Anomalous hydrodynamical dispersion and the transport with multiple families of paths in porous media

Manuel O. Cáceres

doi:10.1103/PhysRevE.69.036302

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Anomalous hydrodynamical dispersion and the transport with multiple families of paths in porous media

Manuel O. Cáceres^*

Comisión Nacional de Energía Atómica, Instituto Balseiro, and CONICET, Centro Atómico Bariloche, Rio Negro, 8400 Bariloche, Argentina
the Abdus Salam International Centre for Theoretical Physics, Strada Costeira, 11-34014 Trieste, Italy

^*Email address: caceres@cab.cnea.gov.ar

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Phys. Rev. E 69, 036302 – Published 23 March, 2004

DOI: https://doi.org/10.1103/PhysRevE.69.036302

Abstract

We investigate a family of probability distributions that shows anomalous hydrodynamic dispersion, by solving a particular class of coupled generalized master equations. The Fourier-Laplace solution is obtained analytically in terms of the matrix Green function method; then the Coats-Smith concentration profile is revisited in a particular case. Two models of disorder are worked out explicitly, and the mean current is asymptotically calculated. We present an approximation method to calculate the first passage time distribution for this stochastic transport process, and as an example an exact Markovian result is worked out; scaling results are also shown. We discuss the comparison with other different methods to work out complex diffusion phenomena in the presence of disordered multiple transport paths. Extensions when the models are nondiffusive can also be solved in the Fourier-Laplace representation.

References (37)

M. Sahimi, Rev. Mod. Phys. 65, 1393 (1993).
H.A. Deans, Soc. Pet. Eng. J. 3, 49 (1963).
K.H. Coats and B.D. Smith, Soc. Pet. Eng. J. 4, 73 (1964).
L.E. Baker, Soc. Pet. Eng. J. 17, 219 (1977).
J.C. Bacri, N. Rakotomalla, and D. Salin, Phys. Fluids A 2, 674 (1990).
G.A. Gist, A.H. Thompson, A.J. Katz, and R.L. Higgins, Phys. Fluids A 2, 1533 (1990).
U. Landman, E.W. Montroll, and M.F. Shlesinger, Proc. Natl. Acad. Sci. U.S.A. 74, 430 (1977).
M.O. Cáceres and H.S. Wio, Z. Phys. B: Condens. Matter 54, 175 (1984).
C.E. Budde and M.O. Cáceres, Phys. Rev. Lett. 66, 2712 (1988).
M.O. Cáceres, H. Schnorer, and A. Blumen, Phys. Rev. A 42, 4462 (1990).
Z.X. Chen and J. Yau, Transp. Porous Media 2, 145 (1987).
K. Hestir and J.C.S. Long, J. Geophys. Res. 95, 21 565 (1990).
B.D. Hughes and M. Sahimi, Phys. Rev. Lett. 70, 2581 (1993).
B.D. Hughes and M. Sahimi, Phys. Rev. E 48, 2776 (1993).
J.W. Haus and K.W. Kerh, Phys. Rep. 150, 263 (1987).
U. Landman and M.F. Shlesinger, Phys. Rev. B 19, 6207 (1979); ibid.19, 6220 (1979).
Note that for a given realization of the disorder, the matrix $Ψ_{ll} {(j, j}^{'}, t)$ characterizes a composite Markovian process; see Ref. [18].
N. G. van Kampen, Stochastic Process in Physics and Chemistry, 2nd ed. (North-Holland, Amsterdam, 1992).
It is interesting to note that the off-diagonal elements $Ψ_{{ll}^{'}} (j, j, t)$ of the supermatrix Ψ are responsible of the exchange matrix $E_{j}$ in Refs. [13][14].
GH. Weiss, Aspect and Applications of the Random Walk (North-Holland, Amsterdam, 1994).
M. O. Cáceres, Elementos de Estadística de no Equilibrio y sus Aplicaciones al Transporte en Medios Desordenados (Reverté S.A., Barcelona, 2003).
H. Scher and E.W. Montroll, Phys. Rev. B 12, 2455 (1975) [see Appendix D].
M.O. Cáceres, Phys. Rev. A 33, 647 (1986).
S. Bustingory, M.O. Cáceres, and E.R. Reyes, Phys. Rev. B 65, 165205 (2002).
B. Berkowitz and H. Scher, Phys. Rev. Lett. 79, 4038 (1997).
Here we have used the result $\int_{- \infty}^{+ \infty} {(p + 2 q k + k}^{2})^{- 1} e^{(- i | a | k)} {d k = e}^{+ i q | a |} π e^{- | a | \sqrt{p - q^{2}}} / \sqrt{p - q^{2}} .$
J.B.T.M. Roerdink and K.E. Shuler, J. Stat. Phys. 40, 205 (1985).
M. O. Cáceres, L. Insua, and E. R. Reyes (unpublished).
Here we have used the theorem of the Laplace shift, and the explicit result $\int_{0}^{+ \infty} e^{- ut} \exp (- x^{2} {/ 4 t) / t}^{3 / 2} d t = 2 \sqrt{π} e^{- | x | \sqrt{u}} / | x | .$
These were numerically computed using the Laplace inversion program LAPIN, see G. Honig and U. Hirdes, J. Comput. Appl. Math. 10, 113 (1984).
E. Hernández-García and M.O. Cáceres, Phys. Rev. A 42, 4503 (1990).
A. Compte and M.O. Caceres, Phys. Rev. Lett. 81, 3140 (1998).
G. Matheron and G. de Marsily, Water Resour. Res. 16, 901 (1980).
T. Odagaki and M. Lax, Phys. Rev. B 24, 5284 (1981).
G. H. Weiss and R. J. Rubin, in Advances in Chemical Physics, editd by I. Progogine and S. A. Rice (Wiley, New York, 1983), Vol. 52.
P.G. de Gennes, J. Fluid Mech. 135, 189 (1983).
J.P. Bouchaud and A. Georges, C. R. Acad. Sci., (Ser. II)307, 1431 (1988); see also p. 245 of the review Phys. Rep. 195, 127 (1990).

