• Nicolae Suciu
Part of the Geosystems Mathematics book series (GSMA)


Stochastic approaches for transport processes in heterogeneous media are motivated by the need to use stochastic parameterizations of the model equations. Essentially, this results in modeling diffusion processes in random fields. For instance, in stochastic subsurface hydrology, random hydraulic conductivity parameters generate random groundwater flow velocity fields and solute transport is modeled by diffusion equations with random drift coefficients. Technically, modeling approaches are based on equivalent Fokker–Planck and Itô representations of the diffusion in random fields, that is, through trajectories of molecules or computational particles and the corresponding continuous fields.


  1. 1.
    Alzraiee, A.H., Baú, D., Elhaddad, A.: Estimation of heterogeneous aquifer parameters using centralized and decentralized fusion of hydraulic tomography data from multiple pumping tests. Hydrol. Earth Syst. Sci. Discuss. 11(4), 4163–4208 (2014)CrossRefGoogle Scholar
  2. 2.
    Bayer-Raich, M., Jarsjö, J., Liedl, R., Ptak, T., Teutsch, G.: Average contaminant concentration and mass flow in aquifers from time-dependent pumping well data: analytical framework. Water Resour. Res. 40(8), W08303 (2004)CrossRefGoogle Scholar
  3. 3.
    Bogachev, L.V.: Random walks in random environments. In: Françoise, J.-P., Naber, G., Tsou, S.T. (eds.) Encyclopedia of Mathematical Physics, vol. 4, pp. 353–371. Elsevier, Oxford (2006)CrossRefGoogle Scholar
  4. 4.
    Bouchaud, J.-P., Georges, A.: Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications. Phys. Rep. 195, 127–293 (1990)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Brunner, F., Radu, F.A., Bause, M., Knabner, P.: Optimal order convergence of a modified BDM1 mixed finite element scheme for reactive transport in porous media. Adv. Water Resour. 35, 163–171 (2012)CrossRefGoogle Scholar
  6. 6.
    Chorin, A.J.: Numerical study of slightly viscous flow. J. Fluid Mech. 57(4), 785–796 (1973)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Chorin, A.J.: Vortex sheet approximation of boundary layers. J. Comput. Phys. 27(3), 428–442 (1978)CrossRefGoogle Scholar
  8. 8.
    Craciun, M., Vamos, C., Suciu, N.: Analysis and generation of groundwater concentration time series. Adv. Water Resour. 111, 20–30 (2018)CrossRefGoogle Scholar
  9. 9.
    de Barros, F.P.J., Fiori, A.: First-order based cumulative distribution function for solute concentration in heterogeneous aquifers: theoretical analysis and implications for human health risk assessment. Water Resour. Res. 50(5), 4018–4037 (2014)CrossRefGoogle Scholar
  10. 10.
    Destouni, G., Graham, W.: The influence of observation method on local concentration statistics in the subsurface. Water Resour. Res. 33(4), 663–676 (1997)CrossRefGoogle Scholar
  11. 11.
    Einstein, A.: On the movement of small particles suspended in stationary liquids required by the molecular kinetic theory of heat. Ann. Phys. 17, 549–560 (1905)CrossRefGoogle Scholar
  12. 12.
    Fiori, A.: On the influence of local dispersion in solute transport through formations with evolving scales of heterogeneity. Water Resour. Res. 37(2), 235–242 (2001)CrossRefGoogle Scholar
  13. 13.
    Gardiner, C.: Stochastic Methods. Springer, Berlin (2009)zbMATHGoogle Scholar
  14. 14.
    Herz, M.: Mathematical modeling and analysis of electrolyte solutions. Ph.D. thesis, Erlangen-Nuremberg University (2014).
  15. 15.
    Karapiperis, T., Blankleider, B.: Cellular automaton model of reaction-transport processes. Physica D 78, 30–64 (1994)CrossRefGoogle Scholar
  16. 16.
    Kloeden, P.E., Platen, E.: Numerical Solutions of Stochastic Differential Equations. Springer, Berlin (1999)zbMATHGoogle Scholar
  17. 17.
    Knabner, P., Angermann, L.: Numerical Methods for Elliptic and Parabolic Partial Differential Equations. Springer, New York (2003)zbMATHGoogle Scholar
  18. 18.
