Transport in Porous Media

, Volume 96, Issue 2, pp 319–336 | Cite as

Diffusion of Tracer in Altered Tonalite: Experiments and Simulations with Heterogeneous Distribution of Porosity

  • Mikko VoutilainenEmail author
  • Paul Sardini
  • Marja Siitari-Kauppi
  • Pekka Kekäläinen
  • Vesa Aho
  • Markko Myllys
  • Jussi Timonen


Numerical time-domain-diffusion simulations were used for studying the diffusion behavior of tracer molecules in rock matrix with homogeneous and heterogeneous porosity. As the heterogeneous sample in these simulations, a 3D tomographic image of altered tonalite was used, in which the mineral components and the pores resolved by X-ray microtomography were represented by their respective intragranular porosities determined previously by the 14C-PMMA method. The apparent diffusion coefficient of a tracer in altered tonalite was determined experimentally, and was then used in the simulations. In the altered tonalite analyzed, inclusion of heterogeneity in the porosity increased the diffusion coefficient by 16 %. Altered and pristine feldspar was the main mineral component in the sample (72 %), and it also provided the dominant contribution to tracer diffusion, explaining alone 52 % of the diffusion coefficient. The large pores resolved by microtomography (6 %) and altered and pristine mica (22 %) gave an equal contribution to the diffusion coefficient. The simulation method applied was also validated by comparing the results to both an analytical and a numerical solution to the diffusion equation in a homogenous medium. In addition, the method was compared to discrete-time random-walk simulations in the case of randomly placed overlapping spheres.


Tracer diffusion Time-domain-diffusion Heterogeneousporosity X-ray microtomography 14C-PMMA method 


