A nodal discontinuous Galerkin finite element method for the poroelastic wave equation

Abstract

We use the nodal discontinuous Galerkin method with a Lax-Friedrich flux to model the wave propagation in transversely isotropic and poroelastic media. The effect of dissipation due to global fluid flow causes a stiff relaxation term, which is incorporated in the numerical scheme through an operator splitting approach. The well-posedness of the poroelastic system is proved by adopting an approach based on characteristic variables. An error analysis for a plane wave propagating in poroelastic media shows a convergence rate of O(h^n+ 1). Computational experiments are shown for various combinations of homogeneous and heterogeneous poroelastic media.

Appendices

Appendix A: Solution of the stiff part

The system of equations represented by (47) is expressed as

$$\begin{array}{@{}rcl@{}} \partial_{t} v_{x}&=-\frac{\eta}{\kappa_{1}} \beta_{12}^{(1)} q_{x} , \end{array} $$

(64)

$$\begin{array}{@{}rcl@{}} \partial_{t} v_{z}&=-\frac{\eta}{\kappa_{3}} \beta_{12}^{(3)} q_{z} , \end{array} $$

(65)

$$\begin{array}{@{}rcl@{}} \partial_{t} q_{x}&=-\frac{\eta}{\kappa_{1}} \beta_{22}^{(1)} q_{x} , \end{array} $$

(66)

$$\begin{array}{@{}rcl@{}} \partial_{t} q_{z}&=-\frac{\eta}{\kappa_{3}} \beta_{22}^{(3)} q_{z} . \end{array} $$

(67)

The solution of Eqs. (64)–(67) is given as

$$\begin{array}{@{}rcl@{}} v_{x}&=&{v_{x}^{n}} + \frac{\beta_{12}^{(1)}}{\beta_{22}^{(1)}}[\exp \left( -\frac{\eta}{\kappa_{1}}\beta_{22}^{(1)} dt\right)-1]{q_{x}^{n}}, \end{array} $$

(68)

$$\begin{array}{@{}rcl@{}} v_{z}&=&{v_{z}^{n}} + \frac{\beta_{12}^{(3)}}{\beta_{22}^{(3)}}[\exp \left( -\frac{\eta}{\kappa_{3}}\beta_{22}^{(3)}dt\right)-1]{q_{z}^{n}}, \end{array} $$

(69)

$$\begin{array}{@{}rcl@{}} q_{x}&=&\exp \left( -\frac{\eta}{\kappa_{1}}\beta_{22}^{(1)} dt\right){q_{x}^{n}}, \end{array} $$

(70)

$$\begin{array}{@{}rcl@{}} q_{z}&=&\exp \left( -\frac{\eta}{\kappa_{3}}\beta_{22}^{(3)} dt\right){q_{z}^{n}}. \end{array} $$

(71)

Appendix B: Computation of λ in (57)

A plane-wave solution for the particle velocity vector V = [v_x,v_z,q_z,q_z]^T is

$$\begin{array}{@{}rcl@{}} \mathbf{V}=\mathbf{V_{0}}\exp[\text{i}(\mathbf{k}.\mathbf{x}-\omega t)], \end{array} $$

(72)

where V₀ is a constant complex vector and k is wave vector. Substituting (72) in (1)–(4) and (20)–(23) , we recover

$$\begin{array}{@{}rcl@{}} \left( \boldsymbol{\Gamma}^{-1} \cdot \mathbf{L} \cdot \mathbf{C} -V \mathbf{I_{4}} \right).\mathbf{V}= 0 , \end{array} $$

