Immiscible viscous fingering in time-dependent fluids: A linear stability analysis

Abstract

When a less viscous fluid displaces a more viscous fluid, the interface between the two fluids becomes unstable. The instability exhibits itself as a large number of fingers of the displacing fluid penetrating the displaced fluid. This phenomenon is called viscous fingering or Saffman–Taylor instability, and plays a key role in oil industry, among others. The present study aims at investigating the immiscible viscous fingering of inelastic time-dependent fluids (i.e., fluids for which viscosity varies with time even at a fixed shear rate) in a homogeneous saturated porous medium. To represent such fluids, in the present work, use is made of the Quemada model. As an inelastic structural-kinetic rheological model, it can capture both thixotropy and anti-thixotropy through proper setting of model parameters. The model incorporates a natural time, which, in dimensionless form, is represented by a nominal Deborah number. Using this rheological model, the generalized Darcy’s law is derived, which is used to derive the basic flow. The interfacial instability is investigated by using the linear temporal stability analysis; that is, infinitesimally-small, normal-mode perturbations are introduced to the basic flow and their evolution in time is monitored. Using the Young–Laplace equation, the generalized dispersion relation is derived, which is solved numerically. The results show that the thixotropy and anti-thixotropy stabilize the Newtonian/time-dependent and time-dependent/Newtonian displacements, respectively. On the other hand, the Newtonian/time-dependent and time-dependent/Newtonian displacements are destabilized by the anti-thixotropy and thixotropy, respectively.