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W01420 BAUER ET AL.: ASSESSING FIRST-ORDER RATES W01420 [34] Results of the sensitivity study are presented in Figure 7. For all degrees of heterogeneity, decreasing median degradation rates are found with increasing aT. Differences between the different aL are most distinct for low values of aT. Considering method 3, it is clearly shown that L3 = L1 only for aL = 0, and L3 > L1 for aL >0. This demonstrates again that method 3 always yields higher estimated degradation rate constants as compared to method 1 (compare equation (7)). Considering that generally l is overestimated in our study, accounting only for aL by using method 3 aggravates this problem. [35] For method 4 it is found that for low and medium heterogeneity values for aL and aT exist, which allow for an optimal estimation of the degradation rate constant, i.e., L4 = 1, and thus L4 L2. For sY 2 = 2.70, L4 > 1.0 always, but L4 < L2 can be achieved for large values of aT and small values of aL. However, for the highest degree of heterogeneity, L4 > L2 > 1 always. For small values of aT, the degradation rate is overestimated for all degrees of heterogeneity, while for low and medium heterogeneity L4 < 1 is possible for large values of aT. [36] For sY 2 = 0.38 and aT = 0.07 m, all medians are larger than L 1 (when a L > 0.8), at a T = 0.3 all medians are 0.5 m (Figure 7c). However, even for an unrealistically large a T of 2 m, L 4 is still significantly larger than L 2. A similar behavior is found for s Y 2 =4.5 (Figure 7d), where only using a L = 9 m and a T >1.15m will result in L 4 being closer to the true rate constant than the corresponding L 1. [37] These results show, that a wide range of dispersivities can and must be used to obtain better estimates using method 4 than using method 1 or method 2. It is also found that the theoretical values (as given above) lead only for the case of low heterogeneity to considerably better estimates than using method 1. Therefore a large fraction of the observed overestimation must result from off center line measurements, as it cannot be corrected for by reasonable values of a T. Especially for the cases of high heterogeneity, the effects of dispersion seem to be minor in comparison to the effect of missing the center line. [38] As shown above, method 4 could yield estimated rate coefficients that are as close or even closer to the true rate constant than rate coefficients estimated with method 2, regardless of the degree of heterogeneity. However, this requires unreasonably low values for a L and very high values for a T, as these parameters would have to correct for the off center line measurement errors. In this case, a L and a T would no longer represent the actual dispersivities, but are lumped fitting parameters. As the magnitude of bias introduced by missing the center line as well as the exact values of s Y 2 or lY are usually not known at a real field site, choosing dispersivities is highly uncertain and may cause over- as well as under-estimation of the degradation rate constant. Thus estimation of dispersivity 12 of 14 introduces an additional error into the estimation of degradation rate constants using methods 3 or 4. Only for aquifers of low heterogeneity method 4 yields better estimates than method 1. Better estimates than using method 2 are only possible by assuming unphysical dispersivity values. 4. Summary and Conclusions [39] In this paper the performance of four different methods for the estimation of degradation rate constants in an aquifer with a heterogeneous distribution of the hydraulic conductivity is studied. All four methods are based on the center line approach. The results demonstrate that a heterogeneous distribution of the hydraulic conductivity may lead to severe overestimation of the ensemble averaged degradation rate constant. Furthermore, the single realizations show a large spread and a large standard deviation, indicating that results obtained from any one estimation are highly uncertain. Mean overestimation as well as spread increase with degree of aquifer heterogeneity. By method comparison, the main reasons are identified as ‘‘measuring off the plume center line’’ and effects of transverse dispersion. Best method performance is observed for method 2, which is based on the one-dimensional transport equation including advection and first order degradation. By normalizing the measured concentrations of the degrading contaminant to a nonreactive co-contaminant emitted by the same source, the above mentioned effects are corrected for. Method 2 thus shows the lowest spread and the lowest overestimation of estimated degradation rate constants of all four methods. However, the presence of the recalcitrant co-contaminant needed for the normalization approach may not always be given at a site. Second best performance is observed for method 1, which yields consistently higher spread and degradation rate constant overestimation as compared to method 2. Methods 3 and 4, although more realistic in the sense that they base on the one-dimensional and two-dimensional transport equation, respectively, show higher spread and larger overestimation. Both methods are prone to errors introduced by estimating longitudinal and transverse dispersivities. The choice of these values introduces additional uncertainty without yielding substantially better results than methods 1 or 2. The ensemble averaged degradation rate constant is highest for method 3, due to the longitudinal dispersivity term in equation (3), while performance of method 4 crucially depends on the choice of an appropriate transverse dispersivity value a T.Ifa T is chosen too small with respect to the real macrodispersivity at the field site under consideration, the degradation rate constant may be overestimated. If a T is chosen too large, the degradation rate may be underestimated. For all methods, a high spread of the results from the single realizations is found, causing a high uncertainty of the estimated degradation rate constant for all methods. For a single realization, the estimated degradation rate may deviate by one or even two orders of magnitude from the correct value. This deviation is caused only by the heterogeneity of the hydraulic conductivity. The spread observed here may contribute to the spread observed in degradation rate constants observed in the field. Wiedemeier et al. [1999] and Aronson and Howard [1997] report measured degradation rate constants for PCE ranging over two orders of
