Effects of tool eccentralization on cement-bond-log measurements: Numerical and experimental results

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1 GEOPHYSICS, VOL. 78, NO. 4 (JULY-AUGUST 2013); P. D181 D191, 19 FIGS., 1 TABLE /GEO Effects of tool eccentralization on cement-bond-log measurements: Numerical and experimental results Ruo-Long Song 1, Ji-Sheng Liu 2, Xiu-Mei Lv 2, Xiu-Tian Yang 3, Ke-Xie Wang 1, and Liang Sun 2 ABSTRACT The cement-bond log (CBL) is a conventional and widely used cement quality evaluation technology for vertical wells. With the increase in horizontal wells around the world, the existing cement evaluation technologies are not appropriate. We have explored the possibilities of utilizing CBL in horizontal wells through investigating the effects of a noncentralized tool on CBL measurements. The parallel finite-difference numerical simulation method and experiments in calibration wells were adopted in the study. The numerical and experimental results matched very well, and indicated that the CBL amplitude decreases linearly with increasing tool eccentricity in a well with free pipe (i.e., a cased but uncemented well). For a standard pipe with a diameter of 5.5 in (139.7 mm) and a thickness of 7.72 mm, an eccentricity of 1 4 in(17% of the maximum eccentricity) could cause the CBL amplitude to be reduced by about 20%. The numerical simulations of CBL in wells with fluid channels in the cement showed that tool eccentralization could either increase or reduce the CBL amplitude relative to a centered tool, depending on the channel azimuth relative to eccentered direction. To explain this phenomenon, we investigated numerically the polarizations of casing waves in a well with free pipe and in a well with a fluid channel, and casing waves at higher frequencies in a well with free pipe. The relationship between the CBL amplitude and the percentage of cemented area for a borehole-centered tool was also studied. Our results provided some insights into understanding CBL measurements in horizontal wells. INTRODUCTION In recent decades, due to the rapid development of horizontal drilling technology, there are more and more horizontal wells around the world. However, the existing cement evaluation technologies, which are designed for vertical wells, would give misleading results if used directly in horizontal wells (Clavier, 1991). The cement-bond log (CBL) is a conventional cement quality evaluation technology. The CBL is an efficient aid in estimating cement-bond quality, and has been found to be reliable, if properly run and interpreted (Fertl et al., 1974). Many new bond-logging tools, such as the radial bond tool (RBT), the radius incremented bond (RIB) tool, and the sector bond tool (SBT), can also perform CBL measurements. New ultrasonic tools that cannot perform CBL measurements, such as the cement evaluation tool (CET), are usually run with CBL on the same trip in the well (Perry and Henry, 1986). As a result, the CBL is now widely used in oil fields around the world. Thus, it is meaningful to explore the interpretation method of the CBL in horizontal wells. The primary problem with the CBL in horizontal wells is that the tool does not center due to gravity. This eccentricity could be overcome with strong centralizers but they would increase the logging difficulty. The other logging tools mentioned above also exhibit the same problem when they are used in horizontal wells. Therefore, the effect of tool eccentralization in horizontal wells should be considered. Schlumberger (1976) recognizes this problem, and investigates the effects of tool eccentralization on CBL amplitude. Smolen (1996) presents the research results in his monograph, which shows that the CBL amplitude decreases exponentially with increasing tool eccentricity. For a standard 5.5 in pipe, a tool eccentricity of only 1 4 in(6.35 mm) is enough to cause the CBL amplitude to be reduced by 50% from what it should be based on the cemented condition. Jutten and Parcevaux (1987) Manuscript received by the Editor 22 October 2012; revised manuscript received 4 February 2013; published online 3 June Jilin University, College of Physics, Changchun, China. songrl@jlu.edu.cn; wangkx@jlu.edu.cn. 2 Daqing Oilfield Company Ltd., Daqing Logging & Testing Services Company, Daqing, China. dlts_liujs@petrochina.com.cn; dlts_lvxm@ petrochina.com.cn; dlts_sunliang@petrochina.com.cn. 3 Daqing Drilling and Exploration Engineering Company, Daqing Drilling Engineering Technology Research Institute, Daqing, China. yangxiutian@ cnpc.com.cn Society of Exploration Geophysicists. All rights reserved. D181

