Abstract

We solve, numerically, the equations of elastodynamics that govern the propagation of waves in a fluid-filled borehole intersected by one or more fluid-filled fractures, thus extending earlier work on formations presumed to be rigid. The model is axis symmetric, and it allows for arbitrary radial and vertical variations in the elastic properties of the formation. We have developed a novel gridding scheme that takes advantage of the assumed thinness of the fracture compared to any relevant wavelength; this technique obviates the necessity for a mesh within the fracture itself and thereby saves computational time and memory. We illustrate our technique by means of examples with single and double fractures with or without washouts, as well as examples with variable width fractures. We compute the frequency-dependent reflection and transmission coefficients of tube waves by fractures from the time waveforms generated by the present method. Major conclusions of this work are (1) reflections of tube waves by fractures can exceed the rigid frame prediction by as much as 50% even for “hard” formations, (2) a washout creates its own distinct signature on tube-wave reflectivity, and (3) reflectivity spectra alone cannot distinguish closely spaced multiple fractures, unless the separation is comparable to the wavelength of the tube wave.

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