Mathematical Modeling of PDC Bit Drilling Process Based on a Single-Cutter Mechanics

A. K. Wojtanowicz Department of Petroleum Engineering,

Louisiana State University, Baton Rouge, LA 70803

E. Kuru Department of Petroleum Engineering,

Middle East Technical University, Ankara, Turkey

Mathematical Modeling of PDC Bit Drilling Process Based on a Single-Cutter Mechanics An analytical development of a new mechanistic drilling model for polycrystalline diamond compact (PDC) bits is presented. The derivation accounts for static balance of forces acting on a single PDC cutter and is based on assumed similarity between bit and cutter. The model is fully explicit with physical meanings given to all constants and functions. Three equations constitute the mathematical model: torque, drilling rate, and bit life. The equations comprise cutter's geometry, rock properties drilling parameters, and four empirical constants. The constants are used to match the model to a PDC drilling process. Also presented are qualitative and predictive verifications of the model. Qualitative verification shows that the model's response to drilling process variables is similar to the behavior of full-size PDC bits. However, accuracy of the model's predictions of PDC bit performance is limited primarily by imprecision of bit-dull evaluation. The verification study is based upon the reported laboratory drilling and field drilling tests as well as field data collected by the authors.

Introduction Since 1960 when Galle and Woods presented their mathe

matical model of rockbits [1,2], there have been several models developed [3-6]. Most of the models were purely empirical, based on arbitrarily selected functions and curve fitting. Later developments included the effect of hydraulics on drilling rate [7, 8], the effect of overburden pressure [4], and the effect of differential pressure on drillability [9]. The number of empirical constants increased from three [1] to eight [4], and both dimensional [10] and multiple regression analyses [4] were used for their estimation. Also, several attempts were made to correlate the behavior of a single tooth with the overall performance of rock bits, but conclusions were mainly qualitative [11, 12]. The stochastic nature of the rockbit cutting action precluded any attempt to formulate a mechanistic model.

The polycrystalline diamond compact bits do not have the difficult kinematics of the rock bits. Also, their cutting action is less complex. Therefore, they can be effectively modeled using a mechanistic rather than an empirical approach. To date, a few predictive models have been proposed [13-18].

Ziaja [13] developed a mathematical model of a single PDC cutter penetration assuming a circular cut and absence of cutter interaction. The model was designed for a core bit. The single proportionality constant was used to up-scale from the penetration and load of a single cutter to the drilling rate and weight on bit. Data from one field run of a core bit was used to verify the model.

Glowka [14] used experimental data from laboratory drilling

Contributed by the Petroleum Division for publication in the JOURNAL OF ENERGY RESOURCES TECHNOLOGY. Manuscript received by the Petroleum Division, February 11, 1990; revised manuscript received June 14, 1993. Associate Technical Editor: J. P. Brill.

in hard rocks and developed a power function correlation between cutter penetration and stress at the wear-flat area. His analytical work also included derivation of the single cutter wear equation as a function of penetration per rotation and footage. He used the model to analyze the response of a single cutter to various input loads in relation to the cutter's position on the bit face. He suggested that the PDC bit performance should be calculated by integrating performance of all individual cutters.

Warren and Sinor [15, 16] developed a computer program that rigorously calculated an individual cutter's performance based on the observed drilling rate, cutter position, and cutter geometry. Wear and load were calculated for each cutter and were then summarized and compared to the observed values of bit weight and torque. To date, the program has been successfully verified by using experimental full-scale drilling results. The program seems to be an effective simulator needed by PDC bit manufacturers to investigate various bit design configurations.

There is still a need, however, for a simple mathematical model of the PDC drilling process that can be used readily in the field to control bit performance. Such control would involve designing optimum drilling programs [17] and interpreting drilling data [18]. The model should be easy to match to the site-specific conditions by adjusting the values of a few empirical constants using past drilling data from typical drilling reports (bit dull, rotating hours, footage, cutter types and their geometries). However, the calculation to match constants requires mathematical formulas. Thus, the approach calls for explicit expressions for all drilling variables measured on the rig site such as bit weight, rotary speed, drilling rate, torque, and bit dull.

Journal of Energy Resources Technology DECEMBER 1993, Vol. 115 / 247

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Fig. 1 Balance of forces acting at PDC cutter

Fc S lna c F„ Cos o 0

/LF COS a . AiFwSlnae

ROCK DEFORMATION

Model Principle and Assumptions The derivation of the model is based on the assumption that

a tandem of cutters rigidly attached to the bit's head would mechanically interact with the rock in a similar way as a single cutter would—given perfect removal of drilled rock particles and complete coverage of the hole bottom with the bit-cutting structure. Such similitude involves the responses of the bit and the cutter to normal and tangential forces (bit weight and torque) and to wear (bit dull and cutter's wear-flat area).

There is one very practical problem with this approach. As the similar responses of the bit and cutter to their forces are easy to perceive and to measure experimentally, a mathematical model can be derived comprising only bit forces. However, the similarity in responding to the wear requires rigorous definition of the bit dull. Such a definition does not exist; most of the definitions in use (IADC system, for example) are descriptive and based on averages. The most rigorous identification of a bit dull for the purpose of mathematical modeling— suggested here—is to use the wear of a selected cutter (or a

FRICTION

— W W — I ; HOOKE'S BODY

ZC 1 SLIDE

Fig. 2 Mechanical analog of PDC cutter's dynamics

few cutters) and consider this wear to be an overall bit dull. Such a "representative" cutter, uniquely defined for each PDC bit type, would ensure that the drilling rate and torque responses to the bit dull are consistent with the mathematical model.

