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Direct simulation of pore-scale two-phase visco-capillary flow on large digital rock images using a phase-field lattice Boltzmann method on general-purpose graphics processing units

  • F. O. AlpakEmail author
  • I. Zacharoudiou
  • S. Berg
  • J. Dietderich
  • N. Saxena
Original Paper
  • 58 Downloads

Abstract

We describe the underlying mathematics, validation, and applications of a novel Helmholtz free-energy—minimizing phase-field model solved within the framework of the lattice Boltzmann method (LBM) for efficiently simulating two-phase pore-scale flow directly on large 3D images of real rocks obtained from micro-computed tomography (micro-CT) scanning. The code implementation of the technique, coined as the eLBM (energy-based LBM), is performed in CUDA programming language to take maximum advantage of accelerated computing by use of multinode general-purpose graphics processing units (GPGPUs). eLBM’s momentum-balance solver is based on the multiple-relaxation-time (MRT) model. The Boltzmann equation is discretized in space, velocity (momentum), and time coordinates using a 3D 19-velocity grid (D3Q19 scheme), which provides the best compromise between accuracy and computational efficiency. The benefits of the MRT model over the conventional single-relaxation-time Bhatnagar-Gross-Krook (BGK) model are (I) enhanced numerical stability, (II) independent bulk and shear viscosities, and (III) viscosity-independent, nonslip boundary conditions. The drawback of the MRT model is that it is slightly more computationally demanding compared to the BGK model. This minor hurdle is easily overcome through a GPGPU implementation of the MRT model for eLBM. eLBM is, to our knowledge, the first industrial grade–distributed parallel implementation of an energy-based LBM taking advantage of multiple GPGPU nodes. The Cahn-Hilliard equation that governs the order-parameter distribution is fully integrated into the LBM framework that accelerates the pore-scale simulation on real systems significantly. While individual components of the eLBM simulator can be separately found in various references, our novel contributions are (1) integrating all computational and high-performance computing components together into a unified implementation and (2) providing comprehensive and definitive quantitative validation results with eLBM in terms of robustness and accuracy for a variety of flow domains including various types of real rock images. We successfully validate and apply the eLBM on several transient two-phase flow problems of gradually increasing complexity. Investigated problems include the following: (1) snap-off in constricted capillary tubes; (2) Haines jumps on a micromodel (during drainage), Ketton limestone image, and Fontainebleau and Castlegate sandstone images (during drainage and subsequent imbibition); and (3) capillary desaturation simulations on a Berea sandstone image including a comparison of numerically computed residual non-wetting-phase saturations (as a function of the capillary number) to data reported in the literature. Extensive physical validation tests and applications on large 3D rock images demonstrate the reliability, robustness, and efficacy of the eLBM as a direct visco-capillary pore-scale two-phase flow simulator for digital rock physics workflows.

Keywords

Free-energy lattice Boltzmann method LBM Phase-field method Cahn-Hilliard equation Navier-Stokes equations Pore-scale flow simulation Two-phase flow Computational fluid dynamics CFD Multiple-relaxation-time MRT Digital rock physics DRP General-purpose graphics processing unit GPGPU Parallel computing Snap-off Haines jumps Forced drainage Forced imbibition Residual oil 

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References

  1. 1.
    Alpak, F.O., Berg, S., Zacharoudiou, I.: Prediction of fluid topology and relative permeability in imbibition in sandstone rock by direct numerical simulation. Adv. Water Resour. 122, 49–59 (2018)CrossRefGoogle Scholar
  2. 2.
    Alpak, F.O., Gray, F., Saxena, N., Dietderich, J., Hofmann, R., Berg, S.: A distributed parallel multiple-relaxation-time lattice Boltzmann method on general-purpose graphics processing units for the rapid and scalable computation of absolute permeability from high-resolution 3D micro-CT images. Comput. Geosci. 22, 815–832 (2018)CrossRefGoogle Scholar
  3. 3.
