Journal of Geochemical Exploration 108 (2011) 196–208

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Journal of Geochemical Exploration

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Fractal models for estimating local reserves with different mineralization qualitiesand spatial variations

Qingfei Wang a,b,⁎, Jun Deng a,b, Huan Liu a,b, Yanru Wang a,b, Xiang Sun a,b, Li Wan a,c

a State Key Laboratory of Geological Processes and Mineral Resources, China University of Geosciences, Beijing 100083, People's Republic of Chinab Key Laboratory of Lithosphere Tectonics and Lithoprobing Technology of Ministry of Education, China University of Geosciences, Beijing 100083, People's Republic of Chinac School of Mathematics and Information Science, Guangzhou University, Guangzhou, Guangdong 510006, People's Republic of China

⁎ Corresponding author at: State Key Laboratory of GeResources, China University of Geosciences, Beijing 1000

E-mail address: wqf@cugb.edu.cn (Q. Wang).

0375-6742/$ – see front matter © 2011 Elsevier B.V. Adoi:10.1016/j.gexplo.2011.02.008

a b s t r a c t

a r t i c l e i n f o

Article history:Received 26 October 2010Accepted 27 February 2011Available online 12 March 2011

Keywords:Ore reserveConcentration-area modelFractal

The concentration-area model, one of the widely applied fractal models, is utilized to describe the spatialdistribution of mineralization variables, i.e., orebody thickness and grade-thickness. And based on theconcentration-area model, a new fractal model for reserve estimation (abbreviated as FMRE-CA) isestablished. Via the demarcation values obtained in the concentration-area model, the orebody is spatiallydivided into several parts with different value ranges and spatial variation of mineralization variable. Based onthe FMRE-CA, the local ore reserves in the parts are estimated, and the global reserve is obtained by additionof the local reserves. In the FMRE-CA, the spatial variation of the mineralization variable in a local reserve ischaracterized by a fractal dimension, and a greater fractal dimension denotes a more variation. Compared totraditional reserve estimations based on discrete and linear functions, the FMRE-CA is established viacontinuous and nonlinear function, and it is easier in calculation process and different in the way in which thelocal reserves are delimitated. Compared to another fractal model for reserve estimation based on number-size model (abbreviated as FMRE-NS), the FMRE-CA is capable of estimating the local reserves. A gold orebodyin Southwest Yunnan and two bauxite orebodies inWestern Guangxi, China, are selected for case study. In thecase study, the global reserve calculated via the FMRE-CA and those derived from the FMRE-NS and traditionalgeometric block method are analogous.

ological Processes and Mineral83, People's Republic of China.

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© 2011 Elsevier B.V. All rights reserved.

1. Introduction

In ore reserve estimation, a crucial topic in the exploration andexploitation of mineral deposits, we need not only to estimate theglobal ore reserve, but also to know the distribution of local reserveand ore quality. The traditional approaches for ore reserve estimationinclude geometric and geostatistic methods, in which the global andlocal reserves are estimated mostly via discrete and linear functionsbased on the mineralization variables, i.e., orebody thickness, gradeand grade-thickness, in the exploration or mining works. For instance,in the geometric polygon method, the orebody is divided into severalpolygons with the exploration works as the vertexes; and the localreserve in the polygon is obtained by multiplying the polygon areaand the average ore grade-thickness or orebody thickness in thepolygon vertexes. In the geostatistic method, the orebody is dividedinto a matrix of rectangular blocks, normally with side of one-half toone-fourth the average spacing of exploration works, and the grade ofeach block is estimated by searching the database for the variablessurrounding each block and computing the weighted average of those

variables via kriging method (Annels, 1991). It is shown that, in thetraditional methods, the local ore reserve is mostly determined by thelocations of exploration works rather than the mineralizationqualities, represented by the size of mineralization variables. Becauseof the complex ore-forming situation and process, the mineralizationvariables in an orebody often display irregular and heterogeneousspatial distributions. The spatial mineralization qualities and theirspatial variations are important for ore mining and ore genesisunderstanding; however they are only roughly analyzed in thetraditional methods. In addition, the traditional methods are oftenbased on linear mathematics and difficult for dealing with theirregular and skewed distributions of mineralization variables(Bárdossy et al., 2003; Wang et al., 2010a), and also require relativelycomplex data processing. A new easy approach of partitioning theorebody into several parts according to the mineralization intensityand spatial variation and calculating the local reserves in the parts ismeaningful and scientific for ore reserve calculation.

