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\begin{document}
\title{Machine Learning Applied to Ultrasonic Flow Meters for measuring Dilute,
Turbulent Water-Bentonite Suspension Flow}
\author[1]{Thiam Wan}%
\author[1]{Hon Chung Lau}%
\author[1]{Wai Lam Loh}%
\affil[1]{National University of Singapore}%
\vspace{-1em}
\date{\today}
\begingroup
\let\center\flushleft
\let\endcenter\endflushleft
\maketitle
\endgroup
\selectlanguage{english}
\begin{abstract}
An ultrasonic flow meter that is calibrated in single phase flow has
inherent errors when applied to measure dilute water-bentonite mixture
flow. This paper endeavors to use artificial intelligence for
recalibration of an ultrasonic flow meter. A commercial ultrasonic
transit time flow meter was tested for measuring dilute water-bentonite
mixture flow of 0.1-1.0 vol\% concentration at room temperature. Results
show the test data had a systematic error of -8.3\% and a random error
of 20.3\%. The machine learning LLS regression,2D interpolation and
Gaussian Na\selectlanguage{ngerman}ïve Bayes methods were considered in this exercise. Finally,
a combined 2D interpolation method and Gaussian Naïve Bayes classifier
approach was preferred. It reduced the systematic error to -0.6\% and
random errors to ±13.7\%. Our study shows a high accuracy ultrasonic
flow meter with systematic errors smaller than 1\% for oil and gas
multiphase application is possible with the aid of artificial
intelligence technology.%
\end{abstract}\selectlanguage{ngerman}%
\sloppy
\textbf{1.0 Introduction}
Artificial Intelligence (AI) will reshape the future the flow metering
industry. The connectivity and flow of information between flow
measuring devices and sensors provide an abundance of available data.
The main goal behind the artificial intelligence research is the use of
technology and data to improve the flow meter accuracy and efficiency.
In recent years, artificial intelligence has been applied to recalibrate
utility ultrasonic flow meters (Yazdanshenashad et al. 2018). This paper
investigated the use of machine learning to address the accuracy problem
of ultrasonic flow meter in multiphase measurement. This includes the
use of available data and extracting only useful information for the
purpose of reducing costs and optimizing capacity.
According to a report by Berrebi et al. (2004), the maximum error of a
typical ultrasonic flow meter is 2\% or 3\% in turbulent flow rate (Re
\textgreater{}4000), and 5\% at laminar (Re \textless{}2000) or
transient flow (20000\textless{}Re\textless{}4000). However, this kind
of accuracy is only valid in single phase flow assuming the flow is
homogeneous. In the case of water-bentonite suspension flow, the mixture
contains particles distributed randomly throughout the fluid. Due to the
nature of the processes, the particle distribution in the fluid may
randomly vary in time and space. As the ultrasound interacts with such
particles, some microscopic phenomena may take place between the waves,
particles and the flow (Eren 1998). For example, a water-bentonite
suspension up to 1 vol\% consists of two or more substances of very
different acoustic impedance that alter the acoustic signal and the way
ultrasonic beam is transmitted through and reflected from multiphase
solid-solid, solid-fluid and fluid-fluid interfaces (Sirmurda et al.
2016). Therefore, the accuracy of ultrasonic flow meter in measuring
particulate flow such as water-bentonite mixture flow is still a
question that needs to be addressed.
In practice, transit time ultrasonic flow meters are sensitive to many
factors. They are sensitive the variation in velocity profile and the
installation effects. For example, all ultrasonic flow meters assume an
ideal flow profile before the disturbing geometry. However, the distance
between the flow meter and elbows in reality is typically not
sufficiently long to redevelop such an ideal profile. In a recent study
(Weissenbrunner et al. 2016), CFD simulation is used to quantify the
uncertainties of ultrasonic flow meter caused by variations of the
inflow profiles. The study revealed a bias of 1.5-4.5\% as flow meter
was installed at a distance smaller than 40 pipe diameters to the double
elbow. Another experiment (Ma et al. 2012) showed that moving the
ultrasonic transducer 0.2 to 0.6 mm axially from the correct position
led to a velocity error as much as 4 - 10\% in a transit time ultrasonic
flow meter. Changes in fluid density, flow viscosity and flow patterns
affected the dynamical characteristics in the ultrasonic flow meter
measurement (Catak and Ergan 2019). Therefore, re-calibration is often
necessary to reduce errors due to the possible change of the fluid
density, viscosity and flow patterns in multiphase flow.
In this study, we have examined the accuracy of transit time ultrasonic
flowmeters in measuring dilute water-bentonite suspension flow up to 1
vol\% bentonite concentration. Error analysis has been conducted to
evaluate the meter performance in terms of systematic errors (indicated
by the mean relative error) and random errors (indicated by the standard
deviation value of relative errors). Next, the study attempted to close
the gap for both type of errors using multiple error reduction
algorithms including LLS regression method, 2D interpolation method, and
various Gaussian Na\selectlanguage{ngerman}ïve Bayes classifier algorithms. The goal is to
reduce the systematic error to less than 1\% and improve random errors
as much as possible without compromising the resolution of data.
Typical results of the measurement errors are presented in section 4.2,
and the results of various machine learning error reduction exercise
will be presented in section 4.3. Finally, the best combination of
machine learning models is trained to reduce both systematic errors and
random errors as presented in section 5.0. This helps to re-calibrate
the transit time ultrasonic flow meters and improve its accuracy for
future use of ultrasonic flow meter measuring drilling fluid flow.
Further evaluation will be continued in higher bentonite concentrations
in the future. Another important issue in the accurate measurement of
multiphase flow is that of temperature compensation. Using machine
learning models for temperature compensation in an ultrasonic flow meter
will be investigated in future.
\textbf{2.0 Literature Survey}
This review introduces the definitions used in error analysis,
summarizes the past activities in flow meters re-calibration and error
reduction exercise for ultrasonic flow meters.