    Alpak, F.O., Riviere, B., Frank, F.: A phase-field method for the direct simulation of two-phase flows in pore-scale media using a non-equilibrium wetting boundary condition. Comput. Geosci. 20, 881–908 (2016)CrossRefGoogle Scholar
  4. 4.
    Alpak, F.O., Samardžić, A., Frank, F.: A distributed parallel direct simulator for pore-scale two-phase flow on digital rock images using a finite difference implementation of the phase-field method. J. Pet. Sci. Eng. 166, 806–824 (2018)CrossRefGoogle Scholar
  5. 5.
    Andrä, H., Combaret, N., Dvorkin, J., Glatt, E., Han, J., Kabel, M., Keehm, Y., Krzikalla, F., Lee, M., Madonna, C., Marsh, M., Mukerji, T., Saenger, E.H., Sain, R., Saxena, N., Ricker, S., Wiegmann, A., Zhan, X.: Digital rock physics benchmarks—part I: imaging and segmentation. Comput. Geosci. 50, 25–32 (2013)CrossRefGoogle Scholar
  6. 6.
    Andrä, H., Combaret, N., Dvorkin, J., Glatt, E., Han, J., Kabel, M., Keehm, Y., Krzikalla, F., Lee, M., Madonna, C., Marsh, M., Mukerji, T., Saenger, E.H., Sain, R., Saxena, N., Ricker, S., Wiegmann, A., Zhan, X.: Digital rock physics benchmarks—part II: computing effective properties. Comput. Geosci. 50, 33–43 (2013)CrossRefGoogle Scholar
  7. 7.
    Anderson, D.M., McFadden, G.B., Wheeler, A.A.: Diffuse-interface methods in fluid mechanics. Annu. Rev. Fluid Mech. 30(1), 139–165 (1998)CrossRefGoogle Scholar
  8. 8.
    Armstrong, R.T., Berg, S.: Interfacial velocities and capillary pressure gradients during Haines jumps. Phys. Rev. E 88(4), 043010 (2013)CrossRefGoogle Scholar
  9. 9.
    Armstrong, R.T., Berg, S., Dinariev, O., Evseev, N., Klemin, D., Koroteev, D., Safonov, S.: Modeling of pore-scale two-phase phenomena using density functional hydrodynamics. Transp. Porous Media 112 (3), 577–607 (2016)CrossRefGoogle Scholar
  10. 10.
    Armstrong, R.T., Georgiadis, A., Ott, H., Klemin, D., Berg, S.: Critical capillary number: desaturation studied with fast X-ray computed microtomography. Geophys. Res. Lett. 41, 1–6 (2014)CrossRefGoogle Scholar
  11. 11.
    Armstrong, R.T., McClure, J.E., Berill, M.A., Rücker, M., Schlüter, S., Berg, S.: Beyond Darcy’s law: the role of phase topology and Ganglion dynamics for two fluid flow. Phys. Rev. E. 94, 043113 (2016)CrossRefGoogle Scholar
  12. 12.
    Armstrong, R.T., McClure, J.E., Berill, M.A., Rücker, M., Schlüter, S., Berg, S.: Flow regimes during immiscible displacement. Petrophysics 58(1), 10–18 (2017)Google Scholar
  13. 13.
    Badalassi, V.E., Ceniceros, H.D., Banerjee, S.: Computation of multiphase systems with phase field models. J. Comput. Phys. 190, 371–397 (2003)CrossRefGoogle Scholar
  14. 14.
    Benzi, R., Succi, S., Vergassola, M.: The lattice Boltzmann equation: theory and applications. Phys. Rep. 222(3), 145–197 (1992)CrossRefGoogle Scholar
  15. 15.
    Beresnev, I.A., Deng, W.: Theory of breakup of core fluids surrounded by a wetting annulus in sinusoidally constricted capillary channels. Phys. Fluids 22, 012105 (2010)CrossRefGoogle Scholar
  16. 16.
