Transport in Porous Media

, Volume 67, Issue 3, pp 413–430 | Cite as

Continuous time random walks and heat transfer in porous media

  • Simon Emmanuel
  • Brian BerkowitzEmail author


An approach to describe heat transfer in porous media is presented on the basis of the continuous time random walk (CTRW) framework. CTRW is capable of quantifying both local equilibrium and non-equilibrium heat transfer in heterogeneous domains, and is shown here to match published experimental data of non-equilibrium thermal breakthrough. It is argued that CTRW will be particularly applicable to the quantification of heat transfer in naturally heterogeneous geological systems, such as soils and geothermal reservoirs.


Anomalous transport Convection Dispersion Non-Fourier Thermal breakthrough 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Abramowitz, M., Stegun, I.: Handbook of Mathematical Functions. Dover Publications Inc., New York (1970)Google Scholar
  2. Alazmi B., Vafai K. (2000). Analysis of variants within the porous media transport models. J. Heat Trans. 122, 303–326CrossRefGoogle Scholar
  3. Amiri A., Vafai K. (1994). Analysis of dispersion effects and nonthermal equilibrium, non-Darcian, variable porosity, incompressible flow through porous media. Int. J. Heat Mass Transf. 37, 939—954CrossRefGoogle Scholar
  4. Berkowitz B., Klafter J., Metzler R., Scher H. (2002). Physical pictures of transport in heterogeneous media: advection-dispersion, random walk and fractional derivative formulations. Water Resour. Res. 38, 1191 doi:10.1029/2001WR001030CrossRefGoogle Scholar
  5. Berkowitz B., Kosakowski G., Margolin G., Scher H. (2001). Application of continuous time random walk theory to tracer test measurements in fractured and heterogeneous porous media. Ground Water 39, 593–604CrossRefGoogle Scholar
  6. Berkowitz B., Scher H., Silliman S.E. (2000). Anomalous transport in laboratory-scale, heterogeneous porous media. Water Resour. Res. 36(1):149–158CrossRefGoogle Scholar
  7. Cortis A., Berkowitz B. (2004). Anomalous transport in classical soil and sand columns. Soil Sci. Soc. Am. J. 68, 1539–1548CrossRefGoogle Scholar
  8. Cortis A., Berkowitz B. (2005). Computing anomalous contaminant transport in porous media: the CTRW MATLAB toolbox. Ground Water 43, 947–950CrossRefGoogle Scholar
  9. Cortis A., Gallo C., Scher H., Berkowitz B. (2004). Numerical simulation of non-Fickian transport in geological formations with multiple-scale heterogeneities. Water Resour. Res. 40, W04209 doi:10.1029/2003WR002750CrossRefGoogle Scholar
  10. Dawe R.A. (1991). Enhancing oil-recovery. J. Chem. Technol. Biot. 51, 361–393Google Scholar
  11. de Hoog F.R., Knight J.H., Stokes A.N. (1982). An improved method for numerical inversion of Laplace transforms. SIAM J. Sci. Stat. Comput. 3, 357–366CrossRefGoogle Scholar
  12. Dentz M., Berkowitz B. (2003). Transport behavior of a passive solute in continuous time random walks and multirate mass transfer. Water Resour. Res. 39(5):1111 doi:10.1029/2001WR001163CrossRefGoogle Scholar
  13. Dentz M., Cortis A., Scher H., Berkowitz B. (2004). Time behavior of solute transport in heterogeneous media: transition from anomalous to normal transport. Adv. Water Resour. 27, 155–173 doi:10.1016/j.advwatres.2003.11.002CrossRefGoogle Scholar
  14. Dixon A.G., Cresswell D.L. (1979). Theoretical prediction of effective heat transfer parameters in packed beds. AIChE J. 25, 663–676CrossRefGoogle Scholar
  15. Hoffman B.T., Kovscek A.R. (2004). Efficiency and oil recovery mechanisms of steam injection into low permeability, hydraulically fractured reservoirs. Petrol. Sci. Technol. 22, 537–564CrossRefGoogle Scholar
  16. Hwang G.J., Chao C.H. (1994). Heat transfer measurement and analysis for sintered porous channels. J. Heat Trans. 116, 456–464CrossRefGoogle Scholar
