Pure and Applied Geophysics

, Volume 175, Issue 8, pp 2987–3001 | Cite as

Analysis of Dynamic Fracture Compliance Based on Poroelastic Theory. Part II: Results of Numerical and Experimental Tests

  • Ding Wang
  • Pin-bo Ding
  • Jing Ba


In Part I, a dynamic fracture compliance model (DFCM) was derived based on the poroelastic theory. The normal compliance of fractures is frequency-dependent and closely associated with the connectivity of porous media. In this paper, we first compare the DFCM with previous fractured media theories in the literature in a full frequency range. Furthermore, experimental tests are performed on synthetic rock specimens, and the DFCM is compared with the experimental data in the ultrasonic frequency band. Synthetic rock specimens saturated with water have more realistic mineral compositions and pore structures relative to previous works in comparison with natural reservoir rocks. The fracture/pore geometrical and physical parameters can be controlled to replicate approximately those of natural rocks. P- and S-wave anisotropy characteristics with different fracture and pore properties are calculated and numerical results are compared with experimental data. Although the measurement frequency is relatively high, the results of DFCM are appropriate for explaining the experimental data. The characteristic frequency of fluid pressure equilibration calculated based on the specimen parameters is not substantially less than the measurement frequency. In the dynamic fracture model, the wave-induced fluid flow behavior is an important factor for the fracture–wave interaction process, which differs from the models at the high-frequency limits, for instance, Hudson’s un-relaxed model.


Fracture compliance poroelastic theory synthetic fractured rock ultrasonic measurement wave-induced fluid flow 



This work is financially supported by the ‘‘Distinguished Professor Program of Jiangsu Province, China’’, the open fund of the State Key Laboratory of the Institute of Geology and Geophysics, CAS (SKLGED2017-5-2E), and the open fund of SINOPEC Key Laboratory of Geophysics. The authors thank Joel Sarout and Yves Gueguen for the helpful comments.


