1

ANALYSİS AND OPTİMİZATİON OF DRİLLİNG OPERATİNG PARAMETERS İN CORİNG AND DRİLLİNG OPERATİONS V.C. Kelessidis

1, H. Ergin

2

1Associate Professor, Mineral Resources Engineering Department, Technical University of Crete,

Greece, kelesidi@mred.tuc.gr 2Mining Engineering Department, Istanbul Technical University, Turkey, hergin@itu.edu.tr

Paper to be presented at the 22nd

World Mining Congress and Expo 2011, Istanbul,

Turkey, Sept. 11-16

ABSTRACT

Drilling and coring operations are very expensive endeavors and efforts are continuous by researchers and engineers to achieve the optimum penetration rate under given conditions, which is defined as the maximum possible rate which will produce safely the most economical borehole or core. To enhance penetration rate and bit life, optimization of bit design and of drilling and coring operations must be realized. In this article, we examine the factors that affect drilling operations related to the bit, the drilling operating parameters and the formation parameters. In rotary drilling, pull-down force, rotational speed, applied torque, bit diameter, circulation method and its efficiency are the important drilling operating parameters. Bit wear rate is a very important issue and it must be investigated deeply and it cannot be determined from rock properties only. We use dimensional analysis to demonstrate the significance of these important parameters, grouped together in dimensionless numbers which will then allow for optimum use of limited field and laboratory data to produce best results. The analysis allows for reduction of total effort in designing laboratory and field experiments, reducing total load and thus cost, permitting variation of the important dimensional groups rather than individual drilling operating parameters, hence a more efficient design of experiments can be realized. We further discuss field results taken with impregnated diamond bits with respect to wear patterns in order to establish the relationship between the wear type and the drilling operational parameters.

1 INTRODUCTION

Drilling may be the most expensive process during the exploration campaign and the ability to predict penetration rates under given subsurface conditions with the various drilling rigs is very essential for the safe design and the accurate cost prediction before the start of the drilling campaign.

Prior experimental and field evidence has already demonstrated that penetration rate, both in drilling and coring operations, depends on two groups of parameters, formation properties and drilling parameters. In the first group, these involve local stresses, rock compaction, mineralogical content, and fluid pore pressure. The most significant drilling parameters are applied weight on bit and torque, rotary speed and hydraulic parameters (flow rate, density and rheology of drilling fluid). Bit condition is equally important because there is blunting while drilling progresses which depends on the formation being drilled. Drilling tools have evolved significantly over the years but modelling of the drilling process and the interaction of bit – formation rock has yet to be modelled adequately, which would allow for better penetration rate prediction.

Rock drillability prediction is a key issue in design and execution of any drilling jobs, e.g. for hydrocarbon, mining or geothermal drilling activities. Many parameters affect rock drillability and industry and research is in constant search for better models as well as experimental data.

Improvements in bit design and on operating procedures could offer, especially in deep wells, improvements of up to 25% in rate of penetration (R) which could translate to significant savings. While drill bit design is at the hands of capable drill bit companies, operating procedures can only be applied by the operator. Availability of models or simulators could allow for better operating procedures, especially in a long drilling campaign in the same field. For e.g. when using PDC bits, one is never sure whether to increase weight on bit (W) or revolutions per minute (RPM). In a field study [Lagnville at al., 2008] it was shown through testing that doubling the bit RPM in 6,000-psi rock while keeping weight on bit (W) constant resulted in 70% increase in the rate of penetration, however, doubling W, with RPM constant, resulted in 300% increase in R.

Various models have been proposed to relate R to drilling and formation parameters. For example, Teale (1965) introduced a rock-bit interaction model with adjustable parameters. In the model Teale introduced the concept of specific energy, the energy required by the rig to drill a unit volume of rock. The model has been used by many researchers and practitioners in the years that followed (for e.g. Bilgin, 1982) and is given as,

( ) ( )( )

b

b

A

W

e

C

AWfDNR

−

=/)(8

(1)

where R is the penetration rate, N is the rpm of

the string, W is weight on bit, bA is the bit

diameter, f is the coefficient of friction between

drill string and formation, converting applied weight

W to torque, D is bit diameter, C is unconfined

compressive strength, and e is the efficiency of

transmitting the rock destruction power of the drilling rig to the rock.

