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Numerical simulation of retention and release of colloids in porous media at the pore scale

TL;DR: In this article, the authors simulated the transport of a solid colloidal particle at the pore scale in presence of surface roughness and particle/pore physicochemical interaction by adopting a "one fluid" approach.
About: This article is published in Colloids and Surfaces A: Physicochemical and Engineering Aspects.The article was published on 2013-06-20 and is currently open access. It has received 43 citations till now. The article focuses on the topics: Particle & Surface roughness.

Summary (4 min read)

2.1. Configuration

  • The geometrical configuration and physical parameters chosen for this study are based on recent experiments [29, 30] .
  • The topographic heterogeneities considered here are surface asperities on the grain surface.
  • Therefore, besides the smooth surface, right triangular prisms of the form of peaks and valleys with two different sizes are considered.
  • The size of the grid blocks are chosen so that the particle diameter contains 16 blocks.
  • Previous works have shown that, with the numerical choices made here which will be presented in the next section, a number of grid blocks between 8 and 16 ensure numerical results in agreement with the physics of particle transport for a large range of Reynolds numbers [34, 35] .

2.2.1. Governing equations

  • Simulations are performed using the numerical code Thetis ® , developed in their lab, in which additional modules have been added in order to take into account particle/particle and particle/grain surface physicochemical interactions.
  • A generalized one-fluid model has been used for the transport of particles [35] .
  • The entire domain is considered as fluid and the solid is a particular fluid with special properties, the two phases being distinguished through a phase indicator function, F c .
  • This force can include gravity, lubrication and DLVO.
  • Gravity is not taken into account and only particle-grain and particle-particle DLVO forces are added in Eq. ( 4).

2.2.2. DLVO forces

  • Physicochemical interactions between particles or a particle and a grain surface include van der Waals (vdW) and electrical double layer (DL) forces, the sum of which is called the DLVO force (DLVO).
  • When the grain surface includes some kind of heterogeneity (chemical or topographic), the analytical expressions introduced above do not hold.
  • In that case, many authors [12, [37] [38] [39] replace the DLVO analytical expressions by using different approximations: Derjaguin Approximation Technique (DAT), Grid Surface Integration (GSI) and Surface Element Integration (SEI).
  • Fig. 5 shows that for an ionic strength of 24 mM, the obtained results are in good agreement using 4500 elements even with separation distances less than the mesh size of a grid block.
  • One must note that the goal of developing the SEI method is to compute DLVO forces for rough surfaces where the analytical formula does not hold anymore (Fig. 6 ).

2.2.3. Lubrication forces

  • At such a distance, the hydrodynamic forces acting on the sphere are underestimated.
  • Multi layer model used for lubrication interaction.
  • This procedure is described in details elsewhere [34] and was proven to predict properly the restitution coefficient for a particle colliding normally a solid wall.
  • Therefore, by doing so the authors account successively for long range hydrodynamic interactions by solving the Navier-Stokes equations, the short range lubrication effect and the solid-solid collision.

2.3. Simulation method

  • A special decomposition of the stress tensor, , is proposed: EQUATION where Ã, and ω are respectively the elongation, the shearing and the rotation viscosities.
  • The choice of the different viscosities ensures physical characteristics of the fluid and the solid phases.
  • The algorithm used is based on the augmented Lagrangian method which allows coupling the velocity and the pressure while keeping an implicit method.
  • The boundary conditions imposed are: constant flow rate at the inlet and a Neumann condition on the velocity at the outlet.
  • Additional data, such as hydrodynamic and DLVO forces at each time step are also computed allowing a more detailed analysis of the particle behavior.

3.1. Flow structure

  • The flow structure for peak geometry is first presented in Fig. 7a where flow pattern and pressure field in the suspending fluid are superimposed.
  • It may be seen that both velocity and pressure fields are disturbed by the solid obstacle representing the roughness and the moving hard spherical particle.
  • This means that during the transport of very small negligibly diffusing particles, their centre of mass would follow the flow streamlines and they would be transported far behind the solid obstacle.
  • It is worth noting that under the same conditions, less flow perturbation is noticed by decreasing the peak height as expected (data not shown) but such a perturbation is strongly changed when the valley form is considered.
  • The flow for this latter geometry will be illustrated and commented in the next section.

