Journal of Petroleum Science and Engineering 30 Ž2001. 129–141
www.elsevier.comrlocaterjpetscieng
Mechanistic model for cuttings removal from solid bed in
inclined channels
A. Ramadan a,) , P. Skalle a , S.T. Johansen b, J. Svein c , A. Saasen d
a
Department of Petroleum Engineering and Applied Geophysics, NTNU, S.P. Andersens Õei 15 A, N-7491 Trondheim, Norway
b
SINTEF Material Technology, Trondheim, Norway
c
SINTEF Industrial Management, Trondheim, Norway
d
Statoil Drilling and Well Fluids, N-4035 StaÕanger, Norway
Received 5 July 2000; accepted 19 April 2001
Abstract
This paper presents the results and analysis of a set of erosion rate experiments, designed to investigate the removal rate
of stationary sand bed particles in an inclined channel. The erosion rate tests of three beds with different bed particle-size
ranges show that beds with intermediate average particle size have the maximum erosion rate. The theoretical analysis using
a mechanistic model supports this observation. The instantaneous acceleration of bed particles at the beginning of
transportation is correlated with particle removal rate. It is shown that the mechanistic model can predict optimum operating
parameters to improve the efficiency of hole cleaning. q 2001 Elsevier Science B.V. All rights reserved.
Keywords: Erosion; Solid bed; Solid–liquid flow; Cuttings transportation; Modeling; Hydrodynamic forces
1. Introduction
Among most important operational problems confronting the drillers of deviated and horizontal wells
is the formation and cleaning of cuttings beds. Many
theoretical and experimental studies related to this
problem have been performed by the industry and
academia. The studies were principally concerned
with modeling of cutting transportation and deposition. Cuttings transportation models are classified
into three categories: Ž1. two-layer, Ž2. three-layer,
Ž3. and mechanistic models. The two- and three-layer
)
Corresponding author. Tel.: q47-7-3594964; fax: q47-73944472.
E-mail address: [email protected] ŽA. Ramadan..
modeling approaches are based on the mass, momentum and energy balances of each layer. The mechanistic modeling approach facilitates the same analysis for a given particle of a solid bed. However, the
results of the three modeling approaches have been
found relatively similar.
In this study, a solid bed erosion model, combining the classical mechanics and hydrodynamics is
presented to analyze the forces acting on a solid bed
particle and to estimate the removal rate of a solid
particle from the surface of a solid bed. A relationship between the removal rate of solid particles from
the surface of a solid bed and the flow parameters in
a circular channel partially filled with solid particles
is developed. This study relates the physics of particle transportation and effective cutting transportation
in highly deviated wells.
0920-4105r01r$ - see front matter q 2001 Elsevier Science B.V. All rights reserved.
PII: S 0 9 2 0 - 4 1 0 5 Ž 0 1 . 0 0 1 0 8 - 5
130
A. Ramadan et al.r Journal of Petroleum Science and Engineering 30 (2001) 129–141
The physics of solid particles transport is quite
complex. The mechanisms of particle transportation
involve a series of processes from initiation to complete entrainment of particles into the flowing fluid.
Laboratory investigations have indicated three different modes of transportation ŽMetha, 1994.: Ž1. surface erosion of a bed, Ž2. mass erosion of a bed, and
Ž3. entrainment of a high-concentration fluidized particle suspension. Forces acting on a particle resting at
the top of the bed are gravity, cohesion, hydrodynamic lift and drag. Surface erosion occurs when the
hydrodynamic lift and drag forces overcome the
gravitational and cohesive force. At this point, particles start vibrating, rolling, lifting and finally resulting in entrainment of the particle as the intensity of
the overcoming forces increase. This type of erosion
involves a micro-level interaction between hydrodynamic and physical forces. Hence, the development
of any useful theoretical basis to explain the removal
of solid particles lies in relating the micro-level
interactions of these forces to experimental observation. Mass erosion occurs when the shear stress on a
bed is high enough to remove all bed materials on
the top plane of the bed. It is the removal of a plane
of solid particles at a time instead of individual
particles. Fluidization of a solid bed occurs, the bed
behaves like a fluid and flow-induced destabilization
of the fluid–bed interface creates interfacial entrainment and mixing as shown in Fig. 1c. However,
there is no clear-cut distinction among the different
modes of transportation.
Theoretical analysis based on surface erosion is
simpler than the other two models and pursued in the
present study.
