PHYSICS OF FLUIDS
VOLUME 11, NUMBER 5
MAY 1999
LETTERS
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Nonlinear interactions of chemical reactions and viscous fingering
in porous media
A. De Wit
Service de Chimie Physique and Centre for Nonlinear Phenomena and Complex Systems, CP 231,
Université Libre de Bruxelles, 1050 Brussels, Belgium
G. M. Homsy
Department of Chemical Engineering, Stanford University, Stanford, California 94305-5025
~Received 23 October 1998; accepted 22 January 1999!
Nonlinear interactions of chemical reactions and viscous fingering are studied in porous media by
direct numerical simulations of Darcy’s law coupled to the evolution equation for the concentration
of a chemically reacting solute controlling the viscosity of miscible solutions. Chemical kinetics
introduce important topological changes in the fingering pattern: new robust pattern formation
mechanisms such as droplet formation and enhanced tip splitting are evidenced and analyzed.
© 1999 American Institute of Physics. @S1070-6631~99!01805-X#
“•uI 50,
The problem of fingering in flows with chemical reactions is of fundamental importance in several applications
ranging from petroleum1 or spill recovery, polymerization
fronts2 and chromatographic separations of concentrated
mixtures3 to deformation of chemical waves by hydrodynamical instabilities.4 Fingering instabilities occur when a
fluid with low mobility displaces another fluid with high mobility and take place in both miscible and immiscible systems. The difference in mobility can originate from a difference in viscosity and/or density. This hydrodynamical
instability has been the subject of numerous studies.1 Despite
the wide number of examples where fingering occurs in systems where chemical reactions take place, very little theoretical work has been devoted to the study of nonlinear interactions of chemical reactions and fingering.
In this letter, we find that viscous fingering of miscible
fluids flowing in porous media is strongly influenced by
chemical reactions, leading to new interactions and pattern
formation mechanisms. In particular, the interplay between
the hydrodynamical instability and chemical reactions leads
to the formation of droplets of one solution disconnecting
from the bulk and invading the other solution. Chemical reaction also catalyzes tip splitting phenomena and maintains
sharp fronts between the miscible fluids.
We consider a homogeneous two-dimensional porous
medium of length L x and width L y with constant permeability K and a base uniform flow of speed U along the x direction. The viscosity m 5 m (c) of the fluid is a function of the
local concentration c(x,y,t) of the chemically reacting solute. Assuming that the fluid is neutrally buoyant and incompressible and that dispersion is isotropic, the equations governing the problem can be written as:
111070-6631/99/11(5)/949/3/$15.00
“p52
~1!
m~ c !
uI ,
K
~2!
] t c1uI •“c5D“ 2 c1 f ~ c ! ,
~3!
where f (c) takes into account chemical reactions. Equations
~1!–~3! can be rewritten by switching to a frame moving
with the mean velocity of the fluid and nondimensionalizing
with dispersive scales.5,6 Introducing the stream function
c (x,y,t) such that u5 d c / d y, w52 d c / d x where u and w
are the longitudinal and transverse velocity components, we
then have:
2 v 5“ 2 c 5R ~ c x c x 1c y c y 1c y ! ,
~4!
c t 1c x c y 2c y c x 5“ 2 c1 f ~ c ! ,
~5!
where v is the vorticity and R52d(ln m)/dc. Important in
what follows is the fact that vorticity is produced whenever
the concentration gradient is not colinear with the velocity
vector. We consider periodic boundary conditions and the
initial condition of a step function between c51 and 0 with
noise added in the front and v50 everywhere. We have here
assumed that m (c)5exp(2Rc) with R5ln@m(0)/m(1)#. If R
.0 the front is viscously unstable and develops fingers even
when f (c)50 as already studied in detail in numerous
works.5–10 We refer to such fingering in absence of chemical
reactions as ‘‘standard fingering.’’ It is known that the nonlinear dynamics of standard fingers involves fading, shielding, and tip splitting.1 Tip splitting occurs only if the Peclet
number Pe5UL y /D or equivalently here the dimensionless
width H is higher than a critical value H c . Figure 1 gives
examples of standard fingering obtained by direct numerical
simulation of Eqs. ~4! and ~5! using spectral methods.5
949
© 1999 American Institute of Physics
950
Phys. Fluids, Vol. 11, No. 5, May 1999
Letters
FIG. 2. Viscous fingering affected by chemical reactions. The parameters
are H5128, L5640, R53 with f (c)52qc(c21)(c21/2) and q50.2.
