Mathematical Geology

, Volume 38, Issue 1, pp 33–50 | Cite as

Effective Permeability Estimation for 2-D Fractal Permeability Fields

  • Tayfun BabadagliEmail author

Hurst exponents (H) of the distribution of permeability at micro (pore) scale were measured as close to 0.1 for sandstone and limestone samples. Based on these observations and previously reported H values for field scale permeability distribution ranging between 0.6 and 0.9, square permeability fields at different scales varying between 1 and 100 ft were generated for the H values of 0.1, 0.5, and 0.9. The study also considered different permeability fields and number of grids ranging from 10 to 500 md and from 8 × 8 to 64 × 64, respectively. The effective permeability of fractally distributed 2-D fields was calculated using different averaging techniques and compared to the actual (equivalent) permeability obtained through numerical simulation. The geometric mean and power averaging techniques as well as the perturbation theory yielded the most reasonable agreement between the actual and calculated effective permeabilities. The accuracy of these techniques increases with increasing average model permeability. It was also observed that as the H decreases, the permeability values obtained were higher than the actual values. Two extreme values of the number of grids (8 × 8 and 64 × 64) yielded the highest error percentages. Thus, the optimum number of grids was found to be 16 × 16 and 32 × 32 depending on the average permeability of the model. The exponent of the power law model was correlated to the fractal dimension of the permeability field for 8 × 8 and 64 × 64 grids. While a good correlation exists for 8 × 8 number of grids, no correlation was obtained for 64 × 64. Hence, an alternate model was proposed for 8 × 8 grids but for grid numbers higher than 32 × 32, no technique was found suitable for averaging of the fractal permeability fields.

Key Words

Effective permeability upscaling averaging techniques fractal distribution of permeability grid size and dimension hurst exponent fractal dimension micro scale 


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The author thanks Schlumberger GeoQuest for providing the ECLIPSE package for research purpose. The partial support from NSERC (Grant No: G121210595) for this research is gratefully acknowledged.


