Pore-scale direct simulation of flow and transport in porous media
1. Pore-Scale Direct Numerical Simulation of Flow and
Transport in Porous Media
Sreejith Pulloor Kuttanikkad
PhD Thesis Defence
(Thursday 15 October, 2009)
Interdisciplinary Centre for Scientific Computing (IWR)
Faculty of Mathematics and Informatics, University of Heidelberg
Sreejith Kuttanikkad (IWR, University of Heidelberg) Pore-Scale Simulation 15.10.2009 1 / 36
2. Outline
1 Introduction - Relevance of the topic
2 Thesis Motivation and Objectives
3 Pore-scale Models for Flow and Transport
4 Unfitted Discontinuous Galerkin (UDG) Method for Complex Domains
5 Random Walk Particle Tracking Method
6 Implementation and Validation of Flow and Transport Models
7 Pore-scale simulation of Dispersion in 2D & 3D
8 Summary and Outlook
3. Introduction - Relevance
Study of flow and transport through porous media has wide practical
applications
Subsurface/Environmental applications (contaminant transport, nuclear waste
disposal, groundwater remediation, oil/gas/petroleum exploration)
Industrial applications (PEM Fuel cells, packed bed reactors, filtration studies)
Studies are being done at various scales using Numerical and Experimental
methods
Numerical simulations are usually done at the continuum scale which require
the knowledge of certain parameters (permeability, dispersion coefficients,
etc.)
4. Introduction - Relevance of the topic
Macroscopic simulations fail to explain certain flow and transport behaviours (e.g,
tailing of BTC, hysteresis in multiphase flow parameters)
Advection dispersion equation
(ADE) is generally used as the tool
for predicting and quantifying solute
transport
∂C 2
+ · (vC) − D. C=0
∂t
The basic assumption of the ADE is
that dispersion follows Fickian
behavior
J = −D C
Numerous experiments have shown
that solute spreading does not
follow a Gaussian distribution
5. Introduction - Relevance
Pore-Scale Simulations
An alternative and more fundamental approach
Provides link between pore-scale properties of the porous medium and large
scale behaviour
Governing flow and transport equations are known at pore-scale
Macroscopic parameters can be obtained using the results of pore-scale
simulations
Pore-Scale Methods
Challenges: require detailed structure of the medium, method should be able
to handle geometry
Pore-scale numerical methods: Pore-Network, LBM, Finite Element, etc.
New and efficient pore-scale simulation methods have its relevance in this
context
6. Motivation and Research Objectives
Motivation
Lack of fundamental understanding of how pore structure controls flow and
transport behaviour at larger scales!
This work is motivated by the need for better understanding of the physical
processes that take place at the pore-scale and to improve the reliability of
numerical models that describe the flow at larger scales
The objectives set for the study are
To develop a model to simulate single phase flow and solute transport
processes through porous media at the pore-scale
In particular, to use a new numerical discretisation approach (called Unfitted
Discontinuous Galerkin UDG) for the solution of partial differential equations
on the pore-scale geometry
And to predict the macroscopic parameters of porous medium based on
pore-scale simulations
7. Pore-Scale Modelling of Flow and Transport
Present approach involve following steps:
1 Compute the pore-scale velocity field (by Solving Stokes equation)
2
−µ u+ p = f; ·u=0
By using a new method called Unfitted Discontinuous Galerkin which requires
Implementation of DG finite element discretisation of Stokes equation in the
framework of unfitted discontinuous Galerkin method
2 Obtain the flow and transport parameters based on the computed pore-scale
velocity field
Permeability is computed by applying the Darcy’s law
Dispersion coefficients are determined by solving the Advection-Diffusion
equation posed at the pore-scale by RWPT method
∂C 2
= −u · C +D C
∂t
Much of the challenge in solving Stokes problem (for velocity and pressure) is how
to account for the complex pore-scale geometry!
