Group Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Title: 3.3.3 - Modeling Asphaltene Deposition in Oil Transport Pipelines
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 Material Information
Title: 3.3.3 - Modeling Asphaltene Deposition in Oil Transport Pipelines Industrial Applications
Series Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Physical Description: Conference Papers
Creator: Eskin, D.
Ratulowski, J.
Akbarzadeh, K.
Pan, S.
Lindvig, T.
Publisher: International Conference on Multiphase Flow (ICMF)
Publication Date: June 4, 2010
 Subjects
Subject: Couette device
particle transport
pipeline
population balance
turbulence
 Notes
Abstract: A possibility of simulating asphaltene deposition in a pipeline by using a Couette device is discussed. Similarity of deposition in both devices is analyzed. The deposition mechanism is understood based on the theoretical and experimental analysis. A concept of the critical particle size limiting the size of particles, which are able to deposit, is introduced. Models of the particle size distribution evolution and particle transport to the wall are developed. The deposition model contains three empirical parameters, which are identified from the Couette device experimental data. The calculation examples show that the dynamics of asphaltene deposit accumulation in a Couette device can be accurately described by the model developed. The computed deposit layer thickness along a vertical production tubing is in a qualitative agreement with the literature data.
General Note: The International Conference on Multiphase Flow (ICMF) first was held in Tsukuba, Japan in 1991 and the second ICMF took place in Kyoto, Japan in 1995. During this conference, it was decided to establish an International Governing Board which oversees the major aspects of the conference and makes decisions about future conference locations. Due to the great importance of the field, it was furthermore decided to hold the conference every three years successively in Asia including Australia, Europe including Africa, Russia and the Near East and America. Hence, ICMF 1998 was held in Lyon, France, ICMF 2001 in New Orleans, USA, ICMF 2004 in Yokohama, Japan, and ICMF 2007 in Leipzig, Germany. ICMF-2010 is devoted to all aspects of Multiphase Flow. Researchers from all over the world gathered in order to introduce their recent advances in the field and thereby promote the exchange of new ideas, results and techniques. The conference is a key event in Multiphase Flow and supports the advancement of science in this very important field. The major research topics relevant for the conference are as follows: Bio-Fluid Dynamics; Boiling; Bubbly Flows; Cavitation; Colloidal and Suspension Dynamics; Collision, Agglomeration and Breakup; Computational Techniques for Multiphase Flows; Droplet Flows; Environmental and Geophysical Flows; Experimental Methods for Multiphase Flows; Fluidized and Circulating Fluidized Beds; Fluid Structure Interactions; Granular Media; Industrial Applications; Instabilities; Interfacial Flows; Micro and Nano-Scale Multiphase Flows; Microgravity in Two-Phase Flow; Multiphase Flows with Heat and Mass Transfer; Non-Newtonian Multiphase Flows; Particle-Laden Flows; Particle, Bubble and Drop Dynamics; Reactive Multiphase Flows
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7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010


Modeling Asphaltene Deposition in Oil Transport Pipelines



Dmitry Eskin, John Ratulowski, Kamran Akbarzadeh, Shu Pan, Thomas Lindvig

Schlumberger DBR Technology Center, 9450 17 Ave., Edmonton, AB T6N 1M9, Canada
E-mail: deskin @slb.com


Keywords: Couette device, particle transport, pipeline, population balance, turbulence



Abstract

A possibility of simulating asphaltene deposition in a pipeline by using a Couette device is discussed. Similarity of deposition in
both devices is analyzed. The deposition mechanism is understood based on the theoretical and experimental analysis. A
concept of the critical particle size limiting the size of particles, which are able to deposit, is introduced. Models of the particle
size distribution evolution and particle transport to the wall are developed. The deposition model contains three empirical
parameters, which are identified from the Couette device experimental data. The calculation examples show that the dynamics
of asphaltene deposit accumulation in a Couette device can be accurately described by the model developed. The computed
deposit layer thickness along a vertical production tubing is in a qualitative agreement with the literature data.


