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 EngineeringThe University of Texas at Austin

2School of Chemical, Biological, and Environmental Engineering, Oregon State University

American Geophysical Union Fall MeetingSan Francisco, CA, Dec 14, 2009

Support & computational resources

“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 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 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

Drainage efficiency (Morrow, ’70.)

Ed< 100% due to irreversible events

Hassanizadeh & Gray, ’93.

increase in surface energy

thermodynamic work done

Interfacial Area Role: Thermodynamics

phase Helmholtz free energy

change in interfacial area term

Experiment: beadssmooth, roundsmooth, round~100% silica ~100% silica (sodalime)(sodalime)low surface arealow surface arealow d60/d10 (1.3)low d60/d10 (1.3)

bead diam: ave 1mm

Voxel length: 17µm

Air-water expts’ available:PI – primary imbibitionPD – primary drainageMI – main imbMD – main draiinSI – secondary imbSD – secondary drain

Culligan, Wildenschild, Christensen, Gray, Rivers & Tompson. Interfacial area measurement for unsaturated flow through a porous medium. WRR04.

Experiments: volcanic tuff

rough, angularrough, angularquartz, feldspar, quartz, feldspar, albitealbitehigh surface areahigh surface areahigh d60/d10 (3.4)high d60/d10 (3.4)

Grain size: ave 2mm

Voxel length: 16.8µm

Air-water expts’ available:Imbibition 1Drainage 1Imbibition 2Drainage 2

Experiments: small beads

Grain size: ave 0.5mm

Voxel length: 17µm

Air-water expts’ available:Imbibition 2Drainage 2Imbibition 3

Level set method (LSM)

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 objectiveo Accurate description of the capillarity dominated fluid

displacemento 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) Drainage

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

Haines

jump

drainage (controlled by throats) imbibition (controlled by pores)

Simulation steps (alternating red and green colors). All <= 2% rel.abs.err.

Melrose

criterionM. 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 example3D packing of equal spheres

Experimental vs. simulated system

Imaged volume

7mm (420 voxels)

LSMPQS simulated volume

200 voxels (3.4mm)

Drainage Pc-Sw comparison: Beads

Experimental setup cross-section

420x420

LSMPQS simulation cross-section

200x200

In all samples, we get LSMPQS curves higher than experimental because we picked inner, tighter subsample.

Simulations in larger samples on the way.

Drainage Pc-Sw: Small beads

Experimental setup cross-section

420x420

LSMPQS simulation cross-section

200x200

Drainage Pc-Sw: Tuff

Tuff grains are larger, boundary/size effects especially severe: this probably affects residual wetting phase.

LSMPQS initial drainage

Tuff is tough… Coarsened image simulation (larger volume)

currently going on We possibly have resolution effects: films

cannot be resolved

dx=2.8µm

LSMPQS SimulationEfficiency (M70) and interfacial area contribution (HG93)

Both theories comparable for the most part in range [0.6,0.8]

Morrow (‘70) Hassanizadeh & Gray ’93.

XCMT ExperimentsEfficiency (M70) and interfacial area contribution (HG93)

Contributions/Ed larger than Pc? Observed by Pyrak-Nolte et al., WRR, 2008 differences in Pc (local, global)? resolution?

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).

Triple contact line measurement

medial axis thinning used to extract length Lc from segmented images (simulation or experiment)

Dimensionless specific length

Initial study done on monodisperse packing

D

LcLc =

R2D

bD b

Lc Lc= R

V V bbD 3

VV =

R

Movie: CL advancement during drainage in a simple

pore

Resolution effects

dx=0.04 dx=0.08

Resolution effects If known that pendular rings not resolved,

double the computed result

Conclusions

Contribution of change interfacial areas to capillary pressure in Hassanizadeh & Gray ’93 computed for the first time in 3D Shown to be sizable in both simulation and

experiments In some experiments even larger

needs investigation Pyrak-Nolte et al. show similar in 2D

Preliminary contact line measurements Objective to determine role in colloid retention in

soils

Thank You!