Ore Reserve for a Beach

Estimation and Depletion Diamond Deposit

Planning

By M. M. OOSTERVELD," M.Sc.

SYNOPSIS

In South West Africa diamonds are mined from ancient balch deposits by the C011$OHdated Diamond Mine8 of South West Africa, Umitcd.

To improve the control of mining production, a computer is used for the calculat ion of statistically more accurate and homogenwus ore blocks. Geological and sampling information has been digitized and stored on magnetic files.

The figures derived for each block entail tbe calcula tion of the diamond SilO frequency distribution, the diamond density distribution and the ore reserve by using the density distribution,

The diamond size frequCIlcy distribution is used for cstln'l:lting the revenue in Dn ore block.

A description is given of how a linear programming system can be used for production planning.

l NTROOUCTION

Diamonds are mined in South West Africa in a narrow coastal belt wbich stretches from the mouth of the Orange River for 100 km to the north. The deposit was discovered in 1928 and mining has taken place since 1937.

There has always been_ a need (0 control the mining production to a larger degree. This required the calculation of statistically more accurate and homogeneous ore blocks which became possibte only nfter a computer was made available.

For the evaluation of a diamond deposit it is necessary to estimate both the diamond content and the size frequency of the diamonds occurring in it. The value of diamonds is a function of the size of the diamonds and this value increase8 grcatly with increasing sire.

In addition to size there are a number of other physical characteristics which influence the value of diamonds, but insufficient information was obtained from sampling to be of use in the evaluation.

Many of the ideas and techniques described in tbis paper have been developed in consultation with Prof. H. S. Sichel, and for the detailed statistics of the various distributions mentioned in this paper reference is made to Sichel (1972).

NATURE OF THE DEPOSIT

Diamonds occur on the South West Africa coast in ancient beach deposits which lie mostly above the present-day sea level. The beaches, which range in age from Middle pteistocene to recent, occur at decreasing elevations and reflect consecu­tive, transgressive, stands of sea level. The diamond deposits arc. o_verlaiu _by marine and terrestrial deposits up to 15 m thick. In Fig. 1 a represeotative cross·section of the deposIt is shown.

A typical beach consists of a layer of gravel between high. and low-water marks, a storm beach or ridge and a marine plalform or shelf with gravel. Diamonds may be found in any portion. However, they occur most abundantly in the beach and shelf gravels.

The shelf gravels lie directly on bedrock which consists mainly of schist. In the bedrock a number of marine abrasion platforms are cut some of which are separated by cliffs. The

65

platforms are uneven and have been eroded deeply by the sea to give gullies and potholes. The gullies and potholes have acted as diamond traps and often contain rich concentrations of diamonds.

In mining the deposit the excavation of the grnve1s above the bedrock peaks and the mining of the gravels in the gullies and potholes are two separate operations and there is a considerable time tag between these. Usually the diamond contel~t of the gravel recovered from tbe gul lies and potboles is much higher than that of the gravel above the bedrock: peaks. For this reason, it is necessary, for control pnrposes, to calculete the diamond contcnt of each zone separately.

The sizc frequency distribution of the diamonds differs from beach to beach. However, within a particular beach the size frequency distributions are similar and only a gradual change occurs from south tQ north, away from the Orange River mouth. While overall trends show up in tbe size frequency, the loca! sile frequencies are influenced to a largo extent by local depositionaJ irregularities. It was found that in geologically homogeneous WlltS, the diamond size frequency distribution is Jognormal.

INFORMATION ON THE DEPOSIT

The deposit has been sampled by 1 m wide trenches which are 500 m apart and run across the strike of the deposit. Each trench was divided into 5 m long sections Ilnd in the economic~ ally important part of the section, samples were taken in 0,5 TO thick laytml.

Nearly al] geological and sampling information is kept on magnetic files. Besides being used for the calculation o f ore reserves tbis information is used for many other applications. The total number of scctioos is approximately 40 000, wbilo thc number of samples is about 280000.

