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Modeling of External Filter Cake Build-up in RadialGeometryF. F. Zinati a , R. Farajzadeh a , P. K. Currie a & P. L. J. Zitha aa Department of Geotechnology, Delft University of Technology, Delft, The Netherlands

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To cite this article: F. F. Zinati, R. Farajzadeh, P. K. Currie & P. L. J. Zitha (2009): Modeling of External Filter Cake Build-up inRadial Geometry, Petroleum Science and Technology, 27:7, 746-763

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Petroleum Science and Technology, 27:746–763, 2009

Copyright © Taylor & Francis Group, LLC

ISSN: 1091-6466 print/1532-2459 online

DOI: 10.1080/10916460802105666

Modeling of External Filter Cake Build-up

in Radial Geometry

F. F. Zinati,1 R. Farajzadeh,1 P. K. Currie,1 and P. L. J. Zitha1

1Department of Geotechnology, Delft University of Technology,

Delft, The Netherlands

Abstract: The problem of formation damage (i.e., permeability reduction due to

injection of particulates), is a matter of interest in several engineering fields. In the

previous attempts to model the external cake formation, cake thickness has been

considered to be only dependent on time; even though in practical applications, the

dependency of the cake profile on space can be important. In this article, a novel

model has been developed to describe the steady state external filter cake thickness

profile along the wellbore. A set of equations is derived from the force balance for a

deposited particle on the cake surface and the volume conservation of the fluid in the

wellbore. These equations are combined with Darcy’s law in radial geometry and the

equation of flow in the wellbore, and solved numerically to obtain the cake thickness

and fluid velocity profiles along the wellbore.

Keywords: cross-flow, external filter cake, filtration, particle deposition, radial

geometry

INTRODUCTION

Many oil and gas reservoirs are connected to water sources connected to

their hydrocarbon reserves. Therefore, the produced fluids are not exclusively

hydrocarbons but contain water, often in very large amounts. Worldwide, 75%

of the production is water (Al-Abduwani, 2005) and in some regions this

fraction may reach to 98% (Busaidi and Bhaskaran, 2003). Produced water

(PW), depending on the geological formation and lifetime of the reservoir,

contains organic and inorganic materials including solid particles and oil

droplets. Disposal of the PW is a challenge for petroleum industry due to the

high costs of the transportation and filtration facilities and the increasingly

stricter environmental regulation.

Address correspondence to Pacelli L. J. Zitha, Delft University of Technology,

Department of Geotechnology, Stevinweg 1, 2628CN, Delft, The Netherlands. E-mail:

p.l.j.zitha@tudelft.nl

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External Filter Cake Build-up in Radial Geometry 747

One economically and environmentally friendly method is to reinject

the produced water into the formation. By reinjecting the produced water,

reservoir pressure can be maintained simultaneously. Nevertheless, a rapid

injectivity decline is usually observed in the PW injection processes due

to the presence of impurities, usually solid particles and oil droplets. The

injectivity decline is caused by the deposition of the particles inside porous

media (internal filtration) (Herzig et al., 1970) or accumulation of the particles

on the surface (external filtration: Ochi et al., 1995). In the latter case, particles

form a cake (porous medium), which is orders of magnitude less permeable

than the reservoir.

Filtration of the particles in the borehole can occur under either static

(dead-end) or dynamic (crossflow) flow conditions. Crossflow filtration refers

to a pressure driven separation process in which the permeate flow is per-

pendicular to the feed flow. In the crossflow filtration, the thickness of the

filter-cake is limited by the shear forces due to the flow of the produced water

containing particles parallel to the borehole surface. A similar phenomenon

happens when drilling mud flows across the borehole surface (Fisher et al.,

2000). The drag force caused by the flux of permeate pushes the suspended

particles toward the cake.

Many authors have investigated the problem of crossflow filtration

(Ferguson and Klotz, 1954; Von Engelhardt and Klotz, 1954). It has been

suggested in the literature that temperature, pressure, shear rate (i.e., flow

rate) and the permeability of the formation influences the filtration process.

However, the effect of individual parameters has remained unclear (Fisher

et al., 2000).