(73)

where

$$\begin{array}{@{}rcl@{}} \boldsymbol{\Gamma}&=&\left[\begin{array}{cccc} \rho & 0 & \rho_{f} & 0\\ 0&\rho & 0 & \rho_{f} \\ \rho_{f} & 0 & \text{i}Y_{1}(-\omega)/\omega & 0\\ 0& \rho_{f} & 0 & \text{i}Y_{3}(-\omega)/\omega \end{array}\right],\\ \mathbf{L}&=&\left[\begin{array}{cccc} l_{x} & 0 & l_{z} & 0\\ 0 & l_{z} & l_{x} & 0\\ 0 & 0& 0& l_{x}\\ 0 & 0& 0& l_{z} \end{array}\right]\\ ~~\\ \mathbf{C}&=&\left[\begin{array}{cccc} l_{x}c_{11}^{u} & l_{z} c_{13}^{u} & \alpha_{1} M l_{x} & \alpha_{1} M l_{z}\\ l_{x}c_{13}^{u} & l_{z} c_{33}^{u} & \alpha_{3} M l_{x} & \alpha_{3} M l_{z}\\ l_{z} c_{55}^{u} & l_{x} c_{55}^{u} & 0 & 0\\ \alpha_{1} M l_{x} & \alpha_{3} M l_{z} & M l_{x} & M l_{z} \end{array}\right],\\ \end{array} $$

with Y_i(ω) = iωm_i + η/κ_i and l_x and l_z being direction cosines and $V=\frac {\omega ^{2}}{k^{2}}$.

Term V in (73) represents the phase velocity of waves and can be computed by adopting the approach for eigenvalue computation. Thus

$$\begin{array}{@{}rcl@{}} \lambda_{i}&=&\left( \text{Re}(1/V_{i})\right)~~~\text{for}~~i = 1...4,\\ \text{and} \ \ \ \lambda&=&\max\left( \lambda_{i}\right) \end{array} $$

Energy velocity V_e can be computed from

$$\begin{array}{@{}rcl@{}} \mathbf{k^{T}}\cdot \mathbf{V_{e}}=\mathbf{V}. \end{array} $$

(74)

Appendix C: System of poroacoustic wave equation

This system is

$$\begin{array}{@{}rcl@{}} \partial_{t} \mathbf{q_{p}} + \mathbf{A_{1p}} \partial_{x} \mathbf{q_{p}} + \mathbf{B_{1p}} \partial_{x} \mathbf{q_{p}}=\mathbf{D_{1p}} \mathbf{q_{p}} + \mathbf{f_{p}} , \end{array} $$

(75)

where

$$\begin{array}{@{}rcl@{}} \mathbf{q_{p}}=[v_{x} ~v_{z}~q_{x}~q_{z}~p~p_{f}]^{T}, \end{array} $$

with p being the bulk pressure, p_f is fluid pressure, v^′s and q^′s are solid and fluid particle velocity (relative to solid). A_1p, B_1p, and D_1p are defined as

$$\begin{array}{@{}rcl@{}} \mathbf{A_{1p}}& =&-\left[\begin{array}{cccccc} 0 & 0 & 0 & 0 & \beta_{11} & \beta_{12} \\ 0&0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -\beta_{21} & -\beta_{22} \\ 0&0 & 0 & 0 & 0 & 0\\ -H & 0 & -C & 0 & 0 & 0\\ -C& 0 & -M & 0 & 0 & 0 \end{array}\right],\\\mathbf{B_{1p}} &=&-\left[\begin{array}{cccccc} 0&0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & \beta_{11} & \beta_{12} \\ 0&0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -\beta_{21} & -\beta_{22} \\ 0 & -H & 0 & -C & 0 & 0\\ 0 & -C & 0 & -M & 0 & 0 \end{array}\right],\\ \mathbf{D_{1p}} &=&-\left[\begin{array}{cccccc} 0&0 & \frac{\eta}{\kappa}\beta_{12} & 0& 0 & 0\\ 0 & 0 & 0 & \frac{\eta}{\kappa}\beta_{12} & 0& 0 \\ 0&0 & -\frac{\eta}{\kappa}\beta_{22} & 0 & 0 & 0\\ 0 & 0 & 0 & -\frac{\eta}{\kappa}\beta_{22} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right], \end{array} $$

where β’s, H, C, and M are dependent on the solid bulk modulus (K_s), the fluid bulk modulus (K_f), the solid density (ρ_s), the porosity (ϕ), the permeability (κ), the fluid density (ρ_f), and the viscosity (η) of the medium, elaborately expressed in [8].