International Journal of Modern Nonlinear Theory and Application, 2012, 1, 113-117
http://dx.doi.org/10.4236/ijmnta.2012.14017 Published Online December 2012 (http://www.SciRP.org/journal/ijmnta)
Size Exclusion Mechanism, Suspension Flow
through Porous Medium
Hooman Fallah, Afrouz Fallah, Abazar Rahmani, Mohammad Afkhami, Ali Ahmadi
Department of Petroleum Engineering, Firoozabad Branch, Islamic Azad University, Firoozabad, Iran
Received August 4, 2012; revised September 25, 2012; accepted October 12, 2012
ABSTRACT
A lot of investigations have been done in order to understand the mechanisms of the transport of particulate suspension
flow through porous medium. In general, Deep Bed Filtration studies have been conducted to analyse the mechanism
involved in the processes of capturing and retaining particles occurs throughout the entire depth of the filter and not just
on the filter surface. In this study, the deep bed filtration mechanism and the several mechanisms for the capture of sus-
pended particles are explained then the size exclusion mechanism has been focused (particle capture from the suspen-
sion by the rock by the size exclusion). The effects of particle flux reduction and pore space inaccessibility due to selec-
tive flow of different size particles will be included in the model for deep bed filtration. The equations for particle and
pore size distributions have been derived. The model proposed is a generalization of stochastic Sharma-Yortsos equa-
tions. Analytical solution for low concentration is obtained for any particle and pore size distributions. As we will see,
the averaged macro scale solutions significantly differ from the classical deep bed filtration model.
Keywords: Transport; Porous Medium; Particulate Suspension Flow; Size Exclusion
1. Introduction
The following model predicts that the particle breakthrough
happens after injection of one pore volume. Nevertheless,
several cases where the break through time significantly
differs from one pore volume injected, have been report-
ed in the literature for particulate and polymer suspen-
sions [1].
That model does not distinguish between different me-
chanisms of formation damage so it can not be used for
diagnostic purposes. Several attempts to correlate the for-
mation damage with sizes of particles and pores were
unsuccessful [2] (A model for average concentrations is
not general enough or may be size exclusion mechanisms
never dominate).
Sharma and Yortsos [3] derived basic population ba-
lance equations for the transport of particulate suspen-
sions in porous media. It is assumed that an overall pore
space is accessible for particles and the particle popula-
tion moves with the averaged flow velocity of the carrier
water. In the case of a porous medium with the uniform
pores size distribution, this assumption results in inde-
pendent deep bed filtration of different particle size popu-
lations. Nevertheless, as we will see, if we consider size
exclusion mechanism, either smaller particles than the
pore or larger particles, do not perform deep bed filtra-
tion.
The pore size exclusion assumes that the particles can
only enter larger pores, so, only a fraction of porosity
will be accessible for particles, i.e. the water flux carry-
ing particles of a fixed size is just a fraction of the overall
water flux via porous media.
Here, analytical solution will shows for a small pore
size variation medium, only the intermediate size parti-
cles perform deep bed filtration. In this case, the popula-
tion velocity is particle size-dependent. The averaged equa-
tions for deep bed filtration of intermediates size parti-
cles differ from the classical deep bed filtration.
2. Deep Bed Filtration
The deep bed filtration system consists of equations for
the particle mass balance, for the particle capture kinetics
and of Darcy’s law [4,5]




 

,,
1,
(,) ,,
,
o
cXT cXTcXT
TX T
XT cXT
T
kk p
ULX




 

 



 
,
(1)
where



 is the dimensionless filtration coefficient,


,cXT is the suspended particle concentration,


,
X
T

C
opyright © 2012 SciRes. IJMNTA

H. FALLAH ET AL.
114
is the deposited particle concentration, and the formation
damage function


,k

shows the permeability declines
due to particle deposition.
If we assume the suspension as an incompressible fluid,
the velocity U is independent of X and we can solve the
third Equation (1) (dynamical model) independently.
In the case of constant filtration coefficient, the parti-
cle penetration depth equals 1

