Elastic Wave Scattering off a Fracture With Non-uniform Stiffness Distribution

Elastic wave scattering by a fracture in rock has been commonly modeled using the seismic displacement discontinuity model (SDDM). This model uses a parameter called fracture stiffness that linearly relates small relative displacement across a fracture and the applied stress. Theoretically the fracture stiffness is frequency-independent, since the SDDM is essentially a static model. However, laboratory measurements on the static and dynamic stiffness of fractures in rocks and synthetic materials often exhibit significant discrepancies, the dynamic stiffness being much higher than the static stiffness. In this study, we hypothesize that these discrepancies occur due to the violation of the assumptions of the SDDM: the characteristic dimension of the fracture such as roughness and contacting asperities should be much smaller than the seismic wavelength. To examine the effect of the random heterogeneous features on a fracture that alter the characteristics of the scattered seismic waves, we model the randomness as a distribution of fracture stiffness on the fracture plane. Using a semi-analytical, frequency-wavenumber domain technique based on the Fourier optics, the dynamic interaction between incoming plane waves and the fracture with random stiffness distribution can be computed to examine the frequency dependent (from static to high frequency dynamic) transmission and reflection coefficient of the fracture. It is noted that this approach is different from the Pyrak-Nolte et al (1990)'s work based on the geometrical optics. The presented technique provides not only statistical guidelines to correlate the dynamically measured stiffness of the fracture to the static stiffness but also a linear inversion scheme that images the distribution of fracture stiffness from measured seismic waves in the far field.