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TECHNICAL REPORT
Vadose Zone Processes and Chemical Transport

Inverse Analyses of Transport of Chlorinated Hydrocarbons Subject to Sequential Transformation Reactions

Francis X.M. Casey^*,a and Jií imnek^b

^a Dep. of Soil Science, North Dakota State Univ., Fargo, ND 58105
^b George E. Brown Jr. Salinity Lab., USDA-ARS, Riverside, CA 92507

* Corresponding author (Francis_Casey{at}NDSU.NoDak.edu)

Received for publication October 23, 2000.

	ABSTRACT

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS NOTES RESULTS AND DISCUSSION CONCLUSION REFERENCES

 
Chemical and biological transformations can significantly affect contaminant transport in the subsurface. To better understand such transformation reactions, an equilibrium–nonequilibrium sorption transport model, HYDRUS-1D, was modified by including inverse solutions for multiple breakthrough curves resulting from the transport of solutes undergoing sequential transformations. The inverse solutions were applied to miscible-displacement experiments involving dissolved concentrations of trichloroethylene (TCE) undergoing reduction and/or transformations in the presence of zero-valent metal porous media (i.e., iron or copper-coated iron filings) to produce ethylene. The inverse model solutions provided a reasonable description of the transport and transformation processes. Simultaneous fitting of multiple breakthrough curves of TCE and ethylene placed additional constraints on the inverse solution and improved the reliability of parameter estimates. Confidence intervals of optimized parameters were reduced significantly in comparison with those obtained by fitting TCE breakthrough curves independently. Further evidence for accurate parameter estimates was given when the parameter values agreed with previously reported values from independent batch and degradation experiments. Optimized values of the normalized degradation rates for the equilibrium (1.4 x 10^-4 to 7.2 x 10^-5 L h^-1 m^-2) and nonequilibrium (1.2 x 10^-4 to 5.5 x 10^-5 L h^-1 m^-2) models compared well with values (0.03 to 6.5 x 10^-5 L h^-1 m^-2) obtained from previous studies. The estimated TCE–iron sorption coefficients (0.52 to 2.85 L kg^-1) were also consistent with a previously reported value (1.47 L kg^-1).

Abbreviations: L, length • M, mass • T, time • TCE, trichloroethylene

	INTRODUCTION

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS NOTES RESULTS AND DISCUSSION CONCLUSION REFERENCES

 
TRANSFORMATION processes, such as biotransformation and radioactive decay, are very common and have major influence on the movement of contaminants in porous media. Examples of solutes undergoing various transformation reactions and migration in the soil include radionuclides (Evans et al., 1997; Viswanathan et al., 1998), nitrogen species (Mishra and Misra, 1991; Misra et al., 1974), organic phosphates (Castro and Rolston, 1977), and organic hydrocarbons (Burt, 1999; Schaerlaekens et al., 1999). These reactions are frequently identified and modeled as linear, instantaneous processes; however, they can often be much more complex. Solutes involved in transformation reactions in addition to non-ideal transport greatly confound the study, modeling, and prediction of solute transport. There is a need for modeling techniques that can help identify these processes by unraveling the complexity.

Several models exist, such as CHAIN (van Genuchten et al., 1985), which can describe the transport of solutes undergoing ideal first-order transformation reactions. CHAIN, an analytical equilibrium, one-dimensional convection–dispersion transport model, has been expanded to include nonequilibrium transport and transient water flow within the numerical model, HYDRUS-1D (imnek et al., 1998a). Inverse methods (van Genuchten, 1981; Kool et al., 1985) have become the standard in soil science to identify solute transport parameters, and have been shown to be very successful. Nonetheless, there still has been little inverse application of transport models to solutes that undergo transformation and to their degradation daughter products. Inverse application of these transformation transport models can lead to better identification of transport processes, which can increase our understanding and lead to better prediction.

