the wavefield of acoustic logging in a … wavefield acoustic...submitted to geophysical journal...

Submitted to Geophysical Journal International, Private manuscripts only for ERL member

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THE WAVEFIELD OF ACOUSTIC LOGGING IN A

CASED-HOLE WITH A SINGE CASING - PART II: A

DIPOLE TOOL

Hua Wang, Mike Fehler, Earth Resources Lab, MIT

Abstract

The acoustic method, being the most effective method for cement bond evaluation,

has been used by industry for more than a half century. However, the methods used

are almost always focused on the first arrival (especially for sonic logging), which has

limitations. We use a 3 dimensional finite difference method to numerically simulate

the wavefields from a dipole source in a singly cased hole with different cement

conditions. By using wavefield snapshots and dispersion curves, we interpret the

characteristics of the modes in the models. We investigate the effect of source

frequency, the thickness and location of fluid columns on different modes. The dipole

wavefield in a single cased-hole consists of a leaky P (only source frequency of more

than 10 kHz), formation flexural, and also some casing modes. Depending on the

mode, their behavior is sometimes sensitive to the existence of fluid between the

cement and formation and sometimes sensitive to that between the casing and cement.

The formation S velocity can be obtained from the formation flexural mode at low

frequency. However, interference from high order casing modes makes leaky P

invisible and the P velocity determination difficult when the casing is not well

cemented. The bonding conditions do not affect the arrival time of the formation

flexural mode although the dispersion curve for this mode is sensitive to the fluid

thickness when fluid exists only at the interface between casing and cement. However,

the dispersion curve cannot be used to determine the fluid thickness when the fluid

columns are present at other places. The casing F1 mode, which is the fundamental

casing mode with non-cutoff frequency (low frequency limit) and effectively excited

by a low frequency source (2 kHz), is only sensitive to the total fluid thickness in the

annulus between casing and formation. Other casing modes have a cutoff frequency

and can only be excited at the high frequency. Casing mode F5 is the first arrival once

the fluid column exists next to the casing and its arrival time is not sensitive to the

fluid column. The amplitudes of the casing modes F3 and F4 could be used to


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determine the fluid thickness when the fluid column is only next the casing. However,

if there is no fluid column next the casing and the fluid column appears in other places,

the first arrival would not be the casing mode F5 and the arrival time varies with

cement thickness next to the casing. The amplitude of modes F3 and F4 are no longer

sensitive to the fluid thickness next to the casing when the fluid also appears in other

places. Based the results, we suggest a data processing flow for field application,

which will highly improve cement evaluation.

Introduction

Characteristics of wavefields have been shown to be sensitive to material condition

and have thus found great utility in nondestructive testing. Wavefield characteristics

have been useful for evaluating the structure of underwater cables, detecting the

locations of leaks in oil and chemical pipelines, and evaluation of cement quality in

cased wells. The acoustic method is the most effective technology for cement bond

evaluation in cased wells and is used to ensure hydraulic isolation between reservoir

layers and aquifers thus guaranteeing production efficiency and safety.

Current acoustic methods for wellbore cement evaluation include sonic and ultrasonic

logging. For the sonic method, most studies are focused on monopole measurements,

such as CBL/VDL (Cement Bond Log/Variable Density Log) (e.g. Pardue et al., 1963;

Walker, 1968) and segmented bond logging (e.g. Tyndall, 1990). These methods can

only evaluate the bonding condition of the interface next to the first casing and have

limitations for evaluating other interfaces such as the one between the cement and

formation. The wavefield from a monopole source in a singly cased wellbore with

different bonding conditions has been evaluated and discussed in a companion paper

to this one (Wang and Fehler, 2017). Further development of cement bonding

evaluation will be facilitated by investigating other methods such as those that use

dipole sources.

There are few studies of dipole measurements in cased holes. Schmitt (1992) focused

on the formation shear velocity measurement in a cased hole. Pampuri et al. (2003)

evaluated the formation compressional wave from the leaky P in the dipole wavefield

in a free casing model. Their results are also reported in Chen et al.(2007). Zhang et al.

(2013) evaluated the modal dispersion curves for dipole measurements in a pipe

immersed in an infinite fluid.


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To better understand the dipole wavefield, we use a 3D finite difference method

(3DFD) to simulate the wavefield in the single casing model with different cement

conditions. We use the pressure snapshots, array waveforms, and dispersion analysis

to investigate the effect of different bonding conditions on different modes that are

excited by the dipole source.

Method and Model

We use a 3DFD code (Wang et al., 2015) to simulate wave propagation in a single

cased borehole with a dipole tool. The code uses a staggered grid with second order

accuracy in both space and time that allows for reliable modeling of the high

impedance contrast between fluid and solid. The model (with good cement) is shown

in Figure 1. The geometries and elastic parameters are listed in Table 1. The model is

identical to the one used in Wang and Fehler (2017).

