Cokrgng Partal Grades - Applcaton to Block Modelng of Copper Deposts Serge Séguret 1, Julo Benscell 2 and Pablo Carrasco 2 Abstract Ths work concerns mneral deposts made of geologcal bodes such as breccas or lenses that contan several categores of grades wth dfferent characterstcs n terms of dstrbuton and varogram. When producton blocks contan few such bodes, estmatng block grades by ordnary krgng may produce unrealstc spatal contnuty. We propose a method based on the ndcators of objects (unts or faces) together wth ther products wth the grade. Ths s llustrated by an applcaton to a porphyry copper depost. Introducton We try to answer ths queston: Gven samples nformed by a categorcal varable and a grade, what s the best way to estmate the average grade at the scale of producton blocks? We propose to splt the grade n a sum of partal grades, whch leads to an sotopc cokrgng system based on the ndcators of the unts and ther products wth the grade. In an applcaton to a porphyry copper depost, we show how we can characterze the geometry of the unts and we buld the cokrgng system. The resultng block model s compared to usual krgng. Ths expanded abstract summarzes an applcaton to a porphyry copper depost located n northern Chle and developed n [1]. The oral presentaton s based on a second, unpublshed, case study. Basc functons used n the sequel are ndcators. The probablstc nterpretaton of a smple or cross varogram and on the rato of a cross varogram by a smple one s presented by Rvorard [2]. A general overvew of these tools s gven by Chlès and Delfner [3]. 1 Mnes ParsTech, Center of Geoscences and Geoengneerng, 35 rue Sant Honoré, 77305 Fontanebleau, France, Serge.seguret@mnes-parstech.fr 2 Codelco, Vce Presdenca Corporatva de Proyecto, Alameda 1449, Edfco Santago Downtown,Torre 2, pso 8, Santago, Chle Nnth Internatonal Geostatstcs Congress, Oslo, Norway June 11 15, 2012
Formalzaton Cokrgng Partal Grades S. Séguret, J. Benscell, P. Carrasco 2 We consder an ore body where the ore s classfed n n subsets or unts (ore types or faces) and denote by 1 (x) the ndcator of unt : 1 ( x)=1 f x unt, 0 f x unt ( 1 ) We consder that we have as many grade varables as unts: The grade Z(x) at pont x s decomposed n partal grades Z ( x)=1 ( x)z( x ) ( 2 ) At pont x, these partal grades are equal to zero, except one whch s equal to Z(x). Ther sum s thus equal to Z(x). Let us now consder a block V. The partal grades of block V are defned by 1 Z( V) Z( x) dx ( 3 ) V V The grade Z(V) of block V s the sum of the partal grades Z (V). When the partal grades do not have the same spatal structure, t s sensble to estmate Z through the Z 's rather than drectly. We are n an sotopc stuaton (the varables are equally sampled) and the cokrgng of the sum equals the sum of cokrgngs CK Z(V) n CK Z (V) ( 5 ) =1 For each grade Z, we estmate by cokrgng ts average over V and we add the estmatons. We have 2n varables to consder for buldng the cokrgng system Tools n unt ndcators 1 (x) (ntroduced for ther major nfluence), n partal grades Z (x) (varables of nterest). Let us denote by the varogram of the ndcator 1 (x), by j the cross varogram of the ndcators 1 (x) and 1 j (x), and by Z the cross varogram of the ndcator 1 (x) and the correspondng partal grade Z (x). Our man tools are the ratos of cross varograms by an ndcator varogram. Table 1 shows ther probablstc nterpretaton. Table 1 Indcator and grade varograms (denoted by Greek letter ) and ther nterpretaton Calculaton Interpretaton Conceptual llustraton j p(x+h j/x, x+h ) Probablty to reach j whle leavng Z E[Z(x+h)/x+h, x ] Average grade when enterng n
Applcaton Cokrgng Partal Grades S. Séguret, J. Benscell, P. Carrasco 3 The volume of the studed doman s approxmately 400x1500x400 m 3 and t contans more than 54000 samples of 1.5 meter length, all nformed n copper grade and coded n 4 unts. (Table 2). Table 2 Man characterstcs of grades Abbr. Color Proporton % Mean grade Std dev. Mn Max All unts 100 0.78 1.54 0 33 Waste W 31.2 0.06 0.17 0 8.1 Low grade C1 27.5 0.31 0.36 0 7.6 Hgh Grade C5 31.7 1.16 0.90 0 23.1 Breccas Bx 9.6 4.27 3.48 0 33 Average grades present mportant dfferences between unts. Even poor unts present hgh grades (8% of copper for waste for example). Table 3 presents the probablty, when leavng a gven unt, to encounter another unt. It must be read together wth the global proportons of Table 2. Table 3 Contact probabltes From To W C1 C5 Bx W 0.8 0.2 0 C1 0.2 0.65 0.15 C5 0.05 0.5 0.45 Bx 0 0.2 0.8 Man comments are: Bx and W are not n contact When leavng W, the probablty to enter n C1 s 0.8, whch s much greater than the global proporton of C1 (less than 0.3). So C1 separates W from C5 and Bx Same remark for the probablty to encounter C5 when leavng Bx: C5 separates Bx from C1 and W When leavng C1, one can encounter Bx wth a low probablty (0.15), whle the probablty to encounter C5 (0.65) s more mportant than the global proporton of C5. Same remark when leavng C5 and encounterng C1. Ths shows that C5 tends to surround Bx.
