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

Abstract

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...

Wave field simulation for heterogeneous transversely isotropic porous media with the JKD dynamic permeability

“…Approximating the integral by a quadrature formula, the fractional derivative can be evaluated by integrating ordinary differential equations for a finite set of auxiliary internal variables. It was shown by Lu and Hanyga [23], direct use of Eq. (25a) to evaluate a fractional derivative will lead to a significant error due to the slow convergence of the integral in Eq.…”

“…In this paper, based on the approach of Lu and Hanyga [23], a new approach for evaluating the shifted fractional derivative will be developed. By using the similar procedure as Lu and Hanyga [23], the asymptotic expressions for /ðy; tÞ when y ! 0 and y !…”

“…(22). The method developed by Lu and Hanyga [23] will be extended to calculate the shifted fractional derivatives involved in the four differential equations. In this way, the four ordinary differential equations will be reduced to four ordinary differential equations supplemented by the ordinary differential equations for the quadrature variables associated with the drag forces.…”

“…Therefore, the method of Lu and Hanyga [23] is extended to evaluate the shifted fractional derivative here without storage and integration of the entire velocity history. By using the new method to calculate the fractional derivative, the governing equations for the 2D transversely isotropic porous medium are reduced to a system of first-order differential equations for velocities, stresses, pore pressure and the quadrature variables associated with the shifted fractional derivatives.…”

Acoustic viscoelastic modeling by frequency-domain boundary element method

“…As an efficient alternative, Varela (1993) proposes a one-way time-domain viscoelastic modeling algorithm with a higher computational efficiency, but involved in a single series expansion that is valid for bigger Q values. To reduce memory requirements, Lu and Hanyga (2004) use an intermediate variable to solve the fractionalorder derivative, which causes huge computations. Chen and Holm (2004) apply a fractional Laplace operator to the calculation of viscous wavefields to improve computational efficiency.…”