Modelling Imbibition Processes in Heterogeneous Porous Media

  • Si Suo
  • Mingchao Liu
  • Yixiang Gan


Imbibition is a commonly encountered multiphase problem in various fields, and exact prediction of imbibition processes is a key issue for better understanding capillary flow in heterogeneous porous media. In this work, a numerical framework for describing imbibition processes in porous media with material heterogeneity is proposed to track the moving wetting front with the help of a partially saturated region at the front vicinity. A new interface treatment, named the interface integral method, is developed here, combined with which the proposed numerical model provides a complete framework for imbibition problems. After validation of the current model with existing experimental results of one-dimensional imbibition, simulations on a series of two-dimensional cases are analysed with the presences of multiple porous phases. The simulations presented here not only demonstrate the suitability of the numerical framework on complex domains but also present its feasibility and potential for further engineering applications involving imbibition in heterogeneous media.


Porous media Imbibition Heterogeneity Interface 



This work was financially supported by Australian Research Council (Projects DP170102886) and The University of Sydney SOAR Fellowship.


  1. Al-Maktoumi, A., Kacimov, A., Al-Ismaily, S., Al-Busaidi, H., Al-Saqri, S.: Infiltration into two-layered soil: the Green-Ampt and Averyanov models revisited. Transp. Porous Media 109(1), 169–193 (2015)CrossRefGoogle Scholar
  2. Alyafei, N., Blunt, M.J.: Estimation of relative permeability and capillary pressure from mass imbibition experiments. Adv. Water Resour. 115, 88–94 (2018)CrossRefGoogle Scholar
  3. Böttcher, C.J.F., van Belle, O.C., Bordewijk, P., Rip, A.: Theory of Electric Polarization. Elsevier, Amsterdam (1978)Google Scholar
  4. Bal, K., Fan, J., Sarkar, M., Ye, L.: Differential spontaneous capillary flow through heterogeneous porous media. Int. J. Heat Mass Transf. 54(13–14), 3096–3099 (2011)CrossRefGoogle Scholar
  5. Bear, J.: Dynamics of Fluids in Porous Media. Courier Corporation, North Chelmsford (2013)Google Scholar
  6. Block, R.J., Durrum, E.L., Zweig, G.: A Manual of Paper Chromatography and Paper Electrophoresis. Elsevier, Amsterdam (2016)Google Scholar
  7. Brooks, R., Corey, T.: Hydraulic Properties of Porous Media, Hydrology Papers, p. 24. Colorado State University, Fort Collins (1964)Google Scholar
  8. Cai, J., Perfect, E., Cheng, C.-L., Hu, X.: Generalized modeling of spontaneous imbibition based on Hagen–Poiseuille flow in tortuous capillaries with variably shaped apertures. Langmuir 30(18), 5142–5151 (2014)CrossRefGoogle Scholar
  9. Cai, J., You, L., Hu, X., Wang, J., Peng, R.: Prediction of effective permeability in porous media based on spontaneous imbibition effect. Int. J. Mod. Phys. C 23(07), 1250054 (2012)CrossRefGoogle Scholar
  10. Cai, J., Yu, B.: A discussion of the effect of tortuosity on the capillary imbibition in porous media. Transp. Porous Media 89(2), 251–263 (2011)CrossRefGoogle Scholar
  11. Conrath, M., Fries, N., Zhang, M., Dreyer, M.E.: Radial capillary transport from an infinite reservoir. Transp. Porous Media 84(1), 109–132 (2010)CrossRefGoogle Scholar
  12. Debbabi, Y., Jackson, M.D., Hampson, G.J., Fitch, P.J., Salinas, P.: Viscous crossflow in layered porous media. Transp. Porous Media 117(2), 281–309 (2017)CrossRefGoogle Scholar
  13. Di Donato, G., Lu, H., Tavassoli, Z., Blunt, M.J.: Multirate-transfer dual-porosity modeling of gravity drainage and imbibition. SPE J. 12(01), 77–88 (2007)CrossRefGoogle Scholar