\textbf{2.1 Operating Principle of Transit Time Ultrasonic Flowmeter}
The single path transit-time ultrasonic flowmeter utilizes two
piezoelectric transducers, which are clamped on the outside of the
closed pipes at a specified distance from each other, as depicted in
Figure 1.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image1/image1}
\end{center}
\end{figure}
\emph{Figure 1: Transit-time ultrasonic flowmeter.}
The flowmeter measures fluid velocity by transmitting acoustic signals
between the two transducers alternatively, first in the opposing
direction of fluid flow, and then in the direction of flow. The transit
time of the ultrasonic signal from the downstream transducer to the
upstream transducer (\(t_{21}\)) and that in the opposite
direction (\(t_{12}\)) are computable using Eq. (1) and (2) as
follows.
\(t_{21}=\frac{L}{\left(c+vcos\theta\right)}\) \ldots{}\ldots{} (1)\(t_{12}=\frac{L}{\left(c-vcos\theta\right)}\)
\ldots{}\ldots{} (2)
where \(c\) is the ultrasound speed in water,
\(L\) is the path travelled by ultrasound, and
\selectlanguage{greek}θ\selectlanguage{ngerman} is the incident angle.
Ultrasonic meters are velocity meters by nature. The fluid flow
velocity\(v\) is calculated from the differences between
the transit times of the signals which are directly proportional to
fluid velocity. The equation for the flow velocity is:
\(v=\frac{L}{2cos\theta}\left(\frac{1}{t_{12}}-\frac{1}{t_{21}}\right)\)\ldots{}\ldots{} (3)
In a single-path flow meter, the ultrasonic beam path is diagonal, and
the fluid flow velocity \(v\) is computed along this path.
However, the velocity needed for computing volume flow rate is the
average fluid velocity \(\overset{\overline{}}{v}\) across the pipe cross
sectional area. Therefore, to convert velocity \(v\) to
average fluid velocity\(\overset{\overline{}}{v}\), an average velocity
correction factor is used that is shown as \(k_{c}\) according
to Eq. (4). This quantity is generally a function of the Reynolds'
number.
\(\overset{\overline{}}{v}=k_{c}v\) \ldots{}\ldots{} (4)
Meter manufacturers have differing methodologies for computing velocity
correction factor \(k_{c}\). Some derive it by using propriety
algorithms. This requires knowledge about the velocity profile patterns
at different Reynolds' numbers. Nonetheless, the flow meter is still
subjected to a certain degree of uncertainty. For example, fluid
velocity profiles in the pipeline are not always uniform. Often there is
swirl and asymmetrical flow profile within the meter (Lansing et al.
2003).
\textbf{2.2 Error Analysis}
All measurements have some degree of uncertainty that may come from a
variety of sources. The process of evaluating the uncertainty associated
with a measurement result is often called error analysis. In flow
measurement terminology, the error is the difference between the true
value of a measurement and the recorded value of a measurement, which
can be classified into two broad categories.
A \textbf{systematic error} (or bias) refers to deviations that are not
due to the changes in flow. It occurs with a measuring device that is
faulty or improperly calibrated so that it consistently overestimates
(or underestimates) the measurements by X units. Systematic errors
cannot be reduced by taking more measurements. To reduce the systematic
error of a data set, researchers must identify the source of the error
and remove it. For example, the errors can be reduced by recalibrating
the flow meter. In other circumstances, the error could be due to the
inherent nature of the measuring technique. Hamouda et al (2016)
reported a scenario where an ultrasound flow meter measured fluid flow
rate based on transit time principle. This difference can be as low as a
few picoseconds, which give rise to technical difficulties in measuring
such a small time-difference with a given accuracy. This type of
systematic errors can be reduced by using more sensitive sensors or
avoiding conducting measurement in low flow range accuracy.
A \textbf{random error} is a deviation that randomly fluctuates over a
mean value. It has no preferred direction. It occurs because there are a
very large number of parameters beyond the control of the instrument
that may interfere with the results of the measurement. For example, a
random error occurs due to the instrument resolution (CDL 2020) and the
way it is affected by changes in the surroundings (Kalla 2009).
Nevertheless, the readings may be imprecise, but not inaccurate, as the
averaging over large number of observations will yield a net effect of
zero deviation.
According to definitions found in literature (CPL
2020),\textbf{accuracy} is defined as the closeness of agreement between
a measured value and a true or accepted value. Accuracy is often
reported quantitatively by using relative error:
\(Relative\ Error=\frac{Measured\ value-Expected\ value}{\text{expected\ value}}\)\ldots{} (5)
Meanwhile, \textbf{precision} is the degree of consistency among
independent measurements of the same quantity; also described as the
reliability or reproducibility of the result (CPL 2020). Precision is
often reported quantitatively by using relative or fractional
uncertainty:
\(Relative\ Uncertainty=\left|\frac{\text{Uncertainty}}{\text{measured\ quantity}}\right|\)\ldots{}. (6)
\textbf{Uncertainty} analyses are essential to determine whether
measurement systems are capable of meeting performance targets.
According to ISO-5168, the uncertainty of a flow measurement should be
specified at a confidence level of 95\%, which corresponds to two
standard deviations.
\(v=\overset{\overline{}}{v}\pm 2\sigma_{v}\) \ldots{}. (7)
where \(\overset{\overline{}}{v}\) is mean velocity and \(\sigma_{v}\)is the
standard deviation of flow data.
In summary, random error corresponds to imprecision (or repeatability),
and bias to inaccuracy.
\textbf{2.3 Error Reduction Algorithm for Self-Calibration of Ultrasonic
Flow Meters}
In recent years, the use of artificial intelligence in flow metering
have attracted researchers' attention. For example, neural networks and
support vector regression algorithms have been applied to the data from
temporal and spatial ultrasonic level measurements of the drilling fluid
in the open channel to estimate the flow rate (Chhantyal et al. 2017).
The Least Square Error Reduction technique and neural networks method
have been used for self-calibration of ultrasonic water flow meter
(Yazdanshenashad et al. 2018; Catak and Ergan 2019). However, none of
these self-calibration exercises involves the use of transit-time
ultrasonic flow meter in multiphase flow such as water-bentonite mixture
flow.
