Copyright 2003, SPE/IADC Drilling Conference This paper was prepared for presentation at the SPE/IADC Drilling Conference held in Amsterdam, The Netherlands, 1921 February 2003. This paper was selected for presentation by an SPE/IADC Program Committee following review of information contained in an abstract submitted by the author(s). Contents of the paper, as presented, have not been reviewed by the Society of Petroleum Engineers or the International Association of Drilling Contractors and are subject to correction by the author(s). The material, as presented, does not necessarily reflect any position of the SPE, IADC, their officers, or members. Electronic reproduction, distribution, or storage of any part of this paper for commercial purposes without the written consent of the Society of Petroleum Engineers or the International Association of Drilling Contractors is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgment of where and by whom the paper was presented. Write Librarian, SPE, P.O. Box 833836, Richardson, TX 75083-3836, U.S.A., fax 01-972-952-9435.

Abstract

With the emergence of rotary steerable systems, the technical issue concerning the bit design for a specific directional application has reappeared. Today, a bit must be specifically designed for use with a particular directional system : rotary bottom hole assembly (BHA), steerable mud motor or rotary steerable system (RSS). The reason is that the bit must have the ability to respond properly and rapidly to a side force applied by the steering system in order to initiate a deviation as requested. To do so, the bit must have a predetermined steerability compatible with the directional system in order to provide the optimum dog leg potential. The new generation of directional drilling systems differentiates pointing the bit" from "pushing the bit". As a consequence, the bit directional response is a key factor that operators and directional drillers need to know to make the good adaptation between the bit and the BHA. However at the moment, there is no standard method that can propose a way to classify bits according to their steerability and walking tendency. Based on a comprehensive analysis of the directional behavior of polycrystalline diamond compact (PDC) bits (numerical simulation, pilot and field tests), a simple methodology has been developed in order to define and evaluate the steerability and the walking tendency of PDC bits. This methodology is used to classify the PDC bits defined with their IADC bit profile codes. As the PDC bit steerability is mainly a function of the bit profile, the gage cutters and the gage pad, some design recommendations are given concerning these three parts. For each IADC bit profile code, the bit steerability and the walking tendency is estimated through some formulas linking

only the heights and lengths of the cutting profile. Some guidelines are also given about the gage pad length and gage cutters characteristics in order to achieve improved steerability. This simple method based on geometrical criteria enables to estimate quickly not only the PDC bit steerability, but also the maximum dog leg potential achievable by the PDC bit, when coupled with the steering system. Introduction It is well recognized today that the directional behaviour of a drilling system is a complex coupling between the bit directional responsiveness and the mechanical behaviour of the directional system, but one also bears in mind a possible rock formation effect1 (anisotropy). This paper is focused on the directional behaviour of PDC bits, characterized by their walk tendency and steerability. After having noticed in a previous paper2 that the bit steerability and walking tendency were mainly a function of the bit profile, gage cutters and gage pad characteristics, we propose in this paper to use a simple methodology to classify PDC bits defined with their IADC bit profile codes (figure 1). This methology is based on a recent study on the directional behavior of PDC bits based on theoretical models, numerical simulation, as well as pilot and field trials3.

Background Definition The directional behaviour of a PDC bit is generally characterized by its walk tendency and steerability. To quantify the walk tendency, Ho4 introduced for PDC bits the walk angle, which is the angle measured in a plane perpendicular to the bit axis, between the direction of the side force applied to the bit and the direction of the lateral displacement of the bit2. The walk angle quantifies the intrinsic azimuthal behaviour of the PDC bit. The bit steerability (BS) corresponds to the ability of the bit, submitted to lateral and axial forces, to initiate a lateral deviation. The bit steerability can be defined as the ratio of the lateral drillability over the axial drillability :

axlat

DDBS = (1)

SPE/IADC 79795

Classification of PDC bits According to their Steerability S. Menand, SPE, and H. Sellami, SPE, Ecole des Mines de Paris/Armines, C. Simon, DrillScan

