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Invasion Percolation

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Encyclopedia of Complexity and Systems Science

Definition of the Subject

Invasion percolation is a simple dynamic process describing the slow displacement of one fluid by another in a porous material. This is a common phenomena with many important applications; these include the penetration of nonaqueous polluting liquids into soil, the penetration of air into porous media such as soil, concrete, wood and ceramic powder during drying and the displacement of water from soil and rocks by gases generated by buried waste. A final important example, which is the primary focus of this review, is the accumulation during initial migration and the subsequent recovery and production of hydrocarbon reservoirs. Experiments on idealized systems have shown that the simple invasion percolation model provides a very realistic description of the slow fluid-fluid displacement processes associated with these important...

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Abbreviations

B :

Bond number

D b :

Backbone

D f :

Fractal dimension

\( { D_{\min} } \) :

Fractal dimension of minimum path

fBm:

Fractional Brownian motion

fLm:

Fractional Lévy motion

H :

Hurst exponent

IP:

Invasion percolation

NTIP:

Non‐trapping invasion percolation

OP:

Ordinary percolation

p c :

Ordinary percolation threshold

S r :

Residual saturation

TIP:

IP with trapping

ξ w :

Correlation length

ν:

Percolation correlation length exponent

ϕ:

Porosity

g :

Field gradient

Z :

Coordination number

σ:

Standard deviation

Defender:

Fluid initially within pore space.

Invader:

Second fluid injected to displace defending fluid (defender).

Drainage:

Displacement of a wetting fluid by a non‐wetting fluid.

Bibliography

  1. Amundsen H, Wagner G, Oxaal U, Meakin P, Feder J, Jøssang T (1999) Slow two-phase flow in artificial fractures: Experiments and simulations. Water Resour Res 35:2619–2626

    Google Scholar 

  2. Arns CH, Sakellariou A, Senden TJ, Sheppard AP, Sok RM, Pinczewski WV, Knackstedt MA (2005) Digital core laboratory: Petrophysical analysis from 3D images. Petrophysics 46(4):260–277

    Google Scholar 

  3. Babadagli T (2000) Invasion percolation in correlated porous media. Physica A 285:248–258

    Article  ADS  Google Scholar 

  4. Babalievski F (1998) Cluster counting: The Hoshen–Kopelman algorithm vs. spanning tree approaches. Int J Mod Phys C 9:43–60

    Google Scholar 

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

    Google Scholar 

  6. Blunt MJ (1997) Pore level modeling of the effects of wettability. SPE J 2:494–510

    Google Scholar 

  7. Blunt MJ, Scher H (1995) Pore-level model of wetting. Phys Rev E 52:6387–6403

    Article  ADS  Google Scholar 

  8. Blunt MJ, Jackson MD, Piri M, Valvatne PH (2002) Detailed physics, predictive capabilities and macroscopic consequences for pore‐network models of multiphase flow. Adv Water Resour 25:1069–1089

    Article  Google Scholar 

  9. Chandler R, Koplik J, Lerman K, Willemsen J (1982) Capillary displacement and percolation in porous media. J Fluid Mech 119:249–267

    Article  ADS  MATH  Google Scholar 

  10. Chen JD, Wada N (1986) Visualisation of immiscible displacement in a three dimensional transparent porous medium. Exp Fluids 4:336–338

    Google Scholar 

  11. Cieplak M, Maritan A, Banavar JR (1994) Optimal paths and domain walls in the strong disorder limit. Phys Rev Lett 72:2320–2323

    Google Scholar 

  12. Dobrin R, Duxbury P (2001) Minimum spanning trees on random networks. Phys Rev Lett 86:5076–5079

    Article  ADS  Google Scholar 

  13. Du C, Xu B, Yortsos YC, Chaouche M, Rakotomalala N, Salin D (1995) Correlation of occupation profiles in invasion percolation. Phys Rev Lett 74:694–697

    Article  ADS  Google Scholar 

  14. Du C, Satik C, Yortsos Y (1996) Percolation in a fractional Brownian motion lattice. AIChE J 42:2392–2394

    Article  Google Scholar 

  15. Dunsmuir JH, Ferguson SR, D'Amico KL (1991) Design and operation of an imaging X-ray detector for microtomography. IOP Conf Ser 121:257–261

