D. L. Johnson
Schlumberger--Doll Res., Ridgefield, CT 06877-4108
A. N. Norris
Rutgers Univ., Piscataway, NJ 08855-0909
The nonlinear characteristics of an acoustic tube wave propagating along the axis of a fluid-filled circular borehole in an elastic solid which is locally isotropic but whose properties may vary radially are considered. The analysis is carried out in the quasistatic limit. All terms are considered through quadratic in the amplitude of the wave and the amplitude of 2nd-harmonic generation is expressed as well as the pressure dependence of the tube wave speed, dV[sub T]/dp, in terms of the fluid and formation of nonlinear parameters. The results show that if there is no radial variation of the shear modulus of the solid then both the amplitude of 2nd-harmonic generation and dV[sub T]/dp are independent of the third-order elastic constants of the solid and nearly equal to those of the fluid alone. If there is a radial variation of the shear modulus then our numerical calculations indicate that both the amplitude of 2nd-harmonic generation and dV[sub T]/dp can be completely dominated by the nonlinear parameters of the solid. A perturbation theory is derived that is valid for the case in which the shear modulus is nearly constant, which demonstrates that the nonlinear response is scaled by the value of the third-order parameters of the solid, leveraged by the degree and depth of alteration of the shear modulus.