The code is the same one that was used in Wang and Fehler (2017), where it was

validated for a monopole source. Here we validate it for a dipole source. For the

code validation, we investigate the model of sonic logging in a free casing hole

(cement outside the steel casing is completely replaced with fluid). A ring source is

approximated by 36 point sources (Ricker wavelet) embedded on the outer boundary

of the casing. Although the source loading is different from that used for sonic

logging in a cased hole, which is a centralized source in the inner fluid, we choose this

source loading because we can easily use the Discrete Wavenumber integration

method (DWM, Byun and Toksöz, 2003) used in an ALWD model (Wang et al., 2015)

for comparison with our 3DFD results.

Figure 2 shows the simulations for a dipole source in a free borehole obtained using

both 3DFD and DWM. Grid sizes of 1 mm in x and y, and 2 mm in z are used in the

3DFD code. Figure 2 shows the comparison between the 3DFD and DWM results for

a source center frequency of 10 kHz. It is easy to identify three modes (marked with

different lines) in time sequence: casing, flexural, casing F1. The comparison is very

good for both the casing and flexural modes and only a difference in the later part of

the waveform (casing F1) due to numerical dispersion of the 3DFD.


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Figure 1 A cased well model with good cement bonding. Right is the top-down view of the left.

Table 1 Elastic parameters of media in the model.

Figure 2 Comparison between the 3DFD and DWM simulations for the dipole measurement (10 kHz

source center frequency) in a free casing model (ring source being loaded on the outer boundary of the

casing).

Media Vp (m/s) Vs (m/s) Density

(kg/m3)

Outer

Radius(mm)

fluid 1500 0 1000 108

Steel 5500 3170 8300 122

Cement 3000 1730 1800 170

Sandstone 4500 2650 2300 300


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Fully Bonded or Fluid filled Annulus

We initially use the 3DFD simulator to investigate the wavefields by examining

pressure snapshots, array waveforms, and dispersion analysis for three different

models: (1) a casing immersed in fluid (no cement or formation outside the casing), (2)

a cased-hole model with cement in the annulus being fully replaced with fluid, (3) a

cased-hole model with perfect cement. In subsequent sections we will consider cases

where the cement is partially replaced by fluid.

Learning from the knowledge in nondestructive testing, there are three types of

guided waves in free pipes: L (longitudinal), F (flexural), and T (torsional) modes

(Cawley et al., 2002). T modes are modes associated with the pipe rotating and these

would be found in drilling pipes. During the acoustic logging in the cased hole, L

modes are the monopole modes in the casing which are similar as the extensional

modes propagating along the collar in acoustic logging-while-drilling (Wang et al.,

2016). F modes are dipole, quadrupole and higher modes on the pipe (similar to the

flexural, screw modes propagating in the collar during acoustic

logging-while-drilling). For convenience, we use the letters „F‟ and „Q‟ denoting

dipole and quadrupole modes in the pipe

(1) Steel casing immersed in fluid (no cement or formation outside the

casing)

This model is used to understand the modes propagating in the pipe and is applicable

to a single cased-hole in addition to underwater cables, and oil and chemical

pipelines.

Figure 3 shows pressure snapshots at 1.0 ms and 2.0 ms on a x-z profile (x and z

directions are marked in Figure 1) for the model of casing immersed in fluid without

cement or formation. There are four white solid lines extending from z = -0.5 m to z =

4.1 m in each image. The two innermost lines around x =108 mm are the inner

boundaries of the casing and the two outermost lines are the casing outer boundaries.

Figures 3a and 3b show wavefields for source center frequencies of 2 kHz and 10 kHz,

respectively. The source is located at x = 0 m and z = 0 m. All the snapshots in this

paper will be at the same source location. It is clear that there are different modes in

the snapshot plots at different times. It is also apparent that fewer modes appear in the

wavefield for the low frequency source (Figure 3a) than that for the high frequency


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source (Figure 3b). In Figure 3a, we can only find one flexural casing mode F1,

propagating on the pipe and leaking energy into the fluid on both sides of the casing

pipe. Figure 4a shows full waveforms along a centralized dense receiver array having

a 0.1 m receiver interval. The full waveforms aid in understanding the wave mode

propagation. Figure 4b shows the dispersion contour plot (Wang et al., 2015) derived

from the array waveforms. The modal dispersion curve derived by following the

monopole modal dispersion curve (Tubman et al., 1984; Zhang et al., 2016; Wang and

Fehler, 2017) is also plotted with white circles. The F1 casing mode propagating with

a slow velocity (less than the fluid compressional velocity) is very clear in both the

array waveforms and the dispersion plot.

