Simplified Neural Network-Based Models for Oil Flow Rate Prediction

Uduak Koffi Umana; Anietie Ndarake Okon; Okorie Ekwe Agwu

doi:doi:10.11648/j.pse.20240802.12

Research Article |

| Peer-Reviewed

Simplified Neural Network-Based Models for Oil Flow Rate Prediction

Uduak Koffi Umana

, Anietie Ndarake Okon

, Okorie Ekwe Agwu^*

Published in Petroleum Science and Engineering (Volume 8, Issue 2)

Received: 2 August 2024 Accepted: 9 September 2024 Published: 23 September 2024

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Abstract

Available neural network-based models for predicting the oil flow rate (q_o) in the Niger Delta are not simplified and are developed from limited data sources. The reproducibility of these models is not feasible as the models’ details are not published. This study developed simplified and reproducible three, five, and six-input variables neural-based models for estimating q_o using 283 datasets from 21 wells across fields in the Niger Delta. The neural-based models were developed using maximum-minimum (max.-min.) normalized and clip-normalized datasets. The performances and the generalizability of the developed models with published datasets were determined using some statistical indices: coefficient of determination (R²), mean square error (MSE), root mean square error (RMSE), average relative error (ARE) and average absolute relative error (AARE). The results indicate that the 3-input-based neural models had overall R², MSE, and RMSE values of 0.9689, 9.6185x10^-4and 0.0310, respectively, for the max.-min. normalizing method and R² of 0.9663, MSE of 5.7986x10^-3 and RMSE of 0.0762 for the clip scaling approach. The 5-input-based models resulted in R² of 0.9865, MSE of 5.7790×10^-4 and RMSE of 0.0240 for the max.-min. scaling method and R² of 0.9720, MSE of 3.7243x10^-3 and RMSE of 0.0610 for the clip scaling approach. Also, the 6-input-based models had R² of 0.9809, MSE of 8.7520x10^-4 and RMSE of 0.0296 for the max.-min. normalizing approach and R² of 0.9791, MSE of 3.8859 x 10^-3 and RMSE of 0.0623 for the clip scaling method. Furthermore, the generality performance of the simplified neural-based models resulted in R², RMSE, ARE, and AAPRE of 0.9644, 205.78, 0.0248, and 0.1275, respectively, for the 3-input-based neural model and R² of 0.9264, RMSE of 2089.93, ARE of 0.1656 and AARE of 0.2267 for the 6-input-based neural model. The neural-based models predicted q_o were more comparable to the test datasets than some existing correlations, as the predicted q_o result was the lowest error indices. Besides, the overall relative importance of the neural-based models’ input variables on q_o prediction is S>GLR>P_wh>T/T_sc>γ_o>BS&W>γ_g. The simplified neural-based models performed better than some empirical correlations from the assessment indicators. Therefore, the models should apply as tools for oil flow rate prediction in the Niger Delta fields, as the necessary details to implement the models are made visible.

Published in	Petroleum Science and Engineering (Volume 8, Issue 2)
DOI	10.11648/j.pse.20240802.12
Page(s)	70-99
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Neural Network, Normalization Methods, Simplified Neural-Based Models, Oil Flow Rate, Niger Delta

1. Introduction

The oil and gas production system has several distinct but interconnected components. These components include a porous medium that accumulates the oil or gas or both, a conduit(s)/well(s) intentionally drilled in the vertical or deviated configuration through the porous medium, a casing to stabilize the well(s), a tubing through which the fluids pass to the surface, a choke to regulate the flow of these fluids, a surface line and a separator to separate the reservoir contents into different phases based on their densities. One factor that links and is common to all the enumerated components is the fluid flow in the flowlines. If the field is subsea, that is, offshore, the flowline links with the inlet separator via a riser

[1]

. If a typical oil and gas production system is unpacked, Figure 1 depicts the interaction interfaces of the system.

Download: Download full-size image

Figure 1. A simple representation of oil and gas production system

[2]

Oil and gas metering, or the measuring of fluid flow rate, is a crucial component used as a standard to judge the economic viability of oil wells

[3]

. Again, understanding the flow rates of gas, oil, and water from various wells enables operators to make crucial decisions regarding rate allocation, production optimization, and future performance projection

[1]

. Beiranvand et al.

