For seismic exploration, the coupling problem between different physical fields is unavoidable. The operation of the coupling interface is very tricky. For this challenge, we propose a unified numerical approach to overcome it. Specifically, we construct a model framework, which contains various physical phenomena, and solve it by using the discontinuous Galerkin (DG) method allowing for the jump of the solution across the interface between the fluids and the solid materials. The viscoelasticity and fracture are first incorporated into the coupled model. The model includes the anisotropy, viscoelasticity, and fracture properties. The Riemann solver is employed to handle the coupling interfaces: the acoustic-elastic contact, the acoustic-viscoelastic contact, and the elastic-viscoelastic contact. Any additional treatment is not required for the complex coupling interface. Several examples are used to demonstrate the validity and flexibility of the proposed numerical scheme and the corresponding results show that our approach can handle more complex coupled models efficiently.

We present an innovative physics-informed numerical flux within the framework of the discontinuous Galerkin (DG) method for solving elastic wave equations in 2-D heterogeneous media that include interfaces between elastic materials. The first-order velocity–stress equations are used, which can be written in the formulation of hyperbolic system and can be easily incorporated into the framework of DG method. Numerical flux is carefully proposed to maintain the physical laws on both sides of the interface where material parameters are discontinuous. We compare the newly suggested numerical flux with both the classic local Lax–Friedrichs flux and the exact upwind flux. The latter is derived from solving the Riemann problem and adheres to the Rankine–Hugoniot condition. In the comparison, we find that our flux is formally similar to the classic local Lax–Friedrichs flux, but the numerical behavior is different from it; its numerical performance is similar to the exact upwind flux. The biggest advantage of the numerical flux we propose is that it does not need to accurately solve the Riemann problem, but it can maintain the physical continuity conditions near the interface with material discontinuities. We present three numerical examples of elastic wave propagation in heterogeneous media, including horizontal and inclined interfaces, and the Marmousi model. The numerical results demonstrate the effectiveness of this flux.

We present a p-adaptive discontinuous Galerkin (DG) approach for solving second-order elastic and acoustic wave equations. An arbitrary anisotropic medium is considered, and the first- to fourth-order polynomials are used. The second-order wave equation is first transformed into a unified first-order hyperbolic system that is suitable for the DG method. In the following, we describe the p-adaptive DG algorithm in detail. The adaptation criterion using the area indicator is adopted, and the order of polynomial applied for each subdomain in the whole computational domain is marked before time evolution. The adaptation used for wave propagation simulation is practical and flexible, and various numerical fluxes can be directly incorporated into this p-adaptive DG algorithm. Two numerical examples are used to demonstrate the performance of the proposed adaptive scheme.

Transversely isotropic (TI) medium is a widely studied anisotropic solid medium in seismology. The numerical simulation of seismic wave propagation in TI media is an effective tool to analyze the mechanism of seismic waves in complex anisotropic media. In this study, we introduce a double-weighted Runge–Kutta discontinuous Galerkin (RKDG) method for numerically solving wave propagation problems in 3-D TI media with surface topography. This method incorporates the discontinuous Galerkin spatial discretization with an explicit double-weighted two-step iteration time discretization. The local Lax–Friedrichs flux is used as the numerical flux in the formulations. This method can solve the first-order velocity–stress seismic wave equations, including TI media and more general anisotropic media. Due to the large scale of 3-D problems, parallel technology is adopted. Regions with irregular boundaries are discretized into unstructured tetrahedral meshes. Numerical experiments for various anisotropic media, including the vertical and tilted TI media, are presented. The results demonstrate the effectiveness of the double-weighted RKDG method in wavefield simulations in 3-D complicated anisotropic media.

In the field of geophysics, the Helmholtz decomposition (HD) formula is mainly numerically discretized by the finite difference method (FDM), which limits its application to a uniform regular grid only. Few scholars note wavefield separation on unstructured grids. In this study, we aim to develop a wavefield separation algorithm to separate P- and S-wavefields on nonuniform grids. Our scheme is based on an isotropic elastic wave equation. We first transform the HD formula into a weak integral form using the discontinuous Galerkin method (DGM). Then, we consider two types of unstructured meshes—triangle and quadrangle, which are more suitable for complex structures. Moreover, to reduce time costs, the single graphic processor unit (GPU) device is used to improve the computational efficiency. We perform a unified DGM operator by transforming unstructured triangles and quadrangles into standard reference elements using coordinate transformation. Our proposed HD operator enables us to effectively separate P- and S-wavefields on unstructured meshes. We carry out the wavefield separation simulation and calculate the numerical solutions in the homogeneous carbonate model, Graben model, and SEG/EAGE model. The homogeneous model verifies the correctness, availability, and superiority of our proposed separated operator, and the other numerical results show excellent performance for P/S-wavefield separation on unstructured meshes.

Elastic wavefield separation has attracted much attention in the field of geophysics, but the focus is mainly on a single solid medium, which is hardly used in the coupled medium. We propose an effective discontinuous Galerkin (DG) approach with the exact Riemann solver for modeling decoupling P- and S- wavefield propagation in coupling solid-fluid media. We apply the selective strong attenuation to P- or S-waves by the viscoelastic theory while keeping the other wave mode. The viscoelastic wave equation is first cast into the decoupling wave equation. Following, we derive a novel exact Riemann solver that can accurately couple with the fluid and the decoupling elastic media. Finally, we use the DG method to solve P/S decoupling wave equations in triangular and tetrahedral meshes. The original wavefield is accurately decomposed into P and S wavefields, whose phases and amplitudes are consistent with those of the elastic data. Several numerical experiments are used to demonstrate the accuracy and capacity of our developed P/S-separated approach.

Poroelastic wave equations incorporating fluid information have been paid much attention to, however, the conventional perfectly matched layer (PML) scheme fails to absorb poroelastic waves in numerical modeling. In this work, we construct a unified framework of the combination of the low- and high-order unsplit complex-frequency-shifted (CFS) -PMLs with the discontinuous Galerkin (DG) method in poroelastic media. We have derived three kinds of PML-corrected poroelastic wave equations, including the split PML formula, the low-order unsplit CFS-PML formula, and the high-order unsplit CFS-PML formula, which are all first-order hyperbolic form. We give a general scheme for transforming the second-order poroelastic equation into a first-order hyperbolic system. These PML-corrected formulations can be numerically solved by the DG method for simulating wave propagation in single-, double-, and triple-porosity media. Several numerical experiments in isotropic and anisotropic multi-porosity media are used to demonstrate the effectiveness of the low- and high-order unsplit auxiliary differential equation (ADE) CFS-PML boundary conditions for wavefield simulation. The model with a sinusoidal irregular interface is discretized into unstructured triangular meshes, and its results are to validate the flexibility of a combination of the low- and high-order unsplit CFS-PMLs with the DG method for simulation of seismic wave propagation in poroelastic media.