Incorporation of ore grade uncertainty into the push back design process

Sy I f gd uy h uh bk dg by J. Ghlmjd*, d M. Ol I m lg d dg, uh bk m qug wdly ud f lg-m du lg. I uh bk dg, f d gd d h dd bl uh bk wh gd h mmum uh bk wdh qum. h d dud by mzg m m lk mmdy, mg g, g,. h m m hz f h uh bk h h m uh bk h h hgh vlu, h d m uh bk h h d hgh vlu, d. h d gd bd h b m f gd wh f h bdy h-dml mx fm. hf, h h uy d wh v m (f u du, gd uy) k u wh uh bk dgd. h mhd d m blh whh f h d lkly b wh m. h du w lghm h gd uy dug h uh bk dg. h uggd gy k hdul ky blk l h x qu. Kywd: m, uh bk, gd uy. Idu Lg-m du lg dg mj m lg, bu dm h m um f j. Al lg-m l gud f mdum d h-m du hdulg. Bu h lg umb f blk wh h ulm lm, h b dvdd f ub- mmly lld uh bk, u bk h. h uh bk dgd wh hul d d gud dug h yly hdulg. hf, uhbk dg ly ky l dfg ul h flw b gd fm mg. h my b my lv uh bk qu f, whh ul bg dff NV fm mg j. S h 1960 my uh bk dg lghm hv b dvld. w g w fllwd hugh h vu uh bk dg lghm: 1. Dmg h uh bk wh h um h h b h h h h hgh gd d d hv vy lw g b xd. Em mz mhd lfd h gu. I h mhd f gd by lyg h ulm lm lghm (Lh d Gm lghm1,2 d wk flw 3) dff m blk mdl f h d. h m mdl b gd by hgg m m m uh mmdy, u-ff gd mg d g. h m wdly ud lghm h gd Whl mhd4,5. Wg d Svm6 d uh bk dg lghm ug v mz d f m mz. h lghm w bd bg mxmum ml g M blk hugh M+N blk h mdl. I 1986 hu h w fmuld by Gh7 whh d h quly f h, h f h blk d h quly f h blk ud h dd blk. Ex hwd h dfg h b bv my ld mxmzg h * Dm f Mg d Mllugl Egg, Yzd Uvy, Yzd, I. Dm f Mg d Mllugl Egg, Amkb Uvy f hlgy, h, I. h Suh Af Iu f Mg d Mllugy, 2007. SA ISSN 0038 223X/3.00 + 0.00. vd O. 2006; vd vd A. 2007. h Jul f h Suh Af Iu f Mg d Mllugy VOLUME 107 REFEREED AER MARCH 2007 177

