Transport of solutes through porous media is generally described by the advection-dispersion equation (ADE). The ADE can be formulated by looking at the mass fluxes of a solute into and out of a control volume in the same way as the subsurface flow equation (see also the derivation of the ADE for transport in a river in Section 14.5). For the transport equation, we have to consider two types of flux, advection and dispersion. The joint effects of advection and dispersion on a point source of solute are shown, in plan, for different times in Fig. 15.12. Real patterns of concentration tend to be much more complicated due to both complex aquifer structures, and complex geochemistry and microbiological effects. There have been a number of large-scale tracer experiments involving the transport of multiple substances in groundwater that have greatly improved process understanding and helped to improve transport models (e.g. the Borden aquifer experiments: Sutton and Barker, 1985; and Cape Cod experiments: Fig. 15.13; Le Blanc et al., 1991).
Fig. 15.12 Effects of advection and longitudinal and transverse dispersion on concentrations of a solute in an aquifer at three different times, tl < t2 < t3. Black circle represents origin, darkness of shading represents concentration.
Advection is the flux due to the mean pore water velocity of the flow (remembering that the mean pore water velocity will be faster than the Darcian velocity or specific discharge of the flow). As already noted, within the pores, solute moving with the water may be subject to a whole range of velocities. This means that some solute will move faster than the mean pore velocity and some more slowly. This gives rise to what is called a dispersive flux of solute. This is generally represented as a form of Fick's law, which describes the expected flux due to simple diffusion of solute molecules as a linear function of a concentration gradient. In dispersion, the process involves difference in flow velocity as well as molecular diffusion but the dispersive flux is also treated as a linear function of a concentration gradient. Thus, the dispersive flux, J, is represented as:
where C is solute concentration, D is called the dispersion coefficient (with units of L2T-1) and x is distance as before. This relationship includes the effect of random molecular diffusion, but this is usually has an effect on mixing that is orders of magnitude smaller than dispersion due to the pore water velocity distribution.
Making use of (15.23), total longitudinal flux of solute per unit area of flow in the direction of flow by both advection and dispersion is then given by
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