Document Type: Research Paper
Authors
^{1} Department of Petroleum Engineering, Petroleum University of Technology, Ahwaz, Iran
^{2} Department of Petroleum Engineering, Petroleum University of Technology, Ahwaz, Iran.
Abstract
Keywords
Iranian Journal of Oil & Gas Science and Technology, Vol. 5 (2016), No. 2, pp. 4565
http://ijogst.put.ac.ir
Asphaltene Deposition Modeling during Natural Depletion and Developing a New
Method for Multiphase Flash Calculation
Gholamreza Fallahnejad and Riyaz Kharrat*
Department of Petroleum Engineering, Petroleum University of Technology, Ahwaz, Iran
Received: August 17, 2014; revised: January 06, 2015; accepted: January 08, 2015
Abstract
The specific objective of this paper is to develop a fully implicit compositional simulator for modeling
asphaltene deposition during natural depletion. In this study, a mathematical model for asphaltene
deposition modeling is presented followed by the solution approach using the fully implicit scheme. A
thermodynamic model for asphaltene precipitation and the numerical methods for performing flash
calculation with a solid phase are described. The pure solid model is used to model asphaltene
precipitation. The transformation of precipitated solid into flocculated solid is modeled by using a first
order chemical reaction. Adsorption, pore throat plugging, and reentrainment were considered in the
deposition model. The simulator has the capability of predicting formation damage including porosity
and permeability reduction in each block. A new set of independent unknowns in a fully implicit
scheme is presented for asphaltene deposition modeling. In order to find the solution of these
variables, the same number of equations is also presented. The description of how to solve the
nonlinear system of equations is also described.
Keywords: Asphaltene Deposition, Composition Simulation, Multiphase Flash, Solid Model,
Modeling
1. Introduction
Asphaltene precipitation and deposition is a very serious problem that occurs during oil production
and processing. Although the problem has been usually observed in the wellbore and the production
system, asphaltene precipitation and deposition may occur anywhere in the reservoirwellbore system
including nearwellbore region and inside the wellbore (Darabi et al., 2014).
Various models have been proposed to describe the phase behavior of asphaltene precipitation.
According to the literature, there are four main groups for asphaltene precipitation modeling. The first
one is the liquid solubility which is based on the FloryHuggins theory (Flory, 1942). The second
models are the solid models which assume that asphaltene is treated as a single component in the solid
phase. The third one is the colloidal solution model proposed by Leontaritis and Mansoori (1987) with
the consideration of asphaltene as solid particles in a colloidal suspension stabilized by adsorbed
resins on asphaltene surface. The fourth one is the thermodynamic micellization model presented by
Firoozabadi and Pan (1998). The minimization of the molar Gibbs free energy is the basis of this
thermodynamic model.
* Corresponding Author:
Email: kharrat@put.ac.ir
46 Iranian Journal of Oil & Gas Science and Technology, Vol. 5 (2016), No. 2
Few mathematical models were correlated to simulate asphaltene deposition. According to the
literature, there are three categories for the asphaltene deposition modeling. The first one is the
Leontaritis’ model (1998) which assumes that asphaltene deposition occurs only around the wellbore
vicinity. The second model is the model of Nghiem et al. (1998) supposing that only adsorption
mechanism contributes to the asphaltene deposition. The third one is the model of Wang and Civian
(2000) which considers primary physical deposition processes including adsorption, pore throat
plugging, and reentrainment to describe the phenomenon of solid particle deposition.
In this study, the pure solid model is used to model asphaltene precipitation. Also, a deposition model
including adsorption, pore throat plugging, and reentrainment is used. The reduction in the rock
porosity and permeability are also included in the asphaltene model. A onedimensional simulation
was carried out for investigating the results of asphaltene deposition model.
2. Mathematical model description
The transportation equations, asphaltene precipitation model, asphaltene deposition model, and the
porosity and permeability reduction are considered as the main part of the mathematical model for
modeling asphaltene deposition. To develop a simulator for asphaltene deposition, the asphaltene
precipitation and deposition models and the porosity and permeability reduction are incorporated into
a compositional simulator.
