Analytical solutions for non-equilibrium solute transport in three-dimensional porous media

The movement of water and chemicals in soils is generally better described with multidimensional nonequilibrium models than with more commonly used one-dimensional and/or equilibrium models. This paper presents analytical solutions for non-equilibrium solute transport in semi-infinite porous media during steady unidirectional flow. The solutions can be used to model transport in porous media where the liquid phase consists of a mobile and an immobile region (physical non-equilibrium) or where solute sorption is governed by either an equilibrium or a first-order rate process (chemical non-equilibrium). The transport equation incorporates terms accounting for advection, dispersion, zero-order production, and first-order decay. General solutions were derived for the boundary, initial, and production value problems with the help of Laplace and Fourier transforms. A comprehensive set of specific solutions is presented using Dirac functions for the input and initial distribution, and/or Heaviside or exponential functions for the input, initial, and production profiles. A rectangular or circular inflow area was specified for the boundary value problem while for the initial and production value problems the respective initial and production profiles were located in parallelepipedal, cylindrical, or spherical regions of the soil. Solutions are given for both the volume-averaged or resident concentration as well as the flux-averaged or flowing concentration. Examples of concentration profiles versus time and position are presented for selected problems. Results show that the effects of non-equilibrium on three-dimensional transport are very similar to those for one-dimensional transport.