Factors Influencing Radiation Sound Fields in Logging While Drilling Using an Acoustic Dipole Source

Abstract

With the increasing number of complex well types in the development stage of oil and gas fields, it is becoming increasingly urgent to use remote detection logging while drilling (LWD) to explore the geological structures in a formation. In this paper, the feasibility and reliability of the dipole remote detection of logging while drilling are demonstrated theoretically. For this purpose, we use an asymptotic solution of elastic wave far-field displacement to derive the calculation formula for the radiation pattern and energy flux of an LWD dipole source. The effects of influencing factors, including the source frequency, formation property, drill collar size, and mud parameter, on the radiation pattern and energy flux are analyzed. The results show that the horizontally polarized shear wave (SH-wave) has a greater advantage in imaging the reflector compared with the cases of the compressive wave (P-wave) and vertically polarized shear wave (SV-wave), which indicates the dominance of the SH-wave in dipole remote detection while drilling. The optimal source excitation frequency of 2.5 kHz and inner and outer radii of the drill collar of 0.02 and 0.1 m, respectively, should be considered in the design of an LWD dipole shear wave reflection tool. However, the heavy drilling mud is not conducive to remote detection during logging while drilling. In addition, the reflection of the SH-wave for the LWD condition is simulated. Under the conditions of optimal source frequency, drill collar size, and mud parameters, the reflection of the SH-wave signal is still detected under the fast formation.

1. Introduction

Recently, there has been a major breakthrough in the exploration of buried hill oil and gas fields such as Bozhong 19-6 and 26-6, indicating that complex oil and gas reservoirs have gradually become an important part of offshore oil and gas production and storage [1,2,3,4]. The reservoir space types for the buried hill reservoir are complex and diverse. In addition to the development of intergranular pores and solution pores, fractures also develop, leading to strong heterogeneity in the reservoir and large differences in the test productivity between different exploration wells. Therefore, it is very important to effectively evaluate the development of reservoir fractures. Acoustic wave remote detection has emerged as a technique for imaging fractures from several to tens of meters outside the borehole, filling the detection gap between cross-well seismic and conventional acoustic well logging [5,6,7,8,9]. This technology has achieved good application results in wireline logging. With the increasing need for LWD measurement in highly deviated or horizontal wells, applying this technology to acoustic logging while drilling has become very significant [10,11,12].

Some studies have been conducted on remote detection during logging while drilling. Zhu studied the azimuthal and reflection wave field performance of monopole acoustic logging while drilling for the evaluation of the geomechanics of inhomogeneous formations and designed an acoustic LWD device [13]. Tan simulated the reflected wave field of geological fractures from an LWD dipole source by using the finite difference method and analyzed the variation in the wave field with source excitation frequency and the distance between the source and receiver [14]. Wang used the finite difference method to study the three-dimensional reflected wave field generated by the phased circular array acoustic source and analyzed the characteristics of the waveform at various azimuth angles [15]. Previous research results show that the reflected wave field mainly consists of source radiation, fracture reflection, and the borehole reception response. We can use the Zoeppritz equation to solve the reflection coefficient at the crack. Moreover, the borehole reception response is equal to source radiation due to reciprocity [16]. Among all the factors listed above, source radiation is key in influencing the reflected wave field [17,18,19].

The radiation property of a cable dipole acoustic source has been studied by many scholars [20,21,22,23,24]. Unlike wireline logging, the radiated wave field generated by an LWD dipole acoustic source is more complex. Tan simulated the radiation directivity of an LWD dipole source at conventional logging frequencies and analyzed the energy distribution range of SH- and SV-waves in far fields [14]. Cao investigated the radiation efficiency of an LWD dipole source with various frequencies and compared it with the case of wireline logging [11]. However, these studies only evaluated the effect of acoustic source frequency on radiation wave fields without considering effects such as the variation in formation elastic properties, drill collar size, mud density, and velocity. In view of the influence of the above factors on the radiation far field of an LWD dipole acoustic source, we refer to the response surface method proposed by Azizollah [25]. Therefore, there is an urgent need to study the influence of the above factors on the radiation characteristics of an LWD dipole source.

In the past, the three-dimensional finite difference method and frequency–wavenumber exact integral method were used to study an LWD dipole-radiated far field. These two methods make it difficult to accurately describe the spatial distribution characteristics and the change law with the source frequency of the radiation energy. In this study, considering that the distance between the source and the field point is much larger than the wavelength, we apply the steepest descent method to the frequency–wavenumber exact integral of the displacement potential and innovatively derive the far-field asymptotic solution of the displacement potential. This new method can be used to study the spatial distribution characteristics and the change law with the source frequency of the radiation energy. The analytical solution of the LWD dipole remote detection sound field can also be derived using this method. Thus, this new method makes up for the shortcomings of the three-dimensional finite difference method and frequency–wavenumber exact integral method.

We first derive the formula for calculating the radiation pattern and energy flux of an LWD dipole source using an asymptotic solution of elastic wave far-field displacement. Then, the radiation patterns of an LWD dipole source with different acoustic source frequencies and formation properties are simulated. Moreover, the source radiation energy fluxes with various formation properties, inner and outer radii of the drill collar, and mud densities and velocities are computed. Based on the simulated results, the change rules of the source radiation pattern and energy flux with the influencing factors consisting of the source frequency, formation property, drill collar size, and mud parameter are investigated. Furthermore, under the conditions of optimal source frequency, drill collar size, and mud parameters, the SH reflection shear wave is simulated and successfully used to image the fractures outside the borehole.

2. Theory and Method

In logging while drilling, the dipole source generates guided and radiated waves. Extensive studies have been made on guided waves, but there are fewer studies on the radiated acoustic field. Therefore, this paper focuses only on radiated waves generated by the LWD dipole source. Figure 1a shows the coordinate system used to study the radiation wave field of an LWD dipole source in the borehole. The model is made up of a drill collar, an infinite elastic formation, and the mud inside and outside the drill collar. r₁ and r₂ are, respectively, the inner and outer drill collar radii. The fluid-filled borehole extends indefinitely in the vertical direction (z-axis), and the radius of the borehole is r₃. R is the distance between the field point and the source. ϕ is the azimuth angle in the horizontal plane. θ is the slant angle in the vertical plane.

