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Proceedings of the 46th Scandinavian Symposium on Physical Acoustics, Geilo, 29 Jan – 1 Feb 2023
From 3D to 1D: Effective numerical modelling of
pulse-echo measurements in pipes
Anja Diez1, Tonni F. Johansen1,2, Erlend Magnus Viggen2
1Acoustics, Sustainable Communication Systems, SINTEF Digital, Trondheim, Norway
2Centre for Innovative Ultrasound Solutions and Dept. of Cirulation and Medical Imaging,
Norwegian University of Science and Technology, Trondheim, Norway
Contact email: anja.diez@sintef.no
Abstract
In the oil and gas industry, pulse echo measurements have for decades been used
in cased holes to estimate the properties of the materials on the outside of the pipe.
To investigate the methods used to analyse pulse echo measurements from pipes,
we use numerical modelling in COMSOL Multiphysics. While 3D models correctly
capture the real geometry, they are computationally heavy and, therefore, not appro-
priate for simulating a large range of geometric and material variations. Analytic 1D
plane wave models are fast to calculate, but we observe large deviations between
the 3D and 1D results, showing that corrections to 1D model results are necessary.
Therefore, we investigate using 2D and axisymmetric 2.5D models instead and find
good agreement between the 2.5D and 3D model results and larger deviations be-
tween the 2D and 3D model results. Further, we find that the time explicit formula-
tion works reliably, with an effective absorbing layer, while using the time domain
formulation requires more care and a larger domain due to the poorer performance
of its perfectly matched layer. Nevertheless, the time domain formulation is prefer-
able when introducing thin domains and additional domain boundaries to keep the
computational time at an acceptable level.
1 Introduction
During well construction and plug & abandonment operations, it is of interest to know
the properties of the pipe in cased boreholes and the material outside of it. It is also im-
portant to know whether this outside material is well-bonded to the pipe, or if a channel
or a micro-annulus exists through which petroleum can escape to the surface [1, 2]. Dif-
ferent techniques exist to determine the bonding state and the material properties, one of
the most common being the ultrasonic pulse-echo (PE) measurement technique [3–5].
The PE measurement principle is typically the same: A transducer inside the pipe
sends an ultrasound pulse perpendicularly onto the pipe wall, and the same transducer
records the echo [3–5]. Various algorithms exist in the industry today to analyse these
recorded echoes. These algorithms typically use a 1D plane wave model in an iterative
process to identify the impedance of the material outside the pipe from parameters de-
rived from the measured signal [5–8]. To compensate for the 1D nature of this model,
corrections are applied to these results afterwards [5].
To better understand the existing algorithms and their corrections and to investi-
gate possibilities of improving these algorithms, we use numerical modelling to gen-
Open Access
1 ISBN 978-82-8123-023-1
erate a catalogue of PE signals for different pipe thicknesses, pipe diameters, materials
behind the pipe, debonding, etc., choosing the finite element method (FEM) simulation
tool COMSOL Multiphysics for our models. However, even when exploiting geometric
symmetries, modelling the full 3D geometry over sufficiently many time steps is compu-
tationally heavy and very time consuming.
In this article, we therefore discuss the possibility of reducing the model dimension
to 2D or 2.5D (i.e., axisymmetric 3D). This reduction introduces errors in the geometry
of either the transducer (2D) or the pipe (2.5D). Our results show that while the errors
are large when using 2D modelling, good agreement can be found between 2.5D and 3D
results. However, results from the 1D model, which is used to derive impedances from
measurement data, deviate significantly from the results of the 3D model.
We first discuss the theory of the PE method and the 1D plane wave model as well as
the setup of the COMSOL model in Sec. 2, before investigating the differences of the 1D
model to the 3D model in Sec. 3. In Sec. 4 this is followed by a comparison of the results
of the 2D and 2.5D models and the full 3D model, and a comparison of the results from
COMSOL’s time domain and time explicit formulations.
2 Theory and Modelling
We will now present the basic principles of the PE measurements, and the parameters
such as reverberation decay and group delay minimum that can be derived from the
measured data. This is followed by a short introduction to the 1D plane wave model
and a presentation of the COMSOL model set up. Here, we also explain the geometric
consequences when reducing the dimensions of the model.
2.1 Pulse-echo analysis
The PE method is based on sending an ultrasound pulse from a transducer inside the
pipe onto the inner pipe surface at normal incidence. The same transducer records the
echo from the pipe, which is then analysed.
Figure 1A shows a sketch of the ray paths in this geometry. When the incoming
pulse p+
in hits the inner pipe surface, most of it is reflected back towards the transducer
(contributing to p−
in), and the other part is transmitted into the pipe wall (contributing
to p+
cas). This transmitted part hits the outer pipe surface, where most of it is reflected
back into the pipe wall (contributing to p−
cas) while part of it is transmitted further into
the outside material (contributing to p+
out). This reflected wave then hits the inner pipe
surface, where again most of it is reflected back into the pipe wall (contributing to p+
cas)
while the other part is transmitted back towards the transducer (contributing to p−
in),
which records the entire echo with all the contributions to p−
in over time.
This cycle of reflections on the inner and outer pipe surfaces, which is called a rever-
beration, then continues. Every time the pulse in the pipe wall hits one of the surfaces,
the relative strength of the reflected and transmitted pulse depends on the contrast be-
tween the acoustic impedance of the pipe material (which is typically very high) and the
acoustic impedance of the other material. Because a lower impedance contrast leads to
more transmission and less reflection, a higher impedance of the material outside the
pipe leads to a faster reverberation decay (also known as ringdown) of the pulse inside
the pipe wall.
2
Figure 1: (A) Sketch of the setup with fluid filled pipe and material outside the pipe. The pipe
(often called a casing) has a thickness dand an impedance Zcas, while the impedances inside and
outside the pipe are Zin and Zout, respectively. The arrows indicate directions of wave propagation
with an incoming wave p+
in, transmitted from the transducer, and corresponding transmission and
reflections. (B) Reflection impulse response with main reflection from the inner wall of the pipe
(t=0) and decreasing reverberation train.
