Compressional Head Wave Research Articles

Rock heterogeneity at the centimeter scale, proxies for interfacial weakness, and rock strength-stress interplay from downhole ultrasonic measurements

Analysis of data acquired with an ultrasonic rotating device lowered inside a vertical well of a highly laminated, kerogen-rich, carbonate source formation reveals centimeter-scale rock heterogeneities. The high-frequency (50–600 kHz) measurement used consists of a pulse-echo modality that yields a high-resolution (millimeter-scale) borehole shape and a pitch-catch modality with one transmitting and two receiving transducers that provide compressional (P) and shear (S) slownesses (depth-versus-azimuth) images estimated from signals propagating in the near-wellbore region as compressional head waves and pseudo-Rayleigh surface waves. The slownesses are compared with their counterparts estimated from a lower-frequency (1–15 kHz) sonic measurement logged in the same well interval. The sonic P and S logs are seen to average the centimeter-scale slowness spatial variation between compliant and stiff laminates at a scale larger than 30 cm (1 ft). This is also accompanied by a reduction in resolution of the slowness contrast, which markedly reduces the spectrum of rock-mechanical property variations that are estimated from the ultrasonic data. Further, ultrasonic images of the P and S slownesses and borehole acoustic caliper reveal a host of features associated with rock geomechanics in the near-wellbore and that inform on a first-order basis the interplay between rock strength and local stress regime, as affected by the lamination. The features include breakouts present in the stiffer limestone layers and their arrest at the intersection with the compliant siltstone layers, as well as azimuthal P and S slowness variations indicative of azimuthal stress concentrations, but without the appearance of breakouts. The ultrasonic borehole shape data also identify compliant thin layers that retract into the formation by amounts that are commensurate with their Young’s modulus, suggesting a proxy for detecting and characterizing thin-layer weakness in situ.

Geometrical spreading factor of head waves in cased and open boreholes

Factors (coefficients) of geometrical spreading of compressional and shear head waves are calculated for an impulse multipole source of elastic oscillations in boreholes. It is shown that the length of the logging tool (i.e., the distance between the source and the nearest receiver) used for sonic measurements and the velocities of elastic waves in the medium both contribute to the factor of geometrical spreading. For a high-velocity formation (the shear wave velocity in the rock is higher than the compressional wave velocity in the fluid that fills the borehole) and a sufficiently long sonic tool with a monopole source, the coefficient of geometrical spreading is approximated by asymptotic formula 1/Z [Roever et al., 1974; Krauklis and Krauklis, 1976], where Z is the length of the tool; i.e., the amplitude of the compressional head wave decreases proportionally to the distance between the source and the receiver. In acoustically soft formations, this approximation is inapplicable even for long tools with length Z > 4 m. Waveforms in cased boreholes have a significant frequency dispersion even in case of good-quality cementing, and the factor of geometrical spreading there depends considerably on the length of the tool and the elastic properties of the rocks.

Leaky modes and their contributions to the compressional head wave in a borehole excited by a dipole source

A method of studying the contributions of leaky modes to the wave field is presented based on the analysis of the Riemann surface structure of the characteristic function, and the sensitivities of contributions to various factors of interest are examimed. Numerical results show that their contributions to the compressional head wave are related to the distributions of complex poles on (−1, −1) and (0, −1) Riemann sheets on the frequency-wavenumber (ω-k) plane. For fast formations, their contributions are small, while for slow formations with large Poisson’s ratio, their contributions are large because of those complex poles with small imaginary parts near the compressional vertical branch cut. The decaying factor of the contributions of leaky modes is approximately proportional to 1/distance2.

Electric arc source for high-frequency seismic measurement

An electric arc source was designed to study high-frequency seismic surface wave propagation. The noncontact nature of this source made it feasible for use in synthetic aperture transmit arrays. The transmit signal, which was not linearly controllable, was found to be predominantly in the 1- to 4-kHz band with a Gaussian spectrum. This is an octave below the simultaneously generated air acoustic pulse. The source was used to create a synthetic line array in conjunction with a stationary receiver. The experiment was conducted in a sand-filled tank. Surface wave speeds in the range of 80 m/s were measured on the resulting seismograms. Significant dispersion occurred in the propagating waveform at distances both near and far from the source. Surface wave arrivals were discernable over 1 m from the source and compressional head waves could be observed within 30 cm. The data were in good agreement with lower frequency measurements made by other techniques. The surface wave generation was studied and found to be a combination of the surface interaction of the arc and the air acoustic interaction. Strong hysteresis was observed in the first arcing event. Later, the signal was smaller but sufficiently stable for averaging. [Work supported by ARO.]

