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Mathematical Geosciences

, Volume 45, Issue 3, pp 321–340 | Cite as

Rock Type Connectivity Estimation Using Percolation Theory

  • Saeid Sadeghnejad
  • Mohsen MasihiEmail author
  • Mahmoudreza Pishvaie
  • Peter R. King
Article

Abstract

Complicated sedimentary processes control the spatial distribution of geological heterogeneities. This serves to make the nature of the fluid flow in the hydrocarbon reservoirs immensely complex. Proper modeling of these heterogeneities and evaluation of their connectivity are crucial and affects all aspects of fluid flow. Since the natural variability of heterogeneity occurs in a myriad of length scales, accurate modeling of the rock type connectivity requires a very fine scheme, which is computationally very expensive. Hence, this makes other alternative methods such as the percolation approach attractive and necessary. The percolation approach considers the hypothesis that a reservoir can be split into either permeable (sand/fracture) or impermeable rocks (shale/matrix). In this approach, the connectivity of the permeable fraction governs the flow. This method links the global properties of the system to the density of the permeable objects distributed randomly in the system. Moreover, this approach reduces many results to some simple master curves from which all-possible outcomes can be predicted by simple algebraic transformations. The current study contributes to extending the applicability of the methodology to anisotropic systems as well as using the complicated and more realistic sandbody shapes (for example, ellipsoids). This enables us to attain a better assessment of the connectivity and its associated uncertainty of the complicated rock types. Furthermore, to validate the approach, the Burgan reservoir dataset of the Norouz offshore oil field in the south of Iran was used. The findings are in conformity with the percolation approach predictions.

Keywords

Percolation theory Rock type Connectivity Ellipsoid sandbodies Anisotropy Validation 

Notes

Acknowledgements

The authors would like to acknowledge useful conversations with Dr Tavakoli. In addition, we are grateful to the editor and reviewers who provided many useful suggestions.

