Groundwater contamination occurs when man-made products such as gasoline, oil, road salts and chemicals get into the groundwater and cause it to become unsafe and unfit for human use. Drinking contaminated groundwater can have serious health effects. Diseases such as hepatitis and dysentery may be caused by contamination from septic tank waste. Poisoning may be caused by toxins that have leached into well water supplies. Wildlife can also be harmed by contaminated groundwater. Other long-term effects such as certain types of cancer may also result from exposure to polluted water. In this study the flow of contaminants in urban ground water is investigated. An analytical model for predicting groundwater contamination in isotropic and homogeneous porous formations was derived. The impact of dispersion and diffusion coefficients was included in the solution of the advection-dispersion equation (ADE), subjected to transient (time-dependent) boundary conditions at the origin. A retardation factor and zero-order production terms are included in the ADE. Analytical solutions were obtained using the Laplace Integral Transform Technique (LITT) and the concept of linear isotherm. Analytical solutions for linearly space- and time-dependent hydrodynamic dispersion coefficients along with molecular diffusion coefficients are presented. Analytical solutions are explored for the Peclet number.