2 SPE/IADC 79795

The lateral drillability (Dlat) is defined as the lateral displacement per bit revolution over the side force. The axial drillability (Dax) is the axial penetration per bit revolution over the weight on bit (WOB). The BS (equivalent to the bit anisotropic index5,4) is generally in the range of 0.001 to 0.1 for most PDC bits, depending on the cutting profile, gage cutters and gage pad characteristics, as evaluated in the present paper. High steerability for a bit implies a strong propensity for lateral deviation, enabling to obtain a maximum dog leg potential. Bit design Besides the fact that the PDC bit should evidently have some stabilization and durability requirements, it should have the ability to respond properly and rapidly to a side force applied by the steering system in order to initiate a deviation as requested. To do so, the PDC bit must have a steerability compatible with the directional system. The design of the bit should consider the three parts (figure 2) which interact with the rock formation : the cutting structure (mainly cutting profile and back rake angle), the active gage (gage cutters or trimmers) and passive gage (conventionally called gage pad). Cutting profile

A recent study2 has shown that the steerability of a PDC cutting structure depends greatly on the bit profile : the flatter the profile is, the more steerable the bit profile is. The authors found also that the walk angle of a PDC cutting structure can be approximated by a simple equation linking the inner cone depth C, the outer structure height G and the PDC back rake angle (c)2 :

)()tan()(2arctan GC

GCfc

CS ++= (2)

Eq. 2 is only appropriate for bits having identical back rake angles along the bit profile, and indicates that the walk tendency (right, neutral and left) of the cutting structure is defined through the inner cone depth C and the outer structure height G:

- G > C : left walking tendency, - G C : neutral walking tendency, - G < C : right walking tendency.

Note that it is easy to extend Eq. 2 to take into account a gradual increase of the back rake from the inner cone to the gage. OHare et al 6 conducted a study to evaluate the directional responsiveness of various bit profiles that were classified according to the IADC codes. The authors give some guidelines on IADC bit profile codes to measure the bits tendency to achieve particular build and walk rates. For example, deep coned PDC bits (IADC bit profile types 1, 4 and 7) tend to be directionally stable, and single cone bits (IADC bit profile types 6 and 9) tend to be directionally

isotropic (responsive in any direction). Even though these general rules are helpful for selecting PDC bits, it is known that the directional behaviour of a PDC bit is not the only parameter in the well deviation process2. Moreover, the bit steerability depends not only on the bit profile, but also on the gage characterictics (gage cutters and gage pad). Barton5 made some numerical simulations on different bit profiles to calculate the anisotropic index (equivalent to the bit steerability BS). The author has observed that the steerability increases as the length of the taper decreases, and that flat bit profiles produce the highest anisotropic index. Back rake angle Some contradictions between authors exist about the rule of the back rake angle in the bit steerability. Some authors6 recommend to choose an aggressive back rake (low angle) to reduce steerability and a less aggressive back rake (high angle) to improve the bits steerability. Barton5 observed from numerical simulations that the anisotropic index of a particular bit profile decreased by 8% when the back rake angle was increased from 20 to 30, meaning that the bit is less steerable with a high back rake angle. In fact, when the back rake angle changes, the lateral and axial drillabilities do not change respectively in the same proportion. The axial drillability being more affected by a back rake angle change, the bit steerability (ratio of the lateral drillability over the axial drillability) thus increases when the back rake angle increases. Gage cutters

The general perception is that the steerability of the bit increases with the number of gage cutters6 (or trimmers). This statement is in contradiction with a recent study2 that showed that, for three different bit profiles, the highest steerability was observed for the bit having fewer gage cutters. However, despite improved steerability, diminishing the number of gage cutters may cause the bit to lose gage, due to higher load per gage cutter7. The active gage formed by the PDCs truncated at bit diameter, constitutes the transition zone between the cutting structure and the passive gage. On most designs, these trimmers are preflatted and provide two contact surfaces with the rock, a cutting face and a frictional surface (figure 3). New designs5 involve full round cylinder PDC cutters (instead of conventional preflatted gage cutters) reducing the friction surface with the borehole and thus increasing the lateral cutting ability of the design. The side cutting ability of an active gage depends greatly on the total friction surface involved during the cutting process. The higher the friction surface is (great number of preflatted trimmers or wear on gage cutters), the less steerable the bit is. The reason is that to penetrate the side of the borehole, an amount of side force corresponding to the friction forces generated on the friction surface has to be consumed before a lateral penetration of the bit occurs.

