Rayleighs, stoneleys, and scholtes interface waves in elastic. A stoneley wave is a boundary wave or interface wave that typically propagates along a solidsolid interface. To validate the equations used here, we included in the appendix, a comparison bet. The roots of the stoneley wave equation for solidliquid interfaces. In many situations, this wave mode is the most recognizable wave in the full waveform acoustic logs because of its relatively large amplitude and slow velocity. The derivation of these formulae shows that if a scholte wave exists, then it is unique. The propagation of the stoneleyscholte wave is analysed at the boundary between an ideal fluid and a viscoelastic solid. An understanding of wave propagation basics is essential for appreciation of modern soniclogging technology. We have applied this hybrid model to interpret laboratory measurements for which the previously suggested choice of the viscous relaxation length does not provide an accurate prediction. Adhesive bondline interrogation using stoneley wave methods.
The string has length its left and right hand ends are held. The method relies on the presence of a stoneley wave travel time curve from a dipole shear or array sonic full waveform acoustic log. The first two of these waves are associated with energy moving along the borehole wall, and the last is a pressure wave that travels through the mud in the borehole. Influence of the rotation and gravity field on stonely. Stoneley wave along vertical fracture 67 previously. This equation indicates that the stoneley wave propagation in a pc. The roots of the equation are related, through the appropriate analysis, to the nature and velocity of the interface waves.
Characterization of surface cracks using rayleighs wave excitations was dealt by an indirect boundary element method. Biots theory of dynamic poroelasticity is used to derive the dispersion equation. The wave equation is a linear secondorder partial differential equation which describes the propagation of oscillations at a fixed speed in some quantity y y y a solution to the wave equation in two dimensions propagating over a fixed region 1. Modified equation and amplification factor are the same as original laxwendroff method.
Chapter 4 the wave equation another classical example of a hyperbolic pde is a wave equation. Stoneleywave attenuation 331 the remainder of the paper considers the dispersion and attenuation of the stoneley wave at higher frequencies. A simplified acoustoelastic formulation of stoneley wave is presented for the parallelism of the borehole axis and the formation axis of symmetry. University of calgary seismic imaging summer school august 711, 2006, calgary abstract abstract. Wave equation the purpose of these lectures is to give a basic introduction to the study of linear wave equation. Stoneleys waves occur at the interface between two solids 6. It is also shown that a scholte wave is always possible. The wave equation is an important secondorder linear partial differential equation for the description of wavesas they occur in classical physicssuch as mechanical waves e. Theoretical study of the stoneleyscholte wave at the interface.
The roots of the stoneley wave equation for solidliquid. There are three wave arrivals that can easily be identified in a full waveform acoustic log. Wave equation in 1d part 1 derivation of the 1d wave equation vibrations of an elastic string solution by separation of variables three steps to a solution several worked examples travelling waves more on this in a later lecture. Permeability from stoneley wave crains petrophysical handbook. The secular equation of stoneley waves is derived in the form of the. Lowfrequency stoneley wave propagation at the interface. Surface and interface waves governed by the membrane equation. A type of largeamplitude interface, or surface, wave generated by a sonic tool in a borehole. Rayleighs, stoneleys, and scholtes interface waves in. Frequencydependent stoneley wave attenuation is extracted by analyzing array sonic measurements.
Stoneley wave, microstretch, micropolar, frequency equation. Viscous seismic wave propagation simulation by using fractional order equation has attracted much recent attention. An understanding of wavepropagation basics is essential for appreciation of modern soniclogging technology. Solution of the wave equation by separation of variables. Stoneleywave attenuation and dispersion in permeable. Omnidirectional rayleigh, stoneley and scholte waves with general. In this paper, we have studied the stoneley wave propagation at the interface of two dissimilar transversely isotropic thermoelastic homogeneous media with two temperature and rotation. Borehole acoustic waves may be as simple or as complex as the formations in which they propagate. Mathematical modelling of stoneley wave in a transversely.
Permeability estimation from the joint use of stoneley wave. Equation 8 has real solutions for conly if the material constants of the two bonded media are suitably related. The simplest example of slow waves may be the plane wave propagation in a medium which consists of a fluid layer sandwiched by. The secular equation of stoneley waves is derived in the form of the determinant by using appropriate boundary conditions i. Tool effects on stoneley wave 73 where c w k is the phase velocity ofthe wave modes that exist in the presence of the rigid tool. As mentioned earlier, the stoneley wave slowness data were only available for well c. The results indicate that the hybrid model can provide another approach to model stoneley wave attenuation and dispersion across the entire frequency range. Stoneleywave attenuation and dispersion in permeable formations. In this section, we present a general derivation of the membrane equation. After them come rayleigh waves, stoneley waves, and mud waves.
Stoneley wave in a fluidfilled pressurized borehole surrounded by a transversely isotropic elastic solid with nine independent thirdorder elastic constants in presence of biaxial stresses are studied. The wave equation is a secondorder linear hyperbolic pde that describesthe propagation of a variety of waves, such as sound or water waves. Lowfrequency stoneley wave propagation at the interface of. Pdf propagation of stoneley waves at an interface between. Rayleigh wave and stoneley wave, as well as the crack wave found by chouet 1986, may be included in this class. With a wave of her hand margarita emphasized the vastness of the hall they were in. Existence and uniqueness of stoneley waves geophysical. Dispersion of stoneley waves through the irregular common.