Introduction

Asphaltenes are molecular substances, which are
components of a crude oil, along with resins, aromatic
hydrocarbons, and alkanes. The asphaltene particles
precipitate from oil due to depressurization below the
so-called asphaltene onset precipitation pressure (AOP). Oil
pipelines usually operate in a regime of a developed
turbulence. The pressure gradually drops along a pipeline;
asphaltene particles precipitate and grow forming
agglomerates, which partially deposit on the pipe walls.
Modeling of asphaltene deposition in oil transport pipelines
is required for more accurate forecasting petroleum
production problems caused by asphaltenes. The asphaltene
deposit layer reduces the pipe cross-section that may lead to
flow rate reduction or increased pressure drop. The deposit
may eventually totally plug a pipeline.
There are only a few papers on asphaltene deposition in
transport pipelines. It is worth to mention here a paper on
modeling the asphaltene deposition in a pipe that was
published recently by Ramirez-Jaramillo et al (2006). The
authors employed a hypothesis that the asphaltene particle
transport to the wall is caused by molecular diffusion. It was
also assumed that the particle concentration gradient in the
wall vicinity is caused by the temperature gradient at the
wall. This modeling approach has no physical ground
because the laboratory experiments have shown that there is
no pronounced effect of the temperature gradient at the wall
on the asphaltene deposition rate (Akbarzadeh et al, 2009).
There are many investigations dedicated to general
approaches to modeling particle deposition in turbulent pipe
flows, which are mainly focused on particle transport to the
wall. Among the latest studies we would like to mention the


papers where the advection-diffusion approach to the
particle deposition was formulated and successfully used
(e.g. Johansen 1991, Guha 2008). According to this
approach the particle transport to the wall, to a great extent,
is caused by the negative particle concentration gradient in
the wall vicinity. The most important driving forces of
particle transport to the wall are (e.g. Johansen 1991, Guha
2008): 1) Brownian motion; 2) turbulent diffusion; 3)
turbophoresis. The majority of the researchers who used the
advection-diffusion technique have employed the standard
boundary condition at the wall: the particle concentration is
zero at the distance of a particle radius from the wall i.e. all
particles reaching the wall stick to it (e.g. Johansen 1991,
Guha 2008).

Nomenclature


fractal dimension
particle size (m)
Fanning friction factor
dimensionless torque
gravity acceleration (m2/s)
Boltzmann constant (W/K)
Couette device height (m)
particle mass (kg)
particle concentration by number (1/m3)
empirical parameter
pressure (Pa)
corrected total deposition particle flux
(kg/(m2.s))
total depositing particle flux (kg/(m2 s))
inner Couette device radius (m)
outer Couette device radius (m)
Reynolds number









S particle fragmentation rate (1/s)
T time, s
Tf temperature, K
Tr torque, N m
Vy flow velocity in the wall vicinity (m/s)
V'2 mean square component of the fluid fluctuation
Y velocity normal to the wall (m/s)
U mean flow velocity in a pipe (m/s)
u. friction velocity (m/s)
x coordinate along a pipeline (m)
y coordinate indicating distance from a wall (m/s)
w0 initial mass concentration of precipitated
particles (kg/m3)
Greek letters
a particle-particle collision efficiency
oa dimensionless parameter
p particle-particle collision frequency (m3/s)
8 deposit layer thickness (m)
E energy dissipation rate per unit mass (W/kg)
F particle breakage distribution function
y particle-wall sticking efficiency
1r inner/outer Couette device radius ratio
(pd deposit layer porosity
X0 inner turbulence scale (m)
1 dynamic viscosity (Pa s)
Vf kinematic viscosity (m2/s)
v, frequency of collisions of i-th size fraction
particle with the wall (1/(m2s))
p density (kg/m3)
't shear stress at the wall (Pa)
) particle velocity in the wall vicinity (m/s)
Q inner cylinder rotation speed (rpm)
wc rotor angular velocity (rad/s)
Subsripts
B Brownian
b bubble point
c Couette
d directed to the wall
f fluid
i particle size fraction number
j particle size fraction number
op operation
s solids
t turbulent


Deposition process Similarity In a Pipeline and a
Couette Device

It is difficult to experimentally investigate the asphaltene
deposition in a laboratory. A flow loop is not suitable for
this purpose. The major reason is a huge volume (up to
hundreds of liters) of a fluid that is needed for running a
flow loop even of a relatively small scale.
To avoid this problem we employed for our experiments a
wide-gap Couette device. The inner cylinder rotates and
particles deposit on the outer immobile wall. The initially
high pressure (above AOP) in this device can be easily
reduced either abruptly or gradually to provide a particle
precipitation scenario needed. Flows of different turbulent
intensities are generated by changing the rotation speed of