    Beresnev, I.A., Li, W., Vigil, R.D.: Condition for break-up of non-wetting fluids in sinusoidally constricted capillary channels. Transp. Porous Media 80, 581–604 (2009)CrossRefGoogle Scholar
  17. 17.
    Berg, S., Armstrong, R., Ott, H., Georgiadis, A., Klapp, S.A., Schwing, A., Neiteler, R., Brussee, N., Makurat, A., Leu, L., Enzmann, F., Schwarz, J.-O., Wolf, M., Khan, F., Kersten, M., Irvine, S., Stampanoni, M.: Multiphase flow in porous rock imaged under dynamic flow conditions with fast X-ray computed microtomography. Petrophysics 55(4), 304–312 (2014)Google Scholar
  18. 18.
    Berg, S., Ott, H., Klapp, S.A., Schwing, A., Neiteler, R., Brussee, N., Makurat, A., Leu, L., Enzmann, F., Schwarz, J.-O., Kersten, M., Irvine, S., Stampanoni, M.: Real-time 3D imaging of Haines jumps in porous media flow. Proc. Natl. Acad. Sci. 110(10), 3755–3759 (2013)CrossRefGoogle Scholar
  19. 19.
    Berg, S., Rücker, M., Ott, H., Georgiadis, A., van der Linde, H., Enzmann, F., Kersten, M., Armstrong, R.T., de With, S., Becker, J., Wiegmann, A.: Connected pathway relative permeability from pore-scale imaging of imbibition. Adv. Water Resour. 90, 24–35 (2016)CrossRefGoogle Scholar
  20. 20.
    Bhatnagar, P.L., Gross, E.P., Krook, M.: A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems. Phys. Rev. 94, 511–525 (1954)CrossRefGoogle Scholar
  21. 21.
    Blunt, M.J., Bijeljic, B., Dong, H., Gharbi, O., Iglauer, S., Mostaghimi, P., Paluszny, A., Pentland, C.: Pore-scale imaging and modelling. Adv. Water Resour. 51, 197–216 (2013)CrossRefGoogle Scholar
  22. 22.
    Blunt, M.J., Jackson, M.D., Piri, M., Valvatne, P.H.: Detailed physics, predictive capabilities and macroscopic consequences for pore-network models of multiphase flow. Adv. Water Resour. 25, 1069–1089 (2002)CrossRefGoogle Scholar
  23. 23.
    Boek, E.S., Zacharoudiou, I., Gray, F., Shah, S.M., Crawshaw, J.P., Yang, J.: Multiphase-flow and reactive-transport validation studies at the pore scale by use of lattice Boltzmann computer simulations. SPE J. 22(3), 940–949 (2017)CrossRefGoogle Scholar
  24. 24.
    Briant, A.J., Yeomans, J.M.: Lattice Boltzmann simulations of contact line motion. II. Binary fluids. Phys. Rev. E 69(3), 031603 (2004)CrossRefGoogle Scholar
  25. 25.
    Briant, A.J., Wagner, A.J., Yeomans, J.M.: Lattice Boltzmann simulations of contact line motion. I. Liquid-gas systems. Phys. Rev. E 69(3), 031602 (2004)CrossRefGoogle Scholar
  26. 26.
    Cahn, J.: Critical-point wetting. J. Chem. Phys. 66(8), 3367 (1977)CrossRefGoogle Scholar
  27. 27.
    Cahn, J.W., Hilliard, J.E.: Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phys. 28(2), 258–267 (1958)CrossRefGoogle Scholar
  28. 28.
    d’Humières, D., Ginzburg, I., Krafczyk, M., Lallemand, P., Luo, L.-S.: Multiple-relaxation-time lattice Boltzmann models in three dimensions. Philosophical Transactions of the Royal Society A – Mathematical. Phys. Eng. Sci. 360(1792), 72 (2002)Google Scholar
  29. 29.
    de Gennes, P.G.: Wetting: statics and dynamics. Rev. Mod. Phys. 57(3), 827–863 (1985)CrossRefGoogle Scholar
  30. 30.
