Analysis of hydrodynamic instabilities in miscible displacement flows in porous media

Abstract

Miscible displacement flows in porous media are of significant importance in many industrial, environmental, and engineering applications, such as enhanced oil recovery, chromatography separation, CO2 sequestration, contaminant transport in geological aquifers, etc. These displacement flows feature different hydrodynamic instabilities depending on the physical properties of the fluids involved and the characteristics of the porous media. In this thesis, we numerically analyze the stability of miscible displacement flows in a homogeneous porous medium. To mathematically model such displacement flows and their stability analysis we assume that the fluids are incompressible, neutrally buoyant, and nonreactive. The fluid viscosity is determined by the presence of a solute concentration dissolved in the solvent. Viscous fingering (VF) is observed when a less viscous fluid displaces a more viscous one in porous media. Perturbations at the fluid-fluid interface protrude, and the more mobile displacing fluid invades through the less mobile resident fluid. Thus, the interface develops finger-like projection at the interface, hence called viscous fingering. The mathematical challenges of a linear stability analysis (LSA) of miscible VF are due, mainly, to the time-dependent base-state flow. These problems feature two different time scales: (a) diffusive-time scale for the evolution of the base state and (b) time scale for the evolution of the perturbations. Over the last few decades’ quasi-steady-state approximation (QSSA) method has been extensively used to characterize the stability in the linear regime. However, QSSA method fails to successfully capture the onset of instability. In this direction, this thesis develops a new linear stability technique, in the similarity transformation domain based on the principle of QSSA method, and abbreviated as SS-QSSA. SS-QSSA successfully captures the onset of instability in miscible VF. SS-QSSA is further used to characterize the stability of a finite fluid slice of more or less viscosity than the displacing fluid. Numerically we have proved the existence of a critical sample width for the onset of instability, and also computed the critical sample width for different flow conditions. Dispersion of the solute is a very essential phenomenon that characterizes the stability of a miscible system. The influence of the solute dispersion on the stability of miscible displacements is analyzed numerically in terms of a dimensionless parameter, the P´eclet number. Steep concentration gradient in miscible system results a nonconventional stress, called the Korteweg stress. The existence of this stress in various mixtures, such as water-glycerine, isobutyric acid-water, water-honey, etc., has been experimentally verified. We analyze the influence of the Korteweg stress on miscible VF. It is shown that the Korteweg stress has a stabilizing influence, which is ensured from the delayed onset of instability measured using SS-QSSA. Critical sample width increases with the Korteweg stress strength. For a given sample size larger than its critical value, instability region shrinks with the strength of the Korteweg stress. Counterintuitive results, the instability enhances with the increasing magnitude of the Korteweg stress, are observed for very large P´eclet values. Numerical simulations of the complete nonlinear problem are performed using a Fourier pseudospectral method to capture the nonlinear dynamics and pattern formation. We showed that the Korteweg stress inhibits the tip-splitting phenomenon, which is important in miscible flow because it increases the contact between fluids, thus enhancing mixing and reaction rates. The inhibition of tip-splitting instability in the presence of these stresses signifies that the Korteweg stress is analogous to the surface tension in immiscible fluids. The growth rate and the wave number of the unstable modes obtained from our nonlinear simulations are in excellent agreement with those obtained from self-similar LSA. Through nonlinear simulations, we have determined three different instability modes of a more viscous circular sample in rectilinear displacement in homogeneous porous media. It is shown that there exists a finite window of log-mobility ratio for VF in a more viscous circular blob, which is completely new in miscible viscous fingering. On the other hand, for a less viscous blob comet shape deformation is observed for viscosity contrasts larger than a critical value. In this case, fingering instability enhances as the viscosity contrast increases beyond the critical value. A linear stability tool is also developed using the same Fourier pseudospectral method used for the nonlinear simulations. This linear stability method is more generic, computationally rigorous but less expensive, and also mathematically robust. Using this tool and a new viscosity scaling we determine the onset of instability in a finite slice of more of less viscosity than the displacing fluid in the presence of the Korteweg stress. It is shown that the instability sets in at the same time both in a less viscous and a more viscous finite slice. Nonlinear simulations support our LSA results. This linear stability technique can be helpful for any hydrodynamic instability problem with unsteady base state. Influence of viscosity contrast on buoyantly unstable miscible layer in vertical porous media is analyzed through nonlinear simulations. It is shown that the viscosity contrast acts against the Rayleigh-Taylor instabilities in vertical porous media. However, when one fluid displaces the other, instability enhances as the viscosity increases with the depth. The findings of this thesis will be helpful for understanding the fluid mechanics aspects inherent in the problems of chromatographic separation, oil recovery, contaminant transport, and geological CO2 sequestration.