modelling two-phase flow and transport in industrial porous …modelling two-phase flow and...

Modelling two-phase flow and transport in industrial porous media

S Majid HassanizadehS. Majid HassanizadehDepartment of Earth Sciences

Ut ht U i itUtrecht UniversityThe Netherlands

C ll bCollaborators: Simona Bottero; Utrecht University, The NetherlandsJenny Niessner; Utrecht University The NetherlandsJenny Niessner; Utrecht University, The NetherlandsVahid Joekar-Niasar; Utrecht University, The NetherlandsCas Berentsen; Delft University of Tech., The NetherlandsRainer Helmig; University of Stuttgart, GermanyMichael A. Celia; Princeton University, USAH l K D hl U i it f B NHelge K. Dahle; University of Bergen, Norway

Outline of presentationp

Introduction: Darcy’s law for two-phase flow in porous mediaCapillary pressure in porous mediaShortcomings of standard capillary pressure theory

Experimental evidencesExperimental evidences New theory of capillaritySignificance of new theoryNew theories of two-phase flow ConclusionsConclusions

INTRODUCTIONINTRODUCTION

Theories of (multiphase) flow in porous media have been too empirical in nature.

They have been developed through “extensions” f i l l ti hi i d h f hiof simple relationships in ad-hoc fashion.

They are not necessarily valid for complexThey are not necessarily valid for complex systems we are considering today.

Take Darcy’s law.

DARCY TESTED HIS FORMULA ON A COLUMN OF SAND IN A LABORATORY AND DETERMINED THESAND IN A LABORATORY AND DETERMINED THE VALUE OF CONDUCTIVITY (K):

hΔLhKAQ Δ=

L

Simple System Complex System

METAMORPHOSIS OF DARCY’S LAW

It is now used in the f f th

Darcy’s law was d f same form for three-

dimensional unsteady proposed for one-dimensional steady-

flow of two or more compressible fluids,

state flow of almost pure incompressible co p ess b e u ds,

with lots of dissolved matter in anisotropic

pure incompressible water in saturated homogeneous matter, in anisotropic

heterogeneous homogeneous isotropic rigid sandy

deformable porous media under non-

soil under constant temperature

isothermal conditions p

METAMORPHOSIS OFDARCY’S LAW

q = Q/AWrite the formula in terms of velocity

hΔhK Δq KL

=qL


FOR ONE-DIMENSIONAL FLOW, Change tod hd x

−hL

Δd xL

dhhΔdhK hq K Δ=q Kd

= −q KL

qdxLdx


For unsteady state conditions: ( , )h h x t=

h∂dhKhK ∂dhq K

d= −q K= −

∂q

dxq

x∂x∂


FOR THREE-DIMENSIONAL FLOW

h∂h∂hK ∂hq K ∂=q K= −∂iq K= −∂x∂ ix∂ i


FOR ANISOTROPIC MEDIA, REPLACE K BY Kij

h∂h∂hK ∂hq K ∂iq K= −

∂i ijq K= −

∂ii

qx∂

jjx∂ij


FOR HETEROGENEOUS MEDIA

KijKijjis assumed to be a function

of position


FOR FLOW IN UNSATURATED ZONE:

h∂hKq ∂−= iji xKq

∂−=

jx∂ j


FOR FLOW IN UNSATURATED ZONE:

K is assumed to be Kij a function of t t t

ijθwater content θ


FOR TWO-PHASE FLOW (say WATER AND OIL):



∂ ww w hK ∂

∂w wi ij

hq = - Kx∂ jx


FOR TWO-PHASE FLOW (SAY WATER AND OIL):

∂ ww w hK ∂

∂w wi ij

hq = - Kx∂ jx

∂ oo o hK

∂o oi ij

j

q = - Kx∂ jx



kw wrkw w

ij ijK = K

ko oK = Krkij ijK K

Capillary Pressure-saturation relationshipp y p

( )c wp S ρ⎛ ⎞ ( )( ) 1o w wo o

p Sh h z hg

ρρ ρ

⎛ ⎞− = + − −⎜ ⎟

⎝ ⎠gρ ρ⎝ ⎠

The form of Darcy’s Law remains unchanged:∂

y g αα α ∂

∂i ijhq = - K

∂i ijjx

W ddi b ll d hi tl t i l f lWe are adding bells and whistles to a simple formula to make it valid for much more complicated situations!

