6/26/18
Transport in Porous Media1/27
M. Quintard
1Institut de Mécanique des Fluides de Toulouse (IMFT) - Université de Toulouse, CNRS-INPT-UPS, Toulouse FRANCE2Total, CSTJF, Avenue Larribau, 64018 Pau, France
Non-Newtonian Flows in Porous Non-Newtonian Flows in Porous Media: upscaling problemsMedia: upscaling problems
Davit Y.1, Zami-Pierre F.1,2 , de Loubens R.2 and Quintard M.1
4th Cargèse Summer School, 2018
https://www.dropbox.com/s/mcgg0ifpogsznv2/non_newtonian_V00.pdf?dl=0
Transport in Porous Media 2/27M. Quintard
Objective/OutlineObjective/Outline
Motivation: flow of polymer solutions, question about heuristic models in Res. Engng
Upscaling
– Introduction (generalized Stokes)
– Transition
– Induced anisotropy, effect of disorder, effect of size of the UC, ...
Further problems: exclusion zone, viscoelastic Conclusions
Transport in Porous Media 3/27M. Quintard
Multi-Scale AnalysisMulti-Scale Analysis
()=0
=g(x)
*(⟨ψ⟩)=0
⟨ψ =⟩= g*(x)
Pore-Scale Darcy-Scale
Sequential multi-scale pattern Used in DRP, Res. Engng,
Hydro., etc... Objectives of macro-scale
theories:– Smoothing operator . → Macro ⟨.⟩ → Macro ⟩ → Macro
variables, Eqs & BCs– Micro-macro link →
Determination of Effective Properties
Needs Scale Separation:
lβ ,lσ REV?≪REV?≪ ≪REV?≪ L
η-region
ω-region
L
Pore-ScaleV
Darcy-Scale
Reservoir-Scale
V∞
β-phase
σ-phaselβ
lσ
lηlω
(process dependent)
Transport in Porous Media 4/27M. Quintard
Multi-Scale Analysis: Upscaling Multi-Scale Analysis: Upscaling TechniquesTechniques Form of the equations?
– averaging and TIP (Marle, Gray, Hassanizadeh, …)– averaging and closure (Whitaker, …)– homogenization (Bensoussan et al., Sanchez-Palencia, Tartar, …), also
“closure”– stochastic approaches (Dagan, Gelhar, ...)
Effective properties calculations?– Assuming the form of Eqs: interpret experiments or DNS– Upscaling with “closure” (averaging, homogenization, stochastic):
provides local Unit Cell problems Many Open Problems: High non-linearities, Strong couplings,
Evolving pore-scale structure, ...
Transport in Porous Media 5/27M. Quintard
A simple introduction to A simple introduction to upscaling with “closure”upscaling with “closure”
x
x
x
DNS
aver.cClosure:
Macro
Micro
Macro-scale Equation
bx
● Tomography● Reconstruction● Geostatistics● ...
Effective property
Transport in Porous Media 6/27M. Quintard
Flow of a non-Newtonian fluidFlow of a non-Newtonian fluid
Pore-Scale problem (Re~0)
Upscaling: (vol. aver. ⟨ψβ⟩=εβ ψ⟨ψ β⟩β with εβ=Vβ/V)?
10-3 10-2 10-1 100 101 102 103
10-1
100
γ̇ / γ̇c
µ/µ
0
plateau + power lawCarreau
cross-fluid
Rheology:
Case of Generalized Stokes equation
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Typical local (over a REV) featuresTypical local (over a REV) features
30°
velocity
viscosity
Pressure dev.
Remark (far from BCs)
⇓
Transport in Porous Media 8/27M. Quintard
Upscaling flow of a non-Upscaling flow of a non-Newtonian fluidNewtonian fluid
Averaging (vol. aver. ⟨ψβ⟩=εβ ψ⟨ψ β⟩β with εβ=Vβ/V)
+...Closure?
macro
micro
⇒ Problem must be solved for each value of ⟨ψvβ⟩β!
