A Hybrid Finite-Element Spectral-Element Domain Decomposition Method for Cross-Well Sensing

A hybrid finite-element spectral-element domain decomposition method (FESE DDM) is proposed for time-domain acoustic cross-well sensing simulations involving elastic waves in boreholes with complex geometries. This method strategically combines the mesh flexibility of finite element (FE) discretization with the computational efficiency of spectral element (SE) discretization to solve the elastic wave equations. Intricate borehole structures, such as grooved drill collars, are discretized using tetrahedral (TET) meshes with second-order FE basis functions, while larger formations outside the boreholes are discretized using hexahedral (HEX) meshes with high-order SE basis functions. To ensure weak continuity of the solution across nonconformal interfaces between FE and SE subdomains, a generalized isotropic Riemann flux is employed, supported by a dual-domain projection algorithm that enables accurate Gauss quadrature at the interface. The implicit-explicit Newmark-β (IMEX-N) temporal discretization scheme is introduced to solve the large-scale FESE DDM system, where the explicit subsystem efficiently handles the SE subdomains, and the implicit subsystem manages the FE subdomains as well as the interfaces between different subdomains. Comprehensive numerical examples demonstrate that the proposed method achieves both high accuracy and computational efficiency.