Post on 18-Feb-2021
Oct 7, 2015
Transport in Porous Media1/36
M. Quintard
D.R. CNRS, Institut de Mécanique des Fluides, Allée Prof. C. Soula, 31400 Toulouse cedex – France
quintard@imft.fr http://mquintard.free.fr
Two-Phase Flow and Heat Two-Phase Flow and Heat Transfer in Highly Permeable Transfer in Highly Permeable
Porous MediaPorous MediaMichel Quintard
mailto:quintard@imft.fr
Transport in Porous Media 2/36M. Quintard
OutlineOutline
Background One-phase flow at high Re number Two-phase flow: quasi-static and dynamic models Hybrid Model: pore-scale / network-scale Macro-Scale Model with phase splitting for
structured porous media Boiling in porous media Conclusions
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BackgroundBackground
Nuclear Engng, Nuclear Safety Chemical Engng: distillation, catalytic columns Flow (Oil,...) in gravel, blasted rocks, … High T geothermy
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Upscaling One-Phase FlowUpscaling One-Phase Flow
Pore-scale equation
Upscaling
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Closure: Stokes problemClosure: Stokes problem
PDEs for deviations
Re~0
(Whitaker, 1986)
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Closure and Macro-Scale Closure and Macro-Scale EquationEquation
See Sanchez-Palencia (homog.), Whitaker, ...
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Non Darcean regimesNon Darcean regimes
Heuristic: Forchheimer, Ergun, …
Upscaling? Darcy Weak Inertial Strong InertialWeak
Turbulent
~10 ~30 ~1000
~Re~Re3
~Re2
~Re2
~Re
Re
(passability)
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Non Darcean regimesNon Darcean regimes
Laminar inertia effects → ~generalized Forchheimer equation
– Re → 0: Darcy– Re ~ 0: weak inertia, F.〈vβ ~ v〉 〈 β〉3 (Levy, Mei
& Auriault, ...)– Re > 0: strong inertia, F.〈vβ ~ v〉 〈 β〉2 (Whitaker,
1996; Lasseux et al., 2011;...)
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Experimental Evidence: Darcy Experimental Evidence: Darcy regimeregime
4×4 mm prisms - Air flow
8.00E-09
8.50E-09
9.00E-09
9.50E-09
1.00E-08
1.05E-08
1.10E-08
1.15E-08
1.20E-08
1.25E-08
0 1 2 3 4 5
Re
µV/ (
-∂P
/ ∂z
-ρg
) (m
²)
PermeabilityDarcyNon Darcy
Clavier et al., 2014
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Experimental Evidence: inertia Experimental Evidence: inertia regimeregime
Clavier et al., 2014
4×4 mm prisms - Water flow
y = 0.0002x2.3831y = 0.0064x1.1916
0.1%
1.0%
10.0%
100.0%
Re
(-∂P
/∂z
-ρg
-µU
/K)
/ (µ
U/K
)
Weak Inertial
Transition
1 10 100
Non-spherical particle beds - Air flow
0.0E+00
1.0E-04
2.0E-04
3.0E-04
4.0E-04
5.0E-04
6.0E-04
7.0E-04
0 500 1000 1500 2000
Re
ρU²
/ (-∂
P/∂z
-ρg
-µU
/K)
(m) c5x5
c5x8c8x12p4x4p6x6
Weak inertia Strong inertia
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Turbulent flows in porous mediaTurbulent flows in porous media
Turbulence: time and spatial averaging (see book De Lemos, 2006; ...)– time and spatial averaging commute!– However: not necessarily the same result if sequential closure!?
• for one-phase flow: scheme “II” seems preferable –contrary to Antohe & Lage (1997), Getachew et al. (2000)–see discussion: Nakayama & Kuwahara (1999), Pedras and de Lemos (2001), etc...
• for multiphase flow?• Legitimacy of seq. Closure? → Simultaneous closure over [R3 R✕ t]? More
complex sequential closures (t → x → t → ...) depending on the hierarchy of scales?
