Decoupling coefficients of dilatational wave for Biot's dynamic equation and its Green's functions in frequency domain

Green's functions for Biot's dynamic equation in the frequency domain can be a highly useful tool for the investigation of dynamic responses of a saturated porous medium. Its applications are found in soil dynamics, seismology, earthquake engineering,rock mechanics, geophysics, and acoustics. However, the mathematical work for deriving it can be daunting. Green's functions have been presented utilizing an analogy between the dynamic thermoelasticity and the dynamic poroelasticity in the frequency domain using the u-p formulation. In this work, a special term "decoupling coefficient" for the decomposition of the fast and slow dilatational waves is proposed and expressed to present a new methodology for deriving the poroelastodynamic Green's functions. The correctness of the solution is demonstrated by numerically comparing the current solution with Cheng's previous solution. The separation of the two waves in the present methodology allows the more accurate evaluation of Green's functions, particularly the solution of the slow dilatational wave. This can be advantageous for the numerical implementation of the boundary element method(BEM) and other applications.

Stabilized Finite Element Methods for Biot’s Consolidation Problems Using Equal Order Elements

Using the standard mixed Galerkin methods with equal order elements to solve Biot’s consolidation problems,the pressure close to the initial time produces large non-physical oscillations.In this paper,we propose a class of fully discrete stabilized methods using equal order elements to reduce the effects of non-physical oscillations.Optimal error estimates for the approximation of displacements and pressure at every time level are obtained,which are valid even close to the initial time.Numerical experiments illustrate and confirm our theoretical analysis.

Wave Propagation Across Acoustic/Biot’s Media:A Finite-Difference Method

Numerical methods are developed to simulate the wave propagation in heterogeneous 2D fluid/poroelastic media.Wave propagation is described by the usual acoustics equations(in the fluid medium)and by the low-frequency Biot’s equations(in the porous medium).Interface conditions are introduced to model various hydraulic contacts between the two media:open pores,sealed pores,and imperfect pores.Well-posedness of the initial-boundary value problem is proven.Cartesian grid numerical methods previously developed in porous heterogeneous media are adapted to the present context:a fourth-order ADER scheme with Strang splitting for timemarching;a space-time mesh-refinement to capture the slow compressional wave predicted by Biot’s theory;and an immersed interface method to discretize the interface conditions and to introduce a subcell resolution.Numerical experiments and comparisons with exact solutions are proposed for the three types of interface conditions,demonstrating the accuracy of the approach.

Normal compression wave scattering by a permeable crack in a fluid-saturated poroelastic solid

A mathematical formulation is presented for the dynamic stress intensity factor(mode I) of a finite permeable crack subjected to a time-harmonic propagating longitudinal wave in an infinite poroelastic solid. In particular, the effect of the wave-induced fluid flow due to the presence of a liquid-saturated crack on the dynamic stress intensity factor is analyzed. Fourier sine and cosine integral transforms in conjunction with Helmholtz potential theory are used to formulate the mixed boundary-value problem as dual integral equations in the frequency domain. The dual integral equations are reduced to a Fredholm integral equation of the second kind. It is found that the stress intensity factor monotonically decreases with increasing frequency, decreasing the fastest when the crack width and the slow wave wavelength are of the same order. The characteristic frequency at which the stress intensity factor decays the fastest shifts to higher frequency values when the crack width decreases.

On Axially Symmetric Vibrations of Fluid Filled Poroelastic Spherical Shells

Employing Biot’s theory of wave propagation in liquid saturated porous media, waves propagating in a hollow poroelastic closed spherical shell filled with fluid are studied. The frequency equation of axially symmetric vibrations for a pervious and an impervious surface is obtained. Free vibrations of a closed spherical shell are studied as a particular case when the fluid is vanished. Frequency as a function of ratio of thickness to inner radius is computed in absence of dissipation for two types of poroelastic materials each for a pervious and an impervious surface. Results of previous works are obtained as a particular case of the present study.

Vibration Analysis of an Infinite Poroelastic Circular Cylindrical Shell Immersed in Fluid

The purpose of this paper is to study the effect of presence of fluid within and around a poroelastic circular cylindrical shell of infinite extent on axially symmetric vibrations. The frequency equation each for a pervious and an impervious surface is obtained employing Biot’s theory. Radial vibrations and axially symmetric shear vibrations are uncoupled when the wavenumber is vanished. The propagation of axially symmetric shear vibrations is independent of presence of fluid within and around the poroelastic cylindrical shell while the radial vibrations are affected by the presence of fluid. The frequencies of radial vibrations and axially symmetric shear vibrations are the cut-off frequencies for the coupled motion of axially symmetric vibrations. The non-dimensional phase velocity as a function of ratio of thickness to wavelength is computed and presented graphically for two different types of poroelastic materials for thin poroelastic shell, thick poroelastic shell and poroelastic solid cylinder.

A model for strong attenuation and dispersion of seismic P-waves in a partially saturated fractured reservoir

In this work we interpret the data showing unusually strong velocity dispersion of P-waves (up to 30%) and attenuation in a relatively narrow frequency range. The cross-hole and VSP data were measured in a reservoir, which is in the porous zone of the Silurian Kankakee Limestone Formation formed by vertical fractures within a porous matrix saturated by oil, and gas patches. Such a medium exhibits significant attenuation due to wave-induced fluid flow across the interfaces between different types of inclusions (fractures, fluid patches) and background. Other models of intrinsic attenuation (in particular squirt flow models) cannot explain the amount of observed dispersion when using realistic rock properties. In order to interpret data in a satisfactory way we develop a superposition model for fractured porous rocks accounting also for the patchy saturation effect.

