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

Abstract:

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 λ_// 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 (ηαγ^n-1). As the flow velocity increases the dispersivity obeys an asymptotic power law: λ_// α Pe^1-n. This asymptotic regime is achieved at moderate Peclet numbers (Pe ≈ 10) with strongly non-Newtonian fluids (n ≤ 0.6) and on the contrary at very large values when n goes to 1 (Pe ≥ 10⁴ for n=0.9). This reflects the cross-over from a scaling behaviour for n≠1 towards a logarithmic behaviour predicted for Newtonian fluids (n=1).