The transducer measures the echoes p−
in inside the pipe. Figure 1B shows a typical
impulse response, under the approximation that the problem is one-dimensional (which
we explain in more detail in Section 2.2). This impulse response contains a strong initial
reflection from the inner pipe surface, followed by a series of impulses that have rever-
berated inside the pipe wall. The decay rate of this impulse series contains information
on the impedance outside the pipe. However, impulse responses like the one shown in
Figure 1B represent the measurement in case of an infinitely-short transducer pulse. In
practice, however, the pulse is significantly longer than the time between each compo-
nent impulse, so that the individual reflections cannot be distinguished.
Figure 2A shows simulated examples of two typical signals s(t)over time trecorded
at the transducer, with mud and cement, respectively, outside the pipe. These signals
are normalised and shifted in time so that the maximum of the pulse is plotted at 0 µs.
The first part of the signal (−7–5 µs) originates from the reflection from the inside surface
of the pipe, and the reverberation signal is visible from 5 µs onward. As the acoustic
impedance of cement is much higher than that of mud, the cement reverberation decays
much faster.
Different analysis techniques exist to estimate the impedance (Zout) of the material
outside the pipe from such signals. One possible technique is based on calculating the
envelope of the time signal (Figure 2B) using the Hilbert transform. The decay of the
reverberation signal can then be estimated as the gradient of the logarithmic envelope.
This gradient is estimated by a curve fit inside a time window covering part of the re-
verberation signal [9, 10]. In this article, we choose a time window that spans the range
from 2.5 to 6 periods (corresponding to 11–27 µs) after the main peak, a choice inspired
by other analysis methods [5, 6]. The faster the decay, the more energy is transmitted
outside the pipe wall during the reverberation, which indicates a material with a higher
impedance behind the pipe.
A very similar analysis technique uses the cumulative sum CS(t)(Figure 2C), which
3
10 0 10 20 30 40 50 60
Time
t
( s)
1.00
0.75
0.50
0.25
0.00
0.25
0.50
0.75
Time signal
s
, normalized
A
3D Mud
3D Cement
10 0 10 20 30 40 50 60
Time
t
( s)
50
40
30
20
10
0
Envelope (dB)
B
10 0 10 20 30 40 50 60
Time
t
( s)
50
40
30
20
10
0
Cumulative sum
CS
(dB)
C
100 150 200 250 300 350
Frequency f (kHz)
3
2
1
0
1
2
Group delay ( s)
D
Figure 2: (A) Example of a time signal from 3D COMSOL modelling (covered in Sec. 2.3) with
mud inside the pipe and either mud (blue) or cement (orange) outside the pipe. Derived from the
time signal are (B) the envelope, (C) the cumulative sum, and (D) the group delay.
sums up the absolute values of the time signal s(t)over the sampling times tn, going
backwards from the end of the recording:
CS(tn) =
n
∑
i=N
|s(tn)|,n∈ {N, . . . , 1}. (1)
The gradient of the reverberation part of the cumulative sum is then used as an indication
for the impedance, similarly to the envelope gradient. A similar technique is extensively
used in room acoustics to study reverberations [11, 12].
A third possibility to estimate the impedance behind the pipe is the analysis of the
group delay (Figure 2D) [5]. To calculate this group delay requires, first, the calculation
of two frequency spectra from the signal, each with a different Hamming time window
applied to the signal. The normalisation spectrum SN(f)(fbeing the frequency) uses a
window that mainly contains the first reflection from the inner pipe surface, while the
processing spectrum SP(f)uses a window that additionally contains a significant por-
tion of the reverberation signal. Both time windows are centred at the reflection’s pulse
peak, and common half-widths of the normalisation and processing windows are 2.5 and
7 periods, respectively [5]. Then, the phase difference ϕ(f) = arg[SP(f)/SN(f)] of these
two spectra is calculated to derive the group delay:
τ(f) = −∂ϕ
∂ω ≈ − ∆ϕ
∆ω, (2)
where ω=2πfis the circular frequency. In practice, this derivative is calculated as
a finite difference approximation from a discrete signal, with ∆denoting the difference
4
between two adjacent points of the function (e.g., ∆ϕ=ϕ1−ϕ2). A clear minimum can
be observed in the group delay function in Figure 2D. The resonance frequency ( fres) of
the reverberation signal, given by the frequency of the group delay minimum, depends
on the pipe thickness (dcas). The pipe thickness can be approximated via a plane-wave
assumption as
dcas ≈vp,cas
2fres
, (3)
where vp,cas is the pipe’s P-wave speed. Sirevaag et al. [9] found that Eq. 3 overestimates
the pipe thickness and suggested a correction factor of 0.95.
2.2 The plane wave model
If we approximate the problem as being one-dimensional, we can straightforwardly de-
rive an analytical PE response. With this assumption, we are really solving the problem
for a plate geometry instead of a pipe geometry, with the inner and outer plate boundaries
at x=0 and x=dcas, respectively. We assume an incoming wave p+
in =Aej(2πf t−kin x)
representing a single frequency component, with kin =2πf/vp,in being the inner-fluid
wavenumber and vp,in the sound speed in the inner fluid. Imposing continuity of pres-
sure and normal displacement (or, equivalently, velocity) at the boundaries, we get a
system of four linear equations for the wave components presented in Figure 1A. These
can be solved to get the frequency-domain reflection coefficient [13]
R(f) = p−
in
p+
in
=1−Zin
Zout cos kcasdcas +jZcas
Zout −Zin
Zcas sin kcasdcas
1+Zin
Zout cos kcasdcas +jZcas
Zout +Zin
Zcas sin kcasdcas
, (4)
where Zcas is the pipe impedance, Zin the impedance of the inner fluid, Zout the impedance
of the material outside the pipe, dcas the pipe or plate thickness, and kcas =2πf/vp,cas
is the pipe wavenumber. Transforming this result into the time domain gives the im-
pulse response (Figure 1B), with the initial reflection from the inside of the pipe and the
decreasing reflections from the reverberation train. To calculate a time signal, the reflec-
tion coefficient is multiplied with a pulse signal in the frequency domain before Fourier
transforming it to the time domain, or equivalently, the impulse response is convolved
with the pulse signal. For the examples here, we use the same Gaussian pulse as in the
COMSOL modelling detailed in the next section.