Acoustic waves in pressurized boreholes: A finite difference formulation

A velocity‐stress, finite difference formulation of acoustic waves in a fluid‐filled, pressurized borehole yields synthetic waveforms for monopole or dipole sources before and after borehole pressurization. Processing of these waveforms using a variation of Prony's procedure isolates dominant arrivals in the full wave field. Differences between the slownesses of individual arrivals before and after pressurization provide stress‐induced changes in propagation characteristics that are of importance in estimating mechanical properties of the formation. The borehole pressurization in an isotropic formation produces insignificant changes in the compressional head wave slownesses; and small changes in the shear head wave slownesses. The most significant changes occur in the Stoneley and flexural slownesses at relatively higher frequencies in the range of 5–10 kHz for a borehole of diameter 20.32 cm (8 inches). These differences in the Stoneley and flexural slowness dispersions for a known increase in the borehole pressure can be used to calculate the acoustoelastic coefficients of the formation. These coefficients are measures of nonlinear elastic parameters of the formation that are generally larger for poorly consolidated slow formations than those of tightly consolidated fast formations.

Compressional head waves in attenuative formations: Forward modeling and inversion

We develop a method of forward modeling and inverting formation attenuation data from sonic compressional head waves in a fluid‐filled borehole using a branch‐cut integration (BCI) technique to calculate individual acoustic arrivals. We validate this approach with a real‐ axis integration (RAI) method that does not separate the individual arrivals. We show that the straightforward application of the original BCI method for lossless media gives erroneous results for attenuative formations. With a choice of the Riemann sheets satisfying the radiation condition, the new BCI method gives correct results for most lossy and lossless formations. However, modeling very slow formations needs to include the contribution of a leaky pole near the vertical branch cut. With a constant‐Q assumption, we develop a simple processing scheme to extract the formation compressional Q factor from the P head‐wave arrivals. We used experimental data from laboratory‐scale borehole measurements to invert for the compressional Q value of a Lucite block. The inverted results agree within 4.5% of an independent ultrasonic transmission measurement of Q.

Spectral ratios of acoustic log in marine sediments

Diagnostic patterns have been noticed in spectral ratios of full waveform acoustic log data taken in slow marine sediments. The spectral ratio is computed with the full 2.56-ms time series after taking the Fourier transform and dividing the 3.05-m (10-ft) source-receiver offset transform by the 2.44-m (8 ft) one. The patterns show less attenuation at certain discrete frequencies in the spectra. The strongest observed peak in the spectral ratio was at slightly less than 7 kHz. Many of these patterns were duplicated by forward modeling consisting of calculating the branch cut integral of the compressional head wave at different source-receiver offsets and computing the spectral ratio. Preliminary analyses show that the presence and location of the peaks in the spectral ratio are controlled by the compressional wave velocity and the Poisson's ratio of the sediments, but the magnitudes of the peaks are controlled by their attenuation. The results were used to invert the full waveform log for shear wave velocity and compressional wave attenuation.