    Demianov, A., Dinariev, O., Evseev, N.V.: Density functional modelling in multiphase compositional hydrodynamics. Can. J. Chem. Eng. 89, 206–226 (2011)CrossRefGoogle Scholar
  31. 31.
    Demianov, A., Dinariev, O., Evseev, N.V.: Introduction to the density functional method in hydrodynamics. Moscow, Fizmatlit (2014)Google Scholar
  32. 32.
    DiCarlo, D.A., Cidoncha, J.I.G., Hickey, C.: Acoustic measurements of pore-scale displacements. Geophys. Res. Lett. 30(17), 1901 (2003)CrossRefGoogle Scholar
  33. 33.
    Dinariev, O., Evseev, N.: Multiphase flow modeling with density functional method. Comput. Geosci. 20, 835–856 (2016)CrossRefGoogle Scholar
  34. 34.
    Ding, H., Spelt, P.D.M., Shu, C.: Diffuse interface model for incompressible two-phase flows with large density ratios. J. Comput. Phys. 226, 2078–2095 (2007)CrossRefGoogle Scholar
  35. 35.
    Dong, B., Yan, Y., Li, W.: LBM simulation of viscous fingering phenomenon in immiscible displacement of two fluids in porous media. Transp. Porous Media 88(2), 293–314 (2011)CrossRefGoogle Scholar
  36. 36.
    Du, R., Shi, B., Chen, X.: Multi-relaxation-time lattice Boltzmann model for incompressible flow. Phys. Lett. A 359(6), 564–572 (2006)CrossRefGoogle Scholar
  37. 37.
    Ferrari, A., Lunati, I.: Inertial effects during irreversible meniscus reconfiguration in angular pores. Adv. Water Resour. 74, 1–13 (2014)CrossRefGoogle Scholar
  38. 38.
    Frank, F., Liu, C., Alpak, F.O., Berg, S., Riviere, B.: Direct numerical simulation of flow on pore-scale images using discontinuous Galerkin finite element method. SPE J. 23(5), 1833–1850 (2018)CrossRefGoogle Scholar
  39. 39.
    Frank, F., Liu, C., Scanziani, A., Alpak, F.O., Riviere, B.: An energy-based equilibrium contact angle boundary condition on jagged surfaces for phase-field methods. J. Colloid Interface Sci. 523, 282–291 (2018)CrossRefGoogle Scholar
  40. 40.
    Georgiadis, A., Berg, S., Makurat, A., Maitland, G., Ott, H.: Pore-scale micro-computed-tomography imaging: non-wetting phase cluster size distribution during drainage and imbibition. Phys. Rev. E 88(3), 033002 (2013)CrossRefGoogle Scholar
  41. 41.
    Ghassemi, A., Pak, A.: Numerical study of factors influencing relative permeabilities of two immiscible fluids flowing through porous media using lattice Boltzmann method. J. Pet. Sci. Eng. 77(1), 135–145 (2011)CrossRefGoogle Scholar
  42. 42.
    Glimm, J., Grove, J.W., Li, X.-L., Zhao, N.: Simple front tracking. In: Chen, G.-Q., DiBenedetto, E. (eds.) Contemporary mathematics. American Mathematical Society, 238, 133–149 (1999)Google Scholar
  43. 43.
    Guangwu, Y.: A Lagrangian lattice Boltzmann method for Euler equations. Acta Mech. Sinica 14(2), 186–192 (1998)CrossRefGoogle Scholar
  44. 44.
    Gunstensen, A.K., Rothman, D.H., Zaleski, S., Zanetti, G.: Lattice Boltzmann model of immiscible fluids. Phys. Rev. A 43(8), 4320 (1991)CrossRefGoogle Scholar
  45. 45.