2.3 COMSOL numerical modelling
The numerical model for the pulse echo measurement is implemented in COMSOL Mul-
tiphysics. Most of the models are implemented using the time explicit formulation in 2D,
2.5D, and 3D with a coupling of the fluid pressure-acoustics and solid elastic-wave do-
mains through an acoustic-structure boundary. The transducer is modelled as a curved
circular transducer with the same surface size and shape as in [14–16]. The initial dimen-
sions of the numerical model are:
• transducer focal length: 20 cm,
• transducer width: 25 mm,
• pipe thickness: 13.06 mm,
5
Figure 3: Example of the 3D COMSOL model with mud inside and outside the pipe, showing a
snapshot of the wave propagation at 70µs (corresponding to −7.5 µs in Figure 2). The pressure
is plotted for the fluid and the solid domain.
• transducer-pipe distance: 45 mm, and
• outer pipe diameter: 13.375 in (33.9725 cm).
In the results presented here, we vary the outer pipe diameter as well as run simulations
with the pipe replaced by an equally thick plate at the same distance to the transducer.
The pulse is defined as a Gaussian pulse with a bandwidth of 0.7 and a centre fre-
quency of 250 kHz. The maximum of the pulse is emitted at 8 µs in the COMSOL model.
Additionally, we use an apodisation over the transducer surface (a von Hann amplitude
weighting as function of the radial coordinate on the transducer surface) when emitting
the signal to ensure high energy in the middle of the transducer surface and lower en-
ergy at the edges, thus reducing edge effects. In this article, we present results with mud
(of impedance 1.63 MRayl) inside the steel pipe (of impedance 46.66 MRayl), and mud or
cement (of impedance 6.63 MRayl) outside the pipe, as examples for materials with low
and high impedance behind the pipe. For more detailed material parameters, compare
Table A.1 in the appendix.
Figure 3 shows the 3D model, which is set up as a quarter of the full 3D domain. It
exploits the symmetries of the geometry to simulate the entire transducer-pipe domain
using symmetry boundaries. All outer boundaries, except for the transducer, are sur-
rounded by an absorbing layer to avoid spurious boundary reflections. The transducer
surface itself is defined on an outer boundary with no absorbing layer outside, such that
total reflection will occur at this boundary.
The model is run using the time explicit interface of COMSOL, which is based on
the discontinuous Galerkin method [17]. With the default settings of this interface, a
rule of thumb says that the mesh size can be as coarse as 1.5-mesh points per shortest
wavelength in each domain of the model while retaining sufficiently high accuracy [17].
6
Figure 4: Model geometry for a 2D (A and B) and a 2.5D model (C and D) for a plate and a pipe.
The plane that is set up as a COMSOL model is indicated in dark blue. In the case of the 2D
model, we assume symmetry at the centre line. In the case of the 2.5D model, axisymmetry is
imposed around the centre line.
Table 1: Simulated geometry of pipe/plate and transducer in models of different dimension and
shape.
Dimension Transducer modelled as Plate modelled as Pipe modelled as
3D circular plate pipe
2D infinite in 3. dimension plate pipe
2.5D circular plate spherical shell
1D infinite plane plate —
Here, we used twice the pulse centre frequency (i.e., 500kHz) and the material properties
to determine the mesh size in each specific domain. Increasing the mesh resolution did
not change the results significantly.
Normally, FEM models require mesh continuity across domain boundaries. For our
model, this would lead to a significantly finer mesh in the pipe domain than what is
required to resolve the wave. However, the time explicit formulation does not require
mesh continuity over the boundaries between different domains. Instead, boundaries
are coupled.
The model is run for 140 µs, with a 0.5 µs sample period. The received signal is calcu-
lated by integrating the pressure over the transducer surface area.
The 2D model is also set up as a time explicit model with the same dimensions as the
3D model. In a 2D model, the geometry and all fields are invariant in the third dimension
(Figures 4A and 4B). Hence, in the plate case, we model an infinitely long plate, and in
the pipe case, we model an infinitely long pipe. In other words, the 2D model correctly
captures the geometry of the pipe. However, the 2D model introduces an error in the
geometry of the transducer, which is modelled as infinitely long in the third dimension
instead of having the true round shape.
The 2.5D model is axisymmetric (Figures 4C and 4D, respectively). The axisymmet-
ric plate model correctly captures the plate geometry. However, the axisymmetric pipe
model does not correctly capture the pipe geometry. Instead, it models the pipe as a
spherical shell with the same inner and outer radii as the pipe (Figure 4D). Neverthe-
less, we will refer to this model as the 2.5D pipe model. Even if an axisymmetric model
7
10 0 10 20 30 40 50 60
Time
t
( s)
1.00
0.75
0.50
0.25
0.00
0.25
0.50
0.75
Time signal
s
, normalized
A
3D pipe
3D plate
1D
10 0 10 20 30 40 50 60
Time
t
( s)
50
40
30
20
10
0
Envelope (dB)
B
10 0 10 20 30 40 50 60
Time
t
( s)
50
40
30
20
10
0
Cumulative sum
CS
(dB)
C
100 150 200 250 300 350
Frequency f (kHz)
5
4
3
2
1
0
1
2
Group delay ( s)
D
Figure 5: Comparison of the 1D plane wave model result (black) with the results from the 3D steel
pipe model (red) with an outer diameter of 13.375 in and 3D steel plate model (cyan), all with a
thickness of 13.06 mm and mud on both sides. Shown are (A) the modelled time signal, (B) the
envelope, (C) the cumulative sum, and (D) the group delay.
cannot capture the true pipe geometry, the 2.5D models capture the circular transducer
geometry correctly. Table 1 summarises how models of different dimension and shape
represent the transducer, the plate, and the pipe.
