What’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications
Upscaling two-phase flowusing pore-network modeling
V. Joekar-Niasar S. M. Hassanizadeh
July 8, 2010
What’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications
Outline
1 What’s a Porous medium
2 Intro to PNMs
3 PNMing technique
4 Two-phase PNMs
5 ApplicationsCase1: idealized porous mediumCase2: micromodelCase3: glass-bead column
What’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications
A porous medium at pore scale: How does it look like?
Glass beads vs. volcanic Tuff
courtesy of D. Wildenschild
What’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications
A porous medium at pore scale: How does it look like?
Sandstonecourtesy of O. Vizika
What’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications
A porous medium at pore scale: How does it look like?
Fibers
www.bazylak.mie.utoronto.ca/research/
What’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications
Characterization of porous media structure
Microscopic: Topology, geometry
Macroscopic: Porosity, descriptive topology functions, geometrystatistics, spatial distribution
What’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications
Quasti-static vs. dynamic PNMs
Quasi-staticComputationally, very cheap.
No pressure field is solved.
Pore-scale geometry andmorphology are only important.
Used extensively, for two-phaseand three-phase flow; Pc
− Sw ,kr
− Sw , Sw− anw , reactive
transport, etc.
Can be used as a predictive tool.
Dynamic
Computationally, expensive.
Pressure field is solved.
Network and fluids properties areimportant.
Not been used as extensively asquasi-static ones; Pc
− Sw ,kr
− Sw , ganglia movement,dynamic pressure field
A long way to go!
What’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications
Investigation toolPore Network Models?
Yes,because:
Physical-based modelsusing pore-scaleinformation.
Application for staticand dynamic processes.
Computationally, not soexpensive (especially forquasi-static ones).
Capability to provideup-scaled information.
But:
Translation of topology andgeometry is inevitable, and notalways straight forward!
No detail information within apore (e.g. pressure field).
Local laws/rules are inevitable,and devil! be careful!!!!
What’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications
Definitions
Geometry
Pore body: large pores among the grains
Pore throat: long narrow pores connectingthe large pores
Shape factor : G = A/P2 :A: area, P:perimeter
Aspect ratio : K = Ri/rij .
Topology
Coordination number: number of porebodies connected to a given pore body
Fixed coordination number: isotropic,variable coordination number: anisotropic
What’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications
Pore-network modeling steps
Step 1. Topological and geometrical data acquisition
Step 2. Pore-network structure
Step 3. Defining governing equations and local laws
Step 4. Numerical experiments
What’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications
Step 1. Topological and geometrical data acquisition
Step 1.a Topological data acquisition
Detailed network acquired by analyzing images of a real porous medium,e.g. 3DMA-Rock software.
Hypothetical to idealized pore-networks. They can be isotropic (fixedcoordination number) or anisotropic (variable coordination number).Center of pore bodies can be located on lattice points (structured) orrandom coordinates (unstructured).
Step 1.b Geometrical data acquisition
Detailed network acquired by analyzing images of a real porous medium,e.g. 3DMA-Rock software.
Statistically equivalent or idealized porous medium. Generating based onstatistical parameters with/without spatial correlations.
What’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications
Step 2. Pore-network structure
Numbering pore bodies, sequentially.
Numbering pore throats, sequentially.
Defining connectivity of pore bodies to pore throats (Pb2Pth).
Defining connectivity of pore throats to pore bodies (Pth2Pb).
What’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications
Step 2. Pore-network structure
Void-Space Network (Pore-Network)
What’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications
Step 2. Pore-network structure
Pore Bodies Numbering
What’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications
Step 2. Pore-network structure
Pore Throats Numbering
What’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications
Step 2. Pore-network structure
Connection Matrices
What’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications
Step 3. Governing equations and local laws
One-phase or multiphase flow?
Drainage or Imbibition process, ...?
Dynamic process or static process?
Probable physical laws: Young-Laplace Equation, Hagen-PoiseuilleEquation, Stokes Equation, Conservation, etc.
Probable local assumptions: phase trapping assumptions,hydraulic conductivites of pores, etc.
