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60 Journal of Canadian Petroleum Technology

Cutting Efficiency of a Single PDC Cutter on Hard Rock

G. Hareland

University of Calgary

W. Yan new Mexico Institute of Mining and Technology

r. nYGaard* University of Calgary

J.l. WISe Sandia national laboratories

*currently with Missouri University of Science and Technology

Peer reviewed PaPer (“review and Publication Process” can be found on our website)

IntroductionPDC bits have gained popularity in drilling for petroleum due

to its long bit life together with its ability to maintain a high rate of penetration (ROP). The shearing action induced by fixed cut-ters has shown to be more efficient for penetrating rock than the crushing effect of the teeth or inserts on the rolling cones of a roller bit(1-4). However, PDC bits have traditionally had limitations when encountering hard formations(5). Therefore, PDC bits are not yet preferred for hard formations encountered with mining, petroleum or geothermal energy. Enhanced geothermal energy, i.e., geo-thermal energy from considerable (+3 km) depths, is seen as one

AbstractPolycrystalline diamond compact (PDC) bits have gained

wide popularity in the petroleum industry for drilling soft and moderately firm formations. However, in hard formation ap-plications, the PDC bit still has limitations, even though recent developments in PDC cutter designs and materials steadily im-proves PDC bit performance. The limitations of PDC bits for drilling hard formations is an important technical obstacle that must be overcome before using the PDC bit to develop competi-tively priced electricity from enhanced geothermal systems, as well as deep continental gas fields. Enhanced geothermal energy is a very promising source for generating electrical energy and, therefore, there is an urgent need to further enhance PDC bit per-formance in hard formations.

In this paper, the cutting efficiency of the PDC bit has been analyzed based on the development of an analytical single PDC cutter force model. The cutting efficiency of a single PDC cutter is defined as the ratio of the volume removed by a cutter over the force required to remove that volume of rock. The cutting efficiency is found to be a function of the back rake angle, the depth of cut and the rock property, such as the angle of internal friction.

The highest cutting efficiency is found to occur at specific back rake angles of the cutter based on the material properties of the rock. The cutting efficiency directly relates to the internal angle of friction of the rock being cut.

The results of this analysis can be integrated to study PDC bit performance. It can also provide a guideline to the application and design of PDC bits for specific rocks.

of the promising energy sources in the U.S.(6) Deep continental gas developments also suffer from a low rate of penetration and high bit wear for wells drilled.

The objective of this paper is to develop an analytical model to study the cutting efficiency of PDC bits in hard rock formations which can improve the future design of PDC bits and, thereby, im-prove the cost efficiency of wells in hard formations.

PDC BitMost PDC bits are composed of a hard matrix body which is

milled out of a solid block of steel. The matrix body is milled out in such a shape that the bits contain blades where the actual PDC cutters are ‘glued’ or ‘braced’ on, and open areas where the cut-tings and mud flow can escape to the annulus (Figure 1). Figure 1

FIGURE 1: Example of an 81/2 inch diameter drag PDC bit(3).

June 2009, Volume 48, No. 6 61

shows a typical 8½ in diameter PDC bit from one of the leading PDC bit manufacturers(3). In Figure 1, the PDC cutters are placed on the gold coloured blades. The flow pathways for mud and cut-tings are coloured blue. Each cutter is fixed on the blade and rock is removed when the cutters are dragged in a circular path when the bit is rotated at the bottom of the hole (Figure 2). Figure 2 shows a sketch of the circular path for a single cutter while rotating the bit. In Figure 3, a sketch in the vertical plane of a single cutter is shown. The free surface is the bottom of the wellbore. The cutter penetrates the rock based on the point load on each cutter origi-nated from the applied weight on the bit (WOB). The cutter is tilted with an angle, φ, with respect to the rock. The angle, φ, is the back rake of the cutter. How efficient the PDC bit is in removing rock is dependant on several factors. Increased rock hardness reduces ROP. Increased WOB and revolutions per minute (RPM) will in-crease the ROP if cuttings are removed efficiently(4). The number of cutters, their back rake and other design features of the bit will also affect ROP(4). And lastly, the PDC cutter material and cutter material design affects ROP throughout the bit cutter life(7). For instance, cutters made up from fine (10 μm) diamond grain size provides higher ROP and less abrasion than those from coarse (70 μm) diamond grains(7). Further, rounding the cutter edges and in-creasing the sintering pressures under cutter production make them more thermally stable and gives dramatically improved bit perfor-mance when drilling in hard formations(1, 7).

