COMPUTING RESERVES MINERAL DEPOSITS: PRINCIPLES CONVENTIONAL · PDF file ·...

COMPUTING RESERVES OF MINERAL DEPOSITS: PRINCIPLES

CONVENTIONAL METHODS

By Constantine C. Popoff

+ 4 + 4 4 + 4 r 4 + r information circular 8283

UNITED STATES DEPARTMENT OF THE INTERICIR Stewart L. Udall, Secretary

Phoenix Training Center

BUREAU OF MINES Walter R. Hibbard, Jr., Director Reprinted by:

Bureau of Land Management

This publicat~on has been cataloged as follows:

Popoff, Constantine C Computing reserves of mineral deposits: principles and

conventional methods. [Washington] U. S. Dept. of the Inte- rior, Bureau of Mines [ 19661

113 p. illus.. tables . (U. 5. Bureau of Mines. Information circular 8283)

Includes bibliography.

1. Mines and mineral resources. I . T i t le . ( S e r i e s )

TN23.U71 no. 8283 622.06173

U. S. Depc. of the Int. Library

A b s t r a c t ................................................................. I n t r o d u c t i o n .............................................................. Acknowledgments .......................................................... P a r t 1 . . P r i n c i p l e s ..................................................... ............................................................. General

S i g n i f i c a n c e of computations ................................... Requirements ................................................... C r i t e r i a f o r method s e l e c t i o n ..................................

Computing r e s e r v e s procedure ........................................ Analysis of exp lora t ion d a t a ................................... Procedure ...................................................... Main elements ..................................................

Bas ic assumptions .................................................... P r i n c i p l e s of i n t e r p r e t a t i o n ........................................ ........................................ Rule of g radua l changes

Mathemat ica l procedure .................................... Graphic procedure .........................................

Rule of n e a r e s t p o i n t s . o r e q u a l i n f l u e n c e . .................... Case of two underground i n t e r s e c t i n g workings ............. Case of two p a r a l l e l workings ............................. General c a s e of underground workings ...................... Case of e q u i l a t e r a l t r i a n g l e .............................. Case of an obtuse t r i a n g l e ................................ General c a s e ..............................................

Geologic and mining i n f e r e n c e .................................. Rule of g e n e r a l i z a t i o n ......................................... V a r i a b i l i t y w i t h i n mineral d e p o s i t s ............................ Weighting ...................................................... .................................................... A p p l i c a t i o n

Computations ........................................................ ................................................ Basic parameters Thickness and a r e a .............................................

Planimeter ing ............................................. Templates ................................................. Geometric.computations .................................... I n d i r e c t methods ..........................................

Volume ......................................................... Weight .........................................................

........................................... Tonnage f a c t o r s S p e c i f i c g r a v i t y .......................................... ....................................... Conversion formulas

Grade .......................................................... E r r o r s ..............................................................

Accuracy v e r s u s p r e c i s i o n ...................................... Errors of i n t e r p r e t a t i o n o r analogy ............................ Tschnical .... ................................................... .................................................... Random

Eiased ....................................................

CONTENTS.. Continued PaRe

..................................................... A n a l y t i c a l P a r t 2 . . Conventional.methods ........................................... .................................. Genera l f e a t u r e s and c l a s s i f i c a t i o n

Average f a c t o r s and a r e a methods .................................... Assumptions and c h a r a c t e r i s t i c s f e a t u r e s ....................... .............................................. Method of analogy ........................................... Average f a c t o r s ....................................... S t a t i s t i c a l f a c t o r s Method of geo log ic blocks ...................................... ..................................................... Advantages Appl ica t ion ....................................................

Mining blocks method ................................................ Block exposed on four s i d e s by underground workings ............ .............. Determining average f a c t o r s f o r each working

Determining average f a c t o r s f o r each b l o c k ................ Block exposed on t h r e e s i d e s by underground workings ........... Block exposed on two s i d e s .....................................

Underground workings on two l e v e l s ........................ I n t e r s e c t i n g underground workings .........................

Block exposed on one l e v e l and i n t e r s e c t e d a t dep th by d r i l l i n g .................................................... Appl ica t ion ............................................... C r o s s - s e c t i o n methods P r i n c i p l e s and requirements .................................... Standard method.. p a r a l l e l s e c t i o n s ............................. ................................................. Procedure

Volume computations ....................................... ...................................... Tonnage computations

The s tandard method f o r n o n p a r a l l e l s e c t i o n s ................... Angle l e s s than 10 degrees ................................ ~ n g l e g r e a t e r than 10 degrees .............................

.................................................. Linear method ..................................................... Advantages .................................................... Appl ica t ion

Method of i s o l i n e s .................................................. P r i n c i p l e s and formulas ........................................ Requirements. advantages. and l i m i t a t i o n s ......................

.................................................... Appl ica t ion Method of t r i a n g l e s .................................................

P r i n c i p l e s and formulas ........................................ Procedure ...................................................... .................................. Laying o u t the t r i a n g l e s

Determining a reas of i n d i v i d u a l t r i a n g l e s ................. ................................... S t u d i e s by d i f f e r e n t a u t h o r s

M o d i f i c a t i o n s .................................................. D i s t i n c t i v e f e a t u r e s ........................................... Appl ica t ion ....................................................

Method of polygons .................................................. P r i n c i p l e s .....................................................

CONTENTS.. Continued Page

Procedure and cons t ruc t ion of polygons ......................... 85 .................................... Case of v e r t i c a l holes 86 Case of l i nea r workings ................................. 87

Pr inc ipa l formulas ............................................. 87 I r r e g u l a r d i s t r i b u t i o n of d r i l l ho les ..................... 87 Regularly spaced d r i l l ho l e s .............................. 88

Requirements. advantages. and l i m i t a t i o n s ...................... 88 Applicat ion .................................................... 89

Combined methods ................................................. 90 Source of e r r o r s i n reserve computations ............................ 92 .................................................................. Sunnnary 92 ............................................................. Bibliography 96

.......... Appendix A . - English and metr ic systems and conversion f a c t o r s 102 Appendix B . - Usage of var ious methods f o r r e se rve computations f o r ....................................... s o l i d mineral depos i t s i n U.S.S.R 103 Appendix C . - A comparison of reserve computations made by va r ious ..................................................... methods (U.S.S.R.). 104 ................................................... Appendix D . - Formulas 105 ................................................... Appendix E . - Glossary 113

ILLUSTRATIONS Fig.

1 . Transforming a t r u e mineral body i n t o an imaginary a u x i l i a r y one ... 2 . True. hor izonta l . and v e r t i c a l thicknesses.. a n a l y t i c a l r e l a t i o n s h i p 3 . Ana ly t i ca l i n t e r p r e t a t i o n of va lues between two ad jo in ing s t a t i o n s . 4 . Finding t h e loca t ion of point D with thickness 5 f e e t by means of

vec tors .......................................................... 5 . Finding thickness td f o r point D ................................... 6 . Finding thickness tc f o r point C and f ind ing point D f o r thickness

t, = 5 f e e t by means of a s p e c i a l template ....................... 7 . I n t e r p r e t a t i o n of values between two adjo in ing holes i n s e c t i o n .... 8 . Template o r guide t o f ind the midpoint between two po in t s .......... 9 . Angle b i sec to r manner of i n t e r p r e t a t i o n of values between two

i n t e r s e c t i n g underground workings ................................ 10 . ~ n t e r ~ r e t a t i o n of va lues between two p a r a l l e l underground workings . 11 . _ I n t e r p r e t a t i o n of values for t h e c a s e of p a r a l l e l and i n t e r s e c t i n g

workings ......................................................... ........ 12 . Construction of a r eas of in£ luence i n equi l a t e r a 1 t r i a n g l e s 13 . Perpendicular b i s e c t o r versus ang le b i sec to r manner of cons t ruc t ing

a r e a s of in f luence i n obtuse t r i a n g l e s ........................... 14 . Correct cons t ruc t ion of areas of in£ luence (polygons) by

perpendicular b i sec to r s . . ........................................ .............. 15 . Inco r rec t cons t ruc t ion of polygons by angle b i s e c t o r s

16 . Areas of inf luence f o r q u a d r i l a t e r a l f i g u r e s . (Rule of g rav i ty . ) . . 17 . Areas of in£ luence f o r q u a d r i l a t e r a l f i gu res . (Rule of nea re s t

po in ts . ) ......................................................... .............................. 18 . Areas of in f luence f o r a square block

ILLUSTRATIONS.. Continued F i g .

............................. Area of inf1uenc.e f o r an i s o l a t e d h o l e Geologic i n t e r p r e t a t i o n of a r e a s o f i n f l u e n c e between two

a d j o i n i n g s t a t i o n s ............................................... C o n s t r u c t i o n of geologic b locks on t h e b a s i s of s t r u c t u r a l changes . Square p a t t e r n template ............... = ............................ Dotted p a t t e r n template ............................................ P a r a l l e l l i n e s template ............................................ Trapezoid formula .................................................. ................................................... Trapezo ida l r u l e Simpson's r u l e f o r determining a r e a s ............................... Accuracy and p r e c i s i o n of chemical a n a l y s e s ........................ ................... Ari thmet ic average method of computing t h i c k n e s s Geologic b locks method ............................................. Mining b locks exposed on f o u r s i d e s ................................ Mining b l o c k s exposed on t h r e e s i d e s ............................... Mining b l o c k s exposed on two s i d e s ( v e i n t h i c k n e s s l e s s than the

width of workings) ............................................... Mining b l o c k s exposed on two s i d e s ( v e i n t h i c k n e s s more than t h e

width of workings) ............................................... Mining b locks exposed by d r i f t and d r i l l h o l e s ..................... Block l a y o u t by c r o s s - s e c t i o n methods .............................. Cross - sec t ion methods.. s t andard and l i n e a r ......................... Standard c r o s s - s e c t i o n method ( p a r a l l e l s e c t i o n s ) .................. Frustum formula versus mean-area formula ........................... Truncated wedge.. s tandard c r o s s - s e c t i o n method ..................... Blocks between p a r a l l e l s e c t i o n s . I n f l u e n c e o f t h e shape of a r e a s

on volume computations ........................................... Cons t ruc t ion of a u x i l i a r y a r e a R f o r Bauman's formula .............. S e c t i o n and l i n e a r reserves..cros s .sect ion methods ................. Standard c r o s s - s e c t i o n method f o r volume computations.. n o n p a r a l l e l

s e c t i o n s ......................................................... Linear method.. b lock between n o n p a r a l l e l s e c t i o n s .................. Method of i s o l i n e s ................................................. Isopach and i sograde maps f o r r e s e r v e computations.- method of

i s o l i n e s ......................................................... Method of t r i a n g l e s ................................................ Two manners of c o n s t r u c t i o n of t r i a n g u l a r pr isms f o r a r e c t a n g u l a r

pr ism ............................................................ ................................................. Method of polygons Regula r ly spaced d r i l l holes.. method of polygons ...................

TABLES * P r i n c i p l e s of i n t e r p r e t a t i o n of e x p l o r a t i o n d a t a used i n

c o n s t r u c t i o n of b l o c k s and r e s e r v e computations .................. 22 Technical e r r o r s i n determining b a s i c parameters ................... 33 Permiss ib le average f o r random t e c h n i c a l e r r o r s i n chemical ......................................................... a n a l y s e s 34 Usage of v a r i o u s methods f o r r e s e r v e computations f o r m e t a l mines

i n Uni ted S t a t e s ................................................. 37 Determination of a v e r a g e th ickness and average g rade f o r a b lock .................................. by a r i t h n e t i c average procedure 40 Determination of thickness-weighted average g rade f o r a b lock ...... 40 .... Computation of r e s e r v e s and average f a c t o r s f o r t h e e n t i r e body 4 1 Reserve computations.. method of analogy ............................ 42 Determinat ion of a r i t h m e t i c average of f a c t o r s f o r i n d i v i d u a 1

b locks ........................................................... 46 R e c a p i t u l a t i o n of r e s e r v e s f o r m i n e r a l body (by c a t e g o r i e s ) and

de te rmina t ion of average grade ................................... 47 Computation o i r e s e r v e s by s t a n d a r d method of c r o s s s e c t i o n s ....... 57 Appl ica t ion of v a r i o u s formulas i n computing volumes of s o l i d

bodies i n s t a n d a r d c r o s s s e c t i o n method .......................... 71 Determination of average a r . i thmet ic grade f o r each t r i a n g l e by ................................................ t r i a n g u l a r method 80 Determination of th ickness-weighted average grade f o r each

t r i a n g l e by t r i a n g u l a r method .................................. 80 Computation of r e s e r v e s by polygonal method ........................ 86

THIS PAQE r INTENT SONALLY

Bh56gEK

COMPUTING RESERVES OF MINERAL DEPOSITS: PRINCIPLES A N D CONVENTIONAL METHODS

Constontine C. Popoff '

ABSTRACT

This r e p o r t reviews and analyzes, by a s imple a n a l y t i c a l and log ica l reasoning, t h e convent ional methods of r e s e r v e computations of mineral depos i t s described i n va r ious domestic and fo re ign pub l i ca t ions . It brings together , formulates , and eva lua tes t h e p r inc ip l e s underlying i n t e r p r e t a t i o n of explora- t i o n da t a ; and t i e s such p r inc ip l e s t o t h e proposed c l a s s i f i c a t i o n of methods. The m a t e r i a l i s discussed i n s u f f i c i e n t d e t a i l t o a l low genera l appl ica t ion .

INTRODUCTION

Computation of reserves i s recognized by t h e minera l industry as a d i s - t i n c t ope ra t ion of increas ing importance i n t h e eva lua t ion of mineral depos i t s i n a l l s t ages of t h e i r dwelopment. Previous ly , va lua t ion was based on f a c t s , experience, and i n t u i t i o n ; methods have inproved because our knowledge of min- e r a l d e p o s i t s , sampling, and mining techniques has increased .

O r i g i n a l l y , computation methods followed p r a c t i c e s of ea r th excavation and road cons t ruc t ion , both standard surveying opera t ions . Advances i n ear th sc iences and engineering r e su l t ed i n t h e modi f ica t ion of o ld and introduct ion of new methods.

The purpose of t h i s i nves t iga t ion i s t o review some of the common methods and t h e i r modi f ica t ions used in r e se rve computations of mineral deposi ts . The scope of t h i s paper i s l imited t o s o l i d minera l depos i t s ( t h a t i s , metal, non- m e t a l l i c , c o a l , and o i l s h a l e ) , because t h e background da ta required, procedure, and methods f o r water , o i l , and gas a r e d i s s i m i l a r . An attempt i s made t o sys temat ize and s tandard ize the methods and terminology.

For convenience, the paper i s d iv ided i n t o two pa r t s . The f i r s t , "Pr inc ip les" , d e a l s with assumptions and s c i e n t i f i c p r inc ip l e s underlying t h e

1 Former mining en.gineer , Bureau of Mines, Area V I I , S e a t t l e Office of Mineral Resources, S e a t t l e , Wash.

Work on manuscript completed February 1965.

use of var ious methods, and provides a general d i scuss ion of t he elements of computations, procedure, and e r r o r s of i n t e r p r e t a t i o n . The second p a r t , "Conventional Methods1', covers the following methods and t h e i r modi f ica t ions : average f a c t o r s and area (analogous and geologic b locks ) , mining b locks , c ross s e c t ions ( standard, l . inear, and i s o l i n e s ) , t r i a n g u l a r and po lygona 1 prisms, and combinations of these.

The t e x t of ten r e f e r s d i r e c t l y t o o r e depos i t s , because the problem of computing t h e i r reserves i s genera l ly more complicated due t o d i v e r s i t y of form and s i z e of mineral bodies and i r r e g u l a r d i s t r i b u t i o n of va lues . The same methods a r e used f o r coa l and nonmetallic depos i t s .

S t a t i s t i c a l ana lys i s is a va luable t o o l of research f o r a l l t h e methods of computations. Application of var ious methods of s t a t i s t i c a l a n a l y s i s t o sampling and explorat ion data i s under cont inuing i n v e s t i g a t i o n by t h e Bureau of Mines. These methods and the use of computers f o r r e s e r v e computations a r e discussed by the Bureau and o ther s c i e n t i s t s i n s e v e r a l r ecen t publ ica t ions . They a r e beyond the scope and i n t e n t of t h i s paper.

Reference i s made t o "explorat ion" workings f o r b r e v i t y ; however, the t e x t app l i e s t o mineral depos i t s i n a l l s tages of exp lo ra t ion , development, and explo i ta t ion . I n the l a t t e r ca se , computations a r e p a r t i c u l a r l y c r i t i c a l from t h e s tandpoint of economics b u t , once accepted, a r e usua l ly subs id ia ry and rou t ine operat ions.

ACKNOWLEDGMENTS

Gratefu l acknowledgment i s made t o J. A. Pa t t e r son , a s s i s t a n t ch ief of Ore reserve Branch, U.S. Atomic Energy Commission, Grand Junc t ion , Colo., f o r the opportunity t o study unpublished ma te r i a l on r e s e r v e computations.

PART 1. - PRINCIPLES

Genera 1

Signi f icance of Computations

The purpose of reserve computations of a mineral body i s t o determine t h e quan t i t y , the q u a l i t y , and t h e amenabili ty t o conrmercial e x p l o i t a t i o n of raw ma te r i a l (o re , rock, coa l , e t c . ) . Computations a r e made during a l l s t ages of the l i f e of a mining en te rp r i s e from discovery t o robbing p i l l a r s and c los ing . They a r e the most responsible and i r r ep l aceab le t a sks i n t he va lua t ion of a mineral deposi t . Efficiency i n ex t r ac t ion and product iveness is impossible without accura te reserve computations.

Reserves a r e computed t o determine the ex t en t of exp lo ra t ion and development; d i s t r i b u t i o n of va lues ; annual ou tput ; probable and poss ib le product ive l i f e of the mine; method of ex t r ac t ion and p lan t des ign ; improvements i n ex t r ac t ion , t reatment , and processing; and requirements f o r c a p i t a l , equipmer Labor, power, and mater ia l s . Such computations a r e used t o a s s i s t developmen. planning; t o determine production cos t s , e f f i c i ency of ope ra t ions , and mining

l o s s e s ; f o r q u a l i t y c o n t r o l ; f o r f i n a n c i n g mining v e n t u r e s ; f o r s a l e , pur- c h a s e , and c o n s o l i d a t i o n o f companies; t o d e t e r m i n e t h e p roduc t ion c o s t per u n i t of a marke tab le p r o d u c t ; f o r a c c o u n t i n g purposes such a s d e p l e t i o n and d e p r e c i a t i o n ; and i n some S t a t e s f o r t a x purposes .

Requirements

No computations a r e j u s t i f i e d u n l e s s c a l l e d f o r and u s e d ; they should be made when r e q u i r e d . The i d e a l method shou ld be s i m p l e , r a p i d , r e l i a b l e , con- s i s t e n t wi th t h e c h a r a c t e r of t h e m i n e r a l body and a v a i l a b l e d a t a , and s u i t a - b l e f o r r a p i d checking. Computations a r e expec ted t o b e inexpens ive when compared w i t h t h e c o s t of e x p l o r a t i o n and development, and t h e r e f o r e , more complex methods a r e sometimes j u s t i f i a b l e , p a r t i c u l a r l y when l abor - sav ing d e v i c e s ( c a l c u l a t o r s and computers) a r e a v a i l a b l e . I n s e l e c t i n g a method, t h e p e c u l i a r i t i e s and conveniences of automat ion should be c o n s i d e r e d , a s w e l l a s t h e magnitude and accuracy requ i red .

The method should b e s e l e c t e d c a r e f u l l y , procedures worked o u t i n d e t a i 1, and computat ions made a c c u r a t e l y . Formulas shou ld be s imple . P roper ly s e l e c t e d procedures w i l l f a c i l i t a t e t h e p r o c e s s o f computat ions and provide t h e same degree of accuracy a s more compl ica ted methods.

O b j e c t i v e t r ea tment o f f a c t u a l d a t a i s c o n s i d e r e d by many e a r t h s c i e n - t i s t s t h e most important requirement . Harding, f o r example, s t a t e s t h a t h i s s t u d i e s and formulas were provided by a d e s i r e t o f i n d "a method of c a l c u l a t - i n g which e l i m i n a t e s a l l f a c t o r s of t e s t and judgment and r e s t s on pure mathemat ics , . . .a method which can be handled a lmost e n t i r e l y by a c a l c u l a t i n g mzxhine" (l3) .>

Computations should a l s o meet t h e purpose of t h e v a l u a t i o n and , when a p p r o p r i a t e , i l l u s t r a t e t h e d i s t r i b u t i o n of v a r i a b l e s .

The r e l i a b i l i t y of r e s e r v e computat ions depends c h i e f l y on t h e accuracy and completeness of our knowledge of t h e m i n e r a l d e p o s i t . It a l s o depends on assumpt ions accep ted f o r i n t e r p r e t i n g t h e v a r i a b l e s , on boundar ies of minera l b o d i e s , on accuracy of a v e r a g e s , and on mathemat ica l fo rmulas . Requirements f o r t h e q u a n t i t y and t h e d e n s i t y of o b s e r v a t i o n s f o r a c e r t a i n ca tegory of r e s o u r c e s depends p r i m a r i l y on t h e s i z e and type of t h e m i n e r a l d e p o s i t .

During t h e l a s t s e v e r a l decades t h e accuracy of computing r e s e r v e s has g r a d u a l l y improved. Th i s was made p o s s i b l e by o u t s t a n d i n g advances i n t h e f i e l d of economic geology; inc reased s p e c i a l i z a t i o n ; improvements i n explora- t i o n , sampl ing, mining, and v a l u a t i o n ; b e t t e r i n t e r p r e t a t i o n of f i e l d informa- t i o n ; use of s t a t i s t i c a l a n a l y s i s ; and more e f f i c i e n t management.

The growing u s e of d a t a - p r o c e s s i n g machines h a s made i t p o s s i b l e t o r e c o r d l a r g e amounts of e x p l o r a t i o n d a t a i n t h e form of punch c a r d s , punched t a p e , magnetic d i s k , or magnet ic t a p e . The computers pe rmi t a p p l i c a t i o n of

s u n d e r l i n e d numbers i n pa ren theses r e f e r t o i t e m s i n b ib l iography a t t h e end of t h i s r e p o r t .

two o r more c o n v e n t i o n a l methods and produce improved a c c u r a c y , i n c r e a s e d speed, and l a b o r and c o s t savings i n r e s e r v e computa t ions . The t e c h n i q u e s and advantages of t h e u s e of computers a r e d i s c u s s e d i n s e v e r a l r e c e n t p u b l i c a t i o n s (2, 2, 23-25, 38).

C r i t e r i a f o r Method S e l e c t i o n

I n g e n e r a l , s e l e c t i n g a method f o r r e s e r v e computa t ions depends upon t h e geology of t h e m i n e r a l d e p o s i t , e x p l o r a t i o n method, a v a i l a b i l i t y and r e l i a b i l - i t y of f a c t u a l d a t a , purpose of computat ions , and t h e r e q u i r e d d e g r e e o f accuracy.

I f computat ions a r e pre l iminary o r a r e r e q u i r e d i m e d i a t e l y , s i m p l e methods , which do no t demand c o n s t r u c t i o n of s p e c i a l maps, a r e s e l e c t e d . I f com- p u t a t i o n s a r e f o r mine des ign , t h e method s e l e c t e d depends on t h e contemplated mining system. The c u t o f f grade, r ecovery , d i l u t i o n , e f f i c i e n c y of equipment and l a b o r , and c o s t per u n i t of ou tpu t v a r y w i t h t h e sys tem of e x t r a c t i o n . A s imple method may b e adequate f o r open p i t o p e r a t i o n s h e n s e l e c t i v e e x t r a c - t i o n of was te o r weakly minera l ized rock i s excluded. Computations of r e s e r v e s f o r a bedded d e p o s i t i s l e s s complex than f o r h i g h - g r a d e , small volume, s t o c k - t y p e d e p o s i t s with i r r e g u l a r l y d i s t r i b u t e d v a l u e s .

E x p l o r a t i o n , whether random, by g r i d , o r by c r o s s - s e c t i o n l i n e s , may a l s o i n £ luence method s e l e c t i o n . It i s o f t e n d e s i r a b l e d u r i n g e x p l o r a t i o n t o use a method p e r m i t t i n g s t ep -by-s tep a d d i t i o n of r e s e r v e s t o p r e v i o u s f i g u r e s i n s t e a d of p e r i o d i c recomputations.

The n a t u r e of t h e v a r i o u s methods should be c a r e f u l l y c o n s i d e r e d . Simple methods a r e p r e f e r r e d , but more compl icated ones may b e j u s t i f i e d . Both extremes, o v e r s i m p l i f i c a t i o n leading t o complete d i s r e g a r d of t h e g e o l o g i c n a t u r e o f t h e d e p o s i t and overcompl icat ion l e a d i n g t o unwarranted p r e c i s i o n , expense, and even i m p r a c t i c a b i l i t y , shou ld be avo ided . The q u e s t i o n of maxi- mal use of a l l f a c t u a l d a t a c o l l e c t e d i n t h e p r o c e s s of e x p l o r a t i o n i s a n important c o n s i d e r a t i o n . Poor p lanning and o v e r e x p l o r a t i o n r e s u l t s i n exces- s i v e d a t a n o t necessa ry f o r t h e accep ted accuracy o f computat ions .

Computing Reserves Procedure

Analysis of E x p l o r a t i o n Data

Reserve computat ions of a minera 1 d e p o s i t i s a t e c h n i c a l t a s k , c o n s i s t i n g of s e v e r a l o p e r a t i o n s . The importance of f o l l o w i n g a d e f i n i t e p rocedure , p roper ly s e l e c t e d f o r a c e r t a i n d e p o s i t , cannot b e overemphasized (5). The o p e r a t i o n s i n o r d e r of t h e i r usua l execu t ion a r e g e o l o g i c a p p r a i s a l , explora- t i o n and sampling methods a p p r a i s a l , e x p l o r a t i o n d a t a a p p r a i s a l , d e l i n e a t i o n of t h e m i n e r a l body, and s e l e c t i o n of an a p p r o p r i a t e method f o r computat ions .

The importance of t h e knowledge of t h e geology of t h e d e p o s i t f o r t h e unders tand ing of t h e s i z e , shape, and g rade d i s t r i b u t i o n , and f o r i n t e r p r e t a - t i o n of e x p l o r a t i o n d a t a has been emphasized by many s c i e n t i s t s (2, 34). Geologic a p p r a i s a l i n c l u d e s o b t a i n i n g , check ing , and p r e s e n t i n g e x p l o r a t i o n

data i n t h e form of graphs, t a b l e s , maps and sec t ions of appropr ia te s c a l e , and assuming a working hypothesis on t h e o r i g i n of mine ra l i za t ion . The explo- r a t i o n method; t h a t i s , t h e kind and dens i ty of workings and sampling, i s s tud ied t o determine the adequacy and accuracy of the da t a from the standpoint of geology, geometric con£ igura t ion of t he rninera 1 bodies , d i s t r i b u t i o n pat- t e r n of v a r i a b l e s , e r r o r s , and category of reserves . Such an a p p r a i s a l ' i s o f t e n supplemented by s t a t i s t i c a l a n a l y s i s and by comparison with other depos i t s s i m i l a r i n type and form.

The a n a l y s i s of explorat ion da t a , o f t en t h e most neglec ted s t e p i n valua- t i o n , i s accomplished by def in ing i n s i d e and ou t s ide parameters of economi- c a l l y minable po r t ions of the mineral body; by determining the prec is ion of measurements and ana lyses ; and by determining whether t h e amount of explora- t i o n of va r ious po r t ions of the mineral body meets t h e requirements f o r comput ing r e se rves of a c e r t a i n category.

Procedure

For r e s e r v e computations the minera l body i s f i r s t de l inea t ed and then subdivided by s e v e r a l methods i n t o segments or blocks of va r ious degrees of r e l i a b i l i t y .

The usua l procedure f o r volume computations i s t o s u b s t i t u t e graphica l ly t he i r r e g u l a r shape of t h e mineral body by an imaginary and a u x i l i a r y one with base s u r f a c e ly ing i n t h e plane of a plan o r l ong i tud ina l s e c t i o n ; the other su r f ace , i r r e g u l a r i n form, shows d i s t r i b u t i o n of th icknesses ( f i g . 1 ) . This a u x i l i a r y body i s then replaced by one o r s eve ra l simple s o l i d f i g u r e s , vo l - umes of which can be computed by geometric formulas.

Divis ion of t h e minera l body i n t o blocks i s done according t o a se lec ted method, so t h a t each block can be d i r e c t l y r e l a t e d t o one or a s u i t e of f a c t u a l explora t ion values.

The r e se rves of t h e e n t i r e body a r e computed by determining areas and volumes f o r each block, convert ing block volumes t o tonnages of raw mineral m a t e r i a l , determining average grades and tonnages of va luab le components, and f i n a l l y , t abu la t ing t h e r e s u l t of blocks of t he same category and, i f poss ib l e , a s se s s ing t h e r e l i a b i l i t y of computations.

Main Elements

Reserve computations r equ i r e a knowledge of the dimensional and q u a l i t a - t i v e f e a t u r e s of t h e mineral body. This knowledge i s gained d i r e c t l y by observa t ions (measurements, chemical ana lyses , and t e s t s ) and i n d i r e c t l y by assump- t i o n s , i n t e r p r e t a t i o n s , and computations. A l l va lues of t h e bas i c block elements, th ickness , length, breadth, weight f a c t o r , and grade, whether they a r e s i n g l e observa t ions o r computed averages, may be presented on maps by numbers pinpointed f o r a d e f i n i t e l oca t ion , or a s a l i n e wi th . the numerical length p l o t t e d t o s c a l e .

X The system s e l e c t e d f o r meas- u r i n g l i n e a r d i s - tances , a r e a s , volumes, and weights should be followed throughout. Uni t s of measure and weight and convers ion f a c t o r s f o r English and met r i c systems a r e given i n appendix A. When s e l e c t i o n is

Y p o s s i b l e , t h e met r i c sys tem i s p r e f e r a b l e ; i t saves time and reduces t h e chance

A X of e r r o r .

