Flow of polymer fluids through porous media

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Eprints ID : 15932

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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 (Unpublished)

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solutions through porous media F. Zami-Pierre Introduction Context Rheology Porous Media Permeability prediction 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

1_{Institut de M´}_{ecanique des Fluides de Toulouse (IMFT) - Universit´}_{e de}

Toulouse, CNRS-INPT-UPS, Toulouse FRANCE

2_{Total, DEV/GIS/SIM}

June 8, 2016

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Overview

Introduction

Context

Rheology

Porous Media

Permeability prediction

Numerical Study

Numerical set up

Numerical results

Modelling

Conclusions

Perspectives

2 / 19

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Scope of work

◮ _{EOR context:}

→ adverse mobility ratio waterflood,

→ hydrodynamic instabilities, → use of polymer.

◮ _{Physics is complex...}

◮ _{Lack of micro-scale observations}

→ real mechanisms remain unclear.

◮ _{Porous media flow is also}

complex, e.g, multi-scale, confinement effects.

˙γ !10−6 s−1

Fig : Power-law fluid flowing through

a Bentheimer sandstone [Zami-Pierre et al., 2016]

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

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Context

µ = µ₀ _⇒ _{hUi = −}K·∇p_µ 0 ⇒ hUi ∝ k∆pk ⇒

1

µ = µ_β ˙γn−1 _⇒ _{hUi = −}K(hUi)·∇p_µ 0 ⇒ hUi ∝ k∆pk 1/n _⇒

2 ˙

γ

c 1

2

˙

γ

µ

Fig : Non-Newtonian rheology

hU

c

i

non-Newtonian regime 2

Newtonian regime 1

?

hUi

k

Fig : k _{versus hUi}

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

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Context

◮ _{Literature: ˙γ} eq = 4α√hUi/φ 8k/φ, [Chauveteau, 1982].

Two semi-empirical parameters: ◮ _R

eq = p8k/φ, comes from an analogy with a single pipe. Is it

still valid in complex multi-scale porous media?

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

◮ _{Our goal: simulate the flow a non-Newtonian fluid over a wide panel of}

porous media. Can we predict and understand the transition velocity, hUci?

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Rheology

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

◮ _plateau _{+ power-law,} µ =    µ₀ if ˙γ < ˙γc, µ₀ _γ_˙γ˙ c n−1 else, (1) ◮ _Carreau, ◮ _{generalized Newtonian.} 10−4₁₀−3₁₀−2₁₀−1 ₁₀0 ₁₀1 ₁₀2 ₁₀3 ₁₀4 10−2 10−1 100

˙

γ/ ˙γ

c

µ/

µ

0 plateau + power-law Carreau generalized Newtonian

Fig : A few rheological model

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

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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 B1

˙γ!s−1

(k) B1

Fig : Investigated porous media.

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

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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 Schemes accuracy (a) Workflow 0 200 400 600 800 1,000 0 0.5 1 1.5

l

_(µm)

φ

∗ B1 B2 C1 C2 S1 S2

(b) φ∗ versus cube length

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

Fig : Permeability prediction issues

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Numerical set up

◮ _{Equations: 0 = −∇p + ∇ · [ν( ˙γ)(∇U + ∇U}T_{)], ∇ · U = 0 .}

◮ _{FVM with OpenFOAM, 2}nd _{order schemes, and the SIMPLE algorithm}

[Patankar, 1980].

◮ _{Permeameter, no-slip conditions.}

◮ _{Grid convergence study, ≃ 100 millions cells, 10}5 _{hours of CPU time,}

use of HPC. Fig : P1 grid 100 101 10−1 100 101 102

1/

_hh 0

e

(h

U

i)

in

%

S C B

Fig : Grid convergence study

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

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Numerical results: at the micro-scale

−10 0

20

40

10

−5

10

−4

10

−3

10

−2

10

−1 1 2 10−1 100 (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 = 1.00 A2 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/hUi)}

for a Newtonian and a non-Newtonian flow.

