# Potential Limitations of Four Domenico-Based Fate and Transport Models

Potential limitations of the Domenico-based fate and transport models have been identified in recently published articles (Guyonnet and Neville 2004; Srinivasan, et al. 2007; West, et al. 2007). CSMoS distributes four software packages (BIOSCREEN, BIOCHLOR, FOOTPRINT, and REMChlor) that use Domenico-based models. BIOSCREEN and REMChlor use the original Domenico model (1987). BIOCHLOR and FOOTPRINT use the modified version of the model (Martin-Hayden and Robbins 1997).

CSMoS acknowledges that the Domenico-based models are approximate analytical solutions of the advective-dispersive solute transport equation; therefore, they could generate an error for a given set of input parameters when compared with the exact solutions. The error is largely sensitive to high values of longitudinal dispersivity (Srinivasan, et al. 2007; West, et al. 2007). However, CSMoS noticed that the error is insignificant when longitudinal dispersion is reasonably low (see Figures 2b and 5b of Srinivasan, et al. 2007).

Furthermore, longitudinal dispersivity is a calibration parameter, not a parameter that is measured in real-world modeling applications. Therefore, CSMoS believes that the Domenico-based models in their current forms are reasonable for screening tools. However, users should be aware of selecting a reasonably low longitudinal dispersivity in order to minimize the potential error.

There are three different approaches outlined in West, et al. (2007) for estimating longitudinal dispersivity (α_{x}):

- α
_{x }= 10% of plume length - α
_{x }= 10% of L, where L is the longitudinal distance to the reference point (i.e., the x distance at any observation point) according to Pickens and Grisak (1981) - α
_{x }= 0.83(log_{10 }L)^{2.414}according to Xu and Eckstein (1995)

In the first method, αx is constant throughout the model domain; in the second and third methods, it increases proportionally with L (i.e., x distance at any observation point). Although the methods stated above would give a reasonable first estimation of αx, in reality, the calibrated value could be even smaller than those estimated from the three methods and, therefore, would minimize the error resulting from the approximation in the Domenico-based models.

Guyonnet and Neville (2004) found from dimensionless analysis that the original Domenico model (1987) shows minimal error when the Peclet Number, Pe, is greater than 6, which represents a relatively permeable aquifer. By definition, Pe is equal to VL/Dx, where V is the seepage or ground water velocity, L is the longitudinal distance to the reference point, and Dx is longitudinal dispersion. Therefore, higher Pe means advection dominated transport, while lower Pe means both mechanical dispersion and molecular diffusion can significantly influence the transport. CSMoS found from West, et al. (2007) that the difference between the Domenico model (1987) and the exact solution by Wexler (1992) is relatively small for Pe = 10 (i.e., when αx ≈ 0.1L) (see Figure 1 of West et al., 2007).

To this end, CSMoS recommends to safely use BIOCHLOR, BIOSCREEN, FOOTPRINT, and REMChlor for advection-dominated transport conditions (i.e., when Pe≥10) and be cautious when the transport processes are highly influenced by dispersion (i.e., when the aquifer is relatively impermeable).

#### References

Domenico, P.A. (1987). “An Analytical Model for Multidimensional Transport of a Decaying Contaminant Species.” *Journal of Hydrology*, 91: 49–58.

Domenico, P.A. and G.A. Robbins. (1985). “A New Method of Contaminant Plume Analysis.” *Ground Water*, 23, 4: 476–485.

Guyonnet, D. and C. Neville. (2004). “Dimensionless Analysis of Two Analytical Solutions for 3-D Solute Transport in Ground Water.” *Journal of Contaminant Hydrology*, 75: 141–153.

Martin-Hayden, J. and G.A. Robbins. (1997). “Plume Distortion and Apparent Attenuation Due to Concentration Averaging in Monitoring Wells.” *Ground Water*, 35, 2: 339–346.

Pickens, J. and G. Grisak. (1981). “Scale-Dependent Dispersion in a Stratified Granular Aquifer.” *Water Resources Research*, 17, 4: 1191–1211.

Srinivasan, V., T.P. Clement, and K.K. Lee. (2007). “Domenico Solution—Is It Valid?” *Ground Water*, 45, 2: 136–146.

West, M.R., B.H. Kueper, and M.J. Ungs. (2007). “On the Use and Error of Approximation in the Domenico (1987) Solution.” *Ground Water*, 45, 2: 126–135.

Wexler, E. (1992). “Analytical Solutions for One-, Two-, and Three-Dimensional Solute Transport in Ground Water Systems With Uniform Flow. Techniques of Water Resources Investigations of the United States Geological Survey.” Chapter B-7 in Book 3, *Applications of Hydraulics*. p. 79.

Xu, M. and Y. Eckstein. (1995). “Use of Weighted Least-Squares Method in Evaluation of the Relationship Between Dispersivity and Field Scale.” *Ground Water*, 33, 6: 905–908.