True Power Law Drilling Fluid Model: Effect of Its Rheological Parameters on Static Particle Settling Velocity

International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395 -0056
Volume: 03 Issue: 01 | Jan-2016 www.irjet.net p-ISSN: 2395-0072
© 2015, IRJET ISO 9001:2008 Certified Journal Page 77
True Power Law Drilling Fluid Model: Effect of Its Rheological
Parameters on Static Particle Settling Velocity
Mazen Ahmed Muherei 1, Saeed Salim Basaleh2
1&2 Assistant Professor, Department of Petroleum Engineering, Hadhramout University, Hadhramout, Yemen
---------------------------------------------------------------------***---------------------------------------------------------------------
Abstract - Adequate cuttings removal from the
bottom hole of an oil well to the surface during a rotary
drilling is critical for cost-effective drilling. The major
factors that describe cuttings transport particularly in
vertical sections are fluid effective viscosity and velocity
which influence particle settling velocity. Rheological
modelling of drilling fluids in oil fields is usually
described by Bingham plastic and power law models.
These models gain popularity because their specific
descriptive parameters are fairly easy to estimate.
Standard methods use Fann V-G Meter dial reading at
600 and 300 rpm to determine these rheological
parameters. Unfortunately, these points correspond to
higher shear rates which seldom prevail during particle
settling. This work aims to investigate different shear
rates to derive power law rheological parameters and
show their influence on the magnitude of effective
viscosity and hence settling velocity. The results show
that data pair of R600/R3 was the best to predict the full
spectrum of the fluid rheogram. Using Chien (1994)
correlation, data pair of R200/R100 give the best results
for predicting observed settling velocity of a non-
spherical particle settling in a static fluid. Data pair of
R200/R100 out performs commonly used data pairs
(R600/R300, R100/R3 and R6/R3). Furthermore, Using
modified Moore correlation with effective viscosity as
suggested by Chien obtained excellent predictions with
error less than 1% particularly with R600/R100.
Key Words: Drilling fluid, Drill cuttings, Hole cleaning,
Rheological model, Settling velocity.
1. INTRODUCTION
Effective removal of cuttings is the primary task of any
drilling fluid. Efficient hole cleaning is essential for the
drilling practices to succeed and are required to complete
the well at lower costs. Inadequate hole cleaning can
trigger other wellbore problems which add to the well cost
significantly. In vertical wells, cuttings tend to settle along
the mud column and if this rate of settling is predicted
accurately it is enough, for good hole cleaning, to pump the
drilling fluid at a velocity greater than this settling
velocity. However, drilling in deep water presents many
challenges and is characterized with narrow margins
between formation pore pressure and fracture pressure
gradients. Estimating cuttings settling velocity to be
considerably higher or lower than actual will result in
circulating the drilling fluid with either higher or lower
velocities than required. Both of these are undesirable,
since higher velocities will increase cost for unnecessary
additional fluid, require higher pumping and fluid
handling capacities, consume more power, increase
erosion of uncased sections and increase frictional
pressure loss [1].
The pressure loss may significantly increase up to the
extent that violates allowable equivalent circulation
density (ECD). As the drilling continues this way, the ECD
becomes more critical and formation fracturing and lost of
circulation (LOC) becomes the role than the exception. On
the other hand, too low circulation velocities will cause
less cuttings to be transported out of the well and more
cuttings to accumulate in the annulus. Accumulated
cuttings could significantly increase fluid column weight
during circulation which may exceed the formation pore
pressure that frequently develop to formation fracture and
complete loss of the drilling fluid. Prediction of down word
velocity of drill cuttings with a considerable degree of
accuracy will assist in determining optimal circulation rate
that will provide sufficient hole cleaning and yet
minimizes fluid volume, power, and equipment
requirements.
The ability of a drilling fluid to lift cuttings is affected by
many factors and there is no universally accepted theory
which can account for all the observed phenomena [2].
Several particle slip velocity correlations are available in
the drilling literature, not the least are Chien [3], Zeidler
[4], Moore [5], Walker and Mayes [6], Peden and Luo [7]
and Chien [8]. These correlations have been developed in
general from Newton’s equation for predicting terminal
velocities of spherical particles in Newtonian fluids. The
principal differences are the way by which the non-
Newtonian nature of the drilling fluids, the irregular shape
of drill cuttings and the drag coefficient are taken into
account in these correlations. Samble and Bourgoyne [9]
have evaluated experimentally the correlations of Chien
[3], Moore [5] and Walker & Mayes [6]. The experimental
data used for this evaluation were obtained in Newtonian
and non-Newtonian fluids for both static and flowing
conditions. Among the evaluated correlations, the
procedure proposed by Moore [5] gave the lowest average