Ground water contamination by chlorinated aliphatic hydrocarbons (CAHs) is an area that reflects the importance of having methods that can lead to deeper understanding of solutes undergoing transport and transformation. The prevalent use of CAHs as industrial solvents and in dry cleaning operations has caused widespread ground water contamination. The National Academy of Sciences identified TCE, a CAH, as the most common contaminant at nearly 300000 to 400000 hazardous waste sites in the USA (National Academy of Science, 1994). Ground water remediation of CAHs by zero-valent metals shows promise because of its relatively low costs and the virtual elimination of the CAHs and their degradation daughter products (Focht et al., 1996; Appleton, 1996). It is theorized that CAHs in the presence of zero-valent metals, such as iron filings, undergo a reduction process where electrons are exchanged between the pollutant molecule and an electron donor (i.e., zero-valent metal). In the reduction of TCE (C₂HCl₃), there is sequential removal of chloride ions, which produces dichloroethylene isomers (C₂H₂Cl₂) that are further reduced to vinyl chloride (C₂H₃Cl), then ethylene (C₂H₄), and finally ethane (C₂H₆).

Numerous degradation experiments of TCE in the presence of zero-valent metals have shown that the process is first order and can be 5 to 15 orders of magnitude greater than natural abiotic rates (e.g., Focht et al., 1996; Gavaskar et al., 1998; Gillham et al., 1997; Gillham and O'Hannesin, 1994; Orth, 1992; Orth and Gillham, 1996). Other batch sorption studies have reported evidence for nonequilibrium sorption (Burris et al., 1995, 1998) that can result in more complex degradation processes and chemical nonequilibrium transport. Casey et al. (2000b) used effluent miscible-displacement experiments to study the fate and transport processes of dissolved concentrations of TCE flowing through water-saturated zero-valent metal filings. They reported breakthrough curves of TCE and its degradation daughter product, ethylene, and described the breakthrough curves by means of equilibrium and nonequilibrium convective–dispersive models. Casey et al. (2000b) did not find other degradation products in their effluent breakthrough curves. Furthermore, in their study they applied the model inversion only to the transport of TCE and then used the optimized transformation coefficients to obtain a reasonable prediction of the ethylene breakthrough curves.

The objective of this study was to modify the equilibrium–nonequilibrium transport model HYDRUS-1D by including an inverse solution that allows simultaneous fit of multiple breakthrough curves resulting from transformation reactions. This modification can improve the interpretation of various transformation reactions, especially when complex transport processes occur. It has been shown previously (e.g., Eching and Hopmans, 1993; Inoue et al., 2000) that simultaneous fit of related information (e.g., pressure heads and outflow during outflow experiments, or pressure heads and concentrations during infiltration experiments) can result in significant reduction of confidence intervals of optimized parameters and thus more reliable determination of governing processes. The HYDRUS-1D model was modified to solve equilibrium and nonequilibrium transport models inversely for multiple solutes subject to sequential transformation reactions. The zero-valent metal miscible-displacement experiments from the Casey et al. (2000b) study were used to demonstrate the inverse application of the modified HYDRUS-1D model. The columns that Casey et al. (2000b) used for the miscible displacement study were packed with iron (Fe) or iron coated in copper (Cu–Fe). It will be shown that the confidence intervals of the optimized parameters improve when information about both TCE and ethylene breakthrough curves is used as opposed to only using information from the TCE breakthrough curve.

	MATERIALS AND METHODS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS NOTES RESULTS AND DISCUSSION CONCLUSION REFERENCES

 
Mathematical Models and Inverse Solution
The following set of coupled differential equations governs the one-dimensional, convective–dispersive, linear-equilibrium transport of a sequence of solutes consecutively undergoing degradation and transformation (van Genuchten, 1985):

[1a]

[1b]

where C_i is the solution concentration (M L^-3), D is the hydrodynamic dispersion (L² T^-1), v is velocity (L T^-1), x is distance (L), t is time (T), _i is the first-order degradation rate constant (T^-1), K_d,i is the distribution coefficient (L³ M^-1), _b is the bulk density (M L^-3), is the volumetric water content (L³ L^-3), and the retardation factor R_i (unitless) is given by:

[2]

where the subscript i denotes the i^th chain member in the transformation reaction. For the purposes of this study, degradation was considered to be a surface process (Burris et al., 1995). Hence, it was assumed in Eq. [1] that degradation occurs only in the sorbed phase.

