Numerical methods to simulate spontaneous imbibition in microscopic pore structures: A review

Yan Zhou, Wei Guan, Changming Zhao, Xiaojing Zou, Zhennan He, Hongyang Zhao

Abstract view|0|times       PDF download|0|times


Spontaneous imbibition, as a fundamental flow phenomenon, is widely utilized in fossil energy production, carbon dioxide and underground hydrogen storage. With the development of computing, the exploration of flow laws of spontaneous imbibition has evolved from macroscopic theoretical models to pore-scale numerical analysis. Currently, the solutions for multiphase flow in pore media mainly consider the volume of fluid and the phase field, and have been classed into level set methods based on macroscopic Navier-Stokes equations and the Shan-Chen, free energy, color gradient, and phase-field methods based on mesoscopic lattice Boltzmann equations. However, no comprehensive review article has summarized the strengths and limitations of these methods. Therefore, this work focuses on critically reviewing and commenting on the fundamentals and limitations of porescale models applied to spontaneous imbibition. In addition, recent works applying these methods are systematically reviewed. Our study aims to provide the scientific community with an expert opinion to understand the basic methods for solving the existing problems of spontaneous imbibition in porous media. Future research directions are suggested, namely, focusing on developing the reconstruction pore medium algorithms, establishing modeling methods for non-stationary states, exploring the flow laws in mixed wetting conditions, linking macroscopic and microscopic flow laws, and developing models for coupled multiphase flow numerical computation with machine learning. Overall, this review provides a comprehensive understanding of spontaneous imbibition simulation methods, promotes a thorough knowledge of spontaneous imbibition in porous media, provides guidance on exploring flow laws, and inspires researchers to give more credit to spontaneous imbibition studies.

Document Type: Invited review

Cited as: Zhou, Y., Guan, W., Zhao, C., Zou, X., He, Z., Zhao, H. Numerical methods to simulate spontaneous imbibition in microscopic pore structures: A review. Capillarity, 2024, 11(1): 1-21.


Numerical method, porous media, spontaneous imbibition, navier-stokes equation, lattice boltzmann method

Full Text:



Abd, A. S., Elhafyan, E., Siddiqui, A. R., et al. A review of the phenomenon of counter-current spontaneous imbibition: Analysis and data interpretation. Journal of Petroleum Science and Engineering, 2019, 180: 456-470.

Abdelaziz, A., Ha, J., Li, M., et al. Understanding hydraulic fracture mechanisms: From the laboratory to numerical modelling. Advances in Geo-Energy Research, 2023, 7(1): 66-68.

Akbarabadi, M., Saraji, S., Piri, M., et al. Nano-scale experimental investigation of in-situ wettability and spontaneous imbibition in ultra-tight reservoir rocks. Advances in Water Resources, 2017, 107: 160-179.

Almutairi, A., Saira, S., Wang, Y., et al. Effect of fines migration on oil recovery from carbonate rocks. Advances in Geo-Energy Research, 2023, 8(1): 61-70.

Alpak, F., 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. Journal of Petroleum Science and Engineering, 2018, 166: 806-824.

AlRatrout, A., Blunt, M. J., Bijeljic, B. Wettability in complex porous materials, the mixed-wet state, and its relationship to surface roughness. Proceedings of the National Academy of Sciences of the United States of America, 2018, 115(36): 8901-8906.

Amaya, B. L., Lee, T. Single bubble rising dynamics for moderate Reynolds number using Lattice Boltzmann Method. Computers & Fluids, 2010, 39(7): 1191-1207.

Ambekar, A. S., Mattey, P., Buwa, V. V. Pore-resolved twophase flow in a pseudo-3D porous medium: Measurements and volume-of-fluid simulations. Chemical Engineering Science, 2021a, 230: 116128.

Ambekar, A. S., Mondal, S., Buwa, V. V. Pore-resolved volume-of-fluid simulations of two-phase flow in porous media: Pore-scale flow mechanisms and regime map. Physics of Fluids, 2021b, 33: 102119.

Amico, S. C., Lekakou, C. Axial impregnation of a fiber bundle. Part 1: Capillary experiments. Polymer Composites, 2002, 23: 249-263.

Amiri, H. A. A. Pore-scale modeling of non-isothermal two phase flow in 2D porous media: Influences of viscosity, capillarity, wettability and heterogeneity. International Journal of Multiphase Flow, 2014, 61: 14-27.

Amiri, H. A. A., Hamouda, A. A. C. Evaluation of level set and phase field methods in modeling two phase flow with viscosity contrast through dual-permeability porous medium. International Journal of Multiphase Flow, 2013, 52: 22-34.

Aronofsky, J. S., Masse, L., Natanson, S. G. A model for the mechanism of oil recovery from the porous matrix due to water invasion in fractured reservoirs. Transactions of the AIME, 1958, 213(1): 17-19.

Ashgriz, N., Poo, J. Y. FLAIR: Flux line-segment model for advection and interface reconstruction. Journal of Computational Physics, 1991, 93(2): 449-468.

Aslannezhad, M., Ali, M., Kalantariasl, A., et al. A review of hydrogen/rock/brine interaction: Implications for hydrogen geo-storage. Progress in Energy and Combustion Science, 2023, 95: 101066.

Ayyaswamy, P. S., Catton, I., Edwards, D. K. Capillary flow in triangular grooves. Journal of Applied Mechanics, 1974, 41(2): 332-336.

Bagheri, M., Mahani, H., Ayatollahi, S., et al. Direct porescale simulation of the effect of capillary number and gas compressibility on cyclic underground hydrogen storage & production in heterogeneous aquifers. Advances in Water Resources, 2023, 181: 104547.

Barati, R., Liang, J. T. A review of fracturing fluid systems used for hydraulic fracturing of oil and gas wells. Journal of Applied Polymer Science, 2014, 131(16): 318-323.

