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Low-frequency, axially-symmetric guided waves which propagate along a fluid-filled borehole (tube waves) are studied in order to characterize the hydraulic fractures intersecting the borehole. We formulate a new equation for the total tube wavefield, which includes simultaneous effects of (1) tube-wave scattering (reflection and transmission) due to wave propagation across hydraulic fractures, and (2) tube-wave generation due to incident plane P waves. The fracture is represented by the nonwelded interface boundary conditions. We use an appropriate form of the representation theorem in order to correctly handle the multiple scattering due to nonwelded interfaces. Our approach can implement any model that has so far been developed. We consider a recent model which includes simultaneous effects of fluid viscosity, dynamic fluid flow, and fracture compliance. The derived equation offers a number of important insights. We recognize that the effective generation amplitude contains the simultaneous effect of both tube-wave generation and scattering. This leads to a new physical understanding indicating that the tube waves are scattered immediately after generation. We show that this scattering is nonlinear with respect to interface compliance. This physical mechanism can be implicitly accounted for by considering more realistic boundary conditions. We also illustrate the application of the new equation in order to predict the complex signature of the total tube wavefield, including generation and scattering at multiple hydraulic fractures. A new formulation for focusing analyses is also derived in order to image and characterize the hydraulic fractures. The obtained results and discussions are important for interpretation, modeling, and imaging using low-frequency guided waves, in the presence of multiple fractures along a cylindrical inclusion.

Low-frequency guided waves in a fluid-filled borehole: Simultaneous effects of generation and scattering due to multiple fractures