Wave Propagation In Poroelastic Media Research Articles

Numerical Modeling of Wave Propagation in Poroelastic Media with a Staggered-Grid Finite-Difference Method

In this paper, wave propagation in poroelastic medium is simulated with a staggered-grid finite-difference method. The formulation is discretized based on the second-order Biot’s equations rather than the corresponding velocity-stress form. In order to eliminate boundary reflections, the PML method is applied. Numerical computations are implemented and the results show the correctness and effectiveness of the schemes presented in this paper.

Vibration analysis of a poroelastic composite hollow sphere

Vibrations in a poroelastic composite hollow sphere are investigated employing Biot’s theory of wave propagation in poroelastic media. A composite hollow poroelastic sphere consists of two concentric poroelastic spherical layers both of which are made of different poroelastic materials with each poroelastic material being homogeneous and isotropic. The boundaries of the composite hollow poroelastic sphere are free from stress. The frequency equations of both radial and rotatory vibrations are obtained each for pervious and impervious surfaces. The frequency equation of vibrations of a poroelastic composite hollow sphere with rigid core is derived as a particular case. The non-dimensional frequency for propagating modes is computed as a function of ratio of thickness to inner radius of core. The results are presented graphically for two types of poroelastic composite spheres and then discussed.

Effects of air voids on ultrasonic wave propagation in early age cement pastes

The objective of this paper is to investigate effects of air voids on ultrasonic wave propagation in fresh cement pastes, and relate ultrasonic wave parameters to cement setting times. First, Biot's theory was used to analyze wave propagation in poroelastic media containing air bubbles. Then, in the experimental study, both the compressional (P) and shear (S) waves were monitored in cement pastes with different water/cement ratios (w/c = 0.4 and 0.5) and various air void content (0.1%–5.3% by cement paste volume). Experimental results indicated that existence of air bubbles in cement paste significantly decreases the P wave velocity, but has little effect on the shear wave propagation. Further analysis shows that the shear wave velocity corresponding to the Vicat initial setting times is a relatively constant value for the investigated air content range. This study shows the potential of using shear waves to monitor setting and hardening process of cement.

DYNAMIC ANALYSIS OF POROUS MEDIA IN TIME DOMAIN USING A FINITE ELEMENT MODEL

The mechanical behavior of porous media is governed by the interaction between its solid skeleton and the fluid existing inside its pores. The interaction occurs through the interface of grains and fluid. The traditional analysis methods of porous media, based on the effective stress and Darcy's law, are unable to account for these interactions. For an accurate analysis, the porous media is represented in a fluid-filled porous solid on the basis of Biot's theory of wave propagation in poroelastic media. Because of irregular geometry, the domain is generally treated as an assemblage of finite elements. In this investigation the numerical analysis for the field equations governing the dynamic response of fluid-saturated porous media is formulated and employed for the study of transient wave motion. A finite element model is developed and implemented into a computer code called DYNAPM for dynamic analysis of porous media. The weighted residual standard Galerkin method is used for development of a finite element model, and the analysis is carried out in time domain considering the dynamic excitation. The 8-node plane strain elements are applied for discretization of domains. A Newmark time integration scheme is developed to solve the time-discretized equations. This scheme is an unconditionally stable implicit method. Finally, some numerical examples are presented to verify its accuracy and show the capability of the developed model for a wide variety of porous media dynamic behaviors. Obtained results show good agreement in comparison with the other numerical and analytical data.

Seismoelectric reflection and transmission at a fluid/porous-medium interface

The dispersion relation for seismoelectric wave propagation in poroelastic media is formulated in terms of effective densities comprising all viscous and electrokinetic coupling effects. Using Helmholtz decomposition, two seismoelectric conversion coefficients are derived, for an incident P-wave upon an interface between a compressible fluid and a poroelastic medium. These coefficients relate the incident P-wave to a reflected electromagnetic wave in the fluid, and a transmitted electromagnetic wave in the porous medium. The dependency on angle of incidence and frequency is computed. Using orthodox and interference fluxes, it is shown that energy conservation is satisfied. A sensitivity analysis indicates that electrolyte concentration, viscosity, and permeability highly influence seismoelectric conversion.

