Analytic solutions to the two-dimensional solute advection-dispersion equation coupled with heat diffusion equation in a vertical aquifer section

Analytical and semi-analytical solutions to solute transport equations are not only used to predict the fate and transport of contaminants in soil or groundwater systems, but also to provide reference standards for validation of numerical models. However, when the solute transport equation is coupled with heat, the corresponding analytical solutions are rarely reported, perhaps due to the complexity of the coupled models. The main purpose of this paper is to present an analytical solution to the solute transport equation coupled with heat for the case of modeling solute transport in a vertical section of a homogenous infinite aquifer with steady uniform groundwater flow under constant-flux solute source. In earlier work (Shan and Javandel, 1997), analytical solutions to the solute transport equation in both finite and infinite aquifer thickness were presented, influence of heat (Soret effect) was generally not considered. Heat can, however, have a significant impact on movement of concentration front. In the present work, we first couple the solute equation with a generalized classic heat diffusion equation. To do this, we assume that the heat is not influenced by the solute (Dufour effect) such that only a one-way coupled model is considered. The repeated integral transformation method (RITM) is utilized to obtain the analytical solutions to the proposed coupled model. As a limiting case, a comparison with analytical solution (Shan and Javandel, 1997), equation (21) and other literature findings is done verify the new proposed solutions and validate our results. The influences of the thermopheresis effect and other model parameters are pictured graphically by the use of MATLAB R2015a. In comparison with the earlier work, the present results show that movement of contaminant in the medium is affected by the presence of heat, in particular along the direction of flow, consistent with the literature. The analysis of longitudinal and transverse concentration profiles suggest that the proposed solutions are applicable for monitoring of groundwater contaminants and risk assessment. Also, the solutions should be useful for comparison with numerical models.