    Kräutle, S., Knabner, P.: A new numerical reduction scheme for fully coupled multicomponent transport-reaction problems in porous media. Water Resour. Res. 41(9), W09414 (2005)CrossRefGoogle Scholar
  19. 19.
    Li, J., Xiu, D.: A generalized polynomial chaos based ensemble Kalman filter with high accuracy. J. Comput. Phys. 228, 5454–5469 (2009)MathSciNetCrossRefGoogle Scholar
  20. 20.
    List, F., Radu, F.A.: A study on iterative methods for solving Richards’ equation. Comput. Geosci. 20(2), 341–353 (2016)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Liu, Y., Kitanidis, P.K.: A mathematical and computational study of the dispersivity tensor in anisotropic porous media. Adv. Water Resour. 62, 303–316 (2013)CrossRefGoogle Scholar
  22. 22.
    Meyer, D.W., Jenny, P., Tchelepi, H.A.: A joint velocity-concentration PDF method for tracer flow in heterogeneous porous media. Water Resour. Res. 46, W12522 (2010)Google Scholar
  23. 23.
    Müller, I.: Thermodynamics. Pitman, Boston (1985)zbMATHGoogle Scholar
  24. 24.
    Naff, R.L., Haley, D.F., Sudicky, E.A.: High-resolution Monte Carlo simulation of flow and conservative transport in heterogeneous porous media 1. Methodology and flow results. Water Resour. Res. 34(4), 663–677 (1998)CrossRefGoogle Scholar
  25. 25.
    Pasetto, D., Guadagnini, A., Putti, M.: POD-based Monte Carlo approach for the solution of regional scale groundwater flow driven by randomly distributed recharge. Adv. Water Resour. 34(11), 1450–1463 (2011)CrossRefGoogle Scholar
  26. 26.
    Radu, F.A., Suciu, N., Hoffmann, J., Vogel, A., Kolditz, O., Park, C.-H., Attinger, S.: Accuracy of numerical simulations of contaminant transport in heterogeneous aquifers: a comparative study. Adv. Water Resour. 34, 47–61 (2011)CrossRefGoogle Scholar
  27. 27.
    Radu, F.A., Nordbotten, J.M., Pop, I.S., Kumar, K.: A robust linearization scheme for finite volume based discretizations for simulation of two-phase flow in porous media. J. Comput. Appl. Math. 289, 134–141 (2015)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Scheidegger, A.E.: General theory of dispersion in porous media. J. Geophys. Res. 66(10), 3273–3278 (1961)CrossRefGoogle Scholar
  29. 29.
    Srzic, V., Cvetkovic, V., Andricevic, R., Gotovac, H.: Impact of aquifer heterogeneity structure and local-scale dispersion on solute concentration uncertainty. Water Resour. Res. 49(6), 3712–3728 (2013)CrossRefGoogle Scholar
  30. 30.
    Suciu, N.: Spatially inhomogeneous transition probabilities as memory effects for diffusion in statistically homogeneous random velocity fields. Phys. Rev. E 81, 056301 (2010)CrossRefGoogle Scholar
  31. 31.
    Suciu, N.: Diffusion in random velocity fields with applications to contaminant transport in groundwater. Adv. Water Resour. 69, 114–133 (2014)CrossRefGoogle Scholar
  32. 32.
    Suciu, N., Radu, F.A., Prechtel, A., Brunner, F., Knabner, P.: A coupled finite element–global random walk approach to advection-dominated transport in porous media with random hydraulic conductivity. J. Comput. Appl. Math. 246, 27–37 (2013)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Suciu, N., Schüler, L., Attinger, S., Knabner, P.: Towards a filtered density function approach for reactive transport in groundwater. Adv. Water Resour. 90, 83–98 (2016)CrossRefGoogle Scholar
  34. 34.
    Vamoş, C., Suciu, N., Vereecken, H.: Generalized random walk algorithm for the numerical modeling of complex diffusion processes. J. Comput. Phys. 186(2), 527–244 (2003)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Vamoş, C., Crăciun, M., Suciu, N.: Automatic algorithm to decompose discrete paths of fractional Brownian motion into self-similar intrinsic components. Eur. Phys. J. B 88, 250 (2015)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Nicolae Suciu
    • 1
    • 2
  1. 1.Department of MathematicsFriedrich-Alexander University of Erlangen-NürnbergErlangenGermany
  2. 2.Tiberiu Popoviciu Institute of Numerical AnalysisCluj-Napoca Branch of the Romanian AcademyCluj-NapocaRomania

Personalised recommendations