As shown in Figure 1b, the dipole acoustic logging tool is mainly composed of one source (represented by a red ring) and eight receivers (represented by blue rings). The radius of the dipole source is r₀. The distance between the source and the first receiver is 3 m, and the distance between the adjacent receivers is 0.1524 m. The working principle of the tool in remote detection logging is that the source radiates elastic waves into the formation. When the radiated wave propagates to a reflector, the reflection effect occurs. Finally, the reflected wave propagates into the borehole and is received by the receivers. Under different working modes, the tool can record single- or four-component data.

The wave field in a fluid-filled borehole is calculated in cylindrical coordinates, and the displacement potentials due to a dipole source are given as follows [26]:

(1)$\left\{\begin{array}{l}\psi (\omega ;r,z,\varphi )={\displaystyle \frac{f{r}_0}{2}}{\displaystyle \frac{\sin \varphi }{4\pi }}{\displaystyle {\int }_{-\infty }^{+\infty }B(\omega ,k)K_1}(pr)e^{ikz}dk\\ \chi (\omega ;r,z,\varphi )={\displaystyle \frac{f{r}_0}{2}}{\displaystyle \frac{\cos \varphi }{4\pi }}{\displaystyle {\int }_{-\infty }^{+\infty }D(\omega ,k)K_1}(sr)e^{ikz}dk\\ \Gamma (\omega ;r,z,\varphi )={\displaystyle \frac{f{r}_0}{2}}{\displaystyle \frac{\sin \varphi }{4\pi }}{\displaystyle {\int }_{-\infty }^{+\infty }F(\omega ,k)K_1}(sr)e^{ikz}dk\end{array}\right.$

where the P-, SH-, and SV-wave potentials are, respectively, denoted by ψ, χ, and Г. K₁ is the second kind of modified Bessel function. p and s represent the radial wavenumbers of the solid compression and shear waves, respectively. f and k denote the axial wavenumbers of the fluid and solid compression wave, respectively. ω is the circular frequency. B(ω,k), D(ω,k), and F(ω,k), respectively, denote the wave amplitude coefficients of P-, SH-, and SV-waves, which are determined by the boundary equation on the borehole wall with an LWD dipole source.

In LWD, the medium space is divided into four parts in the radial direction. As shown in Figure 1c, from the center of the borehole outward are the inner mud, drill collar, outer mud, and formation. The boundary conditions of the liquid–solid interface between the inner mud and the drill collar and the outer mud and the formation require that the radial displacement and radial stress are continuous and the tangential stress is zero. Taking the interface between the outer mud and the formation as an example, the boundary equation can be expressed as follows:

(2)$\left\{\begin{array}{l}{u}_r=u_f\\ {\sigma }_{rr}={\sigma }_{rrf}\\ {\sigma }_{rz}=0\\ {\sigma }_{r\varphi }=0\end{array}\right.,\hspace{1em}\hspace{1em}(r=r_3)$

where u_r and σ_rr denote the radial displacement and the normal stress in the formation, respectively. σ_rz and σ_rΦ are the tangential stresses in the formation. u_f and σ_rrf denote the radial displacement and normal stress in the outer mud, respectively. For the liquid–solid boundary, the tangential displacement does not need to be considered; only the displacement perpendicular to the interface is required to be continuous.

For the interface between the drill collar and the outer mud, the displacement of the sound source should also be considered in the radial displacement conditions at the boundary due to the existence of the source. From the boundary conditions at the above three interfaces, a matrix equation with twelve unknown coefficients can be obtained:

HO = b(3)

where b = [0,0,0,0,u_r^s,0,0,0,0,0,0,0]^T and O = [A_fⁱⁿ,A_co,B_co,C_co,D_co,E_co,F_co,A_f^out,B_f^out,B,D,F]^T.

The superscripts in and out denote the inner and outer mud, respectively, and the subscript co represents the drill collar. The element expressions of matrix H and b are detailed in Appendix A. Finally, the amplitude coefficients B(ω,k), D(ω,k), and F(ω,k) of the P-, SH-, and SV-waves in the formation can be obtained by solving Equation (3). Furthermore, the elastic wave displacement potentials are substituted into the displacement–displacement potential equation, and the exact solution of SH shear wave displacement in the formation outside the borehole is obtained:

(4)$u_{\varphi }={\displaystyle \frac{f{r}_0}{2}}{\displaystyle \frac{\cos \varphi }{4\pi }}{\displaystyle {\int }_{-\infty }^{+\infty }\left[\frac{B(\omega ,k)K_1\left(pr\right)}{r}-D(\omega ,k)\left(\frac{K_1\left(sr\right)}{r}-s{K}_2\left(sr\right)\right)+\frac{ikF(\omega ,k)K_1\left(sr\right)}{r}\right]e^{ikz}dk}$

In an acoustic wave remote detection survey, the geological fracture in the formation is far from the borehole, so the radiation distance from the source to the fracture is much larger than the wavelength. In this case, we approximate the Bessel function with high precision:

(5)$K_1(sr)~{\displaystyle \frac{e^{-sr}}{\sqrt{2\pi sr}}},\hspace{1em}\hspace{1em}\hspace{1em}\left|sr\right|>>1$

such that the k integral of the χ displacement potential in Equation (1) can be written as follows:

(6)$\chi (\omega ;r,z,\varphi )={\displaystyle \frac{f{r}_0}{2}}{\displaystyle \frac{1}{4\pi }}{\displaystyle {\int }_{-\infty }^{+\infty }G(k)}{e}^{\delta (k)}dk$

with

(7)$\left\{\begin{array}{l}G(k)={\displaystyle \frac{D(\omega ,k)\cos \varphi }{\sqrt{2\pi sr}}}\\ \delta (k)=-sr+ikz\end{array}\right.$

