Pore scale study of interfacial
areas at drainage and imbibition
in granular media
Maša Prodanović1, Dorthe Wildenschild2, Elena
Rodriguez Pin1, and Steven L. Bryant1
1Center
for Petroleum and Geosystems Engineering
The University of Texas at Austin
2School
of Chemical, Biological, and Environmental
Engineering, Oregon State University
American Geophysical Union Fall Meeting
San Francisco, CA, Dec 14, 2009
Support & computational resources
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“US Department of Agriculture, grant
"Quantifying the mechanisms of pathogen
retention in unsaturated soils“ (MP, ER,
SLB)
National Science Foundation (EAR 337711
and EAR 0610108) (DW)
Texas Advanced Computing Center (TACC)
Outline
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Motivation:
 Evaluate role of fluid-fluid and fluid-solid interfaces
interfacial areas and triple contact in pore scale
displacements (hypothesized to be missing link in
Pc-Sw relationships)
Thermodynamic theory approaches:
 (M70) Morrow 1970 - drainage efficiency
 (HG93) Hassanizadeh & Gray (’93)
Tools:
 (XCMT) Experiments aided by X-ray computed
microtomography imaging
 (LSMPQS) Simulation of capillarity dominated flow
Using above tools, estimate & compare
 Drainage efficiency
 Interfacial area contribution to capillary pressure
 Contact line measurements
Interfacial Area Importance
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Pc-Sw functional relationship is not sufficient to describe
the state of the system
Processes like mass transfer, filtration etc. depend on
the available area
Area measurement is tricky
 Experimental e.g. BET (based on gas adsorption) for
solid surfaces or interfacial tracers models
assumed
 Image analysis of experiments is an appealing
alternative -somewhat limited by resolution
 LSMPQS simulation offers an independent estimate
of both solid and fluid-fluid areas.
Only recently the technology (XCMT), theory and
modeling/simulation make it possible to compare
all approaches in 3D
Interfacial Area Role: Thermodynamics
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Drainage efficiency (Morrow, ’70.)
increase in
surface energy
thermodynamic
work done
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Ed< 100% due to irreversible events
Hassanizadeh & Gray, ’93.
phase Helmholtz
free energy
change in interfacial area term
Experiment: beads
smooth, round
~100% silica
(sodalime)
low surface area
low d60/d10 (1.3)
bead diam: ave 1mm
Voxel length: 17µm
Air-water expts’
available:
PI – primary imbibition
PD – primary drainage
MI – main imb
MD – main draiin
SI – secondary imb
SD – secondary drain
Culligan, Wildenschild, Christensen, Gray, Rivers & Tompson. Interfacial area
measurement for unsaturated flow through a porous medium. WRR04.
Experiments: volcanic tuff
rough, angular
quartz, feldspar,
albite
high surface area
high d60/d10
(3.4)
Grain size: ave 2mm
Voxel length: 16.8µm
Air-water expts’
available:
Imbibition 1
Drainage 1
Imbibition 2
Drainage 2
Experiments: small beads
Grain size: ave 0.5mm
Voxel length: 17µm
Air-water expts’
available:
Imbibition 2
Drainage 2
Imbibition 3
Level set method (LSM)
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Osher & Sethian, ’88: embed
the moving interface as the
zero level set of function Φ
The evolution PDE:
Physics of the problem
introduced through F
Benefits:
 works in any dimension
 no special treatment
needed for topological
changes
 finding const. curvature
surface by solving a PDE
t=t1
t=t2
LSMPQS pore modeling objective
o Accurate description of the capillarity dominated fluid
displacement
o Equilibrium fluid-fluid interface satisfies Young-Laplace
equation, (const. capillary pressure Pc and interfacial
tension σ)
o Model slow displacement as a sequence of const.
curvature interfaces
Imaged by D. WIldenschild
Progressive quasi-static algorithm (PQS)
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Drainage
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Initialize with a planar front
Solve evolution PDE with slightly
compressible curvature model for F
until steady state:
Iterate
 increment curvature
 Find steady state of prescribed
curvature model
Imbibition starts from drainage endpoint
and decrements curvature
Zero contact angle: wall BC
M. Prodanović and S. L. Bryant. A level set method for determining critical curvatures
for drainage and imbibition. Journal of Colloid and Interface Science, 304 (2006) 442
Simple 2D example for LSMPQS
drainage (controlled by throats)
imbibition (controlled by pores)
Simulation steps (alternating red and green colors).
All <= 2% rel.abs.err.
Haines
jump
Melrose
criterion
M. Prodanović and S. L. Bryant. A level set method for determining critical curvatures
for drainage and imbibition. Journal of Colloid and Interface Science, 304 (2006) 442
Textbook example
3D packing of equal spheres
Experimental vs. simulated system
200 voxels
(3.4mm)
LSMPQS simulated
volume
Imaged volume
7mm (420 voxels)
Drainage Pc-Sw comparison: Beads
In all samples, we get LSMPQS
curves higher than experimental
because we picked inner, tighter
subsample.
Simulations in larger samples on
the way.
LSMPQS simulation
cross-section
Experimental setup cross-section
420x420
200x200
Drainage Pc-Sw: Small beads
LSMPQS simulation
cross-section
200x200
Experimental setup cross-section
420x420
Drainage Pc-Sw: Tuff
LSMPQS
initial
drainage
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Tuff grains are larger, boundary/size effects especially
severe: this probably affects residual wetting phase.
Tuff is tough…
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Coarsened image simulation (larger volume)
currently going on
We possibly have resolution effects: films
cannot be resolved
dx=2.8µm
LSMPQS Simulation
Efficiency (M70) and interfacial area contribution (HG93)
Morrow (‘70)
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Hassanizadeh & Gray ’93.
Both theories comparable
for the most part in range [0.6,0.8]
XCMT Experiments
Efficiency (M70) and interfacial area contribution (HG93)
Morrow (‘70)
Hassanizadeh & Gray ’93.
I plot the change in interfacial area term as a fraction of
capillary pressure so we can compare it directly to the
efficiency (one is integral form of the other).
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Contributions/Ed larger than Pc?
 Observed by Pyrak-Nolte et al., WRR, 2008
 differences in Pc (local, global)?
 resolution?
Triple contact line measurement
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medial axis thinning used to
extract length Lc from
segmented images
(simulation or experiment)
Dimensionless specific
length
Lc D Lc 2
= R
VbD Vb
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Lc D =
Lc
R
VbD =
Vb
R3
Initial study done on
monodisperse packing
Movie: CL advancement
during drainage in a simple
pore
Resolution effects
dx=0.04
dx=0.08
Resolution effects
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If known that pendular rings not resolved,
double the computed result
Conclusions
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Contribution of change interfacial areas to
capillary pressure in Hassanizadeh & Gray ’93
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computed for the first time in 3D
Shown to be sizable in both simulation and
experiments
In some experiments even larger
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needs investigation
Pyrak-Nolte et al. show similar in 2D
Preliminary contact line measurements
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Objective to determine role in colloid retention in
soils
Thank You!