I f gd uy h uh bk dg hvbl NV f mg j, bu h gy d ly k u h g qud g h blk wh h hgh gd8. 2. Dmg h uh bk wh h um h h b h h h hgh gd d x d hv h l mu f w g. Rmz d Dgdl8 d h mmum g uh bk dg lghm bd h gy. h lghm fd h uh bk h h f hm h h mmum g mg ll bl uh bk wh h m z. hy lmd h h hdul bd by h lghm wll h h blk f h vu lghm. h ld g h NV f mg j. h ml f uh bk v h u d wh h u h mz mdl. Dmkul9 lfd h u f mg j fllw: Uy f h bdy mdl d ld u gd vbly d ml y dbu. Uy f hl mg f uh l, xv,. Uy f m u ludg l d g, d mmdy. Amg h u, h uy ld h bdy mdl d u gd vbly h mj bu ug x b m h ly g f j. Vl10 d h h f y f f -u, 60% f h uvyd m hd vg f du l h 70% f h dgd y. All h bv mhd d lghm wk wh h b m f gd gg h uy h m h md gd. h du w mhd h gd uy h uh bk g. Fm h md lghm h vu gh, Whl f h m wdly ud h uh bk dg. Bu u wk bd Whl lghm, h mdl wll b vwd h. dm Whl d, l u df h blk m vlu f h h blk (BEV ))9: O : h l mu f h blk. G : Emd gd f h blk. R: h f h du vd by g h. : h bbl u f du l y dlvy. C: h x f mg d g f blk h h g w. R : h l mu f k ( d w) h h blk. MC: h f mg d mvg f w. Whl dud h umb f m vbl by dvdg Equ [1] by MC: [1] /MC h mu f du h huld b ld y f mg f ml. h h ly gf vbl, bu C/MC xd hg gfly ul h dbl hg h m, w mg g mhd dud5. Equ [2] b w : [2] [3] Wh V h vlu gd u f mg, d qul /MC d C/MC vly. I d g uh bk, hgd h d h h Lh d Gm lghm mlmd h m vd blk mdl. Nw df f h b W u u df bu h b : Ml h h hgh gd, lw mu f uy d d lw w g f x. Adg h bv df h b l h mg uh bk h lkly b wh m. Bf ml xl f h uggd lghm, xml gv. Aum h h w blk bdy blk mdl: blk A d B. Al h bbly dbu f h gd h blk umd b h ll ml dbu wh h m d v f bg qul h Kgg m (G ) d m 2 v ( ) vly (h w vlu l b bd by ug gl mul hqu). h w vlu f blk A d B ld blw: Adg h bbly h wh h ml uv lbl, wh h bbly f 95% h u m gd f h blk (G ) wll fll wh h dd dv f h m ( )11: [4] I h h wd, wh 95% fd lvl h u m gd wll l d : H, h fd vl f h m gd f h w blk b luld : Wh, G A d G B h u bu ukw gd f blk A d B vly. A l fm h bv lul, wh h bbly f 95% h l mu f gd f blk A (1.1) l h h f blk B (3.04). Nw f h u-ff gd umd b 1.5%, h mvg blk A [5] 178 MARCH 2007 VOLUME 107 REFEREED AER h Jul f h Suh Af Iu f Mg d Mllugy

I f gd uy h uh bk dg bf blk B h m k h x f blk B bf A, bu h bbly h blk A w blk. hf, d h blk A h m m gd vlu h blk B, my b b x blk B bf blk A. Nw um h blk A h dd dv f 1.3%. h w hv: I h m fvubl x blk A bf blk B. h uggd lghm wll dm h f. I d dm uh bk wh h bv bjv, f f ll f (u 100) d huld b gd. h mb f h d ld gg bl uh bk f muvg d fw. I h fllwg w uld w mhd g d d wh u df f h b. Nw uggd uh bk dg lghm I d g d wh h md bjv, h f m ly ld. A d bf, h d gd by h gdul mdf f m ky m d h h LG lghm mlmd k h dg ul f h mdf. I h lghm h m vlu f h blk h mdl wuld b dud h m uh h h du l h blk uy. h m h f h w blk h mdl wh h m vg gd d dff dg f uy, h du h m vlu f h blk wh h g mu f uy wll b m h h h. Blk gd dd dv b dd d f blk gd uy. I h h blk m vlu huld b dud h wh mu l h blk gd dd dv. I d du h d blk m vlu w ugg h fllwg fmul lul h blk m vlu: [6] wh: : umb wh v ul gg dff. : dd dv f h blk gd. h h u h blk vlu ug h lw vlu f (z) d h b h ml. h h ulm d ll blk ud b xludd fm fuh lul. If w h x g, h vg gd f blk d, quly, h blk m vlu dud. Hgh dd dv ul g d h vlu f blk. hf, mll d m vlubl d l u blk h bgg. Alyg h fmul ul gg g f ml (d ) vg h lkly g f vlu f vg gd. L u um h w g ml ( I Fgu 1) ug Equ [4]. Nw w w fd h mll wh m vlubl d blk. If w Equ [4] d lul h blk vlu, l h h vlu f vy blk h mdl wll h y h m (w blk), bu h du m h blk wh h g mu f uy (blk wh hgh dd dv). Ah mz ug h w vlu ul h ul (hw II Fgu 1), whh ludd by I. I f II lu m ddl blk d h d w. I huld b d h my b bl h mll vlu d ld g mll. Nw f w h hugh f dg vlu Fgu 1 A - f hwg h lh f d h d h Jul f h Suh Af Iu f Mg d Mllugy VOLUME 107 REFEREED AER MARCH 2007 179

