Transient radiation from an unloaded, finite dipole antenna in a borehole

GEOPHYSICS, VOL. 70, NO. 6 (NOVEMBER-DECEMBER 2005); P. K43–K51, 12 FIGS. 10.1190/1.2106048

Transient radiation from an unloaded, ﬁnite dipole antenna in a borehole: Experimental and numerical results

Sixin Liu1 and Motoyuki Sato2 Sato and Thierbach (1989) use an approximate analytical form for the current distribution on an antenna and calculate the theoretical receiving signals of borehole radar in crosshole measurements. Ebihara et al. (1998) analyze guided waves within a borehole and point out their potential effects on the radiation pattern. Atlamazoglou and Uzunoglu (1998) use a Galerkin moment method to solve the problem of a dielectric coated dipole antenna in a dissipative medium. Chen et al. (2002) use a finite-difference, time-domain (FDTD) technique to model a novel broadband antenna; this can be used in a borehole radar system for oil-field imaging while drilling. Chen and Oristaglio (2002) examine the suitability of borehole radar for near-wellbore imaging. Wang and McMechan (2002) use a 2D finite-difference method to model borehole radar signals and to estimate attenuation and velocity distributions as well as the geometry of reflectors. Holliger and Bergmann (2002) have developed a finitedifference solution of Maxwell’s equations in cylindrical coordinates to model the full electromagnetic (EM) wavefield associated with borehole radar experiments and to assess the adequacy of ray-based methods currently used to interpret the observed data. They find that the radiation pattern of a vertical, infinitesimal electric dipole source located in a water-filled borehole might be distorted relative to the patterns in a homogeneous medium. The authors conclude that some of the basic assumptions of conventional ray-based amplitude tomography often are not fulfilled for borehole radar data. However, it appears that an antenna with real size and geometry has not been considered. Although Ernst et al. (2003) simulate a wire-tape antenna and explore the radiative properties of realistic antenna designs used in borehole radar, taking into account either constant or optimized Wu-King-type resistive loading, the borehole itself is not included in their simulations. In this paper, we consider the dynamics of an antenna of finite size located in a borehole. Our objectives are to quantify the effects of the borehole upon the radiation pattern, the input impedance, and the radiated waveforms and to analyze the

ABSTRACT Dipole antennas in boreholes are used for tomographic imaging or electromagnetic well logging. A cylindrically layered structure within the borehole will change the radiation characteristics of a dipole antenna. Our objective is to understand the effects of the borehole structure upon the impedance, waveform distortion, and directivity patterns of a dipole antenna. We use a finite-difference, time-domain (FDTD) technique to simulate borehole-antenna radiation, while the geometry of both the dipole and the borehole are modeled with a subgrid technique. The simulated input impedances are verified by experimental results. Both the water-filled and the air-filled boreholes distort the radiated wavefronts, waveforms, and resonant frequencies relative to the same characteristics of a dipole in homogeneous media. A water-filled borehole lowers the first resonant frequency, while an air-filled borehole raises it. At high frequencies, the antenna in the waterfilled borehole exhibits radiation side lobes. The borehole effects for water- and air-filled boreholes differ and should not be neglected for borehole antenna design.

INTRODUCTION Borehole radar is important for tomography and reflection profiling. Consequently, antennas with wide radiation patterns are needed, but antennas immersed in media are known to have more narrow patterns (Stutzman and Thiele, 1998). Although only point dipoles have been studied (Sato and Thierbach, 1991; Miwa et al., 1999; Liu and Sato, 2002), we hypothesize that the borehole itself affects the radiation of antennas of finite size. We want to learn more about borehole effects on the impedance, waveforms, and directivity patterns of antennas of finite size.

