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Christina Morency and Jeroen Tromp are using the spectral-element method to model wave propagation in poroelastic media. This research is supported in part by a grant from the Strategic Environment Research Development Program (SERDP) by BAE Systems Advanced Information Technologies (Burlington, MA).

One fundamental result of Biot theory on wave propagation in porous media (Biot 1956, 1962) is the prediction of a second compressional wave, which attenuates rapidly, often referred to as ``type II'' or ``Biot's slow compressional wave'', in addition to the classical fast compressional and shear waves (Figure 1).

Figure1: Schematic representation of the waves generated in a poroelastic material: FP denotes the fast compressional wave, S the shear wave, and SP the slow compressional wave. Also shown is a schematic close-up of the porous medium to define the model volume W , which may be subdivided in terms of its solid and fluid parts W _s and W _f, respectively. The microscopic fluid-solid boundary is denoted by S , and the unit normal to this boundary, pointing from the solid to the fluid, is denoted by n.

The mathematical formulation of wave propagation in porous media developed by Biot is based upon the principle of virtual work, ignoring processes at the microscopic level. Even if Biot formulations are claimed to be valid for non-uniform porosity, gradients in porosity are not explicitly incorporated in the original theory.

Based on recent studies focusing on averaging techniques (e.g., Whitaker 1999) to derive macroscopic equations from the microscale, we have worked on a straightforward derivation of the main equations describing wave propagation in porous media, paying particular attention to the effects of gradients in porosity.

Boundary conditions used to deal with domain heterogeneities - acoustic/poroelastic, poroelastic/elastic, and poroelastic/poroelastic wave interactions - are based upon domain decomposition.

2-D Benchmarks:

Benchmark 1: Acoustic-poroelastic simulation of wave propagation in a water layer over a homogeneous poroelastic half-space (Figure 2-1).

The model dimensions are 4800 m x 4800 m, the source (cross) is located at x_s = (1600,2900) and the receivers (circles) at x_r1 = (2000,2934) and at x_r2 = (2000,1867), the top is a free surface and the remaining three edges are absorbing boundaries. The explosive source has a Ricker wavelet source time function with a dominant frequency of 15Hz.

Figure 2. Acoustic-poroelastic simulation of wave propagation in a water layer over a homogeneous poroelastic half-space. (1) Snapshot of the vertical-component of displacement at t=1.08 s.We can observe the direct P (a) and the reflected P (b) waves in the acoustic domain, while the transmitted fast P (c), the P-to-S converted (d), and the P-to-slow P converted (e) waves are clearly visible in the poroelastic domain. (2) Vertical-component velocity seismograms at receivers 1 & 2 (SEM: solid black line, analytical solution: dashed red line). We use domain decomposition between the fluid and the poroelastic solid.

Figure 2-2 displays SEM synthetic seismograms at the two receivers, which are compared to the analytical solution provided by Julien Diaz (University of Pau, France). The results are in good agreement. 

Benchmark 2: Simulation of wave propagation in a model consisting of two homogeneous poroelastic layers with discontinuous bulk & shear moduli and a jump in porosity (Figure 3-1).

The model dimensions are 4800 m x 4800 m, the source (cross) is located at x_s = (1600,2900) and the receivers (circles) at x_r1 = (2000,2934) and at x_r2 = (2000,1867), the top is a free surface and the remaining three edges are absorbing boundaries. The explosive source has a Ricker wavelet source time function with a dominant frequency of 15Hz.

Figure 3. Simulation of wave propagation in a model consisting of two homogeneous poroelastic layers with discontinuous bulk & shear moduli and porosity. (1) Snapshot of the vertical-component displacement at t=0.9 s. The direct fast P (a), the reflected fast P (b), and the reflected fast P-to-S and fast P-to-slow P converted (c) waves (which overlap because they have similar wave speeds) can be observed in the upper layer, together with the direct slow P (d), the reflected slow P (e), the reflected slow P-to-S converted (f), and the reflected slow P-to-fast P converted (g) waves. We also observe the reflected fast P wave due to the free surface (h). In the lower layer, the transmitted fast P (i) and fast P-to-slow P converted (j) waves can be clearly identified, together with the transmitted slow P (k), slow P-to-S converted (l) and slow P-to-fast P converted (m) waves. Note that the transmitted fast P-to-S converted wave, which presents a low amplitude, is not visible. (2) Vertical-component velocity seismograms at receivers 1 & 2 (SEM: solid black line, analytical solution: dashed red line). We use domain composition to accommodate the first-order discontinuity in porosity in the Biot formulation.

