Stoneley wave generation by an external seismic point source in an infinite fluid-filled borehole embedded in a transversely isotropic formation: Basic relationships

The method of integral transformations is used to obtain a long-wave solution to the problem of tube wave excitation by an external point source in an infinite fluid-filled borehole embedded in a transversely isotropic formation. The external field that occurs in the formation gives rise to waves in the borehole fluid. The waves generated in the borehole include the lowest mode of the Stoneley wave (tube wave), which is the borehole eigenmode, and the qP-and qSV-waves. It is shown that the Stoneley wave is determined by the contributions of two poles in the complex plane of horizontal slowness. According to the asymptotic solution, the Stoneley wave can be described by one of three different waveforms depending on the relationship between the elastic parameters of the surrounding anisotropic formation and the borehole fluid. An analysis of the results of calculations shows that the shape and polarity of the Stoneley wave strongly depend on the sign of the nonellipticity parameter of the elastic medium, which offers a possibility of estimating the anisotropy of the borehole environment from observations of the waveform of the Stoneley wave.     

Exact and Asymptotic Solutions for the Field of a Point Source in a Transversely Isotropic Medium

Exact and asymptotic solutions are obtained for the acoustic field generated by an isotropic pulsed point source in an infinite transversely isotropic elastic medium. The exact solution for the displacement field is obtained in the form of a double integral over the horizontal slowness and the frequency by using the method of integral transforms. The calculation of the integral over the horizontal slowness by the method of stationary phase reduces the exact solution to an asymptotic solution that is convenient for numerical calculations. Formulas are given for calculating the spreading factors and the wave fronts of quasi-longitudinal qP-waves and quasi-transverse qSV-waves. With the formulas obtained, the displacement field of a point source is investigated for a particular transversely isotropic medium.     

Tube-wave propagation in a fluid-filled borehole generated by a single point force applied to the surrounding formation

Propagation of tube waves in an infinite fluid-filled borehole, generated by a single-force point source placed in the elastic surrounding formation, is analyzed in the long-wave approximation. Integral representations of the precise solution are obtained both for fast and slow formations. An asymptotic analysis of tube-wave propagation in the fluid-filled borehole is performed on the basis of these two integral representations. The complete asymptotic wave field in the borehole fluid for a fast formation consists of P and SV phases and the lowest eigenmode of the Stoneley wave (tube wave). For a slow formation the conical Stoneley wave (Mach wave) is generated. It appears only behind the critical angle defined by the ratio of the S wave velocity in the formation to the low-frequency Stoneley wave velocity and decays weakly with an offset. Asymptotic wave forms are in good agreement with wave forms obtained by straightforward calculations.     

Angular spectrum approach for the computation of group and phase velocity surfaces of acoustic waves in anisotropic materials

The decomposition of an acoustic wave into its angular spectrum representation creates an effective base for the calculation of wave propagation effects in anisotropic media. In this method, the distribution of acoustic fields is calculated in arbitrary planes from the superposition of the planar components with proper phase shifts. These phase shifts depend on the ratio of the distance between the planes to the normal component of the phase slowness vector. In anisotropic media, the phase shifts depend additionally on the changes of the slowness with respect to the direction of the propagation vector and the polarization. Those relations are obtained from the Christoffel equation. The method employing the fast Fourier transformation algorithm is especially suited for volume imaging in anisotropic media, based on holographic detection in transmission of acoustic waves generated by a point source. This technique is compared with measurements on crystals performed by phase-sensitive scanning acoustic microscopy.     