2 D182 Song et al. study experimentally the relationship between the CBL output and cement thickness, the casing stand-off, the percentage of cemented area and the shape of the uncemented channels, etc., but for a borehole-centered tool. Hou et al. (2012) present the experimental studies on CBL output in calibration wells with low-density slurry. In this paper, we investigate the effect of tool eccentralization on CBL amplitude in a well with free pipe using the parallel finite-difference (FD) method (Song et al., 2010, 2012). We obtain a linear relationship between CBL amplitude and tool eccentricity. This linear relationship is verified by three different experiments in calibration wells. This paper also presents the numerical results of CBL obtained with borehole-eccentered tools in wells with fluid channels in the cement. FINITE DIFFERENCE SCHEME The 3D stress-velocity time-domain FD scheme in Cartesian or cylindrical coordinates has been used extensively in the simulation of acoustic logging in boreholes (Cheng et al., 1995; Liu et al., 1996; Chen et al., 1998; Liu et al., 2003). Song et al. (2010, 2012) use this FD scheme in cylindrical coordinates in numerical simulation of high-frequency acoustic logging measurements in cased holes. In this paper, we adopt the same parallel FD scheme presented by Song et al. (2010, 2012). Figure 1 shows a cross-sectional view of a 3D FD model of CBL logging in a well with free pipe (i.e., a cased but uncemented well). The model is a hollow cylinder and is homogeneous in the z-direction. The logging tool is treated as a rigid cylinder, so the model is hollowed out. A rigid cylinder is a fair approximation for the logging tool due to the large difference in acoustic impedance between steel and water. With a rigid cylinder replacing a steel cylinder in the model (i.e., computational domain), the singularity of r ¼ 0 in the wave equations is avoided, and the waves along the tool are also eliminated. In the model, the z-axis in cylindrical coordinates coincides with the axis of the tool, which intersects the cross section at O 1,as shown in Figure 1. The casing is always centered in the borehole, whose axis intersects the cross section at O 2. The casing and borehole are eccentered in the model. Thus, the distance d between O 1 and O 2 is equivalent to the eccentricity of the borehole-eccentered tool. The tool is always eccentered downward because of the action of gravity. The downward direction, i.e., direction of O 2 O 1, is taken as a reference for the azimuth (θ ¼ 0 ). The model is truncated with nonsplitting perfectly matched layers (NPML), as presented by Wang and Tang (2003), in the radial and upper axial directions to eliminate artificial reflections. The lower axial boundary located at the central plane of the cylindrical source is set to be a symmetrical boundary instead of NPML for the consideration of computation efficiency. The attenuation coefficient of NPML is chosen the same as that in Wang and Tang (2003), except for α ¼ The radii of tool, casing, and borehole, as shown in Figure 1, are r 0 ¼ 25 mm, r 1 ¼ 62.5 mm, r 2 ¼ 70 mm, r 3 ¼ 100 mm, respectively. The minimum thickness of the formation shell (not including NPML) is set to be 20 mm. so r 4 ¼ r 3 þ d þ 20 mm, r 5 ¼ r 4 þ 50 mm. The size of the computational domain at z-direction is 1600 mm. The NPML is 50 mm thick in the radial direction and 100 mm thick in the axial directions. Thus, the thickness of NPML is about a quarter of the formation compressional wavelength at the central frequency in radial direction. The acoustic parameters are listed in Table 1. Unless stated otherwise, all the geometric parameters and acoustic parameters in FD simulation are as described above. The FD scheme is a second-order central-difference formula in both spatial and temporal domains via staggered grids. A unit cell of the staggered grid is shown in Figure 2. The grid spacings are set to be Δr ¼ 2.5 mm, Δθ ¼ 1.5, Δz ¼ 5.0 mm, and the time step is Δt ¼ s. Then, following Liu et al. (2003), stress and velocity components are defined at discrete space and time domains. Take radial velocity component V r for example, V nþ1 2 rði;jþ1 2;kþ1 2Þ ¼ V rðiδr; ðj þ 1 2ÞΔθ; ðk þ 1 2Þ Δz; ðn þ 1 2ÞΔtÞ: (1) As described above, the casing thickness is 7.5 mm, and the radial grid spacing is 2.5 mm. So, the casing has three cells in the thickness direction. After discretization by cylindrical grid, the eccentered circular casing becomes a rough ring with staircase. Shown in Figure 3 is an example of discretization of casing and borehole with an eccentricity of d ¼ 20 mm. The staircase approximation of the casing and borehole will introduce error and instability in the simulation. We reduce the error and instability by using a material averaging scheme presented by Liu et al. (1996). These Table 1. Acoustic parameters for all the models in this paper. Figure 1. A schematic cross-sectional view of the 3D finite difference model of CBL-VDL logging in a well with free pipe. The z-axis in cylindrical coordinates coincides with the axis of the tool. In a well-bonded well, the water between casing and formation is replaced by cement annulus bonded to both sides. In a well with a channel, there is a sectoral water-filled channel in the cement annulus (see also Figure 12). Layer ρ (kg m 3 ) V P (m s) V S (m s) Water Casing Cement Formation

