Elastic impedance equations for vertically fractured media

Abstract

Equations for elastic impedance (EI) are derived for vertically cracked rocks. Transverse isotropy with horizontal axis of symmetry (i.e., the HTI model) is used for simulating first–order effects of azimuthal anisotropy caused by vertical small fractures. Two distinct approximations for PP–wave reflection (RPP) coefficients in HTI media are introduced into previous isotropic formulation for the EI equation. The general expression for both approaches is presented as a product of two terms. The resulting equations differ only in the correction to anisotropy, i.e., the term which controls incorporation of the effects of anisotropy into the EI technology. Due to the use of distinct anisotropy parameters and the definition of reference isotropic velocities, inaccuracies are found in computing approximate RPP coefficients with the derived EI equations. These inaccuracies are mainly attributed to the use of both the Thomsen’s parameter ± and the S– wave splitting parameter °(H) by one of the derived EI equations. Assumption of invariability for the squared ratio between the average of S–wave to P–wave reference velocities seems to have negligible impact on the above–mentioned inaccuracies.