1. Geometric Analysis of Packings Gady Frenkel, M. Blunt, P. King & R. Blumenfeld

2. Agenda Motivation: Definition of a new model –The balloon algorithm – non negative curvature Results: –2D –3D throats emulation Conclusions and future prospects

3. The Big Picture: Goals: Extracting networks –Robust algorithm –efficient Investigate the wide distribution of permeability –Caused by topology? –Caused by the distribution of the throats cross-section? Can we model and predict connections between electrical conductivity and permeability

4. Granular Packing Characterization Components: Grains, Pores, Throats. Definitions of pores and throats are quite ambiguous. –Two convex pores connected by a wide throat form one concave pore or not? –Example: spheres - poresspheres pores One or Two Pores?

5. Our model of Granular Packing Grains: (transformed) –Straight lines and planes that connect contacts instead of real boundaries PORES: –“Convex” “empty” volumes that are surrounded by transformed grains. THROATS: –the openings that connect two pores:

6. 2D Packing Example: GRAINS: –Straight lines and planes that connect contacts instead of real boundaries

7. 2D Packing Example: GRAINS: –Straight lines and planes that connect contacts instead of real boundaries Contact Points

8. 2D Packing Example: GRAINS: –Straight lines and planes that connect contacts instead of real boundaries Contact Points

9. 2D Packing Example: GRAINS: –Straight lines and planes that connect contacts instead of real boundaries PORES: –“Convex” “empty” volumes that are surrounded by transformed grains. CONTACT POINT: Should I mention here that the pores need to be Convex in 3D (because in 2D it is not true)

10. Obtaining pores 2D Grain Pore Grain: Anti-Clockwise Pore: Clockwise

11. 3D example Packed Spheres Revisited: –Every 3 neighbouring contact points create a plane facet. –Pores spheres - poresspheres pores Need to ask Peter for citation concerning The 3D sphere packing

12. Finding Throats: Facets of Pore are Known Use the 2D algorithm where the radial vector sets the positive edge direction

13. Finding Throats:

14. Implementation: (2D&3D) 2 Step Algorithm: 1.Find contact points – skeletonization 2.Apply algorithm to find the pore-network and the throats characteristics. Benefits: –Easier, Grains are simplified to plane facet. –Less information to deal with –East to extract the throat information Do I need this slide?

15. Growing a deformable object : Inflating balloon inside the pore until it is filled. –Advantage: Fit any pore shape by deforming. –Only one object per pore. Obtaining Pores: Main Idea

16. Question: How can we prevent this balloon from exiting the pore through the throats.? Clue: Balloons tend to be convex. When a balloon expands through the narrower throats it will develop a negative curvature By preferring positive curvature we can prevent the balloon from exiting the pore. Add Picture

17. Algorithm Step 1: –Obtain contact points of grains – Determine the facets of the grains.

18. Algorithm Step 2: –Choose a Facet and put a small balloon at the pore near the facets centre. – Grow the balloon according to the rules: Surface points get further from the centre Curvature is calculated at each point, negative curvature is not allowed. –When balloon is fully grown, find the facets that it touches.

19. Example: Beads in 2D 1.Grains → polygons

20. Example: Beads in 2D 1.Grains → polygons

21. Example: Beads in 2D 1.Grains → polygons 2.Balloons are inflated from each facet

22. Emulating “Throats” in 2D “Throats”

23. Emulating “Throats” in 2D Pores

26. Conclusions New Characterization of pore space. –Step 1: skeletonization –Step 2: non-negative curvature algorithm Algorithm Shows promising results and seems to be applicable in any dimension.

27. Future Prospects 3D Software – is in advanced stages Recognizing the facets that belong to the pore. Combining/dividing pores for the conventional definition. Finding the contact points from real 3D data. Analysis of real systems: –Need Data