Analytical Solution of Advection-Dispersion Equation with Spatially Dependent Dispersivity

Abstract

In the dispersion theory of solute transport in groundwater flow, the dispersion coefficient is regarded as proportional to the nth power of groundwater velocity, where n varies from 1 to 2. The present study derives an analytical solution of a one-dimensional (1D) advection-dispersion equation (ADE) for solute transport for any permissible value of n. For a nonhomogeneous medium, groundwater velocity is considered as a linear function of space and analytical solutions are obtained for n=1, 1.5, and 2.0. For n=1, the dispersivity (ratio of dispersion coefficient and velocity) remains uniform, representing a homogeneous medium, while it varies with position in the finite domain (aquifer) for any other permissible value of n representing the heterogeneity of the medium. From a hydrological point of view, the derived solutions are of significant interest and are of value in the validation of numerical codes. A generalized integral transform technique (GITT) with a new regular Sturm-Liouville problem (SLP) is used to derive analytical solutions in a finite domain. The analytical solutions elucidate the important features of solute transport with Dirichlet-type nonhomogeneous and homogeneous conditions assumed at the origin and at the far end of the finite domain, respectively. The first condition expresses a uniform continuous source of the dispersing mass. The analytical solutions are also compared with numerical solutions and are found to be in perfect agreement. The effect of a Peclet number on the solute concentration pattern is also investigated.