The formulas f o r a l l methods a r e based on corn-'

8 put ing s o l i d s with t h e i r bases con- s t r u c t e d i n t h e

1 plane; v e r t i c a l th ickness i s used f o r h o r i z o n t a l p lan; h o r i z o n t a l th ickness f o r ver- t i c a l o r longi tu- d i n a l s e c t i o n s ; and t r u e th ickness

FIGURE 1. - Transforming a True Mineral Body Into an Imaginary Auxiliary f o r i n c l i n e longi- One. A, Vertical section of true body; B,vertical section of t u d i n a l s e c t i o n s distorted auxiliary body. drawn i n t h e plane

of the d i p of t h e mineral body. The r e l a t i o n s h i p s between t h e t r u e , h o r i z o n t a l , and v e r t i c a l th icknesses a r e :

t t r = th s ing = tocosg ( 1)

where 9 i s t h e t r u e d i p of t h e body ( f i g . 2A).

I n r e s e r v e computations, t h e t r u e s t r i k e and t h e t r u e d i p of t h e d e p o s i t , or i t s p o r t i o n under cons idera t ion , a r e determined f i r s t of a l l . Cor rec t ions f o r d i p taken i n a d i r e c t i o n no t perpendicular t o t h e s t r i k e a r e made from s p e c i a l l y prepared t a b l e s or by a p r o t r a c t o r desc r ibed i n f i e l d geology textbooks (ll, 33).

FIGURE 2 , - w

True, Horizontal and Vert ical Thicknesses-Analytical Relationsh th sin @ = t, cos 8 ; B , strike correction - ttl = t cos a (p = 90° '? p dip correction (a = OO); D, general case - block dtagrarn.

ip. A, t t r =

e = 0); c,

I n e x p l o r a t i o n , th icknesses u s u a l l y a r e measured a t o b l i q u e d i r e c t i o n s t o t h e t r u e s t r i k e and t h e t r u e d i p of t h e body. Such a p p a r e n t t h i c k n e s s e s a r e c o r r e c t e d by g r a p h i c a l means, by t r i g o n o m e t r i c formulas , and by c h a r t s and t a b l e s .

When t h i c k n e s s i s measured a t a n o b l i q u e a n g l e t o t h e s t r i k e , i t 'is co t - r e c t e d by cos a i n a s imple c a s e of a v e r t i c a l o r e body and h o r i z o n t a l work- i n g s , where a i s an a n g l e between t h e a p p a r e n t t h i c k n e s s p l a n e and a p l a n e perpend icu la r t o t h e s t r i k e ( f i g . 22).

When t h e t h i c k n e s s i s measured a t an o b l i q u e a n g l e t o t h e t r u e d i p of the d e p o s i t , t h e t r u e t,. h o r i z o n t a l t h , and v e r t i c a l t, t h i c k n e s s e s i n a s imple c a s e of a equal 0" ( f ~ g . 22) a r e

and

s i n ( B + 8) t h = t a p s i n @ '

s i n (B + 8) tv = t a p

C O S 0 '

where 0 i s an a n g l e i n t e r s e c t i n g t h e body i n t h e p lane of t h e t r u e d i p ; P - t h e d i p of t h e body; and t,, - apparen t th ickness .

I n a g e n e r a l c a s e when t h e d i p of t h e body and h o l e i n c l i n a t i o n a r e uncon- formable ( h o l e c r o s s i n g the body a t s h a r p a n g l e t o t h e s t r i k e and t o t h e d i p ) , t h e th icknesses a r e found by formulas ( f i g . 2 s )

tt, = t,, cos $ c o s 0 (COS a t a n + t a n a ) , o r

= t,, (cos a s i n B cos 0 t cos B s i n a ) , ( 5 )

th = tap (COS a cos 0 + c o t a n i! s i n 0 ) , ( 6 )

and & = t a p cos 0 ( c o s a t a n B + t a n e ) , ( 7 )

where a is an a n g l e between t h e p lane of t h e d i p and t h e p lane of h o l e d i r e c t i o n ;

6 - d i p of body; and 0 - a n g l e of t h e h o l e i n t e r s e c t i n g t h e body.

Bas ic Assumptions

whatever method is used f o r r e s e r v e computations s e v e r a l assumptions a r e taken f o r granted. The chief one i s t h a t t h e b a s i c e lements of a m i n e r a l body observed or e s t a b l i s h e d a t any s t a t i o n ( s u r f a c e exposure, d r i l l h o l e , o r underground workings) change or extend t o t h e a d j o i n i n g a r e a accord ing t o an appro- p r i a t e p r i n c i p l e of i n t e r p r e t a t i o n of e x p l o r a t i o n d a t a . It i s assumed a l s o t h a t obse rva t ions a r e m d e i n conformity with t h e n a t u r e of a g iven depos i t

Hole A C Hole B

4 FIGURE 3. - Analytical Interpretation of Values Between Two

Adjoining Stations (Rule of Gradual Changes).

and t h a t t he samples a r e taken with the same preci- s i o n and a r e representa- t i v e of a se lec ted port ion of t h e mineral body. When observa t ions a r e doubtful o r inadequate i n number, t he r e s u l t s of the compu- t a t i o n s a r e uncertain or erroneous.

Another important presumption i s t h a t the minera l depos i t has been explor ed by an appropr ia te exp lo ra t ion method, and t h a t t h e n e t of workings prove t h e cont inui ty of t h e body. This hypothesis

permits cons idera t ion of any element a s having a cons tan t va lue fo r a block, segment, o r t h e e n t i r e mineral body. Thus, t he problem of computation i s reduced t o determining the volume of a block, segment, o r body by mathematical means.

F i n a l l y , i t i s assumed fo r the purpose of computations, t h a t the t r u e and o f t e n complex form of t h e mineral body can be represented with reasonable accuracy by a hypo the t i ca l body with a base sur face ly ing i n the plan or sec- t i o n . Such an i d e a l i z e d body may embrace t h e e n t i r e d e p o s i t , o r i t may be composed of l a rge segments o r small blocks, each cha rac t e r i zed by a s i n g l e or a s u i t e of recorded v a r i a b l e s .

P r inc ip l e s of I n t e r p r e t a t i o n

The reasoning used i n i n t e r p r e t a t i o n of va r i ab l e s between any two adja- cent observat ions i n a mineral body determines the block cons t ruc t ion and the accuracy of computations. These p r inc ip l e s a r e a n a l y t i c a l , n a t u r a l or i n t r i n - s i c , and empirical. The a n a l y t i c a l group includes t h e r u l e of gradual s t rai 'ght l i n e changes of a l l ba s i c elements of a mineral body and t h e r u l e of nearest po in t s , or equal sphere of i n £ luence. Geologic, techno l o g i c , and economic c r i t e r i a make up t h e n a t u r a l or i n t r i n s i c group, and gene ra l i za t ion the empiri- c a l r u l e of i n t e r p r e t a t i o n .

Applicat ion of t h e a n a l y t i c a l and n a t u r a l p r i n c i p l e s of i n t e r p r e t a t i o n a r e l imi ted t o s p e c i f i c condit ions, necessary and s u f f i c i e n t f o r ce r t a in type and s i z e of depos i t s , and f o r c e r t a i n ca t egor i e s of resources .

Rule of Gradual Changes

Mathematical Procedure

According t o t h e r u l e of gradual changes o r law of l i n e a r funct ion, a l l elements of a mineral body t h a t can be expressed numerical ly change gradually and continuously along a s t r a i g h t Line connecting two ad jo in ing s t a t i o n s ( f i g , 3 ) . Let us consider two adjoining s t a t i o n s o r holes A and B with thicknesses

t, and h. Location of a point C on l i n e AB with a given thickness t, may be found a n a l y t i c a l l y and graphica l ly by t h i s r u l e ; v i c e v e r s a , the th ickness t, may be found f o r a given point C by s imi l a r procedures. To loca t e poin t C with given th ickness t, on l i n e AB, t r i a n g l e s A 1 C 2 q and A 1 & B I a r e s i m i l a r , thus

Al C& = AC and A1 & = AB ,

AC = (tc - t l ) AB; ( t 2 - tl

and t o determine th ickness t, f o r a given poin t C ,

G AlG or - = - ( t , - t l ) AC = - BrBz 1 % ( t z - t i ) AB'

AC t o - tl = ; i i i ( t 2 - t , ) ,

and

I n surveying, equations (8) and ( 9 ) a r e known a s formulas of simple i n t e r p r e t a t i o n .

The r u l e of gradual changes can be appl ied t o other parameters of a mineral body such a s grade and weight f a c t o r s , a s we l l a s t o a r e a s , l i n e a r r e se rves , volumes, and tonnages. It may be used a l s o i n de l inea t ing t h e com- merc ia l por t ion of t he deposit and t o determine a given va lue a t an unknown point on the ex tens ion 'of a l i n e beyond known s t a t i o n s . I n p r a c t i c e , i n t e r - po la t ion and ex t r apo la t ion a r e done by graphic means.

Graphic Procedure

To determine by vec tors po in t D with a given thickness td of 5 f e e t , on a l i n e AB with t, equal t o 4

f ~ - ~ f ' f e e t a t s t a t i o n A and

AwA2 B 5- td

s t a t i o n B equal t o t2 - td ,

f2 -7f f equal t o 7 f e e t a t s t a t i o n B , fd-5fr r a i s e a perpendicular from

o r 2 f e e t , and drop a per- A 2 pendicular from s t a t i o n A

4 0 4 - equal t o t, - tl , or 1 f o o t scots . feet ( f i g . 4 ) . Connect po in t s

A2 and & with a s t r a i g h t FIGURE 4. - Finding the Locotion of Point D With Thickness l i n e ; t he i n t e r s e c t i o n of

Five Feet b y Means of Vectors (Rule of Gradual l i n e s A2 & and AB is Changes). poin t D.

FIGURE 5. - F ind ing Thickness

td for Point D (Rule

of Gradual Changes).

FIGURE 6. - F ind ing Thickness t, for Point C and F ind ing Point D for Thickness td = 5 Feet b y Means of a Special Template (Rule of Graduol Changes).

Find ing t h i c k n e s s td f o r any given p o i n t on l i n e AB i s i l l u s t r a t e d by f i g u r e 5. Raise a pe rpend icu la r A& equa l t o tl from s t a t i o n A and a perpen- d i c u l a r B& equa l t o t2 from s t a t i o n B. Connect t h e p o i n t s A2 and & with s t r a i g h t l i n e . The t h i c k n e s s f o r a given p o i n t D on l i n e AB w i l l b e t h e l eng th of a p e r p e n d i c u l a r DLk r a i s e d from D t o t h e l i n e A2& .

For t h e same purpose s p e c i a l t empla tes may be used, c o n s i s t i n g of a s e r i e s of p a r a l l e l l i n e s drawn t o s c a l e on t r a c i n g c l o t h o r engraved on c l e a r p l a s t i c . The l i n e s a r e e q u i d i s t a n t and may be marked by a p p r o p r i a t e u n i t v a l - ues ( f i g . 6 ) . To f i n d , i n our example, t h i c k n e s s t, f o r p o i n t C on l i n e AB, t h e t empla te i s p laced s o t h a t B w i l l c o i n c i d e wi th t h e l i n e marked 7 f e e t . Let u s t u r n t h e t e m p l a t e around p o i n t B u n t i l A i n t e r s e c t s t h e l i n e marked 4 . A supplementary l i n e drawn p a r a l l e l t o o t h e r l i n e s w i l l show t h i c k n e s s f o r p o i n t C e q u a l t o 6.2 f e e t . A p o i n t D wi th t h i c k n e s s of 5 f e e t between t h e v a l - ues of A and B i s found by a s imple i n t e r s e c t i o n of 5.0 f e e t l i n e and l i n e AB.

?

Rule of Nearest P o i n t s , o r Equal I n f luence

According t o t h e r u l e of n e a r e s t p o i n t s , o r "equal s p h e r e of i n £ luence", t h e v a l u e of any p o i n t between two s t a t i o n s i s cons idered c o n s t a n t , e q u a l t o t h e v a l u e of t h e n e a r e s t s t a t i o n . I n a g e n e r a l c a s e of h o l e s A and B wi th th ickness tl and tz , t h e value of each one extends t o t h e midpoin t X between h o l e s ( f i g . 7 ) . I n p r a c t i c e t h e midpoint i s found by t h e i n t e r s e c t i o n of two a r c s of a c i r c l e wi th r a d i i s l i g h t l y more than h a l f t h e d i s t a n c e between t h e s t a t i o n s , o r by a s p e c i a l template ( f i g . 8). Any p o i n t on l i n e AB, except X , i s i n s i d e t h e " l i n e a r in f luence t t of a s t a t i o n A o r B and n e a r e r t o i t than t o t h e a d j o i n i n g one. Thus, t h i s p roper ty g i v e s t h e r u l e i t s name of n e a r e s t p o i n t s . I n t h e s e c t i o n AB the a r e a s of i n f l u e n c e f o r a g iven t h i c k n e s s tl of h o l e A and f o r a g iven th ickness ta of h o l e B a r e shown by d i f f e r e n t p a t t e r n s - ( f i g . 7 ) .

The r u l e of n e a r e s t spheres of i n f l u e n c e f o r

po in t s i s widely used f o r c o n s t r u c t i o n of e q u a l a reas and volumes of ind iv idua 1 workings. The

FIGURE 7. - lnterpretat ion of Values Between Two Adioining Holes in Section (Rule of Nearest Points).

FIGURE 8. - Template or Guide toFind the Midpoint Between Two Points (Rule of Nearest

u Scale, units

a p p l i c a t i o n v a r i e s due t o t h e type and d i s t r i b u t i o n of workings , and whether t h e s e workings a r e p resen ted on maps i n t h e form of d o t s o r l i n e s . S e v e r a l of t h e more common c a s e s a r e descr ibed below.

Case of Two Underground I n t e t s e c t i n ~ Workings

When a d r i f t and a r a i s e i n t e r s e c t i n a p lane of a map, t h e a r e a s of e q u a l i n f l u e n c e of each working are found by b i s e c t i n g t h e a n g l e between them. Any p o i n t on t h e b i s e c t o r i s e q u i d i s t a n t from both workings, and any po in t w i t h i n each a r e a of i n f l u e n c e i s n e a r e r t o t h e a d j o i n i n g working than t o t h e o t h e r ( f i g . 9 ) .

Case of Two P a r a l l e l Workings

A line c o n s t r u c t e d e q u i d i s t a n t between two p a r a l l e l workings , such a s t r e n c h e s , d r i f t s , c r o s s c u t s , r a i s e s , and d r i l l h o l e s , w i l l d i v i d e t h e i n t e r - vening a r e a i n t o two a r e a s of equal i n f l u e n c e , each s a t i s f y i n g t h e p roper ty of n e a r e s t p o i n t s ( f i g . 10).

General Case of Underground Workings

When a block i s developed by d r i f t s and r a i s e s on a l l f o u r s i d e s , t h e area between them i s d iv ided i n t o f o u r a r e a s of in f luence by a combination of a n g l e b i s e c t o r s and p a r a l l e l l i n e s ( f i g . 11). To s a t i s f y t h e p roper ty of

Raise

,'= D r i f t

FIGUR E 9. - Angle Bisector Manner of Interpreta- tion of Values Between Two Inter- secting Underground Workings (Rule of Nearest Points).

FIGURE 10. - Interpretation of VoluesBe- tween Two Parallel Under- ground Workings (Rule of Nearest Points).

1 Raise

Drif t

FIGURE 11. - lnterpretotion of Values for the Case of Parallel and Intersecting Workings (Rule of Nearest Points).

e q u a l i n f l u e n c e - fo r each pa i r of workings i n t h e b l o c k , only t h e i l l u s t r a t e d c o n s t r u c t i o n i s p o s s i b l e .

Case of E q u i l a t e r a l T r i a n g l e

I n an e q u i l a t e r a l t r i a n g l e t h e a r e a s of i n f l u e n c e of each v e r t e x a r e found by c o n s t r u c t i n g perpendicular b i s e c t o r s from t h e midpoint of each s i d e ( f i g . 12A). The i n t e r s e c t i o n of t h e b i s e c t o r s i s e q u i d i s t a n t from t h e v e r - t e x e s ; i t i s t h e c e n t e r of a c i r c l e pass ing through t h e t h r e e v e r t e x e s .

By c o n s t r u c t i n g a n g l e b i s e c t o r s , d i f f e r e n t shaped a r e a s a r e formed ( f i g . 12;) i n comparison with pe rpend icu la r b i s e c t o r s . Th i s manner of d i v i d i n g a t r i a n g l e i s c a l l e d t h e r u l e of g r a v i t y .

Case of an Obtuse T r i a n g l e

I n an o b t u s e , t r i a n g l e , t h e a n g l e b i s e c t o r s w i l l d i v i d e the f i g u r e i n t o t h r e e a r e a s d i f f e r e n t i n shape, b u t equa l i n s i z e ( f i g . 13). The p o i n t of

B i n t e r s e c t i o n of the a n g l e b i s e c t o r s , o r the c e n t e r of t h e g r a v i t y o f t h e t r i a n g l e , i s much c l o s e r t o t h e v e r t e x of t h e o b t u s e a n g l e than t o t h e o t h e r v e r t e x e s . There fore , t h e c e n t e r as

c w e l l a s o t h e r p o i n t s o f t h e t r i a n g l e , a r e incon- s i s t e n t w i t h t h e r u l e o f n e a r e s t p o i n t s .

Areas of in f luence c o n s t r u c t e d by perpendic-

FIGURE 12. - Construction of Areos of Influence in Equi lateral u l a r b i s e c t o r s i n an o b t u s e t r i a n g l e a r e of

Triangles (Rule of Nearest Points). A, Perpen- erent sires, but the dicular bisector manner of construct ing areas of verrexes are e q u i d i s influence (areas are to vertexes A, 6, and C); B, from t h e i n t e r s e c t i o n of angle bisector manner of constructing areas of t h e p e r p e n d i c u l a r s . The influence (areas ore to l ines AB, BC, and AC). a r e a s of i n £ h e n c e o f

such a c o n s t r u c t i o n s a t i s f y t h e p roper ty of t h e r u l e of n e a r e s t p o i n t s .

Genera l Case

Both manners of c o n s t r u c t i o n of a r e a s of i n f l u e n c e a r e used i n computing r e s e r v e s . The ang le b i s e c t o r manner i s l i m i t e d t o workings p resen ted on plans and s e c t i o n s a s l i n e s , such as i n t e r s e c t i n g underground workings ( f i g . 11) .

FIGURE 13. - Perpendicular Bisector Versus Angle Bisector Manner of Constructing Areos of lnfluence in Obtuse Triangles. 11, Perpendicular bisector manner of constructing areos of influence for vertexes A, B, and C; 8, angle bisector manner of constructing areos of inf luence for l ines AB, BC, and AC; C, angle bisector manner of constructing areas of influence for vertexes A, 6, and C. The last method i s Incorrect from the standpoint of rule of nearest points (0 is closer to 6 than to A and C).

When the workings a r e presented on the map a s dots ( d r i l l h o l e s ) , t h e a r e a s of in f luence of each one a r e found by the perpendicular manner of cons t ruc t ion ( f i g . 14). I n t h e l a t t e r case the ang le b i s e c t o r manner of cons t ruc t ion i s i nco r rec t ( f i g . 15). For the q u a d r i l a t e r a l f i g u r e (ABCD) , t h e ang le b i s e c t o r construct1,on may produce two d i f f e r e n t r e s u l t s , depending on how t h e t r i a n g l e s a r e drawn ( f i g . 16) . Construction of a r e a s by perpendicular b i s e c t o r s ( f i g . 17) produces only one solut ion. For f u r t h e r d i scuss ion see s e c t i o n e n t i t led "Method of Polygons".

An a rea of in f luence fo r the ou t s ide perimeter of t h e mineral body, o r f o r an i s o l a t e d h o l e , can be constructed by t h e r u l e of nea re s t p o i n t s , when

H G

L E G E 0 D r i l l

0 D

Case of exploration by vertical drlll holes. Case of exploration by vertical drill holes

FIGURE 14. -Correct Construction of Areas of FIGURE 15. - Incorrect Consfruction of Poly- Influence (Polygons) by Perpen- gons by Angle Bisectors (Rule dicular Bisectors (Rule of Near- of Gravity). est Points).

A ABC and

h ACD

L E G E N D A,gC,D, dri l l holes

A ABD a n d A BCD

Two solutions - depending on constructlon of t r iangles ( both incorrect 1

FIGURE 16. - Areas of Influence for Quadrilateral Figures (Rule of Gravity).

FIGURE 17. - Areas of Influence for Quadriloterol Figures (Rule of Nearest Points).

O n l y one solution ( correct 1

7 7 - 1 77------ a "s tandard" mean r a d i u s of i n f l u e n c e f o r a c e r t a i n ca te -

Extropolot~on of gory of r e s e r v e s and type of areas of influence d e p o s i t i s a c c e p t e d . Such by rule of nearest a r e a s may be c o n s t r u c t e d by a

c i r c l e equa l t o t h e s t andard r a d i u s of i n f l u e n c e ( f i g s . 18 and 19) .

Geologic and Mining I n f e r enc e

When i n t e r p r e t i n g v a r i a - b l e s between two ad jo ining workings , t h e c o n s t r u c t i o n of

L E O E N O segments or b l o c k s of a min- Armo of intlurnee e r a 1 body may be governed by 7 insid. pwimetrr d i r e c t g e o l o g i c , mining, o r

economic c o n s i d e r a t i o n s . I n Area of ln f lurncr a s imple c a s e o f two d r i l l (-1 outside prr lrnrter h o l e s with corresponding th ick-

n e s s e s ti and t, of o r e and a FIGURE 18. - Areos of Influence for a Square Block fault

(Rule of Nearest Points). between them, t h e sphere of i n f l u e n c e ( a r e a s of in f luences

f o r s e c t i o n and p l a n ) may be assigned on b a s i s of geo log ic i n t e r p r e t a t i o n , a s i l l u s t r a t e d i n f i g u r e 20; t h a t i s , t h e t h i c k n e s s tl i s c o n s i g n e d t o o r e Setween t h e h o l e A and t h e f a u l t , and t h e t h i c k n e s s t2 between t h e f a u l t and h o l e B.

FIGURE 19. - Area of Influence for an Isolated Hole.

Vertical f o u l t

L E G E N D

Area of in f luence for A hole

Area of in f luence for B hole

PLAN

FIGURE 20. - Geologic Interpretation of Areas of Influence Between T w o Adioining Stations.

Motives f o r geologic inference inc lude n a t u r a l geologic boundaries due to s t r u c t u r a l f ea tu re s ( sync l ines , a n t i c l i n e s , f a u l t s , or other d i s l o c a t i o n s , changes i n s t r i k e or d i p ) ; changes i n cha rac t e r of mine ra l i za t ion ; thinning out o r p i tch ing of oreshoots ; zoning; weathering; d i f f e r e n t phys ica l proper- t i e s ; heterogenous composition; var ied a l t e r a t i o n ; and presence of detr imental c o n s t i t u e n t s , such a s ash and su l fu r i n coa l .

C o m n technologic, physiographic, and economic grounds f o r in ference i n cons t ruc t ion of blocks a r e topography, th ickness of overburden, r a t i o of overburden t o thickness of mineral body, depth , water l e v e l , mining methods, proc- e s s ing methods, and c o s t of ex t r ac t ion ; a l s o proper ty , s e c t i o n , township, and s t a t e boundaries. An example of block cons t ruc t ion on the b a s i s of s t r u c t u r a l changes i n a phosphate rock deposi t and a v a i l a b i l i t y of ore f o r open p i t mining i s given i n f i g u r e 21.

Scale, feet

F I G U R E 21. - Construction of Geologic Blocks on the Basis of Structural Changes.

Rule of Genera l iza t ion

The r u l e of genera l iza t ion i s a l s o known as t h e empi r i ca l method and, i n i t s extreme, a s t h e r u l e of thumb. It i s used f r equen t ly f o r i n t e r p r e t a t i o n of explora t ion da t a . .In cont ras t with t h e more o b j e c t i v e i n t e r p r e t a t i o n s descr ibed previously, such a r u l e is used r a t h e r a r b i t r a r i l y . It i s o f t e n adapted f o r lack of o ther c r i t e r i a on the b a s i s of l imi t ed experience, o r a s a mat te r of judgment and general ly r e f l e c t s p a s t experience and opinion. .

I n many cases , the use of t he r u l e i s j u s t i f i a b l e and unavoidable. Adapting a d e f i n i t e weight f ac to r f o r r e s e r v e computations from other s imi l a r mineral depos i t s is probably the most common example. Se l ec t ing s p e c i f i c limits f o r t he s i z e of blocks i n c l a s s i f y i n g r e se rves by c a t e g o r i e s f o r ce r - t a i n mineral depos i t s o r assuming f a c t o r va lues f o r r e se rves on t h e b a s i s of production da t a , r a t h e r than d i r e c t l y from widely spaced d r i l l ho l e s with i r r e g u l a r o r doubt fu l values, a r e gene ra l i za t ions . P ro j ec t ing con t inu i ty of a mineral body beyond the outlying workings along the s t r i k e or a t depth and f i x i n g cu tof f boundaries for computations a r e o the r examples.

The following procedure was used f o r ex t r apo la t ion of boundaries of uranium depos i t s between d r i l l h o l e s , some of which c rossed o r e , s t rong ly min- e r a l i zed ground, weakly mineralized ground, and bar ren rock. The cu tof f boundary between two holes was se l ec t ed a s three- four ths o r two-thirds t h e d i s t ance from o r e t o a strongly mineral ized h o l e , one-half t o a weakly min- e r a l i zed h o l e , and one- th i rd t o a barren ho le (63).

Many e a r t h s c i e n t i s t s exerc ise the above p r i n c i p l e by a r b i t r a r i l y reducing a r e a s , average thicknesses , and grades f o r i nd iv idua l blocks and bodies and even co r rec t ing t h e computed r e se rves by sub jec t ive c o r r e c t i o n f a c t o r s .

Var iab i l i ty Within Minera 1 Deposits

The preceding r u l e s and the inferences based on geologic and mining con- d i t i o n s lead t o well-defined methods f o r r e se rve computation. Commonly , however , t h e information ava i lab le i s such t h a t in ferences and the a p p l i c a b i l i t y of t he preceding r u l e s i s not always c l e a r cu t . This may occur i n t h e e a r l i e r s tages of explora t ion where the amount of information i s sparse . It may a l s o occur i n any s t age of development i f the n a t u r a l v a r i a b i l i t y wi th in the min- e r a l depos i t i s r e l a t i v e l y high. This v a r i a b i l i t y tends t o mask the s i g n i f i - cance and r e l a t i o n s h i p s tha t a r e present i n t he i n f o r m t i o n gathered on a mineral deposi t . It i s the func t ion of s t a t i s t i c a l a n a l y s i s t o remove t h i s mask and t o a s se s s t h e s igni f icance and r e l a t i o n s h i p s t h a t a r e inherent i n a s e t of da ta . I f t he v a r i a b i l i t y within a depos i t i s r e l a t i v e l y h igh , the app l i ca t ion of some s t a t i s t i c a l procedure may be necessary before a n i n t e l l i - gent s e l e c t i o n of method of r e se rve computation can be made. On t h e other hand, i f the v a r i a b i l i t y i s r e l a t i v e l y small a s compared t o meaningful pat- t e rns and t rends i n the information, t h e s e l e c t i o n of a method of r e se rve computation may r e q u i r e no pr ior s t a t i s t i c a l t reatment of t h e data .

I n t h e mining industry s t a t i s t i c a l procedures have been used i n examining mine and explora t ion data t o de t ec t pa t t e rns and t r ends , t o c o r r e l a t e

va r i ab le s , and " to develop numerical da t a from which the r e l i a b i l i t y of e s t i - mates can be assessed" (32). Part ly because these procedures a r e wel l adapted t o e l e c t r o n i c computing, they have been u s e f u l i n obtaining t h e maximum amount of information from sparse explorat ion d a t a and i n handling l a rge amounts of data from opera t ing p rope r t i e s . The a p p l i c a t i o n of s t a t i s t i c a l methods t o . t h e r e s u l t s of sampling and r e se rve computations has been discussed i n many publ ica t ions (4-2, 15-18, 22, 62).

S t a t i s t i c a l procedures a r e general ly u s e f u l i n i s o l a t i n g changes or va r i - a b i l i t y t h a t i s due t o chance from changes t h a t a r e "real." The r e l i a b i l i t y of es t imates i s developed from the v a r i a b i l i t y t h a t i s due t o chance. I n data obtained from mineral depos i t s , t h i s v a r i a b i l i t y may a r i s e from a random dis - t r i b u t i o n of va lues within t h e region considered. As t h e region considered is enlarged, "real" changes o r t rends inva r i ab ly appear and the va lues wi th in t h i s enlarged region can no longer be considered random. It i s genera l ly use- f u l and l o g i c a l t o consider changes i n va lues wi th in a depos i t a s t he super- pos i t ion of changes which a r e r e a l or due t o a t rend and changes which a r e due t o chance and explained by a loca l ly random d i s t r i b u t i o n of va lues .

According t o most e a r t h s c i e n t i s t s , an assumption of random d i s t r i b u t i o n of v a r i a b l e s i s con t r a ry t o the basic geologic hypothesis of the o r i g i n of mineral depos i t s , p a r t i c u l a r l y sedimentary (53). This school considers each depos i t a geochemical f i e l d , a s t r u c t u r a l f i e l d , o r combination of both. Com- mercial concentrat ion and d i s t r i b u t i o n of va luable components wi th in such f i e l d s r e s u l t from the genesis of the depos i t . The n a t u r a l processes govern- ing depos i t ion and migration of minerals may be superimposed upon each o t h e r , or even be adverse t o each o t h e r , thus c r e a t i n g an i n t r i c a t e d i s t r i b u t i o n of valuable components. Advocates of t he geochemical school consider grade and thicknees changes i n a minera l body t o be due t o t h e mode of o r i g i n and hidden i r r e g u l a r i t i e s i n t h e i r d i s t r i b u t i o n . The v a r i a b l e s and t h e i r r e l i a b i l i t y depends on t h e p l ace of observation i n t h e mineral body. Thus, according t o this school , each v a r i a b l e i s a func t ion of space of coordinates XYZ. Grade a t a given point may dev ia t e from mean grade , but the degree of devia t ion depends on the morphology of the body and on the p a r t i c u l a r s of observat ions and sampling.

