On hydrodynamic mixing in a model of a porous medium with stagnant zones

Hydrodynamic mixing in a porous medium is considered with the help of a model containing a series of perfect mixers with stagnant zones. The relationship between this model and the usual diffusion one is studied. Both models are investigated by the device of injection of a delta-shaped tracer at the entrance to the system. For a large enough number of cells and a sufficiently large length of the porous medium, described by a diffusion model, the tracer concentration distribution at the exit as a function of time is shown to be normal in both cases. The effective coefficient of diffusion, or the dispersion, is found from a comparison of the parameters of these two distributions. At small rates of fluid exchange between the flowing and stagnant zones, the dispersion coefficient is very large. This imposes rigorous limitations on the length of the porous medium necessary for the establishment of a normal concentration distribution at the exit. If, during experimental determination of the dispersion coefficient, these conditions are not fulfilled, i.e. if the porous medium does not prove to be long enough the normal distribution at exit is not established. Then the curve is bell-shaped with a long, steady "tail". Calculation shows, that the distribution can then be represented as the sum of two distribution: a normal distribution and an exponentially decaying one. The parameters of the porous medium can be determined from experimental results.