Catak and Ergan (2019) reported using the least square error method for
the self-calibration of ultrasonic water flow meter. Three common least
square errors calibration methods have been employed to the data
obtained from DN-20 type ultrasonic flow meter, namely, Linear Least
Squares (LLS), Weighted Least Squares (WLS), and Piecewise Linear Least
Squares (PLR). The results presented found PLR gave the best results in
all cases, while WLS was the best for higher flow rates. Both WLS and
LLS were especially not adequate for low level of flowrate. For example,
the flowmeter accuracy at 10 L/h (0.167 L/min) was around 5-8\%, but
after calibration, the improvement was only about 0.8-1.3\%. (Catak and
Ergan 2019).
In a recent report, Yazdanshenashad et al. (2018) use the Multi-Layer
Perceptron Neural Network (MLPNN) model to calibrate an ultrasonic flow
meter to achieve an error smaller than 1.5\%. The measured flow range
was from 0.2 to 4 m\textsuperscript{3} per hour. However, it was only
aimed to reduce systematic errors. The authors did not report on the
improvement in random errors which could be revealed by the change in
standard deviation value of the errors. The authors have instead
suggested reducing random errors by averaging large number of data. But
the main drawback of this approach is the loss in data resolution. If it
has to average 1000 data to significantly reduce the random error, it
also means the resolution of data would be compromised by 1000 times.
\textbf{3.0 The Present Work}
\textbf{3.1 Experimental Test Rig}
A set of ultrasonic tests of a concentrated liquid--particle system in
shear flow has been carried out using a transit-time ultrasonic flow
meter. Ca-Bentonite particles (estimated nominal diameter of 60-70 \selectlanguage{greek}μ\selectlanguage{english}m,
dried density \(\rho_{f}\) = 2500 kg/m\textsuperscript{3}) in
water at shear flow of 0-1.5 m/s were studied, for volume fraction up to
1 vol\% (or approximately 2.5 wt\%). Tests were performed for particle
volume fractions ranging from 0 \%, in steps of 0.05 vol\% until 1.0
vol\%.
Figure 2 shows a schematic of the experimental unit used. The test loop
is equipped with a mixing tank, a high-level tank, a mechanical mixer, a
progressive cavity pump, ultrasonic flow meters, a bellow, valves and a
pressure indicator. It consists of a 1 in (24.5 mm) diameter PVC test
section and a 2 in (54.78 mm) diameter stainless steel (SS304) test
section.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image2/image2}
\end{center}
\end{figure}
\emph{Figure 2: Schematic of experimental setup.}
A mixing tank is used to generate a mixture of bentonite powder with
known concentration in water. Dried calcium bentonite powder was
dispersed in water with the assistance of a mechanical stirrer. The
bentonite powder was added at a ratio according to Eq (15) in section
3.2. Before the experiments, the suspension was stirred for 15 minutes.
The homogeneous fluid mixture was then re-circulated by a progressive
cavity pump in a closed loop for ultrasonic flow measurement. The pump
motor speed was regulated by a variable frequency drive (VFD) via a
speed potentiometer. A bellow was installed at the pump discharge side
to minimize the piping vibration which may affect the meter accuracy.
The single path transit-time ultrasonic flowmeter utilized two
piezoelectric transducers, which were clamped on the outside of the
closed pipes at a specified distance from each other. The flowmeter
measured fluid velocity by transmitting acoustic signals between the two
transducers alternatively, first in the opposing direction of fluid
flow, and then in the direction of flow. The differences between the
transit times of the signals were directly proportional to fluid
velocity. The flow output meter signal was then fed to the PC after
being converted to an appropriate digital signal. The mud fluid was
slowly pumped into 2-in (54.78 mm ID) stainless steel test section,
which was controlled by the manual operated ball valve.
Experimentally, the output of transit-time ultrasonic flowmeter alone
was insufficient, and a joint measurement with another reference method
was normally required. Therefore, the desired flow was drained to an
overhead tank for volumetric measurement. Video was recorded and the
video images were processed by an algorithm to acquire the time series
flow rate for comparison with the transit-time ultrasonic flowmeter
readings.
After completion of all measurements, The experimental data were
recorded in room temperature (27\textasciitilde{}33\selectlanguage{ngerman}°C) were analyzed to
obtain relationship between signal patterns in the ultrasonic sensors
and bentonite concentration, flow speed and Reynolds number. Each
experiment was conducted at unsteady flowing condition with flow
accelerating from rest to 1.2 \textasciitilde{}1.5 m/s, and next,
decelerated to rest again, within a duration of 25\textasciitilde{}30
seconds.
\textbf{3.2 Preparation of Water Based Bentonite Fluid Mixture}
Bentonite is naturally occurring clay. It is inorganic, non-toxic, and
non-irritating. It is not considered hazardous on skin contact and is
employed in cosmetics and skin products as a suspender. It is therefore
very safe to use the bentonite clay in the experiment.
In this work, a water-based drilling mud fluid was prepared by using
pure water as the base fluid and bentonite clay as a viscosifier. The
bentonite clay has a density of 2500 kg/m\textsuperscript{3} and the
water density is 1000 kg/m\textsuperscript{3}.
For mud velocity calculations, the following equations are used.
\(V_{s}+V_{m1}=V_{m2}\ \)\ldots{}\ldots{} (8)\(\rho_{s}V_{s}+\rho_{m1}V_{m1}=\rho_{m2}V_{m2}\)
\ldots{}\ldots{} (9)
From equation (8):
\(V_{m1}=V_{m2}-\ V_{s}\) \ldots{}\ldots{} (10)
Substituting Eq (10) into Eq (9) yields:
\(V_{s}\ =V_{m2}\left[\frac{\rho_{m2}-\rho_{m1}}{\rho_{s}-\rho_{m1}}\right]\)\ldots{}\ldots{} (11)
Multiplying Eq (11) by the density of solid yields the corresponding
weight (of solids) required.
