A comprehensive review of pore scale modeling methodologies for multiphase flow in porous media

Amir Golparvar, Yingfang Zhou, Kejian Wu, Jingsheng Ma, Zhixin Yu

Abstract view|2922|times       PDF download|697|times


Multiphase flow in porous media is relevant to amount of engineering processes, such as hydrocarbon extraction from reservoir rock, water contamination, CO2 geological storage and sequestration. Pore scale modeling, as an alternative approach to lab measurement, firstly serves as an effective bridge to link the pore scale properties (pore geometry and wettability) and displacement mechanisms to continuous scale multiphase flow in porous media; and secondly allows us to determine essential flow functions, such as capillary pressure and relative permeability curves, which are required for continuous scale modeling. In the literature, three methodologies, Bundle of Capillary Tube Modeling (BCTM), Direct Pore Scale Modeling (DPSM) and Pore Network Modeling (PNM), have appeared to be mostly widely adopted in the investigation of the pore-scale mechanics of fluid-fluid and fluid-solid interactions in porous media by numerical simulation. In this review article, a comprehensive review is provided to show their strengths and weaknesses and to highlight challenges that are faced in modelling of multiphase flow, key challenges include: are contact angle characterization, validation and upscale pore scale findings to core, or even field scale.

Cited as: Golparvar, A., Zhou, Y., Wu, K., Ma, J., Yu, Z. A comprehensive review of pore scale modeling methodologies for multiphase flow in porous media. Advances in Geo-Energy Research, 2018, 2(4): 418-440, doi: 10.26804/ager.2018.04.07


Multiphase flow, pore scale modeling, bundle of capillary tube modeling, direct pore scale modeling, pore network modeling

Full Text:



Adler, P.M. Fractal porous media. Transp. Porous Media 1991, 13(1): 723-743.

Adler, P.M., Thovert, J.F. Real porous media: Local geometry and macroscopic properties. Appl. Mech. Rev. 1998, 51(9): 537-585.

Aghaei, A., Piri, M. Direct pore-to-core up-scaling of displacement processes: Dynamic pore network modeling and experimentation. J. Hydrol. 2015, 522: 488-509.

Ahrenholz, B., T ¨olke, J., Krafczyk, M. Lattice-Boltzmann simulations in reconstructed parametrized porous media. Int. J. Comput. Fluid D 2006, 20(6): 369-377.

Ahrenholz, B., T ¨olke, J., Lehmann, P., et al. Prediction of capillary hysteresis in a porous material using Lattice-Boltzmann methods and comparison to experimental data and a morphological pore network model. Adv. Water Resour. 2008, 31(9): 1151-1173.

Aker, E., M ˚AlØy, K.J., Hansen, A., et al. A two-dimensional network simulator for two-phase flow in porous media. Transp. Porous Media 1998, 32(2): 163-186.

Albadawi, A., Donoghue, D.B., Robinson, A.J., et al. Influence of surface tension implementation in volume of fluid and coupled volume of fluid with level set methods for bubble growth and detachment. Int. J. Multiph. Flow 2013, 53: 11-28.

Al-Dhahli, A., Geiger, S., Dijke, M.I. Accurate modelling of pore-scale film and layer flow for three-phase EOR in carbonate rocks with arbitrary wettability. Paper SPE 154019 Preseted at the Improved Oil Recovery Symposium, Tulsa, Oklahoma, USA, 14-18 April, 2012.

Al-Gharbi, M.S., Blunt, M.J. Dynamic network modeling of two-phase drainage in porous media. Phys. Rev. E 2005, 71(1): 016308.

Al-Kharusi, A.S., Blunt, M.J. Network extraction from sandstone and carbonate pore space images. J. Pet. Sci. Eng. 2007, 56(4): 219-231.

Amiri, O., A¨ıt-Mokhtar, A., Sarhani, M. Tri-dimensional modelling of cementitious materials permeability from polymodal pore size distribution obtained by mercury intrusion porosimetry tests. Adv. Cem. Res. 2005, 17(1): 39-45.

Andrew, M., Bijeljic, B., Blunt, M.J. Pore-scale contact angle measurements at reservoir conditions using X-ray microtomography. Adv. Water Resour. 2014, 68: 24-31.

Ansari, M.R., Azadi, R., Salimi, E. Capturing of interface topological changes in two-phase gasliquid flows using a coupled volume-of-fluid and level-set method (VOSET). Comput. Fluids 2016, 125: 82-100.

