Generalization of Schoenberg's linear slip model to attenuative media: Physical modeling versus theory

Abstract

We propose a mathematical formalism of introducing attenuation in Schoenberg's Linear Slip model. This formalism consists in replacing the real-valued weaknesses, normal and tangential, entering in the stiffness matrix by the complex-valued quantities. To test the validity of this procedure, an ultrasonic experiment was performed with the use of a plate-stack model made from plexiglass plates. We measured velocities and attenuations of P-, SH-and SV-wave versus the angle between the ray and the symmetry axis of TI model for air-filled and oil-saturated state of the model under uniaxial pressure of 2 and 4 MPa. The data on velocities and attenuations were fitted by the derived theoretical functions. We have estimated the values of the imaginary and real parts of the complex-valued weaknesses. The real parts of the weaknesses are threefold for the air-filled model in comparison with the oil-saturated one. The imaginary parts of the weaknesses, responsible for attenuation, are one order of magnitude less than the values of the real parts of the weaknesses. Both P-wave anisotropies, the velocity anisotropy and the attenuation anisotropy, are greater in the air-filled model than in the oil-saturated one. Besides, in the air-filled model, the symmetry-axis attenuation of the P-wave is much greater than the S-wave symmetry-axis attenuation, whereas in the oil-saturated model these attenuations are similar.

Effective elasticity of a medium with many parallel fractures

“…In the text, we refer to this method as the generalized Schoenberg-Douma approach or, simply, the generalized approach. The linear slip model of Schoenberg and Douma (1988) can be extended to viscoelastic (Chichinina and Obolentseva, 2009) or poroelastic (Rubino et al, 2015) media. Analogously, the extension can be made to the generalized method.…”