Open Archive TOULOUSE Archive Ouverte (OATAO)Zami-Pierre, Frédéric and Davit, Yohan and Loubens,...

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To cite this version : Zami-Pierre, Frédéric and Davit, Yohan and Loubens, Romain de and Quintard, Michel Flow of polymer fluids through porous media. (2016) In: Pore Scale Modeling Total Workshop, 8 June 2016 - 9 June 2016 (Pau, France). (Unpublished)

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Flow of polymersolutions throughporous media

F. Zami-Pierre

Introduction

Context

Rheology

Porous Media

Permeabilityprediction

Numerical Study

Numerical set up

Numerical results

Modelling

Conclusions

Perspectives

Flow of polymer solutions through porous

media

F. Zami-Pierre1,2, Y. Davit1,

R. de Loubens2 and M. Quintard1

1Institut de Mecanique des Fluides de Toulouse (IMFT) - Universite deToulouse, CNRS-INPT-UPS, Toulouse FRANCE

2Total, DEV/GIS/SIM

June 8, 2016

1 / 19


F. Zami-Pierre

Introduction

Context

Rheology

Porous Media


Numerical Study

Numerical set up

Numerical results

Modelling

Conclusions

Perspectives

Overview

Introduction

Context

Rheology

Porous Media

Permeability prediction

Numerical Study

Numerical set up

Numerical results

Modelling

Conclusions

Perspectives

2 / 19


F. Zami-Pierre

Introduction

Context

Rheology

Porous Media


Numerical Study

Numerical set up

Numerical results

Modelling

Conclusions

Perspectives

Scope of work

◮ EOR context:→ adverse mobility ratiowaterflood,→ hydrodynamic instabilities,→ use of polymer.

◮ Physics is complex...

◮ Lack of micro-scale observations→ real mechanisms remainunclear.

◮ Porous media flow is alsocomplex, e.g, multi-scale,confinement effects.

γ(

10−6)

s−1

Fig : Power-law fluid flowing througha Bentheimer sandstone [Zami-Pierreet al., 2016]

Zami-Pierre et al., Transition in the flow of power-law fluids through isotropicporous media, Physical Review Letters, publication under review (2016)

3 / 19


F. Zami-Pierre

Introduction

Context

Rheology

Porous Media


Numerical Study

Numerical set up

Numerical results

Modelling

Conclusions

Perspectives

Context

µ = µ0 ⇒ 〈U〉 = −K·∇p

µ0⇒ 〈U〉 ∝ ‖∆p‖ ⇒ 1©

µ = µβ γn−1 ⇒ 〈U〉 = −K(〈U〉)·∇p

µ0⇒ 〈U〉 ∝ ‖∆p‖1/n ⇒ 2©

γc

1© 2©

γ

µ

Fig : Non-Newtonian rheology

〈Uc〉

non-Newtonianregime

2©

Newtonianregime

1©

?

〈U〉

k

Fig : k versus 〈U〉

Refs: [Sorbie and Huang, 1991; Getachew et al., 1998; Sorbie and Huang,1991; Zitha et al., 1995; Lecourtier and Chauveteau, 1984]

4 / 19


F. Zami-Pierre

Introduction

Context

Rheology

Porous Media


Numerical Study

Numerical set up

Numerical results

Modelling

Conclusions

Perspectives

Context

◮ Literature: γeq = 4α〈U〉/φ√8k/φ

, [Chauveteau, 1982].

Two semi-empirical parameters:

◮ Req =√

8k/φ, comes from an analogy with a single pipe. Is itstill valid in complex multi-scale porous media?

◮ α: fitting parameter coming from core-flood experiments. Manyhidden physical phenomena.

◮ Our goal: simulate the flow a non-Newtonian fluid over a wide panel ofporous media. Can we predict and understand the transition velocity,〈Uc 〉?

5 / 19


F. Zami-Pierre

Introduction

Context

Rheology

Porous Media


Numerical Study

Numerical set up

Numerical results

Modelling

Conclusions

Perspectives

Rheology

EOR polymer solutions: shear-thinning, no yield, assume time-independent. Common models are,

◮ plateau + power-law,

µ =

µ0 if γ < γc ,

µ0

(

γγc

)n−1else,

(1)

◮ Carreau,

◮ generalized Newtonian.

