regular declinein quality outwards from a central axis corresponds to what inactuality is an observed tendency.26Given the indicated degree of correspondence between sampleorebodies and real ones, the question naturally arises as to howclosely we would expect the mine development plans generated by themodel to resemble the operating characteristics of actual mines. Itmust be remembered that the results which are reported pertain to thelong-term, or planning, situation: we are concerned with how a minewould be developed to maximize value under a specified tax regimewhen a given set of expectations about future economic conditions isheld. To make comparisons between model results and real-worldobservations, one must have insight into the conditions posited whenthe real mine was designed. Furthermore, although much of theinvestment required for mineral production is for practical purposesirreversible, alterations in equipment and operating procedures can bemade after production has begun when economic conditions divergefrom earlier expectations. Consequently, a good deal of judgment isrequired in assessing whether model results are a reasonablerepresentation of actual industry operations.In the following paragraphs we report the parameters whichdescribe orebodies in our sample, and then turn to the operating

    25bid. Appendix D.3. conditions which are optimal in various economic circumstances,*Oorrespondence of the type noted specifically, for different price expectations. The reader withexists for these pairs: mine A - Jersey Pit(Bethlehem), mine B - Brenda, mine C - experience in copper mining will wish to consider how these resultsGranisle, mine D - Lornex, mine E - match those of actual operations in comparable circumstances. InSimilkameen. addition to this type of comparison, the knowledgeable reader should

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    Modelling miningTable 1. Physical parameters of sample orebodies.OrebodyDeposit size, lo6 tonsSurface diameter, ftRatio, [ surface diameter I

    maximum diameterDilution, fraction mineral-bearingPit angle, degrees from horizontal

    A B C D E250 900 350 1 800 3501 460 2 775 1 925 3 920 1 9250.80 0.80 1 .o 0.80 1 .o1 .o 1 .o 1 .o 1 .o 0.840 45 38 35 40

    also examine the way in which operating conditions specified for aparticular sample mine change in response to different priceexpectations. Our efforts to model copper mining have alreadybenefitted greatly from criticisms and suggestions elicited by earlierversions of the model reported here.Physical parameters used to describe the sample mines arepresented in Table 1. Although most of the designations appear self-evident, some explanation is desirable. Deposit size denotes thequantity of mineralized rock; this figure together with the gradedistribution determines the amount of copper in a deposit, in total andfor all grade classes. It is greater than either of the two magnitudeswhich are usually reported, amount of material above any specifiedcutoff grade or amount of ore reserves; these figures will be comparedlater. Configuration is described in the table by two numbers,diameter at the surface of the deposit and the ratio of this figure to thelargest diameter. Where the ratio is unity, the deposit is assumed tohave a cylindrical shape, but otherwise the shape resembles that of abarrel. In all cases only material to a depth of 1 440 ft from thesurface of the orebody is considered, and overburden is assumed to be100 ft thick. Dilution indicates the extent to which the deposit con-tains interspersed non-mineralized rock which can be sorted outbefore the concentrating operation. Pit angle refers to the maximumpit wall slope which can be employed. Parameters for the lognormalfunction which specifies grade distribution are not shown in Table 1.For the results reported here the same parameters were used for alldeposits.27The importance of two of these physical parameters, orebodyconfiguration and pit angle, scarcely needs to be pointed out to thosewith experience in open pit mining. It is, however, worth stressing inconnection with ore reserves and mine value. Table 2 providescomparisons where an orebody of a given size is posited and theseparameters varied. In Part A three shapes are considered: cylindrical,tabular (thin, with large area1 extent), and roughly spherical. Thepermissible pit angle is assumed to be 45. In each case figures aregiven for reserves, or usable ore, and value of the mine. If the mostfavourable case is compared with the least favourable (tabular againstcylindrical), reserves are seen to be about 75% greater while value is143% greater. In Part B of Table 2 the sensitivity of reserves andvalue to pit angle is examined. If a 45 angle is possible, compared toa 35 angle, reserves are increased by about 34% while value goes upby 179%. Here the posited orebody has the roughly spherical shape.

    Bradley, op cit. Ref 16, Appendix A.

    Discussion of the results shown in Table 3, mine operatingconditions which maximize the private value of each orebody, will belimited to a few brief comments. We observe that with higher pricessuccessively lower cutoff grades are chosen while mill operating ratesincrease. At the same time mine lives are shortened, although the net

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    Modelling mining

    a Mine value refers to social valueFIRES). Maximization of private value(KRPPNC $) under current tax regime isassumed. Net minesite realization:$0.60/lb ($1976). Pit angle: 45.b Mine value refers to social value(VRESI. Maximization of private value(KRPPNC $) under current tax regime isassumed. Net minesite realization:$0.60/lb ($1976). Configuration:spherical (approximate). Optimal cutoffgrade was 0.35% in each case.