On the o ther hand, t he opponents of t h e above hypothesis be l i eve t h a t adverse geologic processes together with l o c a l and acc iden ta l changes, usua l ly produce no c l e a r order ly r e g u l a r i t i e s i n t h e thickness and grade of the min- e r a l body. Much of t h i s d i f fe rence of opinion might be explained by the s c a l e on which the phenomena i s viewed.

Weighting

Weighting i s t he operat ion of a s s ign ing f a c t o r s t o each of a number of observat ions t o r ep re sen t t h e i r r e l a t i v e va lue , a l l o c a t i o n , or importance when compared with o the r observat ions of t he same s u i t e . In the mining indus t ry , the p r i n c i p l e of weighting i s widely used i n computing averages of var iab les and reserves of mineral bodies. Al loca t ions of weights a r e made i n u n i t s of Length, a r e a , volume, and tonnage on t h e b a s i s of d i f f e r e n t p r inc ip l e s of i n t e r p r e t a t i o n , mainly the r u l e of nea re s t po in t s , geo log ic , mining, and o ther considerat ions.

The use of weighting i n each p a r t i c u l a r case depends on the a n a l y s i s of explora tory da ta . I n s ec t iona l sampling, across a wide mineral body, weight- i ng may be compulsory f o r computation of average grade over t h e e n t i r e width of a ve in with d i f f e r e n t metal va lues near t h e hanging wal l and footwall . I n a l l methods of reserve computations, t h e p r i n c i p l e of weighting i s appl ied t o i nd iv idua l blocks of d i f f e r e n t s i z e s t o determine average thickness and average grade of t he e n t i r e deposi t .

I n c e r t a i n cases , weighting by an a rea of i n f luence i s n o t appropriate . For a reg ion wi th in which the va lues a r e randomly d i s t r i b u t e d , no sample by d e f i n i t i o n has an area of in f luence ; hence, weighting samples within t h i s region by an a rea of inf luence i s no t l o g i c a l i n ob ta in ing an average f o r the region. Thus, t h e r u l e of nea re s t p o i n t s i s no t app l i cab le f o r t h i s case. However, t h i s does not mean t h e samples from such a reg ion should not be weighted f o r some o ther reason when computing t h e r eg ion average.

Appl ica t ion

A l l t he above pr inc ip les of i n t e r p r e t a t i o n a r e used in va lua t ion of min- e r a l depos i t s . A study of the common methods of computations d i s c l o s e s t h a t block cons t ruc t ion i s usual ly based on one d e f i n i t e p r i n c i p l e , and o the r prin- c i p l e s , o f t e n secondary i n importance, appl ied a s supplementary opera t ions ( t a b l e 1 ) . The pr inc ip l e s of s t a t i s t i c a l a n a l y s i s , weighting, and genera l iza- t i o n a r e used i n a l l conventional methods.

TABLE 1. - Pr inc ip les of i n t e r p r e t a t i o n of explora t ion da t a used i n cons t ruc t ion of blocks and r e s e r v e computations

(XXY - Predominant; XX - Supplementary but i n f l u e n t i a l ; X - Secondary)

I and modif icat ions - I economics 1 gradual 1 nea res t 1 i z a t i o n

Rule of general-

Reserve computations: Conventiona 1 methods

Mining blocks............... P I =

I n t r i n s i c I Ana ly t i c a 1 Geologic I Mining and I Rule of ( Rule of

Average f a c t o r s and a rea : ................. Analogous Geologic blocks.. .........

Triangular prisms. .......... I X I -

XXX XXX

Cross s ec t ions : Standard.................. Linear.... . . . . . . . . . . . . . . . , Isolines..................

XX X

XX XX XK

The leading p r inc ip l e i n average f a c t o r s and a rea methods i s based on geologic c r i t e r i a . Mining, economic, and t o l e s s e r e x t e n t , geologic c r i t e r i a

Polvgonal prisms.. . . . . . . . . . . I x -

suppor t t h e mining b locks method. The r u l e of g r a d u a l changes i s b a s i c t o t h e method of t r i a n g u l a r pr isms and the r u l e of n e a r e s t p o i n t s t o t h e method of po lygona l pr isms. The r u l e of gradual changes i s t h e predominant p r i n c i p l e i n t h e s t a n d a r d and i s o l i n e s c r o s s - s e c t i o n methods, and t h e r u l e of n e a r e s t p o i n t s i s used i n t h e l i n e a r c r o s s - s e c t i o n method.

Computations

Basic Parameters

The b a s i c parameters f o r computing r e s e r v e s of a m i n e r a l d e p o s i t inc lude t h i c k n e s s and a r e a - q u a n t i t a t i v e i n d i c a t o r s of form, s i z e , and volume of t h e m i n e r a l body; g r a d e - t h e q u a l i t a t i v e i n d i c a t o r of v a l u e s and t h e i r d i s t r i b u - t i o n i n t h e d e p o s i t ; and weight f a c t o r o r s p e c i f i c g r a v i t v - i n d i c a t o r f o r tonnage computations.

I n most d e p o s i t s t h i c k n e s s and grade v a r y from p l a c e t o p l a c e i n g r e a t e r degree than t h e weight f a c t o r . For s i m p l i c i t y t h e l a t t e r i s considered con- s t a n t i n t h i s r e p o r t .

Thickness and Area

Measurements of t h e th ickness of a m i n e r a l body a r e taken d i r e c t l y by a s e r i e s o f o b s e r v a t i o n s , s c a l e d from maps and s e c t i o n s , or computed, and then a r i t h m e t i c a l l y averaged ,

Area i s measured d i r e c t l y from maps by p l a n i m e t e r i n g , by t h e u s e o f s p e c i a l l y c o n s t r u c t e d t empla tes , by geometric computations , and i n d i r e c t l y by computing from survey da ta .

P lanimet er i n g

A t least two p lan imete r readings taken i n o p p o s i t e d i r e c t i o n s a r e neces- s a r y t o a c h i e v e c o r r e c t r e s u l t s . I f t h e s e r e a d i n g s v a r y by l e s s than 2 per- c e n t , t h e average i s accep ted a s t r u e . The s c a l e of t h e s e l e c t e d maps should meet t h e accuracy requ i rements of t h e s m a l l e s t a r e a measured.

/

Templates

Templates may be of square p a t t e r n , where each s q u a r e h a s a c e r t a i n unit- a r e a v a l u e ; of d o t t e d p a t t e r n , where each d o t i s t h e c e n t e r of a u n i t of equal a r e a ; o r , of p a r a l l e l l i n e s p a t t e r n with a s e r i e s of e q u i d i s t a n t l i n e s drawn t o s c a l e (figs. 22 , 23, 24). The u s e of t empla tes w i t h t h e f i r s t two p a t t e r n s i s s e l f - e v i d e n t . I n t h e c a s e of t h e p a r a l l e l l i n e s t e m p l a t e t h e lengths of a l l l i n e s w i t h i n t h e m i n e r a l body a r e t o t a l e d ; t h e sum of l eng ths m u l t i p l i e d by t h e u n i t v a l u e of t h e s c a l e equa l t h e t o t a l a r e a . Two d i f f e r e n t p o s i t i o n s of a t e m p l a t e a r e t aken f o r p r e c i s e measurements and t h e average accepted a s

Surface Elev.

Scale in units 5-17.5 sq units

FIGURE 22. - Square Pattern Template.

One dot =I0 sq units S = 460 sq units

FIGURE 23. - Dotted Pattern Template.

10 FIGURE 24. - Para l le l L i nes Template.

5

the t r u e area. I n prac- t ice , t h e square pa t t e rn i s used when t h e a r ea i s 50 u n i t s o r less.

h-common interval of off sets Geometric Computations

I r r e g u l a r l y shaped a r e a s may be divided i n t ~ simple geometric f i g u r e s ; t h a t i s , tri- a n g l e s , squares , t e t r a - gons, and t rapezoids . The dimensions of each f i g u r e can be sca led from maps or deduced from survey notes and t h e a r e a computed. The t o t a l a r ea i s equal t o t h e sum of t h e calcu- l a t e d f i gu re s . The most

2 h + . . . + counnon formulas fo r p l a i n f i g u r e s , t r i a n g l e ,

= h - + o2+O3+ . . . + an - 1 squa re , r ec t ang le , and para l le logram a r e wel l

FIGURE 26. - Trapezoidal Rule. known. Formulas f o r the t r apezo id follow.

Trapezoid F o m l a . - An area of a simple t rapezoid i s

where a and b a r e p a r a l l e l s ides of t he f i g u r e and h t h e perpendicular distance ( f i g . 25).

Trapezoidal Rule. - An i r r e g u l a r a r ea may be subdivided i n t o an even number of t r apezo ida l f i g u r e s by a s e r i e s of equ id i s t an t p a r a l l e l l i n e s , o r ordi- n a t e s ( f i g . 26). Assuming t h a t t h e boundaries of t h e s t r i p s between the ordi- na t e s a r e s t r a i g h t l ines , t h e e n t i r e i r r e g u l a r a r ea may be computed by t h e t r apezo ida l r u l e ,

where h is a corrrmon i n t e r v a l between p a r a l l e l l i n e s o r o rd ina t e s , and a1 , +, ... , a, a r e t h e lengths of each o rd ina t e .

It i s obvious t h a t the g r e a t e r t h e number of s t r i p s the g r e a t e r i s t h e prec is ion of t he formula.

Simpson's Rule. - The computation of a n i r r e g u l a r a r e a by Simpson's r u l e ( f i g . 27) i s based on the assumption t h a t t h e curved boundaries of each s t r i p a r e parabolas pass ing through consecut ive points . I f t h e number of o f f s e t s a r e odd and t h e number of s t r i p s even, t he i r r e g u l a r a r e a i s computed by Simpson's formula (42, v. 2 , p. 36-13),

1 S = - h (al + 2 C a o d d + 4 C a,,,, + an 1, 3 ( 13)

where C a,,, - t he sum of odd o f f s e t s

C a,,,, - t h e sum of even o f f s e t s .

If t h e number of o f f s e t s i s even (and number of s t r i p s odd), one of t he end-area t rapezoids i s computed sepa ra t e ly and added t o t h e r e s u l t s computed by t h e formula. Other , and l e s se r known, t r apezo ida l formulas fo r determining a r e a a r e Durand's and Weddle's r u l e s descr ibed i n engineering handbooks.

I

I ) - common interval of offsets

n IS odd

FIGURE 27. - Simpson's Rule for Determining Areas.

I n d i r e c t Methods

Some a r e a s may be computed from survey n o t e s by double meridian d i s t a n c e s o r by t h e c o o r d i n a t e method, desc r ibed i n c i v i l eng ineer ing handbooks.

Volume

The volume of a b lock i s computed from d i r e c t o r i n d i r e c t measurements of l e n g t h ( L ) , b r e a d t h (B), and th ickness (T) by t h e p a r a l l e l e p i p e d formula ,

V = LTB. ( 14)

I n p r a c t i c e m i n e r a l bodies a r e i r r e g u l a r , and i t is necessa ry t o s u b s t i - t u t e t r u e volume by a n equivolume body o f s o l i d geometric c o n f i g u r a t i o n f o r t h e u s e of s imple formulas f o r volume computat ions .

When t h e a r e a S i s d i r e c t l y measured on t h e map and t h e average t h i c k n e s s t,, i s computed methemat ical ly , t h e g e n e r a l formula f o r t h e d e p o s i t i s

V = St, , . ( 15)

I f a m i n e r a l body i s subdivided i n t o segments o r b locks f o r computat ions , t h e volume of t h e e n t i r e body w i l l be

where , % , Sg , . . . , S, a r e block a r e a s and

tl , t2 , tJ , . . . , tn a r e average t h i c k n e s s e s of i n d i v i d u a l b locks .

Various methods of block c o n s t r u c t i o n a r e d i scussed i n p a r t 2 of t h i s r e p o r t . It i s obvious t h a t the s u b s t i t u t i o n e r r o r of t h e t r u e volume of a m i n e r a l body wi th a u x i l i a r y blocks depends on t h e knowledge of t h e form and s i z e of t h e body. I n a d d i t i o n , t h e accuracy of computations depends on t h e number of b l o c k s , v a r i a t i o n s i n t h e size of b locks , and p r e c i s i o n ob ta ined by t h e formulas .

Weight

Tonnaa e Fac t o r s

Conversion of volume t o tonnage of raw minera l m a t e r i a l ( o r e , r o c k , and c o a l ) v a r i e s , depending on t h e system of measures used. Common formulas used i n computing tonnages a r e

I n t h e f i r s t formula F i s the volume-tonnage f a c t o r and i s u s u a l l y expressed i n c u b i c f e e t per ton. I n the second formula f i s t h e tonnage-volume f a c t o r and i s u s u a l l y expressed i n we igh t -un i t s per cub ic f o o t .

Both weight f a c t o r s a r e i n t e r r e l a t e d and a r e determined on the bas i s of pas t product ion, experimental mining, o r adapted from s i m i l a r depos i t s . They a l s o may be determined by measuring excavat ions, o r by s p e c i a l laboratory t e s t s . Techniques of t h e s e determinations a r e descr ibed i n s e v e r a l publica- t i o n s (36, 40, 42). I n some cases tonnage f a c t o r s may be computed from the minera l composition a f t e r cor rec t ions f o r poros i ty and mois ture content of raw m a t e r i a l a r e made.

I n many depos i t s t h e ueight f a c t o r s vary s u b s t a n t i a l l y owing t o the min- e r a l and grade composition. The r e l a t i o n s h i p between weight and grade o f t e n may be expressed graphica l ly ; thus , weight f a c t o r s can be determined f o r app ropr i a t e grade of each individual block.

The o r e tonnage of t h e e n t i r e body i s d e t e d n e d by formula,

Spec i f i c Gravi ty

Conversion of volume to tonnage (met r ic ) i s made by the formula,

where D is s p e c i f i c g rav i ty or densi ty of raw mineral ma te r i a l .

The s p e c i f i c g r a v i t y of the mineral matter can be determined by d i r e c t t e s t s of d r i ed and crushed samples. Spec i f i c g rav i ty of rock i n p lace , o r "rock s p e c i f i c grav i ty-na tura l" may be expressed by

where g is the s p e c i f i c gravi ty of the mineral matter determined by t e s t s of crushed and d r i ed rock.

Po is porosi ty i n percent pore space t o u n i t of volume.

M, i s moisture i n percent weight l o s s upon drying. Mead o f f e r s a conven- i e n t diagram f o r t he English system of weights and measures, showing the in f lu - ence of po ros i ty , moisture, and s p e c i f i c grav i ty on t h e tonnage f a c t o r (36).

Spec i f i c g rav i ty may be ca l cu la t ed t h e o r e t i c a l l y a s an average of the s p e c i f i c g rav i ty of a l l of the minerals i n t he d e p o s i t , o r according t o the weighted average percent of .each mineral i n the rock. For convenience, the ca l cu la t ed devia t ions of spec i f i c grav i ty f o r var ious grades may be presented graphica 1 ly . Conversion Formulas

The volume-tonnage f ac to r , F , i s computed from s p e c i f i c g rav i ty by the fol lowing formulas (42, v. 2 , p. 25-20) :

- For s h o r t t o n F, ., - 2y000 f t 3 / s . t . , and 6 2 . 5 D

For long t o n

where 62.5 l b i s t h e weight of 1 cub ic f o o t of water a t 4" C .

The tonnage-volume f a c t o r , f , i s computed by

Volume i n c u b i c f e e t can be conver ted t o c u b i c mete r s by m u l t i p l y i n g by 0.028 o r d i v i d i n g by 35.3. M e t r i c tons a r e conver ted t o s h o r t t o n s by m u l t i p l y i n g 1.102 o r t o long t o n s by 0.984 (Appendix A).

When t h e d e n s i t y of a p a r t i c u l a r body v a r i e s a p p r e c i a b l y from one p l a c e t o ano the r owing t o t h e r e l a t i v e amounts of m i n e r a l s wi th wide r a n g e s i n spe- c i f i c g r a v i t i e s , such a s ga lena and i r o n o x i d e v e r s u s q u a r t z , more a c c u r a t e r e s u l t s a r e o b t a i n e d by

where 4 , D 2 , ... , D, a r e s p e c i f i c g r a v i t i e s of s e p a r a t e b l o c k s Vl V 2 , V n m

Grade

Grade computat ion of a minera l body i s a c r i t i c a l and impor tan t o p e r a t i o n t h a t c a n be done by v a r i o u s formulas:

1. Simple a r i t h m e t i c mean (unweighted) . 2. Weighted pe r width o r th ickness .

3. Weighted p e r width and l eng th , o r a r e a .

4 . Weighted by f requency of occur rence .

5 . Weighted by t h e square of t h e f requency.

6 , Weighted by f requency and a s s a y ( 3 5 ) .

The problems of sampling of v a r i o u s types of m i n e r a l d e p o s i t s and methods of s t a t i s t i c a 1 a n a l y s i s used i n e v a l u a t i n g e x p l o r a t i o n d a t a , and i n computing average g rade of workings , b locks , and commercial p o r t i o n s of b o d i e s , a r e beyond t h e scope of t h i s paper .

G e n e r a l l y , a v e r a g e g rade of a m i n e r a l body i s computed u s i n g convent iona 1 methods of r e s e r v e computat ions ; the f o r m l a s used a r e

30

Type of problem

~ r i t h m e t i c average.

Thickness- - weighted average.

Area- - weighted average.

Volumetric average (volume-weighted average) .

Gravimetr ic average (tonnage-weighted average) .

Assumption

4 1 1 blocks a r e equal i n a r e a , th ickness , and weight f a c t o r .

411 blocks a r e equal i n a rea and have t h e same weight f a c t o r .

A l l blocks have con- s t a n t th ickness and weight f a c t o r , but d i f f e r e n t a reas .

Weight f a c t o r s of a l l blocks a r e the same.

Equation

e , & , Q3 , . . . , Qn a r e reserves of raw minera 1 m a t e r i a l i n i n d i v i d u a l b locks , i n tons.

Tonnages and grades of blocks a r e d i f f e r e n t .

Reserves of va luab le components a r e determined by formula

where P i s t h e sum of r e se rves of each va luab le component of i n d i v i d u a l blocks PI, P,, P,, ... , Pn and Q i s the sum of r e s e r ~ e s of raw m a t e r i a l Q1, &, Q3, ..., Q, . The average grade of d e p o s i t i s determined by formula

E r r o r s

Accuracy Versus P r e c i s i o n

The terms "accuracy" and "precis ion" a s r e l a t e d t o r e s e r v e computations a r e def ined i n t h i s r e p o r t as fo l lows: The va r i ance between a s i n g l e obse rva t ion o r a computed aver - age of a n element of a mineral body and i t s t r u e v a l u e i n d i c a t e s the degree of accuracy o r exac tness of such observat ions o r es t imates . An a c c u r a t e measurement i s f r e e from a l l e r r o r s . P rec i s ion i n d i c a t e s only t h e degree of f l u c t u a t i o n i n a c e r t a i n s u i t e of v a r i a - b l e s wi th r e spec t t o t h e i r proximity t o each o t h e r .

The d i s t i n c t i o n between accuracy and p r e c i s i o n is w e l l i l l u s t r a t e d g r a p h i c a l l y in t h e fo l lowing example f o r a s u i t e of chemical ana lyses (2 , 57).

1. Accurate and p rec i se ( f i g . 28A).

2. Inaccura te but p rec i se ( f i g s . 282 and 28s).

3. Accurate but n o t precise ( f i g . 28g).

4 . Inaccura te and not precise ( f i g s . 28g and 28E).

The e r r o r s i n r e s e r v e computations may be d iv ided i n t o t h r e e groups: e r r o r s of i n t e r p r e t a t i o n ( o f t e n labeled geologic) , t e c h n i c a l , and a n a l y t i c a l .

Plus l lmit o f permissible deviations

True value

Minus limit of permissible deviations

FIGURE 28. - Accuracy and Precision of Chemical Analyses. : I , Accurote and precise; B and C, inaccurate, precise, large bias errors; 11, accurate, not precise, large random errors; E and F , inaccurate, not precise, large rondom ond biased errors.

E r r o r s of I n t e r p r e t a t i o n o r Analogy

E r r o r s of i n t e r p r e t a t i o n o f t e n c a l l e d e r r o r s of ana logy , r e p r e s e n t a t i o n , d e t a i l s , and geology are due t o t h e a c c e p t e d h y p o t h e s i s of t h e o r i g i n of t h e d e p o s i t , assumpt ion of geo log ic s i m i l a r i t y t o o t h e r d e p o s i t s , i n t e r p r e t a t i o n o r assumtpion of t h e uniform changes of t h e b a s i c e l ements , and t h e c o n t i n u i t y of t h e body a l o n g t h e s t r i k e and a t depth . They a r e e r r o r s of judgment and, consequen t ly , depend on t h e t r a i n i n g and e x p e r i e n c e of t h e person a p p r a i s i n g o r conduc t ing t h e i n v e s t i g a t i o n .

The r e s u l t s o f e x p l o r a t i o n a r e g e n e r a l l y d i s c l o s e d by a s e r i e s of p lans and s e c t i o n s r e p r e s e n t i n g the m i n e r a l body i n g r a p h i c f o r m . Thus, t h e exact - n e s s of our knowledge o f a d e f i n i t e m i n e r a l d e p o s i t depends on t h e c o r r e c t n e s s of t h e maps, which i n t u r n , depends on t h e t y p e of m i n e r a l d e p o s i t , kind and d e n s i t y of workings , and p r e c i s i o n of a l l measurements and q u a l i t a t i v e assays and t e s t s .

Technica 1

Techn ica l e r r o r s a r e those due t o i m p e r f e c t i o n s i n ins t ruments and tech- n iques used i n de te rmin ing a l l v a r i a b l e s . E r r o r s , random, b iased , or both , should be c o r r e c t e d t o prevent downgrading o r upgrading o f ind iv idua 1 observa- t i o n s , s i n c e e r roneous v a r i a b l e s i n f l u e n c e i n t e r p r e t a t i o n of boundaries and

computation of bas ic parameters of the minera l body, and consequently, of the s i z e and value of reserves.

Oversight e r r o r s due t o f a u l t y copying o r typing and recording of samples a r e excluded from t h i s discussion.

Random, casua l , o r acc identa l t e c h n i c a l e r r o r s a r e " those whose causes a r e unknown and undeterminate" (54, p. 454). f i e s e e r r o r s a r e mostly e r r a t i c , and may be of g rea t magnitude ( f ig s . 28g, 28E, 28z) ; they f l u c t u a t e on both s i d e s of t h e t rue va lue and, when t h e ave rage of a s u f f i c i e n t number of va r i a - b l e s i s computed, compensate each other .

Permissible random er rors used i n r e s e r v e computations of v a r i a b l e s a r e given i n t a b l e 2 (31, 46, 57, 59, 6 6 ) , and f o r chemical ana lyses of var ious elements and grades i n t ab l e 3 (57).

To prevent random e r ro r s i n chemical ana lys i s , some l abo ra to r i e s process two port ions of the same sample, regular and dupl ica te . Both determinat ions a r e then compared with permissible dev ia t ion values (such a s l i s t e d i n t a b l e 3) and, i f within range, t he f i n a l r e s u l t i s repor ted a s an average.

To f i n d random e r r o r s fo r a s u i t e of chemical ana lyses , per iodic (monthly o r qua r t e r ly ) con t ro l analyses a r e made. These repeated samples a r e given in code i n the amount of 3 to 5 percent of t he t o t a l number of samples but not l e s s than 30 (67). The magnitude of random e r r o r between con t ro l and regular analyses should not exceed the permissible l i m i t s adopted by the laboratory.

Biased

Biased or systematic e r ro r s a r e " those which a f f e c t a l l measurements a l i k e " (54, p. 454). They a r e due t o imperfect ions of instrument , equipment, and accepted techniques of observations. For chemical ana lyses such e r r o r s may be due t o inexperienced personnel, i n f e r i o r q u a l i t y reagents , and se l ec - t i o n of an improper method fo r a given sample. Most l i k e l y the e r r o r s a r e i n one d i r ec t ion ; t h a t is , a r e p e r s i s t e n t l y e i t h e r more o r l e s s than t h e t r u e value. Such e r r o r s a f f e c t the mean va lues because they a r e not compensating.

The presence and the magnitude of b iased e r r o r s may be d isc losed by s p e c i a l s tud ies . To determine biased e r r o r s i n chemical ana lyses , f o r example (67 ) , con t ro l ana lyses of the same samples a r e made i n a reputable laboratory by s imi la r methods and procedures. Such o u t s i d e laboratory c o n t r o l analyses a r e made f o r l a rge p ro j ec t s a t l e a s t twice a year ; t he number of ana lyses should be 3 to 5 percent of the t o t a l with a minimum of not l e s s than 30 samples . I f a s u b s t a n t i a l e r ror i s found and proved f u r t h e r by a t h i r d pa r ty , a cor rec t ion f ac to r should then be .appl ied t o a l l samples analyzed i n t he f i r s t laboratory.

TABLE 2. - Technical e r r o r s i n de te rmin ing b a s i c parameters

Thickness . , .

Length ....., Angle .......

....... Area

S p e c i f i c g r a v i t y o r weight ,

f a c t o r .

Grade.... . . ,

Random e r r o r s i n de te rmin ing i n d i v i d u a l parameters

O r i R i n a 1 o b s e r v a t i o n s

Tape ( f o r 1 mete r ) . .

......... From p lans .

Surveying. .......... ............. Drawing

Brunton..,. . . . . . . . . .

Surveying data . . . . . .

P lanimeter ( 100-400 cm2 ) .

Template ............ Yaps :

S c a l e 1: 200 Sca le 1:5000

...... I . Sample t ak ing . Sample process ing. .

.. Chemical a n a l y s i s .

' r e c i s i o n . plus-minu!

....... ..O p e r c e n t . .

. ,bove 1.5 percent . .

1.5 pe rcen t and l e s e Ip t o 2.0 percent . . .

1.5 t o 2.0 degrees . .

1.5 percent and l e s s

1.5 t o 0 .3 percent ( a r e a ) . .. Jp t o 3.0 percent .

. 5 pe rcen t . ........ .25 pe rcen t . . ...... t o 10 pe rcen t . . ...

'ar ied. . . . . . . . . . . . . . ......... . .O pe rcen t . t o 20 pe rcen t and more .

Remarks

P r e c i s i o n i n c r e a s e s wit1 o r e t h i c k n e s s and dec reases wi th i r r e g u - l a r i t i e s of o r e body boundar ies .

Depends on s c a l e and drawings.

- Depending on s c a l e

Depends on a n g l e of measurement, exposure , and convenience.

P r e c i s i o n i n c r e a s e s with t h e i n c r e a s e of a r e a s i z e .

Depends on method of de te rmina t ion , type of o r e . e t c .

See t a b l e 3............

Techn ica l e r r o r s i n con- n e c t i o n w i t h r e s e r v e com- p u t a t i o n s of a n o r e body

( r a n ~ e , accuracy , plus-minus)

S u r f a c e and underground work- i n g s : 2.0 t o 3.0 p e r c e n t f o r t h i c k and up t o 10 p e r c e n t f o r t h i n bodies--depending on t h e i r r e g u l a r i t i e s of t h i c k - n e s s e s and grade.

D r i l l h o l e s : s e v e r a l t o 30 pe rcen t and more--depending on e x p l o r a t i o n t e c h n i q u e , c o r e r ecovery , and t y p e of d e p o s i t s .

--1.0 p e r c e n t and above depending on s c a l e .

' 2 t o 3 pe rcen t depending on s c a l e .

I Do.

I Do.

Depending on s c a l e . Do.

For uniform o r e : 3 t o 20 per- c e n t depending on method of d e t e r m i n a t i o n , p o r o s i t y , mois- t u r e c o n t e n t , f i s s u r i n g , e t c . For complex composi t ion o r e s : 20 pe rcen t and more.

Depends on sampling method, sample p rocess ing , chemica 1 a n a l y s e s , t y p e of d e p o s i t , method of computat ion, commodity, e t c .

W W

TABLE 3 . . Permissible averape fo r random technical e r ro rs i n chemical analvses

(The All-Union Committee on Mineral Reserves. U.S.S.R.)

Component and grade range of raw mate-

r i a l . percent. except as noted

Aluminum oxide (A&03 ) : ......... Above 20

5 . 20 ........... 1 - 5 ............

Antimony : Above 2 .......... .......... . 0.2 2

Arsenic : .......... Above 2 0.5 . 2 .......... Below 0.5 ........

Barium s u l f a t e (BaSO, ) : Above 5 .......... 1 - 5 ............

Bery Ilium: 5 . 10 ........... 0.1 . 5 .......... 0.01 . 0.1 .......

Bismuth : ........ Above 0.6 0.2 . 0.6 ........

Cadmium : Above 1 .......... 0.1 . 1 .......... 0.01 . 0.1 ....... Below 0.01 .......

Calcium oxide ( CaO) : Above 25 ......... 5 . 25 ........... 1 . 5 ............

Chromium: Above 10 ......... 1 . 10 ........... Below 1 ..........

Cobalt : Above 0.5 ........ Below 0.5 ........

Co lumb ium : Above 10 ......... 1 . 10 ........... 0.1 . 1 .......... Below 0.1. .......

See footnotes a t end

r i a l . percent.

Component and grade range of raw nrate-

Permissible average e r ro r in percent t o

the grade determined ( p lus-minus)

2 - 4 4 - 8 8 . 20

3 . 12 12 . 20

1 - 5 5 . 7

10

1 - 7 7 . 15

3 - 5 5 . l o

10 . 30

5 . 15 15 . 20

3 - 5 5 . 10

10 . 30 30

3 - 5 5 . 10

10 . 25

1 - 3 3 - 7

7

2 - 6 6

3 - 5 5 . l o

10 . 20 2 0

of table .

except a s noted Copper :

Permissible average e r ro r i n percent t o

Above 3 .......... 0.5 . 3 .......... Below 0.5 ........

