Approximating the wave moduli of double porosity media at low frequencies by a single Zener or Kelvin-Voigt element

The analytic transient acoustic wave 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 means of approximating the double porosity model with a uniform poro-viscoacoustic model based on a single Zener and a single Kelvin-Voigt (KV) element, respectively. We choose the relaxation function of both mechanical elements which just approximates the dispersion behaviour of the double porosity model around source centre frequencies of 5, 50, 200 and 1000 Hz, respectively. The comparison between the results of the three models shows that if the frequency is much lower than the peak attenuation frequency (4470 Hz) of the example double porosity model, then wave propagation can be well described by the poro-viscoacoustic model with a single Zener element. However, if the frequency is less than 50 Hz, then a single KV element gives an even better result. Therefore, this paper investigates the validity and range of applicability of different single mechanical elements in solving for transient acoustic wave modelling in heterogeneous, double porosity media. The primary attraction of using a Zener model or a KV model is that it allows the convolution integral to be replaced by memory equations by which the field quantities calculated at every time step need not be stored.