3.2. Trajectory versus ionic strength for given Re

  • Particle trajectories are studied under fixed hydrodynamic conditions and by changing physicochemical interaction through variation of the ionic strength of the suspending fluid.
  • Indeed, the influence of suspending fluid pH has to be considered as it determines the value of both Debye length and zeta potentials that consequently modify the interaction potential.
  • In practice, pH value is imposed by the porous medium itself (buffering effect), rather than being considered as a variable parameter.

3.3. Smooth flat grain surface

  • First, the reference case of a smooth flat grain surface is considered.
  • In that case, whatever the ionic strength and therefore the magnitude of the physicochemical interaction force, injected particles mainly follow a straight trajectory parallel to the grain surface .
  • The only difference from one situation to another is the initial position of particle versus the interaction intensity.
  • Indeed, for a low ionic strength, the particle-pore surface interaction potential presents a significant energy barrier prohibiting the particle to come in close contact with the solid surface and is pushed away from the surface.
  • When I is increased, the energy barrier decreases enabling particles to remain in the neighborhood of the solid surface leading hence to a lower velocity.

3.4. Peak shaped roughness

  • With the same considerations and in presence of a peak-shaped roughness, the authors may reasonably expect at low ionic strength a moving particle to jump over the roughness and be transported far away from the obstacle.
  • Again by increasing the ionic strength both the energy barrier height and its distance range are reduced and the particle is allowed to approach the solid surface more closely.
  • This is clearly shown by inspecting the trajectory corresponding to medium ionic strength for which the particle is seen to come back in contact with solid surface after over passing the roughness and then continues to move while rotating (Fig. 8 ).
  • This is what is readily observed for the high I value where the particle, after a displacement period, is stopped at the foot of the peak and the particle retained does not move any more.
  • Once again comparable results are obtained for the smaller peak in the sense that for moderate and low ionic strength values, the particle flows over the asperity and it is retained for the high I value.

3.5. Valley shaped roughness

  • Similar in silico experiments have been carried out in case of the valley geometry at the same Reynolds number.
  • Such experiment was repeated as previously for different I values and once again no particle retention was observed for low and intermediate ionic strength.
  • Here colloid retention was only observed for the deeper valley.
  • Such difference arises from the fact that in the case of peaks, the particle interacts with solid wall via two points whereas only one interaction point exist in case of valleys under their hydrodynamic conditions.
  • From these experiments it may be concluded that under given hydrodynamic conditions, the roughness characteristics and physicochemical conditions play major roles in colloid retention in porous media.

3.6. Dimensionless analysis

  • After presentation and general comments of predicted influence of ionic strength on flow structure and colloids retention in presence of roughnesses of different shapes and sizes, in this paragraph the authors compare the results obtained for all considered geometry cases.
  • Up to now the imposed flow rate was often kept constant and consequently the authors still set aside quantitative study of the role of hydrodynamic interactions and their competition with physicochemical interactions.
  • For that purpose, let us first consider the Newton's second law of motion written in a dimensionless form as: EQUATION where EQUATION EQUATION Both of these dimensionless numbers depend on particle radius and, in an identical manner on the mean flow velocity.
  • In Fig. 10 the variation of the dimensionless reciprocal residence time −1* as a function of N 2 is therefore plotted for every topography and ionic strength, the Reynolds number being constant.
  • In such a situation the non deformable colloidal sphere does interact with the solid grain only via a "contact point" and the attractive force is oriented toward the centre of the sphere (and perpendicular to mean flow streamlines).

4. Conclusion

  • Additional modules were implemented in a fluid flow tool in order to simulate colloid transport at the pore scale in presence of rough pore surface while taking into account physicochemical interactions.
  • For non deformable particles it was shown that the existence of surface roughness is a necessary but not sufficient condition for particles retention.
  • For a fixed Reynolds number and under conditions for which no retention occurs, the residence time was found to increase with increasing ionic strength as the particle/pore surface becomes less repulsive.
  • Additional simulations for constant ionic strength of the suspending fluid and various values of Reynolds number must be performed in order to fully validate the analysis.
  • As a long-term perspective and to be representative of real experiments, simulations should also be carried out using a given distribution of true 3D roughness since the characteristic length of the topography (the wave length) is expected to play a major role both in particle retention and release.