\({\rho_{s}V}_{s}\ =\frac{\rho_{s}V_{m2}\left(\rho_{m2}-\rho_{m1}\right)}{\rho_{s}-\rho_{m1}}\)\ldots{}\ldots{} (12)
By re-arranging Eq (11), the density of final mixture is given by:
\(\rho_{m2}=\rho_{m1}+\left(\rho_{s}-\rho_{m1}\right)\left(\frac{V_{s}}{V_{m2}}\right)\ \)\ldots{}\ldots{} (13)
Re-arranging Eq (9) and dividing by \(V_{m2}\) , the volume
percentage of final mixture is given by:
\(\frac{V_{s}}{V_{m2}}=\left[\frac{\rho_{m2}V_{m2}-\rho_{m1}V_{m1}}{\rho_{s}V_{m2}}\right]\)\ldots{}\ldots{} (14)
Bentonite swells when it is wet. This expansion is due to the adsorption
of water. Hence the clay volume increases and the total volume (clay
plus water) is, for practically purposes, unchanged (Othman 2007).
The particle concentration in weight percentage of final mixture is
given by:
\(\frac{w_{s}}{w_{m2}}=\left(\frac{\rho_{s}}{\rho_{m2}}\right)\left(\frac{V_{s}}{V_{m2}}\right)\)\ldots{}\ldots{} (15)
According to Birgersson et al. (2009), Ca-bentonite forms a stable
sediment with water at a solid mass ratio well below 10, which has
sand-like properties. Due to low swelling capacity, Ca-bentonite is
unable to form colloids.
\textbf{4.0 Results and Discussions}
\textbf{4.1 Flow Rate Measurement in 2-in diameter Stainless Steel pipe}
During the experiments, a homogeneous bentonite-water mixture was pumped
through a 2-in pipe. Variable frequency drive (VFD) was installed to
regulate the pump speed. This was to minimize the disturbance resulting
from throttling the valve to improve the flow meter accuracy. Figure 3
shows typical experimental readings acquired from transit-time
ultrasonic flow meter and its comparison to the rate of the flow as
collected in the volume tank. Each experiment was conducted at unsteady
flowing condition with flow accelerating from rest to 1.2
\textasciitilde{}1.5 m/s, and next, decelerated to rest again, within a
duration of 20\textasciitilde{}25 seconds.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image3/image3}
\end{center}
\end{figure}
\emph{Figure 3: Typical time series results of transit-time flow
measurement for various bentonite concentrations.}
The validation was carried out by a comparison between the measurement
made by the device under test and that by a reference method. In this
study, the reference readings were acquired by the `static method' based
on collecting fluid in a volumetric tank and determining its quantity by
a static measurement (measuring the change in fluid level in tank).
Figure 4 shows a scattered plot for the transit-time ultrasonic flow
meter reading versus the actual flow measured by the volume tank. As
expected, the scattered points show a positive linear pattern as they
move from left to right. Transit-time ultrasonic flow meter exhibits
good linearity, giving a predictable response for a known change in the
flow rate. The Pearson coefficient R is 0.9529 indicating a strong
positive correlation.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image4/image4}
\end{center}
\end{figure}
\emph{Figure 4: Scattered plot of transit-time ultrasonic flowmeter
performance on 2-in horizontal bentonite-water mixture flow for
bentonite concentration of 0.1 -- 1.0 vol\%.}
In Figure 5, the scattered points of flow coefficient \(C\)
for transit-time ultrasonic flowmeter have been found to stabilize as
the flow speed exceeds 0.6 m/s. The flow coefficient
C\selectlanguage{ngerman} is defined as the ratio of measured flow rate to the
actual fluid rate. The equation is \(C=Q_{\text{meter}}/Q_{\text{actual}}\)
or\(V_{\text{meter}}/V_{\text{actual}}\). The value of\(C\) should approach
unity (C =1) if the meter is accurate. But the mean flow
coefficient \(\overset{\overline{}}{C_{\text{\ All}}}\) is obtained as 0.9651.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image5/image5}
\end{center}
\end{figure}
\emph{Figure 5: Flow coefficient versus Flow velocity for bentonite
concentration of 0.1 -- 1.0 vol\%}
\textbf{4.2 Error Analysis}
An error is defined as a deviation from its truth value (Abedjan et al.
2016). The relative error is determined by subtracting ultrasonic flow
meter readings from the reference velocity divided by reference
velocity. Figure 6 show the relative error for all experimental data
acquired.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image6/image6}
\end{center}
\end{figure}
\emph{Figure 6: Relative error versus flow velocity for bentonite
concentration of 0.1 -- 1.0 vol\%.}
In this study, a total of 160 sample data were taken from experiments in
a range of flow velocity (0 -1.2\textasciitilde{}1.5 m/s) and bentonite
concentration (0.1-1.0 \% volume). The machine learning models went
under training. About 62\% of the data were used for training, and the
remaining 38\% were used for testing.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image7/image7}
\end{center}
\end{figure}
\emph{Figure 7: The training data set and test data set used to conduct
recalibration exercise.}
From the error analysis in Table 1, the mean relative
error\(\overset{\overline{}}{\epsilon_{\text{All}}}\) is -0.033 (-3.03\%), with a standard deviation
\(\sigma_{\text{All}}\) of 0.3047 (30.47\%). Before applying the error
reduction algorithm, the test dataset has a mean flow coefficient
\(\overset{\overline{}}{C_{\text{Test}}}\) of 0.917, a mean relative error\(\overset{\overline{}}{\epsilon_{\text{Test}}}\) of
-0.0831 (-8.31\%) and a standard deviation \(\sigma_{\text{Test}}\) of \selectlanguage{ngerman}±0.2030
(20.30\%).
\emph{Table 1: Summary of error information of sampled data.}\selectlanguage{english}
\begin{longtable}[]{@{}lllll@{}}
\toprule
Item & Subject & Mean Relative Error, \(\overset{\overline{}}{\epsilon}\) & Std.