3 Cement-bond-log in horizontal wells D183 problems could be further resolved by using finer radial grids or an irregular radial grid, which is currently under investigation. The material averaging scheme does not change the material property of a cell. A casing cell that is adjacent to a water cell would not be treated as water because the density and Lamé coefficients are all located at the center of a cell. The material averaging scheme just guarantees the correct boundary conditions, i.e., the zero-shear stress components, at an interface between casing and water. The Dirichlet boundary condition is imposed on the radial velocity component (V r ) at the cylindrical surface of the rigid tool (i ¼ 10), as shown in Figure 3. First, the center of the cylindrical source which has a height of 40 mm, is located at z ¼ 0. Thus, the V r that are located at discrete positions (i ¼ 10, 1 k 4, 1 j 240), are defined as V nþ1 2 rð10;jþ1 2;kþ1 2Þ ¼ fððn þ 1 2ÞΔtÞ: (2) Variable fðtþ is the time dependence of the source, which is taken as a modification of Tsang-Rader source function (Tsang and Rader, 1979), fðtþ ¼4α 2 t 2 e αt sin ω 0 t; (3) where ω 0 ¼ 2πf 0, f 0 ¼ 20 khz is the central frequency and α ¼ ω is the parameter of pulse width in the time domain. The V r that are located at the remainder of the rigid surface (i ¼ 10, k 5, 1 j 240), are set to zero. Comparing with Song et al. (2012), the FD codes are improved. Figure 4 gives two examples of waveforms obtained from FD method and real axis integration (RAI) method (Tubman et al., 1984; Schmitt and Bouchon, 1985; Dong and Wang, 1985). The tool is centered, i.e., d ¼ 0. It can be seen that the full waveforms obtained from the two methods match each other quite well. The discrepancy in S-waves and Stoneley waves might be caused by high-frequency ringing of the FD method. NUMERICAL SIMULATION OF ECCENTERED CBL IN A WELL WITH FREE PIPE This section presents the effect of tool eccentralization of the CBL for a well with free pipe. The CBL tool typically has a single omnidirectional acoustic transmitter and a 3-ft (1-m) receiver. There is always a 5-ft (1.5-m) receiver for variable density display, which is called a variable density log (VDL). The VDL result is also presented in this section. The model is as shown in Figure 1. First, we investigate the nonaxisymmetric acoustic field caused by tool eccentralization. Figure 5 shows the waveforms obtained at the 3-ft point receivers located at different azimuths for a d = 10-mm eccentricity. There are 12 circumferentially equispaced point receivers around the tool. Seven waveforms are shown in Figure 5 because the acoustic field has plane symmetry about the vertical plane. The sketch shows the configuration of the seven point receivers (black dots). The waveforms are cut off after 0.35 ms to highlight the first arrivals of casing waves. The CBL amplitude is a measure of the amplitude of a specific wave, mostly the first positive or negative arrival of the wavetrain to minimize contamination caused by arrivals of slower modes. In this regard, we focus on the first positive arrival (FPA) of casing waves. It can be seen that the FPA amplitude measured at azimuth 0, which is closest to the casing, has the smallest amplitude and the shortest traveltime. This phenomenon is explained by the following arguments. For the receiver closer to the casing, there is a shorter acoustic path, which implies a shorter traveltime. On the other hand, Figure 2. A unit cell of the finite-difference staggered grid, where τ αβ denotes stress components, and V α denotes velocity components, where α, β ¼ r, θ, z. The density (ρ), Lamé coefficients (λ, μ) and normal stress components τ αα are all located at the center of the unit cell. (modified from Liu et al., 2003). Figure 3. A cross-sectional view of the discretization of an eccentered circular casing and borehole by cylindrical grids. The eccentricity is d ¼ 20 mm. The view zooms in on a quarter of the ring to present a clear view of the staircase caused by the eccentricity. Each small rectangle indicates a unit cell. The black rectangles indicate casing and formation cells, and the remaining rectangles indicate water cells.

