models of multiphase flow in porous media_hassanizadeh

Models of multiphase flow in porous media, including fluid-fluid interfaces

S. Majid HassanizadehDepartment of Earth Sciences, Utrecht University, Netherlands

Soil and Groundwater Systems, Deltares, NetherlandsCollaborators: Vahid Joekar-Niasar; Shell, Rijswijk, The NetherlandsNikos Karadimitriou; Utrecht University, The NetherlandsSimona Bottero; Delft University of Technology, The Neth.Jenny Niessner; Stuttgart University, GermanyRainer Helmig; Stuttgart University, GermanyHelge K. Dahle; University of Bergen, NorwayMichael Celia; Princeton University, USALaura Pyrak-Nolte; Purdue University, USA

“Extended” Darcy’s lawis assumed to apply to 3D unsteady flow of two or more compressible fluids, with any amount of dissolved matter, in heterogeneous anisotropic deformable porous media under non-isothermal conditions

Original Darcy’s lawwas proposed for 1D steady-state flow of almost pure incompressible water in saturated homogeneous isotropic rigid sandy soil under isothermal conditions

- i ijj

hq Kx

αα α ∂=

∂- hq K

x∂

=∂

= /h p g zρ + = /h p g zα α αρ +

Suta Vijaya

Text Box

http://www.uni-regensburg.de/Fakultaeten/nat_Fak_I/fbp2012/FBP2012_files/Talks/Hassanizadeh_two-phase.pdf

“Extended” Darcy’s Lawα

α α ∂∂i ij

j

hq = - Kx

We have added bells and whistles to a simple formula to make it (look more complicated and thus) applicable to a much more complicated system!

hq Kx∂

= −∂

One must follow a reverse process: - develop a general theory for a complex system - reduce it to a simpler form for a less complex system

Standard two-phase flow equations

( )n w wP P f S− =

0Snt

αα∂

+ ∇ • =∂

q

( )rk Pα

α α αα ρμ

=− • ∇ −q K g

( )r r wk k Sα α=

cP=

Measurement of Capillary Pressure-Saturation Curvein a pressure plate

*

Oil chamber

Porous Medium

Porous Plate(impermeable to air)

Water reservoir

*******

*

*

Wetting curve

Drying curve

nextP

S**

Scanning drying curve

Scanning wetting curve

Often it takes more than one week to get a set of wetting and drying curves

P ext

-Pex

t

( )n w c wext ext intP P P f S− = =

wextP

10

Two-phase flow dynamic experiments (PCE and Water)

Selective pressure transducers used to measure average pressure of each phase within the porous medium

PCE

Water

Air chamber

PPT NW back

PPT W front

Soil sample

Hassanizadeh, Oung, and Manthey, 2004

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-2

0

2

4

6

8

10

capi

llary

pre

ssur

e [k

Pa]

water saturation

Static Pc inside


Pc-S Curves

Cap

illar

y pr

essu

re P

c in

kPa

Saturation, S

Primary drainage

Main drainage

Main imbibition


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-2

0

2

4

6

8

10

capi

llary

pre

ssur

e [k

Pa]

water saturation

P.drain Pair~16kPa P.drain Pair~20kPaP.drain Pair~25kPa


Dynamic Primary Drainage Curves

Saturation, S


Flui

ds p

ress

ure

diff.

in k

Pa

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9-2

0

2

4

6

8

10

capi

llary

pre

ssur

e [k

Pa]

water saturation

M.drain PN~16kPaM.drain PN~20kPaM.drain PN~25kPaM.drain PN~30kPa


Dynamic Main Drainage Curves

Saturation, S

Flui

ds p

ress

ure

diff.

in k

Pa


0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-2

0

2

4

6

8

10

capi

llary

pre

ssur

e [k

Pa]

water saturation

M.imb Pair~25kPa PW~0kPaM.imb Pair~16kPa PW~5kPaM.imb Pair~20kPa PW~8kPaM.imb Pair~25kPa PW~10kPaM.imb Pair~30kPa PW~0kPa last


Dynamic Main Imbibition Curves

Saturation, S

Flui

ds p

ress

ure

diff.

in k

Pa

0 0.20 0.40 0.60 0.80 1.00 -2.0

0

2.0

4.0

6.0

8.0

10.0ca

pilla

ry p

ress

ure

[kPa]

water saturation

prim drain PN~16kPa prim drain PN~20kPaprim drain PN~25kPamain imb PW~0kPamain imb PW~5kPamain imb PW~8kPamain imb PW~10kPamain imb PW~0kPa lastmain drain PN~16kPamain drain PN~20kPamain drain PN~25kPamain drain PN~30kPaStatic Pc inside


Saturation, S

S.P.D.