A New Method for Fracture Identification Using Array Sonic Tools

Summary. The development of array sonic tools has enabled the recording of an array of sonic waveforms at each depth location with several receivers. In this paper, we develop a new method to study the sonic amplitude/energy by taking advantage of the redundant information in these waveforms. First, we identify the wave components of interest. Second, we apply windowing procedures to extract the desired wave components. Third, we calculate the energy of the wave component at each receiver depth location. Finally, the calculated energies are displayed as an energy log or a variable-density image as a function of depth. Open fractures are detected when the method is applied to the low-frequency Stoneley wave. Reductions of the Stoneley wave energy caused by open fractures appear as a characteristic pattern on the output. The vertical extent of the pattern provides an estimate of the fracture dip angle, and the magnitude of the energy reduction is related to the hydraulic conductivity of the fractures. We show examples of fracture identification using data from the Conoco Test Well 33–1 located in Oklahoma. Complementary information regarding the location and nature of the fractures is available from full-core and borehole televiewer (BHTV) logs. The Stoneley wave response to different fractures is discussed. Combined with borehole images, this method provides a valuable way for distinguishing between open and closed provides a valuable way for distinguishing between open and closed fractures. Introduction Modern digital sonic tools are capable of recording down-hole sonic waveforms with an array of receivers. The recorded waveforms contain information about the in-situ rock properties, such as compressional and shear slownesses, and attenuation. In this paper, we propose a new method to estimate wave amplitude/energy from recorded array sonic waveforms and to display the information as a well log. Three important contributions are emphasized. First, we discuss ways to extract the sonic amplitude/energy information from array sonic waveforms. Second, we display the information in a meaningful way, taking full advantage of the data redundancy in array sonic waveforms. Third, and most important, we apply the method to formation evaluation, where we have found that the method has great potential for describing the heterogeneous properties of fractured rocks. This is important because the location and evaluation of fractures is critical for oil and gas production in low-permeability formations. Method The acoustic wavefield recorded in the borehole is extremely complex because of the presence of the overlapping wave components: compressional and shear headwaves, trapped-fluid modes, and surface waves. One way to separate the components in the recorded waveforms is to use a sonic tool with a long transmitter-to-receiver (TR) spacing. In this way, wave components that propagate with different velocities can be isolated with a time window. Besides exhibiting a time separation, the wave components may also have harmonic separation in frequency. The frequency content of compressional and shear waves, for example, is generally higher than that of the Stoneley wave. This allows us to apply low-pass filtering to extract the Stoneley wave from the waveforms. Fig. 1 shows an array of sonic waveforms recorded in hard rock with a digital array sonic tool and shows that the different arrivals can be isolated from one another by application of a time-windowing technique. Fig. 2 shows the amplitude spectra of the time-windowed compressional, shear, and Stoneley waves. It is clear that the low-frequency Stoneley wave (2,000 to 5,000 cycles/sec [2 to 5 kHz]) is the strongest event and is separated from the headwaves (approximately 6,000 to 15,000 cycles/sec [6 to 15 kHz]). The filtered output (2,000 to 4,000 cycles/sec [2 to 4 kHz]) of the waveforms is shown in Fig. 3. Note that the filtered waveforms contain only the low-frequency Stoneley wave and not the headwaves. Wave components of interest may be extracted with time windowing or frequency windowing, or with a combination of the procedures. The choice of the windowing procedure and its implementation will depend on the procedure and its implementation will depend on the application. For example, one might be interested in the compressional headwave in the 10,000- to 15.000- cycle/sec [10- to 15-kHz] band. JPT P. 677

The analysis and interpretation of full waveform sonic data. Part II. multiples, mode conversions and reflections

In our previous paper on the analysis of full waveform sonic data (Astbury & Worthington 1986), hereinafter referred to as Part I, discussion was restricted to the dominant phases of a sonic waveform: the compressional head wave, the shear/pseudo-Rayleigh wave, the Stoneley wave and the fluid wave. This paper is devoted to a consideration of other phases that can be identified as multiples, mode conversions and reflections. The utility of full waveform sonic data is greatly increased if such events can be correctly analysed and interpreted.

A numerical investigation of head waves and leaky modes in fluid‐filled boreholes

Although synthetic borehole seismograms can be computed for a wide range of borehole conditions, the physical nature of shear and compressional head waves in fluid‐filled boreholes is poorly understood. This paper presents a series of numerical experiments designed to explain the physical mechanisms controlling head‐wave propagation in boreholes. These calculations demonstrate the existence of compressional normal modes equivalent to shear normal modes, or pseudo‐Rayleigh waves, with sequential cutoff frequencies spaced between the cutoff frequencies for the shear normal modes. Major contributions to head‐wave spectra occur in discrete peaks at frequencies just below mode cutoff for both compressional and shear modes. This result is confirmed by calculations with synthetic waveforms at frequencies corresponding to mode cutoff, and by branch‐cut integrals designed to yield independent spectra for the compressional mode. For soft formations where shear velocity falls below acoustic velocity in the borehole fluid, leaky compressional normal modes attain properties similar to those of shear normal modes in the case of hard rock. In the limit of vanishing S-wave velocity, this result is formally related to a fluid‐fluid waveguide with undamped compressional normal modes. Synthetic waveforms demonstrate that high‐ amplitude arrivals, traveling at velocities less than the acoustic velocity of the borehole fluid and at frequencies above a few kilohertz, represent the Airy phase of the compressional mode and are not a tube wave. Comparison of synthetic waveforms with waveforms obtained in soft sea sediments indicates that the predicted Airy phase arrivals are present in the experimental data.

