Journal of Petroleum Science and Engineering 188 (2020) 106849

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Journal of Petroleum Science and Engineering

journal homepage: http://www.elsevier.com/locate/petrol

Mathematical modelling of performance and wear prediction of PDC drill

bits: Impact of bit profile, bit hydraulic, and rock strength
Ahmed Z. Mazen a, Iqbal M. Mujtaba a, Ali Hassanpour b, Nejat Rahmanian a, *
a
Chemical Engineering Department, Faculty of Engineering and Informatics, University of Bradford, Bradford, BD7 1DP, UK
b
School of Chemical and Process Engineering, University of Leeds, Leeds, LS2 9JT, UK

A R T I C L E I N F O A B S T R A C T

Keywords: The estimation of Polycrystalline Diamond Compact (PDC) cutters wear has been an area of concern for the
PDC cutters wear drilling industry for years now. The cutter’s wear has been measured practically by pulling the bit out for
Mechanical specific energy evaluation at the surface. It is important to find the right time for tripping out as this helps to avoid the fishing
Depth of cut
job and reduces the operational cost significantly. The prediction of the drilling performance is based on the
Effective blades
ROP
interaction of cutter and rock. Several authors focused on the cutter-rock interface but only a few researchers
tried to model the wear of the PDC bit cutters. The aim of this research is to understand the relationships between
the rate of penetration (ROP) and the drilling variables per each foot, and then determine the overall bit effi­
ciency for the whole drilling operation. A new mathematical model is derived to predict the PDC bit performance
by considering the factors that were already not taken into account. These factors include rock strength, bit
design, and bit hydraulic. The model investigates the effect of these parameters to estimate the abrasive cutters
wear on the inner and the outer bit cones by deriving modified equations to calculate the mechanical specific
energy (MSE), torque, and depth of cut (DOC) as a function of effective blades (EB). The model is used to forecast
the bit cutters wear conditions in four wells in the oil fields located in Libya, which were drilled with three
different PDC’s sizes. The model enables the results to be compared to the actual bit cutters wear measured for
inner and outer cones. The results are found that are well in agreement with the actual field data obtained in bit
records.

1. Introduction Kuru and Wojtanowsicz (1988) proposed a model that derived from
torque and ROP equations to prevent early damage of PDC bit by eval­
Drill bits are the main tools that penetrate the formation down to the uating the bit condition and detecting the rock change while drilling.
planned production zone. An efficient PDC bit should have good drill­ The approach required data from bit geometry and its dull condition to
ability by achieving a sufficient ROP, and it should be durable to be used set up software to optimize the drilling parameters.
in other wells. Both ROP and bit life rely directly on bit design, drilling Understanding the breakage process generated at the cutter – rock
parameters, and rock properties (Ersoy and Waller, 1995; Sinor and interface is the key for interpretation of the drilling parameters. There
Warren, 1987). are considerable work in the literature that focused on studying the
Ziaja and Miska (1982) presented a mathematical model for the cutter – rock interface but not to determine the PDC bit wear. Jones
diamond core bit to estimate rock strength index, rock abrasiveness (1990) presented an improved cutter configuration to optimize the
index, and ROP. The model also provided an equation to measure the bit cutter-rock interface area according to the total volume of diamond that
dullness by assuming gradual bit wear while drilling. penetrate the formation along the bit radius. Detournay and Defourny
Warren and Sinor (1987) developed a single PDC cutter model to (1992) studied the cutter-rock interaction as a function of two processes:
predict the cutter temperature, force, and wear. The model assumed cutting of the rock and the friction underneath the cutters. They
constant ROP and considered the mechanical design parameters, and developed a model, which was used as a guide to understand the drilling
results were compared to the laboratory drilling test for four different process by assuming that the cutting component of the torque and the
bits design. weight on bit (WOB) are both proportional to the depth of cut (DOC).

* Corresponding author.
E-mail address: n.rahmanian@bradford.ac.uk (N. Rahmanian).

https://doi.org/10.1016/j.petrol.2019.106849
Received 9 July 2019; Received in revised form 19 December 2019; Accepted 20 December 2019
Available online 25 December 2019
0920-4105/© 2020 Elsevier B.V. All rights reserved.
A.Z. Mazen et al. Journal of Petroleum Science and Engineering 188 (2020) 106849

