Transport in Porous Media

, Volume 123, Issue 1, pp 173–193 | Cite as

Time and Space Fractional Diffusion in Finite Systems

  • R. RaghavanEmail author
  • C. Chen


Spatiotemporal nonlocal diffusion in a bounded system is addressed by considering fractional diffusion in a linear, composite system. By considering limiting conditions, solutions for combinations of Neumann and Dirichlet boundary conditions (either zero or nonzero) at the ends of a finite system are derived in terms of Mittag–Leffler functions by the Laplace transformation. Computational viability is demonstrated by inverting the solutions numerically and comparing resulting calculations with asymptotic solutions. Time and space fractional derivatives, defined by variables \(\alpha \) and \(\beta \), respectively, are employed in the Caputo sense; a single-sided, asymmetric space derivative is used. Inspection of the asymptotic solutions leads to insights on the structure of the solutions that may not be available otherwise; the resulting deductions are verified through the numerical inversions. For pure superdiffusion, characteristics of some of the solutions presented here are very similar to those of classical diffusion but combined effects for the corresponding situation result in power-law behaviors. Incidentally, to our knowledge, the pressure distribution for space fractional diffusion at long enough times in a finite system is derived based on first principles for the first time.


Fine-scale complexity Pressure behavior Fractured rocks Anomalous diffusion Fractional diffusion Finite systems 

List of symbols


Compressibility (L T\(^2\)/M)

\(E_{\alpha , \beta }( z)\)

Mittag–Leffler function; see (Eqs. 4,  5)

\(f(\beta )\)

See Eq. 45


Thickness (L)


Permeability (L\(^2\))

\({k_{\alpha , \beta }}\)

See Eq. 2


Width of linear system (L)


Pressure (M/L/T\(^2\))


Flux (L\(^3\)/T)


Absolute temperature


Time (T)


See Eq. 33


Distance to interface (L)

\(\alpha \)

Exponent, time

\(\beta \)

Exponent, space

\(\varGamma (x)\)

Gamma function

\(\eta \)

Diffusivity; various

\({\tilde{\eta }} \)

‘Diffusivity’; see Eq. 18 (L\(^{\beta +1}\)/T\(^\alpha \))

\(\kappa \)

Second term of Eq. 41

\(\lambda \)

Mobility (L T/M)

\(\mu \)

Viscosity (M/L/T)

\(\phi \)

Porosity (L\(^3\)/L\(^3\))