10−4 10−3 10−2 10−1 100 101 102 103 104

10−2

10−1

100

γ/γc

µ/µ

0

plateau + power-lawCarreau

generalized Newtonian

Fig : A few rheological model

Refs: [Bird and Carreau, 1968; Savins, 1969; Collyer, 1973]

6 / 19


F. Zami-Pierre

Introduction

Context

Rheology

Porous Media


Numerical Study

Numerical set up

Numerical results

Modelling

Conclusions

Perspectives

Porous Media

◮ 2D Arrays media and 3D Packing, Bentheimer and Clashach media.

(a) A1 (b) P1 (c) C1 (d) B1 (e) Mesh P1

(f) A2 (g) P2 (h) C2 (i) B2 (j) Mesh B1γ(

s−1

)

(k) B1

Fig : Investigated porous media.

Keywords: wide panel, complex 3D structure, isotropic, pore size distribution(PSD), local mesh refinment.

7 / 19


F. Zami-Pierre

Introduction

Context

Rheology

Porous Media


Numerical Study

Numerical set up

Numerical results

Modelling

Conclusions

Perspectives

Permeability prediction

Porous media

X-ray micro-tomography

Denoising &Thresholding

Smoothing

Meshing

Solver

Solution

REV? (b)

Resolution?

Subjectivity?(c) and (d)

Error induced?

Grid quality

Schemesaccuracy

(a) Workflow

0 200 400 600 800 1,000

0

0.5

1

1.5

l(µm)

φ∗

B1B2C1C2S1S2

(b) φ∗ versus cube length

(c) Seg. 1 (d) Seg. 2

Fig : Permeability prediction issues

8 / 19


F. Zami-Pierre

Introduction

Context

Rheology

Porous Media


Numerical Study

Numerical set up

Numerical results

Modelling

Conclusions

Perspectives

Numerical set up

◮ Equations: 0 = −∇p +∇ · [ν(γ)(∇U+∇UT )], ∇ · U = 0 .

◮ FVM with OpenFOAM, 2nd order schemes, and the SIMPLE algorithm[Patankar, 1980].

◮ Permeameter, no-slip conditions.

◮ Grid convergence study, ≃ 100 millions cells, 105 hours of CPU time,use of HPC.

Fig : P1 grid

100 101

10−1

100

101

102

1/(

h

h0

)

e(〈U〉)

in%

SCB

Fig : Grid convergence study

◮ Using 1 cell = 1 voxel → factor 2 in final permeability prediction.

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F. Zami-Pierre

Introduction

Context

Rheology

Porous Media


Numerical Study

Numerical set up

Numerical results

Modelling

Conclusions

Perspectives

Numerical results: at the micro-scale

−10 0 20 40

10−5

10−4

10−3

10−2

10−1

1 210

−1

100

(a) PDF of U∗

z

−20 −10 0 10 20

−0.2 0.2

10−1

100

(b) PDF of U∗

y

A2 n = 1.00 B2 n = 1.00 C2 n = 1.00 P1 n = 1.00A2 n = 0.75 B2 n = 0.75 C2 n = 0.75 P1 n = 0.75

Fig : PDF of longitudinal and transverse dimensionless velocity (U∗ = U/〈U〉)for a Newtonian and a non-Newtonian flow.

Keywords: correlation to geometrical features, different distribution (maxvelocities, co- and counter-current flow), exponential decay for P1, isotropy,weak impact of nonlinear effects.

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F. Zami-Pierre

Introduction

Context

Rheology

Porous Media


Numerical Study

Numerical set up

Numerical results

Modelling

Conclusions

Perspectives

Numerical results: at the macro-scale

10−2

103

100

101

〈Uc〉FL

10−2

10−1

1

Fig : Apparent dimensionless permeability k∗ = k/k0 vs. the intrinsic averagevelocity 〈U〉FL

Keywords: Newtonian and non-Newtonian regime, only φPT needed to benon-Newtonian, define a critical velocity 〈Uc〉FL

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F. Zami-Pierre

Introduction

Context

Rheology

Porous Media


Numerical Study

Numerical set up

Numerical results

Modelling

Conclusions

Perspectives

Transition’s model: a remarkable finding

◮ Our goal: predict 〈Uc 〉FL = γcℓeff .

◮ Goal: predict ℓeff .