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““Closure”?Closure”?Under several constraints: scale separation, far from BCs, ...
⇒ Problem must be solved for each value of v⟨vβ⟩β!
Tentatively:
⇒
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A classical story: the linear case A classical story: the linear case and Darcy’s lawand Darcy’s law
Closure (any solution is a linear combination of elementary solutions for ⟨ψvβ⟩β=ei for i=1,2,3)
Macro-Scale equation and effective properties
Important: Proof of symmetry of K0 requires periodicity!
Intrinsic permeability:
Darcy’s law:
(see Sanchez-Palencia, Whitaker, ….)
over a UC!
Transport in Porous Media 11/27M. Quintard
Calculations of the permeabilityCalculations of the permeability
3 possibilities– Initial closure problem
– Transformation of closure problem into ~Stokes with source term and periodic pressure and velocity
– “permeameters”: no-periodicity
Making image periodic? – I: Percolation problem
– II: Loss of anisotropy
– III: potentially various bias
See discussion in Guibert et al., 2015
Case of “diffusion” problem: e.g., permeability, effective diffusion
● thin layers + periodicity
● Eff. Medium
● …..
I II III
Transport in Porous Media 12/27M. Quintard
Calculations over non-periodic Calculations over non-periodic imagesimages
“permeameters”– All methods have bias
– ⟨ψvx⟩β≠0
– Kxy≠Kyx
P1
⇒x
y P2
⇒
P1
P2classical
Bamberger
See discussion in: Manwart et al. 2002; Piller et al. 2009; Guibert et al., 2015; ...
Note: minimal bias if large sample and anisotropy along the axis
Transport in Porous Media 13/27M. Quintard
Non-Linear Case: Non-Non-Linear Case: Non-Newtonian FluidNewtonian Fluid
Fluid rheology
No generic closure independent of fluid velocity! Generic macro-scale law:
Representation as a deviation from Darcy’s law
– kn, P (rotation “matrix”): depend on ⟨ψvβ⟩β
(modulus and orientation)
10-3 10-2 10-1 100 101 102 103
10-1
100
γ̇ / γ̇c
µ/µ
0
plateau + power lawCarreau
cross-fluid
PLCO
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Test casesTest cases
HPC center EOS-Calmip:Typically: 108 mesh cells105 cores×hours
Clashach Bentheimer 2D
Needs very fine grid!
often limited to~ mm3!
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Resolution with OpenFoamResolution with OpenFoam● FVM with OpenFOAM (SIMPLE, second-order scheme)● Use of HPC, calculations up to 100 millions mesh elements● a total of 100000 hours of CPU time.● Conform orthogonal hexahedral elements.● Multi-criteria grid convergence study = OK.
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ResultsResults
Computations allows to analyze various features:– Properties of pore-
scale fields (PDFs)– Transition:
• Starts in a few narrow constrictions
• Scaling for transition?⟨Uc⟩FL
non-Newtonian regime
Newtonian regime
k n= 1
k n≠1
⟨U ⟩FLk
(ap
pare
nt)
⟨ψ.⟩FL
= intrinsic fluid
average
∝U(1
-n)
Transport in Porous Media 17/27M. Quintard
Structure of the Velocity FieldStructure of the Velocity Fieldbackflow
Normalized pdf ~similar between Newtonian and non-Newtonian flow! Not valid for pdf of ∇⟨ψp
β⟩β
z
y
newtonian
non-newtonian
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Transition ScalingTransition Scaling
10-2 103
101
⟨ψU c⟩FL
10-2 10-1
1
(a ) k ∗ v s ⟨ψU ⟩FL µm .s-1
10-1
A1A2C1C2B1B2P 1P 2
(b ) k ∗ v s U∗
100
100 101
Zami-Pierre et al., 2015
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Impact of Domain SizeImpact of Domain Size
16
18
20
22
24
1 2 3 4 5 6 7
L
k n
σ = 0σ = 0.2
− 6
− 4
− 2
0
2
4
6
1 2 3 4 5 6 7
L
α (d
egre
e)
σ = 0σ = 0.2
● Anisotropy induced by non-linear behavior decreases with ↗ L for disordered media
● Effective property variance decreases with ↗ L
θ=22°
~x
~y
⟨vn ≠1β ⟩
α⟨v
n =1β ⟩
θ ∇⟨ p β⟩β-ρβg
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Impact of Domain SizeImpact of Domain Size
0 20 40 60 0
2
4
6
L/Req
k n
B m edium : k n
0 20 40 60− 3
− 2
− 1
0
L/Req
Rot
atio
n A
ngle
s (°
)
B medium: αB medium: β
Disorder → no anisotropy induced by non-linearity if L large enough!
