By Steven Claes (KU Leuven)

1. FACULTY OF SCIENCE
Department : Earth and Environmental Sciences
Geology
A new 3D pore shape classification using
Avizo Fire
Ir. Steven Claes
Dr. A. Foubert
Prof. Dr. M. Ozkül
Prof. Dr. R. Swennen

2. Introduction
Introduction Mathematical
Introduction CT Conclusion
shape description
Overview
1. Introduction
2. CT: Petrography in 3D
3. Mathematical shape description
4. Conclusion

3. Introduction
Introduction Mathematical
Introduction CT Conclusion
shape description
1.. Introduction
A. Heterogeneity
‐ Carbonate reservoirs typically have a complex texture and are very
heterogeneous concerning porosityy measurements
Choquette and Prey,1970
AAPG, 77

4. Introduction
Introduction Mathematical
Introduction CT Conclusion
shape description
1.. Introduction
B. Different scales
‐ Different types of porosity  working on different scales
Rahman, et al 2011

5. Introduction
Introduction Mathematical
Introduction CT Conclusion
shape description
1.. Introduction
B. Different scales
‐ Working on different scales
15 cm
2 cm
0.4 cm
1.5 cm
10 cm
4 cm

6. Introduction
Introduction Mathematical
Introduction CT Conclusion
shape description
2.. CT: Petrography in 3D
A. Workflow
Data acquisition Reconstruction 3D information
‐ 3D information:
‐ Filtering
‐ Pre reconstruction
‐ Post reconstruction
‐ Segmentation
‐ Dual thresholding
‐ Visualization
‐ Avizo
‐ CT‐an / CT‐vox
‐ Calculations
‐ Matlab
‐ Avizo
1979, Houndsfield

7. Introduction
Introduction Mathematical
Introduction CT Conclusion
shape description
2.. CT: Petrography in 3D
B. Principle:

8. Introduction
Introduction Mathematical
Introduction CT Conclusion
shape description
2.. CT: Petrography in 3D
B. Principle:
‐ Advantages:
‐ Non‐destructive
‐ Full 3D information of internal structure
‐ Little sample preparation
‐ Qualitative and quantitative interpretation
‐ Disadvantages:
‐ Limited object size
‐ Relative high recording time
‐ Relative high calculation time

9. Introduction
Introduction Mathematical
Introduction CT Conclusion
shape description
2.. CT: Petrography in 3D
C. Example:
Dolomite cement Dolomite fragment (Fe rich)
Late Calcite vein

10. Introduction
Introduction Mathematical
Introduction CT Conclusion
shape description
3.. Mathematical shape description
A. Form ratio
‐ Pore volume  pore shape

11. Introduction
Introduction Mathematical
Introduction CT Conclusion
shape description
3.. Mathematical shape description
A. Form ratio
‐ Several parameters are defined in the last century:
‐ E.g. : L  I
2S (Wenthworth, 1922)
‐ Most are calculated using L (longest dimension in a shape), I
(longest dimension perpendicular to L) and S (dimension
perpendicular to both L and I) (Krumbein, 1941)
‐ Above definition of L, I and S does not always provide the most
information about a shape e.g. cube
(Blott and Pye 2008)

12. Introduction
Introduction Mathematical
Introduction CT Conclusion
shape description
3.. Mathematical shape description
B. Calculation L, I and S
‐ Individual pores are considered as solid objects
‐ Calculate the mechanical moments of the pore:
 I I xy I xz 
 xx 
 I yx I yy I yz 
 
 I zx

I zy I zz 

‐ Using the spectral theorem for real, symmetric matrices:
 I 0 0 
 1 
 0 I2 0 
 
 0
 0 I3 

‐ I1, I2 and I3 are the principal moments of inertia
 solving an eigenvalue problem

13. Introduction
Introduction Mathematical
Introduction CT Conclusion
shape description
3.. Mathematical shape description
B. Calculation L, I and S
‐ I1, I2 and I3 can be used to calculate L, I and S as the dimensions of the
principal axis of the approximated ellips:
1
I 1  m(I 2  S2 )
5
1
I 2  m(L2  S2 )
5
1
I 3  m(L2  I 2 )
5
‐ Is the fit of an approximating ellipsoid correct?

