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. The main mechanism, wave-induced local fluid flow (WILFF), is affected by the spatial fluid distribution, especially in conditions of 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 and Biot-Rayleigh theories, where the patches are inclusions incrementally added, such that the effective fluid calculated in the current addition serves as host fluid in the next one. The analysis shows that adding identical inclusions in one or several steps produces nearly the same results, but the seismic properties 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 r(0) and variance sigma(2)(r) of the Gaussian distribution. Forced-oscillation experiments were 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 provided a reasonable description of these experimental data as well as other published measurements on tight carbonate and Berea sandstone.