The model is derived from the analysis of forces active at the single cutter, as shown in Figs. 1 and 2. The bit life, drilling rate, and bit torque equations are deduced from the static balance of forces for a cutter moving through the rock with a constant angular velocity. The assumptions made throughout the derivation of the model are as follows:

Nomenclature

Ac = cutting area, sq. in. Af = abrasiveness const, klbf Aw = cutter wear-flat area, sq. in. C4 = constant defined by Eq. (37) Df = bit footage, ft Fc = cutting force, lbfs FfC = friction force acting on cut

ting surface area, klbf FfW = friction force effective on

wear-flat area Fh = horizontal force at cutter,

klbfs F„ = normal force at cutter, klbfs F, = tangential force, klbfs Fw = component of normal force

acting on wear-flat area defined in Fig. 7, klbfs

Gi = unit conversion const, 5 x 10"2 (in./hr)/(ft/min)

G3 = unit conversion const, 105 hr/ min

/ = cutters interference const, un-itless

K = drillability const, in./klbfs L = distance traveled by cutter,

in. N = rotary speed, 1/min Rc = rock resistance to shearing,

klbfs/sq. in. Rp = rock resistance to pressing,

klbfs/sq. in.

R SiA

Tb

uD

vD

W

w0

a « i

b db

dc

f(r)

h

n * i

k2

ki

k4

= drilling rate, ft/hr = spring const, psi = bit torque, ft-lbfs = dimensionless cutter wear

function, unitless = dimensionless volumetric

wear function, unitless = weight on bit, klbfs = threshold weight on bit de

fined by Eq. (2), klbfs = const defined by Eq. (40), in. = rotary speed exponent, unit

less = const defined by Eq. (41), in. = bit diameter, in. = cutter face diameter, in. = rate of cutter's wear at radius

r = cutting depth, in. = dimensionless cutting depth,

unitless = proportionality const between

weight on bit and cutter's normal force, unitless

= proportionality const between volumetric wear and work of friction, inVklbfs

= proportionality const between penetration rate per one rotation and cutting depth, unitless

= proportionality const between

/

n(r) nc

r rb

rc

z

h w

X

y

a ac

P e A

P

<$> 5C

Ts

bit torque and cutter's torque, unitless

= distance defined by Eq. (42), in.

= cutters density, unitless = no. of cutters, unitless = radius, in. = bit radius = radial location of cutter un

der consideration, in. = distance defined by Eq. (43),

unitless = bit rotating time, hr = dimensionless linear cutter

wear, unitless = linear cutter wear, in. = function defined by Eq. (49),

unitless = back rake angle, deg = cutting angle, deg = side rake angle, deg = strain (displacement), in. = coefficient of friction between

cutter surface and rock, unitless

= angle between failure plane and reference (horizontal) plane, deg

= porosity, unitless = cutting stress, lbfs/sq. in. = shear stress, lbfs/sq. in.

248/Vol . 115, DECEMBER 1993 Transactions of the ASME

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1 The formation rock displays plastic behavior (i.e., the rock deforms without losing its cohesion). In other words, the rock resistance to pressing (or cutting) is proportional to the contact area and does not depend upon the cutter's penetration (vertical) or displacement (horizontal).

2 The bottomhole profile is predominantly parallel to the bit profile as a result of the cutter interaction.

3 There is a mechanical similitude between a single cutter and the entire bit as follows:

(a) hole deepening per one bit rotation is proportional to the normal force acting on a single cutter with proportionality constant k\\

(b) the drilling rate is proportional to the cutter penetration with proportionality constant k3;

(c) the bit torque is proportional to the cutter's torque with proportionality constant k4.

4 The cutting angle, ac, is ignored; i.e., the cutter moves in the direction perpendicular to its stud axis.

5 The volumetric wear of the cutter is proportional to friction work with proportionality constant k2.

6 Inadequate bottom-hole cleaning results in a nonlinear response of the drilling rate to rotary speed.

7 Friction at the cutter's side surface is ignored.

The principle of assumption 1 underlies the mechanical model of rock shown in Fig. 2. Many rocks reportedly exhibit ductile behavior at the high confining pressures encountered at typical drilling depths. In general, the rock failure mechanism is brittle at low confining pressures and ductile at high confining pressures, with a transition from predominantly brittle to predominantly ductile behavior at intermediate pressures [19-21]. However, even at low confining pressures, many sedimentary rocks subjected to triaxial stress do not lose cohesion [22]. It is also known that the macroscopic rock failure mode is controlled by nominal effective stresses, i.e., the difference between confining pressure and pore pressure [23-26]. As nominal effective stress increases with depth, the rock deformation mode becomes more plastic. In addition, for soft and medium-soft formations (shales, carbonates), the angle of internal friction decreases at a higher confining stress, thus increasing the contribution of cohesive resistance to the shear strength [27].

Ductile behavior of drilled rock is a simplification which can be inferred from the experimental measurements of forces acting at a single cutter shown in Fig. 3 [28]. The laboratory drilling experiments were performed under atmospheric conditions. The zone of force variation (chipping) represents consecutive cycles of forming new fractures. In the model, the stress fluctuations are averaged by some constant value as shown in Fig. 4. This value represents rock resistance to cutting or pressing.

