Fracture-Induced Anisotropic Attenuation

Abstract

The triaxial nature of the tectonic stress in the earth's crust favors the appearance of vertical fractures. The resulting rheology is usually effective anisotropy with orthorhombic and monoclinic symmetries. In addition, the presence of fluids leads to azimuthally varying attenuation of seismic waves. A dense set of fractures embedded in a background medium enhances anisotropy and rock compliance. Fractures are modeled as boundary discontinuities in the displacement u and particle velocity v as ½j Á u þ g Á v; where the brackets denote discontinuities across the fracture surface, j is a fracture stiffness, and g is a viscosity related to the energy loss. We consider a transversely isotropic background medium (e.g., thin horizontal plane layers), with sets of long vertical fractures. Schoenberg and Muir's theory combines the background medium and sets of vertical fractures to provide the 13 complex stiffnesses of the long-wavelength equivalent monoclinic and viscoelastic medium. Long-wavelength equivalent means that the dominant wavelength of the signal is much longer than the fracture spacing. The symmetry plane is the horizontal plane. The equations for orthorhombic and transversely isotropic media follow as particular cases. We compute the complex velocities of the medium as a function of frequency and propagation direction, which provide the phase velocities, energy velocities (wavefronts), and quality factors. The effective medium ranges from monoclinic symmetry to hexagonal (transversely isotropic) symmetry from the low-to the high-frequency limits in the case of a particle-velocity discontinuity (lossy case) and the attenuation shows typical Zener relaxation peaks as a function of frequency. The attenuation of the coupled waves may show important differences when computed versus the ray or phase angles, with triplication appearing in the Q factor of the qS wave. We have performed a full-wave simulation to compute the field corresponding to the coupled qP-qS waves in the symmetry plane of an effective monoclinic medium. The simulations agree with the predictions of the plane-wave analysis.

Modelling ultrasonic laboratory measurements of the saturation dependence of elastic modulus: New insights and implications for wave propagation mechanisms

“…It should be pointed out that the linear addition of the velocity dispersion and attenuation from different mechanisms is an approximation. However, this approach of combining attenuation and velocity dispersion due to different mechanisms has found success in previous works (Carcione et al, 2012;Chichinina et al, 2009) as well as in the present work.…”

Numerical experiments of fracture-induced velocity and attenuation anisotropy

“…If the background medium is isotropic, with elasticity components (Lamé constants) c 12 and c 55 , and the fracture set is rotationally invariant, we have Z H = Z V ≡ Z T , and the equivalent medium is TIV with a vertical symmetry axis (VTIV), whose stiffness matrix is (Schoenberg 1983; Chichinina et al 2009; Carcione et al 2012), where c 11 = c 12 + 2 c 55 , The stiffness matrix in Coates & Schoenberg (1995) is equivalent to P of eq. () considering the lossless case (= 0).…”

“…In particular, the components p IJ of matrix P are obtained by assuming a periodic medium composed of two layers, where one of the layers has the Lamé constants λ and μ (the background medium) and the other, representing the fracture, is very thin with Lamé constants μ f = pL β= p / Z T and E f =λ f + 2μ f = pL α= p / Z N , where p ≪ 1 is the volume proportion of fractures, and L is the fracture spacing (constant). The displacement discontinuities (boundary conditions) associated with the fractures are [ u 3 ] = LZ N σ 33 and [ u 1 ] = LZ T σ 13 along the x 3 ‐ and x 1 ‐directions, respectively (Schoenberg 1983, eqs 21–23; Carcione et al 2012).…”

“…The specific viscosity accounts for the presence of a liquid. Chichinina et al (2009) and Carcione et al (2012) obtained analytical solutions for the TIV and more general anisotropic cases, respectively, i.e. the complex and frequency-dependent stiffness components.…”

Numerical simulation of seismic wave propagation in viscoelastic-anisotropic media using frequency-independent Q wave equation

“…, where a ¼ 0.001 s. Based on the fracture-induced anisotropy theory (Carcione et al, 2012b), the complex stiffness matrix components are computed. The resulting seismic viscoelasticanisotropic properties are summarized in Table 3.…”