Transport in Porous Media

, Volume 121, Issue 2, pp 315–332 | Cite as

Effect of Anisotropy Structure on Plume Entropy and Reactive Mixing in Helical Flows

  • Yu Ye
  • Gabriele Chiogna
  • Chunhui LuEmail author
  • Massimo Rolle


Plume dilution and reactive mixing can be considerably enhanced by helical flows occurring in three-dimensional anisotropic porous media. In this study, we perform conservative and reactive transport simulations considering different anisotropy structures of a single inclusion with the objective of exploring the effect of the inclusion’s geometry and orientation on the patterns of twisted streamlines and on the overall dilution and reaction of solute plumes. We analyzed 100 different scenarios by varying key parameters such as the angle of the anisotropic structures with respect to the average flow velocity, the spacing between alternated heterogeneous zones of coarse and fine materials, the permeability contrast between such matrices, and the magnitude of the seepage velocity. Entropy conservation equations and entropy-based metrics for both conservative and reactive species were adopted to quantify dilution, reactive mixing and their interactions with the helical flow patterns in the considered three-dimensional anisotropic setups. The results allowed identifying optimal anisotropic configurations maximizing mixing and reactions, and yielding enhancement factors up to 15 times the outcomes of analogous simulations in homogeneous media. Furthermore, the effects of compound-specific diffusive/dispersive properties of the transported species were found to be relevant for both plume dilution and reactive mixing in helical flows.


Anisotropy Helical flow Entropy Dilution Reactive mixing 



This study was supported by the National Natural Science Foundation of China (51709085). Y. Ye acknowledges the support of “The Fundamental Research Funds for the Central Universities” (2017B00214) and a project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions. G. Chiogna acknowledges the support of the Stiftungsfonds für Umweltökonomie und Nachhaltigkeit GmbH (SUN). C. Lu acknowledges the support of the National Natural Science Foundation of China (51679067) and the “111 Project” (B17015), Ministry of Education and State Administration of Foreign Experts Affairs, P. R. China. M. Rolle acknowledges the support of the Danish Council for Independent Research (DFF) and of the Sino-Danish Center. The authors would like to thank Prof. O. A. Cirpka for discussion on helical flows and for providing an earlier version of the code that has been used in this study.