Assumption 2 is based upon the reported examination of the bottom-hole shape patterns for several different bit types [15]. This examination showed that cutters made continuous, smooth cuts at the bottom without losing contact with the rock surface.

Assumption 3, implying that bit behavior can be inferred from the behavior of a single cutter, has been commonly made in drill bit modeling. However, in view of recent developments the assumption may sound somewhat controversial. As PDC bit cutter placement patterns and bit face profiles are subject to experimentation by bit manufacturers, the most rigorous approach to the modeling of PDC bits has already been developed [15, 16]. The model comprises individual cutters' position and orientation and mathematically integrates all cutters' performances across the bit face. The model considers the uniqueness of bit geometry ("bit fingerprint") and can effectively detect differences between the behavior of individual bits.

In field applications, however, the bit interacts with the rock, and this interaction controls drilling process performance. In

FH d jgg^wÂAAAÂAA^-^^^^^M

TIME

Fig. 3 Single cutter's forces: measured [28], and simplified

(/) cmox LLI

J- °cmln C/>

CD Z

H 3 O

„_/l_V_\/AZ.V *"/

V

/f COHESIVE RESISTANCE

DISPLACEMENT

NORMAL STRESS

Fig. 4 Concept of plastic rock's resistance to cutting and its relation to Mohr's theory

order to quantify the drilling process, both the bit and the rock must be modeled. To date, all existing models of the rock drillability have been simplified. Therefore, the rigorous mathematical treatment of the bit cannot be matched with the same treatment for the rock. Also, in field applications, drill bits are not treated as customized products, but rather are standardized by selecting features which are common for a group of bits. The recent IADC classification of the fixed cutter bit geometry, using a three-by-three matrix system, is a good example of generalization made at the expense of accuracy [29],

In addition, there is already experimental evidence supporting assumption 3. The proportionality between drilling rate and cutter penetration is not just a flat assumption, but it can be deduced from experiments. Assumption 3(a), regarding proportionality between bit weight, W, and the average cutter's normal force, F„, stems from a simple balance of forces. Furthermore, it has been experimentally proven that cutter's penetration, h, is proportional to the normal force, F„. The example of experimental data supporting proportionality between h and F„ is shown in Fig. 5. Moreover, a large body of laboratory and field data indicates a linear relationship between drilling rate, R, and bit weight. An example is shown Fig. 6 [30]. Therefore, it follows that the three proportionalities, {R ~

Journal of Energy Resources Technology DECEMBER 1993, Vol. 115/249


I 3

o CHAMPLAIN BLACK MARBLE 0 WHITE DANBY MAR8LE

0 5 10 15 20 25 DEPTH (in. x I0"3)

Fig. 5 Effect of cutter's thrust force on cutting depth [30]

CARTHAGE MARBLE

200 RPM (432 GPM)

100 RPM (432 GPM)

50 RPM (220 GPM)

50 RPM (432 GPM)

BIT WEIGHT (Lb. x I03)

Fig. 6 Drilling rate dependence on weight on bit [30]

W), (W ~ F), and (F ~ h), are equivalent to the linear relation between drilling rate and cutter penetration, (R ~ h), which results in assumption 3(Z?). Assumption 3(c) implies that as the bit dull develops with drilling time, the bit torque responds to the overall bit dull in the same way as the individual cutter's torque responds to the cutter's dull. It can be mathematically proven that the assumption is fully justified for bits showing a steady increase of wear across the bit face (i.e., the rate of wear is constant in time but may change with location).

Assumption 5, regarding the proportionality between volumetric wear and the work of friction, is commonly made when some grinding mechanism is involved. Specifically for PDC cutters, there is evidence of the frictional nature of their wear. It was experimentally shown that the wear rate of PDC drill blanks was much greater for dry cutting than for wet cutting [30]. This observation implies that the basic cutting mechanism for PDC cutters is shearing action and that the friction work predominantly contributes to cutter wear.

Experimental data on abrasive wear of drag cutters indicated that volumetric wear of a cutter per unit cutting distance is proportional to the penetrating force [30]. It was also found that the linear loss of the cutter's material per unit cutting distance was proportional to the contact stress [31]. All these observations are equivalent to the proportionality between volumetric wear and the work of friction at the area of contact.

Assumption 6 draws on several laboratory and field observations, showing the linear effect of rotary speed on drilling

(a) Complete

(b) Simplified

Fig. 7 Cutter's force balance diagram

rate [30, 32]. It also implies that, for the typical range of rotary speeds, cutter penetration is independent from its linear velocity when bottom-hole cleaning is sufficient. In the full-scale tests, however, the effect of rotary speed on drilling rate was often nonlinear. Such discrepancies were usually attributed to poor hole cleaning in field operations as opposed to laboratory tests.

Assumption 7 is supported by the laboratory experiments, which revealed that values of side forces are smaller than 10 percent of the thrust force values [30]. Thus, the frictional drag resulting from the side forces becomes insignificant. The balance diagram of forces active between the rock and the cutter is presented in Fig. 1(a). For small values of cutting depth, the effect of the cutting angle can be ignored, as shown in Fig. 7(b).

