Combined use of finite volume and network modelling for Stokes flow and permeability tensor computation in porous media

N. Combaret1,2 _{D. Bernard}1 _{E. Plougonven}1

1_ICMCB-CNRS

University of Bordeaux 1

2_{VSG - Visualization Sciences Group}

2nd Conference on 3D-Imaging of Materials and Systems 2010

3D image: direct numerical processing

Network modeling:

Structured network model Unstructured network model

3D image: direct numerical processing

Network modeling:

Structured network model Unstructured network model

3D image: direct numerical processing

Network modeling:

Structured network model Unstructured network model

Stokes equations ( #« ∇. #«v = 0 ∇2#«v −∇p = 0#« B.C.: #«v = 0 on A_fs Linear system:       M X j=0 Qij =0 P − P .Q

Stokes equations ( #« ∇. #«v = 0 ∇2#«v −∇p = 0#« B.C.: #«v = 0 on A_fs Linear system:       M X j=0 Qij =0 P − P .Q

Stokes equations ( #« ∇. #«v = 0 ∇2#«v −∇p = 0#« B.C.: #«v = 0 on A_fs Linear system:       M X j=0 Qij =0 P − P .Q

Graph construction: from the pore space to a graph relating pores

Pore positioning: what is a pore? Pores separation: where are the pores?

Network equations and parameters: what are the relations between the pores?

Stokes equations equivalence on a graph Direct numerical computation of the resistances

Graph construction: from the pore space to a graph relating pores

Pore positioning: what is a pore? Pores separation: where are the pores?

Network equations and parameters: what are the relations between the pores?

Stokes equations equivalence on a graph Direct numerical computation of the resistances

Graph construction: from the pore space to a graph relating pores

Pore positioning: what is a pore? Pores separation: where are the pores?

Network equations and parameters: what are the relations between the pores?

Stokes equations equivalence on a graph Direct numerical computation of the resistances

Skeletonisation: homotopic thinning Digitisation artefact removal method Boundary conditions Skeleton points characterisation: curves and intersection points Nodes fusion criterion: relative volume

Skeletonisation: homotopic thinning Digitisation artefact removal method Boundary conditions Skeleton points characterisation: curves and intersection points Nodes fusion criterion: relative volume

Skeletonisation: homotopic thinning Digitisation artefact removal method Boundary conditions Skeleton points characterisation: curves and intersection points Nodes fusion criterion: relative volume

Skeletonisation: homotopic thinning Digitisation artefact removal method Boundary conditions Skeleton points characterisation: curves and intersection points Nodes fusion criterion: relative volume

Skeletonisation: homotopic thinning Digitisation artefact removal method Boundary conditions Skeleton points characterisation: curves and intersection points

Nodes fusion criterion: relative volume

Skeletonisation: homotopic thinning Digitisation artefact removal method Boundary conditions Skeleton points characterisation: curves and intersection points Nodes fusion criterion: relative volume

Skeletonisation: homotopic thinning Digitisation artefact removal method Boundary conditions Skeleton points characterisation: curves and intersection points Nodes fusion criterion: relative volume

Pore delimitation:

Pore throats positioning

Pore volume invasion: watershed

Unexpected configurations

⇒ Skeleton: good for positioning the pores, bad for representing their interconnexions

Pore delimitation:

Pore throats positioning

Pore volume invasion: watershed

Unexpected configurations

⇒ Skeleton: good for positioning the pores, bad for representing their interconnexions

Pore delimitation:

Pore throats positioning

Pore volume invasion: watershed

Unexpected configurations

⇒ Skeleton: good for positioning the pores, bad for representing their interconnexions

Pore delimitation:

Pore throats positioning

Pore volume invasion: watershed

Unexpected configurations

⇒ Skeleton: good for positioning the pores, bad for representing their interconnexions

Pore delimitation:

Pore throats positioning

Pore volume invasion: watershed

Unexpected configurations

⇒ Skeleton: good for positioning the pores, bad for

Z V_Hi #« ∇. #«v dV = 0 Z A_Hi #« v . # «nHidS = 0 Z A_{Hi H} #« v . # «nHiHdS = 0 M X j=1 " Z A_{Hi Hj} #« v . # «nHiHjdS # =0 M X Qij =0

Z V_Hi #« ∇. #«v dV = 0 Z A_Hi #« v . # «nHidS = 0 Z A_{Hi H} #« v . # «nHiHdS = 0 M X j=1 " Z A_{Hi Hj} #« v . # «nHiHjdS # =0 M X Qij =0

Z V_Hi #« ∇. #«v dV = 0 Z A_Hi #« v . # «nHidS = 0 Z A_{Hi H} #« v . # «nHiHdS = 0 M X j=1 " Z A_{Hi Hj} #« v . # «nHiHjdS # =0 M X Qij =0

Z V_Hi #« ∇. #«v dV = 0 Z A_Hi #« v . # «nHidS = 0 Z A_{Hi H} #« v . # «nHiHdS = 0 M X j=1 " Z A_{Hi Hj} #« v . # «nHiHjdS # =0 M X Qij =0

Z V_Hi #« ∇. #«v dV = 0 Z A_Hi #« v . # «nHidS = 0 Z A_{Hi H} #« v . # «nHiHdS = 0 M X j=1 " Z A_{Hi Hj} #« v . # «nHiHjdS # =0 M X Qij =0

What did we learn?

