Complex fluids: non-Newtonian flows in porous media: upscaling problems

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To cite this version:

Davit, Yohan and Zami-Pierre, Frédéric and Loubens, Romain de and Quintard, Michel Complex fluids: non-Newtonian flows in porous media: upscaling problems. (2018) In: 4th Summer School "Flow and Transport in Porous and Fractured media: Development, Protection, Management ad Sequestration of Subsurface Fluids", 25 June 2018 - 7 July 2018 (Cargèse, France).

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6/26/18

1_{Institut de Mécanique des Fluides de Toulouse (IMFT) -}

Université de Toulouse, CNRS-INPT-UPS, Toulouse FRANCE

2_Total_{, CSTJF, Avenue Larribau, 64018 Pau, France}

Non-Newtonian Flows in Porous

Media: upscaling problems

Davit Y.

1

_{, Zami-Pierre F.}

1,2

_{, de Loubens}

R.

2

_{and Quintard M.}

1

4th Cargèse Summer School, 2018

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Transport in Porous Media 2/27

M. Quintard

Objective/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

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Multi-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-Scale V Darcy-Scale Reservoir-Scale V∞ β-phase σ-phase lβ lσ lη lω

(process dependent)

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M. Quintard

Multi-Scale Analysis: Upscaling

Techniques

 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,

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A simple introduction to

upscaling with “closure”

x

DNS aver.

c

Closure:

Macro

Micro

Macro-scale Equation

b

_x ● Tomography ● Reconstruction ● Geostatistics ● ...

Effective property

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M. Quintard

Flow of a non-Newtonian fluid



Pore-Scale problem (Re~0)



Upscaling: (vol. aver. ⟨ψ

_β

⟩=ε

_β

⟨ψ

ψ

_β

⟩

β

with ε

β

=V

β

/V)?

 10-3 ₁₀-2 ₁₀-1 ₁₀0 ₁₀1 ₁₀2 ₁₀3 10-1 100 ˙ γ / _γ˙ c µ /µ 0

plateau + power law Carreau cross-fluid

Rheology:

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Typical local (over a REV) features

30°

velocity

viscosity

Pressure dev.

Remark (far from BCs)

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M. Quintard

Upscaling flow of a

non-Newtonian fluid

Newtonian fluid



Averaging (vol. aver.

⟨ψ

_β

⟩=ε

_β

⟨ψ

ψ

_β

⟩

β

with ε

β

=V

β

/V)

+...

Closure?

macro

micro

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“

Closure”?

Under several constraints: scale

separation, far from BCs, ...

⇒ Problem must be solved for each value of v

〈v

_β

〉

β

_!

Tentatively:

⇒

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M. Quintard

A classical story: the linear case

and Darcy’s law



Closure (any solution is a linear combination of

elementary solutions for ⟨ψv

_β

⟩

β

=e

i

for i=1,2,3)



Macro-Scale equation and effective properties

Important: Proof of symmetry of K

₀

requires periodicity!

Intrinsic permeability: Darcy’s law:

(see Sanchez-Palencia, Whitaker, ….)

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Calculations 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

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M. Quintard

Calculations over non-periodic

images



“permeameters”

– All methods have bias

–

⟨ψv

_x

⟩

β

≠0

– K

_xy

≠K

_yx P 1

⇒

x

y

P 2

⇒

P 1 P 2 classical 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

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Linear Case:

Non-Newtonian Fluid

Newtonian Fluid



Fluid rheology



No generic closure

independent of fluid

velocity! Generic

macro-scale law:



Representation as a

deviation from Darcy’s

law

– k

_n

, P (rotation

“matrix”): depend on

⟨ψv

β

⟩

β

(modulus and

orientation)

10-3 ₁₀-2 ₁₀-1 ₁₀0 ₁₀1 ₁₀2 ₁₀3 10-1 100 ˙ γ /_γ˙_c µ /µ0

plateau + power law Carreau cross-fluid

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M. Quintard

Test cases

HPC center EOS-Calmip:

Typically: 10

8

_{mesh cells}

10

5

_cores×hours

Clashach Bentheimer

2D

Needs very fine grid!

often

limited to

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Resolution 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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Transport in Porous Media 16/27 M. Quintard

Results



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-Newt onian regime Newt onian regime k n= 1 k n≠1 ⟨U ⟩FL k (a pp are nt)

⟨ψ.⟩

_FL

= intrinsic fluid

average

∝U

(1-n )

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Structure of the Velocity Field

backflow

Normalized pdf ~similar between Newtonian and

non-Newtonian flow! Not valid for pdf of ∇⟨ψp

⟩

β

z

y

newtonian non-newtonian

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Transition Scaling

10-2 ₁₀3 101

⟨ψU c⟩FL

10-2 ₁₀-1 1 (a ) k ∗ v s _{⟨ψU ⟩FL} µm .s-1 10-1 A1 A2 C1 C2 B1 B2 P 1 P 2 (b ) k ∗ v s U∗ 100 100 ₁₀1 Zami-Pierre et al., 2015

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Impact of Domain Size

16 18 20 22 24 1 2 3 4 5 6 7 L kn σ = 0 σ = 0.2 − 6 − 4 − 2 0 2 4 6 1 2 3 4 5 6 7 L α (d eg re e) σ = 0 σ = 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 ≠1_β ⟩

α

⟨vn =1_β ⟩

θ

∇⟨ pβ⟩β-ρβg

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M. Quintard

Impact of Domain Size

0 20 40 60 0 2 4 6 L/Req k n B m ed iu m : k n 0 20 40 60 − 3 − 2 − 1 0 L/Req R ot at io n A ng les (° ) B medium: α B medium: β

Disorder → no anisotropy induced by

non-linearity if L large enough!

θ=22°

R_eq (pore size)

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Impact of disorder and velocity

10−1 100 101 1 2 3 4 5 U∗ kn Aσ=0 Aσ=0.05₁ Aσ=0.10₁ Aσ=0.20₁ Aσ=0.30₁ B 10−2 10−1 100 101 −4 −2 0 U∗ A n g le α o f P Aσ=0 Aσ=0.05₁ Aσ=0.10₁ Aσ=0.20₁ Aσ=0.30₁ B

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M. Quintard

Practical 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)

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Further upscaling

η-region ω-region L Pore-Scale V Darcy-Scale Reservoir-Scale V β-phase σ-phase lβ lσ lη lω 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 upscaling



Viscoelastic fluids



Rheological models

FENE-P:

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⛐

-see previous discussion on “apparent permeability”, etc…

- elastic turbulence?

Deborah number:

Example of results: De et al., soft matter, 2018

...also Weissenberg number ☺

Normal stress along average flow direction

De= 0.001 0.1

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M. Quintard

Further perspectives: N-momentum

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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Conclusions

 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