Wave Propagation In Porous Media Research Articles

Simulation of surface waves in porous media

SUMMARY We present a novel numerical algorithm for the simulation of seismic wave propagation in porous media, which is particularly suitable for the accurate modelling of surface wave-type phenomena. The differential equations of motion are based on Biot's theory of poro-elasticity and solved with a pseudospectral approach using Fourier and Chebyshev methods to compute the spatial derivatives along the horizontal and vertical directions, respectively. The time solver is a splitting algorithm that accounts for the stiffness of the differential equations. Due to the Chebyshev operator the grid spacing in the vertical direction is non-uniform and characterized by a denser spatial sampling in the vicinity of interfaces, which allows for a numerically stable and accurate evaluation of higher order surface wave modes. We stretch the grid in the vertical direction to increase the minimum grid spacing and reduce the computational cost. The free-surface boundary conditions are implemented with a characteristics approach, where the characteristic variables are evaluated at zero viscosity. The same procedure is used to model seismic wave propagation at the interface between a fluid and porous medium. In this case, each medium is represented by a different grid and the two grids are combined through a domain-decomposition method. This wavefield decomposition method accounts for the discontinuity of variables and is crucial for an accurate interface treatment. We simulate seismic wave propagation with open-pore and sealed-pore boundary conditions and verify the validity and accuracy of the algorithm by comparing the numerical simulations to analytical solutions based on zero viscosity obtained with the Cagniard-de Hoop method. Finally, we illustrate the suitability of our algorithm for more complex models of porous media involving viscous pore fluids and strongly heterogeneous distributions of the elastic and hydraulic material properties.

Microbial‐induced heterogeneity in the acoustic properties of porous media

It is not known how biofilms affect seismic wave propagation in porous media. Such knowledge is critical for assessing the utility of seismic techniques for imaging biofilm development and their effects in field settings. Acoustic wave data were acquired over a two‐dimensional region of a microbial‐stimulated sand column and an unstimulated sand column. The acoustic signals from the unstimulated column were relatively uniform over the 2D scan region. The data from the microbial‐stimulated column exhibited a high degree of spatial heterogeneity in the acoustic wave amplitude, with some regions exhibiting significant increases in attenuation while others exhibited decreases. Environmental scanning electron microscopy showed differences in the structure of the biofilm between regions of increased and decreased acoustic wave amplitude. We conclude from these observations that variations in microbial growth and biofilm structure cause heterogeneity in the elastic properties of porous media with implications for the validation of bioclogging models.

Finite-frequency kernels for wave propagation in porous media based upon adjoint methods

We determine finite-frequency kernels for wave propagation in porous media based upon adjoint methods. These sensitivity kernels may be obtained based upon two numerical simulations for each source: one calculation for the current model and a second, ‘adjoint’, calculation that uses time-reversed signals at the receivers as simultaneous, fictitious sources. The adjoint equations for a porous medium are identical to the usual Biot equations, with the exception of the adjoint source term, which involves time-reversed measurements of the differences between simulations and data. Isotropic poroelastic wave propagation is governed by eight primary model parameters which appear in the original Biot equations. These parameters include four moduli: the shear modulus of the frame plus three bulk moduli. We consider two alternative parameterizations: one involving density-normalized moduli corresponding to squared wave speeds, and a second involving the poroelastic shear and compressional wave speeds. The alternative parameterizations lead to density-sensitivity kernels that are small, reflecting the fact that the traveltime of a poroelastic wave is governed by wave speed, not density. We systematically investigate and illustrate the sensitivity of the fast and slow compressional waves and the shear wave to the poroelastic model parameters using a 2-D spectral-element method. The poroelastic sensitivity kernels presented herein form the basis of tomographic imaging and inversion in porous media.