3 Motivation: 3D versus 1D
1D models are commonly used in analysis algorithms to derive the impedance of the
material behind the pipe from measurements, following the steps below:
1. making a guess for the impedance of the material behind the pipe (the impedance
of the pipe and the fluid inside the pipe are assumed to be known),
2. deriving the impulse response of the 1D model,
3. convolving the impulse response of the 1D model (step 2) with the measured re-
flection pulse, and
4. using some metric to quantify the discrepancy between the measured signal and
the 1D model signal (step 3).
This discrepancy is then progressively minimised by refining the guess for the impedance
behind the pipe in the 1D model and repeating steps 1–4 until the discrepancy is satisfac-
torily low [5–8].
8
Figure 5 shows that the 1D and 3D model results are quite different, even though
they use the same Gaussian source pulse. The 3D model with a pipe represents the real-
world geometry the best. As the 1D model corresponds to a plate geometry, the 3D plate
model is added here for comparison. This comparison shows that there is little difference
between the time domain signal (Figure 5A), the envelope (B), the cumulative sum (C),
and the group delay (D) of the 3D pipe and 3D plate models.
Comparing the 1D and 3D model results, we observe that the initial reflections of the
1D and 3D time signals agree well with each other. However, the 1D signal’s resonance
frequency differs from the 3D results. Furthermore, the amplitude of the reverberation
tail is much higher in the 1D result than in the 3D results. This is reflected in the results
of the envelope and cumulative sum, which are normalised to their respective maxima.
These curves show that the ringdown energy, relative to the pulse maximum, is about
3–4 dB higher for the 1D model than for the 3D models. Even though the absolute energy
of the ringdown deviates significantly, we can observe that its gradient (corresponding to
the reverberation decay rate) is steady and similar in the 3D and 1D model results. The
group delay derived from the 1D model results differs significantly from the 3D model
results with the resonance peak being shifted by about 14 kHz to higher frequencies and
the minimum being 0.95 µs deeper.
In short, we observe that the 1D model result does not correspond well with the refer-
ence 3D model results. The close correspondence between the 3D pipe and plate models
indicates that these models’ discrepancy with the 1D model does not occur because the
1D model neglects the pipe curvature, but rather because of its inaccurate approximation
of the incoming pulse as a plane wave of infinite extent. Hence, corrections are neces-
sary when using 1D models to derive the impedance behind the pipe across variations
in, e.g., pipe thickness, pipe diameter, transducer-pipe distance, or eccentering. To in-
vestigate the differences caused by such variations, we work on generating a catalogue
of modelled data. Such a catalogue will, for example, allow deriving correction factors
for 1D model results. However, the 3D models are time consuming to simulate (7-12
hours each). Therefore, we investigate the possibility of reducing the dimensions of the
numerical model from 3D to 2D or 2.5D here while retaining sufficient accuracy.
4 Comparison of results
The various 2D and 3D geometries were modelled for mud and cement behind the pipe
to have a comparison for materials with a slow and fast reverberation decay, respec-
tively. In the following sections, we compare the results from (i) 3D, 2D, and 2.5D models
(Sec. 4.1), from (ii) models with different pipe diameters (Sec. 4.2), and from (iii) time do-
main and time explicit models (Sec. 4.3). In each section, we first present the results and
then discuss them.
4.1 3D, 2.5D, and 2D models
4.1.1 Results
The response for a PE measurement with mud and cement was calculated for the different
dimensions (2D, 2.5D, 3D). Figures 6 and 7 show the results for the models with mud on
both sides of a pipe and a plate, respectively. The 3D results are shown in black and
9
are used as a reference in the following discussion. The cement results in the appendix
(Figures A.1 and A.2) show the same trends with a faster ringdown.
We first discuss the differences between the 2D and 3D results before discussing the
2.5D results. For the results from both the pipe (Figure 6) and the plate (Figure 7), we
observe significant differences between the 2D (red) and 3D (black) model results:
• a difference in the phase and height of the peaks of the first reflection in the time
signal (A),
• an energy plateau for the envelope ringdown (B) up to 20 µs in the 2D data which
is not visible in the 3D result,
• a significantly faster ringdown (larger gradient) in the envelope (B) above 20 µs for
the 2D model result in comparison to the 3D model result,
• good agreement between the 2D and 3D pipe model cumulative sum results (C),
up to 20 µs
• a significantly faster ringdown (larger gradient) in the cumulative sum (C) above
20 µs for the 2D result compared to the 3D result,
• a shift to higher resonance frequencies in the group delay (D) for the 2D model
result (up to 8 kHz), and
• deviations (up to 0.35 µs) in the depth of the group delay minimum (D).
Much better agreement can be found between the 2.5D (cyan) and 3D (black) model
results both for the case with a pipe and a plate:
• good agreement for the first pulse and the reverberation part in the time signal (A),
• small ringdown variations can be observed in the envelope (B), with a slight over-
estimation of ringdown energy for the result of the 2.5D pipe model,
• similar observations for the cumulative sum (C) as for the envelope, with a slight
overestimation in energy for the ringdown derived from the 2.5D pipe model,
• small deviations in the resonance frequencies (up to 2kHz) from the group delay
minimum for the 2.5D pipe model result (D),
• small deviations in the depth (up to 0.2 µs) of the group delay minimum for the
2.5D pipe model result (D), and
• excellent agreement between result from the 2.5D plate and 3D plate models for the
envelope (B), the cumulative sum (C) and the group delay (D).
4.1.2 Discussion
The 2D model results deviate significantly from the results of the 3D model, while both
2.5D model results, pipe and plate, show very good agreement with the 3D model results.