The steepest descent method [27] is applied to the integral expression of the SH-wave displacement potential, obtaining the far-field expression:

(8)$\chi =i\sqrt{2\pi }G(k_0){\displaystyle \frac{\exp \left\{\delta (k_0)-i\arg [{\delta }^{″}(k_0)]/2\right\}}{\sqrt{|{\delta }^{″}(k_0)|}}}$

where k₀ = ωcosθ/β and δ″(k₀) = irβ/(ωsin³θ). G(k₀) and δ(k₀) can be obtained by substituting k₀ into Equation (7), and the far-field asymptotic solution of the SH shear-wave displacement potential can be obtained by substituting them into Equation (8). According to similar methods and steps, we also obtain the far-field asymptotic solution of the P-wave and SV-shear wave displacement potentials as follows:

(9)$\left\{\begin{array}{l}\psi (\omega )~{\displaystyle \frac{f_{p0}{r}_0}{2}}B(\omega ,k_{p0}){\displaystyle \frac{e^{i\omega R/\alpha }}{4\pi R}}\sin \varphi \\ \chi (\omega )~{\displaystyle \frac{f_{s0}{r}_0}{2}}D(\omega ,k_{s0}){\displaystyle \frac{e^{i\omega R/\beta }}{4\pi R}}\cos \varphi \\ \Gamma (\omega )~{\displaystyle \frac{f_{s0}{r}_0}{2}}F(\omega ,k_{s0}){\displaystyle \frac{e^{i\omega R/\beta }}{4\pi R}}\sin \varphi \end{array}\right.$

where k_p0 = ωcosθ/α and k_s0 = ωcosθ/β are the steepest descent solutions for the axial wavenumbers of the compression and shear waves, respectively. $f_{p0}=\sqrt{k_{p0}^2-k_f^2}$ or $f_{s0}=\sqrt{k_{s0}^2-k_f^2}$ denotes the steepest descent solution for the axial wavenumber of the fluid compression wave. α and β are the velocities of the formation compression and shear waves, respectively. ρ is the formation density. Substituting Equation (9) into the relationship of the displacement and the displacement potential, the P-, SH-, and SV-wave displacement components are calculated as follows:

(10)$\left\{\begin{array}{l}{u}_R~[{\displaystyle \frac{f_{p0}}{2}}i\rho \alpha \omega B(\omega ,k_{p0})\sin \varphi ]{\displaystyle \frac{e^{i\omega R/\alpha }}{4\pi \rho {\alpha }^{2}R}}\\ u_{\varphi }~[-{\displaystyle \frac{f_{s0}}{2}}i\rho \beta \omega D(\omega ,k_{s0})\sin \theta cos\varphi ]{\displaystyle \frac{e^{i\omega R/\beta }}{4\pi \rho {\beta }^{2}R}}\\ u_{\theta }~[{\displaystyle \frac{f_{s0}}{2}}\rho {\omega }^{2}F(\omega ,k_{s0})\sin \theta \sin \varphi ]{\displaystyle \frac{e^{i\omega R/\beta }}{4\pi \rho {\beta }^{2}R}}\end{array}\right.$

The P-, SH-, and SV-wave radiation patterns shown in Equation (11) are defined by the expressions in the brackets of Equation (10), which reflect the radiation coverage of the three kinds of waves in the far field.

(11)$\left\{\begin{array}{l}{\Re }_P\left(\omega ;\theta ,\varphi \right)=i{\displaystyle \frac{f_{p0}}{2}}\rho \alpha \omega B\left(\omega ,k_{p0}\right)\sin \varphi \\ {\Re }_{SH}\left(\omega ;\theta ,\varphi \right)=-i{\displaystyle \frac{f_{s0}}{2}}\rho \beta \omega D\left(\omega ,k_{s0}\right)\sin \theta \cos \varphi \\ {\Re }_{SV}\left(\omega ;\theta ,\varphi \right)={\displaystyle \frac{f_{s0}}{2}}\rho {\omega }^{2}F\left(\omega ,k_{s0}\right)\sin \theta \sin \varphi \end{array}\right.$

The radiation energy flux is another important parameter to describe the radiation performance of an LWD dipole source, showing how much wave energy is radiated into the far field. The radiation energy fluxes of P-, SH-, and SV-waves in the far field are calculated by the components of the stress and particle motion velocity as follows:

(12)$\left\{\begin{array}{l}{W}_P=-{\displaystyle \frac{\pi }{2}}{\displaystyle {\int }_0^{\pi }[Re(v_R^{\ast }\cdot {\sigma }_{RR})\cdot R^2\cdot \sin \theta ]d\theta }\\ W_{SH}=-{\displaystyle \frac{\pi }{2}}{\displaystyle {\int }_0^{\pi }[Re(v_{\varphi }^{\ast }\cdot {\sigma }_{R\varphi })\cdot R^2\cdot \sin \theta ]d\theta }\\ W_{SV}=-{\displaystyle \frac{\pi }{2}}{\displaystyle {\int }_0^{\pi }[Re(v_{\theta }^{\ast }\cdot {\sigma }_{R\theta })\cdot R^2\cdot \sin \theta ]d\theta }\end{array}\right.$

where the energy flow densities of the P-, SH-, and SV-waves are denoted by v_R*σ_RR, v_Φ*σ_RΦ, and v_θ*σ_Rθ, respectively. The symbol * denotes the complex conjugate value. In addition, the components of the velocity v_R, v_Φ, and v_θ can be obtained by multiplying the corresponding components of the displacement in Equation (10) with a factor iω:

