Elastic-Wave Imaging and Characterization of Fractures With Interfacial Stiffness
This project is an extension of the researchers' recent work on the shape reconstruction of convex obstacles in the high-frequency regime via the method of topological sensitivity (TS). The purpose of this investigation is to not only reconstruct the shape of fractures embedded in a three-dimensional elastic (or acoustic) medium, but also characterize their interfacial condition from far-field (or boundary) measurements. The effect of the surface roughness, fluid, or proppant at the fracture interface is modeled by a distribution of normal and tangential stiffnesses according to the Schoenberg’s linear slip model. This investigation aims at finding these spatial distributions in addition to the geometry of the hidden fracture. Based on their previous work, the reseachers believe this goal is plausible in high excitation frequencies. Thus, in the first step, proper asymptotic analysis will be performed on the TS formula based on the method of multipole expansion, the geometric theory of diffraction, and its generalization where necessary, asymptotic approximations for the scattered field (and its derivatives) on the surface of the fracture will be developed. This gives rise to a TS expression involving Fourier-type surface integrals with highly-oscillatory kernels over the surface of the crack. To evaluate these integrals, in the second step, the method of stationary phase along with the asymptotic expansions from the catastrophe theory are invoked. The quality of both geometrical and interfacial reconstructions is examined in the single incident to full source aperture cases. The final closed-form approximation sheds light on the ability of the TS as a non-iterative imaging tool. Such results are of interest in many research areas such as hydraulic fracturing and non-destructive testing of materials. Numerical experiments are mandatory complements to the entire process from observation-inspired derivations to validation of the final results and due to the high-frequency nature of the problem considerable computational efforts is naturally involved in this project.
A bibliography of this group’s publications is attached.
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