The normal and tangential components are expressed by Eqs. (1) and (2), respectively, as

F„ = Fc sina + FfC cosa - Fw cosac +f/w sinac (1)

F, = Fc cosa - Ffc sina + Ffw cosac - Fw sinac (2)

where

/Yc = KCAC

Ffc = fiFc = fiRcAc

F« = RpA wFj„ = ixRpA „ (3)

For small values of cutting depth (h < 0.1 in.), the value of the cutting angle, ac, is very small. After ignoring the cutting angle and substituting (3) for (1) and (2), we obtain

F„ = RcAc(sma + [i cosa) + RpA„ (4)

F, = RcAc(cosa - ix sina) + fiRpAw (5)

Also, for typical values of the side rake angles, /3, smaller than 5 deg, horizontal force value, Fh, is close to the tangential force value, so that

Fh = Fl(cos0)-l=Ft (6)

Bit Life Equation It is assumed that drilling parameters, weight on bit, rotary

speed, and mud flow rate are constant and that the formation is homogeneous. As drilling continues, cutting depth decreases while the linear wear of the PDC drill blank increases. The

250 / Vol. 115, DECEMBER 1993 Transactions of the ASME


0.8

0.7

0.3

0.2

0.1

0

Vo

/ v o

/

0.1 0.2 0.3 CUTTER WEAR, w ( Unltless )

Fig. 8 Dimensionless functions for PDC bit

0.8

0.7

0.6

0.3

0.2

0.1

0

linear wear on the cutter stems from the work of friction depending on the cutting path, bit weight, and the coefficient of sliding friction between rock and cutter.

L = 2fircNtb (7)

where

tb = time, hr

rc = cutter's radius, in.

Following assumption 5, volumetric wear is given as

V= 2irk2rciiFnNtb (8) As normal force acting on a single cutter is proportional to the weight on bit (assumption 3(a): W = kiF„), Eq. (8) becomes

W V=2irrck2ix — Ntb

(9)

Volumetric cutter wear can also be calculated from geometrical considerations as

V--f Aw(x)dx

and

dx=dc cosa dw

(10)

( ID

where

w, x = dimensionless and linear cutter wears, respectively, (Appendix A)

A„ = wear-flat area derived in Appendix A.

V= cosa

or

dl 4 sina

1 (ArccosA/1-4< w—w

- 2 A / W - 5 W 2 + 8 W 3 - 4 W V W (12)

V= 4 tana

(13)

where Vj = dimensionless volumetric cutter wear function defined by the integral in Eq. (12).

A plot of VD versus w is shown in Fig. 8. Physically, the dimensionless volumetric wear function is a ratio of the actual volumetric cutter wear to the volume of a cube having each side equal to the cutter radius. Combining Eqs. (8) and (11) and solving for bit life, tb, gives

h = Vn

or

4 tana pj/ 2irrck2tJ. — N

k\

f" = A ^

(14)

(15)

where Af is the abrasiveness constant expressed in field units as

j 3

Af=kl &irfirck2 tana

(16)

and G3 is units conversion factor (G3 = 105 hr/min). The physical meaning of the abrasiveness constant, A/, is

the weight-on-bit (in thousands of pounds) necessary to cause volumetric cutter wear equal to 0.1 volume of a cube having each side equal to the cutter radius, dc/2, after 100 hr of drilling with rotary speed 100 min"1. In practical applications, Af is determined by matching Eq. (15) to the reported bit performance data, tb, W, N, and w, as

A^hWNiVnG,)-' (17)

where VD is determined from Fig. 8 for the reported value of w.

Drilling Rate Equation The normal force balance equation is solved for the cutting

surface as

Ac-F„-RPA»

(18) 7?c(sina + fi cosa)

Also, as shown in Appendix 3, cutting area is a function of cutter's penetration and its wear expressed by the dimensionless wear function, UD. A plot of UD versus w is shown in Fig. 8. Substituting for Ac and solving for h gives

h_ AUD(FN-RPAW)

dcRc(sina + fx cosa)cos/3

As the drilling rate is proportional to h (assumption 3(b)), the bit progress per one full rotation is /-fold greater than h— where /represents the number of interfering cutters (i.e., cutters located at the same radius, rc). Thus, the drilling rate is

R = kihNaiI (20)

where exponent at represents the effect of bottom-hole clean-



Table 1 PDC bit field record and values of dimensionless functions

Bit run no.

32 33(0)

35 361'" 38

PDC bit

SX-3 SX-3 SX-3 SX-3 SX-3

Depth out, ft

13335 15044 15992 16302 17175

Footage ft

165 709 792 310 569

Time, hr

13 45 31 30 33

ROP, ft/hr

12.7 15.8 25.5 10.3 17.0

WOB, 103 lbf

10 15 10 15 15

RPM 1/min

90 60 90 90 90

Wear, w

0.05 0.10 0.10 0.10 0.10

D-less functions

VD UD

0.005 0.63 0.02 0.50 0.02 0.50 0.02 0.50 0.02 0.50

'Bit runs used for prediction

ing, and its value can be determined from a drill-off test. After substituting Eq. (19) with (20) and using assumption 3(a) regarding proportionality between bit weight and the cutter's normal force, the drilling rate equation becomes

where

R=KGl(W-W0)UDiYl

W0 = klRpAv

and K is the drillability constant defined as

4/t37 K=

k\dc cos/3 Rc(sma + p. cosa)

(21)

(22)

(23)

