Nonlocal dispersion in media with continuously evolving scales of heterogeneity
Authors
 Received:
DOI: 10.1007/BF00613273
 Cite this article as:
 Cushman, J.H. & Ginn, T.R. Transp Porous Med (1993) 13: 123. doi:10.1007/BF00613273
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 convectivedispersive equation are introduced to derive linearized inverse solutions for transport coefficients. The development is focused on deriving relations between the frequencyand wavevectordependent 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 mechanicsNomenclature
 C _{ v }(t)

particle velocity correlation function
 C _{ v }′,(t)

particle fluctuation velocity correlation function
 C _{ j }(x,t)

current correlation function
 D(x,t)

dispersion tensor
 D′(x,t)

fluctuation dispersion tensor
 f _{0}(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
 P _{α} = α(0)( , α(0))

projection operator
 Q _{α} =IP _{α}

projection operator
 s

real Laplace space variable
 S(k, Ω)

timeFourier transform ofĝ(k,t)
 t

time
 v(t)

particle velocity vector
 v′(t)

particle fluctuation velocity vector
 V

phase space velocity
 Ω

timeFourier 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 phasespace average
 〈 〉

Nonequilibrium phasespace average
 (,)

L ^{2} inner product with respect tof _{0}