Numerical simulation of retention and release of colloids in porous media at the pore scale
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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Cites background from "Numerical simulation of retention a..."
...Particle transport in the boundary layers, and in the bulk of carrier fluid, is reflected by a two-velocity model [Yuan and Shapiro, 2011; Bradford et al., 2012, 2014; Sefrioui et al., 2013]....
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91 citations
Cites background from "Numerical simulation of retention a..."
...The particle drift near the rough pore walls as modelled by Navier–Stokes equations has the speed significantly lower than the injected water velocity (Sefriouri et al., 2013)....
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Frequently Asked Questions (15)
Q2. What are the important phenomena that should be considered in interpreting field and laboratory data?
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.
Q3. What is the effect of DLVO forces on the particle?
The existence of DLVO forces at short separation distances, with high intensities can lead to large particle velocities and therefore very small time steps.
Q4. What are the hydrodynamic forces that can be computed?
When the particle–grain or particle–particle distance is less than a grid size, hydrodynamic forces cannot be computed correctly.
Q5. What is the particle-pore interaction potential for a low ionic strength?
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.
Q6. What are the physicochemical interactions between particles and a grain 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).
Q7. how many simulations are needed to validate the analysis?
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.
Q8. how long does the simulation take to be representative of real experiments?
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.
Q9. What mechanism is used to re-enter the particles?
Similarly colloid particles may be re-entrained by escaping from these retention regions going back to the bulk flowing suspension (re-suspension mechanism).
Q10. What is the normal component of the force involved in particle contact with the wall?
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.
Q11. What is the extent of the interaction between a colloidal particle and a solid surface?
The extent of this interaction depends mainly on the asperities characteristic height H, their form and the inter-asperities distance relative to particle size.
Q12. How is the particle allowed to approach the solid surface?
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.
Q13. What are the forces of a particle near an infinite flat plate?
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.
Q14. What are the main factors that were emphasized in the study?
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].
Q15. What is the effect of the valley form on the flow structure?
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.