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A new incompressible gas-liquid two-phase flow model for non-Newtonian power-law fluid is proposed based on an incompressible lattice Boltzmann model. And the fundamental physical mechanism of Newtonian fluid displacing non-Newtonian power-law fluid liquid in porous medium is studied by using the proposed model. The effects of capillary number Ca, dynamic viscosity ratio M, surface wettability θ, porous medium geometry, and power law index n on the displacement process are investigated. The comprehensive results show that with the increase of capillary number, the displacement process turns faster, the fingering phenomenon becomes more obvious and the displacement efficiency decreases. However, for different values of power-law index n, the effects of the Ca on the displacement process have some differences. Specially, the decrease rate of displacement efficiency becomes slow if the displaced fluid is shear thickening fluid as compared with that if the displaced fluid is shear thinning fluid. On the other hand, the displacement efficiency decreases as dynamic viscosity ratio M increases. And the effect of the viscosity ratio on the displacement process becomes more obvious for the low value of the power-law index n. Moreover, the effect of the surface wettability of the porous medium on the displacement process is also related to the size of the power-law index. With the increase of the contact angle of the porous medium, the fingering phenomenon turns less obvious, and the displacement efficiency increases. However, with the increase of power-law index n, the influence of the contact angle on the displacement process decreases. Besides, the displacement processes with different geometric types of the porous media are also studied in the work. The results show that comparing with the case of porous medium denoted by circle shape and square shape, the fingering phenomenon obtained by the case of triangular shape is most obvious, and the displacement efficiency is lowest.
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Keywords:
- power-law two-phase fluid /
- lattice Boltzmann model /
- immiscible displacement /
- porous media
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Lou Q, Li T, Yang M 2018 Acta Phys. Sin. 67 234701
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Google Scholar
Zang C Q, Lou Q 2017 Acta Phys. Sin. 66 134701
Google Scholar
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Google Scholar
[11] Lou Q, Guo Z L 2015 Phys. Rev. E 91 013302
Google Scholar
[12] 娄钦, 李涛, 李凌 2018 上海理工大学学报 40 13
Lou Q, Li T, Li L 2018 J. Univ. Shanghai Sci. Technol. 40 13
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Google Scholar
Xie C Y, Zhang J Y, Wang M R 2016 Chin. J. Computat. Phys. 33 147
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
[18] Ba Y, Wang N N, Liu H H, Li Q, He G Q 2018 Phys. Rev. E 97 033307
Google Scholar
[19] Halliday I, Law R, Care C M, Hollis A 2006 Phys. Rev. E 73 056708
Google Scholar
[20] Halliday I, Hollis A P, Care C M 2007 Phys. Rev. E 76 026708
Google Scholar
[21] 闵琪, 段远源, 王晓东, 吴莘馨 2013 热科学与技术 12 335
Min Q, Duan Y Y, Wang X D, Wu X X 2013 J. Therm. Sci. Technol. 12 335
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Google Scholar
[23] Shan X W, Chen H D 1993 Phys. Rev. E 47 1815
Google Scholar
[24] Nourgaliev R R, Dinh T N, Theofanous T G, Joseph D 2003 Int. J. Multiphase Flow. 29 117
Google Scholar
[25] Huang H B, Sukop M, Lu X Y 2015 Multiphase Lattice Boltzmann Methods: Theory and Application (USA: WILEY Blackwell) pp7−10
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Google Scholar
[27] He X Y, Chen S Y, Zhang R Y 1999 J. Comput. Phys. 152 642
Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Guo Z L, Zheng C G 2008 Theory and Applications of Lattice Boltzmann Method (Beijing: Science Press) p244 (in Chinese)
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Google Scholar
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Google Scholar
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Google Scholar
[42] Ansarinasab J, Jamialahmadi M 2017 J. Pet. Sci. Eng. 156 748
Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
[47] 黄海波 2009 第六届全国流体力学青年研讨会 中国杭州 2009年10月10日 第27页
Huang H B 2009 The 6th National Youth Workshop on Fluid Mechanics Hangzhou, China October 10, 2009 p27 (in Chinese)
[48] Shiri Y, HassaniH,Nazari M, Sharifi M 2018 Mol. Simul. 44 708
Google Scholar
[49] Liu H H, ValocchiAJ, Kang Q J, Werth C 2013 Transp. Porous Media 99 555
Google Scholar
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Google Scholar
[51] Ferer M, Anna S L,Tortora P, Kadambi J R, Oliver M, Bromhal G S, Smith D H 2011 Transp. Porous Media 86 243
Google Scholar
[52] Dong B, YanY Y, Li W Z, Song Y C 2011 J. Bionic. Eng. 7 267
Google Scholar
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图 1 液滴内外压力差
${P_{\rm{i}}} - {P_{\rm{o}}}$ 和半径倒数1/r之间的关系Figure 1. Relationship between pressure jump across the droplet interface
${P_{\rm{i}}} - {P_{\rm{o}}}$ and inverse of droplet radius 1/r.