7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010

the inner cylinder. The volume of fluid needed for running
our laboratory Couette device is relatively small (150 cm3)
that is convenient for testing numerous fluid samples.
To confidently employ a Couette device for deposition
studies it was important to show that the deposition
conditions in a pipeline can be imitated in a Couette device.
One of the major requirements of deposition similarity in a
Couette device and a pipeline is the same conditions of
particle transport to the wall.
A flow structure in both a pipeline and a Couette device can
be considered as composed of a boundary layer flow and a
core flow. Asphaltene particles are assumed to be well
dispersed in a turbulent core flow. This is explained as
follows. The density of asphaltene particles (aggregates) is
usually evaluated as ps=1200 kg/m3 while the fluid density
is usually in the range of pf=700-1000 kg/m3 i.e. the
particle-fluid density difference is relatively small. Since
our observations show that sizes of largest asphaltene
aggregates in a turbulent Couette flow do not exceed tens of
microns, the particle relaxation times are so small that
particles almost follow fluid streamlines. Thus, the particle
transport to the wall is mainly determined by the structure of
a boundary layer. It is easy to show (Eskin et al 2009) that
the hydrodynamic similarity of two near-wall flows (flow
patterns in the boundary layers) is obtained if the shear
stress at the wall and the wall temperature, determining the
fluid viscosity are the same.
The shear stress at the pipe wall is calculated as (e.g. Bird et
al 2002):


where f is the Fanning friction factor that is a function of the
pipe Reynolds number and the wall surface roughness; U is
the superficial flow velocity.
The maximum roughness of the transport pipe walls is
usually below 50pm. Our calculations showed that in the
majority of flow regimes the transport pipelines are
hydraulically smooth. Moreover, at the initial stage of the
deposition process the cavities between asperities forming
roughness are filled with deposit material, i.e., after a
relatively short time the pipe surface is covered with a
deposit. A newly formed surface is hydraulically smooth
therefore for calculation of the Fanning friction factor the
Blausius correlation (e.g. Bird et al 2002) for a smooth pipe
can be employed (f=0.316/Repipe, where Repipe is the pipe
Reynolds number).
Based on the experimental velocity distribution in both the
laminar and buffer layer of a Couette flow, and the velocity
distribution in the core flow obtained by using the Prandtl
mixing length approach, we derived the analytical
expression relating the non-dimensional torque G applied to
the Couette device rotor and the Reynolds number Rec
(Eskin 2010):

1.103 1r Re -n 21 1l+1 (2)
-2 XI _V = In +-- -In +0.406 (2)
+ 1) C)V 1+112 Yl1)

where G=Tr/(pfv2L) is the dimensionless torque, L is the
Couette device height; ir=ro/R is the Couette device radius


c = 0. 125 p'ftj2









ratio; ro, R are the inner and outer radii of a Couette device
respectively; Re,=wro(R-ro)/vf is the Couette device
Reynolds number; Tr=Tc27R2L is the torque; w the inner
cylinder angular velocity, vf is the fluid kinematic
viscosity.
Equation 2 is in a good agreement with the experimental
data for Rec>13000. In contrast to the other G(Re) relations
known from the literature Eq.2 doesn't contain any
empirical parameters and is not limited to a single radius
ratio.
To evaluate an effect of the centrifugal force on particle
dynamics in a Couette device a model for calculating the
particle concentration distribution along the Couette device
radius was developed (Eskin et al 2009). The model
involves solving the convection-diffusion equation for
particles moving in a turbulent flow. Numerous calculations
allowed us to conclude that because only small asphaltene
particles (usually, of sizes below 1lm) mainly contribute
into the deposition, the particle centrifugal stratification can
be ignored in deposition studies using a Couette device.

Understanding the Deposition Mechanism Basics
by Analyzing the Couette Device Experimental Data

In its batch mode, the Couette device is initially filled with a
hydrocarbon fluid at the pressure well above the detected
asphaltene onset pressure (AOP) at the desired temperature.
Then the system pressure is abruptly (during several
minutes) reduced to a value that is slightly higher than the
bubble point pressure of the fluid where the maximum
amount of precipitated asphaltenes takes place. After that
the Couette device is run with a certain rotation speed for a
given time. At the end of the test, the oil inside the device is
drained at the test pressure and the asphaltene deposit on the
outer Couette device wall is recovered and accurately
analyzed (Akbarzadeh et al 2009).
Deposition experiments in the batch system were performed
at different run times, while maintaining other conditions
the same. It was found that the deposit mass initially grew
slowing down with time and eventually reached a plateau
i.e. the deposition process stopped after a few hours. Thus,
the experiments clearly demonstrated a depletion effect. It is
important to emphasize that the total mass of precipitated
particles is usually two orders of magnitude higher than the
deposit mass i.e. the system is getting depleted only of
particles, which are able to deposit.
In contrast to the batch setup in the flow-through Couette
system a continuous flow of a fresh fluid passes through the
Couette device. The hydrocarbon fluid is initially placed in
a one liter sample storage bottle at a high pressure. The
sample bottle is connected to the Couette device inlet by a
tube. The Couette device outlet is connected to a collecting
cylinder. A back pressure regulator and two automatic
pumps are used to maintain the pressure of the sample
storage bottle above AOP while the pressure of the fluid
entering the Couette device is reduced down to the value
that is slightly higher than the bubble point pressure, and the
pressure in the collecting cylinder is maintained close to the
pressure in the Couette device. The flow-through rate is low
therefore the mean fluid residence time in the Couette


