Shear dispersion in a porous medium. Part 2. Growing immiscible intrusion.

The dispersion of a pulse of tracer through a liquid injected into a confined aquifer with vertically varying permeability is studied theoretically. The injected fluid is buoyant and of low viscosity relative to the original fluid in the aquifer. The nose region of the flow, where the thickness of the injected fluid is less than the thickness of the aquifer, grows in proportion to time T and as a result the tracer continually migrates further into the nose where it has a progressively smaller vertical extent. The spreading of the nose stretches a pulse of initial length, L 0 , longitudinally to have a length proportional to L 0 (T /T E ) 1/2 in the absence of diffusion, where T E is the nose entry time. Diffusion acts at the same rate and the combination of the two processes results in the tracer spreading longitudinally with a length proportional to (DT log T ) 1/2 at long times after entering the nose. The results are generalised to consider the case in which the permeability in the aquifer varies with depth. The shearing of the tracer associated with the permeability variation is eventually balanced by vertical homogenisation. As the tracer migrates into continually thinner regions of the growing nose, the permeability contrast sampled by the tracer rapidly decays. The role of the shear becomes dominated by the stretching of the nose and ultimately the late-time behaviour is as in a uniform aquifer. However, the effective pulse length of the tracer upon asymptoting to the stretching regime is now given by L 0 = ∆U T E , where ∆U is the magnitude of the shear. The spreading in the stretching regime then has a lengthscale of ∆U (T E T ) 1/2 , which may be much faster than in the case of a uniform aquifer. If the diffusion is sufficiently fast, there may be an intermediate regime in which Taylor dispersion is important prior to the stretching dominating. The results have significant implications for tracer tests that are used in CO 2 sequestration because the influence of the early shear associated with any ...