Skip to main content
SpringerLink
Log in

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

  • Published:
Pure and Applied Geophysics Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  • Ass’ad, J. M., Tatham, R. H., & McDonald, J. A. (1992). A physical model study of microcrack-induced anisotropy. Geophysics, 57(12), 1562–1570.

    Article  Google Scholar 

  • 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 

  • 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.

    Article  Google Scholar 

  • Bear, J. (1972). Dynamics of fluid in porous media. New York: Elsevier.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • Biot, M. A., & Willis, D. G. (1957). The elastic coefficients of the theory of consolidation. Journal of Applied Mechanics, 24, 594–601.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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 

  • Chapman, M., Zatsepin, S. V., & Crampin, S. (2002). Derivation of a microstructural poroelastic model. Geophysical Journal International, 151(2), 427–451.

    Article  Google Scholar 

  • Chapman, M. (2003). Frequency-dependent anisotropy due to meso-scale fractures in the presence of equant porosity. Geophysical Prospecting, 51(5), 369–379.

    Article  Google Scholar 

  • Crampin, S. (1985). Evaluation of anisotropy by shear-wave splitting. Geophysics, 50(1), 142–152.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Gassmann, F. (1951). Über die Elastizität poröser Medien. Vierteljahrsschrder der Naturforschenden Gesselschaft in Zürich, 96, 1–23.

    Google Scholar 

  • Grecheka, V., & Kachanov, M. (2006). Effective elasticity of rocks with closely spaced and intersecting cracks. Geophysics, 71(3), D85–D91.

    Article  Google Scholar 

  • 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.

  • 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.

    Article  Google Scholar 

  • Guéguen, Y., & Sarout, J. (2011). Characteristics of anisotropy and dispersion in cracked medium. Tectonophysics, 503(1), 165–172.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Hudson, J. A., Pointer, T., & Liu, E. (2001). Effective-medium theories for fluid-saturated materials with aligned cracks. Geophysical Prospecting, 49(5), 509–522.

    Article  Google Scholar 

  • Jakobsen, M., & Chapman, M. (2009). Unified theory of global flow and squirt flow in cracked porous media. Geophysics, 74(2), WA65–WA76.

    Article  Google Scholar 

  • Jakobsen, M., Hudson, J. A., & Johansen, T. A. (2003). T-matrix approach to shale acoustics. Geophysical Journal International, 154(2), 533–558.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Liu, E., Hudson, J. A., & Pointer, T. (2000). Equivalent medium representation of fractured rock. Journal of Geophysical Research: Solid Earth, 105(B2), 2981–3000.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Mavko, G., Mukerji, T., & Dvorkin, J. (1998). The Rock physics handbook: Tools for seismic analysis of porous media. Cambridge: Cambridge University Press.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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 

  • Sayers, C. M., & Kachanov, M. (1995). Microcrack-induced elastic wave anisotropy of brittle rocks. Journal of Geophysical Research: Solid Earth, 100(B3), 4149–4156.

    Article  Google Scholar 

  • Schoenberg, M. (1980). Elastic wave behavior across linear slip interfaces. The Journal of the Acoustical Society of America, 68(5), 1516–1521.

    Article  Google Scholar 

  • Schoenberg, M., & Sayers, C. M. (1995). Seismic anisotropy of fractured rock. Geophysics, 60(1), 204–211.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Thomsen, L. (1995). Elastic anisotropy due to aligned cracks in porous rock. Geophysical Prospecting, 43(6), 805–829.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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 

  • Wang, H. F. (2000). Theory of linear poroelasticity with application to geomechanics and hydrogeology. USA: Princeton University Press.

    Google Scholar 

  • White, J. E. (1983). Underground sound: Application of seismic waves (Vol. 253). Amsterdam: Elsevier.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • Winterstein, D. F. (1990). Velocity anisotropy terminology for geophysicists. Geophysics, 55(8), 1070–1088.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

Download references

Acknowledgements

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.

Author information

Authors and Affiliations

  1. Center of Rock Mechanics and Geohazards, Shaoxing University, Shaoxing, 312000, China

    Ding Wang

  2. State Key Laboratory of Petroleum Resource and Prospecting, China University of Petroleum, Beijing, 102249, China

    Pin-bo Ding

  3. School of Earth Sciences and Engineering, Hohai University, Nanjing, 211100, China

    Jing Ba

Authors
  1. Ding Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pin-bo Ding
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jing Ba
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jing Ba.