In most instances where more lhan one diamond was found in a sampling layer, only the combined weight oflhe diamonds recovered was recorded and tlris information cannot be used for the size frequency analysis which requires the weigllts of individual stones.

·Superintendent. Computer Services, Too Consolidated Diamond Mines of South West Africa Umitea, Qranjemund, South West Africa.

o Alolian sand

~Calcrell X,A,B,D,E & F represent beaches

} "

,

D Terrestrial red sand & rubble

1:::1::;1 Clay

~~~:ri MQrine SQnd

1::;':1 Grovel

I:".; ~:.~::I Conglomerate.

~ Schist bedrock

"

"

"

,

o

.,

Fit. 1. Typical cross-section through beach deposits

BASIC REQUIR EMENTS FOR THE ORE RESERV E CALCULATIONS

TIle three most important characteristics for the evaluation ofa diamond deposit are:

(i) The size (weight) frequency of the diamonds. (H) The density distribution of the diamonds, that is, the

number or diamonds per unit area. (ill) Th, p,ico ,tt..:',", of th, diamond •.

The grade can be calculated from the size frequency and the density distributions. The grade as used Cor diamond ore reserve caJculalions can be either the carats per square area (normally square metre) or the carats per volume of ore (normally the cubic metre). The carat is a weight measurement used for diamonds; One carat - 0,2 gram.

Grade - G = J) x0,

66

where fj is an estimate of the average number of stones per unit and ~ is an estimate of the meall value of the stone size.

From lhe size frequency distribution of the diamonds aod from the price structure, the average price per carat can be calculated. The combinalion of grade and average price produces an estimate of the revenue in an ore block.

SELECTIO N OF ORE BLOC KS

To obtain homogeneous ore blocks the delimitation of each block was done on the basis of geology.

Across the deposit the boundaries between beaches were used as block boundaries. Along: the strike oC the deposit a combination of the geology and DC the grade (carats per square metre) and average stone size per individual trench. was used for drawing block boundaries.

•

The fotlowing example illustrates this: The boundcuy between ore block 2 and 3 was drawn as ore block 3 has a higher grade and larger stone sizo. Also, there is a change in the strike direction of the deposit between ore bloch 2 and 3. Trench

Av. grade carau/m'

K' 0,05 K8 •• 04

KI3 0,13 RIB • • 09 K23 0.17 K28 0,19 K33 0,20

K38 ..... K44 0,14 K48 0.38 K'2 0,27 KS7 0,54 K62 0,13 K67 0,39 K72 0.34 K17 0,31

D Beach

Av. stone: size carats/stone .,",

0,39

0.56 O~. 0.79 0,73 0.7.

0,69 0.80 1.02 1,34 0,97 1,07 1.04 0,87 •• 80

Ore block 1

Ore block 2

Ore block 3

Ore block 3 is terminated by an eroded area.

Prom the above example it can be seen that within an ore block there can be a wide variation in grade and average stone size between trenches. By taking an orc block over a number of trenches, smoothing out is done of erratic high and low values and a more realistic result is obtained.

SIZE FREQUE NCY CAL C ULATION

The size frequency for each ore block is ca1culated for the following reasons :

(i) For checking whethec or not the diamond sizes are lognormally distributed. The lack of lognormality means that the block is geolOgically not h omogeneous and that tbe block boundaries need to be reconsidered.