  1. Acharya, R.C., Valocchi, A.J., Werth, C.J., Willingham, T.W.: Pore-scale simulation of dispersion and reaction along a transverse mixing zone in two-dimensional porous media. Water Resour. Res. 43, W10435 (2007)CrossRefGoogle Scholar
  2. Amos, R.T., Berkins, B.A., Delin, G.N., Cozzarelli, I.M., Blowes, D.W., Kirshtein, J.D.: Methane oxidation in a crude oil contaminated aquifer: delineation of aerobic reactions at the plume fringes. J. Contam. Hydrol. 125, 13–25 (2011)CrossRefGoogle Scholar
  3. Aquino, T., Bolster, D.: Localized point mixing rate potential in heterogeneous velocity fields. Transp. Porous Media 119, 391–402 (2017)CrossRefGoogle Scholar
  4. Ballarini, E., Bauer, S., Eberhardt, C., Beyer, C.: Evaluation of the role of heterogeneities on transverse mixing in bench-scale tank experiments by numerical modeling. Ground Water 52, 368–377 (2013)CrossRefGoogle Scholar
  5. Bauer, R.D., Rolle, M., Bauer, S., Eberhardt, C., Grathwohl, P., Kolditz, O., Meckenstock, R.U., Griebler, C.: Enhanced biodegradation by hydraulic heterogeneities in petroleum hydrocarbon plumes. J. Contam. Hydrol. 105, 56–68 (2009)CrossRefGoogle Scholar
  6. Beckie, R.D.: Analysis of scale effects in large-scale solute transport models. In: Sposito, G. (ed.) Scale Dependence and Scale Invariance in Hydrology, pp. 314–334. Cambridge University Press, New York (1998)CrossRefGoogle Scholar
  7. Bennett, J.P., Haslauer, C.P., Cirpka, O.A.: The impact of sedimentary anisotropy on solute mixing in stacked scour-pool structures. Water Resour. Res. 53, 2813–2832 (2017)CrossRefGoogle Scholar
  8. Bianchi, M., Pedretti, D.: Geological entropy and solute transport in heterogeneous porous media. Water Resour. Res. 53, 4691–4708 (2017)CrossRefGoogle Scholar
  9. Cabezas, H., Karunanithi, A.T.: Fisher information, entropy, and the second and third laws of thermodynamics. Ind. Eng. Chem. Res. 47, 5243–5249 (2008)CrossRefGoogle Scholar
  10. Castro-Alcala, E., Fernandez-Garcia, D., Carrera, J., Bolster, D.: Visualization of mixing processes in a heterogeneous sand box aquifer. Environ. Sci. Technol. 46, 3228–3235 (2012)CrossRefGoogle Scholar
  11. Chiogna, G., Eberhardt, C., Grathwohl, P., Cirpka, O.A., Rolle, M.: Evidence of compound-dependent hydrodynamic and mechanical transverse dispersion by multitracer laboratory experiments. Environ. Sci. Technol. 44, 688–693 (2010)CrossRefGoogle Scholar
  12. Chiogna, G., Cirpka, O.A., Grathwohl, P., Rolle, M.: Transverse mixing of conservative and reactive tracers in porous media: quantification through the concepts of flux-related and critical dilution indices. Water Resour. Res. 47, W02505 (2011)Google Scholar
  13. Chiogna, G., Hochstetler, D.L., Bellin, A., Kitanidis, P.K., Rolle, M.: Mixing, entropy and reactive solute transport. Geophys. Rev. Lett. 39, L20405 (2012)CrossRefGoogle Scholar
  14. Chiogna, G., Rolle, M., Alberto, B., Cirpka, O.A.: Helicity and flow topology in three-dimensional anisotropic porous media. Adv. Water Resour. 73, 134–143 (2014)CrossRefGoogle Scholar
  15. Chiogna, G., Cirpka, O.A., Rolle, M., Alberto, B.: Helical flow in three-dimensional nonstationary anisotropic heterogeneous porous media. Water Resour. Res. 51, 261–280 (2015)CrossRefGoogle Scholar
  16. Chiogna, G., Cirpka, O.A., Herrera, P.A.: Helical flow and transient solute dilution in porous media. Transp. Porous Media 111, 591–603 (2016)CrossRefGoogle Scholar
  17. Cirpka, O.A., Valocchi, A.J.: Two-dimensional concentration distribution for mixing-controlled bioreactive transport in steady state. Adv. Water Resour. 30, 1668–1679 (2007)CrossRefGoogle Scholar
  18. Cirpka, O.A., de Barros, F.P.J., Chiogna, G., Rolle, M., Nowak, W.: Stochastic flux-related analysis of transverse mixing in two-dimensional heterogeneous porous media. Water Resour. Res. 47, W06515 (2011)Google Scholar
