A stochastic penetration rate model for rotary drilling in surface mines

Omid Saeidi a,n, Seyed Rahman Torabi a, Mohammad Ataei a, Jamal Rostami b

a Department of Mining Engineering, Geophysics & Petroleum, Shahrood University of Technology, Shahrood, Iranb Department of Energy & Mineral Engineering, Pennsylvania State University, University Park, PA 16802-5000, USA

a r t i c l e i n f o

Article history:Received 10 June 2013Received in revised form31 January 2014Accepted 9 February 2014Available online 22 March 2014

Keywords:Rock mass penetrabilityPenetration rateRotary drillPrincipal Component Analysis (PCA)Monte Carlo simulation

a b s t r a c t

Principal Component Analysis (PCA) is used to determine the most effective parameters on the rock masspenetrability by considering their variance ratio in the first principal component. A model is developedfor the prediction of rotary drills penetration rate using non-linear multiple regression analysis.Distribution functions for the effective parameters are calculated using measured data from two casestudies. Applying the developed penetration rate model, a stochastic analysis is carried out using theMonte Carlo simulation. The proposed method provides a simple and effective assessment of thevariability of the penetration rate model and its dependent parameters. Results showed that the PCAand Monte Carlo are suitable techniques for modeling and assessing the variability of rock masspenetrability parameters. According to the developed distribution model, with 90% of confidence levelthe penetration rate values range 0.2–2.5 m/min, which shows the wide possible range of penetrationrates for rotary drilling especially in sedimentary (limestone and sandstone bearing magnetite mineral ofGolgohar mine) and Sarcheshmeh igneous porphyry rock masses.

& 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Rotary blast hole drills are extensively applied for overburdenreleasing throughout the world in surface mining. Prediction ofpenetration rate for rotary drill rigs is of great importance in therock drilling process, especially in mining and petroleum engi-neering [1–6]. The prediction of penetration rate is essential inmine scheduling. Total drilling costs could be assessed by usingprediction equations. In addition, one could use prediction equa-tion to select the drilling rig type, which is best suited for givenconditions [7]. In large surface mining operations, rotary triconebits using tungsten carbide (WC) inserts are the most populardrilling tools for deep holes with large diameter. Their drilling ratehas increased over the time due to higher powered drills andbetter control of the operational parameters, leading to increase inmining production and reduction in drilling costs.

2. Literature review

Yaşar et al. conducted experimental works on rock physico-mechanical properties in relation to its drilling penetration rate[8]. They found significant relationships between specific energyand penetration rate. Their results of the experimental work also

demonstrate the significance of applied load and torque for bothpenetration rate and specific energy in drilling.

Howarth and Adamson considered the operational conditions ofa diamond drilling machine, incorporating the effect of situation ofcutting tools and UCS values on penetration rates. The results werevaluable for the prediction of optimum cutting conditions for thediamond bits [9]. Experimental studies have been conducted on thecutting operation of a single diamond accepted to be the require-ment of establishing design parameters for manufacturing thediamond bits. Specific energies and scaled particle size distributionswere measured and torque and weight on bit measurements wereperformed. The relations between all these parameters wereanalyzed [10]. From the bit rotations, depth of specific cut couldbe calculated. For a given depth of cut, a particle size predictioncould be established. The largest particle size together with thedepth of cut will dictate the exposure and size of the diamond inthe matrix. The depth and width of cut will also give indicationabout the optimum spacing of the diamonds. Pandey et al. foundthe relationship between penetration rate values obtained frommicro-bit drilling test with compressive strength, tensile strength,shear strength and Protodyakonov index and establish logarithmicrelationships [11]. Bilgin et al. presented a mathematical model ofpredicting the penetration rate of rotary blast hole drills using thedrillability index obtained from the indentation tests [12]. Wijkused the stamp test strength index to derive a penetration ratemodel in the laboratory [13]. Kahraman determined differentbrittleness indices and also drillability and boreability from theexperimental works of other researchers. He found that each

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International Journal ofRock Mechanics & Mining Sciences

http://dx.doi.org/10.1016/j.ijrmms.2014.02.0071365-1609/& 2014 Elsevier Ltd. All rights reserved.

n Corresponding author. Tel.: þ98 9149804239.E-mail addresses: osaeidi914@gmail.com, o.saeidi@yahoo.com (O. Saeidi).

International Journal of Rock Mechanics & Mining Sciences 68 (2014) 55–65

brittleness index has its application depending on the mechanismof rock excavation that is one method of measuring brittlenessbased on impact strength that shows good correlation with thepenetration rate of percussive drills, while the other methods doesnot [14]. Akun and Karpuz developed an empirical penetration ratemodel using RQD, discontinuity frequency, pressure loss, specificdepth of cut and specific energy in a surface set diamond coredrill. They concluded that drilling specific energy as the maindrillability indicator is the most important parameter in drillingrate prediction. However, using RQD and discontinuity frequencysimultaneously which are dependent variables could lead to themulticollinearity problem in the model [15].