    Haines, W.B.: Studies in the physical properties of soils, part V—the hysteresis effect in capillary properties, and the modes of water distribution associated therewith. J. Agric. Sci. 20(1), 97–116 (1930)CrossRefGoogle Scholar
  46. 46.
    He, X., Chen, S., Zhang, R.: A lattice Boltzmann scheme for incompressible multiphase flow and its application in simulation of Rayleigh–Taylor instability. J. Comput. Phys. 152(2), 642–663 (1999)CrossRefGoogle Scholar
  47. 47.
    Hecht, M., Harting, J.: Implementation of on-site velocity boundary conditions for D3Q19 lattice Boltzmann simulations. Journal of Statistical Mechanics: Theory and Experiment P01018 (2010)Google Scholar
  48. 48.
    Hilfer, R., Armstrong, R.T., Berg, S., Georgiadis, A., Ott, H.: Capillary saturation and desaturation. Phys. Rev. E. 92, 063023 (2015)CrossRefGoogle Scholar
  49. 49.
    Huang, H., Huang, J.-J., Lu, X.-Y.: Study of immiscible displacements in porous media using a color gradient-based multiphase lattice Boltzmann method. Comput. Fluids 93, 164–172 (2014)CrossRefGoogle Scholar
  50. 50.
    Humphry, K.J., Suijkerbuijk, B.M.J.M., van der Linde, H.A., Pieterse, S.G.J., Masalmeh, S.K.: Impact of wettability on residual oil saturation and capillary desaturation curves. International Symposium of the Society of Core Analysts held in Napa Valley, California, USA, 16-19 September 2013. Paper SCA2013-025 (2013)Google Scholar
  51. 51.
    Inamuro, T., Ogata, T., Tajima, S., Konishi, N.: A lattice Boltzmann method for incompressible two-phase flows with large density differences. J. Comput. Phys. 198(2), 628–644 (2004)CrossRefGoogle Scholar
  52. 52.
    Jacqmin, D.: Calculation of two-phase Navier-Stokes flows using phase-field modeling. J. Comput. Phys. 155, 96–127 (1999)CrossRefGoogle Scholar
  53. 53.
    Jakobsen, H.A.: Chemical reactor modelling. Springer, Berlin (2008)Google Scholar
  54. 54.
    Joekar-Niasar, V., van Dijke, M.I.J., Hassanizadeh, S.M.: Pore-scale modeling of multiphase flow and transport: achievements and perspectives. Transp. Porous Media 94, 461–464 (2012)CrossRefGoogle Scholar
  55. 55.
    Kendon, V., Cates, M., Pagonabarraga, I., Desplat, J.-C., Bladon, P.: Inertial effects in three-dimensional spinodal decomposition of a symmetric binary fluid mixture: a lattice Boltzmann study. J. Fluid Mech. 440, 147–203 (2001)CrossRefGoogle Scholar
  56. 56.
    Kim, J.: A continuous surface tension force formulation for diffuse-interface models. J. Comput. Phys. 204, 784–804 (2005)CrossRefGoogle Scholar
  57. 57.
    Kim, J.: Phase-field models for multi-component fluid flows. Communications in Computational Physics 12 (3), 613–661 (2012)CrossRefGoogle Scholar
  58. 58.
    Koroteev, D., Dinariev, O., Evseev, N., Klemin, D., Nadeev, A., Safonov, S., Gurpinar, O., Berg, S., van Kruijsdijk, C., Armstrong, R., Myers, M.T., Hathon, L., de Jong, H.: Direct hydrodynamic simulation of multiphase flow in porous rock. Petrophysics 55(4), 294–303 (2014)Google Scholar
  59. 59.
