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The objective of this study is to model the transport of groundwater contamination in one-layered and two-zoned porous medium flows by an analytical approach. The one-dimensional advection–dispersion equation (ADE) has usually been used to describe the problems of pollutant transport in a water environment. This study presents some exact solutions to the one-dimensional ADE to examine the variation of solute concentration with and without the biodegradable effect in an unconfined aquifer of a finite domain by the generalized integral transform method (GITM). The modeling results for the concentration of groundwater contamination show that the pollutant concentration is more sensitive to the Peclect number than to the retardation factor and the first-order decaying coefficient in uniform groundwater flow. In composite soil zones, decaying and diffusion factors have significant effects on the contamination concentration around the interface, especially over a long-term period. The transport flux between the two regions is determined by the concentration gradient at the interface of the two regions. The contaminated concentration decreases as the retardation factor, Peclect number, and the first-order decaying coefficient increase for every location at a fixed duration. Moreover, the contaminated concentration is more sensitive to the Peclect number than to the retardation factor and the first-order decaying coefficient in uniform groundwater flow.