I f gd uy h uh bk dg d h, d mz f h gd m blk mdl, w b f d ul. I my b bl, f xml, h f 100, ly 40 50 d gd. h mll hll h blk h bbly h m vlubl. h hll ul xm wh vu dg f k bu lld uh bk bu f h lk f mhg, bhg d. A f hll h mlly du g f d m m mmum mg wdh v uh bk. Ivl m f h vlu f whh ld f m ly huld b uh h hy fl h blk gd uy ly. I h h wd, h gd blk gd h huld b f h bl gd lz h xd b hd f blk x. Fm Equ [4] l h h u blk gd g h G 1.96. Wh f Equ [6], huld b l h 1.96. Al, huld b d h h bl vg blk gd huld b g h qul z,..: H: F xml, um h f blk w hv G d = 12%, h w hv: If w hv G < 0 h, G z. I m w d hv y fm bu h bbly dbu f blk gd, bu w kw m d v. Wh h lmd fm w l m h vl f by ug Chbyhv hm11: L X b y dm vbl f whh h xd vlu ( ) d v ( 2) my b fud. h: h m h h bbly h dm vbl wll fll wh ±k dd dv f m l. Adg h bv hm, f w u k=4.5, h w hv: [10] [7] [8] [9] Fm Equ [10], l h h blk gd g h h wh bbly g h f 0.95. Wh f h Equ [6], huld b l h 4.5. A d bf, f w u kgg d blk gd m, G kgg m d h mmum m v Kgg v12. Kgg gl mhd h m h blk gd h h m qud mmzd; hf, h v f vlu md by kgg mll h h l bu ukw v. h mhg f u vbly f h gd ld vm f lw gd d udm f hgh gd. Addlly, mhg fu f d dy d fgu: f g dy wll hw m ll vbly whl hvg d wll b m ufm13. H, du hdul h bd kgg u f bbl dv du g d bd l wll vd ly u lu. Mull d kgg wk bbl b df h dbu f gd f ml wh h h wdw, vdg d xm h dl umulv dbu fu f h blk14. Rh h kgg, lhugh h bbl m dud by h mull d kgg mhd f du h mhg ff, bu h ll gd vbly my ll b ly hzd. h b wy qufy gd uy dl mul. Cdl mul glz f h M Cl y mul h, whh d hdml l l12,15. I du dd qully bbl mg f u bdy gd whh hv h fllwg h16: A h mld l, h muld vlu f h vbl h m h mud vlu f h vbl. All h muld vlu f gv vbl hv h m l lh bvd h d vlu. All h muld vlu f y f vbl hv h m l lh bvd h d vlu. h hgm f h muld vlu f ll vbl h m h bvd f h d mg. Eh mul u du mg lz f h d h ly fl h l d l vbly f h l d. fmg vl dd mul qud h m f ll vbly. If w hv h ul f mul, h w hv vl gd ml f h blk wh m d v b ud h k ly. Illu f xml A 2D blk mdl fgu g h vg gd d dd dv f h blk gd hw Fgu 2 d 3 vly. I lulg h vlu f blk, umd h h $1100/, mg $5/ f ml, g $20/ f, h vy f 80% d h f wgh f d w 2.9 d 2.3 /m3 vly. 180 MARCH 2007 VOLUME 107 REFEREED AER h Jul f h Suh Af Iu f Mg d Mllugy