(13)$\left\{\begin{array}{l}{v}_R=-[{\displaystyle \frac{f_{p0}}{2}}\rho \alpha {\omega }^{2}B(\omega ,k_{p0})\sin \varphi ]{\displaystyle \frac{e^{i\omega R/\alpha }}{4\pi \rho {\alpha }^{2}R}}\\ v_{\varphi }=[{\displaystyle \frac{f_{s0}}{2}}\rho \beta {\omega }^{2}D(\omega ,k_{s0})\sin \theta cos\varphi ]{\displaystyle \frac{e^{i\omega R/\beta }}{4\pi \rho {\beta }^{2}R}}\\ v_{\theta }=[i{\displaystyle \frac{f_{s0}}{2}}\rho {\omega }^{3}F(\omega ,k_{s0})\sin \theta \sin \varphi ]{\displaystyle \frac{e^{i\omega R/\beta }}{4\pi \rho {\beta }^{2}R}}\end{array}\right.$

Then, substituting the components of the displacement in Equation (10) into the relationship of the displacement and the strain in spherical coordinates and using Hooke’s law to calculate the components of the stress σ_RR, σ_RΦ, and σ_Rθ, we obtain the following:

(14)$\left\{\begin{array}{l}{\sigma }_{RR}=-{\displaystyle \frac{f_{p0}}{2}}\rho {\omega }^{2}B(\omega ,k_{p0})\sin \varphi {\displaystyle \frac{e^{i\omega R/\alpha }}{4\pi R}}\\ {\sigma }_{R\varphi }={\displaystyle \frac{f_{s0}}{2}}\rho {\omega }^{2}D(\omega ,k_{s0})\sin \theta cos\varphi {\displaystyle \frac{e^{i\omega R/\beta }}{4\pi R}}\\ {\sigma }_{R\theta }=i{\displaystyle \frac{f_{s0}}{2}}\rho {\omega }^{3}F(\omega ,k_{s0})\sin \theta \sin \varphi {\displaystyle \frac{e^{i\omega R/\beta }}{4\pi R\beta }}\end{array}\right.$

For acoustic wave remote detection logging, the reflected wave signal received by the instrument is used to image the reflector in the formation far from the borehole. Based on the propagation path, the reflected wave is mainly affected by the source radiation, the wave reflection at a reflector, the borehole reception response, and the propagation distance. Thus, the reflection wave RWV can be expressed as follows:

(15)$RWV(\omega )=S(\omega )\cdot RD(\omega )\cdot RF(\omega )\cdot RC(\omega )\cdot e^{i\omega L(1+i/2Q_{\beta })/\beta }/L$

where S(ω) is the source spectrum. RF denotes the acoustic reflectivity at the reflector. L is the total propagation distance of the reflected wave. Q_β is the shear-wave quality factor. It is one of the important parameters to describe the attenuation characteristics of formation absorption. The reciprocal of Q_β is used to measure the attenuation of the shear wave energy caused by formation absorption in a cycle time range. RD is the borehole radiation pattern, calculated by Equation (11). RC is the borehole reception pattern, which is equal to the source radiation pattern RD due to the elastic reciprocity theorem [28]. Therefore, we focus on the radiation wave field in this paper.

3. Results and Analysis

Based on the above theory and method, the radiation patterns of P-, SH-, and SV-waves of an LWD dipole acoustic source are simulated. Furthermore, the effects of source frequency and the formation property on the radiation patterns of the three kinds of waves are investigated. Then, the radiation energy fluxes of P-, SH-, and SV-waves in the far field are calculated using Equation (12), while the effects of the formation property, drill collar size, and mud parameter are investigated in detail. Finally, the SH-reflection wave data of three cross-well reflectors are calculated and successfully imaged to indicate the feasibility of dipole acoustic remote detection while drilling. The borehole, mud, drill collar, and formation parameters are listed in Table 1.

3.1. Asymptotic Solution Method Verification

To investigate the validity and reliability of using the far-field asymptotic solution to calculate the radiated sound field of an LWD dipole source, an SH shear wave simulated by the asymptotic solution (Equation (10)) is compared with that simulated by the exact solution (Equation (4)), as shown in Figure 2. Formation 1 in Table 1 is used for calculation, and the source frequency is 3 kHz. An array of 15 receivers is placed on the yoz plane and distributed from bottom to top perpendicular to the y-axis, with the first receiver located on the y-axis and adjacent receivers spaced 1 m apart. The source is placed in the position shown by the red ring in Figure 1a, and the vertical distance from the receiver array is 5 m. As shown in Figure 2, the black waveform is the result of the calculation of the exact solution, and the red waveform is the result of the calculation of the asymptotic solution. The comparison shows that they almost exactly coincide, which indicates that the far-field asymptotic solution is accurate and reliable when the distance between the field point and the source is much larger than the wavelength.

3.2. Radiation Pattern of an LWD Dipole Acoustic Source

To study the spatial distribution characteristics of the radiation energy of P-, SH-, and SV-waves in the far field of an LWD dipole acoustic source, the radiation patterns of the three kinds of waves are simulated using Equation (11) versus the θ-angle in the vertical plane at a source frequency of 3 kHz. Formation 1 in Table 1 is used for the simulation. As shown in Figure 3, the radiation patterns of P-, SH-, and SV-waves are denoted by the blue, red, and black curves, respectively. According to Equation (11), the radiation patterns of the three kinds of waves change as trigonometric functions with ϕ in the horizontal direction. ϕ = 0° is set to calculate the SH-wave radiation pattern, and ϕ = 90° is set to calculate the P- and SV-wave radiation patterns. The radial scale denotes the wave radiation amplitude per unit acoustic source energy. The circumferential scale denotes the θ-angle. The P-wave radiation pattern is symmetrical to the borehole, and the amplitude is relatively large at θ-angles ranging from 45 to 135 degrees, indicating the dominant coverage for high-dip reflectors. In contrast, the SV-shear wave radiation pattern is more complex, and the amplitude is relatively large at θ-angles ranging from 15 to 45 degrees and from 135 to 165 degrees, indicating the dominant coverage for low-dip reflectors. Compared with the cases of P- and SV-waves, the radiation pattern amplitude of the SH-wave is much broader throughout the vertical plane, indicating the best radiation coverage for various dip reflectors.

