SH propagation in rocks with planar fractures–I. Excess slowness

SUMMARY A model for a fractured rock is a transversely isotropic solid containing randomly spaced plane‐parallel fractures with random compliance. A fracture constitutive relation is the linear slip model, in which the jump in motion across the fracture is proportional to the stress. I study the propagation at normal and oblique angles of incidence of a shear wave with polarization parallel to the fractures, obtaining the frequency‐dependent excess fracture‐normal slowness and attenuation. I calculate these quantities using some new methods that are not restricted to weak anisotropy or to low frequency or to any particular fracture constitutive relation. These are Monte Carlo (MC), averaged‐multiple imbedding (AMI), path sum theory (PST) and a primary approximation (PA). Finite‐frequency results from these dynamic theories are compared with results from quasi‐static theory for a variety of rocks, all of which have the same quasi‐static anisotropy of 9.5 per cent. I find that (1) quasi‐static theory works well at finite frequencies for high‐ Q fractures if the fractures are stiff and closely spaced, but not if they are compliant and widely spaced; (2) the primary approximation works well if the interfracture distance distribution is exponential but not if it is narrow (periodic fractures) or broad (clumped fractures); (3) reducing the Q of the interfracture rock slightly increases the apparent anisotropy; (4) reducing the Q of the fracture filling material greatly reduces the apparent anisotropy; and (5) increasing the variance of either the fracture spacing distribution or the fracture compliance distribution increases the attenuation. A consequence of (4) is that estimates of fracturing derived from SH seismic data will be erroneously low if the fractures are low‐ Q. Elementary arguments suggest that overpressuring or lubrication increases the shear Q of natural fractures, thus making them more detectable with SH.