Spectral-element simulations of wave propagation in porous media

Abstract

S U M M A R YWe present a derivation of the equations describing wave propagation in porous media based upon an averaging technique which accommodates the transition from the microscopic to the macroscopic scale. We demonstrate that the governing macroscopic equations determined by Biot remain valid for media with gradients in porosity. In such media, the well-known expression for the change in porosity, or the change in the fluid content of the pores, acquires two extra terms involving the porosity gradient. One fundamental result of Biot's theory is the prediction of a second compressional wave, often referred to as 'type II' or 'Biot's slow compressional wave', in addition to the classical fast compressional and shear waves. We present a numerical implementation of the Biot equations for 2-D problems based upon the spectralelement method (SEM) that clearly illustrates the existence of these three types of waves as well as their interactions at discontinuities. As in the elastic and acoustic cases, poroelastic wave propagation based upon the SEM involves a diagonal mass matrix, which leads to explicit time integration schemes that are well suited to simulations on parallel computers. Effects associated with physical dispersion and attenuation and frequency-dependent viscous resistance are accommodated based upon a memory variable approach. We perform various benchmarks involving poroelastic wave propagation and acoustic-poroelastic and poroelastic-poroelastic discontinuities, and we discuss the boundary conditions used to deal with these discontinuities based upon domain decomposition. We show potential applications of the method related to wave propagation in compacted sediments, as one encounters in the petroleum industry, and to detect the seismic signature of buried landmines and unexploded ordnance.

A discontinuous Galerkin method for poroelastic wave propagation: The two-dimensional case

“…In these preliminary papers we addressed some restricted inverse problems and dealt mainly with the computational challenges associated with poroelastic inverse problems. For the 25 forward model we used the SPECFEM2D code based on the spectral element method due to Morency and Tromp, [11]. However issues with the SPECFEM2D code (discussed later in the paper) led us to developing the discontinuous Galerkin (DG) formulation discussed in this paper.…”

“…However, Morency and Tromp [11] and Carcione [43] show that the same equations hold for variable porosity. We note that our numerical implementation accurately resolves discontinuous porosities as well as material discontinuities in general, the former being a problem for the spectral element method; see Section 13.3.3 in [11]. At this point there are several options for reformulating and generalising the second equation to include the high-frequency regime.…”

“…We have had trouble understanding the various treatments of the viscous high-frequency dissipative regime [14,11,43]. Since, in applications to groundwater tomography, due to the relatively large permeabilities of aquifers, one often has to work in the high-frequency regime, this takes on a particular significance for us.…”

“…On the one hand there appears to be a lack of consistency with the modelling of the high frequency viscodynamic operator. Morency and Tromp [11] state that they consider a high-frequency viscodynamic 160 operator of the form b * ∂w ∂t where b is given in terms of a relaxation function (see below), although we have been unable to follow the derivations in Section 8.3.2 of their paper, nor the implementation in the SPECFEM2D code, which we have used extensively; whereas de la Puente [14] considers the convolution with acceleration η k ∂ 2 w ∂t 2 , which has the merit of being dimensionally consistent, and is the convention we adopt here. However, as Carcione points out in [43] the modelling of the high-frequency regime is purely 165 phenomenological.…”

“…Morency and Tromp, [11], work with a second-order frequency-dependent symmetric tensor b = b(t)a n d write the second equation in the form…”