Introduction: Effective permeability of a field compaction band formed in an Aztec sand-stone is computed via a hierarchical lattice Boltzmann/finite element (LB/FE) scheme. As observed in reconstructed 3D images (N. Lenoir, et al., GeoX, 2010), the formation of compaction bands may reduce the number of flow channels and cause a significant portion of pore space become isolated. This isolated pore space is the focus of this study. In particular, we show that (1) isolated pore space is the main cause of the sharp permeability difference inside/outside the compaction band and (2) the speed and accuracy of the multi-scale framework can be significantly improved if the isolated pore space is captured prior to permeability calculations. Method: The calculation of effective permeability involves three steps. First, the topology of the isolated pore space is captured via a level set evolution scheme (Chunming, Li, et al., IEEE, 2005). The key idea is to obtain the signed distance function and use it to separate the micro-channel from the isolated pore space. As pore-fiuid trapped inside the pore space remains undrained, size of the pore-scale LB simulation can be reduced by neglecting the discrete distribution function inside the isolated pore space. Furthermore, by knowing the topology of the isolated pore space, we eliminate any chance of mistaking an isolated pore space as a flow channel in a representative elementary volume and hence enhance the accuracy of the permeability calculation. Secondly, LB simulation is conducted in a representative elementary volume to compute local permeability. Finally, permeability acquired from the pore-scale LB simulation is extracted as input for the macro-scale Darcy\'s flow problem solved via finite element. Components of the macroscopic effective permeability tensor are then inversely computed from the known homogenized Darcy\'s velocity and pressure fields. Result: LB simulations and LB/FEM hybrid simulations are both run on images reconstructed from Aztec sandstone specimens inside and outside a field compaction band. The effective permeability computed via both simulations are in agreement. Although the difference on porosity inside/outside compaction band is only about 5%, we observe a dramatic difference on permeability inside/outside compaction band. This change is nevertheless consistent with the increase of proportion of isolated pore space inside the compaction band. Conclusion: We propose a multi-scale framework to capture the permeability of a field compaction band. Unlike the pore space numerically constructed by randomly distributed disk, real pore space inside field compaction band is narrower and less interconnected. This feature severely limited the number of possible paths pore-fluid can travel in and therefore makes the isolated pore space significant on permeability calculations.

1. Capturing Effective Permeabilities of
Field Compaction bands
with Level Set and Hybrid Lattice Boltzmann/Finite Element
Simulations
WaiChing Sun, Northwestern University
Prof. Jose E. Andrade, California Institute of Technology

2. 2
Motivation
Significant porosity and
permeability reductions are
observed in compaction bands
Are compaction bands
efficient flow barriers?
If so, What are the
geometrical features that lead
to permeability reductions?
Aztec Sandstone, Holcomb et al 2007
7/24/2010 WCCM/APCOM 2010, Sydney, Australia

3. Effective Permeability Measurement
via Lattice Boltzmann method
Flow outside CB Flow in transition zone Flow inside CB
k11  8.6 1013 m 2 k11  1.8 1013 m 2 k11  2.4 1013 m 2
 f  0.21  f  0.13  f  0.15
 intrinsic permeability tensor of REV is computed by finding volume average velocity of Lattice
Boltzmann cube with prescribed hydraulic gradient on a 150x150x150 voxels

kij   vi ( x)
h j
 Reynold’s number must be less than 1 to ensure the validity of Darcy’s law
7/24/2010 WCCM/APCOM 2010, Sydney, Australia 3

4. Question:
1. How does the micro‐structural
change cause permeability
reduction?
2. How about scale effect?
3. How can we verify our results?
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5. 5
Outline
Raw Tomography Images
1. Background
2. Tools used to study Image Segmentation
characteristics of Binary Images
compaction band Level Set Method
i. Medial Axis (use level set)
Medial Axis
ii. Tortuosity (use weight
graph) SPS Algorithm
iii. Isolated Pore Space (use Shortest Flow Path
graph)
iv. Effective permeability (use Recursive Function
FEM/LBM) Isolated Pore Space
3. Results on Aztec FEM/LBM
sandstone
4. Conclusion Effective Permeability
7/24/2010 WCCM/APCOM 2010, Sydney, Australia