We need something on the network to support the flow rate through the surfaces

What did we learn?

We need something on the network to support the flow rate through the surfaces

Z Cij #« ∇p. #«c dC | {z } ¬ − Z Cij ∇2#«v . #«c dC | {z } ­ =0 ¬: Z Cij #« ∇p. #«c dC = p Hj
− p (Hi) ­:

independent from the path (see ¬) Q ∝ #«v ∝ ∇2#«v ⇒ Z Cij ∇2#«v . #«c dC = λijQij
− p (H

Z Cij #« ∇p. #«c dC | {z } ¬ − Z Cij ∇2#«v . #«c dC | {z } ­ =0 ¬: Z Cij #« ∇p. #«c dC = p Hj
− p (Hi) ­:

independent from the path (see ¬) Q ∝ #«v ∝ ∇2#«v ⇒ Z Cij ∇2#«v . #«c dC = λijQij
− p (H

Z Cij #« ∇p. #«c dC | {z } ¬ − Z Cij ∇2#«v . #«c dC | {z } ­ =0 ¬: Z Cij #« ∇p. #«c dC = p Hj
− p (Hi) ­:

independent from the path (see ¬) Q ∝ #«v ∝ ∇2#«v ⇒ Z Cij ∇2#«v . #«c dC = λijQij
− p (H

Z Cij #« ∇p. #«c dC | {z } ¬ − Z Cij ∇2#«v . #«c dC | {z } ­ =0 ¬: Z Cij #« ∇p. #«c dC = p Hj
− p (Hi) ­:

independent from the path (see ¬) Q ∝ #«v ∝ ∇2#«v ⇒ Z Cij ∇2#«v . #«c dC = λijQij
− p (H

Z Cij #« ∇p. #«c dC | {z } ¬ − Z Cij ∇2#«v . #«c dC | {z } ­ =0 ¬: Z Cij #« ∇p. #«c dC = p Hj
− p (Hi) ­:

independent from the path (see ¬) Q ∝ #«v ∝ ∇2#«v ⇒ Z Cij ∇2#«v . #«c dC = λijQij
− p (H

Z Cij #« ∇p. #«c dC | {z } ¬ − Z Cij ∇2#«v . #«c dC | {z } ­ =0 ¬: Z Cij #« ∇p. #«c dC = p Hj
− p (Hi) ­:

independent from the path (see ¬) Q ∝ #«v ∝ ∇2#«v ⇒ Z Cij ∇2#«v . #«c dC = λijQij
− p (H

What did we learn?

We need something on the network to support the flow rate through the surfaces

⇒ one edge per surface separating two pores

We need something on the pores to support the pressure values

What did we learn?

We need something on the network to support the flow rate through the surfaces

⇒ one edge per surface separating two pores

We need something on the pores to support the pressure values

Extracting the pores connected to a surface (⇔ an edge)

Finite volume computation of local Stokes flow Resistance determination from the second Stokes equation in a linear system:

λij =

P_ilocal− Plocal j Qlocal

ij

Linear system solution: pressure value at nodes, flow rate through edges

Extracting the pores connected to a surface (⇔ an edge) Finite volume computation of local Stokes flow

Resistance determination from the second Stokes equation in a linear system:

λij =

P_ilocal− Plocal j Qlocal

ij

Linear system solution: pressure value at nodes, flow rate through edges

Extracting the pores connected to a surface (⇔ an edge) Finite volume computation of local Stokes flow

Resistance determination from the second Stokes equation in a linear system:

λij =

P_ilocal− Plocal j

Qlocal ij

Linear system solution: pressure value at nodes, flow rate through edges

Extracting the pores connected to a surface (⇔ an edge) Finite volume computation of local Stokes flow

Resistance determination from the second Stokes equation in a linear system:

λij =

P_ilocal− Plocal j

Qlocal ij

Linear system solution: pressure value at nodes, flow rate through edges

Strong graph construction method

Digitisation artefacts removal prior to skeletonisation Nodes merging criterion

Pores delimitation methods comparison

Network settings

Linear system construction

Strong graph construction method

Digitisation artefacts removal prior to skeletonisation Nodes merging criterion

Pores delimitation methods comparison

Network settings

Linear system construction

Strong graph construction method

Digitisation artefacts removal prior to skeletonisation Nodes merging criterion

Pores delimitation methods comparison

Network settings

Linear system construction

Strong graph construction method

Digitisation artefacts removal prior to skeletonisation Nodes merging criterion

Pores delimitation methods comparison

Network settings

Linear system construction

Strong graph construction method

Digitisation artefacts removal prior to skeletonisation Nodes merging criterion

Pores delimitation methods comparison

Network settings

Linear system construction

Strong graph construction method

Digitisation artefacts removal prior to skeletonisation Nodes merging criterion

Pores delimitation methods comparison

Network settings

Linear system construction

Strong graph construction method

Digitisation artefacts removal prior to skeletonisation Nodes merging criterion

Pores delimitation methods comparison

Network settings

Linear system construction

Compare the linear system solution to the real numerical computation

Finalise the implementation of the permeability tensor computation

( #«

∇.D# «k =0 ∇2D# «k−∇d#« k = −I#«k B.C.:D# «k =0 on Afs

Compare the linear system solution to the real numerical computation

Finalise the implementation of the permeability tensor computation

( #«

∇.D# «k =0

∇2D# «k−∇d#« k = −I#«k

B.C.:D# «k =0 on Afs