Request pdf on formulas for the velocity of stoneley waves propagating along the loosely bonded interface of two elastic halfspaces. A cylindrical fluid filled tunnel, with a radius of 1. Existence and uniqueness of stoneley waves geophysical. The relation between the attenuation of the stoneley wave and permeability of the formation was predicted theoretically by rosenbaum 1974. Complex roots of the stoneleywave equation bulletin of the. Since the stoneley wave in a slow formation tends to travel at a lower frequency than the body p and s waves e. For the enumeration of the lorentzs force besmeared in the structure, generalized ohms law and maxwells equation have been. The propagation of stoneley waves in a fluidfilled borehole with a vertical fracture is. The resulting development is appreciably simpler than previous treatments of the theory. Stoneleywave attenuation and dispersion in permeable formations. Although the stoneleywave equation 1 has been the subject of much discussion in recent. Following radiation conditions in the media, the particular solutions are obtained, which satisfied the appropriate boundary conditions at an interface to obtain the secular equations of the stoneley wave in media. Elastodynamic simulation of tunnel detection experiments. The dynamics of viscous compressible fluid in a fracture.
Lowfrequency stoneley wave propagation at the interface of. The propagation of the stoneley scholte wave is analysed at the boundary between an ideal fluid and a viscoelastic solid. The range of existence in terms of material parameters for the real root corresponding to the propagation of stoneley. Asymptotic theory for rayleigh and rayleightype waves.
Stoneley waves and which propagate with a phase velocity lower than that of sound in the water. Identification of mixed acoustic modes in the dipole full. Owen 12 showed that of 900 isotropic materials, only 30 pairs would support stoneley waves. Elastodynamic simulation of tunnel detection experiments in. Stoneley is the most important mode affecting the transmitted shear wave.
The earths internal layer is made of various types. For a particular problem in which the body wave velocities in the solid are cq and31 and the density is 41, the body wave velocity in the liquid is c 2 and the density is 0z, and the angular frequency of the waves is m, we can relate the appropriate stoneley wave equation to equation 1. Attenuation for each specimen pair was determined by measuring input and output sa w amplitudes as well as stoneley wave induced interface particle displacements di rectly on the boundary. Request pdf on formulas for the velocity of stoneley waves. Dispersion of rayleigh, scholte, stoneley and love waves. Microstructurerelated stoneley waves andtheir e ect on the. In many realworld situations, the velocity of a wave. The rs wave is theoretically very interesting and gives.
The equation governing elastic waves propagating along a solidsolid interface is found to have sixteen 16 independent roots on its eight 8 associated riemann sheets. Influence of borehole overpressurization and plastic yielding. Mathematical modelling of stoneley wave in a transversely. Explicit secular equations of stoneley waves in a non. Equations for velocities of all these regimes have explicit forms and are verified by comparisons with exact solutions. Adhesive bondline interrogation using stoneley wave methods.
The decay of stoneley wave amplitude with distance from the interface is also frequency. Pdf explicit secular equations of stoneley waves in a non. By using the obtained formulas, we can easily reproduce the numerical results previously obtained by murty g. Stoneley wave velocity variation is analyzed by solving the modified scholte secular equation for velocity of stoneley waves, allowing to find dependency of the stoneley wave velocity on the. The first is to solve the stochastic wave equation using a. Detection of formation shear wave in a slow formation. The coupling ofwave motions at the boundaries z 0 and z l is now considered. Stoneley wave interacts with the surrounding rock mass, it may radiate energy outwards. The roots of the stoneley wave equation for solidliquid. Stoneley waves and provide permeability comparisons between acoustic and nmr logs. These regimes include biot, stoneley, and narrow channel.
Process array stoneley wave data nondispersively to obtain s. Ginzbarg and strick 11 graphically presented ratios of stoneley wave velocities and shear wae velocities for a wide range of elastic properties. Propagation of stoneley waves at an interface between two. The stoneley wave equation arises in the study of waves on interfaces between either two solid media or solid and liquid media. When found at a liquidsolid interface, this wave is also referred to as a scholte wave. In a slow formation, the stoneley wave attenuation becomes more sensitive to the shear wave attenuation of the formation. The reflection and scattering of stoneley guided waves at the tip of a crack filled. The stoneley wave in the tube can be approximated by a tube wave with uniform pressure distribution across the borehole. Permeability estimation from the joint use of stoneley wave. 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 of rayleigh waves for large wave lengths. Stoneley wave dispersion equation is determined by abodahab 3 in magnetothermoelastic materials. Introduction the study of waves interacting with a plane or curved interface between solid and liquid media gives rise to the well known stoneley wave equation. The secular equation of stoneley waves is derived in the form of the determinant by using. Pdf p532 can we determine permeability with the stoneley.
