Non Linear Three Dimensional Miscible Viscous Fingering in Porous Media

Abstract

Viscous fingering resulting from unstable fluid displacements in porous media has been studied extensively over the last forty years since the pioneering experiments of Hill1. Recent reviews on this subject by Homsy2 and Yortsos3 are available as well as on the special issue of the Saffman-Taylor4 finger in a Hele-Shaw cell by Bensimon et al.5. Most of the papers on viscous fingering deal with immiscible fluids, but indeed the problem involving miscible fluids deserves at least as much attention as the immiscible case: as in the immiscible case, the unfavorable viscosity ratio (displacing fluid less viscous than the displaced one) generates the instability but here the stabilizing effect is due to the hydrodynamic dispersion which tends to spread out growing fingers. Dispersion is more subtle than interfacial tension. Further, it is anisotropic and flow dependent which leads to new predictions6–12 such as a cross-over between diffusive and linear growth regimes7, 12 and an enhancement of the instability due to the interplay between the large viscosity ratio and the velocity dependent hydrodynamic dispersion9. Experiments are scarce1, 13–18 and deal generally with a pseudo 2D geometry involving qualitative visualization. In this paper, we use a newly developed acoustic technique19–21 to carry out the first study of the profiles of viscous fingers in 3D porous media. Our experiments have been performed out on three different porous media with a wide range of viscosity ratios and flow rates. Both the diffusive and the linear growth are observed including the cross-over from one to the other. Taken together, our data are best understood in terms of a new instability parameter, that characterizes the main features of viscous fingering. Our determination of the dependence of this parameter on the viscosity ratio, the flow rate and on the porous medium when placed in the context of existing theory leads to new physical insights on this rich and varied problem.