Models that are most commonly used in analyzing the migration of nonreactive contaminants in ground water are based on the advection-dispersion equation derived by spatial averaging of microscopic processes to represent conditions of advection, dispersion, and diffusion at the macroscopic scale. The advection-dispersion equation provides good representation of the results of tracer experiments in saturated columns of porous, homogeneous geologic materials. In field situations, however, the deposits through which contaminants migrate are invariably heterogeneous at the macroscopic scale; hence the applicability of advection-dispersion models is questionable. In unfractured silty or clayey deposits, diffusion generally controls the migration of ground-water contaminants. In fractured deposits, where advective transport may occur along the fractures, molecular diffusion acts as a mechanism of attenuation, causing transfer of contaminants from the fractures to the porous but relatively impervious matrix. In contrast to the coefficients of hydraulic conductivity and dispersivity, the diffusion coefficient for nonreactive constituents in unconsolidated deposits varies within a narrow range. For these reasons, diffusion-controlled hydrogeologic zones have become of interest because of the potential they offer for long-term subsurface isolation of toxic wastes. At higher ground-water velocities, advection and dispersion become important and the influence of heterogeneities becomes dominant. Descriptions of contaminant migration under these conditions are often based on the assumption that the number of heterogeneities is very large relative to the volume of the contaminated zone and that the complex macroscopic velocity field produces dispersion that can be described by the advection-dispersion equation even though the degree of dispersion is orders of magnitude stronger per unit travel distance than in laboratory columns. In field applications, the advection-dispersion model is commonly applied. In deposits that are more discretely heterogeneous, an alternative conceptualization of the transport process involves rapid migration by advection along the more permeable zones. This results in irregular contaminant patterns with fingers or stringers containing relatively undispersed contaminant concentrations that can extend far in advance of contaminant fronts in zones of lesser permeability. Considerable apparent dispersion can result if pumped wells or piezometers for sampling include water from both the contaminated and uncontaminated heterogeneities. Although some field studies provide evidence that contaminant patterns can be strongly controlled by advection along distinct zones of higher permeability, most documented, intensively monitored occurrences of contaminant zones in heterogeneous sand or gravel aquifers and some field tracer experiments in these deposits show concentration patterns that have regularity, with markedly little or no evidence of appreciable fingering. This is the case even at sites where samples were collected from point samplers yielding water from zones that are small in volume relative to many or most of the distinctive heterogeneities. To account for such smoothly dispersed contaminant zones in distinctly heterogeneous deposits, a third conceptualization of the transport process is suggested. As the contaminants are transported primarily by advection in the more permeable heterogeneities, migration by diffusion occurs into the adjacent heterogeneities of lower permeability, thereby reducing the concentrations in the main zones of advection and increasing the concentrations in the zones of lesser flow. Dispersion at the macroscopic scale and at larger scales within the complex flow system is therefore accomplished primarily by molecular diffusion, which acquires its driving force from the numerous local concentration contrasts continually imposed on the system by advection of contaminants in the more permeable heterogeneities. None of the existing mathematical models for contaminant migration in geologic deposits at the field scale has been developed within this conceptual framework. To account for the behavior of reactive contaminants in porous geologic materials, a reaction term that can be assigned various forms has been used by numerous investigators. The most common form involves a linear sorption isotherm to describe rapid reversible reactions between the solute and the solids. In contrast to nonreactive solutes, solutes that are well described by linear reversible isotherms in batch experiments provide breakthrough results from column experiments that are poorly represented by the advection-dispersion equation with an incorporated linear isotherm. This indicates that the combined processes of advection, dispersion, and sorption are not well represented even at the laboratory scale by the conventional mathematical formulations. Although many contaminant species in ground water are influenced primarily by adsorption-desorption, many others are controlled by precipitation-dissolution or oxidation-reduction. For many years these processes have been a focus of attention by hydrogeochemists, who have concentrated on developing thermodynamically based models for systems in isolation from the effects of dispersion and diffusion, and by soil scientists, who have focused on the behavior of dynamic systems but at the flow-path scales of a metre or two. There is a necessity to incorporate information from these disciplines into concepts and models of more direct relevance to reactive contaminants migrating along flow paths in dispersive, heterogeneous deposits. Integration of the processes of advection, dispersion, diffusion, and chemical reaction into unified models for analysis of contaminant migration in hydrogeologic systems presents problems of scale in measurement of parameters. Chemical and biochemical processes occur in response to very local reaction conditions which are not necessarily best sampled or described at the scale most suitable for measurements of the physical transport parameters such as hydraulic conductivity or the dispersion coefficient. Although considerable research on contaminant migration in ground-water flow systems has been conducted during recent decades, this field of endeavor is still in its infancy. Many definitive laboratory and field tests remain to be accomplished to provide a basis for development of mathematical concepts that can be founded on knowledge of the transport processes that exist at the field scale. Without these steps, the tools for assessing the impact of man’s activities on the quality of the ground-water environment will remain inadequate.

Abstract Vulnerability mapping involves an assessment of the potential for solute migration to the water table. Existing procedures rely on simple qualitative indices to calculate this migration potential. As an alternative to these vulnerability indices this paper presents a simple, analytical solution to the advection-dispersion equation. This solution is physically representative, in that it allows for the effects of root zone processes and climate-induced water movement variation on solute transport. The basis of this analytical solution is a spatially transformed advection-dispersion equation. The root zone is treated in a lumped fashion where a relation is developed for the solute concentration of water that has passed through the root zone that includes retardation, degradation and transpiration. This lumped root zone treatment is used in an existing analytical solution that is adapted so that it can be used for a transformed unsaturated advection-dispersion equation. The procedure is tested against hypothetical simulations with dynamic water movement conditions calculated using SWIMv2. The accuracy of the approach is found to be a function of the soil permeability. Where the dynamic variation is averaged out by the water movement process (such as in soils with moderate permeability) the method can accurately represent the transport process. For soils where surface conditions lead to significant variation in water content throughout the profile the technique is more approximate.