Stoneley wave attenuation 331 the remainder of the paper considers the dispersion and attenuation of the stoneley wave at higher frequencies. Of these, the interface wave the stoneley wave when the phase velocity is real and the rayleigh wave are associated with two separate roots, coexisting for some values of the material parameters. Stoneley waves are a further type of sw that create along a solidsolid interface and which are often exploited in borehole seismics to infer the shearwave. A stoneley wave is a boundary wave or interface wave that typically propagates along a solidsolid interface. The equation of a transverse wave traveling along a very long string is y 6. We denote the pressure averaged over borehole radius for re. First equation is the equation of the pressure continuity at z0 or at the whole volume of this small cylinder see 2,3,4 and. The steps include estimation of the stoneley wave slowness from conventional logs using a. Stoneley wave attenuation and dispersion and the dynamic. Complex roots of the stoneleywave equation bulletin of. The possibility of waves propagating along an interface. The secular equation of stoneley waves is derived in the form of the determinant by using appropriate boundary conditions i. When the socalculated relaxation length is erroneous, the inverted formation permeability from the stoneley wave is not correct either.
Scholteastoneley waves on an immersed solid dihedral. T h e reader who is interested in the derivation of these equations may be. There are two primary strategies for solving the wave equation in heterogeneous media. The basic equations are solved to obtain the general surface wave solutions in the medium in xz plane.
Stoneley wave phase velocities show a significant frequency dependence. To overcome this limitation and to provide a versatile alternative, the dynamic permeability problem is reformulated within the viscosityextended biot framework. There are two equations and there are two unknowns. This yields immediately that if a stoneley wave exists, then it is unique. The wave equation in one dimension later, we will derive the wave equation from maxwells equations. Stoneley wave measurement is a direct index for flow zone. Finiteelement simulations of stoneley guidedwave reflection and. Introduction inthis paper, we demonstrate how the partial wave method, as first introducedby solie and auld1, can be used to gain a new perspective on guided waves. Stoneleywave attenuation and dispersion in permeable.
Three different fluid wave regimes can coexist in such objects, depending on the various combinations of object parameters. Influence of borehole overpressurization and plastic. Schematic showing attenuation of stoneley wave at a fracture intersecting a borehole. 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. Its speed is slower than the shear and mud wave speeds, and it is slightly dispersive, so different frequencies propagate at different speeds. The endresult is a persistent, harmonic tertiary wavefield that may be. Lecture 11 chapter 16 waves i university of virginia. Ada228 119 characterization of elastic properties of. Wu and yin 2010 introduced a reliable method for determining reservoir permeability from the stoneley wave attenuation, extracted from conventional sonic logs, by inversion of the full biot wave equations for a porous medium.
Spectral evolution equations are derived for plane, progressive, finiteamplitude stoneley and scholte waves that propagate along plane interfaces formed by. Pdf rayleighs, stoneleys, and scholtes interface waves in. Biots theory of dynamic poroelasticity is used to derive the dispersion equation. Riemann sheets associated with the stoneleywave equation. Formulas for the slowness of stoneley waves with sliding. Acoustic stoneley wave sensor for towed array applications final report for the period september 17, 1979 through september 16, 1980 general order no.
Iot 1955 has worked out the motion equation valid a. Thus, the processing might either over or underestimate formation slowness. Permeability estimation in carbonate reservoirs using. A stoneley wave index is defined as a ratio of the stoneley wave slowness interpreted in natural conditions and calculated for the non permeable rock.
