Advertisement

Transport in Porous Media

, Volume 130, Issue 3, pp 903–922 | Cite as

Hydrodynamic Dispersion and Lamb Surfaces in Darcy Flow

  • Daniel R. LesterEmail author
  • Aditya Bandopadhyay
  • Marco Dentz
  • Tanguy Le Borgne
Article
  • 85 Downloads

Abstract

Transport processes such as the dispersion and mixing of solutes are governed by the interplay of advection and diffusion, where advection acts to organise fluid streamlines and diffusion acts to randomise solute molecules. Thus, the structure and organisation of streamlines, termed the Lagrangian kinematics of the flow, is central to the understanding and modelling of these transport processes. A key question is whether the streamlines in three-dimensional (3D) Darcy flows can wander freely through the fluid domain, or whether all streamlines of the flow are organised into a series of smooth, non-intersecting two-dimensional (2D) surfaces. The existence of such a foliation of surfaces constrains the Lagrangian kinematics in a manner similar to that of 2D flows, which in turn constrains the allowable transport processes. In a series of pioneering studies, Sposito (Water Resour. Res., 30(8):2395–2401, 1994; Adv. Water Resour., 24(7):793–801, 2001) argues that steady Darcy flow in locally isotropic media gives rise to Lamb surfaces, 2D material surfaces which are spanned by both the streamlines and vortex lines (field lines of the vorticity vector) of the flow. Hence, the existence of these surfaces renders the kinematics of such 3D steady Darcy flow as two dimensions. This topological constraint strongly affects transverse mixing and dispersion because 2D steady flow fields limit the rate of deformation of fluid elements and can only admit zero hydrodynamic transverse dispersion. In this study, however, we show that Lamb surfaces are not ubiquitous to all steady Darcy flows in locally isotropic media. We derive the conditions for when Lamb surfaces exist in such Darcy flows, and discuss the implications of these findings for the transport, mixing, and dispersion of solutes.