Equations [1a,b] and [2] can be extended to include the concept of nonequilibrium two-site adsorption–desorption reactions (Selim et al., 1977; van Genuchten and Wagenet, 1989; Gamerdinger et al., 1990). Sorption on labile exchange sites or Type-1 sites (signified by subscript S1) is assumed to be instantaneous, while on remaining resistant exchange sites or Type-2 sites (signified by subscript S2), sorption is kinetic. The following expressions govern nonequilibrium mass transport for a homogeneous system during steady-state water flow:

[3a]

[3b]

where f is the fraction of exchange sites assumed to be at equilibrium (unitless), _S1,i represents solid phase degradation rate constants associated with the equilibrium sorption sites (T^-1), _S2,i represents solid phase degradation rate constants associated with the nonequilibrium sorption sites (T^-1), and _i is the first-order kinetic rate sorption constant (T^-1). In this study we assumed that the transformation rates were the same on both equilibrium and kinetic sorption sites (_S1,i = _S2,i) and that degradation only occurred in the sorbed phase. Mass conservation equations for the nonequilibrium sites (S_S2,i, M M^-1 soil) are given by (van Genuchten and Wagenet, 1989):

[4]

HYDRUS-1D Version 2.0 (imnek et al., 1998a) numerically solves either the equilibrium model (Eq. [1a,b] and [2]) or the nonequilibrium model (Eq. [3a,b] and [4]) using Galerkin-type linear finite element schemes. Additionally, HYDRUS-1D may be applied inversely so that the model solution can be fit to the measured data. However, Version 2.0 of HYDRUS-1D could inversely solve the equilibrium or nonequilibrium transport models only for the first solute in the chain of transformation reactions (i.e., C₁ in Eq. [1a] or [3a]). Thus modifications were needed to allow HYDRUS-1D to simultaneously fit breakthrough curves of several solutes undergoing sequential first-order transformation reactions.

imnek et al. (1998a) described the method of parameter optimization used in HYDRUS-1D, where the numerical model solution is fit to measured data. This is done by iteratively changing model parameters and thus improving model fits to the measured data until a desired degree of precision is obtained. The simulated results that are optimized are used together with the measured data to create an objective function, (imnek et al., 1998b):

[5]

where the right-hand side represents deviations between the measured and calculated space–time variables (e.g., concentrations at different times in the flow domain). In Eq. [5], m is the number of different sets of measurements, n_k is the number of measurements in a particular measurement set, q^*_k represents specific measurements at time t_j for the k^th measurement set at location x, q_k(x,t_jb) is the corresponding model prediction for the vector of optimized parameters b (e.g., K_d, D, , and ), and y_k and w_j,k are weights associated with a particular measurement set or point, respectively. Minimization of is done by the Levenberg–Marquardt nonlinear minimization algorithm (Marquardt, 1963). HYDRUS-1D was modified to include simultaneous parameter optimization for multiple solute breakthrough curves resulting from transformation reactions. To do this, the code was amended to include parameters from the secondary breakthrough curves (i.e., i > 1 in Eq. [1b, 2, 3b, and 4]) into the (Eq. [5]). The objective function, which includes parameters from multiple breakthrough curves, can then be minimized using the Levenberg–Marquardt algorithm so that the model solution is fit simultaneously to the measured data of TCE and its degradation daughter products. The breakthrough curves of TCE and its degradation products were considered as different measurement sets. Weights y_k were selected such that contributions of residuals of each breakthrough curve were about equal, while weights w_j,k were set equal to one.