The equation for specific energy, SE , the energy

required by the dirlling rig to cut a unit volume of rock, is finally given by (Teale, 1965)

DR

fWNSE = (2)

in consistent units. Equation (2) has been

derived after keeping only the work done by the torque and ignoring the axial work as being very small (Kelessidis, 2010).

Warren (1981) proposed the following equation for the penetration rate, initially for perfect cleaning in front of the bit,

1

2

32−

+=ND

c

WN

DaUR

b (3)

where cba ,, are dimensionless constants and

U is defined as a dimensionless relative drilling

strength of rock, which can be determined via drilling tests and was assigned arbitrary values, ranging around 1.0. All constants were later correlated to rock properties such as unconfined compressive strength, porosity, rock shear strength.

One could relate drilling strength to the rock C but

the type of relationship is really unknown and should be dependent on the rock characteristics.

Much later, Warren (1987), extended the above model to incorporate cleaning efficiency of the drilling fluid by modifying the above equation to give

1

2

32−

++=jm

f

b F

sD

ND

c

WN

DaUR

µγ (4)

where s is a dimensionless constant, fγ is

dimensionless liquid specific gravity, µ is the

plastic viscosity of the drilling fluid and jmF is the

modified impact force. Drilling industry and in particular oil-well industry

has implemented approaches which tended to normalize data rather than using full blown models or dimensional analysis. The d-exponent, defined as

3

=

D

W

N

R

d epx

610

12log

60log

(5)

is based on the definition of rock drillability by

Bingham (1965). The variables in equation (4) are

in the following units, hftR /][= , rpmN ][= ,

fklbW ][= , inD ][= . One can see that in Equation

(5), the nominator is dimensionless, but not the

denominator, and thus expd is dependent on the

specific expressions used for the determining variables. The industry also uses the modified d-exponent, which takes into account mud density changes

m

epxm ddρ

9exp=− (6)

where mρ is the density of the drilling fluid in

gallbm / (1 g/cm3=8.33 gallbm / ). One can thus

consider that both of these parameters could be an approach to normalization rather than the use of dimensionless groups.

Other attempts have opted to create simple models and implement them in simulators which would allow fine tuning of the many constants involved utilizing historical drilling data, such as Payzone simulator (Cooper et al., 1995) and the simulator proposed by Hareland et al. (1994).

The Payzone drilling simulator was developed for teaching and research purposes and it is fairly flexible and simple to use. The main feature is the prediction of drilling rate, using Eqn. (7),

))()()()()(( GtlNaggKffR = (7)

where

( )

−−=

tlDC

WG

curv

*4.0

12exp1

5.2 (8)

where the following modifiable by the user

constants are used: K is a constant; agg is a

formation and bit characteristic constant ranging between 20% and 100% and normally it is given the value of 35%; curv is a formation – bit interaction

constant and it is usually given the value of 1.5; ff

is a constant, ranging between 50% and 100% and defines the capability of the system to adequately

clean the bit front by the cuttings and tl is a factor

characterizing the length of the teeth (short, medium, long). Prior work (Kelessidis and

Dalamarinis, 2009; Kelessidis, 2011) have shown that Payzone simulator could provide fair predictions for new drilling campaigns provided that adequate historical drilling data were made available in order to fine tune the simulator and to determine the values of the several constants used.

The quest though for finding a good model for rock-bit interaction continues, with Kelessidis (2010) assessing that we are still far away from a good prediction model despite several advances from the earlier crude models described above, like for example the newer models proposed by Detournay and Defourny (1992), by Stavropoulou (2006) and by Exadaktylos et al. (2008).

In this paper we try to assess the many parameters that affect drilling performance and use dimensional analysis to create dimensionless groups which govern the phenomenon and we propose techniques for maximizing the impact of such an approach.