, but here, as we focused
on the size exclusion capture, the phenomenological mo-
del (1) does not account for particle size distributions
(the larger the particles, the smaller are the pores, and the
higher is the capture rate).
Particles do not move with the carrier water velocity,
although we have continuity Equation (1). In one dimen-
sional deep bed filtration, suspended concentration shock
that moves with carrier water velocity, the suspended and
captured concentrations are equal to zero ahead of this
shock [4].
3. Advective Velocity
In order to discuss particle transport and determine the
average velocity of particle suspension, the velocity dis-
tribution at the scale of the each pore must be considered.
By approximating each pore as a capillary tube (a rough
analogy), the velocity distribution for the fluid will be
parabolic with a no slip condition at the walls.
A particle will not be able to travel the same pathways
as the carrier water, because the particle center of mass
will be excluded from the immediate region of the wall.
They will also be excluded from pores smaller than the
particle.
The result of this exclusion based upon size is that the
particles will take on an average velocity which is greater
than that of the carrier water [6].
The particle flowing through a capillary tube and sub-
sequent size exclusion is shown in Figure 1.
4. Derive the Equations
In size exclusion mechanism, some particles are captured
by the rock from the suspension, i.e. if the large particle
arrives at a small pore, ps , it is captured and plugs
the pore; and a small particle ps passes the pore
without being captured (both large and small particles, do
not perform deep bed filtration).
rr
rr
The geometric model structure of the pore space is as
follows: the porous space is a bundle of parallel capillary;
the flux through each pore is proportional to the fourth
power of its radius; complete mixing takes place at
length scale, i.e. there is a nonzero probability for a par-
ticle moving through any point x to get into any pore at
the point
x
l.
Figure 1. Graphic representation of the size exclusion prin-
ciple for a particle flowing through a capillary tube.
The complete mixing of different size particles occurs
in the chambers. The capture occurs at the thin pore inlet,
where large particles arrive. So an inlet cross-section of
each parallel capillary section acts as a sieve.
A particle with the radius
s
r passes through the pore
with radius p only if the particle radius is smaller than
the pore radius, ps
r
rr

. Therefore, small pores are inac-
cessible pore volume. We introduce the accessibility fac-
tor

for particles with radius
s
r as a fraction of pore
volume with capillary radii larger than
s
r:



2
2
0
,, d
,,
,, d
spp
r
s
pp
rH rxtr
rxt
rH rxtr





p
p
(2)
Let us define the flux

,, dd
s
sp
J
rxt rr of particles
with specific radius
s
r via pores with a specific radius
p and also the total flux
r 

,, d
s
s
J
rxt r of particles
with radii in the interval


,d
s
ss
rr r:


,, d, ,,dd
s
s
sspp
r
s
J
rxtrJrrxtrr










 (3)
The flux of particles with radius
s
r via pores with
smaller radius


ps
rr equal zero. Therefore, the wa-
ter flux carrying
s
r-particles is lower than the overall
water flux in the porous medium.
 


4
4
0
,, d
,, d,,d
,, d
sppp
r
s
ss
pp
Hr xtrr
s
J
rxt rUCrxtr
Hr xtrr



 (4)
Introducing the fraction of the total flux that carries
particles with radius
s
r:




4
4
0
,, d
,,
,, d
spp
r
s
pp
rH rxtr
rxt
rH rxtr





p
p
(5)
So following formula is the flux of particles with radii

H. FALLAH ET AL. 115

,,
s
Crxt


,,,
sp
rr xt


,,
p
H
rxt
Figure 2. Schema of the large particle entrapment by small
pores.
varying from
s
r to d
s
s
rr:
 
,, d,,,, d
s
ss ss
J
rxt rUrxtCrxt r   (6)
Formula for the flux reduction and accessibility factors
((2) and (5)) can be derived for regular pore networks
using effective medium or percolation theories [7].
4.1. Fraction of Particles Trapped and Retained
According to Sharma and Yortsos (1987)
To derive local rates for particle removal due to mecha-
nical entrapment, they focused on a representative volume
of the porous medium with a statistically large number of
pores. They assume fluid flows through the medium at a
constant superficial velocity q, firstly. Then, they denote
by n the average number of pore throats a fluid particle
encounters in the volume element before emerging from it.
If is the time taken for the fluid to traverse the volume
element, then:
t
p
qt
nl


 (a.1)
where: n = number of steps;

= porpsity; q = fluid
superficial velocity, L·T−1; lp = effective pore length, L.
Pore length


p
l is constant. As the fluid carries sus-
pended particles, a certain fraction of the latter is trapped
by the pore throats at each of the n steps. If the fraction of
particles of size in the interval


d
s
ss
rrr r  trapped
at each step is

s

,
P
r the mass balance on particles of
this size at the conclusion of the step reads as follows, ith
 
11
SS SSS
i
f
fPr 


 

 (a.2)
 

no.of particles in,dtrapped inith step
no.of particles in,dbeforeith step
ss s
s
ss s
rr r
Pr rr r


(a.3)
where: ρ = concentration, no. L−3; f = size distribution,
L−1;

s
Pr
They proceed
= fraction of particles retained per step.
by assuming that the above probability of
trapping is constant at each particle step. At the end of n
steps the fraction of particles trapped by the sequence of
n steps, t
P, assumed independent, is given by
1
nin
 
