Ore Geology Reviews 37 (2010) 2–14

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Ore Geology Reviews

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Fractal models for ore reserve estimation

Qingfei Wang a,b, Jun Deng a,b,⁎, Huan Liu a,b, Liqiang Yang a,b, Li Wan a,c, Ruizhong Zhang a,b

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. State Key Laboratory of GeoResources, China University of Geosciences, Beijing 1000

E-mail address: djun@cugb.edu.cn (J. Deng).

0169-1368/$ – see front matter © 2009 Elsevier B.V. Adoi:10.1016/j.oregeorev.2009.11.002

a b s t r a c t

a r t i c l e i n f o

Article history:Received 9 May 2009Received in revised form 11 November 2009Accepted 11 November 2009Available online 18 November 2009

Keywords:Ore reservesJiaodong gold provinceFractal modeling

Traditional geometric and geostatistic methods for reserve estimation in a single deposit are difficult to use withskeweddistributionmineralization variables includinggrade, orebody thickness and grade–thickness, a commoncharacteristic of most deposits, and require complex data processing. It has been shown that the skewedmineralization variables can be described by the number–size model in a fractal domain. Based on the number–sizemodel, assuming that orebody thickness and grade–thickness are continuous variables, the fractal model forreserve estimation (FMRE) in a single deposit can beestablished. In the FMRE, ore tonnage can beestimatedgiventhe orebody area and the fractal parameters of orebody thickness distribution and metal tonnage can beestimated based on the orebody area and the fractal parameters of grade–thickness distribution. The reserveestimatedby the FMRE candenote actual ore tonnageandmetal tonnage that canbeminedout of the deposit. TheFRMEwas applied to theDayingezhuanggolddeposit in the Jiaodonggoldprovince inChina. Thegold reservesviathe FMREand the traditional geometric blockmethod are similar, with relative errors of 3.11% in ore tonnage and0.29% in metal tonnage. Compared to traditional reserve estimation the FMRE is much easier in calculationprocess and is more reasonable in dealing with the skewed distribution. However, this new method fails tocalculate local reserve, which can be derived via any of the traditional estimation methods.

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

ll rights reserved.

© 2009 Elsevier B.V. All rights reserved.

1. Introduction

Ore reserve estimation has crucial importance for the evaluation andexploitation of mineral deposits. Reserve estimation is based on theresults of exploration, such as drillholes and exploratory drifts, locatedin a grid. Several variables concerning exploration works are proposedas representations for mineralization intensity and for calculating thereserve. These variables generally include the ore thickness in theexploration works and the grade over this thickness. The grade–thickness (accumulation) value, with the units m g/t (Parker, 1991),another important variable for reserve estimation, can be obtained bymultiplying the grade and the corresponding thickness. The threevariables are called ‘mineralization variables’ in this paper.

1.1. Traditional reserve estimation methods

Traditional reserve estimation methods can be roughly classifiedinto two main groups: geometric and geostatistical methods (Henley,2000). The geometric methods comprise the block methods (triangle,regular, squares, rectangular, polygonal and irregular or geometricblocks), the methods of profiles (vertical, horizontal and inclined) andthe isopach method. The geometric block method (GBM) is one of the

most popular methods for reserve estimation. In the GBM, eachexploration work is projected horizontally onto a vertical plane (avertical longitudinal projection, VLP) in the case of that orebody witha dip more than 45° (or project each exploration work vertically ontoa horizontal plane – a horizontal longitudinal projection, abbreviatedas HLP – if the dip is smaller than 45°). Orebody area can then bemanually separated into numerous blocks with exploration works invertices of the VLP plane and the inside average grade, orebodyhorizontal or vertical thickness and the local reserve in each block arecalculated according to the exploration works. Finally, the globalreserve in a deposit is obtained by summing up the local reserve inevery block (Annels, 1991).

The geostatistical methods take into account the spatial autocor-relation of mineralization variables. By themethod of variography, theranges of influence of the variables can be determined in two or threedimensions. Then, the local mineralization variables can be obtainedvia linear interpolation, based on the exploration works in theinfluence range. Geostatistical calculation requires a large sample size.With a small number of exploration works (say, about 20), thecalculation of variograms becomes increasingly uncertain, evenimpossible (Diehl, 1997). Geostatistical calculations also requiresuitable computer programs and a considerable mathematicalbackground (Bárdoossy and Fodor, 2004).

Reserve estimations commonly involve extremely high values ofgrades and thicknesses, designated as outliers, which induce a highlyskeweddistribution andare obstacles tofitting traditionalmathematical

Fig. 1. Geological model for the fractal model for reserve estimation (FMRE). (Theheight w and width h are the exploration work horizontal interval and vertical altitudedifference, respectively.) The different shades of grey of the rectangle plots in (c)represent the variant values of mineralization variables.

3Q. Wang et al. / Ore Geology Reviews 37 (2010) 2–14

models applied in geometric and geostatistical methods to thedistributions of mineralization variables. Outliers affect the precisionsof estimation result greatly (Arik, 1990; Costa, 2003). The properhandling of such outliers is crucial to ore reserve estimation. Differentcorrectionmethods, including simple trimming, ignoring their potentialinfluence and keeping them in the dataset, or adjusting the outliersaccording to the overall trend of the cumulative distribution functions,are proposed for dealing with the grade outliers. And few methodsinvolving complex data processing are proposed for the skewedthickness or grade–thickness distribution (Tutmez et al., 2007). Reliableore reserve estimates for deposits with skewed distributions ofmineralization variables are still difficult to obtain when usinggeometric and geostatistical methods.

