Lateef Akanji
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Transcript Lateef Akanji
Finite-Element-Based Characterisation of Porescale Geometry and its Impact on Fluid Flow
Lateef Akanji
Supervisors
Prof. Martin Blunt
Prof. Stephan Matthai
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
2
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
3
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
4
Development of Single-phase Pore-scale Formulation
and Numerical Model
(1/2)
The general p.d.e. governing fluid flow at pore scale is given by the
u
Navier – Stokes equations as:
2
t
u u u P
For an incompressible fluid conservation of mass takes the form
u 0
For a steady-state system, the substantial time derivative goes to zero i.e.
2 u u u P
For slow laminar viscous flow with small Reynold’s number, the advective
acceleration term drops out and we have the linear Stokes equations:
2u P
p
2
2
h
u x y
y
2
2
x, y, z x 2 y 2 z 2
5
Development of Single-phase Pore-scale Formulation and
Numerical Model
(2/2)
FEM discretisation and solution sequence
Define a function x, y, z that obeys:
1
2
Step 1:
We solve Poisson’s equation for x, y, z with homogeneous b.c.
Step 2:
We compute the pressure field using x, y, z – this ensures that
x, y, z P 0
fluid pressure, P
tetrahedron
Since we define the velocity by:
u
u 0
P
x, y, z
μ
u
Dependent
variables are
placed at the
nodes.
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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
7
Workflow and Model Verification
Task
Tool
Model Generation
CAD ( Rhino )
Meshing
ICEM - CFD Mesher
Simulation
Visualization
(1/7)
CSMP++
MayaVi, vtk, Paraview
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Model Verification, Step1: Porosity
(2/7)
Porosity
Pore Volume / (Grain Volume + Pore Volume)
Vp
Vb
9
Model Verification, Step2: Pore Radius Computation (3/7)
Pore radii
rd
GRAIN
2
3.35 µm
PORES
Derivative of f(x,y)
2 0
3.35 µm
Pore Radius (μm)
0
0.5
1.0
1.5
2.0
2.5
3.0 3.5
10
Model Verification, Step3: Pore Velocity
(4/7)
Placement of 7 FEM
Placement of 14 FEM
Placement of 21 FEM
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Model Verification, Step3: Pore Velocity
(5/7)
Error analysis
Case
a
b
c
9860
9860
9860
Channel length (µm)
30
30
30
Number of Elements
7
14
21
22.62
2.54
0.92
22.8
13.64
2.0
Pressure gradient (Pa-m-1)
Channel velocity
mismatch b/w analytical and
numerical (%)
Volume flux
mismatch b/w analytical and
numerical (%)
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Model Verification, Step3: Pore Velocity
(6/7)
Velocity (µms-1)
13
Model Verification, Step4: Effective Permeability (7/7)
keff
q
AP
14
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)
15
(Validation) Porous Media with Cylindrical Posts (1/10)
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Application to Porous Media
(2/10)
Sample I: Ottawa sandstone
(Talabi et al., SPE 2008)
meshing
simulation
thresholding
4.5mm
Velocity (x 10-5 ms-1)
Micro-CT scan
CAD
Hybrid mesh
0 2
4
6
8
10
12
14
Velocity profile
0.0
2.0
4 .0
Velocity (x 10-5 ms-1)
6.0
8.0
10.0
12.0 1714.0
Application to Porous Media
(3/10)
Pore radius distribution
LV60 Sandstone
Ottawa Sandstone
Sombrero beach carbonate
Pore Radius (μm)
0 10 20
30 40 50
60
Pore Radius (μm)
70 80
0 10 20
30 40 50
60
70 80
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(4/10)
Application to Porous Media
Computed versus Measured Permeability
3D Lab Expt
2D Num. Simulation
Ottawa sand
Dimension (mm)
Porosity (%)
Permeability (D)
4.5 x 4.5 x 4.5
35
45
4.5 x 4.5
39
31
4.1 x 4.1 x 4.1
37
40
4.1 x 4.1
40
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LV60 sand
Dimension (mm)
Porosity (%)
Permeability (D)
Sombrero beach carbonate sand
Dimension (mm)
Porosity (%)
Permeability (D)
-
4.5 x 4.5
36
28
19
(5/10)
Application 3D Granular Packs
Permeability vs. Concentration for Single Sphere Numerical
Experiment
6
Permeability (x 10-14 m2)
r 0.5
0.4764
r 0.5
r 0.45
0.r6 0.55 0.4764
0.618
0.3284
r
r 0.625
0.7
r0.2022
0.15
0.041
0
0.25
0.35
0.45
0.55
0.65
0.75
0.85
0.95
Concentration
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3D Granular Packs
(6/10)
Xavier Garcia
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3D Granular Packs
(7/10)
CAD geometry
Fluid Pressure
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(8/10)
Sample 1
Φ= 32.3
Φ= 33.52
Φ= 35.80
2.4 mm
Φ= 37.02
Φ= 38.43
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(9/10)
Sample 2
Φ= 32.43
Φ= 33.52
Φ= 35.57
Does the detail
really matter?
2.4 mm
Φ= 36.81
Φ= 37.63
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(10/10)
X 10 -5
Permeability versus Porosity
25
Single-phase Advection in Porous Media
(1/2)
Ottawa
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Single-phase Advection in Porous Media
(2/2)
LT-M
27
Conclusions
(1/1)
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
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Future work
(1/1)
Two-phase flow with interface tracking testing for snap-off and phase
trapping using level set method (Masa Prodanovic – University of Texas @
Austin)
drainage
Investigate dispersion in porous mediaimbibition
(Branko Bijeljic)
Capturing snap-off during imbibition
Courtesy: (Masa Prodanovic – University of Texas @ Austin)
Courtesy: (Masa Prodanovic – University of Texas @ Austin)
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Acknowledgements
PTDF Nigeria
CSMP++ Group
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THANK YOU!
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