Benson, A. L. L., Clarkson, C. R. Flowback rate-transient analysis with spontaneous imbibition effects. Journal of Natural Gas Science and Engineering, 2022, 108: 104830.

Brackbill, J. U, Kothe, D. B., Zemach, C. A continuum method for modeling surface tension. Journal of Computational Physics, 1992, 100(2): 335-354.

Broughton, J., Joshi, Y. K. Flow boiling in geometrically modified microchannels. Physics of Fluids, 2021, 33(10): 103308.

Burfeindt, M., Alqadah, H. Linear sampling method imaging of three-dimensional conducting targets from limited apertures via phase-delay-constrained formulations. Progress in Electromagnetics Research, 2023, 178: 63-81.

Cahn, J. W. Free energy of a nonuniform system. II. Thermodynamic basis. The Journal of Chemical Physics, 2004, 30(5): 1121-1124.

Cahn, J. W., Hilliard, J. E. Free energy of a nonuniform system. I. Interfacial free energy. The Journal of Chemical Physics, 2004a, 28(2): 258-267.

Cahn, J. W., Hilliard, J. E. Free energy of a nonuniform system. III. Nucleation in a two-component incompressible fluid. The Journal of Chemical Physics, 2004b, 31(3): 688-699.

Cai, J., Chen, Y., Liu, Y., et al. Capillary imbibition and flow of wetting liquid in irregular capillaries: A 100-year review. Advances in Colloid and Interface Science, 2022, 304: 102654.

Cai, J., Hu, X., Standnes, D. C., et al. An analytical model for spontaneous imbibition in fractal porous media including gravity. Colloids and Surfaces A: Physicochemical and Engineering Aspects, 2012a, 414: 228-233.

Cai, J., Li, C., Song, K., et al. The influence of salinity and mineral components on spontaneous imbibition in tight sandstone. Fuel, 2020a, 269: 117087.

Cai, J., Perfect, E., Cheng, C. L., et al. Generalized modeling of spontaneous imbibition based on Hagen-Poiseuille flow in tortuous capillaries with variably shaped apertures. Langmuir, 2014, 30(18): 5142-5151.

Cai, J., Wei, W., Hu, X., et al. Fractal characterization of dynamic fracture network extension in porous media. Fractals, 2017, 25(2): 1750023.

Cai, J., Xia, Y., Lu, C., et al. Creeping microstructure and fractal permeability model of natural gas hydrate reservoir. Marine and Petroleum Geology, 2020b, 115: 104282.

Cai, J., You, L., Hu, X., et al. Prediction of effective permeability in porous media based on spontaneous imbibition effect. International Journal of Modern Physics C, 2012b, 23(7): 1250054.

Cai, J., Yu, B. Advances in studies of spontaneous imbibition in porous media. Advances in Mechanics, 2012c, 42(6): 735-754. (in Chinese)

Cai, J., Yu, B., Zou, M., et al. Fractal characterization of spontaneous co-current imbibition in porous media. Energy & Fuels, 2010, 24(3): 1860-1867.

Caporin, M., Fontini, F. The long-run oil-natural gas price relationship and the shale gas revolution. Energy Economics, 2017, 64: 511-519.

Chai, Z., Sun, D., Wang, H., et al. A comparative study of local and nonlocal Allen-Cahn equations with mass conservation. International Journal of Heat and Mass Transfer, 2018, 122: 631-642.

Chai, Z., Zhao, T. S. Lattice Boltzmann model for the convection-diffusion equation. Physical Review E, 2013, 87(6): 063309.

Chamakos, N. T., Sema, D. G., Papathanasiou, A. G. Progress in modeling wetting phenomena on structured substrates. Archives of Computational Methods in Engineering, 2021, 28: 1647-1666.

Chen, B., Barboza, B. R., Sun, Y., et al. A review of hydraulic fracturing simulation. Archives of Computational Methods in Engineering, 2022, 29: 1-58.

Chen, J., Yang, Y., Cao, X., et al. Simulation study on the spontaneous imbibition mechanism and influencing factors of laminated shale oil reservoirs in Jiyang depression, eastern China. Energy & Fuels, 2023, 37(17): 13048-13064.

Chen, L., Kang, Q., Mu, Y., et al. A critical review of the pseudopotential multiphase lattice Boltzmann model: Methods and applications. International Journal of Heat and Mass Transfer, 2014, 76: 210-236.

Chen, Y., Shen, J. Efficient, adaptive energy stable schemes for the incompressible Cahn-Hilliard Navier-Stokes phasefield models. Journal of Computational Physics, 2016, 308: 40-56.

Cheng, Z., Gao, H., Ning, Z., et al. Inertial effect on oil/water countercurrent imbibition in porous media from a porescale perspective. SPE Journal, 2022, 27(3): 1619-1632.

Chiappini, D., Bella, G., Succi, S., et al. Improved lattice Boltzmann without parasitic currents for Rayleigh-Taylor instability. Communications in Computational Physics, 2010, 7(3): 423-444.

Comminal, R., Spangenberg, J., Hattel, J. H. Cellwise conservative unsplit advection for the volume of fluid method. Journal of Computational Physics, 2015, 283: 582-608.

Cortazar, G., Gravet, M., Urzua, J. The valuation of multidimensional American real options using the LSM simulation method. Computers & Operations Research, 2008, 35(1): 113-129.

Derijani, H, James, L. A., Haynes, R. D. Evaluation of the interFoam solver in the prediction of immiscible twophase flow in imbibition and drainage on the pore-doublet system. E3S Web of Conferences, 2023, 366: 01017.

Diao, Z., Li, S., Liu, W., et al. Numerical study of the effect of tortuosity and mixed wettability on spontaneous imbibition in heterogeneous porous media. Capillarity, 2021, 4(3): 50-62.

Ding, H., Spelt, P. D. M., Shu, C. Diffuse interface model for incompressible two-phase flows with large density ratios. Journal of Computational Physics, 2007, 226(2): 2078-2095.

