Meshing Isosurfaces of 3D Fields Meshing Isosurfaces of 3D Fields and Volume Calculation for and Volume Calculation for

Estimation of Fragment Size Estimation of Fragment Size Distribution in Underground MiningDistribution in Underground Mining

Rob AlkinsRob Alkins

Stroma Service ConsultingStroma Service ConsultingMay 20, 2011May 20, 2011

Problem

Given a given blast configuration, what is the expected size distribution of the blasted rock?

Consider:● Explosive Properties● Rock Properties● Geometry● Dynamics● Modeling the resulting Fragmentation

Explosive Properties● Detonation Velocity● Density● Detonation Pressure● Weight Strength● Bulk Strength● Volumetric Reaction Extent● Etc.

0.5 kg primer for CHMAN

Velocity/Diameter Curve● Shape of detonation Zone is Determined by

Detonation/Wave Velocity

Particle

Shock Front

▀ Charge Diameter 1

▀ Charge Diameter 2

GAS EXPANSION REGION

Velocity Diameter Curves for Ideal and Non-Ideal Explosives

0

5

10

15

20

250 2 4 6 8 10 12

Charge Diameter (inches)

Deto

natio

n Ve

loci

ty (1

000

ft/se

c)

Grain burning explosives

Microcell, micrograin explosives

Molecular explosives

V1

V2

V3

Ideal

Critical

Detonation VelocityCharge Diameter

Coupling● Confinement affects

Detonation Velocity

● An air gap in a borehole will result in decreased detonation velocity

Coupled and Uncoupled

Cracks

Shock Wave

Unreacted PETN

BoreholeExpansion

Gas Products

● P-Waves● S-Waves● Cracking Radius

Shockwave – 1

Shockwave – 2

Gas Expansion - 1

UnreactedExplosive Charge

Reaction Zone - 2Dynamics

Tangential Stress Cracks – Y and ZRadial Stress Cracks – Z

Fragmentation● Fragmentation is due to

Cracking● Borehole will Crush

very close to the explosive

● Cracking from Compressive Wave

● Returning Tensile Wave● Extends Cracks● Cracks are Expanded

By Gas Action

Rock Properties

Some Properties● Young's Modulus● Poisson's Ratio● Bulk Modulus● Shear Modulus● Rock Density● Dynamic/Static Compressive Strength● Dynamic/Static Tensile strengthThat only describes Homogenous Rock

Fragmentation Modeling

Place process parameters on fragmentation histogram – operating points.

Weibul Distribution

-Mean Fragment Size

-Uniformity Index

Open Pit Mining● Regular

Geometry

● Well Studied

● Very Controlled

Kuz-Ram Model● Empirical Fit Model● Restricted to Regular (rectangular) Geometry

Underground Mining

● Restricted● Irregular Geometry● Not well Studied

● Hole Geometry is Restricted by Drill Position

● Drill Position is Restricted by Stope Dimensions

Underground Fragmentation Modeling

● Cannot use Kuz-Ram(Square Peg, Round Hole)

● Holmberg-Persson model Peak Particle Velocity

Analytic Solution ( )

3D Field Surrounding Charge

● We May Take The Maximum PPV Contribution from Each Hole

● PPV Directly Relates to Stress

Marching Cubes● Iterate Through All Cubes in 3D Grid● Classify Vertices as Above/Below Threshold

● Linear Interpolate For Actual Vertex Position

● Construct Mesh

Reflections

Final Distributon

Determine Relationship

Thank You

robert.alkins@stroma.ca