The physical meaning of K is the hole deepening in inches per 100 revolutions of a new PDC bit under 1000 lbfs active bit weight (i.e., W - W0 = 1000 lbf). Similarly to the ab-rasiveness constant, the drillability constant is determined by matching the drilling rate equation to the field drilling data. The formula is

K= Dr

GÛDNaHb(W-W0) (24)

where Df = bit footage, ft Gx = unit conversion const, 5 x 10~2 (ft»min)/(in«hr)

Bit Torque Equation After introducing Eqs. (4), (5), and (18) into Eq. (6) the

horizontal force at the cutter is

Fh=Fn^_2z(l^mLRpAK (25) /x + tga JX + tga

To convert Eq. (25) into the bit torque equation, the density of cutter placement is considered as a function of bit radius, n(r), and the torque is given as

ix + tga J0

2 -Q* + tga)2

fi + tga Awn(r)rdr (26)

The explicit form of Eq. (26) requires some knowledge of load and wear distributions across the bit face. For drilling with a constant bit weight, W, and using assumption 3(a), normal forces at bit cutters are time-independent. That means bit weight is somewhat distributed and the distribution does not change with time. Distributed wear can be related to a specific cutter situated at radius rc as

Aw(r)=A„(rc) f(r) (27) f(rc)

where/(r) stands for the distributed rate of cutter wear. Also the cutter's normal force is

F„(r) = W

kdr)h{r) (28)

Table 2 model

Bit run no.

32 35 38

Bit performance prediction with P D C drilling

Actual performance

Time, hr Footage, ft

13 165 31 792 33 569

Predicted performance

Time, hr Footage, ft

11.3 184 45 523 30 565

After substituting Eqs. (27) and (28) with (26) and integrating, the bit torque becomes

T urn l - / * t g « . _ „ 2 - Q +tga)2

Tb= WEi -rr—-—--A„RPE2 — db(n + tga) F jx + tga

where constants E\ and E2 are given as f-ft r

dr •>o ki(r)

E2=\f(rc)]~l \"f(r)n(r)rdr •in

(29)

(30)

(31)

and represent bit design features. It can be seen from Eq. (29) that, for a constant weight on

bit, torque becomes a unique function of the wear-flat area.

Verification of PDC Drilling Model The verification included quantitative and qualitative stud

ies. In the quantitative study, the model was first matched using field drilling data, and then used for prediction of PDC bit performance in the same drilling area. The qualitative verification was used to find out if there is a similitude between the model and actual bit responses to the change of selected parameters.

Prediction of PDC Bit Performance. The model was tested using field data from an offshore well in the Louisiana Gulf Coast Region. Table 1 shows bit records for drilling the well-bore section from 13,170 ft to 17,175 ft with 8 1/2-in. Stratapax SX-3 PDC bits. The performance of two bits was used to predict performance of the other three bit runs. The prediction was based on calculated values of constants K = 0.5 in./klbf, Af = 20.25 klbf and threshold bit weight W0 = 0. The unit value was assumed for exponent a\. The predicted and the actual bit performance is shown in Table 2. Two out of three cases proved fairly consistent with the model. Misjudgment of bit wear was the most likely source of error in prediction for the third bit. Table 2 exemplifies the need for reliable bit wear records which greatly affect drilling planning strategies. In all calculations similar to those in Table 2, we encountered problems of unreliable bit wear data. In fact, the bit wear data for large numbers of bits examined in this study were so inconsistent that, without additional inspection of the bits, it was impossible to add more items to Table 2.

Qualitative Verification of Drilling Rate Model. The verification was performed using available laboratory drilling data with PDC bits. First, we examined the linear effect of bit weight

252/Vol . 115, DECEMBER 1993 Transactions of the ASME


I 50

40

n MEDIUM HARD ROCK, WATER BASE MUD o MEDIUM HARD ROCK , OIL BASE MUD a SOFT ROCK , OIL BASE MUD

6 8 10 12 14

WEIGHT ON BIT ( L b s * l 0 3 )

Fig. 9 Linearity of drilling rate versus bit weight [33]

40 Ul H < a: a 30 z

0 NO CUTTERS REMOVED

0 8 CUTTERS REMOVED

D 16 CUTTERS REMOVED

A 24 CUTTERS REMOVED

0 •

0 A

D

A A

A°

a

• i i i

a

8

0

<

a

0

I

0 A

1

0

i

Fig. 10 [33]

2.5 5.0 7.5 I0.0 I2.5 I5.0 I7.5 20.0

WEIGHT ON BIT ( Lbs*IOOO)

Lack of drilling rate response to cutter's removal for new bit

on the drilling rate. With only a few experiments indicating otherwise, the prevailing empirical data confirmed the linearity of the R versus W correlation, particularly at low rotary speeds and for medium-hard formations [13, 31, 34]. Shown in Fig. 9 are results of the laboratory drilling tests performed in medium-hard and soft formations with water-base and oil-base drilling fluids [15, 33]. A threshold value of weight-on-bit, WB, is a well-known phenomenon in rock bit modeling theory [3, 35, 36], The value corresponds to the end of surface grinding and indicates an offset of volumetric chipping. However, as shown in Fig. 9, for drilling with a new bit, there is no threshold weight. A sharp new bit generates rock chips, even with a very small bit weight. As the bit wears out, the threshold weight increases.

