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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.
Citation preview
Capturing Effective Permeabilities of Field Compaction bands
with Level Set and Hybrid Lattice Boltzmann/Finite Element Simulations
WaiChing Sun, Northwestern UniversityProf. Jose E. Andrade, California Institute of Technology
Motivation2
Aztec Sandstone, Holcomb et al 2007
7/24/2010 WCCM/APCOM 2010, Sydney, Australia
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?
Effective Permeability Measurement via Lattice Boltzmann method
7/24/2010 WCCM/APCOM 2010, Sydney, Australia 3
( )ij ij
k v xh
13 211 8.6 10
0.21f
k m
13 211 1.8 10
0.13f
k m
intrinsic permeability tensor of REV is computed by finding volume average velocity of Lattice Boltzmann cube with prescribed hydraulic gradient on a 150x150x150 voxels
Reynold’s number must be less than 1 to ensure the validity of Darcy’s law
Flow outside CB Flow inside CBFlow in transition zone13 2
11 2.4 10
0.15f
k m
Question:
7/24/2010 WCCM/APCOM 2010, Sydney, Australia 4
1. How does the micro‐structural change cause permeability reduction?
2. How about scale effect?
3. How can we verify our results?
Outline1. Background2. Tools used to study
characteristics of compaction band
i. Medial Axis (use level set)ii. Tortuosity (use weight
graph)iii. Isolated Pore Space (use
graph)iv. Effective permeability (use
FEM/LBM)3. Results on Aztec
sandstone4. Conclusion
Raw Tomography Images
Binary Images
Medial Axis
Shortest Flow Path
Isolated Pore Space
Effective Permeability
5
Image Segmentation
SPS Algorithm
FEM/LBM
Level Set Method
Recursive Function
7/24/2010 WCCM/APCOM 2010, Sydney, Australia
Image Segmentation
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Raw Tomography Images
Binary Images
Medial Axis
Shortest Flow Path
Isolated Pore Space
Effective Permeability
Image Segmentation
Dijkstra’s Algorithm
FEM/LBM
Level Set Method
Recursive Function
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.
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Aztec Sandstone Image, Leonir, et al 2010
Work conducted by Dr. Leonir
Level Set Method
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Raw Tomography Images
Binary Images
Medial Axis
Shortest Flow Path
Isolated Pore Space
Effective Permeability
Image Segmentation
Dijkstra’s Algorithm
FEM/LBM
Level Set Method
Recursive Function
Medial Axis of Flow Paths
• 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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Sirjani and Cross, 1991L
Le
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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Locating Medial Axis of Flow Path via Level Set
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Convert binary image into level set via semi‐implicit scheme
Extract Local minimum of level set function
Formulation of Variational Level Set Method
• Action functional (Li et al 2005)
• Governing equation
• Semi‐Implicit Finite Difference Scheme
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Penalize the deviation of from a signed distance function
Drive = 0 at the object boundaries
The discrete Laplacianterm is treated
implicitly
Shortest Path Searching Algorithm
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Raw Tomography Images
Binary Images
Medial Axis
Shortest Flow Path
Isolated Pore Space
Effective Permeability
Image Segmentation
SPS Algorithm (Dijkstra, 1953)
FEM/LBM
Level Set Method
Recursive Function
Graph Theory
Extract topology
info
Each fluid voxel as nodes
Connectednodes with
edges
Assignweight to edges
Form weighted graph
Apply SPS algorithm
Determine shortest Flow Path
Determine Geometrical tortuosity
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Weighted Graph
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Distance =