The radiation patterns of P-, SH-, and SV-waves in the far field at the conventional excitation frequency of a dipole acoustic source are given by the above simulation results. To study the influence of source excitation frequency on the radiation patterns of the three kinds of waves in logging while drilling deeply, we calculate the radiation patterns of P-, SH-, and SV-waves at six discrete source frequencies (1, 2, 3, 4, 5, and 6 kHz) using Equation (11), as shown in Figure 4a–c. The amplitudes of the radiant P-, SH-, and SV-waves increase rapidly when the source frequency is less than 3 kHz, while they decrease significantly when the source frequency is greater than 3 kHz. The radiation patterns of P-, SH-, and SV-waves reach the highest amplitude and the widest angular coverage at the source frequency of 3 kHz, which indicates that the radiation energy of an LWD dipole source in the far field is more favorable around this frequency. The change rule of the radiation pattern of the same wave is similar at different frequencies; however, one should note that the amplitudes of SH- and SV-waves increase significantly near the borehole whose diameter is 0.234 m when the source frequencies are 1 and 2 kHz. This is because the patterns are less modulated by the drill collar and borehole with the condition of long wavelength at low frequencies.

In addition to the source frequency, the formation property also has an important influence on the source radiation. To study the impact of the formation property on the radiation patterns of the three kinds of waves in logging while deep drilling, Figure 5a–c give the radiation patterns of P-, SH-, and SV-waves in six different formations (1 to 6 in Table 1) at a source frequency of 2.5 kHz. The change rule of the radiation pattern of the same wave is similar in the different formations. However, one should note that, with the increase in the velocities of compression and shear waves, the amplitudes of P-, SH-, and SV-waves also increase significantly. The radiation patterns of P-, SH-, and SV-waves reach the highest amplitude and the widest angular coverage in formation 1, which indicates that it is easier for an LWD dipole acoustic source to radiate elastic waves into the faster formation around the source frequency of 2.5 kHz.

3.3. Radiated Wave Energy Flux of an LWD Dipole Acoustic Source

As another important parameter to describe the radiation property of an LWD dipole source, the radiation wave energy flux can represent how much wave energy is radiated into the formation as a function of frequency. The radiation energy fluxes of P-, SH-, and SV-waves are calculated using Equation (12) and denoted by the black, red, and blue curves, respectively. Formation 1 in Table 1 is used for the simulation. The horizontal coordinate represents the source frequency, while the vertical coordinate represents the radiation energy fluxes of the three kinds of waves, which are scaled by the unit power of the dipole source. From Figure 6, we can see that the change rules of P-, SH-, and SV-wave energy fluxes are similar: the energy fluxes increase first and then decrease with increasing frequency. Most of the radiated energy is distributed in the frequency range of 2–4 kHz and reaches a peak value around 2.5 kHz. In the whole frequency range, the source radiates much more energy of SH-waves than those of P- and SV-waves into the formation. In terms of maximum radiant energy flux, the energy of the SH-wave is about 7 times those of P- and SV-waves. The above analysis results show the dominance of SH-shear waves and the optimal source excitation frequency around 2.5 kHz in the LWD dipole remote detection.

It is worth mentioning that the influence of the formation property on the radiated energy fluxes in the far field is unable to be neglected for the investigation of the LWD dipole remote detection. To study the impact of the formation property on the radiated energy fluxes in detail, we calculate the P-, SH-, and SV-wave radiation energy fluxes using Equation (12) in six types of formation, respectively, as shown in Figure 7a–c. Formations 1 to 6 in Table 1 are used for the simulation. The horizontal coordinate represents the source frequency, while the vertical coordinate represents the radiation energy fluxes. The energy fluxes for all types of elastic waves change similarly with the frequency in the fast formations: with increasing frequency, they generally increase first and then decrease. With the formation velocity gradually decreasing, the spectrum of radiation energy flux becomes wider while the maximum decreases. In addition, the optimal excitation frequency corresponding to the maximum radiant energy flux gradually approaches the lower frequency. Compared with the cases of the fast formations, the spectrum of radiation energy flux becomes much wider in the slow formation, especially for P- and SV-waves. This is because as the acoustic impedance difference between the slow formation and the fluid on the borehole wall becomes smaller, P- and SV-waves are more likely to radiate into the formation at high frequencies.

3.4. Influence of Drilling Mud and Drill Collar on Radiant Energy Flux

To ensure the stability of the borehole wall in the drilling, mud of various densities is usually used in sections of different depths considering the changes in formation pressure. Taking drilling in the Bohai Sea as an example, under normal formation pressure conditions, the distribution of mud density ranges from 1000 to 1400 kg/m³ from shallow to deep depths. However, when the formation is overpressured, this range is from 1300 to 1800 kg/m³, respectively. The variation in mud density will cause a difference in the velocity at which the wave travels. Therefore, it is significant to analyze the influence of drilling mud on the energy fluxes of radiant waves. Based on the above results (see Section 3.2 and Section 3.3), the SH shear wave plays a dominant role in the far field radiated by an LWD dipole acoustic source, and the optimal source excitation frequency ranges from 2 to 3 kHz. Taking the SH-shear wave as an example and selecting 2.5 kHz as the source excitation frequency, the radiation energy fluxes are calculated in the fast (formation 1) and slow (formation 6) formations under various mud densities and velocities, as shown in Figure 8a,b. The horizontal coordinate represents the mud density or velocity, while the vertical coordinate represents the SH-wave radiation energy flux. From the simulated results, we can see that the SH-wave energy radiated into the fast and slow formations gradually decreases with the increase in mud density. This result indicates that the heavy drilling mud reduces the radiation energy of the dipole source, which is not conducive to remote detection in logging while drilling. Unlike the case of mud density, the SH-wave radiation energy flux first increases and then decreases with the increase in mud velocity. The energy flux in the fast formation changes sharply with mud velocity and is strongest at the mud velocity of 1400 m/s. However, the energy flux changes gently in the slow formation, indicating less influence with the mud velocity.