Figure 8 compares the ringdown derived from the envelope (A) and the cumulative sum
(B), the group delay minimum (C), and the thickness (D) derived from the resonance
frequency for a more direct comparison of these models’ results. These show that the
10
10 0 10 20 30 40 50 60
Time
t
( s)
1.00
0.75
0.50
0.25
0.00
0.25
0.50
0.75
Time signal
s
, normalized
A
2D pipe
2.5D pipe
3D pipe
10 0 10 20 30 40 50 60
Time
t
( s)
50
40
30
20
10
0
Envelope (dB)
B
10 0 10 20 30 40 50 60
Time
t
( s)
50
40
30
20
10
0
Cumulative sum
CS
(dB)
C
100 150 200 250 300 350
Frequency f (kHz)
3
2
1
0
1
2
Group delay ( s)
D
Figure 6: Comparison of 3D pipe (black), 2D pipe (red), and 2.5D pipe (cyan) results with mud
inside and outside of the pipe. Shown are (A) the time signal, (B) the envelope, (C) the cumulative
sum, and (D) the group delay.
10 0 10 20 30 40 50 60
Time
t
( s)
1.00
0.75
0.50
0.25
0.00
0.25
0.50
0.75
Time signal
s
, normalized
A
2D plate
2.5D plate
3D plate
10 0 10 20 30 40 50 60
Time
t
( s)
50
40
30
20
10
0
Envelope (dB)
B
10 0 10 20 30 40 50 60
Time
t
( s)
50
40
30
20
10
0
Cumulative sum
CS
(dB)
C
100 150 200 250 300 350
Frequency f (kHz)
3
2
1
0
1
2
Group delay ( s)
D
Figure 7: Comparison of 3D plate (black), 2D plate (red), and 2.5D plate (cyan) results with mud
on both sides of the plate. Shown are (A) the time signal, (B) the envelope, (C) the cumulative
sum, and (D) the group delay.
11
3D 2.5D 2D
0.2
0.3
0.4
0.5
0.6
0.7
Ringdown (dB/ s)
A
Envelope
Mud plate
Mud pipe
Cement plate
Cement pipe
3D 2.5D 2D
0.3
0.4
0.5
0.6
0.7
0.8
Ringdown (dB/ s)
B
Cumulative Sum
3D 2.5D 2D
3.25
3.00
2.75
2.50
2.25
2.00
1.75
1.50
1.25
Group delay minimum ( s)
C
Group delay
3D 2.5D 2D
13.4
13.5
13.6
13.7
13.8
13.9
14.0
Thickness (mm)
D
Thickness from resonance frequency
Figure 8: Derived ringdown (A) from envelope and (B) from the cumulative sum (11–27 µs win-
dow), (C) group delay minimum, and (D) thickness from the resonance frequency for 3D, 2.5D,
and 2D models with plate (squares) and pipe (circles) geometry, for models with mud and cement
behind the pipe/plate, respectively. The dashed lines extend the reference values from 3D pipe
simulations for comparison.
ringdown derived from the envelope and cumulative sum of the 3D results agree much
better with those from the 2.5D results than those from the 2D results. The same holds for
the group delay minimum values. For the thickness values derived from the resonance
frequency using Eq. 3, we observe a clear difference between the pipe and plate model
results and a significant difference in the thickness values derived from the 3D and 2.5D
pipe model results, although the results of the 2D models have much larger differences
with those of the 3D model. Overall, these derived values confirm the impression from
the waveform comparisons (Figures 6 and 7) that the 2.5D results have a much better
correspondence with the 3D results than the 2D results.
For the results of the 2D model, the phase of the first reflection does not agree with the
3D results and significant differences can be observed in the ringdown (Figures 6 and 7).
Furthermore, the 2D results shows a significant change in the ringdown gradient over
time (before and after 20 µs), making derived ringdown gradients depend significantly
on the chosen time window. Because the 2D results perfectly captures the pipe and plate
geometries (Figure 4), we conclude that these errors are introduced by modelling the
transducer as infinitely long in the third dimension of the 2D model.
On the other hand, the 2.5D plate and pipe model results both agree well with the
corresponding 3D results in all comparisons, with small deviations. Both 2.5D models
describe the transducer geometry correctly, but only the 2.5D plate model captures the
same geometry as in the equivalent 3D model; the 2.5D pipe model represents the pipe
as a spherical shell. While it is not surprising that the results of the 2.5D and 3D plate
models are in excellent agreement as they simulate identical geometries, the very good
correspondence between results of the 2.5D and 3D pipe models is less expected.
12
0123456
Impedance (MRayl)
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Ringdown (dB/ s)
A
Air
Kerosene
Water
Mud
Ethylene
Glycerol
Epoxy
Epoxy 5%
Shale
Cement
Envelope
2D plate
2.5D plate
0123456
Impedance (MRayl)
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Ringdown (dB/ s)
B
Air
Kerosene
Water
Mud
Ethylene
Glycerol
Epoxy
Epoxy 5%
Shale
Cement
Cumulative Sum
0123456
Impedance (MRayl)
4.0
3.5
3.0
2.5
2.0
1.5
1.0
Group delay minimum ( s)
Air
Kerosene
Water
Mud
Ethylene
Glycerol
Epoxy
Epoxy 5%
Shale
Cement
C
Group delay
0123456
Impedance (MRayl)
13.6
13.7
13.8
13.9
14.0
Thickness (mm)
Air
Kerosene
Water
Mud
Ethylene
Glycerol
Epoxy
Epoxy 5%
Shale
Cement
D
Thickness from resonance frequency
Figure 9: Results for the ringdown derived from (A) the envelope and (B) the cumulative sum, (C)
the group delay minimum, and (D) the thickness from the resonance frequency, all for a range of
materials with different impedances. Modelling was done using the 2D plate model (blue squares)
and the 2.5D plate model (orange circles).
For the 2D results, the ringdown gradients derived from the envelope and cumula-
tive sum in particular deviate so strongly from the 3D results that these gradients should
not be used directly. Figure 9 compares results from the 2D and 2.5D plate model over
a range of materials with parameters listed in Table A.1. While the differences between
the 2D and 2.5D results are not constant, the overall trends of both models are still very
similar. Depending on the application, one might therefore consider using 2D modelling
when comparing relative values and investigating changes due to, e.g., different materi-
als behind the pipe.