2.1. Asphaltene precipitation model
In this study, a pure solid model is used for predicting the amount of precipitation, while a cubic EOS
is applied to the modeling of oil and gas phase behavior. The solid particles are divided into three
parts: precipitated, flocculated, and deposited solid. Kohse and Nghiem (2004) proposed a model
which assumes that the heaviest component of oil can be split into a nonprecipitating and a
precipitating component. The precipitated solid is divided into solid 1 which is in equilibrium with the
heaviest component in the oil phase and solid 2 that is created from solid 1 via a first order chemical
reaction. The fugacity of asphaltene component in the precipitated solid 1 (S1) is given by:
*
*
1 1
( ( ))
ln ln
( ) s s
Vs P P
f f
RT
= +  where, s1 f is the fugacity of solid S1 at reservoir pressure, and *
s1 f is the reference solid fugacity at P*
; Vs is the solid molar volume and P* represents the pressure at which the asphaltene just starts to
precipitate. Under the thermodynamic equilibrium conditions between S1, oil, and gas, the following
equations must be applied:
ln ln , 1, , io ig c f = f i = ¼n (2)
, 1 ln ln
nc o s f = f (3)
The equality of fugacity of ith component in the oil and gas phases is shown in Equation 2. Equation 3
expresses the equality of fugacity of the precipitating component in the oil phase and in the
precipitated solid S1.
Gh. Fallahnejad and R. Kharrat/ Asphaltene Deposition Modeling during … 47
a. Specification of solid molar volume for asphaltene model
For the modeling purpose, solid molar volume is used as a matching parameter. The solid molar
volume is initially calculated at reference pressure (P*) by EOS. Then, we plot the error term versus
solid molar volume to find an optimum fit between the calculated and experimental data. The solid
molar volume which is used in the plot contains the values around the calculated solid molar volume
at the reference pressure. The error term is defined as follows:
2
(i) exp (i)model
1
( )
n
i
error W W
=
=Σ 
(4)
Figure 1 shows the error term as a function of solid molar volume. The estimated molar volume for
the asphaltene component from EOS is 10.5611 ft3 / lbmole . As can be seen, the minimum error for
the given values of solid molar volume is found at 10.81 ft3 / lbmole .
Figure 1
Error term as a function of solid molar volume.
b. Stability test analysis
In multiphase flash calculation, the number of phases is generally unknown beforehand. We use the
phase stability analysis to determine the number of phases in each gridblock.
In this study, stability test analysis is separated into two parts: fluidfluid and fluidsolid stability test.
b.a. Fluidfluid stability test
The following set of equations (Nghiem and Li, 1984) should be solved to check the stability of a
fluid phase (oil) with composition x, pressure P, and temperature T.
( , , ) ln ( , , ) 0 i i i i g ºlnK +lnφ y P T  φ x P T = (5)
0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
10.510.610.710.810.9 11 11.111.211.311.411.511.611.7
Error
Solid molar volume (ft3/lb mole)
48 Iranian Journal of Oil & Gas Science and Technology, Vol. 5 (2016), No. 2
i 1, ,
i c
i
Y
K i n
x
= = ¼
(6)
where
If Equation 5 is solved for at a given P and T, a solution is found that
1
1
nc
j
j
Y
=
Σ >
(8)
The mixture x is unstable at P and T.
Equation 5 can be solved with the QNSS method (Nghiem and Li, 1984). With the QNSS, letbe the
vector with elements and let be the iteration value for the vector. The QNSS iteration
is given by:
And
is a scalar given by:
( ) ( )
( ) ( )
( ) ( )
1 1
1
1 1
Δ .
Δ .Δ
k k
k k
k k
α g
ξ ξ
α g
 

 
=
(11)
where
g=norm(g) (12)
α = norm(α) (13)
The process is initialized with
1 andis started with Wilson equation.
Figure 2 shows a flow chart for this procedure.
1
c
i
i n
j j
Y
y
Y
=
=
Σ
(7)
(k) k 1 k α α α D = + 
(9)
(k ) (k ) (k )
Dα = ξ g (10)
Gh. Fallahnejad and R. Kharrat/ Asphaltene Deposition Modeling during … 49
Start
Input initial guess
=0
Usingto evaluateand
Calculation of̅
Convergence
check
Yes
End of QNSS
iteration
No
Calculation of
()
Update
+1 = −
()̅()
Figure 2
QNSS iteration for fluidfluid stability test.
b. Fluidsolid stability test
To test the existence of solid phase, the following criteria should be satisfied. The solid phase exists
if:
1 ln
nco s f ³ ln f (14)
This implies that if the fugacity of the asphaltene component in the solid phase is less than the
fugacity of the component in the oil phase, more asphaltene precipitation will occur until both
fugacities become equal. Figure 3 depicts the stability test calculation procedure.