Spectral-element simulations of wave propagation in porous media

We 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 spectral-element 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 Description of Seismic Acoustic Wave Propagation in Porous Media via Homogenization

We consider a linear system of differential equations describing the joint motion of an elastic porous body and a fluid occupying the porous space. A rigorous justification is performed for the homogenization procedures under various conditions imposed on the physical parameters as the dimensionless size of the pores tends to zero, while the porous body is geometrically periodic and the process's characteristic time is sufficiently small. Such models describe the propagation of seismic acoustic waves. In the present paper, we derive the homogenized equations, which are different types of nonstandard wave equations depending on the relations between the physical parameters. The proofs are based on Nguetseng's two-scale convergence method of homogenization in periodic structures.

AGILD Seismic Modeling For Double-porosity Media

In this paper, we present the seismic fleld modeling for double-porosity media using AGILD method. The seismic difierential integral equations are discretized in the boundary strip zone by collocation FEM method. The seismic Galerkin equations are discretized by FEM method in internal domain. Both integral discretization on the strip zone and Galerkin FEM in the internal domain are used to construct AGILD seismic wave fleld modeling. The three phases' waveflelds are simulated through AGILD method in double-porosity media. 1. INTRODUCTION Studies on wave propagation in porous media have been carried out for decades. Recently, the major cause of ∞uid-induced porous rocks is analyzed in three difierent scales: macroscopic, mesoscopic, and microscopic (22{24). White (25) was the flrst to introduce the mesoscopic-loss mechanism based on the framework of Biot's theory. A patch saturation model was presented to describe the wave propagation character- istics in porous rocks partially saturated with water and gas. Carcione (8) numerically modeled the wavefleld in a porous media with mesoscopic gas pockets, and predicted higher attenuation than White's model due to multiple scattering and wave conversing. Carcione (9) investigated the White model and modeled difierent mesoscopic-scale heterogeneities and concluded that the loss because of ∞uid-modulus and porosity variations was the most important. Pride (24) used a unifled theoreti- cal treatment to describe three P wave attenuation models, which included a double-porosity model with a single ∞uid saturating, a patch-saturation model with two immiscible ∞uid and a microscopic squirt ∞ow model. Either of the two mesoscopic models produced attenuation in seismic band that was comparable to what was measured in the fleld. In this work, we investigate the double-porosity rocks (2,22{24) with a AGILD method calcu- lating the MFF pressure relaxation. Firstly, we simplify the double-porosity model with the same solid skeleton coupling with the two pore phases, and derive the displacement wave equations. Sec- ondly, we analyze the mesoscopic ∞uid ∞ow theory presented by Pride (22,23), derive the integral discretization to numerically solve the divergency increments caused by MFF. Finally, waveflelds are simulated.

On wave propagation in porous media with strain gradient effects

A porous material with large irregular holes is here studied as a hyperelastic continuum with latent microstructure, and the mechanical balance equations are derived from the general ones for a medium with ellipsoidal microstructure. This is done by imposing the kinematical constraint of microstretch bounded to the macrodeformation: in this case, the microstructure disappears, apparently, and the response of the material involves higher gradients of the displacement without incurring known constitutive inconsistencies. An application to the propagation of asymptotic waves compatible with such a model is also considered. Physical situations corresponding to an axisymmetric motion either in spherical or cylindrical symmetry are considered, and it is shown that the time evolution of the wave amplitude factor is governed by the spherical and cylindrical Korteweg–deVries equations, respectively.

Predictions of the modified Biot–Attenborough model for the dependence of phase velocity on porosity in cancellous bone