Armstrong, R.T., Georgiadis, A., Ott, H., et al. Critical capillary number: Desaturation studied with fast X-ray computed microtomography. Geophys. Res. Lett. 2014, 41(1): 55-60.

Arns, C.H., Bauget, F., Limaye, A., et al. Pore scale charac-terization of carbonates using X-ray microtomography. SPE J. 2005, 10(4): 475-484.

A¨ıt-Mokhtar, A., Amiri, O., Dumargue, P., et al. A new model to calculate water permeability of cement-based materials from MIP results. Adv. Cem. Res. 2002, 14(2): 43-49.

A¨ıt-Mokhtar, A., Amiri, O., Sammartinoj, S. Analytic modelling and experimental study of the porosity and permeability of a porous. Mag. Concr. Res. 1999, 51(6): 391-396.

Bakke, S., Øren, P.E. 3-D pore-scale modelling of sandstones and flow simulations in the pore networks. SPE J. 1997, 2(2): 136-149.

Bandara, U.C., Tartakovsky, A.M., Palmer, B.J. Pore-scale study of capillary trapping mechanism during CO2 injection in geological formations. Int. J. Greenhouse Gas Control 2011, 5(6): 1566-1577.

Bartley, J.T., Ruth, D.W. Relative permeability analysis of tube bundle models. Transp. Porous Media 1999, 36(2): 161-188.

Bartley, J.T., Ruth, D.W. Relative permeability analysis of tube bundle models, including capillary pressure. Transp. Porous Media 2001, 45(3): 445-478.

Bedrikovetsky, P., Vaz, A.S.L., Furtado, C.J.A., et al. Formation-damage evaluation from nonlinear skin growth during coreflooding. SPE Reserv. Eval. Eng. 2011, 14(2): 193-203.

Blunt, M.J. Flow in porous mediapore-network models and multiphase flow. Curr. Opin. Colloid Interface Sci. 2001, 6(3): 197-207.

Blunt, M.J. Multiphase flow in permeable media: A pore-scale perspective. Oxford, UK, Cambridge University Press, 2017.

Blunt, M.J., Bijeljic, B., Dong, H., et al. Pore-scale imaging and modelling. Adv. Water Resour. 2013, 51: 197-216.

Blunt, M.J., Jackson, M.D., Piri, M., et al. Detailed physics, predictive capabilities and macroscopic consequences for pore-network models of multiphase flow. Adv. Water Resour. 2002, 25(8-12): 1069-1089.

Blunt, M.J., King, P. Relative permeabilities from two-and three-dimensional pore-scale network modelling. Transp. Porous Media 1991, 6(4): 407-433.

Blunt, M.J., Scher, H. Pore-level modeling of wetting. Phys. Rev. E 1995, 52(6): 6387.

Boek, E.S., Venturoli, M. Lattice-Boltzmann studies of fluid flow in porous media with realistic rock geometries. Comput. Math. Appl. 2010, 59(7): 2305-2314.

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

Bryant, S.L., Blunt, M. Prediction of relative permeability in simple porous media. Phys. Rev. A 1992, 46(4): 2004.

Bryant, S.L., King, P.R., Mellor, D.W. Network model evaluation of permeability and spatial correlation in a real random sphere packing. Transp. Porous Media 1993a, 11(1): 53-70.

Bryant, S.L., Mellor, D.W., Cade, C.A. Physically representa-tive network models of transport in porous media. AIChE J. 1993b, 39(3): 387-396.

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

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., Yu, B., Zou, M., et al. Fractal analysis of invasion depth of extraneous fluids in porous media. Chem. Eng. Sci. 2010, 65(18): 5178-5186.

Chao, J., Mei, R., Singh, R., et al. A filter-based, mass-conserving lattice Boltzmann method for immiscible multiphase flows. Int. J. Numer. Methods Fluids 2011, 66(5): 622-647.

Cheng, S., Fu, M., Kulacki, F.A. Characterization of a porous transducer using a capillary bundle model: Permeability and streaming potential prediction. Int. J. Heat Mass Transf. 2018, 118: 349-354.

Dahle, H.K., Celia, M.A. A dynamic network model for two-phase immiscible flow. Comput. Geosci. 1999, 3(1): 1-22.

Derksen, J.J. Droplets sliding over shearing surfaces studied by molecular dynamics. AIChE J. 2015, 61(11): 4020-4027.

Dong, H., Blunt, M.J. Pore-network extraction from micro-computerized-tomography images. Phys. Rev. E 2009, 80(3): 036307.

