Analysis of the anomalous scale-dependent behavior of dispersivity using straightforward analytical equations: Flow variance vs. dispersion Page: 11 of 34
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where C(Lt) = concentration at reference location (L) at time t
C(0,t) = concentration at x - 0 at time t
The concentration at any receptor location is determined by assuming that the
governing processes between x = 0 and x = L are the same between all other
(x,t) (0,(t - v' L (vt'L/x) d(vt') (7)
where C(xt) is the concentration at location x based on flow variance in
the subsurface. Thus, a reference calibration function fL allows statistical
description of transport to all receptor locations.
Several measurements of flow velocity or hydraulic conductivity varia-
tions have been reported in the literature [Jury and Stolzy, 1982; White
et al., 1986; Molz et al., 1986; Nielson et al., 1973; Van de Pol et al.,
1977; Bigger and Neilson, 1976; Sharma et al., 1980; Smith et al., 1985;
White et al., 1984]. These studies suggest that the calibration function,
fl is a lognormal distribution. Thus the travel time density function may
be written as:
fL(vt) exp=- (pn vt - u)2/a2 (8)
/ 2 n v vt
where v is the mean of the distribution of in(ht) and a2 is the corresponding
variance. The standard deviation, a, is a descriptor of the variance that is
used in the model solutions. In a fully saturated flow system, u will be
approximately equal to ln(L). Several analytical solutions of (7) and (8) are
available for various assumed input functions [Jury et al., 1982]. These
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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/m1/11/: accessed May 20, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.