longitudinal dispersion in nonuniform isotropic porous media

2.2.2.1 Random Capillary Tube Models
The first models of dispersion in porous media based on pore
structure are those of de Josselin de Jong (1958) and Saffman (1959,
1960). The Saffman (1959) model and the de Josselin de Jong (1958)
model use a similar approach, but Saffman's work is more general and
includes the effects of molecular diffusion on dispersion. A separate
review of the de Josselin de Jong (1958) model will not be given.
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Saffman's (1959) model assumes a porous medium to consist of a
random interconnected network of capillary tubes. He considers a
passive tracer particle which proceeds through the capillary tube
network by making a series of random, independent steps, in which the
probability for making a given step is governed by the geometry of the
random network and the physics of laminar flow in a tube. Each step
consists of passing from one junction to another through a capillary
tube. The mean and variance for the motion of a single particle is
calculated and, using the ergodic hypothesis, determines the properties
for a "cloud." of particles. The variance about the mean is then
related to a dispersion coefficient through the Einstein relation
where longitudinal variance of position about the mean (L2)
transverse variance of position about the mean (L2)
Through an analysis of the statistics of particle trajectories in the
network, Saffman was able to compute the mean and variance of the