The simplest example of slow waves may be the plane wave propagation in a medium which consists of a fluid layer sandwiched by two elastic halfspaces ferrazzini and aki, 1987. Keyboards click, crickets chirp, telephones ring and people laugh. Adhesive bondline interrogation using stoneley wave methods richard o. However, conventional finitedifference fd methods of fractional partial difference equation adopt a global difference operator to. These waves appear at small wave lengths, and it is shown further on that their velocity increases continuously with the wave length, finally corresponding to that. However, its implementation on actual field data would be erroneous, as waveforms are recorded in the presence of an acoustic tool of finite size in the borehole. For the enumeration of the lorentzs force besmeared in the structure, generalized ohms law and. Lowfrequency stoneley wave propagation at the interface of two porous halfspaces. Mathematical modelling of stoneley wave in a transversely isotropic thermoelastic media rajneesh kumar1, nidhi sharma2, parveen lata3 and s. The velocity and attenuation of the stoneley surface wave are determined. The basic existenceuniqueness theory for stoneley waves propagating along a plane interface between different isotropic elastic media is reexamined, using a matrix formulation of the secular equation. Abodahab4 1department of mathematics kurukshetra university kurukshetra, haryana, india 2department of mathematics mm university mullana, ambala, haryana, india 3department of basic and applied sciences. These interfaces excited by dynamic loads cause the emergence of rayleighs, stoneleys, and scholtes waves, respectively.
Permeability from stoneley wave this method is reproduced from a schlumberger petrophysical analysis report, edited fpr clarity and consistency with computerready math concepts. Wave equation in 1d part 1 derivation of the 1d wave equation vibrations of an elastic string solution by separation of variables three steps to a solution several worked examples travelling waves more on this in a later lecture dalemberts insightful solution to the 1d wave equation. The mathematics of pdes and the wave equation michael p. Keywords stoneley wave, microstretch, micropolar, frequency equation, unbonded interface, bonded interface abbudi, m. The first wave front is associated with the speed of compressional waves of the limestone, the second is close to the shear waves velocity of limestone, and the last shows a velocity of less than the shear wave velocity of sandstone and can be associated with the propagation of stoneleys waves.
Acoustoelastic effects of stoneley waves in a borehole. The resulting development is appreciably simpler than previous treatments of the theory. Specifically, for a hard formation, an infinite number of pseudorayleigh or normal modes exist biot, 1952. Stoneley wave propagation in transversely isotropic. Propagation of stoneley waves at an interface between two microstretch elastic halfspaces. Stoneley wave dispersion equation is determined by abodahab 3 in magnetothermoelastic materials. In dealing with the interaction of a stoneley wave with a porous formation, we decompose the problem into two parts. Thesecond is the interaction of the stoneley wave with the slow wave. When found at a liquidsolid interface, this wave is also referred to as a scholte wave.
The first is the in teraction of the stoneley with an equivalent elastic formation without fluid flow. Numerical analysis of the propagation characteristics of. Then, based on biots full theory applied to a borehole model and the standard. T h e reader who is interested in the derivation of these equations may be.
Stoneley waves can propagate along a solidfluid interface, such as along the walls of a fluidfilled borehole and are the main lowfrequency component of signal generated by sonic sources in boreholes. Depending on the medium and type of wave, the velocity v v v can mean many different things, e. The second is the interaction with the flow that is governed by the dynamic permeability. Pdf propagation of stoneley waves at an interface between two. It arises in fields like acoustics, electromagnetics, and fluid dynamics. In order of arrival time, they are the p wave packet, the s wave packet, and the stoneley wave. The wave equation governs a wide range of phenomena, including gravitational waves, light waves, sound waves, and even the oscillations of strings in string theory. Jan 17, 2020 in this paper, we have studied the stoneley wave propagation at the interface of two dissimilar transversely isotropic thermoelastic homogeneous media with two temperature and rotation. Keeping in view of this, dispersion equation for stoneley waves at the interface of two different tit mediums with two temperature and rotation have been derived. The pressure wave time series received by a receiver at a position z z 0 is the source location on the tool is given by. Chapter 2 the wave equation after substituting the. In this paper, the eight roots of the stoneley wave equation for solidliquid interfaces are investigated analytically for all values of the.
Analysis of stoneley waves can allow estimation of the locations of. The first two are analogs of the compressional and shear head waves refracted along the borehole wall. Solution of the wave equation by separation of variables the problem let ux,t denote the vertical displacement of a string from the x axis at position x and time t. Based on field ob servations, paillet and white 1982 have suggested that the stoneley wave may be. The secular equation for the rayleigh wave is obtained as a limiting case. In particular, we examine questions about existence and. They can be high amplitude, but always arrive after. The wave is of maximum intensity at the interface and decreases exponentially away from it. Highlights formulae for the velocity of scholte waves are derived using the complex function method. The present work deals with the mathematical inspection of stoneley wave propagation through the corrugated irregular common interface of two dissimilar magnetoelastic transversely isotropic mti halfspace media under the impression of hydrostatic stresses. I show that the dispersion equation coincides with the equation for the stoneley wave at the interface of two elastic halfspaces in the lowfrequency range. Influence of the rotation and gravity field on stonely waves in a nonhomogeneous orthotropic elastic medium article in journal of computational and theoretical nanoscience 102. Numerical examples of calculations are presented for two important cases. The stoneley wave equation arises in the study of waves on interfaces between either two solid media or solid and liquid media.
Pdf frequency equations for stoneley waves at unbonded and bonded interfaces. When applied to linear wave equation, twostep laxwendroff method. Derivation of the wave equation in these notes we apply newtons law to an elastic string, concluding that small amplitude transverse vibrations of the string obey the wave equation. The dispersion curves giving the phase velocity and attenuation coefficients with wave number are computed numerically.
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