Finite-difference modeling of the monopole acoustic logs in a horizontally stratified porous formation

Monopole acoustic logs in a homogeneous fluid-saturated porous formation can be simulated by the real-axis integration (RAI) method to analytically solve Biot's equations [(1956a) J. Acoust. Soc. Am. 28, 168-178; (1956b) J. Acoust. Soc. Am. 28, 179-191; (1962) J. Appl. Phys. 33, 1482-1498], which govern the wave propagation in poro-elastic media. Such analytical solution generally is impossible for horizontally stratified formations which are common in reality. In this paper, a velocity-stress finite-difference time-domain (FDTD) algorithm is proposed to solve the problem. This algorithm considers both the low-frequency viscous force and the high-frequency inertial force in poro-elastic media, extending its application to a wider frequency range compared to existing algorithms which are only valid in the low-frequency limit. The perfectly matched layer (PML) is applied as an absorbing boundary condition to truncate the computational region. A PML technique without splitting the fields is extended to the poro-elastic wave problem. The FDTD algorithm is validated by comparisons against the RAI method in a variety of formations with different velocities and permeabilities. The acoustic logs in a horizontally stratified porous formation are simulated with the proposed FDTD algorithm.

Discontinuous Galerkin methods for wave propagation in poroelastic media

We have developed a new numerical method to solve the heterogeneous poroelastic wave equations in bounded three-dimensional domains. This method is a discontinuous Galerkin method that achieves arbitrary high-order accuracy on unstructured tetrahedral meshes for the low-frequency range and the inviscid case. By using Biot’s equations and Darcy’s dynamic laws, we have built a scheme that can successfully model wave propagation in fluid-saturated porous media when anisotropy of the pore structure is allowed. Zero-inflow fluxes are used as absorbing boundary conditions. A continuous arbitrary high-order derivatives time integration is used for the high-frequency inviscid case, whereas a space-time discontinuous scheme is applied for the low-frequency case. We conducted a numerical convergence test of the proposed methods. We used a series of examples to quantify the quality of our numerical results, comparing them to analytic solutions as well as numerical solutions obtained by other methodologies. In particular, a large scale 3D reservoir model showed the method’s suitability to solve poroelastic wave-propagation problems for complex geometries using unstructured tetrahedral meshes. The resulting method is proved to be high-order accurate in space and time, stable for the low-frequency case, and asymptotically consistent with the diffusion limit.

An unsplit convolutional perfectly matched layer improved at grazing incidence for seismic wave propagation in poroelastic media

The perfectly matched layer (PML) absorbing technique has become popular in numerical modeling in elastic or poroelastic media because of its efficiency in absorbing waves at nongrazing incidence. However, after numerical discretization, at grazing incidence, large spurious oscillations are sent back from the PML into the main domain. The PML then becomes less efficient when sources are located close to the edge of the truncated physical domain under study, for thin slices or for receivers located at a large offset. We develop a PML improved at grazing incidence for the poroelastic wave equation based on an unsplit convolutional formulation of the equation as a first-order system in velocity and stress. We show its efficiency for both nondissipative and dissipative Biot porous models based on a fourth-order staggered finite-difference method used in a thin mesh slice. The results obtained are improved significantly compared with those obtained with the classical PML.

The role of surface tension in elastic wave scattering in an inhomogeneous poroelastic medium

It is well known that many porous media such as rocks have heterogeneities at nearly all scales. We applied Biot's poroelastic theory to study the propagation of elastic waves in isotropic porous matrix with spherical inclusions. It is assumed that the heterogeneity dimension exceeds significantly the pore size. Modified boundary conditions on poroelastic interface are used to take into account the surface tension effects. The effective wavenumber is calculated using the Waterman and Truell multiple scattering theory, which relates the effective wave number to the amplitude of the wave field scattered by a single inclusion. The calculations were performed for a medium containing fluid-filled cavities or porous inclusions contrasting in saturating fluid elastic properties. The results obtained show that when we consider elastic wave propagation in poroelastic medium containing soft inclusions, it is necessary to take into account the capillary pressure. The influence of the surface tension depends on the diffraction parameter and it is a maximum in the low frequency range.

