The Finite Difference （FD） method is an important method for seismic numerical simulations. It helps us understand regular patterns in seismic wave propagation, analyze seismic attributes, and interpret seismic data. However, because of its discretization, the FD method is only stable under certain conditions. The Arbitrary Difference Precise Integration （ADPI） method is based on the FD method and adopts an integration scheme in the time domain and an arbitrary difference scheme in the space domain. Therefore, the ADPI method is a semi-analytical method. In this paper, we deduce the formula for the ADPI method based on the 3D elastic equation and improve its stability. In forward modeling cases, the ADPI method was implemented in 2D and 3D elastic wave equation forward modeling. Results show that the travel time of the reflected seismic wave is accurate. Compared with the acoustic wave field, the elastic wave field contains more wave types, including PS- and PP- reflected waves, transmitted waves, and diffracted waves, which is important to interpretation of seismic data. The method can be easily applied to elastic wave equation numerical simulations for eoloical models.
Tiltmeter mapping technology infers hydraulic fracture geometry by measuringfracture-induced rock deformation, which recorded by highly sensitive tiltmeters placed atthe surface and in nearby observation wells. By referencing Okada＇s linear elastic theory andGreen＇s function method, we simulate and analyze the surface and downhole deformationcaused by hydraulic fracturing using the homogeneous elastic half-space model and layeredelastic model. Simulation results suggest that there is not much difference in the surfacedeformation patterns between the two models, but there is a significant difference in thedownhole deformation patterns when hydraulic fracturing penetrates a stratum. In suchcases, it is not suitable to assume uniform elastic half-space when calculating the downholedeformation. This work may improve the accuracy and reliability of the inversion results oftiltmeter monitoring data.