2. Model development
In surface erosion, particle removal is achieved
when net lifting force or rolling moment on the
particle becomes positive. This removal process,
however, is probabilistic due to the stochastic nature
of the interacting forces. Some of the factors, which
contribute to the stochastic nature of the forces, are
bed particle size, shape, orientation to the direction
of fluid flow, rearrangement of the particles, and
fluctuations of local fluid flow velocity.
Mathematical modeling of solid bed erosion phenomenon requires idealization of the hydrodynamics
in the channel flow and mechanics of the bed particles. The stochastic interaction forces must also be
replaced by mean Žor representative values. to
achieve mathematical simplicity. The following assumptions are made to estimate hydrodynamic, physical and cohesive forces acting on a particle:
1. Void-free spherical bed particles
2. Negligible collision between the particles
3. Uniform bed particle size distribution, density,
angle of friction, and rearrangement as shown
in Fig. 2
4. AThe law of the wallB, described in Appendix
B, applies for any velocity distribution.
Fig. 1. Three modes of solid bed erosion: Ža. surface erosion—entrainment of a simple particle, Žb. mass erosion—removal of a
plane of particles, Žc. fluidization—interfacial entrainment of
particles.
Clark and Bickham Ž1994. suggested that removal
of cuttings in highly deviated wells is dominated by
rolling or lifting of the cutting particles. Fig. 2
illustrates the interacting forces acting on a particle
sitting at a stationary cuttings bed. The forces acting
on the particle are gravity Žweight of the particle in
A. Ramadan et al.r Journal of Petroleum Science and Engineering 30 (2001) 129–141
Fig. 2. Forces acting on particle at an active erosion site of a solid
bed.
One way of evaluating drag force is to use average pressure and shear stress instead of distributions.
In Eq. Ž2., in physical terms, drag force is the sum of
the pressure forces and shear forces over the surface
area. If spherical-shaped sand particles are assumed
and average values of pressure and shear stress are
taken, then the drag force on a solid particle during
erosion of the bed becomes:
FD s
the fluid, W ., lift force Ž FL ., drag force Ž FD ., and
the cohesive force Ž FC .. For simplicity, consider that
only the rolling and lifting phenomena are taking
place during the removal process. Then the net rolling
moment and net lift force acting on the particle can
be formulated if the lift, drag and cohesive forces
acting on the particle can be determined by reliable
means of estimation.
The resultant of various pressure and shear stress
on the surface of a particle can be expressed as drag
and lift forces acting on the particle. The resultant of
these forces can be obtained by integration, for
known pressure and shear stress distributions over
the surface of a particle. Experiment or calculation
can help determine the distribution. The net hydrodynamic force acting on a particle can be calculated by
integrating the pressure and shear stress over the
surface ŽPhilip et al., 1992.:
F s yEpVn d A q Etdis Vt d A
Ž 1.
where Vn and Vt are the unit vectors perpendicular
and tangential to the particle surface, respectively.
The pressure distribution on the surface of a particle
is p, and tdis is the wall shear stress distribution on
the surface of a particle. Drag and lift are the components of this force in the flow direction and perpendicular to it, respectively, given by:
FD s E Ž ypcos u q tdis sin u . d A
Ž 2.
and
FL s E Ž ypsin u q tdis cos u . d A
Ž 3.
where u is the angle between the Vn vector and local
flow velocity u. These equations are valid for any
body in a fluid. But, obtaining the appropriate shear
stress and pressure distributions over the surface of a
particle is complex.
131
1
2
dp Ap
dp
ž /
dx
q tw A s
Ž 4.
where d prd x is the pressure gradient across the
channel, d p is the particle diameter, A p is the projection area of a particle perpendicular to the flow
Žfrontal area., tw is the average wall shear stress on
the bed, and A s is the surface area of the particle
exposed to the shearing action of the fluid. To use
this formula, one must determine pressure drop across
the channel and the average shear stress on the bed
surface. Therefore, accurate determination of pressure drop is necessary to increase the reliability of
this method. This is the most commonly used method
for determining particle drag force when the pressure
drop is measured.
Another method for estimating drag force on a
particle is to use Stokes drag coefficient. When a
particle is close to a solid boundary, this method
should be modified to include the effect of the wall
on the drag coefficient of a sphere. Goldman Ž1967.
presented a formula for drag force on spherical
particle for Stokes flow. But the formula is complex
and applicable only when a particle is within a few
particle diameters distance from the wall. Kim and
Karrila Ž1991. developed the coefficient for wall
effect for a particle moving parallel to a wall as:
f Ž H . s 1y
9 dp
162 H
1
q
dp
ž /
8 2H
3 y1
Ž 5.
where H is the distance between the wall and the
center of the particle. For a solid bed particle, H is
assumed to be half of the bed particle diameter. Then
d prH is equal to 2 and f is 0.986. In addition to the
wall effect, the drag behavior of a single particle in
the presence of other particles should also be taken
into account. Fig. 3a presents the velocity profile for
three particles arranged along a channel centerline.
