Stochastic transport in heterogeneous media with multiple families of transport paths

Barry D. Hughes; Muhammad Sahimi

doi:10.1103/PhysRevE.48.2776

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Stochastic transport in heterogeneous media with multiple families of transport paths

Barry D. Hughes

Muhammad Sahimi

Department of Mathematics, University of Melbourne, Parkville, Victoria 3052, Australia

Department of Chemical Engineering, University of Southern California, Los Angeles, California 90089-1211
Hochstleistungsrechenzentrum Super Computer Center, c/o Kernforschungsanlage Jülich G.m.b.H., D-52452 Jülich 1, Germany

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Phys. Rev. E 48, 2776 – Published 1 October, 1993

DOI: https://doi.org/10.1103/PhysRevE.48.2776

Abstract

We examine in detail a model for transport in heterogeneous solids and porous media which contain N distinct families of transport paths (with N≥2), recently proposed by the authors [Phys. Rev. Lett. 70, 2581 (1993)]. The model is relevant to transport in metals, polycrystals, porous catalysts, coalbed methane reservoirs, and geological systems with fractures and pores. We develop a number of exact results for the one-dimensional case and, more generally, study the behavior of its effective transport properties using an effective-medium approximation, which yields several exact results in one spatial dimension.

References (22)

For a recent review, see M. Sahimi, Rev. Mod. Phys. (to be published). A more extensive mathematical treatment of some aspects may be found in B. D. Hughes, Random Environments and Random Walks (Oxford University Press, London, 1993).
Z.-X. Chen, Trans. Porous Media 4, 147 (1989).
J. C. S. Long and D. Billaux, Water Resour. Res. 23, 1201 (1987); K. Hestir and J. C. S. Long, J. Geophys. Res. B 95, 21 565 (1990).
D. Abdassah and I. Ershaghi, SPE Formation Evaluation 1, 113 (1986).
S. Whitaker, Trans. Porous Media 1, 3 (1986); Chem. Eng. Sci. 41, 2029 (1986).
M. Sahimi and V. L. Jue, Phys. Rev. Lett. 62, 629 (1989); M. Sahimi, J. Chem. Phys. 96, 4718 (1992).
E. C. Aifantis, J. Appl. Phys. 50, 1334 (1979); E. C. Aifantis and J. M. Hill, Q. J. Mech. Appl. Math. 33, 1 (1980).
B. D. Hughes and M. Sahimi, Phys. Rev. Lett. 70, 2581 (1993).
M. F. Shlesinger and U. Landman, in Applied Stochastic Processes, edited by G. Adomian (Academic, New York, 1980), p. 151.
J. M. Hill and B. D. Hughes, J. Aust. Math. Soc., Ser. B 27, 73 (1985).
S. Alexander, J. Bernasconi, W. R. Schneider and R. Orbach, Rev. Mod. Phys. 53, 175 (1981).
T. Odagaki and M. Lax, Phys. Rev. B 24, 5286 (1981); I. Webman, Phys. Rev. Lett. 47, 1496 (1981).
M. Sahimi, B. D. Hughes, L. E. Scriven and H. T. Davis, J. Chem. Phys. 78, 6849 (1983).
B. D. Hughes and M. Sahimi, J. Stat. Phys. 29, 781 (1982); B. D. Hughes, M. Sahimi and H. T. Davis, Physica 120A, 515 (1983).
Omitted end note.
W. Feller, An Introduction to Probability and its Applications, 2nd ed. (Wiley, New York, 1971), Vol. 2.
This EMA is the time-dependent multiple transport path analog of the EMA for steady-state transport in single-path systems, developed by S. Kirkpatrick, Rev. Mod. Phys. 45, 574 (1973).
For refined EMA's with one family of transport paths, based on clusters of bonds, see J. A. Blackman, J. Phys. C 9, 2049 (1976), and G. Ahmed and J. A. Blackman, ibid. 12, 837 (1979) for the steady-state transport; and M. Sahimi, 17, 3957 (1984) for the transient case.
See M. Sahimi, B. D. Hughes, L. E. Scriven and H. T. Davis, Phys. Rev. B 28, 307 (1983) for steady-state transition disorder with one family of transport paths.
K. H. Coats and B. D. Smith, Soc. Pet. Eng. J. 4, 73 (1964).
M. Sahimi, M. C. Robertson and C. G. Sammis, Phys. Rev. Lett. 70, 2186 (1993).
M. Sahimi, B. D. Hughes, L. E. Scriven and H. T. Davis, J. Phys. A 16, L67 (1983); J. A. Given and G. Stell, ibid. 24, 3369 (1991).