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Turbulent flows in porous media (continued)Turbulent flows in porous media (continued)
Localized Turbulence: i.e., nearly periodic (see Jin et al., 2015 for DNS results)– Spatial averaging of RANS models → generalized
Forchheimer equation, F not necessarily ~ v〈 β〉 or v〈 β〉2
Porous media turbulence models (i.e., modified k-ε, k-ω, etc...)?– Pedras & De Lemos, Nakayama & Kuwahara (1999), ...– note: useful for fluid/porous medium interface (D'Hueppe et
al., 2012)
Soulaine and Quintard, 2014
Example: structured packings
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Two-Phase FlowTwo-Phase Flow
Pore-scale
lβ
L
β-phase
averaging volume V
l γ
γ-phase
σ-phase
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Upscaling, quasi-static theoryUpscaling, quasi-static theory
Case of B.C. 4Whitaker, 1986; Auriault, 1987; Lasseux et al., 1996; ...
+ Re numbers + Dynamic Bond numberif ≈0
⇒ ?
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Macro-Scale Models: quasi-Macro-Scale Models: quasi-staticstatic Heuristic (Muskat)
Upscalingimbibition
w = wetting phase
drainage
Pc
1- Sor 1Swi
Sw Sw
kr
1- Sor 1Swi
Phase interaction
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Inertia EffectsInertia Effects
Ergun (Heuristic)
Schulenberg and Muler (1987) (Heuristic and ⛐)
Upscaling (Lasseux et al., 2008)
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Importance of Cross-Terms, and Importance of Cross-Terms, and Non-Linear EffectsNon-Linear Effects
from Clavier et al. (2015)
see also Taherzadeh & Saidi (2015)
Case Vβ=0 :
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Comparison various models: dPComparison various models: dP
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8
4 mm beads 〈 vl =0〉
Exp (Clavier et al.)
Clavier et al.
Lipinski
Reed
Hu&Theofanous
Schulenberg
Tung&Dhir
〈 vg (m/s)〉
Models without cross-terms
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Comparison various models: Comparison various models: SaturationSaturation
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1
0 0.2 0.4 0.6 0.8
Exp (Clavier et al.)
Clavier et al.
Lipinski
Reed
Hu&Theofanous
Schulenberg
Tung&Dhir
4 mm beads 〈 vl =0〉
〈 vg (m/s)〉
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More: Dynamic ModelsMore: Dynamic Models
impact of ∂S/∂t, Vα, av...:– Pseudo-functions in Pet. Engng– Quintard & Whitaker (1990, from large-scale
heterogeneity effects and multi-zone)– Hilfer (1998, multi-zone)– Panfilov & Panfilova (2005, meniscus)– Hassanizadeh and Gray (Irr. Therm., av as state
variable, 1993), also Kalaydjian (1987)...– ...
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Examples of dynamic equationsExamples of dynamic equations
Quintard & Whitaker, 1990
...see also Petroleum Engng literatureon pseudo-functions!
ω
η
η
ω
0 0.5 1
Ωβ=0 Pa/m
Sβ
10-1
10-2
10-3
10-4
10-5
Ωβ=10-4Pa/m
Ωβ=10+4Pa/m
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Mixed or Hybrid ModelsMixed or Hybrid Models Motivation: highly
permeable media, trickle beds
Example of challenging problem: jet dispersion
Trickle Bed (X-ray, IFP)
Horgue et al., 2013
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Mixed or Hybrid ModelsMixed or Hybrid Models
Network model Dynamic rules (may come from local VOF
simulations)
Melli & Scriven, 1991; results from Horgue et al. (PhD CIFRE/IFP/IMFT), 2012
VOF
Dynamic Network Simulation
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Application to tomographic Application to tomographic imagesimages
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Macro-Scale Models with Phase Macro-Scale Models with Phase “Splitting”“Splitting” Example: Flow through Structured
Media
Upscaling with phase splitting (Soulaine et al., 2014): role of momentum exchange term
Mahr and Mewes (2007) Alekseenko (2008)
Model with liquid phase splitting
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Macro-Scale Models with Phase Macro-Scale Models with Phase “Splitting”“Splitting” Example: Soulaine et al. Exp.
