Scale Analysis of the Significance of Dispersion in Mixing-transport in Conduits

Mixing-transport of solute entering at sinkholes or from within the limestone matrix in cavernous conduits is an important process for contaminant migration in karst aquifers. This process may be described with a one-dimensional advection-dispersion equation incorporating the fluxes of solute and water across the conduit wall. For the dilution-dispersion equation, which does not include solute flux across the wall but has the flux of water through the wall, the sufficient and necessary condition for neglecting conduit dispersion is showed by scale analysis to be LP&a, where a is the conduit radius, and LP is the spatial scale of the solute plume. A straightforward necessary and practically, though not strictly, sufficient condition is a=WTB%1, where W is the mean velocity of conduit flow, and TB is the time scale of the breakthrough curve. For the releasing-dispersion equation, which includes the fluxes of water and solute across the wall, LP&a is still a sufficient condition, but no longer a necessary one. The inequality a%WTB is neither a necessary condition nor a sufficient condition.