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Numerical simulation of retention and release of colloids
in porous media at the pore scale
Nisrine Sefrioui, Azita Ahmadi, Aziz Omari, Henri Bertin
To cite this version:
Nisrine Sefrioui, Azita Ahmadi, Aziz Omari, Henri Bertin. Numerical simulation of retention and
release of colloids in porous media at the pore scale. Colloids and Surfaces A: Physicochemical and
Engineering Aspects, Elsevier, 2013, 427, pp.33-40. �10.1016/j.colsurfa.2013.03.005�. �hal-01081224�

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To cite this version :
Nisrine SEFRIOUI, Azita AHMADI, Aziz OMARI, Henri BERTIN - Numerical simulation of
retention and release of colloids in porous media at the pore scale - Colloids and Surfaces A:
Physicochemical and Engineering Aspects - Vol. 427, p.33-40 - 2013
Any correspondence concerning this service should be sent to the repository
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Numerical simulation of retention and release of colloids in porous
media at the pore scale
Nisrine Sefrioui, Azita Ahmadi, Aziz Omari, Henri Bertin
I2M-TREFLE-UMR CNRS 5295, University of Bordeaux, Arts et Metiers ParisTech, Esplanade des Arts et Metiers, 33405, Talence, France
h i g h l i g h t s
Direct Numerical Simulation has
been performed.
Different roughness types have been
investigated.
Physicochemical interactions have
been included in the model.
Residence time depend on roughness,
hydrodynamics and physicochemical
interactions.
g r a p h i c a l a b s t r a c t
Flow structure and particle trajectory for a valley geometry roughness (dashed line corresponds to ideal
trajectory).
Keywords:
Colloid
Porous media
Rough surface
Retention
Numerical simulation
Ionic strength
a b s t r a c t
Transport of a solid colloidal particle was simulated at the pore scale in presence of surface roughness
and particle/pore physicochemical interaction by adopting a “one fluid” approach. A code developed in
our laboratory was used to solve equations of motion, while implementing additional modules in order
to take into account lubrication and physicochemical forces. Particles were recognized through a phase
indicator function and the particle/fluid interface position at each instant was obtained by solving a trans-
port equation. Roughnesses of different shapes were considered and the magnitude of the particle/pore
physicochemical interaction was monitored through the change of the ionic strength of the suspending
fluid. We first show that if pore surface is smooth no retention of the transported particle occurs whether
the particle/pore surface is attractive or repulsive. However for shape roughnesses of “peak” or “valley”,
particles may be retained inside pores or not depending on the considered ionic strength. In absence of
particle retention, the residence time (the time needed for a particle to travel a characteristic pore dis-
tance) is finite and was found to be an increasing function of ionic strength for every considered roughness
at fixed hydrodynamic conditions.
1. Introduction
Flow of reactive solutions or suspensions in saturated porous
media is of great interest in many environmentally relevant
applications such as contaminant dissemination, filtration and
remediation processes. In such processes, the determination of the
Corresponding author. Tel.: +33 556 84 54 06; fax: +33 556 84 54 36.
E-mail address: h.bertin@i2m.u-bordeaux1.fr (H. Bertin).
concentration of species as a function of time and space is of major
concern [1]. It is admitted that besides porous media structure,
transport and chemical aspects are the most important phenomena
that should be considered in interpreting field and laboratory data
or in analyzing modeling results.
Considering chemical aspects, when reactive species
are charged colloid particles of finite size, the DLVO
(Derjaguin–Landau–Verwey–Overbeek) theory is often put
forward to describe particle/pore surface interaction. If such an
interaction is purely attractive the adsorption conditions are called
http://dx.doi.org/10.1016/j.colsurfa.2013.03.005

