Geomaterials, 2012, 2, 57-62
http://dx.doi.org/10.4236/gm.2012.23009 Published Online July 2012 (http://www.SciRP.org/journal/gm)
The Mathematical Model for Particle Suspension Flow
through Porous Medium
Hooman Fallah, Hosein Barzegar Fathi, Hamed Mohammadi
Department of Petroleum Engineering, Firoozabad Branch, Islamic Azad University, Firoozabad, Iran
Received May 12, 2012; revised June 14, 2012; accepted June 24, 2012
ABSTRACT
Transport of suspensions and emulsions in porous media occurs in numerous processes of environmental, chemical,
petroleum and civil engineering. In this work, a mass balance particle transport equation which includes filtration has
been developed. The steady-state transport equation is presented and the solution to the complete advective-dispersion
equation for particulate suspension flow has been derived for the case of a constant filter coefficient. This model in-
cludes transport parameters which are particle advective velocity, particle longitudinal dispersion coefficient and filter
coefficient. This work recommends to be investigated by particle longitudinal dispersion calculation from experimental
data, directly. Besides, the numerical model needs to be developed for general case of a transition filter coefficient.
Keywords: Particulate Suspension Flow; Filtration Theory; Filtration Coefficient; Transport; Porous Medium;
Mathematical Model
1. Introduction
The transport of particulate suspensions in porous me-
dium occurs in a variety of industrial and natural process
such as wastewater treatment, propagation of pollutants
in subsurface environment, fouling of membranes and
seawater injection in oil reservoirs. Flow of solid and
liquid particles with particle capture by the rock takes
place during injection of seawater and produced water in
oil fields.
Particle transport and retention are mostly important
for the environmental processes, where the particle con-
centration must not exceed a safety value, while the per-
meability changes is important for petroleum production
due to its effect on well productivity and injectivity [1].
The study of particle retention in porous medium can
be dated back to the work of Iwasaki [2] who proposed
the following basic empirical for deep bed hydrosol fil-
tration.
cu
t



 (1)
where:

is the specific volume of deposited material
(particle retained concentration), is the fluid superfi-
cial velocity, is the particle suspended concentration
and
u
c

is called the filtration coefficient.
The must used approach for evaluating colloid migra-
tion, retention and detachment is solute transport mass
balance equation with th e sink term for particle retention
[3,4].

2
2
o
cc
cUD
tx


x

 
  (2)

det
cU k
t



 
 (3)
The term

in Equation (3) is called filtration coeffi-
cient. Equations (1) and (2) together with the formula for
coefficient

are called the classical filtration theory in
above refere n ces.
In the current work, a theoretical investigation of par-
ticulate suspension flow through porous media has been
done. The steady-state transport equation is presented
and the solution has been derived. The filtration coeffi-
cient

will be discussed as a dominant parameter in
the particle transport and retention through porous me-
dium.
2. Particulate Suspension Flow through
Porous Medium
2.1. Filtration Theory
When fluid containing particles reaches a porous medium,
the liquid and the solid phase in the suspension can be
separated, either by depositing in the pore or accumulat-
ing in front of the surface. This is similar to what hap-
pens in the filtration process. Then, the retention process
of particles when flowing through a porous media is
called Deep Bed Filtration [5]. Th e deep filtration occurs
because of several mechanisms: the contacting of parti-
C
opyright © 2012 SciRes. GM

H. FALLAH ET AL. 58
cles with the retention site, the fixing of particle sites and
the breaking away of previously retained particles [6].
2.2. Mechanisms of Particles Retention
The particle capture in porous media can be caused by
different mechanisms [7].
• Size exclusion (large particles are captured in small
pores and pass through large p ores );
• Electrical forces (London-Van der Waals, double
electrical layer, etc.);
• Gravity segregation;
• Multi particle bridging.
3. Basic Formulation of Particles Transport
through Porous Medium
The model includes transport parameters which are parti-
cle advective velocity, particle longitudinal dispersion
coefficient and filter coefficient. These parameters have
been defined by dimensional analysis using the pertinent
variables of the porous medium system.
3.1. Advective Velocity
The averaged particle velocity in the porous media, has
been found to be either the same or slightly higher than
that of the carrier fluid. This deviation is caused by the
particle’s size. The expected difference can be deter-
mined by analyzing the velocity profiles of both the fluid
and the particles in a pore. The model has been formu-
lated for a capillary tube which has a constant rate with
the following assumption: No interactions between the
particles and the wall, suspension is well-mixed with a
constant concentration across the cross section .There is
no tran sverse flow.
2
0
(0 )rr






