Wave propagation in anisotropic, saturated porous media: Plane-wave theory and numerical simulation

Abstract

Porous media are anisotropic due to bedding, compaction, and the presence of aligned microcracks and fractures. Here, it is assumed that the skeleton (and not the solid itself) is anisotropic. The rheological model also includes anisotropic tortuosity and permeability. The poroelastic equations are based on a transversely isotropic extension of Biot’s theory, and the problem is of plane strain type, i.e., two dimensional, describing qP−qS propagation. In the high-frequency case, the (two) viscodynamic operators are approximated by Zener relaxation functions that allow a closed differential formulation of Biot’s equation of motion. A plane-wave analysis derives expressions for the slowness, attenuation, and energy velocity vectors, and quality factor for homogeneous viscoelastic waves. The slow wave shows an anomalous polarization behavior. In particular, when the medium is strongly anisotropic the polarization is quasishear and the wave presents cuspidal triangles. Anisotropic tortuosity affects mainly the slow wavefront, and anisotropic permeability produces strong anisotropic attenuation of the three modes. The diffusive characteristics of the slow mode are predicted by the plane-wave analysis. As in the single-phase case, it is confirmed that the phase velocity is the projection of the energy velocity vector onto the propagation direction. Moreover, some fundamental energy relations, valid for a single-phase medium, are generalized to two-phase media. Transient propagation is solved with a direct grid method and a time-splitting integration algorithm, allowing the solution of the stiff part of the differential equations in closed analytical form. The snapshots show that the three waves are propagative when the fluid is ideal (zero viscosity). It is confirmed that, when the fluid is viscous, the slow wave becomes diffusive and appears as a static mode at the source location. The modeling confirms the triplication (cusps) of the slow wave and the polarization behavior predicted by the plane analysis.

On reconstruction of dynamic permeability and tortuosity from data at distinct frequencies

“…Suppose we have values of K(ω j ) = K(is j ) = P (s j ) for different nonzero real-valued frequencies ω 1 , ω 2 , · · · , ω M , then we can generate another M -interpolation points by using the symmetry of (19) …”

“…There have been a few papers which proposed different methods for handling the memory terms. Among them, the most popular ones are the fractional derivative approach for Biot-JKD model, which requires complicated quadrature rules [47] and the phenomenological one which proposed to approximates the memory terms with sums of exponential decay kernels [64], [19]. The latter is more computationally efficient but it is not clear how the weights and decay rates of the exponential decay kernels can be found in a systematic way.…”

A Review of the State of Art in Applying Biot Theory to Acoustic Propagation through the Bone

“…Attenborough et al [29] presented tortuosities deduced from audio-frequency measurements in air-filled cancellous bone replicas and showed that there was strong anisotropy. The Biot theory has been further developed including semi-analytical approach that allows for transverse anisotropy in the frame elastic moduli, tortuosity and permeability for geophysical applications [30]. A modified Biot-Attenborough (MBA) model has also been proposed for acoustic wave propagation in a non-rigid porous medium with circular cylindrical pores starting from a formulation for a rigid-framed porous material [31]- [33].…”

First‐order perturbations of the seismic response of fluid‐filled stratified poro‐elastic media

“…Since the pioneering work of Biot (1956), many authors (de la Cruz and Spanos, 1985;Johnson et al, 1994;Geerits and Kedler, 1997) have contributed to improve the poro-elastodynamic equations, either by averaging or by integrating techniques. The forward problem, i.e., the computation of synthetic seismograms in poro-elastic media has been developed and solved with several techniques (Dai et al, 1995;Carcione, 1996;Haartsen and Pride, 1997;Garambois and Dietrich, 2002). The model involves more parameters than the elastic case, but on the other hand, the wave velocities, attenuation and dispersion characteristics are computed from the medium's intrinsic properties without resorting to empirical relationships.…”