Extension of the kinematic approximations to the multilayered elastic orthorhombic medium
Understanding the kinematics of horizontally-layered reservoir rocks is important to their proper characterization and to accomplish this it is necessary to specify the explicit model for these kinematic properties. The accurate approximations for traveltime and relative geometrical spreading in an elastic homogeneous orthorhombic (ORT) have been investigated with different forms: Shifted Hyperbola Form (SHF), Taylor Series (TS) and the Rational Form (RF). This paper extends these approximations to the multi-layered ORT model by adopting composite coefficients and effective model parameters. The multi-layered model is characterized without and with the azimuthal variation among layers. There is an overdetermined problem when the azimuthal variation exists; and to address that case, the Least Squares Method (LSM) is adopted. To check the feasibility of the expansion, we select the SHF (Shifted Hyperbola Form) approximation specified in the homogeneous elastic ORT model for the calculation in the numerical example. Four groups of examples are analyzed to investigate the influence on the accuracy of the approximation with the change in rotation angle, degree of anisotropy, and the direction of the orientation. The results indicate that, for the multi-layer, the accuracy of the approximation is proportional to the degree of anisotropy and the value of the angle of rotation. The relative errors in travel time and relative geometrical spreading in this multi-layered extension are very small and can be implemented in practical applications