miscible viscous fingering of pushed versus pulled interface · 2013-11-14 · pulled interface...

Miscible viscous fingering of pushed versus pulled interface

Satyajit Pramanik and Manoranjan Mishra

Department of MathematicsIndian Institute of Technology Ropar

Rupnagar – 140001, Punjab, India

Viscous fingering (VF) instability

Miscible viscous fingering of pushed versus pulled interface (Satyajit Pramanik)

• Viscous fingering instability is specialclass of hydrodynamic instability whichleads to the dispossession of the initialinterfacial shape between the fluids withdifferent viscosity.

• Less mobile fluid lag behind the highmobile fluid which penetrates throughthe former in a porous media.

Rectilinear displacement

Radialdisplacement

Viscous fingering (VF) instability


Viscous fingering can be observed in

immiscible fluids where surface tension acts as the most importantfactor miscible fluids where the diffusion plays the key role

AIM

Here we focus on the rectilinear displacement of miscible solutions.

Two most important applications of viscous fingering instability are1. Chromatographic separation and2. Pollutants dispersion of in aquifers.

In both the cases one fluid is localized within a finite region and isdisplaced by another carrying fluid. Sometimes, the finite fluid can be confined within a circular region andhence single interface model is not appropriate for these cases.

Dispersion pollutant in aquifers


Gi cxi

, i 1, 2

g

Fig. Schematic of hypothetical solute body with signs of concentration gradients indicated in

each quadrant.

Welty et al., Water Resour. Res. 39(6), 2003

Heavy sample ofhigher viscosity isdisplaced by waterin a porous media

Viscous fingering enhances thedispersion of pollutant in aquifers

Modeling VF with COMSOL Multiphysics


In version 4.2a:o Two-phase Darcy’s lawo Free triangular meshingo Reproduce the work of Mishra et al. by

keeping the unstable interface at sameposition for both R > 0 and R < 0.

Fingers are shown at t = 10 sec

x

ys2=1

s1=0

s1=0

Pramanik et al. “Miscible viscousfingering: Application in chromatographiccolumns and aquifers”, Proceedings of theCOMSOL conference 2012 Bangalore.

Pramanik et al. Proceeding of the COMSOL conference 2012, Bangalore

Velocity contours


Pramanik et al. Proceeding of the COMSOL conference 2012, Bangalore

Axial velocity contours at time t = 10 sec

VF with circular sample


Chen et al. Phys. Fluids 13, 2001 Maes et al. Phys. Fluids 22, 2010

Theoretical modeling, less viscous sample

Experimental investigation, more viscous sample

12.18 7.61

Theoretical model


u

: 2D velocity fieldk : permeabilityp : pressurec : concentration of the solute driving viscosityD : dispersion coefficient

Governing equations

COMSOL

COMSOL Multiphysics modeling


1 2

1 2

constant, constant

1p

r r

Two-phase Darcy’s law

Governing equationspt

u 0, u

p

1s1 2s2, 1 s1

r1

1

s2 r2

2

,

pt

c1u Dcc1, c1 1s1

1 2 01eRs1

x

y0U s1=1

s1=0

2r

Theoretical



2 2 20 0

1 2 2 20 0

1, ( - ) ( - )0, ( - ) ( - )

x x y y rs

x x y y r

Initial condition

Two-phase Darcy’s law

Normal inflow velocity U0

Boundary conditions

No flux

Outflow

Inflow

n u 0

nDcc1 0

n u (s11 s22 )U0

s2 1 s1

x

y0U s1=1

s1=0

2r

(x0, y0 ) being the center of the circle



Extra fine free triangular mesh (# of elements 64852)

Extremely fine mapped mesh (# of elements 91196)



Comparison of two linear solversMUMPS and PARDISO

Results


U0 =1 × 10-3 m/s, R = -3, Ly =0.08 mm. and r = 0.15 × Ly.

VF at pushed interface

t = 0

t = 8

t = 10

t = 12

U0 =1 × 10-3 m/s, R = 3, Ly =0.08 mm. and r = 0.15 × Ly.

VF at pulled interface

t = 0

t = 100

t = 150

t = 200

Results


U0 =1 × 10-3 m/s, R = -3, Ly = 0.08 mm. and r = 0.15 × Ly

Results


U0 =1 × 10-3 m/s, R = 3, Ly = 0.08 mm. and r = 0.15 × Ly

Conclusions


• COMSOL Multiphysics has been used to model miscibleviscous fingering of pushed and pulled interfaces.

• The classical fingering phenomena like tip splitting, mergingetc. are captured through COMSOL Multiphysics.

• Results are reproducible and are in well accordance with theexperimental findings.

• Differences in the fingering patterns for pushed and pulledinterfaces are explained through the streamlines.

• VF modeling with COMSOL Multiphysics strongly dependson the meshing.

Acknowledgements


• COMSOL Multiphysics

• Financial support• DST Govt. of India

• NBHM (DAE) Govt. of India

Thank you

miscible viscous fingering of pushed versus pulled interface · 2013-11-14 · pulled interface...

Documents