This is an author-deposited version published in: Eprints ID : 15932

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1 Open Archive TOULOUSE Archive Ouverte (OATAO) OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible. This is an author-deposited version published in: Eprints ID : To cite this version : Zami-Pierre, Frédéric and Davit, Yohan and Loubens, Romain de and Quintard, Michel fluids through. (2016) In: Pore Scale Modeling Total Workshop, 8 June June 2016 (Pau, France). (Unpublished) Any correspondence concerning this service should be sent to the repository administrator: staff-oatao@listes-diff.inp-toulouse.fr

2 porous media 1,2, Y. Davit 1, R. de Loubens 2 and M. Quintard 1 1 Institut de Mécanique des Fluides de Toulouse (IMFT) - Université de Toulouse, CNRS-INPT-UPS, Toulouse FRANCE 2 Total, DEV/GIS/SIM June 8, /19

3 Overview 2/19

4 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, Physical Review Letters, publication under review (2016) 3/19

5 µ = µ 0 U = K p µ 0 U p 1 µ = µ β γ n 1 U = K( U ) p µ 0 U p 1/n 2 µ k 1 2? Newtonian regime 1 non-newtonian regime 2 γ c γ U c U Fig : Non-Newtonian rheology 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

6 Literature: γ eq = 4α U /φ 8k/φ, [Chauveteau, 1982]. Two semi-empirical parameters: R eq = 8k/φ, comes from an analogy with a single pipe. Is it still valid in complex multi-scale? 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. Can we predict and understand the transition velocity, U c? 5/19

7 EOR polymer solutions: shearthinning, no yield, assume timeindependent. Common models are, plateau + power-law, µ = Carreau, µ 0 if γ < γ c, ( ) n 1 µ 0 γ γ else, c generalized Newtonian. (1) µ/µ plateau + power-law Carreau generalized Newtonian γ/ γ c Fig : A few rheological model Refs: [Bird and Carreau, 1968; Savins, 1969; Collyer, 1973] 6/19

8 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 (k) B1 γ ( s 1) Fig : Investigated. Keywords: wide panel, complex 3D structure, isotropic, pore size distribution (PSD), local mesh refinment. 7/19

9 Porous media REV? (b) 1.5 X-ray microtomography Denoising & Thresholding Smoothing Resolution? Subjectivity? (c) and (d) Error induced? φ 1 B1 B2 0.5 C1 C2 S1 0 S ,000 l(µm) (b) φ versus cube length Meshing Grid quality Solver Schemes accuracy Solution (c) Seg. 1 (d) Seg. 2 (a) Workflow Fig : issues 8/19

10 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. e( U ) in % S C B Fig : P1 grid ( ) 1 h 1/ h 0 Fig : Grid convergence study Using 1 cell = 1 voxel factor 2 in final permeability. 9/19

11 : at the micro-scale (a) PDF of Uz (b) PDF of Uy 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/ U ) 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. 10/19

12 : at the macro-scale U c FL Fig : Apparent dimensionless permeability k = k/k 0 vs. the intrinsic average velocity U FL Keywords: Newtonian and non-newtonian regime, only φ PT needed to be non-newtonian, define a critical velocity U c FL 11/19

13 Transition s model: a remarkable finding Our goal: predict U c FL = γ c l eff. Goal: predict l eff. Model for l eff : We have tried: 8k 0 /φ, 32k 0 /φ, k 0 /φ (with k 0 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, V medium, S part ) [Ozahi et al., 2008]. Best results using simply l eff = k 0. Leading to: U c FL = γ c k0. (2) 12/19

14 Transition s model: a remarkable finding U c FL 1 A1 A2 C1 C2 B1 B2 P1 P ( (a) k vs U ) FL µm.s 1 (b) k vs U Fig : Apparent dimensionless permeability k = k/k 0 against (a): the average velocity U FL and (b): the dimensionless velocity U = U FL / ( γ c k0 ). Keywords: Newtonian and non-newtonian regime, only φ PT needed to be non-newtonian, define a critical velocity, predict simple U c FL as γ c k0 works! 13/19

15 Let s get back to micro-scale results... U = 0.1cm.D 1 U = 1cm.D 1 U c U = 8cm.D 1 U = 22cm.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. 14/19

16 Transition s model: a derivation U c FL γ c U c FL = γ c l eff U c FL n The macro-scale behaviour is sensitive to φ PT β = U c PT / U c k 0 = µ 0 U 2 FL E The viscous dissipation mainly occurs in φ PT U c FL = γ c l PT β k0 φ FL φpt l PT β l eff k 0 Keywords: small sub domain φ PT, viscous dissipation, non-newtonian transition. 15/19

