Hmzh Ghorbni, Dvi A.Woo, Abouzr Choubinh, Ashin Ttr,Pjmn Ghzipour Abrghoyi, Mohmm Mni, Nim Mohmin

a Young Researchers and Elite Club, Ahvaz Branch, Islamic Azad University, Ahvaz, Iran

b DWA Energy Limited, Lincoln, United Kingdom

c Petroleum Department, Petroleum University of Technology, Ahwaz, Iran

d Young Researchers and Elite Club, North Tehran Branch, Islamic Azad University, Tehran, Iran

e National Iranian South Oil Company (NISOC), Ahvaz, Iran

f Young Researchers and Elite Club, Omidiyeh Branch, Islamic Azad University, Omidiyeh, Iran

Keywords: Orifice flow meters Flow-rate-predicting virtual meters Multiple machine-learning algorithm comparisons Metrics influencing oil flow Flow-rate prediction error analysis

ABSTRACT Fluid-flow measurements of petroleum can be performed using a variety of equipment such as orifice meters and wellhead chokes.It is useful to understand the relationship between flow rate through orifice meters (Qv) and the five fluid-flow influencing input variables: pressure (P), temperature (T), viscosity (μ), square root of differential pressure (ΔP^0.5), and oil specific gravity (SG).Here we evaluate these relationships using a range of machine-learning algorithms applied to orifice meter data from a pipeline flowing from the Cheshmeh Khosh Iranian oil field.Correlation coefficients indicate that (Qv) has weak to moderate positive correlations with T, P, and μ, a strong positive correlation with the ΔP^0.5, and a weak negative correlation with oil specific gravity.In order to predict the flow rate with reliable accuracy, five machine-learning algorithms are applied to a dataset of 1037 data records (830 used for algorithm training; 207 used for testing) with the full input variable values for the data set provided.The algorithms evaluated are: Adaptive Neuro Fuzzy Inference System (ANFIS), Least Squares Support Vector Machine (LSSVM), Radial Basis Function (RBF), Multilayer Perceptron (MLP), and Gene expression programming (GEP).The prediction performance analysis reveals that all of the applied methods provide predictions at acceptable levels of accuracy.The MLP algorithm achieves the most accurate predictions of orifice meter flow rates for the dataset studied.GEP and RBF also achieve high levels of accuracy.ANFIS and LSSVM perform less well, particularly in the lower flow rate range (i.e., ＜ 40,000 stb/day).Some machine learning algorithms have the potential to overcome the limitations of idealized streamline analysis applying the Bernoulli equation when predicting flow rate across an orifice meter, particularly at low flow rates and in turbulent flow conditions.Further studies on additional datasets are required to confirm this.

1.Introduction

Accurate fluid throughput flow-rate information is essential for maintaining oil and gas process plant efficiency and productivity [1].A range of metering equipment and techniques are available to provide high-accuracy flow-rate measurements under a wide-range of conditions.For steady flow conditions the most commonly used equipment includes flow meters of the following types: differential-pressure, hot wire anemometers; electromagnetic; ultrasonic; and, vortex-shedding [2].Multi-phase flow meters are expensive to install and operate, but they provide more valuable information by determining the flow rates of different phase components in a flow stream through one device [3].Ismail et al.[4] illustrate the various techniques employed by oil-field multi-phase flow meters, which have been widely deployed by oil and gas companies in offshore and onshore applications since the 1980s [5].

Measuring the pressure difference across a flow-restricted area in pipework is the principle employed by the majority of multi-phase flow meters, including those based on nozzles, orifice plates and venturi systems [6].Their simple, mechanical components and robust design make orifice meters the most widely used fluid-flow-measurement devices [1].An orifice meter creates an unsteady-state flow regime downstream of the plate involving regions of flow reversal, shear flow, the formation of vortices and layer separation in three dimensions [7], as illustrated in Fig.1.

The coefficients associated with orifice-plate meters tend to be empirical [1] due to the geometrical complications and flow separation regions downstream of the plate (Fig.1).These flow complications lead to higher pressure losses than other meters, which most obviously manifests itself in additional costs related to fluid pumping costs.Despite this orifice-plate meters represent the most commonly used petroleum-field flow meters their simple robust components (Fig.2) with no moving parts [1,8].They offer the flexibility of measuring flow over a wide range of flow rates and climatic/temperature conditions.They are also effective for liquids of varying densities and viscosities, gases and single-phase flow systems [8].