2 USE OF DIMENSIONAL ANALYSIS

Dimensional analysis is a technique used by many engineering and scientific disciplines which enables researchers to take into account several parameters affecting a particular process or phenomenon and when detailed modeling is not available. Dimensional analysis is used most often when the process is very complex. Drilling is one such complex process and in this paper we will try to use dimensional analysis to enable us to point out the most significant parameters, using the dimensionless groups derived from the analysis.

This approach has been suggested in the past for drilling applications where some dimensionless groups have been proposed with limited success, mainly because the full spectrum of parameters has not been taken into account. For example, Warren (1981) mentions that he derived the model (eqn. 3) using dimensionless analysis, which though it gives some of the dimensionless groups. In fact there are three groups in equation (3), however, there must be something wrong with the derivation, as in the first term in the right hand side, the parameters do not form a dimensionless number because the parameter N is raised to an exponent.

As mentioned above, there are many formation and drilling parameters that affect the main parameter of interest, the penetration rate. Based on published theoretical and experimental work, the rate of penetration R (in dimensions of length over time, L/T) is dependent in a strong or a weak manner on the 17 parameters indicated in Table 1, where the dimensions are also shown, thus making a total of 18 parameters describing the process.

All 18 parameters that affect the drilling process have 3 major units, mass, length and time (MLT), thus, according to Buckingham Π-theorem

(Buckingham, 1915), there should be (18-3)=15 dimensionless groups which describe the process.

According to the process for developing the Π-groups, we choose 3-repeating variables, which among themselves do not form a dimensionless group, and these chosen are, W – D – N. Following standard procedures of dimensional analysis, the 15 groups can be constructed which are given in Table 2.

Thus, based on the above analysis the following very general equation can be proposed

=

W

NaD

d

d

D

df

WD

T

E

E

W

ED

C

S

W

CD

W

ND

DN

V

W

ND

fDN

R

c

gcs

f

ss

222

2224

,,,,,,,

,,,,,,

φ

µ

ρ

ρρ

(9) Depending on the situation at hand, several

simplifications can be made. For example, if one ignores fluid effects, for e.g. when drilling with air and having sufficient air flow rate so that the bottom hole is cleaned continuously from the generated cuttings, then this can be reduced by three groups, as we remove fluid velocity, density and viscosity, thus the reduced equation can be

=

W

NaD

d

d

D

df

WD

T

E

E

W

ED

C

S

W

CD

W

ND

fDN

R

c

gc

ss

22

2224

,,,,,

,,,,,

φ

ρ

(10).

It should be noted that if the coring process is

modeled, one has to incorporate the two diameters,

21 , DD , the inside and the outside bit diameters

respectively, instead of D , thus introducing another dimensionless group, which will be

2

116

D

D=Π (11)

while also in all above equations and

dimensionless groups, the diameter D should be

replaced by 2D . Similar simplifications can be

derived making appropriate assumptions. It should be stressed however, that eqn. (9) is the full equation when taking all parameters into account

3 RESULTS

The main idea is then, using the data collected, plot them with respect to the dimensionless parameters (e.g. Π1 vs Π11, trying to keep other Π’s constant, etc.). Thus, instead of the typical data

representation of R-W at many different values of all other parameters, like different rocks, etc., one can use the dimensionless groups space. For example, the main parameter space would be to plot Π1

versus Π6, the group of penetration rate but

normalized for the bit diameter and the rotational speed versus weight on bit normalized by the unconfined compressive strength, keeping the other groups constant. This approach then reduces considerable the number of experiments that could be done to verify specific trends and also to verify the importance of some parameters on the drilling process for particular situations. The difficult issue of course is gathering of data, which would require extra work compared to standard practice because of the inclusion of the many parameters. It is expected, however, that the value of the information gathered will pay itself considerably and thus it is an approach that is suggested, especially when embarking into new and expensive drilling and coring campaigns.