[c1] M. Sahimi, Rev. Mod. Phys. 65, 1393 (1993).

[c2] H.A. Deans, Soc. Pet. Eng. J. 3, 49 (1963).

[c3] K.H. Coats and B.D. Smith, Soc. Pet. Eng. J. 4, 73 (1964).

[c4] L.E. Baker, Soc. Pet. Eng. J. 17, 219 (1977).

[c5] J.C. Bacri, N. Rakotomalla, and D. Salin, Phys. Fluids A 2, 674 (1990).

[c6] G.A. Gist, A.H. Thompson, A.J. Katz, and R.L. Higgins, Phys. Fluids A 2, 1533 (1990).

[c7] U. Landman, E.W. Montroll, and M.F. Shlesinger, Proc. Natl. Acad. Sci. U.S.A. 74, 430 (1977).

[c8] M.O. Cáceres and H.S. Wio, Z. Phys. B: Condens. Matter 54, 175 (1984).

[c9] C.E. Budde and M.O. Cáceres, Phys. Rev. Lett. 66, 2712 (1988).

[c10] M.O. Cáceres, H. Schnorer, and A. Blumen, Phys. Rev. A 42, 4462 (1990).

[c11] Z.X. Chen and J. Yau, Transp. Porous Media 2, 145 (1987).

[c12] K. Hestir and J.C.S. Long, J. Geophys. Res. 95, 21 565 (1990).

[c13] B.D. Hughes and M. Sahimi, Phys. Rev. Lett. 70, 2581 (1993).

[c14] B.D. Hughes and M. Sahimi, Phys. Rev. E 48, 2776 (1993).

[c15] J.W. Haus and K.W. Kerh, Phys. Rep. 150, 263 (1987).

[c16] U. Landman and M.F. Shlesinger, Phys. Rev. B 19, 6207 (1979); ibid.19, 6220 (1979).

[c17] Note that for a given realization of the disorder, the matrix $Ψ_{ll} {(j, j}^{'}, t)$ characterizes a composite Markovian process; see Ref. [18].

[c18] N. G. van Kampen, Stochastic Process in Physics and Chemistry, 2nd ed. (North-Holland, Amsterdam, 1992).

[c19] It is interesting to note that the off-diagonal elements $Ψ_{{ll}^{'}} (j, j, t)$ of the supermatrix Ψ are responsible of the exchange matrix $E_{j}$ in Refs. [13][14].

[c20] GH. Weiss, Aspect and Applications of the Random Walk (North-Holland, Amsterdam, 1994).

[c21] M. O. Cáceres, Elementos de Estadística de no Equilibrio y sus Aplicaciones al Transporte en Medios Desordenados (Reverté S.A., Barcelona, 2003).

[c22] H. Scher and E.W. Montroll, Phys. Rev. B 12, 2455 (1975) [see Appendix D].

[c23] M.O. Cáceres, Phys. Rev. A 33, 647 (1986).

[c24] S. Bustingory, M.O. Cáceres, and E.R. Reyes, Phys. Rev. B 65, 165205 (2002).

[c25] B. Berkowitz and H. Scher, Phys. Rev. Lett. 79, 4038 (1997).

[c26] Here we have used the result $\int_{- \infty}^{+ \infty} {(p + 2 q k + k}^{2})^{- 1} e^{(- i | a | k)} {d k = e}^{+ i q | a |} π e^{- | a | \sqrt{p - q^{2}}} / \sqrt{p - q^{2}} .$

[c27] J.B.T.M. Roerdink and K.E. Shuler, J. Stat. Phys. 40, 205 (1985).

[c28] M. O. Cáceres, L. Insua, and E. R. Reyes (unpublished).

[c29] Here we have used the theorem of the Laplace shift, and the explicit result $\int_{0}^{+ \infty} e^{- ut} \exp (- x^{2} {/ 4 t) / t}^{3 / 2} d t = 2 \sqrt{π} e^{- | x | \sqrt{u}} / | x | .$

[c30] These were numerically computed using the Laplace inversion program LAPIN, see G. Honig and U. Hirdes, J. Comput. Appl. Math. 10, 113 (1984).

[c31] E. Hernández-García and M.O. Cáceres, Phys. Rev. A 42, 4503 (1990).

[c32] A. Compte and M.O. Caceres, Phys. Rev. Lett. 81, 3140 (1998).

[c33] G. Matheron and G. de Marsily, Water Resour. Res. 16, 901 (1980).

[c34] T. Odagaki and M. Lax, Phys. Rev. B 24, 5284 (1981).

[c35] G. H. Weiss and R. J. Rubin, in Advances in Chemical Physics, editd by I. Progogine and S. A. Rice (Wiley, New York, 1983), Vol. 52.

[c36] P.G. de Gennes, J. Fluid Mech. 135, 189 (1983).

[c37] J.P. Bouchaud and A. Georges, C. R. Acad. Sci., (Ser. II)307, 1431 (1988); see also p. 245 of the review Phys. Rep. 195, 127 (1990).

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Anomalous hydrodynamical dispersion and the transport with multiple families of paths in porous media

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