The perfectly stratified aquifer has often been investigated as a simple, tractable model for exploring new theoretical issues in subsurface hydrology. Adopting this approach, we show that steady groundwater flows in the perfectly stratified aquifer are always confined to a set of nonintersecting permanent surfaces, on which both streamlines and vorticity lines lie. This foliation of the flow domain exists as well for steady groundwater flows in any isotropic, spatially heterogeneous aquifer. In the present model example it is a direct consequence of the existence of a stream function. We then demonstrate that tracer plume advection by steady groundwater flow in a perfectly stratified aquifer is never ergodic, regardless of the initial size of the tracer plume. This nonergodicity, which holds also for tracer advection in any isotropic, spatially heterogenous aquifer, implies that stochastic theories of purely advective tracer plume movement err in assuming ergodic behavior to simplify probabilistic calculations of plume spatial concentration moments. ¿ 1998 American Geophysical Union