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Description
The displacement of a wetting fluid by a non-wetting fluid in porous media is an ubiquitous process in multi-phase flow and typically gives rise to a transient propagating interface referred to as the drainage front. Such fronts occur in transient settings, including the injection of supercritical CO2 into brine-saturated geological formations and rapid drying of water-saturated clayey materials. Under certain regimes, these drainage fronts may become unstable and develop finger-like patterns [1], whose morphology depend on the prevailing flow regime. The stability of a drainage front is generally agreed [1, 2, 3, 4] to be controlled by the interplay among capillary, viscous, and gravitational forces.
In this study, we focus on the interplay between capillary and viscous effects in a regime of practical interest where the invading phase is much less viscous and less dense than the displaced phase, while staying within a continuum-scale modeling framework. In classical poromechanics, the capillary pressure difference between immiscible pore fluids is represented as a local, bijective function of the wetting-phase saturation, $P_c(S_w)$. To improve upon the coarse up-scaling inherent in this description, two extended models have been proposed in the literature ([5] and [6]). While the application of either of these formulations has demonstrated the ability to reproduce macroscopic fingering-like flow instabilities, it has been largely limited to imbibition scenarios ([7, 8]) under the strong simplifying assumption of neglecting the non-wetting phase pressure; an assumption appropriate only for specific contexts such as soil hydrology. Their applicability and relative performance in drainage processes remain unexplored.
Bearing in mind the current context, in this study we first restore the non-wetting phase pressure as an independent variable and derive the dimensionless formulations of both extended models. Using one-dimensional numerical simulations, we then demonstrate the formation of self-similar traveling-wave solutions (TWs) during drainage under different parameter regimes. Subsequently, we perform linear stability analysis (LSA) of these solutions with respect to transverse perturbations, thus assessing their tendency towards long-term amplification or decay. This allows us to identify conditions under which the enriched capillary models can reproduce physically meaningful fingering instabilities. Further we demonstrate using LSA against longitudinal perturbations ability of the Cahn-Hilliard like model, presented in [6], to reproduce pinch-off effects. Overall, this work advances the stability analysis of drainage fronts towards more realistic scenarios involving compressible multi-phase flow.
| References | [1] Lovoll, G., Méheust, Y., Maloy, K.J., Aker, E., & Schmittbuhl, J. (2005). Competition of gravity, capillary and viscous forces during drainage in a two-dimensional porous medium, a pore scale study. Energy, 30, 861-872. [2] Saffman, P.G., & Taylor, G.I. (1958). The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 245, 312 - 329. [3] Homsy, G. (1987). Viscous fingering in porous media. Annual Review of Fluid Mechanics, 19, 271-311. [4] Wang, Z., Feyen, J., & Elrick, D.E. (1998). Prediction of fingering in porous media. Water Resources Research, 34, 2183 - 2190. [5] Hassanizadeh, S.M., & Gray, W.G. (1990). Mechanics and thermodynamics of multiphase flow in porous media in- cluding interphase boundaries. Advances in Water Resources, 13, 169-186. [6] Sciarra, G. (2016). Phase field modeling of partially saturated deformable porous media. Journal of The Mechanics and Physics of Solids, 94, 230-256. |
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| Country | France |
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