Extracting Heterogeneous Compliance of a Single Fracture from Seismic Scattering Coupled with Perturbation Theory

We have presented a new methodology to model the scattered wavefield due to a heterogeneous distribution of compliance along a single fracture and then to invert this compliance distribution or its power spectral density (PSD) from the scattered seismic response. We illustrate the validity through numerical examples using Gaussian PSD. First, we show that the perturbation theory offers very close result to that obtained from exact relation (wdSDD approach), by considering only the first 3 orders of perturbation. By solving the inverse scattering problem we can estimate the heterogeneous compliance distribution quite accurately. Next, in order to estimate the PSD of the heterogeneous compliance distribution, the normalized power of the stress field is used. We use the result from the wdSDD approach to represent the observed data. We show the possibility of successful extraction of the correct PSD by curve-fitting. The techniques presented here can lead to a conceptually new and accurate approach for estimating heterogeneous compliance distribution and the corresponding spatial variation in roughness and aperture along the fracture plane. This may, in turn, be related to the crucial fluid flow properties in the fractured subsurface.