The development of acoustic source technology has been an important task for acoustic logging while drilling (LWD) and various source designs have been implemented. Using a multipole wave expansion theory, this study demonstrates that a LWD acoustic source can be represented as a combination of monopole, dipole, and quadrupole constituents and characterized by the contribution of each constituent. The theoretical analysis is experimentally demonstrated with a cylindrical pipe simulating the LWD collar. The result of this study can be used to provide a method for evaluating the performance of a LWD acoustic source.

## 1. Introduction

With the development of the logging while drilling (LWD) acoustic technology in past decades, the study of acoustic wave propagation along cylindrical pipes has received significant interest (Plona *et al.*, 1992; Drumheller and Knudsen, 1995; Rama Rao and Vandiver, 1999; Pan *et al.*, 2003; Sinha *et al.*, 2009). Following the convention of wireline acoustic logging (White, 1967; Zemanek *et al.*, 1984), LWD monopole, dipole, and quadrupole acoustic sources have been designed and implemented (Varsamis *et al.*, 1999; Leggett *et al.*, 2001; Tang and Cheng, 2004). Worthwhile to mention is that in LWD, the quadrupole, instead of dipole, is a better technology for measuring formation shear velocity (Tang *et al.*, 2003). In addition, other types of acoustic sources, known as the unipole (Minear *et al.*, 1995; Wang *et al.*, 2011; Mickael *et al.*, 2012) for anisotropy measurement (Xu *et al.*, 2018) and bipole, have also been used. The latter is also called the bimodal source in the LWD service sector (Market and Bilby, 2012; Wang, 2014). Through numerical modeling, Wang *et al.* (2011) found that the acoustic response of the unipole LWD tool consists of monopole, dipole, and quadrupole responses. The main objective of this study is to provide a unified approach for the LWD acoustic source characterization using the multipole expansion theory for a general acoustic source. Moreover, the theoretical results will be demonstrated using a laboratory experiment on a physical model. The analysis and procedure can also be used to provide a method for characterizing an actual LWD acoustic source.

## 2. Theory

A LWD acoustic transducer is typically an arcuate-shaped shell mounted on a portion or entire exterior circumference of the steel pipe (see Fig. 1). Using the cylindrical coordinates (*θ*, *r*, *z*) to express the radiation of a point source (Tang and Cheng, 2004), the acoustic wave pressure from the source radiation is given by the following surface integral:

where *ω* is angular frequency; *k* is axial wave number, $kv=[k2\u2212(\omega /v)2]1/2$ is radial wavenumber, with *v* being the acoustic velocity of the surrounding medium; *I _{n}* and

*K*are the

_{n}*n*th order modified Bessel functions of the first and second kind, respectively,

*ε*is 1 for

_{n}*n*= 0, and 2 for

*n*> 0. The source is a cylindrical shell strip of radius

*r*′ localized around

*z*′, vibrating with the particle velocity

*V*on the source surface area

*S*. The source's

*z*-dimension

*h*is small compared to wavelength. The azimuthal variation of

*V*with

*θ′*is the basis for generating multipole acoustic waves in the LWD system. Characterizing the azimuthal variation and determining the resulting multipole components are a major task in the LWD acoustic source design.

A LWD source usually has an azimuthal periodicity. Let the source be symmetric with respect to a designated azimuth *θ*. Equation (1) then becomes an azimuthal cosine Fourier series expansion,

where *n* is the azimuthal order number, with $n=0,\u20091,\u20092,\u2009...$, representing monopole, dipole, quadrupole, and higher order multipoles. The Fourier coefficient of the expansion is given by

Equation (3) denotes the contribution of a particular component (i.e., monopole, dipole, and quadrupole, etc.) to the source. Assigning the multipole source to the rim of the drill pipe, various wave modes can be excited in the LWD system (Tang and Cheng, 2004). For example, the *A*_{0}, *A*_{1}, and *A*_{2} terms of Eq. (2) will excite, respectively, the extensional, flexural, and screw wave modes in the cylindrical pipe. Therefore, by measuring the propagation of these wave modes along the pipe and determining their contribution, the characteristic of a LWD acoustic source can be evaluated.

The full-circle integration of *V* over $\theta \u2032$ in Eq. (3) determines the relative amplitude of each term of Eq. (2). In practice, however, instead of measuring $V(\theta \u2032)$ over the source surface, the far-field source radiation directivity $\u211c(\theta )$ on a circle around the source is often measured (in a water tank or pool). On the circle with a large radial distance *r*, the radiated wave pressure is

On the other hand, by using the steepest descend method (Aki and Richards, 1980), the far-field solution of Eq. (1) in the radial plane containing the source (*z* = *z′*) is expressed as

where the coefficient *C* is unrelated to *θ*. Multiplying Eqs. (4) and (5) with $cos\u2009m\u2009\theta \u2009\u2009(m=0,\u20091,\u20092,...)$, integrating over a full circle, and equating the results, one gets

The above equivalence relation implies that one can use the far-field radiation directivity $\u211c(\theta )$ in Eq. (3) to replace *V* for the full circle integration.