the depth.
To evaluate the bit performance accurately, the bit profile needs to
be considered along with the cutter-rock interface. This has been proved
by Glowka (1987) who developed a method to predict the cutter force
and bit performance. He confirmed that the bit profile can significantly
affect PDC wear. His model was based on the assumption that cutter
wear will be uniform regardless of its location.
A number of researches aimed to design the crown shape of the PDC
bit by employing the principles of equal wear, equal volume of rock
removed and power (Glowka, 1985; Hibbs and Flom, 1978). However,
they ignored the effect of cutter-rock interaction; therefore, they did not
efficiently reflect the force and the cutting wear. Wang et al. (2018)
studied the cutter-rock interaction and concluded that the bottom hole
shape of the PDC bit is one of the essential factors that affect the force
Fig. 1. PDC cutters wear prediction model. and the wear of PDC cutters.
Most of the available models in the literature are not so efficient,
albeit are helpful, as they do not address the real operating conditions of
the rig site. These models were developed using a laboratory scale and
Wojtanowicz and Kuru (1993) proposed a model to control the under atmospheric conditions. This paper proposes a new model that
performance of a spherical PDC bit cutters in order to maintain the can be applied on rig site and it is a function of all drilling parameters
balance between the WOB distribution and bit wear uniformity. The that include the rock-bit interaction, bit profile, rock strength, and bit
results were obtained by adjusting empirical constants while drilling in hydraulic as shown in (Fig. 1). The results enable the model to compare
homogenous formation to match bit recorded data from offset well. The the cutters wear on both inner and outer cones with the actual bit wear
model assumed that WOB, rotary speed (RPM), mud flow are constant according to the standard bit record form. Full description of the ana­
and the model also ignored the cutting angle. lyses of the drilling parameters is given by adjusting some factors for
Gerbaud et al. (2006) developed a new cutter-rock interface by rock hardness and mud hydraulic, rock friction, and DOC to understand
taking into account the latest technology of the PDC cutters shape and prediction of the bit performance.
investigated its effects on the ROP, bit stability, and bit wear. Detournay
et al. (2008) presented a model to extend their work of 1992. The model 1.1. Computational model of PDC bit life
introduced the contact length, a measure of bit dullness, and the contact
strength to the frictional process. The model was developed under the The design of the PDC profile was based on the assumption of equal
same assumption that the drilling response of PDC bit followed a linear wear and equal volume and energy (Wang et al., 2018). However, the
constraint between torque, WOB, and DOC. distribution and density of cutters along the bit profile are not the same.
Tulu and Heasley (2009) investigated numerically a 3D single cutter Therefore, the degree of wear is not the same for inner and outer cutters
to analyse the cutter-rock interaction and compared their work with a due to the rotating radius. In this work, bit profile is considered and the
single cutter on a laboratory scale. The objective of their work was to analyses are divided into inner and outer cones.
analyse the vertical and horizontal forces based on the cutting depth, The force applied on PDC is transferred to all cutters, and the cutter
however, they ignored the effect of mud hydraulic. Gouda et al. (2011) breaking mechanism differs because of bit design and cutters location,
developed a mathematical model to estimate the PDC cutters wear by therefore, the analyses of the forces on the single cutter as proposed in
including the torque obtained from drill pipe, string stabilizer, and the previous work done by Glowka (1985), Hibbs and Flom (1978) is not
bit. The model was valid for certain back and side rack angles. accurate. Huang et al. (2017) added that the force conditions are more
Yahiaoui et al. (2011) studied the cutter-rock interaction, by testing related to cutting arc length and wear degree. In this model, the drilling
six different manufacturers’ cutters in the laboratory scale to compare impedance (DRIMP) as a wear evaluation index will be used instead of
the wear rate. The tests were carried out under atmospheric conditions using cutting forces.
in which no drilling fluid was pumped into the contact surface. Patil and The previous studies (Gouda et al., 2011; Pryhorovska, 2017; Woj­
Teodoriu (2013) studied the cutter-rock interaction and introduced a tanowicz and Kuru, 1993) carried out on the analysis of rock-bit inter­
mathematical model to analyse the stick-slip vibration (Alkaragoolee, action suggested good hole cleaning. Besides, Kuru (1990) confirmed
2018) by considering the PDC cutter wear. that the diamond compact material loses strength at a temperature
Liu et al. (2014) introduced an analytical model coupled with above 350 � C resulting from high friction generated at the interface.
real-time gamma-ray data to predict the PDC bit wear. The model Therefore, drilling fluid has to be pumped in during drilling operations
assumed that the volume loss of cutters is proportional to WOB. Chen to maintain and keep the operation temperature below 350 � C. Addi­
et al. (2014) presented a new cutter-rock interaction to analyse the tionally, it was shown experimentally that the wear rate of PDC cutter
cutting force. They concluded that the cutting arc length plays a major was much greater for dry shearing than wet cutting (Gray, 1967). Hence,
role in the cutting force calculation. Doshvarpassand et al. (2017) con­ to address the real conditions of the drilling operation, this model con­
ducted the experimental work to evaluate the effects of cutter-rock siders the influence of the bit hydraulic.
interaction on the cutting action of different sizes of PDC cutters. They It has been confirmed that abrasive wear is a function of DOC;
found that the effect of the cutters edge may differ based on the rock therefore, to control DOC of PDC cutters, the volume of rock removed
type. and the torque generated have to be controlled (Sinor et al., 2001). DOC
Micro- and nano-scratch tests wear applied to estimate the wear is modified in this work based on the active cutters for both cones.
volume of the PDC cutters diamond layers. The experimental results Tian et al. (2015) concluded that the drilling efficiency and the
were compared to the results obtained from wear models in the past volume wear rate of PDC bit is dependent on the cutting angle, while
(Abbas and Hassanpour, 2018). Hankins et al. (2015) proved that the back and side – rack angles have
Yang et al. (2019) presented a model as a combination of mechanical the minor effect on the performance of drilling. As a result, this model
specific energy, principal component analysis, and wavelet analysis to ignores the effect of back and side – rack angles and consider the cutting
decide when to pull the bit out of the hole and predict cutters wear of the angle.
Kymera PDC bit. The model suggested a constant wear increment with Hareland and Rampersad (1994), Maurer (1966), and Teale (1965)

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A.Z. Mazen et al. Journal of Petroleum Science and Engineering 188 (2020) 106849

Fig. 2. Diagram of PDC drill bit cutting element DC , DOC, AC ; ro ; ri ; Ct ; and C.

Fig. 3. Updated bit selection chart (Bourgoyne et al., 1986).

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A.Z. Mazen et al. Journal of Petroleum Science and Engineering 188 (2020) 106849

Fig. 4. Inner and outer cones (Brandon et al., 1992).

Fig. 5. Various cutters geometry due to different locations.