This approach has been tested for the data of Tsoutrelis (1969), which can be represented in the Π1

- Π6 space as shown in Figure 1. We do see a

specific trend of the two dimensionless groups and of course, if all other Π’s were the same, then the two curves would essentially coincide. This cannot be really determined from the standard representation in the R-W space, as it is shown for the same data of Tsoutrelis in Figure 2.

Of course data of Figure 1 are for single bit diameter and single rotational speed. Utilizing to the fuller extent the power of proposed analysis one should have different values for the influential parameters. Such a situation can be derived when utilizing for example some of the data of Ersoy (2003). If we plot in the Π1

- Π6 space the data for

sandstone, hard sandstone and limestone, we derive Figure 3. It is very interesting to note that for different rocks and different rpm, we see similar trend in the Π1

- Π6

space, which is not the case if

one uses the standard representation of R-W, as shown in Figure 4. The trend in Figure 4 is not evident because different rpm values were used for different weights on bit, thus one cannot really assess the significance of any of the parameters in the standard R-W space.

4 DISCCUSION

Other authors have also tried dimensional analysis. For example, Tu (1988) has used dimensional analysis utilizing though several sets of drilling data from oil fields which could not have incorporated rock parameters that are usually determined in the lab, like petrophysical parameters. Tu has used 11 parameters, which, taking into account the three dimensions in the parameters produces a total of 11 dimensionless groups.

5

Similarly, Kramadibrata et al. (2001) have used dimensional analysis to relate specifically the penetration rate of jack hammer to rock properties and operational parameters. They used 9 parameters to describe the process thus giving in total 6 dimensionless groups, but they did not include W nor abrasivity, while also used energy output, which though is a dependent variable and thus their analysis could not be considered as general.

Yin and Liu (2001) have also used dimensional analysis for relating drilling data during drilling and blasting in mining applications. They have designated only 5 parameters, penetration rate (R), rotational speed N, pulldown force (the weight, W), torque T and a parameter which they called ‘rock quality index’, r. Hence, Yin and Liu have lumped all formation parameters into this ‘r’ parameter which could then be determined from specific tests. Thus, they derived only two dimensional parameters and proposed a relationship of the form

=

2W

rNTf

NT

RW (12)

Finally they proposed to combine the suggested

dimensionless equation with the equation for specific energy. Specific energy, given by Eqn. (2) is not an independent variable, is rather a dependent variable and one can show that it can be related to the derived Π’s by

61ΠΠ=

f

C

SE (13).

It should be noted that torque, T, is related to the weight on bit through the friction coefficient, f, by (Kelessidis, 2010)

3

WDT

µ= (14)

for drilling applications and

3

eqWDT

µ= (15)

for coring applications with a coring bit of internal

diameter 1D and external diameter 2D , and

21

2

121

2

2

2

1

DD

DDDDDeq

+

++= (16).

What is then proposed in this paper is to take

into account all relevant parameters, listed in Table 1, and utilize all 15 dimensionless groups listed in Table 2. Collection of extensive sets of data, covering a wide range of conditions will then allow one to determine the various relationships among different Π’s and will also help identify the significance of each Π for different situations.

5 BIT WEAR

Wear of drilling bits is an inevitable phenomenon and it will occur during the course of the short life of the drilling or coring bit. The challenge for the engineer and the driller is to design and maintain such drilling operating parameters and choose the right bit for the job at hand so that wear occurs slowly and uniformly affecting in the same way all cutting edges of the bits and of the bit matrix. Bit wear has been studied in the past and continues to be studied because it is very important phenomenon and early detection is of importance to drilling industry.

Several abnormal wear patterns can occur from incorrect drilling practices and improper bit selection. For coring bits this may include concave face wear, gauge loss ID or OD, cracked waterways, face glazing, excessive diamond exposure, burnt bits from loss of fluid or excessive weight. Drillers should be aware of these potential problems and should look for early warning signals, which are usually given, so that the problems can be remedied early. Monitoring of the major drilling parameters and provision online of a drilling log will allow the driller to identify early enough possible problems. Drilling is really a ‘correlative’ effort and the more data is available, the better the correlation can be and the better will be the identification of abnormal trends, in all aspects of drilling and certainly on anything regarding bit wear.

