Computational Geosciences

, Volume 23, Issue 5, pp 953–967 | Cite as

Modeling mass transfer in fracture flows with the time domain-random walk method

  • J. KuvaEmail author
  • M. Voutilainen
  • K. Mattila
Open Access
Original Paper


The time domain-random walk method was developed further for simulating mass transfer in fracture flows together with matrix diffusion in surrounding porous media. Specifically, a time domain-random walk scheme was developed for numerically approximating solutions of the advection-diffusion equation when the diffusion coefficient exhibits significant spatial variation or even discontinuities. The proposed scheme relies on second-order accurate, central-difference approximations of the advective and diffusive fluxes. The scheme was verified by comparing simulated results against analytical solutions in flow configurations involving a rectangular channel connected on one side with a porous matrix. Simulations with several flow rates, diffusion coefficients, and matrix porosities indicate good agreement between the numerical approximations and analytical solutions.


Matrix diffusion Advection Porous media Solute transport Breakthrough curve Simulation 


Funding information

Financial support from the Finnish Research Programme on Nuclear Waste Management (KYT2018) is gratefully acknowledged.


  1. 1.
    Aromaa, H., Voutilainen, M., Ikonen, J., Yli-Kaila, M., Poteri, A., Siitari-Kauppi, M.: Through diffusion experiments to study the diffusion and sorption of HTO, 36Cl, 133Ba and 134Cs in crystalline rock. J. Contam. Hydrol. 222, 101–111 (2019). CrossRefGoogle Scholar
  2. 2.
    Berkowitz, B., Kosakowski, G., Margolin, G., Scher, H.: Application of continuous time random walk theory to tracer test measurements in fractured and heterogeneous porous media. Ground. Water. 39(4), 593–604 (2001). CrossRefGoogle Scholar
  3. 3.
    Bodin, J., Delay, F., de Marsily, G.: Solute transport in a single fracture with negligible matrix permeability: 1. Fundamental mechanisms. Hydrogeol. J. 11(4), 418–433 (2003). CrossRefGoogle Scholar
  4. 4.
    Bortz, A., Kalos, M., Lebowitz, J.: A new algorithm for Monte Carlo simulation of Ising spin systems. J. Comput. Phys. 17(1), 10–18 (1975). CrossRefGoogle Scholar
  5. 5.
    Cvetkovic, V., Frampton, A.: Transport and retention from single to multiple fractures in crystalline rock at äspö (Sweden): 2. Fracture network simulations and generic retention model. Water Resour. Res. 46(5), w05506 (2010). Google Scholar
  6. 6.
    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
  7. 7.
    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(13), 967–973 (2002). CrossRefGoogle Scholar
  8. 8.
    Dentz, M., Scher, H., Holder, D., Berkowitz, B.: Transport behavior of coupled continuous-time random walks. Phys. Rev. E. 78(4), 041,110 (2008). CrossRefGoogle Scholar
  9. 9.
    Dentz, M., Gouze, P., Russian, A., Dweik, J., Delay, F.: Diffusion and trapping in heterogeneous media: An inhomogeneous continuous time random walk approach. Adv. Water. Resour. 15(1), 13–22 (2012). CrossRefGoogle Scholar
  10. 10.
    Farah, N., Delorme, M., Ding, D.Y., Wu, Y.S., Bossie Codreanu, D.: Flow modelling of unconventional shale reservoirs using a DFM-MINC proximity function. J. Petrol. Sci. Eng. 173, 222–236 (2019). CrossRefGoogle Scholar
