Analysis of the anomalous scale-dependent behavior of dispersivity using straightforward analytical equations: Flow variance vs. dispersion Page: 12 of 34
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for a step function input -
C(O,t) in t > 0 Boundary Conditions
C(Ot) = 0 t 0
- Cin 1 + erf ln(vt L/x) - In (L)
(x,t) 2 /2 0 9
= Cin erfc ln(vt /x)
2 L dv' aJ
Based on the previous studies, a values should be in'the range of 0.5 [Jury
et al., 1986; Jury et al., 1982; Moltz et al., 1986]. A comparison of this-
approach, Equation (9), to the traditional advection dispersion approach,
Equation (4), along with a straightforward transformation of the data set
documented in Gelhar et al. [19851, provides an intuitive explanation of the
observed dispersivity scale effect and provides a semiquantitative estimate
of field values for a.
Comparison of Alternate Models
Equation (9) defines the concentration at a receptor based on solute
transport in a one-dimensional flow system in which the spreading of solute
between the source and receptor is described by media/flow rate heterogeneity.
Equation (4) defines the receptor concentration for this same system in which
the spreading of solute is described by conventional advection and dispersion
assumptions. By equating the underlying equations, the relationship between
the descriptors (dispersion and variance) can be evaluated:
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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/m1/12/: accessed May 23, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.