50 Iranian Journal of Oil & Gas Science and Technology, Vol. 5 (2016), No. 2
Figure 3
Stability test analysis.
c. Multiphase flash calculations
The set of independent variables involved for the flash calculation are ( ), , , and i Ln K Ks Lg Ls , which
gives nc+3 unknowns. The following set of equations should be used for solving these variables:
lnKig +lnjig lnjio =0 (15)
ln ln ln 0
ncs s nco K + φ  φ = (16)
With
ig
ig
io
y
K
y
= (17)
Gh. Fallahnejad and R. Kharrat/ Asphaltene Deposition Modeling during … 51
ij
ij
ij
f
y P
j = j=o and g
(18)
1
ncs
nco
K
y
= (19)
s
s
f
P
j = (20)
In conjunction with the above equations, the following material balance equations can be derived:
( )
( ) ( ) 1
1
0
1 1 1
nc
ig i
i g ig s is
K z
= L K L K

=
+  + 
Σ (21)
( )
( ) ( ) 1
1
0
1 1 1
nc
is i
i g ig s is
K z
= L K L K

=
+  + 
Σ (22)
The oil and gas phase composition can be calculated from the following equations:
1 ( 1) ( 1)
i
io
g ig s is
z
y
L K L K
=
+  +  (23)
ig ig io y =K x (24)
There are nc+3 nonlinear equations and unknowns for multiphase flash calculations. The unknown
vector is considered as follows:
( ( ), , , ) i ig s g s X = Ln K K L L (25)
To solve the nonlinear system of equations, the Newton method is used, where the residual vector ( R
) is computed. The residual vector consists of Equations 15, 16, 21, and 22.
The detailed multiphase flash calculation procedure is given in Figure 4.
52 Iranian Journal of Oil & Gas Science and Technology, Vol. 5 (2016), No. 2
START
Input initial guess
=
0(ln , , !, ! )
Using
to evaluate dependent variables
Calculation of ",, #!, #$
Convergence check Yes
No
Calculation of residual vector, %&&('&&()
Calculation of Jacobean matrix
)*
+, 
Solve the system )*
+, Δ
= −/&(
+, )
Update the independent variables
01 =
+, + Δ
Next time step
Figure 4
Flow chart for flash calculations procedure.
Gh. Fallahnejad and R. Kharrat/ Asphaltene Deposition Modeling during … 53
2.2. Asphaltene deposition model
In this study, the model of Wang and Civian (2000) is used to simulate the formation damage caused
by asphaltene deposition during natural depletion. This model has three terms, including adsorption,
pore throat plugging, and reentrainment to describe asphaltene deposition process. The deposition
equation is presented as follows:
, A ( )
A A L cr L L A
E
C E v v u C
t
¶ =af  b  +g
¶
(26)
2.3. Porosity and permeability reduction model
Once asphaltene deposition occurs, porosity alteration is modeled by the following equation:
0 A f =f E (27)
To update the permeability values, a power law relationship is used as given below:
3
0 0
f
K
R
K
f
f
= =
(28)
2.4. Compositional simulator equations
Compositional simulator includes component molar flow equations, phase equilibrium equations,
multiphase flash equations, and saturation constraint equations, which are solved simultaneously.
a. Component molar balance equations
The following material balance equation with the consideration of asphaltene can be written:
( ) ( ) ( ) j
o jo o g jg g j
N
y y q
t
f
x x
¶
Ñ ÑF +Ñ ÑF + =
¶
1, , 1 c j = ¼n  (29)
( ) ( ) ( )
1 2
( ) c
c c c
n
o n o s s o g n g g n
N
y y y y q
t
f
x x
¶
Ñ + + ÑF + Ñ ÑF + =
¶
(30)
where,
Ni is the moles of component i per pore volume (lbmole/ft3);
is the mole fraction of component i in phase k (k = o,g);
23 is the mole fraction of suspended solid in oil phase (j = 1, 2);
is the molar density of phase k (k = o,g) (lbmole/ft3);
A material balance on solid S2 completes transport equations:
( ) ( )
2
2
o s o b o s2
s N
y V S r q
t
f
x f
¶
Ñ ÑF + + =
¶
(31)
54 Iranian Journal of Oil & Gas Science and Technology, Vol. 5 (2016), No. 2
1
'
T s
s
o
N L
y
N
=
(32)
2
2
s
s
o
N
y
N
=
(33)
12 s1,o 21 s2,o r = K C K C (34)
where,
1,
'
T s
s o
o
N L
C
S
=
(35)
2
1,
s
s o
o
N
C
S
=
(36)
b. Phase equilibrium equations
The following equations for thermodynamic equilibrium between S1, oil, and gas must be applied:
ln ln , 1, , io ig c f = f i = ¼n (37)
, 1 ln ln
nc o s f = f (38)
c. Phase composition constrains
The following equations are used to consider the phase composition constrains:
1 1
0
nc nc
io ig
i i
y y
= =
Σ Σ =
(39)
1 1
0
nc nc
io is
i i
y y
= =
Σ Σ =
(40)
d. Volume constraint equation
The volume constraint equation expresses that the pore volume in each grid must be occupied by the
total fluid volume. This equation can be written as follows:
3. Solution approach
After finite differencing, there are (2+ 6)6 nonlinear equations for 6 number of gridblocks.