K8! Beach removed by erosion (ii) For the calculation of the Cltpected revenue from the diamonds in tbe block.

The boundary between ore block I and ore block: 2 was drawn on the grounds that ore block 2 has a larger stone size and higher grade,.

In Table I, a computer print-out of the size frequency distribution calculation of ore block Kl07/KI22, D-beach, is given as an example.

TABLE I

UH~.{jN O sae U lSTRlbUfl Df<l S FD!I. COM SA MPLE TII Et«:HES ...................... " .......... , .............. . IIL Ct K , ~1(1-H.u. i, I:II: ACt< TR l/OC" ~H: II(~S

~l C H C ~1l ~ C iJ l(l lH 0 U 2H'" C

{ CllO- ( OU t. {(J~ O- lGl <!O O(llCO - 10C<;0 (C12 0 - 1 0l 6 (

cu ~ ~ IHltl( \AL ! I( 1.:i1

O. U -0.10 • 0 . 20 -O. ; {j -(.~O -0 . 50 -0. 60 -0 . 10 -O . ~ C -0.'10 -1. 00 -1.10 -1 .~C -I .H' -1.'10 -1.!>CI -2. ce -3 . 00 -5 . 00 -

l a . GO -2 e.GO -40. 00 • 80. 0 0 -

O. le C.le 1J • .lC (;.4C o.~ o (. b C 0 . 10 O. dO 0.90 1. ( .( I . U l.i( 1.3C 1.<' 0 1.~ O 2. .• co .1 .00 :O . UO

1 0 . 0 0 zo.OO ~O .GC d O.OO

Ita. 00

~I hGIE lell 1(' 1"-LS

s tu. DI S Til I II UT ) ON Of HH!>lU " " S -------------------------

2 . 0 ,., 0 . 0 , ., ,., •• S .. , ,. , ,., '" '. S ,., '. S ,., S., ••• , ~

(JutS

0."00 1.9 0 ~.70 .... 20 3 .10 4.20 ~ .'O 5.10 6.70 ,.10 1.10 1.25 .<: .0 5 ... 15 5.85

HI.20 .. ,.

£6.60

Cl S/~IH

o. a. o 0. :i7 1 0 .B5 0 .'1"00 O. St. "O 0.6U 0 . 1'$0 0. 8' 0 0 . '1!S 1 .1 .020 1. J):! .1. 2.5 0 .1. 36 1 1."On 1.770 2. 2(01 "0.>00

0 .900

CUM. I Hllll ,~'" • f ol. EQ LSI'" VARI ANei:

0 .1<;1 2. .103 2 . 70) '2. H61l 9 .<'5<; 1 ' .16 l 1.0"02

10 . ll ll 22 . '113 9 . ' 91 6.151 ~9.730 10.306 1."032 31.lf>l 9.H89 8 .13 "0 4S. '1~ 6 11.9310 8 .10 8 5 .... 05 "0 1. 805 11.108 62.llol 6.l>d5 ~ ."O59 1l.6 ~ 1 5. 662 6.15 7 11S . HII ~ . 76"0 4::.021 8 C. 'I05 3 . 9 97 1.3!>1 81 . 156 3. 350 :l .Oll 83 .183 2 . 8 07 l . O27 e!>.lUo l.355 6.157 92. ~61 1. 26 1) -6.081 9 6 .6<08 "0 . 754 1.3!>1 9<,1.'19'1 1. 50 28

55. '199 0 . 2.05 9 '1. <,1 99 O.OO!> 59. 9 99 99. 999 <j9. 99~

SL.O'''' T 01 us -- -------

S TIl~ES

CloUTS C U/SlN

" 66.60 0 . 9 00

ALL !iTOHES

'" 301 . TO C.S llt

VAR.IANCES ~ I'IEAN S USED IN CALCULAn ONS

oeSERVEO SAMPLE VARIANCE Of LN SHE ONIlUSl£ SMPLE VARIANCE Of LN SIZE .lliUTNlfT IC.IllAH 01- 1..11 01' SIZES S ItHlS t ' - EST I"ATOR fOR MEAN OF Slle

67

o.~!!glt

0 ."O62111 - 0,. nUl

O.B'Jl0'

CUM. S faEQ , t'_eSTIMATOR R/ur -------O.l'H "A 3.0~9 11.0

10.101 HA 19.1>92 15. 0 2~ .~9 8 11.0 39 . 387 19.0 ~!i.lI Zl Zl.O S6 . 6 21 2!j.0 63 . 3U 211.0 66 . 915 30.0 1).139 32.0 17.736 34.0 81.086 3 1>. 0 83. 893 '" ., 116 .2~8 ~O.O

'11.508 so.o 98.21>2 60.0 '1~.790 110 . 0 '.19.99 5 J 20.0

100. 0 0 0 IBO.O

~9.1

Flc tltl ..... II/CIU' 1"lQ'1ru u . ,,_ .

To arrive &tlhesizc frequency distribution only the diamonds for which the individual weight is known (caLled 'singletons') are used. In the example it can be seen that of the 345 diamonds recovered during sampling only 74 are singletons.

The steps in the calculations are as follows:

(i) The diamonds are split into size (weight) classes. In the past, different levels for rounding weights were used and this has caused problems in determining class boundaries. These were overcoUle by taking the class boundaries at the preferred sizes, and splitting the diamonds of the preferred s.ize half over tbe higher and half over the lower class. 10 addi tion the total carats and carats per slone for each class were calculated to check for bias. Thereafter the percentage number frequency and the cumulative pen:oolage number frequency were calculated.

