Transport in Porous Media

, Volume 13, Issue 1, pp 123–138

Nonlocal dispersion in media with continuously evolving scales of heterogeneity

  • John H. Cushman
  • T. R. Ginn
Article

Abstract

General nonlocal diffusive and dispersive transport theories are derived from molecular hydrodynamics and associated theories of statistical mechanical correlation functions, using the memory function formalism and the projection operator method. Expansion approximations of a spatially and temporally nonlocal convective-dispersive equation are introduced to derive linearized inverse solutions for transport coefficients. The development is focused on deriving relations between the frequency-and wave-vector-dependent dispersion tensor and measurable quantities. The resulting theory is applicable to porous media of fractal character.

Key words

Nonlocal dispersion Lagrangian dynamics memory function heterogeneous porous media statistical mechanics 

Nomenclature

Cv(t)

particle velocity correlation function

Cv′,(t)

particle fluctuation velocity correlation function

Cj(x,t)

current correlation function

D(x,t)

dispersion tensor

D′(x,t)

fluctuation dispersion tensor

f0(x,p)

equilibrium phase probability distribution function

f(x, p;t)

nonequilibrium phase probability distribution function

G(x,t)

conditional probability per unit volume of finding a particle at (x,t) given it was located elsewhere initially

ĝ(k,t)

Fourier transform ofG(x,t)

G′(x,t)

fluctuation conditional probability per unit volume of finding a particle at (x,t) given it was located elsewhere initially

k

wave vector

K(t)

memory function

L

Liouville operator

m

mass

p(t)

particle momentum coordinate

Pα = α(0)( , α(0))

projection operator

Qα =I-Pα

projection operator

s

real Laplace space variable

S(k, Ω)

time-Fourier transform ofĝ(k,t)

t

time

v(t)

particle velocity vector

v′(t)

particle fluctuation velocity vector

V

phase space velocity

Ω

time-Fourier variable

Ω(itn)(k)

frequency moment ofĝ(k,t)

x(t)

particle displacement coordinate

x′(t)

particle displacement fluctuation coordinate

ξ

friction coefficient

ψ(t)

normalized correlation function

General Functions

δ()

Dirac delta function

г()

Gamma function

Averages

〈 〉0

Equilibrium phase-space average

〈 〉

Nonequilibrium phase-space average

(,)

L2 inner product with respect tof0

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References

  1. Afrken, G., 1985,Mathematical Methods for Physicists, 3rd edn., Academic Press, San Diego, CA.Google Scholar
  2. Boone, J. P. and Yip, S. 1980,Molecular Hydrodynamics, McGraw-Hill, New York.Google Scholar
  3. Cushman, J. H., 1991, On diffusion in fractal porous media,Water Resourc. Res. 27, 643–644.Google Scholar
  4. Dieulin, A., Matheron, G. and de Marsily, G. 1981a, Growth of the dispersion coefficient with the mean travelled distance in porous media,Sci. Total Environ.,21, 319–328.Google Scholar
  5. Dieulin, A., Matheron, G., de Marsily, G. and Beaudoin, B., 1981b, Time dependence of an ‘equivalent dispersion coefficient’ for transport in porous media, in A. Verruijt and F. B. J. Barends (eds.),Proc. Euromech 143, Delft 1981, Balkema, Rotterdam, The Netherlands, pp. 199–202.Google Scholar
  6. Kinzelback, W. and Uffink, G. 1991. The random walk method and extensions in groundwater modelling, in J. Bear and M. Y. Corapcioglu (eds.),Transport Processes in Porous Media, Kluwer Academic Publishers, The Netherlands, pp. 761–787.Google Scholar
  7. Scheibe, T. and Cole, C., 1993, Non-Gaussian particle tracking: application to scaling of transport processes in heterogeneous porous media, unpublished manuscript, submitted toWater Resour. Res. Google Scholar
  8. Schiedegger, A. E., 1958, The random-walk model with autocorrelation of flow through porous media,Can. J. Phys. 36, 649–658.Google Scholar
  9. Van Hove, L., 1957, Non-Markovian many-body kinetics from perturbation technique,Physica,23 441–444.Google Scholar
  10. Zwanzig, R., 1960, Ensemble method in the theory of irreversibility,J. Chem. Phys. 33, 1338–1341.Google Scholar
  11. Zwanzig, R. W., 1961, Statistical Mechanics of Irreversibility, in W. E. Brittin, B. W. Downs, and J. Downs (eds.),Lectures in Theoretical Physics, Vol. III, Interscience, New York, pp. 106–141.Google Scholar
  12. Zwanzig, R. W., 1964, Incoherent inelastic neutron scattering and self-diffusion,Physical Review 133(1A) A50-A51.Google Scholar

Copyright information

© Kluwer Academic Publishers 1993

Authors and Affiliations

  • John H. Cushman
    • 1
  • T. R. Ginn
    • 2
  1. 1.1150 Lilly HallPurdue UniversityWest LafayetteUSA
  2. 2.Pacific Northwest LaboratoryRichlandUSA

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