Wear rate of bits has been found to be related to abrasivity of the rocks, which can then be correlated with the Π15 variable shown above and through the dimensionless analysis one can then relate it to the rest of the variables. Such a conclusion has been derived by Ersoy and Miller (1995) who found that abrasive wear volume loss is directly proportional to the weight and torque, a result directly evident from the value of Π15

..

Attempts to predict bit wear, particularly for oil-well drilling but not so much for coring applications have been made in the past as it is always good to know when one would need to pull the bit for replacement (Miller and Ball, 1991; Ersoy and Miller, 1995) while detection patents have been also filed in the past (for e.g., Jardine, 1988) and also very recently (Teodorescou and Hunt, 2010). Recent attempts have used the simulators described before to predict bit wear (Rashidi et al., 2008), however this requires the applicability of the drilling rate model used in such a simulator and of course good knowledge of the formation characteristics. One can then drop into a vicious circle because, provided that a good understanding of the formation to be drilled exists, then, in order to have a workable drilling rate model, one should

know about the wear characteristics of the bit, which though can be determined from the Rashidi et al. approach, only if a drill rate model is available.

Thus it becomes evident that we are still far away from the fundamentals, that is, to be able to predict wear characteristics of given bits from first principles, and thus we should strive for acquiring continuously data while drilling or coring, so that data can become available to research community in order to enhance our understanding of the rock bit interaction and be able to suggest better equations for modelling this interaction.

6 CONCLUSIONS

Prediction of drilling rate is a very important task for the drilling engineer because it allows him to choose the right drilling bit and the optimal rig operating parameters in order to successfully deliver the borehole or the proper cores, either drilling for oil and gas, or for mining exploitation and development or for geothermal resources. Penetration rate is dependent on a number of drilling and formation parameters. Several phenomenological or semi-empirical models have been suggested in the past, but drilling rate is still an elusive parameter.

An approach is proposed in this paper, utilizing dimensional analysis, used very often in other fields like fluid mechanics but not extensively in drilling applications. This approach allows one to relate the main parameters affecting the drilling or coring process. The total number of parameters affecting the drilling or coring processes has been identified as 18 (or 19 if coring process is modeled). Applying the Buckingham-Π theorem gives a total of 15 (or 16 if coring is modeled) dimensionless groups. This analysis then allows utilizing or getting appropriate data in the proper form and use them so that trends of the process can be identified, minimizing the amount of experimental or field work which comes from the reduction of the number of parameters to change in order to see any trends. There has not been much prior work using dimensional analysis in drilling and prior research has utilized fewer parameters than the ones presented here.

Some examples have been shown of how one could use this approach, using literature data. One can see that the proposed technique allows for correct trend identification which of course is rather limited with the data analyzed because the data lacked the monitoring of all parameters of interest. Work is underway for constructing a lab-rig at Technical University of Crete where the full data set can be monitored for a variety of rocks and full utilization and verification of the suggested approach can be examined. Furthermore, work is also underway with a full scale horizontal rig at Istanbul Technical University (Ergin et al., 2000)

and it is expected that the data gathered will provide further verification of this approach.

REFERENCES

Bilgin N., The cuttability of evaporates, Bull. IAEG, 25, 85-90, 1982.

Bingham, MG, 1965. A new approach to interpreting rock drillability, Technical Manual Reprint, Oil And Gas Journal

Buckingham, E. The principle of similitude. Nature 96, 396-397 (1915).

Cooper, GA, AG Cooper, G Bihn, 1995. An interactive drilling simulator for teaching and research, paper SPE 30213 presented at the Petroleum Computer Conference, Houston, TX, 11-14 June.

