Numerical study of the effect of Péclet number on miscible viscous fingering with effective interfacial tension

Abstract

Viscous fingering or Saffman–Taylor instability shows fingering interfacial patterns when a more mobile fluid displaces a less mobile one in porous media. The effective interfacial tension (EIT) is like capillary force, acting at the miscible interface on time scales shorter than interface relaxation. It has been numerically reported so far that the fingers are wider with EIT compared with those without EIT. A recent experiment observed finger widening with increasing flow rate in a miscible system with EIT, which is not observed in classical immiscible and miscible systems. In this study, we have numerically investigated the effect of $Pe$ (which corresponds to the injection flow rates in the experiment) on miscible viscous fingering with/without EIT. We have shown that the fingers are monotonically thinner with an increase in $Pe$ in the system without EIT, while finger widening with increasing $Pe$ is observed in the system with EIT. Furthermore, we have also examined a one-dimensional underlying concentration profile and EIT profile by using a one-dimensional diffusion–convection model because EIT is proportional to the squared concentration gradient. We have then shown that the concentration gradient is steeper and, thus, the EIT is larger as $Pe$ is larger. Therefore, this is the first numerical study that can theoretically verify finger widening with increasing flow rate, which occurs only in a miscible system with EIT to the extent of our targeting EIT values, and explain the mechanism by one-dimensional analysis.

Interfacial instabilities in confined displacements involving non-Newtonian fluids

“…The interfacial velocity u of a Newtonian fluid with viscosity η in a porous medium of permeability κ is described by Darcy's law [8], u = − κ η ∇P , where P is the pressure field. An interfacial instability characterised by the emergence [3,12,13] and (e) is reproduced with permission from [14].…”

“…(c) The length and width, λc, of a viscous finger. Formation of complex interfacial instabilities with increasing (d) capillary number Ca and (e) Péclet number P e. (a), (c), (d) are adapted from[3,12,13] and (e) is reproduced with permission from[14].…”