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  1. Bagtzoglou A.C., Kim D.D.: Modeling contaminant transport at the pore-micropore interface. Water Air Soil. Poll. 6(1–2), 207–225 (2006)Google Scholar
  2. Bear, J.: Dynamics of fluids in porous media, pp. 764. Dover, New York (1988)Google Scholar
  3. Berkowitz B., Scher H.: Anomalous transport in laboratory-scale, heterogeneous porous media. Water Resour. Res. 36(1), 149–158 (2000)CrossRefGoogle Scholar
  4. Berkowitz B., Cortis A., Dentz M., Scher H.: Modeling non-Fickian transport in geological formations as a continuous time random walk. Rev. Geophys. 44, RG2003 (2006)CrossRefGoogle Scholar
  5. Bijeljic B., Mostaghimi P., Blunt M.J.: Signature of non-fickian solute transport in complex heterogeneous porous media. PRL 107, 204502 (2011)CrossRefGoogle Scholar
  6. Bradbury M.H., Green A.: Investigations into the factors influencing long range matrix diffusion rates and pore space accessibility at depth in granite. J. Hydrol. 89(1–2), 123–139 (1986)CrossRefGoogle Scholar
  7. Brusseau M.L.: The influece of solute size, pore water velocity, and intraparticle porosity on solute dispersion and transport in soil. Water Resour. Res. 29(4), 1071–1080 (1993)CrossRefGoogle Scholar
  8. Carslaw, H.S., Jaeger, J.C.: Conduction of Heat in Solids, pp. 510. Clarendon Press, Oxford (1965)Google Scholar
  9. Delay F., Porel G., Sardini P.: Modelling diffusion in a heterogeneous rock matrix with a time-domain Lagrangian method and an inversion procedure. C. R. Geosci. 334, 967–973 (2002)CrossRefGoogle Scholar
  10. Delay F., Porel G.: Inverse modeling in the time domain for solving diffusion in a heterogeneous rock matrix. Geophys. Res. Lett. 30, 1147–1150 (2003)CrossRefGoogle Scholar
  11. Dogu G., Smith J.M.: A dynamic method for catalyst diffusivities. AIChE J. 21(1), 58–61 (1975)CrossRefGoogle Scholar
  12. Dullien, F.A.L.: Porous Media: Fluid Transport and Pore Structure, pp. 396. Academic Press Inc., San Diego (1979)Google Scholar
  13. Epstein N.: On tortuosity and the tortuosity factor in flow and diffusion through porous media. Chem. Eng. Sci. 44, 777–779 (1989)CrossRefGoogle Scholar
  14. Ewing R.P., Horton R.: Diffusion in sparsely connected pore spaces: temporal and spatial scaling. Water Resour. Res. 38, 1285–1297 (2002)CrossRefGoogle Scholar
  15. Foster S.S.D.: The chalk groundwater tritium anomaly: a possible explanation. J. Hydrol. 25(1–2), 159–165 (1975)CrossRefGoogle Scholar
  16. Garrouch A.A., Ali L., Qasem F.: Using diffusion and electrical measurements to assess tortuosity of porous media. Ind. Eng. Chem. Res. 40(20), 4363–4369 (2001)CrossRefGoogle Scholar
  17. Gonzalez, R., Woods, R.: Digital Image Processing, pp. 793. Prentice-Hall, New Jersey (2002)Google Scholar
  18. Gueudré L., Jolimaîte E., Bats N., Dong W.: Diffusion in zeolites: is surface resistance a critical parameter?. Adsorption 15, 17–27 (2010)CrossRefGoogle Scholar
  19. Hellmuth K.-H., Siitari-Kauppi M., Lindberg A.: Study of porosity and migration pathways in crystalline rock by impregnation with l4C-polymethylmethacrylate. J. Contain. Hydrol. 13, 403–418 (1993)CrossRefGoogle Scholar
  20. Hua Q.H., Möri A.: Radionuclide transport in fractured granite interface zones. Phys. Chem. Earth 33, 1042–1049 (2008)CrossRefGoogle Scholar
  21. Hunter J.R., Craig P.D., Phillips H.E.: On the use of random walk models with spatially variable diffusivity. J. Comput. Phys. 106, 366–376 (1993)Google Scholar
  22. Hölttä P., Hakanen M., Hautojärvi A., Timonen J., Väätäinen K.: The effects of matrix diffusion on radionuclide migration in rock column experiments. J. Contam. Hydrol. 21(1–4), 165–173 (1996)CrossRefGoogle Scholar
  23. Hölttä P., Siitari-Kauppi M., Hakanen M., Huitti T., Hautojärvi A., Lindberg A.: Radionuclide transport and retardation in rock fracture and crushed rock column experiments. J. Contam. Hydrol. 26(1–4), 135–145 (1997)CrossRefGoogle Scholar
  24. Jeong N., Choi D.H., Lin C.-L.: Estimation of thermal and mass diffusivity in a porous medium of complex structure using a lattice Boltzmann method. Int. J. Heat Mass Tran. 51(15–16), 3913–3923 (2008)CrossRefGoogle Scholar
