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

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 ina heterogeneous anisotropic triple-porosity medium. We also summarize poroelastic wave equations in single- and double-porosity media. The traditional two-phase theory is a spe- cial case of the porous theory. Compared with single- or dual- porosity wave equations, our triple-porosity wave equation gen- erates more accurate wavefield information. The isotropic and anisotropic cases are considered. Subsequently, we formulate the new poroelastic equation into a first-order hyperbolic conser- vation system, which is suitable to be solved by DGM. An opti- mized 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 unstruc- tured 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 quantita- tively 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. Numeri- cal 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 poroelas- tic media.