Penetration rate models of rocks are among those practiceswith high uncertainties based upon rock mass and machineproperties. When applying these models to rock drilling problems,most users consider only the “average” or mean properties. Rockproperties, themselves, have uncertainties and exhibit a distribu-tion about the mean, even under the ideal conditions, where thesedistributions can have a significant impact upon the designcalculations [16]. Moreover, most of the previous developedmodels use exclusively intact rock properties or machine opera-tional factors rather than comprising rock mass properties as jointcharacteristics (e.g. joint spacing, direction and aperture & fillings).Another drawback may go back to the deterministic values of theprovided penetration rate by these models rather than theirprobabilistic values. Nevertheless, most of the involved rockparameters show wide range of values at the field. Accordingly,establishing a stochastic model might provide better estimation ofthe penetration rate of rotary drills.

PCA is a classical method that provides a sequence of the bestlinear approximations to a given high dimensional observationand it has received much more attentions in many literatures[17,18]. For multi-dimensional systems, factory analysis based onPCA is the most suitable technique to reduce system dimensions tothe least effective one.

Probabilistic analysis has obtained significant attention in manyengineering practices in relation to deterministic methodologies.Deterministic models apply single values for parameters to obtainthe results. However it is well known that parameters are notreliable and are all associated with a level of uncertainty. Theprobabilistic method has also been used as an influential tool forrepresenting uncertainty in the failure model and in the materialcharacteristics.

According to the literature, the stochastic modeling by MonteCarlo method, which is the most common sampling technique, hasgained many advocates among researchers [16,19–23]. In meaning,the Monte Carlo simulation technique is trying all valid combina-tions of the values of input variables to simulate all possibleoutcomes for output variable. Benardos and Kaliampakos defined avulnerability index to identify risk-prone areas in TBM tunnelingand finally they used the Monte Carlo technique to address theuncertainty in the parameters' values [24]. Ghasemi et al. devel-oped an empirical model for predicting fly rock distance in acopper mine using regression analysis. They used the Monte Carlomethod to simulate the distribution of fly rock distance at thatmine and found that Monte Carlo simulation could predict fly rockdistance relatively well near to the real data [25]. Park et al. usedthe fuzzy set theory together with the Monte Carlo technique toevaluate the probability of failure in rock slopes. They used theMonte Carlo simulation technique and reliability index approachwith the fuzzy set theory in order to take into account the fuzzyuncertainties in the evaluation of the probability of failure. Theyfound that the application of the fuzzy set theory shows consistentanalysis results and can obtain reasonable results [26].

In this study, firstly, an attempt has been made to determinethe most effective parameters on the rock mass penetration rate in

rotary drilling by using the PCA technique. Finally, the Monte Carlosimulation was used to determine probabilistic distribution of rockmass penetration rate variables.

3. Effective parameters on the rock mass penetrability

3.1. Uniaxial compressive strength (UCS)

One of the most important properties in rock engineering andits related design parameters is the Uniaxial Compressive Strength(UCS). Rock material strength is used as an important parameter inmany rock classification systems. UCS is influenced by manycharacteristics of rocks such as constitutive minerals and theirspatial positions, weathering or alteration rate, micro cracks andinternal fractures, density and porosity [6,27]. In addition, UCS ofrocks can be considered as representative of rock strength, density,weathering and matrix type. It has been shown that rock drill-ability and penetrability will decrease when its UCS increases [4].

3.2. Rock hardness

Hardness is defined as a mineral or rock's resistance to toolpenetration. Rock hardness is the first strength that has to over-come during drilling. Intuitively, the rock hardness depends on thehardness of the constitutive minerals, cohesion forces, homo-geneity and the water content of rock [6]. Many different methodshave been used to obtain rock hardness by using different testingmachines [28,29]. It was shown that by increasing rock hardness,its drillability will decrease. Schmidt hammer rebound value,Shore hardness, Moh's scale and Vickers indentation hardnessare among the most common methods to determine rock hardness[30]. In this study, Moh's scale was measured for different rocktypes at the case studies.

3.3. Rock abrasiveness

The term “abrasiveness” describes the resistance of a rock orsoil to wear on a tool. Consequently, abrasivity is an importantrock parameter to be determined and to be described in the courseof any larger road, tunnel or mining project in order to allow thecontractor to assess economic aspects of excavation methods [31].

Abrasivity investigation can be based on a wide variety oftesting procedures and standards. Widely used geotechnical wearindices based on these systems included the Abrasive MineralContent (AMC), also referred to as “Mean Hardness”, which usesMoh's hardness, the Equivalent Quartz Content (EQC), which usesRosiwal grinding hardness and the “Vickers Hardness Number ofthe Rock” (VHNR), which is very common in Scandinavia andrefers to Vickers indentation hardness [31]. Rock abrasivity tendsto the bit wear and deformation of bit shape and causes significantdecreasing of rock drillability. Rock Abrasivity Index (RAI) is thenew geotechnical index defined to predict drill bit wear. Plinningerand Thuro found a good logarithmic relationship between the RAIand Cerchar Abrasivity Index (CAI) [32]. The index can be calcu-lated as follows:

RH¼ exp½ðMH�2:12Þ=1:05� ð1Þ

EQC ¼ ∑n

i ¼ 1RHiAi ð2Þ

RAI¼UCS � EQC ð3Þwhere RH is the Rosiwal grinding hardness (%), MH is the Moh'shardness, EQC is the equal quartz content, Ai is the mineralpercentage (%), n is the number of minerals, which contribute in

O. Saeidi et al. / International Journal of Rock Mechanics & Mining Sciences 68 (2014) 55–6556

the rock texture, RAI is the rock abrasivity index and UCS is theuniaxial compressive strength.