## 3. Modeling and experimental results

We first demonstrate the above multipole source expansion principle using a numerical modeling example. Figure 1(a) shows the geometry of a cylindrical pipe mounted with an arcuate-shaped piezoelectric ceramic that occupies a quarter of the exterior circumference of the steel pipe. (In LWD acoustics, this is called the unipole transducer design, Mickael *et al.*, 2012; Wang *et al.*, 2011; El Wazeer *et al.*, 2016.) For the piezoelectric and geometric parameters of the design in Table 1, the radiation of the transducer at 13 kHz is numerically simulated using a finite-element technique (Qiao *et al.*, 2008; Fu *et al.*, 2014; Chen *et al.*, 2017). The radiation directivity pattern of the simulated acoustic wavefield in the surrounding fluid is shown in Fig. 1(b) (black circle). The directivity pattern is then used in the Fourier analysis of Eq. (3) to determine the amplitude of each wave mode in Eq. (2). If the maximum amplitude of the directivity pattern is 100%, then the resulting amplitudes for monopole, dipole, and quadrupole are 27.48%, 31.65%, and 40.13%, respectively. The amplitudes of the higher order (*n* > 2) terms are much smaller and are thus omitted. Using the determined amplitudes and summing the first three terms in Eq. (2), the resulting directivity pattern (solid curve) compares very well with the original pattern [see Fig. 1(b)]. This shows that the unipole transducer source consists primarily of the monopole, dipole, and quadrupole constituents, in agreement with the analysis result of Wang *et al.* (2011).

. | Material . | Young's modulus (N/m^{3})
. | Poisson ratio . | Density (Kg m^{−3})
. | Size (mm) . |
---|---|---|---|---|---|

Piezoelectric ceramics | PZT-5A^{a} | 5.6 × 10^{10} | 0.36 | 7750 | Φ204 × Φ192 × 88 |

Isolate potting | Epoxy resin | 1.0 × 10^{10} | 0.32 | 1500 | Φ208 × Φ188 × 120 |

Drill collar | Steel | 21.6 × 10^{10} | 0.28 | 7840 | Φ215 × Φ145.2 × 300 |

Fluid | Water | 2.25 × 10^{9} | 0.50 | 1000 | 500 |

. | Material . | Young's modulus (N/m^{3})
. | Poisson ratio . | Density (Kg m^{−3})
. | Size (mm) . |
---|---|---|---|---|---|

Piezoelectric ceramics | PZT-5A^{a} | 5.6 × 10^{10} | 0.36 | 7750 | Φ204 × Φ192 × 88 |

Isolate potting | Epoxy resin | 1.0 × 10^{10} | 0.32 | 1500 | Φ208 × Φ188 × 120 |

Drill collar | Steel | 21.6 × 10^{10} | 0.28 | 7840 | Φ215 × Φ145.2 × 300 |

Fluid | Water | 2.25 × 10^{9} | 0.50 | 1000 | 500 |

^{a}

The parameters of the piezoelectric ceramics, including stiffness matrix, piezoelectric matrix and dielectric matrix can be found in Chen *et al.* (2017).

To experimentally verify the theoretical results, an actual unipole transducer was built using the parameters in Table 1. To check the conformance of the transducer to the design, the electrical conductance of the transducer was measured by a precision impedance analyzer (Agilent E4990A, Hewlett-Packard, Palo Alto, CA) in the LWD acoustic frequency range around 10 kHz. Figure 1(c) compares the experimental conductance curve (dashed curve) with its numerically calculated counterpart (solid curve). The theoretical and experimental conductance curves agree well, both showing that the design has a resonant frequency around 13 kHz. The transducer and its combination will then be used in the following experiment.

A physical model was built to demonstrate the above source expansion principle and method. The measurement system in conjunction with the physical model (made of a steel pipe) is shown in Fig. 2(a). The cylindrical source transmitter of the system consists of four sectors, designated as S_{A}, S_{B}, S_{C}, and S_{D}, each being a unipole design described in Fig. 1. The transmitter in connection with a signal generator is placed at the rim of the model to generate wave signals in the model. Using only one sector, the transmitter is a unipole source. Using different combinations of the sectors and actuation schemes (Tang *et al.*, 2002), the transmitter can operate as a monopole, or a dipole, or a quadrupole source. A 10 kHz burst signal is used to actuate the transmitter. Four receivers, placed 2.7 m away from the source, are evenly mounted on the circumference of the model (designated as R_{A}, R_{B}, R_{C}, and R_{D} in the model) to record the source-generated wave signal. The signal is sent to a signal recorder, where the wave data are acquired for further analysis. The measurement with the system, which is taken in the air, is controlled by a computer.

