## 1. Introduction

[2] The transport of chemicals, contaminants and particulates in the geosphere due to groundwater movement is an important component in the assessment of geoenvironmental hazards associated with human activity in the form of pollutant spills, release of leachates from contaminated landfill, discharge of toxic materials from mine waste tailings ponds, agricultural pollution due to fertilizer use and aquifer contamination due to migration of bacteria and fine particles [*Ogata and Banks*, 1961; *Lindstrom et al.*, 1967; *Bear*, 1972; *Greenkorn*, 1983; *Rubin*, 1983; *McDowell-Boyer et al.*, 1986; *de Marsily*, 1986; *Bear and Verruijt*, 1990; *Bear and Bachmat*, 1992; *Fetter*, 1993; *Appelo and Postma*, 1993; *Banks*, 1994; *Vukovich*, 1997; *Ingham and Pop*, 1998; *Domenico and Schwartz*, 1998; *Bedient et al.*, 1999; *Massel*, 1999; *Charbeneau*, 2000; *Vulliet et al.*, 2002; *Selvadurai*, 2006]. In addition, natural release of chemicals and their resulting transport can result from changes in groundwater flow patterns, created by construction of hydro-reservoirs, influences of climate change and from the extraction of water and energy resources, including oil, natural gas and coal [*Philips*, 1991; *Oelkers*, 1996; *Ingebritsen and Sanford*, 1998]. The complete study of chemical and contaminant transport in fluid-saturated media becomes a very complex problem when aspects of chemical influences on the transport and mechanical properties are taken into consideration. In many instances, the interactions and the resulting changes cannot be defined with precision, largely owing to the wide variety of geochemical reactions that can take place at different rates, an absence of information on the migrating chemical species, the geochemistry of the porous medium and the attenuation characteristics of the porous medium itself. The prudent approach has been to consider canonical models of chemical and contaminant transport through porous media so that the parameters required to obtain results from the modeling are kept to a minimum. This allows more attention to be focused on the variabilities in the basic parameters that are perhaps more relevant for practical use and environmental decision making. The typical processes of advection, diffusion, attenuation, etc., are clearly such canonical representations of complex transport processes of chemicals and contaminants in the geosphere. When such basic processes concerning the transport of chemicals in a porous medium are identified for a particular geoenvironmental problem, the issue of hydro-mechanical movement within the domain of practical interest has to be examined by recourse to reliable computational techniques. In this regard, the purely advective transport of the species in the absence of diffusive-dispersive phenomena governed by concentration gradients or hydrodynamic effects, presents an extreme test case for assessing the reliability and accuracy of any sophisticated computational scheme. The research dealing with the computational modeling of the purely advective transport problem is extensive and no attempt will be made to provide a comprehensive survey of the field. The reader is referred to the volumes by *Huyakorn and Pinder* [1983], *Bear and Bachmat* [1992], *Sun* [1996] and *Zheng and Bennett* [2002] and the articles by *Yu and Heinrich* [1986], *Oňate* [1998], *Pereira and Pereira* [2001], *Hauke and Doweidar* [2005] and *Selvadurai and Dong* [2006a, 2006b] for further articles dealing with the computational modeling of both advective and advective-diffusive transport problems. In addition, finite volume methods and particle tracking techniques [*Neuman*, 1981, 1984; *Douglas and Russell*, 1982; *Morton*, 1998; *LeVeque*, 2002] and boundary element methods [*Taigbenu*, 1999; *Young et al.*, 2000; *Singh and Tanaka*, 2000; *Boztosun and Charafi*, 2002] have also been applied to investigate both advective and advective-diffusive phenomena.

[3] The standard analytical solution used for purposes of calibrating computational schemes is usually the advective transport in a one-dimensional region of finite extent, in the presence of a flow velocity that is temporally constant and a boundary concentration that remains time invariant. Such a standard analytical solution has been used by *Noorishad et al.* [1992] and *Wang and Hutter* [2001] to test the accuracy of several computational schemes proposed for analyzing the purely advective transport problem. Only recently have analytical solutions been developed to examine two-dimensional and axisymmetric situations involving advective transport problems [*Selvadurai*, 2002, 2003, 2004a, 2004b, 2006]. Even with these solutions, simplifications have to be made with respect to the time-dependency in the flow velocities and boundary concentrations to enable the development of the relevant analytical solutions. The rationale for focusing on the purely advective transport problem is clear; the absence of a diffusive term in the governing partial differential equation implies the presence of discontinuous chemical concentration profiles as the front moves through the porous medium. The performance of a computational scheme is judged in particular by its ability to model this discontinuous propagation front without numerical diffusion, oscillations and, most importantly, without the development of negative concentrations in the numerical solution. The standard one-dimensional solution used to calibrate numerical schemes is open to improvement, particularly through the incorporation of features that can account for time-dependent flow velocities and concentration boundary conditions that can exhibit time variability. Such extensions are considered useful additions to the catalog of benchmarking exercises, particularly since the time discretization algorithms in the numerical schemes have also to account for time variability in the dependent variable, namely the chemical concentration. Surprisingly such extensions are not available in the literature. The purpose of this paper is to present a catalog of exact closed form analytical solutions that can be used to test the accuracy of computational algorithms available in the literature for the solution of advection dominated transport phenomena in porous media.