17 Partial Studying cut-off phenomena due to non-newtonian fluid through isotropic. 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 k 0. Used in many applications (core-flood experiments, Bingham fluids, inertial regime) meaningful length (topology+flow), universal role? 16/19

18 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. b-phase p-phase e-phase s-phase Fig : Sketch of polymer repartition at the pore-scale. Adoption of a continuum approach? Several conceptual difficulties... 17/19

19 Two major mechanisms [Chauveteau, 1984; De Gennes, 1981]: Sorption [Hatzikiriakos, 2012]. Repulsion [Sorbie and Huang, 1991]. Need of practical micro-scale observations 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]. p (a) b p (b) Approach (c) and (d): effective surface concept [Achdou et al., 1998; Introïni et al., 2011]. (c) (d) Fig : Various models for describing transport near a solid wall. 18/19

20 This work was performed using HPC resources from CALMIP (Grant ). This work was supported by Total. Thank you for your attention. Any question is welcome! PS: I am looking for a job next year :) 19/19

21 Is it new? Is l eff = k 0 so surprising and new? γ eq = α 4 U /φ 8k0 /φ U c = 1 2φα φ γ c k0 (3) Thanks to core-flood experiments, we can calculate the pre-factor, Ref erence Medium 1 2φα [Lecourtier and Chauveteau, 1984] P [Chauveteau, 1982] P [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, l eff = k 0 is a good estimation for the l eff. Plus, we added a model that explains by a simple approach why this length works. 20/19

22 Independence with n? Fig : Critical velocity U c FL (µm.s 1 ) as a function of γ c (s 1 ) for medium C1 and different values of n. 21/19

23 Transition s model: a derivation U c FL γ c U c FL n U c FL = γ c l eff The macro-scale behaviour is sensitive to the PT volume U c PT = γ c l PT β = U c PT / U c FL > 1 U c FL = γ c l PT β With E = 2µe : e, k 0 = µ 0 U 2 E, k 0 = U 2 γ 2 k0 φ FL φpt l PT β γ 2 φ PT γ 2 PT = φ PT U 2 PT l 2 PT φ FL φpt 1 l eff k 0 Keywords: small sub domain φ PT, viscous dissipation, non-newtonian transition. 22/19

24 : at the micro-scale 10 2 PDF 10 2 γ/ γ c (a) U = 1µm.s 1 A1, A2, B1, B2, C1, C2, P1, P2 PDF 10 3 γ/ γ c (b) U ( 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). 23/19

25 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): Bird, R. B. and Carreau, P. J. (1968). A nonlinear viscoelastic model for polymer solutions and melts. Chemical Engineering Science, 23(5): Chauveteau, G. (1982). Rodlike polymer solution flow through fine pores: influence of pore size on rheological behavior. Journal of (1978-present), 26(2): 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: 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): Collyer, A. (1973). Time independent fluids. 24/19

26 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): De Gennes, P. (1981). Polymer solutions near an interface. adsorption and depletion layers. Macromolecules, 14(6): 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. 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, 1(3): Hatzikiriakos, S. G. (2012). Wall slip of molten polymers. 25/19

27 Progress in Polymer Science, 37(4): Introïni, 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): 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): Kozeny, M. (1927). Uber kapillare leitung des wassers im boden. Sitzber. Akad. Wiss. Wein, Math-naturw, 136:Abt. II a, P Lecourtier, J. and Chauveteau, G. (1984). Xanthan fractionation by surface exclusion chromatography. Macromolecules, 17(7): 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): 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. 26/19

28 Rouse Jr, P. E. (1953). A theory of the linear viscoelastic properties of dilute solutions of coiling polymers. The Journal of Chemical Physics, 21(7): Sadowski, T. (1963). Non-Newtonian flow through. PhD thesis, Univiersity of Wisconsin. Savins, J. (1969). Non-newtonian flow through. Industrial & Engineering Chemistry, 61(10): Sorbie, K. and Huang, Y. (1991). Rheological and transport effects in the flow of low-concentration xanthan solution through. Journal of Colloid and Interface Science, 145(1): Wood, B. D., Quintard, M., and Whitaker, S. (2004). Estimation of adsorption rate coefficients based on the smoluchowski equation. Chemical engineering science, 59(10): 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). 27/19

29 Zitha, P., Chauveteau, G., and Zaitoun, A. (1995). dependent propagation of polyacrylamides under near-wellbore flow conditions. In SPE International Symposium on Oilfield Chemistry. 28/19