Gold. glm ton ( 1 g = 15.432 gra ins ) :

Below 0.1 m:l Above 64 ....... 16 . 64 ........ 4 . 16 ......... Below 4 ........

Below 0.6 nrm:" Above 64 ....... . ........ 16 64 4 . 16 ......... ........ Below 4

Above 0.6 nnn:3 ....... Above 64 ........ . 16 64 4 . 16 ......... Below 4 ........

I ron : ......... Above 30 . .......... 10 30

5 . 10 ........... Iron oxide (FeO) :

Above 5 .......... 1 . 5 ............

Lead : ......... Above 15 6 . 15 ........... 0.5 . 6 .......... ....... Below 0.5.

Magnesium oxide (MgO) : .......... Above 5

Manganese : Above 5 ..........

Mercury : Above 2 .......... 0.25 . 2 ......... Below 0.25. ......

the grade determined ( p lus-minus)

TABLE 3. - Permissible averane f o r random t e c h n i c a l e r r o r s i n c h w i c a 1 analyses--Continued

(The All-Union Camnittee on Minera l Reserves, U.S.S.R.)

Component and grade r ange of raw mate-

r i a l , percent , except a s noted

Molybdenum : .......... Above 1 0.25 - 1 ......... ..... Below 0.25..

Nicke l : 1 - 5 ............ 0.2 - 1 .......... ........ Below 0.2

Phosphorus: ....... Above 3... 0.03 - 0.3.......

S i l i c o n d ioxide (SiO,): ... 30 - 5O.......

10 - 30.......... 3 - lo...........

S i l v e r , gra ins per t on : ...... Above 100.. ... 30 - loo......

. 10 30.......... S u l f u r : ......... Above 20

1 - 20........... 0.05 - I.... .....

Tantalum: Above 10 ......... 1 - 10.. ......... 0 . 1 - 1 .......... ....... Below 0.1.

Fermissible average t r r o r i n percent t o t h e grade de t e r - mined (plus-minus)

Samples Samples

with with

Component and grade range of raw mate-

r i a l , pe rcen t , except a s noted

Tin : Above I... ....... 0.25 - 1 ......... 0.05 - 0.25 ......

Titanium dioxide (TiO, ) : 2 - 15.. . . . . . . . . . 0.1 - z.. . . . . . . . .

Tungsten t r i o x i d e (WO, : ........ Above 1..

0.25 - 1 ......... 0.05 - 0.25......

Vanadium: ...... Above 0.5..

- ..... 0.06 0.5.. Zinc :

Above 25......... 10 - 25.. ........ 0.5 - 10 .........

........ Below 0.5 Zirconium:

Abwe 3 . . . . . . . . . . 1 - 3 ............ 0.1 - 1 .......... Below 0.1.. ......

i n s u l f i d e s . .If i d e s and quar tz .

?ermiss ib le average s r ror i n percent t o t h e grade de t e r - mined ( plus-minusl

3Samples w i t h l a rge g r a i n s i z e , o f t e n v i s i b l e , go ld ; mainly i n quar tz .

Source: Reference (57), t a b l e 7 , pp. 67-68.

The c o r r e c t i o n f a c t o r may be computed a s a r a t i o of t he average grade of the c o n t r o l t o t h e average grade of t h e r e g u l a r a n a l y s e s ; t h a t i s

The f a c t o r E i s appl ied t o r egu la r samples t o r e c e i v e t h e c o r r e c t r e s u l t s . For example, a regular s u i t e of copper samples averaged 0.80 percent . Control samples averaged 1.0 percent copper , The c o r r e c t i o n f a c t o r i s

The con t ro l ana lyses a r e 25 percent higher than t h e r egu la r .

Analy t ica l

Some ana ' l y t i ca l e r r o r s of r e se rve computations w i l l be discussed i n p a r t 2. In genera l , t h e accuracy of computations inc reases with t h e number of blocks d iv id ing t h e mineral body, p r w i d e d , t h e same accuracy i s maintained i n cons t ruc t ion of each block. The e r r o r of a s epa ra t e block may be h igh , but f o r a group of blocks represent ing the e n t i r e body t h e r e l a t i v e e r r o r s a r e balanced according t o t he law of averages;

blocks a r e equal i n tonnage,

y + M, + %+ ... + % ma^ a N ; and

blocks a r e unequal i n tonnage of va luable c o n s t i t u e n t

where Ma, i s average r e l a t i v e e r r o r of minera l body, and M., , %, M3, ... , % a r e r e l a t i v e e r r o r s of individual blocks ( i n pe rcen t ) .

PART 2. - CONVENTIONAL METHODS

General Features and C l a s s i f i c a t i o n

For r e se rve computations t h e mineral depos i t , reduced and d i s t o r t e d by mapping, i s converted t o an analogous geometric body composed of one, s eve ra l , o r an aggregate of close-order s o l i d s , t h a t b e s t express t h e s i z e , shape, and d i s t r i b u t i o n of t he var iab les . Construction of t hese blocks depends on the method se lec ted . Some methods o f f e r two o r more manners of block cons t ruc t ion , thus introducing sub jec t iv i ty . I n such a c a s e a c e r t a i n manner of construc- t i on i s accepted a s appropriate , p referab ly on the b a s i s of geology, mining, and economics.

Numerous methods-of reserve computations a r e descr ibed i n the l i t e r a t u r e ; some a r e only s l i g h t modifications of t he most common ones. Depending on t h e c r i t e r i a used i n s u b s t i t u t i n g the explored bodies by a u x i l i a r y blocks and on the manner of computing averages f o r v a r i a b l e s , t h e convent ional methods may be c l a s s i f i e d i n t o four groups.

Group I,, average fac tors and area methods, embraces analogous and geologic blocks methods. Areas a r e de l inea ted by geologic and, i n p a r t , by mining and economic c r i t e r i a , and the bas i c elements ( t h i ckness , grade, and weight f a c t o r s ) a r e determined d i r e c t l y , computed, or taken from o the r por- t i ons of the same o r s imi la r depos i t s .

Group 2 , mining blocks method, involves de l inea t ion of block areas by underground workings and by geologic and economic cons idera t ions ; the f a c t o r s f o r each block a r e computed i n var ious ways. As the name implies the method i s used mainly f o r ex t rac t ion .

Group 3, c ross -sec t ion methods, includes s tandard, l i n e a r , and i s o l i n e s . The minera l body i s de l inea ted and the blocks a r e constructed on the bas i s of c e r t a i n p r inc ip l e s of i n t e r p r e t a t i o n of explora t ion data ; t h e parameters of blocks and t h e e n t i r e body a r e determined i n var ious ways.

Group 4, a n a l y t i c a l methods, d iv ides t h e mineral body graphica l ly i n t o blocks of simple geometric forms--triangular o r polygonal prisms. The f a c t o r s f o r each block a r e determined d i r e c t l y , computed a s an a r i t hme t i c average, or i n o ther ways.

Spec ia l s t u d i e s of t h e usage of var ious methods were made i n the U. S.S.R. Thousands of minera 1 depos i t s were explored and reserves computed and approved by the All-Union Committee on Mineral Reserves. The r e s u l t s f o r metal, non- metal , and c o a l and o i l - s h a l e depos i t s f o r t he years 1941-61 and for s o l i d min- e ra 1 B- 1)

depos i t s f o r years 1941-47, 1951, and 1954- a r e given i n appendix I3 ( t a b l e (57). The predominant methods were--

Methods, percent Average f a c t o r s ( Cross s ec t ions l ~ o l ~ ~ o n s

It i s a l s o repor ted i n the U.S.S.R. t h a t i n computing reserves the use of

Coal and o i l s h a l e depos i t s ................

Nonmetallic depos i t s ..... ............. Ore depos i t s

average f a c t o r s and. a r ea and cross s e c t i o n methods t i g e t h e r had increased from 30 (1941-47) t o a t o t a l of 82 percent (1954) of a l l p ro j ec t s recorded (appen- d ix B , t a b l e B-2).

A comparison of the use of var ious methods by 44 metal mines, described by Jackson and Knaebel i n "Sampling and Estimation of Ore Deposits" (28) pub- l i shed i n 1932, shows t h a t t he mining blocks and cross-sec t ion methods were predominant i n the mining industry ( t a b l e 4) .

"Including mining blocks method.

and a r e a

6 9 4 6 37

TABLE 4. - Usage of var ious methods f o r reserve computations f o r metal mines i n U.S. (1932)~

Methods : Percent Methods : Percent Average f a c t o r s and area . . . 20 Polygons ................... 4 Mining blocks.. . . . . . . . . . . . . 42 Triangles .................. 2 ................ Cross sect ions. . . . . . . . . . . . . 32 Tota 1. 100

lFo r 44 a c t i v e mines descr ibed by Charles F. Jackson and John B. Knaebel.

- 3 7 4 8

-

Sampling and Estimation of Ore Deposi ts , BuMines Bull . 356, 1932, pp. 125-249.

3 0 14 14+:

Average F a c t o r s and Area Methods

Assumptions and C h a r a c t e r i s t i c s F e a t u r e s

Average f a c t o r s .and a rea methods of r e s e r v e computat ions have been v a r i - o u s l y d e s c r i b e d a s a r i t h m e t i c average , weighted a v e r a g e , average dep th and a r e a , s t a t i s t i c a 1, analogous (by ana logy) , geo log ic b l o c k s , and g e n e r a l out- l i n e (27-28, 46, 57, 63). I n t h i s r e p o r t t h e s e methods a r e d i s c u s s e d under t h e t i t l e s of analogous and geo log ic blocks .

Average f a c t o r s and a rea methods a r e a l l based on t h e assumption t h a t c e r t a i n segments o r b locks of t h e m i n e r a l body b e i n g c o n s i d e r e d a r e s i m i l a r i n geology and technology t o s e c t i o n s p r e v i o u s l y s t u d i e d , o r t o b locks o r even bod ies t h a t have been explored o r mined o u t . For computat ions t h e body is d i v i d e d i n t o se,sments o r b locks c o n s t r u c t e d on t h e b a s i s of geology, mining, and economic; t h a t i s , s t r u c t u r e , t h i c k n e s s , g r a d e , v a l u e , dep th , and overburden. I n some c a s e s , t h e q u a l i t a t i v e c h a r a c t e r i s t i c s found i n one p a r t of t h e body may be accepted, f o r t h e purpose of computa t ions , a s r e p r e s e n t a t i v e of t h e block o r t h e e n t i r e mineral body.

I f t h e blocks a r e of equal s i z e each o b s e r v a t i o n and sample a n a l y s i s h a s an e q u a l in£ h e n c e i n determining a v e r a g e f a c t o r s . If t h e number of v a r i a b l e s i n a block a r e i n s u f f i c i e n t q u a n t i t y , average f a c t o r s may be computed and s t u d i e d by s t a t i s t i c a l a n a l y s i s ; on t h e o t h e r hand, t h e method of analogy may' be used where only one observa t ion i s a v a i l a b l e . A number of segments o r b locks wi th d i f f e r e n t c o n t r o l l i n g f a c t o r s r e q u i r e s t h e u s e o f t h e method of geo log ic blocks .

Formulas f o r computations of r e s e r v e s range from s imple t o complex equa- t i o n s . Aside from t h e usua l v a r i a b l e s o f g rade , t h i c k n e s s , and d e n s i t y , more complex f a c t o r s such a s , tons recovered per u n i t of a r e a , volume, o r weight may b e used.

Method o f Analogy

Analogy i s t h e in fe rence t h a t c e r t a i n a d m i t t e d p a r t i a l resemblances prob- a b l y imply f u r t h e r s i m i l a r i t y . The method emphasizes q u a l i t a t i v e s i m i l a r i t y of t h e geology of a g iven.block t o a n analogous and b e t t e r known block of t h e same o r s i m i l a r body.

V a r i a b l e s f o r computations may be t aken from a s i n g l e o r a number of o b s e r v a t i o n s , o r computed from d a t a ga thered from t h e same o r s i m i l a r d e p o s i t s . Such v a r i a b l e s may be accepted a s c o n s t a n t f a c t o r s f o r o t h e r p a r t s of t h e same body, o t h e r d e p o s i t s , o r even d i s t r i c t s . When t h e geology of a g iven a r e a o r d e p o s i t i s considered analogous t o a n o t h e r a r e a o r d e p o s i t , a s i n g l e observa- t i o n may be adequate f o r r e s e r v e computat ions of a c e r t a i n commodity. Reserves computed may belong t o any ca tegory . The method i s widely used i n e x t r a c t i c n o p e r a t i o n s h e n o t h e r s a r e d i f f i c u l t t o a p p l y . - I n r e s e r v e computa- t i o n s of n i c a i n p e g n a t i t e s , f o r example, product ion r e c o r d s may b e considered s u f f i c i e n t and a c c u r a t e f o r a s s i g n i n g mica g rade t o t h e unmined p o r t i o n of t h e v e i n below and between mined-out b locks .

Average F a c t o r s

The a r i t h m e t i c average i s t h e s i m p l e s t v a r i a t i o n of t h e analogous method. No a u x i l i a r y f i g u r e s a r e cons t ruc ted ; t h i c k n e s s and g rade a r e determined by a s imple average of a v a i l a b l e da ta ( f i g . 29 and t a b l e 5 ) . Grade may be d e t e r - mined a l s o by th ickness-weight ing of i n d i v i d u a l g rade o b s e r v a t i o n s from ore i n t e r s e c i i n g workings i n t h e mineral body and even from a d j o i n i n g p a r a l l e l bodies and by ex tend ing t h e r e s u l t s t o t h e unexplored block o r t o t h e e n t i r e m i n e r a l body ( t a b l e 6 ) . I n case of numerous samples ( o b s e r v a t i o n s ) t h e average g rade may be determined by s t a t i s t i c a l a n a l y s i s (15-17). Reserve ccnnputa- t i o n s and t h e d e t e r m i n a t i o n of t h e block-weighted a v e r a g e f a c t o r s f o r t h e e n t i r e body i s i l l u s t r a t e d by t a b l e 7 .

The formulas used a re average t h i c k n e s s (formula l o ) , average grade ( formula 24) , thickness-weighted average g r a d e ( formula 2 5 ) , volume of minera 1 body ( formula 1 5 ) , tonnage of raw m a t e r i a l (formula 1 8 ) , and tonnage of valua- b l e component ( formula 29a).

0 0 0 Inside perimeter of

mineral body

L E G E N D 0 Drill hole showing

thickness and grade

o Blank drill hole P L A N

S E C T I O N A - A ' S E C T I O N A-A' afier computing overage thickness

( equat ion 101

FIGURE 29. - Aiithmetic Average Method of Computing Thickness. (For recapitulation of reserves for mineral body and determination of overoge grade, see table 5.)

TABLE 5. - Determination of average th ickness and average grade f o r a block by a r i t hme t i c average procedure

TABLE 6. - Determination of thickness-weighted average grade f o r a block

Grade ( c ) , percent

C1 Cz c3

Workings (n) , number's

l.............................. 2.............................. 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

N..............................

Total.....................

Average ...................

Thickness ( t ) , f t

t 1

t2 t 3

t n

n C t

i=l

n taV = C t / n

i=l

The a r i t hme t i c average procedure i s the s implest and most r ap id method of computation ; accuracy depends on the q u a l i t y , quant i ty , densi ty , and d i s t r i bu - t i o n of observat ions; i n turn these f a c t o r s depend on the genet ic type and s i z e of the deposi t .

C,

n C c i=l

n c., = c c / n

i=l

Product , t x c . tl cl % % t3 C3

................................ N................................

Average (block A). ..........

The system i s accura te i n uniform depos i t s ; accuracy decreases i n nonuniform depos i t s , w e n i f the d i s t r i b u t i o n of observat ions i s done by a regular p a t t e r n but with i n s u f f i c i e n t dens i ty . ' I n complex deposi ts accuracy i s g r e a t e r i n r egu la r ly d i s t r i bu ted workings than i n i r r egu la r . An important disadvantage of t h i s system i s t h e lack of qua l i t y and quan t i t y d i s t r i b u t i o n of valuable components i n space.

Grade ( c ) , percent

C.

c; 3

Workings ( n ) , numbers

I....... ......................... z................................ 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Thickness ( t ) , f t t,

tz t3

n Note: Thickness-weighted average grade (c,, ), percent : c,., = C t ,ca

i=l i=l

tn n

.z ta i=l

C n

n

c '=&I

i=l

t n Cn

n

z LC, i-1

TABLE 7 . - Computation of r e s e r v e s and a v e r a g e f a c t o r s

Block

A....... . . . B.. . . . . . . . . .......... .......... N..........

T o t a l . . .

Average.

f o r t h e e n t i r e body

S t a t i s t i c a l F a c t o r s

i r e a ) , s q f t

Sci Sb

S, n C S i=l

F a c t o r s f o r r e s e r v e computations of a c e r t a i n m i n e r a l commodity a r e determined i n terms of product ion o r v a l u e y i e l d on t h e b a s i s o f e x p l o r a t i o n , p a s t mining e x p e r i e n c e , o r smel ter r e t u r n s g a t h e r e d f o r t h e same minera l body, o r even s i m i l a r ones . These f a c t o r s a r e u s u a l l y expressed a s pe rcen t of component o r v a l u e recovered per u n i t of a r e a , volume, o r w e i g h t ; t h a t i s

P l a c e r d e p o s i t s : Gold and p r e c i o u s

meta 1s.


t a t b

t n

n c v i=l

t,, = - n z s

i=l

... Heavy minera 1s.

Coal depos i t s . . . ....

Base meta l s and many nonmeta l l i c s .

Volume (V),

cu f t

V, V b

Vn n C V

i=l

English s v s t em

Ounces per cub ic y a r d o r con ( s h o r t , l ong) ; c e n t s pe r c u b i c ya rd , s q u a r e f o o t o r s q u a r e ya rd ; m i l l i g r a m s per cub ic yard o r ton .

Pounds per cub ic y a r d o r ton. .

Weight f a c t o r

( F ) , cu f t / t o n

F F

F

Tons pe r square f o o t , a c r e o r s e c t i o n .

Pe rcen t of weight o r pounds Per ton.

Raw m a t e r i a l r e s e r v e s

(Q) , t o n s

Qa

Qb

Q n n C Q i=l

M e t r i c svstem

Grams p e r cub ic meter or m e t r i c ton .

Va luab le Average

g r a d e ( c ) , p e r c e n t

Ca

b

n c p

i=l c a v = - n

c Q i=l

Kilograms pe r cubic meter o r m e t r i c ton.

M e t r i c t o n s per square mete r o r s q u a r e k i lome t e r .

P e r c e n t of . weight o r k i lograms per ton.

component Reserves

( P I , tons

'a

b

'n

n C P i =l

The u s e of s t a t i s t i c a l f a c t o r s o f t e n may be t h e o n l y p r a c t i c a l way of computing p o t e n t i a l r e s o u r c e s f o r a mine o r d i s t r i c t . The accuracy depends on the geo log ic i n t e r p r e t a t i o n of the m i n e r a l d e p o s i t , a s w e l l as on computed f a c t o r s .

Reserves of uniform bedded d e p o s i t s , c o a l , phosphate rock , and c l a y have been computed by t h e method of analogy from the r e s u l t s of s p o t d r i l l ho l e s and exposures i n t renches and o t h e r sur face workings. Reserves of phosphate rock a v a i l a b l e f o r open p i t mining i n Idaho were computed by t h e Bureau of Mines on t h e b a s i s of d e t a i l e d geo log ic s e c t i o n s , sample ana lyses , geologic maps, and o the r publ i shed da t a . I n f e r r e d r e s e r v e s of a s y n c l i n e , f o r example, i l l u s t r a t e d i n f i g u r e 21 i n p a r t 1 of t h i s r e p o r t were based on s e c t i o n s measured about 1 m i l e from t h e a r e a ( t a b l e 8 ) .

TABLE 8. - Reserve computations--method of analogy1

Sec t ion i s composed bf 2 acid grade zones (+31 percent P,o,), 2 fu rnace grade seams (24 t o 31 percent P,O,), and t h e remainder b e n e f i c i a t i o n grade ( 1 8 t o 24 percent P20, ). R u e bed thickness i s 49.8 f e e t .

Pz Os , s h o r t tons

Bene f i c i a t i on grade T o t a l o r

Method of Geologic Blocks

Block

570;000 1 13.4 1 7;638;000 1 13.0 1 587;5001 19.2 1112,800 ave rage570 ,0001 49.8 128 ,386,0001. - 12,360,4001 27.3 1644,700

Although t h e method of geologic blocks has been widely used by e a r t h s c i e n t i s t s f o r many y e a r s , i t was no t u n t i l 1950 t h a t i t s p r i n c i p l e s were f i r s t d i s c u s s e d and i t s name accepted (57) ; t h e procedure i s a l s o known a s t h e method of analogy and gene ra l o u t l i n e (63).

lComputations made f o r po t en t i a l s u r f a c e r e sou rces i n a phosphor i te d e p o s i t i n Idaho.

A geologic block may be the e n t i r e minera l depos i t o r a r e l a t i v e l y small por- t i o n of i t , o u t l i n e d on a map by i n t e r p r e t a t i o n of explora tory data . The block s i d e s may c o i n c i d e with t h e n a t u r a l boundaries of t h e depos i t , o r be d e l i n e a t e d on t h e b a s i s of g e o l o g i c a l f e a t u r e s , s t r u c t u r a 1 deformations, o r v a r i a t i o n s i n th ickness and grade. I n a d d i t i o n , blocks a l s o may be ou t l i ned on the b a s i s of physiographic f a c t o r s ; a d a p t a b i l i t y t o c e r t a i n mining methods; a v a i l a b i l i t y of mine ra l raw m a t e r i a l a t dep th ; p o s s i b i l i t i e s of u t i l i z a t i o n ; requirements f o r b e n e f i c i a t i o n and process ing; o r p rope r ty , s ec t ion , township, or S t a t e boundaries.

Block Acid grade .........

Do.. .......... Furnace grade.. ....

Do.. .......... Benef i c i a t i o n grade

T o t a l o r a v e r a a e 7 1 5 , O O O

Middle s e c t i o n

( S ) , s q f t

Volume (v),

cu f t

The f a c t o r s a r e determined from a v a i l a b l e explora tory data, o r may b e adapted from r e s u l t s of s p o t sampling, product ion averaging , o r da ta from o t h e r p a r t s of t h e same depos i t . Cutoff grade i s determined by geologic and mining cons ide ra t i ons and process ing . I n t e r p r e t a t i o n of d a t a may be by t h e r u l e s of gradual changes, n e a r e s t p o i n t s , or g e n e r a l i z a t i o n . The parameters of each geologic block and t h e e n t i r e

Phosphate rock ( Q ) ,

s h o r t t ons

Thickness accepted

( t ) , f t

Weight f a c t o r

(F) 3

cu f t l t o n

Grade ( c ) ,

pe rcen t

Acid grade.. ....... Do.. ..........

Furnace grade.. .... Do.. ..........

A , f i x . 715,000 715,000 715,000 715,000 7 15,000

over turned s e c t i o n , 4,260 f e e t lone 316,400 626,400 387,300 893,700 737,000

2,960,800

21, 5.0 9.9 6.5

15.0 13.4 49.8

3,575,000 7,078,000 4,648,000

10,725,000 9,581,000

35,607,000

11.3 11.3 12.0 12.0 13.0 -

Block B, f i g . 21 , normal d i p , 2,050 f e e t long

34.2 32.5 26.9 28 .1 19.2 27.3

570,000 570,000 570,000 570,000

108,200 203,600 104,200 251,100 141,500 808,600

5.0 9.9 6.5

15.0

2,850,000 5,643,000 3,705,000 8,550,000

34.2 32.5 26.9 28 .1

86,300 162,300- 83,100

200,200

11.3 11.3 12.0 12.0

252,200 499,400 308,800 712,500

body a r e determined by procedure d e s c r i b e d f o r t h e method of analogy. The average g rade of an i n d i v i d u a l block i s computed e i t h e r by t h e a r i t h m e t i c average ( t a b l e 5 ) , weighted-average ( t a b l e 6 ) , o r by s t a t i s t i c a l a n a l y s i s . Reserves of each block a r e computed a s t h e p roduc t of a r e a and average f a c t o r s ; t o t a l r e s e r v e s a r e t h e sum of a l l i n d i v i d u a l b locks ( t a b l e 7 ) .

Depending on t h e e x t e n t of the geo log ic knowledge of t h e d e p o s i t , a l l c a t e g o r i e s of m i n e r a l r esources may be computed by t h e geo log ic b locks method. Accuracy o f computations depends, e s s e n t i a l l y , upon t h e accuracy of f a c t o r s accep ted f o r each block and , t o a l e s s e r e x t e n t , on t h e accuracy of b lock a r e a d e t e r m i n a t i o n s . They may be as a c c u r a t e a s any o t h e r method, when a p roper number of o b s e r v a t i o n s support t h e computat ion of f a c t o r s f o r a c e r t a i n d e p o s i t . On t h e o t h e r hand, the computations by t h i s method may be specu la - t i v e o r p u r e l y academic, when t h e f a c t o r s a r e based on an i n s u f f i c i e n t number and d e n s i t y of o b s e r v a t i o n s .

Examples of computations by t h i s method a r e q u i t e common i n t h e e a r l y s t a g e s of e x p l o r a t i o n of bedded d e p o s i t s ; t h a t i s , phosphate r o c k , l imes tone , gypsum, and c o a l ( f i g . 30) . It i s o f t e n t h e only method t h a t can be used when t h e d e p o s i t i s i r r e g u l a r .

An e x c e l l e n t example of resource computat ions by geo log ic b locks h a s been p u b l i s h e d by t h e Geological Survey f o r uranium and vanadium d e p o s i t s of t h e Colorado P l a t e a u (8 ) . The o r e bodies a r e roughly t a b u l a r and g e n e r a l l y p a r a l - l e l i n g t h e bedding of t h e sandstone h o s t rock . They a r e i r r e g u l a r , o f t e n smal l i n s i z e ( l e s s than 5,000 t o n s ) , of v a r i a b l e t h i c k n e s s wi th uranium and vanadium v a l u e s d i s t r i b u t e d e r r a t i c a l l y . Computations have been based on d r i l l h o l e s , underground openings, o b s e r v a t i o n s of n a t u r a l o u t c r o p s , and pro- d u c t i o n r e c o r d s . Often t h e number of o b s e r v a t i o n s f o r i n d i v i d u a l d e p o s i t s were r e s t r i c t e d .

L E G E N O

o D r l l l holes

FIGURE 30. - Geologic Blocks Method.

A t Lvov-Katin (near Moscow, U.S.S.R.) c o a l r ese rv ' e computat ions were made by both geo- l o g i c b locks and polygon methods (64). Geologic b locks were s e l e c t e d on t h e b a s i s of bed t h i c k - n e s s , as t h e r e were only s m a l l v a r i a t i o n s i n c o a l q u a l i t y ( a s h , s u l f u r , e t c . ) . A t o t a l of 15 p r i n c i p a l and 80 supplementary geo log ic blocks were used a s compared wi th 260 b locks by t h e polygon method. The g e o l o g i c b locks method r e v e a l e d t h e p resence of a r e a s of v a r i e d and

sharp ly reduced c o a l thickness. As the geologic and mining condit ions were d i f f e r e n t from o t h e r blocks, t h i s a rea requi red a d d i t i o n a l explora t ion t o per- m i t r e se rve computations of the same category a s t h e r e s t o.f t he depos i t .

Advantages

The average f a c t o r s and area methods of r e se rve computations a r e r e l a - t i v e l y simple; t h e i r u s e , however, r equ i r e s t r a i n i n g and experience. Areas a r e measured by planimeter , computed, or sca led from maps. I n general , the f a c t o r s a r e d e t e d n e d by a minimum number of simple c a l c u l a t i o n s ; no spec i a l o r d e t a i l e d maps a r e needed. The procedure i s f l e x i b l e and r equ i r e s no com- p lex formulas; computations can be made f o r i nd iv idua l b locks , panels, l eve l s , segments, o r f o r t h e e n t i r e mineral body.

These methods a r e adaptable t o a l l types of depos i t s and t o a l l s tages of development; they a l low rapid and continuous eva lua t ion of f a c t u a l da t a , thus permi t t ing improved engineering planning.

Indiv idua l observat ions of thickness and grade a r e o f t e n unconfirmed with r e spec t t o t h e i r l o c a l i t i e s ; therefore , computations of average f a c t o r s usu- a l l y do not r e q u i r e a r ea weighting. Changes i n r e se rves of a mine, whether due t o ex t r ac t ion o r continuing explora t ion , can be e a s i l y made by sub t r ac t ing o r adding r e s p e c t i v e a reas , or by determining new o r co r r ec t ed a reas .

Accuracy of t h e computations v a r i e s depending on t h e type of depos i t , number of blocks, and density of observations. When a d e p o s i t i s q u i t e uniform and average f a c t o r s a r e computed on t h e bas i s of a s u f f i c i e n t number of observat ions, r e s u l t s a r e accurate.

Applicat ion

Analogous and geologic blocks methods a r e widely used f o r a l l types of mineral depos i t s . For successful app l i ca t ion t h e i r p r i n c i p a l requirement i s a geologic and geochemical s imi l a r i t y between t h e segment o r block being consid- e red , and a more thoroughly s tudied por t ion of the same o r s imi l a r deposi t . Both methods a r e convenient f o r rapid approximations t o support explorat ion and everyday mining decisions. They o f t e n can be used when o the r methods f a i l because of lack of s u f f i c i e n t data .

Cer ta in types of mineral depos i t s , such a s t a b u l a r , bedded, and la rge p l ace r depos i t s , a r e pa r t i cu l a r ly su i t ed t o these methods. Their bas ic param- e t e r s vary only s l i g h t l y from one point t o another , and t h e average f a c t o r s may be determined with su f f i c i en t accuracy by a simple ave rag ing j regard less of whether observa t ions a re d i s t r i b u t e d i n a systematic o r unsystematic pat- t e rn . Another u se of these methods i s when the thickness of a mineral body can be accu ra t e ly measured, but the high c o s t or t e c h n i c a l d i f f i c u l t i e s nnke i t impossible t o sample raw ma te r i a l i n place. Grade i n such a case may be computed from pas t production, other por t ions of t h e d e p o s i t , or even m i l l and smelter r e tu rns .

