Dynamic Response of a Fractured Tunnel to Seismic Waves, 14-9116

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Principal Investigators
Jorge O. Parra
Chris L. Hackert
Amitava Ghosh
Pei-cheng Xu

Inclusive Dates: 01/15/99 - 01/15/2000

Background - An underground excavation such as tunnel, mine opening or permanent geological repository for nuclear wastes in a rock mass has to contend with the presence of prominent joints or joint sets or faults. The strength, integrity, and stability of the excavation depend significantly on the characteristics of the fracture pattern as well as its strength and stiffness properties. In particular, the response of an excavation subjected to dynamic events such as earthquakes, underground explosions, blasts, or rock bursts can be critically controlled by nearby fractures in the rock. The dynamic interaction between the cavity wall and fractures causes increased displacements and stresses in the rock in a complex manner. Thus, a thorough understanding of the dynamic response of fractures intersecting the excavation to incoming waves in various frequency ranges is important for the excavation's safety, including preventing rock falls and keeping the excavation stable and usable.

The static analysis of the combined response of fractured tunnels has already been reported. However, the reduction of the stiffness and integrity of the excavation due to fractures is far more significant under dynamic loads than static loads. In particular, upon oblique incident waves, the cavity acts as a wave guide, longitudinally, circumferentially, or both. Studies by the Lawrence Berkeley National Laboratory have shown that fractures also serve as hosts for guided waves. Guided waves are known to be energy focused and less attenuated than bulk waves. The composite guide waves due to both the excavation and fracture are more pronounced and complicated then that due to the excavation or fracture alone. The dynamic response of cavities and fractures to incoming waves has been studied extensively but separately in seismic literature. To the author's best knowledge, no quality models prior to SwRI work currently exist to predict the dynamic response of an excavation in a fractured host medium to incoming waves.

Approach - The research team model the fracture by a slip interface that has been extensively analyzed, verified by experiments, and documented in the literature. Across the interface, the stresses are nonzero and continuous, and the normal and shear displacement jumps are proportional to the normal and shear stresses, respectively. The slip stiffnesses are normalized to include the influence of the wavelength, and a boundary integration equation (BIE) approach is adopted. In the BIE method, only the boundaries of the cross section of the excavation and the fracture are discretized. Although the resulting matrix is not banded or symmetric, its size is several orders of magnitude smaller than that in the finite element method (FEM). Another advantage of the BIE method is its efficiency in solving problems involving a larger number of source-detector configurations. In this case, the total computational cost is merely slightly higher than that for a single source-detector pair. Furthermore, in the BIE model, the host medium surrounding the excavation and fracture is unbounded. Therefore radiation conditions at infinity are automatically satisfied, in contrast to the artificial radiation conditions imposed in the finite difference method (FDM) and FEM. In addition, certain frequency-dependent properties, such as viscoelasticity, poroelasticity and permeability are of interest. FDM is incapable or inefficient in these aspects.

In the BIE, the unknown displacement components on the excavation boundary are coupled in boundary integral equations. The coefficients of the unknown quantities are the stress tensors associated with the Green's functions of the unbounded host rock in the absence of the tunnel and fractures. The displacements along the free surface of the excavation are then solved by the linear equations resulting from discretization of the integral equation. Finally, displacement and stresses at any given field location can be determined through direct evaluation of the boundary integral using the previously obtained boundary displacements.

Accomplishments - Using the above approach, the team has implemented a robust computer algorithm for calculating the dynamic displacements and stresses of intact and fractured excavations of various shapes impinged by plane seismic waves in an otherwise uniform unbounded medium. Benchmark analytical solutions were used to verify the accuracy of this algorithm.

Numerical results were obtained for different excavation shapes such as circle, slot, loaf, and dome intercepted by a fracture or fractures. Hoop stress on the surface of the excavation, as well as displacement and stress fields around the excavation, were computed and plotted in various forms. These results involve typical values of slip stiffness, fracture location, and orientation, frequency, and incident angle. The team found that an intersecting fracture re-distributes and usually amplifies the stress along and around the surface of the cavity. For certain combinations of the above parameters, the stress amplification and complication can be significant.

An example is given in the illustration below. A 0.25-kilohertz P-wave is impinging, at 45°, a dome cavity (5 meters in height and 3 meters in width) with a horizontally crossing fracture at a height of 3 meters. The parameters of the host medium include: P wave speed , 1.754 kilometers per second; S wave speed, 1.0 kilometer per second; and mass density, 1.0 gram per cubic centimeter. Hoop stress is plotted on the boundary. The normalized normal and shear slip stiffness values used were 0 (free fracture), 2, 16, 400, and infinity (nonslip). It can be seen that the strongest stress concentration occurs not when the fracture is free, but when the slip stiffness is moderate. For very high values of slip stiffness, the hoop stress profile is approaching back toward that of the nonslip case.

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