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Mathematical Model for Dynamic Adsorption with Immiscible Multiphase Flows in Three-dimensional Porous Media

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Abstract

In this paper, we present the mathematical model describing the dynamic adsorption processes in three-dimensional porous media. The novelty of this model lies in the ability to study the mass transfer processes with immiscible multiphase flows in porous media. The governing equations describing fluid flow and convective-diffusion of the active component are based on the lattice Boltzmann equations. The phenomena on the interface between two fluids and between fluids and solid phase, including interfacial tension and wetting effects, are described using the most modern version of the color-gradient method. The kinetic of the mass transfer between active component and adsorbent particles is described using the Langmuir adsorption equation. The numerical algorithm has been validated on two benchmarks including the immiscibility of the active component and the displaced fluid, as well as the problem of mass conservation of the active component during its adsorption and transport in porous media. The mathematical model has been adapted for porous media presented by X-ray computed tomography images of natural porous media.

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REFERENCES

  1. T. A. Amirianshoja, R. Junin, A. K. Idris, and O. Rahmani, ‘‘Comparative study of surfactant adsorption by clay minerals,’’ J. Pet. Sci. Eng. 101, 21–27 (2013).

    Article  Google Scholar 

  2. P. E. G. Idahosa, G. F. Oluyemi, M. B. Oyeneyin, and R. Prabhu, ‘‘Rate-dependent polymer adsorption in porous media,’’ J. Pet. Sci. Eng. 43, 65–71 (2016).

    Article  Google Scholar 

  3. X. Zhang, S. Zhang, Z. Du, G. G. X. Wang, S. Heng, X. Liu, and J. Lin, ‘‘CO2 and N2 adsorption/desorption effects and thermodynamic characteristics in confined coal,’’ J. Pet. Sci. Eng. 207, 109166 (2021).

  4. L. Ma and Q. Yu, ‘‘Dynamic behaviors of methane adsorption on partially saturated shales,’’ J. Pet. Sci. Eng. 190, 107071 (2020).

  5. C. Yuan, W. F. Pu, M. A. Ifticene, S. Zhao, and M. A. Varfolomeev, ‘‘Crude oil oxidation in an air injection based enhanced oil recovery process: Chemical reaction mechanism and catalysis,’’ Energy Fuels 36, 5209–5227 (2022).

    Article  Google Scholar 

  6. L. Zhou, Z. G. Qu, L. Chen, and W.-Q. Tao, ‘‘Lattice Boltzmann simulation of gas–solid adsorption processes at pore scale level,’’ J. Comput. Phys. 300, 800–813 (2015).

    Article  MathSciNet  Google Scholar 

  7. T. R. Zakirov and M. G. Khramchenkov, ‘‘Effect of pore space heterogeneity on the adsorption dynamics in porous media at various convection-diffusion and reaction conditions: A lattice Boltzmann study,’’ J. Pet. Sci. Eng. 212, 110300 (2022).

  8. H. Wang, Z. G. Qu, and L. Zhou, ‘‘A combined GCMC and LBM simulation method for CH4 capture in Cu-BTC particle adsorption bed,’’ Int. Commun. Heat Mass Transfer 88, 48–53 (2017).

    Article  Google Scholar 

  9. S. Succi, The Lattice Boltzmann Equation for Fluid Dynamics and Beyond (Oxford Univ. Press, UK, 2001).

    Book  Google Scholar 

  10. Q. Kang, P. C. Lichtner, and D. Zhang, ‘‘Lattice Boltzmann pore-scale model for multicomponent reactive transport in porous media,’’ J. Geophys. Res. 111, B05203 (2006).

  11. Q. Kang, P. C. Lichtner, and D. Zhang, ‘‘An improved lattice Boltzmann model for multicomponent reactive transport in porous media at the pore scale,’’ Water Resour. Res. 43, W12S14 (2007).

  12. T. Reis and T. N. Phillips, ‘‘Lattice Boltzmann model for simulating immiscible two-phase flows,’’ J. Phys. A: Math. Theor. 40, 4033–4053 (2007).

    Article  MathSciNet  Google Scholar 

  13. T. R. Zakirov and M. G. Khramchenkov, ‘‘Wettability effect on the invasion patterns during immiscible displacement in heterogeneous porous media under dynamic conditions: A numerical study,’’ J. Pet. Sci. Eng. 206, 109049 (2021).

