The tube wave as a Biot slow wave

Abstract

The tube wave speed in a simple fluid‐filled circular bore reduces to [Formula: see text] as the frequency goes to zero, where [Formula: see text] is the acoustic sound speed in the fluid, [Formula: see text] is the fluid density, [Formula: see text] is the fluid bulk modulus, and μ is the formation shear modulus. Biot (1952) deduced this simple relation by considering the low‐frequency asymptotic expansion of the exact dispersion relation. In 1956, Biot proposed a theory (Biot, 1956a, b) that predicts a new type of compressional bulk wave in fluid‐saturated porous media. This “slow wave” is associated mainly with the motion of pore fluids. It appears that Biot never related this theory to his previous work on the bore problem, although the connection is apparent if the bore is considered as a pore. Typically, the bore radius is about 10 cm, while the relevant acoustic logging frequency is on the order of 1 kHz. With water as the bore fluid, the viscous skin depth is on the order of 100 μm. Therefore, if the bore is to be considered as a pore, the relevant form of Biot’s theory is the limit in which the pore radius is large relative to the viscous skin depth of the fluid. This form is the high‐frequency limit, in which the effects of the fluid viscosity are negligible and the slow‐wave dissipation is relatively low.