A phase field model for partially saturated geomaterials describing fluid–fluid displacements, Part II: Stability analysis and two-dimensional simulations

Flows involving immiscible displacement of one fluid by another in a porous media are known to destabilize and form fluid fingering. When the non-wetting fluid is a highly mobile gas (air) and the wetting fluid is an in-compressible liquid (water) the classical macroscopic theory is unable to describe the fingered flow. In Part I of this study we have introduced a model that interprets the mixture of wetting and non-wetting fluids within the pore space as a single saturating non-uniform pore fluid characterized by a phase field parameter, which is considered to be the saturation degree of the wetting fluid. In the current study we present a linear stability analysis of its solutions which describe both imbibition and drainage. The analysis sheds light on the sensitivity of the flow stability on injection flux, imposed pressure gradient and initial saturation degree. Two-dimensional numerical simulation results are as well presented which verify the stability analysis and reveal the rich structure of the fluid fingering realized by this model. While these results are found to be in qualitative agreement with experimental observations, they also warrant further experimentation to explore the additional features predicted by the model.