An Anisotropic Cole–cole Model of Seismic Attenuation

S U M M A R YAn inclusion-based model for the long-wavelength equivalent medium properties of a rocklike composite should not only be consistent with the Brown-Korringa relations for the (zerofrequency) dependence of the effective compliance tensor on the bulk modulus of the saturating fluid, but also allow for non-dilute concentrations of communicating cavities (pores, cracks, fractures, etc.) and even multiple solid constituents. As discussed by Jakobsen et al. in the proceedings of the 10th International Workshop on Seismic Anisotropy, the standard (stiffness-based) T-matrix approach to wave-induced fluid flow (which represents an extension of the ideas of Hudson and his associates) appears to be consistent with the Brown-Korringa relations if and only if the spatial distributions of inclusions/cavities are the same for all pairs of interacting inclusions/cavities. In the 'revised' (compliance-based) T-matrix approach to wave-induced fluid flow (presented in this paper), however, one can estimate the effective compliances as complex-valued functions of frequency in a manner which satisfies all these criteria (independent of the aspect ratios of the two-point correlation functions of ellipsoidal symmetry) and leads to rather simple analytical results in the limits of extremely low and high frequencies, provided that a certain quantity (depending on the structural correlations among the inclusions) is small. By 'revised' it is meant that a higher-order expression (which takes into account the effects of spatial distribution and strain propagation between two inclusions only) has been used when incorporating the effects of dynamic fluid pressure communication into modelling the effective viscoelastic behaviour of cracked/porous rocks. In other words, the response of a single inclusion/cavity is coupled to the overall properties in the present formalism. Numerical results suggest that the compliance-based formulation of the present paper gives the same results as the stiffness-based T-matrix approach of a previous paper if and only if the frequency is below a critical limit associated with a peak in the attenuation spectrum for a porous medium with embedded cracks that are fully aligned.

Section: Saturat E D R E S U Lt S V I a D Ry R E S U Lt Smentioning

confidence: 94%

The interacting inclusion model of wave-induced fluid flow

Jakobsen¹

2004

“…Besides, the conditions 0 < α < 1 and a ≤ 1 follow from thermodynamics argument (Bagley & Torvik 1983). The stress relaxation function ψ (t) and the derivative of the stress relaxation function ψ ,t (t) for the Cole-Cole law have the following expressions (Hanyga 2003a)…”

Section: The Col E -C O L E M O D E L F O R a V I S C O E L A S T I Cmentioning

confidence: 99%

“…Also, extension of our method to the 3-D wave propagation in a viscoelastic medium with a singular memory is straightforward. Moreover, material anisotropy can be easily incorporated in our method (Hanyga 2003a). The proposed method can be used in solving the Biot's poroelastic medium with a singular memory involved in the drag force.…”

Section: Conc L U S I O N Smentioning

confidence: 99%

Numerical modelling method for wave propagation in a linear viscoelastic medium with singular memory

Hanyga

2004

Self Cite

S U M M A R YA numerical modelling method for wave propagation in a linear viscoelastic medium with singular memory is developed in this paper. For a demonstration of the method, the ColeCole model of viscoelastic relaxation is adopted here. A formulation of the Cole-Cole model based on internal variables satisfying fractional relaxation equations is applied. In order to avoid integrating and storing of the entire history of the variables, a new method for solving fractional differential equations of arbitrary order based on a set of secondary internal variables is developed. Using the new method, the velocity-stress equations and the fractional relaxation equations are reduced to a system of first-order ordinary differential equations for the velocities, stresses, primary internal variables as well as the secondary internal variables. The horizontal spatial derivatives involved in the governing equations are calculated by the Fourier pseudospectral (PS) method, while the vertical ones are calculated by the Chebychev PS method. The physical boundary conditions and the non-reflecting conditions for the Chebychev PS method are also discussed. The global solution of the first-order system of ordinary differential equations is advanced in time by the Euler predictor-corrector methods. For the demonstration of our method, some numerical results are presented. Key words:Cole-Cole law, fractional derivative, pseudo-spectral method, singular memory, viscoelastic, wave-propagation modelling. INTROD U C T I O NThe real medium of the Earth is different from an ideal elastic solid in many respects. Consequently, seismic wave propagation in the Earth has been considered to be anelastic for a very long time. Compared with the elastic case, the wave motion in an anelastic medium has many different properties. For example, wave attenuation and dispersion significantly affect the amplitude and the traveltime of the wave field. Also, the physical characteristics of the plane wave on the plane boundary separating two viscoelastic media are different from those of the elastic case: a special type of wave, called a generalized inhomogeneous wave can be generated near the interface (Borcherdt 1982;Borcherdt & Wenneberg 1985). Therefore, an accurate wavefield simulation method should be able to account for the effects of attenuation and dispersion.Up to now, there have been many papers concerning numerical wavefield modelling in a heterogeneous viscoelastic medium (Day & Minster 1984;Emmerich & Korn 1987;Carcione et al. 1988;Xu & Mcmechan 1995). However, the investigations are mainly limited to the generalized standard linear viscoelastic solid. For the generalized standard linear viscoelastic solid, the complex modulus belongs to the rational function. By using Padé approximation, the time convolution involved in the constitutive relation can be evaluated by a series of internal variables satisfying the first-order differential equation (Day & Minster 1984;Emmerich & Korn 1987). Alternatively, because the stress relaxation function o...

“…It follows that power laws with exponents exceeding 1 can only hold for a finite frequency band. As an example, the Cole-Cole law (Hanyga 2003a), known to satisfy the causality requirement, has a low-frequency asymptotic behaviour corresponding to a power law with 1 < α < 2, while for high frequencies it behaves like a power law with 0 < α < 1:…”

Section: I M I T S F O R T H E P O W E R L Aw I M P L I E D B Y C Amentioning

confidence: 99%

“…It has been shown that a more sophisticated model, known as the Cole-Cole or Bagley-Torvik model of viscoelastic response, known in polymer rheology (Rouse 1953;Ferry et al 1955;Bagley & Torvik 1983b;Torvik & Bagley 1983;Bagley & Torvik 1986;Friedrich & Braun 1992), rock mechanics (Batzle et al 2001), seismic wave propagation (Hanyga 2003a) and in the context of dielectric properties of geological materials (Cole & Cole 1941), is very successful in problems of creep and wave attenuation. It is, however, easy to see that power-law models provide a high-frequency approximation for this model.…”

Section: Introductionmentioning

confidence: 99%

Power-law attenuation in acoustic and isotropic anelastic media

Hanyga

Seredyńska

2003

S U M M A R YPower attenuation laws were originally introduced for phenomenological descriptions of attenuation in a large variety of materials. More recently they have been applied in viscoelastic inversion. Power-law models can be readily expressed in terms of scalar acoustic partial integro-differential equations. The restrictions on the exponent of the power law imposed by the assumptions of causality and dissipativity are discussed. An extension of power-law attenuation models from acoustic to isotropic anelastic media is proposed. The guiding principle is that the attenuation of longitudinal and transverse waves satisfies the power law. Restrictions imposed by thermodynamics and well-posedness considerations are discussed.