Displacement of Fluids in Permeable Media
PARDEEP KUMAR
Department of Mathematics, ICDEOL, Himachal Pradesh University,
Summer-Hill, Shimla-171005 (HP) INDIA
Abstract: Viscoelastic (Maxwellian) slow, immiscible liquid-liquid displacement in a permeable medium
is considered. The necessary and sufficient criteria for stability are that the displacing fluid is denser and
less mobile than the displaced fluid. The instability criteria and critical wave length are found to be the
same as those for ordinary viscous liquid-liquid displacements in permeable media.
Keywords: Immiscible Displacements, Maxwellian Viscoelastic Fluids, Porous Medium
Received: July 5, 2021. Revised: February 12, 2022. Accepted: March 16, 2022. Published: April 19, 2022.
1 Introduction
Immiscible fluid displacement in
porous media is fundamental for many
environmental processes, including
infiltration of water in soils, groundwater
remediation, enhanced recovery of
hydrocarbons and carbon geosequestration.
Microstructural heterogeneity, in particular of
particle sizes, can significantly impact
immiscible displacement. For instance, it
may led to unstable flow and preferential
displacement patterns. The displacement
process involving two immiscible fluids is of
considerable importance in ground water
hydrology and reservoir engineering. The
nature of immiscible flow in a porous
medium is different from that in the Hele-
Shaw cell. Unlike the Hele-Shaw cell, where
the interfacial tension acts at a single
interface (Homsy [1]), the capillary forces in
a porous medium act on a multitude of
microscopic interfaces, giving rise to a single
dispersed interface at the macroscopic level
(Lake [2], Yortsos and Hickernell [3]).
Additionally, the mobility of individual
phases within the pore space is governed by
the wettability properties of the porous
medium (Bear [4], Scheidegger
[5]). Therefore, the dynamics of immiscible
displacements in porous media is represented
by Darcy’s law for each phase with an
associated relative permeability function
(Lake [2]). In the case in which the injected
fluid is of higher mobility than the resident
fluid, the displacement becomes unstable and
results in macroscopic viscous fingers. These
fingers are different from the microscopic
fractal structure that occurs in the limit of
vanishing capillary number and
infinite viscosity ratio, modeled, respectively,
by invasion percolation and diffusion-limited
aggregation (Sahimi [6]). The regime of
geological fluid flow, which falls between
these limiting cases, e.g.,
with viscosity ratio 100−102100−102 and
capillary number 100−103100−103, can
be represented by the Darcy equation based
on local averages (Blunt and King [7],
Yortsos et al. [8].
A theoretical and experimental
investigation of the occurrence of
macroscopic instabilities in the displacement
of one viscous liquid by another immiscible
one through a uniform porous medium was
made by Chuoke et al. [9]. The theoretical
description of the instability of fluid
displacements in permeable media is
immensely complex due to gross
inhomogeneities of porous media. However,
a quantitative prediction of finger-spacing is
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possible in a porous medium known to be
macroscopically homogeneous and isotropic
throughout. The fluid displacement between
closely spaced parallel plates is two-
dimensional, involves only microscopic
fluid-fluid interface and has been shown to be
mathematically analogous to two-
dimensional flow in a porous medium, by
Saffman and Taylor [10]. The displacements
in a porous medium are three-dimensional
and the macroscopic interface represents
many moving microscopic fluid-fluid
interfaces. The results for the parallel-plate
system can be considered a specialization of
those for porous media, as the formal
mechanics of theory are the same.
Certain assumptions and limitations
of Chuoke et al. [9] theory should be
mentioned. In analogy with the actual
interfacial tension, the assumption of
`effective interfacial tension` across the
macroscopic interface has been made. No
good agreement has been found between
calculated and observed finger spacing
between parallel plates, as well as transparent
glass powder pack experiments. When pure
water was injected to displace oil from
preferentially water-wet porous medium
containing connate water, it was found that
the fingering observations could not test the
theory quantitatively. Scheideggar [11]
studied the stability of displacement fronts in
porous media. Payatakes et al. [12] studied
oil ganglion dynamics during immiscible
displacements. Ekwere and Donald [13]
studied the onset of instability during two-
phase immiscible displacements in porous
media. They extended the work of Chuoke et
al. [9] by means of a stability analysis, a
universal dimensionless scaling group and its
critical value for predicting the onset of
instability during immiscible displacement in
porous media. Stability analysis of
immiscible displacement problems has been
carried out among others, by Cruz and
Spanos [14], Maloy et al. [15], Lenormand et
al. [16], Hilfer and Oren [17] and Rao et al.