Manuscript received by the Editor May 8, 2003; revised manuscript received February 2, 2005; published online September 19, 2005. 1 Jilin University, Department of Geophysics, 6 Ximinzhu, Changchun 130026, China. E-mail: [email protected]. 2 Tohoku University, Center for Northeast Asian Studies, Kawauchi, Sendai 980-8756, Japan. E-mail: [email protected]. c 2005 Society of Exploration Geophysicists. All rights reserved. K43

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transient process of the field less than 3.5 m from the antenna. Because of the existence of a cylindrically layered medium, the mechanism of borehole antenna radiation is more complex than for a free-space antenna and needs much more investigation. We approach this problem with both laboratory and finite-difference studies. First, we consider a realistic dipole antenna size and typical media parameters that correspond with field situations. We then verify our algorithms by comparing calculated input impedances with values we measured for a monopole antenna in a water-filled cylinder. Although a resistively loaded antenna is often used (Wu and King, 1965; Sengupta and Liu, 1974; Arcone, 1995), in this paper we treat the simpler case of a nonresistively loaded dipole antenna to understand the effect of natural dielectric loading on impedance and pulse propagation speed. This can lead to a better understanding of antenna-ground impedance and can explain why the pulse-center frequency decreases for a larger ground dielectric constant. We use a subgrid-based FDTD approach to simulate antenna radiation in a homogeneous medium and then in waterand air-filled boreholes to visualize the propagating wavefronts and the change in waveforms caused by natural loading. Although 3D FDTD modeling with Cartesian coordinates is less efficient than a 2D cylindrical scheme, we use a Cartesian coordinate system rather than a cylindrical one because it is more suitable for cases with nonaxis symmetry. Additionally, Cartesian coordinates must be used to model real targets such as orebodies and fractures.

FDTD SIMULATION FDTD methods for computing EM scattering and propagation problems are well known (Yee, 1966; Maloney et al., 1990; Kunz and Luebbers, 1993; He and Liu, 1999). The perfectly matched layer (PML) technique is used to construct an absorbing boundary. We follow Chevalier et al. (1997) and develop a subgrid-based FDTD code. The fine structure of a borehole and an antenna is modeled by subgrids, while the other regions are modeled by coarse main grids (Figure 1a). The perfectly (electric-field) conducting (PEC) antenna is modeled by setting the tangential electrical field on the metal surface to zero. For simplicity, the electrical fields within the cylinder are also set to zero; the cylinders can be regarded as tapered solid cylinders (Figure 1b). We carefully maintain the continuity and the stability of the FDTD computations between the main grid regions and the subgrid regions. The dipole antenna (Figure 1b) feed point is modeled by a delta gap excited by a Gaussian pulse, such that 2 /T 2

V (t) = e−(t−t0 )

,

(1)

where T denotes the width of the pulse and t0 is the time delay relative to t = 0 ns. We follow Luebbers and Langdon (1996) to model the feed point using a voltage source including an internal resistance. The z-component of the electric field Ez at the feed point can be expressed as

Ez (t) = −

V (t) − RI (t) , dz

(2)

where V is the incident voltage, R is the internal resistance for which R = 50 ohms is chosen, I is the current at the feed point, and dz is the size of a grid element in the z-direction. The current is formulated as

I (t) =

H × ds,

(3)

C

where C is a closed path for integration.

EXPERIMENTAL VALIDATION OF FDTD RESULTS: STEADY-STATE CONDITION

Figure 1. The numerical model of a dipole antenna within a borehole. (a) Horizontal cross section of a dipole located in a borehole; (b) vertical cross section of a dipole with a taper. The positive z-axis points outward from the page.

