Elastic scattering and inversion for the spatially heterogeneous distribution of compliance of a single fracture

The elastdynamic response of a fracture is often modeled using the linear-slip model (LSM) for the fracture compliance. In earlier theoretical and laboratory studies, the distribution of compliance along the plane of a fracture has generally been assumed to be homogeneous. However, naturally occurring fractures are spatially heterogeneous, with the microscale properties varying along the fracture plane. The spatial heterogeneity of the microscale parameters along the fracture plane, e.g., roughness, contact area and distribution of fluid filled aperture, controls significantly the mechanical and hydraulic response of a fracture. When the fracture compliance is spatially heterogeneous, an incident elastic wavefield will be scattered at the fracture plane. This scattered wavefield contains information of the spatial heterogeneity of fracture compliance. In this study, we show through numerical modeling that the scattered elastic wavefield is sensitive to the spatial heterogeneity in compliance distribution. We find that the back-scattered elastic wavefield from a spatially heterogeneous fracture appears as the coda of the specular reflection, with amplitude differing from that for a homogenous fracture compliance. An analysis of the scattered wavefield does reveal the spatial heterogeneity along the fracture plane. In order to estimate the spatially heterogeneous compliance distribution, we have developed an inversion scheme. The scheme has the following two steps: (1) extrapolating the recorded back-scattered elastic wavefield and estimating the stress field at the fracture plane, and (2) solving the boundary condition of LSM using the estimated stress field. We illustrate this new method through numerical examples mimicking laboratory-scale measurements (Figure). In the low frequency, the estimated compliance distribution is smooth and inaccurate because of the presence of the evanescent waves. However, at the peak frequency, the compliance distribution can be accurately estimated (Figure). We find that this approach is equally effective also in the field scale, giving larger-scale spatial variation of the fracture compliance. These results illustrate the possibility of characterizing the spatial heterogeneity along a single fracture from the back-scattered elastic wavefield. Result of inversion (on the right) for spatially heterogeneous normal compliance distribution for a geometry shown on the left side. Δx is receiver spacing.