4 D184 Song et al. each received waveform is a summation of waves that come from a circular casing. One can imagine that a receiver closer to the casing will receive waves with larger phase difference, which implies a smaller summation. The first negative arrival (FNA) and the other later arrivals show complex behavior due to interference effects and are hard to explain. Figure 4. Comparisons of full waveforms obtained from the FD method and the RAI method in an axisymmetric (a) well with free pipe, and (b) a well-bonded well. The tool is centered. Figure 5. Casing waves acquired at 3-ft point receivers at different azimuths in a well with free pipe. There is a d ¼ 10 mm eccentricity. The sketch shows the configuration of point receivers (black dots). The formation is omitted in the sketch. FPA and FNA indicate the first positive arrival and first negative arrival, respectively. There are 240 grid points at the θ direction, so 240 waveforms can be obtained at r ¼ r 0, z ¼ 3ft. Only seven of them are shown in Figure 5. CBL measurement presents a circumferential average result of cement quality. We take the average of the 240 waveforms as the waveform acquired by a 3-ft cylindrical receiver. The same average is applied to waveform obtained at the 5-ft cylindrical receiver. Figure 6 shows the average acoustic pressure waveforms at 3-ft and 5-ft cylindrical receivers obtained for various values of tool eccentricity. The eccentricity d is listed next to the corresponding waveforms. Eccentricity d ¼ 0 indicates the borehole-centered tool case. The waveforms from 0.21 to 0.6 ms in Figure 6a, and the waveforms from 0.3 to 1.0 ms in Figure 6b, are both casing arrivals. Though the source pulse has only three cycles, the casing waves have a very long tail. This tail is due to dispersion and reflection of the first casing arrival by the rigid tool and then by the inner surface of the casing. It can be seen from Figure 6 that the FPA amplitude decreases with the increasing eccentricity. The FPA amplitude in a fixed period ( ms, as shown in Figure 6a) at the 3-ft waveform is measured as the CBL amplitude. The extracted CBL amplitudes are shown in Figure 7. All the CBL amplitudes are normalized by the amplitude corresponding to the centered tool. Figure 7 indicates that the CBL amplitude decreases linearly with increasing eccentricity. An eccentricity of 1 4 in (17% of the maximum eccentricity) causes the CBL amplitude to be reduced by about 20%. It is less than the reduction by 50% presented in Smolen (1996), which was obtained for the same case of the problem, except that a point source and a point receiver were used instead of a cylindrical source and a cylindrical receiver. The source and receiver used in this paper are more similar in geometry to the CBL tool. Thus, the result we obtained is more practical, and is verified by experimental results in the following section. The absolute amplitudes of the FNA extracted from the fixed period ( ms) show similar behavior, though they decay more slowly than those of the FPA amplitude with increasing eccentricity. As mostly used in field CBL, we define amplitude of the FPA to be the CBL amplitude. This last statement is an important definition that will be assumed throughout the remainder of this paper. We further investigate the relationships between the CBL amplitude and tool eccentricity for other sizes of casing and show them in Figure 8. Similar linear relationships are obtained for a thicker casing and a larger casing. We define the percentage of the tool eccentricity to be d ðr 1 r 0 Þ. Figure 8 shows that the same percentage of the tool eccentricity for different sizes of casing causes the CBL amplitude to be reduced by roughly the same amount. Figure 9 shows the variable density display of the waveforms shown in Figure 6b. It indicates that tool eccentralization is often accompanied by shortening the of traveltime of 5-ft receiver signal. The traveltime is measured using a preset amplitude detection level, which is set to be 2% of the FPA amplitude measured for the centered tool. The detected traveltime is listed on the left side of Figure 9. For eccentricity larger than 10 mm, the shortening of traveltime is visible on the VDL display, so it can be used as an indicator of tool eccentricity. A fast formation signal may arrive earlier than the casing signal. However, the formation signal indicates a good cement bond, and the shear waves coming from the formation will be seen on the VDL display. Conversely, the presence of a casing signal

5 Cement-bond-log in horizontal wells D185 indicates a bad cement bond and the shear waves from the formation will be very weak and not visible on the VDL display. From Figure 6, we can also see that the black strips (i.e., positive peaks) of casing waves are equal in width for the borehole-centered tool; however, they become irregular and their amplitudes decrease as the tool becomes more and more eccentered. For the experiment, the centralizers of the SBT were removed and replaced with a pair of tailor-made disks. The disks were made of steel plate, and had a diameter matching the inner diameter of the casing. A hole with the same diameter as that of SBT was punched in a disk to hold the SBT. The eccentricity was determined by the distance between centers of the hole and the disk. Five pairs of disks EXPERIMENTS IN CALIBRATION WELLS To verify the linear relationship between the CBL amplitude and tool eccentricity shown in Figure 7, we performed three experiments in different calibration wells using different methods and tools. Experimental methods The first experiment (experiment 1) was performed with a laboratory CBL tool, coupled to an oscilloscope and recording unit. The laboratory CBL tool has a transmitting transducer and a 3-ft receiving transducer. Both transducers have a diameter of 50 mm and