Both methods should be used wi th d i s c r e t i o n , because t h e accu-

*----- racy f o r a depos i t

54 may depend on 'per- - s o n a l i n t e r p r e t a -

B r4 t i o n , r a t h e r than o b j e c t i v e geologic

I observa t ions and samp l i n g .

1 I The mining

I blocks i s a l s o

I known i n t h e min- e r a l i n d u s t r y a s

----A longi tudina 1 sec- t i o n , mine e x t r a c - t i o n , and mine e x p l o i t a t i o n (11, 2 8 , 2). A mining - block map be def ined a s a por- t i o n of a mineral body d e l i n e a t e d on four s i d e s by workings, o r bounded by work-

FIGURE 31, - Mining Blocks Exposed on Four Sides. A, Vertical section of a vein developed by underground workings; B , isometr~c drawing of three mining blocks a, b , and c (Note-vein th~ckness less than width of workings); C, geometric interpretation of the same blocks for computotlons. ,

i n g s on t h r e e or l e s s s i d e s , and by survey or a r b i - t r a r y l i n e s on the remaining s i d e s ( f i g . 31) . The s i z e and form of t h e mining b lock is determined by exp lora t ion and development work- i n g s , geologic f e a t u r e s , tech- n i c a l , and economic considera- t ions .

I n p r a c t i c e , mining blocks a r e genera l l y

r ec t angu la r i n shape with the bases ly ing i n t he p lane of a p lan , v e r t i c a l , o r i n c l i n e long i tud ina l s ec t ion , depending on the geologic c h a r a c t e r i s t i c s of the depos i t . The most common form of a mining block i s pa ra l l e l l ep iped ; block volume of o re i s computed a s a product of a rea and average thickness (formula 1 5 ) , o r e tonnage a s a product of volume and weight f a c t o r (formula 17) , and meta l tonnage a s a product of ore tonnage and average grade (formula 29). The usua l form f o r reserve computations by t h i s method i s given i n t a b l e s 9 and 10.

, Mining blocks del ineated by a combination of underground openings and

d r i l l ho l e s a r e s p e c i a l cases. Assignment of r e se rves i n t o ca t egor i e s depends on the type of depos i t , kind and dens i ty of mine workings, and economics. Accuracy depends on the way the blocks were de l inea t ed and on t h e method and accuracy of the sampling. Assuming t h a t a l l w r k i n g s a r e s tud ied with t h e same accuracy, s eve ra l t yp ica l cases a r e discussed f u r t h e r .

TABLE 9 . - Determination of a r i t hme t i c average of f a c t o r s f o r individua 1 blocks

Block I Name of ( Thickness ( t ) ,l I Grade (c) ,'

Total. . . . . . . . . . . . .

Average.. .........

I......................

2 . .....................

Total . . . . . . . . .....

Raise A1B1 Raise A ~ F ? Dr i f t A I B ~ Dr i f t BI

workings Raise XB Raise AI B1 Dr i f t A A ~ Drift BB1

f t

:1 t z t '3 t 1 4

Average..... ......

percent C

I 1

C / 2 ' 13

c +

= ( t ) a r e determined a s average a r i t hme t i c th ickness by formula ( 10). ( c ) a r e determined a s average a r i t hme t i c grade by formula (24 ) , o r th ickness-

weighted average grade by formula ( 2 5 ) , or area-weighted average grade by formula (26) .

4 t = C t H / 4

i=l

4 c f l a y = 1 c r ' / 4

i=l

TABLE 10. - R e c a p i t u l a t i o n of r e s e r v e s f o r m i n e r a l body (by c a t e g o r i e s ) and de te rmina t ion of average g r a d e

Block

T o t a l . .

Average

Average :hickness tav 1 , f t

Raw m a t e r i a l

vo lume V ) , cu f t

Vl v2

vn

Weight f a c t o r (a,

:u f t l t o n

F

F

F

Raw n a t e r i a l r ese rves :Q) , t o n s

4 Q

Qn

[ Valuab le com .ve rage g rade

( c a v 1 r p e r c e n t

lnents Leserves PI tons

Block Exposed on Four S ides by Underground Workings

Reserve computations of a number of mining blocks opened on a l l sides by underground workings is made by determining average f a c t o r s f o r each workings; de te rmin ing average f a c t o z s f o r each block; computing volume, o r e , and metal tonnages f o r each b l o c k ; and s u m r i z i n g t h e r e s e r v e s of a l l b l o c k s of t h e same ca tegory and computing weighted average f a c t o r s f o r each c a t e g o r y and f o r t h e e n t i r e m i n e r a l d e p o s i t .

Determining Average F a c t o r s f o r Each Working

When t h e t h i c k n e s s of a mineral body i s uniform and l e s s than t h e width of underground openings , average f a c t o r s f o r each working a r e u s u a l l y found by a s imple a r i t h m e t i c average of a p p r o p r i a t e v a r i a b l e s . Average t h i c k n e s s may be computed by weigh t ing a r e a s of i n f l u e n c e of each t h i c k n e s s accord ing t o the r u l e of n e a r e s t p o i n t s , depending on t h e form and s i z e o f t h e m i n e r a l body, i r r e g u l a r i t i e s i n v a l u e s , and d e n s i t y and d i s t r i b u t i o n of o b s e r v a t i o n s . In i r r e g u l a r bod ies average grade f o r a working i s computed by weigh t ing each sample by a p p r o p r i a t e a r e a s , volumes, and tonnages .

Determining Average F a c t o r s f o r Each Block

When t h e t h i c k n e s s of mineral body i s l e s s than t h e width of t h e opening, average f a c t o r s a r e computed a s f o l l o w s :

lengths of a l l s i d e s a r e equal,

where t, , t, , t,, and t, a r e thicknesses measured o r computed f o r each working; el , c2 , C, , anc c, a r e grades f o r the same workings.

When lengths of s ides a r e unequal and t h e r e i s no r e l a t i o n s h i p between thickness and grade, average f a c t o r s of a mining block may be computed by weighting each working according t o i t s length , 4 , & , L,, and L,,

and

When lengths of s ides a r e unequal and thicknesses and grades of workings vary considerably ( f i g . 3lA), average f a c t o r s a r e

and c,, = formula 26,

h e r e sl , s2 , s, , and si a r e a r eas of in f luences of each working found by the r u l e of nea re s t po in ts .

When the o r e thickness i s more than t h e width of t h e mine openings and t h e blocks a r e developed by c rosscuts on two l e v e l s , r e se rves may be computed from the c ros scu t d a t a , a s discussed f u r t h e r f o r a block exposed on two s ides . D r i f t and r a i s e samples between c ros scu t s a r e not r e q u i r e d , i f t h e r e a r e evi- dences of good o r e cont inui ty and grade uniformity. I n t h e case of an i r r egu - l a r body t h e block may be divided i n t o a r e a s of i n f luence t o check r e s u l t s of o r ig ina 1 computations.

Block Exposed on Three Sides by Underground Workings

Average f a c t o r s f o r a mining block exposed on t h r e e s ides by underground workings a r e computed s imilar t o the previous cases : ( a ) a s an a r i t h m e t i c average of t h r ee s i d e s (disregarding length of workings); (b) by weighting va r i ab l e s of each s i d e of the block according t o the 1ength.of each working; ( c ) by computing f i r s t the f ac to r s f o r t he fou r th s i d e from end-samples a and b of the e x i s t i n g s i d e s ( f i g . 32& l e f t ) and, then , averaging the v a r i a b l e s of - a l l s ides ( t h i s , however, increases the importance of t h e end-samples i n

Sample a

Sample b

comparison with o t h e r s ) ; and ( d ) by weight ing t h e a r e a s o f i n f l u e n c e of

I.\- i each o f t h e t h r e e e x i s t - \ , - - -; i n g workings ( f i g . 324 I I r i g h t ) .

I A block d e l i n e a t e d by a n a d i t , r a i s e , and s u r f a c e workings, t h a t i s t r e n c h e s o r p i t s , i s a s p e c i a l case and may be subd iv ided a s above ( f i g . 322 r i g h t ) , o r on t h e b a s i s of accuracy o f computat ions (ca tegory) ( f i g . 322 l e f t ) . It may a l s o b e subdivided i n t o b l o c k s based on geologic e v i d e n c e s , such a s d e g r e e of o r e a l t e r a t i o n , t h i c k n e s s , g rade , zoning, o r number of observa- t i o n s (32C).

Block Exposed on Two Sides

Underground Workings on Two Levels

FIGURE 32. -

Oxidation zone When a mining block - i s developed on two lev- e l s ( f i g . 33, block 1) a v e r a g e f a c t o r s f o r each l e v e l a r e found f i r s t ' . I n uniform bodies t h e block f a c t o r s a r e computed a s t h e average of both l e v e l s ; o therwise

Mining Blocks Exposed on Three Sides. A , Block they a r e computed a s area-weighted averages

exposed b y two drifts and a raise; B , block exposed of both levels, by adit, raise, and surface workings, no change in

geology wi th depth; C, block exposed by adit, raise, I n t e r s e c t i n g Under- and trenches, natural changes in th~ckness and g round Workings grade with depth due to oxidation.

A mining block developed on two s i d e s by underground i n t e r s e c t i n g workings; t h a t i s d r i f t and raise, i s a t r i a n g u l a r pr ism ( f i g . 33, b lock 2 ) . Block f a c t o r s may be d e t e r - mined as t h e average of both workings, a s t h e length-weighted average of both

Block I

Block 2

FIGURE 33. - Mining Blocks Exposed on Two Sides (Vein Thickness ' ~ e s s Than the Width of Workings). -4, Blocks exposed between two parallel workings (1 ) and between two intersecting workings (2); B , isometric drawing of the samemining blocks.

workings , or a s t h e area-weighted average of bo th workings wi th a r e a s of i n f l u - ence found according t o t h e r u l e of n e a r e s t p o i n t s (b locks 2a and 2b) .

I n t h e f i r s t and second examples, volumes a r e computed a s t h e product of average th ickness ( t a v ) and a rea ( S ) ; i n t h e t h i r d example a s a sum of two a u x i l i a r y b locks , each c a l c u l a t e d a s t h e product of t h e average t h i c k n e s s ' o f t h e working and i t s a r e a o f in f luence .

When t h e t h i c k n e s s of t h e minera l body i s g r e a t e r t h a n t h e width of the workings, t h e mining block may be developed on bo th l e v e l s by c r o s s c u t s o r by c r o s s c u t s and d r i f t s ( f i g . 34) . I n t h e f i r s t example r e s e r v e s a r e computed from c r o s s c u t d a t a by t h e v e r t i c a l c r o s s - s e c t i o n methods. h%en d r i f t samples between c r o s s c u t s a r e a v a i l a b l e , t h e f a c t o r s and a r e a r e s e r v e s may be computed by t h e r u l e of n e a r e s t p o i n t s f o r each l e v e l . Block r e s e r v e s a r e computed by one o f t h e formulas d i s c u s s e d f u r t h e r i n t h e s e c t i o n on c r o s s - s e c t i o n methods.

Block Exposed on One Level and I n t e r s e c t e d a t Depth by D r i l l i n g

A minera l body exposed by workings on one l e v e l and i n t e r s e c t e d a t depth ( o r above) by one o r more d r i l l ho les may be d i v i d e d i n t o mining blocks by c o n s t r u c t i n g p e r p e n d i c u l a r s from t h e d r i l l h o l e o r e i n t e r s e c t i o n l e v e l a , b , and c t o t h e underground workings l e v e l a: , b= , c l ( f i g . 35~). Boundaries of b locks a l s o may be determined on the b a s i s of geo log ic c r i t e r i a , such a s zon- i n g o r r a k e of m i n e r a l i z a t i o n , and on b a s i s of mining d e s i g n and economics. I f t h i c k n e s s and grade of t h e body a r e uniform, b lock f a c t o r s a r e computed by

and

where tl and c: a r e a v e r a g e th ickness and average grade of o r e i n each block on t h e d r i f t l e v e l ;

and c, a r e average th ickness and average grade of each two a d j o i n i n g d r i l l h o l e s l i m i t i n g t h e s i d e of b lock ; and

4 and a r e t h e block lengths on both l e v e l s .

When t h e m i n e r a l body i s i n t e r s e c t e d by one d r i l l h o l e , t h e mining b Lock may be d i v i d e d a c c o r d i n g t o t h e r u l e of n e a r e s t p o i n t s i n t o two a u x i l i a r y b locks of v a r y i n g accuracy and, t h e r e f o r e , of d i f f e r e n t c a t e g o r i e s ( f i g . 35g).

The average f a c t o r s may be determined by formulas (57, p. Z l g ) ,

and

where 5 and c, a r e average th ickness and a s s a y f o r t h e d r i f t and t, and c; a r e t h i c k n e s s and assay f o r t h e d r i l l h o l e .

When a l a r g e number of samples a r e a v a i l a b l e from t h e d r i f t , d r i l l h o l e i n f o r m a t i o n of t e n may serv.e only t o d e l i n e a t e t h e block and e v a l u a t e t h e min- e r a l i z a t i o n . I f each o b s e r v a t i o n , whether i t i s from t h e d r i f t o r d r i l l h o l e , i s cons idered on an e q u a l b a s i s , then t h e computat ions of average t h i c k n e s s and g rade are made by formulas .

I n t h e example o f numerous h o l e s t h e formulas a r e

and

With one h o l e they a r e

and C1 i C 2 t... + C , + c L 1 - - n + l

I n t h e s e computa t ions ,

5 , t2 , . . . t, a r e th icknesses observed i n t h e d r i f t , c1 , c2 , . . . C, a r e corresponding a s s a y s , tl, , t12, .. . tl, th icknesses observed i n d r i l l h o l e s ,

ell , cl, , . . . clm corresponding a s s a y s ,

n - number of samples i n t h e d r i f t , and

m - number of d r i l l ho les .

A p p l i c a t i o n

I n o r d e r t o apply t h e mining blocks method i t i s necessa ry t o develop t h e minera l body i n t o b l o c k s ( f o r e x t r a c t i o n ) by a s u f f i c i e n t d i s t r i b u t i o n of workings . I n g e n e r a l , r e s e r v e s f o r uniform bodies a r e of t h e h i g h e s t c a t e g o r y ; t h a t i s , proved o r semiproved. computat ions a r e r e l a t i v e l y s imple . Block r e s e r v e s may be c l a s s i f i e d f o r mining purposes accord ing t o t h i c k n e s s , g r a d e , and e x t r a c t i o n c o s t . Thus, t h e method a l lows t h e o p e r a t o r t o c o n t r o l t h e q u a l i t y and t h e c o s t of product ion.

The method i s f l e x i b l e and r a y be used i n a l l t y p e s of minera l d e p o s i t s , The degree of e r r o r depends , t o a g r e a t e x t e n t , on t h e g e n e t i c . type of t h e d e p o s i t , on t h e d e n s i t y of workings, and on t h e d i s t r i b u t i o n of o b s e r v a t i o n s . It i s n a t u r a l l y adap ted t o sedimentary beds such a s c o a l , t o v e r t i c a l and s t e e p l y d i p p i n g v e i n s of t h i n and medium t h i c k n e s s , and t o thin-bedded t a b u l a r

o r e b o d i e s , where grade and t h i c k n e s s undergo g r a d u a l changes and where mining and g e o l o g i c f e a t u r e s a r e s i m i l a r t o b l o c k s a l r e a d y e x t r a c t e d . When weighted averag ing of t h i c k n e s s and grade can be e l i m i n a t e d from t h e computat ions , i t becomes a simple opera t ion .

I n n e s t l i k e , broken, o r i n t e r r u p t e d b o d i e s , o r where minera l v a l u e s are d i s t r i b u t e d e r r a t i c a l l y , t h e r e l a t i v e e r r o r of t h i s method may be excess ive .

Cross-Sect ion Methods

P r i n c i p l e s and Requirements

The i n i t i a l s t e p i n t h e a p p l i c a t i o n of c r o s s - s e c t i o n methods i s t o d i v i d e t h e minera l body i n t o blocks by c o n s t r u c t i n g g e o l o g i c s e c t i o n s a t i n t e r v a l s a l o n g t h e t r a n s v e r s e l i n e s o r a t d i f f e r e n t l e v e l s i n conformity w i t h explora- t i o n workings, purpose of computat ions , and t h e n a t u r e o f t h e d e p o s i t ( f i g s . 36 and 3 7 ) . The i n t e r v a l between t h e s e c t i o n s may b e c o n s t a n t o r may vary t o s u i t t h e geology and mining requ i rements . When t h e i n t e r v a l s a r e unequal , formulas f o r computations a r e s l i g h t l y more compl ica ted .

Depending on t h e manner o f t h e b lock c o n s t r u c t i o n t h e r e a r e t h r e e modif i - c a t i o n s of c r o s s - s e c t i o n methods :

1. Standard method based on t h e r u l e of g r a d u a l changes ( f i g . 374). Each i n t e r n a l block i s confined by two s e c t i o n s and by a n i r r e g u l a r l a t e r a l s u r f a c e , and each end block by a s i n g l e s e c t i o n and by a n uneven l a t e r a l

FIGURE 36. - Block Layout b y Cross-Section Methods (Block Diagram). .-I, Rule of graduol changes-standard cross-section method; B , rule of nearest points-linear cross-section method.

P L A N

L E G E N D Vert ical dr i l l holes

crossing ore

o Blank vertlcol drlll holes

P L A N

FIGURE 37. - Cross-Section Methods-Standard and Linear. A, Laying out blocks according to

the rule of gradual changes; B , laying out blocks according to the rule of nearest points.

su r f ace . Sec t ions m y be p a r a l l e l o r n o n p a r a l l e l , v e r t i c a l , h o r i z o n t a l , o r i n c l i n e d ;

2. Linear method based on t h e r u l e of n e a r e s t p o i n t s (fig, 37B). Each block i s def ined by a s e c t i o n and a length equa l t o one-half the d i s t ance to t he ad jo in ing s e c t i o n s ; and

3. Method of i s o l i n e s based on t h e r u l e of g radua l changes ( s e e sec t ion e n t i t l ed Method of I s o l i n e s ) .

. For a c c u r a t e r e s u l t s cross-sectionmethodsrequire

( a ) t h a t a s u f f i c i e n t number of workings completely c r o s s i n g t h e minera l body and an adequate number of obse rva t ions and samples taken from each sec- t i o n t o r e p r e s e n t t h e ' qua l i t y of t h e raw m a t e r i a l ;

( b ) that t h e w r k i n g s l i e i n o r n e a r t h e s e c t i o n s ; and

( c ) t h a t a l l workings a r e d i s t r i b u t e d more o r l e s s equa l ly between t h e s e c t i o n s .

Standard Method--Paral le l Sec t ions

Procedure

Some e a r t h s c i e n t i s t s d i s t i n g u i s h two v a r i a b l e s of t h e s t anda rd method: v e r t i c a l o r f ence used mainly i n e x p l o r a t i o n ; and h o r i z o n t a l o r l e v e l used i n mining. The u s u a l procedure f o r computing r e se rves by t h i s method i s ( a ) determining t h e a r e a s of a l l s e c t i o n s , ( b ) computing average f a c t o r s f o r each s e c t i o n ; ( c ) computing volume, and o r e and meta l tonnages f o r each b lock; and f i n a l l y , ( d ) su-rizing the r e s u l t s f o r a l l blocks by c a t e g o r i e s and comput- i ng average f a c t o r s f o r the e n t i r e body ( t a b l e 11). Measurement of i n d i v i d u a l a r e a s has been d iscussed i n pa r t 1 of t h i s r e p o r t . I

Average f a c t o r s f o r each s e c t i o n may be determined by t h e r u l e of gradual changes, r u l e of n e a r e s t po in ts ( length-weighted average and area-weighted ave rage ) , and a s an a r i t hme t i c average. When i n the c a s e of i r r e g u l a r d i s - t r i b u t i o n of workings, there i s a d i r e c t o r i nve r t ed r e l a t i o n s h i p between th i ckness and grade , average grade may be computed a s a thickness-weighted average.

Volume Computations

The c o n f i g u r a t i o n of a r e a s and t h e l a t e r a l shape of t h e b locks a r e u s u a l l y i r r e g u l a r and, i n order t o compute volume by s o l i d geometry, a r e a s a r e cons idered t o be of equal s i z e c i r c l e s o r polygonal f i g u r e s ; l a t e r a l s u r f a c e s of t h e blocks a r e disregarded.

Mean-Area Formula. - The s imp les t formula f o r a volume between two p a r a l l e l s e c t i o n s with a r eas S1 and S2 and a perpendicular d i s t a n c e , L , between them i s

This mean-area formula i s p r e c i s e when both a r e a s a r e n e a r l y s i m i l a r i n s i z e and shape.

Blocks

I . . . . . . . . . . .

...........

...........

........... N...........

Tota l . . ..

Average. a

TABLE 11. - Computation of reserves by standard method of cros s - sec t ions

zc t ions n t erva 1 etween ec t ions L), f t

) 4

1

Volume (V) , cu f t

Weight factor

( 0 , u f t l t o n

Raw material reserves (Q) ,

tons

I Valuable

percent I Grade ( c ) , :omponen t

Reserves ( P ) , tons

.A s o l i d mine ra l body t h a t h a s been d iv ided i n t o b locks by a s e r i e s of evenly spaced p a r a l l e l s ec t ions i s computed by t h e "end-area!' formula de r ived from formula ( 4 6 ) ,

where L equals t h e d i s t a n c e between sec t ions . I f t h e s e c t i o n s spaced t h e formula f o r the volume of the e n t i r e body w i l l b e

(47)

a r e unevenly

Z 9 ( 4 8 )

where & , Le , La, . . . L, a r e perpendicular d i s t a n c e s between t h e a d j o i n i n g s e c t i o n s with a r e a s S1, &, Sg ... Sn.

Wedge and Cone (F'yramid) Formulas. - End b locks of l e n s l i k e mine ra l bodies may be conver ted t o a wedge o r cone (pyramid) wi th t h e l a r g e r a r e a s S i n one s e c t i o n , t a p e r i n g t o a l i n e o r a po in t i n t h e a d j o i n i n g s e c t i o n ( f i g . 384). I f t he block t a p e r s t o a l i n e , volume i s computed by wedge formula,

This formula, however, i s p r e c i s e only when t h e base i s r e c t a n g u l a r and t h e l a t e r a l f a c e s a r e i s o s c e l e s t r i a n g l e s and t r apezo ids . A more p r e c i s e formula f o r t he wedge i s (42)

V = & ( 2 a + a ~ ) b s i n a , 6

where a and b a r e t h e ' lengths of s i d e s of t h e base a - a n g l e between a and b , and al i s t h e l a r g e r s i d e of t h e t r apezo id (38A r i g h t ) .

I f t h e block t a p e r s t o a po in t ( f i g . 38g), volume is computed by cone formu l a ,

Volume computed by t h e wedge formula i s 50 percent l a r g e r t h a n volume computed by t h e cone formula.

Frustum Formula. - When S1 and & vary i n s i z e , bu t a r e s i m i l a r ( f i g . 38C), f rustum of a cone o r pyramid formula i s used t o compute t h e b lock volume ,

FIGURE 38. - Standard Cross-Section Method (Parallel B, cone (pyramid) formula; C, frustum of

Sections). A, Common wedge formula; a cone formula.

I n p r a c t i c e , t h e frustum formula i s avoided because of complications involved i n computing square r o o t s and i n c e r t a i n cases i t i s l e s s accura te than t h e prism formula. Let us cons ider a block of a r e g u i a r prism (volume of which i s V = S x L) and d i v i d e it i n t o two a u x i l i a r y prisms i n the f o w of t runcated wedges with a reas S, and % equal t o 0.9s and a r e a s S , and Ss equal t o 0.1s ( f i g . 39) (30).

FIGURE 39. - Frustum Formula Versus Mean-Area Formula.

According t o f ru s tum formtila t h e volume of each a u x i l i a r y b lock i s

V, = V2 = (0.1 + 0.9 + ((0.1) (0 .9) ) = 0.433 L, 3

and the volume of t h e r egu la r pr i sm i s

I n t h i s example t h e t o t a l block volume, by f rus tum formula , i s 13.4 percent l e s s than t h e volume computed by the r e g u l a r pr ism formula. It i s concluded, t h e r e f o r e , t h a t t h e f rus tum formula i s i n a c c u r a t e i n wedgel ike bodies . Thus, when t h e a r e a s d e l i n e a t i n g the t runca ted wedge b locks have equal s i d e s , such a s h e i g h t s q and & between two l e v e l s ( f i g . 40A), o r t h i cknesses al and

FIGURE 40. - Truncated Wedge-Standard Cross-Section Method (Parallel Sections).

between p a r a l l e l s e c t i o n s ( f i g . 402), t h e mean-area formula i s more p r e c i s e t han t h e frustum.

Pr i smoida l Formula

Many m i n e r a l bod ie s swe l l , c o n t r a c t , p inch , and i n gene ra l , have i r r egu - l a r l a t e r a l s u r f a c e s t h a t may have a profound in f luence on t h e accuracy of volume computations. The pr ismoidal formula i s based on the assumption t h a t t h e enc los ing l a t e r a l , curved and warped s u r f a c e s can b e a c c u r a t e l y rep laced by p l ane t r i a n g l e s , t r apezo ids , o r para l le lograms bounded by s t r a i g h t Lines and c o n s t r u c t e d from one p a r a l l e l s e c t i o n t o t h e ad jo in ing one (10, l4, 20, 40, 42, 6 2 ) . S e l e c t i o n of t h e p lane f i g u r e s i s c o n t r o l l e d by t h e shape of the - minera l body and i t s enc los ing su r f ace .

The pr i smoida l formula i s de r ived from imp son's r u l e f o r i r r e g u l a r a r e a s .

where M i s t h e a r e a of a n a u x i l i a r y p l a n e s e c t i o n p a r a l l e l t o and midway between s e c t i o n s S1 and &. Const ruc t ion of t h i s a u x i l i a r y s e c t i o n i s based on i n t e r p o l a t i o n of l ong i tud ina l and c r o s s s e c t i o n s and by i n t e r p r e t a t i o n of the geology of t he mine ra l body (39, p. 581, 28, p. 117). Only i n except ional c a s e s i s M a n ave rage of Sl and & . A u x i l i a r y a rea c o n s t r u c t i o n s r e q u i r e a d d i t i o n a l work.

Th i s formula i s advantageous when a mine ra l body i s d iv ided i n t o blocks by a s e r i e s of c l o s e l y spaced c r o s s s e c t i o n s . A l t e r n a t e o r odd number sec- t i o n s may b e regarded a s end block s e c t i o n s , and each i n t e r v e n i n g o r even

numbered s e c t i o n a s a midsection M. The formula i s recommended when t h e cross s e c t i o n s a r e of d i f f e r e n t conf igura t ion and more a c c u r a t e computations a r e d e s i r e d ; i t i s cormnonly used i n c i v i l engineering f o r e a r t h work and i s descr ibed i n f i e l d surveying handbooks (39, p. 582, 57, v. 1, p. 231).

Volume computations, with the p r e c i s i o n of t h e prismoidal formula, may be made by reducing the r e s u l t s computed by the mean-area formula by a "prismoi- d a l c o r r e c t i o n fac tor" ,

p r i s m o i d a l ' V m e a n - a r e a - C p r i s m o i d a l c o r r e c t i o n

where C , t h e f a c t o r f o r any t r i a n g u l a r prismatoid, i s equal t o

and is expressed i n cubic f e e t .

BLOCK A

FIGURE 41. -Blocks Between Parallel Sections. Influence of the

BLOCK 8 shape of areas on

yu4- + volume computa- t ions.

where al and a r e s i d e s of area S1 and a2 and b2 a r e s i d e s of area & ( f i g . 41, Block A ) . Tables of values f o r t r i angu la r prisms f o r a d is tance of 100

. f e e t between sec t ions , and fo r prismoidal co r rec t ions a r e a v a i l a b l e in p u b l i - ca t ions r e l a t e d t o -railway and other ea r th excavation problems . The o b e l i s k formula, described i n some publicat ions as a separa te formula (30), i s a mod- i f i c a t i o n of a prismoidal formula derived by s u b s t i t u t i n g equation (55) f o r M.

The inf luence of t h e shape of p a r a l l e l a reas on volume computations i s w e l l demonstrated by t h e following. Comparing t h e r e s u l t s of computations by mean-area, frustum, and prismoidal formulas f o r two bodies with the same value base a r e a s S1 , & , S3, and Sq , but of d i f f e r e n t shape ( f i g . 4 l ) , gives

Dimensions of bodies, f e e t : I I

1 Block A I Block B Distance between sections...... . .feet. . 1 15

Thus, i n our example the volumes of blocks A and B computed by the mean- a r e a and frustum formulas a r e 13.0 and 50.5 percent less than t h e va lues computed by prismoidal formula.

15

b4... ... .....;....................+.. S4.. ................................. .. Mean- a r ea ......................... f t3

Frustum ................ .. ......... f t 3 . .. Prismoidal.. ..................... .f t3 Rela t ive e r r o r s of mean-area and

frustum formulas.............percent..

I f values and a r e equal, the prismoidal ( o b e l i s k ) formula converts t o the mean-ar ea formula.

- -

1,200 1,200 1,380

13.0

2 80

1,200 1,200 2,425

50.5

FIGURE 42. - Construction of Auxiliary Area R for Bauman's Formula. A, Block between . parallel sections with irregular lateral surface; B, some block with linear

lateral surface and construction of generators; C, intermediate drawing showing how to find proiections of generators; D, construction of auxiliary area R.

Thus, h e n any p a i r of s i d e v a l u e s of a and b a r e equa l o r n e a r l y equa l , t h e mean-area formula i s accu ra t e . This i s i n g e n e r a l a connnon c a s e of volume computations, such as a block having equal h e i g h t o r th ickness .

The pr i smoida l formula h a s been used i n computations of complex v e i n o r e bodies i n t h e Coeur d'Alene mining d i s t r i c t , Idaho (10).