Of all the different areas mentioned above, to improve PDC bit performance in hard rock applications, the question we would ad-dress here is how the orientation of the cutter can improve the PDC bit performance.

Cutting Efficiency

A convenient way to improve PDC bit performance is to re-late the cutting efficiency to the volume of rock removed by the cutter and the force spent on that removing action. A term, spe-cific volume, is introduced here to describe the cutting efficiency. It is defined as:

Specific Volume =Volume of rock removed in oone major chip

Maximum force required to remmove that volume of rock

Volume of Rock Removed By the Cutter in One Chip

The first requirement needed for analyzing the cutting effi-ciency is the volume of rock removed by the cutter in one major

cycle (thus, the major cutting chip). The cutting shape is defined by two curves (in two dimensional view, see Figure 4):

y k x x y

y k x

a a

b b

= −( ) +

=0

2

0

.............................................................................. (1)

where ya is the failure surface of the rock assumed to follow a parabolic representation and yb represents the rock surface of the wellbore. The axes in Figure 4 are oriented such that the y-axis is parallel to the cutter face and the x-axis is normal to the cutter face.

The way the rock is fragmented into cutting chips by the PDC cutter, and thereby controls the shape of the curve ya, is a two stage phase process: crack initiation and crack propagation, shown in Figure 4. The position of the crack initiation is determined pri-marily by the magnitude of stress which will be infinite on the cutter tip if we assume a linear elastic homogenous rock mate-rial and that the PDC cutter is a rigid body. Linear elasticity is an acceptable assumption for hard rocks(8, 9). Since the compressive strength of the rock is orders of magnitude higher than the tensile strength, the crack initiation is caused by tension. The direction of the initial crack at the cutter tip is ψ0 (crack initial angle). Once the crack is initiated, it will continue to propagate as a shear frac-ture until the chip is formed. The crack angle (ψ) will change since the failure stress state is changing from tensile to maximum shear failure. The coefficients in Equation (1) for ya are derived in Yan(9) and are given as:

FIGURE 2: Sketch showing the cutting path, cutting direction and side force on the cutter when bit rotates.

FIGURE 3: Sketch showing the PDC cutter and parameters acting on it.

x

y

( )x y0 0,

( )x y1 1,

ya

yb

dx

y yb a−

FIGURE 4: Sketch showing the chip dimension.


xh

yh

kh

c

k

c

c

ac

b

0

0

0

2

20 25

1

=

= −

= +( )=

tan

.sin

tan

α

ψψ

αα......................................................................... (2)

The failure curve, ya, initiation point (x0, y0) is a function of the depth of cut, hc, the back rake angle, α, and the crack initiation angle. The shape of the failure surface, ya, is made more realisti-cally by introducing a function for the constant, ka, in the parabolic Equation (1). The failure surface, ka, is a function of the depth of the cut, hc, fitting constant, cψ, and crack initiation angle, ψ0. The free surface, yb, is only a function of the back rake angle, α, and the length of the cutting chip along the x-axis.

The shaded area in Figure 4 which represents a cross sectional area of the cuttings at an arbitrary distance from the cutter is given as:

ds y y dxb a= −( )..................................................................................... (3)

The whole cross sectional area of the cutter can be obtained by integrating Equation (3) from x0 to x1.

S y y dxb ax

x

= −( )∫0

1

................................................................................... (4)

Carrying out the integration, notice that:

xe

hc1

2

01 252

1= −( ).tan

sinα

αψ

................................................................. (5)

and rearrange to get the area of the cutter shown in Figure 4 as:

S c h c hc c= +1 22

........................................................................................ (6)

where

c ca b

1 00 0 00 0833 1

2= +( ) −

. sin

sin

tanψ ψψ

α

22

2 0 0 0 0 0

214

1 1c a b b= − − + −( ) −

tansin sin

αψ ψ

................................ (7)

and

a e

b e

02

02

1 25 1

1 25

= −

=

.