error for all fluids studied. Furthermore, Skalle et al. [10]
pointed that Chien [8] and Walker Mayes [6] correlations
still have a good sound in petroleum industry.
Furthermore, Chien [8] correlation was adopted by the
API as the recommended procedure for drilling fluids [11].
Of most importance is the prediction of the drilling fluid’s
effective viscosity during the settling process. The
apparent or effective viscosity represents the viscosity at
the specific shear rate pertinent to that annular location in
an annular flow situation, and does not necessarily
represents the viscosity around the settling particles [7].
When fluid velocity approaches zero and the fluid
becomes stagnant, effective viscosity will approach
infinity.
Moore [5] suggested use of an effective viscosity term that
is derived from equating pressure drop equation of
Newtonian and power law fluids, while Chien [3] proposed
use of an effective viscosity that is derived from equating
pressure drop of Newtonian and Bingham plastic fluids.
Accordingly, Moore [5] correlation employed the power
law model parameters (K and n) that are basically
determined at Fann viscometer readings of R600 and R300
to determine effective viscosity. Similarly Chien [3]
correlation made use of Bingham plastic parameters (p
and YP) that are again determined from viscometer
readings of R600 and R300. However, API [12] methods for
power law fluids recommend use of Fann viscometer
readings of R100 and R3 for calculating pressure loss inside
the annulus. Becker and coworkers [13] stated that Fann
rotary speed of 300 and 600 rpm induce shear rates
higher than those that typically occur in actual drilling.
Chien [8] emphasized that the effective viscosity working
on the settling particles should be determined at the
settling shear rate which is basically unknown. He
recommended use of rheological parameters that are
determined with a viscometer at low shear rates.
However, Skalle et al. [10] stressed that the relative error
is large at such low shear-stress readings.
There is thus a great distortion as which of the Fann
viscometer readings should be used to determine the
more representing rheological parameters. In this paper,
the theory behind slip velocity correlations is presented.
An example of drilling fluid was selected from literature.
The Fann 35 VG meter 6 speed data were plotted and the
fluid model identified. The rheological parameters are
calculated using all shear stress/shear rate data pairs that
can be obtained from a Fann 35 VG meter. The fluid
rheogram was predicted again using previously calculated
rheological parameters. Results were statistically
compared. Finally the settling velocity according to Chien
[3,8] are computed using different previously estimated
rheological parameters. Settling velocity values are plotted
and compared with regard to observed settling velocity.
2. THEORY OF SETTLING VELOCITY
2.1 Settling Velocity Concept
Settling velocity is commonly used as a measure of the
ability of the drilling fluid to transport drill cutting in
vertical wells. Traditionally, terminal velocity has been
used in place of slip velocity; however, terminal and slip
velocities are two different concepts. Terminal velocity is
measured by dropping a single particle into a fluid; it does
not account for particle interference or for accumulation
of cuttings. It refers to the final constant falling velocity of
a single particle in a calm fluid. Constant falling is achieved
when buoyancy and resistance forces of the fluid balances
the gravity of the particle. Terminal velocity is measured
in a vertical direction, inaccuracy increases with
increasing wellbore inclination [1].
Slip velocity refers to the velocity difference between the
cuttings and the circulation fluid. Circulation fluid velocity
varies across the flow area (depending on the flow
regime). Fluid velocity near the wall is much smaller than
the velocity near the centre of the flow area; cuttings and
slip velocities varies accordingly across the flow area.
However, slip velocity term is used to refer to the
difference between the average fluid velocity and the
average cuttings velocity rather than to the real slippage
between cuttings and circulation fluid. Unlike terminal
velocity which is always measured in the vertical
direction, slip velocity is measured in the direction of flow.
2.2 Drag Coefficient Correlations
If one considers the forces acting on a discrete particle
falling through a quiescent liquid, Newton’s law for
terminal velocity can be reached. Initially the particle will
accelerate then at some point rather quickly the forces
causing motion and those impeding motion will balance,
thus the acceleration of the particle will become zero and
will move with a constant velocity, called the terminal
velocity. Ideally there are two forces acting on such a
particle; the force developed as a result of friction between
the particle and the liquid (FD) and the force due to the
effective weight of the particle which is the difference
between gravity and buoyancy forces (Fg and FB,
respectively).
pg VgF 
lB VgF 
)( lpeff VgF   (1)
The frictional or drag force is less easy to quantify.
Generally, it was found to be a function of the properties of
both the liquid and the solid and can be expressed
mathematically as:

2
2
P
lPDD
U
ACF  (2)
In accord to Newton’s second Law, the forces should set
equal to each other; that is:
2
)(
2
p
lPDlp
U
ACVg   (3)
By re-arranging of equation 3, for Up gives:
Pl
lp
D
p
A
V
C
g
U

 )(2 
 (4)
For spherical particles V and AP can be expressed in terms
of particle diameter as:
6
3
pD
V


 &
4
2
p
P
D
A


 then p
P
D
A
V
3
2

Substituting the value of V/AP in equation (4) gives:
lD
plp
P
C
Dg
U


.3
)(4


 (5)
Equation (5) is the Newton’s law for terminal settling
velocity of a spherical particle.
However, the drag coefficient in equation (5) has been
found to be a function of the Reynolds Number:
0.5
24 3
0.34D
RP RP
C
N N
  
For very slow particle fall and laminar slip regime (NRP <
0.5), Stokes reduced drag coefficient expression to [7]:
RP
D
N
C
24

Equation (5) becomes
l
Plplp
P
UDg
U


243
100)(4
2


 (6)
Squaring equation (6) and re-arranging for Up again:


18
)(
100
2
plp
P
Dg
U

 (7)
Equation (7) is the Stokes law for the terminal settling
velocity of a spherical particle under laminar flow
conditions. In turbulent flow (NRP >1000) the drag
coefficient becomes constant and Rittinger’s equation (CD
= 0.5) may be used:
l
plp
P
D
U

 )(
15.51

 (8)
2.3 Moore Settling Velocity Correlation
Moore [5] suggested use of an apparent Newtonian
viscosity obtained by equating the laminar flow frictional
pressure losses in the Newtonian model with frictional
pressure losses in the power law fluid model; the
following expression was resulted using c.g.s units:
 
n
n
a
PH
a
n
U
ddK


















 