Weight
Fluid voxel = node
[74]
Shortest Path Searching Algorithm
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1
24
3
7 86
5
9
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15
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1311
14
1816
19
[19][19+14=33] [19+6=25]
[35] [37]
[46]
[69]
[45] [45]
[51]
[59] [55]
[66]
[70][75]17
20[78]
[56]
Shortest Flow Path Inside and Outside Compaction Bands
0.10.2
0.30.4
0.5
0.10.2
0.30.4
0.5
0.1
0.2
0.3
0.4
0.5
x, mmy, mm
z, m
m
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0.10.2
0.30.4
0.10.2
0.30.4
0.5
0.1
0.2
0.3
0.4
0.5
x, mmy, mm
z, m
m0.1
0.20.3
0.40.5
0.10.2
0.30.4
0.5
0.1
0.2
0.3
0.4
0.5
x, mmy, mm
z, m
m
0.10.2
0.30.4
0.5
0.10.2
0.30.4
0.5
0.1
0.2
0.3
0.4
0.5
x, mmy, mm
z, m
m
0.10.2
0.30.4
0.5
0.10.2
0.30.4
0.1
0.2
0.3
0.4
0.5
x, mmy, mm
z, m
m
0.10.2
0.30.4
0.10.2
0.30.4
0.1
0.2
0.3
0.4
0.5
x, mmy, mm
z, m
m
INSIDE CB
= 0.14
OUTSIDE CB
= 0.21
= 2.79K= 3.4e‐13 m2
= 2.15K= 5.3e‐13 m2
= 2.56K= 4.4e‐13 m2
= 1.77K= 1.3e‐12 m2
= 1.76K= 1.2e‐12 m2
= 1.81K= 1.3e‐12 m2
Recursive Function
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Raw Tomography Images
Binary Images
Medial Axis
Shortest Flow Path
Isolated Pore Space
Effective Permeability
Image Segmentation
Dijkstra’s Algorithm
FEM/LBM
Level Set Method
Recursive Function
Connected Porosity vs. Porosity• Only the connected pore space affects the
effective permeability
• Isolated pore space should be
treated as inactive voxels in LBM simulation
to reduce computational
cost
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Isolated Pore Space
Connected Pore Space
Flow Direction
Recursive Functions
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1
24
3
7 86
5
9
12
15
10
1311
14
1816
19
17
20
PROGRAM MAIN1. Activate all vertices along the flow path
as active nodes and mark them as visited vertices
2. While there exists at least one active node
3. call the recursive function MARKNEIGHBOR
EXIT
FUNCTION MARKNEIGHBOR1. IF at least one neighbors of the active
nodes has not yet been visited1. Activate the unvisited neighbor vertices2. Mark them as visited vertices. 3. Deactivate the old active nodes with
unvisited neighbor(s). 4. Call the recursive function
MARKNEIGHBOR2. ELSE
1. Deactivate the active nodes with no unvisited neighbor.
3. EXITEXIT
Connected and Isolated Pore Space of Compaction Bands
7/24/2010 WCCM/APCOM 2010, Sydney, Australia 21
Inside CB
Outside CB0.00E+00 5.00E‐02 1.00E‐01 1.50E‐01 2.00E‐01 2.50E‐01
OUTSIDE3
OUTSIDE2
OUTSIDE1
CB3
CB2
CB1
CONNECTED POROSITY OCCLUDED POROSITY
Finite Element/Lattice Boltzmann Hybrid Method
7/24/2010 WCCM/APCOM 2010, Sydney, Australia 22
Raw Tomography Images
Binary Images
Medial Axis
Shortest Flow Path
Isolated Pore Space
Effective Permeability
Image Segmentation
Dijkstra’s Algorithm
FEM/LBM
Level Set Method
Recursive Function
Homogenization of Effective Permeability Across scale
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LBMFEM
Lattice Boltzmann/Finite Element Flow Transport Simulation
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• Statistical ensemble– Equation of state
– Balance of momentum
0f v ft
( ) 0pp F f
t m
• Macroscopic Continuum– Balance of mass
– Balance of momentum
0vt
1( ) 0u u u pt
Kij1 Kij2
Kij3 Kij4
P1
P2
Size of Representative Elementary Volume
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7/24/2010 WCCM/APCOM 2010, Sydney, Australia 26
Effective Permeabilities Inside and Outside Compaction Bands
K zz = 3.5e‐13 m2
K zz = 14.0e‐13 m2
Inside CB
Outside CB
FEM/LBM Scheme
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.
7/24/2010 WCCM/APCOM 2010, Sydney, Australia 27
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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Thank you for your attention!
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