On the Use and Error of Approximation in the Domenico (1987) Solution

Christopher Neuzil, Discussion Editor “On the Use and Error of Approximation in the Domenico (1987) Solution,” by M.R. West, B.H. Keuper, and M.J. Ungs, Ground Water, March–April 2007 issue, 45, no. 2: 126–135. West et al. (2007) provide a thoughtful analysis of error in the Domenico solution. One of the main conclusions of their paper is that screening models such as BIOSCREEN and BIOCHLOR, which are based on various forms of the Domenico equation, should not be used because of error associated with the Domenico solution. Although we agree that the concentrations predicted by the Domenico solution can differ from concentrations predicted by an exact solution under certain conditions, we assert that the range of errors presented in West’s paper either are not directly relevant to BIOSCREEN v1.4 and BIOCHLOR v2.2 (distributed since 1997 and 2002, respectively) or are not presented in the context of using these screening models for natural attenuation evaluations. It should be noted that BIOSCREEN v1.4 and BIOCHLOR v2.2 do not employ variable longitudinal dispersivity values—only fixed longitudinal dispersivity values. Thus, most of the error analysis in the paper using the Domenico (1987) model with variable longitudinal dispersivity (i.e., runs 1 through 8 in table 4) is not directly applicable to either of the current versions of BIOSCREEN or BIOCHLOR. In addition, the mention of BIOSCREEN as an example of the Domenico solution with variable dispersivity in table 5 of West et al. (2007) is not accurate. The curves for “3D αL= 10 m” in figures 1 and 2 of West et al. (2007) represent the only calculations that compare the exact Wexler (1992) solution to the Domenico (1987) solution with a fixed longitudinal dispersivity value, thus making the comparison to BIOSCREEN and BIOCHLOR valid. West et al. (2007) conclude that the error in concentration at a given location ranges from +2.5% to −24% near the source. However, as discussed subsequently, these concentration errors do not translate to similar errors in estimates of steady-state plume length. For natural attenuation applications, BIOSCREEN and BIOCHLOR are typically used to estimate plume lengths by fitting the Domenico solution to constituent concentrations along the plume centerline, with longitudinal dispersivity and half-life (or biodegradation rate constant) used as fitting parameters. These parameters are then used to estimate the ultimate plume length. At issue is whether fitting the Domenico solution to centerline concentrations rather than using the exact solution results in unacceptable errors in plume length estimates. To determine the significance of errors in plume length based on the Domenico solution, we used the parameters in table 2 of West et al. (2007) in an example calculation for the 3D case. We first defined plume length as the distance from the source to a concentration of 0.005 mg/L. Using the West et al. (2007) parameter values, including a 5-year half-life and a 10-m longitudinal dispersivity, the exact solution yielded a steady-state plume length of 1976 m. Again, using the exact solution, we calculated the concentration of a constituent along the plume centerline at four locations (corresponding to four hypothetical monitoring wells) spaced evenly from near the source to 1000 m (the maximum distance shown in figure 1 of West et al. 2007). We then fitted a steady-state Domenico solution curve to this dataset by adjusting the half-life and longitudinal dispersivity using the Solver routine in Excel. We used the fitted longitudinal dispersivity and half-life thus obtained to predict the plume length using the Domenico solution evaluated at steady state. After completing this exercise, the Domenico fit to the exact solution resulted in a half-life of approximately 4.86 years and a longitudinal dispersivity of 9.19 m. Using these values of half-life and longitudinal dispersivity, the Domenico solution predicted a plume length of 1961 m, an error of −1%. In the context of the other uncertainties in predicting transport in ground water, these differences are not significant. Others have reviewed the Domenico model and have concluded that it yields reasonably accurate concentration estimates along the centerline when the flow regime is dominated by advection and mechanical dispersion rather than by diffusion (Guyonnet and Neville 2004; http://www.epa.gov/ada/csmos/domenico.html). Error in the Domenico solution will be low when solving transport problems that have low dispersivity values relative to the travel distance, high advection velocities, and long simulation times (Srinivasan et al. 2007). If wells too close to the source are used in an analysis of plume length or if the Domenico solution is applied to recent spills, significant errors could result. However, natural attenuation modeling typically is conducted for plumes with solvent or petroleum releases that occurred decades ago, thus requiring long simulation times (McGuire et al. 2003; Parsons Engineering Science 1999). In conclusion, when employed with the model limitations in mind, BIOSCREEN v1.4 and BIOCHLOR v2.2 should yield reasonable estimates of plume lengths and continue to be useful, freely available tools for natural attenuation screening.

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