The PDC drilling model was also tested using data from experiments with PDC cutter removal [15, 33]. The experiments examined the behavior of PDC bits after some of their cutters were removed. The results of laboratory drilling with constant weight and rotary speed are: 1) the drilling rate did not respond to the reduced number of cutters for new PDC bits (Fig. 10); and 2) the drilling rate increased with a decreased number of cutters for worn bits (Fig. 11).

To be successfully verified, the new PDC model should

100

• NO CUTTERS REMOVED O 12 CUTTERS REMOVED

- o 20 CUTTERS REMOVED

20

ROCK: BEREA SANDSTONE

4 6 8 10 12 14 16 18

WEIGHT ON BIT ( L b s * l 0 0 0 )

Fig. 11 Reduced drilling rate after cutter's removal for dulled bit [33]

respond in the same way as the actual PDC bits tested. The drilling rate for a new PDC bit (w = 0, UD = 1, a{ = 1, W = const, N = const) is

R,= (const) (£,)i/i (32)

After reducing the number of cutters from l\ to I2, the proportionality constant k\ assumes a new value

Wm ( * l ) 2 = -

w W I2 11

and the drilling rate becomes

# 2

which confirms the experimental results

(const)(£i), - I2 = Ri

(33)

(34)



Table 3 Interpretation of cutter's removal test for drilled bits using PDC drilling model

WOB, 103 lbf

7.5

10

Carthage limestone

h/h 10/19 19/39 19/27 27/39 19/39 19/27

ROP2/ROP,

130/50 35/14 50/33 60/30

100/30 100/60

C4

1.29 1.34 1.57 1.30 1.22 1.44

WOB, 103 lbf

12.5

17.5

h/h 27/39 29/39 10/19 19/39 23/39 31/39

Berea sandstone

ROP2 /ROP,

23/14 29/14 50/29 59/23 40/23 37/23

C4

1.47 1.24 1.65 1.32 1.55 1.33

In the case of a dulled PDC bit (w = 1,UD< 1), the drilling rate Eq. (20) can be written as

R = (const), ki h - (const)2/2 (35) and the ratio of drilling rates before and after cutter removal is calculated as

* 2 _ C 4 - ( / 2 / / i )

where C - l

V3 G, W

(36)

8.3lRcRpAdcAw sina cos/3 (37)

Constant C4 is a function of PDC bit geometry, rock properties, the bit weight, and bit wear. Its value should be constant in these experiments.

Equation (36) was used for the quantitative interpretation of the worn PDC bit cutter removal test data shown in Fig. 11. Constant C4 was calculated from Eq. (37) using the measured values of drilling rate ratios and the respective numbers of cutters before and after removal. The calculations were performed for two different rocks and two different values of weight on bit. The results are shown in Table 3. It is noted that the calculated values of constant C4 do not display significant variation, thus providing further verification for the model.

Qualitative Verification of Torque Model. The torque Eq. (29) indicates that as the wear increases, the bit torque should decrease, which is supported by laboratory experiments, as shown in Fig. 12. Field data collected for this research from PDC drilling in the Gulf Coast area further support this observation regarding linearity of torque versus bit weight shown in Fig. 13. However, all the field data from seven wells analyzed here consistently indicated lower sensitivity of torque to bit weight as compared to the laboratory data and expected from Eq. (29).

Theoretically, in homogeneous rock, torque and the penetration rate should both respond to bit weight similarly. In real drilling conditions, however, the correlation between drilling rate and torque is much stronger than correlation between each of them and the weight on bit. This phenomenon is well documented in recent research [16], and was also qualitatively observed in previous works [34].

Conclusions The mathematical model of PDC drilling developed in this

study has been derived by examining the effect of a cutter's wear upon static balance of forces acting on a cutter during a steady-state motion of the cutter through the rock. The "up-scaling" of the cutter's behavior for the whole bit required making several assumptions regarding similitude between the cutter and the bit. Despite the simplification resulting from the assumptions used, the PDC drilling model responds similarly to drilling parameters and variables to actual bits. It is believed that the model provides a simple arid useful analytical tool to plan and control PDC bit use in the field. There are,

WEIGHT ON BIT (Lbf. xlO°)

Fig. 12 Effect of PDC bit wear on torque; laboratory data [33]

"

-

WELL # 4 - — 0 WORN SIT BIT DIAMETER' 9 % In. o NEW BIT DEPTH IN ' 5 9 I 0 Fl. DEPTH OUT ' 8 5 8 6 Fl.

~--p~~""a~

- - S i r

^ ^ °- ®T^~

i i i i i i i i i •

WEIGHT ON BIT (Lbs. X I03)

Fig. 13 Effect of PDC bit wear on torque versus weight; field data

however, concerns resulting from this research and regarding the model's applicability. These are summarized as follows:

1 The new model applies to drag bits having cutting structures comprising spherical PDC compacts (cutters). Because the number, size, and configuration of cutters may vary from bit to bit, the model must be matched to PDC bit type by determining empirical constants Aj, K, Eu and E2.

2 The model is designated for field use and represents the PDC drilling process resulting from the bit-rock interaction of a commercial PDC bit. Therefore, the bit's design must have already ensured a balance between the bit load distribution and uniformity of bit wear. The model cannot be used for testing prototypes where bit geometry and placement of cutters is subject to experimentation.