Polarization method for the determination of Poisson’s ratio in boreholes

Conventional methods for determining Poisson’s ratio from full‐waveform sonic logs rely on measurement of the shear velocity. This measurement is subject to significant error from contamination by events which do not travel at the shear velocity; it fails in soft formations where the shear head wave and pseudo‐Rayleigh waves are not excited. A new method is presented for determining Poisson’s ratio and shear velocity. This method can be implemented by recording the components of particle displacement on the borehole wall. The particle trajectory in the compressional head wave is rectilinear and is deflected with respect to the direction of propagation. The deflection is a sensitive function of Poisson’s ratio, varying by more than 30 degrees over the range of Poisson’s ratios encountered in rocks. In the polarization method all the measurements are made on the compressional head wave, the spatial resolution is greater than for velocity measurements, and the deflection is greatest in soft formations.

Modeling shear and compressional head waves at velocity steps and inversions

Accurate modeling of head waves (lateral waves or refractions) enables improved estimation of sound velocities at and below the ocean bottom, [A.D. McAulay, in Int. Conf. Acoustics Speech and Sig. Proc. 1980, 1981]. Head waves arise when a homogeneous wave strikes an interface with a medium of greater velocity (step) at greater than the critical angle or, less well known, when an inhomogeneous wave strikes an interface with a medium of lower velocity (inversion). In practice head waves are caused by an inhomogeneous wave either when a point source is located near a velocity inversion, or at the exit of a tunnel through a high velocity region. A computer program was developed using the Thompson-Haskel formulation and the Delta matrix method to obtain fast accurate modeling of shear and compressional waves for a plane layered model and a point source. Various velocity models were used to illustrate the accurate modeling of shear and compressional head waves. In particular the accurate modeling of head waves at a velocity inversion is illustrated for the first time.

Numerical evaluation of the transient acoustic waveform due to a point source in a fluid‐filled borehole

A key measurement employed in oil well wireline logging is the acoustic wave traveltime over a specified formation interval, typically 1 ft. In the traditional measurement, only the compressional head wave is monitored, but for some time it has been obvious that there is significant additional information, such as the shear head wave arrival, in the received waveform. We describe two numerical methods for computing the profile and parameter dependence of the transient waveform based on a model of the acoustic logging problem consisting of a point source on the axis of a fluid‐filled cylindrical borehole. The response to this excitation is determined at a distance from the source, generally on the borehole axis. In the first of the two numerical methods, called “real axis integration”, the complete acoustic waveform is obtained. The second method, called “branch‐cut integration”, evaluates the first compressional and shear‐pseudo‐Rayleigh arrivals individually with much less computation time than the first method. The validity and accuracy of the two methods are demonstrated by their close agreement within appropriate time windows. It is also shown that the results from the ordinary asymptotic method that exist in the literature predict different behavior. The dependence of the transient arrivals on formation parameters is illustrated by various numerical results in both time and frequency domains.

Asymptotic methods for the first compressional head wave arrival in a fluid-filled borehole

Asymptotic results are obtained for the first compressional head wave arrival in a fluid-filled borehole with a modified asymptotic technique. The results are verified with direct numerical integrations and shown to be quite different from those obtained with the ordinary asymptotic techniques. The modified asymptotic technique accounts for the effect of an extraneous pole, which is neglected in the ordinary asymptotic method. The basic assumption in the modified asymptotic method is that the observation point must be far away from the source point, whereas in the ordinary asymptotic technique, the frequency is assumed to be large. The pressure responses of the first compressional arrival are illustrated and compared. It is found that the amplitudes of the compressional arrivals are strongly dependent on the Poisson’s ratio and the compressional velocity of the formation.

THE CONVERTED-WAVE METHOD OF SEISMIC REFRACTION INTERPRETATION FOR INTERFACES WITH SEVERE DIP AND CURVATURE

In the converted-wave method of seismic refraction interpretation, travel times for compressional, converted, and transverse head waves are used in combination to obtain velocities and layer thicknesses. The applicability of the method over interfaces with high dip and curvature is investigated, treating the two- and three-layer cases for layers of uniform velocity, with upper-layer velocity known, where the problem is to calculate velocities at depth and delineate the interfaces. It is shown that although dip and curvature affect the calculation, procedures can be devised to obtain solutions in the presence of these factors. Examples of application of the method in the two-layer case to a plane interface dipping at 30°, and to a curved interface with dip changing linearly from 0° to 27° are given.Depth calculation using intervals between the travel times of converted waves are shown to be independent of dip for a plane interface, but sensitive to curvature for a curved interface. It is shown that in these cases, a satisfactory depth calculation is possible.Ratios of these time intervals are shown to produce depth ratios that are either independent of velocity or with low dependence upon this parameter.A number of capabilities peculiar to the converted-wave method are noted.