SPE/IADC 79795 3

Gage pad

Very often the critical and controversial factor for the PDC bit selection is gage length5,7,8,9. What gage length should be selected ? The general perception is that long gage is not favourable to deviate a well, and that a conventional short and aggressive gage (less than 1.5-in.) is often preferred to provide side cutting ability. These short gage designs may lead to poor borehole quality, with irregularities, ledges and hole spiraling10,11. On the contrary, long gage designs prevent hole spiraling and bit whirl with improved stability and produce a perfect hole quality, but to the detriment of side cutting ability. Nevertheless, recent studies have shown that good steerability could be obtained even with a long gage bit with an appropriate placement of stabilizer, bend angle and motor distance12 in the case of steerable BHA and also point-the-bit systems used generally with an extended gage bit. The design of a gage pad for a rotary steerable system depends on the principle of the drilling system : point-the-bit or push-the-bit. Generally, a short gage is used as standard for the push-the-bit systems, although a long gage is preferred for the point-the-bit systems, requiring less side cutting action13. Synthesis It emerges from these last considerations that there does not yet exist any quantified guidelines for optimization and selection of bit steerability. Even though the directional behaviour of a drilling system is a complex coupling between the directional system and the bit, it is well accepted that the steerability of the bit plays a great role in the deviation process. That is why we have developed a simple methodology to estimate the steerability of the bit, through its profile and gages. We propose also in the following paragraph a classification of PDC bit profiles according to their steerability, and some guidelines for gage design. Methodology Geometrical description The cutting structure of a PDC bit (see figure 2) defines the cutting profile which can be divided into two parts according to the IADC Classification14: the inner cone (height C) and the outer structure (height G). Lets define LIC and LOG the lengths of the inner cone and the outer structure profile, cC and cG the average back rake angle respectively. LC and LG are the lengths that locate the position of the nose from the bit axis (figure 2). The active gage (figure 3) is defined by its length LAG, its trimmers number NAG, its trimmer back rake angle cAG and its rock-friction surface SfAG equal to the sum of the individual preflatted trimmer friction surfaces. The passive gage (figure 4) can have many design features. The main passive gage characteristics are the length LPG, the circumferential coverage CovPG (depending on the number of blades and the blade spiral angle), the surface roughness

(smooth gage pads such as the low-friction gage pads or aggressive gage pads depending on the carbide or diamond insert type for protection), and the gage diameter PG (fullgage or undergage); all these parameters define a friction surface SfPG with the borehole. This friction surface SfPG is characterized in the laboratory by the use a directional drilling bench2 enabling to drill a sample of rock with a full scale bit under both WOB and lateral force. This surface can be related to the gage pad length :

PGfPG LkS = (3) where k is a function of circumferential coverage CovPG, the number of blades, the gage diameter PG and the surface roughness. Warning The steerability of a PDC bit depends mainly on the side force applied to the bit2 and on the rock strength3. Due to the frictional forces generated on the gages (active and passive gage) the side force has to overcome a given friction force before making the bit penetrate laterally. A complete 3D rock-bit interaction model has been developed and validated in numerous laboratory tests using full scale PDC bits. We present below a simplified model using only geometrical criteria. For that, lets assume the following points :

- during the drilling process, the lateral penetration of the bit per revolution is very small,

- based on a comparison with the 3D rock-bit interaction model, the forces generated on the chamfered edge of the PDC cutters are negligible.

As the steerability is assumed to be constant when the side force changes, the simple method developed can only be used for relative comparison of PDC bit designs. An accurate calculation of the bit steerability through the 3D rock-bit model is recommended if a trajectory prediction is to be made. However, when considering only the cutting structure of the bit (bit profile), the steerability of the bit can be considered as independent of the side force and the rock strength. The formula presented below for the walking tendency evaluation can be considered as a good approximation. Formula As discussed before, the walk angle of a PDC cutting structure can be approximated by a simple equation linking the inner cone depth C, the outer structure height G and the PDC back rake angle c :

)()tan()(2arctan GC

GCfc

CS ++=

(4)

4 SPE/IADC 79795

If the back rake angle is not identical along the bit profile (for example in the case of a gradual increase of the back rake from the inner cone to the gage), the walk angle can be approximated by :