There are well-established international standards defining geometry, positioning within pipework and pressure regimes appropriate for orifice meters [9].These typically refer to steady-state flow conditions for fluids with Reynolds numbers above 3150 and pipework internal diameters of greater than 50 mm.Such limitations are typically overcome by placing flow rectifiers at various distances upstream of the orifice meters [10].There have been many attempts to develop prediction models that overcome the inaccuracies of orifice meters [11,12].These mainly involve empirical relationships applied to steady-state flow conditions to provide more-accurate discharge coefficients.Comparisons are made to standard flow conditions for a range of flow rates, differential pressures, etc.[10].For steady-state and turbulent-state flow conditions, the discharge coefficients of orifice- plate meters are typically calibrated with the corrected Bernoulli equation.This requires experimental measurements of discharge coefficients under standard conditions.However, even with such calibration the empirical methods typically result in unacceptable levels of inaccuracy for many flow conditions [12,13].

Velocity profiles associated with specific flow rates in steady-state conditions influence the discharge coefficients of most flow meters.The vena contracta effects downstream of the orifice make orifice-plate meters highly sensitive to changes in fluid velocity.For fluid flow with low Reynolds number (Re), pressure losses through an orifice plate are dependent on fluid viscosity, which determines a fluid's internal shear forces.On the other hand, for higher Re conditions separation/divergence is more likely to occur downstream of the orifice plate, with inertia being the overriding force.Most simple empirical formulas are unable to derive reliable discharge coefficients across the full range of laminar flow in such variable conditions.Indeed, the relationship between Re and orifice-plate-discharge coefficients is highly non-linear [9].

Fig.1.Two-dimensional, diagrammatic representation of flow regions associated with an orifice plate.Modified after Shan et al.[7].

Fig.2.Image of an orifice-plate meter with a beta ratio (β) of 0.65.β is the ratio between the inner diameter of the orifice to pipeline diameter.The discharge coefficients are typically stable for 0.2 ≤ β ＜ = 0.7.Outside of that range errors in flow measurements tend to increase.

Significant research has been conducted on multi-phase flow in orifice-plate meters [3,14,15] and orifice-plate discharge coefficients [16,17].Artificial intelligence (AI) and machine learning techniques have also been applied to predict discharge coefficient in rectangular side orifices [18].Cioncolini et al.[19] described pressure decline across micro-orifice plates associated with single-phase liquid flow.The impacts of density, pressure drop, viscosity, wall mass transfer and orifice area on fluid-flow characteristics have also been evaluated [20-22].

Other studies have focused on the impacts of β ratio on flow conditions through orifice plates [23-26] and pulsating flow conditions [2,27,28].Several studies have applied computational fluid dynamics (CFD) models to analyze flow through orifice meters [1,29-33].Cristancho et al.[34] derived a customized formula for natural gas distinct from the standard orifice equation.Flow-accelerated corrosion (FAC), a phenomenon that causes thinning and weakening of pipelines and can lead to premature component failure and accidents [35,36], has been extensively studied downstream of orifice-plates [21,37,38].

It is a valuable capability to be able to accurately predict flow rate through a system based on a range of input variables.This is because the placement and operation of flow meters is often limited and constrained, meaning that they only provide limited data at specific points in a system.Moreover, it is often useful to be able to predict, for planning and evaluation purposes, what the flow rates might be at various points in a system (with and without flow meters) in the future, as the values of some of the underlying input variables change.Flow meters cannot provide such future flow forecasts, whereas accurately calibrated AI algorithms can provide that valuable information, effectively acting as virtual flow meters.

The principal purpose of this work is to optimize the production flow rate prediction of orifice measurement using five different machine-learning methods: Adaptive Neuro Fuzzy Inference System (ANFIS); Least Squares Support Vector Machine (LSSVM); Radial Basis Function (RBF); Multilayer Perceptron (MLP); and, Gene Expression Programming, (GEP).The input variables: pressure (P); temperature (T); viscosity (μ); square root of differential pressure (ΔP^0.5); and, oil specific gravity (SG), are used to determine fluid flow.The prediction accuracies of the five machine-learning methods applied to orifice-plate meter flow are compared using a range of statistical-accuracy metrics.This analysis makes it possible to determine the machine-learning methods that are most suitable for this application.

