A high order discontinuous Galerkin method for the symmetric form of the anisotropic viscoelastic wave equation

TL;DR

This paper introduces a high-order discontinuous Galerkin method for simulating wave propagation in complex anisotropic viscoelastic media, ensuring symmetry and accuracy in modeling stress-strain relationships with dissipation.

Contribution

It presents a novel symmetric formulation and numerical scheme for anisotropic viscoelastic wave equations, improving accuracy and stability in seismic modeling.

Findings

Method achieves high accuracy verified by convergence studies.

Effective in modeling complex inhomogeneous media.

Applicable in 2D and 3D simulations.

Abstract

Wave propagation in real media is affected by various non-trivial physical phenomena, e.g., anisotropy, an-elasticity and dissipation. Assumptions on the stress-strain relationship are an integral part of seismic modeling and determine the deformation and relaxation of the medium. Stress-strain relationships based on simplified rheologies will incorrectly predict seismic amplitudes, which are used for quantitative reservoir characterization. Constitutive equations for the rheological model include the generalized Hooke's law and Boltzmann's superposition principal with dissipation models based on standard linear solids or a Zener approximation. In this work, we introduce a high-order discontinuous Galerkin finite element method for wave equation in inhomogeneous and anisotropic dissipative medium. This method is based on a new symmetric treatment of the anisotropic viscoelastic terms, as well as an appropriate memory variable treatment of the stress-strain convolution terms. Together, these result in a symmetric system of first order linear hyperbolic partial differential equations. The accuracy of the proposed numerical scheme is proven and verified using convergence studies against analytical plane wave solutions and analytical solutions of viscoelastic wave equation. Computational experiments are shown for various combinations of homogeneous and heterogeneous viscoelastic media in two and three dimensions.