Stress-dependent acoustic propagation and dissipation in granular materials

Dr. David Johnson, SchlumbergerDr. Jian Hsu, SchlumbergerProf. Hernan Makse, CCNY Ping Wang, CCNYChaoming Song, CCNYDr. Nicolas Gland, CCNY IFPCollaborations:Prof. Jim Jenkins, CornellProf. Luigi Laragione, Bari, Italy

Computational Geosciences Symposium, DOE-BES Geosciences Program

Outline1. Motivation: Sonic logging application Fundamental understanding of mechanics of unconsolidated granular materials

2. Non-linear elasticity of unconsolidated granular materials: pressure dependence of sound speeds

3. Failure of Effective Medium Theory 4. Molecular Dynamics Simulations or Discrete Elements Methods Two limits: low and large volume fraction: RLP-RCP Large and small coordination number.

5. Beyond Effective Medium Theory

Motivation

2. Application: Sonic logging. Acoustic measurements of shear and compressional sound speeds in hard and unconsolidated formations. Sonic tools provides the axial, azimuthal, and radial formation sound speed information for near-field and far-field surrounding the wellbore.

Determine the stress distribution from field accousticmeassurements.

1. A fundamental understanding of micromechanics of granular materials

Sound speeds in unconsolidatedgranular materials

GK

vp

3/4

Gvs

Compressional sound speed Shear sound speed

K: bulk modulusG: shear modulus

pressure Experiments at SchlumbergerDomenico, 1977

Comparison with Effective Medium Theory

3/1pG

3/1pK

Data contradicts EMT predictions:

Experiments seem to beconsistent with:

2/1pGK

Domenico, 1977Walton, 1987Goddard, 1990Norris and Johnson, 1997

Hertz-Mindlin theory of contact mechanics

2/32/1

3

2wRCF nn

)mod(29 ulusshearGPaGg

Normal force (Hertz)

)1/(4 ggn GC

Tangential force (Mindlin)

w: normal displacement

s: shear displacement

The shear force depends on the path taken in {w,s}:If C = 0 then Path independent modelsIf C = 0 then Path dependent models

tt

sRwCF tt 2/1

)2/(8 ggt GC

)'(2.0 ratiosPoissong

Glass beads

3/12/1

2/3

~~/~

~~)(

ppK

pwstrain

Scaling argument (de Gennes)

Effective Medium TheoryOf Contact Elasticity

1. Assumes the existence of an Energy density function U depending on the current reference state of strain { For an isotropic system:

2. Two approximations: a) Affine approximation: the grains move according to the macroscopic strain tensor:

b) Statistically all the grains are the same:

)()3/2(2

1)( 322

0 ijijiiiiij OGGKpUU

jiji Xu

)(11

uWNZduFV

WV

U ccontactscontacts

c

Average are taken over uniform distribution of contacts

singlegrain

EM

Effective Medium Theorypredictions

3/1

3/23/2 6

12

n

n

C

pZ

CK

3/1

3/23/2 6

20

3/2

n

tn

C

pZ

CCG

P = pressureZ = average coordination number (number of contacts per grain)solid volume fraction

grain properties reference state pressure dependence

Effective Medium Theorypredictions

The Poisson ratio of a granular assembly:

2/11

K=0 G=0, K

According to Experiments: (K/G~1.1)

According to EMT (K/G~0.7 if v = 0.2)

18.015.0

!!!02.0)35(2

g

g

For glass beads

Equivalently, assuming v=0.15

11

22

g

3/42

G

K

v

v

s

p

)3/1/(2

3/2/

22

11

GK

GK

!!2/179.0 g

Why Effective Medium TheoryFails?

1. EMT assumes homogeneous distribution of forces on the grains: Role of disorder and force chains

2. EMT predicts well K but not G: Role of transverse forces

3. EMT assumes affine motion of the grains according to the macroscopic deformation: Role of relaxation and non-affine motion of grains

4. Going beyond EMT: relaxation dynamics

Molecular Dynamics simulationsof granular matter

Hertz-Mindlin contact forcesCoulomb friction and dissipative forces

Makse, Gland, Johnson, Schwartz, PRL (1999)Makse, Johnson, Schwartz, PRL (2000)Johnson et al, Physica B (2000)Makse, Gland, Johnson, Phys Chem Earth (2001)Jenkins, et al, J. Mech. Phys. Sol (2004)Makse, et al. PRE (2004)Zhang, Makse, PRE (2005)Brujic, Wang, Johnson, Sindt, Makse, PRL (2005)Gland, Wang, Makse Eur. Phys. J (2006)J. Hsu, Johnson, Gland, Makse, PRL (submitted)Magnanimo, Laragione, Jenkins, Makse, PRL (sub)

Preparation protocol

Start with a gas of spheres and compress anduncompress isotropically until a desired pressure andcoordination number