25.0
1
2
)(
975.11
1
 (9)
This correlation is for spherical particles and use average
laminar flow velocity but does not account for the effect of
particle slippage upon fluid shear stress. The friction
factor (drag coefficient) in this correlation was obtained
from slip-velocity experiments conducted with actual
limestone and shale cuttings in mixtures of water and
glycerine. Moore [5] pointed that for fully turbulent flow
(NRP = 2000), the friction factor remains constant at a
value of about 1.5. Substituting this value in Newton’s
equation (eq. 5), the following relation was obtained:





 

l
lp
pp DU


53.29 (10)
For laminar flow (NRP ≤ 1) the drag coefficient, according
to Moore [5], was approximated to:
RP
D
N
C
40

Substitution in equation (5) gives:
 lp
a
p
p
D
U 


2
3270 (11)
However, Moore [5] has pointed that equation (11) has a
limited application since in most cases the particle
Reynolds number will exceed 1.0 when the flow around
the particle is laminar. Instead, Moore [5] has developed
an equation for the drag coefficient that should be applied
whenever the particle Reynolds number falls in the range
of 10 to 100:

RP
D
N
C
22

Similarly by substituting this value in equation (5), the
following equation was attained:
 
333.0
667.0
333.0
233.15
l
lp
a
p
p
D
U




 (12)
2.4 Chien Settling Velocity Correlations
Chien [3,8] presented two empirical correlations for the
settling velocity of drill cuttings for rotary drilling
operations. Both for determination of the settling velocity
of cuttings in all slip regimes. In 1972, Chien suggested use
of a drag coefficient slightly different from that of equation
(7); CD = 24/NRP + 1.714. Substituting these values in
equation 5, we approached at the following equation:























 


















 11
155741
07.0 2
l
lp
pl
e
p
pl
e
p
D
D
D
U





……………………………………………………………………..……….. (13)
However the equation published by Chien expressed in
field units was [14]:























 


















 11
36800
0075.0 2
l
lp
pl
e
p
pl
e
p
D
D
D
U





……………………………………………………………………..……….. (14)
Where
Vs = Particle slip velocity, ft/sec
Dp = Particle diameter, in
ρp = Particle density, ppg
ρf = Fluid density, ppg
µe = Fluid effective viscosity, mPa.sec (cp)
Chien [8] emphasized on particle shape factor or
sphericity (). Particle sphericity is the ratio between the
surface area of a sphere of volume equal to that of the
particle and the surface area of the particle. Because a
sphere has the minimum possible surface area of any
shaped particle of the same volume, the value of sphericity
is always less than or equal to 1. For a spherical particle,
=1; the greater the difference between the particle's
shape and a spherical shape, the smaller the value of .
Chien [8] stressed that majority of particles involved are
irregularly shaped and the conveying fluid is commonly
non-Newtonian in nature so neither Stoke’s law nor
Rittinger’s equations are valid. Accordingly, he developed
a correlation to predict settling velocity of irregularly
shaped particles in Newtonian and non-Newtonian fluids
for all types of slip regimes. In this correlation, the drag
coefficient is approximated by: CD = 30/NRP + 67.289/
e5.03
. Substituting this values in equation 5, we
approached at the following equation:










 11
3911734
00223.0 2
03.5
03.5
Y
X
De
XeU p
p


……………………………………………………………………………….. (15)
Where
pl
e
D
X



l
lp
Y

 

The equation published by Chien [8] took the following
form where viscosity is in Pa.sec and =0.7924:























 


















 11
0727.0
120 2
l
lp
pl
e
p
pl
e
p
D
D
D
U





………………………………………………………………………………. (16)
Equation 16 has the same parameters as his previous
equation (eq. 14) except for the numerical coefficients.
Similarly, the equation adopted by API expressed in field
units and took the following form:








 11
.920790
.002403.0 2
03.5
03.5
Y
X
De
XeU p
p


……………………………………………………………………………….. (17)
Where;
pl
e
D
X



l
lp
Y

 

2.5 Effective Viscosity
The settling problem of drilled cuttings in drilling fluids is
complicated by the non-Newtonian behavior of these
fluids, i.e. their shear-dependent viscosities. Theoretically
the viscosity affecting the particle settling velocity in a
non-Newtonian fluid should be that of the fluid envelop
surrounding the particle and this viscosity depends on the
shear rate distribution around the particle. Chien [3]
suggested that, for water-based bentonite suspension, the