The concentration field is shown at successive times t5200, 250, 300, 350,
and 400.
FIG. 1. Viscous fingering in absence of chemical reactions with R53,
f (c)50. The initial condition is a step function between c51 ~black! and 0
~white! with noise added in the front. ~a! H5128,H c and dimensionless
length L5280. The asymptotic state is one single finger and no tip splittings
occur. The concentration field is shown at successive times t5200, 260, and
320. ~b! Tip splittings in a system of size H51024.H c and L54096 shown
at time t51400.
We now focus on the changes induced to this situation
when chemical reactions occur and discuss our choice of
f (c). In many applications concerning viscous fingering,
chemistry is expected to occur only at the front and not to
affect the base state of the invading and displaced fluids.
Plausible choices for f (c) must therefore maintain a front
between the two stable chemical states c51 and 0.11 If we
consider one variable kinetic f (c), it is hence necessary to
consider at least a cubic chemical scheme, such as that describing the chemical waves studied in Ref. 4, i.e.,
f ~ c ! 52qc ~ c21 !~ c2d ! ,
~6!
where the dimensionless rate constant q.0 is the Damkholer number appropriate to ~6!. d is the unstable steady state
and the limit between the basins of attractions of the two
stable steady states. It is known that the reaction diffusion
system ~5! and ~6! with c50 admits planar soliton fronts
between the two stable states given by the following analytical expression:12
c ~ x,t ! 5
1
11e
6 b ~ x2 v t ! ,
~7!
where b 5 Aq/2 and the front velocity v 5 Aq/2(122d) vanishes if d51/2. The competition between dispersion and
nonlinear reaction thus leads to a sharp steady front with a
constant width in the course of time. If both q and R are
positive, this stationary reaction-diffusion front becomes viscously unstable and interaction between fingering and chemical reactions occurs.
We have conducted a series of numerical simulations of
Eqs. ~4!–~6! over a range of parameters (q, Pe,d), details of
which will be reported elsewhere.13 We focus here on the
case d51/2 for which the chemical front has zero velocity
postponing the discussion of the competition between viscous fingering and chemical waves existing when dÞ1/2.13
The typical behaviors shown here in Fig. 2 are robust, occuring as soon as q.0 and for any value of Pe. Comparison of
Figs. 1 and 2 shows important differences between fingering
with or without chemistry, the most striking of which is the
formation of droplets of the less viscous phase that disconnect from the bulk and invade the more viscous solution
when chemical reactions take place. The reverse is also observed, i.e., droplets of the more viscous phase are trapped
into the less viscous region. In addition, tip splitting is
strongly enhanced when q is nonzero and occurs even for
systems with H,H c as is the case in Fig. 2. The mixing
zone between the two pure fluids also increases much more
rapidly in presence of chemical reactions. The nonlinear
chemical scheme chosen here leads thus to a destabilizing
effect of chemistry on the viscous fingers. In addition, the
overall aspect of the pattern is quite different: the characteristic length scale is smaller and the number of fingers is
higher. We also note that because of dispersion, the interface
in standard fingering is smoothly joining the c51 and 0
states while the interface between the two steady states remains sharp at any time when reaction and diffusion mechanisms are competing.