  14. Durlofsky, L.J.: Numerical calculation of equivalent grid block permeability tensors for heterogeneous porous media. Water Resour. Res. 27(5), 699–708 (1991)CrossRefGoogle Scholar
  15. Elizalde, E., Urteaga, R., Berli, C.L.: Rational design of capillary-driven flows for paper-based microfluidics. Lab Chip 15(10), 2173–2180 (2015)CrossRefGoogle Scholar
  16. Ern, A., Mozolevski, I., Schuh, L.: Discontinuous Galerkin approximation of two-phase flows in heterogeneous porous media with discontinuous capillary pressures. Comput. Methods Appl. Mech. Eng. 199(23–24), 1491–1501 (2010)CrossRefGoogle Scholar
  17. Fries, N., Dreyer, M.: An analytic solution of capillary rise restrained by gravity. J. Colloid Interface Sci. 320(1), 259–263 (2008)CrossRefGoogle Scholar
  18. Fries, N., Odic, K., Conrath, M., Dreyer, M.: The effect of evaporation on the wicking of liquids into a metallic weave. J. Colloid Interface Sci. 321(1), 118–129 (2008)CrossRefGoogle Scholar
  19. Guerrero-Martínez, F.J., Younger, P.L., Karimi, N., Kyriakis, S.: Three-dimensional numerical simulations of free convection in a layered porous enclosure. Int. J. Heat Mass Transf. 106, 1005–1013 (2017)CrossRefGoogle Scholar
  20. Hall, C.: Barrier performance of concrete: a review of fluid transport theory. Mater. Struct. 27(5), 291–306 (1994)CrossRefGoogle Scholar
  21. Hanžič, L., Kosec, L., Anžel, I.: Capillary absorption in concrete and the Lucas–Washburn equation. Cement Concr. Compos. 32(1), 84–91 (2010)CrossRefGoogle Scholar
  22. Hashin, Z., Shtrikman, S.: A variational approach to the theory of the elastic behaviour of multiphase materials. J. Mech. Phys. Solids 11(2), 127–140 (1963)CrossRefGoogle Scholar
  23. Helmig, R., Weiss, A., Wohlmuth, B.I.: Dynamic capillary effects in heterogeneous porous media. Comput. Geosci. 11(3), 261–274 (2007)CrossRefGoogle Scholar
  24. Huinink, H.: Fluids in porous media: Transport and phase changes, pp. 1–116 (2016)Google Scholar
  25. Jin, Y., Li, X., Zhao, M., Liu, X., Li, H.: A mathematical model of fluid flow in tight porous media based on fractal assumptions. Int. J. Heat Mass Transf. 108, 1078–1088 (2017)CrossRefGoogle Scholar
  26. Kun-Can, Z., Tong, W., Hai-Cheng, L., Zhi-Jun, G., Wen-Fei, W.: Fractal analysis of flow resistance in random porous media based on the staggered pore-throat model. Int. J. Heat Mass Transf. 115, 225–231 (2017)CrossRefGoogle Scholar
  27. Lewandowska, J., Szymkiewicz, A., Auriault, J.-L.: Upscaling of Richards’ equation for soils containing highly conductive inclusions. Adv. Water Resour. 28(11), 1159–1170 (2005)CrossRefGoogle Scholar
  28. Liu, M., Wu, J., Gan, Y., Hanaor, D.A., Chen, C.: Evaporation limited radial capillary penetration in porous media. Langmuir 32(38), 9899–9904 (2016)CrossRefGoogle Scholar
  29. Liu, M., Wu, J., Gan, Y., Hanaor, D.A., Chen, C.: Tuning capillary penetration in porous media: combining geometrical and evaporation effects. Int. J. Heat Mass Transf. 123, 239–250 (2018)CrossRefGoogle Scholar
  30. Liu, Z., Hu, J., Zhao, Y., Qu, Z., Xu, F.: Experimental and numerical studies on liquid wicking into filter papers for paper-based diagnostics. Appl. Therm. Eng. 88, 280–287 (2015)CrossRefGoogle Scholar
  31. Lucas, R.: Ueber das Zeitgesetz des kapillaren Aufstiegs von Flüssigkeiten. Kolloid-Zeitschrift 23(1), 15–22 (1918)CrossRefGoogle Scholar
  32. Mendez, S., Fenton, E.M., Gallegos, G.R., Petsev, D.N., Sibbett, S.S., Stone, H.A., Zhang, Y., López, G.P.: Imbibition in porous membranes of complex shape: quasi-stationary flow in thin rectangular segments. Langmuir 26(2), 1380–1385 (2009)CrossRefGoogle Scholar