Dong, M., Chatzis, I. Oil layer flow along the corners of non-circular capillaries by gravity drainage. Journal of Canadian Petroleum Technology, 2003, 42(2): PETSOC-03-02-TN.

Ervik, Å., Lervåg, K. Y., Munkejord, S. T. A robust method for calculating interface curvature and normal vectors using an extracted local level set. Journal of Computational Physics, 2014, 257: 259-277.

Esene, C., Rezaei, N., Aborig, A., et al. Comprehensive review of carbonated water injection for enhanced oil recovery. Fuel, 2019, 237: 1086-1107.

Farhadzadeh, M., Nick, H. M. Pore-level mechanics of modified salinity effects in fractured porous media. ECMOR 2022. Paper Presented at European Conference on the Mathematics of Geological Reservoirs 2022, The Hague, Netherlands, 5-7 September, 2022.

Ferrari, A., Jimenez, M. J., Borgne, T. L., et al. Challenges in modeling unstable two-phase flow experiments in porous micromodels. Water Resources Research, 2015, 51(3): 1381-1400.

Ferrari, A., Lunati, I. Direct numerical simulations of interface dynamics to link capillary pressure and total surface energy. Advances in Water Resources, 2013, 57: 19-31.

Gao, H., Tatomir, A. B., Karadimitriou, N. K., et al. A two-phase, pore-scale reactive transport model for the kinetic interface-sensitive tracer. Water Resources Research, 2021, 57(6): e2020WR028572.

Gao, Z., Fan, Y., Hu, Q., et al. A review of shale wettability characterization using spontaneous imbibition experiments. Marine and Petroleum Geology, 2019, 109: 330-338.

Garoosi, F., Hooman, K. Numerical simulation of multiphase flows using an enhanced Volume-of-Fluid (VOF) method. International Journal of Mechanical Sciences, 2022, 215: 106956.

Geier, M., Fakhari, A., Lee, T. Conservative phase-field lattice Boltzmann model for interface tracking equation. Physical Review E, 2015, 91(6): 063309.

Ghasemi, A. N., Ayatollahi, S., Mahani, H. Pore-scale simulation of the interplay between wettability, capillary number, and salt dispersion on the efficiency of oil mobilization by low-salinity waterflooding. SPE Journal, 2021, 26(6): 4000-4021.

Golparvar, A., Zhou, Y., Wu, K., et al. A comprehensive review of pore scale modeling methodologies for multiphase flow in porous media. Advances in Geo-Energy Research, 2018, 2(4): 418-440.

Gong, R., Li, H., Wang, X., et al. Further study on the effect of interfacial tension on cocurrent spontaneous imbibition based on direct simulation and NMR. Energy & Fuels, 2023, 37: 16396-16409.

Greenwood, M., Shampur, K. N., Ofori Opoku, N., et al. Quantitative 3D phase field modelling of solidification using next-generation adaptive mesh refinement. Computational Materials Science, 2018, 142: 153-171.

Grunau, D., Chen, S., Eggert, K. A lattice Boltzmann model for multiphase fluid flows. Physics of Fluids A: Fluid Dynamics, 1993, 5(10): 2557-2562.

Gu, Q., Zhu, L., Zhang, Y., et al. Pore-scale study of countercurrent imbibition in strongly water-wet fractured porous media using lattice Boltzmann method. Physics of Fluids, 2019, 31: 086602.

Gueyffier, D., Li, J., Nadim, A., et al. Volume-of-fluid interface tracking with smoothed surface stress methods for three-dimensional flows. Journal of Computational physics, 1999, 152(2): 423-456.

Gunstensen, A. K., Rothman, D. H. Microscopic modeling of immiscible fluids in three dimensions by a lattice boltzmann method. Europhysics Letters, 1992, 18(2): 157-161.

Gunstensen, A. K., Rothman, D. H. Lattice-Boltzmann studies of immiscible two-phase flow through porous media. Journal of Geophysical Research: Solid Earth, 1993, 98(B4): 6431-6441.

Gunstensen, A. K., Rothman, D. H., Zaleski, S., et al. Lattice boltzmann model of immiscible fluids. Physical Review A, 1991, 43(8): 4320.

Guo, Q., Wang, S., Chen, X. Assessment on tight oil resources in major basins in China. Journal of Asian Earth Sciences, 2019, 178: 52-63.

Guo, Z., Zheng, C., Shi, B. Discrete lattice effects on the forcing term in the lattice Boltzmann method. Physical Review E, 2002, 65(4): 046308.

Gupta, A., Krishnan, U. M., Mandal, T. K., et al. An adaptive mesh refinement algorithm for phase-field fracture models: Application to brittle, cohesive, and dynamic fracture. Computer Methods in Applied Mechanics and Engineering, 2022, 399: 115347.

Hao, L., Cheng, P. Lattice Boltzmann simulations of water transport in gas diffusion layer of a polymer electrolyte membrane fuel cell. Journal of Power Sources, 2010, 195(12): 3870-3881.

Hatiboglu, C. U., Babadagli, T. Oil recovery by countercurrent spontaneous imbibition: Effects of matrix shape factor, gravity, IFT, oil viscosity, wettability, and rock type. Journal of Petroleum Science and Engineering, 2007, 59(1-2): 106-122.

He, X., Chen, S., Zhang, R. A lattice Boltzmann scheme for incompressible multiphase flow and its application in simulation of Rayleigh-Taylor instability. Journal of Computational Physics, 1999, 152(2): 642-663.

He X., Shan, X., Doolen, G. D. Discrete Boltzmann equation model for nonideal gases. Physical Review E, 1998, 57(1): R13-R16.

He, X., Zou, Q., Luo, L. S., et al. Analytic solutions of simple flows and analysis of nonslip boundary conditions for the lattice Boltzmann BGK model. Journal of Statistical Physics, 1997, 87: 115-136.

