Advances in Acoustics and Vibration

Volume 2018, Article ID 9681465, 9 pages

https://doi.org/10.1155/2018/9681465

## Asymptotic Structure of the Seismic Radiation from an Explosive Column

^{1}CODELCOTECH, Los Jardines 279, Santiago, Chile^{2}CODELCO CHILE, Gerencia Corporativa de Innovación y Tecnología, Huérfanos 1270, Santiago, Chile

Correspondence should be addressed to Marco Rosales-Vera; lc.hcetocledoc@100asorm

Received 27 September 2017; Revised 13 February 2018; Accepted 18 February 2018; Published 29 March 2018

Academic Editor: Emil Manoach

Copyright © 2018 Marco Rosales-Vera et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We study the structure of the seismic radiation in the far field produced by an explosive column. Using an asymptotic solution for the far field of vibration (Heelan’s solution), we find analytical expressions to the peak particle velocity (PPV) diagrams. These results are extended to the case of a charge with finite velocity of detonation.

#### 1. Introduction

The radiation of elastic waves produced by a cylindrical charge in a borehole has important applications in seismic exploration, mining geophysics, and blast vibration prediction. Today it is a well-known fact that, in the blasting engineering, the curves of blasting vibration in the near field and the far field show that vibration amplitudes induced by P wave and S wave are generally in the same magnitude, so the effect of S wave could not be ignored.

A pioneering theoretical work is that developed by Heelan (1953) [1, 2], where he study the radiation from a cylindrical source of finite length, the walls of which are subjected to symmetric lateral and tangential stresses. His papers were credited with demonstrating the generation of shear waves by artificial sources in a borehole; at the time this was a fundamentally surprising result. Subsequently, several authors have been making improvements to the results of Heelan, principally by Jordan [3], Hazebroek [4], and Abo-Zena [5]. One of the main advantages of the Heelan solution is its algebraic simplicity and clear physical interpretation, which allows easy implementation in more complex models.

An exact analytical solution of the problem solved by Heelan [2] was developed by Meredith [6]; Meredith et al. [7] and Meredith et al. [8] analyze the exact full-field solution for the axisymmetric wave radiation caused by any time-varying pressure load source acting over a finite length of a borehole. Tubman [9] and Tubman et al. [10] derive a general expression for the dispersion relations and the impulse response of a radially layered borehole.

Blair and Minchinton [11] analyze the near field of vibration from an explosive column; Blair [12] makes a comparison between Heelan solution and exact solutions for seismic radiation from a short cylindrical charge. Analytical solutions for describing Mach waves produced by infinitely long explosive columns were studied by Rossmanith and Kouzniak [13]. Blair [14] studied the blast vibration dependence on charge length, velocity of detonation, and layered media. Triviño et al. [15] studied seismic radiation patterns from cylindrical explosive charges by analytical and combined finite-discrete element methods. Kumar et al. [16] analyze the blast-induced ground vibration equations for rocks using mechanical and geological properties. Lee and Balch [17] generalize the results of Heelan to a fluid-filled borehole. The interaction between two adjacent blast holes has been studied by Yi et al. [18]. Triviño et al. [19] have studied the seismic waveforms from explosive sources located in boreholes and initiated in different directions. Using the Heelan results, Chen et al. [20] have studied the influence of millisecond time, charge length, and detonation velocity on blasting vibration.

In this work we study the structure of the seismic radiation in the far field produced by an explosive column. Using an asymptotic solution for the far field of vibration (Heelan’s solution), we find analytical expressions to the peak particle velocity (PPV) diagrams. The analysis is extended to the case of a charge with finite velocity of detonation, obtaining a first correction to the Heelan’s solution in the case of finite VOD. The results are compared with those existing in the literature.

#### 2. Mathematical Model

##### 2.1. Short Cylindrical Charge (Heelan’s Solution)

Consider a cylindrical explosive charge length and radius , centered at the origin of coordinates and embedded in an infinite elastic medium (Figure 1). The vertical walls of this cylinder are subjected to stresses which are symmetric about the vertical axis, uniform in the sense that they have the same instantaneous value at all points, and finite in time duration.