Deviation, \(\sigma\) & Uncertainty, \(\overset{\overline{}}{\epsilon}\ \)\selectlanguage{ngerman}±
\(2\sigma\)\tabularnewline
\midrule
\endhead
1 & All Data & -0.033 (-3.3 \%) & \selectlanguage{ngerman}±0.305 (±30.5\%) & \selectlanguage{ngerman}±0.610
(61.0\%)\tabularnewline
2 & Train Data Set & -0.007 (0.7\%) & \selectlanguage{ngerman}±0.344 (±34.3\%) & \selectlanguage{ngerman}±0.688
(68.8\%)\tabularnewline
3 & Test Data Set & -0.0830 (-8.3\%) & \selectlanguage{ngerman}±0.203(±20.3\%) &
\selectlanguage{ngerman}±0.406(±40.6\%)\tabularnewline
\bottomrule
\end{longtable}\selectlanguage{ngerman}
\textbf{4.3 Machine Learning Error Reduction Exercises}
\selectlanguage{english}\textbf{4.3.1 Linear Least Square Regression Technique (LLS)}
The least square approach is employed to determine the parameters for
obtaining the ''best fitting line'' for a series of observations. It
aims to train parameter \(\beta_{o}\) and \(\beta_{1}\) in a
linear equation as given below:
\(y=\beta_{o}+\beta_{1}x\) \ldots{}\ldots{} (16)
Let us consider n observations of x and y to build a mathematical model
using the parameters \(\beta_{o}\)and \(\beta_{1}\), such
that:
\(y_{1}=\beta_{o}+\beta_{1}x_{1}+r_{1}\)\(y_{2}=\beta_{o}+\beta_{1}x_{2}+r_{2}\)\ldots{} \ldots{}\ldots{}
(17)\(y_{n}=\beta_{o}+\beta_{1}x_{n}+r_{n}\)
where \(r_{i}\) is the residual error terms, \(\beta_{o}\)
is the is the intercept of the line with the y axis, and
\(\beta_{1}\) is the slope of the line.
The corresponding cost function is defined as:
\(h\left(\beta_{o},\beta_{1}\right)=\frac{1}{2}\sum_{i=1}^{n}r_{i}^{2}=\frac{1}{2}\sum_{i=1}^{n}\left(y_{i}-\beta_{o}-\beta_{1}x_{i}\right)^{2}\)\ldots{}\ldots{} (18)
As the cost function \(h\left(\beta_{o},\beta_{1}\right)\) is at it minimum, the partial
derivatives of\(h\left(\beta_{o},\beta_{1}\right)\) with respect to\(\beta_{o}\ \)and
\(\beta_{1}\) is equal to zero. This ultimately leads to the
solutions of \(\hat{\beta_{o}}\) and \(\hat{\beta_{1}}\) which satisfies
the least square error to give the best fitting line for a number of
given data.
In this machine learning study, the least square approach is used to
train ''best fitting line'' for the scattered data as depicted in Figure
8 below. The purpose is to obtain the value of \(\beta_{o}\) (the
intercept of the line with the y axis) and \(\beta_{1}\) (the
slope of the line), as previously introduced in Eq. (16).\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image8/image8}
\end{center}
\end{figure}
\emph{Figure 8: Linear least square error reduction for training data.}
Results give \(\beta_{o}\) = 1.2082, and \(\beta_{1}\)=
-0.3241. Eq (16) is then modified to yield \(C_{\text{LLS}}\):
\(C_{\text{LLS}}=\beta_{o}+\beta_{1}{\overset{\overline{}}{v}}_{T\text{est}}\)\ldots{} (19)
where \({\overset{\overline{}}{v}}_{T\text{est}}\) is the test data given by transit time
ultrasonic flow meter.
Next, the revised flow coefficient \(C_{\text{LLS}}\) was applied to the
correction equation, which corrects the value of ultrasonic flow
measurement.
\(v_{\text{Rev}}=\frac{v_{T\text{est}}}{C_{\text{LLS}}}\) \ldots{} (20)
By applying the correction equation, the mean flow
coefficient\(\overset{\overline{}}{C_{\text{\ Rev}}}\) has been revised from 0.917 to 0.957. As
shown in Table 2, the mean relative error has reduced from -0.0830
(-8.3\%) to -0.0422 (-4.2\%). However, the revised standard deviation of
relative error \({\sigma_{\text{Rev}}}_{\text{\ \ }}\)is about the same at \selectlanguage{ngerman}±0.2004 (20.04\%).
This indicates the LLS regression method has reduced both the systematic
error and the relative error only slightly.
\emph{Table 2: Summary of error information before and after the LLS
Regression model was implemented.}\selectlanguage{english}
\begin{longtable}[]{@{}lllll@{}}
\toprule
Item & Subject & Mean Relative Error, \(\overset{\overline{}}{\epsilon}\) & Std.
Deviation, \(\sigma\) & Uncertainty, \(\overset{\overline{}}{\epsilon}\ \)\selectlanguage{ngerman}±
\(2\sigma\)\tabularnewline
\midrule
\endhead
1 & Test Data Set & -0.0830 (-8.3\%) & \selectlanguage{ngerman}±0.203 (±20.3\%) & \selectlanguage{ngerman}±0.406
(±40.6\%)\tabularnewline
2 & LLS regression method & -0.0422 (-4.2\%) & \selectlanguage{ngerman}±0.200 (±20.0\%) & \selectlanguage{ngerman}±0.401
(±40.1\%)\tabularnewline
\bottomrule
\end{longtable}\selectlanguage{ngerman}
\textbf{4.3.2 Linear 2D Interpolation Error Reduction}
In numerical analysis, a multivariate interpolation is interpolation on
a function of more than one variable. The function to be interpolated is
known at given points and the interpolation problem consists of yielding
values at arbitrary points. Multivariate interpolation could be useful
for flow meter self-calibration, where it is used to create a digital
model from a set of error information (such as flow coefficient
\(C\)) based on flowmeter measurements. It can be used to
improve meter output at points where the error information is unknown.