2.Orifice-plate flow theory

Typical assumptions for orifice plate flow analysis is that fluid flow within a pipeline is idealized in a steady-state, incompressible, inviscid (i.e., negligible viscosity), laminar flow condition.Moreover, it is assumed that the pipe is horizontal and negligible friction losses occur.Such flow conditions are illustrated in Fig.3.Two equations apply to flow through orifices in such conditions [20]: The Bernoulli's equation (Eqt.1); and, the continuity equation (Eqt.2).

Bernoulli equation:

Continuity equation:

Note: see Nomenclature section for the definitions of all symbols and acronyms used in this study.

Flow-rate measurements through orifice plates requires the determination of the differential pressure between the pressure taps located on the upstream and downstream sides of the orifice (Fig.3).The measurements are taken during steady-state flow in accordance with the ANSI/API-mpms-14.3 standard guidelines [40].Applying these standards, the flow rate in mass terms, qm, is calculated through Eqt.3:

Orifice-plate meters measure mass-flow rate indirectly.Differential pressure signals are monitored and recorded as a function of fluid velocity as the fluid passes through the bore of the orifice plate.Manipulation of the density variable in Eqt.3 facilitates calculating flow rates in terms of mass or volume.Eqt.4 determines the flow rate in volume terms:

These equations used in idealized streamline analysis to calculate flow rate can be evaluated in comparison with the various machine learning algorithms (which make no assumptions about idealized conditions) to optimize the prediction of both qmand Qv.

Orifice flow meters are not the most accurate of available flow meters.However, they have been widely installed historically in many oil fields around the world and remain part of many ongoing oilfield operations.Consequently, it is often necessary to use and interpret the flow-rate data provided by such meters, because it is the only measured information available for specific field systems.Therefore, using AI algorithms to accurately match orifice flow meter readings, in such circumstances, is useful in providing virtual flow meter prediction capabilities.

3.Machine learning methods applied and compared

Here we provide brief summaries of the five machine-learning algorithms applied.

3.1.Adaptive Neuro Fuzzy Inference System (ANFIS)

Fuzzy interference system (FIS) typically consist of four preliminary processing sections:

(1) Knowledge-based analysis determines the definition of the membership functions (MFs) applied to the dataset plus some fuzzy-logic rules.

(2) Interference engine enables the fuzzy rules to be modified and refined.

(3) Fuzzification takes crisp-numerical input data and transforms it into fuzzy numbers through fuzzy sets defined using linguistic terms.

(4) Defuzzification transforms the fuzzy interference computations into crisp-numerical output.

Considering an FIS containing two input parameters and one output parameter helps to explain the process.Two Takagi and Sugeno fuzzy if-then rules exist within the rule-based FIS structure, defined as:

Ifx1is A andx2is B then y is f(x1, x2)Where:

Terms A and B are the two corresponding fuzzy sets; f(x1, x2) is the crisp function, which is usually of polynomial form involving the input parameters, x1and x2; and, y is the dependent variable.

When f(x1, x2) involves constant values, a zero-order Sugeno-type fuzzy system is generated.On the other hand, if f(x1, x2) is a first-order polynomial function, a first-order Sugeno-type FIS is generated.For example, two first order Takagi and Sugeno type fuzzy if-then rules [41] could be expressed as:

Rule I: if (x1is A1) and (x2is B1) thenf1 = p1x1+q1x2+r1

Rule II: if (x1is A2) and (x2is B2) thenf2 = p2x1+q2x2+r2Where:

p, qandrare coefficients establish a linear relationship involving each of the input variables; and,frepresents different values for the objective function.

FIS of this type are efficient when applied to a wide range of engineering and scientific problems.The input parameters are linearly combined, then summed with a constant value.The FIS output is ultimately established when the average outputs of the diverse rules are calculated based on a weighted-average basis.

