Waves In Fluid-saturated Porous Media Research Articles

Effects of confining pressure and pore pressure on multipole borehole acoustic field in fluid-saturated porous media

In-situ stress is a common stress in the exploration and development of oil reservoirs. Therefore, it is of great significance to study the propagation characteristics of borehole acoustic waves in fluid-saturated porous media under stress. Based on the acoustoelastic theory of fluid-saturated porous media, the field equation of fluid-saturated porous media under the conditions of confining pressure and pore pressure and the acoustic field formula of multipole source excitation in open hole are given. The influences of pore pressure and confining pressure on guided waves of multipole borehole acoustic field in fluid-saturated porous media are investigated. The numerical results show that the phase velocity and excitation intensity of guided wave increase significantly under the confining pressure. For a given confining pressure, the phase velocity of the guided wave decreases with pore pressure increasing. The excitation intensity of guided wave increases at low frequency and then decreases at high frequency with pore pressure increasing, except for that of Stoneley wave which decreases in the whole frequency range. These results will help us get an insight into the influences of confining pressure and pore pressure on the acoustic field of multipole source in borehole around fluid-saturated porous media.

On the seismic response of a periodic sequence of three thin layers saturated by two-phase fluids

The conversion of fast to slow diffusion P waves in fluid-saturated porous media induces attenuation and dispersion of waves at seismic frequencies. This effect, known as wave-induced fluid flow, occurs at mesoscopic scales, which are much larger than the average pore size and much smaller than the average fast P-wave wavelength. When analyzing this mechanism in hydrocarbon reservoirs with the pore space saturated by multiphase fluids, it is important to include capillary pressure effects and flow interaction between fluids, which cause additional attenuation and velocity dispersion of P waves. We have developed a procedure to determine the phase velocities and dissipation factors in a medium composed of a periodic sequence of three poroelastic thin layers saturated by two-phase fluids. The methodology consists of applying compressibility tests to representative samples of the material, which are defined as boundary-value problems solved using a finite-element procedure. First, we analyze the case of two-phase fluid saturation on each layer, and the results are compared with those of the single-phase (effective) fluid case. Then, several cases of patchy saturation are presented, indicating that residual and wetting fluid saturation play an important role in determining the P-wave velocities and dissipation factors.

Influence of interface condition on reflection of elastic waves in fluid-saturated porous media

The characteristics of P-wave reflection coefficients can be dependent on the properties of the interface separating the two dissimilar poroelastic media. To evaluate the influence of the interface condition, we have developed a general theoretical formulation of seismic reflection and transmission of waves at arbitrary incidence angles in fluid-saturated porous media. The formulation is derived based on the quasistatic Biot’s theory and incorporates a more general boundary condition for fluid pressure. Simple analytic expressions are obtained for reflection and transmission coefficients at normal incidence. The equations incorporate an interfacial impedance that can be used to effectively characterize the interface condition. Based on the new formulation, we study the coupled effects of dynamic fluid flow and the interface condition on the reflection and transmission coefficients. Two pertinent reflection scenarios in exploration geophysics are considered in the numerical analysis, i.e., a gas-water contact and a free fluid overlying a gas-saturated medium. We examine the corresponding P-wave reflection coefficient resulting from different interface conditions such as imperfect hydraulic contact, capillary pressure, and open- and sealed-pore interfaces. Our results reveal that the interface condition can affect the frequency dependence and amplitude-versus-angle signatures of the reflection coefficient.

A unified poroviscoelastic model with mesoscopic and microscopic heterogeneities

The wave-induced fluid flow (WIFF) is considered to be the main cause of dispersion and attenuation of seismic waves in fluid-saturated porous media. Among numerous theories, the mesoscopic and microscopic heterogeneities are considered to be the primary mechanisms causing the WIFF. Furthermore, in most rocks, the mesoscopic and microscopic heterogeneities exist simultaneously and can cause obvious transitions of the fast P-wave velocity, which means it is necessary to consider the influence of the two mechanisms on the dispersion and attenuation simultaneously. Numerous results have shown that the dispersions and attenuations caused by these two mechanisms can be approximated in terms of the Zener model. To combine the two mechanisms into a unified model, we introduce a new generalized Zener model into the Biot poroelasticity theory to obtain a new poroviscoelastic model. Comparisons between the numerical results and two groups of experimental data further confirm the validity of our new model.