基于鞍点法与互易性的远探测波场模拟*

声波远探测中波场是非轴对称的,采用数值算法计算波场会耗费大量时间,无法满足实际测井数据实时处理的需求。为了解决这一问题,该文采用解析法分别计算辐射场和井外界面反射波激发的井内响应。首先利用鞍点法获得井内声源的远场辐射波场,并与实轴积分获得的精确结果进行比较验证解的正确性。然后将反射波等效为集中力的辐射波,利用集中力与井内声源的互易关系获得反射波激发的井内波场解,该解答与有限差分模拟结果一致。该方法为远探测的正演模拟和远探测结果的及时评价提供了有效手段。     

孔隙地层震电测井波场分波分析

由于孔道的双电层结构和流体-固体耦合运动, 弹性波在孔隙地层中传播时会引起电磁场. 前人提出了基于这种耦合效应的震电测井方法, 但对井中震电波的特性缺少细致的分析. 本文对孔隙地层充流体井孔中点声源激发的震电耦合波的分波进行分析, 证明了井孔流体电磁波波数不是井内震电波场支点, 计算了地层电磁波支点、横波支点和快纵波支点对应的侧面波. 在利用辐角原理求出震电波场函数的复极点之后, 针对典型砂岩地层计算了震电波场模式波(包括泄漏模式)的频散曲线和声压、电场激发曲线. 计算了各个侧面波和模式波的电场激发强度与声压激发强度的比值(电声激发比), 发现纵波的电声激发比相对于横波和模式波更高, 而Stoneley波电声激发比的辐角对渗透率敏感, 其敏感性会随频率增大而增加, 随孔隙度增大而降低.     

Wave scattering in a multiscale random inhomogeneous medium

In this paper, using the Fock method of the fifth parameter and weighted Fourier-transform with respect to the coordinates of the source and observer, an integral representation is obtained for the wave field in a randomly inhomogeneous medium without invoking the assumption about small-angle propagation. Random trajectory variations to a first approximation are taken into account in calculating the partial wave phase (the expression under the integral sign). The expressions for the field in a medium with different-scale irregularities and for the scintillation index, obtained using this integral representation, are compared with known results. The good agreement with results from the theory of single scattering in a medium with background irregularities, and with investigations of the scintillation index made in terms of Rytov's method and path integrals, indicates that it is possible to use the approach developed in this study to describe the effects of simultaneous influence of different-scale irregularities.     

Determination of crack orientation in the vicinity of a borehole by vertical seismoacoustic profiling

Excitation of a Stoneley wave in a single fluid-filled crack of small wave size in a borehole under an external seismic wave incident on it at an arbitrary angle is considered. On the basis of the solution obtained, the feasibility of solving the inverse problem of reconstructing the crack orientation from the data of vertical seismoacoustic profiling by hydrophones with the use of minimal input data is demonstrated.     

Shear-horizontal transverse-electric seismoelectric waves in cylindrical double layer porous media

The shear-horizontal(SH) waves excited by the shear source in a borehole are easy to analyze due to the simple waveform. The borehole-side structures make the formation properties discontinuous. We consider a cylindrical double layer structure and study the borehole shear-horizontal and transverse-electric(SH-TE) seismoelectric waves. We first derive the expressions of the basic field quantities, and simulate the acoustic field and electric field using the real axis integral method. Compared with the wave fields of an infinitely homogeneous porous medium outside the borehole, the cylindrical layered structure makes the multi-mode cylindrical Love waves and their accompanying electric fields excited.Next, in order to study the interface response law of the inducing electric fields, we use the secant integral method to calculate the interface converted electromagnetic waves and analyze the causes of each component. It is found that an interface response occurs each time the SH wave impinges the interface in the layered porous medium. The results show that the SH-TE mode has a potential application for borehole-side interface detection in geophysical logs.     