Based on exploration works, the geometric and geostatisticalmethods can estimate the discrete local reserve with linear interpola-tion and then calculate the global reserve in a deposit. However, bothmethods involve complexdata processing and can present difficulties interms of dealing with the skewed distribution of mineralizationvariables. In order to achieve reliable reserve estimates, a goodmathematical model must be built to represent adequately the overalldistribution of data.

1.2. Fractal models in geology

Natural variability is an inherent feature of geological processes andgeological objects are neither random nor homogeneous; they showirregular, heterogeneous, and skewed characteristics (Bárdoossy et al.,2003). Fractal models are well-established and have been effectivelyapplied to describe the distributions of geological objects, including thebox counting model (Mandelbrot, 1983; Cheng, 1995; Deng et al., 2001,2006), thenumber–sizemodel (Mandelbrot, 1983; Turcotte, 2002;Wanget al., 2007a), the radial–density model (Mandelbrot, 1983; Feder, 1988;Carlson, 1991; Blenkinsop, 1994; Raines, 2008; Carranza, 2009), thegrade–tonnage model (Turcotte, 1997, 2002; Wang et al., 2007c), theself-affine model (Wang et al., 2007b), and the multifractal model(Agterberg et al., 1996;Cheng, 1999;Denget al., 2007, 2008a;Wanget al.,2008). Of the different fractalmodels, the number–sizemodel is themostpopular and it has been applied to characterize the distribution ofearthquakemagnitudes (Lee and Schwarcz, 1995; Turcotte, 1997; Dimri,2005), fault displacement lengths (Aviles et al., 1987; Okubo and Aki,1987; Walsh et al., 1991; Jackson and Sanderson, 1992; Manning, 1994;Fossen and Rørnes, 1996; Knott et al., 1996; Needham et al., 1996; Nicolet al., 1996; Watterson et al., 1996; Yielding et al., 1996; Pickering et al.,1997), vein thickness and size of ore deposits (Carlson, 1991; Manning,1994; Sanderson et al., 1994; Clark et al., 1995; McCaffrey and Johnston1996; Roberts et al., 1998; Turcotte, 2002), and element concentrations(Monecke et al., 2001, 2005; Deng et al., 2009; Wan et al., 2010).

As the fault displacement in a fault system follows the number–sizemodel and it can be assumed to be a continuous variable, Scholz andCowie (1990) created the total displacementmodel to calculate the totaldisplacement of a fault system; this method is easily applied to theanalysis of a fault system and hydrocarbon accumulations (Walsh et al.,1991; Jackson and Sanderson, 1992; Barton andScholz, 1995). However,because most geological objects cannot be considered continuousvariables, further applications of the method proposed by Scholz andCowie (1990) are limited. For example, because the number of depositsin an ore cluster area is limited and the ore tonnage of the deposits is adiscrete variable; this method cannot be utilized to estimate regionalresources.

1.3. Research objective

In an orebody, mineralization variables can be treated ascontinuous variables. Explorations are random samples of orebody,and data of ore thickness and grade–thickness are used for orebodymodeling. We assume that the distributions of ore thickness and

grade–thickness can be described by the number–size model, andthen deduce the fractal models for reserve estimation, abbreviated asFMRE, via the number–size model. The Dayingezhuang deposit in theJiaodong gold province in China is used in the case study.

2. Mathematical modeling

2.1. Preliminary processing of raw data

In the exploration of a deposit, a series of exploration works areperformed along different exploration lines at various depths. Alongexplorationworks, channel sampleswith constant length are completedand their metal concentrations are analyzed. Outliers in metalconcentration data are replaced by the average grade of an orebody.According to the concentration distributions, cutoff grade and minimalmining thickness, the outlines of an orebody can be delimited, and theorebody thickness and the corresponding mean concentration in eachexploration work are calculated. Exploration works with grades greaterthan cutoff and orebody thickness greater than the minimum miningthickness are categorized as “mineralized” in this paper, and their totalnumber is Ca, otherwise they are categorized as “non-mineralized”.

As with the traditional GBM, if an orebody dip is more than 45°, weproject each exploration work horizontally onto a VLP (Fig. 1). In a VLP,the horizontal thickness of an orebody and the average grade in eachexploration work are calculated, and the corresponding grade–thickness can thus be obtained. The following modeling process iscarried out based on the VLP of exploration works. If an orebody dip isless than 45°, we can project each exploration work vertically onto aHLP, and orebody vertical thickness and grade–thickness in everyexploration work can be obtained. In this case the mathematicalmodeling explained below is still applicable.

Fig. 2. Bifractal model with three segments. Solid circles represent the thresholds.

4 Q. Wang et al. / Ore Geology Reviews 37 (2010) 2–14

2.2. Block modeling

It can be preliminarily assumed that exploration works are evenlydistributedwith constant horizontal intervalw and vertical interval h inthe VLP plane. In the VLP plane, we can use rectangles with the sameheight h and width w in order to cover the mineralized explorationworks and ensure that the center of each rectangle is located in anexploration work. Thus, the total number of the mineralized rectanglesis Ca. The area Ai of each rectangle can be expressed by:

Ai = wh: ð1Þ

The mineralized area or orebody area A can be considered as thesum of the areas of all the mineralized rectangles, thus:

A = CaAi = Cawh: ð2Þ

We can multiply the grade–thickness li of mineralized explorationworks by the widthw and height h in order to obtain the metal tonnageMi in each rectangle. Similarly, we multiply the orebody thickness ti ofexploration works by the width w and height h in order to get the oretonnageOi in each rectangle.We sum the localmetal tonnageMi in all therectangles in order to obtain the global metal tonnage M in the min-eralized area and, similarly, to obtain the global ore tonnage O (Fig. 1c).

