1. Statistical analysis of pore space geometry Stefano Favretto Supervisor : Prof. Martin Blunt Petroleum Engineering and Rock Mechanics Research Group Department of Earth Science and Engineering Imperial College London

2. Imperial College London, January 5 th 2005 Segmentation and images cleaning Segmentation and images cleaning Medial Axis construction Medial Axis construction Pore space analysis: pore bodies and pore throats Pore space analysis: pore bodies and pore throats Geometrical statistical parameters: - pore volume distribution Geometrical statistical parameters: - pore volume distribution - throats area distribution - throats area distribution - channel length - coordination number - pore and channel diameters and shape Introduction In order to construct a realistic model for flow and transport simulations, it is fundamental to analyze in details the geometrical structure of the pore space. Here we verify the use of a software, called 3DMA_Rock, to extract (on the basis of a statistical analysis) some geometrical parameters, later used as input data to network generating code for flow simulations. The 3DMA_Rock code was designed to take as input 3-dimensional digital grey-scale images, as those from  –computed tomography. The main steps of the analysis are:

3. pore space medial axis (MA) throats calculation...throats area distribution...pore volume distribution pores identification 3DMA_Rock analysis approach Imperial College London, January 5 th 2005

4. Image segmentation  - CT data: SAMPLE1, 256x256x256 cube (resolution is 8,683  m ) original segmented intensities occurrences Simple threshold or other more advanced techniques (i.e. Indicator Kriging) which threshold ? Imperial College London, January 5 th 2005

5. Cleaning images after segmentation 1) Remove isolated pore from the grains 2) Remove unphysical grains from the pore void space segmented cleaned cleanedoriginal A - generally, isolated pore in the grains are very small...so we remove them by size criterion B – we are looking at effective porosity...so we preserve a pore only if connected to the border Working on the complementary images, we remove all isolated grains (size criterion, local erosions...) Imperial College London, January 5 th 2005

6. Medial Axis construction 3DMA_Rock thinning algorithm (Lee et. al., 1994) Each voxel is labeled with a “burn number”, depending on its distance from grains surface - LKC [Lee, Kashyap, Chu; 1994] algorithm. Each voxel is labeled with a “burn number”, depending on its distance from grains surface - LKC [Lee, Kashyap, Chu; 1994] algorithm. Medial Axis are produced by a thinning algorithm based on Simple Points (this guarantees topology preservation). Medial Axis are produced by a thinning algorithm based on Simple Points (this guarantees topology preservation). Imperial College London, January 5 th 2005

7. Example : Medial Axis calculation on artificial pore system 2D slice 3D Rendering Throats visualization3D Medial Axis Imperial College London, January 5 th 2005

8. Pore throats Pore bodies Pore bodies and pore throats Imperial College London, January 5 th 2005

9. Medial Axis calculation on Sample 1 3D visualization of a 128x128x128 cube Original with MA cluster C cluster B cluster A path I (leaf-leaf) path II (branch-leaf) path III (branch-branch) Imperial College London, January 5 th 2005

10. Throats calculation on Sample 1 Medial Axis Medial Axis & Throats barriers Imperial College London, January 5 th 2005

11. Statistical results Unit = 1 voxel L = 8.683  m L Equivalent pore radius: R = 3 V 4  1/3 Pore radius Pore radius distribution distribution equivalent pore radius occurrences Imperial College London, January 5 th 2005

12. Statistical results Throats radius distribution Throats radius distribution Unit = 1 voxel L = 8.683  m L Equivalent throat radius: R = A  1/2 Imperial College London, January 5 th 2005 equivalent throats radius occurrences

13. Statistical results Path length & channel length Path length & channel length Channel length [voxels] 3DMA assumes the channel length C L as the distance between two pore center (path length P L ) 3DMA assumes the channel length C L as the distance between two pore center (path length P L ) We could calculate channel length as: We could calculate channel length as: C L = n P L with 0<n<1 Imperial College London, January 5 th 2005 CLCL PLPL

14. Statistical results Throat radius vs pore radius Throat radius vs pore radius throats equivalent radius pore equivalent radius throats / pore radius occurrences Imperial College London, January 5 th 2005

15. Statistical results 1 2 3 N c = 3 Coordination number Coordination number distribution distribution Imperial College London, January 5 th 2005

16. Statistical results Throats shape factor Throats shape factor distribution distribution G = A p2p2 G = 1/4  = 0.0796 G = 1/16 = 0.0625G = √3 / 36 = 0.0481...we can distinguish between different cross section shape G occurrences Imperial College London, January 5 th 2005

17. Statistical results Pore diameters in three orthogonal directions ( X ; Y ; Z ) Pore diameters in three orthogonal directions ( X ; Y ; Z ) z x y Imperial College London, January 5 th 2005