References

  1. Adler PM, Thovert JF (1999) Fractures and fracture networks. Kluwer Academic, London CrossRefGoogle Scholar
  2. Baker DR, Paul G, Sreenivasan S et al (2002) Continuum percolation threshold for interpenetrating squares and cubes. Phys Rev E 66:0546136. doi: 10.1103/PhysRevE.66.046136 CrossRefGoogle Scholar
  3. Belayneh M, Masihi M, Matthäi SK et al (2006) Prediction of vein connectivity using the percolation approach: model test with field data. J Geophys Eng 3:219–229. doi: 10.1088/1742-2132/3/3/003 CrossRefGoogle Scholar
  4. Berkowitz B (1995) Analysis of fracture network connectivity using percolation theory. Math Geol 27:467–483. doi: 10.1007/BF02084422 CrossRefGoogle Scholar
  5. Berkowitz B (2002) Characterizing flow and transport in fractured geological media: a review. Adv Water Resour 25:861–884. doi: 10.1016/S0309-1708(02)00042-8 CrossRefGoogle Scholar
  6. Berkowitz B, Balberg I (1993) Percolation theory and its application to groundwater hydrology. Water Resour Res 29:775–794 CrossRefGoogle Scholar
  7. Bour O, Davy P (1998) On the connectivity of three-dimensional fault networks. Water Resour Res 34:2611–2622 CrossRefGoogle Scholar
  8. Bridge JS, Leeder MR (1979) A simulation model of alluvial stratigraphy. Sedimentology 26:617–644 CrossRefGoogle Scholar
  9. Broadbent SR, Hammersley JM (1957) Percolation processes: I. Crystals and mazes. Math Proc Camb Philos Soc 53:629–641. doi: 10.1017/S0305004100032680 CrossRefGoogle Scholar
  10. Chapra SC, Canale R (2002) Numerical methods for engineers: with software and programming applications. McGraw-Hill, New York Google Scholar
  11. Ewing R, Gupta S (1993) Percolation and permeability in partially structured networks. Water Resour Res 29:3179–3188 CrossRefGoogle Scholar
  12. Flory PJ (1941) Molecular size distribution in three dimensional polymers: I. Gelation. J Am Chem Soc 69:30–35 doi: 10.1021/ja01856a061 CrossRefGoogle Scholar
  13. Haldorsen HH, Brand PJ, Macdonald CJ (1988) Review of the stochastic nature of reservoirs. In: Edwards S, King PR (eds) Proceed math oil prod. Clarendon Press, Oxford Google Scholar
  14. Haldorsen HH, Damsleth E (1990) Stochastic modelling (includes associated papers 21255 and 21299). J Pet Technol 42:404–412. doi: 10.2118/20321-PA Google Scholar
  15. Hoshen J, Berry MW, Minser KS (1997) Percolation and cluster structure parameters: the enhanced Hoshen-Kopelman algorithm. Phys Rev E 56:1456–1460 CrossRefGoogle Scholar
  16. Huerlimann A (2004) MDP report of Shell Company. Appendix seven of Soroosh & Nowrooz Burgan rock properties. Tehran, Iran Google Scholar
  17. Hunt AG (2005) Percolation theory and the future of hydrogeology. Hydrogeol J 13:202–205. doi: 10.1007/s10040-004-0405-6 CrossRefGoogle Scholar
  18. Jensen JL, Hart JD, Willis BJ (2006) Evaluating proportions of undetected geological events in the case of erroneous identifications. Math Geol 38:103–112. doi: 10.1007/s11004-005-9007-7 CrossRefGoogle Scholar
  19. King PR (1990) The connectivity and conductivity of overlapping sandbodies. In: Proceedings 2nd international conference of North Sea oil and gas reservoir. Graham & Trotman, London, pp 353–362 CrossRefGoogle Scholar
  20. Kirkpatrick S (1971) Classical transport in disordered media: scaling and effective-medium theories. Phys Rev Let 27:1722–1725. doi: 10.1103/PhysRevLett.27.1722 CrossRefGoogle Scholar
  21. Kirkpatrick S (1973) Percolation and conduction. Rev Mod Phys 45:574–588. doi: 10.1103/RevModPhys.45.574 CrossRefGoogle Scholar
  22. Knudby C, Carrera J, Bumgardner JD et al (2006) Binary upscaling—the role of connectivity and a new formula. Adv Water Resour 29:590–604. doi: 10.1016/j.advwatres.2005.07.002 CrossRefGoogle Scholar
  23. Koltermann CE, Gorelick SM (1996) Heterogeneity in sedimentary deposits: a review of structure-imitating, process-imitating, and descriptive approaches. Water Resour Res 32:2617–2658. doi: 10.1029/96WR00025 CrossRefGoogle Scholar
  24. Lee SB, Torquato S (1990) Monte Carlo study of correlated continuum percolation: universality and percolation thresholds. Phys Rev A 41:5338–5344 CrossRefGoogle Scholar
  25. Lin CY, Hu CK (1998) Universal finite size scaling functions for percolation on three dimensional lattices. Phys Rev E 58:1521–1527 CrossRefGoogle Scholar
  26. Lorenz CD, Ziff RM (2001) Precise determination of the critical percolation threshold for the three dimensional Swiss cheese model using a growth algorithm. J Chem Phys 114:3659–3661. doi: 10.1063/1.1338506 CrossRefGoogle Scholar
