Journal of Fluid Mechanics

A non-local description of advection-diffusion with application to dispersion in porous media

Donald L.  Koch a1p1 and John F.  Brady a1
a1 Chemical Engineering, California Institute of Technology, Pasadena, CA 91125, USA

Article author query
koch dl   [Google Scholar] 
brady jf   [Google Scholar] 


When the lengthscales and timescales on which a transport process occur are not much larger than the scales of variations in the velocity field experienced by a tracer particle, a description of the transport in terms of a local, averaged macroscale version of Fick's law is not applicable. Here, a non-local transport theory is developed in which the average mass flux is not simply proportional to the average local concentration gradient, but is given by a convolution integral over space and time of the average concentration gradient times a spatial- and temporal-wavelength-dependent diffusivity. The non-local theory is applied to the transport of a passive tracer in the advective field that arises in the bulk fluid of a porous medium, and the complete residence-time distribution - space-time response to a unit source input - of the tracer is determined. It is also shown how the method of moments may be simply recovered as a special case of the non-local theory. While developed in the context of and applied to tracer dispersion in porous media, the non-local theory presented here is applicable to the general problem of determining the average transport behaviour in advection-diffusion-type systems in which spatial and temporal variations are occurring on scales comparable with the scale of interest.

(Published Online April 21 2006)
(Received February 14 1986)
(Revised October 6 1986)

p1 Present address: School of Chemical Engineering, Cornell University, Ithaca, NY 14853, USA.