Mô phỏng lưới Boltzmann đa pha cho các ứng dụng trong môi trường xốp

Computational Geosciences - Tập 20 - Trang 777-805 - 2015
Haihu Liu1, Qinjun Kang2, Christopher R. Leonardi3,4, Sebastian Schmieschek5,6, Ariel Narváez6, Bruce D. Jones4, John R. Williams4, Albert J. Valocchi7,8, Jens Harting6,9
1School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an, China
2Earth and Environmental Sciences Division, Los Alamos National Laboratory, Los Alamos, USA
3School of Mechanical and Mining Engineering, The University of Queensland, St. Lucia, QLD, Australia
4Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, Cambridge, USA
5Centre for Computational Science, Department of Chemistry, University College London, London, UK
6Department of Applied Physics, Eindhoven University of Technology, Eindhoven, The Netherlands
7Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, Urbana, USA
8International Institute for Carbon-Neutral Energy Research (WPI-I2CNER), Kyushu University, Fukuoka, Japan
9Research Centre Juelich, Helmholtz-Institute Erlangen-Nuremberg (IEK-11), Nuremberg, Germany

Tóm tắt

Trong hai thập kỷ qua, các phương pháp lưới Boltzmann đã trở thành một công cụ ngày càng phổ biến để tính toán dòng chảy trong các hình học phức tạp như môi trường xốp. Ngoài các mô phỏng đơn pha cho phép, ví dụ, định lượng chính xác tính thấm của một mẫu xốp, có một số mở rộng cho phương pháp lưới Boltzmann cho phép nghiên cứu dòng chảy đa pha và nhiều thành phần ở cấp độ lỗ chân lông. Trong bài viết này, chúng tôi sẽ cung cấp một cái nhìn tổng quan rộng rãi về một số mô hình giao diện khuếch tán và thảo luận về những ưu điểm và nhược điểm của chúng. Hơn nữa, chúng tôi sẽ báo cáo ngắn gọn về các dòng chảy đa pha chứa các hạt rắn, cũng như các chi tiết thực hiện và các vấn đề tối ưu hóa.

Từ khóa

#lưới Boltzmann #mô phỏng đa pha #môi trường xốp #tính thấm #dòng chảy đa thành phần

Tài liệu tham khảo

Hilfer, R.: Adv. Chem. Phys. XCII (1996) Hilfer, R.: Phys. Rev. B 45, 7115 (1992) Dixon, K.L., Nichol, R.L.: Env. Geosci. 12, 279 (2005) Gee, G.W., Zhang, F.Z., Ward, A.L.: Vadose Zone J. 2, 627 (2003) Koponen, A., Kataja, M., Timonen, J.: Phys. Rev. E 56, 3319 (1997) Garrido, P., Concha, F., Bürger, R.: Int. J. Mineral Process 72, 57 (2003) Balhoff, M.T., Thomas, S.G., Wheeler, M.F.: Comput. Geosci. 12, 15 (2008) Blunt, M., King, P.: Transp. Porous Med. 6, 407 (1991) Bryant, S., Blunt, M.: Phys. Rev. A 46, 2004 (1992) Al-Gharbi, M.S., Blunt, M.J.: Phys. Rev. E 71, 016308 (2005) Valvatne, P., Piri, M., Lopez, X., Blunt, M.: Transp. Porous Med. 58, 23 (2005) Piri, M., Blunt, M.J.: Phys. Rev. E 71, 026301 (2005) Blunt, M.J., Bijeljic, B., Dong, H., Gharbi, O., Iglauer, S., Mostaghimi, P., Paluszny, A., Pentland, C.: Adv. Water Resour. 