We tested two models, one for tube-wave generation and the other for tube-wave attenuation at a fracture intersecting a borehole that can be used to estimate fracture compliance, fracture aperture, and lateral extent. In the tube-wave generation model, we consider tube-wave excitation in the borehole when a P-wave is incident on the fracture. The amplitude ratio of the pressure due to the tube wave to that of the incident P-wave is a function of fracture compliance, aperture, and length. Similarly, the attenuation of a tube wave in the borehole as it crosses a fracture intersecting the borehole is also a function of fracture properties. Numerically solving the dispersion relation in the fracture, we study tube-wave generation and the attenuation coefficient as a function of frequency. We observed that measuring amplitude ratios or attenuation near a transition frequency can help constrain the fracture properties. The transition frequency corresponds to the regime in which the viscous skin depth in the fracture is comparable to its aperture. Measurements in the high-frequency limit can place a lower bound on fracture compliance and lateral extent. We evaluated the applicability of the tube-wave generation model to a previously published VSP data set and found that compliance values of the order 1010109m/Pa are likely in the field. These observations support scaling of fracture compliance with fracture size.

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