8. Unfitted Discontinuous Galerkin (UDG) Method
Method for the solution of the Stokes equation is based on a new numerical
approach which has been specifically developed for applications in complex
domains
UDG introduced by Engwer and Bastian (2005,2008)
Use only a structured grid and based on DG finite element method with trial
and test functions defined on the structured grid
Mesh Construction
Given the pore geometry, a fundamental structured grid is chosen
According to desired accuracy and computational resources
Generally a course mesh can be used
9. Unfitted Discontinuous Galerkin Method
Mesh Construction
Grid intersected by the domain generate arbitrary shaped elements
Support of the trial and test functions are restricted according to the shape
of the elements
Essential boundary conditions are imposed weakly via the DG formulation
Number of dofs is proportional to the number of elements in the grid
10. Unfitted Discontinuous Galerkin Method
Evaluation of surface and volume integrals
Local Triangulation
Local triangulation for assembling Subdivision of elements into
sub-elements which are easily
integrable (“Local Triangulation”)
- Predefined triangulation rules for
a class of similar elements
- Reduce number of different
classes by appropriate bisection of
the element
Use of quadratic transformation for
better approximation of curved
boundaries
Based on the marching cube Use of standard quadrature rules for
algorithms the integration over sub-elements
11. Unfitted Discontinuous Galerkin Method
Appealing things
Underlying DGFE discretisation of the PDE model
It has all benefits of standard finite element methods
Advantages of the DG schemes are naturally incorporated
Allow arbitrary shaped elements
Easy incorporation of the complex geometries via implicit function or level set
methods
Possible to choose the computational grid independent of the pore geometry
Number of unknowns independent of the complex geometry
12. DG Discretization of the Stokes Equation
Find (uh , ph ) ∈ Vh × Qh such that
(
µ(A(uh , vh ) + J0 (uh , vh )) + B(vh , ph ) = F (vh ) ∀vh ∈ Vh
B(uh , qh ) = G(qh ) ∀qh ∈ Qh
X Z X Z X Z
A(uh , vh ) = uh : vh dx − uh · ne [ vh ]ds + vh · ne [ uh ]ds
E∈Th E E E
E∈E I E∈E I
h h
X Z X Z
− ( uh · nb )vh ds + ( vh · nb )uh ds
E E
E∈E B E∈E B
h h
σ σ
X Z X Z
J0 (uh , vh ) = [ uh ] · [ vh ]ds + uh · vds
|e| E |e| E
E∈E I E∈E B
h h
XZ X Z X Z
B(vh , ph ) = − ph · vh dx + ph [ vh · ne ]ds + ph vh · nb ds
E E E E
E∈E I E∈E B
h h
X Z X Z σ
X Z
F (vh ) = f · vh dx + µ ( vh · nb )gds + µ g · vh ds
E∈Th E E∈ED E E∈ED
|e| E
X Z
− p0 vh · nb ds
E∈EDP E
X Z
G(q) = − qh g · nb ds
E∈ED E
13. RW Particle Tracking Methods
Lagrangian based numerical approach for the solution of transport problem
Particle distribution
The trajectory of a tracer particle in an external pore velocity field u is given as
Xi (t + ∆t) = Xi (t) + S(t) + Z(t)
|{z} | {z }
Adv. displacement Diff. displacement
| {z } | {z }
√
S(t)=u(t)∆t Z(t)= 2Dm ∆tξ
Statistical moments
The centre of mass of the solute distribution is approximated by the first
moment as
Np
1 X
x(t) = xi (t)
Np i=1
The spread of mass (spatial variance) around x(t) is approximated by the
second moment as
Np
2 1 X 2 2
σ = var(x(t)) = xi (t) − x(t) .
Np i=1
The dispersion coefficient is determined from the spatial variance as
1 dσ 2 (t)
Deff (t) =
2 dt
14. Code Implementation and Validation
Code implementation was done using the DUNE software framework
Validated by performing standard test problems
Convergence tests for the DG Stokes solution
Mass balance checks
Analytical tests, Poiseuille (channel flow), driven cavity and flow around
cylinder (for flow model)
Computation of permeability for ordered sphere packing
Taylor-Aris dispersion (for transport model)
15. Validation of Flow Model
1 1
Simulation
Analytical
0.8 0.8
0.6
0.6
Y
Y
0.4
0.4
0.2
0.2
0 Simulation
0 0.2 0.4 0.6 0.8 1 Donea & Huerta (2003)
0
Velocity −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
Horizontal Velocity
Poiseuille Flow in a channel
Flow in a driven cavity
16. Validation of The Flow Model
Computation of permeability for ordered packing of spheres
The mean pore velocity is calculated as
¯
u= udx · |Ωp |−1
Ωp
|Ωp |
The porosity is given as φ = |Ω| , where |Ωp | is volume of the pore-space
The xx component of the permeability tensor
¯
uφ
κxx = −µ ,
p
where |Ω| denotes the size/volume of the domain.