Bauman's Formula. - Among t h e l e s s cormnon formulas f o r volume computa- t i o n s is one o f f e r e d by Bauman (2, 2). This g raph ic formula has l i m i t e d use.

When t h e l a t e r a l and i r r e g u l a r s u r f a c e of a mine ra l body between two p a r a l l e l s e c t i o n s is assumed t o be l i n e a r ( f i g . 42), t h e volume of t h e b lock may be computed by

where S1 and S2 a r e a r e a s of cross s e c t i o n s , L is d i s t ance between sect ions , and R i s an a u x i l i a r y a rea , graphical ly const ructed a s fo l lows:

1. Draw both end a r e a s and cons t ruc t p ro jec t ions ( A A ~ ' , BB'II, ...) from t h e genera tors ( M I , BB', ...) of t h e l a t e r a l surface o n . t h e same plane ( f igs . 42g and 42s).

2. From po in t 0 ( f ig . 422) cons t ruc t l i n e s of equal length and d i r e c t i o n a s t h e l ines AA1', B'I' , . . . I f t h e generators a r e taken i n s u f f i c i e n t quant i ty and a l l points A , B , C , ... a r e connected by a curved l i n e , t h e r e s u l t a n t f i g u r e w i l l be t h e a u x i l i a r y a r e a R.

Construction of a u x i l i a r y areas requ i res experience and t i m e , therefore , l i m i t i n g t h e use of t h e formula. The common prism and pyramid forruulas may be der ived from Batman's formula by making appropr ia te assumptions. When area Sl i s equal t o Sz , t h e a u x i l i a r y area R i s z e r o and Bauman' s formula i s the same a s a prism o r cyl inder . When area S2 i s zero , t h e a u x i l i a r y a r e a R w i l l be equal t o q , and Bauman's formula i s t h a t of a pyramid.

Tonnage Computations

The product of block volume and weight f a c t o r s produces t h e tonnage of I raw materia 1; t h e product of the l a t t e r and average grade equa 1s the reserves

of va luab le component. Another manner of tonnage computation f o r each block c o n s i s t s of determining "section reserves" f o r a s l i c e of one u n i t i n width, and computing block rese rves a s the product of hal f the sum of the sec t ion rese rves and t h e block length.

Sect ion rese rves a r e computed a s t h e t o t a l of the reserves between work- i n g s and may be determined by the r u l e of gradual changes ( f i g . 43A) or the r u l e of n e a r e s t po in t s (f ig. .43g). Section reserves a r e o f t e n termed l i n e a r r ese rves when taken along explorat ion l ines . The term " l i n e a r reserves", i n t h i s r e p o r t , i s set a p a r t f o r rese tves computed f o r one square u n i t of a r e a ; t h a t i s , square f o o t , square yard, e t c . ( f i g . 432) , and t h e term "area reserves" i s used f o r a reas of s u b s t a n t i a l s i z e such a s a c r e and square mile.

The Standard Method f o r Nonparallel Sect ions

Sect ions const ructed along explora t ion l i n e s may converge or diverge because of changes i n t h e s t r i k e of t h e mineral body. The angles between sec- t i o n s and exp lo ra t ion l i n e s may range from oblique t o obtuse , depending on s t r i k e v a r i a t i o n s . Formulas offered f o r computing reserves wi th nonpara l l e l s e c t i o n s a r e discussed i n t h e following sect ions .

Angle Less Than 10 Degrees

When t h e ang le of i n t e r s e c t i o n i s l e a s than 10 degrees, Zolotarev offered t h e formula ( f i g . 44) (27, 3l; 48, 57, 66),

Section reserves I linear unit wide

i near reserves per square unit WL

FlGU RE 43. - Section and Linear Reserves-Cross-Section Methods. . A, Layout of auxil iary blocks according to the rule of gradual changes; B , linear reserves per square unit; C, layout of auxil iary blocks according to the rule o f nearest points.

1.. *

(s ,+s2) v = - sin a 2

A \ Center of gravity for ore0 SI

FIGURE 44. - Standard Cross-Section Method for Volume Computations-Nonparallel Sections. A, Graphic representation of mineral body crossed by nonparal l~l sections; B , plan-construction of perpendiculars hl and h2 fromcenter of gravity of one section to the other.

where S1 and & are areas of the mineral body in the sections, and hl and b are the lengths of two respective perpendiculars dropped from the center of gravity of one section to another.

I n p r a c t i c e rese rves computed by the previous formula d e v i a t e s l i g h t l y from those computed by mean-area formula; i n most cases t h e use of the Zolotarev formula f o r a block between sec t ions with an angle of i n t e r s e c t i o n l e s s than 10 degrees is not necessary.

Annle Greater Than 10 Degrees

When the angle of divergence between the s e c t i o n s i s g r e a t e r than 10

degrees, a co r rec t ion f a c t o r -, i s used by Zolotarev. The angle between s i n a s e c t i o n s i s expressed i n radians.

This forrmla and the preceding one a r e considered accura te when t h e s i z e s of t h e s e c t i o n a l a reas do n o t d i f f e r more than four t o s i x times.

Some s c i e n t i s t s consider the Z o l o t a r w formulas a c c u r a t e only i n a case of a fragment of a ring-shaped body with t h e c e n t e r coinciding with t h e i n t e r - s e c t i o n of sec t ions . I n t h e i r opinion the use of t h e above forntulas leads t o systematic e r r o r , more of ten increasing r a t h e r than decreas ing the r e s u l t s (48)

The formulas r e q u i r e location of the cen te r s of g r a v i t y of each a r e a , which i s d i f f i c u l t i n complex geometric f igures . The b e s t procedure of f ind- i n g t h e center of g r a v i t y of an area i s graphical , when t h e coordinates of the cen te r a r e determined by the sum of the s t a t i c moments. Such determinations a r e inconvenient and cumbersome and a more p r a c t i c a l procedure c o n s i s t s of reproducing t h e a rea i n cardboard and locat ing i t s c e n t e r of g rav i ty by hang- i n g with a thread.

Compared with p a r a l l e l sec t ions computations of volumes and tomages by nonpara l l e l formulas may show appreciable d i f fe rences ; v a r i a t i o n s i n average grade, however, a r e r e l a t i v e l y small. A simpler method of computing reserves f o r blocks bounded by nonpara l le l sec t ions is discussed i n t h e next chapter .

Linear Method

I n t h e l i n e a r cross-sect ion method of r ese rve computations, blocks a r e const ructed according t o the r u l e of neares t po in t s ( f i g s . 36; and 37g); each block r e s t s on one s e c t i o n with the length of i n f h e n c e extending hal f the d i s t ance t o t h e adjoining sections. Ore, Q, and meta l r e s e r v e s , P, a r e usua l ly determined a s t h e product of l inea r o r e and m e t a l r e se rves q~ , p~ , and a r e a A,

and

When o r e and metal reserves q, and p, a r e given per cubic u n i t , volume ins tead of a rea i s requi red i n previous formulas,

and P = pyV. ( 6 3 )

For an i l l u s t r a t i o n of t h e use of t h e l i n e a r method let us t ake a block between two nonpara l l e l sec t ions ( f i g . 45). The o r e block, two, f o r t h e a r e a of in f luence 4 of sec t ion 2-2, lying between nonpara l l e l s e c t i o n s I- 1 and 3-3, is found by b i s e c t i n g t h e angles a and B with a u x i l i a r y s e c t i o n s e-e and el -4 . Reserve computations a r e made by formulas,

and P2 = pz ( A ' ~ + A " ~ ) ,

where A ~ ~ + A ~ ~ ~ = A2 ( a rea of inf luence of sec t ion 2-2); - a r e l i n e a r o r e reserves (per square f o o t ) , determined f o r s e c t i o n

2-2 by d iv id ing sec t ion r e s e r v e s on the length of t h e body along t h e s e c t i o n ;

& - a r e l i n e a r metal reserves determined s i m i l a r t o q2 . The l i n e a r method i s s u i t a b l e f o r computing reserves of p l a c e r d e p o s i t s ,

where exp lo ra t ion i s c a r r i e d out i n s t a g e s ; explora t ion l i n e s a r e drawn ac ross t h e changing course of the deposi t , and workings a r e d i s t r i b u t e d equal ly along such a l i n e . I f a d d i t i o n a l l i n e s of explora t ion a r e added between t h e i n i t i a l ones, t h e d i s t ances between sec t ions and t h e a reas of in£ h e n c e w i l l decrease and cons t ruc t ion of t h e appropr ia te b locks w i l l change. Reserves w i l l remain unchanged, unless new workings a r e added.

Advantages

The cross-sec t ion methods graphica l ly por t ray t h e geology of t h e minera 1 deposi t . The genera l procedure is simple and rap id , but formulas producing g r e a t e r p rec i s ion may n e c e s s i t a t e use o f diagrams, a d d i t i o n a l c a l c u l a t i o n s , and cons t ruc t ion of a u x i l i a r y sec t ions . To inc rease the accuracy of computa- t i o n s t h e number of blocks should be a s l a r g e a s poss ib le ; i n o the r words, the s e c t i o n s should be placed c l o s e together .

Care should be exerc ised t o avoid a r b i t r a r y loca t ions and cons t ruc t ion of sec t ions . Distance between sec t ions i s usua l ly governed, i n exp lo ra t ion , by t h e cha rac te r of the mineral body and t h e d i s t r i b u t i o n of minera l va lues . S e l e c t i o n of sec t ions u n j u s t i f i e d by exp lo ra t ion da ta may i n f l u e n c e t h e s i z e of t h e a r e a s and, i n tu rn , t h e computations. Construction should n o t r e l y on i n t e r p r e t a t i o n made w e r dis tances unmerited f o r t h e given type of deposi t . Most of t h e disadvantages i n the use of t h i s method can be avoided by properly p lanned explorat ion.

Computations of two o r more o r e bodies i n t h e sec t ions are poss ib le .

Applicat ion

FIGURE 45. - Linear Method-Block Between Nonparallel Sections.

Cross-section methods a r e being used e f f e c t i v e l y when const ruct ion of sec t ions i s poss ib le with a m i n i m amount of i n t e r p o l a t i o n and ext rapola t ion. The a p p l i c a t i o n of v a r ioua formulas depends on t h e a n a l y s i s of t h e layout of t h e sec- t i o n s , on the r e l - a t i v e s i z e and shape of o r e a r e a s , and on t h e distances between sec t ions .

I

Common formu- l a s f o r volume computations a r e mean-area and frustum. The reg- u l a r prism formula r e q u i r e s equa l i ty i n s i z e and shape of both areas. The mean - a r ea f ormu l a i s accura te when t h e a r e a s i n para l- l e l s e c t i o n s a r e s i m i l a r ; i t should not be used when s i d e dimensions a and b a r e d i f f e r - ent . I f t h e a r e a s

of p a r a l l e l sec t ions d i f f e r by more than 40 percent , t h e frustum formula should be used.

The prismoidal formula i s accura te f o r a l l var ious forms of s o l i d s . ~ a u m a n ' s formula requ i res const ruct ion of an a u x i l i a r y area. The proper use of t h e var ious formulas fo r d i f f e r e n t geometric s o l i d s is given i n t a b l e 12. In genera l , computation of block volumes by t h e methods of c ross sec t ions requ i res a n a l y s i s of the shape and s i z e of s e c t i o n s t o determine t h e bes t formula, p a r t i c u l a r l y i n conjugated blocks.

Source : Reference (30).

TABLE 12. - Applicat ion of var ious formulas i n computing volumes of s o l i d bodies i n s tandard c ross - sec t ion method

Well-defined and large bodies t h a t a r e uniform i n th ickness and grade or have g radua l ly changing values can genera l ly be computed accura te ly by cross- s e c t i o n methods. Sect ions may be v e r t i c a l , i n c l i n e d , o r h o r i z o n t a l , a s in p i p e l i k e o r s tock deposi t s . Two s u i t e s of sec t ions c o n s t r u c t e d a t r i g h t angle t o each o t h e r may be employed for l a rge minera l bodies wi th more o r l e s s evenly d i s t r i b u t e d values , such as s tocks and impregnations. The f i n a l r e s u l t s may then be computed a s an average of both s u i t e s of s e c t i o n s , or one s u i t e may se rve f o r c o n t r o l of the o t h e r .

Name

Mean-area.. ...........

Frustum...............

Prismoidal (Simpson). .

Obelisk...............

The method should be used with d i s c r e t i o n i n a l l c a s e s where t h e bodies a r e i r r e g u l a r , o r where values tend t o concen t ra te i n o r e shoots . When compu- t a t i o n s of s e v e r a l va luab le components a r e r equ i red and t h e mineral body shows grade v a r i a t i o n s f o r each component, i t i s d i f f i c u l t and o f t e n impossible t o apply c ross - sec t ion methods. Horizontal c r o s s s e c t i o n s , cons t ructed along l e v e l s o r hor izons of workings, i s p r e f e r r e d because of t h e s e l e c t i o n or design of mining method.

Qui te o f t e n i t i s necessary t o compute r e s e r v e s of o r e between l eve l s , o r of d i f f e r e n t grade wi th in a block, sepa ra te ly . The sum of such aux i l i a ry computations should be equal t o the block r e s e r v e s ; absence of such con t ro l i n d i c a t e s a poss ib le source of e r ro r .

l ~ e p o r t e d a s a new formula by Kravchenko and Kupfer.

Formu l a

L (S, +%)

2

L - 5 (%+%+ & XS,)

L - (S, +4M+S, ) 6

1: 3 [SI % + ( a l b + a 2 h ) j 2

Cross-sect ion methods a r e e a s i l y adaptable f o r use simultaneously with o the r methods. Reserves, developed i n upper l e v e l s by underground workings, may be computed by t h e mining blocks method, and r e s e r v e s of lower levels , explored by d r i l l i n g , by the standard c ross - sec t ion method. Numerous examples of such computations a r e described i n t h e l i t e r a t u r e (60-61, 65).

S o l i d bodies betweed p a r a l l e l sec t ions .

Prism

X

X

X

Pyramid

-

X

X

X

Frustum

-

X

X

X

Wed~e

-

-

X

X

Method of I s o l i n e s

P r inc ip les and Formulas

I s o l i n e s a r e curved l i n e s jo in ing a l l po in t s of equal u n i t value.. They a r e used t o graphica l ly i l l u s t r a t e n a t u r a l physica l and chemical p r o p e r t i e s or processes t h a t can be expressed by u n i t values. A common example i s a topographic map, where r e l i e f i s expressed by contours of equa l e levat ion . I so- . l i n e s a r e widely accepted i n e a r t h and engineering sc iences f o r v i s u a l de l inea- t i o n and d i s t r i b u t i o n s tud ies of va r ious physica l and chemical phenomena. Well-known app l i ca t ions a r e maps using l i n e s t o dep ic t s i m i l a r r e l a t i o n s h i p s such a s isothermal, i s o s t a t i c , i s m g u m t i c , isopach ( o r i s o t h i c k n e s s ) , i s o c a l ( i s o c a l o r i f i c va lues f o r coal) , i socarb (equal content of f i x e d carbon) , i sograde , and o thers . Less common a r e complex i s o l i n e s such a s l i n e a r r e se rves ( foot -percent , tons, per square f o o t , o r d o l l a r s per square f o o t ) u s e d ' i n computing reserves of mineral deposi t s .

The method i s based on the assumption chat u n i t v a l u e s , from one po in t t o another , undergo continuous and uninter rupted changes according t o t h e r u l e of gradual changes. To const ruct i s o l i n e s , in termedia te v a l u e s a r e determined by i n t e r p o l a t i o n between points of known va lues ; a s a r e s u l t c e r t a i n p r o p e r t i e s of mineral bodies may be presented g raph ica l ly on a plan o r s e c t i o n by a sys- t e m of i so l ines . I n aggregate such a system c o n s t i t u t e s a n imaginary s u r f a c e

! s i m i l a r t o a topographic map. These " t o p o s ~ r f a c e s ' ~ a r e graphic express ions of numbers and, thus , may be used according t o the p r i n c i p l e s of s o l i d and ana ly t - i c a 1 geometry.

The theory of t h e method of i s o l i n e s f o r use i n mining and engineer ing was developed by Sobolevsky (37, 53). H e d isc losed t h a t t h e toposurfaces can be added, subt rac ted , mul t ip l ied , and divided and t h a t even more complex oper- a t i o n s , such a s ex t rac t ion of r o o t s , invo lu t ion , d i f f e r e n t i a t i o n , and in teg ra - t i o n , could a l s o be made. P r a c t i c a l app l i ca t ions of t h i s method i n geology and mining a r e varied. Detailed d i scuss ion of these a p p l i c a t i o n s is beyond the scope of t h i s paper.

Cormon cases a r e computations of average th ickness , average grade, and average va lue of a mineral deposi t from appropr ia te i s o l i n e maps (6 , ll). Only an isopach map i s needed t o compute volume and tonnage of minera l o r e reserves . A u n i t of volume rese rves a t a given point on such a map is a product of he igh t , equal t o t h e th ickness of t h e body, and a r e a , equa l t o a u n i t va lue (square f o o t , square yard , square meter, e tc . ) . A u n i t of tonnage o r e r e se rves i s a d iv i s ion of volume rese rves and volume- tonnage f a c t o r . To compute the weight of metal o r o the r va luable component i n t h e d e p o s i t , i so- l i n e s of l inea r m e t a l reserves (product of l i n e a r o r e r e s e r v e s and grade) a r e constructed.

Let us examine a portion of an isopach map ( f i g . 46). The minera l body, confined i n na tu re by i r r e g u l a r s u r f a c e s , i s transformed f o r computations t o an equivalent body l imited on one s i d e by a f l a t plane base and on t h e o ther by a complex surface represented by a s e r i e s of i s o l i n e s of equal th ickness or he igh t . Thus, t h e isopach map gives a d i s t o r t e d p i c t u r e of t h e minera i

FIGURE 46. - Method of lsolines. A, Cross section of a mineral body along exploration line 10-10; B, isopach plan of the some mineral body along the exploration line 10- 10; C, cross section along explorotion l ine 10-10 made from isopach plan B.

body. Each i s o l i n e on the map may be considered a p r o j e c t i o n of a s l i c e of the mineral body, divided by a s e r i e s of p a r a l l e l equa l ly spaced s e c t i o n s . The volume and tonnage of each s l i c e may be computed by s t andard cross-sec t ion formulas; namely, mean-area (46) and frustum (52). Areas bounded by appropr i - a t e i s o l i n e s a r e measured by planimeter. The d i s t a n c e s between t h e a r e a s a r e cons tant , equal t o t h e isopach i n t e r v a l . The volume of t h e e n t i r e k i n e r a 1 body i s a sum of a l l s l i c e s (47).

When t h e minera l body is i r r e g u l a r i n th ickness , t h e computations may be complicated. I f t h e area of thickness h, i s i n two p o r t i o n s , and %'I ,

the volume of t h e s l i c e between and h, w i l l be ( f i g . 47A)

When th ickness & i s missing from p a r t of the a r e a %, but has ins tead a thickness 4 , t h e volume of a s l i c e between t h e & and h3 i s o l i n e s w i l l be

where h - constant thickness i n t e r v a l between i s o l i n e s ; So - area enclosed by h, contour l i n e (minimum th ickness ) ; S, - a rea enclosed by hl contour l i n e s ; and

S2' , SZl1 , and S , l l - a r e a s enclosed by contour l i n e .

Average o r e grade may be computed i n much the same way by t h e construc- t i o n of isograde nnps ( f ig . 473), and by weighting a r e a s o u t l i n e d by i s o l i n e s f o r each grade; namely

C

where c, i s t h e minimum grade of o re ; c - constant grade i n t e r v a l between i s o l i n e s ;

A, - area of o r e body with grade c, and h igher ; A, - a rea of o r e body wi th grade c, p lus c and h i g h e r ;

4 - a r e a of o r e body with grade c, p lus 2c and h igher , e t c .

The isograde map graphica l ly i l l u s t r a t e s the grade d i s t r i b u t i o n of the ore .

The metal tonnage reserves (o r o the r va luable c o n s t i t u e n t s ) may be found by mult iplying toposurfaces of isopach and isograde maps and cons t ruc t ing l i n e a r metal r e se rves maps (weight f a c t o r cons tan t ) . The geometric meaning of the l i n e a r metal reserves may be wel l i l l u s t r a t e d by a n imaginary i n g o t , received by s e t t l i n g a t r i g h t angles a l l metal p a r t i c l e s ' o n the plane of t h e map. The metal reserves of t h e mineral body then w i l l be confined between . base plane and l i n e a r metal toposurface. By s l i c i n g t h e ingot by s e c t i o n s p a r a l l e l t o the base on equal d i s t a n c e s , t r a c e s of such i n t e r s e c t i o n s with toposurface w i l l y i e l d the i s o l i n e s of l i n e a r metal r e se rves .

Requirements, Advantages, .and Limitat ions

The method of i s o l i n e s , a l s o known a s l w e l plan, is a graphical modifica- t i o n of the hor izon ta l s tandard cross-sect ion method. It i s d i s t i n c t i v e and, the re fo re , is discussed i n a separa te chapter . The method r e q u i r e s a s u f f i -

General formula 64, +f ( A ~ + ~ A , + 2 4 + - . * + A n

FIGURE 47. - lsopoch and lsograde Mops for Reserve Computotions- Method of Isolines. A, lsopach map; B, isograde map.

c i e n t number, appro- p r i a t e dens i ty , and d i s t r i b u t i o n of observations f o r accura te p l o t t i n g of i s o l i n e s . When observations a r e unevenly d i s t r i b u t e d , t h e weight in f h e n c e of one s t z t i o n may not be t h e same a s f o r t h e o the rs ; i n some deposi ts t h e densi ty of observa- t i o n s m y be insuf- f i c i e n t ; and i n o t h e r s i t may exceed t h e required accuracy. The method i s appropr ia te f o r min- e ra 1 bodies where t h e r e a r e c e r t a i n na t u r a 1 r e g u l a r i t i e s i n the v a r i a t i o n s i n th ickness , grade, and va h e .

A major advan- t age of t h e method i s i t s descript iveness. The isopach map gives an ideal - ized l ikeness of the mineral body, second only t o the model. The isograde map shows t h e d i s t r ibu- t i o n of r i c h and poor o r e , and t h e map of l i n e a r reserves i l l u s t r a t e s the d i s t r i b u t i o n of reserves of raw mate r i a l and valua- b l e const i tuents . I s o l i n e maps a r e

easy t o read, measure, and i n t e r p o l a t e ; c a l c u l a t i o n s a r e rep laced by graphic i n t e r p r e t a t i o n s and t h e r e a r e fewer b locks , ins tead of numerous small blocks used i n some methods.

The method permits be t t e r mine planning. The boundaries of cutoff o re a r e e a s i l y const ructed and changed. Volume can be computed by measuring areas of r espec t ive i s o l i n e s without addi t iona 1 drawing. I f t h e requirements fo r minimum grade, th ickness , or va lue of o r e a r e changed, t h e isomaps remain the same; rese rve computations can be made plus or minus one o r s e v e r a l s l i c e s of the mineral body. The method can be automated through c a l c u l a t i n g machines o r computers and, when properly appl ied , prevents d i sc repanc ies between the o r i g i - n a l r e se rve es t imates and those remaining a f t e r p a r t i a l ex t rac t ion .

There a r e s e v e r a l disadvantages i n t h e use of t h e method. F i r s t , the pos i t ion of t h e i s o l i n e s depends on t h e s c a l e of t h e map, i n t e r v a l accepted, densi ty of workings, and accuracy of construction. Second, i n p r a c t i c e , d i s - s imi la r toposurfaces can be const ructed on the b a s i s of t h e same data by d i f - f e r e n t persons. Construction and i n t e r p o l a t i o n of d a t a have an influence on the s i z e of a reas and, therefore, on the f i n a l r e s u l t s of computations. Two o r more so lu t ions a r e possible. When data a r e profuse, cons t ruc t ion and i n t e r - p re ta t ion of i s o l i n e s may be complex o r even cumbersome. Such i s t h e case i n multimetal deposits . Checking computations i s i n t r i c a t e and may take a s much time a s a complete reest imate of reserves. I n genera l , t h e method of i s o l i n e s i s b e s t applied t o deposits where thickness and grade decreases from center t d periphery.

Applicat ion

The method of i so l ines is widely used f o r i l l u s t r a t i o n and ana lys i s of physical and chemical propert ies of mineral deposi ts . I s o l i n e maps a r e o f t en i r r ep laceab le i n s tud ies of the morphology of minera l bodies ; geochemical and geophysical prospecting a re a l s o aided by t h e a p p l i c a t i o n of t h i s method.

The use of i s o l i n e s for r ese rve computations i s l imi ted mainly to depos- i t s showing order ly changing thickness and grade characteristics. The method i s widely used i n computing volumes f o r earthwork opera t ions (39, p. S83), s tockp i les (44) , and computing rese rvo i r reserves of water , n a t u r a l gas, and o i l (2, pp. 91-98). It i s time consuming and nonoperat ive when the gr id of explora t ion workings is sparse. It is impract ica l i n s t r u c t u r a l l y broken and small o re bodies, i n complex multimetal and very i r r e g u l a r mineral deposits .

I s o l i n e s a r e known to be drawn from i n t e r p r e t a t i o n of a e r i a l photographs fo r computing volume of mater ia l removed from the a r e a , a s we l l as o re rese rve i n s tockpi les . Few examples of the use of t h i s method f o r r ese rves computa- t ions a r e described i n l i t e r a t u r e (26, 56).

FIGURE 48. - Method of Triangles. A-B, Laying out triangles by various methods; C-D, Isometric drawing of triangular prisms.

Method of T r i ann le s

P r inc ip l e s and Formulas

A plan ( o r l ong i tud ina l s ec t ion ) of t h e mineral d e p o s i t showing explora- t i o n workings and t h e e n t i r e a rea of t h e mine ra l body can be d iv ided graphi - c a l l y i n t o a system of t r i ang le s by connect ing holes w i t h s t r a i g h t l i n e s ( f i g s . 48A and 2). Each t r i a n g l e on t h e p l an r e p r e s e n t s a h o r i z o n t a l projec- t i o n o r a base a r ea of an imaginary prism with edges--tl , & , t, --equal t o v e r t i c a l thicknesses of t he mineral body ; t h e upper base i s t runca ted . Thus, t h e minera l body is divided i n t o a s e r i e s of c l o s e o rde r t r i a n g u l a r prisms wi th base a r e a s i n a plane of t h e map ( f i g s . 48s and 2) . Hence, t h e name of t h e method of t r i a n g l e s or t r i a n g u l a r prisms. The r u l e of gradual changes of a l l v a r i a b l e s from one working t o another i s t h e main p r i n c i p l e of t h i s method. In f i g u r e 484, t he layout of t h e t r i a n g l e s involves t h e use of da t a from h o l e 1 e igh t t i ~ e s ; data from ho le s 2 through 9 a r e used twice each. The layout i n f i g u r e 482 involves t h e use of d a t a from h o l e s 3 and 6 f i v e t imes , h o l e 9 four t imes, ho le 1 three t imes , h o l e s 5 and 8 two t imes, and h o l e s 2 , 4, and 7 once each.

The formula f o r computing volume of a t runca ted t r i a n g u l a r prism with uneven he igh t s of edges i s

1 V = - ( t l + t, + t,) S. 3

The average grade of each prism i s usua l ly determined a s an a r i t h m e t i c average of cl , c,, and c3 , o r a s t h e thickness-weighted average of t h e same grades ; t h a t is ,

Ore tonnage f o r a t r i a n g u l a r block is computed by

Metal tonnage f o r a t r i angu la r block i s computed by

where ql , qz, and q, a r e l inear o r e r e se rves of t h e edges of t h e t r i a n g u l a r prism ( i n tons) ;

c, , o,, and c3 a r e the grades of t h e edges of each tri-a,ngular pr ism ( i n percent ) ; and

pl , p, , and p3 a r e l i nea r m e t a l r e s e r v e s of t h e edges of t h e t r i a n g u l a r prism ( i n t ons ) .

Volume, ore and meta l tonnages of the e n t i r e mineral body, and average grade a r e computed by the usual formulas : volume - ( 16) , o r e tonnage - ( l 8 ) , metal tonnage - (29a), gravimetric average grade - ( 2 8 ) , and volumetric average grade - (27) .

Some e a r t h s c i e n t i s t s prefer t o compute average grade by the gravimetric formula f o r ind iv idua l blocks, and by the volumetric formula f o r the e n t i r e minera l body. The l a t t e r is convenient when a weight f a c t o r i s accepted a s cons tan t f o r t h e e n t i r e body.

Procedure

Laying out the Tr iangles

A l l workings on t h e maps a r e connected with s t r a i g h t l i n e s and the area of t h e mineral body i s divided i n t o a maximum number of t r i a n g l e s ; no l i n e should be crossed by another . For. accura te computations t h e i d e a l t r i a n g l e i s e q u i l a t e r a l ; a c i r c l e circumscribed through t h e v e r t i c e s of the t r i a n g l e should be the smal les t possible. The common p r a c t i c e i s t o t ake t h e shor tes t d iagonal of each trapezium area. Theore t i ca l i n v e s t i g a t i o n s support such a cho ice ( s e e next sec t ion) .

Some s c i e n t i s t s p r e f e r t o s e l e c t t r i a n g l e s corresponding t o the morphology of t h e mineral body. I n vein deposi t s , f o r example, the bases of the t r i a n g l e s may be extended along the s t r i k e , so tha t a l l t r i a n g l e s have a uniform slope. For p lace r s , Baxter and Park suggest f ind ing the configurat ion of t h e minera l body with isopachs and then cons t ruc t ing t r i a n g l e s with a uniform s l o p e throughout each t r i a n g l e (2, p. 46) .

The number of t r i a n g l e s and l i n e s a r e cons tant f o r each p ro jec t . A sim- p l e way t o check t h e correc tness of layout i s by formulas (27),

f o r t h e number of connecting l i n e s and

f o r t h e number of t r i a n g l e s ,

where q is t h e number of workings i n s i d e t h e perimeter of the mineral body, and

n, - t he number of workings bounding the body.

