Andrew Loris Kurkjian

Calculation of transient tube‐wave signals in cross‐borehole acoustics

Closed‐form expressions are obtained for the transient acoustic pressure in a borehole, due to the action of a volume injection (acoustic monopole) source in another borehole in a typical cross‐well seismic setting with a homogeneous isotropic solid formation. At the relatively low frequencies involved the acoustic wave motion inside a fluid‐filled borehole, which may be surrounded by a structure of perfectly bonded circularly cylindrical solid shells, is dominated by tube waves. The excitation and propagation properties of the tube wave are modeled by regarding the borehole as an acoustic waveguide with a compliant inner wall. The corresponding elastic wave‐field quantities at the outer borehole wall are evaluated through a plane‐strain elastostatic transfer of the stress and the elastic displacement across the shell structure. For the radiation of the wave‐field quantities into the formation, the elastodynamic Kirchhoff–Huygens integral representation is used. The acoustic pressure on the axis of the receiving borehole is evaluated with the aid of the fluid/solid acoustic reciprocity theorem. Various physical phenomena are described by the resulting expressions, including pre‐ and postcritical phenomena (conical waves) for slow formations, and tunnelinglike phenomena for proximate boreholes in fast formations.

Tube Waves, Seismic Waves And Effective Sources

The radiation of waves from a monopole source in a fluid‐filled borehole into an elastic medium is considered. A simple asymptotic analysis, based on the smallness of the ratio of the borehole radius to the wavelength, reveals the interaction between tube waves and seismic waves. The pressure field in a tube wave acts as a secondary source of seismic waves and conversely an incoming seismic wave excites a tube wave. The asymptotic analysis leads to a characterization of these sources in terms of the solution to two‐dimensional elastostatic problems. These may be solved exactly when the borehole has an elliptical cross section even in an anisotropic formation. Also the borehole need not be straight provided that its radius of curvature is large compared with a wavelength. The problem of two boreholes, one containing a source and the other a receiver (crosswell tomography geometry) is analyzed and an explicit expression for the received signal is derived.

Farfield decomposition of acoustic waveforms in a fluid‐filled borehole

In acoustic well logging, a transient acoustic source is placed within a mud‐filled borehole for the purpose of exciting refracted compressional and shear waves in the surrounding formation. We model the downhole environment by an infinite fluid‐filled cylinder surrounded by an infinite solid formation. An isotropic point source is placed on the axis of the hole and an exact solution for the field in the hole is obtained in the frequency‐wavenumber domain. Space‐time waveforms are obtained by numerically transforming this solution. The waveforms consist of compressional and shear refracted waves plus modes, which are guided by the hole. In this paper we numerically compute these individual components of the field. The modes are obtained by locating poles and evaluating residues. The refractions are obtained via numerical branchcut integrations. A vertical branchcut is used which converges rapidly but only gives accurate results in the farfield.

Moving point mechanism representation for low frewuency monopole borehole sensors

In this paper we derive the equivalent line mechanism for the case of low frequency monopole sources and receivers in boreholes. This line mechanism is equal to a point mechanism which propagates at the speed of tube waves. This was first recognized and treated in White and Sengbush (Geophysics, 28, p. 1001-1019). The resulting far-field radiation (or receptivity) pattern is a generalization of that published by Lee and Balch (Geophysics, 47, p 1308-1314) in that it addresses the case where the shear velocity is close to or slower than the tube wave speed. The equivalent moving point representation can be incorporated into existing modeling codes, thereby avoiding the need to directly model the borehole. Furthermore, realistic borehole environments (such as casing, caliper logs, curved welk, anisotropic formations, etc) can be incorporated into the equivalent representation.