Fig. 6. Flow chart illustrating the proposed technique.

where ðVr Þt is the theoretical volume of removed rock (in3 ), ​ T is the

Table 1 torque (lbm inÞ, MSE is the mechanical specific energy (psi), ​ AC is the
UCS for various rocks (Ragan, 2009). cutting area of the cutter (in2 ), and R is the radial or the distance from
Rock Type UCS, MPa
cutter to bit centre (inÞ:
MSE is defined as the ratio between the input energy to the volume of
Sandstone 70
removed rock. One of the major causes of error for this method in esti­
Limestone 25
Shale 20 mating the MSE is that the rock hardness is not considered because of the
Calcilutite 15 variety of rock strength (Tveit and Berg, 2016). Apart from the energy
Anhydrite 25 needed to crush the rock, the energy which is required to transport the
Clay 2
drilling cuts underneath the PDC bit to the surface should be considered
Dolomite 70
Salt 12
as well (Mohan et al., 2009).
Chert 180 In this work, MSE is correlated with rock hardness and mud hy­
Marble 100 draulic as shown in Eq. (2).
MSEmod: ¼ MSER þ MSEH ​ ​ ​ ​ (2)
used Eq. (1) which gives the theoretical volume of the removed rock per
The modified MSE for rock hardness is presented in Eq. (3) as
revolution.
follows:
Xi
ðVr Þt ¼ ​
T
​ ¼ ​ ​ 2π ​ AC � R ​ (1) MSER ¼ MSE � Hardness Ceofficient (3)
MSE i¼n

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A.Z. Mazen et al. Journal of Petroleum Science and Engineering 188 (2020) 106849

where, MSEmod: is the modified mechanical specific energy (psi), MSER ​ Sinor et al. (2001) defined the depth of cut per revolution as a
is the correlated mechanical specific energy for the rock hardness ðpsiÞ, function of ROP and RPM as stated in Eq. (15).
and MSEH is the correlated mechanical specific energy for the mud hy­
ROP
draulic (psi). DOC ¼ ​ ​ ​ (15)
RPM
MSE Chen et al. (2014) modified Eq. (15) by including the number of
Hardness Ceofficient ¼ (4)
Hardness blades as follows.
Jogi and Zoeller (1995) proposed that the rock hardness can be
ROP
estimated by use of Eq. (5). DOC ¼ ​ (16)
RPM ​ � ​ Nb
WOB � RPM
Hardness ¼ (5) where, Nb ​ is the number of blades.
ROP � Db
The PDC bit is designed with a plurality of cutters mounted on every
where, Hardness (psi), WOB ​ is the weight on bit (lbm Þ, RPM is the blade. Once weight applied on the bit and the bit started rotating, the
rotary speed (rpm), ROP is the rate of penetration (ft= hr), and ​ Db ​ is cutter dragged by torque to cut a layer of rock. Based on that, DOC is the
the bit diameter (inÞ: depth to which cutter penetrates the formation as shown in Fig. 2. In this
Kerr (1988) introduced that the specific hydraulic energy can be work, DOC ​ is calculated based on a modified method suggested in
expressed by the hydraulic horsepower efficiency as follows: Spread (2017) (see Eq. (17)). The objective of this technique is to esti­
mate the effective blade (EBÞ which is defined as how partially the
HP
MSEH ¼ HSI ¼ (6) cutters are involved in the drilling mechanism by computing the active
Db 2
cutters width to the blade length at a given ​ ROP. Then ​ DOC is
where, HSI is the bit hydraulic efficiency (psi), HP is the bit hydraulic calculated for the inner and outer cones using Eq. (17).
horsepower (lbÞ. ROP
HP and the bit pressure drop (PÞ can be estimated by applying Eqs. DOC ¼ ​ ​ ​ ​ (17)
RPM ​ � ​ Nb ​ � ​ EB
(7) and (8) as proposed by Kerr (1988).
where, EB is the effective blades (unitless).
P​ ​ � ​Q
HP ¼ ​ (7) EB can be calculated using Eq. (18) as a function of the total cutters’
1714
width on the blade ðCt Þ in both cones (see Fig. 2).
where, ​ P is the bit pressure drop (psiÞ, and Q is the flow rate (gpm). Ct
EB ¼ ​ (18)
Q2 ​ � ​ MW Lb
P¼ ​ ​ (8)
TFA2
where, Ct ​ is the total of cutters width (inÞ, and Lb is the blade length
where, MW is the mud weight (lb=galÞ, and ​ TFA is the bit total flow (inÞ.
Lb ​ can be determined according to the bit design as seen in Fig. 3.
area ( ​ in2 ).
The cone height ðGÞ is determined as shown in Fig. 3.
The torque also is correlated with rock friction in this work as stated
The inner and outer radius as shown in Fig. 4 can be determined
by Jogi and Zoeller (1995).
using Eqs. (19) and (20):
The modified torque is given in Eq. (9).
2
Tmod: ¼ T � ​ μ ​ ​ (9) ri ¼ ​ ​ � rb ​ ​ ​ (19)
3
where, Tmod: is the modified torque (lbm inÞ, and μ is the rock friction 1
(unitless). ro ¼ ​ ​ � rb (20)
3
T
μ¼ (10) where, rb is the bit radius (inÞ, ri is the inner cone radius (inÞ, and ro is
WOB ​ ​ � ​ Db
the outer cone radius ðinÞ:
By rearranging and substituting the parameters, Eq. (1) become: Then, Lb can be calculated in both cones using Eqs. (21) and (22) as
Xi shown in Fig. 4.
Hardness � T
� ​ ðVr Þa ¼ ​ 2π AC � R ​ (11) � �0:5
MSE � WOB ​ ​ � ​ Db i¼n Lb ðinnerÞ ¼ ​ ðri Þ2 þ ðGÞ2 (21)