14. Introduction
Introduction Mathematical
Introduction CT Conclusion
shape description
3.. Mathematical shape description
C. Goodness of fit?
‐ Can be evaluated using the Vs or Es parameter:
en
Es 
S
‐ en: the surface area of the approximating ellipsoid
‐ S: the surface area of the pore
vn
Vs 
V
‐ vn: the volumeof the approximating ellipsoid
‐ V: the volume area of the pore
‐ Es also proofs to be an adequate parameter in order to describe the
sphericity of a pore

15. Introduction
Introduction Mathematical
Introduction CT Conclusion
shape description
3.. Mathematical shape description
C. Goodness of fit?
‐ Histogram of Vs:
Complex pores
‐ Mean: 1.38
‐ Median: 1.08
 Good fit for most pores but some exceptions

16. Introduction
Introduction Mathematical
Introduction CT Conclusion
shape description
3.. Mathematical shape description
C. Goodness of fit?
‐ Complex pores:
‐ Define different pore bodies:
‐ Watershed algorithm

17. Introduction
Introduction Mathematical
Introduction CT Conclusion
shape description
3.. Mathematical shape description
D. Defining pore shapes: based on shapes
‐ Based on L, I and S:
‐ Ratio’s: I/L and S/I
‐ 5 shape classes are defined Equant shape
Plate like shape
Blade like shape
Cuboid shape
Rod like shape

18. Introduction
Introduction Mathematical
Introduction CT Conclusion
shape description
3.. Mathematical shape description
D. Defining pore shapes
‐ Based on L, I and S:

19. Introduction
Introduction Mathematical
Introduction CT Conclusion
shape description
3.. Mathematical shape description
D. Defining pore shapes
‐ Based on L, I and S:

20. Introduction
Introduction Mathematical
Introduction CT Conclusion
shape description
3.. Mathematical shape description
D. Defining pore shapes
‐ Rod like shape:

21. Introduction
Introduction Mathematical
Introduction CT Conclusion
shape description
3.. Mathematical shape description
D. Defining pore shapes
‐ Working with an approximating ellipsoid allows to assess the
orientation of the pores
Tot vol = 58578 mm3 Tot vol = 26061 mm3

22. Introduction
Introduction Mathematical
Introduction CT Conclusion
shape description
3.. Mathematical shape description
D. Defining pore shapes
‐ Allows to differentiate between facies types:
rod blade plate cube cuboid rod blade plate cube cuboid
0,22 0,17 0,35 0,07 0,18 0,14 0,27 0,13 0,15 0,31

23. Introduction
Introduction Mathematical
Introduction CT Conclusion
shape description
3.. Mathematical shape description
D. Defining pore shapes: Compactness
‐ Compactness:

24. Introduction
Introduction Mathematical
Introduction CT Conclusion
shape description
3.. Mathematical shape description
D. Defining pore shapes: clustering
‐ Objective way of defining clusters:
‐ Model based clustering:
‐ Based on Probability methods
‐ Clusters are ellipsoidal
‐ Centered around the mean value
‐ Covariances determine the geometrics
‐ Number of clusters are statistically optimized

25. Introduction
Introduction Mathematical
Introduction CT Conclusion
shape description
3.. Mathematical shape description
D. Defining pore shapes: clustering
‐ Based on L, I and S:
‐ Ratio’s: I/L and S/I
‐ Compactness

26. Introduction
Introduction Mathematical
Introduction CT Conclusion
shape description
4.. Conclusion
A. Computer tomography
‐ Visualizes porosity networks in 3D
‐ Allows Petrography in 3D
B. Mathematical shape description
‐ Establishes a new 3D classification for pores in travertine rocks
‐ Classification is confirmed to be statistically relevant
‐ Allows to define facies types