3.4. Joint spacing

Joint spacing associated with the reciprocal of discontinuityfrequency, is broadly used as a measure of the “quality” of a rockmass for classification schemes. In an individual set of joints, themean normal distance between the neighboring joints, is usuallyused to express the joint spacing. Joint spacing is normallymeasured along a specific direction (scan line) for all disconti-nuities and represented by the mean spacing of all discontinuitiesalong the scan line [6].

Thuro revealed that the influence of discontinuities is notvisible, if the spacing is large against the dimensions of theborehole. When the joints get closer, the drilling velocity increasesup to the double. But the connected problem is borehole instabil-ity, causing hole collapses and time consuming scaling of theestablished blast hole. In that way, the efforts of fast drilling,especially in fault zones, may be intended to useless very soon[33]. However, Hoseinie et al. using physical modeling of the effectof joint spacing on rock drilling obtained that as the joint spacinggets wider the rock drilling rate increases [34].

3.5. Joint dipping in relation to drilling direction

Obviously, rock properties and drilling rates are also highlydependent on the orientation of weakness planes related to thedirection of testing or drilling. Thuro showed that when jointsdipping are parallel to the drilling direction the lowest amount ofpenetration is observed because of the perpendicularity of shearstresses. However, maximum penetration will occur when jointsdipping are in the right angle to the drilling direction where in thiscase shear stresses are parallel to the rock anisotropy [33].

3.6. Joint aperture and fillings

Joint aperture, amount and size of filling are the specification ofjoints space that extremely affects the penetration rate of thedrilling system. Closed cracks and joints have no important effecton the decrease of penetration rate of drilling [34]. Existence ofopen joints, due to escape of flushing air and decrease of exiting ofdrilling pieces from hole, leads to the reduction of penetrationrate, and locking of the drilling system.

3.7. Bit diameter

As the rock strength increases the larger bit diameter will bemore efficiently than the smaller one [35]. Maurer [36] in his

penetration rate prediction model showed that increasing bitdiameter tends to decrease in rock penetration. Therefore, largebit diameters are used in rock masses with low drillability classes(also see [37–39]).

3.8. Weight on bit

Weight on bit or thrust is the most crucial parameter related tothe drill machine on rock mass drillability as stated by manyauthors [7,35,38]. Generally, in any drilling process there is anoptimum weight on bit that depends on drilling conditions.Excessive weight on bit results in insufficient flushing, as the fluidis unable to properly clean the blast hole bottom. In this case,unclean cuttings prevent drilling of fresh rock surfaces, reducingthe energy transfer. Lower weight on bit in addition to decrease indrilling rate causes extensive bit wear and damage to the drillstring because of the highly absorption of heating energy.

3.9. Rotational speed

Rotary speed in terms of revolution per minute, RPM, after theweight on bit maybe is the most effective parameter of rotary drillsin rock drilling. As shown by the researchers the rotational speedhas a direct relation with rock drillability where with increasing drillrotational speed the bit penetration rate will increase [7,35–38].

4. Case studies

4.1. Sarcheshmeh copper mine

Sarcheshmeh copper mine is located 160 km southwest ofKerman, about 50 km south of the city of Rafsanjan, in Kermanprovince. It is the largest open pit mine in Iran (Fig. 1a). The areabelongs to the central part of an elongated NW–SE mountain belt,which is principally composed of folded volcano sedimentaryrocks. The geology of the Sarcheshmeh porphyry deposits are verycomplicated and various rock types are found there. Mineraliza-tion in this deposit is associated with the Late Tertiary period. Theoldest host rock in this mine is Eocene andesite. Other mineralizedrock is Sarcheshmeh granodiorite stock. The waste rocks aremainly granodiorite dykes including porphyry hornblendes, por-phyry feldspar, and porphyry biotite. This study deals mainly withdrillings the mine in the current status, which is characterized byvarious lithological units including the Sarcheshmeh porphyry, thelate fine porphyry, the hornblende porphyry dike, biotite porphyrydyke and andesite.

Fig. 1. (a) A view of the Sarcheshmeh copper mine and (b) the Golgohar Sirjan iron mine.

O. Saeidi et al. / International Journal of Rock Mechanics & Mining Sciences 68 (2014) 55–65 57

4.2. Golgohar Sirjan iron mine

Golgohar Magnetite mine is located near Sirjan city in south ofIran. The mine is placed between Sanandaj–Sirjan zone andmassive salt anticline of Kheir–Abad where the anticline is formedbetween Sanandaj–Sirjan and Uremia dokhtar zones. Fig. 1b showsa view of the Golgohar Sirjan mine during drill and blast operation.Geological formations contain Paleozoic metamorphic rocks insouth, Mesozoic and Cenozoic sedimentary rocks in east of themine. Paleozoic metamorphic rocks comprise Golgohar complex,which is the oldest metamorphic setting and provided iron oredeposit in the area. Bottom division of the complex includesintermittent of gneiss, mica schist, amphibolites and quartz–schist.Our surveys of drilling rate measurements were in the limestoneand sandstone parts.