  25. Kekäläinen P., Voutilainen M., Poteri A., Hölttä P., Hautojärvi A., Timonen J.: Solutions to and validation of matrix-diffusion models. Transp. Porous Med. 87, 125–149 (2011)CrossRefGoogle Scholar
  26. Lever D.A., Bradbury M.H., Hamingway S.J.: The effect of dead-end porosity on rock-matrix. J. Hydrol. 80, 45–76 (1985)CrossRefGoogle Scholar
  27. Lindberg, A., Paananen, M.: Konginkangaan Kivetyn, Sievin Syyryn ja Eurajoen Olkiluodon kallionäytteiden petrografia, geokemia ja geofysiikka kairanreiät KI-KR7, SY-KR7 ja OL-KR6, TVO/ Paikkatutkimukset, Työraportti 92–34 (1992) (in finnish)Google Scholar
  28. Liu, J., Löfgren, M., Neretnieks, I.: Data and uncertainty assessment: matrix diffusivity and porosity in situ. SKB R-06-111 (2006)Google Scholar
  29. Lloyd S.P.: Least squares quantization in PCM. IEEE Trans. Inform. Theory 28(2), 129–137 (1982)CrossRefGoogle Scholar
  30. Lü Y., Bü M.: Analysis of diffusion in hollow geometries. Adsorption 6(2), 125–136 (2000)CrossRefGoogle Scholar
  31. McCaig A.M.: Fluid flow through fault zones. Nature 340, 600 (1989)CrossRefGoogle Scholar
  32. McCarthy J.F.: Contunuos-time random walks on random media. J. Phys. A: Math. Gen. 26, 2495–2503 (1993)CrossRefGoogle Scholar
  33. Metzler R., Klafter J.: The random Walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, 1–77 (2000)CrossRefGoogle Scholar
  34. Michaels A.S.: Diffusion in a pore of irregular cross section. AIChE J. 5(2), 270–271 (1959)CrossRefGoogle Scholar
  35. Mills R.: Self-diffusion in normal and heavy water in the range 1–45. J. Phys. Chem. 77(5), 685–688 (1973)CrossRefGoogle Scholar
  36. Moreno L., Gylling B., Neretnieks I.: Solute transport in fractured media the important mechanisms for performance assessment. J. Contam. Hydrol. 25, 283–298 (1997)CrossRefGoogle Scholar
  37. Nakashima Y., Nakanoa T., Nakamuraa K., Uesugib K., Tsuchiyamac A., Ikedad S.: Three-dimensional diffusion of non-sorbing species in porous sandstone: computer simulation based on X-ray microtomography using synchrotron radiation. J. Contam. Hydrol. 74(1–2), 253–264 (2004)CrossRefGoogle Scholar
  38. Neretnieks I.: Diffusion in the rock matrix: an important factor in radionuclide retardation?. J. Geophys. Res. 85(B8), 4379–4397 (1980)CrossRefGoogle Scholar
  39. Neretnieks I.: Fast method for simulation of radionuclide chain migration in dual porosity fracture rocks. J. Contam. Hydrol. 88, 269–288 (2006)CrossRefGoogle Scholar
  40. Norton D., Knapp R.: Transport phenomena in hydrothermal systems: the nature of porosity. Am. J. Sci. 277, 913–936 (1977)CrossRefGoogle Scholar
  41. Pankina E., Rumynin V., Nikulenkov A., Glukhova M., Epimakhov V., Mysik S., Baev M., Kobekov V., Degtev V.: Anisotropy of clays in diffusion transport of radionuclides. Radiochemistry 52(6), 630–637 (2010)CrossRefGoogle Scholar
  42. Petersen E.E.: Diffusion in a pore of varying cross section. AIChE J. 4(3), 343–345 (1958)CrossRefGoogle Scholar
  43. Reimus P.W., Callahan T.J.: Matrix diffusion rates in fractured volcanic rocks at the nevada test site: evidence for a dominant influence of effective fracture apertures. Water Resour. Res. 43(7), W07421 (2007)CrossRefGoogle Scholar
  44. Rhodes M.E., Bijeljic B., Blunt M.J.: Pore-to-field simulation of single-phase transport using continuous time random walks. Adv. Water Resour. 31, 1527–1539 (2008)CrossRefGoogle Scholar
  45. Ritger P.L., Peppas N.A.: Transport of penetrants in the macromolecular structure of coals, 4. Models for analysis of dynamic penetrant transport. Fuel 66, 815–826 (1987)CrossRefGoogle Scholar
  46. Ritger P.L., Peppas N.A.: Transport of penetrants in the macromolecular structure of coals, 7. Transport in thin coal sections. Fuel 66, 1379–1388 (1987)CrossRefGoogle Scholar
  47. Robinet J.C., Sardini P., Delay F., Hellmuth K.H.: The effect of rock matrix heterogeneities near fracture walls on the residence time distribution (RTD) of solutes. Transp. Porous Med. 72, 393–408 (2008)CrossRefGoogle Scholar
  48. 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