Comparison with Fourati et al. (2012) experiments
Model with liquid phase splitting
1. Identification on 1st stack2. Application to several stacks
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Coupling with Heat Transfer: Coupling with Heat Transfer: boiling in porous mediaboiling in porous media
3-Temperature model:– decoupled 2-phase flow, quasi-steady →
Generalized Darcy-Forchheimer? (time-space ergodicity?)
– 3-T model, extension of 2-T model: Berthoud & Valette (1994); Petit et al. (1999); Duval et al. (2004);...based on quasi-steady approx.
– Heuristic: time averaging of averaged equations → porous media Nukiyama curves (see Sapin et al., 2014)Note: Highly open problem!
App. Nuclear safety
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3T-Model3T-Model
withBoiling rate
Note: two-phase flow model (inertia+cross terms) +
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Heat Exchange Terms: Impact of Heat Exchange Terms: Impact of phase configuration!phase configuration! Quasi-static two-phase flow theory
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.25
0.5
0.75
1
1.25
1.5
1.75
2
h×
10−
6(W
m−
3K
−1 ) Chang SLG
Chang SGLStaggered SLGStaggered SGL
⟨ Tl⟩l: exch. ⟨ Tl⟩l- T sat
Sl
s
g
l
R
εℓ = 0.28, εg = 0.42, Sℓ = 0.4Phase repartition:
VOF or Cahn-Hilliard Duval et al., 2004, ...
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Pore-Scale ExperimentationsPore-Scale Experimentations
Sapin et al., 2014
Nucleate Boiling Film Boiling Intense Boiling
Pt wiring R0=100Ω
ceramic
ceramic coating
Remark Need time-averaging!
Slow motion!
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Porous Media Nukiyama CurvesPorous Media Nukiyama Curves
0
5
10
15
20
25
-20 30 80 130 180
Wall fluxqps (W/cm²)
Tsat = Tp - Tsat (°C)
qchf
qmin
TLeidenfrost
Forced Convection
boiling crisis
Vapor film collapse
Impact on non-linear properties: kr, h, D, etc...?!
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Critical FluxCritical Flux
0
5
10
15
20
25
30
-20 -10 0 10 20 30 40 50 60 70∆Tsat = Tp - Tsat (°C)
Qmp = 20mWQmp = 50 mWQmp = 100 mWQmp = 150 mW
q ps (
W/c
m2 )
Surrounding heating changes critical flux!...but the behavior up to CHF seems to be unaffected (contrary to modeling guess)?!
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Macro-Scale Model?Macro-Scale Model?Bachrata et al., 2013 (model implemented in CATHARE Safety code): work confirms main required features (inertia, NLE,…)
→ ∃ Macro-scale behavior: confirmed by experiments but heuristic time averaging or time-space ergodicity
→ Need for adapted correlations for phase permeabilities and cross terms, inertia terms, heat transfer coefficients, etc...
Example for Solid-fluid exchange: IRSN Experiments:PRELUDE, SYLPHIDE, ...
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Macro-Scale Model: reflooding Macro-Scale Model: reflooding simulationsimulation
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 3998000
99000
100000
101000
102000
103000
104000PRELUDE
mesh number
void fractiontotal heat flux [W]
wall T [°C]
gas vel. [m/s]
liq. vel. [m/s]pressure [Pa]
Qmax = 0.8 10+5
W/m2
Tmax = 414 °CVg,max= 3.9 m/sVl,max= 9.4 10
-3 m/s
dp = 2 mm
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Example of Macro-Scale Example of Macro-Scale Simulations (Quenching)Simulations (Quenching)
15s
25s
15s
25s
Temperature field
550K
750K
steam
Water and steam velocities
4 m
1.2 m
QTregoures et al. (2003)
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Conclusions (high permeable Conclusions (high permeable media)media) Various types of models Momentum equations:
– Generalized Forchheimer equation ~OK up to some high Re numbers, including localized turbulence
– Two-phase flow: need for cross-terms Boiling:
Need for: inertia, non-equilibrium, ... Problem: coupling and time averaging?
Effective properties Experimental procedures and interpretation? DNS?
Numerical aspects
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