Wavefield simulation with the discontinuous Galerkin method for a poroelastic wave equation in triple-porosity media

Abstract

In this paper, we derive a new poroelastic wave equation in triple-porosity media and develop a weighted Runge-Kutta (RK) discontinuous Galerkin method (DGM) for solving it. Based on Biot’s theory and Lagrangian formulas, we obtain 3D Biot’s equations in a heterogeneous anisotropic triple-porosity medium. We also summarize poroelastic wave equations in single- and double-porosity media. The traditional two-phase theory is a special case of the porous theory. Compared with single- or dual-porosity wave equations, our triple-porosity wave equation generates more accurate wavefield information. The isotropic and anisotropic cases are considered. Subsequently, we formulate the new poroelastic equation into a first-order hyperbolic conservation system, which is suitable to be solved by DGM. An optimized local Lax-Friedrichs flux and an implicit-weighted RK time discretization scheme are used for this computation. We use two types of mesh elements — quadrilateral and unstructured triangular elements. We find that there are two kinds of slow P waves, P1 and P2 waves, in double-porosity media, whereas three kinds of slow P waves, P1, P2, and P3 waves, exist in three-porosity media. We also study the analytical and numerical solutions of propagation velocities for different waves in isotropic media without dissipation using the Jacobian matrix of DGM and provide a comparison of field variables about three types of wave equations. Finally, we conduct a series of examples to quantitatively investigate the propagation properties of seismic waves in isotropic and anisotropic multiporosity media computed by DGM. The slow P wave in multiporosity media with dissipation decays rapidly, which also will lead to phase distortion. Numerical results verify the correctness and applicability of our proposed new equation and indicate that the weighted RK DGM is a stable and accurate algorithm to simulate wave propagation in poroelastic media.