Wave Propagation in Infinituple‐Porosity Media

Abstract

Abstract The fractal texture (or fabric) of porous media, which supports fluid flow at different scales, is the main cause of wave anelasticity (dispersion and attenuation) on a wide range of frequencies. To model this phenomenon, we develop a theory of wave propagation in a fluid saturated infinituple‐porosity media containing inclusions at multiple scales, based on the differential effective medium (DEM) theory of solid composites and Biot‐Rayleigh theory for double‐porosity media. The dynamical equations are derived from first principles, that is, based on the strain (potential), kinetic, and dissipation energies, leading to generalized stiffness and density coefficients. The scale of the inclusions can be characterized by different distributions. The theory shows that the anelasticity depends on the size (radius) of the inclusions, parameter θ (exponential distribution), mean radius r 0 and variance σ r 2 (Gaussian distribution) and the fractal dimension D f (self‐similar distribution). When D f = 2, θ = 1 and σ r 2 = 4, the three distributions give the same P‐wave velocities and attenuation, since each added inclusion phase has nearly the same volume fraction. For the modeling results, the range of anelasticity of D f = 2.99/ θ = 1/ σ r 2 = 4 is broader than that of D f < 2.99/ θ < 1/ σ r 2 < 4. To confirm the validity of the model, we compare the results with laboratory measurements on tight sandstone and carbonate samples in the range 1 Hz–1 kHz, Fox Hill sandstone (5 Hz–800 kHz) and field measurements of marine sediments (50 Hz–400 kHz). This comparison shows that the model successfully describes the observed anelasticity.