1
SO
o
r
UU r



0
(0)
22
1
p
PO
o
p
o
a
r
UU rr


 



 




rr a

 
S
V
(4)
As the particle travels through a tube, Brownian mo-
tion and shear action will cause the particle to travel
across the entire cross-section of the tube except that the
center-line of the particle will be excluded from the im-
mediate region of the wall due to its radial dimension.
After the particle has traveled far enough longitudi-
nally through the tube, the particle will have spent equal
amounts of time in all radial position across the capillary
tube. Integration of the velocity profile of Equation (4)
over the range of possible radii shown in Equation (4) for
both the particle and fluid yields the higher average ve-
locity for the particle than that for the carrier flu id.
The av erage flu i d velocity, in a capillary tube is:
2
O
S
U
V (5)
The average velocity of a particle,
P
V in a capillary
tube is:
22
1
11
2
pp
PO
oo
aa
VU rr











(6)
By inspecting Equation (5) and (6), the particles are
expected to have a larger average velocity than the car-
rier fluid velocity. This enhanced velocity of the particle
can be expressed as a fractional difference between the
two average velocities:
2
7
23
pp
PS
Soo
aa
VV
VVrr


 

V
(7)
This equation shows that as the radius of the particle
increase, the difference between the average particle ve-
locity and the average fluid interstitial velocity also in-
crease. This increase is not unbounded but reaches a
maximum

value for 3
7
p
o
a
r; as 3 7
p
o
a
r the
velocity difference decrease. In physical sense, the pore
radius can be estimated to approximately equal to
one-fifth of media grain diameter (1
5
og
rd); therefore
the largest possible particle to be able to fit through the
porous bed has a radius to this pore radius (
p
o
ar).

For a particle with
p
o
ar

, the particles have been
shown to collect on the bed surface in a cake [6]. These
references show that the onset of deep bed filtration oc-
curs for a particle radius
p
a.less than
one-twentieth of the media grain diameter (1
20
p
g
ad
5
).
Particles with the radii larger than this will not transport
into the bed, but will collect on the surface. By lettingc
g
o
dr

, the largest particle which will transport has a
radius equal t o one-fourth of the pore radius (1
4
p
o
ar).
3.2. Longitudinal Dispersion Coefficient
An important element of any dispersion model is the re-
presentation of the geometry of the porous medium.
Houseworth [8] has thoroughly reviewed such longitudi-
nal dispersion model for solute tracers. Instead of mod-
eling the internal structure of a porous medium, dimen-
sional analysis is used to analyze the problem. In this
study, the effect of mechanisms is expected to scale with
the pertinent transport variables.
The pertinent variables for solute dispersion are:
DL = longitudinal dispersion coefficient (L·T–1)
D = free fluid molecular diffusion coefficient of

H. FALLAH ET AL. 59
solute (L2·T–1)
Vs = fluid interstitial velocity (L·T–1)
and dg = media grain diameter (L)
From the Buckingham pi theorem, the following pairs
of groups are formed:
,o
Sg
L
Sg
Vd
DF
Vd D




ralternatively
Sg
Vd
F
DD




L
D (8)
where Pec let nuPe mber Sg
Vd
D


Experimental data for solu te longitudinal dispersion in
uniform media show good correlation with these dimen-
sionless groups [8]. When the Peclet number is grater
than 1, the two groups can be re duced to one:
Constant
Sg
D
L
Vd (9)
where P
D
edynamic Peclet numbeSg
L
Vd
rD


An order to magnitude approximation for the longitu-
dinal dispersion coefficient for solutes can be made with:
g
LS
DdV

(10)
Particle longitudinal disp ersion is expected to be simi-
lar to that of solutes. Currently, no particle breakthro ughs
have been performed by others form which particle lon-
gitudinal dispersion coefficient be determined.
3.3. Filter Coefficient
Two approaches exist for analyzing the filter coefficient.
These are the macroscopic mass balance approach and
the microscopic trajectory analysis approach. Deep bed
filtration studies have been conducted to analyze both th e
system variables and the underlying mechanisms in-
volved in the processes of capturing and retaining parti-
cles in porous media .
3.3.1. Macros c opi c Approach
The macroscopic process of filtration or change in sus-
pension concentration over depth is first-order decay
with distance in steady flow. The filter coefficient may
be expressed as a function of single collector efficiency
and the single collector efficiency may be related to mi-
croscopic filtration mechanism.
Filtration results for steady state flow, neglecting lon-
gitudinal dispersion, gives:
CC
x





