using pore-network modeling V. Joekar-Niasar S. M ... · PDF fileWhat’s a Porous medium...

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)

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0.2

60.2

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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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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.