The global metal tonnage M and global ore tonnage O can beexpressed, respectively as:

O = ∑Ca

i=1Oi = ∑

Ca

i=1whti = wh ∑

Ca

i=1ti = Ai ∑

Ca

i=1ti =

ACa

∑Ca

i=1ti ð3Þ

M = ∑Ca

i=1Mi = ∑

Ca

i=1whli = wh ∑

Ca

i=1li = Ai ∑

Ca

i=1li =

ACa

∑Ca

i=1li: ð4Þ

2.3. Fractal algorithms

2.3.1. Number–size modelThe variables, including grade–thickness and orebody thickness in

exploration works, can be assumed to conform to the followingnumber–size model proposed by Mandelbrot (1983):

Nð≥rÞ = Cr−D ð5Þ

where r represents the orebody thickness ti or grade–thickness li inexploration works. N(≥r) stands for the cumulative number ofexploration works in which the variable is not less than r. D is called afractal dimension. C is a capacity constant and is equal to N(≥1); itrepresents thenumberof objects, the sizesofwhicharenot smaller than1.

Eq. (5) can also be rewritten as:

lnNð≥rÞ = −D ln r + lnC: ð6Þ

In a diagram displaying the cumulative number plotted against thesize in ln–ln coordinates, i.e., ln N(≥r) versus ln r plot, Eq. (6) resultsin a straight line with slope−D. Other common distributions (such asthe normal (Gaussian), log-normal or negative exponential) yielddistinctly curved graphs of ln N(≥r) versus ln r plot.

If the plots ofN(≥r) and r in ln–ln coordinates can be fittedwith onestraight line, the distribution is called a simple fractal; in the case thatthe plots can be fitted with several straight-line segments, the model iscalled a bifractal, which means that the fractal models are different ineach segment. In a bifractal model, the size range is divided into severalsegments, with threshold Ri (i=1, 2, …, n), dimension Di and capacityconstant Ci for the ith segment (Fig. 2).

A threshold Ri is a value of r that demarcates two adjoining straight-line segments. The Ri can denote Ti representing thresholds in theorebody thickness distribution or thresholds Li representing thresholds

in the grade–thickness distribution. The distribution of natural objectsoften shows bifractal characteristics. In a bifractalmodel, there is often asharp change from one segment to the next, and the threshold can beeasily selected; when the change is slow, we can determine thethreshold by adjusting it to achieve the highest sum of the regressioncoefficients of the adjacent fitted lines.

In a fractal model, when theminimum value of the variable (rmin) ofthe dataset is equal to 1, the capacity constant C is equal to the totalnumber Ca of mineralized exploration works. When the inferior limitrmin is not equal to 1, we can use the fractal dimension D and theconstant C to calculate the total exploration works number Ca in asimple fractal model:

Ca = Cr−Dmin ð7Þ

and the total number of exploration works in a bifractal model isexpressed as:

Ca = C1r−Dmin: ð8Þ

2.3.2. Fractal model for ore tonnage estimationWe can first deduce the expression of ore volume V by assuming

that orebody thickness is a continuous variable, in a scenario in whichw and h are small enough.

For a simple fractal, ore volume V can be expressed by:

V = ∫tmaxtmin

whdNð≥tÞ

dttdt =

whCD1−D

½t1−Dmax −t1−D

min �ðD N 0;D≠1Þ ð9Þ

where tmin is the minimum orebody thickness worth mining underthe modern industrial standard and tmax is the maximum orebodythickness.

Then, ore tonnage can be obtained:

O =ρwhCD1−D

½t1−Dmax −t1−D

min �ðD N 0;D≠1Þ ð10Þ

where ρ is the average ore density.

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Considering Eqs. (2) and (7), Eq. (10) is simplified into:

O =ρADtmin

D

ð1−DÞ ½t1−Dmax −t1−D

min �: ð11Þ

Eq. (11) implies that we can estimate the global ore tonnage whenthe orebody area and fractal distribution are given. It shows thatglobal ore tonnage O increases with orebody area A.

For bifractals, ore volume becomes:

V = ∑n−1

i=1

whCi Di

1−DiðT1−Di

i + 1−T1−Dii Þ ð12Þ

where T1 is tmin and Tn means tmax, and Cn is expressed by:

Cn = Ci ∏n−1

i=2T Di−Di−1i : ð13Þ

Ore tonnage in a bifractal model is then expressed by:

O = ρwh ∑n−1

i=1

Ci Di

1−DiðT 1−Di

i+1 −T1−Dii Þ: ð14Þ

We can further re-write Eq. (14) as:

O =ρwhC1D1

1−D1ðT1−D1

2 −T1−D11 Þ + ∑

n−1

i=2

pwhC1D1 ∏n−1

i=2TDi−Di−1i

1−DiðT1−Di

i + 1−T1−Dii Þ:

ð15ÞConsidering Eq. (8), Eq. (15) becomes:

O = ρATD11

D1ðT1−D12 −T1−D1

1 Þð1−D1Þ

+ ∑n−1

i=2

Di ∏n−1

i=2TDi−Di−1i

ð1−DiÞðT1−Di

i + 1−T1−Dii Þ

26664

37775:

ð16Þ

Eq. (16) still denotes that global ore tonnage is proportional toorebody area.

2.3.3. Fractal model for metal tonnage estimationIf the distributionof grade–thickness in explorationworks in the VLP

plane follows the number–sizemodel,we can deduce the fractalmodelsfor the metal tonnage estimation. The fractal models of metal tonnagefor simple fractal andbifractalmodels can be expressed, respectively, as:

M =ρADlmin

D

ð1−DÞ ½l1−Dmax −l1−D

min � ð17Þ

M = ρALD11

D1ðL1−D12 −L1−D1

1 Þð1−D1Þ

+ ∑n−1

i=2

Di ∏n−1

i=2LDi−Di−1i

ð1−DiÞðL1−Di

i + 1−L1−Dii Þ

26664

37775

ð18Þ

where L1 is the minimum grade–thickness lmin and Ln is the maximumgrade–thickness lmax of the dataset.