[4]

submit that when oil and gas flow rate predictions are inaccurate, they may lead to challenges such as sand production, formation damage, excessive pressures at the separator, and water coning or gas cusping

[5]

. Crude oil measurements are done critically at two points: first, at the wellhead to allow for the separation of basic sediment and water (BS&W) and then at the custody transfer point where the crude is measured preparatory for export

[6]

. Therefore, accurately quantifying the flow rate of reservoir streams provides enormous hurdles to production and subsurface engineers. The issue is that the multiphase flow meters do not present reliable production rate calculations for any fiscal purposes

[3]

. Then, one would ask what is challenging in detecting multiphase fluid flow rates in the field. First, the equation for multiphase flow is complex because the choke throat's changing pressure, temperature, gas-liquid ratio, and other fluid variables affect the fluid's properties. Second, multiphase flow is highly challenging because of these fluids' vastly varying densities and viscosities

[7]

Fluid flow through chokes may be critical or sub-critical

[8]

. According to Nasriani and Kalantari

[9]

, when the fluid approaches sonic velocity, the difference between downstream and upstream pressures is less than 0.588, and critical flow occurs in the system. The mass flow rate in this flow is determined by the pressure upstream of the bottleneck - choke. Hong and Griston

[10]

reported that the flow conditions are typically present in producing oil and gas wells and are also favoured for some reservoir flooding-enhanced oil recovery methods

[11]

. Choubineh et al.

[12]

maintained that critical-flow conditions are selected to achieve steady flow rates and prevent frequent changes in equipment performance for wellhead chokes.

On the other hand, when the mass flow of the fluid is lower than the sonic velocity, sub-critical flow conditions develop

[4]

. For the flow conditions, the mass flow rate depends on the pressure drop across any obstructions or restrictions (such as chokes) in the stream. Therefore, variations in the conditions upstream and downstream of the choke combine to affect flow rates. Thus, all the downstream-upstream pressure ratios test data less than 0.5 indicate sub-critical flow conditions.

Numerous critical and subcritical multiphase flow correlations have been available

[13]

. The empirical correlations in the literature include Gilbert

[14]

, Baxendell

[15]

, Ros

[16]

, Achong

[17]

, Omana et al.

[18]

, Pilehvari

[19]

, Owolabi et al.

[20]

, Al-Towailib and Al-Marhoun

[21]

, Beiranvand et al.

[4]

, Khorzoughi et al.

[22]

, Okon et al.

[23]

, Choubineh et al.

[12]

, Ghorbani et al.

[24]

, Alrumah and Alenezi

[25]

, Joshua et al.

[26]

, among others. According to Alarifi

[27]

, the empirical correlations are distinct for particular fields or hydrocarbon types. Therefore, the application of these correlations across all oilfields is limited. On the other hand, some authors have applied the advantage of artificial intelligence (AI) or machine learning (ML) model, that is, the ability to accurately predict oil flow rates with minimal data points, to overcome the limitations of empirical correlations. So far, several works have been reported in the literature by Berneti and Shahbazian

[28]

, Mirzaei-Paiaman and Salavati

[29]

, Al-Khalifa and Al-Marhoun

[30]

, Zangl et al.

[31]

, Al-Ajmi et al.

[32]

, Okon and Appah

[33]

, Choubineh et al.

[12]

, Al-Qutami et al.

[34]

, Al-Kadem et al.

[35]

, Khan et al.

[36]

, Ibrahim et al.

[37]

, Alarifi

[27]

, etc. Regrettably, most of these presented models need to be simplified in terms of architecture to apply. Again, Alarifi

[27]

observed that most AI-based models are developed based on data gathered from a few wells or one field. Therefore, their validity must be improved using large datasets and various possible scenarios. These observations are problematic because the unavailability of the models’ details would limit the reproducibility and validation of the models. For the Niger Delta fields, Okon and Appah

[33]

and Okorugbo et al.

[38]

have reported neural-based models' performance for predicting oil flow rate. These works are with the earlier-mentioned limitations. Therefore, it is expedient to develop simplified neural-based models(s) that are reproducible for oil flow rate prediction, keeping the stated drawbacks in mind, thus, the focus of this paper.

