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Drainage front width in a three-dimensional random porous medium under gravitational and capillary effects

Paula Reis1,* and Knut Jørgen Måløy1,2,†

  • *Contact author: paula.reis@mn.uio.no
  • †Contact author: maloy@fys.uio.no
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Phys. Rev. Research 7, 033244 – Published 12 September, 2025

DOI: https://doi.org/10.1103/5bbz-ksds

Abstract

A theoretical approach to estimating stable drainage front widths in three-dimensional (3D) random porous media under gravitational and capillary effects is presented here. Based on the frontier of the infinite cluster in gradient percolation, we propose an expression for the 3D front width dependent on the pore-network topology, the distribution of capillary-pressure thresholds for the pore throats, the stabilizing capillary-pressure gradient, the average pore size, and the correlation length critical exponent from percolation in three dimensions. Theoretical predictions are successfully compared to numerical results obtained with a bond invasion-percolation model for a wide range of drainage flow parameters.

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References (51)

  1. R. Juanes, C. W. MacMinn, and M. L. Szulczewski, The footprint of the CO2 plume during carbon dioxide storage in saline aquifers: Storage efficiency for capillary trapping at the basin scale, Transp. Porous Media 82, 19 (2010).
  2. S. A. Aryana and A. R. Kovscek, Experiments and analysis of drainage displacement processes relevant to carbon dioxide injection, Phys. Rev. E 86, 066310 (2012).
  3. S. S. Datta and D. A. Weitz, Drainage in a model stratified porous medium, Europhys. Lett. 101, 14002 (2013).
  4. F. Hoogland, P. Lehmann, R. Mokso, and D. Or, Drainage mechanisms in porous media: From piston-like invasion to formation of corner flow networks, Water Resour. Res. 52, 8413 (2016).
  5. F. Hoogland, P. Lehmann, and D. Or, Drainage dynamics controlled by corner flow: Application of the foam drainage equation, Water Resour. Res. 52, 8402 (2016).
  6. F. Hoogland, P. Lehmann, and D. Or, Capillary flows across layers and textural interfaces—Pathways and colloid transport considerations in unsaturated layered porous media, J. Colloid Interface Sci. 504, 294 (2017).
  7. I. Zacharoudiou, E. S. Boek, and J. Crawshaw, The impact of drainage displacement patterns and Haines jumps on CO2 storage efficiency, Sci. Rep. 8, 15561 (2018).
  8. K. J. Måløy, J. Feder, and T. Jøssang, Viscous fingering fractals in porous media, Phys. Rev. Lett. 55, 2688 (1985).
  9. R. Lenormand and C. Zarcone, Invasion percolation in an etched network: Measurement of a fractal dimension, Phys. Rev. Lett. 54, 2226 (1985).
  10. G. M. Homsy, Viscous fingering in porous media, Annu. Rev. Fluid Mech. 19, 271 (1987).
  11. R. Lenormand and C. Zarcone, Capillary fingering: Percolation and fractal dimension, Transp. Porous Media 4, 599 (1989).
  12. R. Lenormand, Liquids in porous media, J. Phys.: Condens. Matter 2, SA79 (1990).
  13. A. Birovljev, L. Furuberg, J. Feder, T. Jøssang, K. Måløy, and A. Aharony, Gravity invasion percolation in two dimensions: Experiment and simulation, Phys. Rev. Lett. 67, 584 (1991).
  14. K. J. Måløy, L. Furuberg, J. Feder, and T. Jøssang, Dynamics of slow drainage in porous media, Phys. Rev. Lett. 68, 2161 (1992).
  15. T. Vincent-Dospital, M. Moura, R. Toussaint, and K. J. Måløy, Stable and unstable capillary fingering in porous media with a gradient in grain size, Commun. Phys. 5, 306 (2022).
  16. P. Meakin, Invasion percolation on substrates with correlated disorder, Physica A 173, 305 (1991).
  17. V. Mani and K. Mohanty, Effect of pore-space spatial correlations on two-phase flow in porous media, J. Pet. Sci. Technol. 23, 173 (1999).
  18. D. Wilkinson and J. F. Willemsen, Invasion percolation: A new form of percolation theory, J. Phys. A: Math. Gen. 16, 3365 (1983).
  19. H. Auradou, K. J. Måløy, J. Schmittbuhl, A. Hansen, and D. Bideau, Competition between correlated buoyancy and uncorrelated capillary effects during drainage, Phys. Rev. E 60, 7224 (1999).
  20. M. Moura, E.-A. Fiorentino, K. J. Måløy, G. Schäfer, and R. Toussaint, Impact of sample geometry on the measurement of pressure-saturation curves: Experiments and simulations, Water Resour. Res. 51, 8900 (2015).
  21. P. Meakin, J. Feder, V. Frette, and T. Jøssang, Invasion percolation in a destabilizing gradient, Phys. Rev. A 46, 3357 (1992).
  22. P. Meakin, A. Birovljev, V. Frette, J. Feder, and T. Jøssang, Gradient stabilized and destabilized invasion percolation, Physica A 191, 227 (1992).
  23. V. Frette, J. Feder, T. Jøssang, and P. Meakin, Buoyancy-driven fluid migration in porous media, Phys. Rev. Lett. 68, 3164 (1992).
  24. A. Birovljev, G. Wagner, P. Meakin, J. Feder, and T. Jøssang, Migration and fragmentation of invasion percolation clusters in two-dimensional porous media, Phys. Rev. E 51, 5911 (1995).