favorable with no energy barrier while unfavorable conditions are
reserved to non-monotonic interaction pr ofiles which generally
present an energy barrier, a deep primary minimum and a shallow
secondary minimum. Under physicochemical conditions such that
unfavorable conditions are expected to prevail, the calculated
energy barrier is usually high enough (several hundreds of kT)
so that colloid adsorption should be precluded. However, this
behavior was reported to be in contrast with experimental evi-
dences showing a significant adsorption. To explain such apparent
discrepancy several possible sources were examined. Among
them, the predominant role of the secondary minimum in the
adsorption process was emphasized [2–4] and it was shown that
on the basis of this, the influence of physicochemical parameters
as ionic strength and pH of the background solution were well
predicted [2]. Nevertheless despite qualitative theory/experiment
agreement, quantitative discrepancies were reported to persist. It
was therefore argued that as particle and pore wall surfaces are not
chemically and/or topographically homogenous, a more precise
calculation of interaction potentials on small scale is needed for
quantitative comparison. Chemical heterogeneity is introduced
through the existence of nano-sized chemical patches whose
composition may locally induce irreversible adsorption of particles
in the primary minimum [5,6]. In that respect, several models
containing two or more classes of adsorption sites were proposed
in the literature [7–11]. The local pore structure and surface
topography heterogeneity considered as asperities were shown to
induce a shift of the actual interaction potential. Indeed, repulsive
interaction between a colloidal particle and a solid surface is
lower on a rough surface compared to a smooth surface [12–14].
The extent of this interaction depends mainly on the asperities
characteristic height H, their form and the inter-asperities distance
relative to particle size. Moreover, these topographic hetero-
geneities play an important role in both particle retention and
its counterpart particle release phenomena through flow pattern
modification. When a porous medium is modeled as a collection of
spherical grains, non-deformable colloid particles are considered
to firstly adsorb onto grain surface and may subsequently roll on
it under hydrodynamic drag force and/or diffusion and then are
retained by accumulation in stagnation regions at grain–grain
contact zone or in the rear of the grain [15–18]. Similarly colloid
particles may be re-entrained by escaping from these retention
regions going back to the bulk flowing suspension (re-suspension
mechanism). This was introduced in the convection–dispersion
equation [19] in order to simulate experimental breakthrough
curves but too many rate dependent constants had to be
adjusted.
Basically, detachment of an adsorbed particle results from a bal-
ance between external forces exerted on it. These are adhesion
(physicochemical interactions), drag and lift forces. So, depending
on whether the particle is hard (slightly deformable) or soft (highly
deformable), an adapted continuum mechanics model is usually
used to predict the physicochemistry/hydrodynamic relationship
that governs particle detachment. For smooth surfaces, major stud-
ies predict that detachment should occur mainly by rolling [20–23].
Bergendahl and Grasso [22] have shown that incipience of particle
rolling is well correlated to a dimensionless parameter N
TFT
; the
ratio of the depth of the primary minimum to the exerted shear
force torque. The roughness of grain surface was earlier considered
by several authors [24–27]. Burdick et al. [26] studied the behavior
of colloids of given sizes in the vicinity of an obstacle. Examina-
tion of balance of forces and torques that are exerted on a retained
deformable colloid showed that its re-entrainment follows a lift
process rather than a rolling which is expected to predominate in
the removal of particles adsorbed on smooth surfaces. By the same
way, Neyland [28] examined the influence of both the height (and
the depth) of asperities and their separation distance on critical
flow rate for particle re-entrainment. However, the problem con-
sidered was only in 2 dimensions and the discussion of results was
only qualitative.
From what is briefly exposed here, it may be seen that represen-
tative simulations of retention and removal of colloidal particles of
given size in presence of a rough surface is still lacking. So, the
goal of this paper is to propose a numerical modeling of retention
and re-entrainment of micron sized colloids in presence of rough-
ness on a grain surface. In this work, a direct numerical simulation
method was adopted including both hydrodynamic and physico-
chemical interactions in the vicinity of asperities of various shapes
and characteristic sizes. In the next section the adopted method
is described before presentation in section 3 of primary results
showing successively the influence of physicochemical conditions
through the ionic strength variations and hydrodynamic conditions
all in presence of a rough or a smooth surface.
2. Numerical model
2.1. Configuration
The geometrical configuration and physical parameters chosen
for this study are based on recent experiments [29,30]. In those
works, deposition and release of negatively charged colloidal latex
particles of a radius, a
p
, of 400 nm in an artificial sintered silicate
porous medium were investigated.
Hydrodynamic parameters are also based on laboratory experi-
ments [29,30]. The inlet velocity used in the simulations is equal to
4 × 10
5
ms
1
leading to the following values of the Péclet number
and the Particulate Reynolds number respectively:
P
e
=
V
p
a
p
D
b
= 9.8 (1)
,
Re
p
=
p
a
p
¯
u
= 1.66 × 10
5
(2)
where V
p
is the particle velocity corresponding to the fluid veloc-
ity evaluated at a distance of 3a
p
from the grain surface, D
b
is the
bulk particle diffusion coefficient.
p
,
¯
u and are respectively the
particle density, the interstitial velocity and the dynamic viscosity
of the fluid.
For given P
e
and Re
p
and for each roughness, three different val-
ues of salinity (0.5, 1.2 and 2 or 3 g/L of NaCl) are considered leading
to three levels of ionic strength, I, designated by weak (I = 3 mM),
medium (I = 7.8 mM) and strong (I = 12 or 18 mM). These values of
I correspond to DLVO forces of significantly different intensities
(Fig. 1).
The domain chosen for this study is a rectangular prism
(5 m × 4 m × 3.8 m), in contact with the grain surface (Fig. 2).
The topographic heterogeneities considered here are surface
asperities on the grain surface. The roughness geometries are sim-
ple and have acute angles, which for a class of porous media such
as sands can be closer to reality than hemispherical plots or asperi-
ties that are mostly used in the literature [12,25,31–33]. Therefore,
besides the smooth surface, right triangular prisms of the form of
peaks and valleys with two different sizes are considered. (Fig. 3)
The heights, H, correspond to one or two times the particle radius,
a
p
and therefore vary from 2a
p
to 2a
p
. Despite the 2D nature of the
asperities, particles are spherical and all simulations are performed
in 3 dimensions. The colloid is placed on the symmetry plane of the
domain in the thickness and the results will be presented in this
plane.
The particle transport is solved by Direct Numerical Simulation
(DNS) with fixed Cartesian grids. The size of the grid blocks are cho-
sen so that the particle diameter contains 16 blocks. Previous works