Dyn.P.D.

S.M.D.

Dyn.M.D.

S.M.I.

Dyn.M.I.


Flui

ds p

ress

ure

diff.

in k

Pa

Saturation, S

Primary Drainage Curve

Primary Imbibition CurveMain Imbibition Curve

Main Drainage Curves

Main Scanning Drainage Curves

Main Scanning Imbibition Curves

Secundary Scanning Drainage Curves

Secundary Scanning Imbibition Curves

Dynamic Drainage Curves

Dynamic Imbibition Curves

There is no unique pc-S curve.

Flui

ds p

ress

ure

diff.

in k

Pa

Relative permeability-saturation curve

0 0

( , medium properties, ,oil/water flow rate ratio,

viscosities ratio, , , , , pressuregradient, flow history)

i i wr r

A R

k k S Ca

Co Boθ θ

=

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; Berg et al. TiPM, 2008

Standard theory does not model the development of vertical infiltration fingers in dry soil

Experiments by Rezanejad et al., 2002

Non-monotonic distribution of saturation during infiltration into dry soil; experiments in our gamma system

Sand Column

dia1 cm

50 c

m

Non-monotonic distribution of saturation during infiltration into dry soil; experiments in our gamma system

Satu

ratio

n at

a d

epth

of 2

0 cm

Time (sec)

At different flow rates q (in cm/min)

Non-monotonic distribution of pressure during infiltration into dry soil; experiments in our gamma system

pres

sure

Time (sec)

Pressure at different positions along the column

Thermodynamic basis for macroscacle theories of two-phase flow in porous media

Experimental and computational determinations of capillary pressure under equilibrium conditions

Experimental and computational determinations of capillary pressure under non-equilibrium conditions

Non-equilibrium capillarity theory for fluid pressures

Truly extended Darcy’s law

Outline

Averaging-Thermodynamic ApproachFirst, a microscale picture of the porous medium is given:

Porous solid and the two phases form a juxtaposed superposition of three phases filling the space and separated by three interfaces: solid-water, water-oil, solid-oil, and a water-oil-solid common line.

There is mass, momentum, energy associated with each domain.



Microscale conservation equations for mass, momentum, and energy are written for points within phases or points on interfaces and the common line.

These equations are averaged to obtain macroscale conservation equations.

No microscopic constitutive equations are assumed.



Microscale conservation equations for mass, momentum, and energy are written for points within phases or points on interfaces and the common line.

These equations are averaged to obtain macroscale conservation equations.

No microscopic constitutive equations are assumed.

Macroscopic constitutive equations are proposed at the macroscale and restricted by 2nd Law of Thermodynamic.

Equations of conservation of mass

( )a a Et

αβαβ αβ αβ∂

+∇• =∂

w

0Snt

αα∂

+ ∇ • =∂

q

( ) ( ) ,nS

rt

α αα α α αβ

β α

ρρ

≠

∂+∇• =

∂ ∑q

( ) ( ) , ,a

a r rt

αβ αβαβ αβ αβ α αβ β αβ

∂ Γ+∇• Γ = +

∂w

Divide by a constant ρ and neglect mass exchange term:

Divide by a constant Γ αβ:

For each phase:

For each interface:

Macroscale capillary pressure; theoretical definition

Hassanizadeh and Gray, WRR, 1990

n w nw ns wsc n w

w w w w w

H H H H HP S SS S S S S

∂ ∂ ∂ ∂ ∂= − − − − −

∂ ∂ ∂ ∂ ∂

( )2 nn n

n

HP ρρ

∂= −

∂( )2 w

w ww

HP ρρ

∂= −

∂

?c n wP P P= −

( , ) or ( , )c w wn wn c wP F S a a F P S= =

H is macroscopic Helmholz free energy, which depends on state variables such as Sw, aαβ, ρα, T, etc.