Figure 3-2 shows the SEM synthetic seismograms at the two receivers, which are in good agreement with the analytical solution provided by Julien Diaz (University of Pau, France).

2-D Sample applications:

Compacted sedimentary layer: Simulation of wave propagation in a water layer over a compacted sedimentary layer with variable porosity (Figure 4).

The model dimensions are 4800 m x 4800 m, the source (cross) is located at x_s = (1600,2900) and the receivers (circles) at x_r1 = (2000,2934) and at x_r2 = (2000,1867).

The top is a free surface and the remaining three edges are absorbing boundaries. The explosive source has a Ricker wavelet source time function with a dominant frequency of 15 Hz.

Figure 4 . Simulation of wave propagation in a water layer over a compacted sedimentary layer with variable porosity. (1) Snapshot of the vertical-component displacement at t=1.08 s. We can observe the direct P (a) and the reflected P (b) waves in the acoustic domain, the transmitted fast P (c), the fast P-to-S (d), and the fast P-to-slow P (e) waves in the poroelastic domain. (2) Porosity profile in the poroelastic layer. (3) Vertical-component velocity seismograms at receivers 1 & 2 (compacted sediment layer: solid black line, constant porosity of~0.4: solid red line).

Figure 4-3 shows SEM synthetic seismograms at the two receivers. A comparison is made when the poroelastic lower layer presents a variable porosity and when porosity is kept constant. The dependency of phase velocity with porosity can be appreciated.

Buried object detection: Acoustic detection of buried objects is of interest in the context of landmine identification (Zeng & Liu 2001, Xiang & Sabatier 2003).

We designed three models to evaluate the signature of a purely elastic buried object in three types of environments (Figures 5-1 to 5-3):

- Model 1: acoustic layer on top of a poroelastic medium with a porosity gradient and no viscous damping.

- Model 2: acoustic layer on top of a poroelastic medium with a porosity gradient and viscous damping.

- Model 3: acoustic layer on top of an elastic medium.

We use a Ricker source time function with a dominant frequency of 5 kHz.

The source is located in the acoustic domain, and we place 20 receivers close to the bottom of this domain. The model dimensions are 10 m x 8m.

The differences between synthetic seismograms for Models 1, 2 and 3 with and without the buried metal object illustrate the corresponding seismic signatures (Figures 5-4 to 5-6).

The differential seismograms for Models 1 and 2 illustrate the impact of viscous damping on the slow compressional waves, which are clearly suppressed in Model 2. The signature of the object in differential seismograms for elastic Model 3 is noticeably different from that in the poroelastic models..

Figure 5 . Simulation of wave propagation in a model consisting of a water layer over a poroelastic layer, or an elastic layer, with a buried metal object (yellow rectangle). The source (cross) is located at x_s = (2.5,4.0) and 20 receivers (circles) are evenly located between x_{r1} = (4.0,3.5) and at x_{r20} = (8.0,3.5). Snapshot of the vertical-component displacement at t=0.002 s for (1) Model 1: porosity gradient and no viscous damping, (2) Model 2: porosity gradient and viscous damping and (3) Model 3: elastic. We can observe the direct P (a) and the reflected P (b) waves in the acoustic domain, the transmitted fast P (c), the P-to-S converted (d), and the fast P-to-slow P converted (e) waves in the poroelastic domain, plus waves reflected by the elastic object (f). (4) Difference between vertical-component velocity seismograms at receivers 1-20 for Model 1 with and without the buried metal object. These differential seismograms highlight the signature of the buried target. (5) Difference between vertical-component velocity seismograms at receivers 1-20 for Model 2 with and without the buried metal object. (6) Difference between vertical-component velocity seismograms at receivers 1-20 for elastic Model 3 with and without the buried metal object.

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