Eqs. (16) and (18) represent that the reserve is associated with theorebody area but is not related to the number of exploration works.

2.3.4. Grade estimationAfter calculating metal tonnage and ore tonnage of deposit, we can

estimate ore grade G in the whole deposit following the Eq. (19):

G =MO: ð19Þ

2.4. Limitations of the FMRE

2.4.1. Exploration grid densityExplorationworksare oftenunevenly distributed in space. In order to

reacha reservewithhigher controls andclassifications, exploration gridsin shallow parts are often made denser than those in deep parts, wheregrids are designed for obtaining the reserve with greater quantity.

However, both sparser and denser grids can represent the overalldistribution of mineralization variables in an orebody area, althoughdenser grids can reflect the actual orebody geometry and fractaldistribution of mineralization variables to a greater extent and thusachieve a more accurate estimation result. Therefore, the number–size model obtained via a sparser grid should be analogous to thatbased on a denser grid; therefore, we can estimate the global reservebased on exploration works distributed in the overall orebody area,with the precondition that the geological mineralization character-istics in the area are similar.

2.4.2. Ore densityIn most deposits, ore density proportionally increases with mineral

grade. If the ore industrial type is the same, ore density varies in a smallrange. Therefore, the FMRE uses the average density of ores to calculatethe reserve, as is the samewith the GBM. Therefore, in the application ofthe FMRE, it is necessary to estimate separately the reserve of ores ofdifferent industrial types.

2.4.3. Variable limitAs metal prices increase, or as technology enables exploitation of

an orebody with lower grades or smaller thicknesses, the rmin declines(Annels, 1991). Because the variables are often clustered in a smallrange, the small variance of rmin can induce a critical change toorebody area and affect the calculation result significantly.

Many fractal models of geological objects show that the fractaldimension often increases from the first segment to the ith segmentin a bifractal model (Roberts et al., 1998; Monecke et al., 2001, 2005;Deng et al., 2009; Wan et al., 2010). Therefore, Dn is often largeenough such that the item (1−Dn) is smaller than zero. For 1−Dnb0,rmax1−Dnb1 (rmax≥1), and the small variance of rmax≥1 produces littleeffect on rmax

1−Dn value, further on the estimation result.

2.5. Steps in the FMRE

Three steps are required in the FMRE. The first step is delimitingorebody area. Two approaches are proposed to delimit orebody area. Inthe first one, we delimit the outlines of orebodies by directly connectingthemarginalmineralized explorationworks. The secondone is the sameas that used in the GBM, in which orebody outlines are extendedoutwards for a certain distance from the marginal exploration workswith greater mineralization intensity, and the inside waste boundariesare also obtained in a similar way; then, orebody area is obtained as thedifference between the area confined by the orebody outlines and thewaste area. Selecting between the two approaches is decided bymineralization continuity as judged by the geologists.

In the next step, we calculate the fractal parameters of mineral-ization variables in the overall orebody area. Finally, according to theFMRE, we estimate the global reserve within the whole explorationarea. A numerical example involving calculations using Eqs. (16) and(18) is presented in the following case study.

3. Case study

3.1. Regional geology and deposit geology

3.1.1. Regional geologyThe Dayingezhuang ore deposit is located in the Jiaodong Peninsula,

China (Fig. 3). The JiaodongPeninsula is famous for its large gold reserve

Fig. 3. Geological sketch map of the northwestern part of the Jiaodong gold province in China.

6 Q. Wang et al. / Ore Geology Reviews 37 (2010) 2–14

(N900 t; ChinaNational Gold Bureau, unpublished data). Severalworld-class (N100 t) gold districts have been discovered in this province (Fanet al., 2003). The Jiaodong Peninsula is situated along the southeasternmargin of theNorth China craton. It is boundedby theN- toNE-trendingTan-Lu fault zone to the west and by the Su-Lu ultrahigh-pressuremetamorphic belt to the south. Supracrustal rocks in the northwesternpart of the Jiaodong Peninsula comprise both metamorphosed Precam-brian sequences and Mesozoic intrusions (Wang et al., 1998; Zhou andLu, 2000). The Precambrian sequence is composed of basement rocks ofthe Late Archaean Jiaodong Group, which consists of mafic to felsicvolcanic and sedimentary rocks metamorphosed to amphibolite togranulite facies. Plutonic rocks, which intruded into the Precambrianbasement in the northwestern part of the Jiaodong Peninsula, have beentraditionally divided into two suites, the Linglong and the Guojialing.The Linglong suite consists ofmedium-grainedmetaluminous to slightlyperaluminous biotite granite, and the Guojialing suite is composed ofporphyritic hornblende-biotite granodiorite. The ages of emplacementof these granitoid suites are 160–156 Ma and 130–126 Ma (SHRIMP U–Pb zircon data; Wang et al., 1998; Qiu et al., 2002), respectively.

The gold province consists of three ore-bearing fault zones, i.e., theSanshandao zone, the Jiaojia zone, and the Zhaoping zone, from west toeast in the northwestern part (Fig. 3). More than half of the gold reservesare concentrated in the Zhaoping gold belt. Gold deposits in thenorthwestern part of the Jiaodong Peninsula are divided into two types:the quartz vein style and the disseminated-veinlet style (Fig. 3). Golddeposits consisting of disseminated veinlets dominate the province (Denget al., 2005, 2006, 2008b; Yang et al., 2006, 2007). Direct Rb–Sr dating ofpyrite from the major Linglong vein-style gold deposit in the Jiaodongdistrict shows that gold mineralization occurred at about 123–122Ma(Yang and Zhou, 2001). Zhang et al. (2003) dated sericite at 121.3±0.2 Ma by 40Ar–39Ar methods in the disseminated-veinlet style Cang-shang gold deposit, which is located in the same fault zone as theSanshandao deposit.