图 2 不同初始静态接触角
${\theta _{{\rm{eq}}}}$ 时得到的稳态接触角$\theta $ (a)${\theta _{{\rm{eq}}}}{\rm{ = }}{60^{\rm{o}}}$ ; (b)${\theta _{{\rm{eq}}}}{\rm{ = }}{90^{\rm{o}}};$ (c)${\theta _{{\rm{eq}}}}={120^{\rm{o}}}$ Figure 2. Steady state contact angles
$\theta $ obtained with the different values of static contact angles${\theta _{{\rm{eq}}}}$ : (a)${\theta _{{\rm{eq}}}}{\rm{ = }}{60^{\rm{o}}}$ ; (b)${\theta _{{\rm{eq}}}}{\rm{ = }}{90^{\rm{o}}};$ (c)${\theta _{{\rm{eq}}}}{\rm{ = }}{120^{\rm{o}}}$ .
图 3 稳态接触角
$\theta $ 与指标参数${\phi _{{\rm{wall}}}}$ 的线性关系Figure 3. Linear relationship between steady state contact angle
$\theta $ and the order parameter of a solid wall${\phi _{{\rm{wall}}}}$ .
图 4 T型通道问题物理模型
Figure 4. Physical model for the case of T shape channel.
图 5 不同Ca数对应的液滴形态 (a) Ca = 0.06370; (b) Ca = 0.06835; (c) Ca = 0.07300; (d) Ca = 0.07750; (e) Ca = 0.0820; (f) Ca = 0.08650; (g) Ca = 0.0910
Figure 5. Droplet morphology obtained under various values of Ca: (a) Ca = 0.06370; (b) Ca = 0.06835; (c) Ca = 0.07300; (d) Ca = 0.07750; (e) Ca = 0.0820; (f) Ca = 0.08650; (g) Ca = 0.0910.
图 6 在剪切变稀幂律流体中, 不同的
$Ca$ 数下形成液滴的无量纲直径(其中D是形成的液滴的直径, H是管径的直径)Figure 6. Droplet dimensionless diameters at different values of
$Ca$ in shear thinning power-law fluid. D is diameters of the droplet and H is width of the main channel.
图 7 多孔介质驱替模型
Figure 7. The model for porous media displacement problem.
图 8 不同的
$Ca$ 数下, 被驱替液为剪切变稀、牛顿与剪切变稠流体时得到的指进形态图 (a)−(c) n = 0.7; (d)−(f) n =1.0; (g)−(i) n = 1.3Figure 8. Final finger patterns obtained under different values of Ca for shear thinning, Newtonian and shear thickening fluids: (a)− (c) n = 0.7; (d)−(f) n =1.0; (g)−(i) n = 1.3.
图 9 驱替完成时, 不同幂律指数情况下得到的气液两相动力黏度示意图
$({\rm{a}})\;n = 0.7$ ;$({\rm{b}})\;n = 1.0$ ;$({\rm{c}})\;n = 1.3$ Figure 9. Schematic diagram of gas-liquid two phase dynamics viscosity obtained under different values of power-law exponent:
$({\rm{a}})\;n = 0.7$ ;$({\rm{b}})\;n = 1.0$ ;$({\rm{c}})\;n = 1.3$ .
图 10
$Ca$ 数和幂律指数n对幂律流体驱替效率的影响Figure 10. Effects of
$Ca$ and power-law exponent n on power-law fluid displacement efficiency.
图 11 不同的动力黏性比M下, 被驱替液为剪切变稀、牛顿与剪切变稠流体时得到的指进形态图 (a)−(c) n = 0.7; (d)−(f) n = 1.0; (g)−(i) n = 1.3
Figure 11. Final finger patterns obtained under different values of viscosity ratios M for shear thinning, Newtonian and shear thickening fluids: (a)−(c) n = 0.7; (d)−(f) n = 1.0; (g)−(i) n = 1.3.
图 12 动力黏度比M和幂律指数n对幂律流体驱替效率的影响
Figure 12. Effects of viscosity ratio M and power-law exponent n on power-law fluid displacement efficiency.