I f gd uy h uh bk dg Fgu 2 C- vw f 2D xml hwg h vg blk gd (%) Fgu 3 C- vw f 2D xml hwg h dd dv (% 2 ) Fgu 4 C- vw f h ml F f ll, h ulm lm huld b dmd. h b fmd by lyg h 2D LG lghm h m blk mdl. h ulm lm f h xml hw Fgu 4. Wh Whl lghm mlmd h xml, uh bk bd ul h 54. N h k 4. h gd uh bk hw Fgu 5. A b fm Fgu 5, h f uh bk gd ug Whl mhd vy lg. I h wd, h bg jum fm h f uh bk. h blm lld h g blm. h blm u m dfful h ffv l f h uh bk h yly du hdulg. By g h -by- f h, ulmly 9 uh bk gd, hw Fgu 6. h w mhd ld l h 2D blk mdl whh w hw Fgu 2. If Equ [4], z, h ulm lm (Fgu 4) gd. h f uh bk gd = 0.2. h hw Fgu 7. Wh uv g f, 12 uh bk gd, whh hw Fgu 8. h mll uh bk d wh = 2.9. A l fm Fgu 6 d 8; h mll uh bk h gd ug h uggd lghm mll h h f Whl lghm. h m h h g blm dud by ug h w uggd mhd. h Jul f h Suh Af Iu f Mg d Mllugy VOLUME 107 REFEREED AER MARCH 2007 181

I f gd uy h uh bk dg Fgu 5 C- vw f h f uh bk gd ug Whl mhd Fgu 6 C- vw f h uh bk gd ug Whl mhd Fgu 7 C- vw f h f uh bk gd ug h uggd mhd Fgu 8 C- vw f h uh bk gd ug h uggd mhd 182 MARCH 2007 VOLUME 107 REFEREED AER h Jul f h Suh Af Iu f Mg d Mllugy

I f gd uy h uh bk dg I d m h k d wh h w mhd, umd h 5 blk md h y. Al f h blk k d huld b dfd. h d h m v f h blk ( 2). Blk wh hgh mu f v m k. h um f h udud uy d wh h uh bk dd uh bk k d (RI) d b luld fllw: wh: RI k : udud k d f kh uh bk; k=1,2,,k. K l umb f uh bk wh ml ul. Cv(,j): h v bw blk d blk j wh h kh uh bk. k : l umb f blk wh h kh uh bk. Bu h k h ly y f h du d m l, du huld b ld h uh bk k d. h dud uh bk k d (DRI) f f uh bk h b luld : [11] [12] wh: k : h y whh h x f h kh uh bk fhd. k : h du f kh uh bk. If w hv h yly du d l h uh bk lf, h w b k ly. h du f 12% dd d lul h DRI d wh h hdul mg fm bh Whl d h uggd lghm. h umb f d w blk wh h uh bk, h lf f h uh bk, h uh bk k d d dud uh bk k d f bh Whl d h uggd lghm hw bl I d II vly. I huld b d h h blk v dd h lul. h umulv mu f h dud uh bk k d h d f h w mhd ld Fgu 9. A l fm Fgu 9, h umulv mu f h dud uh bk k d u mhd l h h f Whl. h d h h w lghm x h m f d l du d d m u lf f l d, wh ddl fm uully bm vlbl. H, mlmg h yly du hdul wh uh bk mg fm h w mhd, bl I h udud d dud uh bk k d ug Whl lghm bl II h udud d dud uh bk k d ug h w lghm h Jul f h Suh Af Iu f Mg d Mllugy VOLUME 107 REFEREED AER MARCH 2007 183

