1. Deformation of Sediments via Grain-Scale Simulations: Variational Algorithm Ran Holtzman, Dmitriy Silin, Tad Patzek U.C. Berkeley holtzman@berkeley.edu

2. 2 Motivation Why micromechanics? –Mechanics of granular matter is controlled by interaction of discrete grains Why numerical simulations? –Enable micromechanical analysis, unavailable from experiments (restricted to 2D or a single grain pair) Existing models: –Spatially-averaged solutions (EMT 1 ) –Dynamic grain-scale simulations (DEM 2 ) 1 – Duffy & Mindlin, 1957 2 – Cundall & Strack, 1979

3. 3 Our Model of Granular Matter 3D heterogeneous, disordered pack Spherical grains, differ in size & properties Bounded by a rigid container (imposing boundary conditions) Contact forces & moments  macroscopic stress

4. 4 Variational Algorithm Quasi-static model: sequence of static equilibrium configurations Equilibrium: minimal-work path Moduli: fit stress-strain to Hooke’s law:

5. 5 P P h Hertz (1882) Normal Compression grain i grain j

6. 6 Q Q Shear

7. 7 Frustrated Rotation Q Q

8. 8 Torsion M tor

9. 9 Challenges in Modeling Friction Loads depend on normal force and load history 1 Implementing M-D theory 1 - cumbersome for multiple contacts Simplified models –Ignoring frictional loads (zero tangential stiffness) –Ignoring partial slip (fixed stiffness) 2 –Simplified treatment of partial slip (variable stiffness) 3-4 1 – Mindlin & Deresiewicz (1953)3 – Walton & Braun (1986) 2 – Jenkins & Strack (1993)4 – Vu-Quoc & Zhang (1999)

10. 10 Linearized Formulation Incremental loading, small perturbations Shear increment decoupled from normal components k utut Q 0(proj) Q0Q0 initial current Q u ||Q||=  P QQ QQ

11. 11 Predicted Moduli vs. Experiments

12. 12 Predicted Moduli vs. Experiments

13. 13 Summary Quasi-static grain-scale simulations of a deforming sediment Physically-based model, no calibration used Macroscopic moduli match experimental data Application: effect of dissociation on hydrate- bearing sediments

14. 14 Extensions Add cement, angular grains, and pore constituents that interact with the solid grains Statistical and qualitative analysis of microscopic parameters – e.g. force chains Reduce computing time by using parallel computing

15. 15 Thank You! Funded by the assistant secretary for fossil energy, office of Natural Gas and Petroleum Technology, N.E.T.L. D.O.E. Contract #DE-FC26-05NT42664 holtzman@berkeley.edu