7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010

device is long enough to treat this device as an ideal mixer.
The flow-through system provides valuable information
about the deposition process because the deposit amount
collected can be much higher than that in the batch system.
This is because during the constant influx of a fresh fluid no
or minimal depletion occurs in the flow-through system.
The following experiment significantly advanced our
understanding of the deposition mechanism. The system,
charged with certain oil, was run for 4 hours at the
flow-through fluid rate of 3 cm3/min. After that the deposit
mass was determined and the deposition rate calculated.
Then the sample collected in the collecting cylinder was
conditioned at reservoir conditions of the fluid for five days
under continuous rocking to make sure that previously
precipitated asphaltenes dissolve back into the fluid. Then
the sample storage bottle was filled with the conditioned
fluid. After that the flow-through deposition experiment
was run again under the same conditions. The experiment
showed that the measured deposition rate 15.6g/m2/day was
comparable to that in the first experiment (13.9g/m2/day).
The result obtained led to understanding that the depletion
of a fluid is not associated with disappearance of particles,
which are able to deposit because their chemical properties
are different from other particles. Actually, the depletion is
caused by the particle growth due to agglomeration. The
large agglomerates, formed in a fluid after a certain time,
are not able to deposit because they are removed from the
wall by shear.
Based on the deposition process understanding we
introduced the critical particle size, which is defined as the
maximum size of a particle that can deposit on the wall
under given flow conditions. As soon as all particles
become larger than this size the deposition completely stops
indicating that the fluid is depleted of particles able to
deposit.

Modeling the Particle Size Distribution in a Couette
Device

Both the particle transport to the wall and the particle ability
to stick to the wall depend on the particle size. The particle
size changes along a pipeline, or during the time in the
Couette device. We employed the population balance
approach for modeling evolution of the particle size
distribution (PSD) in a Couette device as well as in a pipe.
The Hounslow's version of this approach (e.g. Flesch et al
1999) was employed. In this case the continuous size
distribution is represented as a set of im discrete fractions.
The volume of the i-th fraction is twice larger than that of
the i-th size fraction.
The set of equations describing the size distribution
evolution in a batch system (Couette device in our case) is:

dN 2 1 2
dt= N,,12 a ,,, N, +-,a,, p N,2

-N 2 a1,j p, NJ--Nl Ip ,J NJ
ji i


jm
-S,N, + IF," Si NJ
J l+l


i, j=l ....im










where N, is the concentration of particles of the i-th size
fraction by number, 1,, is the collision frequency of a
particle of the i-th size fraction with particles of the j-th size
fraction, o, is the collision efficiency of a particle of the i-th
size fraction with that of the j-th size fraction, S, is the
fragmentation rate of the i-th size particle, F, is the
breakage distribution function indicating how many
particles of the i-th size fraction are produced at breakage of
a particle of the j-th size fraction.
Note that a set of population balance equations for a
flow-through system is easily obtained by a corresponding
modification of the set of Eqs. (3). We do not show those
equations here because of their straightforwardness.
The agglomerates can be treated as fractal objects with the
fractal dimension Df. According to Rastegari et al (2004)
the fractal dimensions of asphaltene agglomerates are about
Df=1.5-1.8.
For calculating the collision frequency we employed the
model, formulated, for example by Flesch et al (1999), that
is valid for a system with particles, which are smaller than
the inner Kolmogoroff scale ko=(Vf3. "', where E is the
energy dissipation rate per unit mass. Note that in practice
this approach can be employed for calculating the systems
with agglomerates, which are slightly larger than Xo. The
scale Xo in the systems we analyze here varies in the range
of ~15-50 pm. The collision frequency is calculated as
superposition of the frequencies caused by both the
Brownian motion and the turbulent fluctuations.
For calculation of the fragmentation rate we used a known
Kuster's model (e.g. Flesch et al 1999). This model has the
only empirical parameter that is a function of the
agglomerate strength. Due to absence of experimental data
we used the value of this parameter from the paper of Flesch
et al (1999) where it was identified for an agglomeration of
polysterene particles in a mixing tank. Note that in our case
an accurate calculation of the fragmentation rate is not a
problem of primary importance because only relatively
large asphaltene particles, concentration of which is small,
can be efficiently broken in a turbulent flow. Under large
particles we understand those, sizes of which are close to or
larger than the inner turbulence scale.
The most difficult problem is evaluation of the
particle-particle collision efficiency oa. As we have already
mentioned an asphaltene agglomerate has a complex fractal
structure. In this case researchers often neglect a collision
efficiency reduction caused by interaction of hydrodynamic
flow fields around approaching particles (e.g. Flesh et al
1999). We also know that the particle-particle attraction
force leading to agglomeration is the Van-der-Waals
attraction force. However, there is no certain knowledge
about a possible steric effect that may prevent agglomerate
formation. Because of a limited understanding of the
particle-particle interactions the particle-particle collision
efficiency has been assumed to be a tuning constant,
independent of the particle size.
The simplest (binary) breakage distribution function was
employed: 1,,=F1,,,i=)+I/ I=2. This means that the i+l-th
fraction particle is broken into two fragments of equal sizes.


7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010

Flesch et al (1999) showed that this assumption provides
accuracy acceptable for engineering purposes.
As an initial condition for calculations we used the
concentration of primary (precipitated) particles by number
that is easily calculated if both the mass concentration of
primary particles and the primary particle size are known.
The mass concentration of primary particles was
determined experimentally for each hydrocarbon fluid by
measuring the total mass of asphaltenes after the deposition
experiment. In accordance with the semi-theoretical
analysis by Betancourt et al (2008) the primary asphaltene
particle size is do=1.6 nm.
We performed a number of calculations of PSD evolution in
the batch Couette system. Note that we considered the
regimes when the pressure in a Couette device was reduced
abruptly at the start. The number of measured parameters
was limited. We could evaluate the maximum particle size
at the end of each experiment of a given duration by using
high-pressure microscope (HPM). We could not monitor
PSD in a Couette device directly because of the high
pressure condition in the Couette cell. For modeling we
used only one fitting parameter: the particle-particle
collision efficiency, a. The results obtained can be
considered as estimations only.
Figs.2 and 3 illustrate particle size distribution for an oil
sample with asphaltene content of 5 wt%, the fluid density
of 760 kg/m3, and the fluid viscosity of 0.8 cp. The spindle
rotation speed was 3000 rpm. The particle fractal dimension
was selected to be Df=1.7. The inner turbulence scale under
such conditions was /.,,=V gnm. The maximum particle
size after T=2.6 h was approximately 25 pm. A reasonable
fit of the simulated results to the experimental data was
obtained at the collision efficiency a=2x.105. In Fig.2 one
can see how the mean particle size increases in time. In
Fig.3 the particle size distributions at the different time
moments are shown. The initially slow particle growth is
observed because the agglomeration of sub-micron particles
is controlled by the Brownian motion. The rapid particle
growth at the 2nd stage is explained by the dominance of the
turbulent mechanism of bringing particles together.
Slowing down and then stopping of the particle growth at
the 3rd stage are caused by the agglomerate fragmentation in
a turbulent flow. In this case the calculated PSD becomes
steady-state after approximately 2 hours.
The most important result of the PSD studies performed is
identification of the very low particle-particle collision
efficiency in all asphaltene systems analyzed. This verifies
an existence of a very strong steric effect preventing
asphaltene particle agglomeration. Because our major
interest is the deposition of particles on the previously
formed particle layer we can conclude that the probability
of particle sticking to the asphaltene coated wall should be
of the same order of magnitude as the particle-particle
collision efficiency. Taking into account this finding and
the previously mentioned result that the centrifugal force
doesn't lead to the noticeable asphaltene particle
stratification over the Couette device radius we may
conclude that there are no mechanisms, which may cause
the non-zero particle concentration gradient at the wall.
Thus, since the particle concentration gradient at the wall is






7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010


zero, the conventional deposition models, which assume
that the concentration gradient at the wall controls the
deposition, cannot be applied.
The developed deposition model is presented below. It is
based on a probabilistic approach, where the particle-wall
sticking probability is a key parameter determining the flux
of particles depositing on the wall.

