PARAMETER ESTIMATION AND FAULT DETECTION BY THREE-COMPONENT SEISMIC AND GEOELECTRICAL SURVEYS IN A COAL MINE*

Three-component seismic and geoelectrical in-mine surveys were carried out in Lyukobanya colliery near Miskolc, Hungary to determine the in situ petrophysical parameter distributions and to detect inhomogeneities in the coal seam. The seismic measurements comprise an underground vertical seismic profile, using body waves, and an in-seam seismic amplitude-depth distribution and transmission survey, using channel waves. The geoelectrical measurements are based on the drift- and seam-sounding method. Interval traveltime-, amplitude-, multiple-filter- and polarization analysis methods are applied to the seismic data. They lead to a five-layer model for the strata including the coal seam. The coal seam and two underlying beds act as a seismic waveguide. The layer sequence supports the propagation of both normal and leaky mode channel waves of the Love- and Rayleigh type. A calculation of the total reflected energy for each interface using Knott's energy coefficients shows that the velocity ranges of high reflection energy and of normal and leaky mode wavegroups coincide. The excitation of wavegroups strongly depends on the seismic source. A simultaneous inversion of a geoelectrical drift- and seam-sounding survey prevents misinterpretations of the seismic data by clearly identifying the low-velocity coal seam as a high-resistivity bed. Calculations of dispersion and sounding curves improve the resolution of the slowness and resistivity in each layer. Both diminished amplitudes and distortions in the polarization of transmission seismo-grams and decreasing resistivities in a geoelectrical pseudosection of the coal seam are related to an inhomogeneity. A calculation of synthetic seismograms for Love and Rayleigh channel waves with the finite-difference and the Alekseev-Mikhailenko method agrees well with the field data for the main features, i.e., particular arrivals in the wave train, wavegroups, velocities and symmetries or asymmetries. This in-mine experiment demonstrates that the simultaneous acquisition, processing and interpretation of seismic and geoelectrical data improve the lithological interpretation of petrophysical parameter distributions. Coal seam inhomogeneities can also be detected more reliably by the two independent surveys than by one alone.     

煤矿井下槽波三维数值模拟及频散分析

采用交错网格高阶有限差分法编制了地震波场三维正演模拟软件,设计了基于镜像法原理处理煤矿井下近水平和起伏巷道特殊空间的算法;模拟了煤矿井下含巷道和不含巷道情况下煤层中传播的地震波场,并分析其频散特征.结果发现:由于巷道的影响,巷道壁上产生很强的巷道振型槽波,煤层中则出现了以Love型为主的槽波,据此分析了实际槽波记录的形成机理,研究结果对今后煤矿井下巷道地震超前探测和工作面弹性波透视等具有重要的理论意义和实际价值.     

含小断层煤层Rayleigh型槽波波场和频散分析

煤层隐伏小断层在煤田地质勘探期间能否查明,是影响煤矿安全生产的主要地质因素.煤层中传播的地震槽波,非常适用于探测煤田异常构造.本文利用谱元法模拟计算了含断层模型Rayleigh型地震槽波的产生、传播过程,对于直达槽波、反射槽波、透射槽波进行了波场特征分析,频散特征分析,频谱特征分析.根据不同小落差断层模型（垂直断距分别为1/4、1/2、3/4、1个煤厚;断层倾角分别为90°、60°、45°）,分析了Rayleigh型槽波传播过程中断层对反射槽波、透射槽波能量、频谱的影响效应.基于以上分析,对于小断层模型中的Rayleigh型地震槽波取得了一些规律性认识,并对Rayleigh型地震槽波的勘探应用做了一些探讨.     