  19. Cirpka, O.A., Rolle, M., Chiogna, G., de Barros, F.P., Nowak, W.: Stochastic evaluation of mixing-controlled steady-state plume lengths in two-dimensional heterogeneous domains. J. Contam. Hydrol. 138–139, 22–39 (2012)CrossRefGoogle Scholar
  20. Cirpka, O.A., Chiogna, G., Rolle, M., Bellin, A.: Transverse mixing in three-dimensional non-stationary anisotropic heterogeneous porous media. Water Resour. Res. 51, 241–260 (2015)CrossRefGoogle Scholar
  21. Crevacore, E., Tosco, T., Sethi, R., Boccardo, G., Marchisio, D.L.: Recirculation zones induce non-Fickian transport in three-dimensional periodic porous media. Phys. Rev. E 94, 053118 (2016)CrossRefGoogle Scholar
  22. de Anna, P., Jimenez-Martinez, J., Tabuteau, H., Turuban, R., Le Borgne, T., Derrien, M., Meheust, Y.: Mixing and reaction kinetics in porous media: an experimental pore scale quantification. Environ. Sci. Technol. 48, 508–516 (2014)CrossRefGoogle Scholar
  23. de Barros, F.P.J., Fiori, A., Boso, F., Bellin, A.: A theoretical framework for modeling dilution enhancement of non-reactive solutes in heterogeneous porous media. J. Contam. Hydrol. 175, 72–83 (2015)CrossRefGoogle Scholar
  24. de Dreuzy, J.R., Carrera, J., Dentz, M., Le Borgne, T.: Time evolution of mixing in heterogeneous porous media. Water Resour. Res. 48, W06511 (2012)Google Scholar
  25. Dentz, M., Le Borgne, T., Englert, A., Bijeljic, B.: Mixing, spreading and reaction in heterogeneous media: a brief review. J. Contam. Hydrol. 120–121, 1–17 (2011)CrossRefGoogle Scholar
  26. Di Dato, M., de Barros, F.P.J., Fiori, A., Bellin, A.: Effects of the hydraulic conductivity microstructure on macrodispersivity. Water Resour. Res. 52, 6818–6832 (2016a)CrossRefGoogle Scholar
  27. Di Dato, M., Fiori, A., Chiogna, G., de Barros, F.P.J., Bellin, A.: Impact of the spatial structure of the hydraulic conductivity field on vorticity in three-dimensional flows. In: Proceedings of the Royal Society, vol. 472. The Royal Society (2016b)Google Scholar
  28. Fox, D.T., Guo, L., Fujita, Y., Huang, H., Redden, G.: Experimental and numerical analysis of parallel reactant flow and transverse mixing with mineral precipitation in homogeneous and heterogeenosu porous media. Transp. Porous Media 111, 605–626 (2016)CrossRefGoogle Scholar
  29. Hazen, A.: Some physical properties of sands and gravels with special reference to their use in filtration. Annu. Rep. State Board Health Mass. 24, 541–556 (1892)Google Scholar
  30. Hemker, K., Baker, M.: Analytical solutions for whirling groundwater flow in two-dimensional heterogeneous anisotropic aquifers. Water Resour. Res. 42, W12419 (2006)CrossRefGoogle Scholar
  31. Hemker, K., van den Berg, E., Bakker, M.: Ground water whirls. Ground Water 42, 234–242 (2004)CrossRefGoogle Scholar
  32. Heinz, J., Kleineidam, S., Teutsch, G., Aigner, T.: Heterogeneity patterns of quaternary glaciofluvial gravel bodies (SW-Germany): application to hydrogeology. Sediment. Geol. 158, 1–23 (2003)CrossRefGoogle Scholar
  33. Herrera, P.A., Valocchi, A.J., Beckie, R.D.: A multidimensional streamline-based method to simulate reactive solute transport in heterogeneous porous media. Adv. Water Resour. 33, 711–727 (2010)CrossRefGoogle Scholar
  34. Hochstetler, D.L., Rolle, M., Chiogna, G., Haberer, C.M., Grathwohl, P., Kitanidis, P.K.: Effects of compound-specific transverse mixing on steady-state reactive plumes: insights from pore-scale simulations and Darcy-scale experiments. Adv. Water Resour. 54, 1–10 (2013)CrossRefGoogle Scholar
  35. Icardi, M., Boccardo, G., Marchisio, D., Tosco, T., Sethi, R.: Pore-scale simulation of fuild flow and solute dispersion in three-dimensional porous media. Phys. Rev. E 90, 013032 (2014)CrossRefGoogle Scholar
