Low‐frequency tube waves in permeable rocks

Abstract

Low‐frequency tube waves in permeable boreholes are of interest because their propagation depends upon in‐situ rock permeability. In this paper, we use low‐ frequency asymptotics to unify two approaches that both describe tube waves. The first approach is due to White, in which the low‐frequency tube‐wave velocity and attenuation are expressed explicitly as functions of frequency. The expressions include contributions from three factors: the borehole fluid compressibility, the wall rigidity, and the wall impedance due to the permeability of the solid. The second approach is the full solution of tube‐wave properties based on Biot theory and the appropriate boundary conditions between the fluid and the porous solid. By taking the low‐frequency approximations to the full solution, we derive an explicit expression for the tube‐wave properties, similar to White’s expression. White’s formulation is hereby corrected to include the elasticity of the solid frame around the borehole. For a 20 cm diameter borehole, the corrected low‐frequency expression is quite accurate below 300 Hz for the four cases shown, when compared with the full solution. Above that frequency, one should use the full solution to obtain an accurate expression of tube‐wave properties in permeable rocks.