Nearly perfectly matched layer implementation for time domain spectral element modelling of wave propagation in 3D heterogeneous and anisotropic porous media

Numerical modelling of wave propagation in the heterogeneous and anisotropic porous media is important for seismic exploration. When modelling time-domain poroelastic waves in an unbounded region, the imple-mentation of perfectly matched layer (PML) is a common technique to absorb the outgoing waves. The PML in terms of first-order equations has been well proposed. However, the first-order PML needs fundamental and nontrivial reconstructions when applied to the second-order systems in displacements, such as finite element method (FEM) and spectral element method (SEM). It is well known SEM is an efficient algorithm for simulating seismic waves for its high accuracy and geometrical flexibility. However, at present, few literature studies the implementation of PML into time-domain SEM for simulating waves in the 3D unbounded heterogeneous porous media. In this work, combining with auxiliary ordinary differential equations (ODEs), we first systematically extend nearly PML (NPML) into time-domain SEM in terms of the second-order poroelastic system. The scheme preserves the original form of wave equations, leading to a simple and flexible incorporation into the existing algorithm. It avoids the temporary convolution through solving a set of first-order ODEs in time domain, resulting in high computational efficiency. Furthermore, to solve the coupling poroelastic/acoustic/elastic waves, we employ domain decomposition technique to handle the different regions independently, and apply the corresponding boundary conditions at the region interfaces to guarantee the coupling. Numerical examples demonstrate the accuracy and stability of the proposed method by comparing the numerical results with the analytical solutions and other independent numerical solutions. Finally, we apply the algorithm to solve wave propagation in a large-scale model involved a fluid-filled borehole embedded in heterogeneous porous formations.