He, Z., Liang, F., Meng, J. Effects of groundwater fluctuation on migration characteristics and representative elementary volume of entrapped LNAPL. Journal of Hydrology, 2022a, 610: 127833.

He, Z., Liang, F., Meng, J. Pore-scale study of the effect of bifurcated fracture on spontaneous imbibition in heterogeneous porous media. Physics of Fluids, 2022b, 34(7): 072003.

He, Z., Liang, F., Meng, J. Effects of injection directions and boundary exchange times on adaptive pumping in heterogeneous porous media: Pore-scale simulation. Science of the Total Environment, 2023, 867: 161427.

Hematpur, H., Abdollahi, R., Rostami, S., et al. Review of underground hydrogen storage: Concepts and challenges. Advances in Geo-Energy Research, 2023, 7(2): 111-131.

Hernández Cid, D., Pérez González, V. H., Gallo Villanueva R. C., et al. Modeling droplet formation in microfluidic flow-focusing devices using the two-phases level set method. Materials Today: Proceedings, 2022, 48: 30-40.

Hirt, C. W., Nichols, B. D. Volume of fluid (VOF) method for the dynamics of free boundaries. Journal of Computational Physics, 1981, 39(1): 201-225.

Holdych, D. J., Rovas, D., Georgiadis, J. G., et al. An improved hydrodynamics formulation for multiphase flow lattice-boltzmann models. International Journal of Modern Physics C, 1998, 9(8): 1393-1404.

Hu, R., Wan, J., Kim, Y., et al. Wettability effects on supercritical CO2-brine immiscible displacement during drainage: Pore-scale observation and 3D simulation. International Journal of Greenhouse Gas Control, 2017, 60: 129-139.

Hu, Y., Zhao, C., Zhao, J., et al. Mechanisms of fracturing fluid spontaneous imbibition behavior in shale reservoir: A review. Journal of Natural Gas Science and Engineering, 2020, 82: 103498.

Huang, H., Sukop, M., Lu, X. Multiphase Lattice Boltzmann Methods: Theory and Application. New York, USA, John Wiley & Sons, 2015.

Huppert, H. E., Neufeld, J. A. The fluid mechanics of carbon dioxide sequestration. Annual Review of Fluid Mechanics, 2014, 46: 255-272.

Hyman, J. D., Winter, C. L. Stochastic generation of explicit pore structures by thresholding Gaussian random fields. Journal of Computational Physics, 2014, 277: 16-31.

Issakhov, A., Zhandaulet, Y., Nogaeva, A. Numerical simulation of dam break flow for various forms of the obstacle by VOF method. International Journal of Multiphase Flow, 2018, 109: 191-206.

Jafari, I., Rokhforouz, M. R. Numerical modeling of water oil two-phase flow during counter-current spontaneous imbibition in porous media at pore-scale. Petroleum Science and Technology, 2020, 38(24): 1040-1053.

Jettestuen, E., Helland, J. O., Prodanović, M. A level set method for simulating capillary-controlled displacements at the pore scale with nonzero contact angles. Water Resources Research, 2013, 49(8): 4645-4661.

Jia, C., Zheng, M., Zhang, Y. Unconventional hydrocarbon resources in China and the prospect of exploration and development. Petroleum Exploration and Development, 2012, 39(2): 139-146.

Karimova, M., Kashiri, R., Pourafshary, P., et al. A review of wettability alteration by spontaneous imbibition using low-salinity water in naturally fractured reservoirs. Energies, 2023, 16(5): 2373.

Kim, E., Whitesides, G. M. Imbibition and flow of wetting liquids in noncircular capillaries. The Journal of Physical Chemistry B, 1997, 101(6): 855-863.

Kim, H. Y., Kim, H. G. A novel adaptive mesh refinement scheme for the simulation of phase-field fracture using trimmed hexahedral meshes. International Journal for Numerical Methods in Engineering, 2021, 122(6): 1493-1512.

Kupershtokh, A. L., Medvedev, D. A., Karpov, D. I. On equations of state in a lattice Boltzmann method. Computers & Mathematics with Applications, 2009, 58(5): 965-974.

Lallemand, P., Luo, L. S. Theory of the lattice Boltzmann method: Dispersion, dissipation, isotropy, galilean invariance, and stability. Physical Review E, 2000, 61(6): 6546.

Latva, K. M., Rothman, D. H. Diffusion properties of gradientbased lattice boltzmann models of immiscible fluids. Physical Review E, 2005, 71(5): 056702.

Lee, T., Lin, C. L. A stable discretization of the lattice Boltzmann equation for simulation of incompressible two-phase flows at high density ratio. Journal of Computational Physics, 2005, 206(1): 16-47.

Lee, T., Liu, L. Lattice boltzmann simulations of micron-scale drop impact on dry surfaces. Journal of Computational Physics, 2010, 229(20): 8045-8063.

Li, C., Singh, H., Cai, J. Spontaneous imbibition in shale: A review of recent advances. Capillarity, 2019, 2(2): 17-32.

Li, K., Horne, R. N. Characterization of spontaneous water imbibition into gas-saturated rocks, SPE Western Regional Meeting. SPE Journal, 2000, 6(4): 375-384.

Li, Q., Luo, K., Li, X. Lattice boltzmann modeling of multiphase flows at large density ratio with an improved pseudopotential model. Physical Review E, 2013, 87(5): 053301.

Li, S., Liu, H., Wu, R., et al. Prediction of spontaneous imbibition with gravity in porous media micromodels. Journal of Fluid Mechanics, 2022, 952: A9.

Li, S., Liu, H., Zhang, J., et al. Modeling of three-phase displacement in three-dimensional irregular geometries using a lattice boltzmann method. Physics of Fluids, 2021, 33: 122108.