  17. Hwang G.J., Wu C.C., Chao C.H. (1995). Investigation of non-Darcian forced convection in an asymmetrically heated sintered porous channel. J. Heat Trans. 117, 725—732Google Scholar
  18. Kenkre V.M., Montroll E.W., Shlesinger M.F. (1973). Generalized master equations for continuous time random walks. J. Stat. Phys. 9, 45–50CrossRefGoogle Scholar
  19. Kim J., Park Y., Harmon T.C. (2005). Real-time model parameter estimation for analyzing transport in porous media. Ground Water Monit. R. 25, 78–86CrossRefGoogle Scholar
  20. Klafter J., Silbey R. (1980). Derivation of continuous time random walk equations. Phys. Rev. Lett. 44, 55–58CrossRefGoogle Scholar
  21. Kosakowski G., Berkowitz B., Scher H. (2001). Analysis of field observations of tracer transport in a fractured till. J. Contam. Hydrol. 47, 29–51CrossRefGoogle Scholar
  22. Levec J., Carbonell R.G. (1985). Longitudinal and lateral thermal dispersion in packed beds, Part II: comparison between theory and experiment. AIChE J. 31, 591–602CrossRefGoogle Scholar
  23. Liang C.Y.. Yang W.J. (1975). Modified single-blow technique for performance evaluation on heat transfer surfaces. J. Heat Trans. 96, 16–21Google Scholar
  24. Malate R.C.M., O’Sullivan M.J. (1991). Modelling of chemical and thermal changes in well pn-26 Palinpinon geothermal field, Philippines. Geothermics 260, 291–318CrossRefGoogle Scholar
  25. Montroll E.W., Scher H. (1973). Random walks on lattices. IV: continuous time random walks and influence of absorbing boundaries. J. Stat. Phys. 9(2):101–135CrossRefGoogle Scholar
  26. Nield, D.A., Bejan, A.: Convection in Porous Media. Springer-Verlag, New York (1992)Google Scholar
  27. Oballa V., Coombe D.A. Buchanan W.L. (1993). Factors affecting the thermal response of naturally fractured reservoirs. J. Can. Petrol. Technol. 32, 31–42Google Scholar
  28. Ochsner T.E., Horton R., Kluitenberg G.J., Wang Q.J. (2005). Evaluation of the heat pulse ratio method for measuring soil water flux, Soil Sci. Soc. Am. J. 69, 757–769CrossRefGoogle Scholar
  29. Oppenheim, I., Shuler, K.E., Weiss, G.H.: The Master Equation. MIT Press, Cambridge (1977)Google Scholar
  30. Paek J.W., Kang B.H., Hyun J.M. (1999). Transient cool-down of a porous medium in pulsating flow. Int. J. Heat Mass Trans. 42, 3523–3527CrossRefGoogle Scholar
  31. Paek J.W., Kang B.H., Hyun J.M. (2001). An experimental study on cool-down of a heterogeneous porous body in throughflow. Int. J. Heat Mass Trans. 44, 683–687CrossRefGoogle Scholar
  32. Ren T., Kluitenberg G.J., Horton R. (2000). Determining soil water flux and pore water velocity by a heat pulse technique. Soil Sci. Soc. Am. J. 64, 552–560CrossRefGoogle Scholar
  33. Scher H., Montroll E.W. (1975). Anomalous transit time dispersion in amorphous solids. Phys. Rev. B 12(6):2455–2477CrossRefGoogle Scholar
  34. Shlesinger M.F. (1974). Asymptotic solutions of continuous time random walks. J. Stat. Phys. 10, 421–434CrossRefGoogle Scholar
  35. Shlesinger, M.F.: Random Processes. Encyclopedia of Applied Physics, vol. 16. VCH Publishers Inc., New York (1996)Google Scholar
  36. Steffanson V. (1997). Geothermal reinjection experience. Geothermics 26, 99–139CrossRefGoogle Scholar
  37. Vafai A., Alkire R.L., Tien C.L. (1985). An experimental investigation of heat transfer in variable porosity media. J. Heat Trans. 107, 642–647CrossRefGoogle Scholar
  38. Wang, Q.J., Ochsner, T.E., Horton, R.: Mathematical analysis of heat pulse signals for soil water flux determination. Water Resour. Res. 38, (1091) (2002)Google Scholar
  39. Wu C.C., Hwang G.J. (1998). Flow and heat transfer characteristics inside packed and fluidized beds. J. Heat Trans. 120, 667–673Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2006

Authors and Affiliations

  1. 1.Department of Environmental Sciences and Energy ResearchWeizmann Institute of ScienceRehovotIsrael

Personalised recommendations