We simulated a steady-state borehole antenna in the laboratory and compared its impedance characteristics with modeling values to verify our subgrid-based FDTD code. We placed a monopole antenna, connected to a coaxial line, in an acrylic cylinder that was fixed on a ground plane (Figure 2). The cylinder was filled with water. The monopole antenna length and radius were h = 15 cm and a = 3 mm, respectively. The outer radius, thickness, and cylinder height were b = 30 mm, c = 3.5 mm, and H = 0.5 m, respectively. The conductivity and dielectric constant of the water were σ1 = 0.0072 S/m and εr1 = 81, respectively, while those of the acrylic were σ2 = 0 S/m and εr2 = 2.5. We measured the input impedances of the antenna in the frequency range of 0.5 to 400 MHz with a Hewlett Packard HP8752A network analyzer. During the FDTD numerical simulation, we allowed the radiation process to continue for more than 80 ns until the transient current at the feed point became insignificant. We calculated the input impedance in the frequency domain from the ratio between the Fourier transformations of the current and the voltage waveforms that we measured at the feed point. The good agreement between the calculated and measured

Borehole Antenna Radiation

values, especially at low frequencies (Figure 3), established the validity of the subgrid-based FDTD code. Numerical dispersion and staircasing errors caused the FDTD results to differ slightly in amplitude from the measurements. The difference for resistance was about 10% at the position of the first peak. The peaks and troughs for the simulated data occurred at higher frequencies than did those for the measurements. Lampe et al. (2003) attribute this discrepancy to the buildup of capacitances on the antenna. The input impedance of an antenna is associated with the feeding structure, antenna dimensions, and medium surrounding the antenna. Reflections from the ends of an antenna and the surrounding structure may cause many resistance peaks. For a simple monopole or dipole of finite length in a homogeneous medium, the first resistance peak at the lowest frequency is often larger than the others (e.g., Elliott, 1981, Figures 7–15a; Kunz and Luebbers, 1993, Figures 14–27; Stutzman and Thiele, 1998, Figure 5-5). In our case, the first resistance peak is smaller than the second and third peaks in Figure 3. We tried to understand this phenomenon by investigating the reflections from the top and the wall of the cylinder with FDTD modeling. In the first simulation, the cylinder diameter was held fixed and the height of the water-filled cylinder was varied from 0.3 to 2.0 m, thereby extending the length of the acrylic cylinder. Figure 4 shows that the first peak of the resistance decreased as the height of the cylinder increased, while all other peaks remained relatively stable. We therefore inferred that the first peak was not caused by reflections from the top of the cylinder. The input impedance was roughly the same for heights of 1.0 and 2.0 m, which meant that when the length of the cylinder was sufficiently long, reflections from the top of the cylinder became negligible.

Figure 2. Experimental monopole antenna standing on a ground plane and immersed in a water-filled acrylic cylinder surrounded by air. The dimensions are h = 15 cm, a = 3 mm, b = 30 mm, c = 3.5 mm, and H = 0.5 m.

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In a second simulation, we fixed the cylinder-wall thickness at c = 3.5 mm and the cylinder height at a relatively long H = 2.0 m. We then increased the cylinder radius from 30 to 100 mm (Figure 5). For a sufficiently large diameter, the first peak grew larger than the other two peaks. We concluded, therefore, that wave interference from the wall of the cylinder caused the first impedance peak to be lower than the other two peaks.

FDTD MODEL OF THE RADIATION PROCESS The FDTD model of a dipole antenna in a borehole is shown in Figure 6. The Cartesian dimensions for this FDTD

Figure 3. Measured and calculated (a) input resistance and (b) reactance for the laboratory monopole.

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simulation are 10×10×10 m. The antenna length 2h = 80 cm, the antenna radius a = 2.25 cm, and the borehole radius b = 5 cm. The antenna is insulated by thin acrylic material for which the dielectric constant equals 2.5 and conductivity equals 0; the outer diameter equals 6.5 cm. The dimensions of the antenna are the same as those used in an actual borehole radar system operated at a bandwidth centered near 80 MHz. The borehole is uncased. We consider three cases, for which σ2 = 0.001 S/m and εr2 = 5.8, typical of salt. In case 1, there is no borehole. In case 2, the borehole is filled with water, for which σ1 = 0.0072 S/m and εr1 = 81. Case 3 is an air-filled borehole. In the following, we analyze the transient radiation process, the radiated waveforms, the input impedance, and the radiation pattern.