Numerical Analysis on Cnoidal Wave Induced Response of Porous Seabed with Definite Thickness

Severe water waves can induce seabed liquefaction and do harm to marine structures. Dynamic response of seabed with definite thickness induced by cnoidal water waves is investigated numerically. Biot's consolidation equations are employed to model the seabed response. Parametric studies are carried out to examine the influence of the air content in the pore water and the soil hydraulic conductivity. It has been shown that the air content and soil hydraulic conductivity can significantly afect the pore pressure in seabed. An increase of air content and/or a decrease of soil hydraulic conductivity can change the pore pressure gradient sharply.

Dynamic response of a poroelastic stratum to moving oscillating load

The dynamic response of a poroelastic stratum subjected to moving load is studied. The governing dynamic equations for poroelastic medium are solved by using Fourier transform. The general solutions for the stresses and displacements in the transformed domain are established. Based on the general solutions, with the consideration of boundary conditions, the final expressions of stresses and displacements in physical domain are put forward for the three-dimensional single-layer medium. Some numerical solutions for the stresses, displacements and pore fluid pressure are presented and reveal that the response of a poroelastic stratum varies obviously with the moving velocity.

Seismic wave equations in tight oil/gas sandstone media

Tight oil/gas medium is a special porous medium,which plays a significant role in oil and gas exploration.This paper is devoted to the derivation of wave equations in such a media,which take a much simpler form compared to the general equations in the poroelasticity theory and can be employed for parameter inversion from seismic data.We start with the fluid and solid motion equations at a pore scale,and deduce the complete Biot’s equations by applying the volume averaging technique.The underlying assumptions are carefully clarified.Moreover,time dependence of the permeability in tight oil/gas media is discussed based on available results from rock physical experiments.Leveraging the Kozeny-Carman equation,time dependence of the porosity is theoretically investigated.We derive the wave equations in tight oil/gas media based on the complete Biot’s equations under some reasonable assumptions on the media.The derived wave equations have the similar form as the diffusiveviscous wave equations.A comparison of the two sets of wave equations reveals explicit relations between the coefficients in diffusive-viscous wave equations and the measurable parameters for the tight oil/gas media.The derived equations are validated by numerical results.Based on the derived equations,reflection and transmission properties for a single tight interlayer are investigated.The numerical results demonstrate that the reflection and transmission of the seismic waves are affected by the thickness and attenuation of the interlayer,which is of great significance for the exploration of oil and gas.

Finite element implementation of coupled hydro-mechanical modeling of transversely isotropic porous media in DEAL.II

In geotechnical engineering,modeling geo-structures is challenging,particularly in cases where the interaction between the structures and soil or rock is complex.Most wellknown commercial modeling software is based on homogenous and isotropic materials.However,soil and rock are often modeled in heterogeneous and anisotropic media because of the inherent anisotropy of sedimentary rock masses and their stratified structure.In recent decades,coupled hydro-mechanical(HM)interactions in isotropic porous media have been studied;however,the behavior of transversely isotropic porous media is rarely considered.In addition,it is difficult for commercial software such as Plaxis and Flac3D to express complex rock formation where the anisotropy of the material and the associated cracks and fractures could be assembled into a single model.In this study,a finite element implementation using Differential Equation Analysis Library(DEAL.II),an open-source library of finite element codes,was developed to model the fully coupled HM behavior of transversely isotropic porous media.The proposed implementation can be applied to both isotropic and transversely isotropic porous media based on Biot’s theory.The developed code can be used to model poroelastic media with(1)equations of linear elasticity for the solid matrix and(2)diffusion equations for fluid flow based on mass and linear-momentum conservation laws.We verified the performance and accuracy of the code through two examples,i.e.,Mandel’s problem with a compared analytical solution and a tunnel excavation process with the Flac3D software.On the basis of these numerical applications,we present the code to model the behavior of various geo-structures such as tunnels and pile–soil interactions with anisotropic materials.

Analytical Solution for Waves Propagation in Heterogeneous Acoustic/Porous Media. Part II: The 3D Case

We are interested in the modeling of wave propagation in an infinite bilayered acoustic/poroelastic media. We consider the biphasic Biot’s model in the poroelastic layer. The first part was devoted to the calculation of analytical solution in twodimensions, thanks to Cagniard de Hoop method. In the first part (Diaz and Ezziani,Commun. Comput. Phys., Vol. 7, pp. 171-194) solution to the two-dimensional problem is considered. In this second part we consider the 3D case.

Analytical Solution for Waves Propagation in Heterogeneous Acoustic/Porous Media. Part I: The 2D Case

Thanks to the Cagniard-de Hoop’s method we derive the solution to theproblem of wave propagation in an infinite bilayered acoustic/poroelastic media, wherethe poroelastic layer is modelled by the biphasic Biot’s model. This first part is dedi-cated to solution to the two-dimensional problem. We illustrate the properties of thesolution, which will be used to validate a numerical code.