� �0:5
X (22)
i
Hardness ​ � ​ T ​ � ​ Db Lb ðouterÞ ¼ ðro Þ2 þ ðGÞ2
¼ ​ ​8​ AC � R ​ (12)
MSE ​ � ​ DRIMP i¼n
Cutting ​ angle ​ ð ∅ Þ
where, DRIMP is the wear evaluation index ðlb =inÞ:
The cross-sectional area of the cutting ðAC Þ is dependent on the depth
The actual volume of removed rock as a function of ROP can be
of cut and cutter arc length, taking into account the distribution of all
defined as reported by Jogi and Zoeller (1995) in Eq. (13).
cutters within the radial (Chen et al., 2014; Glowka, 1985).
π ​ Db 2 ROP Chen et al. (2014) stated that the arc length of cutters varies ac­
ðVr Þa ¼ ​ � ​ ​ (13) cording to its location in the cone and the shape of the cutting area as
4 RPM
shown in Fig. 5. Therefore, the cutting area in this model will be
where, ðVr Þa is the actual volume of rock removed per revolution (in3 ). calculated as a function of cutter width ​ ðC) instead of arc cutter length (
DRIMP can be estimated using Eq. (14) as proposed by De Reynal AC ¼ DOC � C ) as described in Fig. 2 and Eq. (23).
(2011). Gouda et al. (2011) reported that the cutting force is obtained as
WOB follows.
DRIMP ¼ ​ ​ ​ (14)
DOC AC DOC ​ � ​ C
Fc ¼ ​ ​ �δ¼ ​ ​ �δ (23)
cos∅ cos∅
where, ​ DOC is the depth of cut (inÞ:

5
A.Z. Mazen et al. Journal of Petroleum Science and Engineering 188 (2020) 106849

where, ​ Fc ​ is the cutting force (lb), ∅ is the cutting angle (� ), and δ is cutters on both inner and outer cones using Eqs. (26), (27) and (30).
the hardness or the cutting force per unit area (psi). The overall methodology is shown in the flow chart in Fig. 6. The
The cutter width can be calculated as suggested in Spread (2017) steps of the modelling process are as follows:
using Eq. (24). Step 1: A collection of the required data, and apply the model
equations using Eqs. (17), (18), (24) and (25) to determine the cutter-
C ¼ 2 � ½DOC � ðDC DOCÞ� ​ ​ 0:5 (24) rock contact variables on inner and outer cones.
Step 2: The volume of removed rock is estimated on both cones using
Eq. (25) is developed based on Eq. (23) by including the number of
Eq. (13) and rock strength is determined based on cutting percentage as
cutters to compute DOC and C for inner and outer cones.
described in section 1.4.
WOB Step 3: DRIMP, MSE, and torque are calculated based on Eqs. (9),
DOC � C ​ � Nc ¼ � cos∅ ​ ​ (25)
Hardness (12) and (14) in both cones to show the influence of bit forces on bit
wear.
where, Nc is the number of cutters. The model has been applied to four vertical wells in the oil fields of
In the cutting process using PDC bit, the mechanism of the cutting Libya. The candidate wells were drilled by using three different sizes of
element at the bottom hole is complex. To facilitate the analyses of the sharp PDC bits and one used PDC bit. The formation stress ​ ðo Þ ​ is
0

cutting forces, based on bit design, the cone height is assumed to be 3 estimated according to the cutting percentage of the lithology that was
inch for the 8.500 and 12.2500 PDC bits, and 4 inches for the 1600 PDC bit as already provided in the well logging data (see section 1.4). The model
follows: equations are used to determine the PDC cutters wears by calculating the
height of cutters wear using the analyses of the drilling parameters
� Both 8.500 PDC are designed with a parabolic profile with shallow which determined as an average for every foot, then the estimated
cone. Therefore, the cone height value represents the option 6 as cutters wear is compared to the actual cutters wear obtained in the bit
shown in Fig. 3, which is within the range of 1/8 Db < G record.
� 1600 PDC is designed with a parabolic profile with a medium cone.
Therefore, the cone height value represents option 5 as shown in 1.2. Assumptions
Fig. 3, which is within the range of 1/8 Db < G � ​ 1/4 Db .
� 12.2500 PDC is designed with a parabolic profile with a medium cone. The following assumptions were made in this work to estimate the
Therefore, the cone height value represents option 5 as shown in abrasive cutters wear:
Fig. 3, which is within the range of 1/8 Db < G � ​ 1/4 Db .
� It is assumed all cutters that located in the inner cone have equal
Knowing the value of G and ​ Lb , the cutting angle can be easily wear, equal volume, and energy (Wang et al., 2018). The same is also
estimated. Given ​ WOB, ​ RPM, torque, ​ Nb , and density of cutters in applied for the outer cone.
both cones; gPROMS software is used to numerically solve the model � DOC and cutting area AC were assumed constant for all cutters in
and find the cutter width C and DOC for both inner and outer cones by every cone.
using Eqs. (24) and (25). Then, ​ DOC is modified by the use of Eq. (17). � The model considered the PDC cutters as round in shape as the most
It has been reported that the abrasive wear occurred caused by low used shape.
DOC, while high DOC caused an impact wear (Van Quickelberghe et al., � Blades have the same length and are assumed to be straight to
2006). The wear occurs while the bit is rotating but not cutting (Men­ calculate Lb ​ for both cones (see Fig. 4, Eq. 21, and 22). Also, the
sa-Wilmot et al., 2003). This was confirmed by Gouda et al. (2011) who space between cutters is neglected.
stated that no torque leads to wear. On this basis, this model neglects the � Radial location of cutters is equal to the average distance for both
cutting force on the wear effect and heat generated between the cutters – inner and outer. The inner radial ðRi Þ is the distance between cutter
rock interaction to estimate the bit wear. position at the middle of the inner cone to the bit centre (r2i ​ ), and the
Maurer (1966) stated that Eq. (26) gives the energy of rotation per
outer radial ðRo Þ is the distance between cutter position at the outer
revolution:
cone middle to the bit centre (r2o þ ri ) as shown in Fig. 5.
Energyrot: ¼ 2 ​ π � T � RPM (26)
1.3. Unconfined compression strength as a measure of formation strength
The correlated MSE then is defined as follows in Eq. (27) which then
be substituted in Eq. (12).
The model calculates the formation strength which is derived from
Energyrot: the unconfined compressional strength (UCS) values as shown in Table 1
MSE ¼ ​ ​ ​ (27)
ðVr Þa and the formation cutting percentage as shown in Fig. 7, 14, 22 and 27.
The standard lithology column was broken down into sections where the
Maurer (1966) defined the torque due to friction as a function of the
formation strength is measured.
wear cutter area with zero cutting force ​ as shown in Eq. (28).
0
T ¼ Cutting ​ Force þ ​ μ ​ � o � Db � Aw ​ (28) � The formation was divided into different sections as described and
shown in Figs. 7, 14, 22 and 27.
where, ​ Aw is the wear cutter area (mm2 Þ: � The formation strength was estimated by multiplying the cutting
In this work, Aw is estimated based on what was proposed by fraction with the values of UCS shown in Table 1.
Detournay (1993) as follows:
2. Results and discussions
A w ¼ L ​ � rc (29)
Based on the methodology described in the above sections, the re­
where, rc is the cutter radius (mmÞ, and L is the wear cutter height (mmÞ:
sults of analysis for the different wells of 1, 2, 3 and 4 are presented, in
By substituting Eq. (29) into Eq. (28), a new Eq. (30) is obtained to
ranking order, in this part.
calculate the torque as a function of L.