    Koroteev, D., Dinariev, O., Evseev, N., Klemin, D., Safonov, S., Gurpinar, O., Berg, S., van Kruijswijk, C., Myers, M., Hathon, L., de Jong, H., Armstrong, R. T.: Application of digital rock technology for chemical EOR screening. Paper SPE 165258, EORC 2013 – SPE Enhanced Oil Recovery Conference 2-4 July 2013, Kuala Lumpur, Malaysia (2013)Google Scholar
  60. 60.
    Kupershtokh, A.L., Medvedev, D.A., Karpov, D.I.: On equations of state in a lattice Boltzmann method. Computers & Mathematics with Applications 58(5), 965–974 (2009)CrossRefGoogle Scholar
  61. 61.
    Ladd, A.J.C., Verberg, R.: Lattice-Boltzmann simulations of particle-fluid suspensions. J. Stat. Phys. 104(5-6), 1191–1251 (2001)CrossRefGoogle Scholar
  62. 62.
    Lake, L.W.: Enhanced oil recovery. Prentice-Hall Inc, Englewood Cliffs (1989)Google Scholar
  63. 63.
    Lallemand, P., Luo, L.-S.: Theory of the lattice Boltzmann method: dispersion, dissipation, isotropy, Galilean invariance, and stability. Phys. Rev. E. 61(6), 6546–6562 (2000)CrossRefGoogle Scholar
  64. 64.
    Li, J., Sultan, A.S.: Permeability computations of shale gas by the pore-scale Monte Carlo molecular simulations. Paper IPTC-18263-MS presented at the International Petroleum Technology Conference 6-9 December, Doha. Qatar (2015)Google Scholar
  65. 65.
    Li, Q., Wagner, A.J.: Symmetric free-energy-based multicomponent lattice Boltzmann method. Phys. Rev. E 76, 036701 (2007)CrossRefGoogle Scholar
  66. 66.
    Li, X., Wu, S., Song, J., Li, H., Wang, S.: Numerical simulation of pore-scale flow in chemical flooding process. Theor. Appl. Mech. Lett. 2, 022008 (2011)CrossRefGoogle Scholar
  67. 67.
    Liu, H., Valocchi, A.J., Kang, Q., Werth, C.: Pore-scale simulations of gas displacing liquid in a homogeneous pore network using the lattice Boltzmann method. Transp. Porous Media 99(3), 555–580 (2013)CrossRefGoogle Scholar
  68. 68.
    Liu, Z., Wu, H.: Pore-scale modeling of immiscible two-phase flow in complex porous media. Appl. Therm. Eng. 93, 1394–1402 (2016)CrossRefGoogle Scholar
  69. 69.
    Luo, L.-S.: Theory of the lattice Boltzmann method: lattice Boltzmann models for nonideal gases. Phys. Rev. E 62, 4982–4996 (2000)CrossRefGoogle Scholar
  70. 70.
    Meakin, P., Tartakovsky, A.M.: Modeling and simulation of pore-scale multiphase fluid flow and reactive transport in fractured and porous media. Rev. Geophys. 47, RG3002 (2009)CrossRefGoogle Scholar
  71. 71.
    Meldi, M., Vergnault, E., Sagaut, P.: An arbitrary Lagrangian–Eulerian approach for the simulation of immersed moving solids with lattice Boltzmann method. J. Comput. Phys. 235, 182–198 (2013)CrossRefGoogle Scholar
  72. 72.
    Moebius, F., Or, D.: Interfacial jumps and pressure bursts during fluid displacement in interacting irregular capillaries. J. Colloid Interface Sci. 377(1), 406–415 (2012)CrossRefGoogle Scholar
  73. 73.
    Mohanty, K.K., Davis, H.T., Scriven, L.E.: Physics of oil entrapment in water-wet rock. SPE Reserv. Eval. Eng. 2(1), 113–128 (1987)CrossRefGoogle Scholar
  74. 74.
    Morrow, N.R.: Physics and thermodynamics of capillary action in porous media. Ind. Eng. Chem. 62(6), 32–56 (1970)CrossRefGoogle Scholar
  75. 75.