(11)
O
C C expx

 
The filter coefficient theoretically may be expr essed as
the single collector efficiency as follows:


1
3
2
e
cT
g
d





(12)
For T

, the particles under consideration are those
contained in a cylinder of diameter
g
d which is coinci-
dent with the vertical axis through the media grain col-
lector.

can be found either experimentally or from
the individual collector efficiency, T

and system vari-
ables using Equation (12).
The classical filtration theory assumes simultaneous
particle capture and dislodging. On the contrary, the
proposed model assumes that the particle capture takes
place only if the total of torques for electrostatic and
gravity forces prevails over the drag and lifting forces, so
the resulting torque presses the particle towards the ma-
trix or the internal cake.
3.3.2. Microscopic App roach
Microscopic study of particle motion near a collector is
defined as trajectory analysis. In porous media, particle
path far from media grains follow fluid streamline. As a
particle approach a media grain, the motion deviate from
the streamline because of various forces and torques act-
ing on the particles. These forces are presented by trans-
port and attachment mechanisms.
The transport mechanisms are: gravity settling, inter-
ception, Brownian diffusion and advection. The attach-
ment mechanisms are considered to be gravity, Lon-
don-van der Waals att racti on, do u bl e lay er fo rces.
From a combination of the trajectory analysis, the
three major contribution of filtration can be formulated.
These are collection due to Brownian diffusion, intercep-
tion and settling. Here, we assumed double layer repul-
sive forces and hydrodynamic retardation (slow drainage
of fluid from between two closely interacting particle
which occurs before contact of particles) are negligible.
The equations for these collection efficiencies are given
in the following [7].
Collection due t o Br ow ni an di ff usi o n,
12
33
4
D
Sp
A
Pe 

 (13)
1
3
4S
A
is constant. Also, either the particle Peclet number
can be allowed to reach a minimum value of approxi-
mately 1, or as 1
p
Pe

the efficiency reaches an as-
ymptotic value of 1.
Collection due to interception,
22
1.5
I
SR
A
N  (14)


2
1.5 is constant here. Also the relative size group,

R
N, can be allowed to reach a minimum value of ap-

H. FALLAH ET AL. 60
proximately 1 in order for the efficiency to remain less
than or equal to 1.
As we mentioned before, the size of the largest parti-
cles which are able to penetrate and transport through a
porous bed is one twentieth of media grain diameter
(1
20
p
g
da) or one-fourt h of the p ore radius (1
4
p
o
ar).
Collection due to settling,

2
9
2
p
fp
ss
s
G
g
a
VV



TDIG
w
  (15)
The limit for this collection efficiency is the best pos-
sible collection which occurs for the settling velocity
equaling the interstitial velocity. As the settling velocity
becomes grater than the interstitial velocity, the effi-
ciency remains at a value of 1.
The equation for to tal collection efficiency is:

 
T
(16)
In this case,

has an asymptotic maximum value of
1. For Brownian particles (p), there is
good agreement between experimental and theoretical
result but for advective particles (p), it is
not, because of neglecting hydrodynamic retardation and
London-van der Waals attraction.
1.0dmicron
1.0dmicron
So, the exact analytical solution is [9]:
12 115
33 88
4 1
1 2
40.722.4 3
TSp SLORSGR
0
A
PeANNEAN N


 
(17)
The first term represents filtration due to Brownian
diffusion. The second and third terms represent the com-
bined effects of interception and gravity when the retar-
dation and London-van der Waals attraction are incl uded.
The effects of surface double layer forces are ignored
(attractive surface double layer is controlled by transport
processes and not depend on surface chemistry).
4. Modeling of Particle Transport and
Filtration
In this part, the steady-state transport equation is pre-
sented and the solution to the complete advective-dis-
persion equation for particulate suspension flow has been
derived for the case of a constant filter coefficient. This
model includes transport parameters which are particle
advestive velocity and particle longitudinal dispersion
coefficient. These parameters have been defined by di-
mensional analysis using the pertinent variables of the
porous media system.
4.1. Particle Advective Velocity
The result of the size exclusion for particles flowing in
capillary tube can be written as Equation (6) where
2
3