Since the geological objects commonly show irregular, heterogeneous,and skewed characteristics, they have been described by various fractalmodels, including box counting model (Deng et al., 2001, 2006, 2008a;Mandelbrot, 1983), number-size model (Deng et al., 2009, 2010a;Mandelbrot, 1983; Turcotte, 2002; Wan et al., 2010; Wang et al., 2010a,2010c; Zuo et al., 2008), concentration-area model (Cheng et al., 1994),

197Q. Wang et al. / Journal of Geochemical Exploration 108 (2011) 196–208

self-affinemodel (Deng et al., 2008b; Wang et al., 2007; Zuo et al., 2009),and multifractal model (Agterberg et al., 1996; Wang et al., 2008). Thenumber-sizemodel waswidely applied to characterize the distribution ofthe various geological objects (Clark et al., 1995; Mandelbrot, 1983;Manning, 1994; Roberts et al., 1998; Sanderson et al., 1994; Turcotte,2002; Walsh et al., 1991; Watterson et al., 1996; Yielding et al., 1996).Wang et al. (2010a) used the number-size model to describe themineralization variables in the exploration works in a single deposit,and then established a fractal model for estimating global ore reserve(abbreviated as FMRE-NS) based on the number-size model. The FMRE-NS is easier in calculation process and more effective to deal with theirregularity and skewness than the geometric and geostatistical methods.However, the number-size model is not concerned with the spatiallocations of mineralization variables, and thus the FMRE-NS cannotestimate the local reserve.

The concentration-area model was proposed by Cheng et al. (1994)and commonly utilized to deal with the spatial structure of geochemicalmap and determine the thresholds for anomalies (Cheng, 1999). Theconcentration-area model is expert in spatial analysis of data, which isabsent in the number-size model and FMRE-NS. In this paper, theconcentration-areamodel is utilized todescribe the spatial distributionsofmineralization variables in a single orebody; and according to the model,the orebody is divided into several parts with different mineralizationqualities and spatial variations. Furthermore, based on the concentration-area model, we deduce another fractal model for reserve estimation,abbreviated as FMRE-CA., The local reserveswith different mineralizationdistributions and theglobal reserve in the single orebody canbeestimatedvia FMRE-CA. One gold orebody in the Yunnan province and two bauxiteorebodies in the Guangxi province, China, are selected for case study.

2. Mathematical modeling

2.1. Preliminary processing of raw data

The preliminary processing of raw data in FMRE-CA is the same withthe geometric block method (abbreviated as GBM) and FMRE-NS (Wanget al., 2010a). The FMRE-CA is based on themineralization variables in theexploration works in an orebody. The orebody thickness and grade-thickness are used for estimating ore tonnage and metal tonnage,separately. The establishment and application of FMRE-CA are performedin a VLP (a vertical longitudinal projection, i.e. project each explorationwork horizontally onto a vertical plane) if the orebody dip is no less than45° or in a HLP (horizontal longitudinal projection, i.e. project eachexploration work vertically onto a horizontal plane) if the dip is smallerthan 45°. The mathematical modeling in the VLP is the same with that inthe HLP. The following modeling is performed in a VLP.

In a VLP, the orebody horizontal thickness, average grade and grade-thickness in each explorationwork are calculated. According to the cutoffand minimal mining thickness, the orebody outlines are first delimitedand theorebody area is obtained. Basedon thediscrete explorationworks,the contour maps of mineralization variables are obtained via inversedistance weighting (IDW) interpolation, which is one of the mostcommonly used techniques for interpolation of scatter points and widelyapplied in the mineral evaluation and concentration-area model (Annels,1991; Cheng et al., 1994; Deng et al., 2010a). The IDWmethod is based onthe assumption that the interpolating point should be influencedmost bythe nearby variables and less by the more distant ones. The interpolatingsurface is a weighted average of the scatter points in a given search areaand the weight assigned to each scatter point diminishes as the distancefrom the interpolation point to the scatter point increases. In theapplication of IDW interpolation, it uses a moving circular window withadjustable parameters to control the weighting of values at neighboringpoints. The parameters include radius of the circular window, theweighting power, the maximum number of samples to be includedwithin eachwindowand the interpolation interval. The contourmapusedin the concentration-area model is selected as a grid format in this paper.