Li, Y., Kim, J. Phase-field simulations of crystal growth with adaptive mesh refinement. International Journal of Heat and Mass Transfer, 2012, 55(25-26): 7926-7932.

Liang, C., Rou, Y., Ye, C., et al. Material symmetry recognition and property prediction accomplished by crystal capsule representation. Nature Communications, 2023, 14(1): 5198.

Liang, H., Liu, H., Chai, Z., et al. Lattice boltzmann method for contact-line motion of binary fluids with high density ratio. Physical Review E, 2019, 99(6): 063306.

Liang, H., Shi, B. C., Guo, Z. L., et al. Phase-field-based multiple-relaxation-time lattice boltzmann model for incompressible multiphase flows. Physical Review E, 2014, 89(5): 053320.

Lin, W., Xiong, S., Liu, Y., et al. Spontaneous imbibition in tight porous media with different wettability: Pore-scale simulation. Physics of Fluids, 2021, 33: 032013.

Liu, H., Sun, S., Wu, R., et al. Pore-scale modeling of spontaneous imbibition in porous media using the lattice Boltzmann method. Water Resources Research, 2021a, 57(6): e2020WR029219.

Liu, Q., Li, J., Liang, B., et al. Complex wettability behavior triggering mechanism on imbibition: A model construction and comparative study based on analysis at multiple scales. Energy, 2023a, 275: 127434.

Liu, X., Chai, Z., Shi, B. Improved hybrid Allen-Cahn phasefield-based lattice boltzmann method for incompressible two-phase flows. Physical Review E, 2023b, 107(3): 035308.

Liu, X., Chai, Z., Zhan, C., et al. A diffuse-domain phasefield lattice boltzmann method for two-phase flows in complex geometries. Multiscale Modeling & Simulation, 2022a, 20(4): 1411-1436.

Liu, Y., Cai, J., Sahimi, M., et al. A study of the role of microfractures in counter-current spontaneous imbibition by lattice Boltzmann simulation. Transport in Porous edia, 2020a, 133: 313-332.

Liu, Y., Chen, M., Sun, S., et al. Effects of grain shape and packing pattern on spontaneous imbibition under different boundary conditions: Pore-scale simulation. Journal of Hydrology, 2022b, 607: 127484.

Liu, Y., Zou, S., He, Y., et al. Influence of fractal surface roughness on multiphase flow behavior: Lattice boltzmann simulation. International Journal of Multiphase Flow, 2021b, 134: 103497.

Liu, Z., Wang, W., Yang, H., et al. Experimental study on spontaneous imbibition characteristics of coal based on fractal theory. Advanced Powder Technology, 2020b, 31(5): 1994-2004.

Liu, Z., Yang, Y., Yao, J., et al. Pore-scale remaining oil distribution under different pore volume water injection based on CT technology. Advances in Geo-Energy Research, 2017, 1(3): 171-181.

López, J., Hernández, J., Gómez, P., et al. An improved PLICVOF method for tracking thin fluid structures in incompressible two-phase flows. Journal of Computational Physics, 2005, 208(1): 51-74.

Lou, Q., Guo, Z., Shi, B. Effects of force discretization on mass conservation in lattice boltzmann equation for twophase flows. Europhysics Letters, 2012, 99(6): 64005.

Lucas, R. Ueber das zeitgesetz des kapillaren aufstiegs von fl¨ussigkeiten. Kolloid-Zeitschrift, 1918, 23(1): 15-22.

Lyu, C., Ning, Z., Chen, M., et al. Experimental study of boundary condition effects on spontaneous imbibition in tight sandstones. Fuel, 2019, 235: 374-383.

Mason, G., Morrow, N. R. Developments in spontaneous imbibition and possibilities for future work. Journal of Petroleum Science and Engineering, 2013, 110: 268-293.

McCracken, M. E., Abraham, J. Multiple-relaxation-time lattice-boltzmann model for multiphase flow. Physical Review E, 2005, 71(3): 036701.

Meakin, P., Tartakovsky, A. M. Modeling and simulation of pore-scale multiphase fluid flow and reactive transport in fractured and porous media. Reviews of Geophysics, 2009, 47: RG3002.

Meng, Q., Cai, J. Recent advances in spontaneous imbibition with different boundary conditions. Capillarity, 2018, 1(3): 19-26.

Meng, Q., Cai, J., Wang, J. Scaling of countercurrent imbibition in 2D matrix blocks with different boundary conditions. SPE Journal, 2019, 24(3): 1179-1191.

Meng, Q., Liu, H., Wang, J. A critical review on fundamental mechanisms of spontaneous imbibition and the impact of boundary condition, fluid viscosity and wettability. Advances in Geo-Energy Research, 2017, 1(1): 1-17.

Meng, Q., Zhao, L., Li, P., et al. Experiments and phase-field simulation of counter-current imbibition in porous media with different pore structure. Journal of Hydrology, 2022, 608: 127670.

Mohammadi Alamooti, A. H., Azizi, Q., Davarzani, H. Direct numerical simulation of trapped-phase recirculation at low capillary number. Advances in Water Resources, 2020, 145: 103717.

Montgomery, C. T., Smith, M. B. Hydraulic fracturing: History of an enduring technology. Journal of Petroleum Technology, 2010, 62(12): 26-40.

Mora, P., Morra, G., Yuen, D. A., et al. Convection-diffusion with the colour gradient lattice Boltzmann method for three-component, two-phase flow. Transport in Porous Media, 2023, 147(2): 259-280.

Morrow, N. R., Mason, G. Recovery of oil by spontaneous imbibition. Current Opinion in Colloid & Interface Science, 2001, 6(4): 321-337.

Mosser, L., Dubrule, O., Blunt, M. J. Reconstruction of three-dimensional porous media using generative adversarial neural networks. Physical Review E, 2017, 96(4): 043309.

Ning, H., Qian, S., Zhou, T. Applications of level set method in computational fluid dynamics: A review. International Journal of Hydromechatronics, 2023, 6(1): 1-33.