a height of 40 mm. The operating frequency of this laboratory CBL tool is 20 khz. The transducers and circuit are sealed and mounted in a grooved steel shell, which has a length of 1.5 m and a diameter of 72 mm. The calibration well is a cased hole with casingcement interface unbounded. A thin coating of slurry is applied to the outer surface of the casing. After the thin coating has dried, the well is cemented. Thus, there is a microannulus at the casing-cement interface. The casing is filled with water. The casing has an outer diameter of 5.5 in (139.7 mm) and a thickness of 7.72 mm. The formation is made by stacking three sandstone cores. Each core has an inner diameter of 228 mm, an outer diameter of 428 mm, and a height of 500 mm. During logging, the CBL tool is suspended by a wire rope to ensure that it is vertical. The calibration well is placed vertically on the floor. The eccentricity is controlled by displacing the wire rope horizontally, about 3 mm at a time. After each move, once the waveform on the oscilloscope is stable, the waveform data are recorded. The tool is long enough so part of it is out of the casing during logging, and the minimum distance between tool and casing is measured by a vernier caliper such that eccentricity can be calculated. We read the FPA amplitude as the CBL amplitude. Experiment 2 is the same as experiment 1, except that the calibration well is a well with free pipe (uncemented cased hole). The well has drilling mud in the casing-formation annulus. The drilling mud has become jelly-like since it was put into the well two years before the experiment. Experiment 3 differs from the other two experiments, namely, the tool is an SBT. The SBT is combined with CBL-VDL. This tool has a diameter of 70 mm. The operating frequency of CBL-VDL of this tool is also 20 khz. The calibration well is also a well with free pipe. The well has water in the casing-formation annulus. The casing and formation are the same as in experiment 1. The casing is also filled with water. Figure 6. Average acoustic pressure waveforms at the 3-ft and 5-ft cylindrical receivers obtained for various values of tool eccentricity d in a well with free pipe. Here, d ¼ 0mmindicates borehole-centered tool. Figure 7. The effect of tool eccentralization on CBL amplitude in a well with free pipe. The casing has a thickness of 7.5 mm and an inner radius of 62.5 mm. The CBL amplitudes are extracted from the first positive arrival (FPA) of simulated waveforms, and then normalized by the amplitude obtained for the centered tool. The label of the right axis is the CBL amplitude in millivolts (mv), with the assumption that CBL amplitude obtained for the centered tool is calibrated to 72 mv. The label of the upper axis is the percentage of tool eccentricity, which is calculated by the formula d ðr 1 r 0 Þ.

6 D186 Song et al. were made with different eccentricities (including d ¼ 0mm). In addition, several small holes were punched in each disk to reduce the reflection of acoustic waves and enable the water to flow past when the tool was put in the casing. The disks were installed at two ends of the tool, so they would not affect the CBL amplitude. The gated period was set to detect the FPA of the casing waves. Experimental results Figure 8. Same as in Figure 7, except that the casing has a thickness of 12.5 mm and an inner radius of 62.5 mm for (a), and the casing has a thickness of 7.5 mm and an inner radius of 82.5 mm for (b). The three experimental results are shown in Figure 10. They all show that CBL amplitude decreases linearly with increasing eccentricity. Experiments 1 and 2 use the same tool and the two curves are approximately parallel. For the same eccentricity, CBL amplitude in the experiment 2 is always about 10 mv less than that in the experiment 1 because the drilling mud in experiment 2 has become jelly-like and should have a small shear modulus. The curve obtained from the experiment 3 has a slightly larger gradient than those obtained from the experiments 1 and 2. The difference in acoustic sonde structure might account for this phenomenon. For all the three experiments, the casing had a thickness of 7.72 mm and an inner radius of mm; the transducers have a diameter of 50 mm, a height of 40 mm, and a central frequency of 20 khz. Hence, the problem geometries in the numerical simulations shown in Figure 7 are very close to those of the three experiments, except for borehole radius, which has negligible effect on CBL amplitude. It is very hard to adopt the exact parameters in numerical simulations as in the physical experiments because of the constant grid spacing. To compare the experimental results shown in Figure 10 with the simulated results shown in Figure 7, the experimental data were normalized and replotted in Figure 11. It can be seen that the agreement between experimental results and simulated results is very good. Experiments 2 and 3 are in wells with free pipe, though they use different tools, different methods, and different materials Figure 9. The variable density display of waveforms shown in Figure 6b. The waveforms are normalized by the FPA amplitude obtained for the borehole-centered tool. Then the magnitude of amplitude is shown by five-color grayscale. White for (-, 0.02), light gray for (0.02, 0.25), gray for (0.25, 0.5), dark gray for (0.5, 0.75), and black for (0.75, -), respectively. Figure 10. Experimental results of CBL in wells with free pipe for borehole-eccentered tool. Experiment 1 is performed in a calibration well with the casing-cement interface unbounded. Experiment 2 is performed in a free pipe calibration well with drilling mud in casing-formation annulus. Experiment 3 is performed in a free pipe calibration well with water in casing-formation annulus. Experiments 1 and 2 use a laboratory CBL tool. Experiment 3 uses an SBT in service.

