Nonlinear pulse propagation in arbitrarily dispersive media: Tube waves inpermeable formations

An approximate quasistatic equation, analogous to the Burgers equation, is derived to account for the combined effects on tube wave propagation of (a) dispersion/attenuation in permeable formations, and (b) quadratic nonlinea rity of the fluid and of the formation. Numerical results for weak nonlinea rity and narrow-band pulses indicate that pulse self-demodulation does occu r, but over relatively large distances because of the relatively low-freque ncy band relevant for tube wave propagation in characteristic borehole geom etries (f<10 kHz). The self-demodulated pulse shape can be very significant ly distorted from that predicted by the conventional Burgers equation, depe nding upon the choice of relevant parameters such as the permeability, the carrier frequency, and the mudcake membrane stiffness. Numerically exact an alytical formulas for the self-demodulated pulse shape, as well as for the energy in the second harmonic band, are derived for cases in which the puls e duration is long and the nonlinearity is relatively weak. These formulas are valid for any arbitrary dispersion/attenuation mechanism, and not just tube waves in permeable formations, as long as the propagation wave vector may be specified uniquely as a function of frequency. (C) 1999 Acoustical S ociety of America. [S0001-4966(99)03006-4].