3.2.Least Squares Support Vector Machine (LSSVM)

The support vector machine (SVM) was first introduced in 1995 [42] and has since been successfully applied to many scientific problems.In SVM, the function f(x) is defined as Eqt.5:

To obtain optimum values forwandb, it is necessary to minimize the cost function [43] expressed as Eqt.6 under specific conditions:

After applying needed conditions, the final form of the SVM function is then expressed as Eqt.7:

Which is suitable for solving using a quadratic programming procedure to calculate the parameters of b,aiand.

Fig.3.Typical flow pattern through an orifice showing idealized streamlined flow analysis assumptions associated with interpreting pressure differential across an orifice is used to calculate flow rate using the idealized Bernoulli equation (Eqt.1).Modified after Rhinehart et al.[39].

Deriving solutions for SVM is not easy, which prompted the development of the LSSVM method [44].LSSVM is essentially a SVM with equality constraints replacing the inequality constraints.Furthermore,

Where:Cis a tuning parameter; and,eiquantifies the error.

Finally, the regression model can be expressed as Eqt.9 according to the same special restrictions:

The type of kernel function applied to Eqt.20 in this study is the radial basis function (RBF) expressed as Eqt.10:

Where:

σ2is the variance calculated using a Gaussian function.

The LSSVM algorithm, as described, involves just two tuning parameters C and σ, which are determined for each specific data set by minimizing the mean squared error (MSE) of the dependent variable outputs from the LSSVM model versus the measured data, i.e., the optimization objective function expressed as Eqt.11:

LSSVM performance is improved by linking it with an optimization algorithm, such as Coupled Simulated Annealing (CSA) [45] to obtain the optimum values of the two tuning parameters.

3.3.Multilayer Perceptron (MLP)

An MLP is a type of artificial neural network (ANN).The structure of MLP networks consist of an input layer, an output layer, and one or several hidden layers of neurons.The layers may each contain one or more neurons depending upon the number of input and output variables considered.Optimization algorithms or trial and error techniques can be used to determine the optimum number of neurons and hidden layers to include in MLP networks [46].The most well-known and widely applied training algorithm for MLPs is back propagation (BP).The training process requires defined inputs and real measured data with which to calibrate its outputs.For training purposes, the weights applied are varied in order to minimize the network's objective performance function, i.e., mean squared error (MSE) in the case evaluated here.MSE is typically used as the default objective performance function for feed-forward MLP networks [47].

3.4.Radial basis function (RBF)

An RBF learning network is also an ANN, but unlike an MLP, it utilizes a radial basis function to train a multi-dimensional network with just one hidden layer.RBF learning networks consist of just three layers: (1) an input layer; (2) one hidden layer with multiple neurons which works based on a non-linear activation function; and, (3) a linear output layer [48].

3.5.Gene expression programming (GEP)

The original genetic optimization algorithm (GA) was developed in 1975 and has been successfully applied and adapted in most branches of science, including petroleum engineering, since then [49] Genetic programming (GP) [50] developed functions to further refine and improve GA.The alternative method of gene expression programming (GEP) was introduced in 2001 [51-54] to overcome some of the disadvantages of the GA and GP methods.GEP establishes the optimum relationships between independent and dependent variables by developing a series of predetermined algebraic operators.Chromosomes and Expression Tree (ET) are the two main distinguishing features of GEP [54].For problems requiring regression analysis and/or evaluations of functions involving symbolic (generic) equations need to be developed, GEP's calculation efficiency and speed of convergence are reported to be between two and four orders magnitude better than the GP algorithm [55].

Generically, the sequential steps required to apply the GEP algorithm [51] are:

(1) randomly choose chromosomes and their corresponding initial populations;

(2) select the set of terminals and the functions, length of the head, and the number of genes for each chromosome;

(3) define a fitness function to quantify the degree of fitness achieved by each proposed solution;

(4) identify the chromosomes with the best fitness values, and use these as parents for the next generation (iteration)

(5) apply genetic operators to produce populations of new parents from which each new generation evolves; and,

(6) repeat steps 1 to 5 until defined convergence conditions have been achieved or a designated computational time has elapsed.