Detournay, E. and P. Defourny, 1992. A phenomenological model for the drilling action of drag bits, International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts, 29, 13-23

Ergin, H., Kuzu, C., Balci, C., Tunçdemir, H. and Bilgin, N., 2000. Optimum bit selection and operation for the rotary blasthole drilling using a Horizontal Drilling Rig (HDR) - A case study at KBI Murgul Copper Mine, International Journal of Mining, Reclamation and Environment, 14: 4, 295 — 304

Ersoy, A. and Waller, M. D., 1995. Wear characteristics of PDC pin and hybrid core bits in rock drilling Wear, 188, 150-165

Ersoy A., 2003. Automatic drilling control based on minimum drilling specific energy using PDC and WC bits, Mining Technology (Trans. Inst. Min. Metall. A), 112, A86-A96.

Exadaktylos G., M. Stavropoulou, G. Xiroudakis, M. de Broissia and H. Schwarz, 2008. A spatial estimation model for continuous rock mass characterization from the specific energy of a TBM, Journal Rock Mechanics and Rock Engineering, 41, 797-834.

Hareland, G. and Rampersad, P.R., 1994. Drag Bit Model Including Wear, paper SPE 26957 presented at the 1994 SPE LAPEC Conference, Buenos Aires, Argentina, 27-29 April.

Jardine, S., 1988. Method of determining drill bit wear, US Patent 4928521

Kelessidis, VC, 2009. Need for better knowledge of in-situ unconfined compressive strength of rock (UCS) to improve rock drillability prediction, 3

rd

AMIREG Conference, Athens, 7-9 Sept. Kelessidis, V.C., P. Dalamarinis, 2009. Monitoring

drilling bit parameters allows optimization of drilling rates, 9th International Multidisciplinary Scientific Geo-Conference & EXPO SGEM 2009, Albena, Bulgaria, 14-19 June.

Kelessidis VC, 2010. Prediction of rock drillability for exploration drilling for minerals and hydrocarbons

7

– how close are we ? Tech. Chron. Sci. J. TCG, I, No 1, 201-218.

Kelessidis, V. C., 2011. Rock drillability prediction with in-situ determined unconfined compressive strength of rock, The Journal of The Southern African Institute of Mining and Metallurgy, accepted for publication

Kramadibrata, S., Rai, MA, Juanda, J., Simangunsong, GM, Priagung, N., 2001. The Use of Dimensional Analysis to Analyse The Relationship Between Penetration Rate of Jack Hammer and Rock properties and Operational Characteristics, Indonesian Mining Conference and Exhibition, November 7-8, Jakarta.

Langille, P., J Hildebrand and K Massie, 2008. Aggressive Drilling Parameters, PDC-Bit Innovations Cut Run Times in Abrasive Oklahoma Granite Wash, J Petrol. Tech. March, 36-41.

Miller, D. and Ball, A., 1991. The wear of diamonds in impregnated diamond bit drilling, Wear, 141, 311-320

Rashidi, B., G.Hareland, R. Nygaard, 2008. Real-Time Drill Bit Wear Prediction by Combining Rock Energy and Drilling Strength Concepts, Paper SPE 117109 presented at the 2008 Abu Dhabi International Petroleum Exhibition and Conference held in Abu Dhabi, UAE, and 3–6 November.

Stavropoulou Μ., 2006. Modeling of small diameter rotary drilling tests on marbles, Int. J. Rock Mechanics & Mining Sci. 43, 1034-1051

Teale R. 1965. The concept of specific energy in rock drilling, Int. J. Rock Mechanics and Mining, 2, 57-73.

Teodorescu, S.G. and Hunt, T., 2010. Method of monitoring wear of rock bit cutters. US Patent Application 20100139975

Tsoutrelis C.E., 1969. Determination of the compressive strength of rock in situ or in test blocks using a diamond drill, Int. J. Rock Mech. Min. Sci. 6, 311-321.

Tu, SKW. 1988. Application of dimensional and regressional analysis in oil drill bit data, Computers and Geotechnics, 6, 49-64.

Yin, K. and Liu, H. 2001. Using Information Extracted From Drill Data to Improve Blasting Design and Fragmentation, Fragblast, 5, 157 — 179.

