Analytical and numerical investigation of the advective and dispersive transport in Herschel–Bulkley fluids by means of a Lattice–Boltzmann Two-Relaxation-Time scheme

Abstract

Dispersion of a passive tracer in a tube has been extensively studied in the case of Newtonian fluids since the pioneer work of Taylor (1953). However, the influence of more complex rheological behavior on the transport has only be scarcely investigated. Non-Newtonian fluids are increasingly used in the industry and transport in this type of fluid merits therefore thorough investigations. An example of industrial application is Enhanced Oil Recovery, that is based on the injection of non-Newtonian fluids as polymers or surfactant solutions in porous media, which are then submitted to dispersion phenomena.This work deals with transport of a passive tracer in shear thinning fluids with and without yield stress whose constitutive behaviors are representative of a large number of industrial fluids. We focus on transport in capillary tubes, essential for the understanding of dispersion in porous media. Transport is investigated at different time scales by solving the advection–diffusion equation using a Two-Relaxation-Time Lattice–Boltzmann method. We also derived an analytical expression of the Taylor dispersion coefficient for a large range of fluid rheologies. Dispersion coefficients of all fluids described by the Herschel–Bulkley model can now be determined. Analytical and numerical results are compared and very good accordance is obtained.We discuss the characteristic time scales of the transport before reaching steady state as a function of fluid rheology and Péclet number. We show that the time to reach the dispersive regime is nearly independent of the fluid rheology whereas the effective dispersion coefficient is a function of the rheological parameters.We also present the displacement distribution of the tracer molecules (propagators) as a function of time and show that they are strongly conditioned by the fluid rheology. Indeed, propagators give valuable information on the temporal evolution of the concentration profile towards the stationary Taylor regime.