.

α

α

...................................................................................... (8)

To calculate the cutting chip volume, the two dimensional x, y cutting chip area, S, has to be multiplied with the width of the cut-tings in the horizontal, z-direction, given in Figure 5. The cutting chip volume is then given, approximately, by:

V w Se=.................................................................................................. (9)

where we is the equivalent width. The actual integration to obtain the entire three-dimensional surface over the shear plane is too complex to be integrated by any analytical method and some sim-plification is needed. Since the thickness of the chip is not large (small depth of cut), the shear surface can be treated as a plane

without losing too much precision. The plane is further projected onto the rock surface to get a half ellipse with the axis of a0 and b0, as shown in Figure 5. The equivalent width, we, can then be calcu-lated as the area of the ellipse, which is given as the numerator in Equation (10) divided by the half length (see Figure 5).

wa b

ae =

π8

2

0 0

0

....................................................................................... (10)

Since we is then only related to b0 and b0 is directly related to the depth of cut by(3):

b d h hc c c022= −

.................................................................................. (11)

The equivalent width can then be calculated as:

w d h he c c c= −1 57 2. ............................................................................ (12)

The only missing parameter is to calculate the crack initiation angle, ψ0. The crack initial direction is determined by the principal stresses at the crack tip. Since it is assumed that the crack is caused by tension at the cutter tip, the initial crack initiation angle can be calculated by:

ψτ

σ σ012

2=

−

arctan xy

x y ....................................................................... (13)

Now we have the full solution to calculate the volume, V, for a cutting chip based on Equation (9).

Specific Volume

A factor, specific volume, is introduced to measure the cutting efficiency. The specific volume, V0, is defined as the volume of rock removed by the cutter divided by the resultant force on the cutter,

VVF0 =

................................................................................................. (14)

where, F is the resultant force (in pounds-force) of the horizontal and vertical forces acting on the cutter.

F P P Ph v s= + +2 2 2

.............................................................................. (15)

b0

a0

b we

FIGURE 5: Sketch showing the simplified shear plane for calculating we.


V is the volume (in cubic inches) of rock removed in one major chip and is given by Equation (9).

Equation (14) then provides a measure of the cutting efficiency in terms of specific volume. The highest efficiency occurs at the maximum specific volume.

Figure 6 and Figure 7 show the specific volume as a function of back rake angle and depth of cut (in inches), respectively. In Figure 6, the specific volume is calculated for different depths of cuts. All the curves show the similar shape with local maxima at 0 and 25 degrees. In Figure 6, the specific volume results are plotted as a function of depth of cut to identify any local maximum. As stated previously, the specific volume represents the cutting ef-ficiency and two important conclusions can be drawn from these plots. These are:

1. The specific volume reaches its local maximum at 0° and 25° back rake angle. This indicates that the best settings for back rake angle are either at 0° or at 25°. Also, the effect of back rake angle on the specific volume becomes less significant as the depth of cut decreases. However, even when the depth of cut is less than 0.02 in, there is still a noticeable effect by the back rake angle, especially at 0°. This implies that the back rake angle of a PDC cutter becomes more important at a larger depth of cut.

2. The specific volume versus depth of cut did not give any local maximum within the range of 0.01 ~ 0.08 in, except at 0.08 in. This indicates that no optimum depth of cut could be obtained. From the plot, it is clear that the larger the depth of

cut, the more efficient the cutting. However, as the depth of the cut increases, the incremental specific volume decreases. This suggests that a high rate of penetration for a PDC bit is appreciated in term of cutting efficiency only. In practice, the depth of cut cannot be too high since the blades where the cutters are mounted will come into contact and take up some of the weight of the bit and reduce the point load on the indi-vidual cutter. The limitation the blades impose on the cutter suggests that a depth of cut around 0.04 in is optimum.