(30)
0
T ¼ ​ ​ 2 � μ � o � rb � rc � L ​
Finally, L can be estimated by taking into account the number of

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A.Z. Mazen et al. Journal of Petroleum Science and Engineering 188 (2020) 106849

2.1. Well number 1 contact with rock (see Fig. 8d), and as a result, the wear evaluation index
in the inner cone (DRIMP I) measured to 22 T/mm as shown in Fig. 8e.
The 8.500 PDC bit with 8 blades and 13 mm cutter size ran at 12720 While in the outer cone effective blades (EB O) was reduced to 25% and
ft–14343 ft to drill through Lower Sirte, Upper salt, and Mid anhydrite, wear evaluation index in the outer cone (DRIMP O) was estimated to 9
which mainly consist of shale, limestone, anhydrite, and salt (see Fig. 7). T/mm with no damage was estimated using the model equations in both
The bit is pulled out because there was a reduction in penetration rate cones as shown in Fig. 8f.
(PR) and dulled in the bit record for 1-1 (a measure of bit ware- 1 mm However, as the PDC bit penetrated into layers of 50% limestone
lost in inner and 1 mm lost in outer cone out of 8 mm). with 50% shale at 12736 ft, the ROP dropped because EB I was reduced
Table 2 summarizes the obtained average of ROP, UCS, DOC, EB, and to approximately 14.6% indicating the reductions on cutters width
DRIMP values. DRIMP in both cones which reflect the bit efficiency involved in drilling mechanism. In addition, the corresponding DRIMP I
condition increase with UCS and decreases with ROP for Lower Sirte, increased to 50.5 T/mm as shown in Fig. 8e that cause a damage in the
Upper salt, and Mid anhydrite rocks. Fig. 8 shows the profiles of ROP, inner cone (L I) which calculated to 0.87 mm lost (see Fig. 8f). While in
WOB, and RPM along the drilling depth obtained form well logging. The the outer cone DRIMP O increased to reach 20.6 T/mm with minor
figure also correspondingly shows the trends of EB, DRIMP, and L that damage. As the rock structure was changed to Upper salt and rock
were calculated using the model. It can be seen from Fig. 8 that EB for strength reduces to 15.7 MPa at 13360 ft (see Table 2), the DRIMP
both inner and outer cones is increased with ROP while WOB and DRIMP reduced to 43.6 and 17.7 for inner and outer cones, respectively due to
show an inverse proportional to ROP among the whole interval. increment in EB of 18 and 15.6% in both cones as a function of DOC.
The bit took 84.6 h to drill 640 ft while the bit drilling in Lower Sirte Accordingly, the bit turns to be more aggressive and this is accompanied
rock (see Table 2), and the results indicated that the bit received damage by an increment in the cutting area and rock removed. This is the case
in the early stage of the run. ROP has reached 33 ft/h at depth of 12729 for both inner and outer cones.
ft during drilling in 100% shale with WOB of 12 klbs and RPM of 85 rpm At 12750 ft depth, the ROP of the bit decreased from 5.87 to 2.6 ft/h.
(see Fig. 8b and c). 29.6% of the inner cone effective blade (EB I) were in Using the model equations, the L I was calculated to 1.58 mm lost at that

Fig. 7. Estimation of the Lower Sirte formation strength.

Table 2
8.500 PDC bit - rock type with estimated drilling parameters.
Rock Type Thick, ft UCS,MPa Time, hr ROP, ft/hr DOC, mm EB, % DRIMP, T/mm

Inner Cone Outer Cone Inner Cone Outer Cone Inner Cone Outer Cone

Lower Sirte 640 22.4 84.6 7.6 0.18 0.45 14.7 12.9 57.4 24.0
Upper salt 840 15.7 67.0 12.5 0.22 0.55 17.9 15.6 43.6 17.7
Mid anhydrite 143 26.2 33.2 4.3 0.16 0.39 12.8 11.1 81.6 32.9

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A.Z. Mazen et al. Journal of Petroleum Science and Engineering 188 (2020) 106849

Fig. 8. Trend of drilling parameters along the depth of drilling and corresponding estimated drilling parameters for well number 1.