I f gd uy h uh bk dg Fgu 9 h umulv mu f h dud uh bk k d h d f Whl d w mhd Fgu 10 h vg gd f h f d gd ug bh mhd ul dug hdul wh bbly f mg du g hgh h h f Whl. h vg gd f h l mu f h f d gd by ug bh mhd luld d ghlly hw Fgu 10. h vg gd f d gd ug h w mhd l h h gd ug Whl mhd bf y 14. Af h h d gd ug bh mhd hv h m vg gd. h juf f h ly h h m blk h mdl h hv hgh mu f k d mdum hgh gd. h mdl f x lw k d lw mdum blk d f mdum hgh gd blk h hv hgh mu f uy h l du d. A qu, h hl ( hvbl) NV Whl mhd my b hgh h h f h w lghm, bu h hvbl NV ug h w mhd wll b m h h f Whl, bu g h y f h x dly ld g h hvbl NV. Clu I d dg uh bk, m mhd dvld h gd uy dug h uh bk dg. I blvd h h m m hz f h uggd mhd : Hgh-gd z h hv lw mu f uy ld f x h l du d. Hgh-gd z h hv mdum mu f uy my l b ld f x l du d. I h f h d, h vg gd f h blk hgh d h, ubg fm ff f dd dv wll u hgh du h vlu. Hgh-gd z wh hgh mu f uy lf b xd h mddl lf f m wh Equ [4] ll h mddl f g. Lw mdum-gd z wh lw mu f uy my l b vlubl wh xd h mddl lf f m. 184 MARCH 2007 VOLUME 107 REFEREED AER h Jul f h Suh Af Iu f Mg d Mllugy

I f gd uy h uh bk dg Lw mdum-gd z wh hgh mu f uy lf b xd h d f m lf, bu h h du du g f lg d my v b bl h blk gd dud mu l h h u-ff gd. I hw h h umb f uh bk gd ug h mhd m h h f Whl mhd. h d h h g blm lmd l dud ug h uggd mhd. Rf 7. GERSHON, M.E. A du hdul: lghm d mlm. Mg Egg, Augu 1987.. 793 796. 8. RAMAZAN, S. d DAGDELEN, K. A w uh bk dg lghm mg. dg f h 17h l ymum M lg d qum l, Clgy, Cd. 1998. 9. DIMIRAKOOULOS, R. Cdl Smul Alghm f Mdlg Obdy Uy O Omz, Il Jul f Suf Mg, Rlm d Evm, vl. 12, 1998.. 173 179. 10. VALLEE, M. Ml Ru + gg, m d lgl fbly = O v, CIM Bull, vl. 90, 2000.. 53 61. 1. LERCHS, H. d GROSSMAN, L. Omum dg f - m. CIM, vl. LXVIII, 1965.. 17 24. 2. ZHAO, H. d KIM, Y.C. A Nw Omum Lm Dg Alghm. 23d Il Symum h Al f Cmu d O Rh h Ml Idu, 1992.. 423 434. AIME, Ll, C. 3. JOHNSON,.B. d BARNES, J. Al f Mxml Flw Alghm Ulm Dg. Egg Dg: B Rul hugh O Rh Mhd. Nh Hlld, 1988.. 518 531. 4. WHILE, J. A dd f m lg d mz h f f ug lghm kg. dg f h 28h Al f Cmu d O Rh h Ml Iduy, 1999,. 15 23. 5. WHILE, J. Byd mz dg. dg f h f Cd f mu l h ml duy, 1998.. 331 337. 6. WANG, Q. d SEVIM, H. Alv mz fdg f mxmum-ml f du lg. Mg Egg, Fbuy 1995.. 178 182, 11. LAIN, L.L. bbly d f md gg, Sd Ed, WS-KEN ublhg Cmy, 1990. 810. 12. JOURNEL, A.G. d HUIJBREGS, C. Mg G. Nw Yk: Adm, 1978. 600. 13. SMIH, M.L. Igg dl mul d h gmmg: l du hdulg. dg f Al f Cmu d O Rh h Ml Iduy, 2001,. 230 207. 14. DAVID, M.A. Hdbk f ld dvd gl v m. Elv, Nhld. 1988. 15. DOWD,.A. A vw f dvlm g. Cmu d G, vl. 17, 1992.. 1481 500. 16. DOWD,.A. Rk ml j: ly, d mgm.. I M. Mll. (S.A: M. Iduy),vl. 106, Juy Al 1997. h Iu f Mg d Mllugy.. A9 A18. h Jul f h Suh Af Iu f Mg d Mllugy VOLUME 107 REFEREED AER MARCH 2007 185