using pore-network modeling V. Joekar-Niasar S. M ... · PDF fileWhat’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications ... using pore-network modeling

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


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


A porous medium at pore scale: How does it look like?

Glass beads vs. volcanic Tuff

courtesy of D. Wildenschild



Sandstonecourtesy of O. Vizika



Fibers

www.bazylak.mie.utoronto.ca/research/


Characterization of porous media structure

Microscopic: Topology, geometry

Macroscopic: Porosity, descriptive topology functions, geometrystatistics, spatial distribution


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!


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!!!!


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


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


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.



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).



Void-Space Network (Pore-Network)



Pore Bodies Numbering



Pore Throats Numbering



Connection Matrices


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.


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.


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


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!


Quasi-static PNMs

Network: Initial Condition


Quasi-static PNMs


Quasi-static PNMs


Quasi-static example











Pc = 2.0, with trapping



movie


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).



Pc-Sw using two pressure update approaches go back



Compact Row Storage (CRS) for an anisotropic medium or dualporosity medium! Example: dual porosity medium; macro poresconnected to micro pores.

courtesy of M. Prodanovic



Indices of columns and rows represents pore body numbers. Numbersin the matrix show the pore throat numbers.

Yellow colors are indices!



We do not need half of the matrix, since it is symmetric.



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.



Compact Row Storage (CRS), if pore throats are not included incalculations!


Geometrical parameters/issues

Geometry

Cross section geometry

Aspect ratio

Shape factor

Phenomena

Piston-like movement

Snap-off

Capillary diffusion

Cooperative filling, ...


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.


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.


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)


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.


Geometrical parameters: aspect ratio

Overlapping of pore bodies and pore throats size distribution.


What we want:

We are interested in investigating extended theories of two-phase flow.


Objectives

1. Insights into pore-scale dynamics of two-phase flow2. Studying validity of extended two-phase flow equations


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)


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)


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


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


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.


Pc-Sw and a

nw -Sw curves: imbibition

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Pc-Sw and a

nw -Sw curves: drainage

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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


Relative permeabilities

Flux passing through the pore throat ij based on Hagen-Poiseuille’s

law qij =πr4

ij

8µlij(Pi − Pj)




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.




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.


kr -Sw -anw surfaces

Drainage and imbibition

c


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!


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.


Specifications

Topology: unstructured, anisotropic.Geometry: square cross-section

Joekar-Niasar et al.2008,WRR


Challenge

How can we use the concept of pore-network modelling for ahigh-porosity medium? Pore-body and pore throat concept does nothold !!!


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


Medial axis transform


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)


Simulations vs. experiments


Simulations vs. experiments


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.


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


Experiment

Air-water system in glass beads. Image obtained through CT scanner.


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.


Pore throats determination

Determination of pore throats cross section based on shape factordistribution.

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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.


Results:grain size distribution

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Results: Pc -Sw curves

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Results: anw -Sw curves

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Simulations vs. Experiments

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0.1

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0 0.2 0.4 0.6 0.8 1

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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

8 0.2 0

.1

0.2

4

0.3

2

0.22

0.1

40.1

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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


Uniqueness of the surfaces

0.1

1

0.2

0.3 0.4

0.5

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0 0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

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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


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.

Documents

using pore-network modeling V. Joekar-Niasar S. M ... · PDF fileWhat’s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications ... using pore-network modeling