Table 3
1600 PDC bit - rock type with estimated drilling parameters.
Rock Type Thick, ft UCS, MPa Hardness lb/sq. Time, hr ROP, ft/hr DOC, mm EB, % DRIMP, T/mm
mm
Inner Cone Outer Cone Inner Cone Outer Cone Inner Cone Outer Cone

Miocene 646 20.6 18.9 25.0 25.7 0.51 0.83 28.9 60.4 9.8 17.5
Oligocene 600 54.7 26.7 29.1 20.6 0.40 0.68 25.2 52.9 11.3 19.0
Upper Eocene 276 31.3 100.4 30.7 8.9 0.24 0.40 19.3 40.8 31.8 41.1

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A.Z. Mazen et al. Journal of Petroleum Science and Engineering 188 (2020) 106849

Fig. 9. Estimating of the stress of Miocene, Oligocene, and Upper Eocene.

Fig. 10. Damage of the 1600 PDC bit.

9
A.Z. Mazen et al. Journal of Petroleum Science and Engineering 188 (2020) 106849

Fig. 11. Trend of drilling parameters along the depth of drilling and corresponding estimated drilling parameters for well number 2.

depth as shown in Fig. 8f, where around 11% of EB I was involved in (see in the second row in Table 2). According to the model estimation,
drilling as shown in Fig. 8d. On the other hand, the wear in the outer the EB I and EB O were 22% and 19%, respectively.
cone (L O) was calculated to about 0.0045 mm lost (see Fig. 8f). The ROP At depth of 14200 ft during drilling in Mid anhydrite with the cor­
is raised to 20 ft/h as the bit started drilling into Upper salt rock at depth responding rock strength of 26.2 MPa, the PDC bit struggled and ROP
13360 ft where the rock strength is decreased to an average of 15.7 MPa dropped to only 4.3 ft/h, while DRIMP jumped to reach high average

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A.Z. Mazen et al. Journal of Petroleum Science and Engineering 188 (2020) 106849

values of 81.6 T/mm in the inner cone compared to 33 T/mm in the calculated to 5.3 mm as compared to 4.0 mm as an actual dull grading
outer cone (see the third row in Table 2). Based on the above discussions (see Fig. 11f). While the bit record presented that the bit is dulled to 1
and observations, keeping the bit in the hole with a sharp increase of mm lost in the inner cone as compared to 1.4 mm lost L I estimated by
DRIMP at the depth of over 14200 ft may cause more severe damage to model equations. Fig. 10 shows the real photos of damage of the 1600 PDC
the inner cone, so the benefit of pulling the bit out is obvious at this bit at the surface. The difference between the actual and the estimated
depth. wear as can be seen later in Fig. 18 would be justified to the assumptions
made in this model.
2.2. Well number 2
2.3. Well number 3
The 1600 PDC bit, designed with 6 blades - double rows and 16 mm
cutter size was run at a depth of 2014 ft–3536 ft to drill through The 8.500 PDC bit contributed from 6 blades and 19 mm cutter size
Miocene, Oligocene, and Upper Eocene. The interval consists of lime­ was used to drill 248 ft into Upper Sirte formation with the average ROP
stone, clay, dolomite, sand, and marble. The bit was pulled out of the of 11.8 ft/h (see Table 4). The formation mainly consist of shale. Hydro -
hole at the depth of 3536 ft after severe damage at the outer cone and
dulled for 1–4 as reported in the bit record.
The application consists of formations of Miocene, Oligocene, and
Upper Eocene rocks with the UCS of 20.6, 54.7 and 31.3 MPa, respec­
tively (see Table 3). In addition to the rock strength, the interval is
heterogeneous as shown in Fig. 9 which create several challenges for the
PDC to achieve a good ROP. The results of the model show that the PDC
bit had a total wear flat height in the outer cone L O greater than the
actual wear after drilling 1522 ft in 84.7 h (see Fig. 10).
The analysis of Table 3 reveals that the bit operated in good condi­
tions and spent over 50 h to drill nearly 1200 ft including Miocene (646
ft) and Oligocene (600 ft). At the depth of 3260 ft, WOB slightly
increased as shown in Fig. 11b where the bit started to require more
energy to maintain the same ROP. This is accompanied by a drop of EB
to 28.6 and 60%, and that led DRIMP to raise to 5.8 and 12.3 T/mm in
both inner and outer cone respectively.
A sudden increase in WOB from 5.4 to 9 tone was applied to the bit to
increase the ROP at 2187 ft (see Fig. 11b), which resulted in a drop of
35% in ROP (see Fig. 11a) due to formation cutting changed from 85%
clay to 75% limestone as shown in Fig. 9. Accordingly, the bit outer cone
showed a much stronger response as compared to the inner cone, DRIMP
O was nearly doubled its value with damage; L O estimated to equal 1
mm lost (refer to Fig. 11f). The bit design has a close influence on wear,
DOC showed more response in the outer cone because of the cutting
angle that resulted in an increase of cutting area. Based on that EB I of
25% out of total cutters width were involved in drilling (see Fig. 11d),
while EB O was doubled of the inner as shown in Fig. 11d. This is cor­
responding to the DRIMP values obtained in both cones. By using the
information of the Oligocene rock strength and rock hardness (see
Table 3), the UCS was increased from 20.6 to 54.7 MPa; while ROP was
decreased from 25.7 to 20.6 ft/h with no extra wear recorded in both
cones as shown in Fig. 11f. This has occurred as DRIMP increased to an
average of 10% in both cones (i.e. from 9.8 to 11.3 T/mm).
Approximately 31 h were spent to drill 276 ft along Upper Eocene
rock (see Table 3). However, the same period of time was taken to drill
600 ft in Oligocene rock using the same PDC bit. Nevertheless, the for­
mation strength was reduced from 54.7 to 31.3 MPa as shown in Table 3.
The closeness of this result may suggest that ROP is insensitive to rock
hardness. The DOC and EB in both cones as shown in Table 3 decreased
dramatically. This is related to ROP with a noticeable increment in
DRIMP in both cones until the end of the 1600 section.
Wear measurement was made at every foot. However, it is logical to
consider the wear when the bit was pulled out of the hole. To determine
the bit dull condition, the wear model equations were used. L O was Fig. 12. Estimating of the Upper Sirte rocks strength.