Dispersion features of transmitted channel waves and inversion of coal seam thickness

In-seam seismic survey currently is a hot geophysical exploration technology used for the prediction of coal seam thickness in China. Many studies have investigated the relationship between the group velocity of channel wave at certain frequency and the actual thickness of exposed coal beds. But these results are based on statistics and not universally applicable to predict the thickness of coal seams. In this study, we first theoretically analyzed the relationship between the depth and energy distribution of multi-order Love-type channel waves and found that when the channel wave wavelength is smaller than the thickness of the coal seam, the energy is more concentrated, while when the wavelength is greater than the thickness, the energy reduces linearly. We then utilized the numerical simulation technology to obtain the signal of the simulated Love-type channel wave, analyzed its frequency dispersion, and calculated the theoretical dispersion curves. The results showed that the dispersion characteristics of the channel wave are closely related to the thickness of coal seam, and the shear wave velocity of the coal seam and its surrounding rocks. In addition, we for the first time realized the joint inversion of multi-order Love-type channel waves based on the genetic algorithm and inversely calculated the velocities of shear wave in both coal seam and its surrounding rocks and the thickness of the coal seam. In addition, we found the group velocity dispersion curve of the single-channel transmitted channel wave using the time–frequency analysis and obtained the phase velocity dispersion curve based on the mathematical relationship between the group and phase velocities. Moreover, we employed the phase velocity dispersion curve to complete the inversion of the above method and obtain the predicted coal seam thickness. By comparing the geological sketch of the coal mining face, we found that the predicted coal seam thickness is in good agreement with the actual thickness. Overall, adopting the channel wave inversion method that creatively uses the complete dispersion curve can obtain the shear wave velocities of the coal and its surrounding rocks, and analyzing the depth of the abruptly changed shear wave velocity can accurately obtain the thickness of the coal seam. Therefore, our study proved that this inversion method is feasible to be used in both simulation experiments and actual detection.     

LOVE-TYPE SEAM-WAVES IN WASHOUT MODELS OF COAL SEAMS*

The propagation of Love seam-waves across washouts of coal seams was studied by calculating synthetic seismograms with a finite-difference method. Seam interruption, seam end and seam thinning models were investigated. The horizontal offset, the dip of the discontinuities and the degree of erosion served as variable parameters. Maximum displacement amplitudes, relative spectral amplitudes and phase and group slowness curves were extracted from the synthetic seismograms. Both seam interruption and seam thinning reduce the maximum displacement amplitudes of the transmitted Love seam-waves. The degree of amplitude reduction depends on the horizontal offset and the degree of erosion. It is four times greater for a total seam interruption than for an equivalent seam thinning with a horizontal offset of four times the seam thickness. In a seam cut vertically, the impedance contrast between the coal and the washout filling determines the maximum displacement amplitudes of the reflected Love seam-waves. They diminish by a maximum factor of four in oblique interruption zone discontinuities with a dip of maximum 27°, and by a maximum factor of ten in a seam thinning with a degree of erosion of at least 22%. The analysis of the relative spectral amplitudes indicates a preferential transmission of those phases with frequencies below, and a preferential reflection of those phases with frequencies above the first mode Airy-phase. The relative spectral amplitudes of the reflected Love seam-waves show a distinct interference pattern of the waves reflected at both interruption zone discontinuities. The dispersion analysis reveals a flattening of the phase and group slowness curves with increasing frequencies, horizontal offset and degrees of erosion. These results imply that a detection of washouts in-mine will be possible in a frequency range including at least the first mode Airy-phase. An interference pattern and a flattening of the dispersion curve indicate a washout rather than other seam obstructions and leads to an estimate of the washout dimension.     