Figure 3 Pressure snapshots for a model consisting of casing immersed in fluid simulated at different

source center frequencies. (a) Pressure snapshots at 1 ms and 2 ms at source center frequency of 2 kHz;

(b) Pressure snapshots at 1 ms and 2 ms at source center frequency of 10 kHz. Source is located at x =

0 m and z = 0 m. There are four white solid lines in (a) and (c) extending from z = -0.5 m to z = 4.1 m.

The two innermost lines around x =108 mm are the inner boundaries of the casing and the other two

lines are the outer boundary of the casing.


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Figure 4 Waveforms acquired on a centralized receiver array for a 2 kHz centralized dipole source in a

borehole with a casing that is surrounded by fluid (no formation). (a) Array waveforms; (b) Dispersion

analysis. The modal dispersion is plotted with white circles.

Figure 5 Waveforms acquired on a centralized receiver array for a 10 kHz centralized dipole source in

a borehole with a casing that is surrounded by fluid (no formation). (a) Array waveforms. (b)

Dispersion analysis. The modal dispersion curves are plotted with lines.

The modes become complex when the source frequency is high (10 kHz). The

pressure snapshots shown in Figure 3b have higher casing modes such as F2 and F3 in

addition to F1. These modes are marked with bars and lines in the figure. Figure 5

shows the array full waveforms and the extracted dispersion contour plot. The poor

comparison of the calculated dispersion contours with the modal dispersion curves for

this model is caused by the limitation of the dispersion calculation method when

applied to waveform data (Wang et al., 2015). Comparing with Figure 4a, we find

some higher velocity modes (marked with different lines) arriving before mode F1 in

Figure 5a. With the modal dispersion curves being plotted on the dispersion contour


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plot in Figure 5b, we find that the waveforms consist of F1 and also higher casing

modes such as F2, F3, F4, F5 and even F6, in which F1 is the slowest casing mode

and occurs in the latest part of the wave train. This leads us to infer that the source

frequency is a key factor for mode excitation and a higher frequency source will

excite higher order modes that complicate the wavefield. Zhang et al. (2013) also

considered the excitation of modes and found that higher source frequency results

large amplitude for higher modes and small amplitude for lower modes.

Figure 6 Impact of source frequency on casing modes. Traces are shown for different source center

frequencies as labeled to the left of each trace. (a) Waveforms normalized by a common factor. (b)

Waveforms normalized by maximum amplitude of each trace; (c) and (d) the frequency spectra of the

normalized waveforms in (a) and (b), respectively.

We further investigate the influence of the source frequency on the waveforms. Figure

6 shows the waveforms for sources with different center frequencies acquired at a

centralized receiver located 3 m away from the source position. Figures 6a and 6b

show the waveforms normalized in two different ways: by common amplitude for all


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waveforms (6a) and each waveform normalized independently (6b). Figures 6c and 6d

show the frequency spectra of the normalized waveforms in Figures 6a and 6b,

respectively. It is obvious that the higher source frequency produces higher

amplitudes of waves and higher modes. In general, F1 has small amplitude even at a

low frequency.

(2) Free casing model (cement fully replaced with fluid)

Figure 7 Pressure snapshots for a cased-hole with cement being fully replaced with fluid. (a) Pressure

snapshots (x-z profiles) at 1 ms and 2 ms when source frequency is 2 kHz; (b) Pressure snapshots (x-y

profiles at different depths) at 1 ms when source frequency is 2 kHz; (c) Pressure snapshot (x-z profile)

at 1 ms when source frequency is 10 kHz; The snapshots in (a) and (c) are at the same position as those

in Figure 3. The two smaller white circles in the (b) denote the inner and outer boundaries of the casing

and the largest white circle is the boundary of the borehole.

Figure 7 shows the pressure snapshots for different profiles for the free casing model

where cement in the annulus is fully replaced with fluid. Figures 7a (x-z profile) and

7b (x-y profiles at different depths) show the wavefield when source center frequency

is 2 kHz. The two small white circles in the Figure 7b denote the inner and outer


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boundaries of the casing and the largest white circle is the boundary of the borehole.

We clearly find one flexural mode propagating along the casing and another flexural

mode propagating in the formation, which are marked in Figures 7a and 7b (marked

as casing F1 and flexural, respectively, in 7a and casing and flexural in 7b). These

modes exhibit perfect dipole characteristics.

Figure 8 shows array waveforms and dispersion analysis. We find the formation

flexural wave is the first arrival and the casing F1 is the latter mode. Although the

wave front of the casing F1 appears to be faster than that of the flexural in the

snapshot in Figure 7a, it arrives later than the formation flexural wave in the borehole

fluid. The prominence of the formation flexural wave means that a low frequency

dipole tool can be used to measure the formation shear velocity independent of

condition of the cement in the annulus.