1.1. Overview of Neural Network

Artificial neural networks (ANNs) are computational systems inspired by biological processes that learn and use that information to predict the outcomes of complicated systems

[39]

. The neurons are the neural network's fundamental building block. According to Behnoud and Hosseini

[40]

, these neurons link to create a network that can handle complex problems. ANN has three layers: input, hidden, and output layers, and the number of input parameters determines the number of neurons in the input layer

[33]

. Initially, computations in ANN involved connection weights (W_i1, W_i2, W_i3,…….W_iN) assigned to the inputs (x₁, x₂, x₃,……x_N). The individual connection weights of the input neurons are multiplied by their respective input variables. The weighted sum of the inputs and connection weights is combined with the threshold or bias (b_i), and the summation gives the neuron’s output. The purpose of the bias is to change the input to the activation function by either increasing it or decreasing it

[3]

. The output is computed and sent to another neuron through a transfer or activation function. It is good practice to use the sigmoid transfer function for the hidden layer and the linear activation function (purelin) for the output layer

[41]

. A typical architecture that depicts the flow processes in a neural network is visible in Figure 2.

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Figure 2. A typical ANN architecture

[45, 46]

The network training is the beginning of modelling with ANN. Data are processed from the input layer through the hidden layer(s) and finally to the output layer. The predicted datasets and the actual data relate in the output layer. The difference between these datasets is fed back into the model, which updates the weights between each connection (or node) and the biases of each layer. This process is known as an epoch. In this manner, training will continue for the entire dataset until the average error (mean square error, MSE) is reduced to a predefined limit

[42]

. The number of neurons in the hidden layer affects network performance as fewer neurons lead to under-fitting, while too many neurons result in over-fitting. Therefore, optimizing the design of neurons is necessary

[43, 44]

. The several sorts of ANN include Feed-forward back-propagation neural networks (FFBP), Modular neural networks, Radial basis function (RBF), Recurrent neural networks (RNN), Convolution neural networks (CNN), Multilayer Perceptron (MLP), Deep neural network (DNN), among others.

1.2. Overview of Some Existing Neural Network Models for Oil Flow Rate Prediction

Correlations to estimate flow rate from the wellhead or through choke have received attention in the literature. Several empirical correlations have been developed for critical and subcritical flow conditions

[27, 47]

. For the critical flow condition, Gilbert

[14]

presented pioneering work using wellhead pressure (P_wh), choke (bean) size (S) and gas-liquid ratio (GLR)

[48]

. Afterward, other authors like Baxendell

[15]

, Ros

[16]

, Achong

[17]

and

[19]

updated Gilbert’s

[14]

correlation constants. Also, Beiranvand et al.

[4]

, Khorzoughi et al.

[22]

, and Choubineh et al.

[12]

modified Gilbert’s correlation to include basic sediments and water (BS&W), well-flowing temperature (T), oil gravity (γ_o) and gas gravity (γ_g) for liquid flow rate estimation. According to Alarifi

[27]

, empirical correlation accuracy is very low compared to the actual field flow rate datasets. This observation is because of the assumptions applied to developing these correlations. Also, Agwu et al.

[3]

criticized the flexibility and replicability of these correlations and then supported the use of artificial intelligence (AI) models for flow rate prediction. A comprehensive review of the AI-based models for liquid flow rate prediction is visible in Barjouei et al.

[49]

and Agwu et al.

[3]

. An extract of the neural network and its variant models for flow rate prediction is in Table 1. The table indicates that most available ANN models for flow rate prediction are developed using datasets from the Iran oilfields. Regrettably, some of these models are not applicable as their structures are too complex for incorporation into computer programs (software). Earlier, Okon et al.

[23]

opined that the performance of these models is limited by the quantity and data source upon which the models are developed. Again, Table 1 further revealed that Okon and Appah

[33]

and Okorugbo et al.

[38]

works are the only available neural-based model for predicting the oil flow rate in the Niger Delta region. Unfortunately, reproducing these models is doubtful, as the authors did not publish the models’ details: training algorithm, transfer (activation) functions, weights and biases. Therefore, their implementation in any petroleum engineering package is limited or not visible. Hence, the need to develop simplified neural-based models for oil flow rate prediction in the Niger Delta is necessary.

