Solute Transport due to Spatio-Temporally Dependent Dispersion Coefficient and Velocity: Analytical Solutions

Abstract

This paper presents an analytical solution of the advection-dispersion equation (ADE) with the dispersion coefficient and velocity being directly proportional to the spatial linear nonhomogeneous function. It is one of the solutions obtained in two particular cases in which the dispersion coefficient and velocity are (1) spatially dependent and (2) temporally dependent. These analytical solutions, particularly the one addressed here, which are of great importance in the existing hydrological theories, had been elusive. These theories assert the spatio-temporal dependence of the transport coefficients of the ADE to frame pollutant transport in aquifers in real situations. This study is carried out in an infinite medium for instantaneous and continuous point sources. The source of the pollutant’s solute mass is defined through a nonhomogeneous production term in the ADE. Darcy velocity is considered to be spatio-temporally dependent in nondegenerate form. According to hydrodynamic dispersion theory, the dispersion coefficient is proportional to the nth power of the velocity, where n varies from 1 to 2. This paper compares solute transport problems for n=1 and n=2 and obtains expected results. Green’s function method (GFM) is used to obtain an analytical solution in general form, from which those for instantaneous and continuous sources in different combinations of spatial and temporal dependence are derived. The spatial dependence is considered linear, whereas temporal dependence is considered asymptotic, exponential, and sinusoidal. To use GFM, a moving coordinate transformation equation is developed which reduces the ADE into a solvable form. The analytical solutions of known dispersion problems are derived as particular cases. The effect of spatial and temporal dependence of the transport parameters on the solute transport is shown through illustrations.