This paper develops certain exact closed form solutions to one-dimensional problems involving both advective and advective-diffusive transport in a porous medium, particularly in the presence of a Darcy velocity field that is time-dependent. Such situations can occur when the boundary potential inducing flow in the porous medium is time-dependent. A particular form chosen is an exponentially decaying time-dependency in the flow velocity, resulting from a non-replenishing source. The value of the one-dimensional solution stems not only from its potential applicability for the calibration of computational schemes used to examine the advection-diffusion equation in general, but also for the study of the purely advective flow problem with time-dependent velocity that requires sophisticated adaptive computational schemes to ensure numerical stability at a leading front in the form a discontinuity.