  36. Jiménez-Martínez, J., de Anna, P., Tabuteau, H., Turuban, R., Le Borgne, T., Méheust, Y.: Pore-scale mechanisms for the enhancement of mixing in unsaturated porous media and implications for chemical reactions. Geophys. Res. Lett. 42, 5316–5324 (2015)CrossRefGoogle Scholar
  37. Kapoor, V., Kitanidis, P.K.: Concentration fluctuations and dilution in two-dimensionally periodic heterogeneous porous media. Transp. Porous Media 22, 91–119 (1996)CrossRefGoogle Scholar
  38. Kitanidis, P.K.: The concept of dilution index. Water Resour. Res. 30, 2011–2026 (1994)CrossRefGoogle Scholar
  39. Liedl, R., Valocchi, A.J., Dietrich, P., Grathwohl, P.: Finiteness of steady state plumes. Water Resour. Res. 41, 3923–3929 (2005)CrossRefGoogle Scholar
  40. Liedl, R., Yadav, P.K., Dietrich, P.: Length of 3-D mixing-controlled plumes for a fully penetrating contaminant source with finite width. Water Resour. Res. 47, W08602 (2011)CrossRefGoogle Scholar
  41. Muniruzzaman, M., Haberer, C.H., Grathwohl, P., Rolle, M.: Multicomponent ionic dispersion during transport of elctrolytes in heterogeneous porous media: experiments and model-based interpretation. Geochim. Cosmochim. Acta 141, 656–669 (2014)CrossRefGoogle Scholar
  42. Ottino, J.M.: The Kinematics of Mixing. Cambridge University, Cambridge (1989)Google Scholar
  43. Paster, A., Aquino, T., Bolster, D.: Incomplete mixing and reactions in laminar shear flow. Phys. Rev. E 92, 012922 (2015)CrossRefGoogle Scholar
  44. Pedretti, D., Fernàndez-Garcia, D., Sanchez-Vila, X., Bolster, D., Benson, D.A.: Apparent directional mass-transfer capacity coefficients in three-dimensional anisotropic heterogeneous aquifers under radical convergent transport. Water Resour. Res. 50, 1205–1224 (2014)CrossRefGoogle Scholar
  45. Pollock, D.W.: Semianalytical computation of path lines for finite-difference models. Ground Water 26, 743–750 (1988)CrossRefGoogle Scholar
  46. Prommer, H., Anneser, B., Rolle, M., Einsiedl, F., Griebler, C.: Biogeochemical and isotopic gradients in a BTEX/PAH contaminant plume: model-based interpretation of a high-resolution field data set. Environ. Sci. Technol. 43, 8206–8212 (2009)CrossRefGoogle Scholar
  47. Rolle, M., Eberhardt, C., Chiogna, G., Cirpka, O.A., Grathwohl, P.: Enhancement of dilution and transverse reactive mixing in porous media: experiments and model-based interpretation. J. Contam. Hydrol. 110, 130–142 (2009)CrossRefGoogle Scholar
  48. Rolle, M., Hochstetler, D., Chiogna, G., Kitanidis, P.K., Grathwohl, P.: Experimental investigation and pore-scale modeling interpretation of compound-specific transverse dispersion in porous media. Transp. Porous Media 93, 347–362 (2012)CrossRefGoogle Scholar
  49. Rolle, M., Chiogna, G., Hochstetler, D.L., Kitanidis, P.: On the importance of diffusion and compound-specific mixing for groundwater transport: an investigation from pore to field scale. J. Contam. Hydrol. 153, 51–68 (2013a)CrossRefGoogle Scholar
  50. Rolle, M., Muniruzzaman, M., Haberer, C.M., Grathwohl, P.: Coulombic effects in advection-dominated transport of electrolytes in porous media: multicomponent ionic dispersion. Geochim. Cosmochim. Acta 120, 195–205 (2013b)CrossRefGoogle Scholar
  51. Rolle, M., Kitanidis, P.K.: Effects of compound-specific dilution on transient transport and solute breakthough: a pore-scale analysis. Adv. Water Resour. 71, 186–199 (2014)CrossRefGoogle Scholar
  52. Sanchez-Vila, X., Guadagnini, A., Carrera, J.: Representative hydraulic conductivities in saturated groundwater flow. Rev. Geophys. 44, 535–540 (2006)CrossRefGoogle Scholar
  53. Staufer, F.: Impact of highly permeable sediment units with inclined bedding on solute transport in aquifers. Adv. Water Resour. 30, 2194–2201 (2007)CrossRefGoogle Scholar
  54. Singh, V.P.: The use of entropy on hydrology and water resoureces. Hydrol. Process. 11, 587–626 (1997)CrossRefGoogle Scholar