What’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications
Step 4. Numerical experiments
Quasi-Static or dynamic
Quasi-static simulations: capillary forces are dominant, no informationabout time scale of phenomena.Dynamic simulations: solving pressure field, change of saturation withtime.
What’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications
Drainage vs. imbibition
Let’s watch some movies:courtesy of Fabiano G. Wolf (http://www.lmpt.ufsc.br/ fgwolf/)
Drainage
First, larger pore will be filled by the nonwetting phase (blue).
Imbibition
First, smaller pore will be filled by the wetting phase (red).
Simulations done by Lattice-Gas Automata Models
What’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications
Example
Quasi-static simulation, drainage process, circular cross sectionEntry Pc , Pc
i = 2σ
rijcos θ (for circular cross section) .
Lower entry pressure: invaded earlier during drainage!
What’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications
Quasi-static PNMs
Network: Initial Condition
What’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications
Quasi-static PNMs
What’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications
Quasi-static PNMs
What’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications
Quasi-static example
What’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications
Quasi-static example
What’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications
Quasi-static example
What’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications
Quasi-static example
What’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications
Quasi-static example
What’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications
Quasi-static example
Pc = 2.0, with trapping
What’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications
Quasi-static example
movie
What’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications
Computational considerations
Adaptive pressure update. Gains:a) Calculating the exact Pc -Sw curve. figure
b) Saving computational time especially at the initial and residualsaturations of a Pc -Sw curve.
Using flags to mark those pores, which are in contact with theinvading phase, but not filled yet.
For a strongly anisotropic medium: save Pb2Pth matrix differentlyto save memory. The best known suggest approach is compactrow storage (CRS).
Trapping is the most time-consuming part. Robust searchalgorithm to save computational time such as depth-first search(Cormen, Leiserson, Rivest, 2000).
What’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications
Computational considerations
Pc-Sw using two pressure update approaches go back
What’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications
Computational considerations
Compact Row Storage (CRS) for an anisotropic medium or dualporosity medium! Example: dual porosity medium; macro poresconnected to micro pores.
courtesy of M. Prodanovic
What’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications
Computational considerations
Indices of columns and rows represents pore body numbers. Numbersin the matrix show the pore throat numbers.
Yellow colors are indices!
What’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications
Computational considerations
We do not need half of the matrix, since it is symmetric.
What’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications
Computational considerations
Compact Row Storage (CRS), matrix at top: first row shows thecolumn index. second row shows the pore throat number. vector atbottom: it shows how many non-zeros exist in upper diagonal in eachrow.
What’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications
Computational considerations
Compact Row Storage (CRS), if pore throats are not included incalculations!
What’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications
Geometrical parameters/issues
Geometry
Cross section geometry
Aspect ratio
Shape factor
Phenomena
Piston-like movement
Snap-off
Capillary diffusion
Cooperative filling, ...
What’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications
Geometrical evolution???
To have corners, or not to have corners: that is the question!
To mimic the grain geometry,sometimes this can be also the question!!
Wetting phase in the corners will reduce wetting phase trapping duringdrainage significantly, however may contribute to residual saturation.
Wetting phase in the corners can increase snap-off during imbibition.Snap-off happens if wetting film/ corner flow cuts connectivity of thenonwetting fluid.
Including corner flow in dynamic models will increase numericalcomplexity enormously.
What’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications
Geometrical parameters: cross section geometry
(Joekar-Niasar et al,2009)
Entry capillary pressure can be calculated for different cross sectionsusing MS-P (Mayer, Stowe and Princen, J. Colloid Interface Sci, 1968,1969; Ma et al., Colloids Surf, 1996).More complex geometry, more complicated system of equations.
What’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications
Geometrical parameters: shape factor
κ = A/P2
κ: shape factor, A:cross section area, P :cross section perimeter
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Pore-network models for glass beads (Joekar-Niasar et al,2009)
What’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications
Geometrical parameters: aspect ratio
Radius of pore body size to pore throat size
Significant effect during imbibition: larger aspect ratio, moresnap-off!
