We investigate the problem of wave propagation in a porous medium, in the framework of Biot's theory, computing the numerical solution of the differential equations by a grid method. The problems posed by the stiffness of the equations are circumvented by using a partition (or splitting) time integrator which allows for an efficient explicit solution as in the case of nonstiff differential equations. The resulting algorithm possesses fourth-order accuracy in time and "infinite" (spectral) accuracy in space. Alternatively, a second-order algorithm, based on a Crank–Nicolson method, provides similar stability properties, although lower accuracy. The simulations correctly reproduce the wave forms of the fast and slow compressional waves and their relative amplitudes. Moreover, we observe the static slow mode, particularly strong when the source is a bulk perturbation or a fluid volume injection. The numerical results are confirmed by the analytical solution.

[1] The attenuation effects predicted by Hickey's poroelastic theory (Hpt) are quantified by means of seismic modeling in an unbounded, homogeneous, isotropic porous model fully saturated with water and with oil. The numerical results are compared to those predicted by Biot poroelastic theory (Bpt). As opposed to Bpt, Hpt accounts for thermomechanical coupling and viscous fluid relaxation and adequately models the transient fluctuations of porosity and mass densities as the wave compresses and dilates the porous medium during propagation. Despite all these theoretical improvements over Bpt, the numerical results show that both theories produce remarkably similar waveforms. Without considering thermal relaxation, Hpt produces less than 1% higher-amplitude attenuation and velocity dispersion than Bpt. The major contrasts correspond to the slow P wave. Thermomechanical coupling affects the fast P wave: seismic amplitudes are 1% smaller and some dispersion for the oil-permeated case can be observed. It produces no effects on the fast S or the slow P wave. Therefore these numerical experiments appear to substantiate what Biot assumed at the outset of his theoretical developments: that the effects of transient oscillations of porosity during wave propagation are negligible in terms of velocity dispersion and amplitude attenuation. Furthermore it is confirmed that in a homogeneous porous medium, the combined dissipation mechanisms mentioned are not adequate to explain the total amount of energy dissipation observed in the field or laboratory.Citation: Quiroga-Goode, G., S. Jiménez-Hernández, M. A. Pérez-Flores, and R. Padilla-Hernández (2005), Computational study of seismic waves in homogeneous dynamic-porosity media with thermal and fluid relaxation: Gauging Biot theory,

S U M M A R YFollowing the first-principle procedure outlined by Buchen (1971a) and Borcherdt (1973), we describe the derivation of SH-wave propagation in a homogeneous transversely isotropic linear viscoelastic (HTILV) solid. A plane SH wave propagates with the frequency-dependent complex phase velocity:where Ph and p, are complex shear-wave velocities perpendicular and parallel to the axis of symmetry of the medium and b is a complex angle that the complex wave vector makes with the axis. The energy flows in a direction governed by the propagation vector, attenuation vector and the rigidities. The attenuation angle between the propagation vector and the attenuation vector can be uniquely determined by the complex ray parameter at the saddle point of the complex traveltime function. Complex rays can be traced between source and receiver locations with intermediate coordinates being complex. By means of the method of steepest descent, the wavenumber integral representing the exact SH-wave field generated by a line source for the layered-medium problem can be approximated to give complex ray amplitudes for reflected and transmitted body waves. The factor accounting for cylindrical divergence is similar in form to that of the isotropic case. For a simple two half-spaces model, the complex ray result agrees well with the o -k solution in regions away from the critical area. For pure SH-mode propagation through a planar HTILV multi-layered structure with 20 per cent velocity anisotropy in each layer (Q, = Q h ) , the reflected amplitudes in the two cases (transversely isotropic and isotropic) generally do not differ much, but traveltimes differ significantly. This suggests that one can, in the case we considered, neglect the effect of weak anisotropy on amplitudes, but not on propagation phase.

S U M M A R YIt is quantified the properties of seismic waves in fully saturated homogeneous porous media within the framework of Sahay's modified and reformulated poroelastic theory. The computational results comprise amplitude attenuation, velocity dispersion and seismic waveforms. They show that the behaviour of all four waves modelled as a function of offset, frequency, porosity, fluid viscosity and source bandwidth depicts realistic dissipation within the sonicultrasonic band. Therefore, it appears that there is no need to include material heterogeneity to model attenuation. By inference it is concluded that the fluid viscosity effects may be enhanced by dynamic porosity.