Constant rate




Coordinate, initial


Constant pressure

1, 2



Laplace transform


  1. Belayneh, M., Masihi, M., Matthäi, S.K., King, P.R.: Prediction of vein connectivity using the percolation approach: model test with field data. J. Geophys. Eng. 33, 219–229 (2006)CrossRefGoogle Scholar
  2. Bisdom, K., Bertotti, G., Nick, H.M.: The impact of different aperture distribution models and critical stress criteria on equivalent permeability in fractured rocks. J. Geophys. Res. Solid Earth 121(5), 2169–9356 (2016). CrossRefGoogle Scholar
  3. Cacas, M.C., Daniel, J.M., Letouzey, J.: Nested geological modelling of naturally fractured reservoirs. Petrol. Geosci. 7, S43–S52 (2001). CrossRefGoogle Scholar
  4. Camacho-Velázquez, R., Fuentes-Cruz, G., Vásquez-Cruz, M.: Decline-curve analysis of fractured reservoirs with fractal geometry. SPE Reserv. Eval. Eng. 11(3), 606–619 (2008)Google Scholar
  5. Caputo, M.: Linear models of dissipation whose Q is almost frequency independent-II. Geophys. J. R. Astron. Soc. 13(5), 529–539 (1967)CrossRefGoogle Scholar
  6. Carslaw, H.S., Jaeger, J.C.: Conduction of Heat in Solids, 2nd edn, 510 pp. Clarendon Press, Oxford (1959)Google Scholar
  7. Chen, C., Raghavan, R.: Transient flow in a linear reservoir for space-time fractional diffusion. J. Petrol. Sci. Eng. 128, 194–202 (2015)CrossRefGoogle Scholar
  8. Chen, Z., Liao, X., Zhao, X., Dou, X., Zhu, L.: Development of a trilinear-flow model for carbon sequestration in depleted shale. SPE J. Soc. Petrol. Eng. 21(4), 1386–1399 (2016)Google Scholar
  9. Chu, W., Pandya, N., Flumerfelt, R.W., Chen, C.: Rate-Transient analysis based on power-law behavior for permian wells. In: SPE-187180-MS Presented at the 2017 SPE Annual Technical Conference and Exhibition held in San Antonio, Texas, 9–11 October (2017)Google Scholar
  10. Dassas, Y., Duby, Y.: Diffusion toward fractal interfaces, potentiostatic, galvanostatic, and linear sweep voltammetric techniques. J. Electrochem. Soc. 142(12), 4175–4180 (1995)CrossRefGoogle Scholar
  11. Defterli, O., D’Elia, M., Du, Q., Gunzburger, M., Lehoucq, R., Meerschaert, M.M.: Fractional diffusion on bounded domains. Fract. Calc. Appl. Anal. 18(2), 342–360 (2015)CrossRefGoogle Scholar
  12. Erdelyi, A., Magnus, W.F., Oberhettinger, F., Tricomi, F.G.: Higher Transcendental Functions, vol. 3, pp. 206–227. McGraw-Hill, New York (1955). (Chapter 18: Misellaneous Functions)Google Scholar
  13. Fomin, S., Chugunov, V., Hashida, T.: Mathematical modeling of anomalous diffusion in porous media. Fract. Diff. Calc. 1, 1–28 (2011)Google Scholar
  14. Fu, L., Milliken, K.L., Sharp Jr., J.M.: Porosity and permeability variations in fractured and liesegang-banded Breathitt sandstones (Middle Pennsylvanian), eastern Kentucky: diagenetic controls and implications for modeling dual-porosity systems. J. Hydrol. 154(1–4), 351–381 (1994)CrossRefGoogle Scholar
  15. Gorenflo, R., Loutchko, J., Luchko, Yu.: Computation of the Mittag-Leffler function and its derivatives. Fract. Calc. Appl. Anal. 5, 491–518 (2002)Google Scholar
  16. Hansen, E.R.: A Table of Series and Products, p. 523. Prentice-Hall, Englewood Cliffs (1975)Google Scholar
  17. Herrmann, R.: Fractional Calculus: An Introduction for Physicists, p. 261. World Scientific, Hackensack (2011)CrossRefGoogle Scholar
  18. Holy, R., Albinali, A., Sarak, H., Ozkan, E.: Modelling of 1D Anomalous Diffusion in Fractured Nanoporous Media. In: Presented at the Low Permeability Media and Nanoporous Materials From Characterization to Modeling, Can We Do Better?, Rueil-Malmaison, France. Oil & Gas Science and Technology-Revue d’IFP Energies nouvelles 71(4) (2016).
  19. Holy, R.W., Ozkan, E.: A Practical and Rigorous Approach for Production Data Analysis in Unconventional Wells, Presented at the SPE Low Perm Symposium, 5–6 May. Denver, Colorado (2016)Google Scholar
  20. Ilic, M., Liu, F., Turner, I., Anh, V.: Numerical approximation of a fractional-in-space diffusion equation (II)-with nonhomogeneous boundary conditions. Fract. Calcul. Appl. Anal. 9(4), 333–349 (2006)Google Scholar
  21. Kang, P.K., Le Borgne, T., Dentz, M., Bour, O., Juanes, R.: Impact of velocity correlation and distribution on transport in fractured media: field evidence and theoretical model. Water Resourc. Res. 51(2), 940–959 (2015). CrossRefGoogle Scholar
  22. Le Mẽhautẽ, A., Crepy, G.: Introduction to transfer and motion in fractal media: the geometry of kinetics. Solid State Ion. 1(9—-10), 17–30 (1983)Google Scholar