51, 197 (2013) Joekar-Niasar, V., Hassanizadeh, S., Leijnse, A.: Transp. Porous Media 74, 201 (2008) Joekar-Niasar, V., Hassanizadeh, S., Dahe, H.: J. Fluid Mech. 655, 38 (2010) Raoof, A., Hassanizadeh, S.: Transp. Porous Media 81, 391 (2010) Jiang, Z., Wu, K., Couples, G., van Dijke, M., Sorbie, K., Ma, J.: Water Res. Research 43 (2007) Hirt, C., Nichols, B.: J. Comput. Phys. 39, 201 (1981) Rider, W.J., Kothe, D.B.: J. Comput. Phys. 141, 112 (1998) Gueyffier, D., Li, J., Nadim, A., Scardovelli, R., Zaleski, S.: J. Comput. Phys. 152, 423 (1999) Ferrari, A., Lunati, I.: Adv. Water Resour. 57, 19 (2013) Osher, S., Sethian, J.A.: J. Comput. Phys. 79, 12 (1988) Sussman, M., Fatemi, E., Smereka, P., Osher, S.: Comput. Fluids 27, 663 (1998) Osher, S., Fedkiw, R.P.: Level sets methods and dynamic implicit surfaces. Springer (2003) Benzi, R., Succi, S., Vergassola, M.: Phys. Rep. (1992) Qian, Y.H., d’Humières, D., Lallemand, P.: Europhys. Lett. 17, 479 (1992) Succi, S: The lattice Boltzmann equation for fluid dynamics and beyond. Oxford University Press (2001) Cancelliere, A., Chang, C., Foti, E., Rothman, D.H., Succi, S.: Phys. Fluids 2, 2085 (1990) Ferréol, B., Rothman, D.H.: Transp. Porous Media 20, 3 (1995) Briant, A.J., Wagner, A.J., Yeomans, J.M.: Phys. Rev. E 69, 031602 (2004) Briant, A.J., Yeomans, J.M.: Phys. Rev. E 69, 031603 (2004) Chen, H.Y., Jasnow, D., Viñals, J.: Phys. Rev. Lett. 85, 1686 (2000) Kang, Q., Zhang, D., Chen, S.: Phys. Fluids 14, 3203 (2002) Kang, Q., Zhang, D., Chen, S.: Adv. Water Resour. 27, 13 (2004) Kang, Q., Zhang, D., Chen, S.: J. Fluid Mech. 545, 41 (2005) Chen, S., Doolen, G.D.: Annu. Rev. Fluid Mech. 30, 329 (1998) Harting, J., Harvey, M., Chin, J., Venturoli, M., Coveney, P.: Phil. Trans. R. Soc. Lond. A 363 (2005) Harting, J., Zauner, T., Narváez, A., Hilfer, R. In: Nagel, W., Kröner, D., Resch, M. (eds.) : High Performance Computing in Science and Engineering ’08. Springer (2008) Narváez, A., Harting, J.: Advances in Applied Mathematics and Mechanics 2, 685 (2010) Narváez, A., Zauner, T., Raischel, F., Hilfer, R., Harting, J.: J. Stat. Mech: Theor. Exp. 2010, 211026 (2010) Martys, N.S., Hagedorn, J.G., Goujon, D., Devaney, J.E.: in Developments in X-Ray Tomography II. Proceeding of SPIE 3772, 205 (1999) Manwart, C., Aaltosalmi, U., Koponen, A., Hilfer, R., Timonen, J.: Phys. Rev. E 66, 016702 (2002) Harting, J., Venturoli, M., Coveney, P.: Phil. Trans. R. Soc. Lond. A 362, 1703 (2004) Ahrenholz, B., Tölke, J., Krafczyk, M.: Int. J. Comp. Fluid Dyn. 20, 369 (2006) Koponen, A., Kandhai, D., Hellén, E., Alava, M., Hoekstra, A., Kataja, M., Niskanen, K., Sloot, P., Timonen, J.: Phys. Rev. Lett. 80, 716 (1998) Frisch, U., Hasslacher, B., Pomeau, Y.: Phys. Rev. Lett. 56, 1505 (1986) Wolfram, S.: J. Stat. Phys. 45, 471 (1986) McNamara, G.R., Zanetti, G.: Phys. Rev. Lett. 61, 2332 (1988) Higuera, F.J., Jimnez, J.: EPL 9, 663 (1989) Benzi, R., Succi, S., Vergassola, M.: Phys. Rep. 222, 145 (1992) Chen, H., Chen, S., Matthaeus, W.H.: Phys. Rev. A 45, R5339 (1992) Qian, Y.H., d’Humiéres, D., Lallemand, P.: Europhys. Lett. 17, 479 (1992) Bhatnagar, P.L., Gross, E.P., Krook, M.: Phys. Rev. 94, 511 (1954) Liboff, R.: Kinetic theory: classical, quantum and relativistic descriptions. Prentice-Hall (1990) Chapman, S., Cowling, T.G.: The mathematical theory of non-uniform gases, 2nd edn. Cambridge University Press (1952) He, X., Luo, L.S.: Phys. Rev. E 56 (1997) McNamara, G., Zanetti, G.: Phys. Rev. Lett. 61, 2332 (1988) Chen, S., Doolen, G.D.: Annu. Rev. Fluid 30, 329 (1998) Aidun, C.K., Clausen, J.R.: Annu. Rev. Fluid 42, 439 (2010) Lallemand, P., Luo, L.S.: Phys. Rev. E 61, 6546 (2000) Higuera, F.J., Succi, S., Benzi, R.: Europhys. Lett. 9, 345 (1989) d’Humiéres, D.: Rarefied gas dynamics- Theory and simulations, pp 450–458. The American Institute of Aeronautics and Astronautics (1992) Ladd, A.J.C.: J. Fluid Mech. 271, 285 (1994) d’Humiéres, D.: Phil. Trans. R. Soc. Lond. A 360, 437 (2002) Ginzburg, I.: Adv. Water Res. 28, 1171 (2005) Ginzburg, I., dHumiéres, D., Kuzmin, A.: J. Stat. Phys. 139, 1090 (2010) Ladd, A.J.C.: Phys. Rev. Lett. 70, 1339 (1993) Dünweg, B., Schiller, U., Ladd, A.: Phys. Rev. E 76 (2007) Gross, M., Adhikari, R., Cates, M.E., Varnik, F.: Phys. Rev. E 82, 056714 (2010) Kaehler, G., Wagner, A.J.: Phys. Rev. E 87, 063310 (2013) Shan, X., Yuan, X.F., Chen, H.: J. Fluid Mech. 550, 413 (2006) Dubois, F., Lallemand, P.: J. Stat. Mech. 2009, P06006 (2009) Cornubert, R., d’Humires, D., Levermore, D.: Physica D 47, 241 (1991) Chen, S., Doolen, G.D.: Annu. Rev. Fluid Mech. 30, 329 (1998) Zou, Q., He, X.: Phys. Fluids 9, 1591 (1997) Narváez, A., Zauner, T., Raischel, F., Hilfer, R., Harting, J.: J. Stat. Mech. 2010, P11026 (2010) Hecht, M., Harting, J.: J. Stat. Mech.: Theor. Exp. 2010, P01018 (2010) Latt, J., Chopard, B., Malaspinas, O., Deville, M., Michler, A.: Phys. Rev. E 77, 056703 (2008) Gunstensen, A.K., Rothman, D.H., Zaleski, S., Zanetti, G.: Phys. Rev. A 43, 4320 (1991) Reis, T., Phillips, T.N.: J. Phys. A-Math. Theor. 