Dong, M., Dullien, F.A.L., Dai, L., et al. Immiscible displacement in the interacting capillary bundle model part I. Development of interacting capillary bundle model. Transp. Porous Media 2005, 59(1), 1-18.

Dong, M., Dullien, F.A.L., Dai, L., et al. Immiscible displace-ment in the interacting capillary bundle model part ii. applications of model and comparison of interacting and non-interacting capillary bundle models. Transp. Porous Media 2006, 63(2): 289-304.

Dong, M., Zhou, J. Characterization of waterflood saturation profile histories by the completecapillary number. Transp. Porous Media 1998, 31(2): 213-237.

Fatt, I. The network model of porous media. Society of Petroleum Engineers 1956, 207: 144-181

Frette, O.I., Helland, J.O. A semi-analytical model for computation of capillary entry pressures and fluid configurations in uniformly-wet pore spaces from 2D rock images. Adv. Water Resour. 2010, 33(8): 846-866.

Gao, H., Yu, B., Duan, Y., et al. Fractal analysis of dimensionless capillary pressure function. Int. J. Heat Mass Transf. 2014, 69: 26-33.

Garcia-Bengochea, I., Altschaeffl, A.G., Lovell, C.W. Pore distribution and permeability of silty clays. J. Geotech. Eng. Div. 1979, 105(7): 839-856.

Glantz, R., Hilpert, M. Capillary displacement in totally wetting and infinitely long right prisms. Multiscale Model. Simul. 2011, 9(4): 1765-1800.

Guo, Z., Zhao, T.S. Lattice Boltzmann model for incompress-ible flows through porous media. Phys. Rev. E 2002, 66(3): 036304.

Helland, J.O., Skjaeveland, S.M. Physically based capillary pressure correlation for mixed-wet reservoirs from a bundle-of-tubes model. SPE J. 2006, 11(2): 171-180.

Hirasaki, G.J. Thermodynamics of thin films and three-phase contact regions, in Interfacial Phenomena in Petroleum Recovery, edited by N. R. Morrow, Taylor & Francis Inc, Bosa Roca, pp. 23-75, 1991.

Hu, X.Y., Adams, N.A. A multi-phase SPH method for macroscopic and mesoscopic flows. J. Comput. Phys. 2006, 213(2): 844-861.

Hughes, R.G., Blunt, M.J. Pore scale modeling of rate effects in imbibition. Transp. Porous Media 2000, 40(3): 295-322.

Hui, M.H., Blunt, M.J. Effects of wettability on three-phase flow in porous media. J. Phys. Chem. B 2000, 104(16): 3833-3845.

Iglauer, S. CO2 -water-rock wettability: Variability, influencing factors, and implications for CO2 geostorage. Acc. Chem. Res. 2017, 50(5): 1134-1142.

Iglauer, S., Mathew, M.S., Bresme, F. Molecular dynamics computations of brine-CO2 interfacial tensions and brine-CO2 -quartz contact angles and their effects on structural and residual trapping mechanisms in carbon geo-sequestration. J. Colloid Interface Sci. 2012, 386(1): 405-414.

Iglauer, S., Salamah, A., Sarmadivaleh, M., et al. Contamina-tion of silica surfaces: Impact on water-CO2 -quartz and glass contact angle measurements. Int. J. Greenhouse Gas Control 2014, 22: 325-328.

Ioannidis, M.A., Kwiecien, M.J., Chatzis, I. Statistical analysis of the porous microstructure as a method for estimating reservoir permeability. J. Pet. Sci. Eng. 1996, 16(4): 251-261.

Jackson, M.D. Characterization of multiphase electrokinetic coupling using a bundle of capillary tubes model. J. Geophys. Res. Sol. Ea. 2008, 113(B4): B04201.

Jackson, M.D. Multiphase electrokinetic coupling: Insights into the impact of fluid and charge distribution at the pore scale from a bundle of capillary tubes model. J. Geophys. Res. Sol. Ea. 2010, 115(B7): B07206.

Jacquin, C.G., Adler, P.M. The fractal dimension of a gas-liquid interface in a porous medium. J. Colloid Interface Sci. 1985, 107(2): 405-417.

Jadidi, M., Tembely, M., Moghtadernejad, S., et al. A coupled level set and volume of fluid method with application to compressible two-phase flow. Paper Presented at the Proceedings of the 22nd Annual Conference of the CFD Society of Canada, Toronto, ON, Canada, June, 2014.

