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

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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
278 (a) Method 1 (b) Method 2 (c) Method 3 (d) Method 4 S. Bauer et al. Fig. 2 “Measured” first-order degradation rate constants normalized to the true degradation rate constant vs source width (left) and degree of heterogeneity (right) for (a) Method 1, (b) Method 2, (c) Method 3 and (d) Method 4. All figures show results for all observations (small symbols) as well as their mean value (large symbols) and the corresponding standard deviations (error bars).
Page 3 and 4: Applied numerical modeling of satur
Page 5: Abstract Applied numerical modeling
Page 9: Vorwort Die zurückliegenden drei J
Page 13: List of abbreviations and mathemati
Page 16 and 17: for the application studies outline
Page 18 and 19: path lengths of water molecules tha
Page 20 and 21: where vmax [s -1 ] is a maximum gro
Page 22 and 23: domain is discretized along advecti
Page 24 and 25: conceptional model error: wrong pro
Page 26 and 27: The conceptual model used is a rigo
Page 28 and 29: equires therefore a higher degradat
Page 30 and 31: studied for heterogeneous sites wit
Page 32 and 33: Fig. 13: Plume length overestimatio
Page 34 and 35: concentration maximum is for the ca
Page 36 and 37: possibilities can not be provided n
Page 38 and 39: technischen Bauwerken. (Model based
Page 40 and 41: Schäfer, D., Dahmke, A., Kolditz,
Page 43: Enclosed Publication 1 Bauer, S., B
Page 46 and 47: Assessing measurements of first-ord
Page 55 and 56: WATER RESOURCES RESEARCH, VOL. 42,
Page 57 and 58: W01420 BAUER ET AL.: ASSESSING FIRS
Page 69: Enclosed Publication 3 Beyer, C., B
Page 72 and 73: 74 C. Beyer et al. / Journal of Con
Page 95: Enclosed Publication 4 Bauer, S., B
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Einführung In dieser Arbeit wird e
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Entlang der so bestimmten Grundwass
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Für die letztgenannten Methoden m
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Um diesen Effekt zu verdeutlichen s
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Unsicherheit zu. Für Methode 3 ( A
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sagt sowie die zusätzliche Problem
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Enclosed Publication 5 Beyer, C., C
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(1995) yields a “hybrid” rate c
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2 integral scale lY of 2.67 m and l
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The approximate solution employed h
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As Bauer et al. (2006a) demonstrate
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Table 4: Normalized degradation rat
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Comparing results for source widths
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Enclosed Publication 6 Beyer, C., K
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Einleitung und Zielsetzung In Deuts
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die Anzahl Simulationsläufe den Vo
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Grundlage des Korngrößenspektrums
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Dies ist als konservative Abschätz
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Randbedingungen abhängig ist. Aus
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keinen Einfluss auf die abgebaute M
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Carsell, R. F., Parish, R. S.: Deve
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van Genuchten, M.T.: A closed form
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