Frequency-dependent energy attenuation and velocity dispersion in periodic layered media

According to Brajanovski periodic layered model, a fractural medium can be equivalent to layered media with periodic distribution of fractural layers and background layers, but the analytical solution given by Brajanovski can only interpret the dispersion and attenuation effects of single characteristic unit model. In order to study the dispersion and attenuation features of multiple characteristic units, forward modeling methods are needed. Based on the theory of two-phase medium, Biot deduced the propagation equation of longitudinal waves in fluid-saturated porous media. However, there are two problems in the forward modeling using time-domain equation. One is the influences of boundary reflection, and the other is the introduction of cumulative error. For convenience, time-domain equation is rewritten in the frequency domain, thus constructing a one-dimensional rock physics model. Then, forward method is used to study the dispersion and attenuation features of fluid-saturated medium. Numerical simulation results are found to be in good agreement with the analytical solution. Furthermore, the frequency-domain forward method can analyze the velocity dispersion and energy attenuation of longitudinal waves in any multilayered fracture medium. By analyzing those numerical simulation results, it can be obtained that, as the length of characteristic unit increases or the number of characteristic unit decreases, both the starting frequency of dispersion and the peak frequency of attenuation shift to low, whatever the attenuation peaks are equal. In addition, the effects of porosity, permeability and fluid saturation on energy attenuation and velocity dispersion are also studied. Finally, the stress field and displacement field distributions of fluid-saturated fractural medium are given by the frequency-domain forward modeling method.

Features of acoustic waves in media with large porosity values in the framework of the Biot theory

The Biot theory is used to study the properties of acoustic waves in fluid-saturated porous media with large porosity values. Materials are considered for which a large elasticity modulus of the matrix (skeleton) is comparable in size to the large fluid pore saturation modulus.

Numerical Study of Frequency-dependent Seismoelectric Coupling in Partially-saturated Porous Media

The seismoelectric phenomenon associated with propagation of seismic waves in fluid-saturated porous media has been studied for many decades. The method has a great potential to monitor subsurface fluid saturation changes associated with production of hydrocarbons. Frequency of the seismic source has a significant impact on measurement of the seismoelectric effects. In this paper, the effects of seismic wave frequency and water saturation on the seismoelectric response of a partially-saturated porous media is studied numerically. The conversion of seismic wave to electromagnetic wave was modelled by extending the theoretically developed seismoelectric coupling coefficient equation. We assumed constant values of pore radius and zeta-potential of 80 micrometers and 48 microvolts, respectively. Our calculations of the coupling coefficient were conducted at various water saturation values in the frequency range of 10 kHz to 150 kHz. The results show that the seismoelectric coupling is frequency-dependent and decreases exponentially when frequency increases. Similar trend is seen when water saturation is varied at different frequencies. However, when water saturation is less than about 0.6, the effect of frequency is significant. On the other hand, when the water saturation is greater than 0.6, the coupling coefficient shows monotonous trend when water saturation is increased at constant frequency.

Dynamic stress-intensity factors of an in-plane shear crack in saturated porous medium

Based on the classical Biot theory of propagation of elastic waves in fluid-saturated porous media, two classes of plane problem are considered. One is the transient problem of a finite crack subjected to a suddenly applied in-plane shear, and another one is dealing with diffraction of transverse SV-wave by the same crack. The formulation utilizes the Laplace and integral transforms to reduce the mixed boundary-value problems to Fredholm integral equations of the second kind. The equations are then solved numerically for time and frequency dependent dynamic stress-intensity factors (mode II), and the influence of pore fluid on the dynamic stress-intensity factor is investigated. Pore fluid is found to have impact on the magnitude of the stress-intensity factor the extent of which depends on the two fluid parameter values, namely ratio of fluid mass with respect to the whole bulk mass and viscosity-to-permeability ratio. Comparisons of the solutions obtained from the present study with the corresponding known solutions of dry elastic medium are also provided to verify the validity of the present solution scheme.