4.Oil field orifice-plate-meter dataset evaluated with machine- learning algorithms

The flow results for 1037 data records through an orifice meter recording flow in the pipeline from the Cheshmeh Khosh oil field to the Ahvaz-3 production unit are analyzed in this study.The Cheshmeh Khosh field is a mature oil field located in Dasht-e-Abbas province, Ilam province, 52 km south of Dehloran city and 70 km west of Andimeshk city in southwest Iran.It was discovered in 1964 and began production began in 1975.

The oil production from the Cheshmeh Khosh oil field is sent through an 18-inch diameter pipeline to the Ahvaz-3 production unit.In that system the production passes through an orifice flow meter at the Ahvaz-3 production unit.The specifications for that orifice meter are:

Internal pipe diameter = 17.5 inches

orifice diameter = 8.4 inches

weep hole diameter = 0

beta ratio = 0.48.

Each of the data records in the dataset includes one dependent variable, measured oil flow rate in stock tank barrels/day, and five input variables:

Temperature (oC)

Pressure (psig)

Oil specific gravity (relative to water = 1)

Differential pressure across the orifice plate (ΔP) (inches of water)

Fluid viscosity (centiStokes-cSt)

ΔP and square root of ΔP are clearly not independent of each other, and it is the square root of ΔP that is widely used in the calibration of orifice meters [39].The square root of ΔP is also a key component of Eqt.3 and clearly has an influencing relationship on the fluid flow rate.However, due to the non-linear relationship between flow and square root of differential pressure, the accuracy of flow measurement with orifice plate meters using this relationship can be degraded, particularly for the lower range of flow rates.

A statistical summary of the five input variable values (plus square root of ΔP) for the 1037 data records for the flowmeter dataset are provided in Table 1.A full listing of all the variable values in each data record is provided in a supplementary file (see Appendix).

The objective of applying the machine learning algorithms to this data set is to use the values for the six input variables to predict oil flow rate as accurately as possible, by minimizing the error between measured and predicted flow rate values.

5.Results

5.1.Relationships between metered flow rates and input variables

The calculated flow rate through orifice meters is a function of temperature (T), pressure (P), oil specific gravity (SG), square root of differential pressure (ΔP^0.5) and viscosity (μ) as defined in Eqt.3.The relationships between these variables and their relative influence on measured flow rates for the Cheshmeh Khosh oil field dataset are evaluated using the Pearson and Spearman correlation coefficients.A comparison of these two distinct correlation coefficients for the orifice- meter input variables versus the measured (calculated) flow rate is illustrated in Fig.4.

The Pearson correlation coefficient (also referred to as the Pearson torque product coefficient correlation coefficient) measures the linear correlation between two random variables [56].The value of this coefficient varies between -1 and +1: a value of "+1” indicates a perfect positive correlation; a value of “0” indicates no correlation; and, a value of "-1″ indicates a perfect negative correlation.This coefficient, developed by Carl Pearson based on the original idea of Francis Galton [57-60], is calculated by Eqt.12;

The Spearman rank correlation coefficient, ρ, is a nonparametric statistic for measuring the correlation coefficient between two random variables [61].It indicates the ability to express a variable as a function of a single other variable.A complete Pearson correlation (-1 or 1) occurs when the functional variable is a single function of the other variable [62-64].The Spearman correlation coefficient is essentially the Pearson correlation coefficient calculated between ranked data [63].For example, for a pair of data (F, H), we first calculate the rank of each data variable as (Fi, Hi) and then calculate the Spearman correlation coefficient using Eqt.13:

The calculated flow rate through the orifice meter (Qv) for the dataset studied shows weak to moderate positive correlations (bothPearson and Spearman coefficients) with temperature (T), pressure (P) and viscosity (μ) and a strong positive correlation with the square root of differential pressure (ΔP^0.5).On the other hand, it demonstrates a weak negative correlation with oil specific gravity (SG).These relationships are summarized by Eqt.14.

Table 1 Statistical Summary for variables measured for the 1037 data records of the meter recording flow in the pipeline from the Cheshmeh Khosh oil field to the Ahvaz-3 production unit.

Fig.4 highlights the significant influences of pressure and (ΔP^0.5) in determining the calculated flow rate value in the dataset studied.

