Analysis of the anomalous scale-dependent behavior of dispersivity using straightforward analytical equations: Flow variance vs. dispersion Page: 10 of 34

transformation of the solute input function into an output function. The
mathematical transformation used is selected to be consistent with the assumed
governing physical and chemical processes. Thus, while the ultimate model is
mechanistically consistent, it is not fully descriptive (e.g., specific
variations in permeability caused by channels or barriers are not explicitly
described). The transfer function model estimates the average and extreme
behavior of solutes based on the field-measurable (or inferred) distribution
of travel times in a porous medium. The transport of solutes in the sub-
surface is assumed to be entirely defined by the mean flow velocity and a
probabilistic description of the variability in flow velocity. The model
assumes no dispersion, other than that which is implicit in the travel time
variation.
Subject to these assumptions, the probability that a solute entering a
flow system at a location (x = 0) will reach a reference receptor location
(x = L) after a specified time at a mean flow velocity (3) is:
vt
PL(t) -= f fL(t d(vt') (5)
0
where fL(vt) is the probability density function (i.e., the probability
that a solute will arrive between vt and it+d(vt)is fL(vt) d(vt)) [Jury
et al., 1982; Raats, 19781. By representing the transfer as a probability
density, all of the mechanisms that contribute to solute spreading are assumed
to be functions of Vt. The system may be viewed as a bundle of tubes of
different length within which fluid flows by piston flow.
By superposition, the concentration at the reference receptor location
(x = L) for arbitrary variations in source concentrations is
(6)
0

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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/m1/10/ocr/: accessed June 25, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.

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