Python coded 2D grid data algorithm has been used to fix a surface of
the form z = f(x,y) from the training data (\(x,y,z\)). For
example, (\(x,y,z\)) can be information extracted from flow
meter outputs such as x as flow meter readings, y as Reynolds number and
z as flow coefficient\(C\) . (\(x_{q},y_{q}\)). The
grid-data function is then used to interpolate the surface at the query
points or test points (\(x_{q},y_{q}\)) and returns the interpolated
values (\(z_{i}\)). As illustrated in Fig. 9, the returned
interpolated values can be visualized by using 2D Delaunay Triangulation
method.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image9/image9}
\end{center}
\end{figure}
\emph{Figure 9: The interpolated plot as visualized by 2D Delaunay
triangulation method.}
The returned interpolated values \(z_{i}\) based on test point
(\(x_{q},y_{q}\)) are plotted in Figure 10. However, 2D interpolation
method does not extrapolate but fill the values which are not within the
input area with 'nan' values by default, illustrated in Fig. 10 (a) as
the area in white color. As can be seen in Fig. 10(b), these `nan'
values can be replaced by numeric `1', illustrated as the area in blue
color.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image10/image10}
\end{center}
\end{figure}
\emph{Figure 10: The interpolated error information based on training
points.}
The interpolated error information will be used to revise the flowmeter
output as follow:
\(v_{\text{Rev}}=\frac{v_{\text{TTUF}}}{Z_{i}}\) \ldots{} (21)
By applying the correction equation (21), the mean flow
coefficient\(\overset{\overline{}}{C_{\text{\ Rev}}}\) has improved from 0.917 to 1.012. The
machine learning method such as 2D interpolation is inherently
non-linear, so it can detect almost any kind of non-linear behaviors and
produce a better estimate for the output and response. It gives a better
accuracy than linear regression approach such as LLS method.
As summarized in Table 3, the mean relative error\(\overset{\overline{}}{\epsilon_{\text{Rev}}}\) was
revised from -0.0830 (-8.30\%) to +0.0124 (+1.24\%). However, the error
was reduced to less than 1.5\% but at the cost of increased standard
deviation,\(\sigma_{\text{Rev}}\) from \selectlanguage{ngerman}±0.2030 (20.30\%) to ±0.2702
(27.02\%).
\emph{Table 3: Summary of error information before and after
implementing the 2D interpolation model.}\selectlanguage{english}
\begin{longtable}[]{@{}lllll@{}}
\toprule
Item & Subject & Mean Relative Error \(\overset{\overline{}}{\epsilon}\) & Std. Deviation
\(\sigma\) & Uncertainty \(\overset{\overline{}}{\epsilon}\ \)\selectlanguage{ngerman}±
\(2\sigma\)\tabularnewline
\midrule
\endhead
1 & Test Data Set & -0.0830 (-8.3\%) & \selectlanguage{ngerman}±0.203(±20.3\%) &
\selectlanguage{ngerman}±0.406(±40.6\%)\tabularnewline
2 & 2D Interpolation method & +0.0124 (+1.2\%) & \selectlanguage{ngerman}±0.270 (±27.0\%) &
\selectlanguage{ngerman}±0.540 (±54.0\%)\tabularnewline
\bottomrule
\end{longtable}\selectlanguage{ngerman}
\textbf{5.0 Filtering off Data with Large Relative Error}
Data cleaning has played a critical role in ensuring data quality for
industrial applications. With the increasing prevalence of data-centric
approaches to business and scientific problems with data as a crucial
asset, data cleaning has become even more important (Abedjan et al.
2016).
As can be seen in preceding Table 1 and Table 2, the results from both
LLS regression method and 2D interpolative model has failed to reach the
accuracy below 1\% error, moreover, it does not improve on the standard
deviation of the relative errors. As reviewed in preceding section 2.2,
the improvement of mean relative error can be associated with the
improvement of the \textbf{systematic error} (or bias). However, the
failure in reducing the standard deviation of relative error is believed
to be the inherent nature of random errors.
As depicted in Fig. 11, it was clearly shown that standard deviation can
increased drastically due to the presence of random errors.
Yazdanshenashad et al. (2018) believed the random errors cannot be
reduced by flow meter recalibration. They suggested to reduce the errors
by averaging large number of data.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image11/image11}
\end{center}
\end{figure}
\emph{Figure 11: Summary of error information of 160 sampled data.}
In this study, we decided to address the issue by the filtering off
sampled data which had large relative errors. However, these incoming
with random errors were not known in real-time operation. Therefore, it
requires a way to train the artificial intelligence models to predict
data errors accurately before the data are cleaned. A successful
exercise should lead to an outcome of lower standard deviation of
relative errors.
\textbf{5.1 Error Data Prediction using Bayes Theorem Classifier
Algorithm}
Error data prediction can be categorized as a type of anomaly detection
(or outlier detection), which is the identification of data or
observations which differs significantly from majority of the data. The
most common way to perform anomaly detection is using the classification
algorithm.
Naive Bayes is a classification algorithm for binary (two-class) and
multi-class classification problems (Brownlee, 2016). This method is a
set of supervised learning algorithms based on applying Bayes' theorem
with the ``naive'' assumption of conditional independence between every
pair of features given the value of the class variable (Zhang, 2004).
The Theorem was named after English mathematician Thomas Bayes
(1701-1761). Bayes' Theorem is stated as:
\(P(A/B)\ =\frac{P(B/A).P(A)\ }{P(B)}\) \ldots{} (22)
\(P(B/A)\ =\frac{P(B\bigcap A)\ }{P(B)}\) \ldots{} (23)
where \(P(A/B)\ \)is the probability of class (A) given the
provided data (B).
Bayes' theorem allows users to figure out \emph{P(A\textbar{}B)}
from\emph{P(B\textbar{}A).} Rather than attempting to calculate the
probabilities of each attribute value, the data are assumed to be
conditionally independent given the class value. As a result, Na\selectlanguage{ngerman}ïve
Bayes classifier approach requires only a small amount of training data
to predict and classify the outcome. In spite of the assumption that the
attributes do not interact (which is most unlikely in real data), the
approach has worked quite well in many real-world situations.