In the while drilling logging situation, the source is embedded into the drill collar, so the radiated energy of the source from the borehole into the formation will be significantly affected. Thus, it is crucial to study the effect of the drill collar on the source radiation energy flux. Considering the dominance of the SH shear wave in the far field and the optimal excitation frequency of the LWD dipole source, the SH-wave radiation energy fluxes are calculated in the fast (formation 1) and slow (formation 6) formations under various outer radii of the drill collar, as shown in Figure 9a. The horizontal coordinate represents these outer radii, while the vertical coordinate represents the SH-wave radiation energy flux. In the calculation, the inner radius of the drill collar is set to 0.027 m. From the calculated results, we can see that the SH-wave energy radiated into the fast and slow formations gradually increases with increasing outer radius. This is because the larger the drill collar outer radius becomes, the closer the source is to the formation, so the energy excited by the source decays less in the borehole and radiates more into the formation. Moreover, assuming the outer radius of the drill collar to be 0.09 m, the SH shear wave radiation energy fluxes are calculated in the fast (formation 1) and slow (formation 6) formations under various inner radii of the drill collar, as shown in Figure 9b. The horizontal coordinate represents the inner radii of the drill collar, while the vertical coordinate represents the SH-wave radiation energy flux. With increasing inner radius, the SH-wave energy radiated into the fast formation gradually decreases. Distinct from that of the fast formation, the SH-wave energy radiated into the slow formation is almost unaffected under an inner radius less than 0.05 m. The results provide the optimization of drill collar size for the design of an LWD dipole remote detection tool.

4. Imaging Result of Dipole Shear Wave Remote Detection While Drilling

In this section, we further investigate the feasibility of dipole remote detection while drilling. Based on the above theoretical research results, the SH-reflected wave field of three reflectors is calculated using Equation (15), as shown in the left panel of Figure 10. The reflectors used in the simulation are given in the right panel of Figure 10. From top to bottom, the angles of the reflector intersected with the borehole are 70, 50, and 30°, respectively. The distance between the first and second reflectors is 15 m, while that between the second and third is 20 m. Their SH-wave reflectivity is set to 0.1 for normal incidence. Considering the factors that influence the radiation performance of an LWD dipole source, the optimal source excitation frequency, drill collar size, and mud parameters are used in the simulation. The inner and outer radii of the drill collar are 0.02 and 0.1 m, respectively, while the mud density and speed are 1000 kg/m³ and 1400 m/s, respectively. A 2.5 kHz Kelly source and formation 1 are used for the wave data calculation. The SH-reflected wave data are displayed with variable density in the section of 150 m. The depth and time sampling intervals of the data are 0.1524 m and 36 μs, respectively. From Figure 10, we can see that the amplitudes of the SH-reflected wave are significantly smaller than those of a flexural wave propagating along the borehole. Therefore, it is necessary to remove the flexural wave by filtering and enhance the SH-reflected wave signal by stacking for the data imaging processing. It is worth mentioning that the signal of the SH-wave reflected from the high-dip reflector received by the instrument is stronger than that of the low-dip reflector, which is well explained by the SH-wave radiation pattern of the LWD dipole source (see Figure 4b). Finally, we use the SH-reflected wave to successfully image three reflectors, and their maximum extension outside the borehole is about 10 m. Since the medium used to simulate the SH-reflected wave field is a constant velocity formation, the position of the reflectors in the imaging results using the forward modeling reflected wave data is completely consistent with the set position. However, the actual formation is not a constant velocity medium. The reflector position will have an offset error of about 2% when using the actual logging data to image. This is because the acoustic logging instrument measurement formation wave speed error is also 2%.

5. Conclusions

This study explores the feasibility of remote detection in logging while drilling (LWD) using an acoustic dipole source in theory. Upon analyzing the influencing factors on the radiation pattern and energy flux of the source, the main results are as follows:

• (1). The comparison of a horizontally polarized shear wave (SH-wave) simulated by the asymptotic solution with that calculated by the exact solution shows that the far-field asymptotic solution is accurate and reliable when the distance between the field point and the source is much larger than the wavelength. The far-field asymptotic solution provides a convenient and accurate analytical method for calculating the reflected sound field.

• (2). By comparing the radiation patterns and energy fluxes of the compressive wave (P-wave) and horizontally (SH) and vertically (SV) polarized shear waves, it can be seen that the maximum radiant energy flux of the SH-wave is about 7 times that of the P- or SV-wave. Therefore, the SH-wave plays a dominant role in the radiation far field of an LWD dipole source and is most suitable for the remote detection and imaging of the reflector outside the borehole.

• (3). The theoretical and simulated results show that the optimal acoustic source excitation frequency is about 2.5 kHz, and the optimal inner and outer radii of the drill collar are 0.02 m and 0.1 m, respectively.

• (4). As an important influencing factor, the effect of mud on the radiation performance of an LWD dipole source is as follows: the SH-wave radiation energy flux changes sharply with mud velocity and reaches a peak value around 1400 m/s in fast formation, while it changes gently in slow formation. Furthermore, the heavy drilling mud is not conducive to remote detection while drilling logging.

• (5). Using the far-field asymptotic solution, the SH-reflected wave field at 150 m depth is simulated, taking five minutes. On this basis, we use the SH-reflected wave data to image the reflectors in the depth range of 150 m, and the whole process takes about ten minutes. The imaging result demonstrates the feasibility of remote detection in logging while drilling using an acoustic dipole source.

Author Contributions

Methodology, J.C. and R.Q.; code, J.C. and X.L.; software, X.Y. and Z.Y.; validation, C.H. and P.W.; formal analysis, X.Y.; investigation, Z.Y.; resources, P.W.; writing—original draft preparation, J.C. and X.L.; writing—review and editing, J.C. and R.Q. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

The data used to support the results of this study are included within the manuscript.

Acknowledgments

All the authors would like to thank the reviewers and editors for their thoughtful comments.

Conflicts of Interest

Authors Jingji Cao, Ruibao Qin, Xiongyan Li, Xinyu Ye, Zhongxu Yin and Peng Wang were employed by the company CNOOC Research Institute Ltd. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest. The funders played no part in the study design, data collection, analysis and interpretation, writing of the manuscript, or decision to publish the results.

