# Nonlocal dispersion in media with continuously evolving scales of heterogeneity

- 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

- 146 Citations
- 294 Downloads

## 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

*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 of

*G*(**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}=**I**-**P**_{α}projection operator

*s*real Laplace space variable

*S*(*k, Ω*)time-Fourier transform of

*ĝ*(**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 of

*ĝ*(**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

- (,)
*L*^{2}inner product with respect to*f*_{0}