Cokrgng Partal Grades S. Séguret, J. Benscell, P. Carrasco 4 Such consderatons as well as the knowledge of the geologst make t possble to generate a scheme showng the mutual behavors of the unts (Fgure 1). Fg. 1 Schematc representaton of the mutual behavors of the unts deduced from the statstcal analyss of the contacts Fgure 2 shows ratos of cross varograms by a smple varogram. j j Fg. 2 Ratos of ndcator cross varograms by smple a smple varogram. We notce that, apart one excepton, all unts present dfferent spatal correlatons and therefore must be estmated jontly by cokrgng. Exceptons are C1 and C5 where the probabltes do not depend on the dstance. The fronter between these two unts marks the lmt between poor ore (manly to the west) and rch ore (to the east). Let us now consder the ratos of cross varograms between the unt ndcators and the partal grades by the ndcator smple varograms. Fgure 3 presents the most representatve behavors.
Cokrgng Partal Grades S. Séguret, J. Benscell, P. Carrasco 5 Z Z Fg. 3 Ratos of cross varograms between unt ndcators and grades by ndcator smple varograms There are border effects for each unt but ther magntude s not mportant. The upper left varogram of Fgure 3 shows that the copper grade n W decreases when movng away from the boundary of W, but the decrease n copper grade s only 0.05% after 150m. The most mportant gradent s lnked to Bx, the average grade ncreases of 0.40% after 150m (bottom rght varogram of Fgure 3). Globally, one can consder that grade varatons are smaller wthn the unts than between unts. A cokrgng system s bult ncorporatng ndcators and partal grades. Results are compared block by block to krgng wthout dstncton of the unts. Fg. 4 Scatter dagrams between partal grade cokrgng versus usual krgng The scatter dagram between drect krgng and cokrgng shows an mportant correlaton (Fgure 4). The standard devaton of the dfference between the two estmates s 0.1%. Both estmators gve close results. Two reasons can be ponted out:
Conclusons Cokrgng Partal Grades S. Séguret, J. Benscell, P. Carrasco 6 The grades follow the sequence W-C1-C5-Bx, whch leads to the mutual organzaton of the faces. When we estmate Z drectly usng a movng neghborhood, we take nto account ths sequence naturally because low grades manly concern W and hgh grades Bx. All the varograms contan more than 50% of nugget effect and ths reduces the mpact of the estmator choce on the results because an mportant part of the calculaton s just a local average. The nterest of ths approach s not located n the resultng estmaton, but on the analyses that leads to t. Frst ths approach enables us to separate the sole geometry of the unts (modelled by the ndcators) from the behavor of ths geometry together wth the grade nsde the unts (modelled by the partal grades). Ths leads to calculaton prortes lke for example n the present data set where one must focus on the proportons estmaton n each block, and one can then affect to each unt an average grade n the block. Secondly, present calculatons, only based on statstcs, could act as a reference for the usual practce whch conssts n drawng geologcal objects by hand and ntersectng them wth the grd to calculate block-by-block proportons. The dfference between both approaches quantfes the mpact of the geologcal knowledge on the results. Important dfferences may ndcate a lack of data and an mportant uncertanty of the resultng block model. The proposed methodology could act as a handral aganst excess. Bblography [1] S. A. Séguret, Block model n a mult faces context Applcaton to a porphyry copper depost, GEOMIN2011, Antofagasta, Chle, paper at http://www.geomn.cl/evento2011/ndex.php?lang=en [2] J. Rvorard, Introducton to Dsjunctve Krgng and Non-Lnear Geostatstcs, Oxford Unversty Press, Oxford; 180 p. 1994. [3] J. P. Chlès, P. Delfner, Geostatstcs. Modelng Spatal Uncertanty, 2 nd edton. John Wley & Sons, 701 p., 2012.