We compare the amplitude of Ez at 6, 11, 16, 20, and 24 ns for case 1 (Figure 7) and at 6, 11, 16, 20, and 29 ns for cases 2 (Figure 8) and 3 (Figure 9). The electric-field amplitudes and the image dimensions are scaled equally in all three cases. The position of the borehole is drawn in Figures 8 and 9. The antenna is excited by a Gaussian pulse with t0 = 12 ns and T = 4 ns (equation 1). An important constant, the time required for the current pulse to travel from the feed point to the end of the antenna, can be expressed as τa = h/v, where v is the velocity of the current pulse along the antenna. If the antenna is located in a homogeneous medium (King and Smith, √ 1981; Smith, 1997), v = c/ εr2 , where c is the speed of light

Figure 4. Calculated input impedance for different cylinder heights. The dimensions are b = 30 mm, c = 3.5 mm, and H = 0.3, 0.5, 1.0, and 2.0 m.

Figure 5. Calculated input impedance for different cylinder diameters. The dimensions are H = 2.0 m, c = 3.5 mm, and b = 30, 60, and 100 mm.

Transient analysis


in free space. Accordingly, τa = 3.3 if the antenna is located in the homogeneous salt rock. It is difficult to compute τa directly for the antenna in the water- or air-filled borehole because it is affected by the media both inside and outside of the borehole. Let us consider the radiation in case 1. A spherical wavefield W1 radiates from the feed point and propagates radially (Figure 7a). The shape of Ez near the broadside direction is roughly spherical, but on the antenna surface it vanishes. The electric field Er , which is normal to the antenna, propagates along the antenna as charges move along the antenna surface and is reflected at the ends of the antenna (Figure 7b). The reflected field W2 at the end is roughly spherical. The superimposed wavefronts W1 and W2 are shown in Figure 7b and c. As a result, the W1 wavefront exits the antenna primarily in the radial direction. The parts of the W2 wavefronts that move back along the antenna are then reflected at the feed point (W3 ; Figure 7d) and also at the opposite ends of the antenna (W4 ; Figure 7e). Figure 7d and e shows the state of the radiation at 20 and 24 ns, respectively. The propagation of the W3 wavefield is now similar to that of W1 , while the propagation of the W4 wavefield is similar to that of W2 ; these processes repeat. The waves in the surrounding medium propagate outward. The quantity τa is approximately 2.95 ns, which is the time needed for the current pulse to move from the feed point to the end of the antenna. The reason we chose 24 ns for case 1

Figure 6. FDTD geometry of the antenna in a borehole. The dimensions are 2h = 80 cm, a = 2.25 cm, and b = 5 cm.

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and not 29 ns as we did for the other cases is that the wavefront W4 can be distinguished at 24 ns for case 1 (no borehole) but not at 29 ns, for which wavefronts W3 and W4 are clear in the other two cases. In cases 2 and 3, the time delays are not as precisely predictable from the phase velocities of water and air because the pulse is transient and the borehole is layered. The amplitude is attenuated, and its shape changes little. We compare the peaks of the current pulse at the feed point and near the end of the dipole, and we take the time difference between them to be a qualitative estimate of the time delay. In case 2, the dipole antenna is located in a water-filled borehole. The quantity τa is estimated to be 3.70 ns from the

Figure 7. Radiation from a dipole antenna at five different times for no borehole at (a) 6 ns, (b) 11 ns, (c) 16 ns, (d) 20 ns, and (e) 24 ns. Scale shows the amplitude of the electric-field z-component.