Model for ℓeff :

◮ We have tried:√

8k0/φ,√

32k0/φ,√

k0/φ (with k0 obtained fromDNS).

◮ Use of Kozeny-Carman formulation and equivalent diameter, [du Plessisand Roos, 1994; Kozeny, 1927; Sadowski, 1963].

◮ Use of volume or surface of the medium (Vpart , Vmedium, Spart) [Ozahiet al., 2008].

Best results using simply ℓeff =√k0. Leading to:

〈Uc 〉FL = γc√

k0. (2)

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F. Zami-Pierre

Introduction

Context

Rheology

Porous Media


Numerical Study

Numerical set up

Numerical results

Modelling

Conclusions

Perspectives

Transition’s model: a remarkable finding

10−2

103

100

101

〈Uc〉FL

10−2

10−1

1

(a) k∗ vs 〈U〉FL(

µm.s−1

)

10−1 1 10

1

A1

A2

C1

C2

B1

B2

P1

P2

(b) k∗ vs U∗

Fig : Apparent dimensionless permeability k∗ = k/k0 against (a): the averagevelocity 〈U〉FL and (b): the dimensionless velocity U∗ = 〈U〉FL/

(

γc√k0)

.

Keywords: Newtonian and non-Newtonian regime, only φPT needed to benon-Newtonian, define a critical velocity, predict simple 〈Uc〉FL as γc

√k0

works!

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F. Zami-Pierre

Introduction

Context

Rheology

Porous Media


Numerical Study

Numerical set up

Numerical results

Modelling

Conclusions

Perspectives

Let’s get back to micro-scale results...

〈U〉 = 0.1 cm.D−1 〈U〉 = 1 cm.D

−1 ≃ 〈Uc〉

〈U〉 = 8 cm.D−1 〈U〉 = 22 cm.D

−1

µ/µ0

Fig : Viscosity fields at different flow regime for C1 case.

Keywords: domain extension, sources (surfaces and PT), critical region φPT. 14 / 19


F. Zami-Pierre

Introduction

Context

Rheology

Porous Media


Numerical Study

Numerical set up

Numerical results

Modelling

Conclusions

Perspectives

Transition’s model: a derivation

〈Uc 〉FL = γcℓeff〈Uc 〉FL ∝ γc 〈Uc 〉FL ⊥⊥ n

The macro-scalebehaviour is

sensitive to φPT

The viscous dis-sipation mainlyoccurs in φPT

〈Uc 〉FL = γcℓPTβ

√k0 ≃ φFL√

φPT

ℓPTβ

ℓeff ≃√k0

β =〈Uc 〉PT /〈Uc 〉FL k0 =

µ0〈U〉2〈E〉

Keywords: small sub domain φPT, viscous dissipation, non-Newtoniantransition.

15 / 19


F. Zami-Pierre

Introduction

Context

Rheology

Porous Media


Numerical Study

Numerical set up

Numerical results

Modelling

Conclusions

Perspectives

Partial Conclusions

◮ Studying cut-off phenomena due to non-Newtonian fluid throughisotropic porous media.

◮ Nonlinear effects weakly impact the flow statistics (PDF).

◮ A small subdomain φPT controls the macro-scale behaviour. It controlsboth the viscous dissipation and the non-Newtonian transition.

◮ Simple theory provides analytical formulation for the transition velocity.

◮ Explain why ℓeff ≃√k0. Used in many porous media applications

(core-flood experiments, Bingham fluids, inertial regime) → meaningfullength (topology+flow), universal role?

16 / 19


F. Zami-Pierre

Introduction

Context

Rheology

Porous Media


Numerical Study

Numerical set up

Numerical results

Modelling

Conclusions

Perspectives

PerspectivesPrevious work based on a simple view of polymer flow, i.e., generalized N.S.A simplified view leads to split the porous medium domain into 3 area;

◮ bulk phase, p-phase (→ Justify continuum approach?),

◮ excluded zones (where macro-molecules do not enter), e-phase,

◮ boundary zones (specific arrangements), b-phase.

p-phase

b-phase

e-phases-phase

Fig : Sketch of polymer repartition at the pore-scale.

◮ Adoption of a continuum approach? → Several conceptual difficulties...