  1. Ass’ad, J. M., Tatham, R. H., & McDonald, J. A. (1992). A physical model study of microcrack-induced anisotropy. Geophysics, 57(12), 1562–1570.CrossRefGoogle Scholar
  2. Ba, J., Xu, W., Fu, L. Y., Carcione, J. M., & Zhang, L. (2017). Rock anelasticity due to patchy saturation and fabric heterogeneity: A double double-porosity model of wave propagation. Journal of Geophysical Research, 122(3), 1949–1976.Google Scholar
  3. Ba, J., Zhao, J., Carcione, J. M., & Huang, X. X. (2016). Compressional wave dispersion due to rock matrix stiffening by clay squirt flow. Geophysical Research Letters, 43(12), 6186–6195.CrossRefGoogle Scholar
  4. Bear, J. (1972). Dynamics of fluid in porous media. New York: Elsevier.Google Scholar
  5. Biot, M. A. (1956). Theory of propagation of elastic waves in a fluid-saturated porous solid. I. Low-frequency range. The Journal of the Acoustical Society of America, 28(2), 168–178.CrossRefGoogle Scholar
  6. Biot, M. A., & Willis, D. G. (1957). The elastic coefficients of the theory of consolidation. Journal of Applied Mechanics, 24, 594–601.Google Scholar
  7. Brajanovski, M., Gurevich, B., & Schoenberg, M. (2005). A model for P-wave attenuation and dispersion in a porous medium permeated by aligned fractures. Geophysical Journal International, 163(1), 372–384.CrossRefGoogle Scholar
  8. Carcione, J. M., Picotti, S., & Santos, J. E. (2012). Numerical experiments of fracture-induced velocity and attenuation anisotropy. Geophysical Journal International, 191(3), 1179–1191.Google Scholar
  9. Chapman, M., Zatsepin, S. V., & Crampin, S. (2002). Derivation of a microstructural poroelastic model. Geophysical Journal International, 151(2), 427–451.CrossRefGoogle Scholar
  10. Chapman, M. (2003). Frequency-dependent anisotropy due to meso-scale fractures in the presence of equant porosity. Geophysical Prospecting, 51(5), 369–379.CrossRefGoogle Scholar
  11. Crampin, S. (1985). Evaluation of anisotropy by shear-wave splitting. Geophysics, 50(1), 142–152.CrossRefGoogle Scholar
  12. de Figueiredo, J., Schleicher, J., Stewart, R. R., Dayur, N., Omoboya, B., Wiley, R., et al. (2013). Shear wave anisotropy from aligned inclusions: ultrasonic frequency dependence of velocity and attenuation. Geophysical Journal International, 193(1), 475–488.CrossRefGoogle Scholar
  13. Ding, P., Di, B., Wang, D., Wei, J., & Li, X. (2014). P and S wave anisotropy in fractured media: Experimental research using synthetic samples. Journal of Applied Geophysics, 109, 1–6.CrossRefGoogle Scholar
  14. Ding, P., Di, B., Wang, D., Wei, J., & Li, X. (2017). Measurements of seismic anisotropy in synthetic rocks with controlled crack geometry and different crack densities. Pure & Applied Geophysics, 174(5), 1–16.CrossRefGoogle Scholar
  15. Gassmann, F. (1951). Über die Elastizität poröser Medien. Vierteljahrsschrder der Naturforschenden Gesselschaft in Zürich, 96, 1–23.Google Scholar
  16. Grecheka, V., & Kachanov, M. (2006). Effective elasticity of rocks with closely spaced and intersecting cracks. Geophysics, 71(3), D85–D91.CrossRefGoogle Scholar
  17. Guéguen, Y., & Kachanov, M., (2011). Effective elastic properties of cracked and porous rocks. In Mechanics of crustal rocks, CISM Courses and Lectures (Vol. 533). Berlin: Springer.Google Scholar
  18. Guéguen, Y., & Sarout, J. (2009). Crack-induced anisotropy in crustal rocks: predicted dry and fluid-saturated Thomsen’s parameters. Physics of the Earth and Planetary Interiors, 172(1), 116–124.CrossRefGoogle Scholar
  19. Guéguen, Y., & Sarout, J. (2011). Characteristics of anisotropy and dispersion in cracked medium. Tectonophysics, 503(1), 165–172.CrossRefGoogle Scholar
  20. Hall, S. A., & Kendall, J. M. (2003). Fracture characterization at Valhall: Application of P-wave amplitude variation with offset and azimuth (AVOA) analysis to a 3D ocean-bottom data set. Geophysics, 68(4), 1150–1160.CrossRefGoogle Scholar
  21. Hudson, J. A. (1981). Wave speeds and attenuation of elastic waves in material containing cracks. Geophysical Journal of the Royal Astronomical Society, 64, 133–150.CrossRefGoogle Scholar
  22. Hudson, J. A., Liu, E., & Crampin, S. (1996). The mechanical properties of materials with interconnected cracks and pores. Geophysical Journal International, 124(1), 105–112.CrossRefGoogle Scholar
  23. Hudson, J. A., Pointer, T., & Liu, E. (2001). Effective-medium theories for fluid-saturated materials with aligned cracks. Geophysical Prospecting, 49(5), 509–522.CrossRefGoogle Scholar
  24. Jakobsen, M., & Chapman, M. (2009). Unified theory of global flow and squirt flow in cracked porous media. Geophysics, 74(2), WA65–WA76.CrossRefGoogle Scholar
  25. Jakobsen, M., Hudson, J. A., & Johansen, T. A. (2003). T-matrix approach to shale acoustics. Geophysical Journal International, 154(2), 533–558.CrossRefGoogle Scholar
  26. Liu, E., Chapman, M., Zhang, Z., & Queen, J. H. (2006). Frequency-dependent anisotropy: Effects of multiple fracture sets on shear-wave polarizations. Wave Motion, 44(1), 44–57.CrossRefGoogle Scholar
  27. Liu, E., Hudson, J. A., & Pointer, T. (2000). Equivalent medium representation of fractured rock. Journal of Geophysical Research: Solid Earth, 105(B2), 2981–3000.CrossRefGoogle Scholar