))tan(.)tan()..(2arctan

.GcCcGC

GfcGCfcC

cGcCCS

+++= (5)

where cC and cG are the average back rake angle respectively in the inner cone and the outer structure respectively. From Eq. 5, one notices that a less aggressive inner cone (cC high) and an aggressive outer structure (cG low) lead to an increase of the right tendency or to the decrease of the left tendency of the bit cutting structure. Concerning the steerability of the bit profile (cutting structure), if the back rake angles are the same (c = cC = cG), it is possible to derive the simple following equation :

222

12

2

12

2

2

)()(4)(tan

)1616)(tan(

81

CGGC

DG

DC

DBSfc

fc

CS

+++

++++=

(6)

where D is the bit diameter (see appendix for details). From Eq. 6, one notices that the BSCS increases when :

- the back rake angle c increases, - the friction angle f increases, - the inner cone depth C decreases, - the outer structure height G decreases, - the diameter D increases (with fixed C and G).

The rise of the bit profile steerability with an increase of back rake angle c or friction angle f seems not intuitive. As indicated previously, this increase is due to the fact that the lateral and axial drillabilities do not change in the same proportion when the back rake changes. These above formulas concern only the cutting structure of the PDC bit (profile and back rake angles). To consider the whole bit, it is necessary to include the active and passive gages. The walk angle of a complete PDC bit having identical back rake along the bit profile (c = cC = cG =cAG) can be estimated through the expression (see appendix) :

fPGfAGPGAGfc

fPGfAGfPGAG

SSLLGC

SSLLGC++++++

++=

)4()tan(

)(tan4)(2arctan (7)

Note that Eq. 6 and 7 can easily be generalized for the case cC cG cAG. We can easily observe from Eq. 7 that if the friction forces are predominant, the walk angle is very close to the angle of friction between PDC and rock, f. Some directional tests

carried out in the laboratory of Ecole des Mines de Paris have shown that the walk angle of PDC bits with a passive gage length greater than 1-in. remains in the range of 5 to 17 (left tendency) depending on the rock type and mud properties. The steerability of the complete bit is proportional to a geometrical function as follows :

)tan1()(4)()(4)(tan

)1616)(tan(

81

22

2222

2

12

2

12

2

2

fPGAGfc

fc

SfSfCGGC

DGDCDBS

++++++

++++

(8)

One can clearly notice from this formula that the longer the gage pad is (greater friction surface SfGP), the less steerable the bit is. Any friction generated at the gage pad level reduces the steerability of the bit. If the gage pad is undergage, the BS increases since the friction surface SfGP with the borehole is dramatically reduced. The assumptions made allow to formulate simple equations to predict both bit steerability and walk tendency. These assumptions are validated using a 3D rock-bit interaction model. As indicated earlier, the steerability of the bit profile can be considered as independent from the side force and the rock strength. Table 1 presents the measured steerability and the calculated values with the 3D rock-bit model and the geometrical method of three bit profiles (the characteristics of these three profiles are presented in table 2). The steerability calculated with the geometrical method is given by Eq. A.6 and constitutes a good approximation of the steerability of the bit profile. PDC bit classification IADC bit profile codes

The IADC bit profile codes enables to classify profiles according to the relation between the inner cone depth (C), the outer structure (G) and the bit diameter (D). From these 3 parameters, we have seen that we could estimate the steerability and the walking tendency of the bit cutting structure (Eq. 4 and 6).

Bit profile steerability Table 3 presents a classification of PDC bit profiles according to their steerability. The steerability of the bit profiles has been calculated using the following parameters :

- f = 12, - c = cC = cG = 20

For each code, we have varied the inner cone depth (C) and the outer structure height (G) in the range of variation defined in the IADC Classification14. We suggest to attribute a steerability code to each profile code, ranging from 1 (least

SPE/IADC 79795 5

steerable) to 9 (most steerable), depending on BSCS extrema calculated with Eq. 6. One observes that the IADC profile code 9 corresponding to sidetrack designs has the highest steerability, although long taper designs (IADC profile code 1, 2 and 3) have the lowest steerability. Generally, the longer the taper is (or the deeper the cone), the less steerable the profile is. According to Eq. 6, we notice also that an increase of friction angle between PDC and rock leads to a slight increase of bit steerability. As evoked earlier, an increase of back rake angle along the entire bit profile produces an increase of bit profile steerability. Note that the proposed classification of PDC bit profiles according to their steerability does not change if the distribution of back rake angle along the profile is similar on the 9 profiles considered. Bit profile walking tendency