17. Validation of Flow Model
Compared simulated permeability with analytical values given in (Sangani & Acrivos, Int. J.
Multiphase Flow,1982) for different ordered sphere packings (FCC, SC)
Simple Cubic (SC) Face Centered Cubic (FCC)
Type φ κanalytical κsimulated
FCC 0.259 8.68e-05 8.69e-05
SC 0.476 2.52e-03 2.51e-03
19. Validation of Flow Model
Grid Convergence: Permeability computed for FCC converging to the
anlytical value on a relatively coarser grid
1e-02
FCC (φ = 0.26)
1e-03
Permeability, κxx
1e-04
1e-05
1 1/2 1/4 1/8 1/16 1/32
h
20. Permeability for an artificial porous medium (sphere pack)
Grid Convergence
1e-01
φ = 0.768
1e-02
Permeability, κxx
1e-03
1e-04
1 1/2 1/4 1/8 1/16 1/32
h
Permeability of the artificial porous medium computed
on various grid levels
Artificial porous medium made of randomly packed
spheres
21. Porosity Vs Permeability
Varied the radius r of the spheres to change the porosity Φ
r 0.0318 0.0530 0.0742 0.0954 0.1060 0.1166
Φ 0.9886 0.9432 0.8437 0.6732 0.5534 0.4161
1e-01
h = 1/16
h = 1/32
1e-02
Permeability, κxx
1e-03
1e-04
1e-05
0.4 0.5 0.6 0.7 0.8 0.9 1,0
Porosity, φ
22. Validation of Transport Model
30
Dm = 0.35 t=0.0
um = 0.8326
y
Pe = 71.365
Taylor-Aris Dispersion 0
−10 0 10 20
x
30 40 50
30
t=43.5
C 1 x − um t
y
= erfc
C0 2 2(Deff · t)1/2 0
0 20 40
x
60 80 100
30
1 t=217.0
15000
y
10000
0.9 5000
1000 0
0 50 100 150 200 250 300 350 400
Analytical
0.8 x
30
0.7 t=433.7
y
0.6
0
0 100 200 300 400 500 600 700 800
C0
0.5
C
x
30
0.4 t=867.55
y
0.3
0
0 200 400 600 800 1000 1200 1400
0.2 x
30
0.1 t=1301.3
y
0
0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 0
0 500 1000 1500 2000
t∗ = t u/L x
Cumulative breakthrough curve (for various number of particles) for the 30
t=3036.4
Taylor-Aris dispersion compared to the analytical solution.
y
Gaussian
0
0 500 1000 1500 2000 2500 3000 3500 4000
x
23. Validation of Transport Model
Variation of the longitudinal dispersion coefficient Deff with the P´clet
e
Number for the Taylor-Aris dispersion
Deff Pe2
Analytical Solution: =1+
Dm 210
102
Analytical
Computed
Fit
101
eff
Dm
D
100
10−1
100 101 102
Peclet Number (Pe)
25. Pore-scale simulation of Transport in 2D
This work (RWPT) Based on Eulerian method (DGFE) by Fahlke (2008)
Sreejith Kuttanikkad (IWR, University of Heidelberg) Pore-Scale Simulation 15.10.2009 25 / 36
26. Pore-scale simulation of Transport in 2D
This work (RWPT) Based on Eulerian method (DGFE) by Fahlke (2008)
Sreejith Kuttanikkad (IWR, University of Heidelberg) Pore-Scale Simulation 15.10.2009 25 / 36
27. Pore-scale simulation of Transport in 2D
This work (RWPT) Based on Eulerian method (DGFE) by Fahlke (2008)
Sreejith Kuttanikkad (IWR, University of Heidelberg) Pore-Scale Simulation 15.10.2009 25 / 36
28. Pore-scale simulation of Transport in 2D
This work (RWPT) Based on Eulerian method (DGFE) by Fahlke (2008)
Sreejith Kuttanikkad (IWR, University of Heidelberg) Pore-Scale Simulation 15.10.2009 25 / 36
29. Pore-scale simulation of Transport in 2D
Concentration breakthrough curve
0.25 0.25
10,000 Present (RWPT)
25,000 Fahlke, 2008 (DGFEM)
50,000
100,000 0.2
0.2
Normalised Concentration
0.15
Concentration
0.15
0.1
0.1
0.05
0.05
0
0 −0.05
0 5 10 15 20 25 0 5 10 15 20 25
Time Time
Breakthrough curve plotted for different number of Breakthrough curve compared with the result of an
solute particles Eulerian scheme
32. Pore-scale simulation of Transport in 3D