Table 4
8.500 Used PDC bit - rock type with estimated drilling parameters.
Top, ft BTM, ft Thick, ft Rock Type UCS, MPa Hardness lb/sq.mm Time, hr ROP, ft/hr DRIMP, T/mm

Inner Cone Outer Cone

10425 10673 248 Upper Sirte 19.54 281.9 20.97 11.83 19.81 12.37

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A.Z. Mazen et al. Journal of Petroleum Science and Engineering 188 (2020) 106849

Fig. 13. Trend of drilling parameters along the depth of drilling and corresponding estimated drilling parameters for well number 3.

Guard mud, a clay free designed, was pumped into this section as a
solution for reactive shale. The bit was pulled out of the hole at depth of
Table 5
10673 ft to check or change the bottom hole assembly and dulled in the
8.500 Used PDC bit – cutting angle for inner and outer cones.
bit record for 3–4.
Cutting angle, DOC, mm L, mm The PDC drilled till depth of 10629 ft under controlled conditions,
�

Inner cone 43.1 0.422 0.340 with stable DRIMP along an approximate of 70% interval of shale as
Outer cone 25.8 0.650 0.450 shown in Fig. 12. The average ROP estimated to 93 ft/h and DRIMP

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A.Z. Mazen et al. Journal of Petroleum Science and Engineering 188 (2020) 106849

calculated to 15.1 and 9.4 T/mm in the inner and outer cones, respec­
tively as shown in Fig. 13e. During the section from 10629 to 10635 ft,
and as the bit penetrated into traces of calcilutite rock (see Fig. 12), the
driller decided to reduce WOB and increase RPM as shown in Fig. 13b
and c. As a result, the average ROP declined to 87 ft/h and DRIMP was
suddenly raised to 24.5 and 15.2 T/mm on both cones which reflect the
poor bit efficiency with a drop in the effective blade in both inner and
outer cones of 21.2 and 23%, respectively (see Fig. 13d). At depth of
10658 ft, ROP dropped from 8.8 to 3.6 ft/h with an increases in DRIMP
because of the 15% increment of hardness from 18.3 to 22.5 T/mm in
inner cone and from 11.4 to 14 T/mm in the outer (see Fig. 13e).
The model has assumed that the bit is sharp, and hence the cutting
angle, DOC, and wear in both cones were calculated as summarised in
Table 5.
The results of Table 5 show that the outer cone is more responsive to
cutting force caused by the radial distance to the bit centre, which was
assumed constant for all cutters. The bit which used to drill in this
section was a reused bit and maybe of partial integrity, i.e. cutter ele­
ments may be lost. However, the bit was evaluated at the surface, for 3–4
in the bit record, and concluded that the PDC has experienced more
damage as compared to the results obtained from the model. This
explained the difference obtained in the comparison between the esti­
mated bit wear and the actual bit wear as shown later in Fig. 18.

2.4. Well number 4

The 12.2500 PDC bit which designed with 6 blades and 16 mm cutter
size was used to drill in 12.2500 hole. The bit drilled 112 ft in Algata
formation and 1588 ft in Gir formation to the final planned depth of
6650 ft (see Table 6). The bit was pulled out from the hole at 6650 ft and
dulled for 3-X. (The actual dull grading of the outer cone is unknown).
The selected 12.2500 PDC drilled 1700 ft in 38.3 h through inter­
bedded formation of calcilutite, dolomite, shale, and some layers of
chert as shown in Fig. 14. The results of the model show that the bit has
pulled out in very good condition with no abrasive wear in both cones as
shown in Fig. 15f.
The high values of DOC are an indication of the impact wear resulted
from sudden change of rock strength, and that can be seen in Fig. 15d for
both inner and outer cones. This is evidence from the DOC which is
reached to a value of approximately 3 mm for the outer cone. The bit is Fig. 14. Estimating of the Algata formation strength.
considered as efficient if the bit has low DRIMP and ROP is improved.
The previous results for wells 1, 2 and 3 have demonstrated that the ROP
estimates the abrasive cutters wear in wells 1, 2 and 3 based on the
should exhibit an inverse relationship with DRIMP, while in well 4, this
assumptions made earlier for the development of the model. The good
is not the case and ROP is directly proportional to DRIMP in both inner
agreements are found between the estimated and the actual cutters wear
and outer cones (see Fig. 15a, e, 16 and 17). This is clearly against the
for wells 1 and 2. There is a considerable difference between the actual
principles of the abrasive wear model. Additionally, considering the
and the determined cutters wear in wells 3 and 4. This is attributed to
nature of the trend of ROP vs. DOC, ROP was decreased instead of
the case as the PDC bit in well 3 was not a new bit, so the bit may lost
rapidly increasing with DOC (see Figs. 16 and 17). These phenomena
partially cutter elements during drilling in an offset well before rerun
strengthen the idea that the wear which occurred on the 12.2500 PDC is
again to drill the 8.500 section in well 3. Also, the wear occurred to the bit
not abrasive wear but it is an impact wear. Accordingly, bit wear cannot
drilled the 12.2500 section in well 4 was a result of impact wear and not
be estimated by the model equations, and that can be clearly shown with
because of abrasive wear.
poor comparison between the estimated and actual wear in both cones
as summarised later in Fig. 18.
Fig. 18 compares the obtained results of the calculated wear using
the model and actual cutters wear recorded in the field. The model

Table 6
12.2500 PDC bit - rock type with estimated drilling parameters.
Top, ft BTM, ft Thick, ft Rock Type UCS, MPa Hardness lb/sq.mm Time, hr ROP, ft/hr DRIMP, T/mm

Inner Cone Outer Cone

4950 5062 112 Algata 14.46 99.85 2.68 41.79 21.11 11.45
5062 6650 1588 Gir 21.27 136.94 35.66 44.53 23.63 12.87

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A.Z. Mazen et al. Journal of Petroleum Science and Engineering 188 (2020) 106849

Fig. 15. Trend of drilling parameters along the depth of drilling and corresponding estimated drilling parameters for well number 4.

