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

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78 C. Beyer et al. / Journal of Contaminant Hydrology 87 (2006) 73–95 Table 2 Estimation of first order degradation rate constants (methods 1–4) and Michaelis–Menten degradation kinetics parameters (method 5) from center line data Formula Description Parameter estimated Method (equation) λ 1D transport equation with advection and first order degradation; advective velocity, va, source concentration C0, down gradient concentration, C(x), first order degradation rate constant, λ, distance, x (1) k1 ¼ − va CðxÞ ln Dx C0 ! 2 λ 1D transport equation with advection, longitudinal dispersion and first order degradation; longitudinal dispersivity, αL −1 lnðCðxÞ=C0Þ Dx 1−2aL (2) k2 ¼ va 4aL λ ! 2D transport equation with advection, longitudinal and transverse dispersion, source width and first order degradation; transverse dispersivity, αT, source area width, WS 2 −1 lnðCðxÞ=ðC0bÞÞ Dx 1−2aL k3 ¼ va 4aL (3) 4 ffiffiffiffiffiffiffiffiffiffi p aTDx C * 0 CðxÞ * ! WS with b ¼ erf λ Same as method 1; contaminant concentrations normalized with regard to conservative solute concentrations, C0⁎, C(x)⁎ CðxÞ ln C0 (4) k4 ¼ − va Dx k max, M C 1D transport equation with advection and Michaelis–Menten degradation kinetics; maximum degradation rate, kmax, half saturation concentration, MC 1 þ kmax lnðC0=CðxÞÞ C0−CðxÞ MC ¼ kmax Dx ð Þ (5) va C0−CðxÞ
Table 3 Calculation of contaminant plume lengths for first order and Michaelis–Menten degradation kinetics Method Formula Equation 1 L1 ¼ Dx ¼ − va k lnðCðxÞ=C0Þ (7) 2 lnðCðxÞ=C0Þ L2 ¼ Dx ¼ 2aL 1−ð1 þ 4kaL=vaÞ 0:5 ( " sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# ) (8) 3, 4 L3;4 ¼ Dxi; for CðxÞ 5 LMM ¼ Dx ¼ va degradation of a contaminant is not limited by electron acceptor availability and the microbial density is assumed to be constant with time, Eq. (6) applies (Simkins and Alexander, 1984): dC dt C ¼ −kmax C þ MC C. Beyer et al. / Journal of Contaminant Hydrology 87 (2006) 73–95 where kmax is the maximum degradation rate [M L − 3 T − 1 ] and MC is the MM half-saturation concentration [ML − 3 ]. This approximation may be applicable when aquifer sediments have been exposed to contaminants for several years (Bekins et al., 1998). The integral form of Eq. (6) can be rearranged to yield equation (5) of Table 2. According to Robinson (1985), equation (5) is the most reliable of several different formulations of the integrated MM model for estimation of kmax and MC. Both parameters are estimated by a linear least squares fit of Δx/[va(C0−C(x))] vs. ln(C0/C (x))/(C0−C(x)). Thus kmax is obtained as the reciprocal of the intercept of the linear function and MC as its slope multiplied by kmax. Application of method 5 assumes advective transport only (see also Parlange et al., 1984). As Robinson (1985) points out, application of a linearized integrated MM model for least squares estimation of its parameters may be problematic, because measured concentrations C(x) appear in the dependent as well as in the independent variable. A preliminary examination of several nonlinear least squares approaches for fitting the MM parameters to the concentration vs. distance data showed that the parameters determined by method 5 were on average more accurate than those obtained by other methods. 4.2. Estimation of contaminant plume lengths C0 kmax −exp Dxi 2aL 1− 1 þ 4kaL va Given a first order degradation rate constant λ (by estimation with one of the four center line methods presented above), equations (1)–(3) (Table 2) can be rearranged to calculate the length of the steady state contaminant plume (see Table 3). The plume length here is defined as the largest distance between the source and the concentration isoline for concentration CPL [ML − 3 ]. Equation (7) gives this distance for the purely advective case of method 1, equations (8) and (9) correspond with methods 2 and 3, respectively. As the rate constant estimated with method 4 is corrected for dispersion, this process has to be accounted for when the plume length is calculated. Therefore equation (9) is also used for calculation of plume lengths based on λ4. A steady state plume length can also be calculated using the MM parameters estimated using method 5. Rearrangement of equation (5) of Table 2 yields equation (10) and gives the distance L MM at which concentrations fall erf WS 4 ffiffiffiffiffiffiffiffiffiffiffi p ¼ 0 (9) aTDxi MCln C0 CðxÞ þ C0−CðxÞ (10) 79 ð6Þ
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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
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List of abbreviations and mathemati
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for the application studies outline
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path lengths of water molecules tha
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where vmax [s -1 ] is a maximum gro
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domain is discretized along advecti
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conceptional model error: wrong pro
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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,
Page 57 and 58: W01420 BAUER ET AL.: ASSESSING FIRS
Page 69: Enclosed Publication 3 Beyer, C., B
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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
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Page 104 and 105: Um diesen Effekt zu verdeutlichen s
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Page 120 and 121: As Bauer et al. (2006a) demonstrate
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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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