  11. 11.
    Flamm, M., Diamond, S., Sinno, T.: Lattice kinetic Monte Carlo simulations of convective-diffusive systems. J. Chem. Phys. 130(9), 094,904 (2009). CrossRefGoogle Scholar
  12. 12.
    Foster, S.: The chalk groundwater tritium anomaly – a possible explanation. J. Hydrolog. 25(1), 159–165 (1975). CrossRefGoogle Scholar
  13. 13.
    Gjetvaj, F., Russian, A., Gouze, P., Dentz, M.: Dual control of flow field heterogeneity and immobile porosity on non-Fickian transport in Berea sandstone. Water. Resour. Res. 51(10), 8273–8293 (2013). CrossRefGoogle Scholar
  14. 14.
    Haggerty, R., Gorelick, S.M.: Multiple-rate mass transfer for modeling diffusion and surface reactions in media with pore-scale heterogeneity. Water Resour. Res. 31(10), 2383–2400 (1995). wR10583CrossRefGoogle Scholar
  15. 15.
    Ikonen, J., Voutilainen, M., Söderlund, M, Jokelainen, L., Siitari-Kauppi, M., Martin, A.: Sorption and diffusion of selenium oxyanions in granitic rocks. J. Cont. Hydrol. 192, 203–211 (2016). CrossRefGoogle Scholar
  16. 16.
    Ikonen, J., Sardini, P., Siitari-Kauppi, M., Martin, A.: In situ migration of tritiated water and iodine in Grimsel granodiorite, part II: Assessment of the diffusion coefficients by TDD modelling. J. Radioanal. Nucl. Chem. 311(1), 339–348 (2017). CrossRefGoogle Scholar
  17. 17.
    Kekäläinen, P: Analytical solutions to matrix diffusion problems. AIP Conf. Proc. 1618(1), 513–516 (2014). CrossRefGoogle Scholar
  18. 18.
    Kekäläinen, P, Voutilainen, M., Poteri, A., Hölttä, P, Hautojärvi, A, Timonen, J.: Solutions to and validation of matrix-diffusion models. Transp. Porous. Med. 87(1), 125–149 (2011). CrossRefGoogle Scholar
  19. 19.
    Kuva, J., Voutilainen, M., Kekäläinen, P, Siitari-Kauppi, M, Timonen, J., Koskinen, L.: Gas phase measurements of porosity, diffusion coefficient, and permeability in rock samples from Olkiluoto bedrock, Finland. Transp. Porous. Med. 107(1), 187–204 (2015). CrossRefGoogle Scholar
  20. 20.
    Kuva, J., Voutilainen, M., Kekäläinen, P, Siitari-Kauppi, M, Sammaljärvi, J, Timonen, J, Koskinen, L.: Gas phase measurements of matrix diffusion in rock samples from Olkiluoto bedrock, Finland. Transp. Porous. Med. 115(1), 1–20 (2016). CrossRefGoogle Scholar
  21. 21.
    LaBolle, E., Quastel, J., Fogg, G.: Diffusion theory for transport in porous media: Transition-probability densities of diffusion processes corresponding to advection-dispersion equations. Water. Resour. Res. 34(7), 1685–1693 (1998). CrossRefGoogle Scholar
  22. 22.
    Lee, Y., Sinno, T.: Analysis of the lattice kinetic Monte Carlo method in systems with external fields. J. Chem. Phys. 145(23), 234,104 (2016). CrossRefGoogle Scholar
  23. 23.
    Mattila, K., Hegele, Jr, L., Philippi, P.: High-accuracy approximation of high-rank derivatives: Isotropic finite differences based on lattice-Boltzmann stencils. Sci. World. J 142, 907 (2014). CrossRefGoogle Scholar
  24. 24.
    McCarthy, J.: Continuous-time random walks on random media. J. Phys. A-Math. Gen. 26(11), 2495–2503 (1993). CrossRefGoogle Scholar
  25. 25.
    McDermott, C., Walsh, R., Mettier, R., Kosakowski, G., Kolditz, O.: Hybrid analytical and finite element numerical modeling of mass and heat transport in fractured rocks with matrix diffusion. Comput. Geosci. 13(3), 349–361 (2009). CrossRefGoogle Scholar