Figure 8 Waveforms acquired on a centralized receiver array for a 2 kHz centralized dipole source in a

cased-hole with cement being fully replaced with fluid. (a) Array waveforms. (b) Dispersion analysis.

Modal dispersion curves are plotted with curves and formation P and S velocities are shown with

horizontal lines.

The pressure snapshot obtained for the 10 kHz source that is shown in Figure 7c tells

us that the higher casing modes appear at offsets of 1.5 m to 4.1 m at time 1.0 ms and

that they propagate faster than the formation flexural mode (around offset of 1 m). We

also find an obvious leaky P mode (marked in Figure 7c) that is induced by the high

frequency dipole source. This mode is clear in the formation. If this mode can be

discerned in the borehole without any interference from the casing mode, the

formation compressional velocity can be determined (marked as “P” in Figure 9b).

However, it is very hard to observe this mode in the borehole because the wide

velocity range of the strong casing modes submerges the weak leaky P wave in the


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borehole fluid.

The waveforms and corresponding dispersion analysis plots for the 10 kHz source are

shown in Figure 9. We clearly find that the higher casing modes such as F2, F3, F4,

and F5 (arrival times marked with lines) are ahead of formation flexural mode and F1

is the slowest mode. The dispersion analysis with modal dispersion curves overlaid

helps to identify the different modes and we can still find the formation flexural in the

low frequency range (below 5 kHz from Figure 9b). Although two flexural modes that

propagate at the shear velocity at the low frequency cutoffs are clear in the dispersion

analysis in Figure 9b, the complex waveforms would make the formation shear

velocity measurement difficult if the dispersion analysis is not provided. For this case,

a low pass filter at a corner frequency lower than 5 kHz would be helpful for making

the flexural wave visible. Between 10 kHz and 20 kHz, we also find that there is a

modal curve that has the formation P velocity. This mode corresponds to the leaky P

induced by a dipole tool although we cannot find any indication of it in the contour

plots made from the waveforms. Some researchers (Pampuri et al., 2003; Chen et al.,

2007) have argued that the formation compressional wave is present in dipole logging

data in a free casing case and that it can be used to measure the compressional

velocity of the formation. However, we find it is very hard to extract the P velocity

because of the strong interference from casing modes. Our results indicate that we

may mistakenly consider one of the casing modes as a leaky P wave.

Figure 9 Waveforms acquired on a centralized receiver array for a 10 kHz centralized dipole source in

a cased-hole with cement being fully replaced with fluid. (a) Array waveforms. (b) Dispersion analysis.

The modal dispersion curves are plotted using lines.

A brief summary for the results obtained from our modeling of a cased hole with all

cement replaced with fluid is that the low frequency dipole source excites a small


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number of casing modes so that the formation flexural wave can be observed even

when the cement is completely replaced with fluid. However, for a higher frequency

source, the higher casing modes that have higher velocity complicate the wavefield

and hinder the formation shear velocity determination. The dispersion analysis or low

frequency filter would be helpful to make the flexural wave visible. The

compressional velocity determination from the dipole leaky P is very hard to obtain.

(3) Cased borehole with good cement

Figure 10 Snapshots (x-z profiles) of the pressure wavefield at 2 ms for a cased-hole with good cement

for two different source frequencies. (a) Source center frequency of 2 kHz; (b) Source center frequency

of 10 kHz. The position of the snapshots is the same as in Figure 3.

Figure 10 shows x-z snapshots of the pressure wavefield at different source

frequencies (2 kHz in Figures 10a and 10 kHz in Figure 10b) for a cased borehole

with good cement. We find only one mode in Figure 10a, which has perfect

dipole-source induced asymmetry and that penetrates deep into the formation in the

radial (x direction). Figure 11a shows the array waveforms acquired for this model

and the dispersion analysis is given in Figure 11b. The dispersion analysis shows that

this mode is the formation flexural mode. The wavefield for the higher frequency

source, shown in Figure 10b, is more complicated and there are two obvious modes

that are visible. According to their features, we consider that they are the leaky P

induced by the dipole source as the first arrival, and the formation flexural mode.

Figure 12 shows the array waveforms and dispersion analysis for the wavefield from


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the 10 kHz source (the modal dispersion curves are also plotted). From this plot, we

infer that we can use a high frequency dipole tool to determine the formation

compressional and shear velocities when the cement in the borehole is good. The

measurement could also be an additional indicator for good cement and complement

monopole measurements (e.g. Zhang et al., 2013; Wang and Fehler, 2017).

。

Figure 11 Waveforms acquired with a centralized receiver array for a 2 kHz centralized dipole source

in a cased hole with good cement. (a) Array waveforms. (b) Dispersion analysis. Modal dispersion

curves are plotted with dotted lines.

Figure 12 Waveforms acquired on a centralized receiver array for a 10 kHz centralized dipole source

in a cased-hole with good cement. (a) Array waveforms. (b) Dispersion analysis. Modal dispersion

curves are plotted with dotted lines.