    Google Scholar 

  16. Flannery BP, Deckman HW, Roberge WG, D'Amico KL (1987) Three‐dimensional X-ray microtomography. Science 237:1439–1444

    Article  ADS  Google Scholar 

  17. Franzese G, Cataudella V, Coniglio A (1998) Invaded cluster dynamics for frustrated models. Phys Rev E 57:88–93

    Article  ADS  Google Scholar 

  18. Fredrich J, Menendez B, Wong TF (1995) Imaging the pore structure of geomaterials. Science 268:276–279

    Article  ADS  Google Scholar 

  19. Galam S, Mauger A (1996) Universal formulas for percolation thresholds. Phys Rev E 53:2177–2180

    Article  ADS  Google Scholar 

  20. Gaylord RJ, Wellin PR (1994) Computer simulations with mathematica: Explorations in complex physical and biological systems. TELOS/Springer, New York

    Google Scholar 

  21. Glass RJ, Nicholl MJ, Yarrington L (1998) A modified invasion percolation model for low‐capillary number immiscible displacements in horizontal rough‐walled fractures: Influence of local in-plane curvature. Water Resour Res 34(12):3215–3234

    Article  ADS  Google Scholar 

  22. Glass RJ, Nicholl MJ, Rajaram H, Andre B (2004) Development of slender transport pathways in unsaturated fractured rock: Simulation with modified invasion percolation. Geophys Res Lett 31:L06502

    Article  ADS  Google Scholar 

  23. Heiba A, Sahimi M, Scriven L, Davis H (1992) Percolation theory of two-phase relative permeability. SPE Reserv Engin 7:123–132

    Google Scholar 

  24. Hoshen J, Kopelman R (1976) Percolation and cluster sizes. Phys Rev B 14:3438

    Article  ADS  Google Scholar 

  25. Ioannidis MA, Kwiecien MJ, Chatzis I, MacDonald IF, Dullien FAL (1997) Comprehensive pore structure characterization using 3D computer reconstruction and stochastic modeling. In: SPE Annual Technical Conference and Exhibition held in San Antonio, Texas, USA, 1997

    Google Scholar 

  26. Knackstedt MA, Sheppard AP, Pinczewski WV (1998) Simulation of mercury porosimetry on correlated grids: Evidence for extended correlated heterogeneity at the pore scale in rocks. Phys Rev E 58:6923–6926

    Article  ADS  Google Scholar 

  27. Knackstedt MA, Sheppard AP, Pinczewski WV (1998) Simulation of mercury porosimetry on correlated grids: Evidence for extended correlated heterogeneity at the pore scale in rocks. Phys Rev E Rapid Communications 58:R6923–R6926

    Article  Google Scholar 

  28. Knackstedt MA, Sahimi M, Sheppard AP (2000) Invasion percolation with long-range correlations: First-order phase transition and nonuniversal scaling properties. Phys Rev E 61:4920–4934

    Google Scholar 

  29. Knackstedt MA, Marrink S, Sheppard AP, Pinczewski W, Sahimi M (2001) Invasion percolation on correlated and elongated lattices: Implications for the interpretation of residual saturations in rock cores. Transp Porous Media 44:465–485

    Article  Google Scholar 

  30. Larson R, Scriven LE, Davis HT (1977) Percolation theory of residual phases in porous media. Nature 268:409–413

    Article  ADS  Google Scholar 

  31. Larson R, Scriven LE, Davis HT (1991) Percolation theory of two-phase flow in porous media. Chem Eng Sci 36:57–73

    Google Scholar 

  32. Lee J-Y, Robins V, Sok RM, Sheppard AP, Pinczewski W, Knackstedt MA (2004) Effect of topology on relative permeability. Transp Porous Media 55:21–46

    Article  Google Scholar 

  33. Lenormand R, Bories S (1980) Description d'un mecanisme de connexion de liaison destin l'tude du drainage avec pigeage en milieu poreux. CR Acad Sci Paris B 291:279

