The effect of Freundlich adsorption in rectilinear miscible flow in porous media

Solute transport in porous media is strongly influenced by adsorption, particularly when the adsorption is non-linear and concentration dependent. In this work, we discuss the effect of Freundlich adsorption, described by a power law isotherm, on solute propagation and the onset of viscous fingering (VF) in unfavorable viscosity contrast systems. Unlike the Langmuir model, the Freundlich isotherm captures adsorption on a heterogeneous surface without a saturation constraint. A high-resolution Fourier pseudo-spectral method was applied to solve the governing equations and examine the intricate coupling between adsorption and instability. The rarefaction interface broadens with time and shows a non-monotonic dependence on the Freundlich exponent n. The reason is that there is a competition between the upstream-induced advection speed and the non-linear dispersion rate, both depending on n. The largest rarefaction layer width occurs at intermediate n values, where the spreading is most pronounced due to balanced transport dynamics. In the presence of viscosity contrast, we investigated the influence of adsorption on the onset of VF instability. The fingering onset time also varies non-monotonically with n, with the maximum delay occurring at intermediate values. This study shows that the presence of non-linear Freundlich adsorption can significantly alter macroscopic and microscopic solute behavior, with consequences for environmental migration, improved oil extraction, and chemically reactive flows in the subsurface.