  26. 26.
    Montroll, E., Weiss, G.: Random walks on lattices. II. J. Math. Phys. 6(2), 167–181 (1965). CrossRefGoogle Scholar
  27. 27.
    Neretnieks, I.: Diffusion in the rock matrix: An important factor in radionuclide retardation? J. Geophys. Res. 85(B8), 4379–4397 (1980). CrossRefGoogle Scholar
  28. 28.
    Nœtinger, B, Estebenet, T: Up-scaling of double porosity fractured media using continuous-time random walks methods. Transp. Porous Med. 39, 315–337 (2000). CrossRefGoogle Scholar
  29. 29.
    Nœtinger, B, Estebenet, T, Landereau, P: A direct determination of the transient exchange term of fractured media using a continuous time random walk method. Transp. Porous Med. 44, 539–557 (2001). CrossRefGoogle Scholar
  30. 30.
    Nœtinger, B, Estebenet, T, Quintard, M: Up-scaling flow in fractured media: equivalence between the large scale averaging theory and the continuous time random walk method. Transp. Porous Med. 43, 581–596 (2001). CrossRefGoogle Scholar
  31. 31.
    Noetinger, B., Roubinet, D., Russian, A., Borgne, T.L., Delay, F., Dentz, M., de Dreuzy, J, Gouze, P: Random walk methods for modeling hydrodynamic transport in porous and fractured media from pore to reservoir scale. Transp. Porous. Med. 115(2), 345–385 (2016). CrossRefGoogle Scholar
  32. 32.
    Patra, M., Karttunen, M.: Stencils with isotropic discretization error for differential operators. Numer. Meth. Part. Differ. Equ. 22(4), 936–953 (2006). CrossRefGoogle Scholar
  33. 33.
    Posiva: Safety case for the disposal of spent nuclear fuel at olkiluoto — Models and data for the repository system 2012. Tech. rep., Posiva Oy (2003)Google Scholar
  34. 34.
    Rasilainen, K: Matrix diffusion model: In situ tests using natural analogue. PhD thesis, Helsinki University of Technology, Finland (1997)Google Scholar
  35. 35.
    Rhodes, M.: Transport in Heterogeneous Porous Media. PhD thesis. Imperial College, London (2008)Google Scholar
  36. 36.
    Robinet, J., 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(3), 393–408 (2008). CrossRefGoogle Scholar
  37. 37.
    Robinet, J., Sardini, P., Coelho, D., Parneix, J., Prêt, D., Sammartino, S., Boller, E., Altmann, S.: Effects of mineral distribution at mesoscopic scale on solute diffusion in a clay-rich rock: Example of the Callovo-Oxfordian mudstone (Bure, France). Water. Resour. Res. 48(5), W05,554 (2012). CrossRefGoogle Scholar
  38. 38.
    Russian, A., Dentz, M., P Gouze, P.: Time domain random walks for hydrodynamic transport in heterogeneous media. Water. Resour. Res. 52(5), 3309–3323 (2016). CrossRefGoogle Scholar
  39. 39.
    Salamon, P., Fernàndez-Garcia, D, Gómez-Hernández, J: A review and numerical assessment of the random walk particle tracking method. J. Contam. Hydrol. 87(3–4), 277–305 (2006). CrossRefGoogle Scholar
  40. 40.
    Sardini, P., Robinet, J., 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(1–4), 21–37 (2007). CrossRefGoogle Scholar
  41. 41.
    Stumm, W.: Chemistry of the Soil-Water Interface: Processes at the Mineral-Water and Particle-Water Interface in Natural Systems. Wiley, New York (1992)Google Scholar
  42. 42.
    Svensson, U., Löfgren, M, Trinchero, P., Selroos, J.O.: Modelling the diffusion-available pore space of an unaltered granitic rock matrix using a micro-DFN approach. J. Hydrol. 559, 182–191 (2018). CrossRefGoogle Scholar
