Correspondence Comment on “Field-Scale Transport of Nonpolar Organic Solutes in 3-D Heterogeneous Aquifers” SIR: In a recent paper, Cushey and Rubin (1) presented a model that simulates field-scale groundwater transport of nonpolar organic compounds. The model couples a stochastic technique for generating flow fields with a submodel to describe rate-limited sorption due to diffusive mass transfer through domains of immobile water. The authors use their model to simulate first and second spatial moment data from a field experiment previously conducted at Canadian Forces Base in Borden, Ontario (2), and contend that their simulations more accurately capture the characteristics of the data than previous studies. Although the coupled stochastic/ diffusive model development and application are noteworthy, we question the basis for improved success at this site relative to prior efforts using closely-related methods (3). In particular, the success of the model in reproducing the field observations hinges on the independent estimation of the first-order mass transfer parameters, R (dimensional) and ω (dimensionless). We are concerned about the authors’ methods for converting laboratory data to R and ω, and we believe that the close match between model results and field observations was fortuitous. In scenario 2, the authors claim that a diffusive mass transfer parameter, Da/a2 (4), “is directly equivalent to a dimensional form of the parameter ω” (ref 1; p 1263). Starting with the diffusion rate constant (Da/a2) measured by Ball and Roberts (4), Cushey and Rubin (1) derive an equivalent first-order mass transfer parameter using
R)
Da θimRim a2 Rm
(1)
Equation 1 is not explicitly presented in Cushey and Rubin (1) but has been inferred from their parameter values. Following the notation of Cushey and Rubin (1), θim represents the immobile zone porosity, and Rm and Rim are the mobile and immobile zone retardation factors, respectively. Equation 1 converts a coefficient for the sorption-retarded (apparent) rate of diffusion through a sphere of radius a (Da/a2) to a purportedly equivalent first-order (or linear driving force (LDF)) mass transfer coefficient (R). The methodology to convert a spherical diffusion model into an equivalent LDF model has been well established in chemistry and chemical engineering as well as in hydrogeology. The conventionally accepted methodology requires the application of a suitable conversion factor, on the order of 15 (5-9) and substantially greater than 15 (9) in the early stages of uptake, i.e., far short of equilibrium. This value of the conversion factor also is generally accepted for adsorbers in chemical engineering practice (10). Using Cushey and Rubin’s nomenclature, eq 2 represents the accepted conversion methodology:
R)
15DaθimRim a2
(2)
Figure 1 shows the application of two alternative LDF conversion methodologies in relation to simulated batch spherical diffusion results of the type generated by Ball and 2654
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FIGURE 1. Approach to sorption equilibrium for the uptake of solute by a spherical sorbent in a batch system of finite volume with 50% uptake from solution at equilibrium. Solid line: spherical diffusion model. Dashed line: LDF model with r/(θimRim) ) Da/a2. Dotted line: LDF model with r/(θimRim) ) 15Da/a2. For the case shown, Rm ) 1 (no instantaneous sorption). Roberts (4). From Figure 1, it is clear that neglecting the conversion factor of 15 has a significantly adverse impact upon how well the LDF model approximates spherical diffusion. Furthermore, we contend that the additional Rm term in eq 1 is an inappropriate artifact that results from Cushey and Rubins’ arbitrary use of retarded velocity (U/ Rm) in defining ω (ref 1, p 1261). Note that eq 2 appropriately reflects a relation between a sorption-retarded first-order aqueous mass transfer coefficient (R/θimRim) and a sorptionretarded diffusion coefficient, Da/a2 (4) without the inclusion of the Rm factor. The first-order mass transfer coefficient calculated using the customary methodology (eq 