Transport in Porous Media

, Volume 87, Issue 1, pp 125–149 | Cite as

Solutions to and Validation of Matrix-Diffusion Models

  • Pekka KekäläinenEmail author
  • Mikko Voutilainen
  • Antti Poteri
  • Pirkko Hölttä
  • Aimo Hautojärvi
  • Jussi Timonen


A model transport system is considered in which a pulse of tracer molecules is advected along a flow channel with porous walls. The advected tracer thus undergoes diffusion, matrix-diffusion, inside the walls, which affects the breakthrough curve of the tracer. Analytical solutions in the form of series expansions are derived for a number of situations which include a finite depth of the porous matrix, varying aperture of the flow channel, and longitudinal diffusion and Taylor dispersion of the tracer in the flow channel. Novel expansions for the Laplace transforms of the concentration in the channel facilitated closed-form expressions for the solutions. A rigorous result is also derived for the asymptotic form of the breakthrough curve for a finite depth of the porous matrix, which is very different from that for an infinite matrix. A specific experimental system was created for validation of matrix-diffusion modeling for a matrix of finite depth. A previously reported fracture-column experiment was also modeled. In both cases model solutions gave excellent fits to the measured breakthrough curves with very consistent values for the diffusion coefficients used as the fitting parameters. The matrix-diffusion modeling performed could thereby be validated.


