In this paper, we introduce the complex modulus to express the viscoelasticity of a medium. According to the correspondence principle, the Biot-Squirt （BISQ） equations in the steady-state case are presented for the space-frequency domain described by solid displacements and fluid pressure in a homogeneous viscoelastic medium. The effective bulk modulus of a multiphase flow is computed by the Voigt formula, and the characteristic squirt-flow length is revised for the gas-included case. We then build a viscoelastic BISQ model containing a multiphase flow. Through using this model, wave dispersion and attenuation are studied in a medium with low porosity and low permeability. Furthermore, this model is applied to observed interwell seismic data. Analysis of these data reveals that the viscoelastic parameter tan6 is not a constant. Thus, we present a linear frequen- cy-dependent function in the interwell seismic frequency range to express tanG. This improves the fit between the observed data and theoretical results.
Taking into account three important porous media mechanisms during wave propagation （the Biot-flow, squirt-flow, and solid-skeleton viscoelastic mechanisms）, we introduce water saturation into the dynamic governing equations of wave propagation by analyzing the effective medium theory and then providing a viscoelastic Biot/squirt （BISQ） model which can analyze the wave propagation problems in a partially viscous pore fluid saturated porous media. In this model, the effects of pore fluid distribution patterns on the effective bulk modulus at different frequencies are considered. Then we derive the wave dynamic equations in the time-space domain. The phase velocity and the attenuation coefficient equations of the viscoelatic BISQ model in the frequency-wavenumber domain are deduced through a set of plane harmonic solution assumptions. Finally, by means of numerical simulations, we investigate the effects of water saturation, permeability, and frequency on compressional wave velocity and attenuation. Based on tight sandstone and carbonate experimental observed data, the compressional wave velocities of partially saturated reservoir rocks are calculated. The compressional wave velocity in carbonate reservoirs is more sensitive to gas saturation than in sandstone reservoirs.
