Skip to main content
SpringerLink
Log in

Advection–diffusion in a porous medium with fractal geometry: fractional transport and crossovers on time scales

  • Published:
Meccanica Aims and scope Submit manuscript

Abstract

In a porous fractal medium, the transport dynamics is sometimes anomalous as well as the crossover between numerous transport regimes occurs. In this paper, we experimentally investigate the mass transfer of the diffusing agents of various classes in the composite porous particle with fractal geometry. It is shown that transport mechanisms differ at short and long times. At the beginning, pure advection is observed, whereas the longtime transport follows a convective mechanism. Moreover, the longtime transport experiences either Fickian or non-Fickian kinetics depending on the diffusing agent. The non-Fickian transport is justified for the diffusing agent with higher adsorption energy. Therefore, we speculate that non-Fickian transport arises due to the strong irreversible adsorption sticking of the diffusing molecules on the surface of the porous particle. For the distinguishing of the transport regimes, an approach admitting the transformations of the experimental data and the relevant analytic solutions in either semi-logarithmic or logarithmic coordinates is developed. The time-fractional advection–diffusion equation is used on a phenomenological basis to describe the experimental data exhibiting non-Fickian kinetics. The obtained anomalous diffusion exponent corresponds to the superdiffusive transport.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  1. Yu X, Regenauer-Lieb K, Tian FB (2019) A hybrid immersed boundary-lattice Boltzmann/finite difference method for coupled dynamics of fluid flow, advection, diffusion and adsorption in fractured and porous media. Comput Geosci 128:70–78. https://doi.org/10.1016/j.cageo.2019.04.005

    Article  Google Scholar 

  2. Wang W, Fan D, Sheng G et al (2019) A review of analytical and semi-analytical fluid flow models for ultra-tight hydrocarbon reservoirs. Fuel 256:115737. https://doi.org/10.1016/j.fuel.2019.115737

    Article  Google Scholar 

  3. Wang C, Winterfeld P, Johnston B, Wu YS (2020) An embedded 3D fracture modeling approach for simulating fracture-dominated fluid flow and heat transfer in geothermal reservoirs. Geothermics 86:101831. https://doi.org/10.1016/j.geothermics.2020.101831

    Article  Google Scholar 

  4. Vivas-Cruz LX, González-Calderón A, Taneco-Hernández MA, Luis DP (2020) Theoretical analysis of a model of fluid flow in a reservoir with the Caputo-Fabrizio operator. Commun Nonlinear Sci Numer Simul 84:105186. https://doi.org/10.1016/j.cnsns.2020.105186

    Article  MathSciNet  MATH  Google Scholar 

  5. Alotta G, Di Paola M, Pinnola FP, Zingales M (2020) A fractional nonlocal approach to nonlinear blood flow in small-lumen arterial vessels. Meccanica 55:891–906. https://doi.org/10.1007/s11012-020-01144-y

    Article  MathSciNet  Google Scholar 

  6. Failla G, Zingales M (2020) Advanced materials modelling via fractional calculus: challenges and perspectives. Philos Trans R Soc A Math Phys Eng Sci. https://doi.org/10.1098/rsta.2020.0050

    Article  Google Scholar 

  7. Qi H, Liu J (2010) Time-fractional radial diffusion in hollow geometries. Meccanica 45:577–583. https://doi.org/10.1007/s11012-009-9275-2

    Article  MathSciNet  MATH  Google Scholar 

  8. Chang A, Sun HG, Zhang Y et al (2019) Spatial fractional Darcy’s law to quantify fluid flow in natural reservoirs. Phys A Stat Mech its Appl 519:119–126. https://doi.org/10.1016/j.physa.2018.11.040

    Article  MathSciNet  Google Scholar 

  9. Li C, Yi Q (2019) Modeling and Computing of Fractional Convection Equation. Commun Appl Math Comput 1:565–595. https://doi.org/10.1007/s42967-019-00019-8

    Article  MathSciNet  MATH  Google Scholar 

  10. O’Shaughnessy B, Procaccia I (1985) Diffusion on fractals. Phys Rev A 32:3073–3083. https://doi.org/10.1103/PhysRevA.32.3073

