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

Assessing measurements of first-order degradation rates through the virtual aquifer approach 277
the hydraulic conductivity. For each realization, the procedures described above were
followed and a degradation rate was calculated for each method, at each downstream
well and for every source width.
RESULTS AND DISCUSSION
Figure 2 illustrates the results for the calculated first-order rate constants. Calculated rate
constants were reported as normalized rate constants (i.e. the calculated rate constant
was divided by the true rate constant used in the numerical simulation). The normalized
rate constant can thus be interpreted as an overestimated factor or an underestimated
factor. Inspection of Fig. 2 yields that most calculated rates are higher than one (i.e. the
degradation rate is overestimated). This conclusion is quite concerning for single
realizations, where overestimations can be of several orders of magnitude. On the lefthand
side of Fig. 2(a), the variation of the calculated normalized rate with the source
zone width is illustrated. It is clear that for Method 1, the calculated rates improve when
the source zone width is increased; this is because Method 1 does not account for
dilution, dispersion or measurements outside of the plume. These factors become less
relevant with increasing source width since the basic assumptions inherent in Method 1
are better fulfilled, and the overestimation factor drops accordingly. On the right-hand
side of Fig. 2(a), the dependence of the calculated rate on the degree of heterogeneity
(given as variance) is shown. It is obvious that an increase of σ²ln(KF) leads to an
overestimation of the calculated degradation rate. Furthermore, the standard deviation of
the mean calculated degradation rate increases, leading to greater uncertainty in the
calculated rate. For the smallest degree of heterogeneity the mean overestimation is a
factor a bit smaller than 2, which increases to values between 3 and 5 for medium to
high heterogeneity, and 10 for very high heterogeneity.
Figure 2(b) shows the results for Method 2. Degradation rates for this method were
also overestimated. However, when comparing this method to Method 1, the overestimation
factor and standard deviation are generally smaller (i.e. both the error and
the uncertainty are lower compared to Method 1). When examining the left-hand side
of Fig. 2(b), the calculated rates show no dependence on source width. This effect is
inherent to the method, since Method 2 accounts for dispersion, dilution and
measurements taken outside of the plume. Method 3 depicts results similar to Method
1 regarding the rate dependence on source width and on the degree of heterogeneity
(Fig. 2(c)). However, the normalized degradation rates for Method 3 are higher than
for Method 1 (and also higher than Method 2); this is due to the dispersivity term. In
comparison to Method 1, a portion of the concentration reduction from the source
observation well to the downstream observation well is attributed to dispersion and
corrected for, and thus a higher degradation rate is estimated. Method 4 (Fig. 2(d))
displays behaviour similar to Method 3, except that the rate values are slightly lower.
Lower rate values are attributed to the additional term in the rate equation, which
accounts for transverse dispersion. It should also be noted that at the lowest degree of
heterogeneity, the normalized degradation rates were actually underestimated; this is
due to an “over correction” of the effects for transverse dispersion. To effectively
illustrate the over and underestimation of the degradation rates for all four methods,
the degradation rates where calculated (for a homogeneous hydraulic conductivity) and