2.2. Concentration-area model

In the concentration-area model applied in geochemical analysis,the area A(r≥ri) enclosed by contours with a concentration ri has apower–law relationship with the ri as follows (Cheng et al., 1994):

A ≥rð Þ = Cr−Di ð1Þ

where D is called fractal dimension. C is a constant. In this paper, theconcentration-area model is applied to analyze the contour maps ofmineralization variables; and ri represents the orebody thickness ti orgrade-thickness li.

If the plots of A(≥r) and r in ln–ln coordinates can be fitted with onestraight line, the distribution is called a simple fractal; yet in most cases,the plots are fittedwith several straight line segments, themodel is calleda bifractal. In a bifractalmodel, the variable values are divided into severalsegments, with demarcation value Ri (i=1, 2, …, n) (i.e. starting point),dimensionDi and constantCi for the ith segment. AdemarcationvalueRi isa value of r that divides two adjoining straight line segments. The Ri isexpressed as Ti and Li for orebody thickness and grade-thicknessdistribution, respectively.

In the concentration-area model, the fractal dimension canindicate the spatial distribution pattern of the mineralization variable.Greater fractal dimension suggests a faster decease outwards from ahigh value or denser contours around the value, denoting a greaterspatial variation.

2.3. Deduction of FMRE-CA

Assuming the concentration-area model comprises n straightsegments, the ith segment is confined by Ri and Ri+1 (i=1, 2, …, n,R1 is the minimum value of the variable and Rn+1 is the maximumvalue). According to the demarcation values, the contour map isspatially divided into n parts and the mineralization qualities areseparated into n ranks; the variables in ith part range from Ri and Ri+1

and their distribution is characterized by a fractal dimension Di.By assuming orebody thickness is a continuous variable, the local ore

tonnage Oi in the ith part can be expressed as:

Oi = ∫Ti+1Ti

ρdA ≥tð Þdt

tdt =ρCiDi

1−DiT1−Dii+1 −T1−Di

i

h iD≠1ð Þ ð2Þ

where ρ is the average ore density.Then, the global ore tonnage O can be written as:

O = ∑n

i=1Oi = ∑

n

i=1

ρCiDi

1−DiT1−Dii+1 −T1−Di

i

� �� �D≠1ð Þ ð3Þ

Similarly, the local metal tonnage Mi can be obtained andexpressed as:

Mi =ρCiDi

1−DiL1−Dii+1 −L1−Di

i

h iD≠1ð Þ ð4Þ

Then the global metal tonnage is:

M = ∑n

i=1

ρCiDi

1−DiL1−Dii+1 −L1−Di

i

� �� �D≠1ð Þ ð5Þ

The average grade G in the orebody is estimated by the followingEq. (6):

G =MO

ð6Þ

Fig. 1. Geological map of the selected gold deposit in Southwest Yunnan, China.

198 Q. Wang et al. / Journal of Geochemical Exploration 108 (2011) 196–208

3. Applications and advantages of the FMRE-CA

3.1. Model application

Just like GBM and FMRE-NS, the FMRE-CA is only based on theorebody thickness and grade-thickness in exploration works, and has

Fig. 2. Locations of exploration works, orebody thickness contour map and the divisions ofcontour map showing the locations of exploration works, (b) partitioned parts for local res

no relationship with the genetic type of deposits. Therefore, theFMRE-CA can be widely applied to the deposits of various genetictypes with the precondition of that the orebody mineralizationvariables can be described by the concentration-area model. In theFMRE, mineralization variables are considered to be continuous,indicating that the orebody area is filled with exploration works with

the different parts for local reserve estimation in the orebody 1. (a) Orebody thicknesserve estimation.

199Q. Wang et al. / Journal of Geochemical Exploration 108 (2011) 196–208

infinitely small intervals, and thus the estimated results denote actualreserves which can be mined out without consideration of ore lossrate in the orebody.

In FMRE-CA, it is presumed that the data distribution obtainedfrom a sparse grid is analogous to that derived from a dense grid.However, it is necessary to note that the denser exploration grid canprovide more accurate fractal model and orebody area, and thusbetter estimation results. In addition, the ore density is considered tobe a constant in the model; therefore, the reserves of ores of totallydifferent industrial types or densities should be estimated separately.

