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

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W01420 BAUER ET AL.: ASSESSING FIRST-ORDER RATES W01420 as can be seen by expanding the squared brackets in equation (3) and using C(x) C0: l3 ¼ va aL va lnð Cx ðÞ=C0 Dx lnð Cx ðÞ=C0 Dx 2 ¼ l1 ! lnðCx ðÞ=C0Þ Dx [26] If a L = 0, method 3 reduces to the advection only case, i.e., method 1. Method 3 still does not account for transverse dispersion, which is the process causing smaller concentrations on the plume center line. Method 4 is based on a two-dimensional solution to the transport equation including advection, longitudinal and transverse dispersion and first-order degradation. Results from method 4 (Figure 4d) show an underestimation of the true rate constant for homogeneous or slightly heterogeneous conditions (sY 2 1.71), while for high degrees of heterogeneity (sY 2 2.7), the ensemble averages of the estimated rate constants approach the respective values obtained with method 3. The underestimation for low heterogeneities is a consequence of the correction for transverse dispersion, represented by the error function b in equation (4). The effect of lower concentrations along the plume center line due to transverse dispersion is strong for small source widths, large transversal dispersivities and large well spacings. b is always less than 1 and asymptotically approaches unity for arguments of the error function larger than 2, i.e., l4 converges toward l3 for small aT or large WS. For arguments
W01420 BAUER ET AL.: ASSESSING FIRST-ORDER RATES Figure 5. Estimated first-order degradation rate constants versus degree of heterogeneity sY 2 for (a) method 1, (b) method 2, (c) method 3, and (d) method 4 and source widths WS of 4, 8, and 16 m, respectively. Solid lines show ensemble means normalized to the true degradation rate constant (left axis), dashed lines show the corresponding standard deviations (right axis). [29] In the following, the influence of different approaches for estimating K (and thus va) on the determination of li is analyzed. As the plume samples only part of the full domain, the hydraulic conductivity needed is not the effective conductivity for the complete domain, but an equivalent hydraulic conductivity valid for the flow path along the center line wells. This equivalent hydraulic conductivity may differ from the global effective value. During the field investigation by the center line method, only the local hydraulic conductivities measured at the observation wells placed along the inferred plume center line or only the global effective hydraulic conductivity are known. Neither the underlying statistical parameters nor the complete conductivity field are known. The first approach represents the situation, where local hydraulic conductivities have been obtained in the observation wells, e.g., by slug tests or sieve analysis, and are averaged to obtain an estimate of the equivalent hydraulic conductivity between the two observation wells used for rate determination. In addition to the geometric mean Kg, as used so far, also the arithmetic mean Ka and the harmonic mean Kh are used, as they present 9of14 W01420 upper and lower limits for Kef [e.g., Renard and de Marsily, 1997]. As hydraulic conductivity data is available only at four wells, a full characterization of the conductivity field is not possible. Often at a real site, values of hydraulic conductivity are not available at the locations of the observation wells at the plume center line, but only from a single well not on the plume center line and thus not used for rate determination. This case is simulated by using a value KS of hydraulic conductivity from a well at point [55.10 m, 55.20 m] (see Figure 1). In the last approach, the true global effective conductivity KG of the synthetic aquifer is known, i.e., from a large-scale pumping test, and KG is used as an estimator for the local equivalent hydraulic conductivity at the observation wells. [30] Figure 6 shows the ensemble averages (Figure 6a) and medians (Figure 6b) of the degradation rate constants estimated using the above five approaches. Only results for method 2 and WS = 16 m are presented here, because method 2 is only affected by the hydraulic error due to a wrong estimation of flow velocity. Inspection of Figure 6a shows that all approaches lead to an overestimation of the
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Applied numerical modeling of satur
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Abstract Applied numerical modeling
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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
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Page 26 and 27: The conceptual model used is a rigo
Page 28 and 29: equires therefore a higher degradat
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Page 32 and 33: Fig. 13: Plume length overestimatio
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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,
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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
Page 98 and 99: Einführung In dieser Arbeit wird e
Page 100 and 101: Entlang der so bestimmten Grundwass
Page 102 and 103: Für die letztgenannten Methoden m
Page 104 and 105: Um diesen Effekt zu verdeutlichen s
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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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