Effects of multiple parallel fractures on apparent attenuation of stress waves in rock masses

Stress waves propagating through multiple parallel fractures are attenuated (and slowed) due to multiple wave reflections and transmissions at the fra ctures. This paper presents a theoretical study and the UDEC modeling on th e effects of multiple parallel planar fractures on the apparent attenuation of normally incident one-dimensional elastic waves. The case of normal inc idence of waves is studied, because we want to remove the influence of the incident angle and to focus purely on the effects of stiffness, spacing and number of parallel fractures. In the theoretical study, an approach is developed to explicitly take into account the attenuative effect of each fracture with the displacement disco ntinuity model, and to implicitly consider complex interfracture multiple w ave reflections with the method of characteristics. The intrinsic attenuati on mechanisms are neglected so that the attenuative effects of the multiple reflections can be concentrated on. This approach does not lose the discre teness of wave attenuation at individual fractures, and avoids the difficul ty in explicitly determining the complex process of superposition of multip le reflected and transmitted wave fields. With this approach, a set of recu rrence equations with respect to particle velocities before and after the f ractures are established. These equations are numerically solved with suffi cient accuracy. In analysis, there is no restriction to large fracture spac ing or to small fracture spacing compared with wavelength. The magnitude of transmission coefficient (\ T-N\) for waves transmitting normally across a set of multiple parallel fractures is calculated as a function of the rati o (xi) of fracture spacing to wavelength, for different normalized stiffnes s (k/zw) and for a different number of fractures. Parametric studies are co nducted to examine the effects of multiple parallel fractures on wave atten uation, especially in terms of the spacing and the number of fractures. It is shown that the dependence of \ T-N\ on the fracture spacing and the frac ture number is governed by xi. In addition, the effects of multiple wave re flections on \ T-N\ are quantitatively discussed for different values of xi . In the numerical modeling, the same problem is studied with a discontinuum- based numerical method, termed as the Discrete Element Method (DEM). The Un iversal Distinct Element Code (UDEC) is used to model one-dimensional wave propagation in rock masses containing no fractures, a single planar fractur e and multiple parallel planar fractures. The modeling results of the trans mission coefficient are compared with the theoretical solutions. An agreeme nt between them has been achieved. It is verified that the UDEC is capable of modeling one-dimensional wave propagation across multiple parallel fract ures. (C) 2000 Elsevier Science Ltd. All rights reserved.