From this dataset, 80% (830) data records were selected randomly to form the training subset for the five machine-learning algorithms applied (i.e., MLP, RBF, LSSVM, ANFIS and GEP).The remaining 20% (207) data records became the testing subset; used to test the validity and accuracy of the trained networks in calculating flow rates through the orifice meter of the Cheshmeh Khosh oil field production pipeline.

5.2.Statistical accuracy metrics assessed

Here, we assess the performance of the machine-learning algorithms in accurately predicting flow rate through the orifice meter by calculating four commonly used statistical measures of accuracy: correlation factor (R2) (Eqt.15), Average Absolute Relative Deviation (AARD) (Eqt.16), Root Mean Squared Error (RMSE) (Eqt.17) and Standard Deviation (STD) (Eqt.18).

Table 2 shows the results for the five machine-learning models in terms of the four statistical measures of accuracy.Displaying the highest correlation coefficient and lowest values of AARD%, RMSE, and STD applied to the entire data set (all 1037 records) the MLP model achieves superior accuracy in comparison with the other algorithmswhen applied to the dataset studied.

Table 2 Statistical-accuracy values for five machine-learning algorithms applied to the Cheshmeh Khosh oil field dataset (with N(Total Data) = 1037; N(Test Data) = 207; and N(Train data) = 830).

6.Discussions

Fig.5 shows cross plots with correlation coefficients displayed of measured versus predicted flow rate through the orifice meter (Qv) for the entire dataset (1037 data records) for each of the five machine- learning models developed.Acceptable correlation coefficients with values greater than 0.90 were obtained for all five algorithms tested, and the trends for measured versus predicted data approximately follow a 45° line in each case.These results indicate acceptable accuracy is achieved by each of the machine-learning prediction models developed when applied to the full data set.This is useful because it demonstrates that applying accurately-calibrated machine-learning algorithms to predict flow rate across an orifice meters has the potential to overcome the limitations of idealized streamline analysis to predict flow applying the Bernoulli equation, or modification thereof, particularly at low flow rates and in turbulent flow conditions.This potential requires further evaluation and verification by calibrating the machine learning algorithms with data from a range of meter types, including those providing higher accuracy than orifice meters.

Fig.4.Relationships of the calculated flow rate through Orifice meter individually with its input calculation variables: temperature (T), pressure (P), oil specific gravity (SG), square root of differential pressure (ΔP^0.5) and viscosity (μ) for each data record of the Cheshmeh Khosh oil field dataset.

Fig.5.Calculation of flow rate through orifice meter values for (a) MLP, (b) RBF, (c) LSSVM, (d) ANFIS and (e) GEP models versus experimentally measured values.

The MLP, RBF and GEP models achieved the highest correlation coefficients and lowest values for AARD, STD and RMSE error measures (Table 2).The root mean squared error (RMSE) achieved by each of the models for the total dataset was 38.31, 720.94, 2466.74, 2796.35 and 366.79 for MLP, RBF, LLSVM, ANFIS and GEP, respectively.It is clear that the total dataset results for these algorithms display significantly less dispersion in Fig.5 than the other algorithms.For the LSSVM algorithm there is significantly greater dispersion between predicted and calculated flow rate values than for the other algorithms.This is particularly so for the flow rates below 40,000 stb/day.On the other hand, whereas the ANFIS algorithm also shows greater dispersion for flow rates below 40,000 stb/day, the errors between predicted versus calculated flow rates are more systematic at the low end of the flow rates.These findings suggest that further studies are justified with other dataset, particularly in lower flow-rate regimes, to compare the performance of the various machine-learning algorithms in predicting flow rates across orifice plates using differential pressure and the other input variables identified.

Fig.6 indicates the relative deviations of predicted versus calculated flow rates through an orifice meter for the five machine-learning algorithms, for each data record of the training and testing subsets distinguished.These trends highlight the much lower relative deviations achieved by the MLP algorithm than the other four algorithms, particularly for the lower-flow-rate data records.Fig.6 display is revealing as it identifies those few records that are poorly predicted by the respective algorithms.This highlights the flow rate region for which each algorithm achieves its lowest prediction accuracy.

