Inversion for two sets of fracture weaknesses using differences between azimuthal reflection coefficients - Theory and synthetic examples

Huaizhen Chen, Junxiao Li, Kristopher A. Innanen

We first express stiffness parameters in terms of two sets of fracture weaknesses for the case of fractured rock consisting of two orthogonal sets of vertical fractures. Using perturbations in these stiffness parameters for the case of an interface separating an isotropic layer and a fractured layer, we derive a linearized P-to-P reflection coefficient as a function of two sets of fracture weakness, and we present differences in azimuthal reflection coefficients in terms of the tangential fracture weaknesses and the normal fracture weakness intercepts. Using the differences in azimuthal reflection coefficients, we establish an approach of estimating the tangential fracture weaknesses and the normal fracture weakness intercepts following a Bayesian framework. Synthetic tests confirm that the unknown parameter vector involving the tangential fracture weaknesses and the normal fracture weakness intercepts is estimated stably and reliably in the case of signal-to-noise ratio of 2.