    Article  MathSciNet  Google Scholar 

  11. Yu B (2008) Analysis of flow in fractal porous media. Appl Mech Rev 61:0508011–05080119

    Article  Google Scholar 

  12. Butera S, Di Paola M (2014) A physically based connection between fractional calculus and fractal geometry. Ann Phys (N Y) 350:146–158. https://doi.org/10.1016/j.aop.2014.07.008

    Article  MathSciNet  MATH  Google Scholar 

  13. Sandev T, Schulz A, Kantz H, Iomin A (2018) Heterogeneous diffusion in comb and fractal grid structures. Chaos Solitons Fractals 114:551–555. https://doi.org/10.1016/j.chaos.2017.04.041

    Article  MathSciNet  MATH  Google Scholar 

  14. Sandev T, Iomin A, Kantz H (2017) Anomalous diffusion on a fractal mesh. Phys Rev E 95:52107. https://doi.org/10.1103/PhysRevE.95.052107

    Article  Google Scholar 

  15. Huang T, Du P, Peng X et al (2020) Pressure drop and fractal non-Darcy coefficient model for fluid flow through porous media. J Pet Sci Eng 184:106579. https://doi.org/10.1016/j.petrol.2019.106579

    Article  Google Scholar 

  16. Yang X, Liang Y, Chen W (2018) A spatial fractional seepage model for the flow of non-Newtonian fluid in fractal porous medium. Commun Nonlinear Sci Numer Simul 65:70–78. https://doi.org/10.1016/j.cnsns.2018.05.014

    Article  MathSciNet  MATH  Google Scholar 

  17. Balankin AS, Valdivia JC, Marquez J et al (2016) Anomalous diffusion of fluid momentum and Darcy-like law for laminar flow in media with fractal porosity. Phys Lett Sect A Gen At Solid State Phys 380:2767–2773. https://doi.org/10.1016/j.physleta.2016.06.032

    Article  Google Scholar 

  18. Jin Y, Li X, Zhao M et al (2017) A mathematical model of fluid flow in tight porous media based on fractal assumptions. Int J Heat Mass Transf 108:1078–1088. https://doi.org/10.1016/j.ijheatmasstransfer.2016.12.096

    Article  Google Scholar 

  19. Liu R, Jiang Y, Li B, Wang X (2015) A fractal model for characterizing fluid flow in fractured rock masses based on randomly distributed rock fracture networks. Comput Geotech 65:45–55. https://doi.org/10.1016/j.compgeo.2014.11.004

    Article  Google Scholar 

  20. Xie J, Gao M, Zhang R et al (2020) Experimental investigation on the anisotropic fractal characteristics of the rock fracture surface and its application on the fluid flow description. J Pet Sci Eng 191:107190. https://doi.org/10.1016/j.petrol.2020.107190

    Article  Google Scholar 

  21. Yin P, Zhao C, Ma J et al (2020) Experimental study of non-linear fluid flow though rough fracture based on fractal theory and 3D printing technique. Int J Rock Mech Min Sci 129:104293. https://doi.org/10.1016/j.ijrmms.2020.104293

    Article  Google Scholar 

  22. Qi H, Guo X (2014) Transient fractional heat conduction with generalized Cattaneo model. Int J Heat Mass Transf 76:535–539. https://doi.org/10.1016/j.ijheatmasstransfer.2013.12.086

    Article  Google Scholar 

  23. Liu L, Zheng L, Liu F (2018) Research on macroscopic and microscopic heat transfer mechanisms based on non-Fourier constitutive model. Int J Heat Mass Transf 127:165–172. https://doi.org/10.1016/j.ijheatmasstransfer.2018.06.011

    Article  Google Scholar 

  24. Liu L, Feng L, Xu Q et al (2020) Flow and heat transfer of generalized Maxwell fluid over a moving plate with distributed order time fractional constitutive models. Int Commun Heat Mass Transf 116:104679. https://doi.org/10.1016/j.icheatmasstransfer.2020.104679