3 The assessment of bit wear has the most significant effect on the model's predictions. Because the bit wear is never dis-



tributed uniformly across the bit head, the main challenge of using the model is to identify cutters which would represent the bit wear. Then, only the wear of these cutters can be used for the model's input. Identification of "representative" cutters should involve laboratory drilling studies of commercial bits. Also, there is a possibility that the cutters can be identified from analytical studies using numerical simulators of PDC bits similar to that reported in references [15 and 16].

References 1 Galle, E. M., and Woods, H. G., "Variable Weight and Rotary Speed

for Lowest Drilling Cost," presented at the AAODC Annual Meeting, New Orleans, LA, September 25-27, 1960.

2 Galle, E. M., and Woods, H. G., "Best Constant Weight and Rotary Speed for Rotary Rock Bits," API Drilling and Production Practice, Vol. 48, 1968.

3 Chefranov, K. A., Drilling Control, Nedra, Moscow, U.S.S.R., (in Russian), 1972.

4 Bourgoyne, A. T., and Young, F. S., "A Multiple Regression Approach to Optimal Drilling and Abnormal Pressure Detection," Society of Petroleum Engineering Journal, Aug. 1974.

5 Warren, T. M., "Drilling Model for Soft Formation Bits," Journal of Petroleum Technology, Dec. 1981, pp. 963-970.

6 Smalling, D. A., and Myers, R. L., "A Drilling Model for Young Offshore Louisiana and Texas Trends," Paper IADC/SPE 14785, presented at the 1986 1ADC/SPE Drilling Conference, Dallas, TX, February 10-12, 1986.

7 Lummus, Y. L., "Drilling in the Seventies," Petroleum Engineer, Part 2, Mar. 1974.

8 Fullerton, Hal B., Jr., "Constant Energy Drilling System for Well Programming," Sii Smith Tool, Aug. 1973, unpublished.

9 Vidrine, D. Y., and Benit, E. Y., "Field Verification of the Effect of Differential Pressure on Drilling Rate," Journal of Petroleum Technology, July 1968.

10 Reza, M. R., and Alcocer, C. F., "A Unique Computer Simulation Model Well Drilling: Part II—The Drilling Model Evaluation," Paper SPE 15109, presented at the 56th California Regional Meeting of SPE, Oakland, CA, April 2-4, 1986.

11 Miska, S., and Wojtanowicz, A., "Study on the Dynamics of the Rotary Bit Tooth Impact," Mining Archives of the Polish Academy of Science, (in Polish), Bulletin 17, Vol. 1, Warsaw, Poland, 1972.

12 Zlobinsky, B. A., "Dynamic Destruction of Rocks by Pressing," Nedra, Moscow, U.S.S.R., 1970.

13 Ziaja, M. B., "Mathematical Model of the Polycrystalline Diamond Bit Drilling Process and Its Practical Application," Paper SPE 14217, presented at the 60th Annual Technical Conference and Exhibition of SPE, Las Vegas, NV, September 22-25, 1985.

14 Glowka, D. A., "The Use of Single-Cutter Data in the Analysis of PDC Bit Designs," Paper SPE 15619, presented at the 61st Annual Technical Conference and Exhibition of the SPE, New Orleans, LA, October 5-8, 1986.

15 Warren, T. M., and Sinor, A., "Drag Bit Performance Modeling," Paper SPE 15618, presented at the 61st Annual Technical Conferences and Exhibition of the SPE, New Orleans, LA, October 5-8, 1986.

16 Sinor, L. A., and Warren, T. M., "Drag Bit Wear Model," Paper SPE 16699, presented at the 62nd Annual Technical Conference and Exhibition of the SPE, Dallas, TX, September 27-30, 1987.

17 Wojtanowicz, A. K., and Kuru, E., "Dynamic Drilling Strategy for PDC Bits," Paper SPE/IADC 16118, presented at the 1987 SPE/IADCDrilling Conference, New Orleans, LA, March 15-18, 1987.

18 Kuru, E., and Wojtanowicz, A. K., "A Method for Detecting In-situ PDC Bit Dull and Lithology Change," Paper SPE/IADC 17192, presented at the 1988 SPE/IADC Drilling Conference, Dallas, TX, February 29-March 2, 1988.

19 Bredthauer, R. O., "Strength Characteristics of Rock Samples Under Hydrostatic Pressure," TRANS. ASME, Vol. 79, 1957, p. 695.

20 Cheatham, J. B. Jr., and Gnirk, P. F., "The Mechanics of Rock Failure Associated with Drilling at Depth," Proceedings, 8th Symposium on Rock Mechanics, University of Minnesota, Oct. 1958, pp. 410-439.

21 Gray, K. E., "Some Rock Mechanics Aspects of Petroleum Engineering," Proceedings, 8th Symposium on Rock Mechanics, University of Minnesota, Oct. 1958, pp. 405-433.

22 Handin, J., Hager, R. V., Jr., Friedman, M., and Geather, J. N., "Experimental Deformation of Sedimentary Rocks Under Confining Pressing Tests," Bulletin AAPG, May 1963, Vol. 27, p. 717.

23 Garner, E. N., Podio, A., and Gatlin, C , "Experimental Study of Crater Formation in Limestone at Elevated Pressure," Journal of Petroleum Technology, Dec. 1963, Vol. 15, p. 1356.

24 Podio, A., and Gray, K. E., "Single-Blow Bit Tooth Impact Tests on Saturated Rocks Under Confining Pressure: I. Zero Pore Pressure," Society of Petroleum Engineers Journal, Sept. 1965, Vol. 5, No. 3, pp. 211-224.