Naive Bayes can be extended to real-valued attributes by assuming a
Gaussian distribution. This extension of naive Bayes is called Gaussian
Naive Bayes (Brownlee, 2016). A collection of data points typically has
a certain distribution (e.g. a Gaussian distribution). To detect error
data (anomalies), the probability distribution p(x) from the data points
is first calculated. As a new datum, x, comes in, we compare p(x) with a
threshold r. If p(x)\textless{}r, it is considered as an error or
anomaly. This is because normal examples tend to have a large p(x) while
anomalous examples tend to have a small p(x) (Flovik 2019). This is the
easiest way to proceed because users only need to estimate the mean and
the standard deviation from the training data.
In this exercise, 62\% of our experimental data were used as the
training data, while the remaining 38\% were used as the test data. The
filtration task is defined as removing data which contains large errors
(i.e. \(C\) \selectlanguage{english}[?] 0.85 or \(C\) \selectlanguage{english}[?] 1.15). Three
types of Gaussian Naive Bayes classifier algorithms were used to predict
the probabilities of having data with large errors. The probabilities
will be used to classify whether the data is the error data which should
be removed or retained. Three types of Gaussian Naive Bayes algorithm
have been implemented, which are (a) Gaussian NB without calibration,
(b) Gaussian NB based on a non-parametric isotonic regression
calibration, and (c) Gaussian NB based on Platt's sigmoid model
calibration. This probability calibration is reported to improve the
confidence on the prediction (Metzen, 2015).
Figure 12 (a) shows the background information of the test. The
red-colored points represent data with an acceptable error (where 0.85
\textless{} \(C\) \textless{} 1.15), while dark colored
points represent the targeted points to be filtered because of excessive
error (i.e.\(C\) \selectlanguage{english}[?] 0.85 or \(C\) \selectlanguage{english}[?] 1.15).
After the Gaussian Naive Bayes scheme is implemented, the results are
plotted in Figure 12 (b). The dark colored points have been filtered.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image12/image12}
\end{center}
\end{figure}
\emph{Figure 12: Graph (a) shows the flow Coefficient C and Absolute
Relative Error of the test data. Dark colored points indicate the
targeted points to be removed because of excessive errors (i.e. C [?] 0.85
or C [?] 1.15); Graph (b) shows the results after the Gaussian Naïve Bayes
Classifier Scheme was implemented.}\selectlanguage{ngerman}
Comparing the two graphs, most of the data points with significant
errors have been predicted and removed. The exercise was repeated with
Gaussian NB methods based on isotonic calibration and sigmoid
calibration. The final outcomes were summarized in Table 4.
\emph{Table 4: Summary of error before and after implementing the
Gaussian Naïve Bayer Scheme.}\selectlanguage{english}
\begin{longtable}[]{@{}lllll@{}}
\toprule
Item & Subject & Mean Relative Error \(\overset{\overline{}}{\epsilon}\) & Std. Deviation
\(\sigma\) & Uncertainty \(\overset{\overline{}}{\epsilon}\ \)\selectlanguage{ngerman}±
\(2\sigma\)\tabularnewline
\midrule
\endhead
1 & Train Data Set & -0.007 (0.7\%) & \selectlanguage{ngerman}±0.344 (±34.3\%) & \selectlanguage{ngerman}±0.688
(68.8\%)\tabularnewline
\textbf{2} & \textbf{Test Data Set} & \textbf{-0.0830 (-8.3\%)} &
\selectlanguage{ngerman}\textbf{±0.203 (±20.3\%)} & \selectlanguage{ngerman}\textbf{±0.406(±40.6\%)}\tabularnewline
3 & Manual removing of test data with flow coefficient
\(C\leq 0.85\ \ \)or \(C\geq 1.15\) (as depicted in Fig. 12 (a)). &
-0.0588 (-5.9\%) & \selectlanguage{ngerman}±0.073 (±7.3\%) & \selectlanguage{ngerman}±0.146 (±14.6\%)\tabularnewline
\textbf{4} & Filtering Test Data using \selectlanguage{ngerman}\textbf{Gaussian Naïve Bayes
algorithm} \textbf{with no calibration} & \textbf{-0.0945 (-9.5\%)} &
\selectlanguage{ngerman}\textbf{±0.080 (±8.0\%)} & \selectlanguage{ngerman}\textbf{±0.161 (±16.1\%)}\tabularnewline
5 & Filtering Test Data using Gaussian Na\selectlanguage{ngerman}ïve Bayes algorithm with
isotonic calibration & -0.0928 (-9.3\%) & \selectlanguage{ngerman}±0.124 (±12.4\%) & \selectlanguage{ngerman}±0.247
(±24.7\%)\tabularnewline
6 & Filtering Test Data using Gaussian Na\selectlanguage{ngerman}ïve Bayes algorithm with
sigmoid calibration & -0.0839 (-8.4\%) & \selectlanguage{ngerman}±0.117 (±11.7\%) & \selectlanguage{ngerman}±0.234
(±23.4\%)\tabularnewline
\bottomrule
\end{longtable}\selectlanguage{ngerman}
The results in Table 2 indicate the Gaussian Naïve Bayes algorithm (with
no calibration) is good enough to filter raw data with large errors. A
significant reduction of the standard deviation value of relative
error\(\sigma_{\text{Rev}}\) from ±20.3\% to ±8.0 \% shows the Naïve Bayes
method is effective to reduce the random errors. However, it may not be
useful to correct the systematic error, as the mean relative
error\(\overset{\overline{}}{\epsilon_{\text{Rev}}}\ \)after the exercise has increased from -8.3\% to
-- 9.5\%.
\textbf{5.3 Combined Strategies using 2D Interpolation and Bayes Theorem
Classifier Algorithm.}
There are multiple types of errors associated with machine learning and
predictive analytics and different types of errors may coexist in the
same data set (such as systematic errors and random errors). It might be
necessary to run more than one kind of tool. In the final resort to
improve both type of errors, it was decided to implement the 2D
interpolative error correction algorithm on the test data which have
been filtered by the Gaussian Naïve Bayes algorithm. The results are
summarized in Table 5.