3.1.2. Deposit geologyThe Dayingezhuang disseminated-veinlet ore deposit is in the middle

segment of the Zhaoping fault. Wallrocks in the Dayingezhuang depositcomprise the JiaodongGroup in thehangingwall of theZhaoping fault andLinglong granite in the footwall. The Zhaoping faultwith variant dips from20° to 51° controls the alteration and mineralization of the deposit. TheDayingezhuang fault in the central of the deposit crosscuts the Zhaopingfault and divides the deposit into southern and northern parts (Fig. 4).

In the footwall of the Zhaoping fault, a cataclastic altered zonedevelops and contains gold mineralization. The alteration related togold mineralization consists of K-feldspar alteration, silicification,

sericitization, chloritization, phyllic alteration, sericite-quartz alter-ation and carbonatization. The degree of fracturing and alterationgradually weakens outwards from the Zhaoping fracture plane. Thereare more than 100 orebodies of different sizes in the Dayingezhuangore region. Among these orebodies, orebody I-1 in the southern partand orebody II-1 in the northern part of the Dayingezhuang golddeposit are the biggest and take up most of the gold reserves.

3.2. Deposit explorations and raw data

In the Dayingezhuang gold deposit, more than 430 explorationworks (mostly drifts, some drills) at different exploration lines and atdifferent depths have been completed up to the end of 2007.Geological and mineralization characteristics are generally consistentat different depths in orebody I-1 and orebody II-1.

Since the orebody dip of the deposit is around 45°, both the VLP andHVP are applicable. In this paper, we adopt the VLP. In theVLP plane, theexploration works fall into the depths range from −13 m to −830 m.The distribution of exploration works is uneven; it generally becomessparser from top to bottom (Fig. 5).

Via the GBM, the reserves in orebody I-1 and orebody II-1 arecalculated with a cutoff of 1 g/t and a minimal grade–thickness of0.8 m g/t. The orebody area, ore tonnage and metal tonnage of orebodyI-1 are485,434 m2, 9.0 Mtand28.72 t, respectively; those inorebody II-1are 572,457 m2, 22.02 Mt and 78.63 t. About 107 t metal and 31.03 Mtore in the two orebodies have been explored, within an area of1,057,891 m2 (Table 1). In this paper, the global ore reserve in thedeposit is the sumof the reserves in both orebody I-1 and orebody II-1. Itis considered that the outlier in gold assay values in an explorationworkmay represent a high-grade zonewith limited spatial extent. The zone isspatially discontinuous and impossible to predict in productionscheduling (Parker, 1991). Therefore, the outliers are substituted bythe average ore grade of the explorationwork in the reserve calculations.

The data of exploration works come from the Dayingezhuang oredeposit. Total 417mineralized explorationworks in both orebody I-1 andorebody II-1 are utilized in this paper. As in theGBM, thegradeoutliers aresubstituted by the average ore grade during estimation grades in eachexploration work. The histograms of the natural logarithms of grade–thickness and thickness for these exploration works are illustrated inFig. 6a and c, and the ‘quantile–quantile’ plots (Q–Q plot) in Fig. 6b and d.The Q–Q plots show that the variables do not follow a log-normaldistribution, especially in the two tails, indicating skewed distribution. Itwill be shown that the skewed distributions of either grade–thickness orthickness can be described by a bifractal model. It is further shown in

Fig. 4. Geological sketch map of the −175 m level in the Dayingezhuang deposit in China.

7Q. Wang et al. / Ore Geology Reviews 37 (2010) 2–14

Figs. 7 and 8 that changes in thickness and in grade–thickness alongvertical and horizontal directions are both abrupt and irregular.

3.3. FMRE application

3.3.1. Consistence test of the fractal modelBefore calculating the reserves via the FMRE, we test the consistency

of the fractal distribution of the mineralization variables obtained withvarious grid densities. For the test, we can divide the area with unevenexploration grids into several sub-areaswith even exploration grids andcalculate the fractal models in the different sub-areas; alternatively, wecan calculate and compare the fractal models of the different areas atvarious depths, i.e., including several grid areas with different densities,as shown in Fig. 9.

The grade–thickness plots and thickness plots in ln–ln coordinatesboth show bifractal characteristics comprising four or three segments(Fig. 9). With increasing depths, the plots in the fractal models in bothFig. 9a and b become denser, further verifying that the grade–thickness

Fig. 5. Locations of exploration works and the areas of orebodies I-1 and II-1 in the VLP

andorebody thickness canbe considered continuous variableswhen thenumber of explorationworks is sufficiently high. Overlaying the grade–thickness or thickness plots at different depths together reveals thatpatterns of plot distributions are similar (Figs. 9–11).

Fractal parameters for grade–thickness and orebody thicknesses inexplorationworks above−200 m,−300 m and−400 m and−830 m,are listed in Tables 2 and3. Fromthefirst segment to the fourth (Table 2)or the third (Table 3), the fractal dimension increases. In the four fractalmodels for the orebody grade–thickness, the fractal dimension of thefirst segment varies from 0.08 to 0.10; that of the second segment, from0.35 to 0.52; that of the third segment, from 0.97 to 1.45; and that of thefourth segment, from 8.22 to 10.22. In the fractal models for orebodythickness, the fractal dimension of the first segment varies from 0.33 to0.36; that of the second segment, from 0.89 to 0.98; that of the thirdsegment, from 1.75 to 8.37.