Calculated concentration profiles along the porous medium
0.025
0.02
Normalised Concentration
0.015
0.01
0.005
0
0e+00 2e+04 4e+04 6e+04 8e+04 1e+05
Time
33. Pore-scale simulation of Transport in 3D
Dependence of dispersion coefficients on Peclet number
A standard way of describing longitudinal dispersion coefficient as a function of Pe
in the laminar flow condition is by using
DL 1
= + αPe + βPeδ + γPe2
Dm Fφ
mechanical dispersion boundary layer diffusion hold-up dispersion
molecular diffusion
34. Pore-scale simulation of Transport in 3D
Dependence of dispersion coefficient (DL ) on Peclet number
104
103
102
Dm
DL
101
Pfannkuch, 1963
Perkins and Johnston, 1963
Seymour and Callaghan, 1997
Maier et al. (2000),LBM+RWPT
100 Kandhai et al.(2002),NMR
Khrapitchev and Callaghan, 2003
St¨hr (2003), PLIF
o
Bijeljic et al.(2004), Pore-network+RWPT
Freund et al.(2005), LBM+RWPT
UDG+RWPT
10−1
10−2 10−1 100 101 102 103 104
P´clet Number (Pe)
e
35. Pore-scale simulation of Transport in 3D
Dependence of longitudinal dispersion coefficient on Peclet number in
the power law regime
103
Reference β δ
Pfannkuch - 1.2
102 (1963)
Gist et al. 0.46 - 3.9 0.93 - 1.2
(1990)
Dullien - 1.2
(1992)
Dm
101
DL
Coelho et 0.26 1.29
al. (1997)
Pfannkuch, 1963 Manz et al. - 1.12
Perkins and Johnston, 1963
Seymour and Callaghan, 1997 (1999)
100 Maier et al. (2000),LBM+RWPT
Kandhai et al.(2002),NMR Stoehr 0.77 1.18
St¨hr (2003), PLIF
o (2003)
Bijeljic et al.(2004), Pore-network+RWPT
Freund et al.(2005), LBM+RWPT Bijeljic et 0.45 1.19
UDG+RWPT
Fit al. (2004)
10−1 Freund et 0.303 1.21
101 102
P´clet Number (Pe)
e
al. (2005)
Simulated longitudinal dispersion coefficients in a random sphere packing compared to data This work 0.214 1.2033
reported in literature in the power law regime (3 < P e < 300). The line corresponds
DL
to the fit of the data to = βPeδ with β=0.214003 and δ=1.20331.
Dm
36. Pore-scale simulation of Transport in 3D
Least square fit of the simulated DL
DL 1 δ 2
= + αPe + βPe + γPe
Dm Fφ
103
102
Dm
101
DL
Pfannkuch, 1963
Perkins and Johnston, 1963
Seymour and Callaghan, 1997
100 Maier et al. (2000),LBM+RWPT
Kandhai et al.(2002),NMR
St¨hr (2003), PLIF
o
Bijeljic et al.(2004), Pore-network+RWPT
Freund et al.(2005), LBM+RWPT
UDG+RWPT
Fit
10−1
10−2 10−1 100 101 102 103
P´clet Number (Pe)
e
The values of the parameters obtained by fitting are τ = 1 =0.79, β= 0.214, δ=1.203 and γ=1.241e-5.
Fφ
37. Pore-scale simulation of Transport in 3D
Pe vs Transverse dispersion coefficients
102
101
Dm
DT
100
Maier et al. (2000), LBM+RWPT
Freund et al (2005), LBM+RWPT
Bijeljic et al.(2007), Pore-network
UDG+RWPT, DT y
UDG+RWPT, DT z
10−1
10−3 10−2 10−1 100 101 102 103
Peclet Number (Pe)
Simulated transverse dispersion coefficients are compared to data reported in literature
38. Summary
Summary
New numerical method has been used for pore-scale simulation
Method offers a direct discretization of the PDE’s on pore-scale
Retain benefits of the standard finite element methods, offers higher
flexibility in the mesh
Easy incorporation of complex geometries
Studied the dependence of permeability and dispersion coefficients on pore
structure
Outlook
UDG is Computationally demanding, a parallel implementation is necessary
A quantitative comparison with other well known approaches
Application to more realistic geometry
Extension to multiphase flows at pore-scale