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leading edge of the Ez wavefront, which means that the waterfilled borehole decreases the current speed along the antenna. The radiation process is generally the same as that in case 1 (Figure 7), but the shapes of some wavefronts are different. The Ez component of W2 now appears as a distorted circle with a narrow tip pointing in the z-direction. Additionally, wavefronts W1 , W2 , W3 , and W4 appear later than their counterparts in case 1 because the wave velocities inside the borehole for case 2 differ from their no-borehole values in case 1. In case 3, the speed of the electric field in the air-filled borehole is greater than it is outside the borehole. From the simulation, the quantity τa is estimated to be about 1.84 ns. The corresponding wavefronts appear sooner than in cases 1 and

We computed the radiated waveforms at a distance of 8 m = 3 ∼ 4 λ, where λ is the wavelength of the pulse for εr2 = 5.8. Here, the wavelength is defined as the product of one cycle of pulse width and the velocity of the medium. The

Figure 8. Radiation from a dipole antenna at five different times for a water-filled borehole at (a) 6 ns, (b) 11 ns, (c) 16 ns, (d) 20 ns, and (e) 29 ns. Scale shows the amplitude of the electric-field z-component.

Figure 9. Radiation from a dipole antenna at five different times for an air-filled borehole at (a) 6 ns, (b) 11 ns, (c) 16 ns, (d) 20 ns, and (e) 29 ns. Scale shows the amplitude of the electric-field z-component.

2 (Figure 9). Wavefront W2 returns to the feed point more quickly, as shown in Figure 9b, but its shape is similar to that in case 1. The state of the radiation at 29 ns is shown in Figure 9e. In this case, the electric field near the antenna is weaker than in earlier instances.

Radiated waveforms


normalized waveforms for the electric field Eθ (with components Ez and Ey ) at various elevation angles θ are shown in Figure 10 for all three cases. In each case, the pulse arrives at about 60 ns; the amplitudes in various directions are different, and the largest is in the normal direction. In going from one case to another, the radiated waveforms are different even in the same direction, as shown at θ = 30◦ . Since the width of the exciting pulse is longer than τa , the different wavefronts are often superimposed both transiently and spatially and result in radiated waveforms that differ from one case to another. We cannot distinguish between the wave components that arrive from the source location of the feed-point radiation or from the end-point reflections because the duration of the pulse is not sufficiently short. The superposition of different wavefronts often causes strong ringing (e.g., Chen et al., 2002, Figure 6; Ernst et al., 2003, Figure 3) for a nonresistively loaded dipole. However, we observe very little resonance in the waveforms despite the lack of resistive load. This is explained at the end of the next section.

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tenna. Therefore, the water-filled borehole lowers the input impedance. Comparing Figure 11 to Figures 4 and 5, we find the sharp resonances seen in the latter two figures disappear for the naturally loaded cases. This means that different frequencies have different retarding forces, so a natural dispersion is taking place without any need for distributed resistors along the antennas. It appears that the natural dielectric load effectively damps the radiated waveforms. A comparison between the case 1-(non-borehole) result and those of the air- and waterfilled boreholes shows that different surrounding materials have different reactive forces at different frequencies.

Input impedances The input impedance of the antennas strongly affects the antenna’s resonant frequency and the radiation. The resonant frequency is that for which the reactance equals zero or is near zero. We consider dielectric constants that vary from 4.5 (quartzite) to 5.8 (salt) and 9.0 (marble). The calculated input impedances for cases 1–3 are shown in Figure 11. For quartzite (left column), resonant frequencies A1 –A2 are 85 and 130 MHz (case 1), B1 are 77 MHz (case 2), and C1 –C2 are 106 and 155 MHz (case 3). For salt (middle column), resonant frequencies A1 –A2 are 78 and 123 MHz (case 1), B1 are 75 MHz (case 2), and C1 –C2 are 102 and 155 MHz (case 3). For marble (right column), A1 –A2 are 69 and 114 MHz (case 1), B1 –B2 are 67 and 81 MHz (case 2), and C1 –C2 are 100 and 153 MHz (case 3). A comparison among A1 , B1 , and C1 shows that the waterfilled borehole (case 2) lowers the first resonant frequency while the air-filled borehole (case 3) raises it, no matter whether the formation is quartzite, salt, or marble, because the speed slows in the water and accelerates in the air relative to any speed in the rock without the borehole. Therefore, the resonant frequency is changed for the same antenna from the water-filled borehole to the air-filled borehole. Consequently, a trade-off may be necessary to design an antenna that can achieve relatively good performance in both waterfilled and air-filled boreholes. The impedance in the water-filled borehole is always relatively low. Input impedance is determined mainly by the structure of the antenna feed point and the material nearby. The input impedance of an antenna can be expressed as the complex power divided by the square of the current at its feed point (Galejs, 1969, equation 1.29). The complex power radiated by the antenna can be computed by integrating the Poynting vector over a surface that encloses the antenna. Qualitatively, both the electric and magnetic fields near the antenna decrease as the dielectric constant increases; therefore, so does the power near the antenna. This analysis can easily be reached from the formulation for an infinitesimal an-