Based on the above analysis, we get a rough understanding on the dipole wavefield in

the single casing borehole,

(1) The dipole source induces a leaky P induced by a dipole source (present only at

high frequency), flexural, and some casing modes.


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(2) There is no casing mode when casing is perfectly cemented and thus we can use a

high frequency dipole tool to measure formation P and S velocities from the first

arrival leaky P and the latter flexural modes, respectively.

(3) However, the interference from high order casing modes makes the leaky P

invisible when the cement sheath is replaced with fluid. The shear velocity can still

obtained from the flexural velocity at low frequency limit.

(4) Casing modes vary with source frequency. Higher frequency sources excite larger

amplitude higher modes (such as F2, F3, F4, F5 even F6). The amplitude of F1

decreases with increasing source frequency.

Partial Bonding Models

In this section, we investigate the influence of the thickness and position of the fluid

column in the region between the casing and the formation on the different modes.

We study the impact on the F1 and flexural modes at low frequency (2 kHz) because

fewer modes are excited at low source frequency. Then we study the excitation and

properties of higher casing modes such as F2 to F5 caused by a high frequency source

(10 kHz).

(1)Partial bonding: fluid between casing and cement

Learning from our understanding of monopole measurements (Zhang et al., 2013;

Wang and Fehler, 2017), we know the casing mode is the first arrival and hardly

changes with fluid thickness once the cement next to the casing is partially or fully

replaced with fluid. Here, we investigate dipole measurements in models with cement

next to casing (bonding interface I) being partially replaced with fluid.

Figure 13 shows a schematic diagram (top-down view) of a partially cemented

borehole model. R1, R2, RCI, and Rh are the inner radii of casing, fluid column,

cement, and borehole wall. The fluid thickness equals to RCI – R2. We investigate the

wavefields in the models with RCI of 122 mm, 122.5 mm, 123 mm, 124 mm, 126 mm,

130 mm, 138 mm, 154 mm, 162 mm and 170 mm, which correspond to models

having fluid thickness of 0 mm (fully cemented), 0.5 mm, 1 mm, 2 mm, 4 mm, 8 mm,

16 mm, 32 mm, 40 mm, and 48 mm (no cement) next to the casing.


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Figure 13 Schematic diagram (top-down view) of a partially cemented borehole model (bad bonding

for interface I). R1, R2, RCI, Rh are the inner radii of casing, fluid column, cement, and borehole wall.

The fluid thickness equals RCI – R2.

We calculate the modal dispersion curves for the different models. Figures 14a and

14b show the dispersion curves for the F1 mode with the various fluid thicknesses.

Figure 14 Dispersion curves of different modes for models with various thicknesses of fluid

next to the casing (Figure 13). (a) Casing F1 mode. (b) Formation flexural dispersion curves.

(c) Casing modes (F2 to F4).

Figure 15a shows the waveforms for a 2 kHz center frequency source for models with

different fluid column thicknesses at bonding interface I. The waveforms consist of

formation flexural and casing F1 modes, which can be inferred from the dispersion

contour plots for waveforms when the fluid thickness is 16 mm that are shown in

Figure 15b (the modal dispersion curves are also plotted with black lines). Although

the cut-off frequency (the low frequency limit) and velocity of the flexural wave

slightly decreases with the increasing fluid thickness (as shown in Figure 14b), we

find that the arrival time hardly changes with fluid thickness (as shown in Figure 15a).

The amplitude of the flexural wave changes with fluid thicknesses, but it stabilizes


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when the fluid thickness is above 32 mm (Figure 16c). The amplitude of casing F1

has no obvious trend with fluid thicknesses. The F1 arrival time decreases with

increasing fluid thicknesses which is consistent with the dispersion curves in Figure

14a, in which the F1 velocity increases with the fluid thickness. We can use either the

arrival time or velocity analysis (time semblance or dispersion analysis) of F1 to

determine the thickness of fluid next to the casing.

Figure 15 Synthetic waveforms and dispersion analysis for the models with fluid of various

thicknesses next to the casing. (a) Traces for source-receiver spacing of 3 m are plotted for

models with different fluid layer thicknesses. (b) Dispersion analysis (waveforms for the 16

mm fluid thickness case), the modal dispersion curves are also plotted (black lines).

