In some cases, where the time scale of interest is long, diffusion can cause contaminants to move considerable distances, even through low-permeability materials. Whether contaminant migration on this time scale is important depends on the nature of the problem. In the case of subsurface disposal of radioactive wastes or highly toxic inorganic or organic compounds, diffusion can be an important process.

Consider the following problem: high-level radioactive waste is buried in a cavern in unfractured shale at a depth of 1000 m below ground surface. The burial zone is separated from the nearest overlying aquifer by a vertical thickness of 100 m of shale. The shale has a hydraulic conductivity of the order of 10^–12 m/s and vertical hydraulic gradient of about 10^-2 directed upward. In the shale, nonreactive radionuclides have effective diffusion coefficients in the order of 10^–10 m2/s. It is expected that the wastes will become wet at some time during the next 1000 years and will then move slowly out into the shale.

Is it reasonable to expect that radionuclides will remain entirely within the shale during the next 100,000 years? Ignore the potential effects of faulting, glaciation, and so on, as a cause of radionuclide transfer through the shale. Consider only the influence of mechanical dispersion and molecular diffusion.