    Table 2A. Reserves and mine value for different orebody configurations.a

    ConfigurationCylindricalTabularSpherical

    Ore reserves(106 tons)110192137

    Mine value(106$1976)

    60146

    61

    Cutoff grade(%I0.350.150.35

    Table 28. Reserves and mine value for different pit angles.bPit angle Ore reserves(degrees) (106 tons)35.0 10240.0 13145.0 137

    Mine valuef106$1976)224261

    Table 3. Mine operating conditions: private value of orebody maximized under different price expectations.aMine A Mine B Mine C Mine 0 Mine E

    Expected price, cents/lb 100 50 60 75 100 75 100 50 60 75 100 75 100Mill operating rate, lo3 tons/day 12 4 20 55 (1001 5 28 16 50 (100) (100) 3 16Cutoff grade, % 0.20 0.40 0.35 0.25 (0.20) 0.30 0.20 0.40 0.35 (0.25) (0.25) 0.25 0.20Mine life, years 15 30 15 12 19) 18 9 21 12 (12) (15) 27 12a Private value is present discounted design for six years production followed regimes assumed to prevail over the lifevalue of after-tax profits to mine owner by successive three year expansions of the mine. Mine operating rate waswith no other income (KRPPNC$). Price (pushbacks). Where values are not constrained to a maximum of 100 000denotes net minesite realization. Real reported for a particular price a mine tons/day. Where this constraint is bindingprice in 1976 dollars assumed to remain would not have been economic. Current the results are shown in parentheses.constant. Mine life consists of initial pit (1978) British Columbia and federal tax

    a See, for example, H.H. Cox, Definitionof ore and classification of ore reserves inCanadian Institute of Mining andMetallurgy, Ore Reserve Estimation andGrade Control, Special Vol 9, 1968.

    effect is that larger quantities of ore are ultimately produced. We nextexamine how different price levels affect stocks of mineable copper, orreserves, and industry output.There does not appear to be disagreement over the principle thatmineral reserves should be defined with reference only to that portionof a deposit which can be produced at a cost which is covered by therealization, or selling price. 28 In a short-term context, where the onlychoice lies between continuing production and shutting down,reserves include that material for which the realization coversoperating cost. Long-term application of the concept must similarlyfocus on whether increments of material add to profitability. Thus thepresent model, which is designed to evaluate optimal productionplans, will generate reserves estimates: the cumulative output of anoptimal production plan over the lifetime of a mine is the appropriateforecast of reserves. The significance of this is recognition of thedependence of reserves on all aspects of mine development andproduction plans. Such factors as scale of operations and ability toselectively use material according to grade will condition the amountof material which it is profitable to mine.The elasticity of reserves with respect to price, or net minesiterealization, can be seen in Figure 3. For mine A reserves are nil untilthe highest of the several assumed price levels, $l/lb, is reached. Forthe more valuable orebodies, such as B and D, reserves are veryresponsive to price. For example, an anticipated price level of $0.60,rather than $0.50, results in increases of reserves for these two mines

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    Figure 3. Copper reserves: individualmines and industry total.Notes: I = industry total.Operating conditions maximize value ofmine to enterprise with no other incomesources (KRPPNC$). Points are inparentheses where output rate isconstrained.

    0 Orebody

    Figure 4. Ore reserves compared tomaterial above cutoff.

    Modelling m ining--1

    0 800 I600 2400 3200 4 000Copper reserves ( b mll I ton )

    of about 220% and 110% respectively. The kinking upward effectseen, for example, for mine D can be attributed to a constraintimposed on the rate of output. The maximum permissible millthroughput was taken to be 100 000 tons/day, a rate which mayalready be beyond the range for reasonable extrapolation of our costrelationships. Where there is a ceiling on output it will apparently payto select a higher cutoff grade than otherwise. Average grade of oreprocessed will be higher, so that even though less copper is ultimatelyproduced the present value of output will be greater than if theadjustment to a higher cutoff grade had not been made.Additional insight into the growth in reserves with higher mineralprices can be gained by resolving that growth into two components,the increase resulting because a larger portion of the orebody ispotentially usable with a lower cutoff grade, and the increase resultingbecause a larger fraction of this above-cutoff material becomesprofitable to mine. Referring to Figure 4, which depicts a cylindricalorebody, the lightly shaded area represents material with mineralvalues above the chosen cutoff grade. In general, the optimal cutoffgrade falls with a higher price, so a larger fraction of the orebody willbe included in this area. A conical pit is shown in the sketch as itwould exist after the termination of production. Although theincreasing strip ratio as a pit is deepened raises the unit cost ofmaterial delivered to the mill, a larger pit will normally be worthwhilewith a higher price. The darkly shaded area, which shows the amountof material above cutoff which is removed, will typically be largerabsolutely and relative to the potentially usable material.

