Transverserly Isotropic Saturated Porous Formations: 1. Theoretical Developments And (Quasi) Body Wave Properties

Abstract

Using the results of homogenization theory, the constitutive equations for an anisotropic porous formation saturated by a Newtonian viscous fluid are derived. The transverse isotropy situation is investigated in terms of partial stresses in both phases. It allows to take into account a transversely isotropic skeleton and/or a transversely isotropic complex permeability tensor whose axes of symmetry are assumed to coincide. The wavenumbers equations are solved as part of a cylindrical geometry and general solutions for the displacement potentials in each phase are presented. Four dispersive and dissipative waves propagate in a transversely isotropic saturated porous formation: two quasi compressional waves, a quasi SV wave and aSH wave. All three quasi body waves approach the isotropic P[subscript 1] wave, slow P[subscript 2] wave, and S wave only as the degree of anisotropy of both the skeleton and the complex permeability tensor vanishes. Simple analytical formuale are derived for vertical and horizontal propagations. The SH-wave only excites the horizontal permeability, whatever its angle of propagation with respect to the vertical. The velocities and attenuations of these four waves are studied as a function of the only transversely isotropic permeability, the anisotropy of the skeleton, and the anisotropy of the mass coupling coefficient. Whatever the frequency, the quasi P[subscript 1]-wave, quasi SV-wave and SH-wave velocities are primarily governed by the anisotropy of the skeleton. Contrary, the quasi slow P[subscript 2] wave velocity is mostly representative of the complex permeability tensor transverse isotropy. At very high frequencies, when the inertial forces in the saturated porous formation are dominant, an anisotropy of the mass coupling coefficient leads to slightly different quasi BV-wave velocities along the principal directions, contrary to the pure elastic situation. For the quasi PI wave and the quasi BV wave, the degree of anisotropy of the strongly frequency dependent attenuation induced by the two phase character of the material is more important than that of the velocity.