More snap-off, more nonwetting phase trapping (Lenormand andZarcone, 1983, 1984; Wardlaw and Yu, 1988; Ioannidis et al.1991) movie.
What’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications
Geometrical parameters: aspect ratio
Overlapping of pore bodies and pore throats size distribution.
What’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications
What we want:
We are interested in investigating extended theories of two-phase flow.
What’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications
Objectives
1. Insights into pore-scale dynamics of two-phase flow2. Studying validity of extended two-phase flow equations
What’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications
Review of extended theories of two-phase flow
n∂Sα
∂t+ ∇ · qα = 0
qα = −Kα ·(
∇pα − ραg − Ψαa∇anw −ΨαS∇Sα
)
(1)
∂anw
∂t+ ∇ · (anwwnw) = Enw (anw ,Sw ) (2)
wnw = −Knw(
∇anwγnw − ΨwS∇Sw)
(3)
pn − pw = pc − τ∂Sw
∂t(4)
pc = f (Sw , anw ) (5)
What’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications
Extended theories of two-phase flow in porous mediaCapillary pressure
pc = f (Sw , awn) = (pn − pw ) |equil
Difference in fluid’s pressure are equal to static capillary pressure butonly at equilibrium.
Under non-equilibrium conditions, assuming linearity:
pn − pw = pc − τ∂Sw
∂t, pc = f (Sw , awn)
What’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications
Case1: idealized porous medium
Joekar-Niasar, V., Hassanizadeh, S.M., Leijnse A., 2008, Insights intothe relationships among capillary pressure, saturation, interfacial areaand relative permeability using pore-network modeling, Transport inPorous Media, 74:201219, doi 10.1007/s11242-007-9191-7
What’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications
Idealized pore-networkSpecifications
Topology: structured, isotropic, lattice network with coordinationnumber of 6Geometry: sphere pore bodies, circular cross-section pore throatsGoverning equation: Pc = 2σ
rcos θ
Joekar-Niasar et al. 2008,
TiPM
What’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications
Filling process
Drainage: Starting from the largest pore throats. If a pore throat isinvaded, the connected pore body will be fully filled.Imbibition: Starting from the smallest pore throats. If a pore throat isinvaded, the connected pore body will be partially filled.
What’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications
Pc-Sw and a
nw -Sw curves: imbibition
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What’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications
Pc-Sw and a
nw -Sw curves: drainage
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What’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications
Pc-Sw -anw surfaces
Drainage vs. imbibition
a
z=-0.552+7.404*x+6.15e-5*y-6.712*x*x-2.693e-4*x*y-2.089e-9*y*y
b
What’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications
Relative permeabilities
Flux passing through the pore throat ij based on Hagen-Poiseuille’s
law qij =πr4
ij
8µlij(Pi − Pj)
What’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications
Relative permeabilities
Flux passing through the pore throat ij based on Hagen-Poiseuille’s
law qij =πr4
ij
8µlij(Pi − Pj)
Based on volumetric conservation law for all pore bodies:∑Ni
j=1 qij = 0, Ni number of pore throats connected to pore body i.
What’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications
Relative permeabilities
Flux passing through the pore throat ij based on Hagen-Poiseuille’s
law qij =πr4
ij
8µlij(Pi − Pj)
Based on volumetric conservation law for all pore bodies:∑Ni
j=1 qij = 0, Ni number of pore throats connected to pore body i.Linear system of equations will be resulted AP=B.A is a symmetric banded sparse matrix made of conductivities of porethroats.No dynamic effect included in relperms.
What’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications
kr -Sw -anw surfaces
Drainage and imbibition
c
What’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications
Conclusions
We can have unique surfaces for Pc-Sw -anw as well as kr -Sw -anw
for drainage and imbibition using our idealized pore-networkmodel.
Increasing aspect ratio decreases uniqueness of the surface.
Further developments should be done; such as including angularcross sections to allow different mechanisms during imbibition andimplementing for an unstructured anisotropic cross sections.
Continuing for a real porous medium!
What’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications
Case2: micromodel
Joekar-Niasar, V., Hassanizadeh, S. M., Pyrak-Nolte, L. J., Berentsen, C.,
Simulating drainage and imbibition experiments in a high-porosity
micro-model using an unstructured pore-network model, 2009, Water
Resources Research, 45: W02430, doi:10.1029/2007WR006641.