132
A. Ramadan et al.r Journal of Petroleum Science and Engineering 30 (2001) 129–141
where the gap between the particles becomes too
high, the factor becomes unity. The trend of drag
ratio curves in Fig. 4 shows that the particle Reynolds
number will also affect the drag ratio. Tsuji Ž1985.
conducted a similar experiment at higher particle
Reynolds numbers Ž Re p . and obtained drag ratios of
the same order. Therefore, the reasonable value of an
average factor is 1.2.
The widely used equation to estimate the drag
force can be modified by including the factors for
the effect of the wall and neighboring particles as:
FD s
1
2
C D r u 2A p f Ž H .
FD
FD 0
Ž 6.
where C D is the drag coefficient given by ŽWhite,
1991.:
Fig. 3. Ža. Velocity field for three particles arranged along the
channel centerline and Žb. the proposed arrangement of bed
particles for 308 angle of repose.
Hereafter, the drag force for a single particle, with
no other particles in the neighborhood, is designated,
FD0 . Variation in drag force is mainly dependent on
the distance between the particles and particle-to-pipe
diameter ratio. If the angle of repose is assumed to
be 308 as shown in Fig. 3b, then the distance between the particles for uniform arrangement of the
bed particles is about three-fourths of their diameter.
Zhu et al. Ž1994. studied the change in drag on the
wake of neighboring particles and found an empirical relationship that is presented in Appendix C ŽEq.
ŽA-12...
Liang et al. Ž1996. conducted an extensive experimental study of the variation of drag force due to the
presence of other neighboring particles in different
particle rearrangements. Their results for a particle
surrounded by six neighboring particles, arranged in
a hexagonal fashion on the same plane, are presented
in Fig. 5. From this figure, when the distance between neighboring particles is about 75% of their
diameter, the factor ranges from 1.2 to 1.3. In reality,
the gap between the neighboring particles, however,
is not the same during the removal process. If we
assume that erosion occurs layer by layer, then the
gap between the particles increases as more particles
are removed from the same layer. At the extreme
CD s
24
6
q
Re p
1 q Re p2
q 0.4
Ž 7.
For non-Newtonian fluids, such as drilling muds and
polymers, the drag coefficient depends not only on
the Reynolds number but also on Hedstrom
¨ number
that also plays an important role ŽKamp and Rivero,
1999..
Drag force is present in all types of flow around a
solid particle. Lift in a spherical particle, however, is
present only if there is asymmetry in the flow field.
In channel flow, the existence of a no slip condition
at the surface of the bed creates the asymmetry in the
flow that is responsible for lifting a particle. Saffman
Ž1965. showed that a small spherical particle moving
through a very viscous liquid in a slow shear flow
Fig. 4. Relationship between drag force ratio and the particle
separation.
A. Ramadan et al.r Journal of Petroleum Science and Engineering 30 (2001) 129–141
133
experiences a lift force. After minor adjustment for
bed particles, this formula is:
FL s 1.615
m ud p2
n 0.5
0.5
du
ž /
Ž 8.
dy
where m is the dynamic viscosity, n is the kinematic
viscosity, u is the local velocity of the fluid at the
center of the particle, and d urd y is the velocity
gradient. Most calculations of particle lift force in
turbulent boundary layers have been performed using
Saffman’s formula. Both direct and large eddy simulations of lift force on a particle, in some cases, have
shown that the lift force is smaller than predicted by
the Saffman’s formula. Saffman’s lift force formula
is based on singular solutions of Navier–Stokes
equations. As a result, solutions are applicable strictly
when the following condition is fulfilled.
ud p
n
<
Ž d urd y . d p
Ž 9.
This condition cannot be fulfilled as the bed particle
size increases. Therefore, Saffman’s formula should
be modified for higher particle size ranges.
