longitudinal dispersion in nonuniform isotropic porous media

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of the required number of steps for the asymptotic dispersion
coefficient to apply to the actual average number of steps downstream
to each of the probes indicates that the experimental measurements have
not been made sufficiently far downstream to reach this criterion for
the asymptotic condition, particularly for the higher Pec1et number
experiments. This fact may explain the increase in the calculated
dispersion coefficient compared to measured values at the higher Pec1et
numbers. It is difficult, however, to determine the necessary
downstream distance for the longitudinal dispersion coefficient to
approach the asymptotic value. This difficulty centers on the
inability to say what difference is expected between the asymptotic
dispersion coefficient and measured dispersion coefficient when the
asymptotic distance requirement is not met. In the Saffman (1959)
model for uniform media, the relation derived for the "near-field"
longitudinal dispersion coefficient shows a logarithmic growth in time.
This type of slow variation in the dispersion coefficient would suggest
that the measured dispersion coefficients will be close to the
asymptotic value long before the asymptotic distance is reached.
The discrepancy found at the higher Peclet numbers in Figure 5.3
was not expected, based on the experimental results and theoretical
predictions shown in Figure 2.5. In this figure, the asymptotic
dispersion coefficient compares well over a range of Peclet numbers
from 10 to 10 6 , although the typical experiment (using water) will lie
outside Darcy's regime when Pe > 10 4 • Based on equation (3.23), the
required number of steps at Peclet = 10 3 is n > 3.4x10 4 or a distance