    Article  Google Scholar 

  25. Palombo M, Gabrielli A, Servedio VDP et al (2013) Structural disorder and anomalous diffusion in random packing of spheres. Sci Rep 3:2631. https://doi.org/10.1038/srep02631

    Article  Google Scholar 

  26. Molina-Garcia D, Sandev T, Safdari H et al (2018) Crossover from anomalous to normal diffusion: truncated power-law noise correlations and applications to dynamics in lipid bilayers. New J Phys 20:103027. https://doi.org/10.1088/1367-2630/aae4b2

    Article  Google Scholar 

  27. Schieber GL, Jones BM, Orlando TM, Loutzenhiser PG (2020) Advection diffusion model for gas transport within a packed bed of JSC-1A regolith simulant. Acta Astronaut 169:32–39. https://doi.org/10.1016/j.actaastro.2019.12.031

    Article  Google Scholar 

  28. Jannelli A, Ruggieri M, Speciale MP (2018) Exact and numerical solutions of time-fractional advection–diffusion equation with a nonlinear source term by means of the Lie symmetries. Nonlinear Dyn 92:543–555. https://doi.org/10.1007/s11071-018-4074-8

    Article  MATH  Google Scholar 

  29. Mojtabi A, Deville MO (2015) One-dimensional linear advection–diffusion equation: analytical and finite element solutions. Comput Fluids 107:189–195. https://doi.org/10.1016/j.compfluid.2014.11.006

    Article  MathSciNet  MATH  Google Scholar 

  30. Zel’dovich YB, Myshkis AD, (1973) Elements of mathematical physics. Nauka Publishing House, Moscow (in Russian)

    Google Scholar 

  31. Weberszpil J, Lazo MJ, Helayël-Neto JA (2015) On a connection between a class of q-deformed algebras and the Hausdorff derivative in a medium with fractal metric. Phys A Stat Mech its Appl 436:399–404. https://doi.org/10.1016/j.physa.2015.05.063

    Article  MathSciNet  MATH  Google Scholar 

  32. Zhokh A, Strizhak P (2017) Non-Fickian diffusion of methanol in mesoporous media: geometrical restrictions or adsorption-induced? J Chem Phys 146:124704. https://doi.org/10.1063/1.4978944

    Article  Google Scholar 

  33. Schloemer S, Krooss BM (2004) Molecular transport of methane, ethane and nitrogen and the influence of diffusion on the chemical and isotopic composition of natural gas accumulations. Geofluids 4:81–108. https://doi.org/10.1111/j.1468-8123.2004.00076.x

    Article  Google Scholar 

  34. Dong J, Cheng Y, Jiang J, Guo P (2020) Effects of tectonism on the pore characteristics and methane diffusion coefficient of coal. Arab J Geosci 13:1–10. https://doi.org/10.1007/s12517-020-05475-8

    Article  Google Scholar 

  35. Datema KP, Den Ouden CJJ, Ylstra WD et al (1991) Fourier-transform pulsed-field-gradient 1H nuclear magnetic resonance investigation of the diffusion of light n-alkanes in zeolite ZSM-5. J Chem Soc Faraday Trans 87:1935–1943. https://doi.org/10.1039/FT9918701935

    Article  Google Scholar 

  36. Raghavan R, Chen CC (2020) A study in fractional diffusion: Fractured rocks produced through horizontal wells with multiple, hydraulic fractures. Oil Gas Sci Technol 75:68. https://doi.org/10.2516/ogst/2020062

    Article  Google Scholar 

  37. Langtangen HP, Pedersen GK (2016) Basic partial differential equation models. In: Scaling of differential equations. Simula SpringerBriefs on Computing, vol 2. Springer, Cham. https://doi.org/10.1007/978-3-319-32726-6_3

    Chapter  Google Scholar 

  38. Fagan WF, Hoffman T, Dahiya D et al (2020) Improved foraging by switching between diffusion and advection: benefits from movement that depends on spatial context. Theor Ecol 13:127–136. https://doi.org/10.1007/s12080-019-00434-w