Hall Ž1988. measured the lift force acting on
smooth spheres on the wall of a turbulent boundary
layer and it is therefore interesting to compare his
results with Saffman’s predictions. For particles on a
smooth surface, Hall showed that the data were well
correlated by
2.31
Ž 10 .
for measurements in the range of 1.8 - rq- 70,
where rq is the particle radius and Fq is the lift
force, both expressed in wall units ŽWang et al.,
1997.. Comparison of Hall’s result and Saffman’s
predictions is shown in Fig. 5, as presented by Wang
et al. Ž1997.. According to this figure, the lift force
estimated by Saffman’s formula is lower than the
result from Hall. Introducing a correction factor to fit
Saffman formula to Hall’s experimental result may
yield a good estimation method for the lift force
using Saffman’s formula. Thus, the modified
Saffman’s formula is given as:
FL s 1.615
m ud p2
n 0.5
du
ž /
dy
in which function F Ž rq. for 1.8 - rq- 100 is given
by:
0 .5
n
Fqs 20.9 Ž rq .
Fig. 5. Comparison of the Saffman force and the correlation from
Hall.
0.5
F Ž rq .
Ž 11 .
20.9 Ž rq .
q
FŽ r . s
10 1.962 log r
q
2.31
q1.412
Ž 12 .
where rq is given by:
rqs
dp
2
ž
n
d urd y
y0 .5
/
Ž 13 .
Evaluation of both the drag and lift forces requires knowledge of the local velocity profile near
the wall. Rough calculations for extreme conditions
of the flow showed that the particles near the bed
surface are in the turbulent boundary layer. Therefore, the local velocity has been estimated on the
bases of Alaw of the wallB, presented in Appendix B.
Another interacting force that must be included
here is the cohesive force. Clark and Bickham Ž1994.
presented an estimation method for the cohesive
force between bed particles based on slip-line theory:
Fp s
p d p2ty
2
Ž f q Ž pr2 y f . sin2f y cos f sin f .
Ž 14 .
where Fc is the cohesive force, f is the angle
repose, and ty is the yield strength of the fluid.
Although an assumption was made that collision
between particles is negligible, some efforts have
been made to estimate the collision forces, though
they are not promising. Collision between particles is
modeled by an isotropic interparticle stress where the
A. Ramadan et al.r Journal of Petroleum Science and Engineering 30 (2001) 129–141
134
off diagonal elements of the stress tensor are neglected. According to Harris and Crighton Ž1994.,
the interparticle stress is given by:
dp s
Ps upb
Ž ucp y up .
rolling tendency of the particle. Thus, the condition
for rolling and subsequent motion is given by:
dp
Gs
Ž 15 .
FD sin Ž f . q Ž FL y FC . cos Ž f .
2
qWsin Ž ya y f . )0
Ž 17 .
where Ps is a constant with pressure unit, ucp is the
particle volume fraction at close packing, and uc is
the particle volume fraction. Thus, the particle stress
depends only on the particle concentration and neglects the size and velocity of particles. Collision
between particles may additionally be described by
correlating the interparticle stress with other factors,
including particle size, velocity of the flow field and
a collision probability distribution function.
Primary mechanisms of particle transportation are
lifting and rolling into the flowing fluid. The net
force acting on a particle is one of the mechanisms
that determines whether the particle stays at rest or
starts lifting off from the bed surface. Using the free
body diagram presented in Fig. 6, the net force, Fnet ,
acting in the direction of the y-axis is expressed by:
where G is the net torque at point P.
Fnet s FL y Fc y W sin a
Er s rs Ž 1 y fp .
Ž 16 .
where a is the angle of inclination from vertical.
Rolling is another important mechanism of particle transportation accountable for the removal of a
particle on the surface of a solid bed. For a particle
to roll out of its resting position, the moment of the
forces about the contact point P tending to cause
downstream rotation of the particle must exceed the
moments about P tending to prevent downstream
rotation. The net rolling moment or torque at the
contact point P, as shown in Fig. 6, can measure the
3. Particle equation of motion and rate of erosion
In physical terms, the rate of erosion is the amount
of solid removed by the fluid in a given time and
contact area. Mathematically the erosion rate, Er , is
Er s
dm
Ž 18 .
Ad t
where m is the mass, t is the time and A is the
contact area. Substituting r AŽ1 y fp .d S for d m in
Eq. Ž18., the following simple relationship is obtained:
dS
dt
s rs Ž 1 y fp .
dh
ž /
dt
Ž 19 .
bed
where S is the upward displacement of the particle
from the bed surface, rs is the density of the particle, fp is the bed porosity and d Srdt is its velocity.
So the rate of erosion is directly proportional to the
particle’s upward velocity. If the net force applied on
the particle during the erosion process is assumed
constant, then a simplified equation of motion for the
particle is given as:
dS
dS
ž / ž /
y
2
d S
dt
Er s
Fig. 6. Free body diagram of a bed particle at the instant of
inception.
