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 fractures. This paper presents a theoretical study and the UDEC modeling on the effects of multiple parallel planar fractures on the apparent attenuation of normally incident one-dimensional elastic waves. The case of normal incidence 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 discontinuity model, and to implicitly consider complex interfracture multiple wave reflections with the method of characteristics. The intrinsic attenuation mechanisms are neglected so that the attenuative effects of the multiple reflections can be concentrated on. This approach does not lose the discreteness of wave attenuation at individual fractures, and avoids the difficulty in explicitly determining the complex process of superposition of multiple reflected and transmitted wave fields. With this approach, a set of recurrence equations with respect to particle velocities before and after the fractures are established. These equations are numerically solved with sufficient accuracy. In analysis, there is no restriction to large fracture spacing or to small fracture spacing compared with wavelength. The magnitude of transmission coefficient (∣TN∣) for waves transmitting normally across a set of multiple parallel fractures is calculated as a function of the ratio (ξ) of fracture spacing to wavelength, for different normalized stiffness (k/zω) and for a different number of fractures. Parametric studies are conducted to examine the effects of multiple parallel fractures on wave attenuation, especially in terms of the spacing and the number of fractures. It is shown that the dependence of ∣TN∣ on the fracture spacing and the fracture number is governed by ξ. In addition, the effects of multiple wave reflections on ∣TN∣ are quantitatively discussed for different values of ξ. In the numerical modeling, the same problem is studied with a discontinuum-based numerical method, termed as the Discrete Element Method (DEM). The Universal Distinct Element Code (UDEC) is used to model one-dimensional wave propagation in rock masses containing no fractures, a single planar fracture and multiple parallel planar fractures. The modeling results of the transmission coefficient are compared with the theoretical solutions. An agreement between them has been achieved. It is verified that the UDEC is capable of modeling one-dimensional wave propagation across multiple parallel fractures.