Transient solution for poro-viscoacoustic wave propagation in double porosity media and its limitations

The analytical transient acoustic solution and dispersion characteristics for the double-porosity model are obtained over the whole frequency range for a homogeneous medium. The solution is also obtained by approximating the double porosity model with a uniform poro-viscoacoustic model and then constructing the transient response. The comparison between the results of the two models shows the likely validity and limitations of numerical solutions using a poro-viscoacoustic model to represent a double porosity medium in the heterogeneous case. Our calculations show that the dissipation by local mesoscopic flow of the double porosity model is very hard to fit over the entire frequency range by a single Zener element. However, since seismic exploration is normally restricted to a fairly narrow frequency band, this means that for frequency-dependent material properties, such as attenuation, the values around the centre frequency of the source will primarily determine the wave propagation characteristics. We choose the relaxation function that just approximates the dispersion behaviour of the double porosity model around the source centre frequency. It is shown that if the frequency is much lower than the peak attenuation frequency of the double porosity model, then wave propagation can be well described by the poro-viscoaoustic model with a single Zener element. For most water-filled sandstones having a double porosity structure, this holds true across the seismic frequency range. The transient solution for the heterogeneous double porosity medium is numerically obtained by a time splitting and Fourier pseudo-spectral staggered-grid method. As illustrative examples, the 2-D wavefield in a two-layer, water-saturated double porosity model are approximated by poro-viscoacoustic and poro-viscoelastic methods, respectively.