35


25

2


Figure 2: Evolution of the mean particle size in a Couette
device in time.

0 -T=0 278 h
0 7od T=0 833 h
-?-T-2 778 h


I I
05I i
04-

03













We consider that chaotically moving particles reach the
wall being transported by Brownian motion and turbulence.
The turbophoresis is neglected because our calculations
102 101 100 101 102





showed tat particles, able to deposit, are too small to beers
efficiently transported by turbophoretic mechanism. Weution in a Couette device at
different time moments.





also assume that the Transport to mechanisms are mutually Wall




independent.
We consider thanismt chaotically moving particles reach the
wall beiSubmicron particles are involved into Brownan motion and turbulenion i.e.
particles chaotically move due to multiple our calculations wit
showed molecuthat particles, assemble to deposit, e veltoo small to be
efficiently transported by turbophoretic mechanism. We




also assume character the transport mechanisms are mdistributually
independent.

Brownian Mechanism
Submicron particles are involved into Brownian motion i.e.
particles chaotically move due to multiple collisions with
fluid molecules. If we assume that the velocity of fluid
molecules are characterized by the Maxwellian distribution
like in gases then the particle velocity is expected to have
the Maxwellian distribution as well (e.g. Beal 1971). Then
the frequency of particle collisions with the wall per unit
surface area is (e.g. Abramovich 1976):


v, =0.5 Ni,),/nB


where, uml =(2kBTf/m(d,))2 is the most probable
fluctuation velocity of particles of the i-th size fraction (e.g.
Abramovich 1976), Tf is the temperature,
kg =1.3806504-10-23 J/K is the Boltzmann constant,
m(d,) is the particle mass.
Considering that at a given time moment only half of the
particles move towards the wall, one can obtain the
equation for the most probable velocity directed to the wall
from Eq.4 as:


1)dBI = ,mB,/V = (2kBTf/m(dl)/ 7)2


Then the particle deposition flux caused by the Brownian
motion is:


q1 = ym(d )(N /2)udBi


where 71 is the sticking efficiency of a collision of the
i-th size fraction particle with the wall.

Turbulent mechanism
As already mentioned above, small asphaltene particles
follow fluid in fluctuation motion caused by turbulence
with a negligible slippage. Fortunately, there are
experimental data for the distribution of the dimensionless
root mean square fluid velocity component normal to the
wall in the wall vicinity (e.g. Guha 2008):


V 0.005y+2
1+0.002923y+2128


at 0 < y <200


where = 2 u. u. is the friction velocity
The root mean square velocity is different from the most
probable velocity. The latter can be approximately
evaluated as follows. There is the experimental evidence
that in a case of isotropic turbulence the distribution of
turbulence fluctuation velocity can be approximated by the
Maxwellian distribution (Kuboi et al 1972). Then, based
on the well known relations for this distribution (e.g. Bird
2002) we obtain the normal to the wall fluctuation velocity
component of the particle of size d, at the moment of
touching the wall as:

=dtl = V 1 (d /2) (8)

where V' (d /2) is the mean square component of the
fluid fluctuation velocity normal to the wall at the
moment of particle touching the wall.
Application of Eq.8 is not limited to the isotropic
turbulence because this equation contains the normal to the
wall fluctuation velocity component that is known (Eq.7).
Then the deposition rate due to turbulence fluctuations is
calculated as:


qt, = 71m(d1)(N1 /2),jl









The deposition mass flux of the i-th size fraction particles
is:

q, =0.5 y,m(d,)N, [udBi +dl (10)

As a reasonable approximation for modeling such a
complex system we assume that the particle-wall
interaction efficiency is constant ,=7.y
Then the total deposition mass flux is:


q =q1 = 0.5 ;mlmNl[dBi +dtl
1i=1 1=1i


where icr is the size fraction corresponding to the critical
particle size.

Calculation of Deposition in a Couette Device

The mass of the deposit accumulated on the wall during a
given time T is calculated in the same way for both batch
and flow through experiments:
T
M(T)= qJq(t)27RLdt (12)
0
The experimental parameter identification is reduced to
finding such values of the tuning parameters, which
minimize the absolute difference between the measured
and the calculated deposit masses.
Note that at a relatively high shear stress at the wall (high
rotation speed of a Couette device) the shear removal of
the deposit layer may significantly affect the deposition
rate. In such a case the actual flux of depositing particles to
the wall should be adjusted to take the shear removal into
account. A purely empirical approach can be used for this
purpose. We suggested calculating the corrected deposit
flux as:


qa = qz 1- -1 (13)



where n, or are the empirical parameters, Two is the deposit
yield stress defining the incipient shear removal.
Equation (13) approximately agrees with the experimental
observations. The deposition rate non-linearly decreases
with increasing the shear stress at the wall above a certain
threshold defined here as the deposit yield stress Tw0. The
all parameters of Eq. 13 are determined experimentally.