It seems like that the ANFIS and LSSVM models provide less-accurate predictions in the lower flow rate interval than the other algorithms evaluated because there are more data records for flow rates above 40,000 stb/day than for lower flow rates.The predictions made by the ANFIS and LSSVM algorithms seem to be more strongly influenced by the higher density of data records in the higher-flow-rate range than the other algorithms.Consequently, they generate less accurate predictions in the lower-flow-rate range (i.e., ＜ 40,000 stb/day).Further studies are required to confirm this with other datasets.

To an extent many machine-learning algorithms function, at least in part, as black boxes, in that it is not easy to interrogate each individual data point to determine how the prediction for that point is calculated.This is a limitation and makes it difficult to explain exactly why the MLP algorithm outperforms the other algorithms in predicting flow rate for the dataset studied.Nevertheless, the fact that these algorithms (in particular, the MLP, RBF and GEP algorithms for this dataset) can provide highly accurate predictions makes them useful for certain flow measurement requirements.We do not suggest that these AI algorithms should replace accurate meters to measure flow rates through pipelines.Rather, we believe that they have the potential to complement such equipment, by acting as virtual flow meters to estimate flow rates at points in a system where meters are not present, and to predict future likely flow rates when underlying conditions in the system might change.

Fig.6.Relative deviation [(Qvmeasured-Qvpredicted)/Qvmeasured] of calculation of flow rate through orifice meter values achieved by the (a) MLP, (b) RBF, (c) LSSVM, (d) ANFIS and (e)GEP models with data records of the training and testing subsets distinguished.

The different manner in which the five machine-learning algorithms fit the data for Cheshmeh Khosh oil field pipeline orifice meter dataset is clear from Table 2 and Figs.5-7.However, for other field data sets the performance of the MLP, RBF, ANFIS, GEP and LSSVM models might be ranked differently.It is therefore not justified, based on just the results for the Cheshmeh Khosh oil field pipeline orifice meter dataset, to conclude that the MLP model will always outperform the other four machine-learning algorithms in applications to other orifice meter flow rate data sets.Future studies are required to evaluate this with other datasets.

A case can be made to also test several machine-learning algorithms to evaluate the data from other fields.

Fig.7 displays as radar diagrams the results for the R2and RMSE statistical measures of accuracy for the five machine-learning algorithms as applied to the Cheshmeh Khosh oil field dataset.Table 2 and Fig.7 identify that the MLP, GEP and RBF models perform more accurately than the other models in the calculation of flow rate through the orifice meter for the entire dataset.Moreover, the results clearly indicate that MLP also outperforms the GEP and RBF algorithms in terms of accuracy and consistency over the entire flow rate range covered by this dataset.

7.Conclusions

Evaluation of 1037 data records (divided randomly into 80% training and 20% testing subsets) from a pipeline orifice meter monitoring oil flow from the Cheshmeh Khosh oil field into the Ahvaz-3 production unit (Iran) were utilized to determine the orifice meter flow rate (Qv) using five machine-learning algorithms based on five input parameters.The following conclusions are drawn from the results and their analysis:

(1) Relevancy factors identified using Pearson and Spearman correlation coefficients indicate weak to moderate positive correlations between the orifice meter calculated flow rate (Qv) and parameters of pressure, temperature, viscosity, and square root of differential pressure and a weak negative correlation between flow rate and oil specific gravity.However, It is the strong positive Pearson and Spearman correlation coefficients between the square root of differential pressure and Qvthat makes it the most relevant variable.

Fig.7.Statistical accuracy measures: (a) R2 and (b) RMSE for the calculation of flow rate through an orifice meter.These compare the performance of the five developed machine-learning models applied to the entire Cheshmeh Khosh oil field pipeline orifice meter dataset (1037 records).

(2) All five of the machine-learning algorithms evaluated [(Adaptive Neuro Fuzzy Inference System (ANFIS), Least Squares Support Vector Machine (LSSVM), Radial Basis Function (RBF), Multilayer Perceptron (MLP), and Gene Expression Programming (GEP)] were able to predict the orifice meter flow rate with a high accuracy (i.e.R2＞0.99) and could be effectively used to do so.

(3) For the dataset studied the MLP algorithm outperformed the other algorithms in terms of prediction accuracy, particularly so in the lower-flow-rate range (＜ 40,000 stb/day) and the GEP and RBF algorithms also performed well in the lower-flow-rate range, but the ANFIS and LSSVM algorithms did not.

