In this paper, we introduce 2D profile analytical models describing advection solute transport induced by a recharge boundary condition in a catchment-scale aquifer represented by a sediment or rock with permeability changing with depth. The suggested mathematical formulation is based on the concept of transit time distribution (TTD), according to which the concentration trend can be derived directly from the cumulative travel time frequency or from the convolution of TTD (considered as a transfer function) with a solute input function. This approach is shown to be applicable to water flow and solute transport in an aquifer characterized by a vertical hydraulic conductivity profile, k(z), which can be described by either continuous (e.g., massive rock exhibiting an exponential or power-law dependence of k on depth) or discrete (a stratified aquifer) analytical functions. A special focus is on the problem related to advection solute transport in a rectangular anisotropic flow domain bounded by a shallow stream of finite width (downflow-converging area). The suggested analytical models and solutions, utilizing new transfer functions and accounting for different inflow and solute flux conditions, can be involved in several applications, including the assessment of the impacts of spatially variable nonpoint contamination sources on groundwater quality; groundwater dating based on the observation of environmental tracers; and the study of groundwater recharge itself. The models were shown to be applicable not only to simulating the impact on groundwater due to the contaminated recharge, but also to predicting aquifer contamination caused by a source located below the water table within the aquifer domain. Such source can be associated with a radioactive material leachate generated within a deep geological repository of nuclear waste. A relevant modeling example is given.