7 Cement-bond-log in horizontal wells D187 outside the casing, the results matched perfectly. The CBL amplitude is confirmed to be in linear relationship with tool eccentricity. NUMERICAL SIMULATION OF ECCENTERED CBL IN WELLS WITH CHANNELS When the cased hole has a fluid channel in the cement annulus, the effects of tool eccentralization on CBL amplitude can be more complex. Figure 12 shows a schematic cross-sectional view of the 3D FD model of CBL logging in a well with a fluid channel in the cement, i.e., a cased and not completely cemented well. The model is also axially homogeneous. The cement is bonded to the casing and the formation. There is a sectoral water-filled channel in the cement annulus. The channel has an angular size θ S, an azimuth θ A, and the same thickness as the cement. The down direction is also taken as a reference azimuth. It should be emphasized that the vertex of angles θ S and θ A coincide with the center of the casing. In Figure 13 we show the CBL amplitude versus eccentricity in wells with channels. One can see that for a channel with a certain size, the CBL amplitude is sensitive to both tool eccentricity and channel azimuth. For example, for θ S ¼ 120, an eccentricity of d ¼ 5mmreduces the CBL amplitude by 56% and 13% for channel azimuths θ A ¼ 0 and θ A ¼ 90 from the CBL amplitude obtained for the centered tool. However, an eccentricity of d ¼ 5mmcauses the CBL amplitude to increase by 41% for θ A ¼ 180 from that obtained for the centered tool. When d ¼ 10 mm, the difference in CBL amplitude caused by channel azimuth is maximal. If d is large enough (>17 mm), the eccentralization will always cause a CBL amplitude reduction with respect to that obtained for the centered tool, whatever the channel size and azimuth are. In addition, the effect of tool eccentralization on the CBL amplitude extracted from FNA shows similar behavior and is not presented. The most important point is that, unlike the well with free pipe, tool eccentralization can either reduce or increase the CBL amplitude in the presence of channeling. How does this phenomenon happen? To explain this, in Figure 14 we plot waveforms corresponding to casing waves acquired at 3-ft point receivers at different azimuths in wells with channels. In Figure 14a, the tool is centered, a θ S ¼ 90 channel is at azimuth θ A ¼ 0, and the casing waves acquired close to the channel have short traveltime. The FPA amplitudes are all equal; however, the FNA and the second positive arrival close to the channel have small amplitude. This phenomenon will be covered in the following section. Remember that when the tool is eccentered, the FPA measured for the receiver on the close side to casing has short traveltime and small amplitude as shown in Figure 5. When the tool eccentralization and channeling coexist, and they are in the same direction, the FPA measured for the receiver close to the casing and channel should have shorter traveltime, as shown in Figure 14b. The interaction of the two nonaxisymmetric factors increases the phase difference of FPA measured for the receiver at different azimuths. Thus, the average acoustic pressure of FPA at the 3-ft cylindrical receiver will weaken. Conversely, when the tool eccentralization and channeling coexist, but they are in the opposite direction, as shown in Figure 14c, the phase difference of FPA measured for the receiver at different azimuths is reduced. As a result, the average acoustic pressure of FPA at the 3-ft cylindrical receiver will be enhanced. AN INSIGHT INTO THE CASING WAVE Casing waves in a well with a channel shown in Figure 14a are hard to understand. It is traditionally understood that the casing waves measured for the receiver close to the channel should be strong, and it is the basis of SBT (Song et al., 2012). The casing waves received by the receivers located at the tool surface radiate from the casing. In this section, to better understand casing waves, we investigate particle vibration inside the casing, and also the obtained casing waves at higher frequencies. Casing waves in a well with free pipe The casing wave is a kind of guided wave. Casing waves excited by a borehole-centered CBL tool in a well with free pipe are Figure 11. The effect of tool eccentralization on normalized CBL amplitude in a well with free pipe: comparison between experiments and numerical simulation. Figure 12. A schematic cross-sectional view of 3D finite difference model of CBL in a well with a channel. The model is axially homogeneous. The cement is bonded to casing and formation. There is a sectoral water-filled channel in the cement annulus. The channel has an angular size θ S, an azimuth θ A and a thickness same as the cement.