Bingham plastic viscosity could be used as an equivalent
viscosity and for polymer type drilling fluids, the effective
viscosity depends on annular fluid flow. Similarly, Moore
[5] suggested an effective viscosity which depend on
annular fluid flow. Accordingly, these viscosities are
functions of the fluid velocity and are independent of the
particle settling velocity. If the fluid velocity approaches
zero, i.e. the fluid is stagnant, the effective viscosity for
annular pipe flow will approach infinity and the settling
velocity would be zero. According to Luo [15], this is
obviously not true.
The effective viscosity suggested by Chien [8] is
determined at the settling shear rate. The settling shear
rate is the ratio of particle settling velocity to its particle
diameter. The effective viscosity of power law fluid is
equal to shear stress at that shear rate (µe = Kγn-1). Since
the shear rate prevailing while the particle is settling out
of the fluid is not known, a trial and error or numerical
iteration method is required to predict the particle settling
velocity in non-Newtonian fluid.
For non-Newtonian fluids, viscosity depends on the shear
rate and knowledge of the settling shear rate is important
for evaluating the viscous force experienced by the
particle. In the turbulent regime the viscosity has a minor
effect on drag force; therefore the settling shear rate has
no important role in turbulent slip. However, for
Newtonian fluids, viscosity is independent of the shear
rate and the concept of a settling shear rate is not used. It
should be pointed out that, in contrast to the previous
suggestions, Chien [8] suggestion depends only upon the
particle settling velocity and is independent of the fluid
velocity. This is also a drawback that might affect settling
velocity predictions.
3. RHEOLOGICAL MODEL
Rheology is classically defined as the study of the
deformation and flow of matter. From a rheological
perspective, drilling fluids are thixotropic (time-
dependent) as well as temperature and pressure
dependent. These fluids are visco-elatic exhibiting both
viscous and elastic properties [16]. In conventional
drilling, drilling fluids are modelled with classical
rheological models like Bingham plastic or power law
model and fluid behaviour is defined with only two points
of the rheological relation (R600 and R300). With increased
use of polymer-based fluids as drilling fluids, the power
law rheological model receives great attentions because it
describes the behaviour of these fluids better than the
Bingham plastic model [17]. The power law model can be
expressed by:
n
K  (18)
The term "K" is defined as consistency index and describes
the thickness of the drilling fluid. The exponent "n" is
called flow behaviour index. There is no term for yield
point and fluids that follow this model have no shear
stress when shear rate is zero. The flow behavior index "n"
indicate the degree of non-Newtonian. If values of "n"
equal to unity the model reduces to Newtonian fluids. As
the values of "n" become lesser than unity, the fluid shows
shear-thinning properties and defined as pseudo-plastic
fluid. At greater than unity n values the fluid shows shear-
thickening properties and defined as dilatant. These
rheological properties help evaluate consequences of fluid
flow behaviour and critical decisions made continually
based on these parameters [16].
3.1 Rheogram of Example Mud
The example drilling fluid was selected from Guliyev [18]
study on settling velocity. The fluid composed 5% PHPA
plolymer-water mixture. The six Fann 35 VG meter dial
readings and corresponding revolutions per minute as
well as other fluid and particle properties are listed in
table-1. The shear stress/shear rate data was plotted and
fitted to a straight line (Bingham plastic model) and
exponent line equations (power law model). As shown in
Figure-1, the example mud follows power law model.
Table-1: Fluid and particle properties
RPM 600 300 200 100 6 3
Shear Rate, sec-1
1022 511 341 170 10 5
Dial Reading 63 46.5 39 29 8 6
Shear Stress, lb/100ft2
67.2 49.6 41.6 30.9 8.5 6.4
Shear Stress, Pa 32.19 23.76 19.93 14.82 4.09 3.07
Fluid Density, g/cm3
1.001 Particle density, g/cm3
2.672
Particle Size, cm 0.3387 Particle Sphericity 0.91
Particle Settling Velocity in the Stagnant Fluid, cm/sec 0.7
3.2 Calculation of Rheological Parameters
To calculate "n" and "K" values, a mud's Fann 35 VG meter
dial readings and corresponding revolutions per minute
are required. Two data pairs are required for a solution.
Generally, R600/R300 are in common use. Chien [8]
suggested use of R6 and R3 to estimate the settling velocity.
He pointed out that the settling shear rate of the laminar-
slip regime ranges from 0.1 – 50 sec-1 and the fluid
rheological data also should be measured in the same
shear-rate range. However, API [12] in its recommended
practice suggest use of R100/R3 for calculating slip velocity
of drill cuttings as experienced in an oil well annuli. This
study employs all data pairs that could be obtained from
the standard Fann 35 VG meter dial readings and
corresponding revolutions per minute. Equations 19 and
20 are general equations for determining fluid flow
behavior index and consistency index, respectively. The
resulted rheological parameters are shown in Tables 1-5.
Comparisons are then observed and discussed.