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F. Zami-Pierre

Introduction

Context

Rheology

Porous Media


Numerical Study

Numerical set up

Numerical results

Modelling

Conclusions

Perspectives

Perspectives

Two major mechanisms [Chauveteau,1984; De Gennes, 1981]:

◮ Sorption [Hatzikiriakos, 2012].

◮ Repulsion [Sorbie and Huang,1991].

⇒ Need of practical micro-scale ob-servations to quantify slip [Chenneviereet al., 2016]

◮ Approach (a): MDS [Rouse Jr,1953; Joshi et al., 2000; Cuencaand Bodiguel, 2013].

◮ Approach (b): Meso-scalemodel [Wood et al., 2004].

◮ Approach (c) and (d): effectivesurface concept [Achdou et al.,1998; Introıni et al., 2011].

(a) (b)

(c)

p pb

(d)

Fig : Various models for describingtransport near a solid wall.

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F. Zami-Pierre

Introduction

Context

Rheology

Porous Media


Numerical Study

Numerical set up

Numerical results

Modelling

Conclusions

Perspectives

This work was performed using HPC resources from CALMIP (Grant 2015-11).

This work was supported by Total.

Thank you for your attention. Any question is welcome!

PS: I am looking for a job next year :)

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F. Zami-Pierre

Is it new?

Is ℓeff =√k0 so surprising and new?

γeq = α4〈U〉/φ√

8k0/φ⇐⇒ 〈Uc 〉 =

1√2φα

× φγc√

k0 (3)

Thanks to core-flood experiments, we can calculate the pre-factor,

Ref erence Medium 1√2φα

[Lecourtier and Chauveteau, 1984] P1 0.71[Chauveteau, 1982] P1 0.87[Chauveteau, 1984] C 0.52[Fletcher et al., 1991] C 0.46

Tab : Comparison with data found in literature

This would mean that without complex physical phenomena, ℓeff =√k0 is a

good estimation for the ℓeff . Plus, we added a model that explains by a simpleapproach why this length works.

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F. Zami-Pierre

Independence with n?

Fig : Critical velocity 〈Uc 〉FL (µm.s−1) as a function of γc (s−1) for mediumC1 and different values of n.

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F. Zami-Pierre

Transition’s model: a derivation

〈Uc 〉FL = γcℓeff

〈Uc 〉PT = γcℓPT

β =〈Uc 〉PT /〈Uc 〉FL

> 1

〈Uc 〉FL = γcℓPTβ

√k0 ≃ φFL√

φPT

ℓPTβ

With E = 2µe :

e, k0 =µ0〈U〉2〈E〉 ,

k0 =〈U〉2〈γ2〉

ℓeff ≃√k0

〈Uc 〉FL ∝ γc〈Uc 〉FL ⊥⊥ n

The macro-scalebehaviour issensitive to

the PT volume

〈γ2〉≃ φPT〈γ2〉PT= φPT

〈U〉2PTℓ2PT

φFL√φPT

≃ 1

Keywords: small sub domain φPT, viscous dissipation, non-Newtoniantransition.

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F. Zami-Pierre

Numerical results: at the micro-scale

10−3 10−2 10−1 100 101 102 10310−6

10−2

102

γ/γc

PDF

(a) 〈U〉 = 1µm.s−1 A1, A2, B1, B2,C1, C2, P1, P2

10−3

10−2

10−1

100

101

102

103

10−7

10−3

101

γ/γc

PDF

(b) 〈U〉(

µm.s−1)

= 0.3 , 2.2 , 31

Fig : PDF of dimensionless shear rate for all media (Newtonian andnon-Newtonian) flow.

Keywords: geometrical differences & nonlinearities induced by the flow,translating to non-Newtonian regime, pore-throat (PT).

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F. Zami-Pierre

Achdou, Y., Le Tallec, P., Valentin, F., and Pironneau, O. (1998).Constructing wall laws with domain decomposition or asymptoticexpansion techniques.Computer methods in applied mechanics and engineering,151(1):215–232.

Bird, R. B. and Carreau, P. J. (1968).A nonlinear viscoelastic model for polymer solutions and melts.Chemical Engineering Science, 23(5):427 – 434.

Chauveteau, G. (1982).Rodlike polymer solution flow through fine pores: influence of pore sizeon rheological behavior.Journal of Rheology (1978-present), 26(2):111–142.

