Abstract
We develop coupled evolution equations for viscous fingering (VF) and phase separation in partially miscible systems by combining a simple double-well thermodynamic free energy and Korteweg force with a classical miscible VF model for a binary system. The VF pattern transition into a droplet formation pattern by the spinodal decomposition effect is demonstrated, and the simultaneous increases in the depth of the energy minimum, in the difference in the equilibrium concentrations, and in the Korteweg force, enhance the droplet growth. The pattern's interfacial length increases with the spinodal decomposition effects. These results match the corresponding experimental results.
Highlights
A fingering interfacial pattern is observed when a more viscous fluid is displaced by a less viscous fluid in a porous medium or in Hele-Shaw cells
We established a numerical simulation of viscous fingering with a phase separation of the spinodal decomposition type, taking into account the Korteweg force
For the first time, we show mathematically that the fingering pattern transitions to the droplet pattern when the system transitions from fully miscible to partially miscible in the rectilinear geometry, which was observed in the experiment in the radial geometry, and that its origin is the effect of phase separation and Korteweg convection, by using a simple model that can effectively adjust the effects of phase separation
Summary
A fingering interfacial pattern is observed when a more viscous fluid is displaced by a less viscous fluid in a porous medium or in Hele-Shaw cells. Amooie, Soltanian & Moortgat (2017) modelled the mixing and spreading resulting from viscous fingering in porous media for fully miscible (single-phase) CO2 oil and partially miscible (two-phase) CO2 and N2 oil mixtures, and compared them Their simulation showed that CO2 viscous fingering in a partially miscible system was suppressed compared to that in a fully miscible system owing to the interphase mass exchange leading to a diminished contrast in viscosity. An experimental study Suzuki et al (2020) concluded that the origin of the multiple droplet formation is the nature of the phase separation and spontaneous convection induced by the so-called Korteweg force. To prove the topological changes obtained in the experimental study (Suzuki et al 2020), we conduct a numerical simulation of the partially miscible VF considering the influence of the phase separation and the Korteweg force. Numerical simulation of the model equations is performed using a Fourier spectral method and successfully explained the origin of multiple droplets formation, and the results compare well with the experimental study (Suzuki et al 2020)
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