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

Jia, P., Dong, M., Dai, L. Threshold pressure in arbitrary triangular tubes using RSG concept for all wetting conditions. Colloids Surf. A 2007, 302(1-3): 88-95.

Jiang, Z., Van Dijke, M.I.J., Sorbie, K.S., et al. Representation of multiscale heterogeneity via multiscale pore networks. Water Resour. Res. 2013, 49(9): 5437-5449.

Jivkov, A.P., Hollis, C., Etiese, F., et al. A novel architecture for pore network modelling with applications to perme-ability of porous media. J. Hydrol. 2013, 486: 246-258.

Joekar-Niasar, V., Hassanizadeh, S.M. Analysis of fundamen-tals of two-phase flow in porous media using dynamic pore-network models: A review. Crit. Rev. Environ. Sci. Technol. 2012, 42(18): 1895-1976.

Joekar-Niasar, V., Hassanizadeh, S.M., Dahle, H.K. Non-equilibrium effects in capillarity and interfacial area in two-phase flow: Dynamic pore-network modelling. J. Fluid Mech. 2010, 655: 38-71.

Johnson, W.P., Pazmino, E., Ma, H. Direct observations of colloid retention in granular media in the presence of energy barriers, and implications for inferred mechanisms from indirect observations. Water Res. 2010, 44(4): 1158-1169.

Kang, Q., Tsimpanogiannis, I.N., Zhang, D., et al. Numerical modeling of pore-scale phenomena during CO2 seques-tration in oceanic sediments. Fuel Process. Technol. 2005, 86(14-15): 1647-1665.

Kang, Q., Zhang, D., Chen, S. Unified lattice Boltzmann method for flow in multiscale porous media. Phys. Rev. E 2002, 66(5): 056307.

Kang, Q., Zhang, D., Lichtner, P.C., et al. Lattice Boltzmann model for crystal growth from supersaturated solution. Geophys. Res. Lett. 2004, 31(21).

Kim, D., Lindquist, W.B. A semianalytic model for the computation of imbibition through polygonal cross sections. Water Resour. Res. 2012, 48(4): W04529.

Kirchhoff, S. Ueber den Durchgang eines elektrischen Stromes durch eine Ebene, insbesondere durch eine kreisfrmige. Ann. Phys-Berlin. 1845, 140(4): 497-514.

Klise, K.A., Moriarty, D., Yoon, H., et al. Automated contact angle estimation for three-dimensional X-ray microtomography data. Adv. Water Resour. 2016, 95: 152-160.

Knudsen, H.A., Hansen, A. Relation between pressure and fractional flow in two-phase flow in porous media. Phys. Rev. E 2002, 65(5): 056310.

Koplik, J., Lasseter, T.J. Two-phase flow in random network models of porous media. SPE J. 1985, 25(1): 89-100.

Kordilla, J., Tartakovsky, A.M., Geyer, T. A smoothed particle hydrodynamics model for droplet and film flow on smooth and rough fracture surfaces. Adv. Water Resour. 2013, 59: 1-14.

Kovscek, A.R., Wong, H., Radke, C.J. A pore-level scenario for the development of mixed wettability in oil reservoirs. AIChE J. 1993, 39(6): 1072-1085.

Kuznar, Z.A., Elimelech, M. Direct microscopic observation of particle deposition in porous media: Role of the secondary energy minimum. Colloids Surf. A 2007, 294(1-3): 156-162.

Levitz, P. Off-lattice reconstruction of porous media: critical evaluation, geometrical confinement and molecular transport. Adv. Colloid Interface Sci. 1998, 76: 71-106.

Lee, S.H., Padmanabhan, L., Al-Sunaidi, H.A. Simulation of linear displacement experiments on massively parallel computers. SPE J. 1996, 1(3): 327-340.

Li, X., Fan, X., Askounis, A., et al. An experimental study on dynamic pore wettability. Chem. Eng. Sci. 2013, 104: 988-997.

Lindquist, W.B. The geometry of primary drainage. J. Colloid Interface Sci. 2006, 296(2): 655-668.

Lindquist, W.B., Venkatarangan, A. Investigating 3D geometry of porous media from high resolution images. Phys. Chem. Earth 1999, 24(7): 593-599.

Liu, H., Kang, Q., Leonardi, C.R., et al. Multiphase lattice Boltzmann simulations for porous media applications. Comput. Geosci. 2016, 20(4): 777-805.

Liu, M., Liu, G. Smoothed particle hydrodynamics (SPH): An overview and recent developments. Arch. Comput. Method E 2010, 17(1): 25-76.