Calculation example
Let's consider the results of study of the oil B. In the all
experiments the initial pressure was p=15000 Psi and the
temperature To=366 K. The fluid density was pf=882 kg/m3,
the fluid viscosity f= 1.17 cp, and the particle density
ps=1200 kg/m3. The asphaltene particles were generated
due to the rapid pressure drop to 2900 Psi that is slightly
higher than the bubble point pressure pb=2800 Psi. The
onset asphaltene precipitation pressure (AOP) was
determined as I. .., =4' I Psi. The mass concentration of


7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010

precipitated particles was rather small wo=0.0017. The
particle fractal dimension was assumed to be Df=1.6. The
rotation speed was Q=1020 rpm that approximately
corresponds to the yield stress conditions for the deposit
layer therefore there the shear removal effect was not taken
into account in this case.


Oil B
Batch Run


0)
E
S15
E
(r
0 10
C


-Model
Experiment


Critical particle size = 30 nm


Figure 4: Deposit mass vs. time for the batch system
(Oil B).

50
Oil B Experiment
Flow-through run
40 Q=3 cc/min
Rotation speed = 1020 rpm
E
j30
E
20
Critical particle size =30 nm


0


0 0.5 1 1.5 2 2.5 3 3.5
Time, h

Figure 5: Deposit mass vs. time for the flow-through
system (Oil B).

The experimental data (masses of the asphaltene deposit at
the different Couette device running times) were fitted to
the experimental results upon identification of the
following model parameters: the particle-particle collision
efficiency, a=3-10-5, the particle-wall collision efficiency,
T=3.55.10-6, and the critical particle size, dcr=30nm.
Figures 4, 5 show the calculations results: the deposit mass
vs. time for both the batch and the flow-through
experiments. One can see that the fitting is very accurate.
Note that computations of the asphaltene deposition for
other hydrocarbon fluids also demonstrated a good model
performance. We would like to emphasize than those
calculations also showed that the identified critical particle
size was always very small (significantly below 1 Ipm).

Calculation of Deposition in a Pipe

The deposition in a pipeline is calculated at using the
model parameters identified from the Couette device









experiments. The population balance equations for a pipe
flow are based on a well known simple model of an ideal
displacement reactor (Kafarov, 1977) and therefore are not
presented here. The function "concentration of precipitated
particles vs. pressure", which is needed for solving the
population balance equations, can be either modeled (e.g
Du and Zhang 2004) or obtained from the experiment.
Usually, a linear function can be employed as a reasonable
engineering approximation.
The deposit layer thickness along a pipeline as a function
of time is easily calculated by considering the mass
balance for a small layer element as:


(x)h Top
6(x)=q. (x) (14)
Ps(1-tPd)

where Top is the duration of pipeline operation, (pd is the
deposit layer porosity that can be evaluated experimentally
based on the deposit analysis.

Production Tubing Calculation Example
The aphaltene deposition can be especially serious issue
for vertical production tubings. The pressure drop in a
vertical tube is much higher than that in a horizontal one
due to the hydrostatic component. The total pressure
gradient is calculated as superposition of the pressure
gradient caused by the fluid friction and the hydrostatic
pressure gradient (-pfg). We performed the deposition
modeling for the oil C flowing through the production
tubing. Firstly, the experiments in the laboratory Couette
device were conducted. The fluid system parameters were:
the initial pressure p 1040 Psi, the temperature To-342 K,
the fluid density pf 710kg/m3, the fluid viscosity
gf=0.7 cp, the asphaltene particle density ps=1200 kg/m3,
the bubble point pressure Pb=2900 Psi, the onset
asphaltene precipitation pressure was determined as
PAoP 7500 Psi, and the mass concentration of precipitated
particles wo=0.014. The particle fractal dimension was
assumed to be Df=1.6. The deposition model in a Couette
device allowed us to identify the particle-particle collision
efficiency ac=510-6, the particle-wall collision efficiency
y= 110-6 and the critical particle size dcr,72 nm. The
parameters of the empirical shear removal model were
determined for the Couette device rotation speeds range:
2720-1800 rpm. These parameters were: 0wo 0.4406 Pa,
o= 0.4302, n 0.2758.
The oil C is characterized by significantly higher
asphaltene content than the oil B used for the Couette
device experimental example presented above. Therefore,
the deposition at transporting the oil C can lead to a
significant production tubing plugging as one can figure
out from Fig.6. In this figure we showed the deposit layer
thickness distributions, calculated for different flow
velocities, U 1.8, 2.15, 2.5 m/s, along the tubing of a
relatively small diameter (D 62.5 mm) and of the 4 km
length. The operation period is 100 days. The pressure
drop on the distance considered is approximately equal to
the difference between the onset precipitation pressure and
the bubble point pressure for the all considered flow
velocities because contribution of the hydrostatic
component into the total pressure drop is much larger than
that caused by the viscous friction. For an illustration