Phys. Fluids, Vol. 11, No. 5, May 1999
FIG. 3. Schematic showing the operative finger/droplet interaction mechanisms. ~a! Vortex carried by a finger of low viscosity m 1 ~high mobility!
entering into high viscosity m 2 ~low mobility! fluid. ~b! Influence of a pair of
low viscosity droplets on a neighboring finger. The tip of the finger encounters the reverse flow associated with the vortices carried by the droplets
which tends to split it.
Let us now successively detail the new mechanisms of
formation of droplets and of tip splitting enhancement. We
recall that, consistently with Eq. ~4!, each finger always carries a vortex with it.5 This vortex originates in the fact that
fluid preferably flows inside the finger, a region of low viscosity and hence of higher mobility. When the fluid reaches
the tip of the finger, it encounters higher viscosity fluid and
consequently turns sideways @Fig. 3~a!#. Because of incompressibility, this fluid recirculates and enters the finger in its
rear creating a convective pinching of the concentration profile behind the tip. In the absence of chemical reactions, this
mechanism can never lead to a breakup of the finger, as if
this would happen, the finger would no longer be fed by the
bulk, the vortex strength would go to zero, and the tip would
fade away because of dispersion. In the presence of chemical
reactions, each finger still carries a vortex but now the entrainment of the flow to the rear of the tip can locally drop
the concentration below c51/2. The system then spontaneously evolves towards the attracting steady state c50. This
mechanism disconnects the tip of the finger from the low
viscosity bulk leading to a disconnected droplet carrying
some vorticity. The droplet shape can be maintained because
chemistry is available to balance locally the decrease of concentration accompanying dispersion. An analogous mechanism leads also to the reverse formation of droplets of the
more viscous phase invading the displacing fluid ~see the
third panel of Fig. 2!. However, ultimately any droplet will
shrink and disappear. Indeed, in the reaction-diffusion system, the critical radius above which a droplet of one stable
steady state expands into the other state and below which the
droplet shrinks is infinite in the special case of the symmetric
cubic kinetics studied here.12
The combination of droplets and their associated vortex
strength has a significant influence on the scale of the pattern
and on tip splitting. In standard fingering, the vortex carried
Letters
951
by the longest finger shields the neighboring fingers that encounter more viscous fluid flowing in the reverse direction
@Figs. 1~a! and 3~a!#. In the presence of chemical reactions, if
a droplet is formed, it moves ahead with a speed related to
the amount of trapped vorticity. If the droplet moves fast
enough, it does not affect the neighboring fingers, which still
grow even if H,H c as shown in Fig. 2. In some cases,
however, the droplet can influence the adjacent finger inducing it to split. Figure 3~b! shows a finger encountering a pair
of droplet, each of them carrying a vortex. This mechanism
is operative in the second panel in Fig. 2 where the upper
finger feels the presence of a pair of droplets because of the
transverse periodic boundary condition. The tip of the finger
encounters the reverse flow associated with the vortices and
consequently splits to follow higher mobility paths. In this
sense, the enhanced splitting in the presence of chemistry
mimics that induced by permeability heterogeneities.6,10 In
summary, the characteristics of viscous fingering in miscible
systems are profoundly modified when chemical reactions
leading to bistability come into play.
ACKNOWLEDGMENTS
A.D. thanks P. Borckmans and G. Dewel for fruitful
discussions and the Belgian Federal Office for Scientific,
Technical and Cultural Affairs for financial support and the
A. Renard Foundation for partial support of this research.
G.M.H. acknowledges support from the U.S. Department of
Energy, Office of Basic Energy Sciences. We both thank also
the Petroleum Research Fund through Grant No. PRF28774-AO9 for financial support.
1
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3
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~1997!.
4
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5
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6
A. De Wit and G. M. Homsy, ‘‘Viscous fingering in periodically heterogeneous porous media. II: Numerical simulations,’’ J. Chem. Phys. 107,
9619 ~1997!.
7
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8
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9
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10
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11
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12
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13
A. De Wit and G. M. Homsy, ‘‘Viscous fingering in reaction-diffusion
systems,’’ to appear in J. Chem. Phys.
2