For the high order casing modes (Figure 14c), only the dispersion curve of F2 changes

with fluid thickness and other modes‟ curves are not sensitive to fluid thickness

because the high frequency range (more than 10 kHz) means a small radius of

influence on those casing modes (such as F3, F4, and F5). We use the waveforms

(first 2 ms) at 10 kHz source frequency to investigate the high order casing modes

with various fluid thicknesses. Different modes are easy discerned and marked in

Figure 16a due to their features. The first arrival is F5 which can be ensured by

evaluating the dispersion contour in Figure 16b that is made from the first 1 ms

waveform in relation to the mode curve. From the waveforms, it is apparent that the

fluid thickness does not change the arrival time of first arrival. We cannot find

obvious amplitude dependence of F5 and F2 on fluid thickness. However, the

amplitudes of F4 and F3 have an obvious change with fluid thicknesses. We pick the

amplitude of the first tough of the two modes and plot them in Figure 16c. Both the

amplitudes of F3 and F4 decrease with increasing fluid thicknesses and F3 is more


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sensitive than F4 to fluid thickness. We can thus use the amplitude of F3 or F4 to

determine the fluid thicknesses. However, the amplitude picking depends on correct

identification of the mode which is a challenge during real time data processing. In

field applications, we suggest choosing a suitable source frequency from the modal

dispersion curves for the certain geometry and elastic parameters (around 5 kHz in

this model) which let the F3 or F4 be first arrival (e.g. frequency less than the cutoff

frequency of mode F5), and then picking it would be more reliable with real-time data

processing.

Figure 16 Synthetic waveforms (with 10 kHz source center frequency) and dispersion

analysis for the models with fluid of various thicknesses next to the casing. (a) Traces for

source-receiver spacing of 3 m are plotted (normalized by waveforms in the free casing

model). Dashed box indicates the dispersive F2. (b) Dispersion contour for the first 1 ms of

the waveforms when the fluid thickness is 4 mm, and the modal dispersion curves of F2 to F5

are also plotted (black lines). (c) Amplitude of the first tough, except first peak for the flexural

mode, of casing modes F3 and F4 with various fluid thicknesses.

(2)Partial bonding: fluid between cement sheath and formation

Figure 17 Schematic diagram (top-down view) of a partially cemented borehole model. R1, R2,

RCO, and Rh are the inner radii of casing, cement, fluid column, and borehole wall. The fluid


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thickness equals to Rh – RCO.

The bonding condition for the interface (bonding interface II) between the cement and

formation is more critical than for interface I because it is close to the reservoir or

aquifer. Here we investigate the models with a portion of the cement next to the

formation (bonding interface II) being replaced with fluid. Figure 17 shows a

top-down view schematic of a partially cemented borehole model. R1, R2, RCO, and

are the inner radii of casing, cement, fluid column, and borehole wall, respectively.

We investigate the wavefields in the models with RCO of 170 mm, 166 mm, 162 mm,

154 mm, 138 mm, 154 mm, 130 mm and 122 mm, which correspond to models with

fluid thickness of 0 mm (fully cemented), 4 mm, 8 m, 16 mm, 32 mm, 40 mm, and 48

mm (no cement) next to the formation.

The waveforms at 2 kHz source frequency are used for investigating the features of

flexural and F1 waves with various thicknesses of fluid next to formation (Figure 18a).

From waveforms, we find that the flexural mode becomes increasingly dispersive

with increasing fluid thickness. As seen from the dispersion analysis for the case of 16

mm fluid thickness in Figure 18b, in which the contours and blue lines are dispersion

analysis and modal dispersion curves, respectively, and the black lines are the modal

dispersion curves in Figure 15b (16 mm fluid thickness at interface I), we find the

flexural modes are slower than for the corresponding fluid thickness case for interface

I due to a larger interface radius. However, there is nearly no change in the dispersion

curves of flexural modes for different interface II fluid thickness cases (not shown

here) because of the fixed effective borehole radius for this case. F1 keeps the same

trend with fluid thickness as in Figure 14a although it is slightly slower than the mode

at the corresponding interface I fluid thickness case. We cannot find a clear

relationship between the amplitudes of two modes with fluid thickness.

Figure 18 Synthetic waveforms and dispersion analysis for the models with fluid of various

thicknesses next to formation (bonding interface II). (a) Waveforms at source-receiver offset of 3


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m (2 kHz source frequency). (b) Dispersion analysis for the waveforms acquired in the model with

16 mm fluid thicknesses. Array waveforms at distances ranging from 2.6 m and 4 m with interval

of 0.2 m are used. Contour plots are the extracted dispersion. Black and green lines are modal

dispersion curves of models in Figures 12 and 15 (fluid thicknesses of 16 mm), respectively. (c)

Waveforms at source-receiver offset of 3 m (10 kHz source frequency).