The modified Biot–Attenborough (MBA) model for acoustic wave propagation in porous media has been found useful to predict wave properties in cancellous bone. The present study is aimed at applying the MBA model to predict the dependence of phase velocity on porosity in cancellous bone. The MBA model predicts a phase velocity that decreases nonlinearly with porosity. The optimum values for input parameters of the MBA model, such as compressional speed c m of solid bone and phase velocity parameter s 2, were determined by comparing the predictions with previously published measurements in human calcaneus and bovine cancellous bone. The value of the phase velocity parameter s 2 = 1.23 was obtained by curve fitting to the experimental data for 53 human calcaneus samples only, assuming a compressional speed c m = 2500 m/s of solid bone. The root-mean-square error (RMSE) of the curve fit was 15.3 m/s. The optimized value of s 2 for all 75 cancellous bone samples including 22 bovine samples was 1.42 with a value of 55 m/s for the RMSE of the curve fit. The latter fit was obtained by using of a value of c m = 3200 m/s. Although the MBA model relies on the empirical parameters determined from experimental data, it is expected that the model can be usefully employed as a practical tool in the field of clinical ultrasonic bone assessment.

Study on the Biot-Stoll model for porous marine sediments

The Biot-Stoll model is useful for analyzing the acoustic wave propagation in porous marine sediments. However, the practical application of the model is not easy, because 13 physical parameters are required to apply it, and the physical phenomena in this model are difficult to understand. In this study, equivalent circuits for the plane longitudinal wave propagation in porous media are proposed for easy understanding of the physical phenomena. Also, practical approximated equations for the longitudinal wave velocity and attenuation are proposed to simplify practical application of the Biot-Stoll model. Then, simpler models, such as the Wood model, the Gassmann model, and the Biot-Stoll model without frame moduli can be used within limited ranges of frequency and porosity to approximate the exact Biot-Stoll model. Therefore, the ranges of frequency and porosity applicable to some simple acoustic models are obtained to simplify practical application of the Biot-Stoll model.

3-D numerical simulation of wave propagation in porous media: Influence of the microstructure and of the material properties of trabecular bone

Finite-difference time domain simulations coupled to 3-D microstructural models of 30 trabecular bones reconstructed from synchrotron radiation microtomography were employed herein to compare and quantify the effects of bone volume fraction, microstructure, and material properties of trabecular bone on QUS parameters. Scenarios of trabecular thinning and thickening using an iterative dedicated algorithm allowed the estimation of the sensitivity of QUS parameters to bone volume fraction. The sensitivity to bone material properties was assessed by considering independent variations of density and stiffness. The effect of microstructure was qualitatively assessed by producing virtual bone specimens of identical bone volume fraction (13%). Both BUA and SOS show a strong correlation with BV/TV (r2=0.94 p10−4) and vary quasi-linearly with BV/TV at an approximate rate of 2 dB/cm/MHz and 11 m/s per % increase of BV/TV, respectively. Bone alterations caused by variation in BV/TV (BUA: 40 dB/cm.MHz; SOS: 200 m/s) is much more detrimental to QUS variables than that caused by alterations of material properties or diversity in microarchitecture (BUA: 7.8 dB/cm.MHz; SOS: 36 m/s). QUS variables are changed more dramatically by BV/TV than by changes in material properties or microstructural diversity. However, material properties and structure also appear to play a role.

Influence of the surrounding pressure on acoustic properties of slightly compressible porous media permeated with air-filled bubbles

The effective wave velocity, attenuation, and nonlinear properties of slightly compressible porous media permeated with air-filled bubbles are studied numerically by employing the nonlinear Hooke’s law for different surrounding pressures. Numerical simulations show that the acoustic properties of porous media are greatly affected by the surrounding pressure if the shear modulus of the elastic medium is very small due to the fact that the acoustic wave propagation in porous media are strongly influenced by the nonlinear oscillation of bubbles; moreover, the oscillation of a bubble depends on the equilibrium bubble radius, which is affected by the surrounding pressures.

Seismic effects of viscous Biot‐coupling: Finite difference simulations on micro‐scale

This paper is concerned with numerical considerations of viscous fluid effects on wave propagation in porous media. We apply a displacement‐stress rotated staggered finite‐difference (FD) grid technique to solve the elastodynamic wave equation. An accurate approximation of a Newtonian fluid is implemented in this technique by using a generalized Maxwell body. With this approach we consider the velocity predictions of the Biot theory for elastic waves in different digital rock samples. To distinguish between the low and the high frequency range we estimate the effective permeabilities by a flow simulation. Our numerical results indicate that the viscous Biot‐coupling is visible in the numerical experiments. Moreover, the influences of other solid‐fluid interactions (e.g., Squirt flow) are also discussed.