The following four steps are required in the FMRE-CA: (1)delimiting orebody area in a VLP or HLP; (2) making the contourmaps of the orebody thickness and grade-thickness via IDW interpo-lation; (3) applying the concentration-area model to the contour mapsto determine the demarcation values, and dividing the orebody intoseveral parts with various ore quality ranks according to thedemarcation values; (4) calculating the local reserves with variousore quality ranks and the global reserve of the orebody.

3.2. Comparison with traditional methods

In the traditionalmethods, the ore blocks are partitioned accordingto the exploration grid, and the local reserve is estimated via discreteand linear function. However, in the FMRE-CA, the local reserves aredivided in terms of the intrinsic orebody features, i.e., mineralizationqualities and spatial variations. In each local reserve, the orebody areain which the mineralization variable is no less than a given valueshows a fractal (power–law) relationship with the value, and thisrelationship is characterized by a fractal dimension; and the greaterfractal dimension means the orebody area decreases more rapidly asthemineralization variable increases, suggesting a more varied spatial

Fig. 3. Locations of exploration works, orebody grade-thickness contour map and the divisionorebody grade-thickness showing the locations of exploration works, (b) partitioned parts

change of the mineralization quality. Furthermore, since the FMRE-CAis established via nonlinear mathematics and thus is much effective infitting the skewed distribution, which is an obstacle for the traditionalgeometric and geostatistic methods. Therefore, FMRE-CA is consid-ered to obtain more accurate result than the traditional methods.

3.3. Comparison with FMRE-NS

The mineralization variables are assumed to conform to thenumber-size model proposed by Mandelbrot (1983):

N ≥rð Þ = Cr−D ð7Þ

where N(≥ r) stands for the cumulative number of exploration worksin which the variable is not less than r.

For bifractals, the global ore tonnage and global metal tonnage inthe FMRE-NS can be expressed by Eqs. (8) and (9), respectively:

O = ρATD11

D1 T1−D12 −T1−D1

1

� �1−D1ð Þ + ∑

n−1

i=2

Di ∏n−1

i=2TDi−Di−1i

1−Dið Þ T1−Dii+1 −T1−Di

i

� �26664

37775

ð8Þ

M = ρALD11

D1 L1−D12 −L1−D1

1

� �1−D1ð Þ + ∑

n−1

i=2

Di ∏n−1

i=2LDi−Di−1i

1−Dið Þ L1−Dii+1 −L1−Di

i

� �26664

37775ð9Þ

s of the different parts for local reserve estimation in the orebody 1. (a) Contour map offor local reserve estimation.

200 Q. Wang et al. / Journal of Geochemical Exploration 108 (2011) 196–208

Comparison to FMRE-NS, the FMRE-CA can deal with the spatialdistribution of the ore reserves and estimate the local reserves withvarious ore quality ranks.

4. Case study 1

4.1. Deposit geology and raw data

A medium-size gold deposit in the Southwest Yunnan, China isselected for case study 1 (Fig. 1). The ores are mainly hosted in thesericite-quartz schist of the Manlai formation in the Late ProterozoicLancang Group, clastic rocks of the Middle Jurassic Huakaizuoformation and Quaternary lateritic. Most orebodies have the char-acteristics of hot spring-type mineralization, while small part sufferedweathering and transformed into the lateritic-type. The wallrockalterations mainly include silicification, clayzation, sericitization, andpyritization. The ore minerals comprise quartz, albite, plagioclase,kaolinite, mica, sericite, pyrite and limonite.

Orebody 1 in the gold deposit is selected for reserve estimation andthe data of exploration works are from Yunnan Bureau of Geology &Mineral Resources. In orebody 1, total 77 exploration works includingboth drills and wells are carried out and their locations are shown inFigs. 2 and 3. In the exploration, the orebody area, ore density, oretonnage, metal tonnage and orebody grade are estimated via the GBM tobe 263,149.33 m2, 1.75 t/m3, 5.91 Mt, 4.59 t and 0.78 g/t, respectively.

The histograms of the natural logarithms of thickness and grade-thickness in exploration works in the orebody 1 are illustrated inFig. 4a and c, and the ‘quantile–quantile’ plots (Q–Q plot) in Fig. 4band d. The Q–Q plots display that the variables do not follow log-normal distributions, especially in the two tails; the variables haveskewed distributions. The mineralization variables in orebody 1 havemuch varied spatial changes as shown in Fig. 5.