“Extended” Darcy’s Lawy

hK Δq Kx

=Δ

The “extended” Darcy’s Law remains unchanged:α

α α ∂∂i ijhq = - K

∂i ijj

qx

We have added bells and whistles to a simple formula to make it valid for much more complicated situations!

Comparison of theories of Aristotle and Galileo for free fall of objects with hypothetical measurements (after French, 1990)

GALILEO

objects with hypothetical measurements (after French, 1990)

tyl V

eloc

it

ARISTOTLE

erm

inal

DATA POINTS

Te

Time

PhD RESEARCH PROJECTHypothetical plot of terminal velocity

R d h d

Hypothetical plot of terminal velocity as a function of mass density for various shapes

Rod shaped

Roundty Round

PyramidDiskoc

itl V

eloc

it

Disk

min

al V

elo

erm

inal

Term T

e

Mass Density

How many relative permeability-saturation curves have been measured during the past few decades!?have been measured during the past few decades!?

How many capillary pressure-saturation curves have been measured during the past few decades!?

Imbibitionscanning

Drainage scanning

been measured during the past few decades!?

scanning curves

scanning curves

Relative permeability is supposed to be less than 1.

Water and oil relative permeabilities in 43 sandstone reservoirs plotted as a function of water saturation

Problems with current theories of multiphase flowProblems with current theories of multiphase flow

P di t d it d t b th lPressure gradient and gravity are assumed to be the only driving forces. Effects of all other forces are lumped into coefficients.

In particular, interfacial forces and interfacial energies are not explicitly taken into account. This is unacceptable.not explicitly taken into account. This is unacceptable.

There is discrepancy between the theory and measurement.

Capillary pressure curves are measured under equilibrium conditions and then used under non-equilibrium (flow)

diticonditions.

Capillary pressure is believed to go to infinity. Nonsense!

Forces acting within interfaces control fluid di t ib ti ithi th didistribution within the porous medium.

2c σ2c

M

pRσ=

Mass transfer among phases occurs through interfaces.

Where are interfaces in porous media theories!?

Problems with current theories of multiphase flowProblems with current theories of multiphase flow

P di t d it d t b th lPressure gradient and gravity are assumed to be the only driving forces. Effects of all other forces are lumped into coefficients.

In particular, interfacial forces and interfacial energies are not explicitly taken into account. This is unacceptable.not explicitly taken into account. This is unacceptable.

There is discrepancy between the theory and measurement.

Capillary pressure curves are measured under equilibrium conditions and then used under non-equilibrium (flow)

diticonditions.

Capillary pressure is believed to go to infinity. Nonsense!

Measurement of Capillary Pressure-Saturation Curvesp y

Capillary pressure-saturation curves for a given medium are obtained by measurement of saturation inside and fluidobtained by measurement of saturation inside and fluid pressures outside the porous medium under quasi-equilibrium conditions. Time to equilibrium can be 24 hours to 7 days.Hydrophobic and hydrophilic membranes affect the resultsHydrophobic and hydrophilic membranes affect the results.

Air chamberapcp =

Porous Medium

*

pa

pp p−

Medium

P Pl tWater reservoir

***

****

Drying curvepScanning drying curve

*

Porous Plate(impermeable to air)*

*

Wetting curve

*Scanning drying curve

S**

Industrial Porous MediaIndustrial Porous Media

M i d t i l i l di flMany industrial processes involve porous media flows.

