Transport - Transport and Fate of Chemicals in Soils: Principles and Applications - page 85

are nonlinear and time dependent in nature, and analytical solutions are not
available. As a result a number of numerical models using finite-difference
or finite-element approximations have been utilized to solve nonlinear reten-
tion problems of multiple reactions and multicomponent solute transport for
one- and two-dimensional geometries.
3.9 Estimation of D
In a number of field and laboratory miscible displacement studies, the main
purpose of a tracer application is to estimate the apparent dispersion coef-
ficient, D . A commonly used technique for estimating D is to describe tracer
breakthrough results where tritium, chloride-36, bromide, or other tracers
are used. It is common to use one of the above exact solutions or an approxi-
mate (numerical) solution of the CDE. In addition, a least-squares optimiza-
tion scheme or curve-fitting method is often used to obtain best-fit estimates
for D . One commonly used curve-fitting method is the maximum neighbor-
hood method of Marquardt (1963), which is based on an optimum interpola-
tion between the Taylor series method and the method of steepest descent
(Daniel and Wood, 1973) and is documented in a computer algorithm by van
Genuchten (1981).
The goodness-of-fit of tracer BTCs is usually unacceptable when D is the
only fitting parameter. Thus, two parameters are fitted (usually D along
with the retardation factor R ) in order to improve the goodness-of-fit of
tracer BTCs (van Genuchten, 1981). Other commonly fitted parameters
include pulse duration t p and the flow velocity v for solute retention (Jaynes,
Bowman, and Rice, 1988; Andreini and Steenhuis, 1990). However, since
v can be measured experimentally under steady-state flow, it may not be
appropriate to fit v to achieve improved fit of the BTCs. The best-fit veloc-
ity v is often different from that measured experimentally. Estimates for R
values for tritium and chloride-36 tracers are often close to unity for most
soils. R greater than unity indicates sorption or simply retardation, whereas
R less than one may indicate ion exclusion or negative sorption. Similar val-
ues for R for tritium and 36 Cl were reported by Nkedi-Kizza et al. (1983),
van Genuchten and Wierenga (1986), and Selim, Schulin, and Fluhler (1987).
Table 3.1 provides estimates for D obtained from tracer breakthrough results
for several soils. Selected examples of measured and best-fit prediction of
tritium breakthrough results for selected cases are shown in Figures 3.11 to
3.13 for two reference clays (kaolinite and monotorillonite) and a Sharkey
clay soil material (Gaston and Selim, 1990b, 1991). Ma and Selim (1994) pro-
posed the use of an effective path length L e or a tortuosity parameter τ ( L e / L )
where L e was obtained based on mean residence time measurements. They
tested the validity of fitting solute transport length ( L e ) or tortuosity (τ)