As demonstrated in the previous theoretical analysis, the unipole source of Fig. 1(a) is equivalent to a combination of monopole, dipole, and quadrupole sources. This equivalence is schematically illustrated in Fig. 2(b). Multiplying the signal amplitude from the signal generator by a factor of 0.275, 0.315, and 0.401, respectively, as determined from the previous analysis, gives the multipole source excitation signals shown in the upper portion of Fig. 2(c). The monopole, dipole, and quadrupole signals can be generated respectively using the signal generator in the system. The measured signals, together with their respective source signal, are displayed in Fig. 2(c) for the four circumferential receivers. The monopole waveforms in the second panel are similar for all four receivers, apart from minor differences due to measurement noise and errors. In the next panel, the dipole waveforms on receivers R_{B} and R_{D}, which are excited by sectors S_{B} and S_{D}, have about the same amplitude but an opposite polarity; the small waveforms on receivers R_{A} and R_{C}, which should be zero, are caused by the measurement noise and errors. In the last panel, the quadrupole waveforms for the four receivers have similar amplitudes, with the signal polarity alternating between adjacent receivers.

The multipole waveforms from panels 1, 2, and 3 are summed for each receiver. The resulting waveforms (dashed line) are overlain with their respective counterpart (solid line) measured for the unipole source [see panel 1 of Fig. 2(c)]. For all four receivers, the sum of the multipole waves and the unipolar waveform are almost identical to each other, giving an excellent experimental verification of the multipole source expansion principle.

As an application example, we use the above analysis and experiment to evaluate another LWD acoustic source, called the bimodal, or bipole source (Market and Bilby, 2012). In this design, two diametrically-opposed unipole sectors are mounted on the steel pipe, as is schematically illustrated in Fig. 3(a). The radiation directivity pattern of the simulated acoustic wavefield in the surrounding fluid is shown in Fig. 3(b) (black circle). Using this directivity pattern (with normalized amplitude of 100%) in Eq. (3), one finds that the amplitude coefficient for the monopole and quadrupole component is 41.25% and 56.78%, respectively. The dipole coefficient vanishes due to the (left–right) symmetry of the directivity pattern. Similar to the previous example, the amplitudes of the higher order (*n* > 2) terms in Eq. (2) are much smaller and are thus omitted. Figure 3(b) shows the theoretical directivity pattern [red curve, calculated as the sum of monopole and quadrupole components, as in Fig. 3(c)], which agrees quite well with the numerically calculated directivity (markers). For the experimental verification using the physical model in Fig. 2(a), we measure the waveforms of the bipole, monopole, and quadrupole sources by modulating the source signal amplitude by a factor of 1, 0.413, and 0.567, respectively. The measured signals, together with their respective source signal, are displayed in Fig. 3(d) for the four circumferential receivers. The sum (dashed curve of panel 1) of the monopole (panel 2) and quadrupole (panel 3) waves compares very well with the measured bipole waveform (solid curve in panel 1). This again verifies the multipole source expansion principle.

The bipole characteristics can be further demonstrated using the waveforms received by R_{A} and R_{B} [see physical model of Fig. 2(a)], as are displayed by the solid and dashed curves, respectively, in Fig. 4(a). The first arrival portion of the waves is the fast-traveling pipe extensional wave associated with the monopole component of the source. Because the monopole has no azimuthal variation, the extensional waves from the two orthogonal azimuths are almost identical. However, the later portion of the waves shows the large-amplitude quadrupole components, for the polarity of the two waves is opposite each other, exhibiting the typical cos 2*θ* variation of a quadrupole [see Eq. (2)]. To prove this, an additional measurement is made at the receiver position R_{E}, which is along the 45° azimuthal angle from the receiver R_{B}. Because cos 2*θ* = 0 for *θ* = 45°, the quadrupole component from the bipole source vanishes and one should get a pure monopole signal at R_{E}. Indeed, the bipole waveform measured at R_{E} is almost identical to the monopole signal measured at R_{B}, which consists of the fundamental and higher-order pipe extensional waves. (Note both measurements used the source signal of equal amplitude.) This experiment again validates the multipole source expansion theory of Eqs. (1)–(3).

## 4. Conclusion

This study has demonstrated and validated the multipole expansion theory for a LWD acoustic source using theoretical and experimental modeling examples. That is, a LWD acoustic source can be represented as a combination of multipole constituents and characterized by the contribution of each constituent. Through the azimuthal Fourier analysis, the contribution of a constituent can be determined. The analysis procedure also provides an effective method for the source characterization by determining or measuring its radiation directivity. The analysis results for the existing LWD unipole and bipole designs show that the contributions from the monopole, dipole (for unipole), and quadrupole components of the expansion series are sufficient to characterize these sources.

## Acknowledgments

This work is supported by the China Natural Science Foundation (Grant Nos. 41604109, 41774138, 41474092, and 41474101), Shandong Provincial Natural Science Foundation, China (Grant No. ZR2014DL009), the Fundamental Research and Development Program (Grant No. 2014CB239006), and the Fundamental Research Funds for the Central Universities (Grant No. 16CX06040A). The authors thank Dr. Gabriel Gallardo Giozza and Dr. Tianqi Deng for valuable discussions, and also thank COSL (China Oilfield Services Limited) for experimental support.