Biot theory research has been extended to the multi-scale heterogeneity in actual rocks. Focused on laboratory frequency bandwidth studies, we discuss the relationships between double-porosity and BISQ wave equations, analytically derive the degeneration method for double-porosity's return to BISQ, and give three necessary conditions which the degeneration must satisfy. By introducing dynamic permeability and tortuosity theory, a full set of dynamic double-porosity wave equations are derived. A narrow band approximation is made to simplify the numerical simulation for dynamic double-porosity wavefi elds. Finally, the pseudo-spectral method is used for wave simulation within the laboratory frequency band (50 kHz). Numerical results have proved the feasibility for dynamic double-porosity's description of squirt fl ow and the validity of the quasi-static approximation method.

We generalize the classical theory of acoustoelasticity to the porous case (one fluid and a solid frame) and finite deformations. A unified treatment of non‐linear acoustoelasticity of finite strains in fluid‐saturated porous rocks is developed on the basis of Biot’s theory. A strain‐energy function, formed with eleven terms, combined with Biot’s kinetic and dissipation energies, yields Lagrange’s equations and consequently the wave equation of the medium. The velocities and dissipation factors of the P‐ and S‐waves are obtained as a function of the 2nd‐ and 3rd‐order elastic constants for hydrostatic and uniaxial loading. The theory yields the limit to the classical theory if the fluid is replaced with a solid with the same properties of the frame. We consider sandstone and obtain results for open‐pore jacketed and closed‐pore jacketed ‘gedanken’ experiments. Finally, we compare the theoretical results with experimental data.

The attenuation of seismic signal is usually characterized in the frequency domain using Fourier power spectra and is often usefully characterized by average measures, such as the center frequency or spectral mean. Fourier analysis, however, suffers from time-frequency resolution problems. Wavelet analysis has better time-frequency localization and offers superior spectral decomposition. In this paper, we show that seismic attenuation can be characterized by the scalogram (also called energy density) in the wavelet domain. A single scale encompasses a frequency band. The scalogram relates absorption to peak-scale variations. The peak scale is the scale of maximum amplitude in the scalogram. Seismic attenuation can be estimated directly from the scalogram according to the scale shift of the data and can also be described indirectly by the centroid of scale (the mean of a scalogram). In absorbing media, seismic attenuation increases with frequency, i.e., decreases with scale. In the wavelet domain, small-scale energies of the seismic signal are attenuated more rapidly than are large-scale energies as waves propagate. As a result, both the peak scale and the centroid of the signal’s scalogram increase during propagation. Under the assumption of a frequency-independent [Formula: see text] model, these increases of the peak scale and the centroid of scale are inversely proportional to the quality factor, i.e., a lower quality factor results in an upshift of the peak scale in the scalogram and an increase of the centroid of scale. The peak-scale-shift method can be applied to seismic data with sufficiently broad signal bandwidth. The centroid of scale can be used as an attribute to qualitatively characterize seismic attenuation. Examples of gas detection in both synthetic and field data show the value of this technique.