Fig. 1. DLVO Interaction forces for different ionic strength values as a function of surface-to-surface separation distance, h.
Fig. 2. Schematic view of the simulation domain (case of peak geometry).
have shown that, with the numerical choices made here which will
be presented in the next section, a number of grid blocks between
8 and 16 ensure numerical results in agreement with the physics
of particle transport for a large range of Reynolds numbers [34,35].
2.2. Model
2.2.1. Governing equations
Simulations are performed using the numerical code Thetis
®
,
developed in our lab, in which additional modules have been added
in order to take into account particle/particle and particle/grain sur-
face physicochemical interactions. A generalized one-fluid model
has been used for the transport of particles [35]. The entire domain
is considered as fluid and the solid is a particular fluid with special
properties, the two phases being distinguished through a phase
indicator function, F
c
. The evolution of the particle is described by
an advection equation on F
c
. The flow of the incompressible New-
tonian fluid is governed by the Navier–Stokes and the mass balance
equations. The final set of partial differential equations is given by
the one-fluid model given below:
.u = 0 (3)
Fig. 3. Roughness geometries with respect to particle dimension H [[2a
p
; 2a
p
]].

Citations
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Journal ArticleDOI
TL;DR: In this article, the authors present a quantitative analysis of temperature effects on the forces exerted on particles and the resultant fines migration in natural reservoirs, and derive a model for the maximum retention concentration and used it to characterize the detachment of multisized particles from rock surfaces.
Abstract: The fluid flow in natural reservoirs mobilizes fine particles. Subsequent migration and straining of the mobilized particles in rocks greatly reduce reservoir permeability and well productivity. This chain of events typically occurs over the temperature ranges of 20-40 degrees C for aquifers and 120-300 degrees C for geothermal reservoirs. However, the present study might be the first to present a quantitative analysis of temperature effects on the forces exerted on particles and of the resultant fines migration. Based on torque balance between electrostatic and drag forces acting on attached fine particles, we derived a model for the maximum retention concentration and used it to characterize the detachment of multisized particles from rock surfaces. Results showed that electrostatic force is far more affected than water viscosity by temperature variation. An analytical model for flow toward wellbore that is subject to fines migration was derived. The experiment-based predictive modeling of the well impedance for a field case showed high agreement with field historical data (coefficient of determination R-2=0.99). It was found that the geothermal reservoirs are more susceptible to fine particle migration than are conventional oilfields and aquifers.