Rock Properties in the Model

In the analysis above, the cutting chip is independent of any rock mechanical properties. To establish a failure criteria for the rock, the unconfined rock strength and internal friction angle has to be included. This was done by assuming the rock chip to be a linear elastic material with the shape of a wedge. There are two basic pa-rameters in the linear elastic derived force model(9): the half wedge angle, α, and the ratio of the normal stress over shear stress on the failure curve, k0. These parameters were used in the model as con-stants. According to the laboratory observations, the half wedge angle takes the value of 25°. The failure criteria k0 takes the value of 1.5 based on the stress analysis. However, these two parame-ters should be functions of rock properties, such as internal friction angle and grain size. In fact, the ratio of normal stress over shear stress on the failure curve can be approximated as:

0

4

8

12

16

20

0 10 20 30 40

Back Rake Angle

V/F

(x1,

000,

000)

hc = 0.01

hc = 0.02

hc = 0.03

hc = 0.04

hc = 0.05

hc = 0.06

hc = 0.07

hc = 0.08

FIGURE 6: Specific volume versus back rake angle. ψ0 for different depth of cut (hc) given in inches.

0

4

8

12

16

20

24

0 0.02 0.04 0.06 0.08

Depth of Cut

V/F

(x1,

000,

000)

br = 0

br = 5

br = 10

br = 15

br = 20

br = 25

br = 30

br = 35

br = 40

FIGURE 7: Specific volume versus depth of cut.

0

2

4

6

0 10 20 30 40

Back Rake Angle

V/F

(x1,

000,

000)

hc = 0.01

hc = 0.02

hc = 0.03

hc = 0.04

hc = 0.05

hc = 0.06

hc = 0.07

hc = 0.08

FIGURE 8: Specific volume versus back rake angle (α = 20˚, k0 = 2.14).

0

2

4

6

8

0 0.02 0.04 0.06 0.08

Depth of Cut

V/F

(x1,

000,

000)

br = 0

br = 5

br = 10

br = 15

br = 20

br = 25

br = 30

br = 35

br = 40

FIGURE 9: Specific volume versus depth of cut (α = 20˚, k0 = 2.14).

0

10

20

30

40

50

60

70

0 10 20 30 40

Back Rake Angle

V/F

(x1,

000,

000)

hc = 0.01

hc = 0.02

hc = 0.03

hc = 0.04

hc = 0.05

hc = 0.06

hc = 0.07

hc = 0.08

FIGURE 10: Specific volume versus back rake angle (α = 30˚, k0 = 3.73).

0

10

20

30

40

50

60

0 0.02 0.04 0.06 0.08

Depth of Cut

V/F

(x1,

000,

000)

br = 0

br = 5

br = 10

br = 15

br = 20

br = 25

br = 30

br = 35

br = 40

FIGURE 11: Specific volume versus depth of cut (α = 30˚, k0 = 3.73).


k s

s0 =

στ

................................................................................................ (16)

and the wedge angle is a function of internal friction angle also.These two parameters will change with the change of rock type

since the internal friction angle varies according to the rock type. The changes in these two parameters may affect the cutting effi-ciency. In order to exam the effect of rock property, such as in-ternal friction angle on the specific volume, two more calculations were made and the results were plotted in Figure 8 through Figure 11. Figure 8 shows the specific volume versus back rake angle for a half wedge angle of 20° and a failure criteria k0 of 2.14. The curves for different cutting depths have the same local maxima of 0° and 20° back rake. Figure 9 gives the specific volume as a function of depth of cut for the same α and k0 as in Figure 8. In Figure 9, the depth of cut has maximum value of around 0.06 in for the lower 0° to 15° back rake angles, but with a maximum value around 0.04 in for cutters with higher back rake angles. Figure 10 shows the spe-cific volume versus back rake angle for a half wedge angle of 30˚ and failure criteria of 3.73. In Figure 10, the local maximums are for 0° back rake angle and, with a weak local maximum, around 30° back rake angle. In Figure 11, the specific volume versus depth of cut is shown for the same same α and k0 as in Figure 10. Here, the specific volume increases with the depth of cut regardless of back rake angle.