  1. Aasum, Y., Kelkar, M. G., and Gupta, S. P., 1991, An application of geostatistics and fractal geometry for reservoir characterization: Soc. Petrol. Engr. Form. Eval., v. 6, p. 11–19.Google Scholar
  2. Ababou, R., McLaughlin, D., Gelhar, L. W., and Tompson, A. F., 1989, Numerical simulation of three-dimensional saturated flow in randomly heterogeneous porous media: Transp. Porous Media, v. 4, p. 549–565.Google Scholar
  3. Abramovich, B., and Indelman, P., 1995, Effective permittivity of log-normal isotropic random media: J. Phys. A: Math. Gen., v. 28, p. 693–700.Google Scholar
  4. Aharony, A., Hinrichsen, E. L., Hansen, A., Feder, J., Jossang, T., and Hardy, H. H., 1991, Effective renormalization group algorithm for transport in oil reservoirs: Physica A, v. 177, p. 260– 266.Google Scholar
  5. Babadagli, T., 1999, Effect of fractal permeability correlations on waterflooding performance in carbonate reservoirs: J. Petrol. Sci. Engr., v. 23, p. 223–238.Google Scholar
  6. Babadagli, T., 2001, Fractal analysis of 2-D fracture networks of geothermal reservoirs in South-Western Turkey: J. Volcanol. Geotherm. Res., v. 112, p. 83–103.Google Scholar
  7. Babadagli, T., and Develi, K., 2001, On the application of methods used to calculate fractal dimension of fracture surfaces: Fractals, v. 9, p. 105–128.Google Scholar
  8. Beier, R. A., and Hardy, H. H., 1993, Comparison of different horizontal fractal distributions in reservoir simulation, in Linville, B., Burchfield, T. E., and Wesson, T. C., eds., Reservoir characterization III: Penn Well Books, Tulsa, OK, USA, p. 271–296.Google Scholar
  9. Berta, D., Hardy, H. H., and Beier, R. A., 1994, Fractal distributions of reservoir properties and the use in reservoir simulation: Proceedings of the SPE Int. Pet. Conf. and Exh., Veracruz: Soc. Petrol. Engr. paper 28734.Google Scholar
  10. Chang, Y. C., and Mohanty, K. K., 1994, Stochastic description of multiphase flow in heterogeneous porous media: Proceedings of the 69th SPE Annu. Tech. Conf., New Orleans, LA, Soc. Petrol. Engr. paper 28443.Google Scholar
  11. Crane, S. D., and Tubman, K. M., 1990, Reservoir variability and modeling with fractal: Proceedings of the SPE 65th Annu. Tech. Conf., New Orleans, LA, Soc. Petrol. Engr. paper 20606.Google Scholar
  12. Dagan, G., 1993, High-order correction of effective permeability of heterogeneous isotropic formations of log-normal conductivity distributions: Transp. Porous Media, v. 12, p. 279–290.Google Scholar
  13. ECLIPSE, 1996, Reference Manual, Schlumberger GeoQuest, p. 714.Google Scholar
  14. Emanuel, A. S., Alameda, G. K., Behrens, R. A., and Hewett, T. A., 1989, Reservoir performance prediction methods based on fractal geostatistics: Soc. Petrol. Engr. Reservoir Eng., v. 4, p. 311–318.Google Scholar
  15. Feder, J., 1988, Fractals: Plenium, New York, p. 283.Google Scholar
  16. Hewett, T. A., 1986, Fractal distribution of reservoir heterogeneity and their influence on fluid transport: Proceedings of the SPE Annual Technical Conference and Exhibition, New Orleans, LA, Soc. Petrol. Engr, paper 15386.Google Scholar
  17. Hewett, T. A., and Behrens, R. A., 1990, Conditional simulation of reservoir heterogeneity with fractals: Soc. Petrol. Engr. Formation Eval., v. 5, p. 217–226.Google Scholar
  18. Hird, K. B., and Dubrule, O., 1995, Quantification of reservoir connectivity for reservoir description applications: Proceedings of the SPE Annual Technical Conference and Exhibition, Dallas, TX, Soc. Petrol. Engr., paper 30571.Google Scholar
  19. Hurst, A., and Rosvoll, K. J., 1991, Permeability variations in sandstones and their relationship to sedimentary structure, in Lake, L.W., Carroll, H.B., and Wesson, T.C., eds., Reservoir characterization II. Academic, San Diego, CA, p. 726.Google Scholar
  20. Jacquin, C. G., Henriette, A., Guerillo, D., and Adler, P. M., 1991, Heterogeneity and effective permeability of porous rocks: Experimental and numerical investigation, in Lake, L.W., Carroll, H.B., and Wesson, T.C., eds., Reservoir characterization II. Academic, San Diego, CA, p. 726.Google Scholar
  21. Jensen, J. L., and Corbett, P. W. M., 1993, A stochastic model for comparing probe permeameter and core plug measurements, in Linville, B., Burchfield, T. E., and Wesson, T. C., eds., Reservoir characterization III. Penn Well Books, Tulsa, OK, p. 12–22.Google Scholar