θ=22°
Req
(pore size)
L Bentheimer
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Impact of disorder and velocityImpact of disorder and velocity
10−1 100 101
1
2
3
4
5
U∗
kn
Aσ=0
Aσ=0.05 1
Aσ=0.10 1
Aσ=0.20 1
Aσ=0.30 1B
10−2 10−1 100 101
−4
−2
0
U∗
Angle
αof
P
Aσ=0
Aσ=0.05 1
Aσ=0.10 1
Aσ=0.20 1
Aσ=0.30 1B
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Practical ConsequencesPractical Consequences
Eng. Practice: apparent Darcy’s law
Discussion:– P=I for all ⟨vvβ⟩β if isotropic disordered media and REV (→ need tests for
various sizes)!
– Apparent permeability ~ scales with (K0)½ → classical scaling “may” introduce artificial dependence upon parameters such as porosity:
– Description of transition near the critical velocity may not be well described by an apparent viscosity (no observed angle in the apparent permeability in the case of PLCO)
Fitting parameter (rock dependent)
versus
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Further upscalingFurther upscaling
η-region
ω-region
L
Pore-Scale
V
Darcy-Scale
Reservoir-Scale
V
β-phase
σ-phaselβ
lσ
lηlω
cont. DLVO
effective BC
zone model
SubPore-Scale
Depletion layer treated as an effective BC
Zami-Pierre et al., 2017
see Chauveteau (1982), Sorbie & Huang (1991) (double-layer model)
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Further upscalingFurther upscaling
Viscoelastic fluids
Rheological modelsFENE-P:
upper convected Derivative
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⛐ -see previous discussion on “apparent permeability”, etc… - elastic turbulence?
Deborah number:
Example of results: De et al., soft matter, 2018Example of results: De et al., soft matter, 2018
...also Weissenberg number ☺
Normal stress along average flow direction
De= 0.001 0.1
Steady-state!
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Further perspectives: N-momentum Further perspectives: N-momentum equations, multi-component aspects, ...equations, multi-component aspects, ...
Superfluid: 2 momentum equations → complex behavior → macro-scale model?
Polymer solution as multi-component systems:
– Mechanical segregation, degradation (bio., mech.)
– Model?
• Momentum balances:– diffusion theory or– N-momentum equations
• Composition:– Continuous models or– PBM (population balance model), ...
see Allain et al. (2010, 2013, 2015), Soulaine et al. (2015, 2017)
mol. weight
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ConclusionsConclusions
Upscaling tells that this is not always possible to separate in an apparent Darcy’s law permeability and viscosity
Specific anisotropy effects Simplifications arise for disordered media Various results published in the literature for various rheology:
power-law (...), Ellis and Herschel–Bulkley fluids (Sochi & Blunt, 2008), Yield-Stress Fluids (Sochi, 2008), etc…
Additional problems: retention effects, Inaccessible Pore Volume (IPV), mobile/immobile effects
Perspectives: viscoelastic, multicomponent, coupling with other transport problems (transport of species, heat transfer, etc…), ...