Influence of non monotonic viscosity profiles on the stability of miscible displacement flows in a Hele-Shaw cell

Abstract

If a less viscous fluid displaces a more viscous one in a porous medium or Hele-Shaw cell, the unfavorable mobility gradient is known to cause the classical ’viscous fingering instability’. In the past, this instability has been investigated mostly on the basis of Darcy’s law, which involves certain averaging procedures that are usually not valid for the case of a Hele-Shaw displacement. In this diploma thesis, miscible fingering in a horizontal Hele-Shaw cell is studied by means of Stokes simulations and linear stability analysis. The difference from former studies is that a nonmonotonic concentration viscosity profile is used which can exist in nature for specific fluid pairs. Here, particular emphasis is placed on identifying parameter regimes in terms of the Peclet-number Pe, the viscosity contrast R, the value and position of the viscosity maximum μm and cm. Λ is an additional parameter depending on the end-point derivatives. The existing nonlinear Stokes code for monotonic exponential viscosity profiles developed by N. Goyal was modified to incorporate any general viscosity concentration functional relationship. These twodimensional simulations lead to a moving finger with a quasisteady state near the tip of the displacement front for sufficiently large Peclet number and high viscosity ratios. The front thickness d0 and the tip velocity vtip of the quasisteady state is primarily affected by the Peclet number Pe and the viscosity contrast R. The scale of d0 with Pe−1/2 which is found for the monotonic profile is deviating for the nonmonotonic profile. A higher value of the maximum of the nonmonotonic viscosity profile leads to a slower tip and thicker front, while a shift of the maximum from the less viscous fluid to the more viscous fluid leads to a increase of the tip velocity and a decrease of the front thickness. In a second step the results of the direct numerical Stokes flow simulation are used to examine the stability of the quasisteady front to spanwise perturbations. In the linear stability analysis the viscosity contrast R is the dominant factor that affects the stability of the flow. Manickam&Homsy [10] investigated that for the Darcy flow a negative Λ leads to a more stable configuration than a positive Λ, while we see a reversal of this behavior. Only higher values of Λ lead to strong changes for the growth rate σmax of the most dangerous mode and the corresponding wavenumber βmax. Because the shape of the profile is also changing a lot with high values of Λ it is really hard to figure out what leads to this change. The exact reason could not be identified and would need further investigation.