Some examples are processes in fuel cells, paper-pulp drying, food production and safety, filtration, concrete, ceramics, moisture absorbents, textiles, paint drying, polymer composites and detergent tabletspolymer composites, and detergent tablets

There is clear difference between industrial porous media and soils:and soils:- fast flow- significant mass transfer across fluid-fluid interfaces- significant deformations- wettability changes- high porsosityhigh porsosity

Dewatering of paper pulp

press nips The paper pulp layer is transported through the press aperture on a felt. The paper-felt sandwich is thus compressed, so that the water is pressed out of the paper, is absorbed by the felt, and then flows laterally away from the nips.

Pictures courtesy of Oleg Iliev of ITWM, Kaiserslautern

Simulation of dewatering of paper pulpg p p p p

Capillary pressure curves are measured for feltCapillary pressure curves are measured for felt under quasi-static conditions. It takes around 24 hours to obtain them.24 hours to obtain them.

They are then used to model a process thatThey are then used to model a process that takes only seconds!

Air and oil flow in oil filters

Pictures courtesy of Oleg Iliev of ITWM, Kaiserslautern

Proton Exchange Membrane (PEM) Fuel Cells

Pc-Sw measurement of hydrophobic y pgas diffusion layer of a PEM fuel cell

Unsaturated flow in thin fibrous materials

Unsaturated flow in thin fibers

Unsaturated flow in pampers

Measurement of capillaryMeasurement of capillary pressure curves takes about 12 hours.

Unsaturated flow in pampers

Capillary pressure-saturation experiments (PCE and Water)(PCE and Water)

Experiments carried out at GeoDelft; Soil sample: 3 cm high and 6cm in dia. Fluids: PCE and water;Fluids: PCE and water; Primary drainage, Main drainage, Main imbibition; Quasi-static and dynamic experiments

PCE

Air chamber

PPT NW back

3 cmsand

water PCE reservoir

PPT NW back

PPT W frontwater

Water

Soil sample

Water

10 0

12.5

Pc [(in sand]

Pc [in fluids]Pc-S curves based on

d

7.5

10.0 [ ] pressures measured inside the soil sample.

5.0

c [k

Pa] Capillary pressure

DOES NOT go to

2.5

Pc

ginfinity!

A th th0.0

Are these the curves we should use in our i l t ?

-2.50.0 20.0 40.0 60.0 80.0 100.0

Water saturation

simulators?

Water saturation

Measurement of “Dynamic Pressure Difference” CurvesDynamic Pressure Difference Curves

n wP Pn wP PP

− =Δ

10

]

dynPΔ8

ure

[kPa

]

4

6

ry P

ress

u

cP2C

apilla

D ynam ic Prim ary drainage-1

0.2 0.4 0.6 0.8 10W ater Saturation [-]

S tatic prim ary drainage

Hassanizadeh et al., 2004

Two-phase flow dynamic experiments (PCE and Water)

10.0prim drain PN~16kPa

Dyn.P.D.

)

8.0

pprim drain PN~20kPaprim drain PN~25kPamain imb PW~0kPamain imb PW~5kPamain imb PW~8kPa

S.P.D.S M D

Dyn.M.D.

w(k

Pa)

6.0

main imb PW 8kPamain imb PW~10kPamain imb PW~0kPa lastmain drain PN~16kPamain drain PN~20kPamain drain PN~25kPaS.M.D.

S.M.I.Pn-P

w

4.0

main drain PN~25kPamain drain PN~30kPaStatic Pc inside

Dyn.M.I.2.0

-2 0

0

Saturation, S0 0.20 0.40 0.60 0.80 1.00

2.0

Experiments on the uniqueness of curves(drainage experiments by Topp Klute and Peters 1967)

cp S−(drainage experiments by Topp, Klute, and Peters , 1967)

These are all drainage curves

There is no unique pc-S curve.

