Biot's Slow Wave and Effective Hydraulic Conductivity in Random Media

We study Biot’s slow wave propagation in the presence of strong hydraulic conductivity fluctuations in the low-frequency range. The latter condition implies that the slow wave is a diffusion process. To elucidate the characteristics of the diffusion wave in an inhomogeneous medium we perform numerical simulations. These simulations demonstrate that the diffusion wave field does not only depend on the spatial distribution of the in homogeneities but also on the frequency. Therefore, if we seek to replace the inhomogeneous medium by an effective, up-scaled medium the corresponding effective hydraulic conductivity will become frequency-dependent. Based on a strong contrast approximation, suggested in the context of an effective dielectric constant, closed form expressions for the effective, frequency-dependent conductivity are derived. These expressions yield in 1D the exact low- and high-frequency bounds, while in 3D the frequency limits for certain optimal microstructures can be obtained