A computational method for wave propagation from a point load in an anisotropic material

One approach which is employed to solve dynamic point load problems in plates and laminates is to take integral transforms to reduce the governing equations to a system of ordinary differential equations with respect to the depth variable. The solution of this system leads to expressions for the transforms of the displacement and stress components at any level in the plate and the transient response at any location may then be recovered by inversion of the multiple transforms. The formal transform inversion involves a double infinite integral but by making a change of variable this may be replaced by an infinite integral associated with a line source and a finite integral with respect to the orientation of the line. A first attempt at applying this approach to obtain the point load response of quasi-isotropic fibre composite laminate led to a non-causal predicted signal. This paper deals with an investigation of this proposed method applied to the simpler model problem of wave propagation in a two-dimensional anisotropic medium. Results are obtained for two different time histories of point loads, namely: a delta function; and a single period of a sine function. In the case of the delta function source a comparison is made with the analytic solution and the errors arising from the numerical approach are discussed. Graphs are also presented showing the non-causal contributions to the overall response which arise at individual angles of orientation of the line source.     

The group velocity variation of Lamb wave in fiber reinforced composite plate

Experimentally measured Lamb wave group velocities in composite materials with anisotropic characteristics are not the same as the theoretical group velocities which is calculated with the Lamb wave dispersion equation. This discrepancy arises from the fact that the angle between the group velocity direction and the phase velocity direction in anisotropic materials exists. Wave propagation in a composite material with anisotropic characteristics should be considered with respect to magnitude correction in addition to direction correction. In this study, S0 mode phase velocity dispersion curves are depicted with the variation of degree with respect to the fiber direction using a Lamb wave dispersion relation in the unidirectional, bidirectional, and quasi-isotropic composite plates. Slowness surface is sketched by the reciprocal value of the phase velocity curves. The magnitude and direction of the group velocity could be calculated from the slowness surface. The recalculated group velocities with consideration of the magnitude and direction from the slowness surface are compared with experimentally measured group velocities. The proposed method shows good agreements with theoretical and experimental results.     

A uniform asymptotic integral representation for the fourth moment of the field of a wave propagating in a statistically homogeneous medium

In this paper, a uniform integral representation has been obtained for the fourth moment of the field of a wave propagating in a medium with random large-scale irregularities. The solution to the equation was obtained using a method of integral transformations and Maslov's complex WKB method. The representation obtained differs in its form from those reported thus far and in particular from those given by the method of two-scale expansions and the interference integral method. First, the paper considers the case of a plane wave incident on a layer with irregularities, followed by a treatment of the general case of an arbitrary source.     

Scattering and diffraction of a plane wave by a randomly rough half-plane

The scattering and diffraction of a TE (transverse electric) plane wave by a randomly rough half-plane are studied by a combination of three techniques: the Wiener-Hopf technique, the small perturbation method and a probabilistic method based on the shift-invariance of a homogeneous random function. By use of the D^a-Fourier transformation based on the shift-invariance, it is shown that the scattered wave is written by an inverse Fourier transformation of a homogeneous random function with a complex parameter. For a small rough case, such a random function with a complex parameter is expanded in a perturbation series and then the first-order solution is obtained exactly in an integral form. The first-order solution involves two physical processes such that the edge-diffracted wave is scattered by the randomly rough plane and the scattered wave, due to roughness, is diffracted by the half-plane. The solution is transformed into a sum of the Fresnel integrals with complex arguments, an integral along the steepest descent path and a branch-cut integral, which are evaluated numerically. Then, intensities of the coherently scattered wave and incoherent wave are calculated in the region near the edge and illustrated in figures.     

柱面分层流体饱和孔隙地层中的声波测井波场模拟

因钻井作业时泥浆侵入等因素,井外介质沿径向常呈现非均匀性。本文将地层视为柱面分层孔隙介质,采用传递矩阵法模拟声波测井波场。注意到慢纵波波数的实部和虚部都很大,当井外含大厚度层且频率较高时,易因计算大自变量的Hankel函数而溢出,我们在波场表达式中引入了归一化的Hankel函数。论文将前人模拟关于柱面分层固体弹性介质声波测井的广义反射/透射系数计算方法推广到了分层孔隙介质情况,导出了波幅系数的计算公式,并针对含侵入带的柱面分层孔隙地层,计算了声波测井全波波形。      

Effect of a mudcake on the propagation of stoneley waves in a borehole

The problem of detecting a permeable stratum blocked by a mudcake with the help of acoustic measurements inside a borehole is considered. Different physical models of the mudcake are compared: in the form of a highly viscous liquid layer, in the form of a soft elastic shell, and in the form of an elastic shell fixed in an arbitrary way to the borehole walls. Numerical calculations are presented for the wave field in a borehole.     