Abbreviations

Afin	Amplitude coefficient of reflected P-wave in the inner mud
Aco	Amplitude coefficient of reflected P-wave in the drill collar
Afou	Amplitude coefficient of reflected P-wave in the outer mud
b	Vector of 12 × 1 elements
B	Amplitude coefficient of P-wave in the formation
Bco	Amplitude coefficient of radiated P-wave in the drill collar
Bfout	Amplitude coefficient of radiated P-wave in the outer mud
Cco	Amplitude coefficient of reflected SH-wave in the drill collar
D	Amplitude coefficient of SH-wave in the formation
Dco	Amplitude coefficient of radiated SH-wave in the drill collar
Eco	Amplitude coefficient of reflected SV-wave in the drill collar
F	Amplitude coefficient of SV-wave in the formation
f	Radial wavenumber of fluid compressive wave, m−1
fp0	Steepest descent solution for axial wavenumber of fluid compressive wave, m−1
fs0	Steepest descent solution for axial wavenumber of fluid shear wave, m−1
Fco	Amplitude coefficient of radiated SV-wave in the drill collar
H	Matrix of 12 × 12 elements
k	Axial wavenumber of solid compressive wave, m−1
K1	Modified Bessel function
kf	Wavenumber of fluid compressive wave, m−1
kp0	Steepest descent solution for axial wavenumber of solid compressive wave, m−1
ks0	Steepest descent solution for axial wavenumber of solid shear wave, m−1
L	Distance the reflected wave travels, m
O	Vector of 12 × 1 elements
p	Radial wavenumber of solid compressive wave, m−1
r	Radial distance, m
r0	Radius of dipole source, m
r1	Inner radius of drill collar, m
r2	Outer radius of drill collar, m
r3	Radius of the borehole, m
R	Distance the radiated wave travels, m
s	Radial wavenumber of solid shear wave, m−1
S	Source power, W
ur	Radial displacement in the formation, m
uf	Radial displacement in the outer mud, m
urs	Radial displacement of the source, m
uR	Displacement of P-wave, m
uΦ	Displacement of SH-wave, m
uθ	Displacement of SV-wave, m
vR	Polarization velocity of a particle of P-wave, m/s
vΦ	Polarization velocity of a particle of SH-wave, m/s
vθ	Polarization velocity of a particle of SV-wave, m/s
WP	Radiation flux of P-wave, W
WSH	Radiation flux of SH-wave, W
WSV	Radiation flux of SV-wave, W
z	Axial distance, m
α	Formation compressive wave velocity, m/s
β	Formation shear wave velocity, m/s
θ	Slant angle in the vertical plane, °
ρ	Formation density, kg/m3
σrr	Normal stress in the formation, Pa
σrz	Tangential stress of z-direction in the formation, Pa
σrΦ	Tangential stress of Φ-direction in the formation, Pa
σrrf	Normal stress in the outer mud, Pa
σRR	Normal stress of R-direction in far field, Pa
σRΦ	Tangential stress of Φ-direction in far field, Pa
σRθ	Tangential stress of θ-direction in far field, Pa
ϕ	Azimuth angle in the horizontal plane, °
χ	Displacement potential of SH-wave, m
ψ	Displacement potential of P-wave, m
Г	Displacement potential of SV-wave, m
ω	Circular frequency, rad/s
Qβ	Shear-wave quality factor, dB−1
Abbreviations
LWD	Logging while drilling
P	Compressive wave
RC	Borehole reception response
RD	Source radiation pattern
RF	Reflection coefficient
SH	Horizontally polarized shear wave
SV	Vertically polarized shear wave

Footnotes

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Figures and Table

Figure 1. Coordinate system for solving radiated wave field of an LWD dipole source (a); schematic diagram of acoustic logging while drilling (b); diagram of media composition from the center of borehole to infinity formation (c).

Figure 2. Compared result of the SH-wave radiated by an LWD dipole source simulated by using exact and asymptotic solution methods.

Figure 3. P-, SH-, and SV-wave radiation patterns of an LWD dipole acoustic source at source frequency of 3 kHz.

Figure 4. Radiation patterns of P-waves (a), SH-shear waves (b), and SV-shear waves (c) at six discrete source frequencies.

Figure 5. Radiation patterns of P-wave (a), SH-shear wave (b), and SV-shear wave (c) in six different formations.

Figure 6. P-, SH-, and SV-wave energy fluxes radiated by an LWD dipole source into formation 1.

Figure 7. Radiation energy fluxes of P-wave (a), SH-shear wave (b), and SV-shear wave (c) in six different formations.

Figure 8. Radiation energy fluxes of SH-shear wave with various mud densities (a) and velocities (b).

Figure 9. Radiation energy fluxes of SH-shear waves with various outer (a) and inner (b) radii.

Figure 10. The data and image of SH-wave remote detection for an LWD dipole acoustic tool.

Table 1

Model parameters.

Media Parameter	P-Wave Velocity(m · s−1)	Shear Wave Velocity(m · s−1)	Density(kg · m−3)	Quality FactorQp or Qs	Outer Radius (m)
Inner Mud	1500	—	1000	50	0.027
Drill collar	5860	3130	7850	1000	0.090
Outer Mud	1500	—	1000	50	0.117
Formation 1	5000	2800	2600	100	∞
Formation 2	4500	2500	2570	100	∞
Formation 3	4000	2200	2540	100	∞
Formation 4	3500	1950	2510	100	∞
Formation 5	3000	1650	2480	100	∞
Formation 6	2500	1390	2450	100	∞

Appendix A

The element expressions of matrix H and b are detailed as follows:

H11 = −I1(fr1)/r1 − fI2(fr1); H12 = I1(pcor1)/r1 + pcoI2(pcor1);H13 = K1(pcor1)/r1 − pcoK2(pcor1); H14 = I1(scor1)/r1;H15 = K1(scor1)/r1; H16 = ik[I1(scor1)/r1 + scoI2(scor1)];H17 = ik[K1(scor1)/r1 − scoK2(scor1)]; H21 = ρfω2I1(fr1);H22 = ρco(2k2βco2 − ω2)I1(pcor1) − 2ρcoβco2pcoI2(pcor1)/r1;H23 = ρco(2k2βco2 − ω2)K1(pcor1) + 2ρcoβco2pcoK2(pcor1)/r1;H24 = 2ρcoβco2scoI2(scor1)/r1; H25 = −2ρcoβco2scoK2(scor1)/r1;H26 = 2ikρcoβco2[sco2I1(scor1) − scoI2(scor1)/r1]; H27 = 2ikρcoβco2[sco2K1(scor1) + scoK2(scor1)/r1];H32 = −2ρcoβco2pcoI2(pcor1)/r1; H33 = 2ρcoβco2pcoK2(pcor1)/r1;H34 = −ρcoβco2[sco2I1(scor1) − 2scoI2(scor1)/r1]; H35 = −ρcoβco2[sco2K1(scor1) + 2scoK2(scor1)/r1];H36 = −2ikρcoβco2scoI2(scor1)/r1; H37 = 2ikρcoβco2scoK2(scor1)/r1;H42 = 2ikρcoβco2[I1(pcor1)/r1 + pcoI2(pcor1)]; H43 = 2ikρcoβco2[K1(pcor1)/r1 − pcoK2(pcor1)];H44 = ikρcoβco2I1(scor1)/r1; H45 = ikρcoβco2K1(scor1)/r1;H46 = ρcoβco2(k2 + sco2)[ − scoI2(scor1) − I1(scor1)/r1];H47 = ρcoβco2(k2 + sco2)[scoK2(scor1) − K1(scor1)/r1];

H52 = I1(pcor2)/r2 + pcoI2(pcor2); H53 = K1(pcor2)/r2 − pcoK2(pcor2);H54 = I1(scor2)/r2; H55 = K1(scor2)/r2;H56 = ik[I1(scor2)/r2 + scoI2(scor2)]; H57 = ik[K1(scor2)/r2 − scoK2(scor2)];H58 = −I1(fr2)/r2 − fI2(fr2); H59 = −K1(fr2)/r2 + fK2(fr2);H62 = ρco(2k2βco2 − ω2)I1(pcor2) − 2ρcoβco2pcoI2(pcor2)/r2;H63 = ρco(2k2βco2 − ω2)K1(pcor2) + 2ρcoβco2pcoK2(pcor2)/r2;H64 = 2ρcoβco2scoI2(scor2)/r2; H65 = −2ρcoβco2scoK2(scor2)/r2;H66 = 2ikρcoβco2[sco2I1(scor2) − scoI2(scor2)/r2]; H67 = 2ikρcoβco2[sco2K1(scor2) + scoK2(scor2)/r2];H68 = ρfω2I1(fr2); H69 = ρfω2K1(fr2);H72 = −2ρcoβco2pcoI2(pcor2)/r2; H73 = 2ρcoβco2pcoK2(pcor2)/r2;H74 = −ρcoβco2[sco2I1(scor2) − 2scoI2(scor2)/r2]; H75 = −ρcoβco2[sco2K1(scor2) + 2scoK2(scor2)/r2];H76 = −2ikρcoβco2scoI2(scor2)/r2; H77 = 2ikρcoβco2scoK2(scor2)/r2;H82 = 2ikρcoβco2[I1(pcor2)/r2 + pcoI2(pcor2)]; H83 = 2ikρcoβco2[K1(pcor2)/r2 − pcoK2(pcor2)];H84 = ikρcoβco2I1(scor2)/r2; H85 = ikρcoβco2K1(scor2)/r2;H86 = ρcoβco2(k2 + sco2)[ − scoI2(scor2) − I1(scor2)/r2];H87 = ρcoβco2(k2 + sco2)[scoK2(scor2) − K1(scor2)/r2];H98 = −I1(fr3)/r3 − fI2(fr3); H99 = −K1(fr3)/r3 + fK2(fr3);H9,10 = K1(pfmr3)/r3 − pfmK2(pfmr3); H9,11 = K1(sfmr3)/r3;H9,12 = ik[K1(sfmr3)/r3 − sfmK2(sfmr3)]; H10,8 = ρfω2I1(fr3);H10,9 = ρfω2K1(fr3); H10,10 = ρfm(2k2βfm2 − ω2)K1(pfmr3) + 2ρfmβfm2pfmK2(pfmr3)/r3;H10,11 = −2ρfmβfm2sfmK2(sfmr3)/r3; H10,12 = 2ikρfmβfm2[sfm2K1(sfmr3) + sfmK2(sfmr3)/r3];H11,10 = 2ρfmβfm2pfmK2(pfmr3)/r3; H11,11 = −ρfmβfm2[sfm2K1(sfmr3) + 2sfmK2(sfmr3)/r3];H11,12 = 2ikρfmβfm2sfmK2(sfmr3)/r3; H12,10 = 2ikρfmβfm2[K1(pfmr3)/r3 − pfmK2(pfmr3)];H12,11 = ikρfmβfm2K1(sfmr3)/r3;H12,12 = ρfmβfm2(k2 + sfm2)[sfmK2(sfmr3) − K1(sfmr3)/r3];b5 = ε1[K1(fr2)/r2 − fK2(fr2)]I1(fr2);

where the elements of matrix H and vector b that are not listed are 0. pco = (k2 − ω2/αco2)1/2 and pfm = (k2 − ω2/αfm2)1/2 are the radial wave numbers of compressive waves, while sco = (k2 − ω2/βco2)1/2 and sfm = (k2 − ω2/βfm2)1/2 are the radial wave numbers of shear waves of the drill collar and formation, respectively. ρf, ρco, and ρfm are, respectively, the densities of the mud, drill collar, and formation. αf, αco, and αfm are, respectively, the compressive wave velocities of the mud, drill collar, and formation. βco and βfm are, respectively, the shear wave velocities of the drill collar and formation.

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