    Niessner, J., Berg, S., Hassanizadeh, S.M.: Comparison of two-phase Darcy’s law with a thermodynamically consistent approach. Transp. Porous Media 88, 133–148 (2011)CrossRefGoogle Scholar
  76. 76.
    Nourgaliev, R.R., Theofanous, T.G.: High fidelity interface tracking: unlimited anchored level set. J. Comput. Phys. 224, 836–866 (2007)CrossRefGoogle Scholar
  77. 77.
    Oughanem, R., Youssef, S., Bauer, D., Peysson, Y., Maire, E., Vizika, O.: A multi-scale investigation of pore structure impact on the mobilization of trapped oil by surfactant injection. Transp. Porous Media 109, 673–692 (2015)CrossRefGoogle Scholar
  78. 78.
    Pooley, C.M., Furtado, K.: Eliminating spurious velocities in the free-energy lattice Boltzmann method. Phys. Rev. E 77(4), 046702 (2008)CrossRefGoogle Scholar
  79. 79.
    Pooley, C.M., Kusumaatmaja, H., Yeomans, J.M.: Contact line dynamics in binary lattice Boltzmann simulations. Phys. Rev. E 78, 056709 (2008)CrossRefGoogle Scholar
  80. 80.
    Premnath, K.N., Abraham, J.: Three-dimensional multi-relaxation-time (MRT) lattice-Boltzmann models for multiphase flow. J. Comput. Phys. 22(2), 539–559 (2007)CrossRefGoogle Scholar
  81. 81.
    Prodanovic, M., Bryant, S.L.: A level set method for determining critical curvatures for drainage and imbibition. J. Colloid Interface Sci. 304(2), 442–458 (2006)CrossRefGoogle Scholar
  82. 82.
    Raeesi, B., Morrow, N.R., Mason, G.: Contact angle hysteresis at smooth and rough surfaces. Integration Geoconvention, Geoscience Engineering Partnership (2013)Google Scholar
  83. 83.
    Raeini, A.Q., Blunt, M.J., Bijeljic, B.: Modelling two-phase flow in porous media at the pore scale using the volume-of-fluid method. Comput. Phys. 231, 5653–5668 (2012)CrossRefGoogle Scholar
  84. 84.
    Roman, S., Abu-Al-Saud, M., Tokunaga, T., Wan, J., Kovscek, A., Tchelepi, H.A.: Measurements and simulation of liquid films during drainage displacements and snap-off in constricted capillary tubes. J. Colloid Interface Sci. 507, 279–289 (2017)CrossRefGoogle Scholar
  85. 85.
    Roof, J.R.: Snap-off of oil droplets in water-wet pores. SPE J. 10(1), 85–90 (1970)Google Scholar
  86. 86.
    Sedghi, M., Piri, M., Goual, L.: Molecular dynamics of wetting layer formation and forced water invasion in angular nanopores with mixed wettability. J. Chem. Phys. 141, 194703 (2014)CrossRefGoogle Scholar
  87. 87.
    Seth, S., Morrow, N.R.: Efficiency of the conversion of work of drainage to surface energy for sandstone and carbonate. SPE Reserv. Eval. Eng. 10(4), 338–347 (2007)CrossRefGoogle Scholar
  88. 88.
    Shan, X., Chen, H.: Lattice Boltzmann model for simulating flows with multiple phases and components. Phys. Rev. E. 47(3), 1815 (1993)CrossRefGoogle Scholar
  89. 89.
    Silin, D., Patzek, T.: Pore space morphology analysis using maximal inscribed spheres. Physica A – Statistical Mechanics and its Applications 371(2), 336–360 (2006)CrossRefGoogle Scholar
  90. 90.
