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

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

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

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?

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:

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

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

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

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)

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

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

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

Finding Throats:

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?

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

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

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

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.

Example: Beads in 2D 1.Grains → polygons

Example: Beads in 2D 1.Grains → polygons

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

Emulating “Throats” in 2D “Throats”

Emulating “Throats” in 2D Pores

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.

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