图 13 不同的润湿性角度
$\theta $ 下, 被驱替液为剪切变稀、牛顿与剪切变稠流体时得到的指进形态图 (a)−(c)$\theta = {45^{\circ}}$ ; (d)− (f)$\theta = {60^{\circ}}$ ; (g)−(i)$\theta = {120^{\circ}}$ ; (j)−(l)$\theta = {135^{\circ}}$ Figure 13. Final finger patterns obtained under different values of contact angles
$\theta $ for shear thinning, Newtonian and shear thickening fluids: (a)−(c)$\theta = {45^{\circ}}$ ; (d)－(f)$\theta = {60^{\circ}}$ ; (g)−(i)$\theta = {120^{\circ}}$ ; (j)−(l)$\theta = {135^{\circ}}$ .
图 14 润湿性
$\theta $ 和幂律指数n对幂律流体驱替效率的影响Figure 14. Effects of contact angles
$\theta $ and power-law exponent n on power-law fluid displacement efficiency.
图 15 不同的障碍物几何类型, 被驱替液为剪切变稀、牛顿与剪切变稠流体时驱得到的指进形态图 (a)−(c) n = 0.4; (d)− (f) n = 0.7; (g)−(i) n = 1.0; (j)−(l) n = 1.3, (m)−(o) n = 1.6
Figure 15. Final finger patterns obtained under different geometric type for shear thinning, Newtonian and shear thickening fluids: (a)− (c) n = 0.4; (d)−(f) n = 0.7; (g)−(i) n = 1.0; (j)−(l) n = 1.3; (m)−(o) n = 1.6.
图 16 障碍物几何类型和幂律指数n对幂律流体驱替效率的影响
Figure 16. Effects of geometric type and power-law exponent n on power-law fluid displacement efficiency.
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[1] Santvoort J V, Golombok M 2018 J. Pet. Sci. Eng. 167 28
Google Scholar
[2] Fang T M, Wang M H, Gao Y, Zhang Y N, Yan Y G, Zhang J 2019 Chem. Eng. Sci. 197 204
Google Scholar
[3] Xu X F, Zhang J, Liu F X, Wang X J, Wei W, Liu Z J 2017 Int. J. Multiphase Flow. 95 84
Google Scholar
[4] Du W, Fu T T, Duan Y F, Zhu C Y, Ma Y G, Li H Z 2018 Chem. Eng. Sci. 176 66
Google Scholar
[5] Fu T T, Ma Y G, Li H Z 2015 Chem. Eng. Process. 97 38
Google Scholar
[6] Salehi M S, Esfidani M T, Afshin H, Firoozabadi B 2018 Exp. Therm. Fluid Sci. 94 148
Google Scholar
[7] Sontti S G, Atta A 2017 Chem. Eng. J. 330 245
Google Scholar
[8] 娄钦, 李涛, 杨茉 2018 物理学报 67 234701
Google Scholar
Lou Q, Li T, Yang M 2018 Acta Phys. Sin. 67 234701
Google Scholar
[9] 臧晨强, 娄钦 2017 物理学报 66 134701
Google Scholar
Zang C Q, Lou Q 2017 Acta Phys. Sin. 66 134701
Google Scholar
[10] Lou Q, Guo Z L, Shi B C 2012 Europhys. Lett. 99 64005
Google Scholar
[11] Lou Q, Guo Z L 2015 Phys. Rev. E 91 013302
Google Scholar
[12] 娄钦, 李涛, 李凌 2018 上海理工大学学报 40 13
Lou Q, Li T, Li L 2018 J. Univ. Shanghai Sci. Technol. 40 13
[13] 谢驰宇, 张建影, 王沫然 2016 计算物理 33 147
Google Scholar
Xie C Y, Zhang J Y, Wang M R 2016 Chin. J. Computat. Phys. 33 147
Google Scholar
[14] Swift M R, Osborn W R, Yeomans J M 1995 Phys. Rev. Lett. 75 830
Google Scholar
[15] Shi Y, Tang G H 2014 Comput. Math. Appl. 68 1279
Google Scholar
[16] Fakhari A, Rahimian M H 2010 Phys. Rev. E 81 036707
Google Scholar
[17] Shi Y, Tang G H 2016 J. Non-Newtonian Fluid Mech. 229 86
Google Scholar
[18] Ba Y, Wang N N, Liu H H, Li Q, He G Q 2018 Phys. Rev. E 97 033307
Google Scholar
[19] Halliday I, Law R, Care C M, Hollis A 2006 Phys. Rev. E 73 056708
Google Scholar
[20] Halliday I, Hollis A P, Care C M 2007 Phys. Rev. E 76 026708
Google Scholar