基于槽波的TVSP超前探测方法:可行性研究

针对隧道工程中TVSP（Tunnel Vertical Seismic Profiling）方法不适合于煤矿巷道超前探测的问题,介绍了煤矿巷道中的槽波TVSP超前探测方法原理和实现过程.通过理论和数值模拟证明：（1）当采用垂向集中力源（z方向）激发地震波时,地震波场z分量受到巷道影响较小,且主要为Rayleigh型槽波,较为适合槽波超前探测;（2）采用传播相对稳定且易于识别的槽波埃里相峰值时间代替传统TVSP方法中体波的初至走时时间,能够较准确地确定断层在掌子面前方位置和断层角度;（3）当断层与巷道掘进方向夹角较大时,断层交点位置（断层线与巷道掘进方向的交点）具有很好的定位精度,但角度误差则会较大;当实际断层交点位置距离迎头较远时,断层交点位置定位精度会下降,但角度误差则影响较小;（4）基于槽波的TVSP方法对观测系统要求不高,数据处理过程简单,计算量小.     

CALCULATION OF DISPERSION CURVES AND AMPLITUDE-DEPTH DISTRIBUTIONS OF LOVE CHANNEL WAVES IN HORIZONTALLY-LAYERED MEDIA*

We present dispersion curves, and amplitude-depth distributions of the fundamental and first higher mode of Love seam waves for two characteristic seam models. The first model consists of four layers, representing a coal seam underlain by a root clay of variable thickness. The second model consists of five layers, representing coal seams containing a dirt band with variable position and thickness. The simple three-layer model is used for reference. It is shown that at higher frequencies, depending on the thickness of the root clay and the dirt band, the coal layers alone act as a wave guide, whereas at low frequencies all layers act together as a channel. Depending on the thickness, and position of the dirt band and the root clay, in the dispersion curves of the group velocity, secondary minima grow in addition to the absolute minima. Furthermore, the dispersion curves of the group velocity of the two modes can overlap. In all these cases, wave groups in addition to the Airy phase of the fundamental mode (propagating with minimum group velocity) occur on the seismograms recorded in in-seam seismic surveys, thus impeding their interpretation. Hence, we suggest the estimation of the dispersion characteristics of Love seam waves in coal seams under investigation preceding actual field surveys. All numerical calculations were performed using a fast and stable phase recursion algorithm.     

煤巷小构造Rayleigh型槽波超前探测数值模拟

对煤巷小构造地震波场进行了数值模拟研究,分析了层状煤层中地震波的传播特征.研究表明:(1)在煤巷迎头前方煤层内以纵波震源激发的Rayleigh型槽波相对于体波能量较强,波列较长,波速较低.(2)沿煤层传播的Rayleigh型槽波在小构造面上产生Rayleigh型槽波反射波,反射Rayleigh型槽波垂直分量相对于水平分量能量较强.沿煤层反向传播的反射Rayleigh型槽波在煤巷迎头面上转换为沿煤巷底板传播的Rayleigh面波.沿煤巷底板可以接收到能量较强的反射Rayleigh型槽波产生的Rayleigh面波,其可以作为超前探测小构造面的特征波.在地震记录上反射Rayleigh型槽波产生的Rayleigh面波波至最迟,在时间域与其他波列时间间隔较大,其垂直分量能量相对于水平分量较强,在地震记录上容易识别.(3)在相同的地质条件下应用反射地震超前探测方法,标志煤巷迎头前方存在小构造面的反射地震波能量较弱,受煤巷顶、底板界面和采煤迎头面的强反射波干扰,在地震记录中难以识别.     

Oblique wave scattering by undulating porous bottom in a two-layer fluid: Fourier transform approach

The problem involving scattering of oblique waves by small undulation on the porous ocean bed in a two-layer fluid is investigated within the framework of linearised theory of water waves where the upper layer is free to the atmosphere. In such a two-layer fluid, there exist waves with two different wave numbers (modes): wave with lower wave number propagates along the free surface whilst that with higher wave number propagates along the interface. When an oblique incident wave of a particular mode encounters the undulating bottom, it gets reflected and transmitted into waves of both modes so that some of the wave energy transferred from one mode to another mode. Perturbation analysis in conjunction with Fourier transform technique is used to derive the first-order corrections of velocity potentials, reflection and transmission coefficients at both modes due to oblique incident waves of both modes. One special type of undulating bottom topography is considered as an example to evaluate the related coefficients in detail. These coefficients are shown in graphical forms to demonstrate the transformation of water wave energy between the two modes. Comparisons between the present results with those in the literature are made for particular cases and the agreements are found to be satisfactory. In addition, energy identity, an important relation in the study of water wave theory, is derived with the help of the Green’s integral theorem.     

