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Modeling anomalous heat transport in geothermal reservoirs via fractional diffusion equations

  • Yury Luchko
  • Alessandro Punzi
Original Paper

Abstract

The aim of this article is to give an overview of the current research towards applications of fractional partial differential equations for the modeling of anomalous heat transfer in porous media. We start with presenting a physical background behind the anomalous processes described by the Continuous Time Random Walk (CTRW) model and arguing its feasibility for modeling of heat transport processes in heterogeneous media. From the CTRW model on the microscopic level, a macroscopic model in form of a generalized fractional diffusion equation is then deduced. Both mathematical analysis of the generalized fractional diffusion equations and some methods for their numerical treatment are presented. Finally, some open questions and directions for further work are suggested.

Keywords

Geothermal processes Fractional diffusion equation Anomalous heat transfer Continuous time random walk 

Mathematics Subject Classification (2000)

34K37 76S05 

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References

  1. Barenblatt G., Zheltov I., Kochina I.: Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks [Strata]. J. Appl. Math. Mech. 24(5), 1286–1303 (1960)zbMATHCrossRefGoogle Scholar
  2. Bazhlekova, E.: The abstract Cauchy problem for the fractional evolution equation. PhD thesis (1998)Google Scholar
  3. Berkowitz B., Klafter J., Metzler R., Scher H.: Physical pictures of transport in heterogeneous media: Advection-dispersion, random walk and fractional derivative formulations. Water Resour. Res. 38(10), 1191–1203 (2002)CrossRefGoogle Scholar
  4. Blank, L.: Numerical treatment of differential equations of fractional order. NAR Report (1996)Google Scholar
  5. Bondarenko A., Ivaschenko D.: Numerical methods for solving inverse problems for time fractional diffusion equation with variable coefficient. J. Inv. Ill-Posed Problems 17, 419–440 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  6. Chen C.M., Liu F., Turner I., Anh V.V.: A Fourier method for the fractional diffusion equation describing sub-diffusion. J. Comp. Phys. 227, 886–897 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  7. Chen S., Liu F.: ADI-Euler and extrapolation methods for the two-dimensional fractional advection-dispersion equation. J. Appl. Math. Comput. 26, 295–311 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  8. Cheng, J., Nakagawa, J., Yamamoto, M., Yamazaki, T.: Uniqueness in an inverse problem for one- dimensional fractional diffusion equation. Inverse Problems 25, 115002 (16 pp) (2009)Google Scholar
  9. Chowdhury A., Christov C.: Memory effects for the heat conductivity of random suspensions of spheres. Proc. Roy. Soc. A 466(2123), 3253–3273 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  10. Cuesta E., Lubich C., Palencia C.: Convolution quadrature time discretization of fractional diffusion-wave equations. Math. Comput. 75(254), 673 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  11. Dentz M., Cortis A., Scher H., Berkowitz B.: Time behavior of solute transport in heterogeneous media: transition from anomalous to normal transport. Adv. Water Resour. 27, 155–173 (2004)CrossRefGoogle Scholar
  12. Diethelm K.: An investigation of some nonclassical methods for the numerical approximation of Caputo-type fractional derivatives. Num. Alg. 47(4), 361–390 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  13. Diethelm K., Ford N.J., Freed A.D.: A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlin. Dyn. 29(1), 3–22 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  14. Diethelm K., Walz G.: Numerical solution of fractional order differential equations by extrapolation. Num. Alg. 16, 231–253 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  15. Eidelman S., Kochubei A.: Cauchy problem for fractional diffusion equations. J. Differ. Equ. 199(2), 211–255 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  16. Emmanuel S., Berkowitz B.: Continuous time random walks and heat transfer in porous media. Transp. Porous Media 67(3), 413–430 (2007)MathSciNetCrossRefGoogle Scholar