8 D188 Song et al. basically the axisymmetric casing extensional mode. When the casing wave propagates along the casing, i.e., in the z-direction, the vibration of a particle in the casing should have z and r components, which are always coupled. Figure 15 shows the waveforms of the radial velocity component (V r ) and axial velocity component (V z ) of two points inside the casing in a well with free pipe, one at 3 ft and another at 5 ft from the transmitter (black point in the sketch). The tool is centered, so the wavefields are axisymmetric. It can be seen that the axial component dominates as the casing waves pass. There is a phase difference between V r and V z, so the polarization of casing waves is elliptic. In a well with free pipe, the casing is surrounded by water such that only radial vibration radiates energy into both sides. However, from the 3-ft and 5-ft waveforms shown in Figure 15, it can be seen both V r and V z attenuate. This phenomenon indicates that the two vibration components are coupled. In a well-bonded well, both radial and axial vibrations radiate energy into the cement. The casing waves inside the casing attenuate rapidly, hence are nearly invisible in the 3-ft full waveform. As shown in Figure 16a, inside the casing adjacent to the channel, axial vibration does not radiate energy into either side of the casing, and thus is much stronger than that inside the casing adjacent to the cement. The wavefront in Figure 16a is a curve as shown by the dashed line and is not a straight line. It implies two things. First, the casing waves inside the casing adjacent to the cement attenuate so rapidly that they can not arrive at the 3-ft receivers. Second, the casing waves inside the casing adjacent to the channel not only radiate energy into the inner and outer sides but also radiate energy in the circumferential direction, while they are propagating along the casing. As a result, for a θ S ¼ 90 channel, the CBL amplitude is 5% of that in a well with free pipe (see Figure 13), though the percentage of uncemented area is 25%; for a θ S ¼ 120 channel, the CBL amplitude is 9% of that in a well with free pipe, though the percentage of uncemented area is 33.3%, as shown in Figures 13 and 17. In Figure 17, we present the numerical results of the relationship between CBL amplitude and the percentage of uncemented Casing waves in wells with channels In the presence of a water-filled channel, the casing wave is not axisymmetric, hence the three vibration components (r, θ, and z) are coupled. Here, we still focus on the r and z components. Figure 16 shows the snapshots of V z (a) and V r (b) of the casing, and a snapshot of pressure at the surface of the tool (c). Each figure is an unfolded view of a cylindrical surface. The cylinder radius is r ¼ ðr 1 þ r 2 Þ 2 for Figure 16a and 16b, and r ¼ r 0 for Figure 16c (as defined in Figure 1). The time of the three snapshots is at 0.24 ms. The well is the same as that shown in Figure 9a. The tool is centered. There is a θ S ¼ 90 channel at θ A ¼ 0. In the following discussion, we focus on the first and the second arrivals of casing waves. The later arrivals are disturbed by reflected casing waves. Figure 13. Effect of tool eccentralization on CBL amplitude in wells with channels. A channel in the cement has an angular size θ S and an azimuth θ A, which are given out in the legend. The CBL amplitude is normalized by CBL amplitude obtained for centered tool in a well with free pipe. The label of right axis is the CBL amplitude in millivolt (mv), with the assumption that CBL amplitude obtained for centered tool in a well with free pipe is calibrated to 72 mv. Figure 14. Casing waves acquired at 3-ft point receivers at different azimuths in wells with channels. The sketches at the left of each drawing show the bond condition and configurations of point receivers (black dots). The formation is omitted in the sketches. The tool eccentricity, and the channel size (θ S ) and azimuth (θ A ) are given in each plot.

9 Cement-bond-log in horizontal wells D189 area in wells with channels for a borehole-centered tool. For comparison, the experimental results from Hou et al. (2012) are also presented in Figure 17. In the experiments, the calibration wells were cemented by cement slurry with a density of 1.8 g cm 3. The acoustic parameters used in the numerical simulation in Figure 17 are shown in Table 1, except that the parameters of cement are ρ ¼ 1.8 g cm 3, V P ¼ 2800 m s, and V S ¼ 1650 m s. The fact that casing waves radiate energy in the circumferential direction can also explain why the casing waves acquired closer to the channel have shorter traveltime in Figure 14a. In Figure 16b, on the contrary to the V z case, the second arrival of V r inside the casing adjacent to the channel has smaller amplitude. We conjecture that the ellipticity of the particle motion changes while they propagate along circumferential direction. We calculate the vibration trajectories of two opposite points by integrating the waveforms of V r and V z, and show them in Figure 18. Figure 18 shows that the vibration of a point close to the channel (at azimuth 0 ) has smaller radial displacement. Only radial vibration radiates casing waves into water, so the point receiver close to the channel measures smaller casing waves. The snapshot shown in Figure 16c Figure 15. Waveforms of radial (V r ) and axial (V z ) particle velocity obtained at 3-ft and 5-ft point receiver inside the casing, in a well with free pipe. The point receiver is the black point in the sketch. The tool is borehole-centered. Figure 17. The numerical and experimental results of the relationship between CBL amplitude and the percentage of uncemented area in wells with channels for a borehole-centered tool. The experimental results are from Hou et al. (2012). Figure 16. Snapshots of axial velocity (a) and radial velocity (b) of the casing, and snapshot of pressure at the surface of the tool (c). Each figure is an unfolded view of a cylindrical surface. Black and white indicate negative and positive amplitude, respectively. The cylinder radius is r ¼ðr 1 þ r 2 Þ 2 for Figure 16a and 16b, and r ¼ r 0 for Figure 16c. The time of snapshot is 0.24 ms. The well is the same as that shown in Figure 14a. Two vertical lines are plotted in Figure 16a and 16b to separate areas adjacent to the channel and cement. Figure 18. Polarization of casing waves inside the casing in a well with a channel. The two points are 3 ft from the source and in opposite azimuth inside the casing (θ ¼ 0 and 180 ). The time ranges from 0 to 0.35 ms. The arrows point the direction of time. The horizontal axis and vertical axis are radial and axial displacements, respectively. The well is the same as that shown in Figure 14a.

