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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(hn+ 1). Computational experiments are shown for various combinations of homogeneous and heterogeneous poroelastic media.

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Acknowledgements

KS would like to acknowledge the School of Geology, OSU and the MCSS, EPFL Switzerland, for providing the fund to carry out this work. We also acknowledge the OGS, Italy for hosting KS at various occasions. We thank editors and three anonymous reviewers for very useful comments. KS would like to acknowledge Sundeep Sharma at Devon Energy, for various discussions and proof-reading the manuscript. This is Boone Pickens School of Geology, Oklahoma State University, contribution number 2019-100.

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Correspondence to Khemraj Shukla.

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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 = [vx,vz,qz,qz]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 Yi(ω) = iωmi + η/κi and lx and lz 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 Ve 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, pf is fluid pressure, vs and qs are solid and fluid particle velocity (relative to solid). A1p, B1p, and D1p 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 (Ks), the fluid bulk modulus (Kf), the solid density (ρs), the porosity (ϕ), the permeability (κ), the fluid density (ρf), and the viscosity (η) of the medium, elaborately expressed in [8].

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Shukla, K., Hesthaven, J.S., Carcione, J.M. et al. A nodal discontinuous Galerkin finite element method for the poroelastic wave equation. Comput Geosci 23, 595–615 (2019). https://doi.org/10.1007/s10596-019-9809-1

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Keywords

  • Waves
  • Poroelasticity
  • Lax-Friedrich
  • Attenuation
  • Numerical flux

Mathematics Subject Classification (2010)

  • 35L05
  • 35S99
  • 65M60
  • 74J05
  • 74J10
  • 93C20