Nonlinear tube waves in permeable formations: Difference frequency generation

We extend earlier work on nonlinear tube wave propagation in permeable formations to study, analytically and numerically, the generation and propagation of a difference frequency, ∆ω = ω1 − ω2, due to an initial pulse consisting of carrier frequencies ω1 and ω2. Tube waves in permeable formations have very significant linear dispersion/attenuation, which is specifically addressed here. We find that the difference frequency is predicted to be rather easily measurable with existing techniques and could yield useful information about formation nonlinear properties. Introduction A tube wave is an acoustic normal mode in which the energy is confined to the vicinity of a fluid-filled cylinder within an elastic solid. From a practical point of view it is generally the dominant signal which appears in a typical borehole-logging measurement and thus it is important in a variety of contexts in the search for hydrocarbon sources. One of these contexts lies in the fact that the tube wave may couple to fluid flow within the rock formation if the latter is permeable. The linearized tube wave propagation in this regime has been extensively studied both theoretically and experimentally [Winkler, et al., 1989] (see also [Pampuri, et al., 1998] and references therein). In the present article, we use a model of the tube wave due to [Liu and Johnson, 1997]. According to 2 the model, the fluid in the borehole is separated from the porous formation by an elastic membrane (mudcake) of finite thickness. As a tube wave propagates, the membrane flexes in and out of the pores, thus forcing the fluid to flow through the formation. This leads to the coupling between the tube wave and the acoustic slow wave in the formation, which in turn leads to attenuation and dispersion of the tube waves. In formations of moderate to large permeability, this mechanism is the largest known source of attenuation/dispersion of the tube wave and is the reason why it is specifically considered in the present article. Quite apart from this effect it is also known that sedimentary rocks have very large coefficients of nonlinearity and so [Johnson, et al., 1994] developed a theory for nonlinear tube waves neglecting the effects of the permeable formation. Later, [Johnson, 1999] combined this theory with the linearized theory of [Liu and Johnson, 1997] to describe a situation when the two effects are simultaneously present. As a numerical demonstration of the theory, [Johnson, 1999] considered the propagation of a narrow-banded (long duration) pulse consisting of a single carrier frequency. He showed that for realistic system parameters, the main signal (the fundamental) quickly decays, but before completely disappearing it generates a second harmonic and a low-frequency band (the “self-demodulated” pulse) both of which are due to the nonlinearity of the problem. The second harmonic decays even faster than the carrier, with the result that the self-demodulated pulse eventually dominates the entire signal at large enough distances. Because the second harmonic decays so fast, often it is advantageous to determine nonlinear characteristics by using pulses which consist of two different carrier frequencies, ω1 and ω2. In addition to the second harmonics (above) nonlinear effects lead to the generation of a signal centered around the difference frequency ∆ω = ω1 − ω2. This component may be reasonably energetic while at the same time it is not attenuated as much as either second harmonic, or even as much as either carrier frequency. Thus, in this article we are motivated to consider the propagation of two narrow-banded pulses whose frequency separation ∆ω is, say, 10% of the central frequency. Moreover, because ∆ω is not that different from ω1,2, it is often possible to measure its amplitude with the same acoustic transducers as for the fundamental. The organization of the article is as follows. First, we review the theory and derive analytical results for the nonlinear propagation of a pulse consisting of two different frequencies. We derive an approximation for 3 the propagation of the entire signal and we find an analytical form for the energy of the band with frequency ∆ω. Next, we report results of numerical calculations for a few different parameter sets and we show a good agreement between our analytical and our numerical results. In the last section, we make a brief summary of our work. Theory The dispersion and attenuation of the linear tube wave propagation has been studied in [Liu and Johnson, 1997]. To simplify our discussion we use an approximate form of the dispersion relation from [Johnson, 1999]: kz(ω) = ω [