Figure 10. Simulated waveforms in different directions 8 m away from the antenna: (a) case 1, (b) case 2, and (c) case 3.

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Radiation patterns

Figure 11. Calculated input impedance for three cases of dielectric constants: from 4.5 (left column, quartzite) to 5.8 (middle column, salt) and 9 (right column, marble). Top figures are for the resistive component; bottom figures are for the reactive.

We calculated the radiation amplitude directivity patterns for two frequencies within the pulse spectra at a range of 8 m (Figure 12). First we Fourier transformed the transient signals in different directions into the frequency domain and then normalized the amplitudes by the maximum amplitude for a certain frequency. Again, we considered εr2 = 4.5, 5.8, and 9.0. The corresponding theoretical radiation patterns for wire antennas with the same length in homogeneous media εr2 (thick line) are given for comparison in the top row. Here, a wire antenna refers to a dipole antenna consisting of two straight, collinear, infinitely thin conductors of equal length, separated by a small feeding gap. The large differences between our results and those for wire antennas probably exist because our antennas are not wires and our calculations are not for the far-fields, which are used for the theoretical patterns of wire antennas. The calculated radiation patterns at 69 MHz for cases 1–3 (solid line, Figure 12) are similar for any surrounding formations. At 122 MHz (dashed line, Figure 12), the side lobes are more dominant for case 2 regardless of the surrounding formation. This means that a waterfilled or air-filled borehole does not change the radiation pattern of an antenna at the lower frequency, and the antenna exhibits the characteristics of the half-wave dipole. However, a water-filled borehole does change the radiation pattern at the higher frequency because the wavelength is much shorter than the dipole length and the current distribution on the antenna is complicated. Additionally, as the formation εr2 becomes greater, the side lobes become larger at 122 MHz for case 2 because the wavelength becomes much shorter than the antenna in the medium.

CONCLUSIONS

Figure 12. The calculated radiation patterns at 8 m at 69 MHz (solid line) and 122 MHz (dashed line): (top) case 1, (middle) case 2, and (bottom) case 3. Rocks have dielectric constants that vary from 4.5 (left column, quartzite) to 5.8 (middle column, salt) and 9 (right column, marble). The thicker solid and dashed lines of the top figures show the corresponding asymptotic far-field radiation patterns of a wire antenna with the same length (Stutzman and Thiele, 1998, equations 5 and 6.)

The transient radiation of a nonresistively loaded dipole antenna in a borehole filled with water or air was simulated using FDTD. The numerical propagation simulations suggest that the natural dielectric loading of the surrounding material decreases the pulse speed along the antenna and damps the current pulse. Therefore, for a practical resistively loaded antenna used in a fluid-filled borehole, both antenna resistance and natural dielectric loading damp the radiated waveforms and


limit any end reflections. The loading itself can be quantified in a complex impedance analysis at the antenna feed point, which would provide important design procedures for these antennas.

ACKNOWLEDGMENTS We are very grateful to assistant editor Jose´ M. Carcione and associate editor Steve Arcone, who reviewed our manuscript very patiently and gave us many useful opinions. We thank two anonymous reviewers for many constructive comments and suggestions.

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