Rigorous Bounds for Seismic Attenuation and Dispersion in Poroelastic Rocks

The Hashin-Shtrikman (HS) bounds define the range of bulk and shear moduli of an elastic composite, given the moduli of the constituents and their volume fractions. Recently, the HS bounds have been extended to the quasi-static moduli of composite viscoelastic media. Since viscoelastic moduli are complex, the viscoelastic bounds form a closed curve on the complex plane. When the medium is poroelastic (a composite of an elastic solid and a viscous fluid), the viscoelastic bounds for a bulk modulus are represented in the complex plane by a semi-circle and a segment of the real axis, connecting the formal HS bounds. Furthermore, these bounds are independent of frequency. The complex bulk modulus describing attenuation and dispersion due to squirt flow in a porous medium of a particular geometry spans the entire bounding region. This shows that the bounds for the bulk modulus are attainable (realizable). These bounds account for the viscous shear relaxation and squirt-flow dispersion, but not for Biot’s global flow dispersion. This is to be expected, since the bounds are quasi-static whereas the global flow dispersion is largely controlled by inertial forces.