In petroleum engineering, transport of fluid containing particles in the

subsurface is similar to crossflow filtration. For example, in the fractures, the

transport of the particles in the injected fluid results in formation of the cake

on the rock surface. Al-Abduwani (2005) developed a model to obtain the

filter cake profiles in an experimental set-up. Their model can be applied for

the formation of cake in the fractures (linear cases). Another example is the

build-up of mud cake during the water injection or drilling of the well around

the borehole. In this article we adapt the model described in Al-Abduwani

(2005) to obtain the filter cake and velocity profiles in a radial geometry.

MODEL ASSUMPTIONS

The schematic of the domain of interest is depicted in Figure 1. In order to

develop the model the following assumption are made:

� Both suspension and particles are incompressible.� Porosity and permeability of the cake are constant.� The reservoir permeability remains constant during the crossflow filtration

process.

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748 F. F. Zinati et al.

Figure 1. Schematic of the system of interest, depicting the velocity components and

the borehole.

� The change of crossflow velocity in the r-direction is neglected.� The velocity profile in any section of the domain is fully developed.� Shear forces are only in the r -direction.� The injected fluid is dilute and Newton’s law is applicable.

FORCE BALANCE

The starting point of the modeling of the cake formation is to acquire adequate

knowledge about the competing forces that act on a particle. Figure 2 shows

a simplified scheme of such a system in which a particle with a certain

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External Filter Cake Build-up in Radial Geometry 749

Figure 2. Forces acting on a particle deposited on the surface of the horizontal cake

(Al-Abduwani, 2005).

velocity is brought into contact with the cake surface. This individual particle

is in interaction with the particles already deposited on the cake surface

and the neighboring particles floating in the suspension (i.e., the injected

water). The nature and magnitude of different forces acting on a particle

lying on the surface of the external filter cake are illustrated by Al-Abduwani

(2005). These forces include hydrodynamic (crossflow and permeate viscous

drag forces and lift force), thermodynamic (diffusion), body (gravity) and

electrostatic forces. The expressions presented in Al-Abduwani (2005) can

be applied for radial geometry with some modifications. Here we discuss this

further.

Tangential (crossflow) drag force, FD , is the drag force acting on the

particle due to the crossflow flux. This drag force can also be quantified

using Stokes’s law or Poiseuille’s law. Generally we can write

FD D!��a2ucf

.R � h/(1)

where, ! is proportionality factor, � is the viscosity, a is the particle radius,

ucf is the average crossflow velocity at a given cross section, R is the borehole

radius and h is the external cake thickness.

Normal (permeate) drag force, Fp, is the drag force acting on the spher-

ical particle due to the permeate flux in the radial direction. This force can

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750 F. F. Zinati et al.

be quantified using Stokes’s law by utilizing the permeate velocity up which

is assumed to be constant for an increment equivalent to the diameter of the

particle. Thus

Fp D 6��aupˆH (2)

ˆH is the correction factor and can be estimated from work of Sherwood

(1998) as:

ˆH D 0:36

�

kc

a2

�2=5

(3)

where, kc is the cake permeability.

Electrostatic force, Fe , is the sum of multiple physico-chemical forces

between the particle and the neighboring particles. These forces include

(attractive) van der Waals, double electric layer, steric and Born forces

Fe D FvdW C FDEL C Fsteric C FBorn (4)

The electrostatic force is directly proportional to the radius of the particles

and is given by

Fe D aAe (5)

Ae is sum of two constants taking into account the repulsive and attractive

forces and can be found in (Bedrikovesdky et al., 2005, Song and Elimelech,

1995). The range of surface-to-surface distance of two approaching particles

at which these forces are significant appears to be 0.4 nm � 50 nm.

Lift Force, FL, is caused by the crossflow flux and can be expressed as

FL D �a3

s

��u3cf

�3.R � h/3(6)

Different constant values are given for � in different references ranging from

50–1500 (Altmann et al., 2005; Kang et al., 2004).