Main Scanning Drainage Curves

Dynamic Drainage Curves

c P i D i C

Main Drainage CurvesSecundary Scanning Drainage Curves

ssur

e pc Primary Drainage Curve

ary

pres

Cap

illa

Primary Imbibition CurveM i I bibiti CMain Imbibition Curve

Main Scanning Imbibition Curves

Secundary Scanning Imbibition Curves

Saturation, S

Dynamic Imbibition Curves

There is no unique pc-S curve.

Main Scanning Drainage Curves

Dynamic Drainage Curves

Main Drainage Curves

g g

Secundary Scanning Drainage Curves

Primary Drainage Curve

re p

cpr

essu

r

Primary Imbibition Curve

apill

ary

Main Imbibition Curve

Main Scanning Imbibition Curves

Secundary Scanning Imbibition Curves

Ca

Secundary Scanning Imbibition CurvesDynamic Imbibition Curves

Saturation, S

A New Theory of yCapillarity in Porous Media

1 Difference in fluids’ pressures are equal1. Difference in fluids pressures are equal to capillary pressure but only at equilibrium.

2. Capillary pressure is not only a function p y p yof saturation but also of interfacial area

Forces acting within interfaces control fluid di t ib ti ithi th didistribution within the porous medium.

2σ2cp σ=M

pR

=MR

n w cp p p=p p p− =

Microscale Dynamic EffectsConsider two-phase flow in a single tube:

A⎛ ⎞

2

2 An w

M

qp p BR r

γ μ⎛ ⎞− = + ⎜ ⎟⎝ ⎠

Nonwetting phaseWetting phase

NONEQUILIBRIUM CAPILLARY EFFECTLi hLinear theory

Equilibrium capillary theory:Equilibrium capillary theory:

( )n w cp p p S− =Under non-equilibrium conditions:

( ), /n w cp p p f S S t− = + ∂ ∂

Linear approx.:S∂( )n w c Sp p p St

τ ∂− = −∂

where τ is a damping coefficient.

Dynamic capillary pressure curvesDynamic capillary pressure curves

( ) S∂cp = ( ) ( )a w c Sp p p St

τ ∂− − = −∂

Thi ti t th b h in wp p−Dynamic drying curve c Sp τ ΔΔ = −

This equation captures the behavior observed in the experiments

Drying curve cpΔp

tτΔ

Δ

Wetting curve

Dynamic wetting curve

SCurves measured in the labCurves measured in the lab can be used to evaluate τ


dS/dt vs. S - all curves4.0E-02

3.0E-02

1.0E-02

2.0E-02

[s-1

]

0.0E+00

(-)d

s/d

t

-2.0E-02

-1.0E-02

-3.0E-020.00 0.20 0.40 0.60 0.80 1.00

saturation


Determination of τPrimary drainage; S=70%Primary drainage; S=70%

2 .5

y = 6 1 .4 x + 0 .7y = 5 4 .0 x + 1 .22

kPa]

y = 5 0 .0 x + 0 .41

1 .5

) - P

c(st

at)

[k

0 .5

1

Pc(

dyn)

P a ir = 1 6 k P a

00 .0 0 0 0 .0 0 5 0 .0 1 0 0 .0 1 5 0 .0 2 0 0 .0 2 5 0 .0 3 0

P a ir = 2 0 k P a

P a ir = 2 5 k P a

0 .0 0 0 0 .0 0 5 0 .0 1 0 0 .0 1 5 0 .0 2 0 0 .0 2 5 0 .0 3 0(- )d S /d t [s -1 ]

TWO PHASE FLOW EXPERIMENTSUtrecht University

TWO- PHASE FLOW EXPERIMENTSDynamic coefficient, τ

5x 107

4

5x 10

3

4

a.s]

S [ ]2ta

u [P

a Sw [-] τ [Pa.s]

0.80 9.83 * 104

1

0.80 9.83 100.70 1.06* 105

0.60 1.33* 105

0 50 2 61 * 106

0.4 0.5 0.6 0.7 0.80Sw [-]

0.50 2.61 106

0.40 3.62 * 107

Estimation of τ from literature data((Hassanizadeh, Celia, and Dahle, Vadose Zone Journal, 2002)