Nguyen Thanh, N., Li, W., Huang, J., et al. Adaptive higherorder phase-field modeling of anisotropic brittle fracture in 3D polycrystalline materials. Computer Methods in Applied Mechanics and Engineering, 2020, 372: 113434.

Nguyen, V. T., Park, W. G. A volume-of-fluid (VOF) interfacesharpening method for two-phase incompressible flows. Computers & Fluids, 2017, 152: 104-119.

Noh, W. F., Woodward, P. SLIC (simple line interface calculation). Paper Presented at 5th International Conference on Numerical Methods in Fluid Dynamics, Twente University, Enschede, 25 June-2 July, 1976.

Olsson, E., Kreiss, G. A conservative level set method for two phase flow. Journal of Computational Physics, 2005, 210(1): 225-246.

Olsson, E., Kreiss, G., Zahedi, S. A conservative level set method for two phase flow II. Journal of Computational Physics, 2007, 225: 785-807.

Osiptsov, A. A. Fluid mechanics of hydraulic fracturing: A review. Journal of Petroleum Science and Engineering, 2017, 156: 513-535.

Pavuluri, S. Direct numerical simulations of spontaneous imbibition at the pore-scale: Impact of parasitic currents and dynamic capillary barriers. Edinburgh, Heriot-Watt University, 2019.

Peng, X., Wang, X., Du, Z., et al. Phase-field simulations of precursor film in microcapillary imbibition for liquidliquid systems. International Journal of Multiphase Flow, 2021, 144: 103789.

Petersen, K. J., Brinkerhoff, J. R. On the lattice Boltzmann method and its application to turbulent, multiphase flows of various fluids including cryogens: A review. Physics of Fluids, 2021, 33: 041302.

Premnath, K. N., Abraham, J. Simulations of binary drop collisions with a multiple-relaxation-time lattice-boltzmann model. Physics of Fluids, 2005, 17(12): 12205.

Prodanovic, M., Bryant, S. L. Investigating pore scale configurations of two immiscible fluids via Level Set Method. Paper Presented at XVI International Conference on Computational Methods in Water Resources, Copenhagen, Denmark, 18-22 June, 2006.

Prodanovic, M., Bryant, S. L. Investigating spontaneous capillarity-controlled events via the level set method. PAMM: Proceedings in Applied Mathematics and Mechanics, 2007, 7: 1141601-1141602.

Prodanovic, M., Bryant, S. L. Physics-driven interface modeling for drainage and imbibition in fractures. SPE Journal, 2009, 14 (4): 532-542.

Prodanovic, M., Bryant, S. L., Karpyn, Z. T. Investigating matrix/ fracture transfer via a level set method for drainage and imbibition. SPE Journal, 2010, 15(1): 125-136.

Rabbani, H. S., Joekar Niasar, V., Pak, T., et al. New insights on the complex dynamics of two-phase flow in porous media under intermediate-wet conditions. Scientific Reports, 2017, 7(1): 4584.

Ramstad, T., Berg, C. F., Thompson, K. Pore-scale simulations of single- and two-phase flow in porous media: Approaches and applications. Transport in Porous Media, 2019, 130: 77-104.

Ranganathan, P. Pore-scale modelling of immiscible displacement of ScCO2-brine in a homogeneous porous network using direct numerical method. The Canadian Journal of Chemical Engineering, 2022, 100(2): 391-404.

Raza, A., Arif, M., Glatz, G., et al. A holistic overview of underground hydrogen storage: Influencing factors, current understanding, and outlook. Fuel, 2022, 330: 125636.

Reis, T., Phillips, T. N. Lattice Boltzmann model for simulating immiscible two-phase flows. Journal of Physics A: Mathematical and Theoretical, 2007, 40(14): 4033.

Ren, F., Song, B., Sukop, M. C., et al. Improved lattice boltzmann modeling of binary flow based on the conservative Allen-Cahn equation. Physical Review E, 2016, 94(2): 023311.

Renardy, M., Renardy, Y., Li, J. Numerical simulation of moving contact line problems using a volume-of-fluid method. Journal of Computational Physics, 2001, 171(1): 243-263.

Rodriguez, E., Prodanović, M., Bryant, S. L. Contact line extraction and length measurements in model sediments and sedimentary rocks. Journal of Colloid and Interface Science, 2012, 368(1): 558-577.

Rokhforouz, M. R., Akhlaghi, H. A. A. Phase-field simulation of counter-current spontaneous imbibition in a fractured heterogeneous porous medium. Physics of Fluids, 2017, 29: 062104.

Rothman, D. H., Keller, J. M. Immiscible cellular-automaton fluids. Journal of Statistical Physics, 1988, 52: 1119-1127.

Rusche, H. Computational fluid dynamics of dispersed twophase flows at high phase fractions. London, University of London, 2003.

Santra, S., Mandal, S., Chakraborty, S. Phase-field modeling of multicomponent and multiphase flows in microfluidic systems: A review. International Journal of Numerical Methods for Heat & Fluid Flow, 2021, 31(10): 3089-3131.

Sbragaglia, M., Benzi, R., Biferale, L., et al. Surface roughness-hydrophobicity coupling in microchannel and nanochannel flows. Physical Review Letters, 2006, 97(20): 204503.

Scapin, N., Costa, P., Brandt, L. A volume-of-fluid method for interface-resolved simulations of phase-changing twofluid flows. Journal of Computational Physics, 2020, 407: 109251.

Schembre, J. M., Akin, S., Castanier, L. M., et al. Spontaneous water imbibition into diatomite. Paper SPE 46211 Presented at the SPE Western Regional Meeting, Bakersfield, California, 10-13 May, 1998.

Shan, X., Chen, H. Lattice boltzmann model for simulating flows with multiple phases and components. Physical Review E, 1993, 47(3): 1815.

