longitudinal dispersion in nonuniform isotropic porous media

large pore which is connected to smaller pores in the network. It is
quite likely that the smaller pores will not be capable of supplying a
sufficient quantity of fluid to the large pore (note that flux is
proportional to the fourth power of the radius) such that the large
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pore can conduct fluid according to Poiseuille's law and the projection
of the mean pressure gradient. Mass balance would then require a
reduced pressure gradient over the large pore and increased pressure
gradients over the smaller pores. This is equivalent to saying that
the pressure gradient in a nonuniform medium is not necessarily uniform
down to the grain scale, and would expect spatial fluctuations in the
pressure gradient to occur on the grain scale. By definition, however,
the macroscopic pressure gradient must be constant on the macro scale
for a homogeneous medium.
The nature of the random capillary tube model makes it difficult
to describe quantitatively network effects on the pressure gradient.
To include this effect in an ad hoc manner, consider the "unit cell"
shown in Figure 3.2. A hypothetical "cell" of capillary tubes replaces
each original capillary tube in the network, but only for the purpose
of calculating a pressure gradient over the original tube. The
capillary tubes in each cell have radii which vary randomly according
to the measured distribution for pore radii (g(a», but their lengths
are constant. All pores in each cell are oriented in the same
direction (parallel to the original tube) and the number of pores
leading to and from a junction are equal. The pressure gradient over
the entire cell (between locations 1 and 2 in Figure 3.2) is assumed to