  43. 43.
    Tang, D., Frind, E., Sudicky, E.: Contaminant transport in fractured porous media: Analytical solutions for a single fracture. Water. Resour. Res. 17(3), 555–564 (1981). CrossRefGoogle Scholar
  44. 44.
    Tecklenburg, J., Neuweiler, I., Carrera, J., Dentz, M.: Multi-rate mass transfer modeling of two-phase flow in highly heterogeneous fractured and porous media. Adv. Water. Resour. 91, 63–77 (2016). CrossRefGoogle Scholar
  45. 45.
    Toivanen, J, Mattila, K, Hyväluoma, J, Kekäläinen, P, Puurtinen, T, Timonen, J: Simulation software for flow of fluid with suspended point particles in complex domains: Application to matrix diffusion. Lect. Notes Comput. Sc. 7782, 434–445 (2013). CrossRefGoogle Scholar
  46. 46.
    Toivanen, J., Mattila, K., Puurtinen, T., Timonen, J.: LBM simulations of matrix diffusion with sorption. AIP Conf. Proc. 1618(1), 517–520 (2014). CrossRefGoogle Scholar
  47. 47.
    Trinchero, P., Molinero, J., Deissmann, G., Svensson, U., Gylling, B., Ebrahimi, H., Hammond, G., Bosbach, D., Puigdomenech, I.: Implications of grain-scale mineralogical heterogeneity for radionuclide transport in fractured media. Transp. Porous. Med. 116(1), 73–90 (2017). CrossRefGoogle Scholar
  48. 48.
    Voutilainen, M., Kekäläinen, P, Hautojärvi, A, Timonen, J.: Validation of matrix diffusion modeling. Phys. Chem. Earth. 35(6–8), 259–264 (2010). CrossRefGoogle Scholar
  49. 49.
    Voutilainen, M., Sardini, P., Siitari-Kauppi, M, Kekäläinen, P, Aho, V, Myllys, M, Timonen, J: Simulated diffusion of tracer in altered tonalite with heterogeneous distribution of porosity. Transp. Porous. Med. 96(2), 319–336 (2013). CrossRefGoogle Scholar
  50. 50.
    Voutilainen, M., Poteri, A., Helariutta, K., Siitari-Kauppi, M, Nilsson, K, Andersson, P, Byegård, J, Skålberg, M, Kekäläinen, P, Timonen, J, Lindberg, A, Pitkänen, P, Kemppainen, K, Liimatainen, J, Hautojärvi, A, Koskinen, L: In-situ experiments for investigating the retention properties of rock matrix in ONKALO, Olkiluoto, Finland. In: Proceedings of the Annual Waste Management Symposium (WM2014), March 2–6, pp 1969–1981, Phoenix (2014)Google Scholar
  51. 51.
    Voutilainen, M, Kekäläinen, P, Poteri, A, Siitari-Kauppi, M, Helariutta, K, Andersson, P, Nilsson, K, Byegård, J, Skålberg, M, Yli-Kaila, M, Koskinen, L: Comparison of water phase diffusion experiments in laboratory and in situ conditions. J. Hydrol. 575, 716–729 (2019)CrossRefGoogle Scholar
  52. 52.
    Voutilainen, M., Kekäläinen, P, Siitari-Kauppi, M, Sardini, P, Muuri, E, Timonen, J, Martin, A: Modeling transport of cesium in Grimsel granodiorite with micrometer scale heterogeneities and dynamic update of K d. Water. Resour. Res. 53(11), 9245–9265 (2017). CrossRefGoogle Scholar
  53. 53.
    Yan, B., Mi, L., Chai, Z., Wang, Y., Killough, J.E.: An enhanced discrete fracture network model for multiphase flow in fractured reservoirs. J. Petrol. Sci. Eng. 161, 667–682 (2017). CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Geological Survey of FinlandEspooFinland
  2. 2.Department of ChemistryUniversity of HelsinkiHelsinkiFinland
  3. 3.Department of PhysicsUniversity of JyväskyläJyväskyläFinland
  4. 4.Laboratory of PhysicsTampere University of TechnologyTampereFinland

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