Gravitational Force, Fg , is the body force acting on the particle due to

gravity and depends on the orientation of the problem. If �� is the difference

between the density of the suspended particles and the suspending fluid, the

expression for the Fg is given by

FG D4�a3

3��g (7)

Friction Force, Ff , is the Coulomb frictional force, which is proportional to

the normal force acting on the particle

Ff D f .Fp C Fe C FL/ (8)

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External Filter Cake Build-up in Radial Geometry 751

Estimates of the forces reveal that three forces need to be considered for the

force balance (Farajzadeh, 2004): the tangential drag force, FD , the normal

drag force, Fp, and the Coulomb frictional force, Ff . The gravity force is

neglected because in the system of interest it is acting upwards in the opposite

direction to the tangential drag force. For micron-size particles this force is

three orders of magnitude larger than gravity force due to the high velocities

near the wellbore (Farajzadeh, 2004). The equilibrium thickness of the cake

is achieved when the two tangential forces FD and Ff are equal:

FD D Ff (9)

Inserting Equation (8) into Equation (9) yields

FD D f � Fp (10)

Substituting Eq. (1) and (2) into Eq. (10) and gives

ucf

up

D6 .R � h/

a(11)

where Df ˆH

!is a proportionality factor which can be obtained from dif-

ferent experiments utilizing mono-sized particles of radius a and controlling

both the permeate and crossflow velocities (up and ucf ) and measuring cake

thickness (Al-Abduwani, 2005).

SIMULATIONS FOR A WATER INJECTION WELL

We consider an injection well as shown in Figure 1 with radius R, infinite

reservoir radius L, and reservoir pressure PR. The water is injected with a

rate of Q and pressure of Pinj. We model formation of the filter cake in the

perforation zone of the borehole.

Governing System of Equations

For a porous medium in radial geometry, Darcy’s Law reads:

up D �k

�

@p

@r(12)

where k is the permeability of the porous medium within the drainage radius.

Integration of this equation over the external filter cake and porous medium

from the surface of the filter cake to the drainage radius leads to

up D .z/

.R � h.z//ln

�

ˇR

R � h.z/

� ; (13)

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752 F. F. Zinati et al.

where,

.z/ Dkc

�.Pinj.z/ � PR/ (14)

and

ˇ D .L=R/kc =kf (15)

formation permeability (kf) is the permeability of the reservoir. The crossflow

rate can be expressed in terms of the average crossflow velocity as

Q D �.R � h/2ucf ; (16)

The rate of change in volumetric flux along the wellbore, Q, happens due to

the leakoff of permeate flux, up ,

@Q

@zD �2�.R � h/up : (17)

Combination of Eq. (16) and (17) leads to the following relation:

up.z/ D �1

2.R � h.z//

@ucf .z/

@zC ucf

@h.z/

@z: (18)

Applying Navier-Stokes equations in radial geometry and taking into account

the no slip boundary condition at the surface of the cake and the radial

symmetry condition at the centerline, the following expression for the velocity

is obtained:

ucf .r/ D �1

4�

dPinj

dz.R � h/2

�

1 �

� r

R � h

�2�

(19)

The details of the derivation of the Navier-Stokes equations are given in

Appendix. Substituting. Equation (19) in the following mass balance equation

gives the average crossflow velocity

�

Z .R�h/

0

2�rucf .r/dr D ��.R � h/2ucf (20)

Thus,

ucf D �.R � h/2

8�

dPinj

dz: (21)

From Eq. (14) it appears

@ .z/

@zDkc

�

@Pinj.z/

@z: (22)

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External Filter Cake Build-up in Radial Geometry 753

Replacing this expression in Eq. (21) yields

ucf D �.R � h/2

8kc

@ .z/

@z(23)

Equations (13), (18), and (21) involve four basic variables: ucf ; up; and

h on the spatial coordinate z. If we add Eq. (11) derived from force balance

equation to this set, we end up with the system of four equations that describe

our problem completely. This set of governing equations can be solved using

appropriate boundary conditions to obtain, ucf ; up; and h.

Boundary Condition

We take top of the injection zone as reference of the depth, corresponding

to z D 0 (z is directed downwards). At steady state condition we assume to

have a cake thickness of h0 at this depth. Assuming constant injection rate,

Q0, and velocities of up0 and ucf0 at the inlet, we can use Eq. (11), (13),

(14), and (16) to evaluate the cake thickness at z D 0 by

Q0 D6� kc

a�

.R � h0/2

ln

�

ˇR

R � h0

� .Pinj � PR/ (24)

Equation (24) shows that the value of the cake thickness at the boundary

depends on injection rate, injection pressure, and a, cake permeability and

well radius. Open flow boundary is assumed at the end of the perforation

zone, i.e., the build-up of cake at the bottom is neglected.