(4) Future work is required with other dataset, particularly in lower flow-rate regimes, to compare the performance of the various machine-learning algorithms in predicting flow rates across orifice plates using differential pressure and the other input variables identified.

(5) Applying accurately-calibrated machine-learning algorithms (i.e., MLP, RBF and GEP for the dataset studied) to predict flow rate across an orifice can potentially overcome the limitations of idealized streamline analysis to predict flow applying the Bernoulli equation, or modification thereof, particularly at low flow rates and in turbulent flow conditions.This potential requires further evaluation and verification by calibrating the machine learning algorithms with data from a range of meter types, including those providing higher accuracy than orifice meters.

Acknowledgments

We thank National Iranian South Oil Company (NISOC) and Mr.Mohsen Sharifat, Mr.Saeed Kooti, Dr.Abouzar Mirzaei-Paiaman from NISOC for their advice.

Appendix A.Supplementary data

Supplementary data to this article can be found online at https:// doi.org/10.1016/j.petlm.2018.09.003.

Nomenclature

x The Input Vector

cThe Central Point of The Function

Δp Orifice Differential Pressure

A Corresponding Fuzzy Set

A1Cross-Sectional Area Point = 1

A2Cross-Sectional Area Point = 2

ai, ai*Lagrangian Multipliers

ANFIS Adaptive Neuro Fuzzy Inference System

B Corresponding Fuzzy Set

b The Bias Term

BP Back Propagation Algorithm

C Tuning Parameter Related to Structure of LSSVM Model

CdDischarge Coefficient

Cd(FT) The Coefficient of Discharge at a Specified Pipe Reynolds Number for Flange-Tapped Orifice Meter

Ci (CT) The Coefficient of Discharge at Infinite Pipe Reynolds Number for Corner-Tapped Orifice Meter

Ci (FT) The Coefficient of Discharge at Infinite Pipe Reynolds Number for Flange-Tapped Orifice Meter

d Orifice Plate Bore Diameter Calculated at Flowing Temperature (Tf)

D meter Tube Internal Diameter Calculated at Flowing Temperature (Tf)

drReference Orifice Plate Bore Diameter at Reference Temperature (Tr)

DrReference Meter Tube Internal Diameter at Reference Temperature (Tr)

e The Napierian Constant = 2.71828

eiThe Error Value

ET Expression Tree

EvVelocity of Approach Factor

f(x1, x2) Crisp Function

FIS Fuzzy Interference System

GA Genetic Algorithm

gcDimensional Conversion Constant

GEP Gene Expression Programming

h The Head Length

k Isentropic Exponent

L1The Dimensionless Correction For The Tap Location = L2

L2The Dimensionless Correction For The Tap Location = L1

LSSVM Least Squares Support Vector Machine

MF Membership Function

MLP Multilayer perceptron

MMSCFD Million Standard Cubic Feet Per Day

MSE Mean Square Error

n The Largest Arity of The Functions Applied in Head of The Gene

N N Number of Pairs (Fi, Hi)

O Layer Output

? (x) The Kernel Function

P1Upstream Pressure

P2Downstream Pressure

Q, QVVolumetric Flow

Qi/MMeasured Flow Rate

Qi/PPredicted Flow Rate

qmMass Flow

r The Radius of Function

R The Correlation Coefficient

RBF Radial Basis Function

ri, ni, miLinear Variable Which Accompany the First Layer Variable in the ANFIS Model

stb/day Standard Barrels Per Day

t The Tail Length

TfTemperature of The Fluid at Flowing Conditions

TrReference Temperature of The Orifice Plate Bore Diameter

V1Upstream Velocity

V2Downstream Velocity

WiThe Associate Weight (for ANFIS and RBF Components)

WTTranspose Vector of Output Layer

X Differential Pressure to Absolute Static Pressure Ratio

x Matrix (m = Rows, n = Columns)

Y Expansion Factor

Z Gaussian MF

β Diameter Ratio

? Denotes The Flow Rate Through Orifice Meter

μ Absolute Viscosity

π Universal Constant = 3.14159

ρ Pearson Correlation Coefficient

ρLLiquid Density

σ The Variance Parameter

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