The mathematical simulation of advective dispersive contaminant transport in groundwater involving large Péclet numbers is subject to numerical difficulties. For most numerical models the computational cost and computer core requirements escalate as the Péclet number increases. A two-dimensional Galerkin finite element model for flow and Lagrangian mass transport in porous media has been developed to alleviate numerical and computational difficulties. The solution methodology involves a linear triangular mesh which tracks along streamlines calculated from a flow equation having the stream function as the dependent variable. A comparison of this model to Eulerian or fixed coordinate type models showed this model to be accurate and stable at a reduced computational effort. For an actual field contamination problem which contains large vertical concentration gradients, the simulated results compared with observed data.