Method of Solution

The four unknowns ucf ; up; , h and their first order derivatives are related

to each other as expressed in the equations above. Expressing the first order

derivatives of all these variables in terms of the variables we get a system of

non-linear first order differential equations

8

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

<

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

:

@up

@zD F1.up ; ucf ; h; /

@ucf

@zD F2.up ; ucf ; h; /

@h

@zD F3.up ; ucf ; h; /

@

@zD F4.up ; ucf ; h; /

(25)

where F1, F2, F3 and F4 are non-linear functions of ucf , up, , h.

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754 F. F. Zinati et al.

The system of equations is solved based on an explicit Runge-Kutta

formula, the Dormand-Prince pair (Dormand and Prince, 1980).

RESULTS AND DISCUSSION

The permeate flow carries particles toward the porous medium and they

deposit on its surface. However, contrary to the static filtration the cake growth

is not uniform along the borehole (green line in Figure 3). This is a direct

consequence of the shear forces caused by the crossflow velocity which leads

to the erosion of the cake. Therefore, the cake thickness increases along the

borehole (red line in Figure 3). Note that by perforation depth, we refer to the

distance from the top of the perforation zone (z D 0). Since the shear forces

decrease, any increase in the cake thickness will result in higher resistance

for the permeate flow to enter the porous medium. Therefore, with increasing

thickness, the permeate velocity decreases along the borehole (Figure 4). The

growth of the external cake reduces the cross-section for crossflow. Thus,

the shear rate at the cake surface increases. These two effects, reduction

in permeate flow and increase of the shear rate; cause a steady state cake to

eventually be reached when the rate of particle convection to the cake surface

is balanced by shear back-transport of the particles. As can be observed

from Figure 5, crossflow velocity also decreases with distance from the top

of the perforation zone because of the liquid leakoff along the hole. The

parameters to run the simulator for the base case (Figures 3–5) are presented

in Table 1.

Figure 3. Cake thickness (m) vs. perforation depth for the base case. Green line is

when there is no crossflow (static filtration) and red line is for the dynamic (crossflow)

filtration case.

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External Filter Cake Build-up in Radial Geometry 755

Figure 4. Permeate velocity (up) vs. perforation depth for the base case.

In addition to the base case, we run the simulator for different cases to see

the effect of each parameter on the thickness of cake individually. We chose

different parameters (particle size, �P D Pinj �PR , porous medium and cake

permeability and the proportionality coefficient ) by the factors of 0.5 and 2

of the base case. We present the plots of the runs for particle radius. Figures 6–

8 show the sensitivity of the cake thickness and velocity profiles to the particle

size. Figure 6 shows that as particle radius decreases to 0.5 �m, the thickness

of the cake increases. This is quite interesting because it is contrary to what

one should see in the static filtration. In static filtration, the particles with

Figure 5. Crossflow velocity (ucf) vs. perforation depth for the base case.

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756 F. F. Zinati et al.

Table 1. Base case input parameters

Cake permeability 10 �D

Porous media permeability 1.5 D

Injection rate 2 � 10 � 3 m3/s

Perforation depth 20 m

Distance between injection and production wells 100 m

Gamma (f/˛) 0.016

Well radius 25 cm

Particle radius 1 �m

Inlet concentration 40 ppm

Reservoir pressure 2000 psi

Injection pressure 3000 psi

bigger size will form a higher porosity and thicker cake. In the dynamic,

or crossflow filtration, the velocity vector has two components which insert

drag forces on the entering particles. Our model takes into account the drag

forces and frictional force. Besides the velocity, according to Stokes’s law for

laminar flow, the magnitude of the drag force also depends on the radius of the

particles. As particle size increases, the tangential drag force on the particle

increases. This means that the drag force is bigger for the bigger particles.

For an open-end channel, when the drag force overcomes the frictional force,

the particle will go further or out of the channel, but small particles will stay

on the surface of the channel and will form the cake. Another interesting

feature in our results is that for particles larger than 3 �m, the thickness of

Figure 6. Sensitivity of the cake thickness along borehole to particle size.

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External Filter Cake Build-up in Radial Geometry 757

Figure 7. Sensitivity of crossflow velocity to particle size.

the cake becomes as thick as the radius of the well and therefore, the damage

is severe for larger particles.

Figures 9–11 show the sensitivity of the cake thickness to different

parameters at different parts of the cake. We perform sensitivity analysis

for different parts of the cake because the cake profile is not uniform. All

figures indicate that the permeability of the porous medium has no effect on

the thickness of the cake because of the assumption of constant permeability

Figure 8. Sensitivity of permeate velocity along borehole to particle size.