SEISMIC MODELLING WITH CHANNEL WAVES IN SEAM STRUCTURES INFLUENCED BY MYLONITE ZONES1

Channel waves generated in coal-seams and their reflections from discontinuities are widely used to indicate the tectonic and stratigraphic features of coal deposits, resulting in greater efficiency and safety in coal-mining. In the mining area of Ibbenbüren (F.R.G.) seam structures sometimes contain so-called mylonite zones, which are crushed coal deposits capable of binding gas. If mining hits a mylonite zone this would probably not only reduce output of the mine, but could even cause gas explosions. To investigate the influence of a mylonite zone on the propagation of channel waves, Rayleigh channel wave measurements for 2D analogue models were performed and synthetic seismograms of Love channel waves were calculated. Analogue modelling of the mylonite zone using Rayleigh seam waves within the seam was carried out using a perforation technique. Calculations were made to obtain an estimate of velocity reduction due to perforation. The results agree well with velocity values measured up to a perforation of 25% in a 2D epoxy resin model. Reflected channel wave energy was found by applying dispersion analysis in the case where the impedance reduction between the mylonite seam structure and the undisturbed seam was approximately 5%. The horizontal width of the mylonite structure was detectable from the time lag between reflected channel wave signals from both in-seam borders of the mylonite zone. Resolution of two discrete borders was possible for a width of 1.5 λ's. The influence of a vertical fault, positioned within the mylonite zone, could only poorly be resolved. Numerical model investigations of Love seam waves were concerned mainly with the variation of the horizontal width of the mylonite zone. Mylonite zones with dimensions of the order of 0.4 λ's allow definite statements about their widths from dispersion and spectral analyses. For zones with smaller widths down to 0.2 λ's, it was found that reflectivity and transmissivity analyses give a qualitative criterion for distinguishing a mylonite structure surrounding a fault from a pure fault.     

槽波层析成像方法在煤田勘探中的应用——以河南义马矿区为例

在地下煤田的开发中,工作面内的小构造、异常体、煤层厚度变化等是需要解决的关键问题,而槽波探测则为这些问题的解决提供了重要的物探方法.本文对河南义马矿区11061工作面进行槽波透射法测量,在巷道显示的煤层厚度变化为1.5~8 m,从理论频散曲线分析速度与厚度关系,确定125 Hz频率槽波主要用于观测厚度约为2~5 m的煤层厚度变化;有效提取了684个频散曲线,并分别拾取了125 Hz时槽波群速度与走时,采用走时层析成像方法获得工作面内煤层速度、厚度以及高应力区分布特征,回采验证了结果的正确性.     

SHEAR WAVES BY AN EXPLOSIVE POINT-SOURCE: THE EARTH SURFACE AS A GENERATOR OF CONVERTED P-S WAVES*

The most common source of seismic energy is an explosion at some depth in a borehole. The radiated waves are reflected not only at the subsurface layers but also at the free surface. The earth's surface acts as a generator of both P- and S-waves. If the source depth is much less than the dominant wavelength the reflected waves resemble closely the waves generated by a single force. Theoretical seismograms were computed with different methods to look for the relevance of the surface-reflected waves. The numerical experiments show reflected shear waves even for small shotpoint—receiver distances. Due to their polarization these waves can be detected most easily on in-line horizontal geophones. The existence of these waves was examined during a conventional survey in Northern Germany. Conventional data analysis shows a large variability in the ν_p/ν_s ratio. The method used here produced a shear-wave section with a rather good signal-to-noise ratio down to 4 s S-wave reflection time.     