Estimation of critical and viscous frequencies for Biot theory in cancellous bone

The use of Biot theory for modelling ultrasonic wave propagation in porous media involves the definition of a ‘critical frequency’ above which both fast and slow compressional waves will, in principle, propagate. Critical frequencies have been evaluated for healthy and osteoporotic cancellous bone filled with water or marrow, using data from the literature. The range of pore sizes in bone gives rise to a critical frequency band rather than a single critical frequency, the mean of which is lower for osteoporotic bone than normal bone. However, the critical frequency is a theoretical concept and previous researchers considered a more realistic ‘viscous frequency’ above which both fast and slow waves may be experimentally observed. Viscous frequencies in bone are found to be several orders of magnitude greater than calculated critical frequencies. Whereas two waves may well be observed at all ultrasonic frequencies for water-filled cancellous bone at 20 °C, it is probable megahertz frequencies would be needed for observation of two waves in vivo.

A comparative study of Biot's theory and the linear Theory of Porous Media for wave propagation problems

Wave propagation in porous media is an important topic for example in geomechanics or oil-industry. Especially due to the interplay of the solid skeleton with the fluid the so-called second compressional wave appears. The existence of this wave is reported in the literature not only for Biot's theory (BT) but also for theoretical approaches based on the Theory of Porous Media (TPM – mixture theory extended by the concept of volume fractions). Assuming a geometrically linear description (small displacements and small deformation gradients) and linear constitutive equations (Hooke's law) the governing equations are derived for both theories, BT and the TPM, respectively. In both cases, the solid displacements and the pore pressure are the primary unknowns. Note that this is only possible in the Laplace domain leading to the same structure of the coupled differential equations for both approaches. But the differential equations arising in BT and TPM possess different coefficients with different physical interpretations. Correlating these coefficients to each other leads to the well-known problem of Biot's “apparent mass density”. Furthermore, some inconsistencies are observed if Biot's stress coefficient is correlated to the structure arising in TPM. In addition to the comparison of the governing equations and the identification of the model parameters, the displacement and pressure solutions of both theories are presented for a one-dimensional column. The results show good agreement between both approaches in case of incompressible constituents whereas in case of compressible constituents large differences appear.

Assessment of gas hydrate and free gas distribution on the South Shetland margin (Antarctica) based on multichannel seismic reflection data

Processing and interpretation of a grid of intermediate-resolution multichannel seismic reflection profiles collected on the NE sector of the South Shetland continental margin, allowed us to map the lateral extent of a Bottom Simulating Reflector (BSR). The margin, an accretionary wedge located off the northern tip of the Antarctic Peninsula, consists of two distinct and superimposed tectonic regimes: an older regime is related to Mesozoic-Middle Cenozoic subduction-related tectonism; a younger one is associated with a mainly extensional tectonic phase, and related to the Oligocene development of the western Scotia Sea. The occurrence of the BSR appears to be controlled by the geological structure of the margin. The BSR lacks continuity near basement structures, main geological discontinuities and faults. On the other hand, the amplitude and continuity of the BSR are not affected by the presence of folded structures and undeformed sedimentary layering. We found that the BSR is mostly confined to the NE sector of the South Shetland Margin, where propagation of faulting associated with the Shackleton Fracture Zone may have driven migration of natural gas towards the surface and created the conditions for a BSR to appear. The application of reflection tomography techniques allowed us to reconstruct the averaged seismic velocity field between the seafloor and BSR in order to map the depth of BSR. By averaging the observed velocity structure above and below the BSR, and applying a theoretical model of elastic wave propagation in porous media, we attempted as rigorous a quantitative assessment as possible of the natural gas present as gas hydrate above the BSR and as free gas between the BSR and the Base of Gas Reflector (BGR).