Fig. 4.Histograms and Q–Q plots of the exploration data in the orebody 1. (a) Histogram for tgrade-thickness.

4.2. FMRE-CA application

Since the orebody is nearly horizontal, the HLP is applicable. Theparameters in the IDW interpolation were set as: radius of 400 m,weighting power of 2 (corresponding to a linearweighting functionwithweight 0 for the points located at the boundary or outside the movingwindow, and 1 for the point situated at the center), maximum of 15samples per window (if more than 15 samples occur within thewindowfor a given location, only the 15 nearest points are selected) andinterpolation interval of 15 m. The contourmaps of thickness and grade-thickness in orebody 1 are illustrated in Figs. 2a and 3a, respectively.

After the contour maps of the thickness and grade-thickness areobtained, they are analyzed via concentration-area model. Theconcentration-area model for thickness contour map comprises 6segments, as is illustrated in Fig. 6a, the fractal dimensions anddemarcation values for the 6 segments are listed in Table 1. Accordingto the demarcation values, the orebody thickness is divided into 6ranks, and the thickness contour map is correspondingly separatedinto 6 parts as shown in Fig. 2b. Generally, with the raise ofmineralization rank, the fractal dimension increases gradually,suggesting an upgrade of spatial variation. The local ore tonnage ineach part is calculated in light of Eq. (2) and the results are listed inTable 1, and the estimated global ore tonnage is 6.08 Mt. Percentagesof the local ore tonnages with various mineralization qualities areshown in Fig. 7a, and it is revealed that the local ore tonnage withmineralization quality rank III takes up 57.44% of the global reserve.

The concentration-area model for grade-thickness is composed of 7segments (Fig. 6b), and correspondingly, the orebody area ispartitioned into 7 parts (Fig. 6b). As is similar to orebody thicknessdistribution, the spatial variation of grade-thickness grows approxi-mately as the mineralization rank rises. The percentages of local metaltonnages in the 7 parts calculated via Eq. (4) are shown in Fig. 7b, and

hickness; (b) Q–Q plot for thickness; (c) histogram for grade-thickness; (d) Q–Q plot for

Fig. 5. Orebody thickness and grade-thickness curves along sections in the orebody 1. (a) Thickness curve along section AA′; (b) grade-thickness curve along section AA′; (c)thickness curve along section BB′; (d) grade-thickness curve along section BB′.

Fig. 6. Concentration-area models for orebody thickness and grade-thickness distributions in the orebody 1. The different shades of gray represent the different size segments. (a)Concentration-area model of orebody thickness, (b) concentration-area model of orebody grade-thickness.

Table 1Fractal parameters in the concentration-area model for thickness distribution in orebody 1 and the reserves with different ore qualities estimated via FMRE-CA.

Orequalityrank

Orebody area(m2)

Ore density(t/m3)

Fractal parameters Local ore tonnage(Mt)

Global ore tonnage(Mt)

ri Di ri+1 Ci

I 24,356.94 1.75 1.81 0.09 7.58 2.92×105 0.22 6.08II 83,105.00 1.75 7.58 1.22 10.87 2.93×106 1.40III 133,921.30 1.75 10.87 2.83 21.61 1.42×108 3.49IV 14,192.59 1.75 21.61 4.69 27.12 3.92×1010 0.59V 7175.12 1.75 27.12 9.26 36.67 1.40×1017 0.36VI 400.00 1.75 36.67 18.29 39.44 1.64×1031 0.02

201Q. Wang et al. / Journal of Geochemical Exploration 108 (2011) 196–208

Fig. 7. Pie diagrams of local reserves with different mineralization qualities and spatial variations in the orebody 1. (a) Ore tonnage; (b) metal tonnage.

202 Q. Wang et al. / Journal of Geochemical Exploration 108 (2011) 196–208

themetal tonnagewithmineralization quality rank IV occupies 43.1% ofthe global reserve. Based on Eq. (5), the global metal tonnage isestimated to be 4.80 t (Table 2).

4.3. Estimation result comparisons

The number-size models of thickness and grade-thickness dis-tributions in the orebody 1 are illustrated in Fig. 8a and b, respectively.According to the number-size models and based on Eqs. (8) and (9),

Fig. 8. Number-size models for the thickness and grade-thickness distributions in the orebodmodel of orebody thickness, (b) number-size model of orebody grade-thickness.

