Analysis of the anomalous scale-dependent behavior of dispersivity using straightforward analytical equations: Flow variance vs. dispersion Page: 13 of 34
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erfc x - vt 1 = erfc ln(vt /x) (10)
2 2 t J 2 a J
This is equivalent to assuming that the the various descriptors in each
equation must correspond to an actual receptor concentration. If the
resulting relationship between variance and dispersivity demonstrates the
appropriate dispersivity scale effect (for constant values of a), then one can
conclude that the flow variance model is a more appropriate description of
solute transport in subsurface systems. Equation 11 can be simplified to
[ t - x 2 = D x Q 1=xD (11)
~ 2 n(vt/x) t
Table 1 presents the calculated D, a, and 5 values for various velocities,
receptor distances, and times. Table 1 indicates that dispersivity is a
function of receptor location for constant values of v and a. Note also
that the $ value is constant for any given value of a. Figure 2 shows the
relationship between a and a for locations in the vicinity of vt. A dashed
line is plotted on Figure 2 showing that a a 0.45 for 8 = 0.1. Thus, at a
a value of 0.45, the often-used rule of thumb (dispersivity is 0.1 times the
scale of the system) is met for all locations and all times.
Figures 2 and 3 show direct comparisons of the two modeling approaches
for locations both closer than, and more distant than, the reference location.
These figures show the front of the nonreactive solute passing the various
locations as a function of time. The flow velocity (v) in these figures is
10 (length/time), the a is 0.45, and the reference receptor is located at 100
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Looney, B. B. & Scott, M. T. Analysis of the anomalous scale-dependent behavior of dispersivity using straightforward analytical equations: Flow variance vs. dispersion, report, Spring 1988; Aiken, South Carolina. (https://digital.library.unt.edu/ark:/67531/metadc684121/m1/13/: accessed May 26, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.