1. Stress-dependent acoustic propagation and dissipation in granular materials Dr. David Johnson, Schlumberger Dr. Jian Hsu, Schlumberger Prof. Hernan Makse, CCNY Ping Wang, CCNY Chaoming Song, CCNY Dr. Nicolas Gland, CCNY  IFP Collaborations: Prof. Jim Jenkins, Cornell Prof. Luigi Laragione, Bari, Italy Computational Geosciences Symposium, DOE-BES Geosciences Program

2. Outline 1. 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

3. 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

4. Sound speeds in unconsolidated granular materials Compressional sound speed Shear sound speed K: bulk modulus G: shear modulus pressure Experiments at Schlumberger Domenico, 1977

5. Comparison with Effective Medium Theory Data contradicts EMT predictions: Experiments seem to be consistent with: Domenico, 1977 Walton, 1987 Goddard, 1990 Norris and Johnson, 1997

6. Hertz-Mindlin theory of contact mechanics Normal force (Hertz) 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 models If C = 0 then Path dependent models t t Glass beads Scaling argument (de Gennes)

7. Effective Medium Theory Of 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: Average are taken over uniform distribution of contacts single grain EM

8. Effective Medium Theory predictions P = pressure Z = average coordination number (number of contacts per grain)  solid volume fraction grain properties reference state pressure dependence

9. Effective Medium Theory predictions The Poisson ratio of a granular assembly: K=0G=0, According to Experiments: (K/G~1.1) According to EMT (K/G~0.7 if v = 0.2) For glass beads Equivalently, assuming v=0.15 g

10. Why Effective Medium Theory Fails? 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

11. Molecular Dynamics simulations of granular matter Hertz-Mindlin contact forces Coulomb 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)

12. Preparation protocol Start with a gas of spheres and compress and uncompress isotropically until a desired pressure and coordination number 3D 10,000 to 100,000 grains Bernal packings of steel balls fixed by wax (Nature, 1960) Z~6 First focus on reference states with large Z~6 and  RCP Random close packing 

13. Mean coordination number Pressure [Mpa] Frictionless packs Frictional packs to RLP Z=4 to RCP Z=6 Constraint arguments for rigid grains Edwards, Grinev, PRL Isostatic condition of force balance Dense packings Z = 6 (frictionless)  RCP Loose packings Z= 4 (frictional)  RLP RCP limit Soft grain limit The reference state Random close packing  Random loose packing  RLP limit

14. Calculation of K and G From linear elasticity theory: Stress tensor 1. Uniaxial compression: 2. Pure shear deformation:3. Biaxial shear deformation:

15. Numerical results for K and G Crossover behavior: Not a well-defined power law for the entire range of pressures The reference state is changing with pressure. Incorporate the behavior of Z(p) and  (p)

16. Numerical results for K and G Corrected EMT:

17. Numerical results for K/G Corrected EMT captures the trend, but the ratio K/G is still not predicted

18. Role of tangential forces 1. K is captured by EMT 2. EMT drastically fails for G, specially for low friction systems with perfect slip Redefine the transversal force: Perfect slip

19. Role of relaxation of grains Is the affine approximation correct? NO! Non-affine relaxation B C

20. Role of disorder and force chains B C Uniaxial compression of granular materials

21. Reference states with low Z~4 and low density  Preparation protocol for loose packings coordination number Pressure [Mpa] Frictional packs to RLP Z=4 Constraint arguments for rigid grains Edwards, Grinev, PRL. Isostatic limit for frictional grains: Z= D+1 = 4 (frictional)

22. EMT Z G/K =4. Jamming transition at Z=4. EMT completelly fails. No perturbative analysis possible. Collective relaxation ensues In “agreement” with EMT For low Z~4, there is a jamming transition with critical behavior: For large Z>6 Reference states with low Z~4 and low density 

23. Going beyond EMT EMT Z G/K =4. Jamming transition at Z=4. EMT completelly fails. No perturbative analysis possible. Collective relaxation ensues Perform a perturbation around the EMT solution for high coordination number Pair fluctuation theory of Jenkins, Laragione et al. (submitted) EMT Pair relaxation in an effective medium

24. 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.

25. Search for force chains Emulsion Data (Expt.) vs. Hertzian Balls (Simulation) Under isotropic compression No force chains, yet exponential

26. 2D or 3D under Uniaxial Stress Behringer’s exp. Hertzian Frictional Spheres

27. (b) Hertz spheres under isotropic compression (a) Droplets under isotropic compression

28. (d) Hertz spheres under uniaxial compression in 3D (c) Hertz spheres under isotropic compression in 2D

29. JAMMED MATTER Granular Matter Compressed emulsions Colloidal glasses Molecular Glasses Jamming “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 viscous liquid fast enough. Debenedetti and Stillinger Nature (2001) Thermal systemsAthermal systems