2
s
Ž 1 y fp .
2
tsT
dt
ts0
Ž 20 .
T
dt
where Žd Srdt . ts0 is the initial velocity equal to zero
in this case and T is the time of entrainment. From
Eq. Ž20., the average particle velocity Žd Srdt .ave is
estimated as aTr2; where a is the acceleration of
the particle. Combining Eqs. Ž19. and Ž20., the erosion rate becomes:
as
rs aT
Ž 21 .
Using this mechanistic model approach, it is possible
to determine the acceleration of the particle, but the
A. Ramadan et al.r Journal of Petroleum Science and Engineering 30 (2001) 129–141
entrainment time T is difficult to estimate or measure. Thus, an entrainment function ´ is introduced
which can take care of the entraining time and other
idealization used in derivation of the mechanistic
model. Therefore:
Er s
Ž 1 y fp .
2
rs a ´
Fnet
mp
where a l is the lifting acceleration and m p is the
mass of the particle. The particle movement can be
achieved not only through the lifting force but also
through the rolling moment. Therefore, it is more
convenient to evaluate the acceleration at the center
of the particle due to rolling according to:
Ž 22 .
´ can be a function of particle size, geometry,
arrangement, etc. Net force, estimated from Eq. Ž16.,
can be used to calculate the lifting acceleration of the
particle as:
al s
135
Ž 23 .
ar s
10
G
Ž 24 .
7 mp dp
where a r is the rolling acceleration at the center of
the particle due to the torque G . Finding an appropriate entrainment function ´ makes this model
complete and applicable.
4. Model prediction
The model predictions are based on the following base case parameters, unless stated otherwise.
Fluid density
Fluid viscosity
Particle density
Fluid model
Flow velocity
1000 kgrm3
1 cp
2600 kgrm3
Newtonian
0.35 mrs
The model is not closed because the entrainment
function is unknown. As a result, prediction of erosion rate cannot be made; however, the acceleration
of the particle that is related to the erosion rate can
be evaluated.
The results for the base case and mean velocity at
0.6 mrs are presented in Figs. 7a and 9, respectively. Acceleration of the particle due to rolling and
lifting is presented; the positive sign of acceleration
confirms the removal of a particle, zero is the critical
condition Žthreshold of a particle motion. and negative values indicate a state of rest. Moreover, the
results show that both rolling acceleration and lifting
acceleration have peak values, which shift in the
same direction as the particle size increases. The
peak value changes with respect to the mean velocity. In Fig. 7a, the peak values are at 0.5 and 0.8 mm
particle size for the lifting and rolling accelerations
for mean velocity of 0.35 mrs. And in Fig. 7b, the
lifting and rolling accelerations have peak values
near 0.35 and 0.6 mm particle size for a mean
Yield strength
Bed height
Pipe diameter
Flow regime
Bed porosity
0
30%
64 mm
Turbulent
52%
velocity of 0.6 mrs. In addition, the value of rolling
acceleration is greater than the lifting acceleration.
Increasing the mean velocity from 0.35 to 0.6 mrs
has shown a significant increase in the particle acceleration.
At 0.35 mrs mean velocity, the rolling transport
criterion is satisfied for particle sizes ranging from
0.2 to 2.9 mm, while lifting is also satisfied for
particle between 0.2 and 2 mm. For particles ranging
from 0.2 to 1.3 mm, both transport mechanisms are
involved in the process.
The effect of fluid viscosity on the lifting and
rolling accelerations was also investigated in the
model by taking three fluids with different viscosities and the results are shown in Fig. 8. In this
figure, widening and flattening of the acceleration
curves are observed with increasing viscosity, resulting in a higher range of particle size that satisfies the
conditions of transportation; however, a significant
reduction in the acceleration for fine particles and an
increase for the coarse ones has been observed.
136
A. Ramadan et al.r Journal of Petroleum Science and Engineering 30 (2001) 129–141
Assuming a stable bed at a different angle of
inclination of the channel, the effect of angle of
inclination on the acceleration has been analyzed by
examining Eqs. Ž16. and Ž17. for various angles of
inclination at constant flow velocity. The net lift
force ŽEq. Ž16.. has a minimum value at an angle of
908 and maximum value at an angle of 08. Similarly,
the net rolling torque ŽEq. Ž17.. has a minimum
value at a s 90 y f 8 and maximum at an angle of
08. Therefore, for the base case, with angle of repose
equal to 308, the minimum rolling torque occurs at
608 angle of inclination. Eqs. Ž16. and Ž17. indicate
that the influence of angle of inclination is related to
the weight of the particle in the fluid. Therefore, the
density of the fluid plays a great roll in controlling
the effect of hole inclination on cuttings transportation. When the fluid velocity is significantly higher
than the critical fluid velocity, however, the lift and
drag forces become relatively high enough that the
influence of angle of inclination becomes negligible.