. By using the Equations (5), (6) and (7) we

will have:
1VV (18)


*
P
S
V
VV  (19)
As the particle size increase, the difference between
particle velocity and fluid velocity increase.
4.2. Particle Longitudinal Dispersion Coefficient
In modeling particle dispersion, the following variable
substitutions are used:
P
DD
P
VV
L
LP
DD
Particle size variable can be removed by using the
particle properties as shown, provided 1
p
g
d
d
 also the
effect of particle size is included in the enhanced advec-
tive velocity for the particles.
This analysis shows that particle and solute longitudi-
nal dispersion are similar. When the particle Peclet
number (
p
Pe =
P
g
p
P
Vd
Pe D  is grater than 10, the two
groups can be reduced to one :
constant
LP
Pg
D
Vd  (20)
An order of magnitude approximation for the longitu-
dinal dispersion coefficient for particles can be made
with:
L
D
PPg
Vd
1
p
Pe 
10Pe 
(21)
As mentioned before, the dimensional argument for
defining the longitudinal dispersion coefficient is only
valid when . For uniform media, this restriction
is seen to be p. Flow conditions are simultane-
ously limited to the linear, laminar regime for which the
Reynolds number must be less than 10.
4.3. Steady-State Transport Equation and
Solution
Particle removal or filtration occurs as a particle suspen-
sion flows through a porous medium due to the interac-
tion of the advecting particles and grains of the medium.
Iwasaki [1] is credited with being the first to express fil-
tration a first-order decay of particle concentration with
distance:

H. FALLAH ET AL. 61
5. Conclusion CC
x




0
0Cx C

(22) In this work, a mass balance particle transport equation
which includes filtration has been d eveloped. This model
includes transport parameters which are particle advec-
tive velocity, particle longitudinal dispersion coefficient
and filter coefficient. The steady-state transport equation
is presented and the solution to the complete advective-
dispersion equation for particulate suspension flow has
been derived for the case of a constant filter coefficient.
(23)
and a solution in dimensional terms:

0expCC x


***
expCx 
(24)
or in dimensionless terms:


 (25)

This work recommends to be investigated by particle
longitudinal dispersion calculation from experimental
data. Besides, the numerical model needs to be devel-
oped for general case of a transition filter coefficient.
where *
0
C
CC 


00CCx
*
6. Acknowledgements
d

 
g
The authors thank Professor P. Bedrikovetsky, Chair in
Petroleum Engineering, Australian School of Petroleum,
University of Adelaide.
*
g
x
xd

A complete equation of steady-state filtration can be
formulated by using the general steady-state advection-
dispersion equation of transport for particle concentration
with a sink term to describe particle removal due to fil-
tration:
REFERENCES
[1] S. Pang and M. M. Sharma, “A Model for Predicting
Injectivity Decline in Water-Injection Wells,” SPE Paper
28489, Vol. 12, No. 3, 1997, pp. 194-201.
2
2
0LP DV
p p
CC
VC
x
x





0
0Cx C

lim 0
xCx  


 (26) [2] T. Iwasaki, “Some Notes on Sand Filtration,” Water
Works Association, 1937, pp. 1591-1602,
The following semi-infinite medium boundary condi-
tions are: [3] J. W. A. Foppen and J. F. Schijven, “Evaluation of Data
from the Literature on the Transport and Survival of Es-
cherichia coil in Aquifers under Saturated Conditions,”
Journal of Water Research, Vol. 40, No. 3, 2006, pp.
401-426. doi:10.1016/j.watres.2005.11.018
(27) [4] J. F. Schijven and S. M. Hassanizadeh, “Removal of Vi-
ruses by Soil Passage: Overview of Modeling Processes,
and Parameters,” Critical Reviews in Environmental Sci-
ence and Technology, Vol. 30, No. 1, 2000, pp. 49-127
The solution which is shown in dimensional terms is
derived in appendix:


0expC xCx

 (28) [5] R. Farajzadeh, “An Experimental Investigation into In-
ternal Filtration and External Cake Build Up,” MSc. The-
sis, Delft University of Technology, 2004.
where 1114
2
PLP
LP P
VD
DV