6. Image Segmentation
Raw Tomography Images
Image Segmentation
Binary Images
Level Set Method
Medial Axis
Dijkstra’s Algorithm
Shortest Flow Path
Recursive Function
Isolated Pore Space
FEM/LBM
Effective Permeability
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7. Aztec Sandstone Specimen
3D tomographic images are
taken from Aztec sandstone
collected at Valley of Fire
State Park, Nevada.
A threshold is defined to
distinguish pore space and
skeleton.
A binary 3D images are
crated from the original 3D
images.
Connected and isolated pore
space are not distinguished.
Work conducted by Dr. Leonir
Aztec Sandstone Image, Leonir, et al 2010
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8. Level Set Method
Raw Tomography Images
Image Segmentation
Binary Images
Level Set Method
Medial Axis
Dijkstra’s Algorithm
Shortest Flow Path
Recursive Function
Isolated Pore Space
FEM/LBM
Effective Permeability
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9. Medial Axis of Flow Paths
Le
Sirjani and Cross, 1991
L
• Medial axis is the spine of a volume filling object
• Media axis is a union of curves that represent the
topology and geometry of the volume
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10. Relation between Level Set Function
and Medial Axis
• Local minimum of the signed distance function  are
located at medial axis.
• Use level set scheme to obtain signed distance function
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11. Locating Medial Axis of Flow Path via
Level Set
Convert binary
Extract Local
image into level
minimum of level
set via semi‐
set function
implicit scheme
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12. Formulation of Variational Level Set
Method
Penalize the deviation
of from a signed
distance function 
• Action functional (Li et al 2005)
Drive  = 0 at the object
• Governing equation boundaries
The discrete Laplacian
• Semi‐Implicit Finite Difference Scheme
term is treated
implicitly
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13. Shortest Path Searching Algorithm
Raw Tomography Images
Image Segmentation
Binary Images
Level Set Method
Medial Axis
SPS Algorithm (Dijkstra, 1953)
Shortest Flow Path
Recursive Function
Isolated Pore Space
FEM/LBM
Effective Permeability
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14. Graph Theory
Extract Each fluid Connected Assign Form
topology voxel as nodes with weight to weighted
info nodes edges edges graph
Determine Determine
Apply SPS
Geometrical shortest
algorithm
tortuosity Flow Path
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15. Weighted Graph
Distance
=
Weight
Fluid
voxel =
node
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16. Shortest Path Searching Algorithm
1
[19]
[19+14=33] [19+6=25]
2
4
3
[45] [45] [35] [37]
7 8 5
6
[51] [46]
10
9
[59] [55]
[66] 13 [56]
11
[74] 14 12
[70]
17 [75] 16 [69]
[78] 18 20
19 15
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17. Shortest Flow Path Inside and Outside
Compaction Bands
0.5
0.5
0.5
0.4
INSIDE CB
0.4 0.4
0.3
z, mm
0.3
z, mm
0.3
z, mm
0.2
 = 0.14
0.2 0.2
0.1
0.1 0.1
0.5
0.5 0.5
0.4
0.4 0.4 0.5
0.3 0.5 0.4
0.4 0.3 0.3
0.2 0.4 0.3
0.3 0.2 0.3 0.2 0.2
0.1 0.2
0.1 0.2 0.1 0.1
y, mm 0.1 0.1
x, mm y, mm y, mm x, mm
x, mm
= 2.79 = 2.15 = 2.56
K= 3.4e‐13 m2 K= 5.3e‐13 m2 K= 4.4e‐13 m2
0.5 0.5
0.5
0.4 0.4
0.4
OUTSIDE CB 0.3
z, mm
0.3 0.3
z, mm
z, mm
0.2 0.2 0.2
 = 0.21 0.1 0.1 0.1
0.5
0.4 0.4
0.5 0.4 0.5
0.3 0.4 0.3 0.4
0.4 0.3 0.3
0.2 0.3 0.2
0.3 0.2 0.2
0.1 0.2 0.1
0.2 0.1
0.1 0.1
y, mm 0.1 y, mm y, mm
x, mm x, mm x, mm
= 1.77 = 1.76 = 1.81
K= 1.3e‐12 m2 K= 1.2e‐12 m2 K= 1.3e‐12 m2
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18. Recursive Function
Raw Tomography Images
Image Segmentation
Binary Images
Level Set Method
Medial Axis
Dijkstra’s Algorithm
Shortest Flow Path
Recursive Function
Isolated Pore Space
FEM/LBM
Effective Permeability
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19. Connected Porosity vs. Porosity
• Only the
connected pore
space affects the Connected
effective
Flow Direction
Pore Space
permeability
• Isolated pore
space should be
treated as
inactive voxels in Isolated
LBM simulation Pore Space
to reduce
computational
cost
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20. Recursive Functions
1 PROGRAM MAIN
1. Activate all vertices along the flow path
as active nodes and mark them as visited
vertices
2 2. While there exists at least one active
4 node
3 3. call the recursive function
MARKNEIGHBOR
EXIT
7 8 5
6 FUNCTION MARKNEIGHBOR
1. IF at least one neighbors of the active
nodes has not yet been visited
10 1. Activate the unvisited neighbor vertices
9
2. Mark them as visited vertices.
3. Deactivate the old active nodes with
13 unvisited neighbor(s).
11 4. Call the recursive function
MARKNEIGHBOR
14 12
2. ELSE
1. Deactivate the active nodes with no
17 unvisited neighbor.
16
20 3. EXIT
18 15
19 EXIT
7/24/2010 WCCM/APCOM 2010, Sydney, Australia 20

