Numerical simulation of elastic wave propagation in fractured rock with the boundary integral equation method

Abstract

This paper describes a boundary integral equation method for simulating two‐dimensional elastic wave propagation in a rock mass with nonwelded discontinuities, such as fractures, joints, and faults. The numerical formulation is based on the three‐dimensional boundary integral equations that are reduced to two dimensions by numerical integration along the axis orthogonal to the plane of interest. The numerical technique requires the assembly and solution of the coefficient matrix only for the first time step, resulting in a significant reduction in computational time. Nonwelded discontinuities are each treated as an elastic contact between blocks of a fractured rock mass. Across such an elastic contact, seismic stresses are continuous and particle displacements are discontinuous by an amount which is proportional to the stress on the discontinuity and inversely to the specific stiffness of the discontinuity. Simulations demonstrate that such formulated boundary element method successfully models elastic wave propagation along and across a single fracture generated by a line source.

How to Incorporate the Spring-Mass Conditions in Finite-Difference Schemes

“…Coates and Schoenberg incorporate these conditions in a staggered-grid scheme using an equivalent medium theory [5]. Gu, Nihei and Myer follow a boundary integral approach [11], [12]. Delsanto and Scalerandi investigate the spring-mass conditions in the framewok of the local interaction simulation approach (LISA) [7].…”

Numerical modeling of elastic waves across imperfect contacts.

“…To our knowledge, only three approaches have been proposed for this purpose. First, Gu et al developed a boundary integral method which can be applied to arbitrarilyshaped interfaces [7]; but this method requires knowing the Green's functions on both sides of the interface, which complicates the study of realistic heterogeneous media. Secondly, a finite-element method has been proposed by Haney and Sneider [8].…”

Natural fracture characterization using passive seismic illumination

“…If the seismic wavelength is on the order or smaller than the fracture spacing, fractures behave as localized impedance contrasts that can produce a variety of frequency and fracture compliance sensitive waves, including converted waves, fracture interface waves (coupled Rayleigh waves; Pyrak-Nolte et al, 1992;Gu et al, 1996a), and fracture channel waves (Nihei et al, 1999;Nakagawa, 1998). In this study, we focus our efforts on investigation of the effects of discrete fractures on seismic waves generated by local seismicity (i.e., passive seismic illumination).…”

“…Boundary element approaches probably offer the highest accuracy and flexibility of available numerical methods for modeling discrete, finite-length fractures. Fractures can be modeled explicitly as displacementjump boundary conditions (Equation 2) with spatially varying fracture compliances (Gu et al, 1996a). Additionally, special crack tip boundary elements have been developed to preserve the singular nature of the stress field at the tips of the fracture (e.g., Zhang and Achenbach, 1988).…”