When the source center frequency moves to a high frequency (10 kHz), the

waveforms with various fluid thicknesses (Figure 18c) are strikingly different from

those shown in Figure 16a. The mode appearing as the first arrival varies with the

fluid thickness next to formation. F5 is the first arrival when the fluid thickness is

larger than 16 mm (the exact value would between 8 mm and 16 mm). However,

when the fluid thickness is smaller than about 8 mm, the cut-off frequency of high

order modes shifts towards lower frequency than that for the cases with larger fluid

thicknesses. This shift is because the equivalent thickness of the casing plus cement

increases when the fluid thickness becomes smaller. Therefore, the first arrival is no

longer mode F5 when fluid thickness is below a critical thickness between 8 mm and

16 mm. This affects the amplitude and arrival time of the first arrival (marked with

circles). The relationships between them and fluid thickness are not monotonic

functions but rather have “U” shapes. Although it is hard to determine the fluid

thickness by using the high frequency first arrival, we can use this delay in arrival

time or decay in amplitude to differentiate the location of the fluid column relative to

the different bonding interfaces. At the same time, F1 at the low source frequency can

be used to determine the fluid thickness.

(3)Partial bonding: cement inside the fluid columns

It is highly possible to get bad cement at both two bonding interface. Knowing the

distribution of the fluid columns is critical for safety and environmental reasons.

Here we separate the annulus between casing and formation into three parts and each

part consists of a different medium (cement or fluid), each with different thickness.

Then we investigate the dipole wavefield (both high and low source frequencies).

The waveforms for 2 kHz source center frequency are shown in Figure 19a. The

names for different models are listed in the figure. The letters „f‟ and „c‟ are fluid and

cement, respectively. The number before the letter is the thickness (units in mm) of

the medium. For example, „4f16c28f‟ denotes the annulus consists of 4 mm and 28

mm fluid columns next to the casing and the formation, respectively. Also, 16 mm of


20

cement is present between the two fluid columns. The flexural and F1 waves are

marked in the figure. Because the fluid column exists next to the casing, the arrival

time of the flexural mode is the same and independent of the thickness of the fluid

column. The dispersion analysis contours for different cases are shown from Figures

19b to 19f. The blue, green, red, and black lines are the modal dispersion curves for

cases of fluid thickness of 4 mm, 8 mm, 16 mm, and 32 mm next to the casing.

Although we can use the flexural dispersion curves to determine the fluid thickness

next to the casing if only interface I is not cemented (as shown in Figure 14b), here

we cannot use the flexural dispersion curves to determine the fluid thickness next to

the casing because the flexural dispersion curve is affected by fluid columns at two

different interfaces. F1 dispersion has a strong relationship with the total fluid

thickness in the annulus and it is not sensitive to the distribution of fluid columns.


21

Figure 19. Waveforms (2 kHz source center frequency) and dispersion analysis for different

models having cement between two fluid columns. (a) Waveforms at source-receiver offset of 3 m.

(b-f) Dispersion analysis for the different models as labeled in each plot. Array waveforms at

distances between 2.6 m and 4 m with interval of 0.2 m are used. Contour plot is the extracted

dispersion. Blue, green, red and black lines are the modal dispersion curves for cases of fluid

thickness of 4 mm, 8 mm, 16 mm, and 32 mm next to the casing and the casing well bonded to the

formation.

Figure 20. Waveforms (10 kHz source frequency) and dispersion analysis for the models with

cement between fluid columns. (a) Waveforms at the source-receiver offset of 3 m. Names for

different models are listed, following the same rule as Figure 19a. (b) Amplitude of modes F3 and

F4 for different models.

The waveforms for different models simulated with a 10 kHz source center frequency

are shown in Figure 20. Similar as Figure 16a, different modes are picked and marked

in the figure. Independent of its thickness, the fluid column next to the casing makes

the first arrival be F5 and it has a constant arrival time and amplitude. We pick

amplitudes of F4 and F3 in a manner similar to that described during the discussion of

Figure 16c. The pattern in the resulting amplitudes, shown in Figure 20b, is very


22

different from Figure 16c. If we use the amplitude of F4 or F3 to determine the fluid

thickness next to the casing, an erroneously small value of fluid thickness would be

obtained.

In this situation, we can use F1 obtained from the 2 kHz center frequency source to

determine the total fluid thickness and then compare the result with the thickness

estimated from the amplitude of F3 or F4. If there is a difference, we know that both

interfaces are not cemented. Otherwise, only the first interface is unbounded.

However, the distribution of fluid is still unresolved.

(4)Partial bonding: fluid inside cement sheath

Currently research on cement evaluation using acoustic logging focuses only on the

interface bonding condition (as discussed above), but there will be also a problem for

safety and environment if the fluid column appears inside the cement body. Here we

investigate the wavefields in models with fluid column inside the cement sheath with

the varying thickness of cement next to casing and the formation.