5. Experimental equipment and procedures
Fig. 7. Relationship between particle acceleration and particle
diameter Ža. for the base case; Žb. for mean velocity of 0.6 mrs.
Accordingly, keeping all the operating parameters
the same for these three cases, the fluid with viscosity 1, 3 and 6 cp is most advantageous to transport
average solid bed particles with 0.7, 1.5 and 2.75
mm diameters, respectively.
Fig. 8. Rolling acceleration versus particle size for mean velocity
0.6 mrs at different viscosities.
The experiments were conducted in a flow loop
that has a 6.4-cm circular channel with fluid re-circulation facilities. A diagram of the loop is presented in
Fig. 9; the loop consists of the following items: the
channel that is made up of a 2.6-m long transparent
PVC pipe with a diameter of 64 mm; an overhead
tank to maintain constant pressure head at the inlet
Fig. 9. Simplified flow diagram of the flow loop.
A. Ramadan et al.r Journal of Petroleum Science and Engineering 30 (2001) 129–141
137
of the channel; a circulation tank for pumping, collecting, and separating solid–liquid mixture; and a
centrifugal pump to recycle the fluid. The loop is
designed to maintain more or less constant flow rate
of fluid during a test run. Continuous flow of fluid in
the channel is achieved by maintaining constant fluid
level in the overhead tank. Minimum and maximum
level switches are used to regulate the pump in
accordance with the fluid level in the overhead tank.
The magnetic flow meter, placed upstream of the
channel, is used to measure the flow rate and it is
connected to a personal computer for on-line displaying and recording. Temperature measurements of the
fluid were made at the circulation tank. A manually
operated valve, placed downstream of the channel
was used to regulate the flow rate. The main objective of each test run was to measure the time taken to
remove a given amount of sand bed placed in the
channel at a constant flow rate.
6. Experimental results
The experimental runs were conducted at four or
five different sand bed volumes under constant fluid
flow rate. The result from the experiment is presented in Table 1. The experimental equipment is
characterized by its simplicity, being inexpensive,
and producing repetitive, reliable results. Thus, the
erosion rate measurements were done indirectly. The
loop test delivered the time taken to remove a given
bed thickness. In order to estimate the erosion rate,
according to Eq. Ž19., the rate of change of the bed
height is required. The rate of change of the bed
height is determined from the slope of bed height
Table 1
Time required to erode a given type of sand bed under constant
liquid flow rate Žin minutes.
d p Žmm.
0.125–0.500
0.500–1.200
2.000–3.500
Q Žlrs.
1.42
1.35
1.78
Vs Žl.
0.50
1.00
2.00
3.00
4.00
24
–
–
30
18
11
43
21
14
47
27
18
50
30
20
Q is the volume flow rate of the fluid in litersrsecond and Vs is
the volume of sand placed in the channel in liters.
Fig. 10. Erosion rate versus Ža. mean velocity for different particle
sizes; Žb. diameter for different mean velocities.
versus the erosion time curve at different velocities.
The results are presented in Fig. 10a. The erosion
rate shown in this figure for the finest sample is the
lowest of the three samples tested. Moreover, the
sample with a particle size range of 0.5–1.2 mm has
the highest erosion rate at a given velocity.
To show the effect of particle size on the erosion
rate, the experimental data are presented as rate
versus particle size in Fig. 10b. This figure illustrates
the effect of particle size on the removal process.
Intuition might infer that fine particles should be
easier to remove or transport than coarse ones; however, according to this graph, that is not always the
case. A maximum erosion rate is observed at around
0.9 mm.
In Eq. Ž22., the rate of erosion is proportional to
the product of acceleration of the particle and the
entrainment function ´ . If the entrainment function
is assumed to be constant, then the erosion rate and
138
A. Ramadan et al.r Journal of Petroleum Science and Engineering 30 (2001) 129–141
Fig. 11. Entrainment function versus particle sizes for different
mean velocities.
the particle acceleration graphs will have the same
pattern. Thus, comparing the pattern of these graphs
can help to determine the influence of the entrainment function on the removal rate. In order to see
this, compare Figs. 7a and b with Fig. 10b; both
acceleration and erosion rate graphs have a maximum point around the same particle size and their
pattern is similar. As seen from Figs. 7a and 10, the
patterns of erosion rate and acceleration graphs are
not identical. Therefore, it is more reasonable if the
entrainment function is considered as a function of
particle size or a flow parameter.
