Mathematical Modeling of Spatial Wave Processes in Fractured Seismic Media

Abstract

This chapter is devoted to the study of the propagation of elastic waves in a fractured seismic medium using methods of mathematical modeling. The obtained results are compared with the results of physical modeling on similar models. For mathematical modeling, a grid-characteristic method is used with 1-3-order hybrid schemes with approximation on unstructured triangular meshes (2D case) and tetrahedral meshes (3D case). Such meshes make it possible to specify inhomogeneities (fractures) of various complex shapes and spatial orientations. There is a description of developed mathematical models of fractures, which can be used for numerical solution of exploration seismology problems. The base of developed models is the concept of infinitely thin fracture, which aperture does not influence on wave processes in fracture area. These fractures are represented by boundaries and contact boundaries with different conditions on the contact and boundary surfaces. Such approach significantly reduces the computational costs due to the absence of the mesh definition inside the fracture. On the other side, it lets to state the fractures discretely in integration domain. Therefore, one can observe qualitative new effects such as diffractive waves forming and multiphase wave front due to multiple reflections between surfaces of neighbor fractures. These effects are not available to observe using effective models of fractures, actively applied in computational seismology.