4.3. Drill rig specification

In today's mining industry, using large drilling rigs includingrotary drills with tricone bits are popular among other methods likerotary-percussion systems, especially in large surface mining becauseof the demands for high production rate and achieving projectplanning. As rotary drill targets large diameter and deep blast holes,they have considerably attracted mining contractors' attention.However, some drawbacks of using these systems are costly main-tenance and difficult transportation in harsh topography regions. InFig. 2 a view of rotary drill with tricone bit used at Sarcheshmehcopper mine is shown. Most of the tricone bits studied were API-RR321 type which manufactured by Sandvik Co. In Table 1 theconfiguration of all the rotary drill rigs studied is listed.

5. Field investigation and experimental works

Databases consisting of 38 datasets from both sites were mea-sured for rotary drill rigs to be used in the analyses. It containsinformation on rock properties as density, Moh's hardness, Pointload strength, Schmidt hammer rebound value, quartz content,uniaxial compressive strength of the rock material (UCS) andmachine properties as net penetration rate, bit diameter, weighton bit, rotational speed, and operational pressure. To obtain rockphysico-mechanical properties, five to ten representative cubic rocksamples with dimensions of 20�30�20 cm3 from each zone at themine site were transported to the laboratory. Uniaxial compressiontests were performed on truncated core samples, which had a

diameter of NX size (54 mm) and L/D �2–2.5. The stress rate wasapplied within the limits of 0.5–1.0 MPa/s. In addition, indirecttensile strength determination test was carried out according toISRM [40]. Tensile strength was determined by using the Braziliantesting (BTS) method. Disc specimens NX in diameter with athickness to diameter ratio of 1:2 were used. Non-destructive testwas conducted using N-type Schmidt hammer tests in the field. TheSchmidt hammer was held in downward position and twentyrebound values recorded from single impacts was separated by atleast a plunger diameter, and the upper ten values was averaged asfinal rebound value. Net penetration rates have been averaged bydividing hole lengths to the net drilling times in a drilling patternand some data about the drills has been attained from themanufacture catalogs of the industrial corporations. The databasestatistics measured at the both case studies are summarized inTable 2.

Another factor which could affect the rate of drilling is theincrement of wear of tricone bit by time lapse in contacting withabrasive minerals. In this regard, weight loss percentage of studiedbits obtained before and after completion of their useful life. InFig. 3 the relationship between bit weight loss as wear rate andtheir penetration rate is shown but as it is seen no consistentcorrelation was obtained in this case study. It seems that othermachine operational factors as bit diameter, weight on bit androtational speed impose more effects on the penetration raterather than that of bit wear rate.

6. Principal Component Analysis (PCA)

Factor analysis, in which principal component analysis (PCA) isincluded, consists of a family of procedures for removing the

Fig. 2. The view of rotary drill with tricone bit during drilling blast hole.

Table 1The configuration of rotary drills used at this study.

Drill model Bitdiameter(m)

Thrust(MPa)

Rotationpress(MPa)

Rotaryspeed(RPM)

Air-linepressure(MPa)

DMH-Ingersoll-Rand 0.25 0–21 0–35 0–200 0–2.76Bucyrus 45-R-135490 0.23 0–17.23 0–31 0–150 0–2.41DMH-IR-XL 0.26 0–24.13 0–34 0–200 0–2.75D9-K 0.165 0–28 0–24 0–200 0–2.75

O. Saeidi et al. / International Journal of Rock Mechanics & Mining Sciences 68 (2014) 55–6558

redundancy from a set of correlated variables and representing thevariables with a smaller set of “derived” variables or factors [41].With minimal additional effort PCA provides a roadmap for how toreduce a complex data set to a lower dimension.

If there are n characteristic variables X1, X2, …, Xn, PCA meansthe determination of m (on) synthetic variables Z1, Z2, …, Zm; thecorrelation between two of which is 0. Here, amn is the covariancevalue between two variables (features), Z1, is known as the firstprincipal component or first factor, and Z2 as the second principalcomponent or second factor [42]:

Z1 ¼ a11X1þa12X2þ⋯þa1nXn

Z2 ¼ a21X1þa22X2þ⋯þa2nXn

Zm ¼ am1X1þam2X2þ⋯þamnXn ð4Þ

This methodology was selected in order to identify factorsunderlying our set of variables and in order to screen our variablesto obtain relationships among them. As in Fig. 4a, a three variabledata set is demonstrated, which we have scattered in the coordi-nate system. The principal directions in which the data varies isshown by the PC1 axis and the second most important direction isthe PC2 axis orthogonal to it (Fig. 4b). If we transform each datacoordinate into its corresponding (PC1, PC2) values, the data is de-correlated, meaning that the covariance between the PC1 andPC2variables is zero (Fig. 4c). For a given set of data, principal

component analysis finds the axis system defined by the principaldirections of variance (i.e. the PC1–PC2 axis system in Fig. 4c). Thedirections PC1 and PC2 are called the principal components. In thisnew reference frame, note that variance is greater along axis PC1

than it is on axis PC2. PCA computes new variables which areobtained as linear combinations of the original variables. Thesevariables are found by calculating the covariance (or correlation)matrix of the data patterns. On the one hand, the more thecorrelation of original variables is, the more the variance of firstprinciple component is [43].