Table 2Fractal parameters in the concentration-area model for grade-thickness distribution in oreb

Orequalityrank

Orebody area(m2)

Ore density(t/m3)

Fractal parameters

ri Di r

I 26,990.65 1.75 1.35 0.09II 57,018.56 1.75 5.89 0.96III 78,807.10 1.75 7.90 1.74 1IV 85,622.04 1.75 10.98 3.63 1V 10,012.61 1.75 18.20 3.36 2VI 4300.00 1.75 25.35 8.80 3VII 400.00 1.75 33.02 35.43 3

the estimated global ore tonnage and metal tonnage in the orebody 1are 6.37 Mt and 5.10 t respectively (Table 3). The global reserves andore grade obtained via FMRE-CA, GBM and FMRE-NS are compared,and it is shown that the FMRE-CA has similar results to the other twomethods (Fig. 9). The relative errors between the reserve estimated byFMRE-CA and that calculated via GBM are 2.85% for ore tonnage, 4.76%for metal tonnage and 1.25% for ore grade; in addition, those betweenthe FMRE-CA and FMRE-NS are 4.57% for ore tonnage, 5.83% for metaltonnage and 1.32% for ore grade.

y 1. The different shades of gray represent the different size segments. (a) Number-size

ody 1 and the reserves with different ore qualities estimated via FMRE-CA.

Local metal tonnage(t)

Global metal tonnage(t)

i+1 Ci

5.89 2.88×105 0.18 4.807.90 1.32×106 0.700.98 6.58×106 1.278.20 6.36×108 2.075.35 2.53×108 0.363.02 1.03×1016 0.204.18 2.84×1056 0.02

Table 3Fractal parameters in the number-size model for thickness and grade-thickness distributions in orebody 1 and the reserves with different ore qualities estimated via FMRE-NS.

Orebody area(m2)

Density(t/m3)

Fractal parameters Result G(g/t)

r1 D1 r2 D2 r3 D3 r4 D4 r5 D5 r6

O (Mt) 2.63×105 1.75 1.80 0.06 4.35 0.63 13.62 1.07 16.88 2.63 29.35 5.32 44.58 6.37 0.80M (t) 1.20 0.14 3.60 0.53 11.89 1.73 28.14 3.73 37.89 5.10

203Q. Wang et al. / Journal of Geochemical Exploration 108 (2011) 196–208

5. Case study 2

5.1. Deposit geology and raw data

Two typical orebodies in the Xinxu bauxite deposit, WesternGuangxi, China are chosen for case study 2. The deposit strata fromoldest to youngest include Devonian carbonates and shale, Carbon-iferous carbonates, Permian limestone, siliceous limestone andbauxite ores, Triassic sandstone and carbonates and Quaternarylaterite (Fig. 10). The bauxite orebodies with thickness from 1.22 mto 25.3 m are contained in the Quaternary karst depressions, whichare widely developed in the deposit. The Quartenary bauxite oreswere transformed from the Permian bauxite ores during theformation of Quaternary karst terrain (Deng et al., 2010b). The

Fig. 9. Comparisons of the global reserves and ore grades obtained by the FMRE-CA,FMRE-NS and GBM in the orebody 1.

Fig. 10. Geological map of the Xinxu bauxit

Quaternary ores have various colors in surface, such as red, brownishred and dark red, and are mainly characterized by nodular, pseudo-porphyritic and ooidic textures. The ores are mainly composed ofdiaspore, hematite, goethite, gibbsite, amesite, kaolinite, chamositeand small amount of debris minerals, e.g., rutile and zircon (Liu et al.,2010; Wang et al., 2010b).

In the Xinxu deposit, orebody 2 and orebody 3 are selected forreserve estimation and the exploration data are from bureau of geo-exploration and mineral development of Guangxi. Total 87 wells areevenly distributed in the orebody 2, and 35 wells in orebody 3, asshown in Figs. 11 and 12. Since the orebodies are horizontal, the HLP isapplied; and vertical orebody thickness is used for ore tonnageestimation. Via the GBM, the orebody area, ore density and global oretonnage in orebody 2 are calculated to be 155,509 m2, 784 kg/m3 and1.78 Mt, respectively; those in orebody 3 are 56,259 m2, 1054 kg/m3

and 0.56 Mt. The histograms of the natural logarithms of orebodythickness in the orebody 2 and orebody 3 are illustrated in Fig. 13a andc, and the ‘quantile–quantile’ plots (Q–Q plot) in Fig. 13b and d,respectively. The Q–Q plots show that the thickness distributions areskewed. It is further shown in the Fig. 14 that the spatial changes oforebody thickness are irregular in the two orebodies.