Soil Type of Value of τReference

Soil Type

Type of experiment

Value of τ(kg/m.s)

For Air-Water Systems:For Air-Water Systems:Topp et al. (1967)Smiles et al (1971)

SandSand

DrainageDrainage

2*107

5*107Smiles et al. (1971)Stauffer (1978) Nützmann et al (1994)

SandSandSand

DrainageDrainageDrainage

5 103*104

2*106Nützmann et al. (1994)Hollenbeck & Jensen(1998)

SandSand

DrainageDrainage

2*106

2 to 12*106

For Oil Water S stems: S d tFor Oil-Water Systems:Kalaydjian (1992b)

Sandst./limest. Imbibition 2*106

Significance of new theory for unsaturated flow:Development of vertical wetting fingers in dry soilDevelopment of vertical wetting fingers in dry soil

Experiments by Rezanejad et al., 2002

Consequences of new theory for unsaturated flow:Development of vertical wetting fingers in dry soilDevelopment of vertical wetting fingers in dry soil

David A. DiCarlo, “Experimental measurements of saturation overshoot on infiltration” VOL. 40, 2004

Classical unsaturated flow equationqFlow equation:

⎛ ⎞⎛ ⎞( )k wrS k S pn g

t z zρ

μ⎛ ⎞⎛ ⎞∂ ∂ ∂= +⎜ ⎟⎜ ⎟∂ ∂ ∂⎝ ⎠⎝ ⎠

Capillary pressure-saturation relationship:

( )

⎝ ⎠

( )a w cp p p S− =

Richards equation:S S∂ ∂ ⎛ ∂ ⎞⎛ ⎞S Sn D gt z z

ρ∂ ∂ ⎛ ∂ ⎞⎛ ⎞= +⎜ ⎟⎜ ⎟∂ ∂ ∂⎝ ⎠⎝ ⎠

Simulation of vertical wetting fingers in dry soil

Stability analysis by Dautov et al. (2002) has proven that:

Richards equation is unconditionally stable.It does not produce any fingers, but a monotoniclly

M difi d Ri h d i ( i h d i

p y g , ydecreasing saturation profile toward the wetting front.

Modified Richards equation (with dynamic effect) is conditionally unstable.It is able to produce gravity wetting fingers.

Dynamic capillary pressure theoryy p y p yUnsaturated flow equations:

Flow equation:

⎛ ⎞⎛ ⎞( )k wrS k S pn g

tρ

μ⎛ ⎞⎛ ⎞∂ ∂ ∂= +⎜ ⎟⎜ ⎟∂ ∂ ∂⎝ ⎠⎝ ⎠

Capillary pressure-saturation relationship:

t z zμ ⎜ ⎟∂ ∂ ∂⎝ ⎠⎝ ⎠

( )a w c SS ∂( )a w cstat

Sp p p St

τ ∂− = −∂

Development of vertical wetting fingers in dry soil;

Simulations based on new capillarity theory

Dautov et al. (2002)

A New Theory of yCapillarity in Porous Media

1 Difference in fluids’ pressures are equal1. Difference in fluids pressures are equal to static capillary pressure but only at equilibrium.

2. Capillary pressure is not only a function p y p yof saturation but also of interfacial area

Imbibition Drainage

P c/ρ

g

P c/ρ

g

e he

ad,

e he

ad,

pres

sure

pres

sure

pilla

ry p

pilla

ry p

Cap

Cap

Water content Water contentCAPILLARY PRESSURE-SATURATION DATA POINTS MEASURED IN LABORATORY

Imbibition Drainage

P c/ρ

g

P c/ρ

g

e he

ad,

e he

ad,

pres

sure

pres

sure

pilla

ry p

pilla

ry p

Cap

Cap

Water content Water content

CAPILLARY PRESSURE-SATURATION DATA POINTS MEASURED IN LABORATORY

New capillarity theory with no (need for) hysteresis

( ),c c wnp p S a= ( ),p p

where:

is capillary pressurecp is capillary pressure

S is wetting phase saturation

p

awn is the specific interfacial area between wetting & nonwetting phases

pc-S-awn Relationships55.0

8000

9000

53.0

54.0

(kP

a)