In this study, PCA were implemented on a set of output andfeatures (input parameters), listed in Table 2, and the ratio ofvariance of first component to total variance (variance ratio) wascalculated. This statistical multivariate method also tends to theclustering of variables into classes. The variables corresponding toone class are highly correlated with one another, so redundantvariables can be excluded. According to above-mentioned, the ratiocan determine the similarity among the output and a set of features.The analysis was carried out using SPSS ver. 21 [44]. The explainedvariance of the eleven components obtained is shown in Table 3. Infact, the PCA reduced the system from 11 components to three maincomponents (bold numbers in Table 3), which was determined,based on the correlation matrix eigenvalues. It is seen that first threecomponents account for 66.77% of the total explained variance. Sincethe contributions of the remaining components are small, they willbe disregarded in the resultant discussion.

Fig. 5 shows the scree plot of the parameters where 11components plotted in terms of their matrix eigenvalues. In PCAeigenvalue more than one, is acceptable as a principal component.When the polyline breaks (Fig. 5) here in the fourth componentthen previous numbers are considered as principal components.Then in this study, three principal components were determinedbased on the parameters affecting rock mass penetration rate inrotary drilling.

As seen in Fig. 6, the most effective rock mass and drill parameterson the penetration rate of rock mass belong to the first componentand in this component seven out of eleven parameters have highpositive variance ratio. On the first component these parametersincluding bit diameter (D), bit rotational speed (N), weight on bit(W), uniaxial compressive strength (UCS), joint dipping (JD), jointspacing (JS) together with penetration rate (Pr) have positive varianceratio (See bar graphs in Fig. 6). On the other hand, the four remained

Table 2Field and laboratory databases obtained for this study.

Case study No. ofdatasets

Statistics Penetration rate(m/min)

Bit diameter(mm)

Weight on bit(kg)

Rotational speed(rpm)

UCS(MPa)

Tensilestrength (MPa)

Golgohar mine 20 Max 2.86 251 6127 122 70.5 6Min 0.2 165 829 72 10.1 0.9Average 1.2012 235.4 4562 112.12 30.352 2.88

Sarcheshmehmine

18 Max 2.95 195 7250 119 86 8.3Min 0.45 165 1230 71 12.5 1.2Average 1.34 186.3 1536 113.4 53.2 3.2

Statistics Reboundnumber

Abrasivity(RAI)

Hardness(Moh's scale)

Joint spacing(cm)

Aperture andfillings (mm)

Dipping(deg.)

Golgohar mine 20 Max 68 35 6 115 15 90Min 35 10 3 5 0.001 10Average 51.4 19.08 4.8 44.5 4.161 38.24

Sarcheshmehmine

18 Max 78 47 9 113 17 89Min 36 13 3 7 0.05 14Average 54 24.5 5.5 48 5.6 45

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.05 0.1 0.15 0.2

Pene

tratio

n ra

te (m

/min

)

Bit weight loss

Fig. 3. Relationship between weight loss of bits as wear rate and penetration rate.

O. Saeidi et al. / International Journal of Rock Mechanics & Mining Sciences 68 (2014) 55–65 59

i.e. tensile strength (T), hardness (Hrd), rock abrasivity (RAI) and jointaperture and filling (JAF) have negative variance ratio on the firstcomponent.

Additionally, in Fig. 7 the 3-D view of the parameters andcomponents are shown and the most effective parameters can beseen on the first component dimension. It can be seen that the above-mentioned parameters are located in the same class with penetrationrate (Pr) with high correlation on component1 while the rest arelocated in the opposite side. Once the most effective parameters wereobtained, non-linear multiple regression analysis is used to determinetheir relationship with the rock mass penetration rate.

Input parameters as determined in previous section using thePCA were correlated with penetration rate listed in Table 2.

Regression analysis was carried out using the Microsoft Excel soft-ware. Several models have been produced and each model hasstatistically been tested to find the best-fit model. Analysis of

Table 3Percentage of explained variance using PCA.

Component Total % Of variance Cumulative %

1 3.818 34.706 34.7062 2.304 20.945 55.6513 1.223 11.117 66.7684 0.895 8.132 74.9005 0.752 6.834 81.7346 0.671 6.102 87.8367 0.406 3.691 91.5278 0.374 3.397 94.9249 0.293 2.661 97.585

10 0.181 1.646 99.23111 0.085 0.769 100.000

Fig. 5. The scree plot of the all components.

Fig. 6. The variance of parameters in three components; T: tensile strength,AI: abrasivity index, Pr: penetration rate, UCS: uniaxial compressive strength,N: rotational speed, D: bit diameter, JAF: joint aperture and filling, Hrd: hardness,JS: joint spacing, JD: joint dipping in relation to drilling direction, and W: weighton bit.

Fig. 7. The 3-D view of the parameters plotted on three components.

Fig. 4. (a) A 3D scattered dataset, (b) the principal component directions, and (c) the two major principal components.