(H) For the calculation of the menn stone 8i7..6 the method as described by Sichel (1966) was applied. The maximum likelihood estimator (tf -estimator) for the mean of the two­parameter lognormal population was used. By using this estimator the variance of the mean value distribution is minimized.

R - [, i 1(F,S, Vd]/[,E1

(F,Sd] Here R is the average value, Fj is the percentage frequency

in the i-th class, Sf is the average stone size in the i-th class. V, is the average value per carat in the i-th class and n is tin} number of classes.

By lIsing the expected frequencies for the value calculation in the estimate, the value of the large diamonds in the block is included. The probability of recovering the large stones duriug sampling is small. However. ba:a use of their hish value they coutribute considerably 10 the block revenue.

(v) In Fig. 2, the observed and expected cumulative pertentage frequencies are plotted on 10gprobabiJity paper. This plot is used to check the lognormality of tbe dis tribution.

o· ~

, , 2cyel, IDO Drobo:bility DOD,r

99·9

-... 99·8

The observed sample variance. V •• of the logarithm of o . , ,.., the size was calcula ted by using the following fommla:

v, - '.3019 r ~ .:E (x" )' - (! i': X")'] Ln 1=1 ,,1 - 1

In this formula n is the number of singletons, Xf, is log10 (ZI) , where ZI is the weight of the ;-th stone.

To obtain the unbiased sample variance of the logarithm of the size, tJ e2, Bessel's correction for bias was applied.

<!! 2 " " , --- V, n- l

The arithmetic mean of the Logarithmic si'leS. xe. was calculated as follows:

i. "" 2,3026(~Jl xf,) It was found that V~ is usually smaller than unity and that

the number of observations in the snmple usually exceeds 20, Hence it is pennitted to use the simpler t f -estimator instead of the theoretically more correct t-estimator, that is,

t ' zo exp(i,+ !V,).

(iii) From &,2 and i 6 the cumulative percentage frequencies were calculated by using an inverse polynomial [Hastings (1955»), whicb approximates the standard normal integral, that is,

1 u ~ ' (u) ~ <ll(u) = - f exp ( - iz')dz

.j2~ -. Tllis formula is valid for 0 ::;;;; u < co. If - ro < u ~ 0,

then $ (-11) - 1 - 4l (11) and

1 ~' ( - 11) -- 1 - <!) ' (u) - 2[1 + bl" + btu' +, , . beU' ]U

Here u = (/"zc - xe)/&~ Zc is the upper class limit of tbe c-th class, and blJ b'1' etc, are constants.

The percentage frequencies per size class were also calculated.

(iv) Using the percentage frequency per class and the average value of the diamonds per class, the average value can be calculated.

68

, , , 0

N • '0 ~ '0

i 40

5 50 60

~ 70

~ •• 0 v

" ~ 90

• , " .,.

". .,.

, •

5'1 • • 0"

-/

.. " /

~ ~ " J

Ii'

00

80 ~ • n '0 " z

" 60 • a

" 50 ~ 40

~ '0 • '0 " V • z '0 N

, , 0-5

0·, 0 · , , , , ,

" l z CARATS PER STONE

Fig. 2. Obseryed and expected cunl/llal{ye percentage number frequen­des in ore block KI07{KI22 D beach.