21. Connected and Isolated Pore Space of
Compaction Bands
CB1
CB2
Inside CB
CB3
OUTSIDE1
OUTSIDE2
OUTSIDE3
Outside CB
0.00E+00 5.00E‐02 1.00E‐01 1.50E‐01 2.00E‐01 2.50E‐01
CONNECTED POROSITY OCCLUDED POROSITY
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22. Finite Element/Lattice Boltzmann
Hybrid Method
Raw Tomography Images
Image Segmentation
Binary Images
Level Set Method
Medial Axis
Dijkstra’s Algorithm
Shortest Flow Path
Recursive Function
Isolated Pore Space
FEM/LBM
Effective Permeability
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23. Homogenization of Effective
Permeability Across scale
LBM
FEM
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24. Lattice Boltzmann/Finite Element Flow
Transport Simulation
• Statistical ensemble
– Equation of state

f  v f  0
t
– Balance of momentum
 p 
(    F   p ) f  0 P1
t m
Kij1 Kij2
• Macroscopic Continuum
– Balance of mass

  v   0 Kij3 Kij4
t
– Balance of momentum
P2
 1
u  (u )u  p  0
t 
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25. Size of Representative Elementary
Volume
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26. Effective Permeabilities Inside and
Outside Compaction Bands
FEM/LBM Scheme
Inside CB K zz = 3.5e‐13
m2
K zz = 14.0e‐
Outside CB 13 m2
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27. Conclusion
• The geometrical changes of micro‐structure
due to the formation of compaction bands are
examined.
• Level set, graph theory, lattice Boltzmann and
finite element methods are used as tools in
this study.
• Increased tortuosity, reduction of connected
porosity are main factors that lead to
permeability reduction of compaction band.
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28. Acknowledgement
• Dr. Nicolas Leonir from Ecole des Ponts Paris Tech,
Université Paris‐Est
• Dr. David Salac from Northwestern University
• Prof. John Rudnicki from Northwestern University
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29. Thank you for your attention!
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