International Journal of Rock Mechanics and Mining Sciences
Transition from the displacement discontinuity limit to the resonant scattering regime for fracture interface waves
Introduction
Fractures in solid media represent mechanical discontinuities that strongly affect the propagation of elastic waves either across or along the fracture plane. Signatures of the fracture properties, especially fracture specific stiffness, appear in the amplitudes, phases and velocities of the elastic waves. These waves therefore become probes of the fracture, with potential benefits for predicting fracture mechanical stability and fluid flow through the fracture [22]. One of the physical models used to analyze the seismic properties of fractures is the displacement discontinuity model [16], [17], also known as the linear slip interface model [12], [26], or incomplete or imperfect interface model [23]. The displacement discontinuity model assumes that the stresses across a fracture are continuous but that the displacements are not. The discontinuity in displacement is inversely proportional to the quantity called specific stiffness of the fracture.
Specific stiffness plays a purely phenomenological role in the displacement discontinuity model as a proportionality factor that relates the displacement discontinuity to the elastic stress of the propagating wave. However, there is considerable importance in connecting the fracture specific stiffness to the physical properties of the fracture, such as the geometry of the voids and asperities, because ultimately it is the geometry of the voids and asperities that determine the hydraulic–mechanical properties of interest for a fracture [4].
In the effort to connect stiffness with physical fracture properties, quasi-static stiffness of a fracture can be defined by relating far-field fracture displacement with the applied stress across a fracture. The specific stiffness of the fracture in this case depends on the number and distribution of the asperities (points of contact between the two fracture surfaces), [1], [2], [7], [10], [11]. Alternatively, in the displacement discontinuity model, dynamic stiffness can be viewed to be caused by finely distributed springs that couple the two half-spaces on either side of the fracture.
One of the outstanding problems in the study of fracture stiffness and its relation to fracture geometry is understanding the connection between quasi-static mechanical stiffness and the dynamic stiffness extracted from displacement discontinuity theory. Dynamic and quasi-static stiffnesses are not equal [16], [18], with dynamic stiffness always exceeding the magnitude of quasi-static stiffness. Furthermore, even within the dynamic stiffness itself, there is often a frequency dependence in which the stiffness increases with increasing frequency [27]. The mechanisms for this frequency-dependent stiffness are not fully understood, although part of the frequency dependence may not be dynamical in nature, but would be a simple consequence of wavefront averaging over inhomogeneous distributions of local stiffnesses along a fracture [19].
This issue of inhomogeneous distributions of stiffnesses along a fracture interface raises many interesting questions about what seismic propagation experiments actually measure. For instance, the concept of finely distributed asperities, represented as springs coupling the two faces of a fracture, is expected to be valid only if the spacing between the asperities is much smaller than a wavelength. However, if the distribution of the asperities is strongly inhomogeneous, then even if the individual asperity separations satisfy this condition, the correlation length describing the fluctuations in the stiffnesses may be much larger than the average asperity spacing. In addition, short pulses of seismic energy are broad-band waves that contain a spectrum of wavelengths, some of which may be longer than the correlation length, but others may not. This situation would lead to complicated behavior as low frequencies probe large areas of asperities and high frequencies probe smaller areas of asperities, and are therefore each subject to different average values. Added to this complexity is the possibility for resonant scattering when the fluctuation lengths become comparable to a wavelength, perhaps calling into question the validity of the displacement discontinuity theory. For instance, periodic asperity spacings that are on the order of a wavelength can produce resonant (Bragg) reflections that show up clearly in time-frequency (wavelet) analyses of the received waveforms.
