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.
6
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
8
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
rd
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
11
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 (%)
12
Model Verification, Step3: Pore Velocity
(6/7)
Velocity (µms-1)
13
Model Verification, Step4: Effective Permeability (7/7)
keff
q

AP
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)
16
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
18
(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
29
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
20
3D Granular Packs
(6/10)
Xavier Garcia
21
3D Granular Packs
(7/10)
CAD geometry
Fluid Pressure
22
(8/10)
Sample 1
Φ= 32.3
Φ= 33.52
Φ= 35.80
2.4 mm
Φ= 37.02
Φ= 38.43
23
(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
24
(10/10)
X 10 -5
Permeability versus Porosity
25
Single-phase Advection in Porous Media
(1/2)
Ottawa
26
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
28
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)
29
Acknowledgements
PTDF Nigeria
CSMP++ Group
30
THANK YOU!
31