40, 4033 (2007) Liu, H., Valocchi, A.J., Kang, Q.: Phys. Rev. E 85, 046309 (2012) Shan, X., Chen, H.: Phys. Rev. E 47, 1815 (1993) Shan, X., Chen, H.: Phys. Rev. E 49, 2941 (1994) Shan, X., Doolen, G.: J. Stat. Phys. 81, 379 (1995) Swift, M.R., Osborn, W.R., Yeomans, J.M.: Phys. Rev. Lett. 75, 830 (1995) Swift, M.R., Orlandini, E., Osborn, W.R., Yeomans, J.M.: Phys. Rev. E 54, 5041 (1996) He, X., Chen, S., Zhang, R.: J. Comput. Phys. 152, 642 (1999) Lee, T., Liu, L.: J. Comput. Phys. 229, 8045 (2010) Rothman, D.H., Keller, J.M.: J. Stat. Phys. 52, 1119 (1988) Grunau, D., Chen, S., Eggert, K.: Phys. Fluids A 5, 2557 (1993) Tölke, J., Krafczyk, M., Schulz, M., Rank, E.: Phil. Trans. R. Soc. Lond. A 360, 535 (2002) Latva-Kokko, M., Rothman, D.H.: Phys. Rev. E 71, 056702 (2005) Halliday, I., Law, R., Care, C.M., Hollis, A.: Phys. Rev. E 73, 056708 (2006) Leclaire, S., Reggio, M., Trépanier, J.Y.: Appl. Math. Model 36, 2237 (2012) Leclaire, S., Reggio, M., Trépanier, J.Y.: Comput. Fluids 48, 98 (2011) Huang, H., Daniel, J., Thorne, T., Schaap, M.G., Sukop, M.C.: Phys. Rev. E 76, 066701 (2007) Martys, N.S., Chen, H.: Phys. Rev. E 53, 743 (1996) Schmieschek, S., Harting, J.: Commun. Comput. Phys. 9, 1165 (2011) Sbragaglia, M., Shan, X.: Phys. Rev. E 84, 036703 (2011) Nekovee, M., Coveney, P.V., Chen, H., Boghosian, B.M.: Phys. Rev. E 62, 8282 (2000) González-Segredo, N., Nekovee, M., Coveney, P.V.: Phys. Rev. E 67, 046304 (2003) Emerton, A.N., Coveney, P.V., Boghosian, B.M.: Phys. Rev. E 56, 1286 (1997) González-Segredo, N., Coveney, P.V.: Phys. Rev. E 69, 061501 (2004) González-Segredo, N., Harting, J., Giupponi, G., Coveney, P.V.: Phys. Rev. E 73, 031503 (2006) Giupponi, G., Harting, J., Coveney, P.V.: Europhys. Lett. 73, 533 (2006) Saksena, R.S., Coveney, P.V.: J. Phys. Chem. B 112, 2950 (2008) Kunert, C., Harting, J.: Progress in CFD 8, 197 (2008) Harting, J., Kunert, C., Hyväluoma, J.: Microfluid. Nanofluid. 8, 1 (2010) Schmieschek, S., Narváez Salazar, A., Harting, J. In: Nagel, M.R.W., Kröner, D. (eds.) : High Performance Computing in Science and Engineering ’12, p 39. Springer (2013) Harting, J., Kunert, C., Herrmann, H.: Europhys. Lett. 75, 328 (2006) Yuan, P., Schaefer, L.: Phys. Fluids 18(042101), 1 (2006) Zhang, J., Tian, F.: Europhys. Lett. 81, 66005 (2008) Shan, X.: Phys. Rev. E 73, 047701 (2006) Sbragaglia, M., Benzi, R., Biferale, L., Succi, S., Sugiyama, K., Toschi, F.: Phys. Rev. E 75, 026702 (2007) Kupershtokh, A.L., Medvedev, D.A., Karpov, D.I.: Comput. Math. Appl. 58, 965 (2009) Huang, H., Krafczyk, M., Lu, X.: Phys. Rev. E 84, 046710 (2011) Li, Q., Luo, K.H., Li, X.J.: Phys. Rev. E 86, 016709 (2012) Sun, K., Wang, T., Jia, M., Xiao, G.: Physica A 391, 3895 (2012) Yu, Z., Fan, L.S.: Phys. Rev. E 82, 046708 (2010) Porter, M.L., Coon, E.T., Kang, Q., Moulton, J.D., Carey, J.W.: Phys. Rev. E 86, 036701 (2012) Bao, J., Schaefer, L.: Appl. Math. Model 37, 1860 (2013) Yu, Z., Fan, L.S.: J. Comput. Phys. 228, 6456 (2009) Kamali, M.R., Sundaresan, S., Van den Akker, H.E.A., Gillissen, J.J.J.: Chem. Eng. J 207–208, 587 (2012) Kamali, M.R., Van den Akker, H.E.A.: Ind. Eng. Chem. Res. 52, 11365 (2013) Yang, J., Boek, E.: Comp. Math. Appl. 65, 882 (2013) van der Sman, R., van der Graaf, S.: Comput. Phys. Commun. 178, 492 (2008) Hao, L., Cheng, P.: Int. J. Heat Mass Transf. 53, 1908 (2010) Niu, X.D., Munekata, T., Hyodo, S.A., Suga, K.: J. Power Sources 172, 542 (2007) Liu, H., Zhang, Y.: J. Appl. Phys. 106, 034906 (2009) Liu, H., Zhang, Y.: J. Comput. Phys. 229, 9166 (2010) Liu, H., Zhang, Y.: Commun. Comput. Phys. 9, 1235 (2011) van der Graaf, S., Nisisako, T., Schroën, C.G.P.H., van der Sman, R.G.M., Boom, R.M.: Langmuir 22, 4144 (2006) Pooley, C.M., Kusumaatmaja, H., Yeomans, J.M.: Phys. Rev. E 78, 056709 (2008) Lee, T., Fischer, P.F.: Phys. Rev. E 74, 046709 (2006) Jacqmin, D.: J. Comput. Phys. 155, 96 (1999) Jamet, D., Torres, D., Brackbill, J.U.: J. Comput. Phys. 182, 262 (2002) Huang, J.J., Shu, C., Chew, Y.T.: Phys. Fluids 21, 022103 (2009) Fakhari, A., Rahimian, M.H.: Phys. Rev. E 81, 036707 (2010) Yiotis, A.G., Psihogios, J., Kainourgiakis, M.E., Papaioannou, A., Stubos, A.K.: Colloid Surf. A-Physicochem. Eng. Asp. 300, 35 (2007) Meakin, P., Tartakovsky, A.M.: Rev. Geophys. 47, RG3002 (2009) Inamuro, T., Ogata, T., Tajima, S., Konishi, N.: J. Comput. Phys. 198, 628 (2004) Lee, T., Lin, C.L.: J. Comput. Phys. 206, 16 (2005) Zheng, H., Shu, C., Chew, Y.: J. Comput. Phys. 218, 353 (2006) Liu, H., Valocchi, A.J., Kang, Q., Werth, C.: Transp. Porous Med. 99, 555 (2013) Jacqmin, D.: J. Fluid Mech. 402, 57 (2000) Khatavkar, V., Anderson, P., Meijer, H.: J. Fluid Mech. 572, 367 (2007) Wiklunda, H., Lindströmb, S., Uesaka, T.: Comput. Phys. Commun. 