    Sivanesapillai, R., Falkner, N., Hartmaier, A., Steeb, H.: A CSF-SPH method for simulating drainage and imbibition at pore-scale resolution while tracking interfacial areas. Adv. Water Resour. 95, 212–234 (2016)CrossRefGoogle Scholar
  91. 91.
    Succi, S.: The lattice-Boltzmann equation. Oxford University Press, Oxford (2001)Google Scholar
  92. 92.
    Swift, M.R., Orlandini, E., Osborn, W.R., Yeomans, J.M.: Lattice Boltzmann simulations of liquid-gas and binary fluid systems. Phys. Rev. E 54(5), 5041–5052 (1996)CrossRefGoogle Scholar
  93. 93.
    Swift, M.R., Osborn, W.R., Yeomans, J.M.: Lattice Boltzmann simulation of non-ideal fluids. Phys. Rev. Lett. 75(5), 830–833 (1995)CrossRefGoogle Scholar
  94. 94.
    Tölke, J., Freudiger, S., Krafczyk, M.: An adaptive scheme using hierarchical grids for lattice Boltzmann multi-phase flow simulations. Comput. Fluids 35(8), 820–830 (2006)CrossRefGoogle Scholar
  95. 95.
    Yan, G., Dong, Y., Liu, Y.: An implicit Lagrangian lattice Boltzmann method for the compressible flows. Numer. Methods Fluids 51(12), 1407–1418 (2006)CrossRefGoogle Scholar
  96. 96.
    Yang, J.: Multi-scale simulation of multiphase multi-component flow in porous media using the lattice Boltzmann method. PhD dissertation Imperial College, London, UK (2013)Google Scholar
  97. 97.
    Yeomans, J.: Mesoscale simulations: lattice Boltzmann and particle algorithms. J. Phys. A Math. Theor. 369 (1), 159–184 (2006)Google Scholar
  98. 98.
    Youssef, S., Peysson, Y., Bauer, D., Vizitak, O.: Capillary desaturation curve prediction using 3D microtomography images. International Symposium of the Society of Core Analysts held in St. John’s, Newfoundland and Labrador, Canada, 16-21 August 2015. Paper SCA2015-008 (2015)Google Scholar
  99. 99.
    Yuan, H.H., Swanson, B.F.: Resolving pore-space characteristics by rate-controlled porosimetry. Paper SPE-14892. SPE Form. Eval. 4(1), 17–24 (1989)CrossRefGoogle Scholar
  100. 100.
    Zacharoudiou, I., Boek, E.S.: Capillary filling and Haines jump dynamics using free energy lattice Boltzmann simulations. Adv. Water Resour. 92, 43–56 (2016)CrossRefGoogle Scholar
  101. 101.
    Zhang, J., Kwok, D.Y.: A mean-field free energy lattice Boltzmann model for multicomponent fluids. The European Physical Journal Special Topics 171(1), 45–53 (2009)CrossRefGoogle Scholar
  102. 102.
    Zheng, H., Shu, C., Chew, Y.-T.: A lattice Boltzmann model for multiphase flows with large density ratio. J. Comput. Phys. 218(1), 353–371 (2006)CrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • F. O. Alpak
    • 1
    • 2
    Email author
  • I. Zacharoudiou
    • 3
  • S. Berg
    • 4
    • 5
    • 6
  • J. Dietderich
    • 1
  • N. Saxena
    • 1
  1. 1.Shell International Exploration and Production, Inc.HoustonUSA
  2. 2.Department of Computational and Applied MathematicsRice UniversityHoustonUSA
  3. 3.Qatar Carbonates and Carbon Storage Research Centre, Department of Chemical EngineeringImperial College London, South Kensington CampusLondonUK
  4. 4.Shell Global Solutions International B.V.AmsterdamThe Netherlands
  5. 5.Department of Earth Science and EngineeringImperial College London, South Kensington CampusLondonUK
  6. 6.Department of Chemical Engineering, Imperial College LondonSouth Kensington CampusLondonUK

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