1
111
ts s s
i
PPr PrPr









(a.3)
In the case of low concentrated suspensions, the pore
space fraction occupied by retained particles is negligibly
small if compared with the overall pore space. Therefore,
the porosity is assumed to be constant. The population
balance equation is derived as the following form:




,, ,,,, ,,rxtCrxtU rxtCrxt 

,,
ss ss
s
tx
rxt
t






 



 
(7)
The number of particles with size in the interval


,d
s
ss
r r captured in pores with radius in the interval r
,d
pp p
rr r




 per unit of time is called the particle-cap-
rate is proportionality coefficient is called
the filtration coefficient
ture rate. This


,
s
p
rr 
:


,0:
s
pp
r r
s
(8)
Finally, for incompressible aqueous sus
cl
r r 

pension and in a
osed system for three unknowns 
,,
s
Cr xt,


,,
s
rxt
and


,,
p
H
rxt
we will have:



 

4
4
0
,,
,, ,,,
,, dp
p
pp d
s
ps
r
pp p
Hr xt
t
rH rxt
UrrC
rH rxtr






 
s
rxtr
(9)
Introduction of dimensionless variables
,,
xUt
X
TL
LL



  (10)
So:






4
4
0
,,
,, ,,,
,, d p
p
pp d
s
ps
r
pp p
s
r XT
T
rH rXTrr CrXTr
rH rXTr




  

(11)
The boundary condition at the core inlet correspond to
th
H
e injection of water with a given particle size distribu-
tion


0,
s
CrT . The injected
s
r-particle flux is equal to


0
Che inlet core/reservoir cross-section acts
he injected
,
s
rTU . T
as a sieve. T
s
r-particles are carried into the
porous medium by a fraction of water flux via accessible
pores-


0,
s
rTU  (Figure 2). The injected
s
r-par-
ticles carried by r flux via inaccessible pores


0
1,
s
rT U 
wate




 are deposited at the outer surface of
the inlet and form the external filter cake from the very

H. FALLAH ET AL.
116
beginning of injection. For particles larger than any pore,
there are no accessible pores and the flux reduction factor
is zero. So, all these particles are retained at the inlet
cross-section, contributing to external filter cake growth.
On the other hand, for particles smaller than the smallest
pore, they will enter the porous medium without being
captured. (deep bed filtration will not perform in both
condition).The particles retained at the outer surface of the
inlet large particles do not restrict access of newly arriving
particles to the core inlet before the transition time [7]
Finally,

0
,,hXTh XXT  

(12)
Equation (12) shows that one particle can
po
4.2. Filtration in a Single Pore Size Medium
ingle
ution (Dirac’s
de
plug only one
re and vice versa.
Distribution of suspended particles and pores in a s
pore size medium are illustrated bellow.
Figure 3(a) shows the pore size distrib
lta function) at 0T and the particle size distribu-
tion in the injected nsion at 0X. If we consider
the propagation of small particles
suspe
with
s
p
rr

. For this
case, formulae (2) and (5) show that 1

 
; i.e. all
pores are accessible for small particlesre is no
flux reduction.
Therefore, sma
, and the
ll particles are transported with the ve-
lo
ge particles
city of carrier water without being captured (no pores
will be plugged by small particles).
Now consider the propagation of lar
s
p
rr

.
ws thaIn this case, from Equations (2) and (5) it follot
0

 . Therefore, none of the pores is accessible for
icles, and there is no large particle flux. So, all
large particles are deposited in the inlet cross-section (they
never arrive at the core out-let). It was also observed in a
laboratory study [8] where size exclusion was the domi-
nant capture mechanism.
It is important to highli
large part
ght that, depending on the size,
th
5. Highlighted Assumptions
uspension is incom-
were no deposited particles and plugged pores at
th
e particles in uniform pore size medium either pass or
are trapped. Therefore, the deep bed filtration, where
there exists an average penetration length for each size
particle, does not happen in the case of particulate flow
in a single size porous media. The penetration length is
zero for large particles, and is infinite for small particles.
It was assumed that the aqueous s
pressible so the velocity U in Equation (1) is independent
of X and we can solve the third Equation (1) indepen-
dently (dynamical model separates from the kinematics
model).
There
e beginning of deep bed filtration. There are no sus-