  28. Maultzsch, S., Chapman, M., Liu, E., & Li, X. Y. (2007). Modelling and analysis of attenuation anisotropy in multi-azimuth VSP data from the Clair field. Geophysical Prospecting, 55(5), 627–642.CrossRefGoogle Scholar
  29. Mavko, G., Mukerji, T., & Dvorkin, J. (1998). The Rock physics handbook: Tools for seismic analysis of porous media. Cambridge: Cambridge University Press.Google Scholar
  30. Müller, T. M., Gurevich, B., & Lebedev, M. (2010). Seismic wave attenuation and dispersion resulting from wave-induced flow in porous rocks—A review. Geophysics, 75(5), 75A147–75A164.CrossRefGoogle Scholar
  31. Norris, A. N. (1993). Low-frequency dispersion and attenuation in partially saturated rocks. The Journal of the Acoustical Society of America, 94(1), 359–370.CrossRefGoogle Scholar
  32. Pride, S. R., & Berryman, J. G. (2003). Linear dynamics of double-porosity dual-permeability materials. I. Governing equations and acoustic attenuation. Physical Review E, 68(3), 036603.CrossRefGoogle Scholar
  33. Rathore, J. S., Fjaer, E., Holt, R. M., & Renlie, L. (1995). P-and S-wave anisotropy of a synthetic sandstone with controlled crack geometry. Geophysical Prospecting, 43(6), 711–728.CrossRefGoogle Scholar
  34. Sarout, J. (2012). Impact of pore space topology on permeability, cut-off frequencies and validity of wave propagation theories. Geophysical Journal International, 189(1), 481–492.CrossRefGoogle Scholar
  35. Sarout, J., Cazes, E., Delle Piane, C., Arena, A., & Esteban, L. (2017). Stress-dependent permeability and wave dispersion in tight cracked rocks: Experimental validation of simple effective medium models. Journal of Geophysical Research: Solid Earth, 122, 6180–6201.Google Scholar
  36. Sayers, C. M., & Kachanov, M. (1995). Microcrack-induced elastic wave anisotropy of brittle rocks. Journal of Geophysical Research: Solid Earth, 100(B3), 4149–4156.CrossRefGoogle Scholar
  37. Schoenberg, M. (1980). Elastic wave behavior across linear slip interfaces. The Journal of the Acoustical Society of America, 68(5), 1516–1521.CrossRefGoogle Scholar
  38. Schoenberg, M., & Sayers, C. M. (1995). Seismic anisotropy of fractured rock. Geophysics, 60(1), 204–211.CrossRefGoogle Scholar
  39. Smyshlyaev, V. P., Willis, J. R., & Sabina, F. J. (1993). Self-consistent analysis of waves in a matrix-inclusion composite—III. A matrix containing cracks. Journal of the Mechanics and Physics of Solids, 41(12), 1809–1824.CrossRefGoogle Scholar
  40. Thomsen, L. (1995). Elastic anisotropy due to aligned cracks in porous rock. Geophysical Prospecting, 43(6), 805–829.CrossRefGoogle Scholar
  41. Tillotson, P., Chapman, M., Sothcott, J., Best, A. I., & Li, X. Y. (2014). Pore fluid viscosity effects on P-and S-wave anisotropy in synthetic silica-cemented sandstone with aligned fractures. Geophysical Prospecting, 62(6), 1238–1252.CrossRefGoogle Scholar
  42. Tillotson, P., Sothcott, J., Best, A. I., Chapman, M., & Li, X. Y. (2012). Experimental verification of the fracture density and shear-wave splitting relationship using synthetic silica cemented sandstones with a controlled fracture geometry. Geophysical Prospecting, 60(3), 516–525.CrossRefGoogle Scholar
  43. Verdon, J. P., Angus, D. A., Michael, K. J., & Hall, S. A. (2008). The effect of microstructure and nonlinear stress on anisotropic seismic velocities. Geophysics, 73(4), D41–D51.CrossRefGoogle Scholar
  44. Verdon, J. P., & Kendall, J. (2011). Detection of multiple fracture sets using observations of shear-wave splitting in microseismic data. Geophysical Prospecting, 59(4), 593–608.CrossRefGoogle Scholar
  45. Wang, D., Qu, S. L., Ding, P. B., & Zhao, Q. (2017). Analysis of dynamic fracture compliance based on poroelastic theory. Part I: model formulation and analytical expressions. Pure and Applied Geophysics, 174(5), 2103–2120.CrossRefGoogle Scholar
  46. Wang, D., Wang, L., & Ding, P. (2016). The effects of fracture permeability on acoustic wave propagation in the porous media: A microscopic perspective. Ultrasonics, 70, 266–274.CrossRefGoogle Scholar
  47. Wang, D., & Zhang, M. G. (2014). Elastic wave propagation characteristics under anisotropic squirt-flow condition. Acta Physica Sinica, 63(6), 69101. (in Chinese with English abstract).Google Scholar
  48. Wang, H. F. (2000). Theory of linear poroelasticity with application to geomechanics and hydrogeology. USA: Princeton University Press.Google Scholar
  49. White, J. E. (1983). Underground sound: Application of seismic waves (Vol. 253). Amsterdam: Elsevier.Google Scholar
  50. White, J. E., Mihailova, N., & Lyakhovitsky, F. (1975). Low-frequency seismic waves in fluid-saturated layered rocks. The Journal of the Acoustical Society of America, 57(S1), S30–S30.CrossRefGoogle Scholar
  51. Winterstein, D. F. (1990). Velocity anisotropy terminology for geophysicists. Geophysics, 55(8), 1070–1088.CrossRefGoogle Scholar
  52. Zatsepin, S. V., & Crampin, S. (1997). Modelling the compliance of crustal rock—I. Response of shear-wave splitting to differential stress. Geophysical Journal International, 129(3), 477–494.CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Center of Rock Mechanics and GeohazardsShaoxing UniversityShaoxingChina
  2. 2.State Key Laboratory of Petroleum Resource and ProspectingChina University of PetroleumBeijingChina
  3. 3.School of Earth Sciences and EngineeringHohai UniversityNanjingChina

Personalised recommendations