  14. F. Kuznik, C. Obrecht, G. Rusaouen, and J.-J. Roux, ‘‘LBM based flow simulation using GPU computing processor,’’ Comput. Math. Appl. 59, 2380–2392 (2010).

    Article  Google Scholar 

  15. E. Aslan, I. Taymaz, and A. C. Benim, ‘‘Investigation of the lattice Boltzmann SRT and MRT stability for lid driven cavity flow,’’ Int. J. Mater. Mech. Manuf. 2, 317–324 (2014).

    Google Scholar 

  16. C. Pan, L. S. Luo, and C. T. Miller, ‘‘An evaluation of lattice Boltzmann schemes for porous medium flow simulation,’’ Comput. Fluids 35, 898–909 (2006).

    Article  Google Scholar 

  17. H. Huang, J.-J. Huang, and X.-Y. Lu, ‘‘Study of immiscible displacements in porous media using a color-gradient-based multiphase lattice Boltzmann method,’’ Comput. Fluids 93, 164–172 (2014).

    Article  MathSciNet  Google Scholar 

  18. X. Shan and H. Chen, ‘‘Lattice Boltzmann model for simulating flows with multiple phases and components,’’ Phys. Rev. E 47, 1815–1819 (1993).

    Article  Google Scholar 

  19. H. Huang, L. Wang, and X. Lu, ‘‘Evaluation of three lattice Boltzmann models for multiphase flows in porous media,’’ Comput. Math. Appl. 61, 3606–3617 (2011).

    Article  MathSciNet  Google Scholar 

  20. L. Chen, Q. Kang, Y.-L. He, and W.-Q. Tao, ‘‘Pore-scale simulation of coupled multiple physicochemical thermal processes in micro reactor for hydrogen production using lattice Boltzmann method,’’ Int. J. Hydrogen Energy 37, 13943–13957 (2012).

    Article  Google Scholar 

  21. J. Lei, R. Ding, and J. Zhang, ‘‘Study on seepage and adsorption characteristics of porous media containing adsorbent based on lattice Boltzmann,’’ AIP Adv. 11, 045126 (2021).

  22. S. Leclaire, A. Parmigiani, O. Malaspinas, B. Chopard, and J. Latt, ‘‘Generalized three-dimensional lattice Boltzmann color-gradient method for immiscible two-phase pore-scale imbibition and drainage in porous media,’’ Phys. Rev. E 95, 033306 (2017).

  23. A. Riaud, S. Zhao, K. Wang, Y. Cheng, and G. Luo, ‘‘Lattice-Boltzmann method for the simulation of multiphase mass transfer and reaction of dilute species,’’ Phys. Rev. E 89, 053308 (2014).

  24. N. Saxena, S. Kumar, and A. Mandal, ‘‘Adsorption characteristics and kinetics of synthesized anionic surfactant and polymeric surfactant on sand surface for application in enhanced oil recovery,’’ Asia-Pacif. J. Chem. Eng. 13, e2211 (2018).

  25. Q. Zou and X. He, ‘‘On pressure and velocity boundary conditions for the lattice Boltzmann BGK model,’’ Phys. Fluids 9, 1591–1598 (1997).

    Article  MathSciNet  Google Scholar 

  26. T. R. Zakirov, M. A. Varfolomeev, and C. Yuan, ‘‘Characterization of dynamic adsorption regimes in synthetic and natural porous structures using lattice Boltzmann simulations,’’ Chem. Eng. Res. Design 189, 14–29 (2023).

    Article  Google Scholar 

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Funding

The work is carried out under the support of the Russian Science Foundation related to Project no. 23-71-01008.

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Authors and Affiliations

  1. Institute of Geology and Petroleum Technologies, Kazan Federal University, 420008, Kazan, Tatarstan, Russia

    T. R. Zakirov, O. S. Zhuchkova & M. G. Khramchenkov

  2. Kazan Branch of Joint Supercomputer Center, Scientific Research Institute of System Analysis, Russian Academy of Sciences, 420111, Kazan, Russia

    M. G. Khramchenkov

Authors
  1. T. R. Zakirov
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  2. O. S. Zhuchkova
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  3. M. G. Khramchenkov
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Correspondence to T. R. Zakirov, O. S. Zhuchkova or M. G. Khramchenkov.

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Zakirov, T.R., Zhuchkova, O.S. & Khramchenkov, M.G. Mathematical Model for Dynamic Adsorption with Immiscible Multiphase Flows in Three-dimensional Porous Media. Lobachevskii J Math 45, 888–898 (2024). https://doi.org/10.1134/S1995080224600134

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