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© Math2Market GmbH 1 Determination of Two-Phase Flow Properties with SatuDict Jürgen Becker

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Page 1: Determination of Two-Phase Flow Properties with SatuDict · Aim: Determine (saturation dependent) permeability Steps: Segmentation (ImportGeo) Pore size distribution (PoroDict) Capillary

© Math2Market GmbH

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Determination of Two-Phase Flow Properties with SatuDictJürgen Becker

Page 2: Determination of Two-Phase Flow Properties with SatuDict · Aim: Determine (saturation dependent) permeability Steps: Segmentation (ImportGeo) Pore size distribution (PoroDict) Capillary

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Case Study

Input:

Tomogram of Palatine Sandstone (Pfälzer Buntsandstein)

Aim: Determine (saturation dependent) permeability

Steps:

Segmentation (ImportGeo)

Pore size distribution (PoroDict)

Capillary pressure curve (SatuDict)

Permeability, fully saturated (FlowDict)

Relative permeability, partially saturated (SatuDict)

Page 3: Determination of Two-Phase Flow Properties with SatuDict · Aim: Determine (saturation dependent) permeability Steps: Segmentation (ImportGeo) Pore size distribution (PoroDict) Capillary

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Tomogram

F. Enzmann, Inst. for Geosciences, Uni Mainz

Pfälzer Buntsandstein

resolution 0.7 µm

1024³ voxels

Page 4: Determination of Two-Phase Flow Properties with SatuDict · Aim: Determine (saturation dependent) permeability Steps: Segmentation (ImportGeo) Pore size distribution (PoroDict) Capillary

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Segmentation

Porosity 25.7 %

Downscaled to 512³ voxels

Page 5: Determination of Two-Phase Flow Properties with SatuDict · Aim: Determine (saturation dependent) permeability Steps: Segmentation (ImportGeo) Pore size distribution (PoroDict) Capillary

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Pore Size Distribution

Pore space : X

Opening of radius r :

Volume of pores with radius :

(PoroDict - Geometric Pore Size Distribution)

dark grey: light grey:

Page 6: Determination of Two-Phase Flow Properties with SatuDict · Aim: Determine (saturation dependent) permeability Steps: Segmentation (ImportGeo) Pore size distribution (PoroDict) Capillary

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Pore Size Distribution (Sandstone)

Page 7: Determination of Two-Phase Flow Properties with SatuDict · Aim: Determine (saturation dependent) permeability Steps: Segmentation (ImportGeo) Pore size distribution (PoroDict) Capillary

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Capillary Pressure / Pore Morphology Method Young – Laplace equation

(pore radius <-> cap. pressure)

=> Pore size distribution gives saturationat given cap. pressure

But: connectivity of pores ?

Pore Morphology Method: adds connectivity checks to Young-Laplace low numerical cost

non-wetting fluid

wetting fluid

r

Page 8: Determination of Two-Phase Flow Properties with SatuDict · Aim: Determine (saturation dependent) permeability Steps: Segmentation (ImportGeo) Pore size distribution (PoroDict) Capillary

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Drainage I

Hilpert / Miller 2001

SatuDict 2010R2

Guarantees connectivity of NWP to reservoir

Idea: move in spheres

Start: completely wet

Start: large radius (i.e. small pc)

Steps: smaller radius(higher pc)

No residual water

Non-wetting phase (air) reservoir

Wetting phase (water) reservoir

Page 9: Determination of Two-Phase Flow Properties with SatuDict · Aim: Determine (saturation dependent) permeability Steps: Segmentation (ImportGeo) Pore size distribution (PoroDict) Capillary

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Drainage II

Ahrenholz et al. 2008

Additionally: WP must be connected to reservoir

Residual water (orange)

Non-wetting phase (air) reservoir

Wetting phase (water) reservoir

Page 10: Determination of Two-Phase Flow Properties with SatuDict · Aim: Determine (saturation dependent) permeability Steps: Segmentation (ImportGeo) Pore size distribution (PoroDict) Capillary

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Drainage (Sandstone Sample)

Drained pores ( r = 8.4 µm)Drained pores ( r = 14 µm)

Page 11: Determination of Two-Phase Flow Properties with SatuDict · Aim: Determine (saturation dependent) permeability Steps: Segmentation (ImportGeo) Pore size distribution (PoroDict) Capillary

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Drainage - Sandstone Sample

Slice of the 3D result

Residual water: 8.6 %

black: air

red: residual water

white: matrix material

Page 12: Determination of Two-Phase Flow Properties with SatuDict · Aim: Determine (saturation dependent) permeability Steps: Segmentation (ImportGeo) Pore size distribution (PoroDict) Capillary