These figures show some very interesting results. First, at α = 20°, there are two maxima of specific volume at back rake angles 0° and 20°, with the 20° angle being the most significant. This in-dicates that the optimum back rake angles are 0° and 20° instead of 0° and 25°, as shown in Figure 6. At α = 30°, the optimum back rake changed to 0° and 30°, though the optimum value is not sig-nificant at this value of the half wedge angle.

Second, there appears to be a local maximum value of the spe-cific volume versus depth of cut. At low back rake angles, this op-timum depth of cut is about 0.06 in, while at high back rake angles it becomes less (about 0.05 in). This means that if the wedge angle is about 40°, the optimum depth of cut should be considered.

To turn this theory into an applicable tool for the drilling engi-neer to help with bit selection, further investigation is needed to fully understand the relationship between the wedge angle and the internal friction angle.

ConclusionsThe specific volume reaches its maximum at 0° and 25° back

rake angle. This suggests that both 0° and 25° back rake angle are the best settings from the view of cutting efficiency for the hard rock analyzed.

The specific volume has no local maximum when plotted against depth of cut. However, the increasing specific volume with the increasing depth of cut indicates a higher cutting efficiency is obtained at larger depths of cut.

The analysis of the cutting efficiency of a single cutter gives two very important suggestions:

1. Cutter back rake is best for these specific rock properties at 0° and 25°. Since 0° back rake angle is more efficient than 25°, this suggests that 0° ~ 5° back rake angle is best from the standpoint of cutting efficiency.

2. A larger depth of cut yields a higher cutting efficiency. How-ever, too large a depth of cut will cause problems when the blades on the bit come in contact with the formation. A range of 0.04 ~ 0.06 in is suggested.

AcknowledgementsPart of this study was funded under D.O.E. Contract DE-AC07-

05ID14517.

NomENClaTurEcψ = fitting constantdc = diameter of the PDC cutterhc = depth of cut Pv = vertical force on a single PDC cutterPh = drag force on a single PDC cutterPs = side force on a single PDC cutterF = resultant force on a PDC cutter trajectory S = cross sectional area of the chipV = volume removed by a PDC cutter in a major chipya = cuttings failure surfaceyb = cutting rock surfaceV0 = specific volumewe = equivalent width of the cutting chipα = half wedge angleφ = back rake angleψ0 = crack initial angleσs = normal stress on the maximum shear stress trajectoryτs = shear stress on the maximum shear stress

SI Metric conversion factors

in × 2.54 E-03 = mlbs × 4.44822 E+00 = N

rEfErENCEs 1. WISE J.L., GROSSMAN, J.W., WRIGHT, E.K., GRONEWALD,

P.J., BERTAGNOLLI, K. and COOLEY, C.H., Latest Results of Pa-rameter Studies on PDC Drag Cutters for Hard Rock Drilling; GRC Transactions, Vol. 29, 2005.

2. SWENSON, D.V., WESENBERG, D.L. and JONES, A.K., An-alytical and Experimental Investigations of Rock Cutting Using Polycrystalline Diamond Compact Drag Cutters; paper SPE 10150 presented at the SPE Annual Technical Conference and Exhibition, San Antonio, TX, 4-7 October 1981.

3. WARREN, T.M. and ARMAGOST, W.K., Laboratory Drilling Per-formance of PDC Bits; SPE Drilling Engineering, Vol. 3, No. 2, pp. 125-135, June 1988.

4. ZEUCH, D.H. and FINGER, J.T., Rock Breakage Mechanisms with a PDC Cutter; paper SPE 14219 presented at the SPE Annual Tech-nical Conference and Exhibition, Las Vegas, NV, 22-26 September 1985.

5. BRETT, J.F., WARREN, T.M. and BEHR, S.M., Bit Whirl: A New Theory of PDC Bit Failure; paper SPE 19571 presented at the SPE Annual Technical Conference and Exhibition, San Antonio, TX, 8-11 October 1989.