    Article  Google Scholar 

  39. Vilquin A, Bertin V, Soulard P et al (2020) Time dependence of advection-diffusion coupling for nanoparticle ensembles. Arxiv preprint. arXiv:2007.08261

  40. LaBolle EM, Quastel J, Fogg GE (1998) Diffusion theory for transport in porous media: transition-probability densities of diffusion processes corresponding to advection-dispersion equations. Water Resour Res 34:1685–1693. https://doi.org/10.1029/98WR00319

    Article  Google Scholar 

  41. Ito K, Miyazaki S (2003) Crossover between anomalous superdiffusion and normal diffusion in oscillating convection flows. Prog Theor Phys 110:875–887. https://doi.org/10.1143/PTP.110.875

    Article  MATH  Google Scholar 

  42. Zheng L, Wang L, James SC (2019) When can the local advection–dispersion equation simulate non-Fickian transport through rough fractures? Stoch Environ Res Risk Assess 33:931–938. https://doi.org/10.1007/s00477-019-01661-7

    Article  Google Scholar 

  43. Muralidhar R, Ramkrishna D (1993) Diffusion in pore fractals: a review of linear response models. Transp Porous Media 13:79–95. https://doi.org/10.1007/BF00613271

    Article  Google Scholar 

  44. Nizkaya TV, Asmolov ES, Vinogradova OI (2017) Advective superdiffusion in superhydrophobic microchannels. Phys Rev E 96:033109. https://doi.org/10.1103/PhysRevE.96.033109

    Article  Google Scholar 

  45. ten Elshof JE, Abadal CR, Sekulić J et al (2003) Transport mechanisms of water and organic solvents through microporous silica in the pervaporation of binary liquids. Microporous Mesoporous Mater 65:197–208

    Article  Google Scholar 

  46. Aguilar-Armenta G, Patino-Iglesias ME, Leyva-Ramos R (2003) Adsorption kinetic behaviour of pure CO2, N2 and CH4 in natural clinoptilolite at different temperatures. Adsorpt Sci Technol 21:81–92. https://doi.org/10.1260/02636170360699831

    Article  Google Scholar 

  47. Haase F, Sauer J (1995) Interaction of methanol with Broensted acid sites of zeolite catalysts: an ab initio study. J Am Chem Soc 117:3780–3789. https://doi.org/10.1021/ja00118a014

    Article  Google Scholar 

  48. Zamani M, Dabbagh HA (2014) Adsorption behavior of the primary, secondary and tertiary Alkyl, Allyl and Aryl Alcohols over nanoscale (1 0 0) surface of γ-Alumina. J Nanoanalysis 1:21–30

    Google Scholar 

  49. Zhang Y, Yu JY, Yeh YH et al (2015) An adsorption study of CH4 on ZSM-5, MOR, and ZSM-12 zeolites. J Phys Chem C 119:28970–28978. https://doi.org/10.1021/acs.jpcc.5b09571

    Article  Google Scholar 

  50. Sawilowsky EF, Meroueh O, Schlegel HB, Hase WL (2000) Structures, energies, and electrostatics for methane coniplexed with alumina clusters. J Phys Chem A 104:4920–4927. https://doi.org/10.1021/jp9926084

    Article  Google Scholar 

Download references

Acknowledgments

This research was funded by the National Research Foundation of Ukraine (grant 2020.02/0050).

Author information

Authors and Affiliations

  1. L.V. Pisarzhevskii Institute of Physical Chemistry of National Academy of Sciences of Ukraine, Prospekt Nauki, 31, Kiev, 03028, Ukraine

    Alexey Zhokh & Peter Strizhak

Authors
  1. Alexey Zhokh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Peter Strizhak
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alexey Zhokh.

Ethics declarations

Conflict of interest

The authors declare no conflict of interests.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhokh, A., Strizhak, P. Advection–diffusion in a porous medium with fractal geometry: fractional transport and crossovers on time scales. Meccanica 57, 833–843 (2022). https://doi.org/10.1007/s11012-021-01353-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11012-021-01353-z

Keywords

Search

Navigation