  17. Ervin V.J., Roop J.P.: Variational solution of fractional advection dispersion equations on bounded domains. Num. Methods Partial Differ. Equ. 23(2), 256–281 (2006)MathSciNetCrossRefGoogle Scholar
  18. Ford N.J., Simpson A.: The numerical solution of fractional differential equations: Speed versus accuracy. Num. Alg. 26(4), 333–346 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  19. Fulger, D., Scalas, E., Germano, G.: Monte Carlo simulation of uncoupled continuous time random walks yielding a stochastic solution of the space-time fractional diffusion equation. Phys. Rev. E 77(2), 021122 (7 pp) (2008)Google Scholar
  20. Geiger, S., Emmanuel, S.: Non-Fourier thermal transport in fractured geological media. Water Resour. Res. 46, W07504 (2010). doi: 10.1029/2009WR008671
  21. Gorenflo R., Mainardi F.: Some recent advances in theory and simulation of fractional diffusion processes. J. Comput. Appl. Math. 229, 400–415 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  22. Gorenflo, R., Mainardi, F.: Fractional calculus: integral and differential equations of fractional order. In: Carpinteri, A., Mainardi, F. (eds.) Fractals and Fractional Calculus in Continuum Mechanics, pp. 223–276. Springer, Wien, New York (1997)Google Scholar
  23. Gorenflo R., Luchko Y., Zabrejko P.: On solvability of linear fractional differential equations in Banach spaces. Fractional Calculus Appl. Anal. 2, 163–176 (1999)MathSciNetzbMATHGoogle Scholar
  24. Helmig, R., Niessner, J., Flemisch, B., Wolff, M., Fritz, J.: Efficient modeling of flow and transport in porous media using multiphysics and multiscale approaches. In: Freeden, W., Sonar, T., Nashed, M.Z. (eds.) Handbook of Geomathematics, pp. 417–457. Springer, Berlin (2010)Google Scholar
  25. Ilyasov, M., Ostermann, I., Punzi, A.: Modeling deep geothermal reservoirs: recent advances and future problems. In: Freeden, W., Sonar, T., Nashed, M.Z. (eds.) Handbook of Geomathematics, pp. 679–711. Springer, Berlin (2010)Google Scholar
  26. Kochubei A.N.: Fractional-order diffusion. Differ. Equ. 26, 485–492 (1990)MathSciNetzbMATHGoogle Scholar
  27. Langlands T.A.M., Henry B.I.: The accuracy and stability of an implicit solution method for the fractional diffusion equation. J. Comp. Phys. 205(2), 719–736 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  28. Li B., Wang J.: Anomalous heat conduction and anomalous diffusion in one-dimensional systems. Phys. Rev. Lett. 92, 044301 (2004)CrossRefGoogle Scholar
  29. Li B., Wang J., Wang L., Zhang G.: Anomalous heat conduction and anomalous diffusion in nonlinear lattices, single walled nanotubes, and billiard gas channels. Chaos 15(1), 15121 (2005)CrossRefGoogle Scholar
  30. Li X., Xu C.: A space-time spectral method for the time fractional diffusion equation. SIAM J. Num. Anal. 47(3), 2108 (2009)zbMATHCrossRefGoogle Scholar
  31. Lin Y., Xu C.: Finite difference/spectral approximations for the time-fractional diffusion equation. J. Comp. Phys. 225(2), 1533–1552 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  32. Liu F., Zhuang P., Anh V., Turner I., Burrage K.: Stability and convergence of the difference methods for the space-time fractional advection-diffusion equation. Appl. Math. Comp. 191(1), 12–20 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  33. Long J., Remer J., Wilson C., Witherspoon P.: Porous media equivalents for networks of discontinuous fractures. Water Resour. Res. 18(3), 645–658 (1982)CrossRefGoogle Scholar
  34. Lubich C.: Convolution quadrature revisited. BIT Num. Math. 44(3), 503–514 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  35. Luchko Y.: Maximum principle for the generalized time-fractional diffusion equation. J. Math. Anal. Appl. 351(1), 218–223 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  36. Luchko Y.: Some uniqueness and existence results for the initial-boundary-value problems for the generalized time-fractional diffusion equation. Comput. Math. Appl. 59(5), 1766–1772 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  37. Margolin G., Berkowitz B.: Application of continuous time random walks to transport in porous media. J. Phys. Chem. B 104(16), 3942–3947 (2000)CrossRefGoogle Scholar