  25. G. Wagner, A. Birovljev, P. Meakin, J. Feder, and T. Jøssang, Fragmentation and migration of invasion percolation clusters: Experiments and simulations, Phys. Rev. E 55, 7015 (1997).
  26. Y. Méheust, G. Løvoll, K. J. Måløy, and J. Schmittbuhl, Interface scaling in a two-dimensional porous medium under combined viscous, gravity, and capillary effects, Phys. Rev. E 66, 051603 (2002).
  27. B. Sapoval, M. Rosso, and J.-F. Gouyet, The fractal nature of a diffusion front and the relation to percolation, J. Phys. Lett. 46, 149 (1985).
  28. M. Rosso, J. Gouyet, and B. Sapoval, Gradient percolation in three dimensions and relation to diffusion fronts, Phys. Rev. Lett. 57, 3195 (1986).
  29. J.-F. Gouyet, M. Rosso, and B. Sapoval, Fractal structure of diffusion and invasion fronts in three-dimensional lattices through the gradient percolation approach, Phys. Rev. B 37, 1832 (1988).
  30. P. Meakin, G. Wagner, A. Vedvik, H. Amundsen, J. Feder, and T. Jøssang, Invasion percolation and secondary migration: Experiments and simulations, Mar. Pet. Geol. 17, 777 (2000).
  31. Y. Zhang, M. Shariati, and Y. C. Yortsos, The spreading of immiscible fluids in porous media under the influence of gravity, Transp. Porous Media 38, 117 (2000).
  32. G. Lovoll, Y. Méheust, K. J. Måløy, E. Aker, and J. Schmittbuhl, Competition of gravity, capillary and viscous forces during drainage in a two-dimensional porous medium, a pore scale study, Energy 30, 861 (2005).
  33. M. Chaouche, N. Rakotomalala, D. Salin, B. Xu, and Y. Yortsos, Invasion percolation in a hydrostatic or permeability gradient: Experiments and simulations, Phys. Rev. E 49, 4133 (1994).
  34. D. Wilkinson, Percolation model of immiscible displacement in the presence of buoyancy forces, Phys. Rev. A 30, 520 (1984).
  35. D. Stauffer, Introduction to Percolation Theory (Taylor & Francis, London, 1985).
  36. L. Furuberg, K. J. Måløy, and J. Feder, Intermittent behavior in slow drainage, Phys. Rev. E 53, 966 (1996).
  37. Y. Masson and S. R. Pride, A fast algorithm for invasion percolation, Transp. Porous Media 102, 301 (2014).
  38. P. Reis, M. Moura, G. Linga, P. A. Rikvold, R. Toussaint, E. G. Flekkøy, and K. J. Måløy, A simplified pore-scale model for slow drainage including film-flow effects, Adv. Water Resour. 182, 104580 (2023).
  39. K. Khobaib, P. Reis, M. Moura, R. Toussaint, E. G. Flekkøy, and K. J. Måløy, Gravity stabilized drainage in porous media with controlled disorder, Phys. Rev. Res. 7, 023040 (2025).
  40. W. Lindquist and A. Venkatarangan, Investigating 3D geometry of porous media from high resolution images, Phys. Chem. Earth, Part A: Solid Earth Geod. 24, 593 (1999).
  41. M. Ioannidis and I. Chatzis, On the geometry and topology of 3D stochastic porous media, J. Colloid Interface Sci. 229, 323 (2000).
  42. L. Vasilyev, A. Raoof, and J. M. Nordbotten, Effect of mean network coordination number on dispersivity characteristics, Transp. Porous Media 95, 447 (2012).
  43. M. Moura, E. G. Flekkøy, K. J. Måløy, G. Schäfer, and R. Toussaint, Connectivity enhancement due to film flow in porous media, Phys. Rev. Fluids 4, 094102 (2019).
  44. See Supplemental Material at http://link.aps.org/supplemental/10.1103/5bbz-ksds for for extra plots related to the drainage-front critical regions and the maximum trapped-cluster size.
  45. D. Wilkinson and M. Barsony, Monte Carlo study of invasion percolation clusters in two and three dimensions, J. Phys. A: Math. Gen. 17, L129 (1984).
  46. S. J. Breen, S. R. Pride, Y. Masson, and M. Manga, Stable drainage in a gravity field, Adv. Water Resour. 162, 104150 (2022).
  47. K. A. Culligan, D. Wildenschild, B. S. Christensen, W. G. Gray, M. L. Rivers, and A. F. Tompson, Interfacial area measurements for unsaturated flow through a porous medium, Water Resour. Res. 40 (2004).
  48. C. Chen, P. Duru, P. Joseph, S. Geoffroy, and M. Prat, Control of evaporation by geometry in capillary structures. From confined pillar arrays in a gap radial gradient to phyllotaxy-inspired geometry, Sci. Rep. 7, 15110 (2017).
  49. P.-E. Øren and S. Bakke, Reconstruction of Berea sandstone and pore-scale modelling of wettability effects, J. Pet. Sci. Technol. 39, 177 (2003).
  50. S. L. Bryant, P. R. King, and D. W. Mellor, Network model evaluation of permeability and spatial correlation in a real random sphere packing, Transp. Porous Media 11, 53 (1993).
  51. J. Tran, T. Yoo, S. Stahlheber, and A. Small, Percolation thresholds on three-dimensional lattices with three nearest neighbors, J. Stat. Mech. (2013) P05014.

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Phys. Rev. Research 7, 033244– Published 12 September, 2025
  • Received 7 May 2025
  • Accepted 20 August 2025
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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