Table 3 presents the walk tendency of the 9 IADC profile codes. As it is well known, the parabolic profiles (IADC profile code 1, 2 and 3) exhibit a left tendency, although the profiles with deep cone (IADC profile codes 7 and 8) have a right tendency. However, as shown in a previous paper2, the walking tendency of most PDC bits is greatly infuenced by the active and passive gages, and exhibit a left tendency with a walk angle close to the angle of friction between bit metal and the drilled rock. Gages guidelines

The steerability of the bit decreases as the gage length increases. This statement observed in the field and laboratory can be quantified using Eq. 8. Figure 5 shows the steerability of a bit having only a cutting structure (IADC profile code 5) and an active gage. The values of bit steerability are calculated for a given side force and rock strength and should be used only for comparison purpose. Whatever the type of trimmer (rounded or preflatted), the bit steerability decreases as the number of trimmers increases. However, there is a major difference between rounded and preflatted trimmers since the steerability of a bit equipped with 10 trimmers is 6 times higher when the trimmers are rounded than when they are preflatted. As evoked earlier, the high friction surface SfAG generated on preflatted trimmers reduces the bit steerability. The rounded trimmers may be considered as an extension of the taper of the bit profile. On most conventional PDC bit designs, a high back rake angle cAG is generally chosen (typically greater than 30) in order to reduce the risk of cutters failure when the bit is subjected to lateral vibrations. Even though the cutting forces on each trimmer are generally negligible relative to friction forces (during the drilling process, the lateral depth of cut per bit revolution is very small), an increase of the back rake angle cAG reduces slightly the steerability of the bit. Figure 6 shows the bit steerability as a function of gage length for a PDC bit of IADC code profile 5. The values of bit

steerability are calculated for a given side force and rock strength and should be used only for comparison purpose. One notices that the bit steerability highly decreases with an increase of the gage pad length. It is also interesting to note that the bit steerability seems not to depend on the number of trimmers used when the gage pad length is greater than 1-in. This same remark is also true for the bit profile. The reason is that the steerability of a PDC bit is almost unaffected by the profile type, but depends mainly on the gage characteristics as shown in the figure 7 which presents the splitting up of a 500 daN lateral force applied to the Bit C. We observe that an insignificant side force is consumed by the bit profile (cutting structure) compared to the 84% consumed by the gage pad (fullgage).

In order to reduce the friction surface SfGP the diameter of the gage pad may be reduced (undergage gage pad). This leads to a significant increase of the steerability of the bit. However, this kind of design can be detrimental to the bit stability and the borehole quality.

Conclusion

Through a comprehensive analysis of the directional behaviour of PDC bits, a geometrical method has been developed in order to evaluate the steerability and the walking tendency of a given PDC bit. We have presented a classification of PDC bit profiles according to their steerability for the 9 profiles codes defined in the IADC classification, and some guidelines about gage designs allowing to achieve fastly and simply, a desired steerability of the bit. The lateral drillability as well as the steerability of the bit are not constant with the side force. The simple formulas presented in this paper do not take into account this effect of the side force on the steerability of the bit. Thus, these formula can be used to compare different PDC bit designs under the same operating conditions in order to select the most appropriate for a given directional application. The following conclusions can be drawn :

The steerability of a PDC bit is greatly influenced by

any friction surface generated on the gages. The longer the gage is, the less steerable the bit is. The friction surface of the trimmers (preflatted or rounded) influences significantly the steerability of the bit.

An increase of the back rake angle of the PDC cutters along the bit profile increases the steerability of the bit profile.

At last, it should be emphasized that an accurate calculation of bit steerability through the 3D rock-bit model is necessary to predict correctly the directional behaviour of the drilling system. Moreover, one should not forget that the bit tilt angle has also an additional effect on the deviation process, which has not yet completely understood.

