Advertisement

Transport in Porous Media

, Volume 26, Issue 1, pp 1–23 | Cite as

Preferential Flow-Paths Detection for Heterogeneous Reservoirs Using a New Renormalization Technique

  • Yann Gautier
  • Benoit Nœtinger
Article

Abstract

We have devised a renormalization scheme which allows very fast determination of preferential flow-paths and of up-scaled permeabilities of 2D heterogeneous porous media. In the case of 2D log-normal and isotropically distributed permeability-fields, the resulting equivalent permeabilities are very close to the geometric mean, which is in good agreement with a rigorous result of Matheron. It is also found to work well for geostatistically anisotropic media when comparing the resulting equivalent permeabilities with a direct solution of the finite-difference equations. The method works exactly as King's does, although the renormalization scheme was modified to obtain tensorial equivalent permeabilities using periodic boundary conditions for the pressure gradient. To obtain an estimation of the local fluxes, the basic idea is that if at each renormalization iteration all the intermediate renormalized permeabilities are stored in memory, we are able to compute -- ad reversum -- an approximation of the small-scale flux map under a given macroscopic pressure gradient. The method is very rapid as it involves a number of calculations that vary linearly with the number of elementary grid blocks. In this sense, the renormalization algorithm can be viewed as a rapid approximate pressure solver. The ‘exact’ reference flow-rate map (for the finite-difference algorithm) was computed using a classical linear system inversion. It can be shown that the preferential flow paths are well detected by the approximate method, although errors may occur in the local flow direction.

upscaling renormalization permeability local flux heterogeneity. 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Charlaix, E., Guyon, E. and Roux, S.: 1987, Permeability of a random array of fractures of widely varying apertures, Transport in Porous Media 2, 31–43.Google Scholar
  2. Deutsh, C. and Journel, A.: 1992, GSLIB: Geostatistical Software Library and User's Guide, Oxford University Press, New York.Google Scholar
  3. Duquerroix, J. P. et al.: Influence of the permeability anisotropy ratio on large-scale properties of heterogeneous reservoirs, SPE 26612, 28th SPE Ann. Tech. Conf. Exh., Oct. 1993.Google Scholar
  4. Durlofsky, L. J.: 1991, Numerical calculation of equivalent grid block permeability tensors for heterogeneous porous media, Water Resour. Res. 27, 699–708.Google Scholar
  5. King, P.: 1989, The use of renormalization for calculating effective permeability, Transport in Porous Media 4, 37–58.Google Scholar
  6. Mantoglou, A. and Wilson, J. L.: 1982, The turning bands method for simulation of random fields using line generation by a spectral method, Water Resour. Res. 18(5), 1379–1394.Google Scholar
  7. Matheron, G.: 1966, Structure et composition des perméabilités, Revue Inst. Français Pétrole XXI(4), 564–582.Google Scholar
  8. Romeu, R. K. and Noetinger, B.: 1994. Calculation of internodal transmissivities in finite difference models of flows in heterogeneous porous media, Water Resour. Res. 31(4), 943–959.Google Scholar
  9. Stauffer, D.: 1979, Scaling properties of percolation clusters, Phys. Rep. 54, 1.Google Scholar
  10. Zimmermann, D. A. and Wilson, J. L.: Description of and user's manual for TUBA: a computer code for generating two-dimensionnal random fields via the turning bands method, SeaSoft, New Mexico, 1990.Google Scholar

Copyright information

© Kluwer Academic Publishers 1997

Authors and Affiliations

  • Yann Gautier
    • 1
  • Benoit Nœtinger
    • 1
  1. 1.Institut Fran¸ais du Pétrole, Helioparc Pau PyrénéesPauFrance

Personalised recommendations