Analytical solutions for contaminant transport in a semi-infinite porous medium using the source function method

The general solutions for contaminant transport in a saturated semi-infinite porous media are derived by using the Laplace transform and Fourier transform, along with their transform inversions, under the conditions of one-dimensional seepage flow and the three-dimensional dispersive effect. The analytical expression of contaminant concentration in a porous medium, subjected to a local contaminant source with arbitrary geometry, and intensity that varies with time and coordinates is derived by the source function method based on the elementary solution of an instantaneous point contaminant source. The results show that an exponentially degenerated contaminant source injected into the porous medium migrates gradually toward the depth and width of the porous medium due to the convective water flow and diffusion induced by molecular and mechanical movement, along with deposition of the contaminant on the solid matrix surface. The contaminant concentration in a porous medium subjected to a cyclic contaminant source exhibits cyclic fluctuations due to the fluctuation of the contaminant source applied on the porous surface; concentrations reach a quasi-steady state, with the same fluctuation phase as the contaminant source. The hydrodynamic dispersion effect accelerates the migration processes of the contaminant in the vertical direction as well as the diffusion in the horizontal direction, resulting in a dramatic rise in the contaminant concentration in a short period of time.