ABSORBING LAYER VIA WAVE-EQUATION SPLITTING

Modeling acoustic waves generated by a localized source is always vexed by the nagging problem of spurious reflections and wraparound arising when the wavefront reaches the boundary of the numerical mesh. This difficulty may be circumvented by using a very large computational domain, which is very inefficient, or can be tackled by using some kind of absorbing boundary technique, which has not yet found a universally satisfactory solution. In this work, the wave equation is modified by introducing a term that is nonzero only in a narrow strip near the boundary. Then, a splitting technique permits to compute part of the solution analytically (hence, at no computational cost), while an application of Weyl's formula for the exponential of a matrix leads to a second-order accurate scheme that completes the algorithm. An application to SH seismic wave modeling shows that the performance of the present method is competitive with standard ones. Moreover, there is evidence for a potential application to the modeling of wave propagation in porous media, where stiff differential equations arise.

Absorbing Layer via Wave-Equation Splitting

Modeling acoustic waves generated by a localized source is always vexed by the nagging problem of spurious reflections and wraparound arising when the wavefront reaches the boundary of the numerical mesh. This difficulty may be circumvented by using a very large computational domain, which is very inefficient, or can be tackled by using some kind of absorbing boundary technique, which has not yet found a universally satisfactory solution. In this work, the wave equation is modified by introducing a term that is nonzero only in a narrow strip near the boundary. Then, a splitting technique permits to compute part of the solution analytically (hence, at no computational cost), while an application of Weyl's formula for the exponential of a matrix leads to a second-order accurate scheme that completes the algorithm. An application to SH seismic wave modeling shows that the performance of the present method is competitive with standard ones. Moreover, there is evidence for a potential application to the modeling of wave propagation in porous media, where stiff differential equations arise.

OG06 2つの骨格を有する多孔媒質中の音波伝搬(強力超音波,水中音響)

OG06 2つの骨格を有する多孔媒質中の音波伝搬(強力超音波,水中音響)

On the phase speed and attenuation of an interface wave in an unconsolidated marine sediment

The phase speed and attenuation of the interface wave at the seawater–sediment boundary are obtained by solving the characteristic equation for one of its complex roots. The characteristic equation itself is derived on the basis of a recently developed theory of wave propagation in porous media. Central to the theory is the stress relaxation that occurs when mineral grains slide against one another during the passage of a seismic wave. This type of stress relaxation is characterized by material response functions for compressional and shear waves of the form h(t)∝t−n, where t is time since the sliding began and n is a small positive number. The phase speed of the interface wave relative to that of the shear wave depends weakly on the grain size, increasing from about 85% for fine-grained silts and clays to 90% for coarse sands. The loss tangent of the interface wave, βi, is found to be independent of the mechanical properties (grain size, porosity, and density) of the sediment, and is the same as that for the shear wave: βi≈0.04. Since the loss tangent and phase speed are, in effect, independent of frequency, the attenuation coefficient of the interface wave scales as the first power of frequency. It turns out that the characteristic equation for the interface wave, as derived from the intergranular stress-relaxation mechanism, is exactly the same as if the sediment had been treated as an elastic solid. However, the elastic description fails to account for the grain-size dependencies exhibited by the compressional and shear waves. These dependencies emerge naturally from the stress-relaxation model.

Derivation of Forchheimer terms and their verification by application to waves propagation in porous media

Forchheimer terms are developed for the macroscopic momentum and energy balance equations considering saturated thermoelastic porous media, and for the macroscopic momentum balance equations in the case of multiphase porous media. It is shown that these terms represent the exchange between the interacting phases at their common interface. Using these Forchheimer terms, a very good agreement was evident when the 1-D numerical simulation of the propagation of compaction waves in a saturated thermoelastic porous medium was compared with shock tube experiments.