  23. Lenormand, R.: On use of fractional derivatives for fluid flow in heterogeneous media. In: Proceedings 3rd European Conference on the Mathematics of Oil Recovery, Delft, The Netherlands (1992)Google Scholar
  24. Magin, R.L., Ingo, C., Colon-Perez, L., Triplett, W., Mareci, T.H.: Characterization of anomalous diffusion in porous biological tissues using fractional order derivatives and entropy. Microporous Mesoporous Mater. 178(15), 39–43 (2013). CrossRefGoogle Scholar
  25. Metzler, R., Glockle, W.G., Nonnenmacher, T.F.: Fractional model equation for anomalous diffusion. Phys. A 211(1), 13–24 (1994)CrossRefGoogle Scholar
  26. Metzler, R., Chechkin, A.V., Goncharb, VYu., Klafter, J.: Some fundamental aspects of Lévy flights. Chaos Solitons Fract. 34(1), 129–142 (2007)CrossRefGoogle Scholar
  27. Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations, p. 384. Wiley, London (1993)Google Scholar
  28. Molz III, F.J., Fix III, G.J., Lu, S.S.: A physical interpretation for the fractional derivative in Levy diffusion. Appl. Mathe. Lett. 15(7), 907–911 (2002)CrossRefGoogle Scholar
  29. Nigmatullin, R.R.: To the theoretical explanation of the universal response. Phys. Stat. Solidi B Basic Res. 123(2), 739–745 (1984)CrossRefGoogle Scholar
  30. Obembe, A.D., Hasan, M., Fraim, M.: A Mathematical model for transient testing of naturally fractured shale gas reservoirs. In: Presented at the SPE Kingdom of Saudi Arabia Annual Technical Symposium and Exhibition, 24–27 April, Dammam, Saudi Arabia, Society of Petroleum Engineers. (2017)
  31. Oldham, K.B., Spanier, J.: The Fractional Calculus; Theory and Applications of Differentiation and Integration to Arbitrary Order, p. 234. Academic Press, New York (1974)Google Scholar
  32. Ozcan, O., Sarak, H., Ozkan, E., Raghavan, R.: A trilinear flow model for a fractured horizontal well in a fractal unconventional reservoir, SPE 170971, Presented at the SPE Annual Technical Conference and Exhibition, Amsterdam, The Netherlands (2014)Google Scholar
  33. Podlubny, I.: Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, Some Methods of Their Solution and Some of Their Applications, p. 340. Academic Press, New York (1998)Google Scholar
  34. Povstenko, Y.: Linear Fractional Diffusion-Wave Equation for Scientists and Engineers, p. 460. Birkhäuser, Basel (2015)CrossRefGoogle Scholar
  35. Raghavan, R.: Well Test Analysis Prentice Hall, p. 558. Englewoods Cliffs, New Jersey (1993)Google Scholar
  36. Raghavan, R.: Fractional derivatives: application to transient flow. J. Pet. Sci. Eng. 80, 7–13 (2011)CrossRefGoogle Scholar
  37. Raghavan, R., Chen, C.: Fractured-well performance under anomalous diffusion. SPE Res. Eval. Eng. 16(3), 237–245 (2013). CrossRefGoogle Scholar
  38. Raghavan, R., Chen, C.: Rate decline, power laws, and subdiffusion in fractured rocks. SPE Res. Eval. Eng. 20(3), 738–751 (2017)Google Scholar
  39. Raghavan, R., Hadinoto, N.: Analysis of pressure data for fractured wells: the constant-pressure outer boundary. Soc. Petrol. Eng. J. 18(2), 139–150 (1978)CrossRefGoogle Scholar
  40. Raghavan, R., Chen, C., DaCunha, J.J.: Nonlocal diffusion in fractured rocks. SPE Res. Eval. Eng. 20(2), 237–245 (2017). CrossRefGoogle Scholar
  41. Reiss, L.H.: The Reservoir Engineering Aspects of Fractured Formations, p. 108. Editions TECHNIP, Paris (1980)Google Scholar
  42. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications, p. 976. Gordon and Breach Science Publishers, Philadelphia (1993)Google Scholar
  43. Sharp Jr., J.M., Kreisel, I., Milliken, K.L., Mace, R.E., Robinson, N.I.: Fracture skin properties and effects on solute transport: geotechnical and environmental implications. In: Aubertin, M., Hassam, F., Mitri, H. (eds.) Rock Mechanics, Tools And Techniques. Balkema, Rotterdam (1996)Google Scholar
  44. Stehfest, H.: Algorithm 368: numerical inversion of Laplace transforms [D5]. Commun. ACM 13(1), 47–49 (1970a)CrossRefGoogle Scholar
  45. Stehfest, H.: Remark on algorithm 368: numerical inversion of Laplace transforms. Commun. ACM 13(10), 624 (1970b)CrossRefGoogle Scholar
  46. Uchaikin, V.V.: Fractional Derivatives for Physicists and Engineers Volume I Background and Theory, p. 384. Springer, New York (2013)CrossRefGoogle Scholar
  47. van Everdingen, A.F., Hurst, W.: The application of the LaPlace transformation to flow problems in reservoirs. Trans. AIME 186, 305–324 (1949)Google Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.R. Raghavan Inc.TulsaUSA
  2. 2.Kappa EngineeringHoustonUSA

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