The fractal dimensions of the corresponding segments for the fourgrade–thickness fractal models vary in a small range, and the fractaldimensions of the orebody thickness distribution show more variance

plane (vertical longitudinal project plane) in the Dayingezhuang deposit in China.

Table 1Ore tonnage and Au reserve calculated by the GBM in the Dayingezhuang deposit,China.

Orebody number Orebodyarea (m2)

Density(t/m3)

Grade(g/t)

Ore tonnage(Mt)

Metaltonnage (t)

No.I-1 485,434 2.84 3.19 9.00 28.72No.II-1 572,457 2.76 3.57 22.02 78.63Sum 1,057,891 2.78 3.46 31.03 107.35

8 Q. Wang et al. / Ore Geology Reviews 37 (2010) 2–14

than those of the grade–thickness distribution. This suggests that thethickness and grade–thickness distribution is approximately consistentin the deposit, from thewell-explored region to the less-explored region.

Fig. 7. Grade–thickness and orebody horizontal thickness curves at different levelswithin exploration lines 80 and 80.5 in Dayingezhuang ore deposit. (a) Grade–thickness curve; (b) orebody horizontal thickness curve.

3.3.2. Reserve estimationThe orebody area is calculated via the two methods. The area

obtained by directly connecting the mineralized exploration works is880,481 m2, which is smaller than that obtained from the GBM, inwhich the orebody outlines are extended outwards for a certaindistance from the marginal mineralize exploration works.

Based on the orebody area estimated via the GBM and the fractalparameters, the global ore tonnage and global metal tonnage arecalculated and listed in Tables 2 and 3. The fractal model based on thegroup of overall exploration works, i.e., those above −830 m, issuitable for the global estimation of the deposit, since this group canreflect the true mineralization distribution in the global orebody areamore comprehensively than the other three groups, in whichexploration works are only partly distributed.

Fig. 6.Histograms and Q–Q plots of the raw data utilized in the case study. (a) Histogram for gplot for thickness.

As shown in Table 2, the estimated metal tonnage obtained by thefractal model for depth down to −830 m is 107.66 t and thoseaccording to the fractal models for depths down to−200 m,−300 m

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

Fig. 8. Grade–thickness and orebody horizontal thickness curves from exploration line 58.5 to exploration line 83 within−170 m and−176 m levels in Dayingezhuang ore deposit.(a) Grade–thickness curve; (b) orebody horizontal thickness curve.

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and−400 m are 82.35 t, 110.78 t and 111.10 t with the relative errorswith respect to themetal tonnage based on the fractal model for depthdown to −830 m are 23.51%, 2.90% and 3.19%, respectively. The oretonnage obtained by the fractal models for depth down to −830 mis 30.06 Mt. The ore tonnage obtained by the fractal models for

Fig. 9. Comparisons of fractal models of grade–thickness and orebody horizontalthickness at different depths in the Dayingezhuang deposit, China. (a) Fractal model ofgrade–thickness; (b) fractal model of orebody horizontal thickness. Plot series 1, 2, 3and 4 represent ln N–ln r plots of data in exploration works for depths down to−200 m, −300 m, −400 m and −830 m, respectively.

depths down to −200 m, −300 m, −400 m are 25.46 Mt, 30.04 Mt,and 30.16 Mt (Table 3), with relative errors relative to the ore tonnageobtained by the fractal models for depth down to−830 m are 15.30%,0.06% and 0.33%, respectively. The estimated average grade of thedeposit is 3.58 g/t.

It is clear that the estimates based on the fractal models for depthsdown to −300 m, −400 m and −830 m are somewhat larger thanthe estimates derived from the fractal models for depth down to−200 m. This can be explained by the fact that data in explorationworks down to depth of −200 m have less statistical meaning andcannot represent all the mineralization features of the orebodies intheir entirety. Exploration works at depth down to−200 m are in themarginal part of the orebody area, and the mineralization intensity isweaker, thus inducing a higher fractal dimension in the second andthird segments and further resulting in reduced reserve estimates.The estimated ore tonnage is more unstable than the metal tonnage,probably because orebody thickness is more variable than grade–thickness in the disseminated-veinlet deposit.

The fractal models of the grade–thickness in orebody I-1 and inorebody II-1 are illustrated separately in Figs. 12a and 13a, and thefractal models of the orebody thickness are shown in Figs. 12b and 13brespectively. The reserve in orebody I-1 and orebody II-1 is estimatedand listed in Tables 2 and 3. The grades calculated via the FMRE of theorebody I-1 and II-1 are 2.97 g/t and 3.51 g/t, respectively.

The sum of the metal tonnage in orebodies I-1 and II-1 is 107.73 t,which is roughly equal to the global metal tonnage 107.66 t obtainedvia the FMRE, with relative error 0.06% (Table 2). The sum of theglobal ore tonnage in orebodies I-1 and II-1 is 32.17 Mt, which isroughly equal to the ore tonnage 30.06 Mt obtained via the FMRE,with a relative error of 7.01% (Table 3).

4. Discussion

4.1. Estimation result

The ore tonnage and metal tonnage in the Dayingezhuang oredeposit are calculated via the FMRE proposed in this paper, since the

Fig. 10. Fractal models of grade–thickness at different depths in the Dayingezhuang deposit, China. The different colors represent the different size segments. (a), (b), (c) and (d)represent the fractal models of data in exploration works for depths down to −200 m, −300 m, −400 m and −830 m, respectively.

Fig. 11. Fractal models of orebody horizontal thickness at different depths in the Dayingezhuang deposit, China. The different colors represent the different size segments. (a), (b),(c) and (d) represent the fractal models of data in exploration works for depths down to −200 m, −300 m, −400 m and −830 m, respectively.