  49. Sahimi, M.: Flow and Transport in Porous Media and Fractured Rock: From Classical Methods to Modern Approaches, pp. 482. VCH, Weinheim (1995)Google Scholar
  50. Sardini P., Delay F., Hellmuth K.-H., Porel G., Oila E.: Interpretation of out-diffusion experiments on crystalline rocks using random walk modeling. J. Contam. Hydrol. 61, 339–350 (2003)CrossRefGoogle Scholar
  51. Sardini P., Siitari-Kauppi M., Beaufort D., Hellmuth K.-H.: The connected porosity of mineral aggregates in crystalline rocks. Am. Mineral. 91, 1069–1080 (2006)CrossRefGoogle Scholar
  52. Sardini P., Robinet J.C., Siitari-Kauppi M., Delay F., Hellmuth K.H.: Direct simulation of heterogeneous diffusion and inversion procedure applied to an out-diffusion experiment. Test case of Palmottu granite. J. Contam. Hydrol. 93, 21–37 (2007)CrossRefGoogle Scholar
  53. Scher H., Lax M.: Stochastic transport in disordered solid. II. Impurity conduction. Phys. Rev. B 7(10), 4502–4519 (1973)CrossRefGoogle Scholar
  54. 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 on tonalite. J. Contam. Hydrol. 26(1–4), 147–158 (1997)CrossRefGoogle Scholar
  55. Thornton A.W., Hilder T., Hill A.J., Hill J.M.: Predicting gas diffusion regime within pores of different size, shape and composition. J. Membr. Sci. 336(1–2), 101–108 (2009)CrossRefGoogle Scholar
  56. Torstensson G., Eriksson S.: A new method for determining the porosity of the soil. Soil Sci. 42(6), 405–417 (1936)CrossRefGoogle Scholar
  57. van Brakel J., Heertjes P.M.: Analysis of diffusion in macroporous media in terms of a porosity, a tortuosity and a constrictivity factor. Int. J. Heat Mass Trans. 17(9), 1093–1103 (1974)CrossRefGoogle Scholar
  58. van Loon L.R., Soler J., Müller W., Bradbury L.: Anisotropic diffusion in layered argillaceous rocks: a case study with opalinus clay. Environ. Sci. Technol. 38, 5721–5728 (2004)CrossRefGoogle Scholar
  59. Vilarrasa V., Koyama T., Neretnieks I., Jing L.: Shear-induced flow channels in a single rock fracture and their effect on solute transport. Transp. Porous Med. 87, 503–523 (2011)CrossRefGoogle Scholar
  60. Voutilainen M., Kekäläinen P., Hautojärvi A., Timonen J.: Validation of matrix diffusion modeling. Phys. Chem. Earth 35, 259–264 (2010)CrossRefGoogle Scholar
  61. Voutilainen M., Siitari-Kauppi M., Sardini P., Timonen J.: Pore-space characterization of an altered tonalite by X-ray computed microtomography and the 14C-labeled-polymethylmethacrylate method. J. Geophys. Res. 117, B01201 (2012)CrossRefGoogle Scholar
  62. 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
  63. Wu Y.-S., Ye M., Sudicky E.A.: Fracture-flow-enhanced matrix diffusion in solute transport through fractured porous media. Transp. Porous Med. 81, 21–34 (2010)CrossRefGoogle Scholar
  64. Xu S., Wörman A., Dverstorp B.: Heterogeneous matrix diffusion in crystalline rock - implications for geosphere retardation of migrating radionuclides. J. Contam. Hydrol. 47, 365–378 (2001)CrossRefGoogle Scholar
  65. Xuan Y.M., Zhao K., Li Q.: Investigation on mass diffusion process in porous media based on Lattice Boltzmann method. Heat Mass Transf. 46(10), 1039–1051 (2010)CrossRefGoogle Scholar
  66. Zalc J.M., Reyes S.C., Iglesia E.: Monte-Carlo simulations of surface and gas phase diffusion in complex porous structures. Chem. Eng. Sci. 58, 4605–4617 (2003)CrossRefGoogle Scholar
  67. Zhoua Q., Liua H., Molzb F.J., Zhanga Y., Bodvarsson G.S.: Field-scale effective matrix diffusion coefficient for fractured rock: Results from literature survey. J. Contam. Hydrol. 93(1–4), 161–187 (2007)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2012

Authors and Affiliations

  • Mikko Voutilainen
    • 1
    Email author
  • Paul Sardini
    • 2
  • Marja Siitari-Kauppi
    • 3
  • Pekka Kekäläinen
    • 1
  • Vesa Aho
    • 1
  • Markko Myllys
    • 1
  • Jussi Timonen
    • 1
  1. 1.Department of PhysicsUniversity of JyväskyläJyväskyläFinland
  2. 2.IC2MP, University of PoitiersPoitiersFrance
  3. 3.Laboratory of RadiochemistryUniversity of HelsinkiHelsinkiFinland

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