Similar to the previous section, we first investigate the wavefields in the models

simulated using a low source center frequency (2 kHz). Figure 21 shows the

waveforms and dispersion analysis for different models. The naming of the models is

the same as described for Figure 19. The waveforms consist of flexural (first arrival)

and F1 and their dispersion contour analyses are plotted in Figures 21b to 21f. The

modal dispersion curves for cases when the fluid thickness is 16mm at interfaces I and

II are plotted with black and gray lines, respectively. We find the flexural mode is

similar to those in Figure 19a as it keeps the same arrival time for all cases and the

dispersion contours match the gray lines well, which means the placement of the fluid

column does not affect the velocity (or arrival time) of the flexural mode. We cannot

find a dependence of flexural mode amplitude with fluid thickness. For the F1 mode,

the result is the same as in previous discussions. We find that the dispersion curve of

F1 is only sensitive to the thickness of the fluid column rather than to the position of

the fluid.


23

Figure 21. Waveforms and dispersion analysis for the models with fluid of various thicknesses

inside the cement sheath (2 kHz source frequency). Model naming convention is the same as that

used for Figure 19. (a) Waveforms at source-receiver offset of 3 m. (b-f) Dispersion analysis for

the various models. Contour plot is the extracted dispersion. The modal dispersion curves for cases

when the fluid thickness is 16mm at interfaces I (Figure 13) and II (Figure 17) are plotted with

black and gray lines, respectively.


24

Figure 22. Waveforms at source-receiver offset of 3 m simulated using source with a 10 kHz

center frequency (fluid inside cement sheath). Model naming convention is the same as that used

for Figure 19.

When source center frequency moves to10 kHz, we find the arrival time of the first

arrival (marked with circles in Figure 22) increases with increasing thickness of the

cement next to the casing and the amplitude of the first arrival decreases with

increasing cement thickness. However, Figure 18c shows a reverse trend on both

amplitude and arrival time when the cement thickness is above 32 mm (16 mm fluid

thickness at interface II). Therefore, we cannot use the arrival time or amplitude to

determine the cement thickness. Other modes are hard to use since the increasing

cement thickness changes the modes in the waveforms, which increases the difficulty

of identifying other modes.

Conclusions and Discussions

We have simulated the dipole wavefields in a singly cased hole with different cement

conditions. The modal composition and extracted dispersion curves have been used to

analyze the data. After understanding the modes propagating in the models, we

investigated the effect of source frequency, the thickness and position of the fluid

columns on the different modes. Based on the analyses, we conclude as follows,

(1) The dipole wavefield in the singly cased hole consists of a leaky P that is only

visible for high source frequency, formation flexural and casing modes. The

formation shear velocity can be obtained from flexural mode at low frequency.

However, the interference from high order casing modes makes the P velocity


25

determination difficult when the casing is not cemented well.

(2) The bonding conditions do not affect the arrival time of the flexural mode

although the dispersion curve is sensitive to the fluid thickness when fluid is only

present at the interface between the casing and cement. However, the dispersion curve

cannot be used determine the fluid thickness next to the casing when the fluid

columns are present at other places.

(3) The casing F1 mode, which is efficiently excited at low source frequency without

a cutoff frequency, is only sensitive to the total fluid thickness in the annulus between

the casing and the formation. It is not sensitive to the position of the fluid columns.

(4) Casing F5, which can only be excited at the high frequency due to the large cutoff

frequency, is the first arrival and keeps a constant arrival time once there is the fluid

column next to the casing and is independent of the thickness of the fluid column.

However, if the fluid column is not next to the casing, the first arrival will not be F5

and the arrival time is not independent of the thickness of cement next to the casing.

The amplitude and arrival time of the first arrival is not a monotonic function of fluid

(or cement) thickness.

(5) The amplitudes of F4 and F3 can be used to determine the fluid thickness when a

fluid column exists only next to casing. However, they are not reliable indicators of

fluid thickness when fluid is present in other places.

In field applications, we can use the casing F1 excited by a low frequency source to

determine the total thickness of fluid column in the annulus. Then we use the arrival

time of first arrival excited by the high frequency source to determine the bonding

condition of the interface between casing and cement. If the first arrival has the arrival

time of F5 (would be other modes for different source frequency and geometry), we

know that interface I is not well cemented. We can obtain a value of the fluid

thickness next to the casing by using F3 or F4. If the value is very small and there is a

difference between the value and total fluid thickness determined by the

low-frequency F1, we would know that there are fluid columns in other places.

However, we cannot tell if the fluid columns only appear at interface II or also at

other places although it is uncommon for fluid to be present in other places.

If interface I is well cemented, the arrival time and amplitude of first arrival at high

frequency will tell us the thickness of cement next to the casing when then cement


26

thickness is below 32 mm. Comparing with the total fluid thickness, we will know if

there are any other fluids excepting the fluid at interface II. Although this processing

flow is not perfect for telling the distributions of the fluid thickness, it will

significantly improve cement evaluation.

Acknowledgements

This study is supported the Founding Members Consortium of the Earth resources

Laboratory at MIT.

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