  55. Stroock, A.D., Dertinger, S.K.W., Ajdari, A., Mezić, I., Stone, H.A., Whitesides, G.M.: Chaotic mixer for microchannels. Science 295, 647–651 (2002)CrossRefGoogle Scholar
  56. Tartakovsky, A.M., Tartakovsky, G.D., Scheibe, T.D.: Effects of incomplete mixing on multicomponent reactive transport. Adv. Water Resour. 32, 1674–1679 (2009)CrossRefGoogle Scholar
  57. Theis, C.V.: Aquifers and models. In: Symposium on Ground-Water Hydrology, San Francisco, California, vol. 4, pp. 138–148. American Water Resources Association Proc. Ser. (1967)Google Scholar
  58. Thierrin, J., Kitanidis, P.K.: Solute dilution at the Borden and Cape Cod groundwater tracer tests. Water Resour. Res. 30, 2883–2890 (1994)CrossRefGoogle Scholar
  59. Tuxen, N., Albrechtsen, H., Bjerg, P.L.: Identification of a reactive degradation zone at a landfill leachate plume fringe using high resolution sampling and incubation techniques. J. Contam. Hydrol. 85, 179–194 (2006)CrossRefGoogle Scholar
  60. Ursino, N.: Modeling media with oriented structures. Transp. Porous Media 55, 137–151 (2004)CrossRefGoogle Scholar
  61. Ursino, N., Gimmi, T., Flühler, H.: Combined effects of heterogeneity, anisotripy, and saturation on steady state flow and transport: a laboratory sand tank experiment. Water Resour. Res. 37, 201–208 (2001)CrossRefGoogle Scholar
  62. Weiss, J.B., Provenzale, A.: Transport and Mixing in Geophysical Flows. Springer, Berlin (2008)CrossRefGoogle Scholar
  63. Werth, C.J., Cirpka, O.A., Grathwohl, P.: Enhanced mixing and reaction through flow focusing in heterogeneous porous media. Water Resour. Res. 42, W12414 (2006)CrossRefGoogle Scholar
  64. Willingham, T.W., Werth, C.J., Valocchi, A.J.: Evaluation of the effects of porous media structure on mixing-controlled reactions using pore-scale modeling and micromodel experiments. Environ. Sci. Technol. 42, 3185–3193 (2008)CrossRefGoogle Scholar
  65. Ye, Y., Chiogna, G., Cirpka, O.A., Grathwohl, P., Rolle, M.: Enhancement of plume dilution in two-dimensional and three-dimensional porous media by flow focusing in high-permeability inclusions. Water Resour. Res. 51, 5582–5602 (2015a)CrossRefGoogle Scholar
  66. Ye, Y., Chiogna, G., Cirpka, O.A., Grathwohl, P., Rolle, M.: Experimental evidence of helical flow in porous media. Phys. Rev. Lett. 115, 194502 (2015b)CrossRefGoogle Scholar
  67. Ye, Y., Chiogna, G., Cirpka, O.A., Grathwohl, P., Rolle, M.: Experimental investigation of compound-specific dilution of solute plumes in saturated porous media: 2-D vs. 3-D flow-through systems. J. Contam. Hydrol. 172, 33–47 (2015c)CrossRefGoogle Scholar
  68. Ye, Y., Chiogna, G., Cirpka, O.A., Grathwohl, P., Rolle, M.: Experimental investigation of transverse mixing in porous media under helical flow conditions. Phys. Rev. E 94, 013113 (2016)CrossRefGoogle Scholar
  69. Zarlenga, A., Fiori, A.: Advective transport through three-dimensional anisotropic formations of bimodal hydraulic conductivity. Transp. Porous Media 107, 573–593 (2015)CrossRefGoogle Scholar
  70. Zarlenga, A., Janković, I., Fiori, A.: Advective transport in heterogeneous formations: the impact of spatial anisotropy on the breakthrough curve. Transp. Porous Media 96, 295–304 (2013)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2017

Authors and Affiliations

  • Yu Ye
    • 1
  • Gabriele Chiogna
    • 2
    • 3
  • Chunhui Lu
    • 1
    Email author
  • Massimo Rolle
    • 4
  1. 1.State Key Laboratory of Hydrology-Water Resources and Hydraulic EngineeringHohai UniversityNanjingChina
  2. 2.Faculty of Civil, Geo and Environmental EngineeringTechnical University of MunichMunichGermany
  3. 3.Institute of GeographyUniversity of InnsbruckInnsbruckAustria
  4. 4.Department of Environmental EngineeringTechnical University of DenmarkLyngbyDenmark

Personalised recommendations