5.2. FMRE-CA application

The contour maps of the orebody thickness in the orebody 2 andorebody 3 are produced via IDW method and illustrated in Figs. 11aand 12a, respectively. The parameters involved in the IDWmethod areset as: radius of 270 m,weighting power of 2, maximum of 10 samplesper window and interpolation interval of 10 m. Based on the thicknesscontour maps, the concentration-area models have been established.The concentration-area models in the orebodies 2 and 3 are both

e ore deposit, Western Guangxi, China.

Fig. 11. Locations of shallow wells, orebody thickness contour map and the divisions of the parts with different mineralization qualities and spatial variations in the orebody 2 in theXinxu deposit, China. (a) Contour map of orebody thickness showing the locations of shallow wells, (b) partitioned parts for local reserve estimation.

204 Q. Wang et al. / Journal of Geochemical Exploration 108 (2011) 196–208

composed of 5 segments (Fig. 15). According to the demarcationvalues in the concentration-area models, the thickness maps oforebodies 2 and 3 are both divided into 5 parts as shown in Figs. 11band 12b separately. Compared to the orebody 1, the fractal dimensionin the models of the orebodies 2 and 3 changes more abruptly as themineralization quality rank increase. It is shown that the fractaldimensions in segments 4 and 5 in the concentration-area models aremuch greater than those in segments 1 and 2 in the two orebodies,suggesting the orebody thickness in the parts with ranks 4 and 5 have

Fig. 12. Locations of shallow wells, orebody thickness contour map and the divisions of the pXinxu deposit, China. (a) Contour map of orebody thickness showing the locations of shallo

much more variation. Based on Eq. (3), the local ore tonnages in eachpart in orebody 2 and orebody 3 are calculated and listed in Table 4,and percentages of the local ore tonnage in orebody 2 and orebody 3are illustrated in Fig. 16a and b, respectively. Finally, the global oretonnage in orebody 2 is estimated to be 0.551 Mt, and that in orebody3 is 1.869 Mt. The local reserve with mineralization quality rank IV isthe greatest in the five local reserves in both orebody 2 and orebody 3,and it accounts for 42.48% of the global reserve in orebody 2, and itoccupies 37.57% in orebody 3.

arts with different mineralization qualities and spatial variations in the orebody 3 in thew wells, (b) partitioned parts for local reserve estimation.

Fig. 13. Histograms and Q–Q plots of the orebody thickness in the orebodies 2 and 3 in the Xinxu ore deposit. (a) Histogram for thickness in the orebody 2; (b) Q–Q plot for thicknessin the orebody 2; (c) histogram for thickness in the orebody 3; (d) Q–Q plot for thickness in the orebody 3.

205Q. Wang et al. / Journal of Geochemical Exploration 108 (2011) 196–208

5.3. Estimation result comparisons

The number-size models of thickness distribution in the orebody 2and orebody 3 are illustrated in Fig. 17a and b separately. According tonumber-size models and Eq. (7), the global reserves in the twoorebodies are estimated (Table 5). Comparing the global reservescalculated via FMRE-CA, GBM and FMRE-NS, it reveals that the FMRE-CA result is roughly equivalent to those of the other two methods(Fig. 18). The relative errors between the global reserve derived from

Fig. 14. Orebody thickness curves along sections in the orebodies 2 and 3 in the Xinxu ore desection DD′ in the orebody 3.

the FMRE-CA and that calculated by the GBM are 4.84% for orebody 2and 1.24% for orebody 3; and those between the FMRE-CA and FMRE-NS are 6.73% for orebody 2 and 3.91% for the other.

6. Conclusions

In this paper, a new fractalmodel for reserve estimationbasedon theconcentration-areamodel is established. Compared to the FMRE-NS, theFMRE-CAanalyzes the spatial distribution of the ore reserves. Compared

posit. (a) Thickness curve along section CC′ in the orebody 2; (b) thickness curve along

Table 4Fractal parameters in the concentration-area model for thickness distribution in orebodies 2 and 3 and their reserves with different ore qualities estimated via FMRE-CA.