3D10,000 to 100,000 grains

Bernal packings of steel balls fixedby wax (Nature, 1960)

Z~6

First focus on reference states with large Z~6 and RCPRandom close packing

Mea

n co

ordi

nati

on n

umbe

r

Pressure [Mpa]

Frictionless packs

Frictional packs

to RLP Z=4

to RCP Z=6

Constraint argumentsfor rigid grainsEdwards, Grinev, PRLIsostatic conditionof force balance

Dense packingsZ = 6 (frictionless) RCP

Loose packingsZ= 4 (frictional) RLP

RCP limit

Soft grain limit

The reference state

Random close packing

Random loose packing

RLP limit

Calculation of K and G

ijllijijllij GK

3

12

From linear elasticity theory:

Stress tensor

1. Uniaxial compression:

2. Pure shear deformation:

0011 ij

11

113/4

GK

012

12

12

2

1

G

3. Biaxial shear deformation:

0332211

2211

2211

2

1

G

Numerical results for K and G

Crossover behavior:Not a well-defined power law for theentire range of pressures

3/13/23/2 )()( ppZpGK

The reference state is changing with pressure. Incorporate the behaviorof Z(p) and (p)

Numerical results for K and G

3/13/23/2 )()( ppZpGK

Corrected EMT:

RCP

GPa

pp

isostaticZ

MPa

pZpZ

c

c

c

c

64.0

14)(

6

10)(

3/2

3/2

Numerical results for K/G

3/13/23/2 )()( ppZpGK

Corrected EMT captures the trend,but the ratio K/G is still not predicted

71.0)45(3

)2(5

g

g

G

K

Role of tangential forces

sRwCFF ttt 2/1'

1. K is captured by EMT

2. EMT drastically fails for G, specially for low friction systems with perfect slip

Redefine the transversal force:

10

Perfect slip

3/1

3/23/2 6

20

3/2

n

tn

C

pZ

CCG

3/1

3/23/2 6

12

n

n

C

pZ

CK

Role of relaxation of grainsIs the affine approximation correct? NO!

Non-affine relaxation

B

C

Role of disorder and force chains

B

C

Uniaxial compression of granular materials

Reference states with low Z~4 and low density

Preparation protocol for loose packings

coor

dina

tion

num

ber

Pressure [Mpa]

Frictional packs

to RLP Z=4

Constraint argumentsfor rigid grainsEdwards, Grinev, PRL.Isostatic limit forfrictional grains:

Z= D+1 = 4 (frictional)

EMT

Z

G/K

Jamming transition at Z=4. =4. EMT completelly fails. No perturbative analysis possible. Collective relaxation ensuesEMT completelly fails. No perturbative analysis possible. Collective relaxation ensues

In “agreement” with EMT

For low Z~4, there isa jamming transitionwith critical behavior:

2/1)(/ cZZKG

4cZ

For large Z>6

constKG /

Reference states with low Z~4 and low density

Going beyond EMT

EMT

Z

G/K

Jamming transition at Z=4. =4. EMT completelly fails. No perturbative analysis possible. Collective relaxation ensuesEMT completelly fails. No perturbative analysis possible. Collective relaxation ensues

Perform a perturbation around the EMT solution for high coordination numberPair fluctuation theory of Jenkins, Laragione et al. (submitted)

EMT

Pair relaxationin an effectivemedium

Summary

1. EMT captures approximately the behavior of the bulk modulus

2. EMT fails drastically for the shear modulus

3. The elastic moduli depends critically on the reference state.

4. For low coordination number near RLP there is a jamming critical transition

5. No hope for EMT near the jamming point.

6. Perturbative analysis may provide corrections to EMT for high coordination numbers.

7. Future work involves going beyond the EMT.

Search for force chains

Emulsion Data (Expt.) vs. Hertzian Balls (Simulation)

Under isotropic compression

No force chains, yet exponential

2D or 3D under Uniaxial Stress

Behringer’s exp. Hertzian Frictional Spheres

(b) Hertz spheres under isotropic compression

(a) Droplets under isotropic compression

(d) Hertz spheres under uniaxial compression in 3D

(c) Hertz spheres under isotropic compression in 2D

JAMMED MATTER

Granular Matter

Compressed emulsionsColloidal glasses

Molecular GlassesJamming “phase diagram”Liu and Nagel, Nature (1998)

Jamming oil droplets (10 m) by increasing osmotic pressure. Brujic, Edwards, Hopkinson, Makse, Physica A (2003)

Jamming grains (1mm) in a periodic box:Molecular dynamics simulations of sheared granular matter. Makse, and Kurchan, Nature (2002).

Jamming PMMA colloidal particles (3 m) by increasing density.

Glass transition: cooling a viscousliquid fast enough. Debenedetti and Stillinger Nature(2001)

Thermal systemsAthermal systems