Analytical solutions to the transient, unsaturated transport of water and contaminants through horizontal porous media

We present a range of analytical solutions to the combined transient water and solute transport for horizontal flow. We adopt the concept of a scale and time dependent dispersivity used for contaminant transport in aquifers and apply it to transient, unsaturated horizontal flow to develop similarity solutions for both constant solute concentration and solute flux boundary conditions. Through the use of a specific form of the water profile as used by Brutsaert [Water Resour Res 1968:4;785], the solute profiles can be reduced to a simple quadrature. We also derive a solution for the instantaneous injection of water and solute into a horizontal media for an arbitrary dispersivity. It is found that the solute concentration remains constant in both space and time as the water redistributes, suggesting that the solute does not disperse relative to the water.