Elastic wave scattering by a fluid-saturated circular crack and effective properties of a solid with a sparse...

Hydraulic fractures and preexisting cracks in natural aquifers and hydrocarbon reservoirs are often saturated with fluids. Understanding the elastic wave properties in such a cracked fluid-saturated medium is of importance for many physical and engineering applications such as hydrology, petroleum engineering, oil exploration, induced seismicity, and nuclear waste disposal. In this paper, the scattering of a normally incident longitudinal (P-) wave by a fluid-saturated circular crack in an infinite elastic non-porous matrix is studied. In particular, the mechanism of hydraulic conduction (including the effects of the crack permeability and fluid inertia) inside the crack is incorporated. A semi-analytic solution for this scattering problem is derived. Based on the solution and multiple scattering theorem, an effective medium model is developed to determine the velocity dispersion and attenuation due to wave scattering in an elastic matrix with sparse distribution of aligned cracks. It is shown that the effective P-wave velocity is consistent with Gassmann's theory in the low-frequency limit. The effect of crack permeability on scattering is negligible, but the effect of fluid inertia is important. Specifically, it is found that resonance phenomena can take place inside the cracks at frequencies much lower than the scattering characteristic frequency so that rapid velocity variation can occur at relatively low frequencies. The fluid viscosity plays a damping role in weakening the resonance. The effects of crack thickness and fluid compressibility on scattering dispersion are similar to those in the case of plane-strain (two-dimensional) slit crack.