Effects of crack-dilatancy on Rayleigh waves in fluid-saturated porous media

The study considers the propagation of surface waves on the stress-free surface of a porous solid saturated with non-viscous fluid. The surface pores have the option of being either sealed or fully-opened. With the presence of dilatant cracks, the interior of the porous solid is characterised through three different crack-regimes, based on the connections between embedded cracks. Secular equations are derived in closed form for the propagation of Rayleigh waves in the porous media with sealed or fullyopened surface pores. The velocity of non-dispersive surface waves varies significantly with the density of cracks present. However, aspect (thickness to radius) ratio of (circular) cracks may not have much effect on the velocity of Rayleigh waves. The opening of surface pores may be an important reason for a faster propagation of Rayleigh waves in any realistic elastic medium. Finally, the dilatancy due to the growth of cracks up to their interconnection or drainage may be able to affect the velocity of Rayleigh waves quite significantly.Keywords: Rayleigh waves, phase velocity, dilatancy, porous solid, cracksInternational Journal of Engineering, Science and Technology Vol. 6, No. 1, 2014, pp. 77-87

Recent developments in seismic rock physics

In this review I identify major developments in rock physics over the past decade, and discuss how they have been applied in seismic exploration. Reference is made to key publications discussing these developments, both in the laboratory and in modelling, and how they are applied in reservoir characterization and monitoring. I discuss the validity of theories employed for describing the propagation of elastic waves in fluid-saturated porous media, developed originally by Gassmann and Biot in the 1950s. Problems experienced in the use of ultrasonic measurements to predict velocities and attenuation at seismic frequencies are outlined, as are their solutions. Reference is made to a field and laboratory study in which the range of frequencies, seismic through sonic to ultrasonic, was covered and successfully modelled. Procedures developed for interpreting laboratory measurements in terms of pore structure and degree of saturation, based on self-consistent and scattering theories, are described. Laboratory equipment used for determining the elastic properties of anisotropic rocks is described, along with a discussion of the results obtained, including the correlation between both fluid permeability and in situ state of stress prediction with the degree of shear-wave splitting. Finally, I discuss the application of rock physics developments to reservoir characterization and monitoring, in particular to time-lapse seismic monitoring.

IN VITROACOUSTIC WAVE PROPAGATION IN HUMAN AND BOVINE CANCELLOUS BONE AS PREDICTED BY BIOT'S THEORY

Recent in vitro studies have provided evidence of the propagation of two different longitudinal wave modes at ultrasonic frequencies in cancellous bone. The genesis of these two plane waves in fluid-saturated porous media is predicted by the poroelastic approach to wave propagation originally developed by Biot. However, wave velocity is usually analyzed as a function of bone mass density only; therefore, the influence of the cancellous bone microstructure over the wave velocity is not taken into account. In the present study, a descriptor of the microstructure is considered in Biot's theory. This model is used to evaluate the large experimental variability of both fast and slow wave velocities measured on randomly oriented human and bovine cancellous bone samples. The role of the anisotropic solid structure and fluid in the behavior of fast and slow wave velocities is examined. Experimental and theoretically predicted velocities are found in close agreement when analyzed as a function of both porosity and structural index. This model has the potential to be used to determine an acoustically derived structural index in cancellous bone.

Asymptotic Biot's models in porous media

This work deals with a class of evolution problems consisting of a pair of coupled equations for modelling propagation of elastic waves in fluid-saturated porous media. The type of the first equation depends on two physical parameters (density and secondary consolidation) which can vanish while the second one is always parabolic. In case the density never vanishes, the first equation is second-order hyperbolic type and a weak solution to the problem is constructed using a variational method in a Sobolev framework. Next, the proof of uniqueness involves Ladyzenskaja's test-functions used to compensate a lack of regularity that would be required in a standard energy method. This approach gives rise to a priori estimates which are useful to prove that the linearized thermoelasticity and the quasi-static systems are defined as asymptotic models of the Biot problem when the secondary consolidation coefficient or the density is small.

