This paper deals with solute transport phenomena for a one-dimensional finite heterogeneous medium. In the proposed paper, comprehensive analytical solutions are derived for two cases, assuming dispersion as a quadratic function and an exponential function in space. In the first case, the groundwater velocity is a linear function of space, and the dispersion is squarely proportional to the groundwater velocity. In the second case, the groundwater velocity is an exponential function of space, and the dispersion is proportional to the groundwater velocity. Input is being injected in two-stage with the flow from one end of the porous domain. Flux type boundary condition is imposed on the other end of the domain. The variable coefficients of the advection-dispersion equation (ADE) have been converted into constant coefficients using an appropriate transformation so that the solution can be easily obtained. Finally, Laplace Integral Transformation Technique (LITT) is applied to obtain the solution for the proposed problem. The derived results are also verified with the numerical solutions. The effects of various parameters on the contaminant concentrations based on the obtained results are illustrated graphically. The study on two forms of finite heterogeneous media enhanced the importance of the model.