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Capillary Pressure Curve (Drainage, Sandstone) Drainage II

contact angle 0°

0

20000

40000

60000

80000

100000

120000

140000

0 0.2 0.4 0.6 0.8 1

Water Saturation

Cap. P

ressure [Pa]

Page 13: Determination of Two-Phase Flow Properties with SatuDict · Aim: Determine (saturation dependent) permeability Steps: Segmentation (ImportGeo) Pore size distribution (PoroDict) Capillary

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Imbibition I

Hilpert / Miller 2001

SatuDict 2010R2

No connectivity checks

No residual air

Start: completely dry

Start: small radius (i.e. large pc)

Steps: larger radius(smaller pc)

Distribution by pore radius (Young-Laplace)

Page 14: Determination of Two-Phase Flow Properties with SatuDict · Aim: Determine (saturation dependent) permeability Steps: Segmentation (ImportGeo) Pore size distribution (PoroDict) Capillary

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Imbibition IINon-wetting phase (air) reservoir

Wetting phase (water) reservoir

Ahrenholz et al. 2008

WP must be connected to reservoir

No residual air

Page 15: Determination of Two-Phase Flow Properties with SatuDict · Aim: Determine (saturation dependent) permeability Steps: Segmentation (ImportGeo) Pore size distribution (PoroDict) Capillary

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Imbibition IIINon-wetting phase (air) reservoir

Wetting phase (water) reservoir

Ahrenholz et al. 2008

WP must be connected to reservoir

NWP must be connected to NWP reservoir

Residual air (orange)

Page 16: Determination of Two-Phase Flow Properties with SatuDict · Aim: Determine (saturation dependent) permeability Steps: Segmentation (ImportGeo) Pore size distribution (PoroDict) Capillary

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Drainage & Imbibition (Sandstone)

0

20000

40000

60000

80000

100000

120000

140000

0 0.2 0.4 0.6 0.8 1

Drainage

Drainage (res)

Imbibition

Imbibition (res)

Water Saturation

Cap. P

ressure [Pa]

small pores are connected

large pores are not connected

Page 17: Determination of Two-Phase Flow Properties with SatuDict · Aim: Determine (saturation dependent) permeability Steps: Segmentation (ImportGeo) Pore size distribution (PoroDict) Capillary

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Permeability Macroscopic description (homogenized porous media model)

Darcy’s law :

u : average flow velocity : permeability tensor unknown : viscosityp : pressure

Microscopic description (pore structure model)

Stokes equation:

Boundary conditions: no-slip on surface, pressure drop can be determined from the solution!

Page 18: Determination of Two-Phase Flow Properties with SatuDict · Aim: Determine (saturation dependent) permeability Steps: Segmentation (ImportGeo) Pore size distribution (PoroDict) Capillary

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Permeability (Sandstone)

FlowDict Result (EJ-Stokes): 0.966 e-12 m²

Page 19: Determination of Two-Phase Flow Properties with SatuDict · Aim: Determine (saturation dependent) permeability Steps: Segmentation (ImportGeo) Pore size distribution (PoroDict) Capillary

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Relative Permeability

Idea:

Combine

phase distributions from pore morphology

single-phase flow

Advantage:

low computational costs

stability

Page 20: Determination of Two-Phase Flow Properties with SatuDict · Aim: Determine (saturation dependent) permeability Steps: Segmentation (ImportGeo) Pore size distribution (PoroDict) Capillary

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Relative Permeability (Sandstone)

Choose saturation level, choose wetting model

Use PM to find air distribution (here: yellow)

Solve Stokes equation in remaining pore space

Page 21: Determination of Two-Phase Flow Properties with SatuDict · Aim: Determine (saturation dependent) permeability Steps: Segmentation (ImportGeo) Pore size distribution (PoroDict) Capillary

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Relative Permeability (Sandstone)

0

2E-13

4E-13

6E-13

8E-13

1E-12

1,2E-12

0 0,2 0,4 0,6 0,8 1

Perm

ebili

ty [m

²]

Water Saturation

Page 22: Determination of Two-Phase Flow Properties with SatuDict · Aim: Determine (saturation dependent) permeability Steps: Segmentation (ImportGeo) Pore size distribution (PoroDict) Capillary

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Summary: Case Study

Input:

Tomogram of Palatine Sandstone (Pfälzer Buntsandstein)

Calculated properties:

Pore size distribution

Capillary pressure curve

Permeability

Relative permeability

Page 23: Determination of Two-Phase Flow Properties with SatuDict · Aim: Determine (saturation dependent) permeability Steps: Segmentation (ImportGeo) Pore size distribution (PoroDict) Capillary

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Comments

Same ideas can be used for diffusivity or conductivity

Applications:

Gas diffusion layer of fuel cells

Hygiene products

Paper dewatering felts

Thank you !