Shan, X., Chen, H. Simulation of nonideal gases and liquidgas phase transitions by the lattice Boltzmann equation. Physical Review E, 1994, 49(4): 2941.

Sheng, T., Jing, L., Zhang, X., et al. Simulation of dynamic wetting effect during gas-liquid spontaneous imbibition based on modified LBM. Chinese Journal of Theoretical and Applied Mechanics, 2022, 55(2): 355-368.

Shi, J., Cheng, L., Cao, R., et al. Phase-field simulation of imbibition for the matrix-fracture of tight oil reservoirs considering temperature change. Water, 2021, 13(7): 1004.

Sudhakar, T., Das, A. K. Evolution of multiphase lattice Boltzmann method: A review. Journal of The Institution of Engineers (India): Series C, 2020, 101(4): 711-719.

Sun, L., Zou, C., Zhu, R., et al. Formation, distribution and potential of deep hydrocarbon resources in China. Petroleum Exploration and Development, 2013, 40(6): 687-695.

Sussman, M., Smereka, P., Osher, S. A level set approach for computing solutions to incompressible two-phase flow. Journal of Computational Physics, 1994, 114(1): 146-159.

Swift, M. R., Orlandini, E., Osborn, W. R., et al. Lattice boltzmann simulations of liquid-gas and binary fluid systems. Physical Review E, 1996, 54(5): 5041-5052.

Swift, M. R., Osborn, W. R., Yeomans, J. M. Lattice Boltzmann simulation of nonideal fluids. Physical Review Letters, 1995, 75(5): 830-833.

Takada, N., Matsumoto, J., Matsumoto, S. A diffuse-interface tracking method for the numerical simulation of motions of a two-phase fluid on a solid surface. The Journal of Computational Multiphase Flows, 2014, 6(3): 283-298.

Takada, N., Misawa, M., Tomiyama, A. A phase-field method for interface-tracking simulation of two-phase flows. Fluids Engineering Division Summer Meeting, 2005, 41995: 259-264.

Tryggvason, G., Esmaeeli, A., Lu, J., et al. Direct numerical simulations of gas/liquid multiphase flows. Fluid Dynamics Research, 2006, 38(9): 660-681.

Van Der Waals, J. D. The thermodynamic theory of capillarity under the hypothesis of a continuous variation of density. Journal of Statistical Physics, 1979, 20(2): 200-244.

Wang, D., Ma, Y., Song, K., et al. Phase-field modeling of pore-scale oil replacement by spontaneous imbibition in fractured porous media. Energy & Fuels, 2022, 36(24):14824-14837.

Wang, H., Cai, J., Su, Y., et al. Imbibition behaviors in shale nanoporous media from pore-scale perspectives. Capillarity, 2023, 9(2): 32-44.

Wang, H., Chai, Z., Shi, B., et al. Comparative study of the lattice boltzmann models for Allen-Cahn and Cahn-Hilliard equations. Physical Review E, 2016, 94(3): 033304.

Wang, H., Su, Y., Wang, W. Investigations on water imbibing into oil-saturated nanoporous media: Coupling molecular interactions, the dynamic contact angle, and the entrance effect. Industrial & Engineering Chemistry Research, 2021, 60(4): 1872-1883.

Wang, H., Yuan, X., Liang, H., et al. A brief review of the phase-field-based lattice boltzmann method for multiphase flows. Capillarity, 2019, 2(3): 33-52.

Wang, J., Dong, M. Trapping of the non-wetting phase in an interacting triangular tube bundle model. Chemical Engineering Science, 2011, 66(3): 250-259.

Wang, L., Ma, F., He, Y., et al. The prediction of spontaneous oil-water imbibition in composite capillary. Petroleum, 2022, 8(1): 84-91.

Washburn, E. W. The dynamics of capillary flow. Physical Review, 1921, 17(3): 273-283.

Wei, H., Zhu, X., Liu, X., et al. Pore-scale study of drainage processes in porous media with various structural heterogeneity. International Communications in Heat and Mass Transfer, 2022, 132: 105914.

Wei, W., Cai, J., Xiao, J., et al. Kozeny-Carman constant of porous media: Insights from fractal-capillary imbibition theory. Fuel, 2018, 234: 1373-1379.

Welch, S. W. J., Wilson, J. A volume of fluid based method for fluid flows with phase change. Journal of Computational Physics, 2000, 160(2): 662-682.

Worner M. Numerical modeling of multiphase flows in microfluidics and micro process engineering: A review of methods and applications. Microfluidics and Nanofluidics, 2012, 12(6): 841-886.

Wu, Y., Wang, X., Zhang, C., et al. Pore scale numerical investigation of counter-current spontaneous imbibition in multi-scaled pore networks. Petroleum, 2022, 9(4): 558-571.

Wu, Z., Cui, C., Ye, Y., et al. A fractal model for quantitative evaluating the effects of spontaneous imbibition and displacement on the recovery of tight reservoirs. Journal of Petroleum Science and Engineering, 2021, 198: 108120.

Xiao, J., Luo, Y., Niu, M., et al. Study of imbibition in various geometries using phase field method. Capillarity, 2019, 2(4): 57-65.

Xiao, L., Zhu, G., Zhang, L., et al. Effects of pore-size disorder and wettability on forced imbibition in porous media. Journal of Petroleum Science and Engineering, 2021, 201: 108485.

Xin, B., Hao, F., Han, W., et al Paleoenvironment evolution of the lacustrine organic-rich shales in the second member of Kongdian Formation of Cangdong Sag, Bohai Bay Basin, China: Implications for organic matter accumulation. Marine and Petroleum Geology, 2021, 133: 105244.

Xu, A., Gonnella, G., Lamura, A. Phase-separating binary fluids under oscillatory shear. Physical Review E, 2003, 67(5): 056105.

Yang, J., Boek, E. S. A comparison study of multi-component Lattice Boltzmann models for flow in porous media applications. Computers & Mathematics with Applications, 2013, 65(6): 882-890.