Applications of the Boundary Absorption Using A Perfectly Matched Layer for Elastic Wave Simulation in Poroelstic Media

AbstractIn elastic wave numerical modeling for a porous medium, artificial boundaries should be treated so that they have minimum effects on the modeling results. In this paper, a high‐order velocity‐stress staggered‐grid finite‐difference scheme, with a perfectly matched layer (PML) absorbing boundary condition was proposed for simulating wave propagation in poroelastic media. The construction of the perfectly matched layer was discussed in detail, and the implementation of the high‐order finite‐difference scheme of the PML boundary conditions was also studied. The numerical results were validated by using analytical solutions in a homogeneous porous model. The performance of the PML for body waves with various incidence angles and various absorbing thicknesses was demonstrated. Also, free surface Raleigh waves were investigated for their absorptions. The PML was compared with two kinds of absorption boundary conditions to confirm the efficiency of the PML method. Our numerical results show that the PML can greatly absorb or reduce outgoing waves, not only for body waves, but also for surface waves. The PML method is one of the best absorption boundary conditions for porous elastic wave modeling. This work will play an important role in investigating elastic wave response of inhomogeneous porous media, especially in studying borehole sonic logging response of heterogeneous porous media.

Dam–water–sediment–rock systems: Seismic analysis

A computational procedure for two-dimensional finite-element analysis of dam–water–sediment–rock systems subjected to seismic excitations is reviewed. In particular, the semidiscrete approximation of the water–sediment–rock region on the upstream side of the dam by means of a hyperelement is described in detail. The sediment is represented in the hyperelement as a fluid-filled porous solid on the basis of the Biot theory of wave propagation in poroelastic media while the water is taken as a compressible, inviscid fluid and the rock as a viscoelastic solid. An application of the procedure to a study of the effects of sediment porosity and thickness on the response of a model dam to horizontal and vertical ground motions is presented and discussed.

The application of the nonsplitting perfectly matched layer in numerical modeling of wave propagation in poroelastic media

The nonsplitting perfectly matched layer (NPML) absorbing boundary condition (ABC) was first provided by Wang and Tang (2003) for the finite-difference simulation of elastic wave propagation in solids. In this paper, the method is developed to extend the NPML to simulating elastic wave propagation in poroelastic media. Biot’s equations are discretized and approximated to a staggered-grid by applying a fourth-order accurate central difference in space and a second-order accurate central difference in time. A cylindrical two-layer seismic model and a borehole model are chosen to validate the effectiveness of the NPML. The results show that the numerical solutions agree well with the solutions of the discrete wavenumber (DW) method.

A Model for Multiphase Flows through Poroelastic Media

A continuum model for multiphase fluid mixture flows through poroelastic media is presented. The basic conservation laws developed via a volume averaging technique are considered. Effects of phasic equilibrated forces are included in the model. Based on the thermodynamics of the multiphase mixture flows, appropriate constitutive equations are formulated. The entropy inequality is exploited, and the method of Lagrangian multiplier is used along with the phasic conservation laws to derive the constitutive equations for the phasic stress tensors, equilibrated stress vectors, and the interactions terms. The special cases of wave propagation in poroelastic media saturated with multiphase fluids, and multiphase flows through porous media, are studied. It is shown that the present theory leads to the extended Darcy’s law and contains, as a special case, Biot’s theory of saturated poroelastic media.

The application of the perfectly matched layer in numerical modeling of wave propagation in poroelastic media

The perfectly matched layer (PML) was first introduced by Berenger as a material absorbing boundary condition (ABC) for electromagnetic waves. In this paper, a method is developed to extend the perfectly matched layer to simulating seismic wave propagation in poroelastic media. This nonphysical material is used at the computational edge of a finite‐difference algorithm as an ABC to truncate unbounded media. The incorporation of PML in Biot’s equations is different from other PML applications in that an additional term involving convolution between displacement and a loss coefficient in the PML region is required. Numerical results show that the PML ABC attenuates the outgoing waves effectively.

Energy balance and fundamental relations in dynamic anisotropic poro-viscoelasticity

An energy balance equation and fundamental relations are obtained for wave motion in anisotropic poro‐viscoelastic media. The balance of energy identifies the strain and kinetic energy densities, and the dissipated energy densities due to viscoelastic and viscodynamic effects. The relations allow the calculation of these energies in terms of the Umov‐Poynting vector and kinematic variables. The energy balance is obtained for time‐harmonic fields, while the following relations are valid for inhomogeneous body waves. (i) The magnitude of the phase velocity is equal to the projection of the energy velocity vector onto the propagation direction. (ii) The time‐average of the dissipated energy density is obtained from the projection of the average power‐flow vector onto the attenuation direction. (iii) The time‐average energy density (kinetic plus strain) is obtained from the projection of the average power‐flow vector onto the propagation direction. (iv) The strain energy equals the kinetic energy when the medium is lossless. These relations are shown to be valid for anisotropic poro‐viscoelasticity at all frequency ranges. An example of ultrasonic wave propagation in an orthorhombic medium (human femoral bone saturated with water) illustrates the theory. Measurable quantities, like the attenuation factor and the energy velocity, can easily be interpreted in terms of microstructural properties such as tortuosity and permeability.

