Capturing Effective Permeabilities of Field Compaction bands

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:


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



Binary Images

Medial Axis

Shortest Flow Path

Isolated Pore Space


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.


Aztec Sandstone Image, Leonir, et al 2010

Work conducted by Dr. Leonir

Level Set Method



Binary Images

Medial Axis

Shortest Flow Path

Isolated Pore Space


Image Segmentation


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


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


Locating Medial Axis of Flow Path via Level Set


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


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



Binary Images

Medial Axis

Shortest Flow Path

Isolated Pore Space


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


Weighted Graph


Distance =

Weight

Fluid voxel = node

[74]

Shortest Path Searching Algorithm


1

24

3

7 86

5

9

12

15

10

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


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



Binary Images

Medial Axis

Shortest Flow Path

Isolated Pore Space


Image Segmentation


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


Isolated Pore Space

Connected Pore Space

Flow Direction

Recursive Functions


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


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



Binary Images

Medial Axis

Shortest Flow Path

Isolated Pore Space


Image Segmentation


FEM/LBM

Level Set Method

Recursive Function

Homogenization of Effective Permeability Across scale


LBMFEM

Lattice Boltzmann/Finite Element Flow Transport Simulation


• 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



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.


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


Thank you for your attention!


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Capturing Effective Permeabilities of Field Compaction bands

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