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

90 C. Beyer et al. / Journal of Contaminant Hydrology 87 (2006) 73–95
6.3.2. Estimation of plume lengths
For the determination of contaminant plume lengths, the estimated parameters kmax and MC are
evaluated with equation (10) (Table 3). Normalized plume lengths LMM are presented in Fig. 7
(single realization results, ensemble means with corresponding standard deviations as error bars,
medians, coefficients of variation). For the homogeneous case the estimated and true plume
lengths agree well with LMM=0.97. Also for σY 2 =0.38 the mean yields the correct result with
L MM=1.0 and spread is about 0.5 orders of magnitude. For higher heterogeneities, however,
plume lengths again tend to be underestimated with L MM decreasing to 0.80 for σ Y 2 =4.5. Spread
for all degrees of heterogeneity is smaller than one order of magnitude. When the spread of
estimated kmax and MC values is compared to the spread of calculated MM plume lengths, it is
found – as for case A – that the uncertainty in MM parameters is not fully propagated to the
plume lengths. Comparing MM plume lengths with those determined for the plumes subject to
first order degradation in case A (see Fig. 4), the LMM outperform L1, L2 or L3 with regard to
accuracy as well as uncertainty. Only the L4 are closer to the true lengths on average, while the
spread of single realizations is comparable. It is thus found in this study that the proposed method
of estimating MM kinetics parameters is adequate and performs better than the methods for
estimating first order rate constants.
6.4. Comparison and discussion of cases B and C
In this section the plume lengths LMM calculated in case C using the MM model (Table 3,
equation (10)) are compared to the L1 −L4 of case B, where plume lengths were calculated using
the first order kinetics approximations of equations (7) through (9) of Table 3. Table 5 compares
the accuracy of calculated plume lengths using MM parameters estimated by method 5 with the
four different first order approaches. Given are the percentages of realizations for which LMM
constitutes a closer estimate of L than L1, L2, L3, orL4. In comparison to L1, L2 and L3 it is found
that LMM is the closer estimate for 93 up to 99% of all realizations, depending on the degree of
heterogeneity and the first order method used. In comparison with L4 LMM constitutes the closer
estimate still for 53 up to 72% of all realizations.
As a more quantitative criterion, plume length error factors EF are calculated for each
realization and all five estimation methods:
EF ¼ L *
L
a
with
a ¼ −1jL * bL
a ¼þ1jL * zL
where L ⁎ and L are the estimated and true plume lengths, respectively, and the exponent a is used to
obtain a standardized measure for over- as well as for underestimation. Hence, EF gives the degree of
accuracy of the plume length estimate L ⁎. For each ensemble of Li and LMM cumulative empirical
Table 5
Percentage of realizations for which LMM is a closer estimate of L than L1−L4
LMM vs. L1, L2 % LMM vs. L3 % LMM vs. L4 %
0.38 97 99 72
1.71 99 99 57
2.7 98 99 59
4.5 94 93 53
σ Y 2
ð13Þ