6 SPE/IADC 79795

Nomenclature BS - Bit Steerability, dimensionless BSCS - Bit Steerability of cutting structure, dimensionless C - Inner cone depth, L, mm CovPG - Circumferential coverage of passive gage,

dimensionless D - Bit diameter, L, mm Dax - Axial drillability, L/m/rev, (mm/Mg)/rev Dlat - Lateral drillability, L/m/rev, (mm/Mg)/rev G - Outer structure height, L, mm LAG - Active gage length, L, mm LC - Inner cone length, L, mm LG - Outer structure length, L, mm LIC - Inner cone profile length, L, mm LOG - Outer structure profile length, L, mm LPG - Passive gage length, L, mm NAG - Trimmers number, dimensionless SfAG - Total friction surface of the active gage, L2, mm2 SfPG - Total friction surface of the passive gage, L2, mm2 UCS - Uniaxial Compressive Strength, m/Lt2, MPa - Bit walk angle, rad, deg CS - Walk angle of the cutting structure, rad, deg c - Back rake angle, rad, deg cAG - Average back rake angle of the trimmers, rad, deg cC - Average back rake angle in the inner cone, rad, deg cG - Average back rake angle in outer structure, rad, deg f - Friction angle between PDC and rock, rad, deg References

1. Simon, C.: Modelisation of PDC bit directional

behaviour in anisotropic formation, (in French) PhD thesis of Ecole des Mines de Paris, 1996

2. Menand S., Sellami H., Simon C., Besson A. and Da Silva N. : How the Bit Profile and Gages Affect the Well Trajectory, paper SPE 74459 presented at the 2002 IADC/SPE Drilling Conference, Dallas, Texas, Febr.26-28.

3. Menand, S.: Analysis and validation of a PDC drilling bit directional behaviour model, (in French) PhD thesis (confidential) of Ecole des Mines de Paris, 2001

4. Ho, H.S.: Method and System of Trajectory Prediction and Control using PDC Bits, United State Patent 5,456,141, Oct. 10, 1995.

5. Barton S. : Development of Stable PDC Bits for Specific Use on Rotary Steerable Systems, paper SPE 62779 presented at the 2000 IADC/SPE Asia Pacific Drilling Technology, Kuala Lumpur, Malaysia, Sept. 11-13.

6. OHare J. and Aigbekaen O.A. : Design Index: A Systematic Method of PDC Drill-Bit Selection, paper SPE 59112 presented at the 2000 IADC/SPE Drilling Conference, New Orleans, Louisiana, Febr. 23-25.

7. Mensa-Wilmot G., Krepp T. and Hill R. : Specialized PDC Bit Improves Efficiency of Rotary Steering Drilling Tools in Demanding Directional Drilling Programs, paper SPE 62781, presented at the 2000 IADC/SPE Asia

Pacific Drilling Technology, Kuala Lumpur, Malaysia, Sept. 11-13.

8. Poku Ernest K. : Experiences and Learning Points From the Use of Steerable Rotary Drilling Systems on Northern North Sea Platforms, paper SPE 56937 presented at the 1999 Offshore Europe Conference, Aberdeen, Scotland, Sept. 7-9.

9. Gaynor T.M., Chen D.C-K., Stuart D. and Comeaux B. : Tortuosity versus Micro-Tortuosity Why Little Things Mean a Lot, paper SPE 67818 presented at the 2001 SPE/IADC Drilling Conference, Amsterdam, The Netherlands, 27 feb.-1 march.

10. McNair G.A., Cassidy S.D. and Zheng Z. : Technology Applied to Extend the Drilling Reach of a Platform Workover Rig, paper SPE 64620 presented at the 2000 International Oil and Gas Conference and Exhibition, Beijing, China, Nov. 7-10.

11. Norris, J.A., Dykstra, N.W., Beuershausen, C.C., Fincher R.W.and Ohanian, M.P.: Development and successful application of unique steerable PDC bits, paper SPE 39308 presented at the 1998 IADC/SPE Drilling Conference, Dallas, March 3-6.

12. Gaynor T., Chen D. C-K., Maranuk C. and Pruitt J. : An improved Steerable System: Working Principles, Modeling and Testing, paper SPE 63248 presented at the 2000 SPE Annual Technical Conference and Exhibition, Dallas, Texas, Oct. 1-4.