  22. Katz, A. J., and Thompson, A. H., 1995, Fractal sandstone pores: Implications for conductivity and pore formation: Phys. Rev. Lett., v. 54, p. 1325–1328.Google Scholar
  23. Kelkar, M., and Shibli, S., 1994, Description of reservoir properties using fractals, in Yarus, J. M., and Chambers, R. L., eds., Stochastic modeling and geostatistics: Am. Assoc. Petrol. Geol. Publication No. 3, Tulsa, OK, p. 261–272.Google Scholar
  24. King, P. R., 1989, The use of renormalization for calculating effective permeability: Transp. Porous Media, v. 4, p. 37–58.Google Scholar
  25. Krohn, C. E., 1988, Fractal measurements of sandstones, shales and carbonates: J. Geophys. Res., v. 93, p. 3297–3305.Google Scholar
  26. Li, D., Beckner, B., and Kumar, A., 1999, A new efficient averaging technique for scaleup of multimillion-cell geological models: Proceedings of the SPE Annual Technical Conference and Exhibition, Houston, TX, Soc. Petrol. Engr, paper 56554.Google Scholar
  27. Makse, H. A., Davies, G. W., Havli, S., Ivanov, P. C., King, P. R., and Stanley, H. E., 1996, Long-range correlations in permeability fluctuations in porous rocks: Phys. Rev. E, v. 54 no. 4, p. 3129– 3134.Google Scholar
  28. Manrique, J. F., Kasap, E., and Georgi, D. T., 1996, Effects of heterogeneity and anisotropy on probe measured permeabilities: Soc. Petrol. Engr., paper 27650.Google Scholar
  29. Matheron, G., 1968, Composition des permeabilities en milieu poreux heterogene: critique de la regle pour la ponderation geometrique: Revu. l’IFP, v. 23 (Feb.), p. 201–218.Google Scholar
  30. McGill, C., King, P., and Williams, J., 1993, Estimating effective permeability: A comparison of techniques, in Linville, B., Burchfield, T. E., and Wesson, T. C., eds., Reservoir characterization III. Penn Well Books, Tulsa, OK, p. 829–833.Google Scholar
  31. Mukhopadhyay, S., and Sahimi, M., 2000, Calculation of the effective permeabilities of field-scale porous media: Chem. Eng. Sci., v. 55, p. 4495–4513.Google Scholar
  32. Muller, J., and McCauley, J. L., 1992, Implication of fractal geometry for fluid flow properties of sedimentary rocks: Transp. Porous Media, v. 8, p. 133–147.Google Scholar
  33. Neuman, S. P., 1994, Generalized scaling of permeabilities: Validation and effect of support scale: Geophys. Res. Lett., v. 21, p. 349–352.Google Scholar
  34. Noetinger, B., 1994, The effective permeability of a heterogeneous porous medium: Transp. Porous Media, v. 15, p. 99–127.Google Scholar
  35. Noetinger, B., and Jacquin, C., 1991, Experimental tests of a simple permeability composition formula: Proceedings of the 1991 SPE Annual Technical Conference and Exhibition, Dallas, TX, Soc. Petrol. Engr, paper 22841.Google Scholar
  36. Perez, G., and Chopra, A. K., 1991, Evaluation of fractal models to describe reservoir heterogeneity and performance: Proceedings of the SPE 66th Annu. Tech. Conf., Dallas, TX, Soc. Petrol. Engr, paper 22694.Google Scholar
  37. Renard, Ph., and deMarsily, G., 1997, Calculating equivalent permeability: A review: Adv. Water Res., v. 20, p. 253–278.Google Scholar
  38. Richardson, J. G., Sangree, J. B., and Sneider, R. M., 1987, Permeability distributions in reservoirs: J. Petrol. Tech. (October), p. 1197–1199.Google Scholar
  39. Saupe, D., 1988, Algorithms for random fractals, in Saupe, D., and Peitgen, H. O., eds., The science of fractal images. Springer-Verlag, New York, p. 71–113.Google Scholar
  40. Tidwell, V. C., and Wilson, J. L., 1997, Laboratory method for investigating permeability upscaling: Water Res. Res., v. 33, p. 1607–1616.Google Scholar
  41. Tubman, K. M., and Crane, S. D., 1995, Vertical versus horizontal well log variability and application to fractal reservoir modeling, in Barton, C. C., and La Pointe, P. R. eds., Fractals in petroleum geology and earth processes: Plenum, New York, p. 279–293.Google Scholar
  42. Voss, R. F., 1985, Random fractal forgeries, in Earnshaw, R. A., ed., Fundamental algorithms in computer graphics. Springer-Verlag, Berlin, p. 805–835.Google Scholar

Copyright information

© International Association for Mathematical Geology 2005

Authors and Affiliations

  1. 1.Department of Civil and Environmental Engineering, School of Mining and PetroleumUniversity of AlbertaEdmontonCanada

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