4.2 Different pipe diameters
4.2.1 Results
In Section 4.1, we established that the results from the 3D and 2.5D plate models have an
excellent correspondence, and that the correspondence between the 3D and 2.5D results
for a 13.375 in pipe is also very good. However, we also have to investigate whether this
correspondence persists for smaller pipe diameters with higher surface curvature. We
initially tested a large range of pipe diameters using the 2.5D model and some typical
pipe diameters using the 3D model. From Figure 10, which shows in-depth results for a
13.375 in and a 7 in pipe, we can observe that:
• the 3D and 2.5D 13.375 in pipe model results have a very good correspondence in
the time signal (A), envelope (B), cumulative sum (C), and group delay (D), as we
13
10 0 10 20 30 40 50 60
Time
t
( s)
1.00
0.75
0.50
0.25
0.00
0.25
0.50
0.75
Time signal
s
, normalized
A
3D, 13.375 inch
2.5D, 13,375 inch
3D, 7 inch
2.5D, 7 inch
10 0 10 20 30 40 50 60
Time
t
( s)
50
40
30
20
10
0
Envelope (dB)
B
10 0 10 20 30 40 50 60
Time
t
( s)
50
40
30
20
10
0
Cumulative sum
CS
(dB)
C
100 150 200 250 300 350
Frequency f (kHz)
3
2
1
0
1
2
Group delay ( s)
D
Figure 10: Comparison of 3D and 2.5D models for a 13.375 in and a 7in pipe with mud inside and
outside the pipe with pipe thickness of 13 mm. Shown are (A) the modelled time signal, (B) the
envelope, (C) the cumulative sum, and (D) the group delay.
also observed in Figure 6,
• the 3D and 2.5D 7 in pipe model results show significant differences in the time
signal (A), envelope (B), and group delay (D),
• a phase shift of the time signal (A) from the first reflection can be observed between
the 13.375 in pipe results and the 7 in pipe results, with a small difference for the 3D
results and a clear difference for the 2.5D results,
• in the envelope results (B), we observe a significant minimum at around 7.5µs after
the initial reflection for the 7 in 2.5D model result only,
• the cumulative sum results (C) from all models agree very well,
• the group delay result (D) of the 7in 2.5D pipe model has a significantly shifted
minimum (about 0.6 µs smaller, with a frequency shift of 6kHz) compared to those
of the other models, and
• the 7 in 2.5D pipe model results clearly deviates from the 13.375 in 2.5D pipe model
results in the envelope (B) and group delay (D), while the 7 in 3D pipe model result
is very similar to the 13.375 in 3D pipe model result.
To further investigate the effect of the pipe diameter, we simulated a range of 2.5D
models with mud and cement behind the pipe and a range of pipe diameters between 7 in
and 20 in. Figure 11 shows the the ringdown from the envelope and cumulative sum, the
14
8 10 12 14 16 18 20 plate
Diameter (in)
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Ringdown (dB/ s)
A
Envelope
mud 3D
mud 2.5D
cem 3D
cem 2.5D
8 10 12 14 16 18 20 plate
Diameter (in)
0.3
0.4
0.5
0.6
0.7
Ringdown (dB/ s)
B
Cumulative Sum
8 10 12 14 16 18 20 plate
Diameter (in)
3.00
2.75
2.50
2.25
2.00
1.75
1.50
1.25
1.00
Groupd delay minimum ( s)
C
Group delay
8 10 12 14 16 18 20 plate
Diameter (in)
13.60
13.65
13.70
13.75
13.80
13.85
13.90
13.95
14.00
Thickness (mm)
D
Thickness from resonance frequency
Figure 11: Comparison of 3D and 2.5D model results for a plate and pipes with diameters between
7 in and 20 in and thickness 13 mm, for both cement and mud behind the steel. Shown are (A)
the ringdown from the envelope, (B) the ringdown from the cumulative sum, (C) the group delay
minimum, and (D) the thickness derived from the resonance frequency.
group delay minimum, and the pipe thickness (from the resonance frequency) derived
from these simulations. For comparison, 3D simulations for the typical pipe diameters
of 7, 9.625, 13.375 and 20 in were run and the same parameters were derived. From the
comparisons in Figure 11, we make the following observations:
• the results from the 3D models and 2.5D models agree well for the envelope (A)
down to the 9.625 in pipe model,
• for pipe diameters smaller then 9.625 in we observe that the 2.5D models overesti-
mate the ringdown derived from the envelope (A),
• no clear trend can be observed for the results from the cumulative sum (B), even
though some variations can be observed in the results from the 2.5D models with
cement behind the pipe,
• in case of the group delay (C), we observe good agreement between the 3D and
2.5D model results down to the results of the 13.375 in pipe,
• for pipes with a diameter smaller than 13.375 in we observe an underestimation of
the group delay minimum (C), especially for the models with mud behind the pipe,
• in case of the thickness (D) derived from the resonance frequency we observe that
the difference between 2.5D and 3D model results increase with decreasing diame-
ter,
15
Figure 12: COMSOL snapshots for pipes of diameter (A and C) 7 in and (B and D) 13.375 in
from 2.5D models. The snapshots show the wavefield after (A and B) 40µs and (C and D) 70 µs
(corresponding to −37.5 µs and −7.5 µs in Figure 10, respectively).
• it should be noted that the largest difference between the derived thickness (D) and
the real pipe thickness of 13 mm can be observed for the thickness derived from the
3D plate results; with decreasing pipe diameter we observe that this error decreases.
4.2.2 Discussion
The results of the 13.375 in 3D and 2.5D pipe models and the 7 in 3D pipe model are all
similar. The 7 in 2.5D pipe model results, however, differ significantly in the envelope
and the group delay.
Figure 12 shows time snapshots of the wave field for the two 2.5D models, 7 in and
13.375 in pipe, at 40 µs (A and B) and 70µs (C and D). The snapshots at 40 µs show the
wavefront as it reaches the pipe. The 13.375 in pipe has a lower curvature than the wave-
front, so that the centre of the wavefront reaches the pipe first. The 7 in pipe, however,
has a higher curvature than the wavefront, so that the outer edges of the wavefront reach
the pipe before the centre does. The result of this difference can be seen in the snapshots
at 70 µs (C and D), which show that the reflected wavefronts from the two pipes have
different shapes as they reach the transducer.