Numerical simulations of full-wave fields and analysis of channel wave characteristics in 3-D coal mine roadway models

Currently, numerical simulations of seismic channel waves for the advance detection of geological structures in coal mine roadways focus mainly on modeling twodimensional wave fields and therefore cannot accurately simulate three-dimensional (3-D) full-wave fields or seismic records in a full-space observation system. In this study, we use the first-order velocity–stress staggered-grid finite difference algorithm to simulate 3-D full-wave fields with P-wave sources in front of coal mine roadways. We determine the three components of velocity V_x, V_y, and V_z for the same node in 3-D staggered-grid finite difference models by calculating the average value of V_y, and V_z of the nodes around the same node. We ascertain the wave patterns and their propagation characteristics in both symmetrical and asymmetric coal mine roadway models. Our simulation results indicate that the Rayleigh channel wave is stronger than the Love channel wave in front of the roadway face. The reflected Rayleigh waves from the roadway face are concentrated in the coal seam, release less energy to the roof and floor, and propagate for a longer distance. There are surface waves and refraction head waves around the roadway. In the seismic records, the Rayleigh wave energy is stronger than that of the Love channel wave along coal walls of the roadway, and the interference of the head waves and surface waves with the Rayleigh channel wave is weaker than with the Love channel wave. It is thus difficult to identify the Love channel wave in the seismic records. Increasing the depth of the receivers in the coal walls can effectively weaken the interference of surface waves with the Rayleigh channel wave, but cannot weaken the interference of surface waves with the Love channel wave. Our research results also suggest that the Love channel wave, which is often used to detect geological structures in coal mine stopes, is not suitable for detecting geological structures in front of coal mine roadways. Instead, the Rayleigh channel wave can be used for the advance detection of geological structures in coal mine roadways.     

煤层中流体地震可探测性的模拟分析

作为一种典型的强阻抗差低阻抗薄层,煤层中孔隙含流体时是否会引起地震反射产生明显的异常是回答地震检测流体是否可行的根本.为此,本文针对强阻抗差薄层模型,基于Biot双相介质理论,通过弹性波有限差分法数值模拟,与各向同性单相介质假设的煤层反射对比,探讨了反射复合波受煤层孔隙度及流体性质变化的影响程度.模拟分析发现：由于薄层孔隙度和孔隙流体属性的变化在Biot理论中表现为纵波速度的变化,PP波反射AVO（Amplitude Versus Offset,振幅随偏移距变化）特征对薄层是否含流体相对敏感;综合使用PP与PS波对比有利于薄层中流体的预测;孔隙度一定时,PP波反射振幅随着含气饱和度的增加而增大;受薄层调谐作用的影响,孔隙和流体变化对煤层反射的频谱特征影响不大,近似于单相介质时的情况.     

SEISMIC MODELLING OF SEAM WAVES EXCITED BY ENERGY TRANSMISSION INTO A SEAM1

The traditional method of exciting channel waves in coal deposits underground consists of firing explosive sources in a mid-seam position generating seam waves of the Rayleigh and Love type. We investigate various source positions and excitation mechanisms within the bedrock structure surrounding the seam to evaluate the effects of source positions adjacent to the seam. The investigation is based on analogue and numerical modelling of half- and full-space cases, for which the excitation and the nature of Rayleigh channel waves are examined. In the analogue modelling, sources, located from mid-seam out into the bedrock, along the edge of a 2D plate model, excited channel waves through a conversion of the free surface Rayleigh wave at the edge of the plate. The excited channel wave belongs to the normal mode range. Frequency-wavenumber analysis shows that the symmetric 2nd mode of the channel wave is excited with frequencies comparable to the forcing frequency of the source signal. The polarization changes from retrograde to prograde, as the wave develops from the front to the rear of the seam, respectively. The amplitude-depth distribution resembles that of an ordinarily excited seam wave, for the symmetric component. However, the antisymmetric component does not show the characteristic change of sign in amplitudes across the mid-seam axis. Numerical modelling with sources located in the bedrock (full-space case) shows that relocating the source away from the seam lowers the frequency content of the excited channel wave. Based on these investigations, the influence of a lower-frequency source signal on the excitation of the channel wave is examined in an analogue experiment. Sources are sited in the bedrock adjacent to the seam at three locations. A lower-frequency wavelet is calculated for each source location from the results obtained in the numerical analysis. For comparison, a higher-frequency wavelet is also used which is known to be optimal for this model geometry when excited by a mid-seam source location. It is found that in two cases the use of the lower-frequency wavelet improves the channel wave excitation, while no amplification is achieved in one case.     