    Google Scholar 

  34. Lenormand R, Bories S (1985) Fractal patterns from chemical dissolution. Physicochem Hydro 6:497

    Google Scholar 

  35. Lenormand R, Zarcone C (1985) Invasion percolation in an etched netowrk; measurement of a fractal dimension. Phys Rev Lett 54:2226–2229

    Article  ADS  Google Scholar 

  36. Lindquist B, Lee SM, Coker D (1996) Medial axis analysis of void structure in three‐dimensional tomographic images of porous media. J Geophys Res 101B:8297–8310

    Article  Google Scholar 

  37. Lindquist WB, Venkatarangan A, Dunsmuir J, Wong TF (2000) Pore and throat size distributions measured from synchrotron X-ray tomographic images of fontainbleau sandstones. J Geophys Res 105B:21508

    Google Scholar 

  38. Marrink SJ, Paterson L, Knackstedt M (2000) Definition of percolation thresholds on self‐affine surfaces. Physica A 280:207–214

    Google Scholar 

  39. Meakin P (1988) Invasion percolation and invading Eden growth on multifractal lattices. J Phys A: Math Gen 21:3501–3522

    Google Scholar 

  40. Meakin P (1991) Invasion percolation on substrates with correlated disorder. Physica A 173:305–324

    Article  ADS  Google Scholar 

  41. Meakin P, Feder J, Frette V, Jøssang T (1992) Invasion percolation in a destabilizing gradient. Phys Rev A 46:3357–3368

    Google Scholar 

  42. Meakin P, Wagner G, Frette V, Feder J, Jøssang T (1995) Fractals and secondary migration. Fractals 3:799–806

    Google Scholar 

  43. Meakin P, Wagner G, Vedvik A, Amundsen H, Feder J, Jøssang T (2000) Invasion percolation and secondary migration: experiments and simulations. Mar Pet Geol 17:777–795

    Google Scholar 

  44. Molz FJ, Liu HH, Szulga J (1997) Fractional Brownian motion and fractional Gaussian noise in subsurface hydrology: A review, presentation of fundamental properties, and extensions. Water Resour Res 33:2273–2286

    Article  ADS  Google Scholar 

  45. Øren P, Bakke S, Arntzen OJ (1998) Extending predictive capabilities to network models. SPE J 3:324–336

    Google Scholar 

  46. Painter S (2001) Flexible scaling model for use in random field simulation of hydraulic conductivity. Water Resour Res 37:1155–1163

    Article  ADS  Google Scholar 

  47. Paterson L (1998) Trapping thresholds in ordinary percolation. Phys Rev E 58:7137–7140

    Article  ADS  Google Scholar 

  48. Paterson L, Painter S, Knackstedt MA, Pinczewski WV (1996) Patterns of fluid flow in naturally heterogeneous rocks. Physica A 233:619–628

    Article  ADS  Google Scholar 

  49. Paterson L, Painter S, Zhang X, Pinczewski WV (1998) Simulating residual saturation and relative permeability in heterogeneous formations. SPE J 3:211–218

    Google Scholar 

  50. Paterson L, Sheppard AP, Knackstedt MA (2002) Trapping thresholds in invasion percolation. Phys Rev E 66:056122

    Article  ADS  Google Scholar 

  51. Porto M, Havlin S, Schwarzer S, Bunde A (1997) Optimal path in strong disorder and shortest path in invasion percolation with trapping. Phys Rev Lett 79:4060–4063

    Article  ADS  Google Scholar 

  52. Sahimi M (1994) Applications of percolation theory, 1st edn. Taylor Francis, London

    Google Scholar 

  53. Sheppard AP, Knackstedt MA, Pinczewski WV, Sahimi M (1999) Invasion percolation: New algorithms and universality classes. J Phys A Lett 32:L521–L529

    MathSciNet  ADS  MATH  Google Scholar 

  54. Sheppard AP, Knackstedt MA, Pinczewski WV, Sahimi M (1999) Invasion percolation: New algorithms and universality classes. J Phys A: Math Gen 32:L521–L529