7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010

purpose we assumed that the mean flow velocity does not
change with the deposit layer thickness increase and the
corresponding reduction of the tubing cross-section.
Actually, calculating the flow rate and the deposit layer
thickness are coupled problems. Besides the tubing
diameter the flow rate depends also on the bottomhole
pressure (the pressure in the wellbore at the level of
producing formation). In practice the characteristic "the
bottomhole pressure vs. the production rate" has to be
known from the well test. It is clear that the actual flow
rate data are needed to accurately evaluate the all
deposition model parameters. However, taking the
mentioned well characteristic into account is beyond the
scope of this paper.
As one can see from Fig.6 the deposit thickness increases
along a pipe asymptotically approaching a constant value.
The higher the flow velocity the thinner the deposit layer
due to the shear removal effect. It is important to
emphasize that the calculated deposit layer thickness
distributions are in a qualitative agreement with the field
observations published by Alkafeef et al (2005). Those
authors measured the asphaltene deposition layer thickness
distribution after 180 days of production in the production
tubing of the same diameter (D=6.25 mm) that we
employed for our calculations. Though we are not aware of
the oil composition and the fluid flow rate in their
investigations the measured deposit layer thickness
(abound 1/3 of the tubing diameter radius) was close to
that that we calculated at the mean flow velocity
U=2.15 m/s if the production period would be 180 days.
The same deposit layer thickness can be reached in
100 days if the mean flow velocity is U=1.8 m/s (see
Fig.6). Based on the analysis performed we can conclude
that the computed results at least do no contradict to the
Alkafeef's et al (2005) experimental data.


,I I
35 -

3 -

25 :

2-
1 I I

I /
1 /
////
0 5


Oil C
D=2 5"
Top= 100 days


0 05 1 15 2
Deposit layer thickness, mm


-U=1 80 m/s
--U=215 m/s
--,U=250 m/s


25 3


Figure 6: Distribution of the asphaltene deposit layer
thickness along a vertical production tubing.

Thus, the asphaltene deposition may be a serious problem
for production systems where the pressure gradient is large.
Besides the deposition in vertical tubings illustrated above,
the significant asphatene deposition occurs inside and/or
immediately after flow line sections possessing a high
hydrodynamic resistance. The abrupt pressure drop leads
to a massive particle precipitation and deposition









respectively. The asphaltene deposition model developed
allows computing the asphaltene deposition process at any
pressure drop scenarios if both the pressure gradient and
the asphaltene precipitation rate are known.

Conclusions

A possibility of simulating asphaltene particle deposition in
a pipeline by using a Couette device has been analyzed.
Similarity of the shear stresses at the walls of both devices
provides similarity of the particle transport to the wall. Only
particles, which are smaller than a certain (critical) size, are
able to deposit. The population balance approach was
employed for modeling an evolution of the particle size
distribution in time in a Couette device and along a pipeline.
The calculations showed that the particle-particle collision
efficiency is very low. This means that the particle-wall
sticking efficiency has to be small to the same extent and
therefore the particle concentration gradient at the wall is
zero. Since the conventional modeling approach to the
particle transport to the wall could not be employed in this
case the depositing particle flux was modeled by using the
particle-wall sticking efficiency as an empirical parameter.
The calculations showed that the experimental data on
particle deposition in a Couette device can be accurately
described by the model developed at using only 3 fitting
parameters. The computations of the asphaltene deposition
in a vertical production tubing demonstrated that the
calculated deposit thickness distributions are in a qualitative
agreement with the field data found in literature.

References

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(in Russian) (1976).

Alkafeef, S.F., Al-Medhadi F., Al-Shammari, A.D. A
Symplified Method to Predict and Prevent Asphaltene
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7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010


Eskin, D. An Engineering Model of a Developed Turbulent
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