HYDRUS-1D also specifies the upper and lower bounds of the 95% confidence level around each fitted parameter b. It is desirable that the real value of the target parameter always be located in a narrow interval around the estimated mean as obtained with the optimization program. Large confidence limits indicate that the results are not very sensitive to the value of a particular parameter (imnek et al., 1998a).

To compare optimized values with previously reported values, it was important to consider the specific surface area (a_s, L² M^-1) of the zero-valent metal. Johnson et al. (1996) was able to compare observed degradation rates (_obs, T^-1) from several studies by normalizing the _obs values according to the surface area of the zero-valent metals. The following relation was presented by Johnson et al. (1996):

[6]

where _norm is the normalized degradation rate (L³ T^-1 L^-2) and _m is the mass concentration of zero-valent metal (M L^-3 of solution). The terms a_s and _m multiplied together equal the surface area concentration of the zero-valent metal (L² L^-3 of solution; Table 1).

	RESULTS AND DISCUSSION

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS NOTES RESULTS AND DISCUSSION CONCLUSION REFERENCES

 
Figures 1 and 2 present the breakthrough curves of TCE and ethylene from the Casey et al. (2000b) study and the corresponding inverse solutions to the equilibrium and nonequilibrium models from the current study. Trichloroethylene and ethylene were the only products detected in the column effluent. Excellent descriptions of the TCE and ethylene breakthrough curves were obtained when both curves were fitted simultaneously. This is reflected by the relatively high coefficients of determination (r²; ranging from 0.94 to 1.00 for the equilibrium model and 0.96 to 1.00 for the nonequilibrium model; Table 2). These correlations for the combined fit of the TCE and ethylene breakthrough curves were much better than those reported by Casey et al. (2000b) (r² = 0.69 to 0.99 for the equilibrium model, 0.91 to 0.99 for the nonequilibrium model). Casey et al. (2000b) were only able to inversely fit the TCE breakthrough curves and predict, not fit, the ethylene breakthrough curves. Significant improvement in description of the ethylene breakthrough curves in comparison with the Casey et al. (2000b) study was registered, especially for slow water flow rates.

View larger version (22K): [in this window] [in a new window]   Fig. 1. Breakthrough curves from the column filled with iron filings (Fe column) for the fast, intermediate, and slow flow rates. Inverse model solutions to the equilibrium (Eq. [1a,b and 2]) and nonequilibrium (Eq. [2a,b and 3]) models are also plotted.

View larger version (22K): [in this window] [in a new window]   Fig. 2. Breakthrough curves from the column filled with iron fillings coated with copper (Cu–Fe column) for the fast, intermediate, and slow flow rates. Inverse model solutions to the equilibrium (Eq. [1a,b and 2]) and nonequilibrium (Eq. [2a,b and 3]) models are also plotted.

View this table: [in this window] [in a new window]   Table 2. Values of the equilibrium (Eq. [1a,b, and 2]) and nonequilibrium (Eq. [3a,b and 4]) transport model parameters with 95% confidence intervals (in parentheses) for the simultaneous inverse fit of the trichloroethylene (TCE) and ethylene breakthrough curves.

Optimized dispersivities () were about an order of magnitude larger for the equilibrium model than for the nonequilibrium model (Table 2). This is a typical result of fitting breakthrough curves and not taking into account all underlying processes, which may cause the model code to move the effects of the unaccounted processes into another process. In our case, the kinetic sorption was not accounted for in the equilibrium model, which may have moved the kinetic sorption effect into the dispersion term. Values of both and K_d parameters for both soil columns are about the same for the equilibrium and nonequilibrium models.