Green's electron function in a quantized plane wave field

It is shown that the Green's function of an electron that interacts with a quantized plane wave can be expressed in terms of the corresponding Green's function of a scalar particle. By using the known expression for the Green's function of a scalar particle, an integral representation is found with respect to the intrinsic time for the Green's electron function in a quantized plane wave of arbitrary form.     

Multiple monopole scattering of sound waves from spherical particles in liquid and elastic media

A solution to the problem of the mean sound field in liquid and elastic media with spherical particles causing monopole scattering of sound is proposed. The integral equation obtained for the field allows passage to the Helmholtz equation with an effective wave number. The characteristic features of the solution are the absence of radiation loss in the mean field wave and the absence of limitations on the particle concentration. The integral equation is used as the basis for solving the problem of the incidence of a plain sound wave at an arbitrary angle on a plane layer of a medium with particles.     

Scattering and diffraction of a plane wave by a randomly rough half-plane*

Abstract

The scattering and diffraction of a TE (transverse electric) plane wave by a randomly rough half-plane are studied by a combination of three techniques: the Wiener-Hopf technique, the small perturbation method and a probabilistic method based on the shift-invariance of a homogeneous random function. By use of the D^a-Fourier transformation based on the shift-invariance, it is shown that the scattered wave is written by an inverse Fourier transformation of a homogeneous random function with a complex parameter. For a small rough case, such a random function with a complex parameter is expanded in a perturbation series and then the first-order solution is obtained exactly in an integral form. The first-order solution involves two physical processes such that the edge-diffracted wave is scattered by the randomly rough plane and the scattered wave, due to roughness, is diffracted by the half-plane. The solution is transformed into a sum of the Fresnel integrals with complex arguments, an integral along the steepest descent path and a branch-cut integral, which are evaluated numerically. Then, intensities of the coherently scattered wave and incoherent wave are calculated in the region near the edge and illustrated in figures.     

Seismoelectric waves in a borehole excited by an external explosive source

The conversion of energy between seismic and electromagnetic wave fields has been described by Pride’s coupled equations in porous media. In this paper, the seismoelectric field excited by the explosive point source located at the outside of the borehole is studied. The scattering fields inside and outside a borehole are analyzed and deduced under the boundary conditions at the interface between fluid and porous media. The influences of the distance of the point source, multipole components of the eccentric explosive source, and the receiving position along the axis of vertical borehole, on the converted waves inside the borehole are all investigated. When the distance from the acoustic source to the axis of a borehole is far enough, the longitudinal and coseismic longitudinal wave packets dominate the acoustic and electric field, respectively. The three components of both electric field and magnetic field can be detected, and the radial electric field is mainly excited and converted by the dipole component. Owing to the existence of borehole, the electric fields and magnetic fields in the borehole are azimuthal. The distance from the point where the maximum amplitude of the axial components of electric field is recorded, to the origin of coordinate indicates the horizontal distance from the explosive source to the axis of vertical borehole.     

Existence of extraordinary zero-curvature slowness curve in anisotropic elastic media

Acoustic wave propagation in elastic media is characterized by the slowness surface. The slowness surface consists of three sheets associated with three modes of wave propagation and the two outer sheets can have zero-curvature locally. It is shown that the outmost sheet can admit extraordinary zero-curvature and the slowness curve can appear as a straight line locally. Using the perturbation method, the conditions for the extraordinary zero-curvature are derived analytically without violating the thermodynamic condition for elastic media. The results can be applied to crystals with higher symmetry and to the study of phonon focusing and surface waves.