The 2.5D model approximates the pipe geometry as a spherical shell geometry (Fig-
ure 4D), which has the same curvature in all directions perpendicular to the symmetry
axis. In a real pipe geometry, however, only the axial plane (across the pipe) has this
maximum curvature. In all other planes passing through the transducer axis, the pipe
curvature is smaller, and in the sagittal plane (along the pipe) there is no curvature at all.
16
Thus, the axisymmetry of the 2.5D model overestimates the effect of the higher curvature
of small-diameter pipes.
At the same time, we observe that most of the 3D model results are not strongly
affected by the pipe diameter. The variations of the derived parameters over the pipe
diameter are small with standard deviations of only 0.008 dB/µs for the envelope ring-
down, 0.009 dB/µs for the cumulative sum, and 0.04 µs for the group delay. However, for
the thickness, the 3D model results show a clear trend where decreasing diameters lead
to decreasing estimated thicknesses, from 13.93 mm at 20 in to 13.85 mm at 7in.
Hence, we observe that reducing the pipe diameter (increasing their curvature) in-
creases the errors for all parameters derived from the 2.5D pipe models (which represent
the pipe as a spherical shell). In contrast to that, the 2.5D plate model seems to provide
a good approximation for the ringdown and the group delay minimum of pipes, even at
low pipe diameters. However, the 2.5D plate model’s pipe thickness estimates deviate
significantly from those of the 3D pipe model.
4.3 Time domain models versus time explicit models
4.3.1 Results
The modelling presented so far was carried out using COMSOL’s time explicit (TE) for-
mulation. The TE formulation is recommended for domains that contain many wave-
lengths. However, COMSOL’s time domain (TD) formulation can also be used for this
problem.
The TE formulation uses fourth-order accurate elements, while the TD formulation
uses only second-order accurate elements. This means that TE modelling can achieve the
same accuracy as the TD modelling using a much coarser mesh. The recommended grid
size is 1.5 mesh points per wavelength for TE models, compared to 5 mesh points per
wavelength for TD models.
While TD models require mesh continuity across all domain boundaries, TE models
do not require mesh continuity across fluid-solid boundaries. Mesh continuity implies
that the mesh at the domain boundaries must use the finer resolution grid of the domain
with the lower sound speed for two adjacent domains with different sound speeds (and
therefore different spatial resolution requirements). Because the mesh resolution can only
change gradually, this finer resolution at the boundary transforms only gradually into the
desired coarser resolution for the domain with the higher sound speed with increasing
distance from the boundary. This means that TD simulations may be slowed down by
the need to use higher resolutions in part of domains with higher sound speeds, while
TE simulations’ calculations of the wave fields across boundaries without mesh conti-
nuity are computationally heavy. Therefore, the TD formulation may be faster for small
domains with many boundaries, while the TE formulation will be faster for large con-
tinuous domains. Hence, the TD formulation has an advantage when introducing thin
layers, e.g., to represent debonding of the formation from the pipe, which introduces a
very small domain with additional boundaries.
Because the model represents a finite section of the full geometry, boundaries or do-
mains that absorb sound waves without reflecting them back into the model are neces-
sary to simulate the geometry correctly. TE models support absorbing layers for both
acoustic fluid and elastic solid domains. In the TD formulation, we use a combination
of perfectly matched layer regions and low-reflecting and impedance boundary condi-
17
10 0 10 20 30 40 50 60
Time
t
( s)
1.00
0.75
0.50
0.25
0.00
0.25
0.50
0.75
Time signal
s
, normalized
A
3D TE pipe
2.5D TE pipe
2.5D TD pipe
2.5D TD pipe wide
10 0 10 20 30 40 50 60
Time
t
( s)
50
40
30
20
10
0
Envelope (dB)
B
10 0 10 20 30 40 50 60
Time
t
( s)
50
40
30
20
10
0
Cumulative sum
CS
(dB)
C
100 150 200 250 300 350
Frequency f (kHz)
3
2
1
0
1
2
Group delay ( s)
D
Figure 13: Comparison of (A) the time signal, (B) the envelope, (C) the cumulative sum, and (D)
the group delay for a 3D and 2.5D model using the TE formulation and a 2.5D model modelled
using the TD formulation; all models with mud behind the pipe.
tions. Experience shows that this does not work as efficiently in damping the waves at
the model boundaries as the absorbing layer in the TE formulation.
Figure 13 shows results from the 3D and 2.5D TE models and the 2.5D TD model.
While the TD result (green curve) agrees well with the TE results for up to 20 µs, a signif-
icant deviation in the TD result can be observed above 20 µs. This deviation is especially
visible in the envelope (Figure 13B) and cumulative sum (13C). The effect on the group
delay (13D) is weaker, with a reduction in the depth of the group delay minimum of
0.2 µs.
However, increasing the domain width of the TD model moves this deviation and
it appears towards later times in the envelope and cumulative sum. When increasing
the domain width enough, the deviation is no longer visible within the simulated time
interval. The red curves in Figure 13 show results from a model where we increased the
domain width to 70 mm. In this case, the TD result agrees very well with the TE results.
4.3.2 Discussion
The deviation we observe in the envelope and cumulative sum of the TD model (Fig-
ures 13B and C) is visible in the COMSOL snapshots of the wave propagation (Fig-
ure 14A). We can detect a change within the parallel wavefronts marked with an arrow
in Figure 14A. This deviation is not visible in the snapshot corresponding to the wide
model. As increasing the domain width causes the deviation to appear later and even-
tually disappear, we conclude that the deviation is caused by unwanted reflections from
the perfectly matched layer in the TD formulation. When increasing the domain width,
18
Figure 14: Snapshot of the 2.5D COMSOL model after 75 µs wave propagation (corresponding
to −2.5 µs in Figure 13), with mud behind the pipe for a model with (A) 40 mm domain width and
(B) 70 mm domain width. Plotted are the total acoustic pressure for the acoustic domain and the
volumetric strain in case of the solid domain.
the distance to the perfectly matched layer increases, causing this signal to appear later.
