Non-modal stability analysis of miscible displacement flows in porous media

Abstract

The uid displacement in a porous medium applies to a wide variety of industrial and environmental processes, for example, oil recovery is enhanced by displacing the oil with injected water, or rainwater penetrates into soil by displacing air. During these displacement processes, some fascinating patterns are always displayed in the

uid- uid interface. Two prototypes of these phenomena are viscous ngering (VF) instability and density ngering (DF) instability. VF occurs between two uids when the less viscous uid displaces the more viscous one, whereas DF is observed when the heavy uid displaces the lighter one. Mathematical modelling of such miscible displacement ows and their stability analysis are studied under the assumptions that the uids are incompressible and nonreactive. The dynamic viscosity and the density of the uid mixture is determined by the solute concentration. In this thesis, we present a non-modal linear stability analysis of miscible displacement ows in a homogeneous porous medium. The linear stability analysis (LSA) of miscible VF (and DF) is plagued, mainly, due to the time-dependent stability operator. Commonly used techniques such as frozen coe cient analysis, which is also known as quasi-steady state approximation (QSSA) or initial value problem (IVP) approach using ampli cation measure yield substantially di erent results for the onset of instability. To address this disagreement, we developed a novel linear stability analysis known as non-modal analysis (NMA) in a self-similar transformation domain. Further, the time-dependent stability operators are in general fundamentally and irreconcilably non-normal, which leads to transient growth of perturbations. The proposed NMA is in the spirit of Lyapunov stability criterion and singular value decomposition, and precisely addresses the transient behavior rather than the long-time behavior predicted by quasi-static eigenvalues determined from QSSA. The transient behavior of the response to external excitations and the response to initial conditions are studied by examining the structures of spectra and the largest energy growth function. We have shown that the dominant perturbation that experiences the maximum ampli cation within the linear v regime leads to the transient growth. The physical relevance of obtained optimal ampli cation of perturbations has been compared with nonlinear simulations. Further, the e ect of solute dispersion in the stability of a miscible ow has been analyzed in terms of a dimensionless parameter, the P eclet number. It has been shown that within the framework of 2-norm, the stability matrix can be symmetrizable by a similarity transformation and thereby we show that the transient growth of perturbations are norm dependent. Fingering instabilities also observed during geological sequestration of disposal in deep saline aquifers where miscible ngering instabilities are driven by both viscous and buoyancy forces. Hence, a careful analysis is presented using NMA to obtain the onset of instability in the time-varying linearized operator. It is shown that the viscosity contrast acts against the DF in a vertical porous medium. However, when one uid displaces the other, instability enhances as the viscosity increases with the depth. The e ects of non-monotonic viscosity pro les on the viscous ngering instabilities in miscible displacement ows in porous media are also investigated. In the miscible displacement with non-monotonic viscosity pro le develops an unstable region followed downstream by a stable region or vice versa. Instabilities rst set in the unstable region, which then grow and penetrate the stable region. The stable region has the potential to act as a barrier to the growth of the ngering instabilities. The stability analysis presented by NMA suggests that the quasi-static eigenvalues and vectors are not su cient to analyse the ow instability. Finally, a linear stability tool based on an IVP is developed using the Fourier pseudospectral method to address the e ect of a linear adsorption isotherm on the onset of ngering instability in a miscible displacement in a chemical separation process known as liquid chromatography. The proposed linear stability method is helpful to predict the growth rate of each individual ow variables, which is not possible by QSSA and proposed NMA. The presented linear stability analysis based on NMA and IVP based on the Fourier pseudopsectral method brings valuable insights into the mechanics of miscible viscous ngering. This will help in understanding other related areas of the uid mechanics where the ow stability is altered or a ected due to VF.