Seismic wave propagation in partially saturated rocks with a fractal distribution of fluid-patch size

Laboratory experiments on partially saturated rocks show that seismic attenuation can be significant, where the main mechanism is wave-induced local fluid flow (WILFF) due to patchy saturation at different spatial scales. We propose a theory to obtain the seismic properties of partially saturated rocks based on fractal (self-similar) patches, leading to an effective frequency-dependent fluid modulus. The model combines the differential effective medium (DEM) and Biot-Rayleigh theories, where the patches are inclusions incrementally added, such that the effective fluid is assumed the host for the next addition. The analysis shows that the seismic properties are independent on the number of iterations, but depend on the scale range (radius) of the inclusions, fractal dimension D_f of the self-similar distribution, parameter \(\theta\) of the exponential distribution, mean radius r₀ and variance \({\sigma_r}^2\) of the Gaussian distribution. Forced-oscillation experiments are performed on a limestone sample under partial water-saturation conditions at seismic frequencies (2-500 Hz), to obtain the velocity dispersion and extensional attenuation. The proposed theory provides a reasonable description of the experimental data and other existing measurements on Berea sandstone and tight carbonates.