Appendix

Appendix

1.1 Effective Elastic Moduli of a Fractured Medium

By taking \( \varphi \) as the angle between the direction normal to paralleled fractures and the coordinate system \( Y_{1} \)-axis (as shown in Fig. 7), the matrix of fracture compliance (Eq. 10) can be given in the observing coordinate system based on the Band’s conversion (Liu et al. 2006; Winterstein 1990). When \( \varphi = 0 {{^\circ }} \), the off-diagonal terms of the compliance matrix are zero:

$$ \overline{{S^{f} }} = \left[ {\begin{array}{*{20}c} {\overline{{S_{11}^{f} }} } & {\overline{{S_{12}^{f} }} } & 0 & 0 & 0 & {\overline{{S_{16}^{f} }} } \\ {} & {\overline{{S_{22}^{f} }} } & 0 & 0 & 0 & {\overline{{S_{26}^{f} }} } \\ {} & {} & 0 & 0 & 0 & 0 \\ {} & {} & {} & {\overline{{S_{44}^{f} }} } & {\overline{{S_{45}^{f} }} } & 0 \\ {} & {} & {} & {} & {\overline{{S_{55}^{f} }} } & 0 \\ {} & {} & {} & {} & {} & {\overline{{S_{66}^{f} }} } \\ \end{array} } \right], $$
(10)

where

$$ \overline{{S_{11}^{f} }} = \frac{{3Z_{n} + Z_{t} }}{8} + \frac{{Z_{n} \cos (2\varphi )}}{2} + \frac{{(Z_{n} - Z_{t} )\cos (4\varphi )}}{8} $$
(11)
$$ \overline{{S_{22}^{f} }} = \frac{{3Z_{n} + Z_{t} }}{8} - \frac{{Z_{n} \cos (2\varphi )}}{2} + \frac{{(Z_{n} - Z_{t} )\cos (4\varphi )}}{8} $$
(12)
$$ \overline{{S_{44}^{f} }} = \frac{{Z_{t} [1 - \cos (2\varphi )]}}{2} $$
(13)
$$ \overline{{S_{55}^{f} }} = \frac{{Z_{t} [1 + \cos (2\varphi )]}}{2} $$
(14)
$$ \overline{{S_{66}^{f} }} = \frac{{Z_{n} + Z_{t} }}{2} - \frac{{(Z_{n} - Z_{t} )\cos (4\varphi )}}{2} $$
(15)
$$ \overline{{S_{12}^{f} }} = \frac{{(Z_{n} - Z_{t} )[1 - \cos (4\varphi )]}}{8} $$
(16)
$$ \overline{{S_{16}^{f} }} = \frac{{Z_{n} \sin (2\varphi )}}{2} + \frac{{(Z_{n} - Z_{t} )\sin (4\varphi )}}{4} $$
(17)
$$ \overline{{S_{26}^{f} }} = \frac{{Z_{n} \sin (2\varphi )}}{2} - \frac{{(Z_{n} - Z_{t} )\sin (4\varphi )}}{8} $$
(18)
$$ \overline{{S_{45}^{f} }} = \frac{{Z_{t} \sin (2\varphi )}}{2}. $$
(19)
Fig. 7
figure 7

Diagram of paralleled fractures in the observing coordinate system

Full size image

The host porous medium is isotropic and homogeneous, and the corresponding compliance is given by the following:

$$ S_{0} = \left[ {\begin{array}{*{20}c} {\frac{1}{E}} & {\frac{ - \nu }{E}} & {\frac{ - \nu }{E}} & 0 & 0 & 0 \\ {} & {\frac{1}{E}} & {\frac{ - \nu }{E}} & 0 & 0 & 0 \\ {} & {} & {\frac{1}{E}} & 0 & 0 & 0 \\ {} & {} & {} & {\frac{1}{\mu }} & 0 & 0 \\ {} & {} & {} & {} & {\frac{1}{\mu }} & 0 \\ {} & {} & {} & {} & {} & {\frac{1}{\mu }} \\ \end{array} } \right], $$
(20)