5000

6000

7000

8000

1/m

)

ress

ure

ea

51 0

52.0

Pre

ssur

e (

2000

3000

4000

5000

IAV

wn

(1

pilla

ry p

r

rfac

ial a

re

50.0

51.0

0.4 0.5 0.6 0.7 0.8 0.9 1

ImbibitionDrainage

Wetting Phase Saturation

1000

2000

0.4 0.5 0.6 0.7 0.8 0.9 1

IAVndWNIAVndWN

Wetting Phase Saturation

Cap

Inte

r

S t tiS t tig g

8000

9000

SaturationSaturation

5000

6000

7000

n (1

/m)

Figures courtesy of ar

ea

1000

2000

3000

4000

ImbibitionDrainage

IAV

wncourtesy of

Laura Pyrak-Nolte of Purdue te

rfac

ial

1000

51 51.5 52 52.5 53 53.5 54 54.5 55Pressure (kPa)

Purdue University

Capillary pressure

Int

pc-S-Awn Surfacesp

Imbibition Drainage

Figure courtesy of Laura Pyrak-Nolte of Purdue University

Comparison of Drainage & Imbibition Surfaces

Mean Relative Difference: -3.7%Mean Relative Difference: 3.7%Standard Deviation: 10.6%

C ill t lCapillary pressure at macroscale

(data obtained from Vahid Joekar-Niasar et al. 2007)

New theories of two-phase flowExtended Darcy’s law :

( )( )Gα α α αρ= − • ∇ −q K g

where Gα is the Gibbs free energy of a phase:

( )wnG G S Tα α α αρExtended Darcy’s law :

( ), , ,wnG G a S Tα α α αρ=

Extended Darcy s law :

( )a wn Sp a Sα α α α αα αρ σ σ−=− • ∇ ∇ −− ∇q K g( )Equation of motion for interfaces :

( )⎡ ⎤( )wn wn wn wn wn wK a Sγ⎡ ⎤=− ∇ +Ω ∇⎣ ⎦w

New theories of two-phase flowp

Volume balance equation for each phase

0Snα

α∂ + ∇ • =q 0nt

+ ∇∂

q

Area balance equation for fluid-fluid interfaces

( ) ( ),wn

wn wn wn wn wa a r a S∂ +∇• =w( ) ( ),a r a St

+∇∂

w

Summary of two-phase flow equationsy p q

0Snα

α∂ + ∇ • =q 0nt

+ ∇∂

q

( )a wn S Sα α α α αα α∇ ∇ ∇Kwn∂

( )a wn Sp a Sα α α α αα αρ σ σ−=− • ∇ ∇ −− ∇q K g

( ) ( ),wn

wn wn wn wn wa a r a St

∂ + ∇ • =∂

w

( )wn wn wn wn wn wK a Sγ⎡ ⎤= − ∇ + Ω ∇⎣ ⎦w

( ),w

n w w nc c wSp p p p f S aτ ∂−∂

− = =

⎣ ⎦

( ),p p p pt

f∂

CONCLUSIONSDarcy’s law is not really valid for complex porous

media.media.

Difference in fluid pressures is equal to capillary b l d ilib i di ipressure but only under equilibrium conditions.

Under nonequilibrium conditions, the difference inUnder nonequilibrium conditions, the difference in fluid pressures is a function of time rate of change of saturation as well as saturation.of saturation as well as saturation.

Fluid-fluid interfacial areas should be included in lti h fl th imultiphase flow theories.

Hysteresis should be modelled by introducing ys e es s s ou d be ode ed by oduc ginterfacial area into the two-phase flow theory

modelling two-phase flow and transport in industrial porous …modelling two-phase flow and...

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