10 Q. Wang et al. / Ore Geology Reviews 37 (2010) 2–14

Table 2Metal tonnage calculated by the FMRE with the orebody area obtained via the GBM in the Dayingezhuang deposit, China.

Orebodyarea (m2)

Density(t/m3)

Fractal parameters Total metaltonnage (t)

L1 D1 L2 D2 L3 D3 L4(LMax) D4 LMax

Total a 1,057,891 2.78 1 0.1 4.5 0.52 34 1.45 313 82.35b 1,057,891 2.78 1 0.1 3 0.38 28 1.06 261 10.22 319 110.78c 1,057,891 2.78 1 0.08 2.5 0.35 22 0.97 248 8.22 340 111.10d 1,057,891 2.78 1 0.08 2.5 0.35 22 1.01 261 8.22 340 107.66

No.I-1orebody

485,434 2.84 1 0.12 4 0.56 40 3.01 129 28.30

No.II-1orebody

572,457 2.76 1 0.13 10 0.66 248 5.81 340 79.43

a, b, c and d represent the fractal models of grade–thickness comprising the exploration works upon −200 m, −300 m, −400 m and −830 m, respectively.

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distributions of local orebody thickness and grade–thickness can bedescribed by bifractal models. It is documented that the roll-off effectis a common feature in fractal analyses and is usually attributed toneglecting the samples at smaller scales or with smaller sizes (Walshet al., 1991; Blenkinsop and Sanderson, 1999). However, in some caseswith more systematic samples, the roll-off effect can be avoided. Forinstance, the element distribution along the drifts with variedmineralization intensity in the Dayingezhuang deposit invariablyfollows the bifractal model. Because, in the exploration grid, all theexploration works are carefully analyzed, we can consider that thefractal analysis can diminish the roll-off effect and reflect the truedistribution, as is confirmed by the consistency of the fractal models invarious areas of the Dayingezhuang deposit.

The estimation result with the fractal parameters of the overallexploration works is accepted for the global reserve in the Day-ingezhuang deposit. The calculated results via the FMRE and theresults of the GBM block were compared and reveal that the FMRE hasapproximately equivalent results to the GBM (Fig. 14). The relativeerrors between the global reserve estimated by the FMRE and thatcalculated by the GBM are 0.29% for metal tonnage, 3.11% for oretonnage and 3.51% for grade. In addition, those of the orebody I-1 are1.45%, 5.86% and 6.90% for metal tonnage, ore tonnage and grade,respectively; those of orebody II-1 are 1.01%, 2.79% and 1.73%.

It is mentioned that the small variance of rmax produce little effecton the results of reserve estimation. We substitute the maximumgrade–thickness 319 in Table 4 by 340, 360 and 400, and it yieldsrelative errors less than 2.27% (Table 4). To determine the effect oferrors in threshold selections of the bifractal analysis, we alsoperformed a sensitivity analysis. We changed the L3 value 28 inTable 4 from 22 to 34, yet find few effects on D2 and D3; the resultranges from 106.04 t to 111.79 t, with relative errors less than 4.28%with respect to the primary result. Since the grade–thickness and thethickness were assumed to be continuous variables, meaninginnumerable exploration works should be completed to ensure asufficiently high exploration density, the reserves estimated via theFMRE represent ore or metal quantities that can be mined in theorebody without consideration of ore loss rate.

Table 3Ore tonnage calculated by the FMRE with the orebody area obtained via the GBM in the Da

Orebodyarea (m2)

Density(t/m3)

Fractal parameters

T1 D1

Total a 1,057,891 2.78 1 0.33b 1,057,891 2.78 1 0.34c 1,057,891 2.78 1 0.36d 1,057,891 2.78 1 0.33

No.I-1 orebody 485,434 2.84 1 0.41No.II-1 orebody 572,457 2.76 1 0.3

a, b, c and d represent the fractal models of orebody thickness comprising the exploration

4.2. Comparison between FMRE and traditional methods

The geometric and geostatistical methods, which are based onlinear mathematics, assume that mineralization variables vary evenlythroughout an orebody. However, the local outliers or anomalies areinvariably developed, and produce great effects on the estimationprocess and result of the traditional methods. The traditional methodsneed to replace the outliers via different ways even if they are the truevalues, or to apply complex data processing to diminish the influenceof the outliers (Arik, 1990). The FMRE utilizes the nonlinear fractalmodel to fit the skewed distributions of mineralization variablesinstead of changing the outliers or utilizing complex data processing.Hence, the FMRE can deal with the natural variability and skewnessmore effectively, which is the essential difference between the FMREand the traditional methods. In the case study of the FMRE, thedistributions of orebody thickness and grade–thickness are skewed,and they are directly fitted by a bifractal model.

Preliminary data processing is the same in the FMRE and GBM. Inthe twomethods, we need to deal with the outliers that should not beinvolved in reserve estimation; then project exploration works onto avertical or horizontal plane and delimit orebody area according toorebody thickness and grade in each exploration work. Yet, the latersteps of data processing in the FMRE are much easier than thecorresponding later steps in the GBM.

The obvious disadvantage of the FMRE is that it can only estimateglobal reserve in a deposit; in contrast, the geometric and geostatis-tical methods can estimate both local reserve and global reserve.

4.3. FMRE application scope

Just like the GBM, the FMRE application is based on the VLP ofexploration works. Since the exploration process in deposits ofdifferent genetic types, e.g., porphyry deposit, quartz vein deposit,SEDEX deposit, VMS deposit, etc., are almost similar, a system ofexploration works is required. The FMRE proposed in this paper dealswith the distribution of orebody thickness and grade–thickness inexploration works and have no relationship with the genetic type of

yingezhuang deposit, China.

