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 - poresOne 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 mentio

n here that th

e pores need to

be

Convex i

n 3D (beca

use in

2D it is

not true)

Obtaining pores 2D

Grain

Pore

Grain: Anti-ClockwisePore: Clockwise

3D example

• Packed Spheres Revisited:– Every 3 neighbouring

contact points create a plane facet.

– Pores

• spheres - pores

Need to ask

Peter fo

r cita

tion co

ncerning

The 3D sphere pack

ing

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 – skeletonization2. 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 th

is sli

de?

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

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 Pict

ure

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 2D1. Grains → polygons

Example: Beads in 2D1. Grains → polygons

Example: Beads in 2D1. Grains → polygons2. 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