[21] 闵琪, 段远源, 王晓东, 吴莘馨 2013 热科学与技术 12 335
Min Q, Duan Y Y, Wang X D, Wu X X 2013 J. Therm. Sci. Technol. 12 335
[22] Shan X W, Chen H D 1994 Phys. Rev. E 49 2941
Google Scholar
[23] Shan X W, Chen H D 1993 Phys. Rev. E 47 1815
Google Scholar
[24] Nourgaliev R R, Dinh T N, Theofanous T G, Joseph D 2003 Int. J. Multiphase Flow. 29 117
Google Scholar
[25] Huang H B, Sukop M, Lu X Y 2015 Multiphase Lattice Boltzmann Methods: Theory and Application (USA: WILEY Blackwell) pp7−10
[26] Yu Z, Fan L S 2009 J. Comput. Phys. 228 6456
Google Scholar
[27] He X Y, Chen S Y, Zhang R Y 1999 J. Comput. Phys. 152 642
Google Scholar
[28] Fakhari A, Bolster D 2017 J. Comput. Phys. 334 620
Google Scholar
[29] Fakhari A, Rahimian M H 2011 Comput. Fluids 40 156
Google Scholar
[30] Lou Q, Guo Z L, Shi B C 2013 Phys. Rev. E 87 063301
Google Scholar
[31] Sadeghi R, Shadloo M S 2017 Numer. Heat Transfer Part A 71 560
Google Scholar
[32] Kano Y, Sato T 2017 Energy Procedia 114 3385
Google Scholar
[33] Ye F, Di Q F, Wang W C, Chen F, Chen H J, Hua S 2018 J. Appl. Math. Mech. 39 513
Google Scholar
[34] Huang H B, Huang J J, Lu X Y 2014 J. Comput. Phys. 269 386
Google Scholar
[35] Chao J H, Mei R W, Singh R, Shyy W 2011 Int. J. Numer. Methods Fluids 66 622
Google Scholar
[36] Chen Y P, Deng Z L 2017 J. Fluid Mech. 819 401
Google Scholar
[37] Fu Y H, Bai L, Jin Y, Cheng Y 2017 Phys. Fluids 29 032003
Google Scholar
[38] 郭照立, 郑楚光 2008 格子Boltzmann方法的原理及应用 (北京: 科学出版社) 第244页
Guo Z L, Zheng C G 2008 Theory and Applications of Lattice Boltzmann Method (Beijing: Science Press) p244 (in Chinese)
[39] Guo Z L, Zheng C G, Shi B C 2011 Phys. Rev. E 83 036707
Google Scholar
[40] Davies A R, Summers J L, Wilson M C T 2006 Int. J. Comput. Fluid. D 20 415
Google Scholar
[41] Shi Y, Tang G H 2015 Commun. Comput. Phys. 17 1056
Google Scholar
[42] Ansarinasab J, Jamialahmadi M 2017 J. Pet. Sci. Eng. 156 748
Google Scholar
[43] Basirat F, Yang Z B, Niemi A 2017 Adv. Water Resour. 109 181
[44] Zheng X L, Mahabadi N, Yun T S, Jang J 2017 J. Geophys. Res.: Solid Earth 122 1634
Google Scholar
[45] Xu Z Y, Liu H H, Valocchi A J 2017 Water Resources Res. 53 3770
Google Scholar
[46] Soulaine C, Roman S, Kovscek A, Tchelepi H A 2018 J. Fluid Mech. 855 616
Google Scholar
[47] 黄海波 2009 第六届全国流体力学青年研讨会 中国杭州 2009年10月10日 第27页
Huang H B 2009 The 6th National Youth Workshop on Fluid Mechanics Hangzhou, China October 10, 2009 p27 (in Chinese)
[48] Shiri Y, HassaniH,Nazari M, Sharifi M 2018 Mol. Simul. 44 708
Google Scholar
[49] Liu H H, ValocchiAJ, Kang Q J, Werth C 2013 Transp. Porous Media 99 555
Google Scholar
[50] Dong B, YanY Y, Li W Z, Song Y C 2010 Comput. Fluids 39 768
Google Scholar
[51] Ferer M, Anna S L,Tortora P, Kadambi J R, Oliver M, Bromhal G S, Smith D H 2011 Transp. Porous Media 86 243
Google Scholar
[52] Dong B, YanY Y, Li W Z, Song Y C 2011 J. Bionic. Eng. 7 267
Google Scholar
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