Hence, the results gained through the TE modelling are generally more reliable due to
the better performance of its absorbing layer formulation. However, it might be beneficial
to use the TD formulation in certain circumstances, especially when introducing debond-
ing and, thus, thin acoustic layers which increase the modelling time for the TE model
significantly. When using the TD formulation, special care should be taken when setting
up the model to ensure that it is wide enough to avoid unwanted boundary effects.
5 Conclusion
When modelling PE measurements in a pipe, only 3D models can correctly capture the
geometry of both the pipe and the transducer. However, simulating 3D models is not
always feasible due to high computational power requirements for 3D modelling. How
to reduce the dimension of the model while keeping deviations of the results at an ac-
ceptable level depends on the application, the pipe diameter, and the required precision
of the study.
2.5D models can correctly capture the axisymmetric geometries of the transducer and
the plate, but represent the pipe as a spherical shell. We found that results from the
2.5D plate models, as well as the 2.5D models with large-diameter (i.e., small-curvature)
pipes show very good agreement with 3D results. However, when the pipe diameter
decreases (and the curvature increases) we observe increasingly larger discrepancies with
the 3D results, especially for the values derived from the group delay curve (group delay
minimum and pipe thickness). If the diameter is not a parameter of interest, precise
results can be found by simply modelling a plate in 2.5D.
2D models can correctly capture the geometries of the plate and the pipe, but rep-
19
resent the transducer as being infinitely long in the 3rd dimension. We found that 2D
models introduce significant errors in the results compared to 3D models. However,
the observed trends from 2D modelling are similar to those from 2.5D modelling over
a range of different impedances behind the plate. While using 2D modelling cannot be
recommended to obtain absolute values, 2D models might potentially be useful for in-
vestigating relative values, i.e., changes in the derived parameters due to changes of the
impedance outside the pipe.
We found that using the TE formulation generally generates more reliable results
than the TD formulation. TE models’ results contain less artefacts due to their better-
performing absorbing layer in the elastic and fluid-acoustic wave domains. However,
computation times can increase significantly when introducing additional boundaries,
e.g., from a thin domain to represent debonding between the pipe and the outer material.
In this case, the TD formulation is preferable, although it is important to ensure that the
model domain is wide enough to avoid unwanted edge effects.
While different approaches can be used to compute PE data of sufficient accuracy us-
ing reduced degree modelling, we observe a large difference between the analytical 1D
result and higher-dimensional numerical results. Hence, substantial corrections are re-
quired when using 1D models in an iterative process to determine the impedance behind
the pipe.
Acknowledgements
This work was performed through the Centre for Innovative Ultrasound Solutions, which
is funded by the Research Council of Norway under grant no. 237887.
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21
Appendix
Table A.1 contains the material parameters used for the modelling in this publication.
Table A.1: Material parameters sorted by P-wave impedance. Numbers with large uncertainty are
given in brackets.
Material P-wave v. S-wave v. Density Poisson’s r. Young’s m. Bulk m. Impedance
(m/s) (m/s) (kg/m3) (GPa) (GPa) (MRayl)
air 340 — 1.2 0.5 0 0.0001 0.0004
kerosene 1320 — 770 0.5 0 1.34 1.02
water 1480 — 1000 0.5 0 2.19 1.48
mud 1301 — 1250 0.5 0 2.12 1.63
ethylene 1670 — 1115 0.5 0 3.11 1.86
glycerol 1950 — 1260 0.5 0 4.79 2.46
epoxy 2510 1120 1150 0.3757 3.97 5.32 2.89
epoxy 5% 1818 (900) 1940 0.3377 4.2 4.32 3.53
shale 2100 1000 2110 0.3657 5.41 9.31 4.43
cement 3350 1920 1980 0.2554 18.33 12.49 6.63
steel 5847 3160 7980 0.2937 206 158 46.66
Figures A.1 and A.2 show the same comparison as Figures 6 and 7. The main fea-
tures discussed in the text in Sec. 4.1 apply as well to the results using cement. In the
case of cement the reverberation decays faster, reaching the noise floor, leading to larger
differences in the cumulative sum result between the 2.5D plate result and the 3D plate
result.
22
10 0 10 20 30 40 50 60
Time
t
( s)
1.00
0.75
0.50
0.25
0.00
0.25
0.50
0.75
Time signal
s
, normalized
A
2D pipe
2.5D pipe
3D pipe
10 0 10 20 30 40 50 60
Time
t
( s)
50
40
30
20
10
0
Envelope (dB)
B
10 0 10 20 30 40 50 60
Time
t
( s)
50
40
30
20
10
0
Cumulative sum
CS
(dB)
C
100 150 200 250 300 350
Frequency f (kHz)
3
2
1
0
1
2
Group delay ( s)
D
Figure A.1: Comparison of 3D pipe (black), 2D pipe (red), and 2.5D pipe (cyan) results with mud
inside and cement outside of the pipe. Shown are (A) the time signal, (B) the envelope, (C) the
cumulative sum, and (D) the group delay.
10 0 10 20 30 40 50 60
Time
t
( s)
1.00
0.75
0.50
0.25
0.00
0.25
0.50
0.75
Time signal
s
, normalized
A
2D plate
2.5D plate
3D plate
10 0 10 20 30 40 50 60
Time
t
( s)
50
40
30
20
10
0
Envelope (dB)
B
10 0 10 20 30 40 50 60
Time
t
( s)
50
40
30
20
10
0
Cumulative sum
CS
(dB)
C
100 150 200 250 300 350
Frequency f (kHz)
3
2
1
0
1
2
Group delay ( s)
D
Figure A.2: Comparison of 3D plate (black), 2D plate (red), and 2.5D plate (cyan) results with mud
inside the plate and cement outside the plate. Shown are (A) the time signal, (B) the envelope,
(C) the cumulative sum, and (D) the group delay.
23