)/log(
)/log(
12
12


n (19)
n
K
2
2
511.0


 (20)
y = 0.028x + 6.7045
R² = 0.8999
y = 1.4707x0.4466
R² = 0.9999
0
5
10
15
20
25
30
35
40
0 500 1000 1500
ShearStress,Pascal
Shear Rate, sec-1
Fig-1: Example drilling fluid rheogram.
4. RESULT AND DISCUSSION
4.1 Predicting Mud Rheogram
Tables 2-7 are a statistical comparison of the accuracy of
each specific couple of dial readings and their
corresponding shear rates in predicting shear stress for
the example fluid. Tables 2-6 contain the values of the
example fluid's rheological parameters calculated using
different pairs of dial readings and their corresponding
shear rates. The shear stress at six Fann 35 VG rpms are
calculated again using these rheological parameters.
Table 7 contain the values of the example fluid's
rheological parameters calculated from average of the
previously calculated values from all different pairs of dial
readings as well as rheological parameters estimated from
non-linear regression using all available shear
stress/shear rate data.
Table-2
Data Pair R600/R300 R600/R200 R600/R100
n 0.4381 0.4364 0.4329
K, Pa.secn
1.5464 1.5647 1.6031
Measured
Shear Stress, Pa Calculated Shear Stress, Pa
rpm τ τ %E τ %E τ %E
600 32.19 32.19 0 32.19 0 32.19 0
300 23.76 23.76 0 23.79 0.13 23.85 0.38
200 19.93 19.89 -0.2 19.93 0 20.01 0.4
100 14.82 14.68 -0.94 14.73 -0.61 14.82 0
6 4.09 4.28 4.65 4.31 5.38 4.38 7.09
3 3.07 3.16 2.93 3.19 3.91 3.25 5.86
Average Error, % 1.07 1,47 2.29
Standard Deviation 2.2 2.52 3.27
Table-3
Data Pair R600/R6 R600/R3 R300/R200
n 0.448 0.4435 0.4338
K, Pa.secn
1.4439 1.4896 1.5885
Measured
600 32.19 32.19 0 32.19 0 32.09 -0.31
300 23.76 23.6 -0.67 23.67 -0.38 23.76 0
200 19.93 19.68 -1.25 19.78 -0.75 19.93 0
100 14.82 14.43 -2.63 14.54 -1.89 14.75 -0.47
6 4.09 4.09 0 4.18 2.2 4.35 6.36
3 3.07 3 -2.28 3.07 0 3.22 4.89
Average Error, % 1.14 0.14 1.75
Table-4
n 0.4298 0.4499 0.4447
K, Pa.secn
1.6287 1.4368 1.4841
Measured
Shear Stress, Pa Calculated Dial Reading
600 32.19 32.01 -0.56 32.46 0.84 32.34 0.47
300 23.76 23.76 0 23.76 0 23.76 0
200 19.93 19.96 0.15 19.8 -0.65 19.84 0.45
100 14.82 14.82 0 14.49 -2.23 14.58 -1.62
6 4.09 4.42 8.07 4.09 0 4.17 1.96
3 3.07 3.28 6.84 2.99 -2.61 3.07 0
Average Error, % 2.42 0.78 0.21
It is clear that using rheological parameters from linear
regression show improvement in re-estimating the fluid's
rheogram over those values calculated from only two data
pairs. Among all other rheological parameters derived
from only two data pairs, the parameters obtained from
R600/R3 was able to accurately predict the said rheogram
with the least average error followed by the parameter
obtained from R300/R3. Rheological parameters obtained
from R6/R3 was the worst to predict the fluid rheogram
followed by those obtained using R200/R100, R300/R100 and
R600/R100, respectively.
Table-5
n 0.4274 0.4518 0.4457
K, Pa.secn
1.6489 1.4303 1.4821
Measured
600 32.19 31.87 -0.99 32.74 1.71 32.52 1.03
300 23.76 23.7 -0.25 23.94 0.76 23.88 0.51
200 19.93 19.93 0 19.93 0 19.93 0
100 14.82 14.82 0 14.57 -1.69 14.63 -1.28
6 4.09 4.45 8.80 4.09 0 4.18 2.2
3 3.07 3.31 7.82 2.99 -2.61 3.07 0
Average Error, % 2.56 0.31 0.41

Table-6
n 0.4578 0.4493 0.4150
K, Pa.secn
1.4105 1.4734 1.5582
Measured
600 32.19 33.66 4.57 33.15 2.98 27.65 -14.1
300 23.76 24.5 3.16 24.28 2.19 20.74 -12.71
200 19.93 20.35 2.11 20.23 1.51 17.53 -12.04
100 14.82 14.82 0 14.82 0 13.15 -11.27
6 4.09 4.09 0 4.19 2.44 4.09 0
3 3.07 2.98 -2.93 3.07 0 3.07 0
Average Error, % 1.15 1.52 8.35
Table-7
Ave. Plot
n 0.4403 0.4466
K, Pa.secn
1.5193 1.4707
Measured
rpm τ τ %E τ %E
600 32.19 32.11 -0.25 32.47 0.87
300 23.76 23.67 -0.38 23.83 0.29
200 19.93 19.8 -0.65 19.88 -0.25
100 14.82 14.59 -1.55 14.59 -1.55
6 4.09 4.23 3.42 4.15 1.47
3 3.07 3.12 1.63 3.05 -0.65
Average Error, % 0.37 0.03
Standard Deviation 1.82 1.08
4.2 Settling Velocity
Equations 13 and 15 are used to calculate the settling
velocity of the example particle. The result are shown in
Figures 2-5. It is obvious that both equations
overestimates the particle settling velocity of the example
particle (0.7 cm/sec). However equation 13 significantly
over estimates the settling velocity (142% average error;
SD = 18) more than twice the value observed by Guliyev
[18]. While equation 15 shows better predictions as the
estimated settling velocity are slightly higher than
observed settling velocity (48% average error; SD = 12).
This is not strange, as equation 13 do not take into account
the irregularity of the particle. According to Wadell [19], if
the volume of a particle remains constant, the smaller the
value of "", the larger the fluid drag force on the particle
during settling and thus the smaller the settling velocity of
the particle. Though the sphericity of the example particle
is not significantly less than 1 (0.9), it is however one
reason for equation 13 to yield higher predictions.
Moreover, the drag coefficient used to derive equation 13
(CD = 24/NRP + 1.714) was lower than that used to derive
equation 15 (CD = 30/NRP + 67.289/ e5.03
). This is another
reason for equation 13 to yield higher predictions.
It is clear that the effect of different rheological
parameters that have been used to determine effective
viscosity is similar for both equations. In Figure 2,
rheological parameters derived from R600/R100 (32%
error) followed by R600/R200 (39% error) produced the
lowest settling velocities compared to R600/R300 (42%
error), R600/R3 (53% error) and R600/R6 (61% error).
Though rheological parameters of R600/R3 predicts well
the full rheological spectrum of example fluid, it over
estimates the settling velocity.
eq. 13 eq. 15
R600&R300 1.628 0.992
R600&R200 1.6 0.971
R600&R100 1.532 0.927
R600&R6 1.827 1.127
R600&R3 1.743 1.068
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
SettlingVelocity,cm/sec
Fig-2: Settling velocities using n & K derived from R600-
R300, R600-R200, R600-R100,R600-R6 and R600-R3.
In Figure 3, rheological parameters derived from R300/R100
shows the good predictions with approximately 28% error
followed by R300/R200 (35% error). Rheological
parameters derived from R300/R6, generated the highest
value followed by R300/R3 (53% error). As has been
mentioned earlier, rheological parameters derived from
R300/R3 ranked second in performance for predicting the
example fluid rheogram. Similar, to R600/R3, rheological
parameter of R300/R3 over predicts the settling velocity
compared to other rheological parameters.

