Numerical study on viscous fingering with a phase separation and the Korteweg force in a partially miscible system

Viscous fingering(VF) or Saffman-Taylor instability occurs during the displacement of one viscous fluid by another less viscous fluid in porous media. We have established a numerical simulation of viscous fingering with a phase separation of the spinodal decomposition type, taking into account the Korteweg force. The simulation reproduced successfully the experimental observation in Suzuki et al. (J. Fluid Mech., 898, A11, 2020) of the transition from viscous fingering to a droplet pattern owing to the increase in the phase separation effects. We show that the time derivative of the mixing length is a property that can qualitatively distinguish three systems: systems with strong phase separations, a weak phase separation, and no phase separation. Furthermore, the interfacial length as a function of the mixing length obtained in the simulation is in good agreement with the experiments of the previous study (Suzuki et al. 2020).