Finite-Element-Based Characterisation of Pore-scale Geometry and its Impact on Fluid Flow

Lateef Akanji

Supervisors

Prof. Martin Blunt

Prof. Stephan Matthai

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Outline

1. Research Objectives

2. Development of Single-phase Pore-scale Formulation and Numerical Model

3. Workflow and Model Verification

4. Validation: Application to Porous Media

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Research Objectives

To characterize pore-scale geometries and derive the constitutive relationship governing single and multiphase flow through them

To contribute to a better understanding of the physics of fluid flow in porous media based on first principle numerical approach

To investigate the dependency of fluid flow on the pore geometry which is usually neglected on the continuum scale

To develop a constitutive relationship which allows a more rigorous assessment of fluid flow behavior with implications for the larger scale

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Outline

1. Research Objectives

2. Development of Single-phase Pore-scale Formulation and Numerical Model

3. Workflow and Model Verification

4. Validation: Application to Porous Media

5

Development of Single-phase Pore-scale Formulation and Numerical Model

The general p.d.e. governing fluid flow at pore scale is given by theNavier – Stokes equations as:

For an incompressible fluid conservation of mass takes the form

For a steady-state system, the substantial time derivative goes to zero i.e.

For slow laminar viscous flow with small Reynold’s number, the advective acceleration term drops out and we have the linear Stokes equations:

P u2

P

uuuu 2 t

P uuu2

0u

22

x 2hy

2

pyu

(1/2)

222,, zyxzyx

FEM discretisation and solution sequence

Define a function that obeys:

Step 1:

We solve Poisson’s equation for with homogeneous b.c.

Step 2:We compute the pressure field using – this ensures that

Since we define the velocity by:

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Development of Single-phase Pore-scale Formulation and Numerical Model

,,u zyxP

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0,, Pzyx

(2/2)

zyx ,,

μ

u

fluid pressure, P

Dependent variables are placed at the nodes.

zyx ,,

zyx ,, 0 u

tetrahedron

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Outline

1. Research Objectives

2. Development of Single-phase Pore-scale Formulation and Numerical Model

3. Workflow and Model Verification

4. Validation: Application to Porous Media

8

Workflow and Model Verification

Model Generation

Meshing

Simulation

Visualization

Task Tool

CAD ( Rhino )

ICEM - CFD Mesher

CSMP++

MayaVi, vtk, Paraview

Model Generation

Meshing

Simulation

Visualization

Task Tool

CAD ( Rhino )

ICEM - CFD Mesher

CSMP++

MayaVi, vtk, Paraview

(1/7)

9

Model Verification, Step1: Porosity

Porosity

Pore Volume / (Grain Volume + Pore Volume)

b

p

VV

(2/7)

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Model Verification, Step2: Pore Radius Computation

Pore radii

Derivative of f(x,y)

2dr

02

Pore Radius (μm)

0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

3.35 µm

3.35 µm

GRAIN

PORES

(3/7)

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Model Verification, Step3: Pore Velocity

Placement of 7 FEM

Placement of 14 FEM

Placement of 21 FEM

(4/7)

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Model Verification, Step3: Pore Velocity

Error analysis

Case a b c

Pressure gradient (Pa-m-1) 9860 9860 9860

Channel length (µm) 30 30 30

Number of Elements 7 14 21

Channel velocity

mismatch b/w analytical and

numerical (%)

22.62 2.54 0.92

Volume flux

mismatch b/w analytical and

numerical (%)

22.8 13.64 2.0

(5/7)

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Model Verification, Step3: Pore Velocity

Velocity (µms-1)

(6/7)

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Model Verification, Step4: Effective Permeability

PA

qkeff

(7/7)

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Outline

1. Research Objectives

2. Development of Single-phase Pore-scale Formulation and Numerical Model

3. Workflow and Model Verification

4. Validation: Application to Porous Media (Results)

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(Validation) Porous Media with Cylindrical Posts (1/10)

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(Talabi et al., SPE 2008)

(2/10)Application to Porous Media

Sample I: Ottawa sandstone

Micro-CT scan CAD Hybrid meshVelocity profile

Velocity (x 10-5 ms-1) 0 2 4 6 8 10 12 14

simulationthresholding

meshing

4.5mm

Velocity (x 10-5 ms-1) 0.0 2.0 4 .0 6.0 8.0 10.0 12.0 14.0

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Pore Radius (μm)

0 10 20 30 40 50 60 70 80

Ottawa Sandstone

Application to Porous Media

Pore radius distribution

(3/10)

Pore Radius (μm)

0 10 20 30 40 50 60 70 80

LV60 Sandstone Sombrero beach carbonate

Application to Porous Media

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3D Lab Expt 2D Num. Simulation

Ottawa sand

Dimension (mm) 4.5 x 4.5 x 4.5 4.5 x 4.5Porosity (%) 35 39 Permeability (D) 45 31

LV60 sand

Dimension (mm) 4.1 x 4.1 x 4.1 4.1 x 4.1Porosity (%) 37 40 Permeability (D) 40 29

Sombrero beach carbonate sand

Dimension (mm) - 4.5 x 4.5Porosity (%) - 36 Permeability (D) - 28

Computed versus Measured Permeability

(4/10)

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Application 3D Granular Packs (5/10)

4764.0

5.0

r

618.0

45.0

r

4764.0

5.0

r

3284.0

55.0

r

2022.0

6.0

r

15.0

625.0

r

041.0

7.0

r

0

6

0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95

Pe

rme

ab

ilit

y (x

10-

14

m2 )

Concentration

Permeability vs. Concentration for Single Sphere Numerical Experiment

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3D Granular Packs (6/10)

Xavier Garcia

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3D Granular Packs

Fluid Pressure

(7/10)

CAD geometry

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Sample 1

2.4 mm

Φ= 33.52

Φ= 37.02 Φ= 38.43

Φ= 32.3

Φ= 35.80

(8/10)

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Sample 2

2.4 mm

Φ= 32.43Φ= 33.52

Φ= 36.81

Φ= 35.57

Φ= 37.63

Does the detail really matter?

(9/10)

Permeability versus Porosity

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X 1

0 -5

(10/10)

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Single-phase Advection in Porous Media (1/2)

Ottawa

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Single-phase Advection in Porous Media (2/2)

LT-M

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Conclusions

I have presented a Finite-Element-Based numerical simulation work flow showing pore scale geometry description and flow dynamics based on first principle

This is achieved by carrying out several numerical simulation on micro-CT scan, photomicrograph and synthetic granular pack of pore scale model samples

In order to accurately model fluid flow in porous media, the φ, r, pc, k distribution must be adequately captured

(1/1)

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Future work

Two-phase flow with interface tracking testing for snap-off and phase trapping using level set method (Masa Prodanovic – University of Texas @ Austin)

Investigate dispersion in porous media (Branko Bijeljic)drainage imbibition

Courtesy: (Masa Prodanovic – University of Texas @ Austin)Capturing snap-off during imbibitionCourtesy: (Masa Prodanovic – University of Texas @ Austin)

(1/1)

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Acknowledgements

PTDF Nigeria

CSMP++ Group

THANK YOU!

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