D e t e r m i n i n ~ Areas of Individual Tr iangles

I f t h e a r e a of any t r i a n g l e i s computed a s one-half t h e product of the b a s e l i n e and h e i g h t , i t i s des i rab le t o use the common l i n e of two contiguous t r i a n g l e s a s a base f o r computing t h e o the r area .

The procedure of computing volume, o r e and metal tonnages, and average grade i s i l l u s t r a t e d i n t ab les 13 and 14.

Studies by Different Authors

For many years t h e t r iangular method was considered s tandard , although e r r o r s i n r e s u l t s due t o t h e manner of d iv iding the a r e a i n t o t r i a n g l e s were recognized. The u n r e l i a b i l i t y of t h i s method was discussed by Harding i n . 1921 (13-14) and by Zhuravsky in 1934 (68-69). The l a t t e r s tudied the r e l a - t i v e e r r o r f o r a volume of a block, explored by four v e r t i c a l ho les , with thicknesses tl , $ , tg , and t, and base a rea S ( f i g . 49). The volume of the right prism may be computed i n two ways: Vl --as t h e volumes of two t r iangular prisms with bases ABD and BDC; or V,--as volumes of two t r i a n g u l a r prisms with bases ABC and ADC; t h a t is

I n t h e f i r s t case t, and t, , and i n t h e second case tl and t3, a r e taken twice i n each formula. Graphical ly, the lower su r face of t h e block i n the f i r s t c a s e i s convex and t h e volume is overestimated, and i n t h e second, the block

I i s concave and t h e volume i s underestimated.

Lower surface convex Lower surface concove

FIGURE 49. - Two Manners of Construction of Triangular Prisms for a Rectangular Prism.

The volume of t h e prism is computed by h a l f t h e sum of both c a s e s ,

The l a t t e r is a s tandard rec tangular p r i s m formula.

The r e l a t i v e e r r o r between volumes Vl and V2 is

I f A V i s equal t o zero, the volumes Vl and V2 a r e equa 1, and

t, + t, = t, + t,. (75)

I n o the r words, the method of t r i a n g l e s is accura te only i f t h e sum of t h e two opposi te edges, tl and t, , of each rec tangu la r prism a r e equa l t o the two remaining edges, t, and t, . Assuming t h a t ( t , + t, ) i s two t imes l e s s than (b + t 4 ) ; t h a t i s 2 ( t ,+t , ) = ( k t ' t , ) , t h e volume of Vl w i l l be more than volume V2 f o r the value of 6 (Q + t 4 ) , a r e l a t i v e e r r o r of 20 percent .

Mod i f i c a t ions - In t h e t r i a n g u l a r method equal weights a r e given t o each h o l e i n the

t r i a n g u l a r prism; t h i s is c o r r e c t only i n a prism with an e q u i l a t e r a l base. I n s e v e r a l modificat ions of t h i s method t h e average grades of each t r i a n g u l a r prims a r e determined by weighting o re th icknesses ( f o r m l a 25), a n g l e s of the t r i a n g l e (13-3, s i d e lengths of each t r i a n g l e (57), d i s t a n c e s of each hole from t h e cen te r of gravity (l5), and a r e a s of in f luence on each h o l e , con- s t r u c t e d by r u l e of nearest po in t s .

The "Harding angular system" of computations was an a t tempt t o reduce t h e r e l a t i v e e r r o r inherent in t h e method by applying a f a c t o r equal t o anp le of ho le

180 O , i n t h e volume formula f o r each ho le of the t r i a n g l e i n s t e a d of

a cons tan t f a c t o r of 1/3. I n o ther words, he subdivided t h e t r u n c a t e d triangular prism i n t o three a u x i l i a r y prisms with he igh t s equal t o t h e edges of the b a s i c prism and areas weighted according t o the magnitude of t h e angle of the b a s i c p r i s m .

La te r , Harding found the system t o be i n c o r r e c t i n computing average grade, p a r t i c u l a r l y when the prism i s apexed on a h o l e having a z e r o value th ickness . The computed grade may be of higher tenor than any of t h e o ther va lues from t h e holes of the prism. To c o r r e c t such e r r o r s t h e fo l lowing formulas a r e given by Harding (2, p. 124) :

When a 11 ho les have p o s i t i v e va lues ,

Similar formulas a r e used f o r holes 2 and 3.

When t, has ze ro value ,

3t, + t, t . ~ = 6 '

i f 5 and tg have zero va lues ,

Addi t ional s t u d i e s of the t r i a n g u l a r method led Harding and others t o t h e development of the po lygona 1 method.

I When t h e workings a r e d i s t r i b u t e d i n a regular g r id and the a reas of the

t r i a n g l e s a r e equal or nea r ly equal, r e se rves may be computed by these s i m p l i - f i e d formulas (27, 3 l ) ,

and

1 N V = - s C t k ,

3 i=l

1 N Q = - s c t f k ,

3

1 N P = - s C t f c k ,

3 i=l

S where s = - N

S - t o t a l a r e a of a l l t r i ang les N - number of t r i a n g l e s t and f - th icknesses and weight f a c t o r s of t r i a n g l e s k - c o e f f i c i e n t determined by t h e number of t r i a n g l e s s t a r t i n g a t

each hole.

This v a r i a t i o n may be used t o f a c i l i t a t e computations i n cases of large numbers of t r i a n g l e s ; e r r o r s connected with t h e const ruct ion of t r i a n g l e s and t h e i r measurements a r e eliminated and the r e s u l t s do not depend on the manner of const ruct ing the t r i a n g l e s .

D i s t i n c t i v e Features

The method of t r i a n g l e s is bas ica l ly formal and withdrawn from geologic and mining considera t ions . The i n s i d e perimeter of t h e n e t of t r i a n g u l a r prisms may be i n c o n f l i c t with t h e physica l boundaries of t h e body, and t h e prism s i d e s may c r o s s the boundaries of i n d i v i d u a l o r e types . It is o f t e n

.

d i f f i c u l t o r even impossible t o subdivide t h e o r e body i n t o segments of simi- l a r th ickness o r grade. Triangles may conceal t h e d i s t r i b u t i o n of va r i ab les .

The procedure f o r reserve computations by the method is r e l a t i v e l y s i m - p l e , when t h e f o r m ~ l a f o r t runcated prisms is used. Modificat ions of t h e method requ i re more elaborate computations. The r e l a t i v e e r r o r depends on the manner i n which t h e area is divided i n t o t r i a n g l e s , t h e i r form, and t h e t o t a l number of t r i a n g l e s .

I n comparison with other methods t h e t r i a n g l e method requ i res cons t ruc- t i o n of a g rea te r number of blocks u l t ima te ly r e s u l t i n g in labor and time consuming computations. When a mineral body con ta ins s e v e r a l va luable components, camputations may a l s o be cumbersome.

In t h e t r i a n g u l a r method of computations t h e use of explora t ion data con- cerning ind iv idua l mine workings may not be cons tan t , f o r example, i n f i g u r e 4% data f o r ho les on the perimeter of t h e body are used two times i n compari: son with e igh t t imes within t h e body ( h o l e 1 ) . I n c a s e s of i r r e g u l a r sharp changes in v a r i a b l e s , ins ide workings may have a d i sp ropor t iona te in f luence on the computations.

The method i s no t exact when v a r i a b l e s decrease from t h e cen te r t o the o u t s i d e boundaries, such a s the thickness of l e n s l i k e bodies. According t o Zhuravsky the volume reserves of a l e n t i c u l a r body computed by t h i s method w i l l be underestimated (68).

Thus, e r r o r s i n computing reserves may be s u b s t a n t i a l , p a r t i c u l a r l y when f luc tua t ions of v a r i a b l e s a r e la rge and t h e number of t r i a n g l e s i s small. When mine workings a r e numerous and c l o s e l y spaced, e r r o r s f o r each t r i a n g l e tend t o compensate each other. Even i n t h e most f avorab le d i s t r i b u t i o n of workings, such a s square s e t , t h e t r i a n g u l a r method may produce an apprec iab le e r r o r .

Applicat ion

The uniform and gradual changes of v a r i a b l e s a r e c h a r a c t e r i s t i c f e a t u r e s of only a few minera l deposi t s , predominantly sedimentary. Natura l ly , t h e t r i angu la r method, based on the r u l e of gradual changes of i n t e r p r e t a t i o n of explorat ion d a t a , i s most app l i cab le t o such depos i t s . Large sedimentary and large disseminated o r e deposi t s , explored by r e g u l a r l y spaced d r i l l holes , have been computed by t h i s method (58).

Method of Polynons

P r i n c i p l e s

The method of polygons, a l s o known as polygonal pr i sms , equal spheres of i n f l u e n c e , a r e a s of equa 1 inf luence , and a r e a s of n e a r e s t p o i n t s , is based on t h e concept t h a t a l l f a c t o r s , determined f o r a c e r t a i n p o i n t of a minera l body, extend h a l f t h e d i s t a n c e t o ad jo in ing and surrounding p o i n t s , t hus forming a n a r e a of i n f luence . The r u l e of n e a r e s t p o i n t s was d i s c u s s e d i n p a r t 1 of t h i s r e p o r t . B r i e f l y , a r e a s of equa l in£ luence a r e found f o r workings symbolized on t h e aaap a s p o i n t s by us ing perpendicular b i s e c t o r s and f o r t hose symbolized a s l i n e s by a n g l e b i s e c t o r s .

The f i r s t d e s c r i p t i o n of t h e method was given a s e a r l y as 1909 by Boldyrev (57). I n t h e United S t a t e s t h e method was developed independently from t h e t r i a n g u l a r method by Davis (l3, p. 122) and Harding (l4) during t h e 1920's . A concept of areas of equal i n f l u e n c e was in t roduced s t e p by s t e p , and i t w a s accepted and developed a s a new p r i n c i p l e f o r t h e polygon method, where t r i a n g l e s a r e used a s a u x i l i a r y cons t ruc t ions . The f i r s t a p p l i c a t i o n of t h i s method i n t h e United S t a t e s was i n computing r e s e r v e s of extremely i r r egu - lar bodies of t h e J o p l i n and Wisconsin z i n c depos i t s i n 1920 (2).

Procedure and Cons t ruc t ion of Polygons

In t h i s method, t h e explored p o r t i o n of t h e minera l body i s s u b s t i t u t e d by a s e r i e s of polygonal pr isms, t h e p l ane bases being equal t o a r e a s of i n f luence of a p p r o p r i a t e workings ( f i g . 50 ) . Each such pr i sm assumes t h e t h i c k n e s s , weight f a c t o r , and grade determined f o r such workings.

FIGURE 50. - Method of Polygons. Plan and polygonal prisms.

The usual s teps in computing reserves by t h i s method a r e

1. Construction of auxi l iary t r i ang l e s , when necessary; the manner of const ruct ion of t r i ang les has no inf h e n c e on the f i n a l shape of polygons ;

2. Construction of polygons by following a d e f i n i t e order ; f o r . example, clockwise and frum periphery t o the center of t h e deposi t ;

3. Computing reserves fo r each block ( t a b l e 15) ;

4. Grouping of blocks on the basis of evaluat ion of grade, thickness, l inear reserves , r e l i a b i l i t y , e tc . , and sumrrarizing and c l a s s i fy ing reserves i n t o various categories.

TABLE 15. - Computation of reserves by polygonal method

Polygon number

I...........

........... N...........

Total..

Average

-


n C v

i-1 t,, = - n

C s i=l

Weight f ac to r (F),

:u f t l t o n F F

F

F

mate r i a l Grade ( c ) , I Raw I-- monent .eserves P) , tons : es erves

Q) , tons Ql Qa

Q3

Q,

Case of Vertica 1 Holes

percent

5 Cz c3

The polygons around ver t i c a 1 ho les a r e constructed by t he i n t e r s ec t ion of perpendicular b i sec tors erected from the middle of t h e s ides of t he t r i ang les . The c r i t e r i a f o r correctness i s tha t a l l points , i n a c e r t a in polygon, a r e nearer t o the r a l l y ing hole than t o others. I n polygon construction some perpendiculars may not be used a t a l l ( t r i a n g l e ABD, f i g . 14) and others may a s s i s t construction by thei r continuation outs ide the boundaries of the obtuse t r i ang l e ( t r i a n g l e BCD, f ig . 14).

When t h e i n t e r s e c t i o n of perpendiculars f o r two ad jo in ing t r i a n g l e s forms a t e t r agon , t h e diagonal l i n e connecting each p a i r of perpendiculars becomes a s i d e of a polygon. Point 0 i s equ id i s t an t from A , D , and C and point 0' i s e q u i d i s t a n t from A , B , and C. The diagonal i s e q u i d i s t a n t from A and C. The d i s t i n c t i v e f e a t u r e of a co r rec t ly cons t ructed polygon is t h a t each of the i n s i d e ang les between t h e s ides a r e always l e s s than 180".

I f t h e ang le b i s e c t o r manner of cons t ruc t ion i s used i n the a b w e example, the property of nea res t points is no t f u l f i l l e d , because most of the aux i l i a ry t r i a n g l e s a r e no t e q u i l a t e r a l ( f i g . 14). I n comparison with t h e perpendicular b i s e c t o r manner of cons t ruct ion , angle b i s e c t o r s produce polygons with twice t h e number of s i d e s . Polygons may be i r r e g u l a r in shape and have i n t e r n a l angles of more than 180'; therefore , they a r e more d i f f i c u l t t o measure with a planimeter . Fur ther a n a l y s i s shows t h a t polygons cons t ructed by angle bisec- t o r s a r e only another graphic expression of t h e method of t r i a n g l e s .

I n s h o r t , cons t ruc t ion of polygons by perpendicular b i s e c t o r s f o r v e r t i - c a l workings s a t i s f i e s t h e p r inc ip le of , n e a r e s t po in t s , i s s impler , and always the same; a r e a measurements a r e more accura te .

It is poss ib le t o construct an a rea of in f luence f o r a given point by a c i r c l e when workings a r e too widely spaced t o s a f e l y assume con t inu i ty of the mineral body. C i r c l e r ad ius i s chosen as optimum f o r a c e r t a i n category of r e se rves and a given type of deposi t . I n such cases the block i s in the form of a cy l inder i n s t e a d of a polygonal prism.

Case of Linear Workinps

When a minera l body i s explored by workings represented a s l i n e s on a p lan , t h a t is , d r i f t s , ho r i zon ta l d r i l l h o l e s , and t r enches , t h e angle bisec- t o r manner of cons t ruc t ion of a reas of in f luence i s used. A rec tangular block i s d iv ided i n t o four a r e a s of in£ luence, o r four elementary prisms, each one cha rac te r i zed by appropr ia t e workings. The block volume i s the sum of a1 1 elementary prisms, and t h e thickness and average grade a r e computed by weighting t h e volumes and tonnages of t h e a u x i l i a r y prisms. This modificat ion of t h e polygon method was previously described as a common case of t h e mining blocks method, when a r e a s of in£ luence a r e determined f o r four s i d e s of a block ( f i g . 3 lA , block A).

Pr inc ipa l Formulas

I r r e g u l a r Dis t r ibu t ion of Drill Holes

Le t US f i r s t cons ider a common c a s e where a mineral body i s explored by i r r e g u l a r l y spaced v e r t i c a l d r i l l holes . The genera l formulas f o r a group of polygonal prisms ( f i g . 50) a r e f o r volume - formula 16, f o r average thickness - s i m i l a r t o formula 37, f o r ore tonnage - formula 18, f o r metal tonnage-

p = 5 % + GQ + c3q3 + '" + c,qn ¶ (82)

f o r average grade - formula 29.

Regularly Spaced Drill Holes

More simple formulas m y be der ived from p r i n c i p a l ones when workings a r e l a i d down i n a r egu la r g r i d t o form simple polygons, such a s squares , rec tan- g l e s , o r hexagons. The comnon n a t u r e of these modif ica t ions i s t h a t t h e a reas of inf luence f o r each ho le , except those lying on t h e boundary of t h e body, a r e equal i n s i z e .

Square Net of Workings. - The volume of an a r e a , s , explored by four ho les with th icknesses $ , ta , t3 , and t, is

The area of mineral bodies t h a t have been explored by numerous ho les located i n the corners of squares w i l l be divided by perpendicular b i s e c t o r s i n t o squares with equal a reas , s ; t h e formula f o r volume computation w i l l transform t o

where 5 , 5, t,, . . . , t, a r e th icknesses of holes , and

When the quant i ty of holes is l imi ted o r more accura te r e s u l t s a r e d e s i r - a b l e , the above formula may be improved by adding t o each v a r i a b l e a f a c t o r based on t r u e a r e a s of influence i n a r b i t r a r y u n i t s , termed "weight." Thus, in a square network, the area of inf luence of a ho le f o r a complete square must have a weight of four , f o r a s i d e h o l e a weight of two, and f o r a corner hole a weight of one ( f i g . 5%).

Chessboard Grid. - I n a chessboard o r t r i a n g u l a r g r i d map, t h e e n t i r e area of the mineral body w i l l be d iv ided by perpendicular b i s e c t o r s i n t o hem- gons with equal a r e a s ( f i g . 51;). Formulas f o r computing volume w i l l be the same a s a square g r i d , except-s w i l l equal the a rea of a hexagon. When the number of workings i s l imited, t h e formula should have a weight of s i x f o r a complete, t h r e e f o r a ha l f , and one and a ha l f f o r a corner of a hexagon.

~ e q u i r ement s , Advantages , and Limitat ions

The method of polygons is based on t h e o r e t i c a l assumptions r a t h e r than on geologic and mining considera t ions and requ i res a s u i t a b l e p lan o r longi tudi- n a l sec t ion . The c o r r e c t mnner of cons t ruct ing a reas of i n f h e n c e r e q u i r e s experience; however, there is only one way t o do i t , and t h e r e s u l t s do not depend on personal judgment. I n comparison with o the r methods t h e n a t u r e of the minera 1. depos i t i s poor ly i 1 lus t r a t ed , a lthough , polygons may, under appropr ia te pa t t e rn of explorat ion workings fo r a given type of d e p o s i t , i n d i c a t e reasonably well the d i s t r i b u t i o n of th ick and narrow and high- and low-grade port ions of the body.

L E G E N D

rrA Weight

FIGURE 51. - Regularly Spaced Dr i l l Holes-Method of Polygons. A, Square net pattern of drilling; R , chessboard or triangular pattern of drilling.

The f a c t o r s , t h i ckness , g rade , and weight , a r e cons idered cons tan t f o r each b lock . Hence, each block i s computed without in£ luence from any ad jo in - i ng b l o c k s , and it i s p o s s i b l e t o add and compute r e s e r v e s f o r new blocks a s e x p l o r a t i o n progresses . I n o ther methods new d a t a o f t e n c a l l s f o r a complete r e c a l c u l a t i o n of r e s e r v e s .

When t h e workings a r e i n a r e g u l a r g r i d p a t t e r n , r e s e r v e computations a r e s i m p l i f i e d . The s i z e of t h e polygons v a r i e s when workings a r e unevenly spaced. More widely spaced holes may have a n undue i n f l u e n c e on the s i z e of b locks and on t h e average grade. Any one block may have an unreasonably la rge in£ luence on t h e f i n a l computations, i f t h e v a r i a b l e s of t h i s block vary s t r o n g l y from t h e v a r i a b l e s of t h e o t h e r s .

I n t h e c a s e of i r r e g u l a r d i s t r i b u t i o n of workings it i s necessary t o measure each polygon with a planimeter . When t h e blocks a r e numerous such measurements may be time consuming.

App l i ca t ion

Favorable c r i t e r i a f o r t he use of t h e method of polygons a r e t h e proven c o n t i n u i t y of a minera l body between workings and t h e g radua l changes of a l l v a r i a b l e s . The method i s b e s t app l i ed when t h e workings a r e numerous and in g r i d p a t t e r n . The g r e a t e r t he number of b locks and t h e more r e g u l a r t he g r i d , t h e more a c c u r a t e a r e t h e computations. Reserves of t a b u l a r bodies (beds , b l a n k e t s , and t h i c k ve ins ) , l a rge l e n s e s , and s tocks a r e s u c c e s s f u l l y computed by t h i s method (43, 45).

Polygons can be used wi th d i s c r e t i o n i n c a s e s of nonuniform and i r r e g u - l a r l y shaped mine ra l bodies ; they a r e i n c o r r e c t when t h e bodies cannot b e c o r r e l a t e d s a t i s f a c t o r i l y between workings, when they a r e small and d i s t r i b - u t ed e r r a t i c a l l y ( s m a l l s t o c k s , o r e shoo t s , chimneys, and p i p e s ) , o r when horses of waste a r e present . I n t h e c a s e of small l e n s l i k e bodies t h e middle b locks may show a n unduly l a r g e i n f h e n c e on t h e f ina 1 r e s u l t s . I n minera l d e p o s i t s composed of s e v e r a l bodies ove r ly ing each o t h e r s e p a r a t e s e t s of polygons may be cons t ruc t ed f o r each one.

Combined Methods

The u s e of two o r more methods t o compute r e s e r v e s f o r t h e same d e p o s i t i s a common p r a c t i c e . Various methods may be a p p l i e d f o r d i f f e r e n t p a r t s of a body depending on t h e geology, mine des ign , t y p e and d e n s i t y of e x p l o r a t i o n workings, and ca tegory of r e s e r v e computations. Mining b l o c k s , f o r example, may be used f o r h igh category r e s e r v e s and geologic b locks f o r lower c a t e - go r i e s . A second method may o f t e n b e used f o r c o n t r o l of t h e computations made by the p r i n c i p a l method, s o t h a t no crude e r r o r s may occur .

A common c a s e of combined methods i s when one method i s app l i ed t o out- l i n e and d i v i d e t h e minera l body i n t o blocks and ano the r t o determine t h e parameters of each block.

Methods of mining b locks , geologic b locks , and c r o s s s e c t i o n s have been ' used f o r high-grade nonferrous v e i n s developed by underground workings, and t h e polygon method f o r disseminated o r e bodies (copper -z inc d e p o s i t s i n Bu t t e , and copper-nickel d e p o s i t s i n N o r i l , U.S.S.R.). The Corrigan-McKinney S t e e l Company, Michigan and Minnesota, computed the r e s e r v e s of No. 1 o r e body by t h e average f a c t o r s and a rea methods and t h e No. 2 o r e body by t h e c r o s s - s e c t i o n methods (=, p. 147).

A t t h e In sp i r a t i ' on mine, Arizona, disseminated copper o r e r e s e r v e s based on d r i l l i n g were computed by polygons ( q u a d r i l a t e r a l p r i sms) and checked by t h e c ros s - sec t ion methods (60). A t t h e Ray Copper mine, Ar izona , l a r g e r e s e r v e s of i r r e g u l a r bodies of disseminated o r e , were computed by t h e mining blocks method f o r t h e po r t ion of t h e o r e body developed f o r underground mining and by v e r t i c a l c r o s s s e c t i o n s f o r t h e po r t ion explored by churn and diamond d r i l l i n g (61) . -

A t t h e Kennecott Copper mine, Bingham Canyon, Utah, t h e method of ho r i - z o n t a l c r o s s s e c t i o n s was used t o s e p a r a t e each proposed bench of t h e open p i t development of a l a r g e low-grade disseminated copper body. Mining blocks were used f o r computing r e s e r v e s developed by underground workings and po lygon method f o r a r e a s explored b.y v e r t i c a l churn d r i l l h o l e s . The t r i a n g u l a r pr i sm method was used f o r computing r e s e r v e s explored by churn d r i l l s below underground workings (58) .

A t t h e Copper Queen mine, Bisbee, Ar i z . , "ore i n s i g h t 1 ' r e s e r v e s of an i r r e g u l a r and l e n t i c u l a r l imestone-replacement body were computed by horizon- t a l c r o s s s e c t i o n s and probable r e s e r v e s by two s e t s of v e r t i c a l c r o s s sec- t i o n s cons t ruc t ed a t r i g h t ang le s t o each o the r . A low-grade copper-porphyry

d e p o s i t , explored by c h u m d r i l l ho l e s , was computed by t h e v e r t i c a l c ross - s e c t i o n method and checked by t h e t r i a n g u l a r method (50).

A combination of h o r i z o n t a l c r o s s - s e c t i o n and polygon methods was used f o r t h e des ign of open p i t ' opera t ions f o r s e v e r a l l a r g e disseminated copper d e p o s i t s (Berkeley p i t , Bu t t e , and Konrad, U .S .S .R . ) . Hor izonta l c r o s s sec- t i o n s were drawn f o r each proposed bench, and r e s e r v e s wi th in each of t he two l e v e l s were computed by t h e polygon method. A combination of hor izonta 1 c r o s s s e c t i o n s and geologic b locks f o r each l e v e l a l s o h a s been used f o r mul t imeta l d e p o s i t s (Kadain, U.S.S.R.).

Owing t o t h e s t r i c t a p p l i c a t i o n of t h e r u l e of n e a r e s t p o i n t s t o t he con- s t r u c t i o n of polygons, t h e morphology and o t h e r p e c u l i a r i t i e s of mineral bod ie s , such a s p a t t e r n s o r r e l a t i o n s h i p s between t h e elements , may be over- looked. A minera l body may show s t a b i l i t y i n t h i ckness , un i formi ty of grade, o r d e f i n i t e r e l a t i o n s h i p s between grade and th i ckness a long s t r i k e b e t t e r than down t h e d i p . Gold and o the r heavy mine ra l s i n p l ace r d e p o s i t s may show t o a g r e a t e r degree gradual changes i n d i s t r i b u t i o n and grade i n one d i r e c t i o n than i n another . Ash con ten t i n c o a l d e p o s i t s may inc rease o r decrease i n a ce r - t a i n d i r e c t i o n owing t o paleographic c o n d i t i o n s .

The grade of t h e o r e i n replacement d e p o s i t s o f t e n depends on s t r u c t u r a l and l i t h o l o g i c a l c h a r a c t e r i s t i c s of t he count ry rock. In such c a s e s t h e method of polygons may be modified; ho le s may be f i r s t connected with a u x i l - i a r y l i n e s on the b a s i s of geomorphology and o the r p e c u l i a r i t i e s , such a s s t r i k e , d i p , or r ake of t he ore body. Square and r e c t a n g u l a r blocks may then be formed by c o n s t r u c t i o n of l i n e s p a r a l l e l t o and/or perpendicular t o a u x i l - i a r y l i n e s . This mod i f i ca t ion i s o f t e n desc r ibed i n l i t e r a t u r e a s a method of r e c t a n g l e s , or v a r i o u s a r e a s of i n f luence i n c o n t r a s t with t h e polygon or equa 1 a r e a s of in£ h e n c e method.

Examples of r e s e r v e c ~ m p u t a t i o n s publ i shed a s r e c t a n g u l a r methods a r e c l a s s i f i e d i n t h i s r e p o r t according t o t h e accepted p r i n c i p l e s of i n t e r p r e t a - t i o n of e x p l o r a t i o n d a t a ; a s mining blocks method, i n underground mining; and a s a s imple mod i f i ca t ion of t he polygon method, i n a r e g u l a r l y spaced g r i d of d r i l l ho l e s .

In i r r e g u l a r l y spaced d r i l l h o l e g r i d s , t h e r e c t a n g u l a r block cons t ruc- t i o n i s found t o be sub jec t ive , ' o r a f f e c t e d by personal opinion. The use of such cons t ruc t ion shows t h e same d isadvantages and t h e same l i m i t a t i o n s a s the method of t r i a n g l e s .

Thus, t h e r e c t a n g u l a r system may be cons idered a s a combined method, i f block cons t ruc t ion i s made on the b a s i s of geologic , mining, and economic con- s i d e r a t i o n s , r a t h e r than on p l a i n geomet r i ca l po in t s of view, and t h e f a c t o r s a r e computed by a r i t h m e t i c average, by thickness-weighted average , o r by a rea - weighted average , determined according t o t h e r u l e of n e a r e s t po in t s .

Source of Er rors i n Reserve Computations

Combined average e r ro r i n any r e s e r v e computations may be expressed by

where Ma, - average r e l a t i v e e r r o r of r e s e r v e computations;

aq, - average r e l a t i v e e r r o r of t e c h n i c a l e r r o r s ;

- average r e l a t i v e e r r o r of t h e method and formula used;

m, - average r e l a t i v e e r r o r due t o t h e i n t e r p r e t a t i o n of explora t ion da ta .

Random e r r o r s due t o precis ion of obse rva t ions compensate each o the r ; they a r e smal l i n comparison with e r r o r s of i n t e r p r e t a t i o n and may be d i s - regarded. Biased e r r o r s , due t o inexperienced personnel , equipment d e f e c t s , and improper techniques of observat ions and ana lyses , may be apprec i ab le and should be compensated f o r by co r r ec t ion f a c t o r s p r i o r t o computations ( p a r t 1).

The comparative accuracy of var ious methods of computing r e se rves has been discussed i n many publ icat ions (2, 46, 49, 53). According t o fo re ign sources the va r i ance i n reserves of the same h igh ca tegory computed by d i f f e r - en t methods f o r a depos i t ra re ly exceed 10 percent (appendix C ) .

When t h e r e s u l t s of computations by a s e l e c t e d method a r e within 1 t o 5 percent of r e s u l t s received by o ther methods, they a r e cons idered by Stammberger t o be accu ra t e (59).

In genera l , t h e average r e l a t i v e e r r o r of t h e method and formula used f o r reserve computations should l i e within t h e same l i m i t s and should not exceed t h e average r e l a t i v e e r r o r f o r determining grade , a s w e l l a s o the r f a c t o r s ; otherwise such e r r o r s w i l l d i s t o r t t he p r e c i s i o n of t h e r e s u l t s of explorat ion.

In most ca ses t h e s e r e l a t i v e e r r o r s due t o method a r e neglected, because t h e e r r o r s of i n t e r p r e t a t i o n a r e much l a r g e r and determine t h e accuracy of computations. The l a t t e r e r ro r s depend on t h e type and form of t he depos i t , on degree of v a r i a t i o n i n thickness , grade, and o ther f a c t o r s , on t h e kind of explora t ion workings, t h e i r dens i ty , and on sample technique.