182, 2192 (2011) Liu, H., Valocchi, A.J., Zhang, Y., Kang, Q.: J. Comput. Phys. 256, 334 (2014) Li, Q., Luo, K., Gao, Y., He, Y.: Phys. Rev. E 85, 026704 (2012) Liu, H., Valocchi, A.J., Zhang, Y., Kang, Q.: Phys. Rev. E 87, 013010 (2013) Zheng, R., Sun, J., Liu, H.: Comput. Fluids 73, 175 (2013) Zhang, T., Wang, Q.: Commun. Comput. Phys. 7, 362 (2010) Ahlrichs, P., Dünweg, B.: J. Chem. Phys. 111, 8225 (1999) Ahlrichs, P., Everaers, R., Dünweg, B.: Phys. Rev. E 64(4), 040501 (2001) Lobaskin, V., Dünweg, B.: N. J. Phys. 6, 54 (2004) Ladd, A.: J. Fluid Mech. 271 (1994) Ladd, A.: J. Fluid Mech. 271 (1994) Ladd, A., Verberg, R.: J. Stat. Phys. 104(5/6), 1197 (2001) Nguyen, N., Ladd, A.: Phys. Rev. E 66 (2002) Aidun, C., Lu, Y., Ding, E.: J. Fluid Mech. 373 (1998) Komnik, A., Harting, J., Herrmann, H.J.: J. Stat. Mech.: Theor. Exp., P12003 (2004) Janoschek, F., Toschi, F., Harting, J.: Phys. Rev. E 82, 056710 (2010) Jansen, F., Harting, J.: Phys. Rev. E 83, 046707 (2011) Frijters, S., Günther, F., Harting, J.: Soft Matter 8(24), 6542 (2012) Stratford, K., Adhikari, R., Pagonabarraga, I., Desplat, J.C., Cates, M.E.: Science 309, 2198 (2005) Joshi, A.S., Sun, Y.: Phys. Rev. E 79, 066703 (2009) Noble, D.R., Torczynski, J.R.: Int. J. Mod. Phys. C 9, 1189 (1998) Cook, B.K., Noble, D.R., Williams, J.R.: Eng. Comput. 21, 151 (2004) Owen, D.R.J., Leonardi, C.R., Feng, Y.T.: Int. J. Numer. Meth. Engng. 87, 66 (2011) Heemels, M., Hagen, M., Lowe, C.: J. Comp. Phys. 64 (2000) Satofuka, N., Nishioka, T.: Comput. Mech. 23, 164 (1999) Donath, S., Zeiser, T., Hager, G., Habich, J., Wellein, G. In: Frontiers in Simulation: Simulationstechnique - 18th Symposium in Erlangen, pp. 728–735 (2005) Chen, L., Kang, Q., Robinson, B.A., He, Y.L., Tao, W.Q.: Phys. Rev. E 87, 043306 (2013) Ma, J., Wu, K., Jiang, Z., Couples, G.: Phys. Rev. E 81, 056702 (2010) Pohl, T., Kowarschik, M., Wilke, J., Iglberger, K., Rde, U.: Parallel Processing Letters 13, 2003 (2003) Holmes, D.W., Williams, J.R., Tilke, P.: Comput. Phys. Commun. 181, 341 (2010) Leonardi, C.R., Holmes, D.W., Williams, J.R., Tilke, P.G. In: L.T.W. et al. (ed.) : Spring Simulation Multiconference-2011, vol. 43, vol. 43, pp 166–174. Red Hook: Simulation Councils (2011) Desplat, J., Pagonabarraga, I., Bladon, P.: Comp. Phys. Comm. 134, 273 (2001) Feichtinger, C., Donath, S., Köstler, H., Götz, J., Rüde, U.: J. Comp. Sci. 2, 105 (2011) Bernaschi, M., Fatica, M., Melchionna, S., Succi, S., Kaxiras, E.: Concurrency Computat.: Pract. Exper. 22, 1 (2010) Coon, E., Porter, M., Kang, Q.: Comput. Geosci. 18, 17 (2014) Swift, M.R., Osborn, W.R., Yeomans, J.M.: Phys. Rev. E 75, 830 (1995) Krüger, T., Frijters, S., Günther, F., Kaoui, B., Harting, J.: European Phys. J. Special Topics 222, 177 (2013) ’lb3d open source version 7’ available online (2012). Http://mtp.phys.tue.nl/lb3d Groen, D., Henrich, O., Janoschek, F., Coveney, P., Harting, J. In: Bernd Mohr, W.F. (ed.) : Jülich Blue Gene/P Extreme Scaling Workshop 2011. Jülich Supercomputing Centre, Jülich (2011). FZJ-JSC-IB-2011-02 Chun, B., Ladd, A.J.C.: Phys. Rev. E 75, 066705 (2007) d’Humières, D., Ginzburg, I., Krafczyk, M., Lallemand, P., Luo, L.: Phil. Trans. R. Soc. A 360, 437451 (2002) Mawson, M., Revell, A.: ArXiv e-prints (2013) Obrecht, C., Kuznik, F., Tourancheau, B., Roux, J.J.: Parallel Comput. 