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

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Numerical results: at the macro-scale

10 −2

10

3

10

0

10

1

hU

c

i

FL

10−

2 10−

1

Fig : Apparent dimensionless permeability k∗ = k/k0 vs. the intrinsic average

velocity hUiFL

Keywords: Newtonian and non-Newtonian regime, only φ_PT needed to be non-Newtonian, define a critical velocity hUciFL

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Transition’s model: a remarkable finding

◮ _{Our goal: predict hU}

ci_FL = ˙γcℓeff .

◮ _{Goal: predict ℓ}

eff .

Model for ℓeff :

◮ _{We have tried:} p8k

0/φ, p32k0/φ, pk0/φ (with k0 obtained from

DNS).

◮ _{Use of Kozeny-Carman formulation and equivalent diameter, [du Plessis}

and Roos, 1994; Kozeny, 1927; Sadowski, 1963].

◮ _{Use of volume or surface of the medium (V}

part, Vmedium, Spart) [Ozahi

et al., 2008].

Best results using simply ℓ_eff = √k₀. Leading to: hUci_FL = ˙γc

p

k₀. (2)

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Transition’s model: a remarkable finding

10

−2

10

3

10

0

10

1

hU

c

i

FL 10−2 10−1 1 (a) k∗ vs hU i 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 average

velocity hUiFL and (b): the dimensionless velocity U∗ = hUiFL/! ˙γc√k0.

Keywords: Newtonian and non-Newtonian regime, only φ_PT needed to be non-Newtonian, define a critical velocity, predict simple hUciFL as ˙γc√k0

works!

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

hUi = 0

_.1

cm

.

D

−1

hUi = 1 cm

.

D

−1

≃ hU

c

i

hUi = 8 cm

_.

D

−1

hUi = 22 cm

.

D

−1

µ/µ

0

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

Keywords: domain extension, sources (surfaces and PT), critical region φ_PT.

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Transition’s model: a derivation

hUciFL = ˙γcℓeff

hUciFL ∝ ˙γc hUciFL ⊥⊥ n

The macro-scale behaviour is sensitive to φ_PT

The viscous dis-sipation mainly occurs in φ_PT hUciFL = ˙γc ℓ_βPT √k₀ ≃ √φFL φ_PT ℓ_PT β ℓ_eff _≃ √k₀ β = hUciPT/hUciFL k0 = µ₀_hUi2 hEi

Keywords: small sub domain φ_PT, viscous dissipation, non-Newtonian transition.

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Partial Conclusions

◮ _{Studying cut-off phenomena due to non-Newtonian fluid through}

isotropic porous media.

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

◮ _{A small subdomain φ}

PT controls the macro-scale behaviour. It controls

both the viscous dissipation and the non-Newtonian transition.

◮ _{Simple theory provides analytical formulation for the transition velocity.}

◮ _{Explain why ℓ}

eff ≃ √k0. Used in many porous media applications

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

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Perspectives

Previous 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-phase

s-phase

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

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

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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 [Chenneviere et al., 2016]

◮ _{Approach (a): MDS [Rouse Jr,}

1953; Joshi et al., 2000; Cuenca and Bodiguel, 2013].

◮ _{Approach (b): Meso-scale}

model [Wood et al., 2004].

◮ _{Approach (c) and (d): effective}

surface concept [Achdou et al., 1998; Intro¨ıni et al., 2011].

(a)

(b)

(c)

p

b

(d)

Fig : Various models for describing transport near a solid wall.

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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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solutions through porous media

F. Zami-Pierre

Is it new?

Is ℓeff = √k0 so surprising and new?

˙γeq = α 4hUi/φ p8k₀/φ ⇐⇒ hUci = 1 √ 2φα × φ ˙γc p 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 simple

approach why this length works.

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F. Zami-Pierre

Independence with n?

Fig : _{Critical velocity hU}ci_FL (µm.s−1) as a function of ˙γc (s−1) for medium

C1 and different values of n.