  27. de Marsily G, Delay F, Gonsalves J et al (2005) Dealing with spatial heterogeneity. Hydrogeol J 13:161–183. doi: 10.1007/s10040-004-0432-3 CrossRefGoogle Scholar
  28. Masihi M, King PR (2007) A correlated fracture network: modeling and percolation properties. Water Resour Res 43:W07439. doi: 10.1029/2006WR005331 CrossRefGoogle Scholar
  29. Masihi M, King PR, Nurafza P (2006) Effect of anisotropy on finite-size scaling in percolation theory. Phys Rev E 74:042102. doi: 10.1103/PhysRevE.74.042102 CrossRefGoogle Scholar
  30. Meester R, Roy R (1996) Continuum percolation. Cambridge University Press, London CrossRefGoogle Scholar
  31. Mukhopadhyay S, Sahimi M (2000) Calculation of the effective permeabilities of field-scale porous media. Chem Eng Sci 55:4495–4513. doi: 10.1016/S0009-2509(00)00098-1 CrossRefGoogle Scholar
  32. Mutti E, Normark WR (1991) An integrated approach to the study of turbidite systems. In: Weimer P, Link MH (eds) Seismic facies and sedimentary processes of submarine fans and turbidite systems. Springer, New York, pp 75–106 Google Scholar
  33. Nurafza P, King PR, Masihi M (2006) Facies connectivity modelling: analysis and field study. In: Proceedings annual conference and exhibition of SPE Europec/EAGE, Vienna, Austria, SPE 100333 Google Scholar
  34. Ozkaya SI, Mattner J (2003) Fracture connectivity from fracture intersections in borehole image logs. Comput Geosci 29:143–153 CrossRefGoogle Scholar
  35. Perram JW, Rasmussen J, Præstgaard E et al (1996) Ellipsoid contact potential: theory and relation to overlap potentials. Phys Rev E 54:6565–6572 CrossRefGoogle Scholar
  36. Prakash S, Havlin S, Schwartz M et al (1992) Structural and dynamical properties of long-range correlated percolation. Phys Rev A 46:R1724–R1727. doi: 10.1103/PhysRevA.46.R1724 CrossRefGoogle Scholar
  37. Ronayne MJ, Gorelick SM (2006) Effective permeability of porous media containing branching channel networks. Phys Rev E 73:026305. doi: 10.1103/PhysRevE.73.026305 CrossRefGoogle Scholar
  38. Sadeghnejad S, Masihi M, King PR et al (2010a) Effect of anisotropy on the scaling of connectivity and conductivity in continuum percolation theory. Phys Rev E 81:061119. doi: 10.1103/PhysRevE.81.061119 CrossRefGoogle Scholar
  39. Sadeghnejad S, Masihi M, King PR et al (2010b) Reservoir conductivity evaluation using percolation theory. Pet Sci Technol 29:1041–1053. doi: 10.1080/10916460903502506 CrossRefGoogle Scholar
  40. Sadeghnejad S, Masihi M, Shojaei A et al (2011) Field scale characterization of geological formations using percolation theory. Transp Porous Media 92:357–372. doi: 10.1007/s11242-011-9907-6 CrossRefGoogle Scholar
  41. Sahimi M (1994) Applications of percolation theory. Taylor and Francis, London Google Scholar
  42. Sahimi M, Mukhopadhyay S (1996) Scaling properties of a percolation model with long-range correlations. Phys Rev E 54:3870–3880. doi: 10.1103/PhysRevE.54.3870 CrossRefGoogle Scholar
  43. Stauffer D, Aharony A (1994) Introduction to percolation theory. Taylor & Francis, London Google Scholar
  44. Stockmayer WH (1944) Theory of molecular size distribution and gel formation in branched polymers. J Chem Phys 12:125–132. doi: 10.1063/1.1723803 CrossRefGoogle Scholar
  45. Tóth TM, Vass I (2011) Relationship between the geometric parameters of rock fractures, the size of percolation clusters and REV. Math Geosci 43:75–97. doi: 10.1007/s11004-010-9315-4 CrossRefGoogle Scholar
  46. Wilke S, Guyon E, de Marsily G (1985) Water penetration through fractured rocks: test of a tridimensional percolation description. J Int Assoc Math Geol 17:17–27. doi: 10.1007/BF01030364 CrossRefGoogle Scholar
  47. Xia W, Thrope MF (1988) Percolation properties of random ellipses. Phys Rev A 38:2650–2656. doi: 10.1103/PhysRevA.38.2650 CrossRefGoogle Scholar

Copyright information

© International Association for Mathematical Geosciences 2013

Authors and Affiliations

  • Saeid Sadeghnejad
    • 1
  • Mohsen Masihi
    • 2
    Email author
  • Mahmoudreza Pishvaie
    • 2
  • Peter R. King
    • 3
  1. 1.Department of Chemical EngineeringTarbiat Modares UniversityTehranIran
  2. 2.Department of Chemical and Petroleum EngineeringSharif University of TechnologyTehranIran
  3. 3.Earth Science & Engineering DepartmentImperial CollegeLondonUK

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