Simultaneous inversion places additional constraint on the inverse solution, which improved the stability and reliability of the parameter estimates by reducing the confidence intervals (imnek et al., 1998b). Casey et al. (2000b) obtained significant cross-correlations between optimized parameters and their correspondingly large confidence intervals when they optimized nonequilibrium model solute transport parameters (, f, K_d, , and ) to the TCE breakthrough curves only. The 95% confidence intervals of optimized parameters from the Casey et al. (2000b) study often spanned physically nonrealistic values, such as negative values of K_d and f. Fitted TCE breakthrough curves obviously did not contain enough information to allow simultaneous fitting of five parameters of the nonequilibrium transport model. This was the case especially for the slow flow rate experiments, when most TCE was already transformed before reaching the end of the soil column and fitting was applied to the breakthrough curves with relatively small concentrations (less than one hundredth of the initially applied concentration). As a result, optimized parameters were hardly identifiable, confidence in them was relatively low, and also the predictions of the ethylene breakthrough curves were not very good. This problem was overcome when both TCE and ethylene breakthrough curves were fitted simultaneously. Confidence intervals of all optimized parameters, as well as their cross-correlations, were reduced dramatically for both equilibrium and nonequilibrium models.

The nonequilibrium model described the TCE and ethylene breakthrough curves significantly better for the fast flow rates for both Fe and Cu–Fe columns than the equilibrium model. However, as the flow rate decreased, differences between the equilibrium and nonequlibrium model fits decreased until they were nearly identical for the slow flow rate. This trend was more prevalent when the inverse model solution was applied to both TCE and ethylene, and not just to TCE, as was done by Casey et al. (2000b). This trend also suggested that some type of mass transfer limitation existed at the faster flow rates, such as diffusion to the iron surface (Burt, 1999), a process that exhibits itself as a kinetically controlled sorption. Burris et al. (1998) suggested that a rate-limiting step may be present in the transformation of TCE as it desorbs from reactive sites on zero-valent metal. Although the simultaneous inverse fit to the TCE and ethylene breakthrough curves can provide more insight into which models best describe the data, more experiments need to be performed to verify this assertion.

The more complex nonequilibrium transport model has more parameters, which should improve the fit of the model to the data. However, good model fits do not necessarily mean more accurate identification of the underlying transport processes because the model solutions may not be unique, resulting in unrealistic parameter estimates or large confidence intervals. Therefore, complex transport models need to be used with discretion, by using as many as possible independent estimates of unknown transport parameters.

To verify the results found from this study, some of the model parameters were compared with independently estimated values from previous batch and degradation experiments. Numerous zero-valent metal degradation experiment studies exist where first-order degradation rates for TCE have been reported. For example, Tratnyek et al. (1997) summarized a collection of degradation rates (Fig. 3) that were normalized according to surface area (Johnson et al., 1996). The normalized TCE degradation rates from this study fell within the range of previously reported values (Fig. 3), but they were at the low end of the range. Burt (1999) used batch and column experiments to study the degradation of another chlorinated aliphatic hydrocarbon, perchloroethylene, in the presence of zeolites impregnated with zero-valent metals. He found that the degradation rate values estimated from the column experiments were lower than values estimated from batch experiments and suggested that the difference was caused by physical mass transfer limitations. This may suggest why the degradation rate values from this study fell at the lower end of the range reported by previous researchers. Furthermore, a linearized K_d value (1.47 L kg^-1) from Burris et al. (1995) was calculated and found to be within the range of K_d values obtained in this study (Table 2).

View larger version (16K): [in this window] [in a new window]   Fig. 3. A comparison of normalized degradation rates from this study to previous studies. The normalized degradation rates were obtained from curve fits using the (a) equilibrium and (b) nonequilibrium sorption models. Figure adapted from Tratnyek et al. (1997) and is reprinted by permission of Ground Water Monitoring & Remediation (Copyright 1997).

	CONCLUSION

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS NOTES RESULTS AND DISCUSSION CONCLUSION REFERENCES

	NOTES

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS NOTES RESULTS AND DISCUSSION CONCLUSION REFERENCES

 
¹ Names of all products or brands are provided solely for the reader's information and do not imply endorsement of individual brands, nor criticism of similar suitable products.

	REFERENCES

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS NOTES RESULTS AND DISCUSSION CONCLUSION REFERENCES

 

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