[6] J. P. Herzig, D. M. Leclerc and P. Le Goff, “Flow of sus-
pension through porous media-application to deep filtra-
tion,” Journal of Industrial and Engineering Chemistry,
Vol. 65, No. 5, 1970, pp. 8-35
2*
*2
0Dp
CC
Pe
***
*Dp
Pe C
x
x



 

**01Cx
(29)
[7] M. Elimelech and O’Melia, “Effect of Particle Size on
Collosion Efficiency in the Deposition of Brownian Par-
ticles with Electrostatic Barriers,” 1990, pp. 1153-1163
With the same boundary conditions:



**
m 0Cx
 
***
expCx 
(30) [8] J. E. Houseworth, “Longitudinal Dispersion in Nonuni-
form, Isotropic Porous Media,” Ph.D. Thesis, W.M. Keck
Laboratory of Hydraulics and Water Resources, Califor-
nia Institute of Technology, Report No. KH-R-45, 1984.
*
li
x (31)
and a solution in dimensionless terms: [9] A.C. Paytakes and C. Tien, “Particle Deposition in Fi-
brous Media with Dendritic Pattern: Apreliminary
model,” Journal of Aerosol Science, Vol. 7, No. 2, 1976,
pp. 85-94. doi:10.1016/0021-8502(76)90067-7


 (32)


*
4
11
P
DP
Pe


 


*1
2
gD
dPe   (33)

H. FALLAH ET AL. 62
Appendix. Solution Derivation
Consider the one-dimensional steady-state particle ad-
vective-dispersion equation which includes the removal
term to account for filtration effects:
2
2
0LP
DV
P P
CC
VC
x
x





0
0Cx C

lim 0
xCx  
 (A.1)
With the following boundary conditions:
(A.1.1)
For convenience, the x-variab le is allowed to range from
negative to positive infinity


x

 , although the
equation are only applied for x > 0 this avoids difficulty
at x = 0, because small dispersion is allowed. In dimen-
sionless form, the transport equ ation becomes:
2*
*2
0LP Dp
CC
DPe
***
*Dp
PeC
x
x






**01Cx

**
m 0Cx
 
**
exp x 
(A.2)
With the same boundary conditions:
*
li
x (A.2.1)
In order to derive a solution, try the following as a solu-
tion:

**
Cx





*
(A.3)
Check the equation (A.3) by substituting into Equation
(A.2), this results in second- deg ree po lyno mial in ter m of

and two roots of this polynomial are:
*
14
11
DP
Pe






*
12DP
Pe  
*
14
11
DP
Pe





**** **
12
Bexp x 



**01AB
 
B 
1and 0AB
*
12DP
Pe  
Using these two roots, Equation (A.3) becomes:

C xAexpx


 (A.4)
The constants of this Equation can be determined by ap-
plying the boundary conditions:
Cx

**
Cx 

00A



By substituting these constants into Equation (A.4), the
solution to Equation (A.2) becomes:
*
** *
14
exp1 1
2DP
DP
Cx Pex
Pe





 




(A.5)


Nomenclature
dg= media grain diameter (L)
dp= particle diameter (L)
c = suspended particle concentration in carrier fluid
σ = particle retained concentration
kdet = detachment rate coefficient
U = flow velocity
US = fluid velocity
UP = particle velocity
UO = fluid centerline velocity
r = radial distance
ro = capillary radius
ap = particle radius
p = dynamic pressure
x = longitudinal distance
DL = longitudinal dispersion coefficient (L·T–1)
D = free fluid molecular dispersion coefficient of solute
(L2·T–1)
VS = fluid interstitial velocity (L·T–1)
Pe = Peclet number = Sg
V.d
DL
PeD = dynamic Pec let number = Sg
V.d
DL
C = particle concentration (M·L–3)
X = longitudinal position (L)
λ = filter coefficient (L–1)
WS = particle settling velocity
VS = fluid interstitial velocity
Ρpf = densities of particle and fluid, respectively
g = gravitational acceleration
H = Hamakar constant (e r gs)
NG = gravitional group = ηG
DLP = particle longitudinal dispersion coefficient (L2·T–1)
DP = particle molecular diffusion coefficient in a free
fluid (L2·T–1)
VP = particle velocity (L2·T–1)