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

W01420 BAUER ET AL.: ASSESSING FIRST-ORDER RATES
(equation (3)). It can be clearly seen, that an increase of
sY 2 leads to an increase in spread of the single realizations.
Quite a number of realizations exhibit normalized rate
constants L1 of more than 10 up to about 100, i.e., the
true rate constant is severely overestimated by a factor of
10 to 100. Also, L1 increases with increasing sY 2 as well
as the standard deviations of the ensemble means. This
points to an increase in uncertainty and reflects the spread
of the single realizations. Mean overestimation increases
from a factor of about 1.6 for sY 2 = 0.38 to about 10.2 in
the case of sY 2 = 4.50. In the homogeneous case (sY 2 =0),
method 1 yields the correct result. Although on average l
is overestimated, even for high values of sY 2 in single
realizations l may actually be underestimated by L1. For
the highest degree of heterogeneity, the L1 of the single
realizations span about three orders of magnitude. Comparison
of ensemble means with the corresponding
medians shows that in all cases the medians are significantly
lower. The populations of estimated l1 are positively
skewed as some exceedingly large values of l1
shift the means to high values. However, the general trend
of increasing overestimation with heterogeneity is also
distinct for the medians.
[21] Figure 4b shows the corresponding results obtained
with method 2, i.e., using the one dimensional advectiondegradation
equation with normalization to a recalcitrant cocontaminant
(equation (4)) [Wiedemeier et al., 1996]. As for
method 1, the spread of the single realizations increases
with increasing s Y 2 . Compared to method 1, however, the
spread is smaller and more equally distributed about L 2 =1.
Therefore the average overestimation factors as well as the
standard deviations are much smaller than for method 1 and
both the error and the uncertainty are lower. Average rate
constants L 2 are 1.1 for s Y 2 = 0.38, increasing to 3.3 for sY 2 =
4.50. For homogeneous conditions, also method 2 yields the
correct result, i.e., L 2 = 1 for s Y 2 = 0. Ensemble medians are
just slightly above the true l, i.e., deviating less than a
factor of 2.
[22] Degradation rate constants calculated with method
3, i.e., the one-dimensional method introduced by
Buscheck and Alcantar [1995] (equation (5)), are displayed
in Figure 4c. They exhibit a similar general
behavior as found with method 1, i.e., increasing spread
and increasing overestimation of the true rate constant for
higher s Y 2 . However, mean L3 are significantly higher
than the corresponding L 1, with ensemble means of 2.0
for s Y 2 = 0.38 increasing to 29.4 for sY 2 = 4.50. In the
homogeneous case, the true rate constant is slightly
overestimated, i.e., L 3 =1.25fors Y 2 = 0. Spread in the
single realizations is higher compared to method 1, now
spanning nearly four orders of magnitude for the largest
variance value of s Y 2 .
[23] Results for method 4, i.e., the two-dimensional
solution (equation (6)) suggested by Zhang and Heathcote
[2003], are depicted in Figure 4d. Compared to the other
methods, method 4 displays the largest spread of calculated
L 4 around the mean values. For the highest degree of
heterogeneity, the spread of the single realizations covers
nearly five orders of magnitude. Ensemble means of L 4
increase from 0.6 for s Y 2 = 0.38 to about 23.0 for sY 2 = 4.5,
i.e., for low s Y 2 the normalized rate constant L4 is actually
underestimated in most realizations, while for larger varian-
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W01420
ces the ensemble averages approach the results of method 3.
In the homogeneous case (sY 2 = 0), the estimated rate
constant l4 is about two orders of magnitude lower than
the true rate constant l. Ensemble medians for method 4
show a similar behavior as the ensemble means, with their
values closer to the true rate constant than for methods 1
and 3 for large heterogeneities (sY 2 = 1.71). This reflects the
fact, that the spread of the single realizations is distributed
symmetrically around L4 = 1, however, the spread of the
populations and thus the uncertainty is significantly larger
than for methods 1 and 3.
[24] Method 1 is based on the one-dimensional solution
to first-order biodegradation and advection. Therefore it is
expected that rate constants estimated with method 1
overestimate the true rate constant due to two effects.
Firstly, method 1 does not account for measuring off the
center line. So if an observation well is placed off the plume
center line, concentrations sampled there will be smaller
than on the center line and therefore the degradation rate
constant will be estimated too high. Secondly, as method 1
is based on a one-dimensional solution of the transport
equation, it does not account for transverse dispersion,
which lowers concentrations on the plume center line. This
second effect also causes an overestimation of the rate
constant. Both effects together cause the overestimation
of the rate constant as shown in Figure 4a. Method 2
tries to overcome these two problems by normalization
to a conservative tracer. Both above effects also determine
the concentration of the nonreactive component, and are
thus corrected for by the normalization. As is shown in
Figure 4b, results of method 2 are considerably better than
of method 1, both considering spread and ensemble averages.
However, as can be seen from Figure 4b, method 2
does not correct for all effects, as overestimation is observed
with increasing s Y 2 . Effects of dispersion and measuring off
the center line are accounted for by method 2, so the
deviation seen for method 2 has to have a hydraulic cause.
This deviation is introduced by the determination of the
average flow velocity between the observation wells, which
is calculated using an averaged value of the hydraulic
conductivity at the two observation wells. This averaged
value may not be representative of the flow path between the
two wells and bias may be introduced into the calculation of
degradation rate constants. This effects is studied closely
below in section 3.3.
[25] Method 3 is based on the one-dimensional transport
equation including advection, degradation and longitudinal
dispersion. Results from method 3, as shown in Figure 4c,
display a higher spread and higher ensemble averages
compared to method 1. Because method 3 includes longitudinal
dispersion, it should be closer to reality and advantageous
over method 1. The differences in estimated rate
constants between method 1 and 3 are therefore due to the
longitudinal dispersivity a L in method 3. With the onedimensional
transport model used, pronounced longitudinal
dispersion of a degrading contaminant results in a stronger
spreading of the solute downstream and thus in higher
concentrations along the plume center line compared to an
advection only case. Therefore a larger rate constant is
calculated to accomplish a given concentration decrease
between the upgradient and the downgradient observation
well. l 3 grows linearly with a L and is always larger than l 1,