Steady-state distributions of water potential and salt concentration in coastal aquifers are typically modelled by the Henry problem, which consists of a fully coupled system of flow and transport equations. Coupling arises from the dependence of water density on salt concentration. The physical behaviour of the system is fully described by two dimensionless groups: (i) the coupling parameter $\alpha$, which encapsulates the relative importance of buoyancy and viscous forces, and (ii) the Péclet number $\mbox{\textit{Pe}}$, which quantifies the relative importance of purely convective and dispersive transport mechanisms. We provide a systematic analytical analysis of the Henry problem for a full range of the Péclet number. For moderate $\mbox{\textit{Pe}}$, analytical solutions are obtained through perturbation expansions in $\alpha$. This allows us to elucidate the onset of density-driven vertical flux components and the dependence of the local hydraulic head gradients on the coupling parameter. The perturbation solution identifies the regions where salt concentration is most pronounced and relates their spatial extent to the development of a convection cell. Next, we compare our solution to a solution of the pseudo-coupled model, wherein flow and transport are coupled only via the boundary conditions. This enables us to isolate the effects caused by density-dependent processes from those induced by external forcings (boundary conditions). For small $\mbox{\textit{Pe}}$, we develop a perturbation expansion around the exact solution corresponding to $\mbox{\textit{Pe}}\,{=}\,0$, which sheds new light on the interpretation of processes observed in diffusion experiments with variable-density flows in porous media. The limiting case of infinite Péclet numbers is solved exactly for the pseudo-coupled model and compared to numerical simulations of the fully coupled problem for large $\mbox{\textit{Pe}}$. The proposed perturbation approach is applicable to a wide range of variable-density flows in porous media, including seawater intrusion into coastal aquifers and temperature or pressure-driven density flows in deep aquifers.