This work deals with a comparison of different numerical schemes for the simulation of contaminant transport in heterogeneous porous media. The numerical methods under consideration are Galerkin finite element (GFE), finite volume (FV), and mixed hybrid finite element (MHFE). Concerning the GFE we use linear and quadratic finite elements with and without upwind stabilization. Besides the classical MHFE a new and an upwind scheme are tested. We consider higher order finite volume schemes as well as two time discretization methods: backward Euler (BE) and the second order backward differentiation formula BDF (2). It is well known that numerical (or artificial) diffusion may cause large errors. Moreover, when the Péclet number is large, a numerical code without some stabilising techniques produces oscillating solutions. Upwind schemes increase the stability but show more numerical diffusion. In this paper we quantify the numerical diffusion for the different discretization schemes and its dependency on the Péclet number. We consider an academic example and a realistic simulation of solute transport in heterogeneous aquifer. In the latter case, the stochastic estimates used as reference were obtained with global random walk (GRW) simulations, free of numerical diffusion. The results presented can be used by researchers to test their numerical schemes and stabilization techniques for simulation of contaminant transport in groundwater.