00
,
ss
f
rTcT



00
00pp
f
rh
0p
r

r
(a)
0p
r

r




00
,
pp
f
rThT

,, ,
ss
f
rXTcXT




00
,
ss
f
rT XcT X
(b)
Figure 3. (a) initial and boary concentration distribu-
ended particles ahead of the injected water front.
f low
co
articles retained at the outer
su
6. Conclusions
ptured during flow through pore sys-
are smaller then
th
analytical solution for flow in a single pore size
und
tions for pores and suspended particles; (b) particle distri-
bution for any X and T; pore distribution at the inlet cross
section for T > 0.
p
The porosity is assumed constant in the case o
ncentrated suspensions.
We assumed that the p
rface of the inlet large particles do not restrict access of
newly arriving particles to the core inlet before the tran-
sition time [9]. The external cake does not form a solid
matrix before the transition time and cannot capture the
particles from the injected suspension.
Particles are not ca
tem, but there is a sequence of particle capturing sieves
perpendicular to the flow direction.
Absence of particles in the pores that
e particles, results in reduction of the particle carrying
water flux if compared with the overall water flux. So,
only a fraction of the pore space is accessible for parti-
cles.
The
p
r

medium shows that capture free advection of small
rticles pa


s
p
rr

 takes place, and large particles


s
p
rr

 netrate into the porous medium (there
p bed filtration in a uniform pore size medium).
Large particles never arrive at the core outlet. It was
do not pe
is no dee
ob
REFERENCES
[1] R. Dawson ansible Pore Volume
served in a laboratory study [8] where size exclusion
was the dominant capture mechanism.
d R. B. Lantz, “Inacces

H. FALLAH ET AL.
117
F. G. van Velzen and K. Leerlooijer, “Im-
. C. Yortsos, “Transport of Particu-
in Polymer Flooding,” SPE Journal, Vol. 12, No. 5, 1972,
pp. 448-452.
[2] E. V. Oort, J.
pairment by Suspended Solids Invasion: Testing and Pre-
diction,” SPE Production & Facilities, Vol. 8, No. 3,
1993, pp. 178-184.
[3] M. M. Sharma and Y
late Suspensions in Porous Media: Model Formulation,”
AIChE Journal, Vol. 33, No. 10, 1987, pp. 1636-1643.
doi:10.1002/aic.690331007
[4] J. P. Herzig, D. M. Leclerc and P. Le Goff, “Flow of Sus-
Filtration,” American
ar, “Particle Transport in Flow through Porous
d V. V. Kadet, “Percolation Models in
and M. M Sharma, “A Model for Predicting In-
ort of
pension through Porous Media—Application to Deep Fil-
tration,” Journal of Industrial and Engineering Chemis-
try, Vol. 62, No. 5, 1970, pp. 8-35.
[5] T. Iwasaki, “Some Notes on Sand
Water Works Association, Vol. 29, No. 10, 1937, pp. 1591-
1602.
[6] R. Edg
Media: Advection, Longitudinal Dispersion and Filtra-
tion,” Ph.D. Thesis, California Institute of Technology,
Pasadena, 1992.
[7] V. I. Seljakov an
Porous Media,” Kluwer Academic Publishers, Dordrecht,
1996.
[8] S. Pang
jectivity Decline in Water-Injection Wells,” SPE Forma-
tion Evaluation, Vol. 12, No. 3, 1997, pp. 194-201.
[9] N. Massei, M. Lacroix and H. Q. Wang, “Transp
Particulate Material and Dissolved Tracer in a Highly
Permeable Porous Medium,” Journal of Contaminant
Hydrology, Vol. 57, No. 1-2, 2002, pp. 21-39.
Table of Symbols dynamic Peclet numberSg
D
L
V
Pd eD


suspended particle concentration in carrier fluid;c

3
particle concenterationCM


p
article retained concentration;

L;


longitudinal position;andxL
detachment rate coefficient;
det
k
flow velocity;U


1
filter coefficient.L 


p
article settling velocit
s
w
fluid velocity;
S
Uy;

p
article velocityU
fluid interstitial velocity;V
;
P
fluid center line vU

densities of particle and f
s
elocity;
O
radial distance;r
luid, respectively;
pf


gravitational accelerationg

;



capillary radius;
o
r
p
article radiusaHamakar constantergs;H
gravitational groupN
p
dynamicpressurep
; GG





21
spersion coefficientDLparticle longitudinal di
LP
;
longitudinal distance;x


1
longitudinal dispersioncoefficient;
L
D
free fluid molecular diffusion coefficient of solD
LT
T


particle molecular diffusi
P
D
;
on coefficient in a



21
free fluid;LT


21
ute
21
p
article veloci
Pty ;VL


;LT


1
interstitial velocity; andVL

T
T


media grain diameter;
g
dL and


particle diameterdL
fluid
Sp
Peclet numberSg
Vd
Pe D 