jats:pWe study the migration of a tracer within an injection-driven flow in a horizontal aquifer in which the permeability varies with depth. The permeability gradient produces a shear and this leads to lateral dispersion of the tracer. In the high permeability regions, the tracer moves substantially faster than the mean flow and eventually enters the nose region of the flow where the depth of the current is less than the depth of the aquifer. Depending on the influence of (i) the viscosity contrast between the injected fluid and the original fluid, and (ii) the vertical permeability gradient, the nose of the current may be of fixed shape or may gradually lengthen with time. This leads to a variety of patterns of dispersal of the tracer, which may either remain in the nose or cycle through the nose and be left behind. Our results illustrate the complexity of the migration of a tracer in a heterogeneous aquifer which has important implications for interpreting the results of tracer tests as may be proposed for monitoringjats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112018008911_inline1" />jats:tex-math$CO2$</jats:tex-math></jats:alternatives></jats:inline-formula>or gas injected into subsurface reservoirs.</jats:p>