Liu, Z., Wu, H. Pore-scale modeling of immiscible two-phase flow in complex porous media. Appl. Therm. Eng. 2016, 93: 1394-1402.

Luo, L., Yu, B., Cai, J., et al. Numerical simulation of tortuosity for fluid flow in two-dimensional pore fractal models of porous media. Fractals 2014, 22(4): 1450015.

Løvoll, G., M ´eheust, Y., M ˚aløy, K.J., et al. Competition of gravity, capillary and viscous forces during drainage in a two-dimensional porous medium, a pore scale study. Energy 2005, 30(6): 861-872.

Ma, J., Couples, G.D., Jiang, Z., et al. A multi-scale framework for digital core analysis of gas shale at millimeter scales. Paper 1063-1070 Presented at Unconventional Resources Technology Conference held in Denver, Colorado, USA, 25-27 August, 2014.

Ma, S., Mason, G., Morrow, N.R. Effect of contact angle on drainage and imbibition in regular polygonal tubes. Colloids Surf. A 1996, 117(3): 273-291.

Man, H.N., Jing, X.D. Network modelling of wettability and pore geometry effects on electrical resistivity and capillary pressure. J. Pet. Sci. Eng. 1999, 24(2-4): 255-267.

Man, H.N., Jing, X.D. Pore network modelling of electrical resistivity and capillary pressure characteristics. Transp. Porous Media 2000, 41(3): 263-285.

Man, H.N., Jing, X.D. Network modelling of strong and intermediate wettability on electrical resistivity and capillary pressure. Adv. Water Resour. 2001, 24(3-4): 345-363.

Mason, G., Morrow, N.R. Capillary behavior of a perfectly wetting liquid in irregular triangular tubes. J. Colloid Interface Sci. 1991, 141(1): 262-274.

Mayer, R.P., Stowe, R.A. Mercury porosimetry-breakthrough pressure for penetration between packed spheres. J. Colloid Sci. 1965, 20(8): 893-911.

McDonald, P.J., Turner, M.N. Combining effective media and multi-phase methods of Lattice Boltzmann modelling for the characterisation of liquid-vapour dynamics in multi-length scale heterogeneous structural materials. Model. Simul. Mat. Sci. Eng. 2015, 24(1): 015010.

McDougall, S.R., Sorbie, K.S. The prediction of waterflood performance in mixed-wet systems from pore-scale modelling and simulation. Paper SPE 25271 Presented at the SPE Symposium on Reservoir Simulation, New Orleans, LA, USA, 28 February-3 March, 1993.

McNamara, G.R., Zanetti, G. Use of the Boltzmann equation to simulate lattice-gas automata. Phys. Rev. Lett. 1988, 61(20): 2332.

Meakin, P., Tartakovsky, A.M. Modeling and simulation of pore-scale multiphase fluid flow and reactive transport in fractured and porous media. Rev. Geophys. 2009, 47(3): RG3002. Mohammadmoradi, P., Kantzas, A. Pore scale investigation of wettability effect on waterflood performance. Paper SPE 181309 Presented at the Annual Technical Conference and Exhibition, Dubai, UAE, 26-28 September, 2016.

Monaghan, J.J. Smoothed particle hydrodynamics. Annu. Rev. Astron. Astrophys. 1992, 30(1): 543-574.

Morrow, N.R. Physics and thermodynamics of capillary action in porous media. Ind. Eng. Chem. 1970, 62(6): 32-56.

Ning, Y., He, S., Liu, H., et al. A rigorous upscaling procedure to predict macro-scale transport properties of natural gas in shales by coupling molecular dynamics with lattice Boltzmann method. Paper SPE 181689 Presented at the Annual Technical Conference and Exhibition, Dubai, UAE, 26-28 September, 2016.

Noh, W.F., Woodward, P. SLIC (simple line interface calculation). Paper 330-340 Presented at the Proceedings of the Fifth International Conference on Numerical Methods in Fluid Dynamics, Springer, Berlin, Heidelberg, June 28-July 2, 1976.

Okabe, H., Blunt, M.J. Prediction of permeability for porous media reconstructed using multiple-point statistics. Phys. Rev. E 2004, 70(6): 066135.

Olsson, E., Kreiss, G. A conservative level set method for two phase flow. J. Comput. Phys. 2005, 210(1): 225-246.

Olsson, E., Kreiss, G., Zahedi, S. A conservative level set method for two phase flow II. J. Comput. Phys. 2007, 225(1): 785-807.