In this paper, we study the physical validity of the displacement discontinuity model as it relates to the question of asperity spacing, and we explicitly observe the transition from the displacement discontinuity limit to the resonant scattering regime. We study the seismic response of elastic interface waves for all three polarizations (Sv, Sh and P-waves) propagating along synthetic fractures. The fractures are engineered to give increasing asperity separation, and we monitor the spectral response of the elastic waves to the changing separation. We first describe the physical properties of fracture interface waves into which energy from Sv and P-waves are partitioned, then describe the experimental set-up for studying the effect of asperity spacing on these interface waves. The waveforms are analyzed using the Nolte–Hilbert wavelet that balances the time-bandwidth product for time-frequency analysis without violating the admissibility condition that has previously impeded the use of the Morlet wavelet transform. Clear evidence for both fracture interface waves as well as resonantly scattered Rayleigh waves is observed in the wavelet transforms, and the wavelet transforms are used to extract the fracture wave dispersion. The paper finishes by revisiting the conditions for validity of the displacement discontinuity model in view of the new results presented here.
Section snippets
Fracture interface waves
Fracture interface waves have been demonstrated to propagate along natural and synthetic fractures at the laboratory scale [5], [6], [14], [20], [24], [25]. Fracture interface waves can be viewed as coupled Rayleigh waves in which Rayleigh waves travel along each surface of the fracture and are coupled through the points of contact between the two surfaces. The analytic and numerical solutions have shown that this coupling produces two dispersive waves that exhibit both longitudinal and
Experimental set-up
Shear (S) wave and compressional (P) wave piezoelectric transducers were used to send and receive waves propagated along the synthetic fracture. Two polarizations of the S-wave were used, one with the transverse particle motion oriented perpendicular to the interface (Sv) and the other transverse particle motion parallel to the interface (Sh). The P-wave has longitudinal particle motion oriented parallel to the direction of propagation along the interface. The transducers with a central
Seismic waveforms
Fig. 2a shows the normalized received waveforms for Sv waves propagated parallel to the asperities and Fig. 2b shows the normalized received waveforms for Sv waves propagated perpendicular to the asperities along the synthetic fractures with 1×1 mm, 2×2 mm, 4×4 mm, and 8×8 mm asperity periodicities (Table 1). Each data set also includes the propagated waveform through the bulk sample. The bulk S-wave is observed to arrive at 96 μs in the signals recorded for the intact sample, as well as for
Nolte–Hilbert wavelet
All time-varying signals can be viewed as combinations of waveforms that are localized in time and frequency. Each frequency component carries dynamic information about the system under study, while each time component carries kinetic information. Together, they contain the information necessary to extract dispersive properties, as well as the ability to identify distinctly different eigenmodes that co-propagate. The challenge is to find an optimal decomposition into time-frequency space to
Wavelet analysis
To quantify the frequency components of the signals, the Nolte–Hilbert wavelet was applied to the time-domain signals of Fig. 2, Fig. 3, Fig. 4 to obtain the time-frequency information from the received signals. In Fig. 6a and 6b, the wavelet transforms for synthetic fractures 1×1, 2×2, 4×4, and 8×8 are shown for Sv waves propagating parallel and perpendicular, respectively, to the lead strips that define the fracture asperities. All of the wavelet transforms from the synthetic fracture show a
Interface-wave dispersion
The use of the time-frequency balanced Nolte–Hilbert wavelet makes it optimally suited to extract wave dispersion. This analysis was performed on each of the wavelets of Fig. 6, Fig. 7, Fig. 8. The results are shown in Fig. 9, Fig. 10, Fig. 11. For the Sv waves, the dispersion shows the traditional frequency-dependent signature of the Rayleigh-mode interface wave [17], [21], with higher velocities at low frequencies trending towards the Rayleigh velocity at high frequencies. For both
Summary and conclusions
In this paper, we have specifically tracked the evolution of the seismic wave behavior as a function of asperity spacing for waves of all polarizations propagating along the interface. Several clear signatures were observed that help distinguish between the long-wavelength limit, in which the displacement discontinuity model is valid, and the short-wavelength limit, which is characterized by scattered Rayleigh waves.
In the case of Sv waves propagating perpendicular to the asperities, a clear
Acknowledgements
The authors wish to acknowledge support of this research by Department of Energy — Office of Basic Energy Science (DE-FG02-97ER14785) and the Nation Science Foundation — Young Investigator Award (9896057).
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