10 D190 Song et al. gives out a clear view of pressure at the tool surface. The above discussion gives the explanation of the waveforms shown in Figure 14a. High-frequency casing waves This subsection investigates the casing waves in a well with a channel excited by acoustic sources with higher central frequencies. Figure 19 shows the casing waves obtained at 3-ft point receivers at different azimuths in the same well as that shown in Figure 14a. The central frequency is 40 khz for Figure 19a and 100 khz for Figure 19b. Notice that the central frequency is 20 khz for Figure 14a. In Figure 19, the casing waves acquired at a point receiver close to the channel have a shorter traveltime, the same as those shown in Figure 14a. However, unlike the case with 20 khz excitation, all the casing waves acquired close to the channel have larger amplitude. The higher the central frequency, the more rapidly the casing waves decay versus offset from the channel. For the case with 40 khz excitation, the FPA amplitude at 180 is 43% of that at 0 ; For the case with 100 khz excitation, this ratio is only 15%. The small difference of casing waves amplitude between different azimuths shown in Figure 14a, is due to the diffraction effect because the wavelength of 20 khz casing waves (about 275 mm) is comparable to the circumference of the casing (about 440 mm). It can be inferred that, for sector cement bond logging, high source central frequency means high circumferential resolution of cement bond. In Figure 19b, the primary casing waves (arriving at about 0.21 ms) are separated in time from the reflected casing waves by the rigid tool and then by the inner surface of the casing (arriving at about 0.26 ms). Using the time difference (0.05 ms) and velocity in water (1500 m s), it can be calculated that the distance between receiver and inner surface of the casing is 37.5 mm, which agrees with the model parameters used (r 1 r 0 ¼ 37.5 mm). If several receivers are spaced circumferentially, just like SBT, the tool eccentricity could be obtained using this method. CONCLUSIONS We have simulated the effect of tool eccentralization on CBL amplitude in a well with free pipe using parallel 3D stress-velocity FD scheme in cylindrical coordinates. A linear relationship between the CBL amplitude and tool eccentricity is obtained. This result is verified by three experiments in different calibration wells using different acoustic logging tools. The effect of tool eccentralization on CBL amplitude in wells with channels has also been simulated. It is found that if the tool eccentralization and channeling are in the opposite direction, the CBL amplitude will first increase and then decrease with the increasing eccentricity. However, if they are in the same direction, i.e., the tool is located closer to the channel, the CBL amplitude will always decrease rapidly with the increasing eccentricity. Polarization of casing waves inside the casing is investigated carefully to explain this phenomenon. Presented results can be helpful in understanding CBL measurements in horizontal wells. If properly interpreted, CBL might be a transitional technology in cement bond evaluation in horizontal wells, however, a reliable CBL interpretation requires precise knowledge of the amount of tool eccentricity. Although the tool eccentricity might be estimated from the traveltime at the 5-ft receiver, it is not reliable for eccentricities less than 10 mm. The other acoustic cement bond tools, which need to be centered while logging, will also have the problem of eccentralization. A new generation of cement bond tool for horizontal wells needs to be developed. ACKNOWLEDGMENTS This work was supported by the National Natural Science Foundation of China (Grant nos and ). We thank the four anonymous reviewers and the editors for their earnest help in corrections for language usage and their constructive criticisms. REFERENCES Figure 19. Same problem as in Figure 14a, except that the central frequency f 0 of acoustic source is 40 khz and 100 khz for (a) and (b), respectively. Chen, Y. H., W. C. Chew, and Q. H. Liu, 1998, A three-dimensional finite difference code for the modeling of sonic logging tools: Journal of the Acoustical Society of America, 103, Cheng, N. Y., C. H. Cheng, and M. N. Toksöz, 1995, Borehole wave propagation in three-dimensions: Journal of the Acoustical Society of America, 97, , doi: / Clavier, C., 1991, The challenge of logging horizontal wells: The Log Analyst, Dong, Q. D., and K. X. Wang, 1985, Numerical evaluation and analysis of the acoustic pressure waveform in cylindrical multilayer quasi elastic media: Acta Geophysica Sinica, 28, Fertl, W. H., P. E. Pilkington, and J. B. 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