Matrix-diffusion Diffusion in porous medium 


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  1. Aris R.: On the dispersion of a solute in a fluid flowing through a tube. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 235, 67–77 (1956)CrossRefGoogle Scholar
  2. Bodin, J., Delay, F., de Marsily, G.: Solute transport in a single fracture with negligible matrix permeability: 1. Fundamental mechanisms. Hydrogeol. J. 11, 418–433 (2003). Solute transport in a single fracture with negligible matrix permeability: 2. Mathematical formalism. Ibid. 434–454Google Scholar
  3. Buckley R.L., Loyalka S.K.: Numerical studies of solute transport in a fracture surrounded by rock matrix: effect of lateral diffusion and chemical reactions on the overall dispersion. Ann. Nucl. Energy 21(8), 461–494 (1994)CrossRefGoogle Scholar
  4. Chatwin P.C., Sullivan P.J.: The effect of aspect ratio on longitudinal diffusivity in rectangular channels. J. Fluid Mech. 120, 347–358 (1982)CrossRefGoogle Scholar
  5. Chittaranjan R., Ellsworth T.R., Valocchi A.J., Boast C.W.: An improved dual porosity model for chemical transport in macroporous soils. J. Hydrol. 193(1–4), 270–293 (1997)Google Scholar
  6. Cvetkovic V., Selroos J.-O., Cheng H.: Transport of reactive tracers in rock fractures. J. Fluid Mech. 378, 335–356 (1999)CrossRefGoogle Scholar
  7. Cvetkovic V., Cheng H., Widestrand H., Byegård J., Winberg A., Andersson P.: Sorbing tracer experiments in a crystalline rock fracture at Äspö (Sweden): 2. Transport model and effective parameter estimation. Water Resour. Res. 43, W11421 (2007)CrossRefGoogle Scholar
  8. Delay F., Kaczmaryk A., Ackerer P.: Inversion of a Lagrangian time domain random walk (TDRW) approach to one-dimensional transport by derivation of the analytical sensitivities to parameters. Adv. Water Resour. 31(3), 484–502 (2008)CrossRefGoogle Scholar
  9. Doughty C.: Investigation of conceptual and numerical approaches for evaluating moisture, gas, chemical, and heat transport in fractured unsaturated rock. J. Contam. Hydrol. 38(1–3), 69–106 (1999)CrossRefGoogle Scholar
  10. Foster S.S.D.: The chalk groundwater tritium anomaly—a possible explanation. J. Hydrol. 25(1–2), 159–165 (1975)CrossRefGoogle Scholar
  11. Guimera J., Carrera J.: A comparison of hydraulic and transport parameters measured in low-permeability fractured media. J. Contam. Hydrol. 41(3–4), 261–281 (2000)CrossRefGoogle Scholar
  12. Hadermann J., Heer W.: The Grimsel (Switzerland) migration experiment: integrating field experiments, laboratory investigations and modelling. J. Contam. Hydrol. 21(1–4), 87–100 (1996)CrossRefGoogle Scholar
  13. Hodgkinson D., Benabderrahmane H., Elert M., Hautojärvi A., Selroos J., Tanaka Y., Uchida M.: An overview of Task 6 of the Äspö Task force: modelling groundwater and solute transport: improved understanding of radionuclide transport in fractured rock. Hydrogeol. J. 17(5), 1035–1049 (2009)CrossRefGoogle Scholar
  14. Hölttä, P.: Radionuclide migration in crystalline rock fractures, PhD Thesis, University of Helsinki (1992)Google Scholar
  15. Hölttä P., Hautojärvi A., Hakanen M.: Transport and retardation on non-sorbing radionuclides in crystalline rock fractures. Radiochim. Acta 58/59, 285–290 (1992)Google Scholar
  16. Kennedy C.A., Lennox W.: A control volume model of solute transport in a single fracture. Water Resour. Res. 31(2), 313–322 (1995)CrossRefGoogle Scholar
  17. Korevaar J.: Tauberian Theory, a Century of Developments. Springer, Berlin (2004)Google Scholar
  18. Lobo V.M.M., Ribeiro A.C.F., Verissimo L.M.P.: Diffusion coefficients in aqueous solutions on potassium chloride at high and low concentrations. J. Mol. Liq. 78, 139–149 (1998)CrossRefGoogle Scholar
  19. Małoszewski, P., Zuber, A.: Interpretation of artificial and environmental tracers in fissured rocks with a porous matrix. In: Isotope Hydrology 1983. Int. At. Energy Agency (I.A.E.A.), Vienna, pp. 635–651 (1983)Google Scholar
  20. Mills R.: Self-diffusion in normal and heavy water in the range 1–45°. J. Phys. Chem. 77(5), 685–688 (1973)CrossRefGoogle Scholar
  21. Neretnieks I.: Diffusion in the rock matrix: an important factor in radionuclide retardation?. J. Geophys. Res. 85(B8), 4379–4397 (1980)CrossRefGoogle Scholar
  22. Neretnieks I.: A stochastic multi-channel model for solute transport: analysis of tracer tests in fractured rock. J. Contam. Hydrol. 55(3–4), 175–211 (2002)CrossRefGoogle Scholar
  23. Norton D., Knapp R.: Transport phenomena in hydrothermal systems: the nature of porosity. Am. J. Sci. 277, 913–936 (1977)CrossRefGoogle Scholar
  24. Painter S., Cvetkovic V., Mancillas J., Pensado O.: Time domain particle tracking methods for simulating transport with retention and first-order transformation. Water Resour. Res. 44, W01406 (2008)CrossRefGoogle Scholar
  25. Perez N.: Electrochemistry and Corrosion Science. Kluwer Academic Publishers, Boston (2004)CrossRefGoogle Scholar
  26. Ryan D., Carbonell R.G., Whitaker S.: Effective diffusivities for catalyst pellets under reactive conditions. Che. Eng. Sci. 35(1–2), 10–16 (1980)CrossRefGoogle Scholar
  27. Sahimi M.: Flow phenomena in rocks: from continuum models to fractals, percolation, cellular automata, and simulated annealing. Rev. Mod. Phys. 65(4), 1393–1534 (1993)CrossRefGoogle Scholar
  28. Shapiro A.M.: Effective matrix diffusion in kilometer-scale transport in fractured crystalline rock. Water Resour. Res. 37(3), 507–522 (2001)CrossRefGoogle Scholar
  29. Siitari-Kauppi M., Lindberg A., Hellmuth K.-H., Timonen J., Väätäinen K., Hartikainen J., Hartikainen K.: The effect of microscale pore structure on matrix diffusion—a site specific study of tonalite. J. Contam. Hydrol. 26(1–4), 147–158 (1997)CrossRefGoogle Scholar
  30. Skagius K., Neretnieks I.: Porosities and diffusivities of some nonsorbing species in crystalline rocks. Water Resour. Res. 22(3), 389–398 (1986)CrossRefGoogle Scholar
  31. Tang A., Sandall O.C.: Diffusion coefficient of chlorine in water at 25–65° C. J. Chem. Eng. Data 30, 189–191 (1985)CrossRefGoogle Scholar
  32. Tang D.H., Frind E.O., Sudicky E.A.: Contaminant transport in fractured porous media: analytical solutions for a single fracture. Water Resour. Res. 17(3), 555–564 (1981)CrossRefGoogle Scholar
  33. Taylor G.I.: Dispersion of soluble matter in solvent flowing slowly through a tube. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 219, 186–203 (1953)CrossRefGoogle Scholar
  34. Voutilainen M., Kekäläinen P., Hautojärvi A., Timonen J.: Validation of matrix diffusion modeling. Phys. Chem. Earth 35(6–8), 259–264 (2010)Google Scholar
  35. Webster D.R., Felton D.S., Luo J.: Effective macroscopic transport parameters between parallel plates with constant concentration boundaries. Adv. Water Resour. 30, 1993–2001 (2007)CrossRefGoogle Scholar
  36. Wood W.W., Kraemer T.F., Hearn P.P.: Intergranular diffusion: an important mechanism influencing solute transport in classic aquifers?. Science 247, 1569–1572 (1990)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  • Pekka Kekäläinen
    • 1
    Email author
  • Mikko Voutilainen
    • 1
  • Antti Poteri
    • 2
  • Pirkko Hölttä
    • 3
  • Aimo Hautojärvi
    • 4
  • Jussi Timonen
    • 1
  1. 1.Department of PhysicsUniversity of JyväskyläJyväskyläFinland
  2. 2.VTT Research CenterEspooFinland
  3. 3.Laboratory of RadiochemistryUniversity of HelsinkiHelsinkiFinland
  4. 4.Posiva OyOlkiluotoFinland

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