# Ellipsoidal impulse responses in fractured media

## TL;DR

This paper introduces an ellipsoidal approximation method for monoclinic fractured media to characterize azimuthal anisotropy in elastic-wave propagation, validated through numerical comparisons with exact solutions.

## Contribution

It develops a two-step procedure to estimate phase and group velocities in monoclinic media using ellipsoidal functions, applicable near the vertical axis of symmetry.

## Key findings

- Ellipsoidal functions in phase and group domains correspond in monoclinic media.
- The approximation accurately matches exact responses for different polar angles.
- Validation shows effectiveness for homogeneous fractured media.

## Abstract

Multiple vertical fracture sets, combined with horizontal fine layering produce an equivalent medium of orthorhombic or monoclinic symmetry. This is particularly important in fracture reservoir characterization. Fractured reservoirs are azimuthal anisotropic with respect to elastic-wave propagation. In this work we introduce an ellipsoidal approximation for monoclinic media that is able to characterized fractured media near the vertical axis of symmetry. The procedure is basically two-fold. First, we estimate phase velocities near the vertical axis using an expansion of the slowness. Secondly, the phase velocities are used to build the group velocities near the vertical axis. We particularly establish that for monoclinic media ellipsoidal functions in the phase domain correspond to ellipsoidal functions in the group domain. Finally, in order to validate the approximation, the $P$, $S_1$ and $S_2$ ellipsoidal impulse responses are compared for different polar angles with the exact responses obtained by solving numerically the eigenvectors problem from the Christoffel equation. Examples are shown for monoclinic media, and are validated showing results from a previous work for orthorhombic media. The whole procedure is valid for homogeneous media.