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758 F. F. Zinati et al.

Figure 9. Sensitivity of the cake thickness at the top of the perforation zone (z D 0)

to different parameters.

Figure 10. Sensitivity of the cake thickness at the middle of the perforation zone

(z D 10 m) to different parameters.

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External Filter Cake Build-up in Radial Geometry 759

Figure 11. Sensitivity of the cake thickness at the end of the perforation zone (z D

20 m) to different parameters.

for the porous medium. Moreover, we assume that the external cake starts to

build when the formation reaches its critical porosity. Therefore, the formation

of the cake on the surface does not affect the reservoir permeability and vice

versa. Moreover, since the cake has much lower permeability compared to

the porous medium the main resistance to the fluid is in the cake itself.

The three figures clearly show that cake thickness has a linear relationship

with the difference between the injection pressure and reservoir pressure

(i.e., the thickness of the cake will be changed by the factor that the base

�P D Pinj � PR is multiplied to). Pressure difference is related to the

permeate velocity with Darcy’s law (Eq. (13)). Therefore, when the pressure

difference is higher, the permeate velocity is higher and so is the probability

of contact of particle with the cake surface. This increases the chance of

particle attachment to the neighboring particles and will result in higher cake

thickness growth. A similar behavior is true for gamma ( ).

The cake thickness decreases when the cake permeability decreases and

increases when the cake permeability increases. However, the relationship is

different in different parts of the perforation zone. It should be noted that

the cake permeability is assumed constant in this study, which might not be

completely true in the reality.

The sensitivity to the particle size is more complicated. The changes in

its value show different effects in different parts of the cake. Particle size

plays a significant role in the force balance equations as well as the external

cake model equations derived in the previous section.

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760 F. F. Zinati et al.

CONCLUSION

In this article, we developed a simple model which predicts the cake thickness

and velocity profiles in a radial geometry for a suspension containing mono-

sized particles.

This model is based on the Navier-Stokes equations for laminar flow and

is developed by considering the hydrodynamic forces which act on a particle.

The sensitivity analysis of the cake thickness was performed to different

parameters.

The particle radius and pressure difference between the injection and

reservoir pressure appear to have significant roles in the formation of the

filter cake around the borehole.

Moreover, the simulation results show that simplifying assumptions may

lead to errors in predicting the cake profiles.

ACKNOWLEDGMENT

The authors of this article are gratefully thankful to Professor P. Bedrikovetsky

for fruitful comments.

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Song, L., and Elimelech, M. (1995). Particle deposition onto a permeable

surface in laminar flow. J. Colloid. Interf. Sci. 173:165–180.

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imental study of drilling fluids dynamic filtration. Paper no. 15412, SPE

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57–68.

APPENDIX A

Writing the force balance equation for a fluid element shown in Figure A1

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762 F. F. Zinati et al.

Figure A1. Derivation of the velocity profile in laminar radial flow (Al-Abduwani,

internal report, Delft, 2004).

leads to

2�r � p ��r �

�

2�r � p ��r Cd

dx.2�r � p ��r/�x

�

C 2�r � � ��x �

�

2�r � � ��x Cd

dr.2�r � � ��x/�r

�

D 0 (A-1)

and in differential form it becomes:

�rd

dx.p/�

d

dr.� r/ D 0 (A-2)

Newton’s law of viscosity states that:

� D ��du

dr(A-3)

Therefore:

d

dr

�

�du

dr� r

�

D rdp

dx

�

r�d

dr

�

rdu

dr

�

Ddp

dx(A-4)

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External Filter Cake Build-up in Radial Geometry 763

The right hand side of the equation is independent of r and can therefore be

integrated twice with respect to r to yield:

u.r/ Dr2

4�

dp

dxC C1 ln.r/C C2 (A-5)

Given the following boundary conditions:

u.R � h/ D 0 (A-6)

du

dr

ˇ

ˇ

ˇ

ˇ

rD0

D 0 (A-7)

where Eq. A-6 is the no slip boundary condition at the surface of the cake

and Eq. A-7 is the radial symmetry condition at the centerline, then Eq. A-5

reduces to:

u.r/ D �1

4�

dp

dx.R � h/2

�

1 �

� r

R � h

�2�

(A-8)

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