O. Saeidi et al. / International Journal of Rock Mechanics & Mining Sciences 68 (2014) 55–6560

variance (ANOVA) including F-test, has been carried out to determinethe validity of the model (Table 4). An F-test is any statistical test inwhich the test statistic has an F-distribution under the null hypoth-esis. It is most often used when comparing statistical models thathave been fit to a data set, in order to identify the model that best fitsthe population from which the data were sampled. Exact F-testsmainly arise when the model has been fit to the data using leastsquares [45]. Using the null hypothesis, H0: no relationship betweenresponse and predictor variables against alternative hypothesis, Ha:relationship between response and predictor variables one wouldcompare calculated F-value, F-test with tabulated F-value, and F-tab.Since, F-test4F-tab¼3.32 and significance of F-test is smaller thanP-value¼0.05 at 95% confidence level then null hypothesis will berejected and it can be inferred that the proposed model is valid. Themodel developed for rotary drill in this study is

Pr¼ 0:57W0:6N1:28JD0:27

D1:7UCS0:47JS0:14ð5Þ

where Pr is the penetration rate in m/min,W is theweight on bit in kg,N is the rotational speed in rpm, JD is the joint dipping relative to thedrilling direction in degree, D is the bit diameter in meter, UCS isthe uniaxial compressive strength of rock in MPa and JS is the jointspacing in cm.

In addition to the statistical test, to check the validation of themodel it was compared to the real data (remainder eighteendatasets) measured in the field (other than data in Table 2) forrotary drill. The cross-correlation between estimated penetrationrate from Eq. (5) and real one from field showed reasonable results(see Fig. 8). The model for rotary drills is valid for the sedimentaryand porphyritic igneous rock masses, and for air-operated rotarydrills having tricone bit with tungsten carbide insert. According toTable 4 and Fig. 8 the presented model for predicting rock masspenetration rate demonstrated a good agreement with field data.In addition, statistical analysis showed that all the variables in the

model with confidence level of 95% are good predictors where allP-values in Table 4 are less than 5%.

7. Monte Carlo simulation of the penetration rate model forrock masses using rotary drill

Risk analysis as a probabilistic approach has achieved remark-able attention in many engineering practices with respect todeterministic approaches. Rock mass penetrability models, aredeterministic models where single values are applied for para-meters to obtain the results. However it is well known that rockmass and drill machine parameters are not consistent andare all associated with a level of uncertainty. Therefore theresults will suffer from uncertainties, which, in this study,are applied to estimate their probability ranges. Quantitativeanalysis techniques have gained a great deal of popularity withdecision makers and analysts in recent years. The most commonmethod to assess project uncertainty is the Monte Carlo stochasticsimulation [16].

At this stage, the developed model (Eq. (5)) in the previoussection is used to simulate the distribution of penetration ratemodel for rotary drills. The simulation process was conductedusing the @Risk software [46]. This software uses probabilitydistributions to describe uncertain values in the Excel worksheetsand to present results. It also uses a basic data fitting by the use ofMaximum Likelihood Estimators to estimate the distributionparameters (i.e. to determine the parameters that maximize thelikelihood of the sample data). Furthermore, the goodness of thedata fit is achieved by Chi-squared statistics, determining the sumof differences between the observed and expected sample out-comes [46]. In the stochastic simulation of rock mass penetrationrate model, the subsequent steps are taken into account:

1. The data for penetration rate dependent parameters includingrock mass and rotary drill rig parameters were measured at thefield and also in the laboratory.

2. Probability distribution functions for all the input parametersthat represent the range of these parameters in a drillingproject were defined for each parameter.

3. A penetration rate model based on Eq. (5) was constructedusing the defined distribution functions.

4. Setting Monte Carlo simulation and running it at 5000 itera-tions to obtain a statistical demonstration of the penetrationrate risk in the worksheet model. By running the program at5000 iterations, output distributions became more stablewhere the statistics presenting a distribution vary less withadditional iterations. It is essential to run enough iteration sothat more reliable outputs will be achieved.

The data set in Table 2 (excluding bit diameter due to limitednumber of drill rigs) is used to obtain the best-fitted distributionfunction of the input parameters to the simulation. In a typical

Table 4ANOVA test of the proposed penetration rate model.

Variables Coefficients Standard error P-value R2 Standard error Observations F F-significance

Intercept �0.557 1.728 0.7503 0.95 0.166463 18 43.59 4.89�10�7

D �1.701 0.510 0.0030 – – – – –

W 0.603 0.128 0.0001 – – – – –

N 1.279 0.331 0.0008 – – – – –

UCS �0.466 0.040 0.0000 – – – – –

JD 0.273 0.110 0.0212 – – – – –

JS �0.142 0.052 0.0130 – – – – –

D: bit diameter, W: weight on bit, N: rotational speed, UCS: uniaxial compressive strength, JD: joint dipping, JS: joint spacing.

R² = 0.829

0

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5 2 2.5 3 3.5

Mea

sure

d pe

netr

atio

n ra

te

(m/m

in)

Estimated Penetration rate (m/min)

Fig. 8. The cross-correlation between estimated and measured penetration rate.

O. Saeidi et al. / International Journal of Rock Mechanics & Mining Sciences 68 (2014) 55–65 61

simulation by the Monte Carlo method a random value is selectedfrom each input's distribution function according to the definedrange for the inputs. The model is calculated based on the randomvalues. This process is repeated n-times and the results (output)themselves now described a statistical distribution of the penetra-tion rate model.

