PEDRO ALVAREZ: In this lecture, we would describe an analytical model to predict the fate and transport of contaminants in groundwater. This model can be a very useful tool to assess how long will the plume be, and whether in situ biodegradation is proceeding to a significant extent. Let us start with some basic concepts. Considering the various processes that affect the migration of a groundwater contaminant from a constant source. First, is advection, which refers to migration with the bulk flow in a plug flow manner. Then there is dispersion, which is caused by the tortuosity of the porous medium and the associated differences in the magnitude and local direction of groundwater velocity vectors. This mechanical dispersion is orders of magnitude more important than molecular diffusion due to Brownian motion. And it causes some molecules to migrate faster or slower than the center of mass of the plume. Then there is sorption of the contaminant onto aquifer solid surfaces which can be important as it slows down the migration velocity of the pollutant relative to the groundwater. This is a phenomenon known as retardation. Sorption is more prominent when the organic contaminant is hydrophobic, and when the aquifer material has a height content of organic matter, which represents a stickier surface. And then there's biodegradation, which is generally a very important sync and can be a major attenuation mechanism. Mathematically, we can account for all of these processes in a mass balance equation. Physically, the first term here on the left hand side represents dispersion using Fick's law. The second term is advection, and the third term is biodegradation, which is here assume to follow first order kinetics. The fourth term is sorption of the contaminants from the aqueous phase to a solid phase, assuming a linear partitioning ISO term. Here, row b represents the bulk density of the soil. And n is the total porosity. And the final term is accumulation, which is of course 0 as steady state. The solution to these differential equation, assuming again a constant rectangular source, is the advection dispersions sorption equation that was developed by Patrick Domenico. This analytical model, which includes a complimentary error function, now available in commercial spreadsheets such as Excel, can be used to estimate the concentration of a contaminate at different times and locations, as well as the lens of the plume. And this model, as you can see, has a lot of parameters. But which ones are the most important so that we know where to focus our attention in the more carefully estimating them? Well, a sensitivity analysis all this model also predicts the length of benzene plumes using the full parameters used for risk based corrective action as baseline, shows that the parameters that have the greatest influence on model simulations or plume lens are the biodegradation rate coefficient, lambda, and the groundwater flow velocity. So we need to pay special attention to these parameters. Of course groundwater velocity can be measure accurately in relatively homogeneous formations. The first order coefficient lambda, on the other hand, is subject to considerable variability in both time and space. Therefore, we're going to focus a little bit more on this parameter. And one important question we should ask is how variable are these lambda values, for example, for benzene degradation in the field, and what are reasonable values? So we estimated values from 79 different sites impacted by benzene, and found that the distribution of this value was low normal. And that the lambda values varied widely. Over three to four orders of magnitude with a mean value of 0.0112 per day. That corresponds to a half life of 62 days. Now although these models assume that lambda and the groundwater velocity are independent, our analysis shows that they could be significantly correlated at the 95% confidence level. And perhaps this is due to the fact that higher conductivity formations, sandy aquifers for example, experience more oxygen replenishment, which accelerates benzene degradation. So this suggests that we perhaps should use smaller lambda values for modeling clay ridge formations than for more permeable sandy aquifers. To wrap it up, the key points of this lecture are that first, analytical models can be very useful tools to assess if natural attenuation is occurring. For example, by comparing this simulated plume lens, with no natural attenuation, that would be simulations with lambda equals 0, to those assuming a real value more reasonably from the literature. One caveat is that this commonly used model is based on simplifying assumptions that do not account for the complexity and heterogeneity that you may find at a given site. Consequently, the predictive capabilities of these models may be limited to order of magnitude accuracy. It seems ironic that the main advantage of these models, which is their simplicity, is also their main disadvantage.