2) is thus larger than the coefficient calculated by Cushey and Rubin (eq 1) by a factor of 15Rm. For Cushey and Rubin’s scenario 2, that corresponds to a factor of 33. Previous authors who have attempted to simulate the Borden data (3, 11, 12) have shown that incorporating grain-scale spherical diffusion into a transport model using laboratory-determined rate parameters does not adequately explain the field results, but that the diffusion model achieves good agreement if the diffusion rate constant is reduced by a factor of 15-100, as Cushey and Rubin effectively have done in their improper conversion of the rate term using eq 1. In the previous studies (3, 11, 12), it was hypothesized that diffusion to and from comparatively immobile water at scales larger than the grain-scale (e.g., low permeability lenses) may be responsible for the smaller diffusion rate constants that are inferred from the field observations. More recently, Brusseau and Srivastava (13) have made similar inferences, while also invoking processes of nonlinear, spatially variable sorption. In conclusion, it appears to us that the authors (1) have simulated the Borden spatial moment data by using an erroneously low value for their first-order rate constant inferred from the laboratory data. We have limited the above discussion to scenario 2 of Cushey and Ruben (1) since these are the laboratory results with which we are most familiar. However, we are also concerned that an absence of lengthscale correction in the conversion of the scenario 1 data may have resulted in an R estimate that is roughly 50 times too low. Overall, we submit that the authors’ improper translaS0013-936X(98)00264-8 CCC: $15.00
1998 American Chemical Society Published on Web 07/15/1998
tions of the laboratory-determined diffusional rate parameters into the framework of a first-order model may have fortuitously compensated for larger scale effects that influenced the field data but were not represented in their model. We suggest that Cushey and Rubin recompute their results using eq 2 to assess whether their simulations still represent the field data without further parameter adjustments.
Literature Cited (1) Cushey, M. A.; Rubin, Y. Environ. Sci. Technol. 1997, 31, 12591268. (2) Roberts, P. V.; Goltz, M. N.; Mackay, D. M. Water Resour. Res. 1986, 22, 2047-2058. (3) Quinodoz, H. A. M.; Valocchi, A. J. Water Resour. Res. 1993, 29, 3227-3240. (4) Ball, W. P.; Roberts, P. V. Environ. Sci. Technol. 1991, 25, 12371249. (5) Glueckauf, E. Trans. Faraday Soc. 1955, 51, 1540-1551. (6) Parker, J. C.; Valocchi, A. J. Water Resour. Res. 1986, 22, 399407. (7) van Genuchten, M. Th.; Dalton, F. N. Geoderma 1986, 38, 165183. (8) Goltz, M. N.; Roberts, P. V. Water Resour. Res. 1987, 23, 15751585. (9) Young, D. F.; Ball, W. P. Water Resour. Res. 1997, 33, 21812192. (10) LeVan, M. D.; Carta, G.; Yon, C. M. in Chemical Engineers’ Handbook, 7th ed.; Perry, R. H., Green, D. W., Eds.; McGrawHill: New York, 1997; Chapter 16, pp 16-22-16-23. (11) Goltz, M. N. Three-Dimensional Analytical Modeling of Diffusion-Limited Solute Transport. Ph.D. Dissertation, Stanford University, Stanford, CA, 1986. (12) Ball, W. P. Equilibrium Sorption and Diffusion Rate Studies with Halogenated Organic Chemicals and Sandy Aquifer Material. Ph.D. Dissertation, Stanford University, Stanford, CA, 1989. (13) Brusseau, M. L.; Srivistava, J. Contam. Hydrol. 1997, 28, 115155.
William P. Ball Department of Geography and Environmental Engineering Johns Hopkins University Baltimore, Maryland 21218
Mark N. Goltz Department of Engineering and Environmental Management Air Force Institute of Technology Wright-Patterson AFB, Ohio 45433
Paul V. Roberts Department of Civil and Environmental Engineering Stanford University Stanford, California 94305
Albert J. Valocchi Department of Civil Engineering University of Illinois Urbana-Champaign, Illinois 61801
Mark L. Brusseau Departments of Soil, Water & Environmental Science and Hydrology & Water Resources University of Arizona Tucson, Arizona 85721 ES980264+
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