6. MASSACHUSETTS INSTITUTE OF TECHNOLOGY, The Future of Geothermal Energy, An assessment by an MIT-led interdiscipli-nary panel; DOE Contract DE-AC07-05ID14517, US Department of Energy, Office of Geothermal Technologies, 2006.

7. WISE, J.L., Geometry and Material Choices Govern Hard-Rock Drilling Performance of PDC Drag Cutters; paper presented at The 40th U.S. Symposium on Rock Mechanics, Anchorage, AK, 25-29 June 2005.

8. NYGAARD, R. and HARELAND, G., Calculating Unconfined Rock Strength from Drilling Data; 1st Canada-U.S. Rock Mechanics Sym-posium, Vancouver, BC, 27-31 May 2007.

9. YAN, W., Single PDC Cutter Force Modeling for Hard Rock Cutting; Ph.D. dissertation, New Mexico Institute of Mining & Technology, Socorro, NM, 1997.

Provenance—Original Petroleum Society manuscript, Cutting Efficiency of a Single PDC Cutter on Hard Rock (2007-082), first presented at the 8th Canadian International Petroleum Conference (the 58th Annual Tech-nical Meeting of the Petroleum Society), June 12-14, 2007, in Calgary, Alberta. Abstract submitted for review December 1, 2006; editorial com-ments sent to the author(s) February 22, 2008; revised manuscript received February 17, 2009; paper approved for pre-press February 17, 2009; final approval April 28, 2009.


Authors’ BiographiesDr. Geir Hareland is a Professor in the Chemical and Petroleum Engineering De-partment and the CAODC/NSERC Chair in Drilling Engineering at the University of Calgary. He got his B.Sc. in mechanical engineering from the University of Min-nesota (1981), M.Sc. in petroleum engi-neering from the University of Tulsa (1985) and Ph.D. in mechanical engineering from Oklahoma State (1991). Dr Hareland was an Associate Professor at the New Mexico

Institute of Mining and Technology. He has consulted with various oil companies and founded Drops Technology AS in Norway. Dr. Hareland’s research focuses on working closely with all aspects of the Canadian drilling industry, including contractors, service com-panies and operators. His main research interests are centred on the various aspect of drilling engineering optimization with an em-phasis on ‘engineering.’ The specific research tasks include pro-cesses/factors affecting the actual and trouble time during drilling of oil or gas wells.

Dr. Wenzhuo Yan has Bachelor’s and Master’s degrees in petroleum drilling en-gineering (rock mechanics and rock/drill bit interaction mechanism) from China. He ob-tained his Ph.D. degree in petroleum engi-neering from the New Mexico Institute of Mining and Technology in 1997. He worked briefly as a software engineer on petroleum drilling-related software. He then worked as a senior R&D engineer in the high pressure waterjet industry in Canada. His major re-

search interests are high frequency pulse waterjets, cavitation and abrasive waterjets and their application in cleaning and coating re-moval.

Dr. Runar Nygaard is an Assistant Pro-fessor in Petroleum Engineering at the Department of Geological Sciences and Engineering at Missouri University of Sci-ence and Technology. He got his B.Sc. and M.Sc. degrees in geosciences from the Uni-versity of Oslo (both 1996), and his Ph.D. in geomechanics from the same institution in 2004. Runar was the general manager for Drops Technology AS. He also held a position as a Management Consultant with

McKinsey & Company until he started as Research Associate at the University of Calgary, a position he held until he started at Mis-souri S&T in 2007. His research interests are in geomechanics ap-plied to drilling, well design and long-term integrity of wells.

Jack L. Wise, Ph.D., is a Principal Member of the Technical Staff at Sandia National Laboratories in Albuquerque, NM, where he has been engaged in a variety of research activities since 1979. His work has in-cluded drilling studies with Sandia’s Geo-thermal Research Department, as well as investigations of material response to high-rate loading in support of programs con-ducted by the Dynamic Material Properties Department. He is an active member of the

Society of Petroleum Engineers (SPE), the American Physical So-ciety, Sigma Xi - The Scientific Research Society and Tau Beta Pi - The National Engineering Honor Society.