Transport in Porous Media

, Volume 103, Issue 3, pp 401–419 | Cite as

Mathematical Modeling of Colloidal Particles Transport in the Medium Treated by Nanofluids: Deep Bed Filtration Approach

  • Danial Arab
  • Peyman Pourafshary
  • Shahaboddin Ayatollahi


A deep bed filtration model has been developed to quantify the effect of nanoparticles (NPs) on mitigating fines migration in porous media. The filtration coefficients representing the total kinetics of particles capture were obtained by fitting the model to the laboratory data. Based on the optimum filtration coefficients, the model was utilized to history match the particle concentration breakthrough profiles observed in twelve core flood tests. In the flooding experiments, the effect of five types of metal oxide NPs, \(\upgamma \hbox {-Al}_{2}\hbox {O}_{3}\), CuO, MgO, \(\hbox {SiO}_{2}\), and ZnO, on migrating fines were investigated. In each test, a stable suspension was injected into the already NP-treated core and effluents’ fines concentration was measured based on turbidity analysis. In addition, zeta potential analysis was done to obtain the surface charge (SC) of the NP-treated medium. It was found that the presence of NPs on the medium surface results in SC modification of the bed and as a result, enhances the filter performance. Furthermore, the ionic strength of the nanofluid was recognized as an important parameter which governs the capability of NPs to modify the SC of the bed. The remedial effect of NPs on migrating fines is quantitatively explained by the matched filtration coefficients. The SC of the medium soaked by \(\upgamma \hbox {-Al}_{2}\hbox {O}_{3}\) nanofluid is critically increased; therefore, the matched filtration coefficient is of remarkably high value and as a result, the treated medium tends to adsorb more than 70 % of suspended particles. The predicted particle concentration breakthrough curves well matched with the experimental data.


Colloidal particles transport Deep bed filtration  Fines migration in porous media Mathematical model Nanoparticles 

List of symbols


Suspended particle concentration (NTU)


Diffusion (dispersion) coefficient (\(\hbox {L}^{2}\hbox { T}^{-1}\))


Defined as \(\lambda /\lambda _{0}\)


Index of summation existing in Eq. 11


Index of summation existing in Eq. 14


Matching parameter


Filter depth (L)


Upper bound of the summation existing in Eq. 14


Upper bound of the summation existing in Eq. 11


Time (T)


Superficial velocity (\(\hbox {LT}^{-1}\))


Axial distance (L)

\(\theta \)

Corrected time defined by Eq. 3

\(\lambda \)

Filtration coefficient (\(\hbox {cm}^{-1}\))

\(\lambda _{0}\)

Initial filtration coefficient (\(\hbox {cm}^{-1}\))

\(\sigma \)

Concentration of retained fine particles (NTU)

\(\phi \)


\(\psi \)

An objective function defined by Eq. 14









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Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • Danial Arab
    • 1
  • Peyman Pourafshary
    • 2
  • Shahaboddin Ayatollahi
    • 3
  1. 1.Institute of Petroleum Engineering, School of Chemical EngineeringUniversity of TehranTehranIran
  2. 2.Department of Chemical and Petroleum EngineeringSultan Qaboos UniversityMuscatOman
  3. 3.Enhanced Oil Recovery Research Center, School of Chemical and Petroleum EngineeringShiraz UniversityShirazIran

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