Orebody Orequalityrank

Orebody area(m2)

Density(kg/m3)

Fractal parameters Global ore tonnage(Mt)

r1 D1 r2 C Local ore tonnage(Mt)

Orebody 2 I 7084.93 784 1.62 0.03 11.15 1.61×105 0.03 1.87II 33,194.31 784 11.15 1.18 13.88 2.64×106 0.34III 55,683.59 784 13.88 6.06 15.50 1.03×1012 0.69IV 58,646.64 784 15.50 11.44 22.42 2.50×1018 0.79V 900.00 784 22.42 19.59 25.09 2.53×1029 0.01

Orebody 3 I 7702.46 1054 1.37 0.13 6.38 6.44×104 0.04 0.55II 18,291.09 1054 6.38 1.37 9.11 6.41×105 0.16III 12,841.65 1054 9.11 4.12 10.39 2.80×108 0.13IV 16,230.64 1054 10.39 8.34 14.43 5.52×1012 0.21V 1195.30 1054 14.43 20.53 16.22 7.98×1026 0.02

Fig. 15. Concentration-area models for the thickness distributions in the orebodies 2 and 3 in the Xinxu deposit, China. The different shades of gray represent the different sizesegments. (a) orebody 2, (b) orebody 3.

206 Q. Wang et al. / Journal of Geochemical Exploration 108 (2011) 196–208

to the traditional geostatistical and geometric methods, the FMRE-CA isestablished based on continuous and nonlinear function, and it is notonly easier in calculation process, but also prominent in delimitating thelocal reserves in termsof thedifferent size ranges and spatial variation ofthemineralization variables. In theFMRE-CA, the variabledistribution ina local reserve is characterized by a size range and a fractal dimension,and a greater fractal dimension denotes a more varied spatial change of

Fig. 16. Pie diagrams of local ore tonnages with different mineralization qualities and spatial v

themineralization quality. One gold orebody and twobauxite orebodiesin China were used in the case studies. It is revealed that the estimatedglobal reserves via FMRE-CA and those derived from traditional GBMand from FMRE-NS are similar, with relative errors less than 6.73% and5.83% for global ore tonnage and metal tonnage respectively. The casestudies further approve that the FMRE-CA can be utilized in the depositswith various genetic types.

ariations in orebodies 2 and 3 in the Xinxu deposit, China. (a) Orebody 2; (b) orebody 3.

Fig. 18. Comparisons of global ore tonnages and ore grades obtained by the FMRE-CA,FMRE-NS and GBM for the orebody 2 and orebody 3 in the Xinxu deposit, China.

Table 5Fractal parameters in the number-size model for thickness distribution in orebodies 2 and 3 and their reserves with different ore qualities estimated via FMRE-NS.

Orebody Orebody area(m2)

Density(kg/m3)

Fractal parameters Total ore tonnage(Mt)

r1 D1 r2 D2 r3 D3 r4 D4 r5 (rmax) D5 r6 (rmax)

Orebody 2 1.56×105 784 1.22 0.04 9.94 0.57 11.90 1.94 14.10 4.27 17.75 7.82 25.30 1.75Orebody 3 5.63×104 1054 1.30 0.12 4.79 0.73 10.96 4.00 15.97 21.01 16.83 0.53

Fig. 17. Number-size models for the orebody thickness distributions in the orebodies 2 and 3 in the Xinxu deposit, China. The different shades of gray represent the different sizesegments. (a) Orebody 2, (b) orebody 3.

207Q. Wang et al. / Journal of Geochemical Exploration 108 (2011) 196–208

Acknowledgements

We appreciate the valuable comments from associate editorAntonella Buccianti, Alexis Navarre-Sitchler and the other anonymousreviewer. This research is jointly supported by National Basic ResearchProgram (Nos. 2009CB421006 and 2009CB421008), the key project ofthe resource exploration bureau in Guangxi province, the Program forNew Century Excellent Talents, the National Science and TechnologySupport Program (Grant No. 2011BAB04B09), National NaturalScience Foundation of China (Grant nos. 40872194, 40872068,40572063, and 40672064), the 111 project (No. B07011).

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