[c1] For a recent review, see M. Sahimi, Rev. Mod. Phys. (to be published). A more extensive mathematical treatment of some aspects may be found in B. D. Hughes, Random Environments and Random Walks (Oxford University Press, London, 1993).

[c2] Z.-X. Chen, Trans. Porous Media 4, 147 (1989).

[c3] J. C. S. Long and D. Billaux, Water Resour. Res. 23, 1201 (1987); K. Hestir and J. C. S. Long, J. Geophys. Res. B 95, 21 565 (1990).

[c4] D. Abdassah and I. Ershaghi, SPE Formation Evaluation 1, 113 (1986).

[c5] S. Whitaker, Trans. Porous Media 1, 3 (1986); Chem. Eng. Sci. 41, 2029 (1986).

[c6] M. Sahimi and V. L. Jue, Phys. Rev. Lett. 62, 629 (1989); M. Sahimi, J. Chem. Phys. 96, 4718 (1992).

[c7] E. C. Aifantis, J. Appl. Phys. 50, 1334 (1979); E. C. Aifantis and J. M. Hill, Q. J. Mech. Appl. Math. 33, 1 (1980).

[c8] B. D. Hughes and M. Sahimi, Phys. Rev. Lett. 70, 2581 (1993).

[c9] M. F. Shlesinger and U. Landman, in Applied Stochastic Processes, edited by G. Adomian (Academic, New York, 1980), p. 151.

[c10] J. M. Hill and B. D. Hughes, J. Aust. Math. Soc., Ser. B 27, 73 (1985).

[c11] S. Alexander, J. Bernasconi, W. R. Schneider and R. Orbach, Rev. Mod. Phys. 53, 175 (1981).

[c12] T. Odagaki and M. Lax, Phys. Rev. B 24, 5286 (1981); I. Webman, Phys. Rev. Lett. 47, 1496 (1981).

[c13] M. Sahimi, B. D. Hughes, L. E. Scriven and H. T. Davis, J. Chem. Phys. 78, 6849 (1983).

[c14] B. D. Hughes and M. Sahimi, J. Stat. Phys. 29, 781 (1982); B. D. Hughes, M. Sahimi and H. T. Davis, Physica 120A, 515 (1983).

[c15] Omitted end note.

[c16] W. Feller, An Introduction to Probability and its Applications, 2nd ed. (Wiley, New York, 1971), Vol. 2.

[c17] This EMA is the time-dependent multiple transport path analog of the EMA for steady-state transport in single-path systems, developed by S. Kirkpatrick, Rev. Mod. Phys. 45, 574 (1973).

[c18] For refined EMA's with one family of transport paths, based on clusters of bonds, see J. A. Blackman, J. Phys. C 9, 2049 (1976), and G. Ahmed and J. A. Blackman, ibid. 12, 837 (1979) for the steady-state transport; and M. Sahimi, 17, 3957 (1984) for the transient case.

[c19] See M. Sahimi, B. D. Hughes, L. E. Scriven and H. T. Davis, Phys. Rev. B 28, 307 (1983) for steady-state transition disorder with one family of transport paths.

[c20] K. H. Coats and B. D. Smith, Soc. Pet. Eng. J. 4, 73 (1964).

[c21] M. Sahimi, M. C. Robertson and C. G. Sammis, Phys. Rev. Lett. 70, 2186 (1993).

[c22] M. Sahimi, B. D. Hughes, L. E. Scriven and H. T. Davis, J. Phys. A 16, L67 (1983); J. A. Given and G. Stell, ibid. 24, 3369 (1991).

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