Two approaches can be used to obtain distribution functions forinput parameters, first prior knowledge of the range of inputparameter based on the experience at field and second using themeasured data and fitting probability distribution function to them.

In this study, the data in Table 2 was used to obtain best-fittedprobability distribution functions. In Table 5 distribution functionsand their dependent variable amounts are seen.

In Fig. 9 the frequency histograms together with the best-fittedfunction also are shown for input parameters to be used in thesimulation. The bit diameter is a certain value with fewer changesamong other data according to the limited number of drill rigs atthe field. Thus, it will be substituted with a constant value in thesimulation process. In other words, during the process just fourinput parameters W, N, UCS, JS, JD and an output parameter PR arecalculated, iteratively. However, rotational speed which dependson the rock mass strength and operational conditions couldsignificantly vary through the drilling process.

In addition to being certain or uncertain, parameters in a RiskAnalysis model can be either “independent” or “dependent”. Anindependent parameter is entirely unaffected by any other para-meter within your model. It is extremely important to correctlyrecognize correlations between parameters, or your model mightgenerate nonsensical results. However, in Section 6 using principalcomponent analysis we determined the most effective parameters,which affect the penetration rate model without any correlationto other parameters and to avoid multicollinearity between

Table 5Probability distribution functions of input parameters used in Monte Carlosimulation.

Input variable Function

W Smallest extreme value (4809.2126, 981.5708)N Logistic (120.7765, 9.1295)UCS Exponential (77.064, Shift (7.4426))JD Normal (50.138, 18.409)

W: weight on bit, N: rotational speed, UCS: uniaxial compressive strength, JD: jointdipping, JS: joint spacing.

600050004000300020001000

9

8

7

6

5

4

3

2

1

0

Weight on Bit (Kg)

Freq

uenc

y Smallest Extreme Value

16014012010080

16

14

12

10

8

6

4

2

0

Freq

uenc

y

Logistic

360300240180120600

10

8

6

4

2

0

UCS

Freq

uenc

y

Exponential

1209060300

7

6

5

4

3

2

1

0

Joint Spacing (cm)

Freq

uenc

y

Normal

80604020

9

8

7

6

5

4

3

2

1

0

Joint Dipping (degree)

Freq

uenc

y

Normal

Rotational speed(RPM)

Fig. 9. The frequency histograms and best-fitted probability distribution functions for input parameters of the Monte Carlo simulation.

O. Saeidi et al. / International Journal of Rock Mechanics & Mining Sciences 68 (2014) 55–6562

parameters, those in the first principal component were selectedfor developing the model. Moreover, in the Monte Carlo simulationit is important to obtain correlations between probability distribu-tion functions of the input parameters using correlation matrixand apply the relationships in the simulation unless no correlationwould exist.

Once the probability distribution functions were defined forinput parameters, by carrying out 5000 iterations with the Latinhypercube sampling, probability distribution function is definedfor the output (penetration rate model). However, in the @Risksoftware in addition to the mentioned technique, Monte Carlosampling can be used for sampling from the input distributions. Incomparison with the Monte Carlo sampling technique, LatinHypercube sampling will accurately re-establish the probabilitydistributions specified by distribution functions in less iteration. Atthe final stage of simulation, running of the simulation produces5000 different possible patterns of input parameters, sampledrandomly from the defined distributions.

In Fig. 10, simulated distribution model compared with itsdistribution model based on the measured penetration rate fromthe field for the rotary drill penetration rate in the rock masses. Asit can be seen there is a good agreement between simulated modelby the Monte Carlo method and probability distribution ofmeasured penetration rates.

The probability distribution function for measured penetrationrates was found to be Inversed Gaussian with a mean of μ¼1.038and shape parameter of λ¼3.78. For simulated penetration ratesthe probability distribution function was found to be Lognormalwith a mean of μ¼1.03 and standard deviation of s¼0.72. Averagepenetration rate was simulated as 1.05 m/min with a standarddeviation of 0.72 m/min. The maximum and minimum penetrationrates were calculated as 2.86 and 0.2 m/min, respectively. It isevident from the results that the model can predict a wide range ofpenetration rate for rotary drilling in the rock masses. According tothe distribution model in Fig. 10 with 90% confidence level mostvalues are located between two delimiters 0.2 and 2.5 m/minwhich shows the wide possible range of penetration rates forrotary drilling especially in sedimentary (limestone and sandstonebearing magnetite mineral) and Sarcheshmeh igneous porphyryrock masses.

8. Sensitivity analysis

To identify which distribution is the most significant amonginput distributions in determining output values (penetration rate)a sensitivity analysis can be done in the @Risk environment. Thesensitivity analysis performed on the output variables, and their

associated inputs, uses either a change in output statistical analy-sis, multivariate stepwise regression analysis, or a Spearman rankcorrelation analysis. In the regression analysis, the coefficientscalculated for each input variable measure the sensitivity of theoutput to that of particular input distribution. The sensitivityanalysis using rank correlations is based on the Spearman rankcorrelation coefficient calculations. With this analysis, the rankcorrelation coefficient is calculated between the selected outputvariable and the samples for each of the input distributions. Thehigher the correlation between the input and the output, the moresignificant the input is in determining the output's value.