eq. 13 eq. 15
R300&R200 1.555 0.94
R300&R100 1.492 0.897
R300&R6 1.833 1.133
R300&R3 1.743 1.068
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
R200, R300-R100, R300-R6 and R300-R3.
The best settling velocity predictions obtained by equation
15 obtained by rheological parameters obtained from
R200/R100 with approximately 26% error. As seen in Figure
4, among all rheological parameters, those derived from
R200/R100 shows the nearest settling velocity value to the
observed one (0.879 vs 0.7). Therefore, the best data pairs
to predict settling velocity are R200/R100 followed by
R300/R100 (28% error) and R300/R200 (35% error).
Figure-5 shows the settling velocities from data pairs as
suggested by Chien [8] and API RP 13D [12] together with
those derived from average of all rheological parameters
derived from all possible data pairs and those obtained by
non-linear regression of rheological data. It is clear that
approximately all rheological parameters of Figure-5
yielded similar settling velocities. The settling velocities
approximately equal to 1 cm/sec which is higher than
observed settling velocity (0.7 cm/sec). As seen in Figure-
5, the highest percent error was obtained from rheological
parameters derived from non-linear regression (55%)
followed by those derived from R100/R3 (53%) compared
to those of R6/R3 (47%) and average (46%). Therefore, it
clear that, rheological parameter from data pairs of
R200/R100 followed by R300/R100 and R300/R200 out performs
those in Figure-5. Generally, all data pairs that do not
include R6 & R3 have less average % error (34% vs 55%)
compared to those that include R6 & R3.
eq. 13 eq. 15
R200&R100 1.464 0.879
R200&R6 1.837 1.137
R200&R3 1.743 1.069
R100&R6 1.853 1.154
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
R100, R200-R100, R200-R3 and R100-R6.
eq. 13 eq. 15
R100&R3 1.743 1.072
R6&R3 1.743 1.031
Average 1.682 1.025
Plot 1.77 1.086
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Fig-5: Settling velocities using n & K derived from R100-R3,
R6-R3, ave., and plot.
To ensure that the obtained results are because of the
effect of different rheological parameters not an error of
used equations, published equations (equations 14, 16 and
17) have been used to confirm results. Data pairs that
showed improved performance to predict settling velocity
have been used to calculate settling velocity by different
equations (Figure-6). The results have been repeated with

API equation returned slightly good predictions. Again
settling velocities from equation 13 and 14 significantly
over estimates the settling velocity. The effect of
rheological parameters shows similar trends in all
equations.
eq. 13 eq. 14 eq. 15 eq. 16 eq. 17
R300&R200 1.559 1.52 0.94 0.955 0.85
R300&R100 1.492 1.46 0.897 0.903 0.82
R200&R100 1.464 1.43 0.879 0.884 0.79
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
R200, R300-R100, and R200-R100.
Furthermore, we suggested use of effective velocity at
settling shear rate in Moore's correlation instead of his
apparent viscosity. Hence, it was possible to make Moore's
correlation applicable for stagnant fluids. Data are
compared with data pairs that showed best predictions to
observed settling velocity, i.e. R300/R200, R300/R100, and
R200/R100 (Figure 7). As depicted in Figure-7, modified
Moore correlation give overall better predictions at all
data pairs of this figure with R600/R300 and R200/R100 give
the best estimates. Modified Moore correlation was used
later to calculate settling velocity using all data pairs
(Figures 8 and 9). Generally, all data pairs of Figure-8 give
excellent predictions to the observed settling velocity
(1.17 average %error; SD=4.62) compared to those of
Figure-9 (17.25 average %error; SD = 4.53). Overall, this
method obtained approximately 12 average percent error
using all data (SD=9) which is significantly less than Chien
correlations. The best data pairs to predict the observed
settling velocity are R600/R100 (0.77 %error), R300/R200
(1.55 %error), R300/R100 (-2.38 %error), R600/R200 (4.72
%error), R200/R100 (-5.2 %error), respectively.
modified
Moore
eq. 15 eq. 16 eq. 17
R200&R100 0.664 0.879 0.884 0.79
R600&R300 0.753 0.992 1 0.91
R100&R3 0.806 1.072 1.077 0.98
R6&R3 0.78 1.031 1.024 0.94
All Ave. 0.778 1.025 1.024 0.94
Plot 0.824 1.086 1.091 1.01
0
0.2
0.4
0.6
0.8
1
1.2
Fig-7: Settling velocities using modified Moore and Chien
correlations.
0.6
0.65
0.7
0.75
0.8
Fann 35 VG Data Pair
Fig-8: Settling velocities using modified Moore (A).
0.7
0.75
0.8
0.85
0.9
R600/R6
R600/R3
R300/R6
R300/R3
R200/R6
R200/R3
R100/R6
R100/R3
R6/R3
Ave.
Plot
Fann 35 VG Data Pair
Fig-9: Settling velocities using modified Moore (B).
Finally, effects of the rheological parameters and Reynolds
number on settling velocity were illustrated. In Figure-10,