    A numerical illustration of how reserves increase with a higherprice level is provided in Table 4, which pertains to mine B, a minecapable of profitable operations at all price levels which wereexamined. Suppose, for example, that the mine was developed inanticipation of a $0.75, rather than a $0.60, net minesite realization.At the higher price about 36% of the material in the orebody is abovethe optimal cutoff grade of 0.25% copper, and this material contains62% of the total amount of copper in the deposit. The comparablefigures at the $0.60 level, for which the optimal cutoff grade is 0.35%,are 19% and 41% respectively. At the higher price it pays to mine73% of the material above cutoff instead of 63% as at the $0.60 price,so that 41% of the copper in the above-cutoff material is taken rather

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    Modefling miningTable 4. Elasticity of reserves with respect to price.Mine 9Price (cents/lb) 50 60 75 (100)Cutoff grade (%) 0.40 0.35 0.25 (0.20)Ore

    Fraction of orebody above cutoff 0.14 0.19 0.36 (0.50)Fraction mined of material above cutoff 0.35 0.63 0.73 (0.73)Fraction of orebody utilized 0.048 0.12 0.27 (0.37)

    Contained coooerNote: Values shown in parentheses wheremine operating rate constrained by100 000 ton/day ceiling.

    Fraction inmaterial above cutoff 0.33 0.41 0.62 (0.74)Fraction mined in material above cutoff 0.12 0.32 0.41 (0.46)Fraction in deposit utilized 0.041 0.13 0.25 (0.34)

    than 32%. Combining these two effects, the fraction of copper in thedeposit which is utilized is 0.25 at the higher price compared with0.13 at the lower. This corresponds to a price elasticity of reserves ofroughly 2.8 in this range.29Conventional market analysis relies on comparisons of demandand supply, supply referring to the outputs forthcoming at differentprice levels. In Figure 5 the simulation results are applied to portraysupply curves for the different mines in the sample and thecorresponding aggregate supply curve. These curves relate to thelong-term, or planning, situation: outputs are those corresponding tooptimal production plans for the price levels indicated with the taxregime assumed to remain unaltered.

    It is evident from Figure 5 that industry output is responsive toprice both at the intensive and extensive margins. As with the case ofreserves, when output is constrained the curves are kinked. Here, inthe case of mine D, the supply curve is actually backward-bending.As already noted, an output ceiling leads to a higher cutoff gradechoice, which curtails the growth of reserves in response to higherprices. Moreover, with the constraint and higher prices it becomesattractive to operate the mine longer, with the result that averageannual copper output falls.

    a This value is computed as an arc It appears that-if the expected price level were a step higher thanelasticity with reference to the midpoint that experienced recently, considerably larger copper output would beof the $0.60-0.75 range. forthcoming from British Columbia deposits. Consider the effect of a

    Figure 6. Supply curves: individualmines and industry total.Notes: I = industry total.Operating conditions maximize value ofmine to enterprise with no other incomesources (KRPPNC$). Points are inparentheses where output rate isconstrained.

    200 400 600 800 1000Average copper output (IO3 lb/day 1

    1

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    Modelling mining$0.75 rather than a $0.60 net minesite realization. At the intensivemargin, mines B and D would be designed to produce at substantiallyhigher levels, their combined percentage increase in output beingalmost 85%. At the extensive margin, the $0.75 price makes itprofitable to mine orebodies C and E. The total increase in supplyrepresents a price elasticity of about 3.2 in this range.30Determinations of reserves and rate of use are fundamental tomineral economics. The responsiveness of either of these magnitudesto mineral value is the resultant of many interrelated factors. We haveattempted in the present model to incorporate within an overallframework of value maximization the economic and physicalrelationships and the design possibilities that must be considered. Abalance has been struck between simplicity and abstraction, on theone hand, and elaboration and particularization, on the other.Little has been said about returns generated by mining or thedistribution of these returns between private capital and federal andprovincial tax collectors. In fact, the questions of primary interest tous have to do with the consequences of different tax policies. Howmight changes in taxation alter the development and productionstrategy chosen by a private operator, and how might they alter hisexpected returns, and hence incentive to invest at all? The model isdesigned to facilitate comparisons of economic efficiencv in differentd

    30This value is also computed as an arc tax settings, and it is, we believe, far enough advanced in its evolutionelasticity with reference to the midpoint to be capable of providing valid insights. Aspects of this subject willof the $0.60-0.75 range. be considered in a subsequent article.

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