Numerical examples of calculations are presented for two important cases. On formulas for the velocity of stoneley waves propagating along. There have been many attempts to use the stoneley waves for reservoir characterization. Stoneley and rayleigh waves in thermoelastic materials. On formulas for the velocity of stoneley waves propagating. Here it is, in its onedimensional form for scalar i. Mechanical continuity equations at the solidliquid interface applied to calculate analytically scholte stoneley wave properties. Riemann sheets associated with the stoneley wave equation.
Nov 01, 2011 the derivation also shows that if a stoneley wave exists, then it is unique. Stoneley wave mode to locate permeable fractures intersect ing a borehole and to estimate their effective apertures. The roots of the equation are related, through the appropriate analysis, to the nature and velocity of the interface waves. Diffraction of an incident scholte stoneley wave at the corner of a solid dihedral. Permeability estimation from the joint use of stoneley wave velocity. Using the partial wave method for wave structure calculation. This equation determines the properties of most wave phenomena, not only light waves. On the other hand, if the processing window width is expanded too much, the stoneley wave will contribute to the final slowness value. The static bias refers to the prestress and associated elastic and plastic strains in the formation prior to any tensile fracture. 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. Mathematical modelling of stoneley wave in a transversely isotropic.
When the elasticity k is constant, this reduces to usual two term wave equation u tt c2u xx where the velocity c p k. In this paper, the eight roots of the stoneley wave equation for solidliquid interfaces are investigated analytically for all values of the equation parameters and the general pattern of the roots is elucidated. In this paper, the eight roots of the stoneley wave equation for solidliquid interfaces are investigated analytically for all values of the. Specifically, for a hard formation, an infinite number of pseudorayleigh or normal modes exist biot, 1952. Formulas for the slowness of stoneley waves with sliding contact. In this study support vector regression svr was used as a technique to analyze the stoneley waves in wells no. This rs wave is essentially a stoneley type wave29 propagating along the geometry described above. Con sequently, in the limiting case where one medium becomes a vacuum, the interface. The wave is of maximum intensity at the interface and decreases exponentially away from it. Download free pdf explicit secular equations of stoneley waves in a nonhomogeneous orthotropic elastic medium under the influence of gravity applied mathematics and computation, 2010. The shear wave ti parameter, v sh or c 66, can be estimated using the following procedure. The second equation is the equation of the pressure uniform. Theoretical study of the stoneleyscholte wave at the interface. Other examples and tests, such as nmr permeability estimation in.
Fracture characterization from attenuation of stoneley waves. Dispersion of stoneley waves through the irregular common. Rayleighs, stoneleys, and scholtes interface waves in. Explicit secular equations of stoneley waves in a nonhomogeneous orthotropic elastic medium under the influence of gravity. A stoneley wave propagating along the interface is assumed to have the form, u e ikbx 3 e ikx 1 vt 9 where uj are particle displacement vectors along xjaxes. In terms of the potential 7j, the fluid pressure p and axial displacement ofthe stoneley wave are given by 3 4 where pf is fluid density and w is the angular frequency. Pdf explicit secular equations of stoneley waves in a. The roots of the stoneley wave equation for solidliquid interfaces. Let us first derive the original surface wave equation discovered by lord rayleigh 1885. Models for single and double fractures in the pres. Rayleighs, stoneleys, and scholtes interface waves in elastic.
The tunnel is characterized by a p wave velocity of 0. This can be studied by analyzing the stoneley slowness in the low and high frequency limits. Stoneley wave velocity variation is analyzed by solving the modified scholte secular equation for velocity of stoneley waves, allowing to find dependency of the stoneley wave velocity on the wiechert parameter and construct a set of inequalities that confines region of existence for the appropriate root of the secular equation. Fracture characterization from attenuation of stoneley. 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. Stoneley wave along vertical fracture 67 previously. Various forms of secular equations for stoneley wave velocity were constructed. Thus, the processing might either over or underestimate formation slowness depending on the selected processing parameters. Microstructurerelated stoneley waves andtheir e ect on the scattering propertiesof a 2d cauchyrelaxedmicromorphic interface alexios aivaliotisa, ali daouadjia, gabriele barbagallo a, domenico tallarico, patrizio ne b. From the derivation of them, it is shown that if a stoneley wave exists. Stoneley wave velocity variation journal of theoretical. Fivesolutionsarecompared,thebiotsolution equations4346,thenonviscous.
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