Wave Propagation in Poroelastic Media Saturated by Two Fluids

A theory of wave propagation in isotropic poroelastic media saturated by two immiscible Newtonian fluids is presented. The macroscopic constitutive relations, and mass and momentum balance equations are obtained by volume averaging the microscale balance and constitutive equations and assuming small deformations. Momentum transfer terms are expressed in terms of intrinsic and relative permeabilities assuming the validity of Darcy’s law. The coefficients of macroscopic constitutive relations are expressed in terms of measurable quantities in a novel way. The theory demonstrates the existence of three compressional and one rotational wave. The third compressional wave is associated with the pressure difference between the fluid phase and dependent on the slope of the capillary pressure-saturation relation.

Displacement-based finite element method for guided wave propagation problems: Application to poroelastic media

Poroelastic materials are used extensively in the literature for sound absorption and attenuation. Guided wave propagation in these materials is therefore of considerable interest to researchers. In the present paper, a finite element method is presented for guided wave propagation in poroelastic media. For harmonic vibrations, the finite element method gives rise to a quadratic eigenvalue problem in the phase speed. Solution of the standard finite element procedure is shown to excite modes with spurious phase speeds. It is shown that the real and spurious modes can be identified using the penalty constraint approach. For this, one needs to compute the rate of change of phase speed with the penalty constraint constant α. An expression for computing this rate of change of phase speed has been derived. It is shown that in most cases the rate of change of phase speed needs to be evaluated only at α=0. The results apply to a pure fluid and also to poroelastic media. In the process, analytical expressions are also derived for wave propagation in poroelastic cylinders. The use of reduced order integration to eliminate spurious modes is also discussed in this paper. Dispersion curves for the phase speeds are presented for the first few breathing modes of a poroelastic cylinder. Finally, results are presented for the case of a rigid, lined ducted system.

Analytical Solutions for Harmonic Wave Propagation in Poroelastic Media

This paper presents several analytical solutions for problems of harmonic wave propagation in a poroelastic medium. The pressure‐solid displacement form of the harmonic equations of motion for a poroelastic solid are developed from the form of the equations originally presented by Biot. Then these equations are solved for several particular situations. Closed‐form analytical solutions are obtained for several basic problems: independent plane harmonic waves; radiation from a harmonically oscillating plane wall; radiation from a pulsating sphere; and the interior eigenvalue problem for a sphere, for the cases of both a rigid surface and a traction‐free surface. Finally, a series solution is obtained for the case of a plane wave impinging on a spherical inhomogeneity. This inhomogeneity is composed of poroelastic material having different properties from those of the infinite poroelastic medium in which it is embedded, and the incident wave may be composed of any linear combination of Biot “fast” and “slow”...

FFT-based spectral analysis methodology for one-dimensional wave propagation in poroelastic media

The general one-dimensional equilibrium equations describing the dynamic behaviour of a porous medium form a system of coupled hyperbolic partial differential equations. A transition from the time to the frequency domain is made by spectral decomposition of the displacements. The equations simplify to a set of coupled ordinary differential equations. A solution can be obtained by solving a frequency-dependent eigenvalue problem. The characteristic equation clarifies the double wave-pattern and the attenuation of each wave. A spectrally formulated element uses the frequency-dependent eivenvectors as shape functions. The mass distribution is treated exactly without the need of subdividing a member into smaller elements and therefore wave propagation within an element is also treated exactly.

A time domain integral formulation of dynamic poroelasticity

In this paper, a direct boundary element approach for solving three-dimensional problems of dynamic poroelasticity in the time domain is developed. Based on Biot's theory of wave propagation in poroelastic media, and on Cleary's reciprocal theorem, the complete set of necessary time-dependent fundamental solutions has been derived. Then, after an analytical time integration, a time stepping Boundary Element procedure for the numerical solution of wave propagation problems is established.