Computational study of convection {diffusion{reaction equations apropos of fingering instabilities

Abstract

Fingering instabilities are ubiquitous in porous media ows like enhanced oil recovery, CO2 sequestration, contaminant transport in aquifers. The mathematical modelling and the visualisation of these instabilities in terms of the solutions of the governing equations is important to understand the instabilities, to control them and to predict the transition from one ow regime to other, if exists. Thus, the use of proper governing equations is necessary to capture all the underlying mechanisms. We deal with miscible, incompressible, Newtonian uids owing through a porous medium. Hence the continuity equation for conservation of mass and the Darcy's law for conservation of momentum of ow through porous medium are the apt choices. The miscibility of the uids demands another equation required for the conservation of the mass of the species (concentration of solute in the solvent), for which we use the Convection{ Di usion{Reaction (CDR) equation to take into account two physical processes of convection and di usion interplaying with the chemical reaction. The CDR equation is further coupled to the aforementioned equations with the help of the equation of state. We solve the system of equations numerically by converting the equations dealing with uid ow to stream function-vorticity formulation, while the CDR equation is solved without any modi cation, that is, we deal with the computational study of CDR equation in order to understand various aspects of a few ngering instabilities. Fingering instabilities are observed when the uid- uid interface deforms into nger like patterns due to some variation in the physical property during the ow. The ngering instability arising due to change in viscosity, in particular, when a less viscous uid displaces a more viscous one, is termed as viscous ngering (VF), while another kind of instability also arises due to a change in the permeability. We discuss the numerical solution of CDR equations coupled to the equations of uid ow to discuss these two ngering instabilities. We consider two kinds of displacements viz., radial and rectilinear. As the name suggests, the uid displaces the other uid radially in the former and linearly in the latter. Another di erence is the unperturbed velocity which is non-uniform spatially varying for radial displacement and is considered only uniform for the other displacement. We perform a linear stability analysis (LSA) considering a radial displacement of a nite source of one uid which is being continuously injected and is also displacing the surrounding more viscous and non-reactive uid. The base state concentration is thus a solution of the Convection{Di usion equation obtained numerically using the method of lines approach. The growth/decay of the perturbations is analysed using the energy ampli cation. The LSA hints the use of a nite source in controlling the VF instability. Further non-linear simulations are performed using compact nite di erence and pseudo-spectral methods. In the parameter space spanned by the P eclet number and log-mobility ratio, we obtain a stable zone in which no VF occurs. In the unstable zone of the mentioned parameter space, we explore full VF dynamics and capture a convection dominated region sandwiched between two di usion dominated regimes. VF with reactive uids is understood both in the linear and the non-linear regime. The developed LSA is extended to understand the reactive VF in the linear regime. The onset time of instability is found to be a function of the viscosity contrast between the reactants and the product; being larger for the reaction developing a minimum in the viscosity pro le. This is in contrast to existing LSA with rectilinear displacement signifying the e ect of the kind of displacement on the dynamics. The properties of the chemical reaction are explored as a function of the Damk ohler number by choosing proper non-dimensionalisation and solving the resulting dimensionless CDR equation using method of lines approach. We solve the governing non-linear partial di erential equations to understand the ngering instability induced solely by the chemical reaction when (a) viscosity is modi ed by the chemical reaction, (b) precipitation reaction modi es the permeability of the porous medium. Same set of governing equations with suitable constitutive relation are used to model both ngering instabilities and our model is found to capture the underlying dynamics as well as the di erences between the two instabilities. The results in both the cases are in agreement with the experimental studies existing in literature. The results of the non-linear simulations of reactive VF are in agreement with the LSA we did. Further, it is found that a critical viscosity contrast between the products and reactants is required to trigger the VF instability for a given Damk ohler number and P eclet number. We obtain a stable zone sandwiched between two unstable zones in the parameter space spanned by the Damk ohler number and log-mobility ratio.This hints the use of chemical reaction in controlling the VF instability by suitably choosing the reactants. During a porous medium ow, one uid may be con ned between the layers of other uid. Motivated by this, we explore the case when one of the uid is con ned between the layers of the other uid but by considering the uids to be non-reactive. The VF of the annulus is found to be a persistent dynamics in contrast to the transient VF dynamics of a slice sandwiched between layers of other uid and displaced linearly. The non-linear simulations performed using COMSOL Multiphysics to obtain the exact conditions used in the existing experiments and a good agreement exists between our numerical study and the experiments. The e ect of niteness is also explored by considering a highly viscous blob in the ambient uid undergoing rectilinear displacement. For a large viscosity contrast between the uids, no VF occurs and a comet shaped instability is observed. The numerical method developed to understand LSA in this thesis provides a new approach to understand time-dependent linear systems arising in miscible VF. The hybrid numerical scheme used to solve CDR equations can be applied to visualise pattern formation in various elds. The ndings of this thesis will be helpful in controlling the instability during porous media ows like enhanced oil recovery along with providing a numerical insight into CO2 storage.