Fluid and solid motion in the neighborhood of a fluid‐filled borehole due to the passage of a low‐frequency elastic plane wave

Abstract

The exact problem of the interaction of an arbitrary plane wave in a homogeneous elastic medium with a fluid‐filled circularly cylindrical borehole is formulated to analyze three‐axis, full‐waveform borehole seismic data. The plane elastic wave (which may be incident upon the borehole from any direction) is partially scattered around the hole and an acoustic field is induced in the borehole fluid. Since seismic wavelengths tend to be much larger than diameters of boreholes, the exact solution may be expanded in powers of Ω (a small number), the incident wavenumber times the borehole radius. In the limit as Ω→0, the scattered field in the elastic medium is of O(Ω); in this limit the borehole crosssection is undistorted and moves with the velocity of the incident wave as if no borehole were present. Neglecting terms proportional to [Formula: see text] or higher leaves three kinds of θ‐dependent terms, where θ is the azimuthal angle about the borehole axis measured with respect to the wavenumber vector of the incident wave. Terms independent of θ (n = 0) give radial motion for incident P- or SV-waves, or rotational motion for incident SH-waves, and fall off as inverse distance from the borehole axis, i.e., 1/r. Terms proportional to cos θ for incident P- or SV-waves or sin θ for incident SH-waves (n = 1) give axial motion that falls off as 1/r. Terms proportional to sin 2θ or cos 2θ (n = 2) give motion in the plane perpendicular to the borehole axis, part of which falls off as 1/r and part as [Formula: see text]. At the borehole wall, the n = 2 terms express the deformation of the borehole cross‐section into an ellipse whose major and minor axes interchange as the wave passes. In the limit as Ω→0, the fluid velocity is constant over the borehole cross‐section. In the plane perpendicular to the borehole axis the fluid is constrained to move with the solid. However, in the axial direction the fluid velocity differs from that of the borehole wall for incident P- or SV-waves (for incident SH-waves the axial fluid and solid velocities both vanish as Ω→0). The ratio of axial fluid velocity to axial solid velocity at the borehole wall, for given elastic medium and acoustic fluid properties, is a function of both the wave type and the angle between the incident wavenumber vector and the borehole axis. Further, for incident P-waves, the axial fluid and solid velocities are in‐phase while for incident SV-waves, they are 180 degrees out‐of‐phase. Thus, as a plane wave passes a given level in a fluidfilled borehole, the plane‐wave type and (for P- or SV-waves) its angle of propagation relative to the borehole axis are theoretically obtainable by measuring fluid and solid axial velocities at that level.