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

W01420 BAUER ET AL.: ASSESSING FIRST-ORDER RATES W01420
[34] Results of the sensitivity study are presented in
Figure 7. For all degrees of heterogeneity, decreasing
median degradation rates are found with increasing aT.
Differences between the different aL are most distinct for
low values of aT. Considering method 3, it is clearly shown
that L3 = L1 only for aL = 0, and L3 > L1 for aL >0.
This demonstrates again that method 3 always yields
higher estimated degradation rate constants as compared
to method 1 (compare equation (7)). Considering that
generally l is overestimated in our study, accounting only
for aL by using method 3 aggravates this problem.
[35] For method 4 it is found that for low and medium
heterogeneity values for aL and aT exist, which allow for
an optimal estimation of the degradation rate constant, i.e.,
L4 = 1, and thus L4 L2. For sY 2 = 2.70, L4 > 1.0
always, but L4 < L2 can be achieved for large values of
aT and small values of aL. However, for the highest
degree of heterogeneity, L4 > L2 > 1 always. For small
values of aT, the degradation rate is overestimated for all
degrees of heterogeneity, while for low and medium
heterogeneity L4 < 1 is possible for large values of aT.
[36] For sY 2 = 0.38 and aT = 0.07 m, all medians are larger
than L 1 (when a L > 0.8), at a T = 0.3 all medians are 0.5 m (Figure 7c). However, even
for an unrealistically large a T of 2 m, L 4 is still significantly
larger than L 2. A similar behavior is found for s Y 2 =4.5
(Figure 7d), where only using a L = 9 m and a T >1.15m
will result in L 4 being closer to the true rate constant than
the corresponding L 1.
[37] These results show, that a wide range of dispersivities
can and must be used to obtain better estimates using
method 4 than using method 1 or method 2. It is also found
that the theoretical values (as given above) lead only for the
case of low heterogeneity to considerably better estimates
than using method 1. Therefore a large fraction of the
observed overestimation must result from off center line
measurements, as it cannot be corrected for by reasonable
values of a T. Especially for the cases of high heterogeneity,
the effects of dispersion seem to be minor in comparison to
the effect of missing the center line.
[38] As shown above, method 4 could yield estimated
rate coefficients that are as close or even closer to the true
rate constant than rate coefficients estimated with method 2,
regardless of the degree of heterogeneity. However, this
requires unreasonably low values for a L and very high
values for a T, as these parameters would have to correct for
the off center line measurement errors. In this case, a L
and a T would no longer represent the actual dispersivities,
but are lumped fitting parameters. As the magnitude of
bias introduced by missing the center line as well as the
exact values of s Y 2 or lY are usually not known at a real
field site, choosing dispersivities is highly uncertain and
may cause over- as well as under-estimation of the
degradation rate constant. Thus estimation of dispersivity
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introduces an additional error into the estimation of degradation
rate constants using methods 3 or 4. Only for aquifers
of low heterogeneity method 4 yields better estimates than
method 1. Better estimates than using method 2 are only
possible by assuming unphysical dispersivity values.
4. Summary and Conclusions
[39] In this paper the performance of four different
methods for the estimation of degradation rate constants
in an aquifer with a heterogeneous distribution of the
hydraulic conductivity is studied. All four methods are
based on the center line approach. The results demonstrate
that a heterogeneous distribution of the hydraulic conductivity
may lead to severe overestimation of the ensemble
averaged degradation rate constant. Furthermore, the single
realizations show a large spread and a large standard
deviation, indicating that results obtained from any one
estimation are highly uncertain. Mean overestimation as
well as spread increase with degree of aquifer heterogeneity.
By method comparison, the main reasons are identified as
‘‘measuring off the plume center line’’ and effects of
transverse dispersion. Best method performance is observed
for method 2, which is based on the one-dimensional
transport equation including advection and first order degradation.
By normalizing the measured concentrations of the
degrading contaminant to a nonreactive co-contaminant
emitted by the same source, the above mentioned effects
are corrected for. Method 2 thus shows the lowest spread
and the lowest overestimation of estimated degradation rate
constants of all four methods. However, the presence of the
recalcitrant co-contaminant needed for the normalization
approach may not always be given at a site. Second best
performance is observed for method 1, which yields consistently
higher spread and degradation rate constant overestimation
as compared to method 2. Methods 3 and 4,
although more realistic in the sense that they base on the
one-dimensional and two-dimensional transport equation,
respectively, show higher spread and larger overestimation.
Both methods are prone to errors introduced by estimating
longitudinal and transverse dispersivities. The choice of
these values introduces additional uncertainty without yielding
substantially better results than methods 1 or 2. The
ensemble averaged degradation rate constant is highest for
method 3, due to the longitudinal dispersivity term in
equation (3), while performance of method 4 crucially
depends on the choice of an appropriate transverse dispersivity
value a T.Ifa T is chosen too small with respect to the
real macrodispersivity at the field site under consideration,
the degradation rate constant may be overestimated. If a T is
chosen too large, the degradation rate may be underestimated.
For all methods, a high spread of the results from the
single realizations is found, causing a high uncertainty of
the estimated degradation rate constant for all methods. For
a single realization, the estimated degradation rate may
deviate by one or even two orders of magnitude from the
correct value. This deviation is caused only by the heterogeneity
of the hydraulic conductivity. The spread observed
here may contribute to the spread observed in degradation
rate constants observed in the field. Wiedemeier et al. [1999]
and Aronson and Howard [1997] report measured degradation
rate constants for PCE ranging over two orders of