91 citations


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TL;DR: In this article, a model for predicting the Derjaguin−Landau−Verwey−Overbeek (DLVO) interaction energy between colloidal particles and rough membrane surfaces is developed.
Abstract: Recent experimental investigations suggest that interaction of colloidal particles with polymeric membrane surfaces is influenced by membrane surface morphology (roughness). To better understand the consequences of surface roughness on colloid deposition and fouling, it is imperative that models for predicting the Derjaguin−Landau−Verwey−Overbeek (DLVO) interaction energy between colloidal particles and rough membrane surfaces be developed. We present a technique of reconstructing the mathematical topology of polymeric membrane surfaces using statistical parameters derived from atomic force microscopy roughness analyses. The surface element integration technique is used to calculate the DLVO interactions between spherical colloidal particles and the simulated (reconstructed) membrane surfaces. Predictions show that the repulsive interaction energy barrier between a colloidal particle and a rough membrane is lower than the corresponding barrier for a smooth membrane. The reduction in the energy barrier is ...

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Frequently Asked Questions (15)
Q1. What have the authors contributed in "Numerical simulation of retention and release of colloids in porous media at the pore scale" ?

In this paper, Bertin et al. used the DLVO ( physicochemical interactions ) theory to describe particle/pore surface interaction. 

It is admitted that besides porous media structure, transport and chemical aspects are the most important phenomena that should be considered in interpreting field and laboratory data or in analyzing modeling results. 

The existence of DLVO forces at short separation distances, with high intensities can lead to large particle velocities and therefore very small time steps. 

When the particle–grain or particle–particle distance is less than a grid size, hydrodynamic forces cannot be computed correctly. 

for a low ionic strength, the particle–pore surface interaction potential presents a significant energy barrier prohibiting the particle to come in close contact with the solid surface and is pushed away from the surface. 

Physicochemical interactions between particles or a particle and a grain surface include van der Waals (vdW) and electrical double layer (DL) forces, the sum of which is called the DLVO force (DLVO). 

Additional simulations for constant ionic strength of the suspending fluid and various values of Reynolds number must be performed in order to fully validate the analysis. 

As a long-term perspective and to be representative of real experiments, simulations should also be carried out using a given distribution of true 3D roughness since the characteristic length of the topography (the wave length) is expected to play a major role both in particle retention and release. 

Similarly colloid particles may be re-entrained by escaping from these retention regions going back to the bulk flowing suspension (re-suspension mechanism). 

The normal component of this force which is involved in particle contact with the wall is written as:Flub.n = −6 ap[ s(εh) − s(εh0 )]vp.n for h ≤ h0 (12)where εh = h/ap is a dimensionless surface-to-surface distance between neighboring particles or between a particle and a grain surface, εh0 = h0/ap, is the dimensionless critical distance for activation or deactivation of lubrication, vp is the particle velocity, n is the normal unit vector directed from the particle towards the plane or another particle and s is the Stokes correction (amplification) factor. 

The extent of this interaction depends mainly on the asperities characteristic height H, their form and the inter-asperities distance relative to particle size. 

Again by increasing the ionic strength both the energy barrier height and its distance range are reduced and the particle is allowed to approach the solid surface more closely. 

For the case of a spherical particle near an homogeneous infinite flat plate (smooth grain surface in their case), the approximate analytical expressions of these forces denoted respectively FSPvdW, F SP DL and F SP DLVO are given by Prieve and Ruckenstein [36]:FSPvdW = − 2AHa3P3h2(h + ap)2 (6)FSPDL = 2 ε0εrap 1 − e−2 h (2 P Se − h − ( 2P + 2S )e−2 h) (7)FSPDLVO = FSPvdW + FSPDL (8) where AH is the particle/water/solid Hamaker constant; h is the minimum separation distance between the particle and the flat plate; is the inverse Debye screening length; ε0 is the dielectric permittivity of vacuum; εr is the relative dielectric constant of water and P and S are the surface zeta potentials of the particle and the grain respectively. 

Among them, the predominant role of the secondary minimum in the adsorption process was emphasized [2–4] and it was shown that on the basis of this, the influence of physicochemical parameters as ionic strength and pH of the background solution were well predicted [2]. 

It is worth noting that under the same conditions, less flow perturbation is noticed by decreasing the peak height as expected (data not shown) but such a perturbation is strongly changed when the valley form is considered.