In fact using the mentioned techniques one can rank the inputvariables based on their importance in predicting output variable.In Table 6 input parameters for the penetration rate prediction modelare ranked using both stepwise regression and correlation coefficientanalyses. The results of both the methods show agreement indetermining importance of the parameters as UCS, weight on bit,rotational speed, joint dipping in relation to drilling direction andjoint spacing.

Sensitivity analysis also can be shown in spider graphs. Thesegraphs are created using results of the change in output statisticsensitivity analysis. The spider graph shows how the output statisticvalue changes as the sampled input value changes. The steeper theline, the impact of input variables will be greater on the outputvariables.

Fig. 11 shows spider graph for the input parameters of penetrationrate model in rotary drilling. Among the inputs, UCS displays thehighest impact on the penetration rate where by decreasing 10% itcauses 35% increase in output in relation to other inputs.

In accordance with previous studies in this field [7,30,35,47]it is obvious that among the rock properties UCS and among drillmachine parameters, weight on bit and rotational speed are themost effective factors in rotary drilling in rock masses.

X <= 2.50094.8%

X <= 0.2000.7%

0

0.2

0.4

0.6

0.8

1

1.2

-1 0 1 2 3 4 5 6

Penetration rate (m/min)

Pro

babi

lity

MeasuredSimulation

Fig. 10. The comparison between probability distribution of simulated and mea-sured penetration rates for rotary drills in rock masses.

Table 6Ranking of input parameters using Spearman rank correlation and stepwiseregression.

Ranking Parameters Correlation coefficient(Spearman rank)

Regression coefficient(stepwise regression)

1 UCS �0.625 (1) �0.592 (1)2 Weight on

bit (W)0.325 (2) 0.31 (2)

3 Rotationalspeed (N)

0.297 (3) 0.308 (3)

4 Joint dipping(JD)

0.192 (4) 0.197 (4)

5 Joint spacing(JS)

�0.135 (5) �0.157 (5)

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%

Input percentile

Pene

trat

ion

rate

(m/m

in)

W N UCS JD JS

Fig. 11. The spider graph of the input parameters for penetration rate model inrotary drilling.

O. Saeidi et al. / International Journal of Rock Mechanics & Mining Sciences 68 (2014) 55–65 63

9. Conclusions

Rock mass penetrability is one of the most important topics indrilling project for civil, petroleum and mining engineering.Many parameters affect rock mass penetrability including rockmass properties, machine and operational parameters. At presentin productive surface mining, rotary drilling plays importantrole in planning and cost estimation. Prediction penetration rateof rotary drills is very crucial in selecting bit type, miningplanning, total drilling costs and sometime controlling rockproperties.

In this study, a set of effective parameters from intact and rockmass properties to drill machine parameters was studied forpossible presentation in a penetration rate model. Field data wasobtained from two case studies, Sarcheshmeh copper mine mostlyporphyritic rocks and Golgohar Sirjan mine mostly sedimentaryrocks bearing magnetite mineral.

A dimension reduction method, Principal Component Analysis(PCA) was applied to realize effective parameters on the firstprincipal component in rotary drilling. It was observed that,among eleven parameters only six parameters including, bitdiameter, bit rotational speed, weight on bit, rock uniaxial com-pressive strength, joint spacing and joint dipping in relation todrilling direction showed positive and high variance ratioalong with penetration rate on the first principal component.Once the parameters are chosen, non-linear multiple regressionanalysis has been used to establish relationship between thepenetration rate and selected parameters. Good correlation wasobtained between measured data from field and predicted ratesusing the model.

Due to uncertainties in both rock mass properties as well asmachine parameters in predicting penetration rate, a stochasticsimulation was performed using the Monte Carlo method. How-ever, the bit diameter was substituted with a deterministic valuebecause of limited number of drill machines which shows fewerchanges naturally. Thus, it was excluded from probabilistic calcu-lations. Probability distribution functions were defined for inputparameters in the proposed equation. Running the program,iteratively, resulted in a probability distribution model for output(penetration rate) parameter. The model presented wide range ofuncertainty for penetration rate of rotary drills.

Results showed that in the distribution model of the penetra-tion rate with 90% of confidence level most values are locatedbetween two delimiters 0.2 and 2.5 m/min which shows thewide possible range of penetration rates for rotary drilling espe-cially in sedimentary (limestone and sandstone bearing magnetitemineral) and Sarcheshmeh igneous porphyry rock masses.

In addition, sensitivity analysis was carried out to rank theinput parameters of the model based on their significance inpredicting the penetration rate. It was observed that rock uniaxialcompressive strength, weight on bit and bit rotational speedacquired high ranks among the parameters.

It should be noticed that the presented model in this study isjust applicable for the mines with same conditions as our casestudies, and more in depth investigations are needed to present acomprehensive model.

Acknowledgements

The authors wish to thank Sarcheshmeh copper mine staff,especially, R&D department for their kind cooperation and supportduring this research. Also, authors would proudly like to thankthree unknown reviewers for their useful comments to improvethe content of this paper.

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