Transverserly Isotropic Saturated Porous Formations: II. Wave Propagation And Application To Multipole Logging

Abstract

The wavefields generated by monopole and dipole sources in a fluid filled borehole embedded in multilayered transversely isotropic saturated porous formations are studied. The layers are modeled following Biot theory modified in accordance with homogenization theory. It allows to take into account a transversely isotropic skeleton and/or a transversely isotropic complex permeability tensor. Their axes of symmetry are assumed to coincide, parallel to the vertical axis of the borehole. A general formulation, valid for any order of multipole source and based on the Thomson Haskell method, allows to take into account any combimi.tion of elastic and saturated porous layers, either isotropic or transversely isotropic. The presence of an external fluid layer is also possible. The study focuses on the modes behavior. It is achieved through the computation of dispersion and attenuation curves, sensitivity coefficients with respect to the stiffness constants of the skeleton(s), and full waveform synthetic microseismograms using the discrete wavenumber method. In the simple hole model with an impermeable borehole wall, whatever the type of the formation (fast or slow), the behavior of the modes is analogous to that in the presence of simple elastic formations with body wave attenuations added. The phase velocity of the Stoneley wave generated by a monopole source is sensitive to the horizontally propagating SH-wave velocity. Such a coupling decreases with increasing frequency and stiffness of the formation. The low frequency part of the zero-th order (Le., flexural) mode generated by a multi(di)pole source measures the vertically propagating SV-wave velocity. The shear wave anisotropy may then be evaluated. With a fast formation, the vertically propagating SV-wave velocity can also be obtained from the low frequency (high velocity) part of the pseudo-Rayleigh mode generated by the monopole source. The anisotropy of the complex permeability tensor cannot be detected. Moreover, only the attenuation of the vertically propagating P wave is sensitive to the only vertical permeability. Any anelastic (anisotropic) attenuation will supersede the latter. When the borehole wall is permeable, the fluid flow which takes place at the interface refers to the horizontal mobility (I.e., horizontal permeability/saturant fluid viscosity). Assuming greater horizontal velocities, the decrease of the Stoneley wave phase velocity and the increase of its low frequency attenuation are enhanced. The shear wave transverse isotropy cannot be anymore detected and any estimation of the horizontal permeability based on Stoneley wave characteristics may become questionable with a high anisotropy degree of the skeleton. However, detection of permeability variation may still be reasonably performed. In the presence of an invaded zone, whatever the boundary conditions at the borehole wall, Stoneley wave integrates the properties of the inner layer in the entire frequency range. This coupling phenomenon increases with increasing thickness and decreasing body wave velocities of the inner layer. As a result, both estimations of the shear wave transverse anisotropy and the permeability of the virgin formation from the Stoneley wave characteristics are ill posed. Of course, such a result hold true in a cased borehole, whatever the quality of the bonding. In any of the multilayered configuration, the low frequency part of both the flexural mode and the pseudo-Rayleigh mode, when it exists, measures the characteristics of the vertically propagating SV wave of the virgin formation. Such an interesting information may be however difficult to extract.