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.