39, 259 (2013) Latt, J.: Technical report: How to implement your DdQq dynamics with only q variables per node. Tech. rep., Tufts University, Medford (2007) Bailey, P., Myre, J., Walsh, S., Lilja, D., Saar, M.: International Conference on Parallel Processing pp. 550–557 (2009) Wittmann, M., Zeiser, T., Hager, G., Wellein, G.: CoRR abs/1111.0 (2011) Tölke, J., Krafczyk, M.: International Journal of Computational Fluid Dynamics 22, 443 (2008) Rinaldi, P., Dari, E., Vénere, M., Clausse, A.: Simul. Model. Pract. Theory 25, 163 (2012) Latva-Kokko, M., Rothman, D.H.: Phys. Rev. E 72, 046701 (2005) Liu, H., Zhang, Y.: Phys. Fluids 23, 082101 (2011) Pan, C., Hilpert, M., Miller, C.: Water Resour. Res. 40, W01501 (2004) Sukop, M.C., Or, D.: Water Resour. Res. 40, W01509 (2004) Li, H., Pan, C., Miller, C.T.: Phys. Rev. E 72, 026705 (2005) Huang, H., Lu, X.Y.: Phys. Fluids 21, 092104 (2009) Porter, M., Schaap, M., Wildenschild, D.: Adv. Water Resour. 32, 1632 (2009) Parmigiani, A., Huber, C., Bachmann, O., Chopard, B.: J. Fluid Mech. 686, 40 (2011) Dong, B., Yan, Y.Y., Li, W.Z.: Transp. Porous Med. 88, 293 (2011) Ghassemi, A., Pak, A., Pet, J.: Sci. Eng. 77, 135 (2011) Mukherjee, P.P., Kang, Q.J., Wang, C.Y.: Energy Environ. Sci. 4, 346 (2011) Middleton, R.S., Keating, G.N., Stauffer, P.H., Jordan, A.B., Viswanathan, H.S., Kang, Q.J., Carey, J.W., Mulkey, M.L., Sullivan, E.J., Chu, S.P.P., Esposito, R., Meckel, T.A.: Energy Environ. Sci. 5, 7328 (2012) Dou, Z., Zhou, Z.F.: Int. J. Heat Fluid Flow 42, 23 (2013) Pan, C., Miller, L.S.L.C.T.: Comput. Fluids 35, 898 (2006) Hao, L., Cheng, P.: J. Power Sources 186, 104 (2009) Premnath, K.N., Abraham, J.: J. Comput. Phys. 224, 539 (2007) Fakhari, A., Rahimian, M.H.: Int. J. Numer. Methods Fluids 64, 827 (2010) Lenormand, R., Touboul, E., Zarcone, C.: J. Fluid Mech. 189, 165 (1988) Zhang, C., Oostrom, M., Wietsma, T., Grate, J., Warner, M.: Energy Fuels 25, 3493 (2011) Leonardi, C., Owen, D., Feng, Y.: Eng. Comput. 29, 366 (2012) Leonardi, C., Owen, D., Feng, Y.: Eng. Comput. 29, 392 (2012) Boutt, D.F., Cook, B.K., Williams, J.R.: Int. J. Numer. Anal. Methods Geomech. 35, 997 (2011) Yang, J., Crawshaw, J., Boek, E.: Water Resources Res. 49, 8531 (2013)