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F. Zami-Pierre

Transition’s model: a derivation

hUciFL = ˙γcℓeff hUciPT = ˙γcℓ_PT β = hUciPT/hUciFL > 1 hUciFL = ˙γc ℓPT_β √ k₀ _≃ _√φFL φ_PT ℓ_PT β With E = 2µe : e, k₀ = µ0hUi2 hEi , k₀ = hUi2 h ˙γ2i ℓ_eff _≃ √k₀ hUciFL ∝ ˙γc hUciFL ⊥⊥ n The macro-scale behaviour is sensitive to the PT volume h ˙γ2i ≃ φPTh ˙γ2iPT = φ_PThUi2PT ℓ2_PT φ_FL √ φ_PT ≃ 1

Keywords: small sub domain φ_PT, viscous dissipation, non-Newtonian transition.

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F. Zami-Pierre

Numerical results: at the micro-scale

10−3 10−2 10−1 100 101 102 103 10−6 10−2 102 ˙γ

/

_˙γ_c P D F

(a) hU i = 1 µm.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

/

_˙γ_c P D F (b) hU i! µm.s−1 = 0.3 , 2.2 , 31

Fig : PDF of dimensionless shear rate for all media (Newtonian and

non-Newtonian) flow.

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

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Flow of polymer solutions through

porous media

F. Zami-Pierre

Achdou, Y., Le Tallec, P., Valentin, F., and Pironneau, O. (1998).

Constructing wall laws with domain decomposition or asymptotic expansion 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 size on rheological behavior.

Journal of Rheology (1978-present), 26(2):111–142.

Chauveteau, G. (1984).

Concentration dependence of the effective viscosity od polymer solutions in 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 on grafted brushes.

Macromolecules, 49(6):2348–2353.

Collyer, A. (1973).

Time independent fluids.

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porous media

F. Zami-Pierre

Physics Education, 8(5):333.

Cuenca, A. and Bodiguel, H. (2013).

Submicron flow of polymer solutions: Slippage reduction due to confinement.

Physical review letters, 110(10):108304.

De Gennes, P. (1981).

Polymer solutions near an interface. adsorption and depletion layers.

du Plessis, J. and Roos, L. (1994).

Predicting the hydrodynamic permeability of sandstone with a pore-scale model.

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 of Petroleum 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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porous media

F. Zami-Pierre

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

Intro¨ıni, C., Quintard, M., and Duval, F. (2011).

Effective surface modeling for momentum and heat transfer over rough surfaces: 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.

Ozahi, E., Gundogdu, M. Y., and Carpinlioglu, M. . (2008).

A modification on ergun’s correlation for use in cylindrical packed beds with non-spherical particles.

Advanced Powder Technology, 19(4):369 – 381.

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Numerical heat transfer and fluid flow.

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porous media

F. Zami-Pierre

Rouse Jr, P. E. (1953).

A theory of the linear viscoelastic properties of dilute solutions of coiling polymers.

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Sadowski, T. (1963).

Non-Newtonian flow through porous media.

PhD thesis, Univiersity of Wisconsin.

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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-concentration xanthan solution through porous media.

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Estimation of adsorption rate coefficients based on the smoluchowski equation.

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Zami-Pierre, F., de Loubens, R., Quintard, M., and Davit, Y. (2016).

Transition in the flow of power-law fluids through isotropic porous media.

Physical Review Letters (publication under review).

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porous media

F. Zami-Pierre

Zitha, P., Chauveteau, G., and Zaitoun, A. (1995).

Permeability dependent propagation of polyacrylamides under near-wellbore flow conditions.

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Flow of polymer fluids through porous media

Flow of polymer solutions through porous

media

Overview

Introduction

Context

Rheology

Porous Media

Permeability prediction

Numerical Study

Numerical set up

Numerical results

Modelling

Conclusions

Perspectives

2 / 19

Scope of work

Context

1

2

˙

γ

˙

γ

µ

hU

i

?

hUi

k

Context

Rheology

˙

γ/ ˙γ

µ/

µ

Porous Media

Permeability prediction

l

(µm)

φ

Numerical set up

1/





e

(h

U

i)

in

%

Numerical results: at the micro-scale

−10 0

20

40

10

10

10

10

10

−20 −10

0

10

20

Numerical results: at the macro-scale

10

−2

10

3

10

0

10

1

hU

c

i

FL

10−

2

10−

_(µm)

_.1

_.