Analytical solutions for solute transport in three-dimensional semi-infinite porous media

This paper presents several analytical solutions for three-dimensional solute transport in semi-infinite porous media with unidirectional flow using first-type (or concentration) and third-type (or flux) boundary conditions at the inlet location of the medium. The solutions may be used for predicting solute concentrations in homogeneous media, verification of more comprehensive numerical models, and laboratory or field determination of solute transport parameters. The transport equation incorporates terms accounting for advection, dispersion, zero-order production, and first-order decay. General solutions were derived for an arbitrary initial distribution and solute input with the help of Laplace, Fourier, and Hankel transforms. Specific solutions are presented for rectangular and circular solute inflow regions, as well as for solutes initially present in the form of parallelepipedal or cylindrical regions of the medium. The solutions were mathematically verified against simplified analytical solutions. Examples of concentration profiles are presented for several solute transport parameters using both first- and third-type boundary conditions. A mass balance constraint is defined based on a prescribed solute influx; the third-type condition is shown to conserve mass whereas the first-type condition was found to always overestimate resident solute concentrations in the medium