Propagation of longitudinal elastic waves in a fluid-saturated porous medium with spherical inclusions

The Frenkel-Biot theory is used to study the propagation of a longitudinal harmonic wave of the first kind in an isotropic porous matrix with inclusions contrasting in elastic properties and hydrodynamic permeability. The generation of elastic waves of the second kind at the boundaries of inclusions is taken into account. The effective wave number of the longitudinal wave is calculated using the equations of multiple scattering theory. The characteristic size of inhomogeneities is assumed to be much greater than the size of pores. The parameters of the model used for calculations correspond to sandstone with centimeter-scale inhomogeneities. The presence of such inhomogeneities is typical of sedimentary rocks. Calculations show that, in the frequency range of acoustic logging, the effective attenuation factor of the longitudinal wave may noticeably exceed the attenuation factors of longitudinal waves of the first kind in both matrix and inclusions. From the results obtained, it follows that, when studying the propagation of elastic waves in fluid-saturated porous media, it is necessary to take into account the hydrodynamic effects associated with the filtration overflows that arise at the boundaries of inhomogeneities.

Rock-physics developments in seismic exploration: A personal 50-year perspective

I present an historical overview of the major developments in rock physics over the past half-century. Key developments are described for both modeling and laboratory experiments and how they have been applied in seismic exploration, reservoir characterization, and monitoring. In particular, I chronicle the development of laboratory methods for determining the velocity and attenuation of elastic waves in rock samples at sonic and ultrasonic frequencies. This is followed by a description of the validity of theories originally developed by Gassmann and Biot in the 1950s and employed for describing the propagation of elastic waves in fluid-saturated porous media. I outline problems and solutions in the use of ultrasonic measurements to predict velocities and attenuation at seismic frequencies. Field and laboratory studies are described in which a range of frequencies — seismic through sonic and ultrasonic — was covered and successfully modeled. I describe procedures developed during the 1950s and 1960s for interpreting laboratory measurements in terms of pore structure and degree of saturation, based on self-consistent and scattering theories. Laboratory equipment used for determining the elastic properties of anisotropic materials is described along with a discussion of the results obtained, including the correlation between both fluid permeability and prediction of in situ state of stress with the degree of shear-wave splitting. Finally, I discuss the application of rock-physics developments to reservoir characterization and monitoring, in particular to time-lapse seismic monitoring.

流体飽和多孔質媒質における骨格硬度の低下に伴うＢｉｏｔ波特性の変化

In this paper, various characteristics of the Biot waves in fluid-saturated porous media are examined numerically and analytically based on the relations in the previous paper (Kitsunezaki, 2004a), whose notation is also kept here. The Biot waves mean elastic waves in the Biot theory (Biot, 1956), which consist of two longitudinal waves (I-and II-waves) and a transverse wave. The main aim of this study is to reveal general variation tendency of characteristics of the longitudinal waves in wide range of sediments, typically from consolidated stiff sandstone to unconsolidated loose sand, mainly in connection with variation of skeleton-stiffness, which is represented by the velocity ratio in P-wave of skeleton to sound wave in fluid, VPb/Vƒ. In the wave characteristics, special attention is paid to displacement ratios and stress ratios in fluid to solid, as well as velocities and attenuation (logarithmic decrements). The both ratios are key factors to understand propagation mechanism of the two longitudinal waves. Main points of this study are as follows. First, general feature of the above properties is viewed as functions of frequency and skeleton-stiffness. The frequency characteristics are examined for typical two models of media with hard and soft skeletons. The properties in the low and high frequency limits are remarked to examine the effects of skeleton-stiffness. Second, approximate expressions of the characteristics are derived for media with very low skeleton-stiffness (VPb/Vƒ<<1), which almost corresponds to loose alluvial sand, in order to clarify factors controlling wave properties. Third, dynamic process in the media with very low skeleton-stiffness is analyzed. Then the results are represented as schematic models of stress-strain relations which demonstrate clearly the contrastive properties of I-and II-waves. In all examinations mentioned above, such a reciprocal relation between the two longitudinal waves is remarked as a useful general law, that the displacement ratio of I-wave in fluid/solid is equal to the opposite value of stress ratio of II-wave in solid/fluid, where I and II are mutually exchangeable.