Yang, Z., Liu, X., Li, H., et al. Analysis on the influencing factors of imbibition and the effect evaluation of imbibition in tight reservoirs. Petroleum Exploration and Development, 2019, 46(4): 779-785.

Yildiz, H. O., Gokmen, M., Cesur, Y. Effect of shape factor, characteristic length, and boundary conditions on spontaneous imbibition. Journal of Petroleum Science and Engineering, 2006, 53(3-4): 158-170.

Yu, Z., Fan, L. S. Multirelaxation-time interaction-potentialbased lattice boltzmann model for two-phase flow. Physical Review E, 2010, 82(4): 046708.

Yuan, P., Schaefer, L. Equations of state in a lattice boltzmann model. Physics of Fluids, 2006, 18(4): 042101.

Yue, P., Feng, J. J., Liu, C., et al. A diffuse-interface method for simulating two-phase flows of complex fluids. Journal of Fluid Mechanics, 2004, 515: 293-317.

Yue, P., Zhou, C., Feng, J. J., et al. Phase-field simulations of interfacial dynamics in viscoelastic fluids using finite elements with adaptive meshing. Journal of Computational Physics, 2006, 219(1): 47-67.

Zacharoudiou, I., Boek, E. S. Capillary filling and Haines jump dynamics using free energy lattice boltzmann simulations. Advances in Water Resources, 2016, 92: 43-56.

Zacharoudiou, I., Chapman, E. M., Boek, E. S., et al. Porefilling events in single junction micro-models with corresponding lattice boltzmann simulations. Journal of Fluid Mechanics, 2017, 824: 550-573.

Zhang, A., Guo, Z., Jiang, B., et al. Numerical solution to phase-field model of solidification: A review. Computational Materials Science, 2023, 228: 112366.

Zhang, S., Pu, H., Zhao, J. X. Experimental and numerical studies of spontaneous imbibition with different boundary conditions: Case studies of middle bakken and berea cores. Energy & Fuels, 2019, 33(6): 5135-5146.

Zhang, S., Zhu, G. Natural gas origins of large and mediumscale gas fields in China sedimentary basins. Science in China Series D: Earth Sciences, 2008, 51(1): 1-13.

Zhao, J., Qin, F., Fischer, R., et al. Spontaneous imbibition in a square tube with corner films: Theoretical model and numerical simulation. Water Resources Research, 2021, 57(2): e2020WR029190.

Zheng, H., Shu, C., Chew, Y. Lattice boltzmann interface capturing method for incompressible flows. Physical Review E, 2005, 72(5): 056705.

Zheng, H., Shu, C., Chew, Y. A lattice boltzmann model for multiphase flows with large density ratio. Journal of Computational Physics, 2006, 218(1): 353-371.

Zheng, J., Chen, Z., Xie, C., et al. Characterization of spontaneous imbibition dynamics in irregular channels by mesoscopic modeling. Computers & Fluids, 2018a, 168: 21-31.

Zheng, J., Ju, Y., Wang, M. Pore-scale modeling of spontaneous imbibition behavior in a complex shale porous structure by pseudopotential lattice boltzmann method. Journal of Geophysical Research: Solid Earth, 2018b, 123(11): 9586-9600.

Zheng, J., Lei, W., Ju, Y., et al. Investigation of spontaneous imbibition behavior in a 3D pore space under reservoir condition by lattice boltzmann method. Journal of Geophysical Research: Solid Earth, 2021, 126(6): e2021JB021987.

Zhong, J., Duan, H., Wang, J., et al. Simulation of water self-imbibition in nanometer throat-pore structure filled with oil. Geoenergy Science and Engineering, 2023, 221: 211370.

Zhou, Y., Guan, W., Cong, P., et al. Effects of heterogeneous pore closure on the permeability of coal involving adsorption-induced swelling: A micro pore-scale simulation. Energy, 2022, 258: 124859.

Zhou, Y., Guan, W., Zhao, C., et al. Spontaneous imbibition behavior in porous media with various hydraulic fracture propagations: A pore-scale perspective. Advances in Geo-Energy Research, 2023a, 9(3): 185-197.

Zhou, Y., Guan, W., Zhao, C., et al. A computational workflow to study CO2 transport in porous media with irregular grains: Coupling a fourier series-based approach and CFD. Journal of Cleaner Production, 2023b, 418: 138037.

Zhu, C., Ma, F., Lei, P., et al. Comparison between level set and phase field method for simulating bubble movement behavior under electric field. Chinese Journal of Physics, 2021a, 71: 385-396.

Zhu, G., Chen, H., Yao, J., et al. Efficient energy-stable schemes for the hydrodynamics coupled phase-field model. Applied Mathematical Modelling, 2019, 70: 82-108.

Zhu, G., Zhang, L., Yao, J. Energy capillary number reveals regime transition of imbibition in porous media. Physics of Fluids, 2021b, 33: 123111.

Zhu, X., Wang, S., Feng, Q., et al. Pore-scale numerical prediction of three-phase relative permeability in porous media using the lattice boltzmann method. International Communications in Heat and Mass Transfer, 2021c, 126: 105403.

Zhu, Z., Song, Z., Shao, Z., et al. Numerical investigation of single-and two-phase flow in porous media with a bifurcated fracture. Physics of Fluids, 2021d, 33(5): 052117.

Zou, X., He, C., Wei, G., et al. Reservoir tortuosity prediction: Coupling stochastic generation of porous media and machine learning. Energy, 2023, 285: 129512.

Zu, Y., He, S. Phase-field-based lattice boltzmann model for incompressible binary fluid systems with density and viscosity contrasts. Physical Review E, 2013, 87(4): 043301.


  • There are currently no refbacks.

Copyright (c) 2024 The Author(s)

Creative Commons License
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Copyright ©2018. All Rights Reserved