Finite-Element-Based Characterisation of Pore-scale Geometry and its Impact on Fluid Flow
Lateef Akanji
Supervisors
Prof. Martin Blunt
Prof. Stephan Matthai
2
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
3
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
4
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:
6
Development of Single-phase Pore-scale Formulation and Numerical Model
,,u zyxP
12
0,, Pzyx
(2/2)
zyx ,,
μ
u
fluid pressure, P
Dependent variables are placed at the nodes.
zyx ,,
zyx ,, 0 u
tetrahedron
7
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)
10
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)
11
Model Verification, Step3: Pore Velocity
Placement of 7 FEM
Placement of 14 FEM
Placement of 21 FEM
(4/7)
12
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)
15
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)
17
(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
18
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
19
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)
20
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
24
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)
28
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)
29
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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