(vi) The average monetary value of the diamonds (money per carat) in an ore block is a function of the mean stone size and the variance of the size. This is illustrated in Fig. 3. where. for a number of ore blocks. the relationships between the , '-estimator for the mean size and the average value are shown. In Fig. 4, the relationship between the sample variance of the logarithmic size and tho average value is illustrated. For the same mean size, a higher variance means a higher monetary value.

I ,

t

• o '0 >

•

•

•

• • • •

• • .. .

• • •

•

f '- estimotor 01 the mE!'On site====C>

Fig. 3. rim average vaTue as function of the t' ·estimator oftlie mean size.

•

• • • 0 .. >

• • • ~ e • > 0 • • •

• . ' •

• •

c::::::::= unbiased somple voriance of Ln sill! ==C> Fig. 4. The average value as/unction of the unbJ(/$Cd sample lIarionce

of tile logarithm a/size.

DENSITY DISTRI BUTION

The density distribution is discrete and shows the number of stones per sample unit.

In Fig,S, !l oomputer-caJculatcd density distribution in orc block K.38/K77 i~ pIQued.

69

80

70

60

~ z 0

50 ~ u I:! ~ 40 0

'" ~ 30 • • 0 z

20

10

0 K)

ORE BLOCK K38 / 1(77

20 30 40

NUMBER OF STONES

Fig. 5.

..

50 60

The distrib ution is reverse J-shaped aDd has an extremely long tail. The mathematical representation of this type of distribution is difficult and no satisfactory mathematical model. has been suggested lUltil recently, Siebe] (1972) .

ORE BLOCK CALCULATION

For the calculation of the average grade in carats per &quare metre and the average stone size, Mocks were selected which encompass a number of trenches and which may be up to S km in length. T he thickness of the ore can vary considerably within these blocks. However, it was found that the diamond content in general is oot related to the thickness of ore, but to the area.

For Ihis reason the area grade and lhe stone size over the whole of the block are calculated, smoothing out irregular low and high values. The block is then divided into sub-blocks and for each sub-block the average thickness of ore is used to obtain volume grades. The sub-blocks are situated between trenches and are 500 m in leugth and 100 to 300-m wide .

'The calculations are done separately for ore above the be<lrock peaks and ore in lhe gullies and pothoJes.

The block grade in carats per square metre, G, is given by

m ·J:1 ,-

where 81 and Cl are the number of stones and the number of carats in the j-th section, and al is the area of the same section. Also, n denotes the number of sections, A, is the area of the ;·th sub-block a nd III is the number of sub-blocks

In this case the estimate of the average stone size is not the I'-estimator, which is related to the singletons only. hut is based on all the diamonds. In general, the singletons form only a fraction of the total number of stones recovered from the block during sampling.

The average stone size for the block, Z, in carats per stone, is expressed as

m .~ ,~1 (J~ l " . J~ l ")

J~l a, j~ l sf

Using the avorage block grade (area) G and average stone size Z, the following culculations are done for each sub·block :

Carats

Stones

Ore volume

Grade (volume) ... G A. M,

in carats per cubic metre, where dj is the thickness of ore in thej-th section.

For calculation of the bulked volume and volume grade, a bulking factor of 1,5 is used. In Table 11, the computer print-out of the result for ore block K57-KTI, O-beach. i! given.