A CROSS-HOLE AND FACE-TO-BOREHOLE IN-SEAM SEISMIC EXPERIMENT AT INVINCIBLE COLLIERY,AUSTRALIA*

An experimental cross-hole and face-to-borehole in-seam seismic survey was carried out at Invincible Colliery in the Western Coalfield of New South Wales. Objectives were to determine propagation characteristics of the Lithgow seam and to establish the infrastructure for seismic mapping hole-to-hole in Australia. The seam supports leaky P-, S- and P-SV-modes. These modes propagate with group velocities (at 60 Hz) of 3.1, 1.5 and 1.2 km/s respectively. Particle motion polarization is well developed, as is dispersion of the SH-mode. Attenuation rates are high. The seam is lossy (Q of approximately 20). Two prominent structures were mapped by mode conversion. One is believed to be a fracture zone, the other a zone of intense roof thrusting. The old workings and a minor strike-slip fault, which intersected raypaths, were found to be relatively transparent to P- and S-waves at 60 Hz. Telemetry delay and shot-break timing errors of the exploder box are significant. The resulting traveltime scatter is reduced by means of a least-squares “statics” procedure. The group velocity estimation algorithm (based on Fourier transform) yields dispersion characteristics which can be matched with theoretical results for a simple model of a coal seam waveguide. The experiment demonstrates the capability to retrieve in-seam seismic data of diagnostic quality over an appreciable distance (2 km). The experience gained in both survey layout and data processing will be beneficial to future seam wave surveying of Australian coal mines.     

THE INFLUENCE OF AN ASYMMETRY IN THE SEQUENCE ROCK/COAL/ROCK ON THE PROPAGATION OF RAYLEIGH SEAM WAVES*

Small offsets in hard coal seams can be detected with the aid of seam (channel) waves. Transmission and reflection of seam waves depend, among other parameters, upon the symmetry properties of the sequence rock/coal/rock. Two typical unsymmetrical sequences are found in European coal deposits: (i) coal seams with roof and floor of differing acoustic impedance and (ii) coal seams interlayered with rock and soil. Two-dimensional analog models with appropriate impedance contrasts are used to study the effect of the unsymmetrical layers upon the propagation of Rayleigh seam waves. Data analysis is based upon amplitude measurements both parallel and perpendicular to the layers and dispersion curves. The effect of unsymmetrical roof (rock 1) and floor (rock 2) was studied with models containing homogeneous coal seams. Leaky mode wave groups with phase velocities (c_R) in the range between the SV-wave velocities (β_{r1, 2}) of the two rock materials, i.e. β_r1≥c_R > β_r2, form a characteristic part of the Rayleigh seam wave signal. Using Knott's energy coefficient calculations it is shown that in that range energy leakage into the surrounding rock by refracted SV-waves is restricted to only one of the two interfaces, namely coal/rock 2. At the other interface, coal/rock 1, all waves are totally reflected. Thus, the high amplitudes of these leaky mode wave groups are explained by “quasi-normal mode” features. The influence of a dirt bed on wave propagation was studied in models where the roof and the floor have the same elastic properties. The maximum thickness of the dirt bed did not exceed 20% of the total seam thickness. The effect of the bed's location within the seam was also investigated. For all recorded normal-mode wave groups either the total seam or the coal layers could be regarded as wave guides. This was shown by the fact that the phases could be associated with the phase velocity dispersion curves calculated for the symmetrical sequence rock/coal/rock. These curves are relevant under the condition that the thickness of the coal layer assumed under the calculation coincides with the thickness of the effective wave guide of the respective wave groups. Wave groups guided in the total seam are not influenced by either the thickness or the position of the dirt bed. On the other hand, for wave groups guided in the coal layers, the quotient of signal amplitudes in the coal layers is influenced by the position of the dirt bed.     

