Linear stability analysis of time-dependent immiscible flow displacements in porous media

Flow displacements in porous media are encountered in a wide variety of fields, in particular in the energy and environmental sectors. Such flows are prone to a hydrodynamic instability that develops as a result of differences in the physical properties of the different fluids, namely their viscosity and/or density. A simple and inexpensive way to attenuate the instabilities is to implement time-dependent injection schemes. In this study, flows in radial twodimensional homogeneous porous media are modelled. A low-viscosity fluid is injected radially at the centre of the cell to displace a more viscous one. The model equations are developed and a linear stability analysis is carried out. Different time-dependent velocity profiles are considered and the effects of different flow parameters are analysed and compared with the corresponding constant injection velocity flows. For consistency, the comparisons are conducted on the basis that all injection schemes including the constant one, result in the same total amount of injected fluid. Qualitative analyses are conducted using different measures of the growth of the instability and are used to determine the changes induced by the time-dependent injections. These changes in the flow dynamics can be used to control the degree of mixing between the two fluids and to optimize a variety of flow displacements in porous media. INTRODUCTION Hydrodynamic instabilities can develop at the interface between two fluids flowing in a porous medium. These instabilities manifest themselves in the form of finger-like structures at the interface between the two fluids. In the case of homogeneous porous media, the instability develops as a result of viscosities mismatch and is referred to as viscous fingering and/or densities mismatch where it is known as the RayleighTaylor instability. Both instabilities are encountered in a variety of processes that span a wide range of fields that include chemical, petroleum and environmental sectors. Hence a better understanding of these instabilities is crucial for the design, optimization and improvement of numerous processes encountered in the operation of packed bed reactors, production and enhanced recovery of oil, decontamination of soil as well as the sequestration of carbon dioxide Extensive studies have been conducted to analyze the effects of different factors on the instability including the viscosity ratio [1], dispersion [2], non-Newtonian rheological behavior of the fluids [3], chemical reaction [1], heat transfer [4] and flow configuration [5] among others. Detailed literature reviews that date back to the 1980s and 1990s are found in [6] and [7]. The vast majority of existing studies have focused on miscible displacements when dispersion plays an important role in addition to the viscosity/density ratios and the injection rates. However in numerous practical applications, the flow displacements are immiscible and surface tension forces become a major factor in the development and subsequent fate of the instability. Immiscible flows were studied experimentally. Paterson [8] investigated the width of fingers and developed equations to fit the radius of the tip and base of fingers based on the injection flow rate, time and arbitrary parameters. In a later study, Chen [9] analyzed the effect of plate roughness and flow rate as well as interfacial tension. It was observed that for smooth plates, fingering patterns are strongly influenced by the flow rate, with narrower fingers occurring as the flow rate increases. In rough plates, fingers tend to branch more easily. Low flow rates for both smooth and rough plates enable the fingers to be more compact. The same author later observed the development of fingers at later long times after multiple splitting, suggesting that immiscible patterns can be similar at the same dimensionless time [10]. In a subsequent study, Praud and Swinney [11] found an asymptotic behavior for the ratio of the fingers’ radius over the gap thickness in a radial Hele-Shaw cell. Many miscible displacement experiments have also been performed under a variety of conditions. Chen demonstrated that fractal dimensions remain constant [5]. Much of the experimental work on miscible flow has however been conducted for rectilinear geometries. The stability of immiscible displacement in a radial geometry has been studied by Wilson [12] and later by Paterson [8]. An equation relating the growth rate of the instabilities with the radius of the interface, the surface tension, the mobility ratio as well as the injection rate was derived. Later variations on this equation describing the growth rate of instabilities were developed. In particular Park and Homsy [13] derived an equation for the pressure jump at the interface. Their study was based on a three dimensional analysis, where the displaced fluid leaves a thin layer in the Hele-Shaw cell. Experiments validated the relationship developed to predict the number of fingers more accurately. Subsequently Martyushev and Birzina [14] used this pressure jump expression to develop a new equation for the growth rate of instabilities in the front for radial instabilities flows in the case of constant injection rate and constant pressure. 12th International Conference on Heat Transfer, Fluid Mechanics and Thermodynamics