In most cases there is reasonable agreement between the t'-estimalor of the mean size, based on the singletons only, and the mean size found by using the arithmetic mean of all diamonds recovered during sampling in the block. If there is an appreciable difference, it would mean either that tho I'·estimator is based on too smaU a number of stones or, alternatively, that tbere is a significant departure fro m the logllOnnal distribution. In both cases an adjustment has to be made to obtain a more realistic size frequency.

TABLE 11

c.u. ~ [~Jo IIl:HRVH I>LOCIl ':: 51-1(11 D tiEl.tH IlAH ~'11l011 1 ...... t.u ... n .... ........ u . ................ .." ........ ,.

51.HJ 8LU U; K!il -KU Il RE l CH ......................... 111. ~J;C H t SH.Tl(l<l 4~U T1i/;S vc~ DUN VOl ()f,P ClRATS nOHES YlfLO I N CARUS '" t.lUllY ""U"Ht.U ... H2X 1000 , H ~X 1000 MJ X1000 ;0;1000 X1000 " HJ .~O " snmE AV DTU .......... •• u ....... .. u, •• . ... ., .u.n .. • t'·· .. .. , .... .",. ~o~nGco'no ~U~100002&C ~Ol>.O OIlOJ tU Koo~o,)(10220

DVHWi<,LH n . ~ , .. ~ 41>. ! ••• ••• 0.:11 0 . 02 '. 00 ••• m #ltVE BtU£CI< 81.' • •• H I .l ,., • •• O. Od O.O~ a . 08 0.1 '1 , .. '" I " CUll US ~7 . S .. , 118. L l ... , <:11 . 2- O.H 0.21 o.ze U. IIJ , .. TOUl /57 . ~ , .. ~~6.~ ZH . " 31.S 31.1 0.1' Ooll 0 . 36 0.85 , ..

"" Bl OCK K61-Kll U S ~~C~ ••• ~ ... * ......... u ..... .....

lUNCH t SE CTIONS MI U H.., ~UL U61! ~I,ll u tI' CAIIA t ~ STOHES Yi ELD [N CAUTS '" GUL LY .... H.UU.U ••• IIU I OCO , NlX I OOO 10);.\000 )(l OOO J. l ooo '" /0 . '0 " HONE AOV TH " ...... ...... •• uu • U""n ..... ,. .. .... •• . .... •• ••••• n ..... ~0l>1tCnO .. 2G ~Ot. lcOc02 .. a '::0 12COC02L O II012H,,-,UJC

OVEH~oJ;I.D9I 85 . 0 ,., lH . 3 .. , .. , 0 . 0 1 0 . 112 1.00 .. , '" ~I(~f hftROCK /1 5 . 0 , .. 11'0.8 ~ . ••• 0 . 09 O.VO 0 . 08 O.lg ,., '" H GUl1.HS a5 .a .. , 153 .0 23. , Z7 • • n.ll 0 .10 0.Z8 0.117 .. , TeUL 85.0 ,., 3H.J <67.8 30.1> 3 6.0 0.17 O. 1l O .3~ 0.d5 .. ,

'"' tllO (~ ~n • aUCH ........ n.nn ........... .

III EHCH t SEtTI (/~S .~U rh/>5 VOL uerl ~I)l. DU' (AlIA I S Sl ONES YlHD III (A1l"S '" CNlLY .. uH ...... t u •• " 2);.1000 • II X) I OOO 113X1000 )1. 1000 ..... '" 113 . ~ '" SI ON E AVD1H

• n . .... •••• ........ ......... .n .... ...... •• .. ... .. ...... .40" lCl1(t(UI>C l<.Oll(C(e05U

OHR~IJRO~H 1 6 .0 , .. "10 .4 I. ~ 1. 5 0.01 0.02 '.00 .. , Il ~.e: A~ GVf: II E~RUCK 16.0 ,., 171.0 o •• ,. , o.~ 0.0. :I . Ud 0 .19 . ., O~11 11\ GUllHS 70.0 .. , UIo.O 2 1. ) 2~ . ~ 0.26 0 .19 0.l8 0.81 ••• TOHl 10.0 , .. HO.4 2115.0 21 •• 3Z.2 a.H 0 .10 0 . 36 O.ij, I.'

8~OCK )(~l-lnl D 6EHk .... ...... .,."... ................ lltl, NCII t SEC 1 H Io S A!l U fh"S VOl \,ItN .. , ~. UR"U Sl('.NES YIEtO I N ("""A,S m wu.y ...... HU ........ "2~ 1 000 • !lHI OOC 1.,11 0011 HOOD UOOO " 1'11.)0 " STUNE A\l Of H ........ . .... . ...... ....... • un .. .. ..... •• . .... .. .. e • • .....

UVH BURDEN ~48.~ ,., 120 1.0 ,. , , .. 0.01 0 . 02 1.00 .. , O~r, ~~(V ~ B <H(iU 2~8 . ~ ••• Hr.1 1 ... 'I Z ~ .3 0.07 0 . 0' 0 .08 0.19 .. , ,, ~ ~ l~ GULLIES H8.S .. , 385 .1 M .c 80.0 o.n 0. 18 0.l8 0.81 .. , U! UL Z4~ .~ ,., 1201. 0 802.Z 019. , 105.3 O.ll 0 .11 0.36 O.lI!i .. ,

70

D EPLETION PLANNING