\emph{Table 5: Summary of error information before and after
implementing the combined strategies.}\selectlanguage{english}
\begin{longtable}[]{@{}lllll@{}}
\toprule
Item & Subject & Mean Relative Error \(\overset{\overline{}}{\epsilon}\) & Std. Deviation
\(\sigma\) & Uncertainty \(\overset{\overline{}}{\epsilon}\ \)\selectlanguage{ngerman}±
\(2\sigma\)\tabularnewline
\midrule
\endhead
1 & Test Data Set & -0.0830 (-8.3\%) & \selectlanguage{ngerman}±0.203 (±20.3\%) &
\selectlanguage{ngerman}±0.406(±40.6\%)\tabularnewline
2 & Test Data after being filtered using Gaussian Na\selectlanguage{ngerman}ïve Bayes algorithm
with no calibration & -0.0945 (-9.5\%) & \selectlanguage{ngerman}±0.080 (±8.0\%) & \selectlanguage{ngerman}±0.161
(±16.1\%)\tabularnewline
3 & Test Data after being filtered using Gaussian Na\selectlanguage{ngerman}ïve Bayes algorithm
with no calibration+ followed by 2D Interpolative Error Reduction. &
-0.0059 (-0.6\%) & \selectlanguage{ngerman}±0.137 (±13.7\%) & \selectlanguage{ngerman}±0.274 (±27.4\%)\tabularnewline
\bottomrule
\end{longtable}\selectlanguage{ngerman}
Results show the combined 2D interpolative method and Gaussian Naïve
Bayes algorithm has reduced both types of errors. From Table 5, it was
obvious that the mean relative error\(\overset{\overline{}}{\epsilon_{\text{Rev}}}\) was revised from
-8.3\% to -0.6\%. The standard deviation \(\sigma_{\text{Rev}}\) has
decreased from ±20.3\% to ±13.7\%. It has improved the mean flow
coefficient\(\overset{\overline{}}{C_{\text{\ Rev}}}\) from 0.900 to 0.994.
\textbf{6.0 Conclusions}
The following conclusions may be drawn from our study.
\begin{itemize}
\tightlist
\item
The linear least square error method (LLS) and 2D interpolation method
exhibit lower mean relative error (-4.2\% and +1.2\%) compared to that
of the test dataset before calibration (-8.3\%). Either modeling
approaches was able to improve accuracy (indicated by the improvement
in systematic error) but not the precision (indicated by the standard
deviation of relative errors, which is associate with the distribution
patterns of random errors).
\item
Three types of Gaussian Naïve Bayes modeling approaches were tested.
The best result has succeeded to reduce the standard deviation value
of relative error from ±20.3\% (of test data) to ±8.0 \%. However, the
reduction is achieved at the cost of increased mean relative error
from -8.3\% (of test data) to -- 9.5\% (after data cleaning). Results
also show the Gaussian Naïve Bayes method is useful to improve random
errors, but not the systematic errors.
\item
Since different types of errors may coexist in the same data set (such
as systematic errors and random errors), we show that use of multiple
modeling approaches can successfully reduce both types of errors. When
the 2D interpolation method was applied to the test data using the
Gaussian Naïve Bayes algorithm, the mean relative errors were reduced
from -8.3\% (of test dataset) to -0.6\% (after data cleaning), and
standard deviation of the relative errors was reduced from ±20.3\% to
±13.7\%.
\item
These results proved that multiple machine learning models can be
trained to evaluate the sampled data from flow meter, thus
re-calibrating both the systematic errors and random errors.
\item
Our study shows a high accuracy ultrasonic flow meter with systematic
errors less than 1\% for oil and gas multiphase application is
possible with the aid of artificial intelligence technology.
\end{itemize}
\textbf{Nomenclature}
Latin Characters
\(C\) = flow coefficient.
\(C_{\text{LLS}}\) = predicted flow coefficient based on LLS method.
\(\overset{\overline{}}{C_{\text{Rev}}}\) = revised mean flow coefficient after machine
learning algorithm was implemented.
\(c\) = ultrasound speed in water (m/s).
\(D\) = pipe internal diameter (m).
\(k_{c}\) = velocity correction factor.
\(L\) = the path travelled by ultrasound (m).
\(m_{p}\) = total mass of suspended particles (kg)
\(u\) = flow velocity (m/s).
\(u_{f}\) = fluid velocity (m/s).
\(V_{s}\) = volume of solids (m\textsuperscript{3}).
\(V_{m1}\) = volume of initial mud or fresh water
(m\textsuperscript{3}).
\(V_{m2}\) = volume of final mixture (m\textsuperscript{3}).
\(\overset{\overline{}}{v}\) = mean flow velocity (m/s)
\(v_{T\text{est}}\) = flow velocities from test data set.
Re = Reynolds number.
\(t_{21}\),\(\ t_{12}\) = The transit time of the
ultrasonic signal from downstream transducer to the upstream transducer
and that is in opposite direction (s).
\(w_{b}\) = weight of bentonite particles (kg).
\(w_{w}\) = weight of water (kg).
\emph{Greek Characters}
\(\beta_{o},\beta_{1}\) = coefficient in the least square error method.
\(\overset{\overline{}}{\epsilon_{\text{Rev}}}\) = revised mean relative error after machine learning
algorithm was implemented.
\selectlanguage{greek}θ\selectlanguage{ngerman} = the incident angle of ultrasound wave.
\(\mu_{f}\) = fluid viscosity (Pa.s)
\(\nu\) = kinematic viscosity (m\textsuperscript{2}/s).
\(\rho_{f}\) = density of fluid (kg/m\textsuperscript{3}).
\(\rho_{s}\) = density of solids (kg/m\textsuperscript{3}).
\(\rho_{p}\) = density of particle (kg/m\textsuperscript{3}).
\(\rho_{m1}\) = density of initial mud or fresh water
(kg/m\textsuperscript{3}).
\(\rho_{m2}\) = density of final mixture (kg/m\textsuperscript{3}).
\(\sigma_{v}\) = standard deviation of flow velocities (m/s).
\(\sigma_{\text{Test\ \ }}\) = standard deviation of relative error of test data
set.
\(\sigma_{\text{Rev\ \ }}\) = revised standard deviation of relative error after
machine learning algorithm was implemented.
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\selectlanguage{english}
\FloatBarrier
\end{document}