Estimation of normal and shear compliance for inclined fractures from full-waveform sonic log data

Fractures are ubiquitous throughout the Earth’s upper crust and represent localized zones of mechanical weakness as well as preferential pathways for fluid flow. Correspondingly, their detection and characterization is vital for a wide range of pertinent applications in geological, civil, and environmental engineering, hydrocarbon exploration, nuclear waste and carbon dioxide storage, as well as geothermal energy production. Particularly important mechanical characteristics of fractures are their normal and shear compliances, which relate the displacement perpendicular and parallel to the fracture plane, respectively, to the corresponding components of the prevailing stress tensor. Based on the linear slip model, previous works developed a phase delay method to estimate the normal compliance of individual fractures using the P-wave first-arrivals in full-waveform sonic (FWS) log data. This approach is viable for a quasi-normal incidence scenario of the sonic wavefield. However, the conditions under which this technique remains valid at oblique P- and S-wave incidence angles as well as the role played by the combined effects of the normal and shear compliances remains enigmatic. To alleviate this problem, we have extended the phase delay technique to allow for non-normally-incident P- and S-waves. In addition to improving the accuracy of the normal compliance estimates with respect to the results computed under a normal incidence assumption, this method allows for a simultaneous estimation of the normal and shear compliances. The proposed approach has been validated through analytical tests and numerical simulations of wave propagation in a hard-rock-type borehole environment intersected by a single fracture with dip angles of 0, 30, and 40 degrees with regard to the horizontal. For fracture compliance values typical of mesoscale fractures (10^-14 to 10^-12 m/Pa), the effects associated with oblique incidence become significant for dip angles larger than 50 and 30 degrees for P- and S-waves, respectively. However, our results also demonstrate that the normal incidence assumption can produce similar errors at even lower fracture dip angles in the presence of larger fracture compliance values and/or shear-to-normal compliance ratios. Finally, we apply the method to observed FWS data acquired in granitic rocks where the considered boreholes intersect fractures at a range of oblique angles. Direct in-situ estimates of compliances for discrete individual fractures are scarce, but essential to bridge the scale gap between laboratory estimates and input data for reservoir scale models. While recent studies show the feasibility of estimating normal compliances from FWS data, this study aims to explore whether and to what extent this approach can be practically extended to shear compliances and to the corresponding shear-to-normal compliance ratios.