MODELLING CHANNEL WAVES WITH SYNTHETIC SEISMOGRAMS IN AN ANISOTROPIC IN-SEAM SEISMIC SURVEY1

In-seam seismic surveys with channel waves have been widely used in the United Kingdom and elsewhere to map coal-seams and to detect anomalous features such as dirt bands, seam thinning and thickening, and particularly in-seam faulting. Although the presence of cleat-induced anisotropy has been recognized in the past, almost all previous analyses have assumed homogeneous isotropic or transversely isotropic coal-seams. Channel waves, however, exhibit properties which cannot be fully explained without introducing anisotropy into the coal-seam. In particular, Love-type channel waves are observed for recording geometries where, in a homogeneous isotropic or transversely isotropic structure, the source would not be expected to excite transverse motion. Similarly, modes of channel-wave propagation display the coupled three-component motion of generalized modes in anisotropic substrates, which would not be expected for Rayleigh and Love wave motion in isotropy or in transversely isotropic media with azimuthal isotropy. We model the observed in-seam seismic channel waves with synthetic seismograms to gain an understanding of the effects of cleat-induced anisotropy on the behaviour of channel waves. The results show a reasonable good match with the observations in traveltime, relative amplitudes, dispersion characteristics and particle motions. We demonstrate that anisotropy in the surrounding country rocks contributes significantly to the coupling of channel wave particle motion, although its effect is not as strong as the anisotropy in the coal-seam. We conclude that the effects of cleat- and stress-induced anisotropy are observed and can be modelled with synthetic seismograms, and that anisotropy must be taken into account for the detailed interpretation of channel waves.     

Analytical expressions for Fresnel volumes and interface fresnel zones of seismic body waves. Part 2: Transmitted and converted waves. Head waves

Summary Fresnel volumes and interface Fresnel zones of transmitted and head waves are studied. The relation derived for transmitted waves may also be used for converted reflected waves. Considerable attention is devoted to the penetration of Fresnel volumes across structural interfaces, particularly for head waves.     

Flexural waves in drill‐string tubulars with variable loads

We revisit the equations governing the bending motions in thin rods and analyse the filtration of flexural waves in vertical drill strings pre‐stressed by gravity. The aim is to study transverse drill‐string vibrations at seismic frequencies for acoustic communication purposes and provide an algorithm for processing reflected and transmitted bending motions generated by downhole lateral vibrations. We obtain the dispersion equation, including attenuation due to a gravity pre‐stress gradient and frequency‐dependent reflection and transmission coefficients at the interface between subsequent tube intervals. We then develop a propagation‐matrix algorithm to simulate flexural waves in a drill string consisting in an assembly of multiple tube sections of different dimensions. The deflection vibrations are obtained at any arbitrary recording point in the drill string. The modelling is cross‐checked with a full‐wave grid algorithm. The analysis shows that the waves produced by a concentrated force are partitioned in standing and propagating modes, which are calculated by using the flexural impedance of the drill string. Moreover, the reflection coefficients weakly depend on the pre‐stress conditions and pre‐stress has important effects for far‐field signal transmission with variable weight on bit (WOB). We discuss the approximations and limits of the method with respect to realistic drilling conditions.