Cap

illar

y pr

essu

re h

ead,

Pc/ρ

g

Saturation

Cap

illar

y pr

essu

re h

ead,

Pc/ρ

g

Saturation

Imbibition Drainage

Capillary pressure-saturation data points measured in laboratory

Capillary pressure-saturation curve is hysteretic

Imbibition Drainage

Cap

illar

y pr

essu

re

Cap

illar

y pr

essu

re

Saturation

Capillary pressure and saturation are two independent quantities

Saturation

Macroscale capillary pressure; theoretical definition

( , ) or ( , )c w wn wn c wP F S a a F P S= =

So, microscale pc is proportional to awn.

Macroscale capillary pressureDynamic Pore-network modeling (Joekar-Niasar et al., 2010, 2011)

Pore body

Pore throat

Joekar-Niasar et al. 2008

R2=0.98

Pc-Sw-awn relationship; imbibition


Equilibrium Pc-Sw-awn surfacesimbibition drainage


Pc-Sw-awn SurfaceResults from Lattice-Bolzmann simulations

Porter et al., 2009

This has been shown by:- Reeves and Celia (1996); Static pore-network modeling- Held and Celia (2001); Static pore-network modeling- Joekar-Niasar et al. (2007) Static pore-network modeling- Joekar-Niasar and Hassanizadeh (2010, 2011)

Dynamic/static pore-network modeling

- Porter et al. (2009); Column experiments and LB modeling- Chen and Kibbey (2006); Column experiments

- Cheng et al. (2004); Micromodel experiments- Chen et al. (2007); Micromodel experiments- Bottero (2009); Micromodel experiments- Karadimitriou et al. (2012); Micromodel experiments

Capillary Pressure-Interfacial Area-Saturation data form a (unique) surface

A micro-model (made of PDMS) with a length of 35 mm and width of 5 mm; It has 3000 pore bodies and 9000 pore throats with a mean pore size of 70 μm and constant depth of 70 μm

A new generation of micro-model experiments

Karadimitriou et al., 2012

Micromodel experimental setup

A new generation of micro-model experiments

Visualization of interfaces in a micromodelDrainageDrainage

Sn = 24%Pc = 4340 PaAwn = 18.32 mm2


Visualization of interfaces in a micromodelDrainage Imbibition


Visualization of interfaces in a micromodel

Sn = 24%Pc = 4340 PaAwn = 18.32 mm2

ImbibitionDrainage

Sn = 24%Pc = 2200 PaAwn = 30.56 mm2Karadimitriou et al., 2012

Capillary pressure-saturation points


Spec

ific

inte

rfac

ial a

rea


Capillary pressure-saturation-interfacial area SurfaceFitted to drainage points

Capillary pressure-saturation-interfacial area SurfaceFitted to imbibition points

Spec

ific

inte

rfac

ial a

rea


Capillary pressure-saturation-interfacial area Surface

The average difference between the surface for drainage and the surface with all the data points is 9.7%.

The average difference between the surface for imbibition and the surface with all the data points is -5.77%.

2 2

18070 151500* 1.075*

137300* 2.577* * 0.0001104*

wnc

c c

a S P

S S P P

= − + −

− + −


Capillary Pressure-Saturation-Interfacial Area data points fall on a (unique) surface, which is a property of the fluids-solid system.

Fluids Pressure Difference, Pn-Pw

is a dynamic property which depends on boundary conditions and fluid dynamic properties (e.g. viscosities).

Macroscale capillarity theory

n w cw

Pt

P P Sτ ∂−=∂

−

Non-equilibrium Capillary Equation:

The coefficient τ is a material property which may depend on saturation.

It has been determined through column experiments, as well as computational models, by many authors.