\( E \), \( \nu \), and \( \mu \) are the Young modulus, Poisson ratio, and shear modulus of the saturated host frame, according to the Gassmann’s equations (Gassmann 1951), respectively. Then, the effective compliance of the fractured medium is derived as follows:

$$ S = S^{0} + \frac{{\overline{{S^{f} }} }}{H}. $$
(21)

The effective elastic stiffness matrix of a fractured medium is as follows:

$$ C = S^{ - 1} , $$
(22)

where

$$ C_{11} = \frac{{S_{22} S_{33} S_{66} - S_{26}^{2} S_{33} - S_{23}^{2} S_{66} }}{Q} $$
(23)
$$ C_{12} = \frac{{S_{13} S_{23} S_{66} + S_{16} S_{26} S_{33} - S_{12} S_{33} S_{66} }}{Q} $$
(24)
$$ C_{13} = \frac{{S_{26}^{2} S_{13} + S_{12} S_{23} S_{66} - S_{13} S_{22} S_{66} - S_{16} S_{23} S_{26} }}{Q} $$
(25)
$$ C_{16} = \frac{{S_{23}^{2} S_{16} + S_{12} S_{26} S_{33} - S_{13} S_{23} S_{26} - S_{16} S_{22} S_{33} }}{Q} $$
(26)
$$ C_{22} = \frac{{S_{11} S_{33} S_{66} - S_{16}^{2} S_{33} - S_{13}^{2} S_{66} }}{Q} $$
(27)
$$ C_{23} = \frac{{S_{16}^{2} S_{23} + S_{12} S_{13} S_{66} - S_{11} S_{23} S_{66} - S_{13} S_{16} S_{26} }}{Q} $$
(28)
$$ C_{26} = \frac{{S_{13}^{2} S_{26} + S_{12} S_{16} S_{33} - S_{11} S_{26} S_{33} - S_{13} S_{16} S_{23} }}{Q} $$
(29)
$$ C_{33} = \frac{{S_{11} S_{22} S_{66} + 2S_{12} S_{16} S_{26} - S_{12}^{2} S_{66} - S_{16}^{2} S_{22} - S_{26}^{2} S_{11} }}{Q} $$
(30)
$$ C_{36} = \frac{{S_{11} S_{23} S_{26} + S_{13} S_{16} S_{22} - S_{12} S_{13} S_{26} - S_{12} S_{16} S_{23} }}{Q} $$
(31)
$$ C_{44} = \frac{{S_{55} }}{{S_{44} S_{55} - S_{45}^{2} }} $$
(32)
$$ C_{45} = \frac{{ - S_{45} }}{{S_{44} S_{55} - S_{45}^{2} }} $$
(33)
$$ C_{55} = \frac{{S_{44} }}{{S_{44} S_{55} - S_{45}^{2} }} $$
(34)
$$ C_{66} = \frac{{S_{11} S_{22} S_{33} + 2S_{12} S_{13} S_{23} - S_{12}^{2} S_{33} - S_{13}^{2} S_{22} - S_{23}^{2} S_{11} }}{Q} $$
(35)
$$ \begin{aligned} Q & = 2(S_{12} S_{16} S_{26} S_{33} + S_{12} S_{13} S_{23} S_{66} - S_{13} S_{16} S_{23} S_{26} ) + S_{11} S_{22} S_{33} S_{66} + S_{13}^{2} S_{26}^{2} \\ & \quad + S_{16}^{2} S_{23}^{2} - S_{12}^{2} S_{33} S_{66} - S_{16}^{2} S_{22} S_{33} - S_{23}^{2} S_{11} S_{66} - S_{13}^{2} S_{22} S_{66} - S_{26}^{2} S_{11} S_{13} . \\ \end{aligned} $$
(36)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, D., Ding, Pb. & Ba, J. Analysis of Dynamic Fracture Compliance Based on Poroelastic Theory. Part II: Results of Numerical and Experimental Tests. Pure Appl. Geophys. 175, 2987–3001 (2018). https://doi.org/10.1007/s00024-018-1818-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00024-018-1818-9

Keywords

Search

Navigation