Equations 28, 29, 30, 36, 37, 38, 39, 40, and 25 provide a system of nonlinear equations that can be
solved for (7, ! ,7 8,7 9,:, , !, ! for each gridblock. A fully implicit technique is used
1
1 0
np
j
j
S
=
Σ  = (41)
Gh. Fallahnejad and R. Kharrat/ Asphaltene Deposition Modeling during … 55
to solve the governing equation for these independent variables. The unknown vector is considered as
follows:
( ( ) ) 2 3 , , , , , , , i = N Ln Ki Ns Ns P Ks Lg Ls i X
(42)
A Newton method is used to linearize the governing equations in terms of the independent variables.
The solution procedure is described as follows:
Input initial
1. The solution vector at the old time step is usually set as the initial guess for the new time step;
2. Using initial guess vector to compute dependent variables such as porosity, molar density,
relative permeability, molar composition, phase viscosities, and source or sink;
3. Calculate the residual vector and check for the convergence. If convergence is achieved, the
guessed solution vector is considered as the true solution for the new time step. If a tolerance
is exceeded, another solution vector is guessed and the independent variables are updated.
The new solution vector is obtained by solving the system )*
;<=Δ
/&
;<=where the
Jacobian matrix (J) and the residual vector are computed using the previous solution vector.
Independent variables are updated as follows:
X&&?@A
X&&BCD
ΔX&& (
43)
4. Results and discussion
In this section, the results of the implemented asphaltene model which is obtained from a 1D
simulation case are presented. Table 1 shows the composition of oil used for studying asphaltene
precipitation, which is obtained from the work of Burke et al. (1990). The reservoir properties are
given in Table 2.
Table 1
Modeled fluid composition.
Component mole % MW
CO2 2.46 44.010
N2 0.57 28.013
C1 36.37 16.043
C2 3.47 30.070
C3 4.05 44.097
iC4 0.59 58.124
nC4 1.34 58.124
iC5 0.74 72.151
nC5 0.83 72.151
FC6 1.62 86.000
C7C15 19.66 147.272
C16C25 12.55 279.2
C26C30 4.00 389.52
C31A+ 7.42 665.624
C31B+ 4.32 665.624
56 Iranian Journal of Oil & Gas Science and Technology, Vol. 5 (2016), No. 2
Table 2
Simulation input data.
Parameter Value
No. of gridblocks 21
Reservoir size 500×10×10 ft3
Reservoir Temperature 212 °F
Initial water saturation 0
Initial reservoir pressure 6014.7 psi
Porosity 0.2
Permeability 30 md
Production constraint 2.5 bbl./day
Rock compressibility 5 F 10GH psi1
Ref. P for rock compressibility 14.7 psi
The reservoir is onedimensional homogenous with the size of 500F 10 F 10 ft3 and divided into 21
gridblocks. The porosity and permeability are 0.2 and 30 mD respectively. The initial reservoir
pressure is 6014.7 psia and reservoir temperature is 212 I. The well is located at the center of the
reservoir at the constant bottomhole reservoir fluid rate (BHF) of 2.5 bbl./day. The input parameters
for the precipitation, deposition, and flocculation models, which are taken from Kohse and Nghiem
(2004), are shown in Table 3. In this study, it is assumed that the interstitial velocity is less than the
critical interstitial velocity; as a result, the entrainment rate coefficient J is set to zero. The liquidgas
relative permeability curves and the liquidgas capillary pressure curve for the simulation case are
shown in Figures 5 and 6.
Table 3
Asphaltene precipitation, deposition, and flocculation model parameters.
