Rock fracture compliance derived from time delays of elastic waves

Möllhoff, Martin; Bean, Christopher J.; Meredith, P. G.

doi:10.1111/j.1365-2478.2010.00887.x

Cited by 17 publications

(17 citation statements)

References 19 publications

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“…Previous laboratory and numerical works have shown that the P wave transmission coefficient of a fracture can be linked to its mechanical compliance through the linear slip theory (Möllhoff et al, ; Pyrak‐Nolte & Nolte, ; Schoenberg, ). That is, fractures are modeled as nonwelded interfaces, across which traction is continuous but seismic displacement is not.…”

Section: Effect Of Individual Fractures On the Attenuation And Phase mentioning

confidence: 99%

Estimation of Fracture Compliance From Attenuation and Velocity Analysis of Full‐Waveform Sonic Log Data

et al. 2019

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In fractured rocks, the amplitudes of propagating seismic waves decay due to various mechanisms, such as geometrical spreading, solid friction, displacement of pore fluid relative to the solid frame, and transmission losses due to energy conversion to reflected and transmitted waves at the fracture interfaces. In this work, we characterize the mechanical properties of individual fractures from P wave velocity changes and transmission losses inferred from static full‐waveform sonic log data. The methodology is validated using synthetic full‐waveform sonic logs and applied to data acquired in a borehole penetrating multiple fractures embedded in a granodioritic rock. To extract the transmission losses from attenuation estimates, we remove the contributions associated with other loss mechanisms. The geometrical spreading correction is inferred from a joint analysis of numerical simulations that emulate the borehole environment and the redundancy of attenuation contributions other than geometrical spreading in multiple acquisitions with different source‐receiver spacing configurations. The intrinsic background attenuation is estimated from measurements acquired in the intact zones. In the fractured zones, the variations with respect to the background attenuation are attributed to transmission losses. Once we have estimated the transmission losses associated with a given fracture, we compute the transmission coefficient, which, on the basis of the linear slip theory, can then be related to the mechanical normal compliance of the fracture. Our results indicate that the estimated mechanical normal compliance ranges from 1 × 10−13 to 1 × 10−12 m/Pa, which, for the size of the considered fractures, is consistent with the experimental evidence available.

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Section: Effect Of Individual Fractures On the Attenuation And Phase mentioning

confidence: 99%

Estimation of Fracture Compliance From Attenuation and Velocity Analysis of Full‐Waveform Sonic Log Data

et al. 2019

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“…Assume that hypotheses (6)(7)(8)(9) hold and ∈ [0, 1). Then, for initial data (u 0 , u 1 , g 0 ) ∈ , the Lamé system (5) possesses a unique weak solution (u, t u, z) ∈ C(R + ; ).…”

Section: Well-posednessmentioning

confidence: 99%

“…Assume that assumptions (6)(7)(8)(9)(10)(11) hold. Given a sequence { n } ⊂ (0, 1), let (u n , t u n , z n ) be the corresponding weak solution of (5) with n and initial data (u 0 , u 1 , g 0 ) ∈ .…”

Section: Theorem 32 (Limit Of Solutions)mentioning

confidence: 99%

Global attractors for a system of elasticity with small delays

Araújo

Bocanegra‐Rodríguez

Calsavara

et al. 2021

Math Methods in App Sciences

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This paper is concerned with dynamics of elasticity systems featuring a nonlinear foundation of critical growth and delay effects. The existence of global attractors for such systems has been studied recently. Our main contribution establishes the upper-semicontinuity of attractors with respect to a small parameter multiplying the delay term. Our results are new even for the analogous scalar wave equation.

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“…The group arrival time was then used to calculate the group velocity. For a transmitted signal, the group time delay is always different from the first-break/onset delay (Möllhoff et al, 2010). First break indicates the first recorded transmitted signal attributed to the energy generated by the source, and group delay refers to the delay of the wave packet.…”

Section: Wavefront Imagingmentioning

confidence: 99%

Measurements of elastic constants in anisotropic media

2014

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Simulation of elastic-wave propagation in rock requires knowledge of the elastic constants of the medium. The number of elastic constants required to describe a rock depends on the symmetry class. For example, isotropic symmetry requires only two elastic constants, whereas transversely isotropic symmetry requires five unique elastic constants. The off-diagonal elastic constant depends on a wave velocity measured along a nonsymmetry axis. The most difficult barrier when measuring these elastic constants is the ambiguity between the phase and group velocity in experimental measurements. Several methods to eliminate this difficulty have been previously proposed, but they typically require several samples, difficult machining, or complicated computational analysis. Another approach is to use the surface (Rayleigh) wave velocity to obtain the off-diagonal elastic constant. Rayleigh waves propagated along symmetry axes have phase and group velocities that are equal for materials with no frequency dispersion, thereby eliminating the ambiguity. Using a theoretical secular equation that relates the Rayleigh velocity to the elastic constants enable determination of the offdiagonal elastic constant. Laboratory measurements of the elastic constants in isotropic and anisotropic materials were made using ultrasonic transducers (central frequency of 1 MHz) for the Rayleigh-wave method and a wavefront-imaging method. The two methods indicated agreement within 1% and 3% for isotropic and transversely isotropic samples, respectively, demonstrating the ability of the Rayleigh-wave method to measure the off-diagonal elastic constant. The surface-wave approach eliminates the need for multiple samples, expensive computational calculations, and most importantly, it removes the ambiguity between the phase and group velocity in the measured data for materials with no frequency dispersion because all measurements are made along symmetry axes.

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Rock fracture compliance derived from time delays of elastic waves

Cited by 17 publications

References 19 publications

Estimation of Fracture Compliance From Attenuation and Velocity Analysis of Full‐Waveform Sonic Log Data

Estimation of Fracture Compliance From Attenuation and Velocity Analysis of Full‐Waveform Sonic Log Data

Global attractors for a system of elasticity with small delays

Measurements of elastic constants in anisotropic media

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