[18].
In many reservoirs, the oils naturally
occurring beneath the surface of the earth are
found to exhibit some non-Newtonian
behavior (Allen and Boger [19]). The
consideration of the viscoelastic nature of
fluid to be displaced is closer to field
reservoirs and therefore to primary oil
recovery processes. Oldroyd [20] proposed a
theoretical model for a class of viscoelastic
fluids. Since viscoelastic fluids play an
important role in polymers and
electrochemical industry, the studies of
waves and stability in different viscoelastic
fluid dynamical configuration has been
carried out by several researchers in the past.
The nature of instability and some factors
may have different effects on viscoelastic
fluids as compared to the Newtonian fluids.
For example, Bhatia and Steiner [21] have
considered the effect of a uniform rotation on
the thermal instability of a Maxwell fluid and
have found that the rotation has a
destabilizing influence, for a certain
numerical range, in contrast to the stabilizing
effect on Newtonian fluid. In another study,
Bhatia and Steiner [22] have studied the
problem of thermal instability of a
viscoelastic fluid in hydromagnetics and have
found that the magnetic field has the
stabilizing influence on Maxwell fluid just as
in the case of Newtonian fluid. The
thermosolutal instability in a Maxwellian
viscoelastic fluid in porous medium to
include the Hall effect has been considered
by Sharma and Kumar [23]. In another study,
Kumar and Singh [24] have considered the
instability of the plane interface between two
viscoelastic (Maxwellian) superposed fluids
in porous medium in the presence of uniform
rotation and variable magnetic field. For
stable density stratification, the system is
found to be stable for disturbances of all
wave numbers and the magnetic field
stabilizes the potentially unstable
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stratification for small wave-length
perturbations which are otherwise unstable.
Kazachkov [25] has studied the development
and analysis of the mathematical model for
mixing and heat transfer in the two-fluid
turbulent heterogeneous jet of mutually
immiscible liquids.
Keeping in mind the immiscible
displacements process for primary oil
recovery in field reservoirs and the
viscoelastic nature of the fluid to be displaced
(e.g. oil), a theoretical study has been made
of the immiscible displacement of
viscoelastic (Maxwellian) fluid by another
viscoelastic fluid of similar nature in
permeable media.
2 Formulation of the Problem and
Basic Equations
Here we consider the configuration of
two Maxwellian viscoelastic fluids, labeled 1
and 2, each of infinite extent, with a plane
macroscopic interface moving slowly
through a uniform permeable medium with
velocity , normal to the interface. A
viscoelastic fluid 1 displaces another
viscoelastic fluid 2 with velocity in the
positive direction, which is normal to the
plane, macroscopic interface between the
two. The vertically upward direction is
chosen as the axis of a fixed system of
co-ordinates. If fluid 1 is displacing fluid 2,
is a positive quantity. The -component
of the gravitational acceleration is
. The porous medium is assumed
to be homogeneous and isotropic.
The Maxwellian viscoelastic fluid is
described by the constitutive relations:
where
j
i
j
i
j
i
e
j
i
p vi xi and
dt
d
denote, respectively the stress tensor, the
shear stress tensor, the rate-of-strain tensor,
the Kronecker delta, the viscosity, the stress
relaxation time, the isotropic pressure, the
velocity vector, the position vector and the
convective derivative.