Parameter Value
:∗ 4014.7 psi
LMN2
∗ 25.223 psi
Vs 10.812 lbm/ft3
k 12 (day1 ) 0.1
k 21 (day 1 ) 0.08
α (day1) 0.01
β (ft1) 0.0
γ (ft1) 0.05
σ 150
Gh. Fallahnejad and R. Kharrat/ Asphaltene Deposition Modeling during … 57
Figure 5
The liquidgas relative permeability curves.
Figure 6
The liquidgas capillary pressure curve.
Figure 7 presents pressure versus time for the gridblock (11,1,1). As can be seen, the initial reservoir
pressure, the onset pressure (P*) of asphaltene, and saturation pressure are 6014.7, 4550, and 2950
psia respectively. The onset pressure for asphaltene precipitation will be reached after approximately
13 days of production time. The saturation pressure will be reached after 30 days of production time.
The pressure depletion continues to reach a value of 2223.4 at the end of simulation time.
0.0000
0.2000
0.4000
0.6000
0.8000
1.0000
1.2000
0.0000 0.2000 0.4000 0.6000 0.8000 1.0000
Relative permeability
Gas saturation
Krg Krog
0.0000
5.0000
10.0000
15.0000
20.0000
25.0000
30.0000
35.0000
0.0000 0.2000 0.4000 0.6000 0.8000 1.0000
Capillary pressure (psi)
Gas station
58 Iranian Journal of Oil & Gas Science and Technology, Vol. 5 (2016), No. 2
Figure 7
Pressure profile at gridblock (11,1,1) after 80 days.
Oil and gas saturation are depicted in Figure 8. As can be seen, the oil saturation remains constant
until the bubble point pressure is reached. After 30 days of production time the oil saturation
gradually decreases and will be reached the value of 0.9 at the end of simulation time.
Figure 8
Oil and gas saturation at gridblock (11, 1, 1).
Porosity versus time is presented in Figure 9. As this figure shows, the porosity reduction shows a
sharper trend after saturation pressure is reached. Both pressure depletion and asphaltene deposition
are the causes for porosity alteration.
0
1000
2000
3000
4000
5000
6000
7000
0 20 40 60 80
Pressure (psi)
t (Day)
0
0.2
0.4
0.6
0.8
1
0 2 0 4 0 6 0 8 0 1 0 0
So or Sg t
(
DAY)
So Sg
Gh. Fallahnejad and R. Kharrat/ Asphaltene Deposition Modeling during … 59
Figure 9
Porosity at gridblock (11, 1, 1).
Permeability reduction (K/Ki) is reported in Figures 10. This parameter has an important role in
asphaltene precipitation studies. A uniform decline of permeability can be seen in this plot.
Figure 11 shows how the productivity index varies with time. This parameter also shows the same
trend as permeability reduction. As can be seen, after 30 days of production time, in addition to
asphaltene deposition, oil relative permeability reduction plays an important role in the productivity
index reduction; as a result, this parameter starts decreasing sharply after 30 days.
Figure 10
Ratio of the damage permeability to the original permeability.
0.2015
0.202
0.2025
0.203
0.2035
0.204
0.2045
0.205
0.2055
0.206
0.2065
0 20 40 60 80
Porosity
t (Day)
0.995
0.996
0.997
0.998
0.999
1
1.001
0 20 40 60 80
K/Ki
t (Day)
60 Iranian Journal of Oil & Gas Science and Technology, Vol. 5 (2016), No. 2
Figure 11
Productivity index versus time at wellblock.
Figure 12 illustrates the amount of precipitated asphaltene (mole per pore volume lbmole/ft3) versus
time during the simulation time. As can be seen, after 12 days of simulation time, the precipitation is
started. The maximum value of precipitation occurs around the saturation pressure and decreases as
pressure drops further.
Figure 12
Asphaltene precipitated mole per pore volume at gridblock (11,1,1).
Figure 13 shows the amount of flocculated asphaltene (mole per pore volume lbmole/ft3) versus time
during the simulation time. As can be seen, the trend of plot is the same as the plot of asphaltene
0.0065
0.007
0.0075
0.008
0.0085
0.009
0.0095
0.01
0 20 40 60 80
Productivity index (bbl./day/psi)
t (Day)
0
0.0001
0.0002
0.0003
0.0004
0.0005
0.0006
0 20 40 60 80
Precipitated mole per PV
t (Day)
Gh. Fallahnejad and R. Kharrat/ Asphaltene Deposition Modeling during … 61
precipitation, but there is a time interval between the maximum value of the precipitation and
flocculation.
Figure 13
Asphaltene flocculated mole per pore volume at gridblock (11,1,1).