  33. 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. Adv. Geo-energy Res. 1, 1–17 (2017)CrossRefGoogle Scholar
  34. Morrow, N.R., Mason, G.: Recovery of oil by spontaneous imbibition. Curr. Opin. Colloid Interface Sci. 6(4), 321–337 (2001)CrossRefGoogle Scholar
  35. Navarro, V., Yustres, A., Cea, L., Candel, M., Juncosa, R., Delgado, J.: Characterization of the water flow through concrete based on parameter estimation from infiltration tests. Cem. Concr. Res. 36(9), 1575–1582 (2006)CrossRefGoogle Scholar
  36. Nguyen, T.H., Fraiwan, A., Choi, S.: Based batteries: a review. Biosens. Bioelectron. 54, 640–649 (2014)CrossRefGoogle Scholar
  37. Patel, H.S., Meher, R.: Modelling of imbibition phenomena in fluid flow through heterogeneous inclined porous media with different porous materials. Nonlinear Eng. 6(4), 263–275 (2017)CrossRefGoogle Scholar
  38. Perez-Cruz, A., Stiharu, I., Dominguez-Gonzalez, A.: Two-dimensional model of imbibition into paper-based networks using Richards’ equation. Microfluid. Nanofluid. 21(5), 98 (2017)CrossRefGoogle Scholar
  39. Pettersen, Ø.: Simulation of two-phase flow in porous rocks on a laboratory scale: diffusion operator splitting and consistency. Comput. Methods Appl. Mech. Eng. 65(3), 229–252 (1987)CrossRefGoogle Scholar
  40. Quéré, D.: Inertial capillarity. EPL (Europhys. Lett.) 39(5), 533 (1997)CrossRefGoogle Scholar
  41. Reyssat, M., Sangne, L., Van Nierop, E., Stone, H.: Imbibition in layered systems of packed beads. EPL (Europhys. Lett.) 86(5), 56002 (2009)CrossRefGoogle Scholar
  42. Rokhforouz, M., Akhlaghi Amiri, H.: Phase-field simulation of counter-current spontaneous imbibition in a fractured heterogeneous porous medium. Phys. Fluids 29(6), 062104 (2017)CrossRefGoogle Scholar
  43. Schneider, M., Köppl, T., Helmig, R., Steinle, R., Hilfer, R.: Stable propagation of saturation overshoots for two-phase flow in porous media. Transp. Porous Media 121(3), 621–641 (2018)CrossRefGoogle Scholar
  44. Spaid, M.A., Phelan Jr., F.R.: Modeling void formation dynamics in fibrous porous media with the lattice Boltzmann method. Compos. A Appl. Sci. Manuf. 29(7), 749–755 (1998)CrossRefGoogle Scholar
  45. Tang, R., Yang, H., Gong, Y., Liu, Z., Li, X., Wen, T., Qu, Z., Zhang, S., Mei, Q., Xu, F.: Improved analytical sensitivity of lateral flow assay using sponge for HBV nucleic acid detection. Sci. Rep. 7(1), 1360 (2017)CrossRefGoogle Scholar
  46. Warren, J., Price, H.: Flow in heterogeneous porous media. Soc. Pet. Eng. J. 1(03), 153–169 (1961)CrossRefGoogle Scholar
  47. Washburn, E.W.: The dynamics of capillary flow. Phys. Rev. 17(3), 273 (1921)CrossRefGoogle Scholar
  48. Xiao, J., Cai, J., Xu, J.: Saturated imbibition under the influence of gravity and geometry. J. Colloid Interface Sci. 521, 226–231 (2018)CrossRefGoogle Scholar
  49. Xiao, J., Stone, H.A., Attinger, D.: Source-like solution for radial imbibition into a homogeneous semi-infinite porous medium. Langmuir 28(9), 4208–4212 (2012)CrossRefGoogle Scholar
  50. Zhuang, L., Hassanizadeh, S.M., Kleingeld, P.J., van Genuchten, M.T.: Revisiting the horizontal redistribution of water in soils: Experiments and numerical modeling. Water Resour. Res. 53(9), 7576–7589 (2017)CrossRefGoogle Scholar
  51. Zienkiewicz, O.C., Taylor, R.L., Zienkiewicz, O.C., Taylor, R.L.: The Finite Element Method. McGraw-hill, London (1977)Google Scholar
  52. Zimmerman, R.W.: Thermal conductivity of fluid-saturated rocks. J. Pet. Sci. Eng. 3(3), 219–227 (1989)CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.School of Civil EngineeringThe University of SydneySydneyAustralia
  2. 2.Department of Engineering Mechanics, CNMM and AMLTsinghua UniversityBeijingChina

Personalised recommendations