Keywords

Darcy flow Fluid deformation Hydrodynamic dispersion 

Notes

References

  1. Aref, H.: Stirring by chaotic advection. J. Fluid Mech. 143, 1–21 (1984)CrossRefGoogle Scholar
  2. Arnol’d, V.I.: Sur la topologie des écoulments stationnaires des fluids parfaits. Comptes Rendus Acad. Sci. Paris 261, 312–314 (1965)Google Scholar
  3. Attinger, S., Dentz, M., Kinzelbach, W.: Exact transverse macro dispersion coefficients for transport in heterogeneous porous media. Stoch Environ Res Risk Assess 18(1), 9–15 (2004)CrossRefGoogle Scholar
  4. Balay, S., Abhyankar, S., Adams, M.F., Brown, J., Brune, P., Buschelman, K., Dalcin, L., Eijkhout, V., Gropp, W.D., Kaushik, D., Knepley, M.G., McInnes, L.C., Rupp, K., Smith, B.F., Zampini, S., Zhang, H.: PETSc Web page. http://www.mcs.anl.gov/petsc (2015)
  5. Batchelor, G.K.: An Introduction to Fluid Dynamics. Cambridge Mathematical Library. Cambridge University Press, Cambridge (2000)CrossRefGoogle Scholar
  6. Bear, J.: Dynamics of Fluids in Porous Media No. 1 in Dover Classics of Science and Mathematics. Dover, Mineola (1972)Google Scholar
  7. Boyland, P.L., Aref, H., Stremler, M.A.: Topological fluid mechanics of stirring. J Fluid Mech 403, 277–304 (2000)CrossRefGoogle Scholar
  8. Brand, L.: Vector and Tensor Analysis. Wiley, Hoboken (1947)Google Scholar
  9. Cho, M.S., Solano, F., Thomson, N.R., Trefry, M.G., Lester, D.R., Metcalfe, G.: Field trials of chaotic advection to enhance reagent delivery. Groundw. Monit. Remediat. 39(3), 23–39 (2019).  https://doi.org/10.1111/gwmr.12339 CrossRefGoogle Scholar
  10. Danckwerts, P.: The effect of incomplete mixing on homogeneous reactions. Chem. Eng. Sci. 8(1), 93–102 (1958)CrossRefGoogle Scholar
  11. Dentz, M., Carrera, J.: Effective dispersion in temporally fluctuating flow through a heterogeneous medium. Phys. Rev. E 68, 036310 (2003)CrossRefGoogle Scholar
  12. Dentz, M., LeBorgne, T., Englert, A., Bijeljic, B.: Mixing, spreading and reaction in heterogeneous media: a brief review. J. Contam. Hydrol. 120–121, 1–17 (2011).  https://doi.org/10.1016/j.jconhyd.2010.05.002 CrossRefGoogle Scholar
  13. Dentz, M., Lester, D.R., Borgne, T.L., de Barros, F.P.J.: Deformation in steady random flow is a Lévy walk. Phys. Rev. E 94, 061102 (2016)CrossRefGoogle Scholar
  14. Hénon, M.: Sur la topologie des lignes de courant dans un cas particulier. Comptes Rendus Acad. Sci. Paris 262, 312–314 (1966)Google Scholar
  15. Kelvin, W.: Papers on Electrostatics and Magnetism. Macmillan & Company, London (1884)Google Scholar
  16. Kozlov, V.: Notes on steady vortex motions of continuous medium. J. Appl. Math. Mech. 47(2), 288–289 (1983)CrossRefGoogle Scholar
  17. Lamb, H.: Hydrodynamics. The University Press, Cambridge (1932)Google Scholar
  18. Le Borgne, T., Dentz, M., Villermaux, E.: Stretching, coalescence, and mixing in porous media. Phys. Rev. Lett. 110, 204501 (2013)CrossRefGoogle Scholar
  19. Le Borgne, T., Dentz, M., Villermaux, E.: The lamellar description of mixing in porous media. J. Fluid Mech. 770, 458–498 (2015)CrossRefGoogle Scholar
  20. Lester, D., Trefry, M., Metcalfe, G.: Chaotic advection at the pore scale: Mechanisms, upscaling and implications for macroscopic transport. Adv. Water Resour. 97, 175–192 (2016)CrossRefGoogle Scholar
  21. Lester, D.R., Dentz, M., Borgne, T.L., Barros, F.P.J.D.: Fluid deformation in random steady three-dimensional flow. J. Fluid Mech. 855, 770–803 (2018)CrossRefGoogle Scholar
  22. Lester, D.R., Dentz, M., Le Borgne, T.: Chaotic mixing in three-dimensional porous media. J. Fluid Mech. 803, 144–174 (2016)CrossRefGoogle Scholar
  23. Lester, D.R., Metcalfe, G., Trefry, M.G.: Is chaotic advection inherent to porous media flow? Phys. Rev. Lett. 111, 174101 (2013)CrossRefGoogle Scholar
  24. Lester, D.R., Metcalfe, G., Trefry, M.G.: Anomalous transport and chaotic advection in homogeneous porous media. Phys. Rev. E 90, 063012 (2014)CrossRefGoogle Scholar
  25. Lester, D.R., Rudman, M., Metcalfe, G., Trefry, M.G., Ord, A., Hobbs, B.: Scalar dispersion in a periodically reoriented potential flow: acceleration via Lagrangian chaos. Phys. Rev. E 81, 046319 (2010)CrossRefGoogle Scholar
  26. Metcalfe, G., Lester, D., Ord, A., Kulkarni, P., Trefry, M., Hobbs, B.E., Regenaur-Lieb, K., Morris, J.: A partially open porous media flow with chaotic advection: towards a model of coupled fields. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. 368(1910), 217–230 (2010)CrossRefGoogle Scholar
  27. Moffatt, H.K.: The degree of knottedness of tangled vortex lines. J. Fluid Mech. 1, 117–129 (1969)CrossRefGoogle Scholar
  28. Ottino, J.M.: The Kinematics of Mixing: Stretching, Chaos, and Transport. Cambridge University Press, Cambridge (1989)Google Scholar
  29. Piaggio, H.: An Elementary Treatise on Differential Equations and Their Applications. Bell’s mathematical series : Advanced section. G. Bell and Sons, London (1952)Google Scholar
  30. Poincaré, H.: Théorie des tourbillons. Réimpressions (Editions Jacques Gabay). Jacques Gabay, Paris (1893)Google Scholar
  31. Sposito, G.: Steady groundwater flow as a dynamical system. Water Resour. Res. 30(8), 2395–2401 (1994)CrossRefGoogle Scholar
  32. Sposito, G.: On steady flows with Lamb surfaces. Int. J. Eng. Sci. 35(3), 197–209 (1997)CrossRefGoogle Scholar
  33. Sposito, G.: A note on helicity conservation in steady fluid flows. J. Fluid Mech. 363, 325–332 (1998)CrossRefGoogle Scholar
  34. Sposito, G.: Topological groundwater hydrodynamics. Adv. Water Resour. 24(7), 793–801 (2001)CrossRefGoogle Scholar
  35. Sposito, G., Weeks, S.W.: Tracer advection by steady groundwater flow in a stratified aquifer. Water Resour. Res. 34(5), 1051–1059 (1998)CrossRefGoogle Scholar
  36. Trefry, M.G., Lester, D.R., Metcalfe, G., Ord, A., Regenauer-Lieb, K.: Toward enhanced subsurface intervention methods using chaotic advection. J. Contam. Hydrol. 127(1–4), 15–29 (2012)CrossRefGoogle Scholar
  37. Trefry, M.G., Lester, D.R., Metcalfe, G., Wu, J.: Temporal fluctuations and poroelasticity can generate chaotic advection in natural groundwater systems. Water Resour. Res. 55(4), 3347–3374 (2019)CrossRefGoogle Scholar
  38. Truesdell, C.: The Kinematics of Vorticity. Indiana University Publications: Science Series. Indiana University Press, Indiana (1954)Google Scholar
  39. Villermaux, E.: Mixing by porous media. Comptes Rendus Mécanique 340(11–12), 933–943 (2012)CrossRefGoogle Scholar
  40. Ye, Y., Chiogna, G., Cirpka, O.A., Grathwohl, P., Rolle, M.: Experimental evidence of helical flow in porous media. Phys. Rev. Lett. 115, 194502 (2015)CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.School of EngineeringRMIT UniversityMelbourneAustralia
  2. 2.Department of Mechanical EngineeringIndian Institute of Technology KharagpurKharagpurIndia
  3. 3.Spanish National Research Council (IDAEA-CSIC)BarcelonaSpain
  4. 4.Geosciences Rennes, UMR 6118Université de Rennes 1, CNRSRennesFrance

Personalised recommendations