  38. McLean W.: Discretization with variable time steps of an evolution equation with a positive-type memory term. J. Comp. Appl. Math. 69(1), 49–69 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  39. McLean W., Mustapha K.: A second-order accurate numerical method for a fractional wave equation. Num. Math. 105(3), 481–510 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  40. McLean W., Mustapha K.: Convergence analysis of a discontinuous Galerkin method for a sub-diffusion equation. Num. Alg. 52, 69–88 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  41. McLean W., Thomee V.: Maximum-norm error analysis of a numerical solution via Laplace transformation and quadrature of a fractional-order evolution equation. IMA J Num. Anal. 30(1), 208–230 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  42. Meerschaert M., Nane E., Vellaisamy P.: Fractional Cauchy problems on bounded domains. Ann. Probab. 37, 979–1007 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  43. Metzler R., Klafter J.: The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics. J. Phys. A 37(31), 161–208 (2004)MathSciNetCrossRefGoogle Scholar
  44. Montroll E., Weiss G.: Random walks on lattices. II. J. Math. Phys. 6, 167 (1965)MathSciNetCrossRefGoogle Scholar
  45. Mustapha K., McLean W.: Discontinuous Galerkin method for an evolution equation with a memory term of positive type. Math. Comp. 78, 1975–1995 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  46. Neuman, S.: Stochastic continuum representation of fractured rock permeability as an alternative to the REV and fracture network concepts. In: The 28th U.S. Symposium on Rock Mechanics (USRMS), pp. 331–362 (1988)Google Scholar
  47. Ochsner T.E.: Evaluation of the heat pulse ratio method for measuring soil water flux. J. Soil Sci. Soc. Am. 69(3), 757–765 (2005)CrossRefGoogle Scholar
  48. Podlubny I., Chechkin A., Skovranek T., Chen Y., Jara B.M.V.: Matrix approach to discrete fractional calculus II: partial fractional differential equations. J. Comp. Phys. 228(8), 3137–3153 (2009)zbMATHCrossRefGoogle Scholar
  49. Pruess, J.: Evolutionary integral equations and applications, 1st edn, pp. 392. Birkhäuser, Basel (1993). ISBN-10: 3764328762, ISBN-13: 978-3764328764Google Scholar
  50. Scalas E., Gorenflo R., Mainardi F.: Uncoupled continuous-time random walks: solution and limiting behavior of the master equation. Phys. Rev. E 69(1), 011107 (2004)MathSciNetCrossRefGoogle Scholar
  51. Scalas E.: Five years of continuous-time random walks in econophysics. Lect. Notes Econ. Math. Sys. 567, 3–16 (2006)MathSciNetCrossRefGoogle Scholar
  52. Schneider W., Wyss W.: Fractional diffusion and wave equations. J. Math. Phys. 30, 134–144 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  53. Valdes-Parada, F., Alberto, O., et al.: Effective medium equation for fractional Cattaneo’s diffusion and heterogeneous reaction in disordered porous media. Phys. A Stat. Mech. Appl. 369(2), 318–328 (2006)Google Scholar
  54. Yang Q., Liu F., Turner I.: Numerical methods for fractional partial differential equations with Riesz space fractional derivatives. Appl. Math. Model. 34(1), 200–218 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  55. Yuan L., Agrawal O.: A numerical scheme for dynamic systems containing fractional derivatives. J. Vib. Acoust. 124(2), 321–325 (2002)CrossRefGoogle Scholar
  56. Yuste S.: Weighted average finite difference methods for fractional diffusion equations. J. Comput. Phys. 216(1), 264–274 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  57. Zacher R.: Boundedness of weak solutions to evolutionary partial integro-differential equations with discontinuous coefficients. J. Math. Anal. Appl. 348(1), 137–149 (2008)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Beuth Hochschule für Technik BerlinFachbereich II Mathematik-Physik-ChemieBerlinGermany
  2. 2.Fraunhofer ITWMKaiserslauternGermany

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