In Fig. 11, the entrainment function versus particle diameter is presented for different mean velocities. These data were calculated from the erosion rate
determined experimentally and particle acceleration
that is estimated from the model. From the graph, the
entrainment function increases with increasing particle size.
7. Conclusions
Ž1. Although many idealization steps are taken in
its derivation, a mechanistic model based on momentum balance of the fluid and the particle describes
the physical problem encountered during the removal
of solid particles and cuttings in highly deviated
wells and channels.
Ž2. The model prediction and experimental results
confirm that there is a direct relationship between
particle acceleration and particle removal rate.
Ž3. In transportation of solid particles and cuttings, there is an optimum particle size that has the
highest rate of removal. Thus, exploitation of the
optimum particle size to improve the efficiency of
hole cleaning is possible by prescribing the optimum
operating parameters that include the mean velocity,
viscosity of the fluid, and angle of hole inclination so
that it will be in the range of the optimum value.
Ž4. The results show that both rolling and lifting
of bed particles, caused by drag and lift forces, exist
simultaneously during solid bed erosion; however,
one mechanism may dominate over the other one.
The process parameters such as the angle of inclination, flow velocity, and viscosity of the fluid determine the dominating mechanism.
Nomenclature
n
kinematic viscosity
u
angle between the n vector and the local
flow velocity
a
angle of inclination
b
constant
´
entrainment function
m
plastic viscosity
particle volume fraction
uc
particle volume fraction at close packing
ucp
angular bed height
uh
wall shear distribution on the surface of a
tdis
particle
density of the fluid
rf
bed porosity
fp
density of the particle
rs
wall shear stress
tw
yield strength of the fluid
ty
f
angle of repose
G
rotating torque on a particle
a
acceleration of a bed particle
A
contact area between the fluid and the bed
cross-sectional area of the channel
A ch
lifting acceleration
al
projection area of a particle perpendicular to
Ap
the flow
acceleration at the center of a bed particle
ar
due to rolling
surface area of the particle exposed to the
As
shearing action of the fluid
drag coefficient
CD
hydraulic diameter of the channel
D hyd
particle diameter
dp
A. Ramadan et al.r Journal of Petroleum Science and Engineering 30 (2001) 129–141
d prd x pressure gradient across the channel
d Srdt velocity of a bed particle during the process
of removal
d urd y local velocity gradient
erosion rate
Er
f
drag force variation factor to presence of a
wall
f
friction factor
lift force expressed in wall units
Fq
average friction factor of the channel
fave
average friction factor of the bed
f bed
cohesive force
FC
drag force
FD
drag force on a particle without the presFD0
ence of a neighboring particle
lift force
FL
net force on the particle acting in the direcFnet
tion of the y-axis
H
distance between the wall and the center of
the particle
l
interparticle distance
m
mass
mass of a particle
mp
unit vector perpendicular to the particle surVn
face
p
pressure distribution on a particle
perimeter of the channel
Pch
constant with pressure unit
Ps
Q
volume flow rate
particle radius expressed in wall units
rq
Re
pipe Reynolds number
particle Reynolds number
Re p
S
upward displacement of the particle from
the bed surface
T
entrainment time
t
time
unit vector tangential to the particle surface
Vt
u
local flow velocity
U
mean flow velocity of the fluid
friction velocity
ut
dimensionless local velocity
uq
volume of sand bed
Vs
W
weight of the particle in the fluid
bed width
W bed
x
axial coordinate
length fraction of the bed
X bed
y
coordinate normal to the flow
dimensionless distance from the wall
yq
139
Acknowledgements
The authors express their appreciation to the staff
of the workshop and the laboratory at the Institute of
Petroleum Technology of NTNU for their assistance
in building the flow loop. The work was financed by
Statoil, and we thank them for their support.