Osher, S., Sethian, J.A. Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 1988, 79(1): 12-49.

Ovaysi, S., Piri, M. Direct pore-level modeling of incompress-ible fluid flow in porous media. J. Comput. Phys. 2010, 229(19): 7456-7476.

Øren, P.E., Bakke, S., Arntzen, O.J. Extending predictive capabilities to network models. SPE J. 1998, 3(4): 324-336.

Øren, P.E., Billiotte, J., Pinczewski, W.V. Mobilization of waterflood residual oil by gas injection for water-wet conditions. SPE Form. Eval. 1992, 7(1): 70-78.

Peng, X., Qiao, X., Qiu, S., et al. Radial permeability of fractured porous media by Monte Carlo simulations. Int. J. Heat Mass Transf. 2013, 57(1): 369-374.

Pinilla, J., Bruneau, C.H., Tancogne, S. Front-tracking by the level-set and the volume penalization methods in a two-phase microfluidic network. Int. J. Numer. Methods Fluids 2016, 80(1): 23-52.

Piri, M., Blunt, M.J. Three-phase threshold capillary pressures in noncircular capillary tubes with different wettabilities including contact angle hysteresis. Phys. Rev. E 2004, 70(6): 061603.

Piri, M., Blunt, M.J. Three-dimensional mixed-wet random pore-scale network modeling of two-and three-phase flow in porous media. I. Model description. Phys. Rev. E 2005a, 71(2): 026301.

Piri, M., Blunt, M.J. Three-dimensional mixed-wet random pore-scale network modeling of two-and three-phase flow in porous media. II. Results. Phys. Rev. E 2005b, 71(2): 026302.

Piri, M., Karpyn, Z.T. Prediction of fluid occupancy in fractures using network modeling and X-ray microto-mography. II: Results. Phys. Rev. E 2007, 76(1): 016316.

Princen, H.M. Capillary phenomena in assemblies of parallel cylinders: I. Capillary rise between two cylinders. J. Colloid Interface Sci. 1969a, 30(1): 69-75.

Princen, H.M. Capillary phenomena in assemblies of parallel cylinders: II. Capillary rise in systems with more than two cylinders. J. Colloid Interface Sci. 1969b, 30(3): 359-371.

Princen, H.M. Capillary phenomena in assemblies of parallel cylinders: III. Liquid columns between horizontal parallel cylinders. J. Colloid Interface Sci. 1970, 34(2): 171-184.

Prodanovi ´c, M., Bryant, S.L. A level set method for deter-mining critical curvatures for drainage and imbibition. J. Colloid Interface Sci. 2006, 304(2): 442-458.

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

Purcell, W.R. Capillary pressures-their measurement using mercury and the calculation of permeability therefrom. J. Pet. Technol. 1949, 1(2): 39-48.

Rabbani, H.S., Joekar-Niasar, V., Shokri, N. Effects of intermediate wettability on entry capillary pressure in angular pores. J. Colloid Interface Sci. 2016, 473: 34-43.

Raeini, A.Q., Bijeljic, B., Blunt, M.J. Numerical modelling of sub-pore scale events in two-phase flow through porous media. Transp. Porous Media 2014, 101(2): 191-213.

Raeini, A.Q., Bijeljic, B., Blunt, M.J. Generalized network modeling: Network extraction as a coarse-scale dis-cretization of the void space of porous media. Phys. Rev. E 2017, 96(1): 013312.

Raeini, A.Q., Bijeljic, B., Blunt, M.J. Generalized network modeling of capillary-dominated two-phase flow. Phys. Rev. E 2018, 97(2): 023308.

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. J. Comput. Phys. 2012, 231(17): 5653-5668.

Ramstad, T., Øren, P.E., Bakke, S. Simulation of two-phase flow in reservoir rocks using a lattice Boltzmann method. SPE J. 2010, 15(4): 917-927.

Raoof, A., Hassanizadeh, S.M. A new method for generating pore-network models of porous media. Transp. Porous Media 2010, 81(3): 391-407.

Reeves, P.C., Celia, M.A. A functional relationship between capillary pressure, saturation, and interfacial area as revealed by a pore-scale network model. Water Resour. Res. 1996, 32(8): 2345-2358.

Rhodes, M.E., Bijeljic, B., Blunt, M.J. Pore-to-field simulation of single-phase transport using continuous time random walks. Adv. Water Resour. 2008, 31(12): 1527-1539.