Two-phase flow equations with dynamic capillarity effect

n w cw

Pt

P P Sτ ∂−=∂

−

0Snt

αα∂

+ ∇ • =∂

q

( )1 Pα α α αα ρμ

=− • ∇ −q K g

2u uDt x xx

ux t

τ∂ ∂ ∂⎛ ⎞= • +⎜ ⎟∂ ∂⎛ ⎞∂ ∂

• ⎜ ⎟∂ ∂ ∂⎝∂⎝ ⎠⎠

Combine the three equations for unsaturated flow:

Dautov et al. (2002)

Development of vertical wetting fingers in dry soil;

Simulations based on dynamic capillarity theory

Differential pressure

PPT1 NW-W

PPT2 NW-W

PPT3 NW-W TDR3

TDR2

TDR1

Water Pump

PCE Pump

Pressure Regulator

PCEWaterAir

Valves

Experimental Set-up for measurement of dynamic capillarity effect; Bottero et al., , 2011

Pressure transducer

Switch on/off Water Pump

Switch on/off PCE Pump

Brass filter

Differential pressure gauge

No membranes

Steady-state flowof invading fluidwith incremental pressure increase

Primary drainage

Main drainage

Main imbibition

Transient drainage with large injection pressure

Non-equilibrium primary drainage; LocalFluids Pressure Difference vs Saturation at position z1

Pres

sure

diff

eren

ce (

kPa)

Bottero et al., 2011

n w cw

Pt

P P Sτ ∂−=∂

−

Value of the damping coefficient τ as a function of saturation; local scale

Bottero et al., 2011

Simulation of non-equilibrium primary drainage;Local pressure difference vs time; Injection pressure, 35kPa

Bottero, 2009

Conclusions on Capillarity Theory:Capillary Pressure is not just a function of saturation.

Capillary Pressure -Saturation-Interfacial Area form a (unique) surface, which is a property of the fluids-solid system.

Fluids Pressure Difference, Pn-Pw is a dynamic property which depends on boundary conditions and fluid dynamic properties (e.g. viscosities)

Fluids Pressure Difference, Pn-Pw, is equal to capillary pressure but only under equilibrium conditions. Otherwise, it depends on the rate of change of saturation.

where Gα is the Gibbs free energy of a phase:

Extended Darcy’s law (linearized equation of motion ):

Extended theories of two-phase flow

( )Gα α α αρ= − ∇ −q K gi

Extended Darcy’s law :

( )r

a wn Sk P a Sα

α α αα

α α αρ ψμ

ψ= ∇∇ − − ∇− −q K gi

( ), , ,wnG G a S Tα α α αρ=

where are material coefficients.a Sα αψ ψand

Extended theories of two-phase flow

Simplified equation of motion for interfaces (neglecting gravity term):

where is a material coefficient and is macroscale surface tension.

wnΩ

Linearized equation of motion for interfaces:

( )wn wn wn wn wnK a G=− Γ ∇ −w gwhere Gwn is the Gibbs free energy of wn-interface:

( ), , ,wn wn wnG G a S Tα α= Γ

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

wnγ

Summary of extended two-phase flow equations

( )wn

wn wn wna a Et

∂+∇• =

∂w

n w cw

Pt

P P Sτ ∂−=∂

−

0Snt

αα∂

+ ∇ • =∂

q

( )1 a wn SP a Sα α α αα

α α αψ ψρμ

− ∇ ∇• ∇ −=− −q K g


( ),c w wnP f S a=

Simulation of redistribution moisture; a numerical example

Equilibrium moisture distribution from standard two-phase flow equations with no hysteresis

x

Sw

interface

Initial distribution

Final distribution

Simulating horizontal moisture redistribution with standard two-phase flow equations+hysteresis

( ) 0n w wd

SP P f S ift

∂− = <

∂

0Snt

αα∂

+ ∇ • =∂

q

1 Pα α ααμ

=− •∇q K

( ) 0n w wi

SP P f S ift

∂− = >

∂

Solving moisture redistribution problem with standard two-phase flow equations

Equilibrium result from standard two-phase flow equations with hystresis

x

Sw

2p

interface

2p withhysteresis

Solving a problem with extended two-phase flow equations

( ) ( ),wn

wn wn wn wn wa a E a St

∂+∇• =

∂w

n w cw

pt

p P Sτ ∂−=∂

−

0Snt

αα∂

+ ∇ • =∂

q

( )1 a wn Sp a Sα α α αα

α α αψ ψρμ

− ∇ ∇• ∇ −=− −q K g


( ),c w wnP f S a=

There is a self-similar solution for this problem:

With a semi-analytical solution.