The depletion planning is complicated due to the large number of plants, the large number of ore blocks and the large number of constraining factors involved. For long-term as well as for short-term planning the following variables have to be included.

For the whole mine:

(I) TOO requ;,,,d amouot or "'ca~ to '" produ",d. (ii) The average diamond size to be produced.

(iil) Ore hawing requirements. For the minimizing of equip­ment requirements it is necessary to control the amount of ore hauling. Ore has to be transported over distances of up to 20 kilometres.

With regard to individual plants the variables to be considered are:

0) The capadty of the p",tiou1", p1'nt.

(ii) Only ore blocks within the plant ore reserve boundaries can be mined. (Otherwise the hawing cost increases.)

(ili) Only the mining of available ore blocks is possible as prior stripping of overburden is required.

(iv) FOI' each individual ore block: there is a physical liurit of the amount of ore which can be mined in a period.

Cv) There is a limit on certain types of 'difficult' ore which can be mined per period. For instance. there is a limit on the treatment of 'wet ore', as wet ore must be mixed with 'dry ore' to facilitate treatment. Another constraint is the amount of conglomerate which a plant can treat.

(vi) Any preference, for practical reil sons, to mine certain ore blocks.

It is possible to express the variables in a system of linear equations and linear inequalities and a linear progmmmiog technique can be used for finding the best solution. In the objective function deviations from certain values, as, for instance, the carat production, (he avera~ s tOI)C s ize or

71

average hauling distance or a combiDation of these, are minimized.

The calculation of statistically IDOre accurate ore blocks, as described earlier in tbis paper, comhioed with the linear progranuning technique for production planning, will make it possible to control mining production more precisely than W1IS possibLe in the past.

ACKNOWLEDGE MENTS

The author is grntefuilo:

(i) The Consulting Engineer and the Consulting GeoLogist of the Angle American Corporation of South Africa Limited, and the General Manager of the Consolidated Diamond Mines of South West Africa, Limited, for granting permission to publish this paper.

(ii) Professor H. S. SicheL for his invaluable advice and assistance in the p~paration of this paper. M r W. K. Hartley and Mr C. B. &!wards for cons tructive criticism and Dr G. C. Stocken for discussion of the geology.

( iii) Mr R. 1. Nell for the programming, Mr I. Morrow for the compilation of tbe drawings aed Mrs Y. Roberts for preparing the manuscript,

REFERENCES

HALLAM, C. D. (1964). The geology of the coastal diamond deposif! of Southern Africa. The geology 0/ some ore depos(ts in Southern A/rica. vol. H, pp. 671-728. HA5TINGS, C. (1955). Approximations/or digital computers. pp. 187. Princeton University Press, Princeton, New Jersey. SIa-J1lL, H. S. ([966). The estimate of ~ans and associated confidence limits for small samples from lognonna! populatiol'L'l. Symposium on Mathematical Statistics Rnd Computer ApplJcations in Ore Valuation, JohanllC!lbura. March, 1%6. pp. 105-122. SICHEL, H. S. (1972). Statistical valuation of diamondiferoU:!l deposil8. Tenth mternational Symposium on the Application of Computers in the Mineral Industry, Johannesburg. 1972.

72