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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