    Article  MathSciNet  ADS  MATH  Google Scholar 

  55. Sheppard AP, Sok RM, Averdunk H (2005) Improved pore network extraction methods. 19th International Symposium of the SCA, SCA, Toronto, 2005

    Google Scholar 

  56. Sok RM, Knackstedt MA, Sheppard AP, Pinczewski W, Lindquist WB, Venkatarangan A, Paterson L (2002) Direct and stochastic generation of network models from tomographic images; effect of topology on two phase flow properties. Transp Porous Media 46:345–372

    Article  Google Scholar 

  57. Spanne P, Thovert J, Jacquin J, Lindquist WB, Jones K, Adler PM (1994) Synchotron computed microtomography of porous media: Topology and transports. Phys Rev Lett 73:2001–2004

    Article  ADS  Google Scholar 

  58. Stark C (1991) An invasion percolation model of drainage network evolution. Nature 352:423

    Article  ADS  Google Scholar 

  59. Stauffer D, Aharony A (1994) Introduction to percolation theory, 2nd edn. Taylor Francis, London

    Google Scholar 

  60. Stokes JP, Weitz D, Gollub J, Dougherty A, Robbins M, Chaikin P, Lindsay H (1986) Interfacial stability of immiscible displacement in a porous medium. Phys Rev Lett 57:2226–2229

    Article  Google Scholar 

  61. Suding PN, Ziff RM (1999) Site percolation thresholds for archimedean lattices. Phys Rev E 60:275–283

    Article  ADS  Google Scholar 

  62. Thovert J-F, Salles J, Adler P (1993) Computerised characterization of the geometry of real porous media: Their description, analysis and interpretation, J Microsc 170:65–79

    Google Scholar 

  63. Tsimpanogiannis IN, Yortsos YC (2004) The critical gas saturation in a porous medium in the presence of gravity. J Colloid Interface Sci 270:388–395

    Article  Google Scholar 

  64. Vedvik A, Wagner G, Oxaal U, Feder J, Meakin P, Jøssang T (1998) Fragmentation transition for invasion percolation in hydraulic gradients. Phys Rev Lett 80:3065–3068

    Google Scholar 

  65. Wagner G, Amundsen H, Oxaal U, Meakin P, Feder J, Jøssang T (2000) Slow two-phase flow in single fractures: Fragmentation, migration, and fractal patterns simulated using invasion percolation models. Pure Appl Geophys 157:621–635

    Google Scholar 

  66. Wagner G, Meakin P, Feder J, Jøssang T (1997) Buoyancy‐driven invasion percolation with migration and fragmentation. Physica A 245:217–230

    Google Scholar 

  67. Wagner G, Meakin P, Feder J, Jøssang T (1997) Invasion percolation on self‐affine topographies. Phys Rev E 55:1698–1703

    Google Scholar 

  68. Wagner G, Meakin P, Feder J, Jøssang T (1999) Invasion percolation in fractal fractures. Physica A 264:321–337

    Google Scholar 

  69. Wilkinson D (1984) Percolation model of immiscible displacement in the presence of buoyancy forces. Phys Rev A 30:520–531

    Google Scholar 

  70. Wilkinson D (1986) Percolation effects in immiscible displacement. Phys Rev A 34:1380–1391

    Article  ADS  Google Scholar 

  71. Wilkinson D, Willemsen JF (1983) Invasion percolation: A new form of percolation theory. J Phys A: Math Gen 16:3365–3376

    Article  MathSciNet  ADS  Google Scholar 

  72. Xu B, Yortsos YC, Salin D (1998) Invasion percolation with viscous forces. Phys Rev E 57:739–751

    Article  ADS  Google Scholar 

  73. Yuan H (1991) Pore-scale heterogeneity from mercury porosimetry data. SPE Form Eval 6:233–242

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Applied Maths, RSPhysSE, Australian National University, Canberra, Australia

    Mark Knackstedt

  2. CSIRO Petroleum, Clayton, Australia

    Lincoln Paterson

Authors
  1. Mark Knackstedt
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  2. Lincoln Paterson
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Editor information

Editors and Affiliations

  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Knackstedt, M., Paterson, L. (2009). Invasion Percolation . In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_294

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