Applied numerical modeling of saturated / unsaturated flow and ...

C. Beyer et al. / Journal of Contaminant Hydrology 87 (2006) 73–95
and large source widths WS. A surprising result is that method 1, despite its simplicity, yields closer
estimates of the true rate constant than the more comprehensive description by method 3. Since
method 3 depends on longitudinal and transverse dispersivities, an adequate parameterization is
crucial for its success. From stochastic hydrogeology it is known that αL as well as αT strongly
depends on travel time and distance as well as on the correlation structure of hydraulic conductivity
and flow velocity (e.g. Dagan, 1989). Consequently, a uniform parameterization solely based on the
scale of the contaminant problem as used in this study (and with many field applications) is not
adequate. A detailed sensitivity study on the influence of dispersivity parameterization on the
performance of methods 2 and 3 is presented in Bauer et al. (2006). It is found that for method 2 no
value of αL and for method 3 only very high and thus unphysical values of αT yield the correct
degradation rate constant. The required values, however, cannot be deduced from aquifer heterogeneity
σY 2 alone, as the other errors also influence the estimated degradation rate constant.
Method 4 circumvents this problem and corrects for transverse dispersion as well as for measuring
off the center line by normalizing concentrations to a conservative tracer. The bias towards too large
degradation rate constants observed for the other methods is significantly reduced yielding the closest
estimates of λ of the four methods. The remaining deviation from the true rate constant is due to the
hydraulic error introduced by the approximation of va. For low heterogeneities there is no evidence for
a systematic bias towards either too high or too low rate constants. A prerequisite which may limit the
applicability of method 4 is the presence of a suited normalization compound. A discussion of potential
normalization compounds is provided in U.S. EPA (1998) and Wiedemeier et al. (1996, 1999).
6.1.2. Estimation of plume lengths
During site characterization, estimating degradation rate constants rarely is a goal per se. Here, the
kinetics of contaminant degradation are quantified to be used for prediction of the steady state length
of the plumes. In site assessment, such information could be used to identify potential receptors and
exposure levels. Rate constants λ1, λ2 and λ3 are evaluated using the respective corresponding
equations (7), (8) and (9) of Table 3 yielding plume length estimates L1, L2 and L3. Forλ4 also
equation (9) is used yielding L4. To be able to compare the results of all realizations, the Li are
normalized by the respective true length L read from the model output. Resulting over- respectively
underestimation factors against aquifer heterogeneity σY 2 are presented in Fig. 4 (single realization
results, ensemble means with standard deviations as error bars, medians, coefficients of variation).
Plume lengths L 1 and L 2, calculated from λ 1 and λ 2, show exactly identical results, although
the λ2 show a stronger overestimation than the corresponding λ1 for all realizations. The reason
for the equivalence of L1 and L2 is that the bias introduced by estimating λ2 with a one
dimensional model accounting for longitudinal dispersion only is reversed by using the same
transport equation to calculate the plume length. While for homogeneous conditions the true
plume length is obtained, L is underestimated in most realizations for all degrees of heterogeneity.
Mean L1 and L2 decrease to 0.59 for σY 2 =4.5. As for λ, spread and uncertainty of L increase with
σ Y 2 . The overestimation of the degradation rate constant is thus reflected in an underestimation of
the plume length.
Plume lengths L3 calculated with the two-dimensional transport equation on average are lower
than the corresponding L1 and L2. This is a consequence of the β term in equation (5) (Table 2),
which is used to correct down gradient concentrations for transverse dispersion. When estimating
the rate constant with method 3, each down gradient concentration C(x) is scaled by a different
value β, as the correction factor is dependent on the distance from the source. This scaling is not
fully reversed when the plume length is calculated using equation (9) (Table 3), as then only one
single Δx is used.
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