25 Yang, J. H., and Gray, K. E., "Single-Blow Bit-Tooth Impact Tests on Saturated Rocks Under Confining Pressure: II. Elevated Pore Pressure," Society of Petroleum Engineers Journal, Dec. 1967, Vol. 7, No. 4, pp. 389-408.

26 Myers, G. M., and Gray, K. E., "Rock Failure During Tooth Impact and Dynamic Filtration," Society of Petroleum Engineers Journal, June, 1968, Vol. 8, No. 2, pp. 163-173.

27 Winters, W. J., and Warren, T. M., "Roller Bit Model with Rock Ductility and Cone Offset," Paper SPE 16696, presented at the 62nd Annual Technical Conference and Exhibition of the SPE, Dallas, TX, September 27-30, 1987.

28 Zeuch, D. H., and Finger, J. T., "Rock Breakage Mechanisms with a PDC Cutter," Paper SPE 14219, presented at the 60th Annual Technical Conference, September 22-25, 1985.

29 Winters, W. J., and Doiron, H. H., "The 1987 IADC Fixed Cutter Bit Classification System," Paper SPE/IADC 16142, presented at the 1987 SPE/ IADC Drilling Conference, New Orleans, LA, March 15-18, 1987.

30 Hibbs, L. E., Jr., and Sogoian, G. C , "Wear Mechanisms for Polycrystalline Diamond Compacts as Utilized for Drilling in Geothermal Environments," Report SAND 82-7213, Sandia National Laboratories, Albuquerque, NM, May, 1983.

31 Glowka, D. A., "Implications of Thermal Wear Phenomena for PDC Bit Design and Operation," Paper SPE 14222, presented at the 60th Annual Technical Conference and Exhibition of the SPE, Las Vegas, NV, September 22-25, 1985.

32 Clark, D. A., and Walker, B. H., "Comparison of Laboratory and Field Data for a PDC Bit," Paper SPE/IADC 13459, presented at 1985 SPE/IADC Drilling Conference, New Orleans, LA, March 6-8, 1985.

33 Warren, T. M., and Armagost, W. K., "Laboratory Drilling Performance of PDC Bits," Paper SPE 15617, presented at the 61st Annual Technical Conference and Exhibition of the SPE, New Orleans, LA, October 5-8, 1986.

34 Cheatham, C. A., and Loeb, D. A., "Effects of Field Wear on PDC Bit Performance," Paper SPE/IADC 13464, presented at the 1985 SPE/IADC Drilling Conference, New Orleans, LA, March 6-8, 1985.

35 Bingham, M. G., "A New Approach to Interpreting Rock Drillability," Oil and Gas Journal, 1965.

36 Wojtanowicz, A. K., "Drilling Optimization," (in Polish), Technical University of Mining and Metallurgy, University Press, Krakow, Poland, 1975.

37 Kuru, E., "Effect of Rock-Cutter Friction on PDC Bit Drilling Performance," Chap. 1, Ph.D. dissertation, Louisiana State University, Baton Rouge, LA, May 1990, pp. 18-32.

A P P E N D I X A

Cutter Wear-Flat Area

The shape of cutter wear-flat is elliptical. The ellipse is described using PDC cutter terminology defined in Fig. 14. The dimensionless linear wear measured across the cutter's face is

dc cosa

The equation of ellipse is

z2 I2 ,

b2 + b2=l

(38)

(39)

where z, I, a, and b are shown in Fig. 14, and calculated as

2 a = dc .

2 sma b = 0.5dr

1= d, cosa cos a

Z = a .n—ff.

(40)

(41)

(42)

(43)

Fig. 14 Nomenclature of cutter's geometry: cutting area Ac

wear-flat area A„, and

Journal of Energy Resources Technology DECEMBER 1993, Vol. 115/255


The wear-flat area A„ is a part of the ellipse as

Aw = ab Arccos(- ) - z l W

or, combining (38) through (44) gives

(44)

Funct ion/^w) is defined as

/ i (w) = dc cos/3

4t / n

where UD is a dimensionless wear function as

(47)

UD = dA, dh

1 2 . o„3 ^,,4-, ^ + : / ( .y-5^+8.y J-4jO

M + ^ j V j ' - / + 8j'3-4.y4-(l-l()y + 24/-16>>3) J ;

(48)

y+^y

-Aw — 4 sina

[VArccos(l —4(w- w2))]

- 2 h / w - 5 w 2 + 8w 3 -4w 4 ] (45)

Note that Eq. (45) is valid for x < dc cos a.

A P P E N D I X B

Cutting Surface and Wear Function A rigorous derivation of the expression for cutting surface,

Ac, is given in reference [37]. It was proved that for practical range of the single cutter's penetration, h < 0.1, cutting surface is almost linear with h; thus,

Ac=Mw)h (46)

The variable y in Eq. (48) is

y=w + h

(49) dc cosa

and is not dependent upon h because for typical geometries and loads on PDC cutters, the penetration h can be approximated as [37]

h

dr cosa

h

dc cosa exp(0.05w) (50)

where

[h/dc cosa]o = dimensionless penetration for w = 0, depending upon cutter's normal load

The plot (UD versus w) in Fig. 8 is made for (h/dc cosa) = 0.028.

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Mathematical Modeling of PDC Bit Drilling Process Based on a Single-Cutter Mechanics