models of multiphase flow in porous media_hassanizadeh
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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 +
“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
Two-phase flow dynamic experiments (PCE and Water)
Pc-S Curves
Cap
illar
y pr
essu
re P
c in
kPa
Saturation, S
Primary drainage
Main drainage
Main imbibition
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
P.drain Pair~16kPa P.drain Pair~20kPaP.drain Pair~25kPa
Two-phase flow dynamic experiments (PCE and Water)
Dynamic Primary Drainage Curves
Saturation, S
Hassanizadeh, Oung, and Manthey, 2004
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
Two-phase flow dynamic experiments (PCE and Water)
Dynamic Main Drainage Curves
Saturation, S
Flui
ds p
ress
ure
diff.
in k
Pa
Hassanizadeh, Oung, and Manthey, 2004
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
Two-phase flow dynamic experiments (PCE and Water)
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
Two-phase flow dynamic experiments (PCE and Water)
Saturation, S
S.P.D.
Dyn.P.D.
S.M.D.
Dyn.M.D.
S.M.I.
Dyn.M.I.
Hassanizadeh, Oung, and Manthey, 2004
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.
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.
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.
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.
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
Joekar-Niasar et al. 2008
Equilibrium Pc-Sw-awn surfacesimbibition drainage
Joekar-Niasar et al. 2008
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
Karadimitriou et al., 2012
Visualization of interfaces in a micromodelDrainage Imbibition
Karadimitriou et al., 2012
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
Karadimitriou et al., 2012
Spec
ific
inte
rfac
ial a
rea
Karadimitriou et al., 2012
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
Karadimitriou et al., 2012
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
= − + −
− + −
Karadimitriou et al., 2012
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
wn wn wn wn wn wK a Sγ⎡ ⎤= − ∇ +Ω ∇⎣ ⎦w
( ),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
wn wn wn wn wn wK a Sγ⎡ ⎤= − ∇ +Ω ∇⎣ ⎦w
( ),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
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
Material coefficients as functions of Sw
for a wide range of capillary numbers
wa wSψ ψand
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
Material coefficients as functions of Sw
for a wide range of capillary numbers
na nSψ ψand
simulating two-phase flow in a long domain
Interfacial permeability as a function of saturation for a range of capillary numbers
wn wn wn wn wn wK a Sγ⎡ ⎤= − ∇ +Ω ∇⎣ ⎦w
simulating two-phase flow in a long domain
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)
Development of vertical wetting fingers in dry soil;
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
Development of vertical wetting fingers in dry soil;
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.
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