Total oretonnage (Mt)

T2 D2 T3 D3 TMax

4 0.89 16 1.75 67 25.464 0.94 53 8.37 70 30.044 0.94 50 4.66 93 30.164 0.98 50 4.66 93 30.064 0.98 14 2.83 48 9.535.5 0.8 53 4.82 93 22.64

works upon −200 m, −300 m, −400 m and −830 m, respectively.

Fig. 13. Fractal models of the grade–thickness and horizontal orebody thickness fordepth down to −830 m in orebody II-1 of the Dayingezhuang deposit in China. Thedifferent colors represent the different size segments. (a) Fractal model of grade–thickness; (b) fractal model of orebody horizontal thickness.

Fig. 14. Comparison of the results obtained by the FMRE and those derived from theGBM. (a) Metal tonnage; (b) ore tonnage; (c) ore grade. Total means the mineralizationarea containing both orebody I-1 and orebody II-1.

Fig. 12. Fractal models of the grade–thickness and horizontal orebody thicknessfor depth down to −830 m in orebody I-1 of the Dayingezhuang deposit in China.The different colors represent the different size segments. (a) Fractal model of grade–thickness; (b) fractal model of orebody horizontal thickness.

12 Q. Wang et al. / Ore Geology Reviews 37 (2010) 2–14

deposits. Therefore, the FMRE can bewidely applied to various genetictypes of deposits if orebody mineralization variables in explorationworks conform to a fractal model.

Compared to conventional estimation methods, the FMRE is moresuitable for deposits explored with irregular drillhole densities or withhighly skewed distributions of mineralization variables or lacking ofproper geologic modeling.

5. Conclusions

The traditional geometric and geostatistic methods based on linearmathematics are difficult for dealing with the skewed distribution ofmineralization variables, which is the common characteristic of miner-alization, and thus require complex data processing. Fractal modelsbelonging to nonlinear mathematics are effective tools for describing thenatural variability and skewed distribution of geological objects. Based onthe fractal number–size model, we established a new reserve estimationtool, called fractal models for reserve estimation (abbreviated as FMRE),in this paper. In the FMRE, we first calculate orebody area according to

Table 4Sensitivity analysis of the metal tonnage estimated by the FMRE with fractal model upon −300 m in the Dayingezhuang deposit, China.

Level(m)

Orebodyarea (m2)

Density(t/m3)

Fractal parameters Total metaltonnage (t)

Relativeerror

L1 D1 L2 D2 L3 D3 L4 D4 LMax

300 1,057,891 2.78 1 0.1 3 0.36 22 1.03 261 10.22 319 106.04 4.28%300 1,057,891 2.78 1 0.1 3 0.37 24 1.04 261 10.22 319 107.35 3.10%300 1,057,891 2.78 1 0.1 3 0.37 26 1.05 261 10.22 319 110.14 0.58%300 1,057,891 2.78 1 0.1 3 0.38 28 1.06 261 10.22 319 110.78300 1,057,891 2.78 1 0.1 3 0.39 30 1.06 261 10.22 319 111.76 0.89%300 1,057,891 2.78 1 0.1 3 0.4 32 1.07 261 10.22 319 111.79 0.92%300 1,057,891 2.78 1 0.1 3 0.41 34 1.08 261 10.22 319 111.60 0.74%300 1,057,891 2.78 1 0.1 3 0.38 28 1.06 261 8.31 340 111.83 0.95%300 1,057,891 2.78 1 0.1 3 0.38 28 1.06 261 6.88 360 112.42 1.48%300 1,057,891 2.78 1 0.1 3 0.38 28 1.06 261 5.04 400 113.29 2.27%

The italic and bold numbers represent the adjusted values.

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mineralized exploration works projected onto a VLP plane, and thenestimate the fractal parameters of the distribution of mineralizationvariables. After obtaining orebody area and the fractal parameters oforebody thickness, global ore tonnage can be derived by the FMRE;similarly, global metal tonnage can be estimated based on the orebodyarea and the fractal parameters for grade–thickness. In the FMRE,mineralization variables are considered to be continuous, which meansthat distributions of data obtained from a sparse grid are analogous tothose derived fromadense grid and can represent the overall distributionof the variables; this indicates that the orebody area is filled withexploration works with infinitely small intervals, and thus the estimatedresults denote the actual ore tonnage and metal which can be mined outwithout consideration of the ore loss rate in the deposit. The Day-ingezhuang gold deposit in China was used as a case study; this revealedthat the estimated reserves via the FMRE and those derived fromtraditional geometric block methods are very similar, with relative error3.11% in ore tonnage and 0.29% in metal tonnage. The case study furtherdemonstrates that the fractal distribution obtained via a sparse grid issimilar to that derived from a dense grid.

For proper application of the FMRE, we should be aware that: (1)denser exploration grids can give more accurate fractal model andorebody area, and thus better estimation results; (2) that reserves ofores of totally different industrial types or densities should beestimated separately; and (3) that the selection of the lower limit ofthe variables based on cutoff grade and minimum mining thicknesscan influence orebody area and estimated reserves to a greater extent.

The FMRE can be applied not only to gold deposits but also to othertypes of mineral deposits. In addition, the FMRE are much easier thantraditional estimationmethods, and aremore effective in dealing withthe skewed distribution of mineralization variables. However, thisnew method fails to calculate local reserve as traditional estimationmethods allow.

Acknowledgements

We appreciate the valuable and careful comments from the twoanonymous reviewers and Ore Geology Reviews Editor-in-Chief Nigel J.Cook. This research is jointly supported by National Basic ResearchProgram (Nos. 2009CB421000 and 2003CB214600), National NaturalScience Foundation of China (Grant Nos. 40572063, 4067206440872194, and 40872068), the 111 project (No. B07011) and theProgram for Changjiang Scholars and Innovative Research Team inUniversity (PCSIRT).

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