Governing eqs for moisture redistribution

Long-term result from extended two-phase flow equations with no hystresis

CONCLUSIONSUnder nonequilibrium conditions, the difference in

fluid pressures is a function of time rate of change of saturation as well as saturation.

Resulting Eq. will lead to nonmonotonic solutions.

Fluid-fluid interfacial areas should be included in multiphase flow theories.

Hysteresis can be modelled by introducing interfacial area into the two-phase flow theory

simulating two-phase flow in a long domain

( )r

a wn Sk p a Sα

α α α α α αα ψμ

ψρ − ∇= • ∇ − ∇− −q K g

Total permeability as a function of saturation for a range of capillary numbers

rk αK

( )q K grk pα

α α αα ρμ

=− • ∇ −

Transient relative permeability curves

Total permeability as a function of saturation for a range of capillary numbers

/Kr qk

p x

α αα

α

μ= −

∂ ∂

Krk α


( )r

a wn Sk p a Sα


ψρ − ∇= • ∇ − ∇− −q K g

Material coefficients as functions of Sw

for a wide range of capillary numbers

wa wSψ ψand


( )r

a wn Sk p a Sα


ψρ − ∇= • ∇ − ∇− −q K g

Material coefficients as functions of Sw

for a wide range of capillary numbers

na nSψ ψand


Interfacial permeability as a function of saturation for a range of capillary numbers



Rate of generation/destruction of interfacial area as a function of saturation and rate of change of saturation

( ) ( ),wn

wn wn wn wn wa a E a St

∂+∇• =

∂w

CONCLUSIONSThe driving forces in Darcy’s law should be gradient

of Gibbs free energy and gravity.

Difference in fluid pressures is equal to capillary pressure but only under equilibrium conditions.

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

Fluid-fluid interfacial areas should be included in multiphase flow theories.

Hysteresis can be modelled by introducing interfacial area into the two-phase flow theory

Dautov et al. (2002)


Simulations based on new capillarity theory

Dynamic Capillary Pressure Mechanism for Instability in Gravity-Driven Flows; Review and Extension to Very Dry Conditions; JOHN L. NIEBER, RAFAIL Z. DAUTOV, ANDREY G. EGOROV, and ALEKSEY Y. SHESHUKOV; TiPM

Stability analysis of gravity-driven infiltrating flow; Andrey G. Egorov, Rafail Z. Dautov, John L. Nieber, and Aleksey Y. Sheshukov


Simulations based on new capillarity theory

pc - Sw measurement: the measurement cell

pamb

pwater

indicating calliper

green: hydrophilic membranegrey: sealingdotted: sample (GDL)red: hydrophobic membraneblue: syringe pump

sample diameter: 25 mmflow rates for every step:- pumping: 3 µl / min- withdrawing: 3 µl / min

volume infused / withdrawn- each step: 3 µl

0 20 40 60 80 100 120 140 160 180

0

1

2

3

capillary pressure, infused volume vs. time

Time [s]

Infu

sed

volu

me

[µl]

0 20 40 60 80 100 120 140 160 180

0

200

400

600

capi

llary

pre

ssur

e [P

a]

infused volumecapillary pressure

period II

period III

period I

pc - Sw measurement: operation strategy

period I: initial stateperiod II: pumping water

into the sampleperiod III: relaxation period

(a) (b)

Micromodel experiments

J.-T. Cheng, L. J. Pyrak-Nolte, D. D. Nolte and N. J. Giordano,Geophysical Research Letters, 2004

Dark area is solid and light area is pore space

Measuring awn interfaces in a micromodel

J.-T. Cheng, L. J. Pyrak-Nolte, D. D. Nolte and N. J. Giordano,Geophysical Research Letters, 2004

(a) (b)

ImbibitionDrainage

Equilibrium drainage and imbibition experiments*

*Experiments performed by S. Bottero at Purdue Univ.

models of multiphase flow in porous media_hassanizadeh

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