Finite Element Methods for the Equations of Waves in Fluid-Saturated Porous Media

In this paper, finite element methods for the problems of wave propagation in fluid-saturated porous media are discussed. Particular emphasis is laid on the problems of infinite media. For this case, a new kind of non-reflecting boundary conditions on artificial boundaries is developed. It is of a simple form and is convenient to be used. Then finite element formulas for the equations of waves in fluid-saturated porous media with the non-reflecting boundary condition mentioned above are given. The results of numerical experiments are presented, which have shown that the given non-reflecting boundary condition and numerical methods are effective.

Ultrasonic wave propagation in bovine cancellous bone.

The acoustic properties of bovine cancellous (spongy) bone have been experimentally studied in vitro by the pulse transmission technique. Fast and slow longitudinal waves have been clearly identified when the acoustic wave propagates parallel to the direction of the trabeculae. Propagation speeds and attenuation of the fast and slow waves were observed in the frequency range of 0.5-5 MHz. Theoretical discussion is given to Biot's theory and the propagation of sound waves in fluid-saturated porous media.

The biot drag and virtual mass coefficients for face centered cubic granular materials

The Biot equations model the propagation of acoustic waves in fluid-saturated porous media. The “drag” and “virtual mass” coefficients in the equations depend on the physical properties of the fluid and solid constituents, the frequency and the microstructure of the porous medium. Recently, a method has been developed for determining the drag and virtual mass coefficients as functions of frequency. The method requires solving for the motion of the fluid in the pores when the pore walls are subjected to a spatially uniform, oscillatory motion. In this work we use the finite element method to compute the fluid motion and evaluate the drag and virtual mass coefficients for granular media consisting of spherical and polyhedral grains with face centered cubic (fcc) packings. The materials we consider have porosities that range from 0.261 to 0.480. We conclude that an empirical approach based on Biot's results for cylindrical pores can adequately approximate our computations for an fcc packing of spheres.

Reflection and transmission of elastic waves from a fluid-saturated porous solid boundary

Based on the classical work of Biot [J. Acoust. Soc. Am. 28, 168 (1956)] which predicts that three different kinds of bulk waves may propagate in the fluid-saturated porous solid, the wave equation is solved to determine the energy reflection and transmission coefficients of plane elastic waves at oblique incidence on an interface between a fluid and a fluid-saturated porous solid. For this purpose, the necessary formalism of the energy equation, the Poynting energy flux vector, and the sound intensity of elastic waves in fluid-saturated porous media are presented. Two general cases of mode conversion have been investigated: (1) The initial wave is incident from the fluid to the interface and generates three transmitted bulk waves in the fluid-saturated porous solid, and (2) the initial wave is incident from the fluid-saturated porous solid to the interface and generates three reflected bulk waves in the same medium. Furthermore, the transmission of sound through a fluid-saturated porous solid plate immersed in fluid is calculated. Measurement of sound transmission through a porous plate was carried out using experimental techniques suggested by Plona [Appl. Phys. Lett. 36, 259 (1980)]. Good correlation between measured and calculated values of the angular behavior of the transmission coefficients for fast, shear, and slow waves was obtained after adjusting the theory for experimentally obtained attenuation.

Computation of the biot drag and virtual mass coefficients

The Biot equations model the propagation of acoustic waves in fluid-saturated porous media. The equations contain coefficients which depend upon the frequency and the fabric, or microstructure, of the solid constituent. Recently, a method has been developed for determining the drag and virtual mass coefficients in Biot's equations as functions of frequency. The method requires solving for the motion of the fluid in the pores when the pore walls are subjected to a spatially uniform, oscillatory motion. To determine the fluid motion in realistic pore spaces, a numerical method must be used. In this paper the finite element method is used to determine the fluid motion. The drag and virtual mass coefficients are determined for several two-dimensional pore spaces. It is concluded that the drag coefficient is very insensitive to the pore geometry, while the virtual mass coefficient is sensitive to the pore geometry. It is also shown that the results can be expressed in nondimensional forms which permit the coefficients to be determined for different values of a characteristic linear dimension of the pore space.