Unrealistic solution for advection-diffusion-reaction PDE with heterogeneous porous media
It's not clear exactly what is being modelled here, but it seems to me that there are two ways in which the concentration can 'go negative'. Firstly, the rate of change of concentration can be massive, in which case see what happens when modelling with much smaller time steps. Or, the diffusion term substantially exceeds the advection term, which is physically unlikely. However, you are modelling a large (more than 10-fold) anisotropy, in which almost all the flow will be in the x, horizontal direction - it is almost a one-dimensional problem in fact. So how can the advection be less than the diffusion? In the case of a constant concentration source, such as a saline water body, the negative results don't make sense, in which case the calculations have gone unstable, probably from using too large a time step. In the case of a pulsed source, as the concentration peak passes (along the x direction), the advected concentration rises and falls, such that the lateral (y axis) diffusion increases, then decreases, i.e. the concentration gradient changes direction, becoming negative.

As if this wasn't hard enough, in real life applications the 'sink' terms probably far exceed any back-diffusive effects.