The amount of deposited asphaltene (mole per pore volume lbmole/ ft3) with respect to production
time is shown in Figure 14. The gradual increase of deposited asphaltene at gridblock (11,1,1) is
shown in this figure.
Figure 14
Asphaltene deposited mole per pore volume at gridblock (11,1,1).
Figures 15 and 16 show the comparison of the precipitated and deposited asphaltene profile between
Matlab code and CMG GEM during 80 days of simulation time respectively. As can be seen, because
0
0.00005
0.0001
0.00015
0.0002
0.00025
0.0003
0.00035
0.0004
0.00045
0 20 40 60 80
Flocculated mole per PV
t (Day)
0
0.00002
0.00004
0.00006
0.00008
0.0001
0.00012
0.00014
0.00016
0 20 40 60 80
Deposited Mole per PV
t (Day)
62 Iranian Journal of Oil & Gas Science and Technology, Vol. 5 (2016), No. 2
of considering new multiphase flash calculation approach, the deposited asphaltene profile gives a
better match than the precipitated profile. As a matter of fact, in the deposited profile, in addition to
thermodynamic equations, material balance equations play a great role, and since it regards the new
approach to multiphase flash calculation, the material balance equations would change. As a result, a
clear difference between Matlab code and CMG GEM can be seen for the deposited asphaltene profile
compared to the precipitated profile.
Figure 15
Asphaltene precipitated mass per bulk volume at gridblock (11,1,1).
Figure 16
Asphaltene deposited mass per bulk volume at gridblock (11,1,1).
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0 20 40 60 80
Precipitated mass per BV (lb./ft3)
t (Day)
Matlab CMG GEM
0
0.005
0.01
0.015
0.02
0.025
0 20 40 60 80 Deposited mass per BV (
lb./ft3)
t (DAY)
Matlab CMG GEM
Gh. Fallahnejad and R. Kharrat/ Asphaltene Deposition Modeling during … 63
5. Conclusions
This paper presents the modeling of the phase behavior and dynamic aspects of asphaltene
precipitation and deposition for natural depletion. A solid model is used to model asphaltene phase
behavior. The thermodynamic model for asphaltene precipitation has been successfully incorporated
into a fully implicit compositional simulator, where the phase equilibrium equations, the saturation
constraint equation, the component transport equations, the multiphase flash equations, and the
deposition equation are solved simultaneously for each gridblock. Just a part of the precipitated
asphaltene is considered to adsorb to the rock and the remaining part is considered to stay suspended
in the oil phase. A resistance factor model was introduced for the estimation of the effect of
asphaltene precipitation on relative permeability.
Nomenclature
O2P; : Concentration of suspended solid s1 in oil phase [lbmole/ft3]
O2Q; : Concentration of suspended solid s2 in oil phase [lbmole/ft3]
O2Q
R : Volumetric concentration of flowing solid 8 per volume of oil []
fig : Fugacity of component i in gas phase [psi]
L; : Fugacity of component i in oil phase [psi]
L2S
∗ : Reference solid fugacity [psi]
L2S : Fugacity of solid S [psi]
6 : Total number of gridcells
: Number of hydrocarbon components
Ni : Moles of component i per pore volume [lbmole/ft3]
Ns1 : Moles of solid S per pore volume [lbmole/ft3]
Ns2 : Moles of solid 8 per pore volume [lbmole/ft3]
Ns3 : Moles of deposited solid per pore volume [lbmole/ft3]
NT : Total number of moles per pore volume [lbmole/ft3]
7T U : Total number of moles without flocculated and deposited solid per pore volume
[lbmole/ft3]
V; : Oil phase Darcy velocity [ft/day]
V : Gridblock volume [ft3]
$W,; : Critical oil phase interstitial velocity [ft./day]
$2Q
X : Volume of deposited solid 8 per gridblock volume []
: Mole fraction of component i in phase k (k = o,g) []
23 : Mole fraction of suspended solid in oil phase (j = 1, 2) []
Y : Global mole fraction of component i in feed []
64 Iranian Journal of Oil & Gas Science and Technology, Vol. 5 (2016), No. 2
Greeks
: Surface deposition rate coefficient [day1]
J : Entrainment rate coefficient [ft1]
Z : Pore throat plugging rate coefficient [ft1]
#3 : Fugacity coefficient of component i in phase j
: Molar density of phase k (k = o,g) [lbmole/ft3]
[; : Oil phase interstitial velocity [ft./day]
[2 : Solid molar volume [ft3/lbmole]
Subscripts
g : Gas
o : Oil
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