For flows through porous media, the
Brinkman and viscous derivative terms are
very small in magnitude as compared to the
Darcian term, which is retained here. Using
the volume averaging procedure, a derivation
of the equations of motion and continuity for
viscoelastic fluids and the dominance of
Darcian term as the resistance term has been
shown by Slattery ([26], [27]). The linearized
macroscopic equations of motion and
continuity for incompressible, Maxwellian
viscoelastic fluid, a set for each fluid, are
or
where
and
where p g,
, and
with
the subscripts 1 and 2 distinguishing the two
fluids, stand for density, pressure, viscosity,
acceleration due to gravity, medium porosity,
effective medium permeability and
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perturbation in velocity, respectively and
t
T
.
The above equations describe the viscoelastic
fluid motion in a coordinate system moving
with velocity W in which the unperturbed
interface is at rest. Equations (2) and (3)
possess the integrals
where and are arbitrary functions
of time.
3 Criteria for Stability and
Discussion
Let us take to be the
arbitrary deformation of the macroscopic
interface and assume its Fourier
decomposition of the form
where are wave numbers along
directions;
is the resultant
wave vector of magnitude
,
is in general complex and .
The kinematic conditions to be satisfied at
the interface are
Since
1
and
2
must satisfy (8) and since
the -components of the perturbation velocity
must vanish at , the solutions of (4)
are
for fluid 1, and
for fluid 2. Here
k
is assumed small under
the first-order theory.
At each point of the macroscopic interface
there is conceived to be a pressure
discontinuity consisting of two types of
terms, i.e.,
where is independent of curvature of
the macroscopic interface but may be a
function of time and is related to the capillary
pressure drops across the microscopic fluid-
fluid interfaces underlying the macroscopic
interface, whereas is an effective
interfacial tension and are the signed
principal curvatures of the macroscopic
interface, to be taken as negative when the
respective center of curvature falls in the
domain of fluid 2.
Using (5) and (6) in relation (11), we obtain
where
Using (7), (9) and (10), (12) yields
and the characteristic equation
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Equation (14) determines as a function of
wave number and yields the kinematics of
early growth.
For , it is evident from equation (14)
that the necessary and sufficient criterion for
instability, i.e., for to be positive, is given
by
Now, introducing the volumetric velocity ,
we may say that instability will occur for all
velocities , where is a critical
velocity defined by
provided the perturbation contains
wavelength
greater than a critical
wavelength , defined by
Equation (14) admits of no positive root if
the constant term and the coefficients of
and are all positive. The stability criteria
(i.e. conditions to check fingering phenomena
on the macroscopic scale), which are of
fundamental importance in oil recovery
processes, are then
Physically, the Necessary and Sufficient
Criteria for Stability are:
(i) the displacing fluid is denser than the
fluid to be displaced,
and (ii) the less mobile fluid displaces the
more mobile one.
The criteria for instability and critical
wavelength for the case of Maxwellian
viscoelastic liquid-liquid displacements in
permeable media remain the same as those
for ordinary viscous liquid-liquid
displacements in permeable media.
4 Conclusions
The flow of viscous incompressible
fluids in the presence of porous bodies seems
to have generated a great deal of interest
among researchers, because of its
applications in numerous Scientific and
Industrial fields like Lubrication of Porous
Bearings, Ground Water Hydrology,
Petroleum Industries, Industrial filteration
and Agricultural Engineering etc. Stability
analysis of immiscible displacement
problems has been carried out by different
researchers in the past and in particular, a
theoretical and experimental investigation of
the stability of two slow, immiscible, viscous
liquid-liquid displacements in porous media
has been given by Chouke et al. [9]. Since in
many reservoirs, the oils naturally occurring
beneath the surface of the earth are found to
exhibit some non-Newtonian behavior and
keeping in mind the immiscible
displacements process for primary oil
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recovery in field reservoirs and the
viscoelastic nature of the fluids, a theoretical
study of the generalization of Chouke et al.’s
work has been made in the present paper by
considering slow, immiscible viscoelastic
(Maxwellian) liquid-liquid displacements in
permeable medium.
The necessary and sufficient stability
conditions which are of fundamental
importance in oil recovery processes are
obtained and are that the displacing fluid is
denser and less mobile than the displaced
fluid. The instability criteria and critical wave
length are found to be the same as those for
ordinary viscous liquid-liquid displacements
in permeable media.
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