Roberts, A.P., Torquato, S. Chord-distribution functions of three-dimensional random media: Approximate first-passage times of gaussian processes. Phys. Rev. E 1999, 59(5): 4953.

Rothman, D.H., Zaleski, S. Lattice-gas models of phase separation: Interfaces, phase transitions, and multiphase flow. Rev. Mod. Phys. 1994, 66(4): 1417.

Sahimi, M. Flow and transport in porous media and fractured rock: From classical methods to modern approaches. New Jersey, USA, John Wiley and Sons, 2011.

Scanziani, A., Singh, K., Blunt, M.J., et al. Automatic method for estimation of in situ effective contact angle from X-ray micro tomography images of two-phase flow in porous media. J. Colloid Interface Sci. 2017, 496: 51-59.

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

Sheng, Q., Thompson, K. Dynamic coupling of pore-scale and reservoir-scale models for multiphase flow. Water Resour. Res. 2013, 49(9): 5973-5988.

Shigorina, E., Kordilla, J., Tartakovsky, A.M. Smoothed particle hydrodynamics study of the roughness effect on contact angle and droplet flow. Phys. Rev. E 2017, 96(3): 033115.

Silin, D.B., Jin, G., Patzek, T.W. Robust determination of the pore space morphology in sedimentary rocks. Paper SPE 84296 Presented at the Annual Technical Conference and Exhibition, Denver, Colorado, 5-8 October, 2003.

Silin, D., Patzek, T. Pore space morphology analysis using maximal inscribed spheres. Phys. A 2006, 371(2): 336-360.

Silin, D., Tomutsa, L., Benson, S.M., et al. Microtomography and pore-scale modeling of two-phase fluid distribution. Transp. Porous Media 2011, 86(2): 495-515.

Soll, W.E., Celia, M.A. A modified percolation approach to simulating three-fluid capillary pressure-saturation relationships. Adv. Water Resour. 1993, 16(2): 107-126.

Soll, W.E., Celia, M.A., Wilson, J.L. Micromodel studies of three-fluid porous media systems: Pore-scale processes relating to capillary pressure-saturation relationships. Water Resour. Res. 1993, 29(9): 2963-2974.

Sun, D.L., Tao, W.Q. A coupled volume-of-fluid and level set (VOSET) method for computing incompressible two-phase flows. Int. J. Heat Mass Transf. 2010, 53(4): 645-655.

Sun, X., Yao, Y., Liu, D., et al. Investigations of CO2 -water wettability of coal: NMR relaxation method. Int. J. Coal Geol. 2018, 188: 38-50.

Sussman, M., Puckett, E.G. A coupled level set and volume-of-fluid method for computing 3D and axisymmetric incompressible two-phase flows. J. Comput. Phys. 2000, 162(2): 301-337.

Tartakovsky, A.M., Meakin, P. Pore scale modeling of immiscible and miscible fluid flows using smoothed particle hydrodynamics. Adv. Water Resour. 2006, 29(10): 1464-1478.

Tartakovsky, A.M., Tartakovsky, D.M., Scheibe, T.D., et al. Hybrid simulations of reaction-diffusion systems in porous media. SIAM J. Sci. Comput. 2008, 30(6): 2799-2816.

Thompson, K.E. Pore-scale modeling of fluid transport in disordered fibrous materials. AIChE J. 2002, 48(7): 1369-1389.

Tokan-Lawal, A., Prodanovi ´c, M., Landry, C.J., et al. Influence of numerical cementation on multiphase displacement in rough fractures. Transp. Porous Media 2017, 116(1): 275-293.

Tryggvason, G., Bunner, B., Esmaeeli, A., et al. A front-tracking method for the computations of multiphase flow. J. Comput. Phys. 2001, 169(2): 708-759.

Valvatne, P.H., Blunt, M.J. Predictive pore-scale network modeling. Paper SPE 84550 Presented at the SPE Annual Technical Conference and Exhibition, Denver, Colorado, 5-8 October, 2003.

Van Dijke, M.I.J., Lago, M., Sorbie, K.S., et al. Free energy balance for three fluid phases in a capillary of arbitrarily shaped cross-section: Capillary entry pressures and layers of the intermediate-wetting phase. J. Colloid Interface Sci. 2004, 277(1): 184-201.

Verma, R., Icardi, M., Prodanovi ´c, M. Effect of wettability on two-phase quasi-static displacement: Validation of two pore scale modeling approaches. J. Contam. Hydrol. 2018, 212: 115-133.