W01420 BAUER ET AL.: ASSESSING FIRST-ORDER RATES magnitude from 0.07 a 1 to 1.2 a 1 and for TCE ranging from 0.05 a 1 to 4.75 a 1 . For benzene, toluene and xylene rate constants of 0.07–3.0 a 1 , 0.36–21.0 a 1 and 0.32– 76.0 a 1 , respectively, are reported [Wiedemeier et al., 1999]. Method performance increases with increasing source width for all methods. For sources very wide with respect to the integral scale of the hydraulic conductivity field, all methods yield reasonable results. In reality, however, when sources are heterogeneous or formed by a complex combination of a number of zones, the total source width may be difficult to estimate. If degradation rates are used for assessing the NA potential at a contaminated site, overestimation of the degradation rates is a critical point. Overestimation of the degradation rate constant leads to an overestimation of the overall natural attenuation potential. If plume lengths are calculated with too high degradation rate constants, then estimated plume lengths are too short. Remediation times as well as downgradient concentrations may be underestimated. The results presented show that determination of degradation rate constants suffers from two main sources of error, i.e., sampling off the plume center line and an incorrect estimate of the average transport velocity. The first can be overcome by using method 2, the second can be resolved by conducting tracer tests or additional measurements of the hydraulic conductivity. A tracer test would furthermore prove, that the observation wells under consideration are sampling the same flow path. Further work on this subject will include the effects of measurement error on the estimated degradation rates, both in measuring hydraulic head and contaminant concentration. Also effects of different formulations of the kinetic reactions used to simulate the plume will be investigated. [40] Acknowledgments. This work is funded by the German Ministry of Education and Research as part of the KORA priority program, subproject 7.1 Virtual Aquifer. We would like to acknowledge the thoughtful reviews of three anonymous reviewers. Their comments have greatly improved this manuscript. References Ababou, R., D. McLaughlin, A. L. Gutjahr, and A. F. B. Tompson (1989), Numerical simulation of three-dimensional saturated flow in randomly heterogeneous porous media, Transp. Porous Media, 4, 549–565. Aronson, D., and P. H. Howard (1997), Anaerobic biodegradation of organic chemicals in groundwater: A summary of field and laboratory studies, SRC TR-97-0223F, Sci. Cent. Rep., Syracuse Res. Corp., New York. Bauer, S., and O. Kolditz (2006), Assessing contaminant mass flow rates obtained by the integral groundwater investigation method by using the virtual aquifer approach, IAHS Publ., in press. Bauer, S., C. Beyer, and O. Kolditz (2005), Assessing measurements of first order degradation rates by using the Virtual Aquifer approach, IAHS Publ., 297, 274–281. Bear, J. (1972), Dynamics of fluids in Porous Media, Elsevier, New York. Beinhorn, M., O. Kolditz, and P. Dietrich (2005), 3-D numerical evaluation of density effects on tracer tests, J. Contam. Hydrol., 81, 89–105, doi:10.1016/j.jconhyd.2005.08.001. Bockelmann, A., D. Zamfirescu, T. Ptak, P. Grathwohl, and G. Teutsch (2003), Quantification of mass fluxes and natural attenuation rates at an industrial site with a limited monitoring network: A case study, J. Contam. Hydrol., 60, 97–121, doi:10.1016/S0169-7722(02)00060-8. Buscheck, T. E., and C. M. Alcantar (1995), Regression techniques and analytical solutions to demonstrate intrinsic bioremediation, in Intrinsic Bioremediation, edited by R. E. Hinchee, T. J. Wilson, and D. Downey, pp. 109–116, Battelle Press, Columbus, Ohio. Chapelle, F. H., P. M. Bradley, D. R. Lovley, and D. A. Vroblesky (1996), Measuring rates of biodegradation in a contaminated aquifer using field and laboratory methods, Ground Water, 34(4), 691–698. 13 of 14 W01420 Dagan, G. (1989), Flow and Transport in Porous Formations, Springer, New York. Diersch, H.-J., and O. Kolditz (1998), Coupled groundwater flow and transport: 2. Thermohaline and 3-D convection processes, Adv. Water Resour., 21(5), 401–425. Diersch, H.-J., and O. Kolditz (2002), Variable-density flow and transport in porous media: Approaches and challenges, Adv. Water Resour., 25(8– 12), 899–944. Domenico, P. A. (1987), An analytical model for multidimensional transport of a decaying contaminant species, J. Hydrol., 91, 49–59. Dykaar, B. B., and P. K. Kitanidis (1992), Determination of the effective hydraulic conductivity for heterogeneous porous media using a numerical spectral approach: 2. Results, Water Resour. Res., 28(4), 1167– 1178. Goovaerts, P. 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Res., 28(12), 3309–3324. Renard, P., and G. de Marsily (1997), Calculating equivalent permeability: A review, Adv. Water Resour., 20(5–6), 253–278. Rubin, Y. (2003), Applied Stochastic Hydrogeology, Oxford Univ. Press, New York. Schäfer, D., A. Dahmke, O. Kolditz, and G. Teutsch (2002), ‘‘Virtual Aquifers’’: A concept for evaluation of exploration, remediation and monitoring strategies, IAHS Publ., 277, 52–59. Schäfer, D., B. Schlenz, and A. Dahmke (2004), Evaluation of exploration and monitoring methods for verification of natural attenuation using the virtual aquifer approach, Biodegradation J., 15(6), 453–465, doi:10.1023/b:biod.0000044600.81216.00. Stenback, G. A., S. K. Ong, S. W. Rogers, and B. H. Kjartonson (2004), Impact of transverse and longitudinal dispersion on first-order degradation rate constant estimation, J. Contam. Hydrol., 73, 3 – 14, doi:10.1016/ j.jconhyd.2003.11.004. Suarez, M. P., and H. S. 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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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2 integral scale lY of 2.67 m and l
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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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