SUMMARY

Reserve computations of a mineral depos i t a r e a geologic and engineering problem; i t is o f t e n an i n t r i c a t e task . S e l e c t i o n of a method aepends on the geology of t h e mine ra l deposi t , t he kind and d e n s i t y of workings, the a p p r a i s a l of geologic and explorat ion d a t a , and t h e accuracy required. Time and c o s t of computations a re o f t en important cons ide ra t ions .

Knowledge of t h e mineral d e p o s i t ' s geology i s a p r e r e q u i s i t e t o any r e l i a b l e computation, This knowledge inc ludes space l o c a t i o n , s i z e , shape, environment, country rock, overburden, and hydrology; average grade and d i s - t r i b u t i o n of va luab le and detr imental c o n s t i t u e n t s ; and minera l , chemical and phys ica l c h a r a c t e r i s t i c s of the raw ma te r i a l .

Accurate computations of a c e r t a i n depos i t r equ i re a proper ly s e l e c t e d and executed explora t ion program; t h a t i s kind of workings, d r i l l i n g o r underground system, number and densi ty of observat ions , sampling procedure ( loca- t i o n , sample i n t e r v a l , and weight of samples) , and accura te measurements, analyses , and t e s t s .

To s e l e c t the b e s t method c a r e f u l a n a l y s i s of geology and explora t ion should be made. I n genera l , the method ( o r combination of methods) s e l e c t e d should s u i t the purpose of computations and t h e required accuracy; i t should a l s o b e s t r e f l e c t t h e cha rac te r of t h e mineral deposi t and t h e performed explorat ion. I n a complex o r i r r e g u l a r d e p o s i t , it i s adv i sab le t o use two or more methods f o r b e t t e r accuracy and self-confidence. An average of these methods may be accepted a s a f i n a l r e s u l t , o r the values of one method may be considered a s a c o n t r o l of o thers .

The purpose of r e s e r v e computations i s one of the most important consid- e r a t i o n i n s e l e c t i n g a method. For prel iminary explora t ion the method should bes t i l l u s t r a t e t h e depos i t , the opera t ions , and permit s e q u e n t i a l computa- t i o n s and a p p r a i s a l . On t h e other hand, time-consuming procedures should be avoided i f r e se rves a r e being computed f o r prospect ive planning.

The system of mining, o r the problem of s e l e c t i n g one, may i n f h e n c e the preference. A c e r t a i n method of computation may f a c i l i t a t e more so than o thers t h e design of development and e x t r a c t i o n operat ions owing t o t echn ica l and economic f a c t o r s (mining by l e v e l s , average grade, d i f f e r e n t c u t o f f , e t c . ) This expla ins why, i n p rac t i ce , methods of mining blocks and c r o s s sec t ions a r e p re fe r red .

The p r i n c i p l e s of i n t e r p r e t a t i o n of explora t ion data and t h e a n a l y t i c a l pe r fec t ion of formulas a r e a l s o considered i n method s e l e c t i o n . The p r inc i - p les t h a t e s s e n t i a l l y uphold the described conventiona 1 methods a r e :

Method : P r i n c i p l e s

............................. Analogous Geologic inference . Geological blocks................. .... Do.

Mining blocks. ........................ Mining and o t h e r cons idera t ions .

Standard c r o s s s e c t i o n and isol ines. . .Rule of gradual changes. Tr iangular prisms.... ................. Do.

Linear c ross section..................Rule of nea res t po in t s . Polygonal prisms ...................... Do.

A 1 1 formulas f o r computing volumes , tonnage, and average f a c t o r s a r e approximate, because of t h e i r r e g u l a r s i z e and shape of t h e mineral body, e r r o r s i n s u b s t i t u t i n g n a t u r a l bodies by more simple geometric ones, geologic i n t e r p r e t a t i o n , assumptions, and inconsistency i n the v a r i a b l e s . Accuracy of the f i n a l r e s u l t s usual ly depends more on geologic i n t e r p r e t a t i o n and assump- t ions r a t h e r than on t h e method used. Systematic explora t ion and uniform

sampling genera l ly s implify the s e l e c t i o n and t h e use of the conventional methods and produce g r e a t e r accuracy. Reserves of t h e same category computed by d i f f e r e n t methods and based on the same d a t a , usua l ly d i f f e r s l i g h t l y .

The average f a c t o r s and area methods a r e widely used by e a r t h s c i e n t i s t s . I n the analogous method, reserves of a block of a depos i t can be computed with reasonable accuracy by analys is of r e s u l t s of exp lo ra t ion , development, and e x t r a c t i o n ( p a s t production) from ad jo in ing blocks of t h e same o r even geolog- i c a l l y s i m i l a r depos i t s . In t h i s method an i n d i v i d u a l block of t h e same or geologica i ly s i m i l a r deposi t may be computed on the b a s i s of l imi ted , o r even a s i n g l e observat ion properly taken. I n t h e geologic block method, t h e min- e r a l body is subdivided i n t o segments and blocks e s s e n t i a l l y on t h e b a s i s of geology; average f a c t o r s f o r each segment or block a r e determined according t o a v a i l a b l e da ta by va r ious methods.

The mining blocks method r e q u i r e s adequate da ta t o allow subd iv i s ion of t h e mineral body i n t o blocks e i t h e r proved or semiproved fo r e x t r a c t i o n . Most o f t en it i s used i n mining t h i n and medium-thick v e i n s and t abu la r bodies.

The c ross - sec t ion methods a r e the most convenient ways f o r computing rese rves of unif o m mineral deposi t s . In the s tandard c ross - sec t ion method the mean-area formula of a p r i s m with base a r e a s i n p a r a l l e l s e c t i o n s i s t h e most common one; i t i s prec ise when t h e r e i s no l a rge d i f fe rence i n s i z e aq shape of base a r e a s . I n case of d i s p a r i t y between base areas of more than ' percent , frustum or prismoida 1 formulas a r e used.

In underground mining, hor i zon ta l c ross s e c t i o n s cons t ructed a long t h e proposed mining l e v e l s a r e of ten p re fe r red i n mine design. Two s e t s of ver- t i c a l sec t ions a t r i g h t angles t o each other would b e t t e r i l l u s t r a t e s t o c k l i k e bodies than any o the r method. Computations may be made with t h e f i n a l r e s u l t s taken a s ha l f the sum of both s u i t e s .

The l i n e a r cross-sect ion method, where r e s e r v e s a r e f i r s t determined f o r a u n i t of a r e a , u n i t of volume, o r f o r the s e c t i o n s , i s used wi th advantage i n bedded and p l ace r deposi t s .

The method of i s o l i n e s r equ i res numerous observat ions with d a t a more o r l e s s r egu la r ly d i s t r i b u t e d i n the v e r t i c a l or h o r i z o n t a l plane of the minera l deposi t s . It i s app l i cab le t o depos i t s of gradual physica l and chemical changes, such a s sedimentary depos i t s . Large p l ace r gold d e p o s i t s , explored by hundreds of shal low p i t s o r d r i l l ho les , may be w e l l i l l u s t r a t e d and evaluated by t h e method of i s o l i n e s .

The ana l y t i c a 1 methods ( t r i a n g u l a r and polygona 1 prisms) a r e d e f i c i e n t i n exposing the morphology of the mineral body and t h e f l u c t u a t i o n s of v a r i a - b l e s within t h e individual blocks. Although average thickness and grade a r e computed, t h e p a t t e r n of t h e i r space d i s t r i b u t i o n i s no t revealed.

The method of t r i a n g l e s i s app l i cab le t o a few predominantly sediment& depos i t s , t h e minera l i za t ion of which i s c o n s i s t e n t with the r u l e of gradual

changes. The method must be careful ly app l i ed . Er ro r s of computations may be very high owing t o i r r e g u l a r i t i e s i n v a r i a b l e s and unsystematic explora t ion .

The polygonal method' i s successful ly used i n computing rese rves of tabu- l a r d e p o s i t s , such a s sedimentary beds of c o a l , phosphate rock, and o i l sha les ; blanket - type , l a r g e lenses; and t h i c k vein bodies. The accuracy of the r e s u l t s inc reases with the number of b locks and t h e d e n s i t y of the g r i d of workings and d r i l l ho les .

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28. Jackson, C. F., and J. B. ICnaebel. Sampling and Estimation of Ore Deposits. BuMines Bull. 356, 1932, pp. 114-155.

29. Joralemon, I. B. Sampling and Estimating Disseminated Copper Deposits. Trans. AIME, v. 72, 1925, pp. 607-620.

30. Kravchenko, A. Ya., and S. M. Kupfer. Ratsional 'nyye Fonrmly Dlya Oprede leniya Obl'yemov B lokov Pr i Podschete Zapasov Metodom Razrezov (Rational Formulas for Computing Block Volumes i n Reserve Estimates by Method of Cross Sections). Razvedka i Okhrana Nedr (Exploration and Conservation of Mineral Resources), No. 8 , August 1962, pp. 15-19.

31. Kre i te r , V. M. Poiski i Razvedka Mestorozhdenii Poleznykh Iskopaemykh (Prospecting and Exploration of Mineral Deposits). Gosudarsvennoe Nauchno-Tekhnicheskoye Izda te l ' s tvo Li tera tury po Geologii i Okhrane Nedr ( S t a t e Sc ien t i f i c and Technical Publishers of L i t e r a tu r e in Geology and i n Conservation of Mineral Resources), Moscow, U.S.S.R., v. 1, 2d ed., 1960, 322 pp.; v. 2, 2d ed., 1961, 389 pp.

32. Koch, George S., and Richard F. Link. Accuracy i n Estimating Metal Content and Tonnage of an Ore Body From Diamond-Drill Hole Data. BuMines Rept. of Inv. 6380, 1964, 24 pp.

33. Iahee, F. H. Field Geology. McGraw-Hill Co., Inc., New York, 4th ed., 1941, 853 pp.

34. McLaughlin, D. H. Geologic Factors i n the Valuation of Mines. Econ. Geol., v. 34, No. 6 , 1939, pp. 589-621.

35. McKinstry, H. E . , Stanley A. Tyler, E. N. Pennebaker, and Kenyon E. Richard. Mining Geology. Prentice-Hall , Inc., New York, 1948, 680 pp.

36. Mead, W. J. The Relation of Density, Porosity, and Moisture t o the Specif ic Volume of Ores. Econ. Geol., v. 3, No. 4 , 1908, pp. 319-325.

37. Ogloblin, D. N. Podschet Zapasov Metalla v Rossipi po Metodu I z o l i n i i prof. P . K. Sobolevskogo (Reserve Estimates of Metals i n Placers by Method of I so l ines , proposed by Prof. P. K. Sobolevsky). Tswetnyye Metally (Nonferrous Metals) , No. 7 , 1931, pp. 926-949.

38. Olk, J . F. An Open P i t Application of a Digi ta l computer. Min. Cong, J . , v. 48, No. 1, January 1962, pp. 35-37.

OIRourke, C. E. General Engineering Handbook. McGraw Book Co., New York, 2d ed., 1940, 1120 pp.

Parsons, A. B. The. Porphyry Coppers. American I n s t i t u t e of Mining and Meta l lu rg ica l Engineers, New York, 1933, 581 pp.

Pa t t e r son , J . A. Estimating Ore Reserves Follows Logical Steps. Eng. and Min, J. , v. 160, No. 9 , September 1959, pp. 111- 115.

Peele, R. Mining ~ n g i n e e r ' s Handbook. V. 1 and 2, John Wiley & Sons, Inc. , New York, 3d ed., 1947.

PodWyakonw, S. A. Metody Podscheta Plastovykh i Rossipnykh Mestorozhdenii (Methods of Computing Gold Reserves f o r Bedded and Placer Deposits). Gorn. Zhur. (Mining Journal ) , No. 10, 1928, pp. 729-791.

PohLman, W. E., and H. Jacoby. New Stockpi le Measuring Method. P i t and Quarry, Apr i l 1959 , pp. 88-9 1.

Poston, R. H. Method and Cost of Mining a t No. 8 Mine, St . Louis Smelting & Refining Co. , Southeast Missouri D i s t r i c t . BuMines I n f . Circ . 6160, 1929, 22 pp.

Prokof 'yev, A. P. Prakticheskiye Metody Podscheta Zapasov Rudnykh Mestorozhdeniy (Prac t i ca 1 Methods f o r Computing Reserves of Ore Deposits). Gosgeolizdat ( S t a t e Publ ishers f o r Geologic L i t e r a t u r e ) , Moscow, U.S.S.R., 1953, 135 pp.

- . Sravnenie Dannykh Geologorazvedochnykh Rabot s Resu 1 ' t a tami Eksp lua ta t s i i Mestorozhdeniy (A Comparison of Geologic and Exploration Data With t h e Resul ts of Exp lo i t a t ion Mineral Deposi ts) . Razvedka i Okhrana Nedr (Exploration and Conservation of Mineral Resources), No. 11, 1958, pp. 25-28.

Prokof'yev, A. P., and V. V. Stefanovich. 0 Podschete Zapasw v Blokakh Mezhdu Neparallel'nymi Secheniyami Sposobom A. S. Zolotareva (Reserve Computations f o r Blocks Between Non-Para l l e l Sect ions by Zolotarev Method). Razvedka i Okhrana Nedr (Explorat ion and Conservation of Mineral Resources), No. 2 , 1960, pp. 10-14.

Prokop'yev, E. P. Sravni te l 'naya Otsenka Raznykh Metodw Podscheta Zapasw Zolota v Rossipnykh Mestorozhdeniyakh (A Comparative ~ p p r a i s a l of Various Methods of Gold Reserve Estimates i n Placer Deposits) . Gorn. Zhur . (Mining Journal ) , No. 10, 1925, pp . 837-845.

Prouty, R. W . , and R. T. Green. Methods of Sampling and Est imating Ore i n Underground and Steam-Shovel Mines of Copper Queen Branch, Phelps Dodge Corp. Trans. AIME, v. 72, 1925, pp. 628-639.

Rickard, T. A. The Sampling and Estimation of Ore i n a Mine. The Engineering and Mining Journa 1, New York , 1905, 229 pp. .

Ryzhw, P. A. Matematicheskaya Otsenka Tochnosti Podscheta Zapasov Metodami Sredne-Arifmeticheskim, ~ n o g o u g o l 'nikov i Treugol nikov (Mathematical Evaluation of t h e Accuracy of Reserve Est imates by Aver- a g e Ar i thmet ic , Polygon and Tr i ang le Methods). Tsvetnyye Metal ly (Nonferrous Meta 1s) , Nos. 4-5, 1940, pp. 26- 35.

- . Geometriya Nedr (Subsurface Geometry). ~ z d a t e l ' s t v o Nedra (Mineral Resources Publishing Co.), Moscow, U.S.S.R., 1964, 500 pp.

Scarborough, J. B. Numerical Mathematical Analysis . The Johns Hopkins Press , Bal t imore, Md., 5 t h ed., 1962, 594 pp.

Sear les , W. H., and H. C. Ives. F i e l d Engineering Handbook, John Wiley & Sons, Inc. , New York, 21s t ed., 1945, v. 1, 366 pp.; v. 2, 403 pp.

Shklyarskiy , F. N. Odin i z Sposobov Opredeleniya Zapasov Gnezdovykh Mestorozhdeniy (A Method of Computing Reserves f o r Nes t l ike Ore Bodies). Gorn. Zhur. (Mining Jou rna l ) , Nos. 4-5, 1921, pp. 158-160.

Smirnov, V. I., A. P. Prokof 'yev, and o thers . Podschet Zapasov Mestorozhdenii Poleznykh I skopaeqkx (Reserve Computations of Mineral Deposi ts) . Gosgeoltekhizdat ( S t a t e S c i e n t i f i c and Technical Publishers! f o r Mining L i t e r a tu re ) , Moscow, U.S.S .R., 1960, 672 pp.

Soderberg, A. Mining Methods and Costs a t t h e Utah Copper Co., Bingham Canyon, Utah. BuMines I n f . C i r c . 6234, 1930, 23 pp.

Stammberger , Friedrich. Uber Ungenauigkeiten und "er l aub te ~ e h l e r g r e n s e n " b e i Vorratsberedhnungen ( Inexac t i t ude and "Permissible Limits of Error" i n the valuation of Reserves). Z tschr . f i r angewandte Geologic, No. 4 , 1956, pp. 180-189.

Stoddard, A. C. Mining P r a c t i c e and Methods a t I n s p i r a t i o n Consolidated Copper Co., In sp i r a t ion , A r i z . BuMines I n f . C i r c . 6169, 1929, 23 pp.

Thoubas, R. W. Mining P r a c t i c e a t Ray Mines, Nevada Consolidated Copper Co., Ray, Ariz. BuMines I n f . Circ. 6167, 1929, 27 pp.

Truscot t , S. J. Mine Econon;ics. Rev. and enl . by J a m e s Russel l . Mining Publ ica t ions , Ltd., 3d ed., London, 1962, 47 1 pp.

U. S. Atomic Energy Coarmission. Manual on Ore Reserve Procedures. (For o f f i c i a l use only), 1958, 99 pp.

Vinogradov, B. C. Znacheniye Metoda Podscheta Zapasov Geo logicheskimi Blokami v Uslwiyakh Podmoskovskogo Basseina (The Value of Reserve Computations by Geologic Blocks Under Conditions i n Moscow Coa l f i e ld s ) . Razvedka i Okhrana Nedr (Explorat ion and Conservation of Mineral Resources) , No. 4, 1958, pp. 14- 15.

65. Wolff, J. W . , E. L. Derby, and W. A. Cole. Sampling-and Estimating Lake Superior Iron Ores. Trans. A M , v. 72, 1925, pp. 641-652,

66. Yakzhin, A. A. Oprobovaniye i Podschet Tvedykh Poleznykh Iskopaemykh (Sampling and ~ e s k e Estimates f o r So l id Mineral Deposi ts) . Gosgeolizdat ( S t a t e Publishers f o r Geologic L i t e r a t u r e ) , Moscow, U.S.S.R., 1954, 296 pp.

67. Yufa, B. Ya. Metodika Otsenki Sluchainykh Pogreshnostey Analizov P r i Podschete Zapasov Mineral ' nogo Syr ' ya (Methods of Appraisa 1 Random E r r o r s of Chemical Analyses When Computing Reserves of Mineral Raw Mate r i a l s ) . Razvedka i Okhrana Nedr (Explorat ion and Conservation of Mineral Resources), No. 12, 1960, pp. 7-14.

68. Zhuravskiy, A. M. Obshchiye Metody Podscheta Zapasov Rudnykh . . Mestorozhdeniy (General Methods f o r Computing Ore Reserves). Trudy

Glavnogo Geologo-Razvedochnogo Upravleniya V. S . N. Kh. S . S . S .R. (Transact ions o t t h e Geological and Prospecting Serv ice of U.S.S.R.). I s s u e 116, Moscow-Leningrad, 1931, 40 pp. (Surrmary i n English.)

69. . 1. Ob Odnm Obobshchenii formuly V. I. Baumana (On Generalized Bauman's Formula). 2. K Metodike Podscheta Zapasw Rudnykh Tel (Method

, of O r e Reserve Computations). Trudy Glavnogo Geologo-Razvedochnogo Uprableniya V.S.N. Kh. S.S.S.R. (Transact ions of t h e Geological and Prospecting Serv ice of U.S.S.R.) I s s u e 201, Moscow-Leningrad, 1932, 25 PP.

APPENDIX A. - ENGLISH AND METRIC SYSTEMS AND CONVERSION FACTORS

Linear Measure

Foot.................... ........... Yard ( 3 f t ) . . Mi le (5,280 f t ) . . .......

English ( s t anda rd ) system

Meter........... Kilometer. ......

Met r i c system Units

Square foot........ ..... Square yard ( 9 sq f t ) ... Acre (43,560 s q f t ) ..... Square m i l e (640 a c r e s ) .

Un i t s Conversion f a c t o r t o metric system

3.2808 f t 0.6214 m i l e

Conversion f a c t o r t o English system

- -- ~- -

Land o r Area Measures

10.764 sq f t 0.3861 sq m i 2.4710 a c r e s

0.0929 sq m 0.8361 sq m 0.4047 h e c t a r e 2.5900 sq km

Volume Measure

Square meter. . . . Square k i lome te r Hectare. . . . . . . . .

............ Cubic f e e t . . .............. Cubic ya rd

- -

... Cubic meter . .

Ounc el .................. Pound ................... Ounce ( t r o y ) ............ Pound ( t r o y ) . ........... Ton, s h o r t (2,000 lb ) . .. Ton, long (2,240 lb ) ....

Weight Measure

Gram............

1.0160 me t r i c ton I

-

31.103 g 0.3732 kg 0.9072 m e t r i c t on

--

l Avoirdupois weights when not noted.

Kilogram ........ .... Met r i c ton..

0.0353 oz 0.0322 oz (troy) 2.2046 l b 1.1023 s h o r t ton 0.9842 long ton

Source: Pee le , Mining Engineers ' Handbook, v. 2 , pp. 45-49.

APPENDIX B. - USAGE OF VARIOUS METHODS FOR RESERVE COMPUTATIONS FOR SOLID MINERAL DEPOSITS I N U.S.S.R.

(percentage of t o t a 1s )

........ Cross sections................. I 48 1

TABLE B-1. - Computation by type of s o l i d minera l deposi ts1

I s o l i n e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I - Polygons... ............................ I 1

Coal and o i l sha le deposi ts

69

-

............................ Triangles . .

Total.............................

Nonmetallic d e p o s i t s

46

-

Method

Geological blocks......................

Mining blocks..........................

I I I

'AS considered by t h e A l l Union Comnittee of Mineral Reserves, U.S.S.R., f o r

Ore depos i t s

37

12

Source: Reference 57, t a b l e 22, p. 205.

TABLE B-2. - Solid mineral depos i t s by s e l e c t e d yea r s

19 54

36

15

46

2

1

- -

100

Source: Reference 57, t a b l e 23, p. 206.

1951

12

2 4

45

15

1

1

2

100

Method

Geological blocks.....................

Mining blocks.........................

Cross sections.........................

Polygons ............................... Triangles. . ........................... Rectangular ........................... Others................................

Total............................

1941-47

16

34

14

22.5

2.5

- 11.0

100

APPENDIX C. - A COMPARISON OF RESERVE COMPUTATIONS MADE BY VARIOUS METHODS (U.S.S.R.)

TABLE C-1. - Polymetal d e p o s i t i n A l t a i , U.S.S.R.~

I Ore i

Average f a c t o r s and a r e a : Thickness-weighted average...........

Average f a c t o r s and a r e a ( a r i t h m e t i c average) . .................

Cross sections......................... Polygon ................................ ...................... Triangles.... . . . . Tr iangles modified by t h e a r e a s of .......... in f luence f o r each working..

Reserves perc Zinc - 100

102 87 94 94

87 -

Gold S i l v e r !7=- LOO 1 100

' ~ a t c f o r 26 holes--3 ho le s c ros sed high-grade lead o r e of narrow width.

Source: Reference 46, t a b l e 3, p. 29. -

Bauxite d e p o s i t i n Tichvin d i s t r i c t , U.S.S.R.1

Average f a c t o r s and a r e a ( a r i t h m e t i c ave rage ) : ....................................... Index....... 100 Cross sections....................................... 103.1 Polygons .................. ........................... 99.3 Tr i ang le s ............................................ 9 7 . 2

lData f o r 3 baux i t e d e p o s i t s explored by 4 1 h o l e s .

Source: R e f e r e n c e s , t a b l e 4 , p. 29.

APPENDIX D. - FORMUlAS Main Elements

1. Relationship between t rue , horizonta 1, and v e r t i c a l thicknesses.

2. h u e thickness - correct ion, when a = 0.

t t r = t., s i n ($ + 8)

3. Horizontal thickness - correction, when a = 0.

s i n (B + 0) % = P s i n $

4. Ve r t i c a l thickness - correct ion, d e n a = 0.

s i n ( $ + 0) tv = cos

5. General case - t r u e thickness

' t t = t a p (cos a s i n $ cos 0 + s in fl .s in 0 )

6. General case - hor izon ta l thickness

th = t a p (COS a cos 0 - cotan B s i n 0 )

7 . General case - vert icH1 thickness

t~ ' t.p cos 0 (cos a tan + t an 0)

Rule of Gradual Changes

Computations

10. Average thickness

11. Area - Trapezoid formula

12. Trapezoidal ru le

13. imps son's rule

1 S = - h (a, + 2 C aodd + 4 C a,,,, 3 + a,)

14. Volume for a block

V = LBT

V = St,, 15.

16. Volume for the entire body

v = v , + v, + v, + ... + vn = ys, + Gs, + t,s, + * . * + tnsn

17. - Weight - ore tonnage

v Q = - F and Q = Vf

Q = V , 4 + V 2 4 + V,D, + ... + V,D,

Grade

24. Arithmetic average

Page

25. Thickness - Weighted average

26. Area - Weighted average

27. Volumetric average

28. Gravimetric average

29a. Weight-meta 1 tonnage

29b.

30. Correction factor for grade

E = 5 c,

Errors

31. Errors for the entire body

Mining B locks Method

Cross Sections Method

46. Mean- ar ea f ormu la

47. End-area formula (equal distances between sect ions)

L v = ( S 1 i 2% + 2S, + ... + S,) - 2

48. Volume for ent ire body (unequal distances between sect ions)

49. Wedge formula

v I- S L 2

50. More accurate wedge fb-la

51. Cone formula

52. Frustum formula

L V = - (2a + a l ) b s i n a 6

53. Prismoidal formula L V = ( % + 4 M + & ) x

54. Prismoidal correction factor - C ( f o r triangular prism)

55. Value of M for prismoidal formula

56. Obelisk formula

57 . ~auman's formula

The Standard Method for Nonparallel Sections

58. Angle less than 10'

L L

59. Angle greater than 10"

Linear Method Pane

60. Ore tonnage (based on l i n e a r o r e r e se rves )

Q = Q L *

61. Metal tonnage (based on l i n e a r meta l r e s e r v e s )

P = P L A

6 2 . Ore tonnage (based on ore r e se rves per cubic u n i t )

Q = qvv

6 3 . Metal tonnage (based on metal r e se rves per cubic u n i t )

Method of I s o l i n e s

64. Volume

66. Grade c cOA, + z ( & +2A1 + 2 & + ... +A, , )

C.v = G,

Method of t r i a n g l e s

67. For t r i angu la r prism - volume

68. For t r i angu la r prism - grade

6 9 .

70. Ore tonnage

Metal tonnage

72. Number of connecting l ines

NL = 3 4 + 2% + 3, or NL = 3 (q - 1) + 2%

7 3 . Number of t r iangles

N,, = 2n, + Q - 2 = 2 (q - 1) + ZIE

7 4 . The r e l a t i v e e r ro r fo r rectangular prism

A V = Vl - V2 = 2 (t l+2e+t3+2t4) S - 1 (2t1+&+2t3+4) s 6 6

75. Condition for precise volume computations in rectangular prism

I tl + tS = tZ + tq

76. General case - average thickness

77. Case of tS = 0

78. Caseof and t3 = O

tl t.v = - 3

7 9 . Volume for the e n t i r e body ( s = 2) N

1 N P C - - a C t k

i=1

80. Ore reserves f o r the en t i re body ( e = 2) N

S 81. Metal reserves for the entire body (s = E)

82. Metal tonnage

83. Square net of workings - one block

S 84. Square net of workings for the e n t i r e body ( s = -) N

t, + t2 + t, + ... + t, v = N

S = ( t , + t, + tg + ... + t,) S

85. Total average re la t ive error

APPENDIX E. - GLOSSARY

Se lec t ed terms used i n t h i s r e p o r t a r e included i n t h e fol lowing d e f i n i t i o n s .

Area r e se rves . - Reserves computed f o r a c e r t a i n a r ea .

Ari thmetic average o r mean. - Simple average of a s u i t e of q u a n t i t i e s , measurements, ana lyses , e t c . ; sum of a s u i t e d iv ided on t h e number of q u a n t i t i e s .

Block. - A u n i t of minera l body de l inea t ed by va r ious p r i n c i p l e s of i n t e r p r e t a - - t i o n of explora t ion d a t a ; va r i e s with t h e method of computations. .

I n s i d e perimeter . - The por t ion of a minera l depos i t de l inea t ed by out ly ing mine workings.

Linear meta l reserves . - Metal r e se rves f o r an a r e a u n i t , such a s square foo t f o o t and square meter (product of l i n e a r o r e r e s e r v e s and average grade) , i n tons o r o the r weight u n i t s .

Linear o r e reserves . - Ore reserves computed f o r a n a rea u n i t , such a s square f o o t o r square meter , i n tons o r o ther weight u n i t s .

Mine workings. - A l l s u r f ace and underground exp lo ra t ion , development, and I- e x p l o i t a t i o n exposures of a miner2 1 body ; d r i l l i n g included.

Ore. - A mineral substance tha t can be mined a t p resent a t p r o f i t t o t h e - opera tor o r t o t he b e n e f i t of t h e n a t i 0 n . l

Outside perimeter. - Portion of a minera l depos i t extended beyond the out ly ing mine workings ; del inea ted according t o geologica 1 evidence o r c e r t a i n p r i n c i p l e s .

Parameters of a mineral bodv. - A s e r i e s of phys ica l and chemical cons tan ts which express t h i s body.

Reserves. - Mineral ma te r i a l considered e x p l o i t a b l e under e x i s t i n g condi t ions ; inc luding cos t , p r i c e , technology, and loca 1 circumstances (1).

Resources. - Reserves p lus p o t e n t i a l raw m a t e r i a l ; inc ludes marginal , submar- g i n a l , and l a t e n t ca tegor ies ( I ) .

Sect ion reserves . - Reserves along a sect ion--one u n i t of length wide.

Sezment. - A large por t ion of a rninera 1 body.

Underground mine workings. - Explorat ion and development s h a f t s , a d i t , d r i f t s , c r o s s c u t s , r a i s e s , and winz es ; d r i l l i n g exc luded.

Unit volume reserves . - Reserves computed f o r one u n i t of volume, such a s cubic f o o t , cubic yard , or cubic meter.

De f in i t i on approved by the United Nations Educational , S c i e n t i f i c and Cul- t u r a 1 Organizations.

I N T . - B U . O F M I N E S . P G ~ . . P A . 8999

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