flow behavior index was plotted against settling velocities.
The variations in magnitude of "n" due to different data
pairs from which they are derived spans between 0.42 and
0.46. One single value of "n" departs from this range
particularly that derived from R6/R3. The effect was
consistent with the settling velocities decreasing when "n"
decreased (shear rates > 1 sec-1: 2.59-3.39 sec-1). This
contradicts with results obtained by other researchers [8].
Chien [8] pointed out that "the effective viscosity will
decrease with an increase in i (n) value" particularly for
shear rates > 1 sec-1, therefore, "the settling velocity
decreases as the value of i (n) increases".
Therefore it is postulated that lower values of "n" will
facilitate hole cleaning at vertical and near vertical
sections of the well particularly at laminar flow regimes
since the rheological properties of the mud plays a
dominant role. However, at higher angles where the
drillstring rests against the low side of the well, higher
values are always welcomed. Several researchers [20-23]
stated that "increasing values of n promote higher fluid
velocities under the eccentric drillpipe". Therefore, a
compromise must be reached at to balance between these
different trends.
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0.4 0.42 0.44 0.46
Power Law Flow behavior index "n"
Equation 15
Fig-10: Effect of flow behavior index on settling velocity.
The effect of Reynolds number on settling velocity showed
similar trends as "n". As seen in Figure-11, particle
Reynolds number varies between 0.02 and 0.04 with
different settling velocities and effective viscosities. It is
clear that lower values of Reynolds number ensures lower
settling velocities. Reynolds number is a ratio between the
inertial and viscous forces of liquid. Lower Reynolds
number implies a relatively high viscous force of the fluid.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.02 0.03 0.04 0.05
Particle Reynolds Number "NRP"
Modified Moore
Fig-11: Effect of Reynolds number on settling velocity.
The effect of consistency index, K, on settling velocities has
been demonstrated in Figure-12. The effect was
completely opposite to the effect of "n". Settling velocities
decreased as "K" values increased. This result is in
agreement with results of other investigators [8,24]. Chien
[8] pointed out that an increase in "K" at a constant value
of "n" increases the effective viscosity for a given shear
rate and therefore decrease the settling velocity. To
elaborate further on this, effective viscosities were plotted
against settling velocities and presented in Figure-13. A
similar trend as for K values has been obtained.
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.3 1.4 1.5 1.6 1.7
Power Law Consistency index "K", Pa.secn
Equation 15
Fig-12: Effect of consistency index on settling velocity.

