Tracer dispersion in power law fluids flow through porous media: Evidence of a cross-over from a logarithmic to a power law behaviour

An analytical model is presented to describe the dispersion of tracers in a power-law fluid flowing through a statistically homogeneous and isotropic porous medium. The model is an extension of Saffman's approach to the case of non-Newtonian fluids. It is shown that an effective viscosity depending on the pressure gradient and on the characteristics of the fluid, must be introduced to satisfy Darcy's law. An analytical expression of the longitudinal dispersivity lambda(//) is given as a function of the Peclet number Pe and of the power-law index n that characterizes the dependence of the viscosity on the shear rate (eta proportional to gamma(n-1)). As the flow velocity increases the dispersivity obeys an asymptotic power law: lambda(//) proportional to Pe(1-n). This asymptotic regime is achieved at moderate Peclet numbers (Pe approximate to 10) with strongly non-Newtonian fluids (n less than or equal to 0.6) and on the contrary at very large values when n goes to 1 (Pe greater than or equal to 10(4) for n = 0.9). This reflects the cross-over from a scaling behaviour for n not equal 1 towards a logarithmic behaviour predicted for Newtonian fluids (n = 1).