What’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications
Specifications
Topology: unstructured, anisotropic.Geometry: square cross-section
Joekar-Niasar et al.2008,WRR
What’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications
Challenge
How can we use the concept of pore-network modelling for ahigh-porosity medium? Pore-body and pore throat concept does nothold !!!
What’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications
Challenge
How can we use the concept of pore-network modelling for ahigh-porosity medium? Pore-body and pore throat concept does nothold !!! Skeletonization using medial axis: defining pixels and bottlenecks
What’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications
Medial axis transform
What’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications
Governing equation
Pe = σnw
−(a + b)cosθ +√
(a + b)2cos2θ + 4ab(π
4 − θ −√
2cos(π
4 + θ)cosθ)
4(π
4 − θ −√
2cos(π
4 + θ)cosθ)
What’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications
Simulations vs. experiments
What’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications
Simulations vs. experiments
What’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications
Conclusions
Given geometry, pore-network model can be used to simulate notonly Pc-Sw but also anw -Sw curves.
Pc -Sw -aw surfaces resulted from simulation and experimentsshowed good agreement.
This predictive tool can be useful for assessing the micromodelbefore manufacturing.
It is very hard to simulate imbibition process in high porositymedia.
What’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications
Case3: glass-bead column
Joekar-Niasar, V., Prodanovic, M., Wildenschild, D., Hassanizadeh, M.,
Translation of a Granular Porous Medium into a Pore-Network Model:
Application for Capillary Pressure-Saturation and Interfacial Area-Saturation
Relationships
What’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications
Experiment
Air-water system in glass beads. Image obtained through CT scanner.
What’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications
Specifications
Topology: unstructured, anisotropicGeometry: mixed hyperbolic polygonal cross sections for pore throatsand prolate spheroid for pore bodies.Image analysis and data acquisition using 3DMA-Rock : skeleton,shape factor, pore unit volume, mean inscribed circle at pore throatsand pore bodies as well as mean radius of pore bodies.
What’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications
Pore throats determination
Determination of pore throats cross section based on shape factordistribution.
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What’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications
Numerical simulation
Determination of cross sections types.
Determining pore body geometries based on two min. and meaninscribed radii.
Solving a nonlinear system to determine geometry of a crosssection based on area and shape factor.
Solving a nonlinear system to determine entry capillary pressure.
Determining fluid occupancy during drainage and imbibition basedon imposed capillary pressure.
What’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications
Results:grain size distribution
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What’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications
Results: Pc -Sw curves
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What’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications
Results: anw -Sw curves
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What’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications
Simulations vs. Experiments
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Capilla
ry P
ressure
(kPa)
Saturation (Sw
drainage_final
Value
High : 0.39
Low : 0.02
)
Specific Interfacial Area (1/mm)
a)
0.3
0.2
60.2
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.1
0.2
4
0.3
2
0.22
0.1
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2
0.1
6
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8 0.0
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6
0.3
4
0.1
0.3
4
0.2
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4
0.2
6
0.28
0.16
0.2
0.3
2
0.28
0 0.2 0.4 0.6 0.8 1
0.2
0.4
0.6
0.8
1
Capilla
ry P
ressure
(kPa)
Saturation (Sw
imbibition_final
Value
High : 0.36
Low : 0.02
)
Specific Interfacial Area (1/mm)
b)
Joekar-Niasar et al. 2010, WRR
What’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications
Uniqueness of the surfaces
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Capilla
ry P
ressure
(kPa)
Saturation (Sw
normalized difference
abs(D-I)/D
Value
High : 1.38
Low : 0.00
)
Normalized Error
c)
Joekar-Niasar et al. 2010, WRR
What’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications
Conclusions
Given geometry, pore-network model can be used to simulate notonly the Pc -Sw but also anw -Sw curves.
At least for granular porous medium, including shape factor togenerate hyperbolic polygonal cross sections works well.
There is enough consistency between grain size distribution andcross sections of pore throats.