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A.Z. Mazen et al. Journal of Petroleum Science and Engineering 188 (2020) 106849

Fig. 16. Average. DOC - ROP and DRIMP (Inner cone).

Fig. 17. Average. DOC - ROP and DRIMP (outer cone).

3. Conclusion 4. Effective blades, as well as DOC, are shown to be inversely pro­

portional to the cutting angle.
The following conclusions can be drawn in this work: 5. Optimizing the drilling parameters i.e. WOB and RPM by moni­
toring the effective blade and DRIMP are key to improve bit
1. The model presented in this paper is developed to aid PDC bit performance.
design and evaluation. The model also serves as a methodology to 6. The actual bit cutters wear for wells no 1 and 2 were in good
identify additional factors such as bit design and bit hydraulic agreement with the estimated wear by the model equations based
that were not included in the past. on the model assumptions.
2. The concept of equal cutter wear and volume across the bit face 7. The actual bit cutters wear for wells no 3 and 4 exceeds the
reported in the literature is incomplete. This model does not estimated bit wear. The bit ran in well 3 was a used bit, while the
consider the single cutter test, and accordingly, the model enables damage occurred to bit drilled the 12.2500 section in well 4 was
to estimate of the cutters wear in the inner and outer cones. due to impact wear (see Fig. 18).
3. One of the most important outcomes of this work is the inclusion 8. This model has given satisfactory results only for sharp PDC bits
of the cutter width and the estimation of the effective cutters per and can only be applied to estimate the abrasive cutters wear. In
blade which are involved in the drilling mechanism. This prin­ addition, DRIMP changed in agreement with the bit life and is
ciple can account for designing a PDC bit with varied wear. mainly depend on the bit geometry.

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A.Z. Mazen et al. Journal of Petroleum Science and Engineering 188 (2020) 106849

Fig. 18. Comparing the actual and the estimated bit wear for wells 1, 2, 3, and 4.

9. The proposed model in this paper is valid and can be applied for Nejat Rahmanian (Principal superviosr) and Prof. I.M. Mujatba (2nd
homogeneous and heterogeneous formation under the assump­ supervisor) and Dr A. Hassanpour (external collaborator). Original
tions made in this work. Furthermore, the input data are standard draft was prepared by the student and revised and commented by the
and readily available or easy to obtain. superviors and the collaborator.
10. It would be most desirable if the model can be extended to apply
to different shapes of cutters such as oval cutters. This is the Acknowledgment
ongoing research and will look into this in the future.
The financial support from ministry of higher education in Libya is
Author’s contributions gratefully acknowledged.

Mr Ahmed Mazen is a PhD student works under supervision of Dr

Nomenclature

AC Cutting area of the cutter, in2

Aw Wearflat area, in2
C Cutter width, in2
Ct Total of cutters width, in
Db Bit diameter, in
DC Cutter diameter, in
DOC Depth of cut, in
DRIMP Wear evaluation index, lb=in
DRIMP I Wear evaluation index in inner cone, lb=in
DRIMP O Wear evaluation index in outer cone, lb=in
EB ​ Effective blades, unitless
EB I Effective blades in inner cone, unitless
EB O Effective blades in outer cone, unitless
Fc Cutting force, lb
g gage height, in
G Cone height, in
HP Bit hydraulic horsepower; lb
HSI Bit hydraulic efficiency, psi
L Wear bit height, mm
Lb ðinnerÞ Inner blade length, mm
Lb ðouterÞ Outer blade length, mm
L I ¼ Wear bit height in inner cone, mm
L O ¼ Wear bit height in outer cone, mm
Lb Blade length, in
MSE Mechanical specific energy, psi
MSEH Correlated mechanical specific energy for mud hydraulic, psi

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A.Z. Mazen et al. Journal of Petroleum Science and Engineering 188 (2020) 106849

MSEmod: Modified mechanical specific energy, psi

MSER Correlated mechanical specific energy for rock hardness, psi
MW Mud weight, lb=gal
Nb Number of blades, unitless
Nc Number of cutters, unitless
P Bit pressure drop, psi
PR Penetration rate, ft=hr
Q Flow rate, gpm
R Distance from cutter to bit centre, in
Ri Inner radial distance, in
Ro Outer radial distance, in
ROP Rate of penetration, ft=hr
RPM Rotary speed, rpm
rb Bit radius, in
rc Cutter radius, in
ri Inner cone radius, in
ro Outer cone radius, in
T Torque, lbm in
TFA Bit total flow area, in2
Tmod: Torque, lbf in
UCS Unconfined compressional strength, psi
ðVr Þa Actual volume of rock removed, in3
ðVr Þt Theoretical volume of rock removed, in3
WOB Weight on bit, lbm
δ Hardness or the cutting force per unit area, psi
∅ Cutting angle, �
o Formation stress, psi
0

μ Rock friction, unitless

Appendix A. Supplementary data

Supplementary data to this article can be found online at https://doi.org/10.1016/j.petrol.2019.106849.

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