Appendix A. Pressure drop calculation
The flow of a fluid in a channel may be laminar
or turbulent. Turbulent flow is more likely to occur
than laminar flow in practical situations. Pressure
drop calculation for laminar flow in the channel can
be made using the smooth pipe flow equation. Turbulent flow is a very complex process and surface
roughness must be included in the analysis. The
surface of a pipe may be treated as smooth when it is
made from plastic materials or glass. But, for this
work, the channel surface is considered as smooth
and the bed as a rough surface because the channel is
made of PVC. Therefore, the calculating begins with
the pressure drop from this assumption. Hence:
dp
dx
s fave
rf U 2
Ž A-1.
2 D hyd
The formula for the average friction factor Ž fave . of
the channel is presented here. The Colebrook formula for estimating the friction factor of smooth and
rough surfaces that works for both fully turbulent
and transition flow in channels is:
0.25
fs
' //
2
Ž A-2.
log d pr3.7D hyd q 2.51r Ž Re f .
ž ž
Average friction factor for the channel shown in Fig.
A-1 is:
fave s f bed X bed q f pipe Ž 1 y X bed .
Ž A-3.
Length fraction of the bed
X bed s
W bed
Pch
Ž A-4.
Bed width
W bed s Dsin u h
Ž A-5.
A. Ramadan et al.r Journal of Petroleum Science and Engineering 30 (2001) 129–141
140
serious. It may be considered permissible to use the
Alaw of the wallB with some degree of limitations.
Several forms of the Alaw of the wallB have been
suggested; however, the law is often represented by
different expressions in different regions of flow.
Near the wall, there exists a laminar sublayer, where
the conditions are similar to laminar flow. A formulation of the Alaw of the wallB that is valid throughout the laminar sublayer as well as through the
turbulent boundary layer is ŽLeif, 1972.:
Fig. A-1. Section of a channel with sand bed.
q
yqs uqq A e k u y 1 y k uqy
Angular bed height
1
u h s cosy1 Ž 1 y 2 hrD .
Ž A-6.
Channel perimeter
Pch s D Ž p y u h q sin u h .
Ž A-7.
Channel cross-sectional area
A ch s
p D2
4
ž
1y
Ž u h y cos u h sin u h .
p
/
Ž A-8.
Appendix B. ALaw of the wallB
The Alaw of the wallB is still used to this day as a
boundary condition in the more sophisticated turbulence models for which it is either difficult or too
computationally expensive to integrate directly to a
solid boundary. It was remarkably successful in collapsing the experimental data for turbulence pipe and
channel flows for a significant range of distances
from the wall ŽSpeziale, 1991..
The Alaw of the wallB, however, is supported by
experimental results obtained on flat plates with zero
pressure gradient and in tubes. Although a pressure
gradient is present in tubes, some doubt may exist on
the applicability of the Alaw of the wallB for a
turbulent boundary layer with non-zero pressure gradients. One may argue that a proper description of
separation of the turbulent boundary layer may not
be feasible if the Alaw of the wallB is maintained.
Noting that in the neighborhood of the separation
point, the boundary layer assumptions are not very
well satisfied anyway, but this objection is not too
y
6
3
Ž k uq . y
1
24
Ž k uq .
1
2
4
Ž k uq .
2
Ž A-9.
where A is 0.1108, k is 0.4, yq is the dimensionless distance from the wall and uq is the dimensionless local velocity. Dimensionless distance and velocity are calculated as follows:
yqs
yut
n
uqs
u
Ž A-10.
ut
where y is the distance from the wall, u is the local
velocity and ut is the friction velocity that is given
by Žtw rr f . 0.5. The above formulas are not only used
to calculate the velocity profile of turbulent boundary layers but also the velocity gradient. The velocity
gradient can be calculated as follows:
du
s
dy
ut2 d uq
Ž A-11.
n d yq
Appendix C. Empirical relation
An empirical relation is obtained to describe both
effects of the interparticle distance and Re p on the
drag force of the trailing particle. The empirical
equation takes the exponential form ŽLi et al., 1999.:
FD
FD 0
l
ž /
s 1 y Ž 1 y A . exp B
dp
Ž A-12.
where FD0 is the drag force of a single non-interacting particle, and l is the interparticle distance.
A. Ramadan et al.r Journal of Petroleum Science and Engineering 30 (2001) 129–141
The coefficients A and B are both functions of Re p
and are empirically correlated as
A s 1 y exp Ž y0.483 q 3.45 = 10y3 Re p
y1.07 = 10y5 Re p2 .
Ž A-13.
B s y0.115 y 8.75 = 10y4 Re p q 5.61 = 10y7 Re p2
Ž A-14.
Li et al. Ž1999. noted that the application of Eqs.
ŽA-13. and ŽA-14. should be limited to 20 - Re p 150.
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