Chauveteau, G. (1984).Concentration dependence of the effective viscosity od polymer solutionsin small pores with repulsive or attractive walls.Journal of Colloid and Interface Science, 100:41–54.

Chenneviere, A., Cousin, F., Boue, F., Drockenmuller, E., Shull, K. R.,

Leger, L., and Restagno, F. (2016).Direct molecular evidence of the origin of slip of polymer melts ongrafted brushes.Macromolecules, 49(6):2348–2353.

Collyer, A. (1973).Time independent fluids.

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F. Zami-Pierre

Physics Education, 8(5):333.

Cuenca, A. and Bodiguel, H. (2013).Submicron flow of polymer solutions: Slippage reduction due toconfinement.Physical review letters, 110(10):108304.

De Gennes, P. (1981).Polymer solutions near an interface. adsorption and depletion layers.Macromolecules, 14(6):1637–1644.

du Plessis, J. and Roos, L. (1994).Predicting the hydrodynamic permeability of sandstone with a pore-scalemodel.Journal of Geophysical Research.

Fletcher, A., Flew, S., Lamb, S., Lund, T., Bjornestad, E., Stavland, A.,

Gjovikli, N., et al. (1991).Measurements of polysaccharide polymer properties in porous media.In SPE International Symposium on Oilfield Chemistry. Society ofPetroleum Engineers.

Getachew, D., Minkowycz, W., and Poulikakosi, D. (1998).Macroscopic equations of non-newtonian fluid flow and heat transfer.Journal of Porous Media, 1(3):273–283.

Hatzikiriakos, S. G. (2012).Wall slip of molten polymers.

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F. Zami-Pierre

Progress in Polymer Science, 37(4):624–643.

Introıni, C., Quintard, M., and Duval, F. (2011).Effective surface modeling for momentum and heat transfer over roughsurfaces: Application to a natural convection problem.International Journal of Heat and Mass Transfer, 54(15):3622–3641.

Joshi, Y. M., Lele, A. K., and Mashelkar, R. (2000).Slipping fluids: a unified transient network model.Journal of non-newtonian fluid mechanics, 89(3):303–335.

Kozeny, M. (1927).Uber kapillare leitung des wassers im boden.Sitzber. Akad. Wiss. Wein, Math-naturw, 136:Abt. II a, P. 277.

Lecourtier, J. and Chauveteau, G. (1984).Xanthan fractionation by surface exclusion chromatography.Macromolecules, 17(7):1340–1343.

Ozahi, E., Gundogdu, M. Y., and Carpinlioglu, M. . (2008).A modification on ergun’s correlation for use in cylindrical packed bedswith non-spherical particles.Advanced Powder Technology, 19(4):369 – 381.

Patankar, S. V. (1980).Numerical heat transfer and fluid flow.Series in computational methods in mechanics and thermal sciences.Hemisphere Pub. Corp. New York, Washington.

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F. Zami-Pierre

Rouse Jr, P. E. (1953).A theory of the linear viscoelastic properties of dilute solutions of coilingpolymers.The Journal of Chemical Physics, 21(7):1272–1280.

Sadowski, T. (1963).Non-Newtonian flow through porous media.PhD thesis, Univiersity of Wisconsin.

Savins, J. (1969).Non-newtonian flow through porous media.Industrial & Engineering Chemistry, 61(10):18–47.

Sorbie, K. and Huang, Y. (1991).Rheological and transport effects in the flow of low-concentrationxanthan solution through porous media.Journal of Colloid and Interface Science, 145(1):74 – 89.

Wood, B. D., Quintard, M., and Whitaker, S. (2004).Estimation of adsorption rate coefficients based on the smoluchowskiequation.Chemical engineering science, 59(10):1905–1921.

Zami-Pierre, F., de Loubens, R., Quintard, M., and Davit, Y. (2016).Transition in the flow of power-law fluids through isotropic porousmedia.Physical Review Letters (publication under review).

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F. Zami-Pierre

Zitha, P., Chauveteau, G., and Zaitoun, A. (1995).Permeability dependent propagation of polyacrylamides undernear-wellbore flow conditions.In SPE International Symposium on Oilfield Chemistry.

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Open Archive TOULOUSE Archive Ouverte (OATAO)Zami-Pierre, Frédéric and Davit, Yohan and Loubens,...

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