13. Von Flatern R. : Extending the reach, Offshore Engineer, pp. 30-35, february 2002.

14. Winters, W.J., Doiron H.H.: The 1987 IADC Fixed Cutter Bit Classification System, paper SPE 16142 presented at the 1987 SPE/IADC Drilling Conference, New Orleans, March 15-18.

Appendix

Constant back rake angle (c = cC = cG =cAG) The bit steerability of the cutting structure can be approximated by the following expression :

222

22

)()(4)(tan

)()tan(2

CGGC

LL

LL

BSfc

OG

G

IC

Cfc

CS

+++

++=

(A.1)

With the following simplifying assumption :

)()1616(1622

12

2

12

22

LL

LLDGDCD

OG

G

IC

C ++++

(A.2)

Eq. A.1 becomes :

SPE/IADC 79795 7

222

12

2

12

2

2

)()(4)(tan

)1616)(tan(

81

CGGC

DG

DC

DBSfc

fc

CS

+++

++++=

(A.3)

Without taking into account the cutting forces on the active and passive gages and the side force effect, the overall bit steerability is proportional to a geometrical function as follows:

)tan1()(4)()(4)(tan

)1616)(tan(

81

22

2222

2

12

2

12

2

2

fPGAGfc

fc

SfSfCGGC

DGDCDBS

++++++

++++

(A.4)

The global walk angle of the bit can be approximated by the following expression :

fPGfAGPGAGfc

fPGfAGfPGAG

SSLLGC

SSLLGC++++++

++=

)4()tan(

)(tan4)(2arctan (A.5)

We can easily check that if the friction forces are predominant, the walk angle is very close to f, the angle of friction between PDC and rock. Note that Eq. A.3, A.4 and A.5 can easily be generalized for the case cC cG cAG.

8 SPE/IADC 79795

Table 3 : Classification of PDC bit profile according to their steerability

Table 2 : Characteristic dimensions of Bit A, Bit B and Bit C

Dimensions Bit A Bit B Bit C D (mm) 216 216 216 C (mm) 24.1 20.3 36.7 G (mm) 19.1 41.6 19.4 LPC (mm) 77.6 66.5 84.5 LPG (mm) 37 62.1 32.3 LC (mm) 73.5 63 75 LG (mm) 34.5 45 33 cC (deg.) 21 20 22 cG (deg.) 27 27 28 NAG 7 15 4 LAG (mm) 18.9 30.5 14.7 SfAG (mm2) 176 365 105 IADC profile code 9 5 8

Table 1 : Comparison of experimental drilling results using full scale PDC bits with calculated values using both geometrical method and 3D rock-bit interaction model

Bit profile steerability Laboratory tests 3D rock-bit model Geometric method

Bit A 9.6 9.2 9.6 Bit B 3.2 3.5 3.4

Bit profile

Bit C 6.7 5.4 6.5

IADC bit profile code Walking tendency1 Steerability code Steerability1

1 Left 2 0.4 < BSCS < 1.7 2 Left 3 0.4 < BSCS < 1.7 3 Left 1 0.4 < BSCS < 1.5 4 Right / Neutral 5 1.1 < BSCS < 2.9 5 Right / Neutral / Left 7 1.5 < BSCS < 7.1 6 Left 6 1.2 < BSCS < 7.1 7 Right 4 0.9 < BSCS < 2.9 8 Right 8 2.0 < BSCS < 7.1 9 Right / Neutral / Left 9 4.5 < BSCS

1 : Calculated with f=12 and c = cC = cG = 20

SPE/IADC 79795 9

Figure 1 : IADC bit profile codes

10 SPE/IADC 79795

Figure 2 : Geometrical characterization of a PDC bit

SPE/IADC 79795 11

Figure 3 : Description of the active gage (trimmers)

Figure 4 : Description of the passive gage (gage pad)

12 SPE/IADC 79795

Figure 5 : Steerability of cutting structure + active gage as a function of trimmers number

Figure 6 : Bit Steerability (complete bit) as a function of gage pad length

SPE/IADC 79795 13

Figure 7 : Theoretical splitting up of 500 daN lateral force applied to the Bit C (gage pad 2-in.) in the Vosges sandstone (UCS = 40 MPa)