Warren, T.M., 1981. Drilling model for soft-formation bits, J Petrol. Technology, June, 963-970

Warren, T.M., 1987. Penetration-Rate Performance of Roller-Cone Bit, SPEDE (March 1987) 9.

Figure 1. Presentation of Tsoutrelis (1969) data in the Π1 - Π6 space. Data of drilling in granite and marble

with 36.4 mm bit at 260 rpm.

Fıgure 2. Data of Tsoutrelis (1969) represented in the R-W space, for drilling granite and marble with 36.4

mm bit at 260 rpm

Fıgure 3. Presentation of Ersoy (2003) data in the Π1 - Π6 space. Data of drilling in various rocks with 50 mm PDC bit, at different N values, with the following C values: for Sandstone C=85.2 MPa, for Hard Sandstone C=175.1 MPa and for Limestone C=59.7 MPa.

0

200

400

600

800

1000

0 0.2 0.4 0.6

Π6

Π1*1000

sandstone

hard sandstone

limestone

0

5

10

15

20

25

0 2000 4000 6000

R (

m/h

)

W (N)

granite

marble

0.0

0.2

0.4

0.6

0.8

0.0 0.5 1.0 1.5Π

6*

10

00

Π1*1000

granite

marble

9

Fıgure 4. Data of Ersoy (2003) presented in the standard R-W space, for PDC drilling in various rock types with 50 mm bit, at different N values: for sandstone N values of 150, 240, 320, 550 rpm; for Hard Sandstone N values of 240, 550, 750 and 1150 rpm, and for Limestone N values of 150, 240, 320, 750 and 1150 rpm.

Table 1. Parameters affecting drilling process.

Name Symbol Dimensions 1 Rate of

penetration R

LT

-1

2 fluid density fρ ML

-3

3 solid density sρ ML

-3

4 fluid velocity V MT-1

5 fluid viscosity µ ML-1T

-1

6 rock compressive

strength

C ML-1T

-2

7 rock tensile strength

S ML-1T

-2

8 rock porosity φ -- 9 bulk modulus

elasticity E ML

-1T

-2

10 shear modulus Es ML-1T

-2

11 bit diameter D L 12 bit rotational

speed N T

-1

13 weight on bit W MLT-2

14 torque on bit T ML2T

-2

15 friction coefficient

f --

16 cuttings size dc L 17 grain size dg L 18 rock abrasivity a ML

-1

0

10

20

30

40

50

0 5000 10000 15000

R (

m/

h)

W (N)

sandstone

hard sandstone

limestone

Table 2. Dimensionless groups.

NAME Estimation Coefficients Dim. Group

Π1 R*WaD

bN

c a=0, b= -1,

c= -1 DN

R=Π1

Π2 fρ* W

aD

bN

c a=-1, b= 4, c=2

W

NDf

24

2

ρ=Π

Π3 sρ* W

aD

bN

c a=-1, b= 4, c=2

W

NDs24

3

ρ=Π

Π4 V*WaD

bN

c

a= 0, b= -1, c= -1

DN

V=Π 4

Π5 µ*WaD

bN

c a= -1, b= 2,

c= 1 W

ND 2

5

µ=Π

Π6 C*WaD

bN

c a= -1, b=

2, c= 0 W

CD 2

6 =Π

Π7 S*WaD

bN

c a= -1, b= 2,

c= 0 W

SD2

7 =Π

Π8 φ φ=Π8

Π9 Ε* WaD

bN

c a= -1, b= 2,

c= 0 W

ED2

9 =Π

Π10 Εs* WaD

bN

c a= -1, b= 2,

c= 0 W

DEs2

10 =Π

Π11 T* WaD

bN

c a= -1, b= -1,

c= 0 WD

T=Π11

Π12 f f=Π12 Π13 dc* W

aD

bN

c a= 0, b= -1,

c= 0 D

dc=Π13

Π14 dg* WaD

bN

c a= 0, b= -

1, c= 0 D

d g=Π14

Π15 a* WaD

bN

c a= -1, b= 2,

c= 2 W

NaD 22

15 =Π