Vinogradov, J., Jackson, M.D. Multiphase streaming potential in sandstones saturated with gas/brine and oil/brine during drainage and imbibition. Geophys. Res. Lett. 2011, 38(1): 121-133.

Virnovsky, G.A., Lohne, A., Frette, O.I. Modeling capillary pressure using capillary bundles with arbitrary cross-sections obtained from photomicrographs. J. Pet. Sci. Eng. 2009, 69(12): 117-128.

Vogel, H.J., Roth, K. Quantitative morphology and network representation of soil pore structure. Adv. Water Resour. 2001, 24(3): 233-242.

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

Wells, J.T., Janecky, D.R., Travis, B.J. A lattice gas automata model for heterogeneous chemical reactions at mineral surfaces and in pore networks. Phys. D 1991, 47(1-2): 115-123.

Wildenschild, D., Sheppard, A.P. X-ray imaging and analysis techniques for quantifying pore-scale structure and processes in subsurface porous medium systems. Adv. Water Resour. 2013, 51(1): 217-246.

Xu, P., Qiu, S., Yu, B., et al. Prediction of relative permeability in unsaturated porous media with a fractal approach. Int. J. Heat Mass Transf. 2013, 64(3): 829-837.

Yang, J., Bondino, I., Regaieg, M., et al. Pore to pore validation of pore network modelling against micromodel experiment results. Comput. Geosci. 2017, 21(5-6): 849-862.

Yerramilli, R.C., Zitha, P.L.J., Yerramilli, S.S., et al. A novel water-injectivity model and experimental validation with CT-scanned corefloods. SPE J. 2014, 20(6): 1200-1211.

Yong, W., Zhou, Y., Derksen, J. Molecular dynamic simulation of wettability in the context of EOR process. Msc Dissertation, University of Aberdeen, 2017.

Yu, B., Cheng, P. A fractal permeability model for bi-dispersed porous media. Int. J. Heat Mass Transf. 2002, 45(14): 2983-2993.

Yu, B., Li, J., Li, Z., et al. Permeabilities of unsaturated fractal porous media. Int. J. Multiph. Flow 2003, 29(10): 1625-1642.

Yu, B., Liu, W. Fractal analysis of permeabilities for porous media. Aiche J. 2004, 50(1): 46-57.

Yuster, S.T. Theoretical considerations of multiphase flow in idealized capillary systems. Proceedings of the 3rd World Petroleum Congress, Section II, The Hague, The Hague, the Netherlands, 28 May-6 June, 1951

Zhang, B., Yu, B., Wang, H., et al. A fractal analysis of permeability for power-law fluids in porous media. Fractals 2006, 14(3): 171-177.

Zhou, Y., Hatzignatiou, D.G., Helland, J.O. On the estimation of CO2 capillary entry pressure: Implications on geological CO2 storage. Int. J. Greenhouse Gas Control 2017, 63: 26-36.

Zhou, Y., Helland, J.O., Hatzignatiou, D.G. A dimensionless capillary pressure function for imbibition derived from pore-scale modeling in mixed-wet-rock images. SPE J. 2012, 18(2): 296-308.

Zhou, Y., Helland, J.O., Hatzignatiou, D.G. Pore-scale modeling of waterflooding in mixed-wet-rock images: Effects of initial saturation and wettability. SPE J. 2014, 19(1): 88-100.

Zhou, Y., Helland, J.O., Hatzignatiou, D.G. Computation of three-phase capillary pressure curves and fluid configurations at mixed-wet conditions in 2D rock images. SPE J. 2016a, 21(1): 152-169.

Zhou, Y., Helland, J.O., Hatzignatiou, D.G., et al. Experi-mental validation of a pore-scale derived dimensionless capillary pressure function for imbibition under mixed wet conditions. Paper Presented at the 78th EAGE Conference and Exhibition 2016, Vienna, Austria, 30 May-2 June, 2016b.

Zhou, Y., Helland, J.O., Jettestuen, E. Dynamic capillary pressure curves from pore-scale modeling in mixed-wet-rock images. SPE J. 2013, 18(4): 634-645.

Zhu, J., Ma, J. An improved gray lattice Boltzmann model for simulating fluid flow in multi-scale porous media. Adv. Water Resour. 2013, 56: 61-76.

Zhu, J., Ma, J. Extending a gray Lattice Boltzmann model for simulating fluid flow in multi-scale porous media. Sci. Rep. 2018, 8(1): 5826.


  • There are currently no refbacks.

Copyright (c) 2018 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