Tube wave attenuation and dispersion in permeable formations

Theoretical calculations show that borehole Stoneley waves, can be significantly attenuated in the presence of a permeable formation. Attenuation is greatest in gaseous pore fluid in highly permeable sandstone, and occurs in the range 1-10 kHz. Numerical simulations also show that water saturation can produce significant attenuation. The problem consists of a fliud-filled bore in a permeable, porous formation. Using the Biot theory of dynamic poroelasticity, a dispersion equation is obtained for the complex phase speeds of the guided waves as functions of frequency. In certain cases the Q value for the tube wave exhibits a pronounced maximum as a function of frequency. This is due to a transition from a quasi-static problem in a cylindrical geometry at low frequency to a dynamic problem for a plane interface at high frequencies. Analyzing these simpler problems individually shows that the Q value increases with frequency in the quasi-static regime, but decreases with frequency in the high-frequency regime.The quasi-static problem reduces to a relatively simple equation for the complex tube wave speed, similar in form to one derived by White. Comparisons are also made with the theory of Mathieu and Toksoz. However, these low-frequency approximate theories do not produce the Q peak. Numerical results are presented for three sandstones with widely differing permeabilities. Three pore fluids are considered: oil, water, and gas.