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Penny-shaped fractures revisited

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Abstract

All methods of seismic characterization of fractured reservoirs are based on effective media theories that relate geometrical and material properties of fractures and surrounding rock to the effective stiffnesses. In exploration seismology, the first-order theory of Hudson is the most popular. It describes the effective model caused by the presence of a single set of thin, aligned vertical fractures in otherwise isotropic rock. This model is known to be transversely isotropic with a horizontal symmetry axis (HTI). Following the theory, one can invert the effective anisotropy for the crack density and type of fluid infill of fractures, the quantities of great importance for reservoir appraisal and management.

Here I compute effective media numerically using the finite element method. I deliberately construct models that contain a single set of vertical, ellipsoidal, non-intersecting and non-interconnected fractures to check validity of the first-order Hudson’s theory and establish the limits of its applicability. Contrary to conventional wisdom that Hudson’s results are accurate up to crack density e ≈ 0.1, I show that they consistently overestimate the magnitudes of all effective anisotropic coefficients ε(V), δ(V), and γ(V). Accuracy of theoretically derived anisotropy depends on the type of fluid infill and typically deteriorates as e grows. While the theory gives | ε(V)|, |δ(V)|, |γ(V)| and close to the upper bound of the corresponding numerically obtained values for randomly distributed liquid-filled fractures, theoretical predictions of ε(V), δ(V) are not supported by numerical computations when the cracks are dry. This happens primarily because the first-order Hudson’s theory makes no attempt to account for fracture interaction which contributes to the final result much stronger for gas- than for liquid-filled cracks. I find that Mori-Tanaka’s theory is superior to Hudson’s for all examined crack densities and both types of fluid infill.

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References

  1. Bakulin A., Grechka V. and Tsvankin I., 2000. Estimation of fracture parameters from reflection seismic data. Part I: HTI model due to a single fracture set. Geophysics, 65, 1788–1802.

  2. Christensen R.M., 1979. Mechanics of Composite Materials. Wiley, New York.

  3. Douma J., 1988. The effect of the aspect ratio on crack-induced anisotropy. Geophys. Prospect., 36, 614–632.

  4. Eshelby J.D., 1957. The determination of the elastic field of an ellipsoidal inclusion and related problems. Proc. Roy. Soc., A, 241, 376–396.

  5. Eshelby J.D., 1959. The elastic field outside an ellipsoidal inclusion. Proc. Roy. Soc., A, 252, 561–569.

  6. Femlab Reference Manual, 2003. Comsol, www.femlab.com.

  7. Grechka V., 2003. Effective media: A forward modeling view. Geophysics, 68, 2055–2062.

  8. Hudson J.A., 1980. Overall properties of a cracked solid. Math. Proc. Camb. Philos. Soc., 88, 371–384.

  9. Hudson J.A., 1981. Wave speeds and attenuation of elastic waves in material containing cracks. Geophys. J. Roy. Astr. Soc., 64, 133–150.

  10. Hudson J.A., 1991. Overall properties of heterogeneous material. Geophys. J. Int., 107, 505–511.

  11. Hudson J.A., 1994. Overall properties of a material with inclusions or cavities. Geophys. J. Int., 117, 555–561.

  12. Hudson J.A., Pointer T. and Liu E., 2001. Effective-medium theories for fluid-saturated materials with aligned cracks. Geophys. Prospect., 49, 509–522.

  13. Jakobsen M. and Hudson J.A., 2003. Visco-elastic waves in rock-like composites. Stud. Geophys. Geod., 47, 793–826.

  14. Jakobsen M., Hudson J.A. and Johansen T.A., 2003. T-matrix approach to shale acoustics. Geophys. J. Int., 154, 533–558.

  15. Kachanov M., 1993. Elastic solids with many cracks and related problems. Advances in Applied Mechanics, 30, 259–445.

  16. Kachanov M., Shafiro B. and Tsurkov I., 2003. Handbook of Elasticity Solutions. Kluwer Academic Publishers, Dordrecht, The Netherlands.

  17. Kanit T., Forest S., Galliet I., Mounoury V. and Jeulin D., 2003. Determination of the size of the representative volume element for random composites: statistical and numerical approach: Int. J. Solids Struct., 40, 3647–3679.

  18. Levin V., Markov M. and Kanaun S., 2004. Effective field method for seismic properties of cracked rocks. J. Geophys. Res., 104, B08202.

  19. Liu E., Hudson J.A. and Pointer T., 2000. Equivalent medium representation of fractured rock. J. Geophys. Res., 105(B2), 2981–3000.

  20. Mori T. and Tanaka K., 1973. Average stress in matrix and average energy of materials with misfitting inclusions. Act. Metall., 21, 571–574.

  21. Mura T., 1987. Micromechanics of Defects in Solids. Martinus Nijhoff Publishers, Leiden, The Netherlands

  22. Ponte Castaneda P. and Willis J.R. 1995. The effect of spatial distribution on the effective behaviour of composite materials and cracked media. J. Mech. Phys. Solids, 43, 1919–1951.

  23. Ruger A., 1997. P-wave reflection coefficients for transversely isotropic models with vertical and horizontal axis of symmetry. Geophysics, 62, 713–722.

  24. Russel W.B., 1973. On the effective moduli of composite materials: Effect of fiber length and geometry at dilute concentrations. J. Appl. Math. Phys. (ZAMP), 24, 581–600.

  25. Saenger E.H., Kruger O.S. and Shapiro S.A., 2004. Effective elastic properties of randomly fractured soils: 3D numerical experiments. Geophys. Prospect., 52, 183–195.

  26. Sayers C.M. and Kachanov M., 1991. A simple technique for finding effective elastic constants of cracked solids for arbitrary crack orientation statistics. Int. J. Solids Struct., 27, 671–680.

  27. Schoenberg M., 1980. Elastic wave behavior across linear slip interfaces. J. Acoust. Soc. Am., 68, 1516–1521.

  28. Schoenberg M. and Douma J., 1988. Elastic wave propagation in media with parallel fractures and aligned cracks. Geophys. Prospect., 36, 571–590.

  29. Tsvankin I., 1997. Reflection moveout and parameter estimation for horizontal transverse isotropy. Geophysics, 62, 614–629.

  30. Tucker C.L. and Liang E., 1999. Stiffness predictions for unidirectional short-fiber composites: Review and evaluation. Compos. Sci. Technol., 59, 655–671.

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Additional information

The paper was presented at the 11th International Workshop on Seismic Anisotropy (11IWSA) held in St. John’s, Canada in 2004.

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Grechka, V. Penny-shaped fractures revisited. Stud Geophys Geod 49, 365–381 (2005). https://doi.org/10.1007/s11200-005-0015-3

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Keywords

  • effective media
  • fractures
  • single set
  • horizontal transverse isotropy