0
0.2
0.4
0.6
0.8
1
1.2
1.4
0.6 0.7 0.8 0.9 1
Chien Effective Viscosity "µeff",Pa.sec
Equation 15
Fig-13: Effect of effective viscosity on settling velocity.
5. CONCLUSIONS
(1) Rheological parameters derived from linear regression
was the best to re predict the full rheogram of the fluid.
While rheological parameter obtained from R6/R3 are the
worst to predict the said rheogram.
(2) Among all other rheological parameters derived from
only two data pairs, the parameters derived from R600/R3
was able to accurately describe the fluid rheogram.
(3) Using different data pairs to calculate rheological
parameters will definitely affect effective viscosities and
settling velocities. The variance in predicted setting
velocity values is not negligible.
(4) For Chien correlations, good settling velocity
predictions were obtained using rheological parameter
obtained from R200/R100 followed by R300/R100 and
R300/R200.
(5) For Chien correlations, commonly used data pairs
(R600/R300) as well as data pairs suggested by Chien [8]
and API [12] recommended practice resulted in
rheological parameters that over predicts settling
velocities compared to their counter parts, i.e. R200/R100,
R300/R100.
(6) Settling velocities obtained from modified Moore
correlation were nearly identical to observed velocity
particularly with R600/R200, R600/R100, R300/R200 R300/R100
and R200/R100.
(7) Settling velocity where found to decrease with
decreased "n" values, something which is not welcomed at
highly deviated and horizontal wellbore. Lower value
impede flow diversion below the eccentric drillpipe.
(8) Higher values of K promotes higher effective
viscosities and results in lower settling velocities. This
trend is opposite to that of "n".
NOMENCLATURE
Ap = Projected area of Particle, cm2
AS = Projected area of a sphere of same vol., cm2
CD = Drag Coefficient, dimensionless
Dp = Particle diameter, cm
dH = Hole inside diameter, cm
dp = Drillpipe outside diameter, cm
FD = Drag force on particle, dyne
Fg = Gravitational force on particle, dyne
FB = Buoyancy force on particle, dyne
Feff = Effective force on particle, dyne
g = Acceleration due to gravity, 981 cm/sec2
K = Power law consistency index, Pa.secn
n = Power law flow behaviour index, dimensionless
NRP = Particle Reynolds number, dimensionless
R3 = Fann dial reading at 3 rpm
Up = Particle terminal velocity, cm/sec
Ua = Fluid annular velocity, cm/sec
V = Particle volume, cm3
Greek Symbols
γ = Shear rate, sec-1
γ1 = Lower shear rate, sec-1
γ2 = Higher shear rate, sec-1
τ = Shear stress, Pa
τ1 = Lower Shear stress, Pa
τ2 = Higher Shear stress, Pa
π = Constant = 3.14
ρl = Liquid density, g/cm3
ρp = Particle density, g/cm3
a = Moore's apparent viscosity, mPa.sec = cP
e = Chien's effective viscosity, mPa.sec = cP
 = Particle sphercity, dimensionless ( = AS / AP)
REFERENCES
[1] S. Tian and G. H. Medly, "Re-Evaluating Hole Cleaning
in Underbalanced Drilling Applications," IADC
Underbalanced Drilling Conference and Exhibition,
Aug. 28-29, Houston, Texas, 2000.
[2] M. N. Belavadi and G. A. Chukwu, "Experimental
Study of the Parameters Affecting Cutting
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Western Regional Meeting, Mar. 23 – 25, Long Beach,
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403, 1972.
[4] H. U. Zeidler, "An Experimental Analysis of the
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[5] P. L. Moore, "Drilling Practice Manual," PennWell
Publishing Co., Tulsa, pp. 268-276, 1974.
[6] R. E. Walker and T. M. Mayes "Design of Muds for
Carrying Capacity," JPT, vol. 27 (7), July, pp. 893, SPE
4975, 1975.
[7] J. M. Peden and Y. Luo, "Settling Velocity of Various
Shaped Particles in Drilling and Fracturing Fluids,"
SPEDE, vol. 2 (4), Dec, pp. 337-343, SPE 16243, 1987.
[8] S. F. Chien, "Settling Velocity of Irregularly Shaped
Particles," SPEDC, vol. 9 (4), Dec, pp. 281-289, SPE
26121, 1994.
[9] K. J. Sample, and A. T. Bourgoyne, "An Experimental
Evaluation of Correlations Used for Predicting
Cutting Slip Velocity," SPE Annual Technical
Conference and Exhibition, Oct. 9-12, Denver,
Colorado, SPE 6645, 1977.
[10] P. Skalle, K. R. Backe, S. K. Lyomov, and J. Sveen,
"Barite Segregation in Inclined Boreholes," Journal of
Canadian Petroleum Technology, special vol., PETSOC
97-76, 1999.
[11] T. Hemphill, "Hole-Cleaning Model Evaluated Fluid
Performance in Extended Reach Wells," Oil & Gas
Journal, Jul. 14, pp. 56-64, 1997.
[12] American Petroleum Institute, "Recommended
Practice on the Rheology and Hydraulics of Oil-Well
Drilling Fluids," API Recommended Practice 13D,
Third Edition, pp. 20-21, June, 1995.
[13] M. S. Bizanti and S. Robinson, "PC Program Speeds
Slip-Velocity Calculations," Oil & Gas Journal, Nov. 7,
pp. 44-45, 1988.
[14] T. E. Becker, J. J. Azar and S. S. Okrajni, "Correlations
of Mud Rheological Properties with Cuttings-
Transport Performance in Directional Drilling,"
SPEDE, vol. 6 (1), Mar, pp. 16-24, SPE 19535, 1991.
[15] Y. Luo, "Cuttings Transport through Drilling Annuli at
Various Angles," PhD dissertation, Heriot-Watt Uni.,
1988.
[16] M. Zamora and F. Growcock, "The Top 10 Myths,
Misconceptions and Mysteries in Rheology and
Hydraulics," AADE Fluid Conference and Exhibition,
Apr. 6-7, Houston, Texas, AADE-10-DF-HO-40, 2010.
[17] T. Hemphill, W. Campos and A. Pilehavari, "Yield-
Power Law Model Accurately Predicts Mud
Rheology," Oil& Gas Journal, Aug. 23, pp. 91-49.
[18] E. Guliyev, "The Importance of Low-end-rheology and
its Influence on Particle Slip Velocity," Master thesis,
Norwegian University of Science and Technology,
2013.
[19] H. Waddell, "The Coefficient of Resistance as a
Function of Reynolds Number for Solids of Various
Shape," J. Franklin Inst., pp. 459, 1934.
[20] T. Hemphill and T. Larsen, "Hole Cleaning Capabilities
of Water-Based and Oil-Based Drilling Fluids: A
comparative, Experimental Study," Annual Technical
Conference and Exhibition, Houston, Texas, Oct. 3-6,
SPE 26328, 1989.
[21] P. Kenny and T. Hemphill, "Hole-Cleaning Capabilities
of an Ester-Based Drilling Fluid System," SPEDE, vol.
11 (1), March, SPE 28308, 1996.
[22] P. Kenny, E. Sunde and T. Hemphill, "Hole Cleaning
Modelling: What's "n" Got To Do With it?," IADC/SPE
Drilling Conference, Mar. 12-15, New Orleans,
Louisiana, IADC/SPE 35099, 1996.
[23] M. Haciislamoglu, "Non-Newtonian Flow in Eccentric
Annuli and Its Application to Petroleum Engineering
Problems," PhD dissertation, Louisiana State U.,
Baton Rouge, 1989.
[24] R. A. Eltilib, H. H. Al Kayiem and A. Jaafar,
"Investigation on the Particle Settling Velocity in
Non-Newtonian Fluids," Journal of Applied Science,
vol. 11 (9), pp. 1528-1535, 2011.
BIOGRAPHIES
Mazen A. Muherei is an assistant
professor in Petroleum Eng. Dept.
at Hadhramout Uni., Yemen. He
obtained his BSc in Petroleum Eng.
from Faculty of Petroleum and
Mining Eng., Suez Canal Uni. in
1997. He get his MSc and PhD
degrees in Petroleum Eng. in 2003
and 2009 respectively, from
Universiti Teknologi Malaysia,
Johor Bahru, Malaysia.
Saeed S. Basaleh is an assistant
professor in Petroleum Eng. Dept.
at Hadhramout Universit, Yemen.
He holds MSc in oil well
excavation Eng. Technology and a
PhD in Technology and Eng. of
Geological explorations, both from
Russian State Geological
Prospecting University n. a. Sergo
Ordzhonikidze.

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