Table 1. Some existing works on artificial neural networks and their hybrids for oil flow rate prediction.

Author(s)

Datasets Source / Points

Artificial Intelligence Approach

Input Variables

Output Variable

Model Performance

Model / Architecture Pitfall

Berneti and Shahbazian

[28]

Source: Iran Oilfield

Datasets: 31 oil wells

ANN and ICA

Architecture:

2-7-1

P_wh, T

MSE = 0.0123

RMSE = 0.1109

R² = 0.9703

R = 0.9850

The models’ structures are simple and applicable for development.

Mirzaei-Paiaman and Salavati

[29]

Source: Iran Oilfield

Datasets: 134

ANN

Architecture:

3-4-1

P_wh, S, GOR

AARE = 2.110

ARE = -0.330

R² = 0.9998

R = 0.9999

Ahmadi et al.

[

50 ]

Source: Iran Oilfields

Datasets: 50 oil wells; 1600

ANN

Architecture:

N/A

P_wh, T

MSE = 0.0913

RMSE = 0.3022

R² = 0.9391

R = 0.9691

Models’ structures are not provided to assess their simplicity and applicability.

ANN-ICA

Architecture:

N/A

MSE = 0.0030

RMSE = 0.0551

R² = 0.9951

R = 0.9976

Al-Khalifa and Al-Marhoun

[30]

Source: Middle East fields

Datasets: 4031

ANN

Architecture:

6-9-5-8-1

P_wh, T, S, GOR, ,

MSE = 110.25

RMSE = 10.50

R² = 0.9860

R = 0.9930

The model architecture is complex and not applicable.

Zangl et al.

[31]

Source: N/A

Datasets: 258

ANN

Architecture:

N/A

P_wh, , ,

R² = 0.9308

R = 0.9648

Models’ structures are not provided to assess their simplicity and applicability.

Gorjaei et al.

1 ]

Source: Iran Oilfields

Datasets: 276

PSO-LSSVM

Architecture:

N/A

S, P_wh, GLR

AARE = 0.70

R² = 0.9935

R = 0.9967

Hasanvand and Berneti

2 ]

Source: Iran Oilfields

Datasets: 600

ANN

Architecture:

2-7-1

MSE = 0.0094

RMSE = 0.097

R² = 0.9874

R = 0.9937

The model architecture is simple and applicable for development.

Al-Ajmi et al.

[32]

Source: N/A

Datasets: 174

ANN

Architecture:

N/A

P_wh, S, T, GOR, WCT

MAPE = 15.15

R² = 0.890

R = 0.9434

Model architecture is not provided to assess its simplicity and application.

Okon and Appah

[33]

Source: Niger Delta Oilfields

Datasets: 64

ANN

Architecture:

3-6-5-1

P_wh, S, GLR

AARE = 0.1920

R² = 0.9653

R = 0.9825

Models’ structures (two hidden layers) could be more complex.

ANN

Architecture:

5-6-6-1

P_wh, S, GLR, T, BS&W

AARE = 0.1045

R² = 0.9951

R = 0.9976

Baghban et al.

3 ]

Source: Iran Oilfields

Datasets: 100

SVM

Architecture:

N/A

P_wh, S, GOR

R² = 0.9998

R = 0.9999

Models’ structures are not provided to assess their simplicity and applicability.

Buhulaigah et al.

4 ]

Source: Middle East Oilfields

Datasets: 174

ANN

Architecture:

N/A

P_wh, L_e, S, , k, ,

R² = 0.9140

R = 0.9560

Choubineh et al.

[12]

Source: Iran Oilfields

Datasets: 113

ANN-TLBO

Architecture:

N/A

P_wh, S, GLR, T, ,

AARE = 6.50

ARE = 2.09

R² = 0.9810

R = 0.9905

Al-Qutami et al.

[34]

Source: N/A

Datasets: 238

NN ensemble and ASA

Architecture:

N/A

T, P_wh, BHP, S

MAPE = 4.70

MSE = 0.0034

RMSE = 0.0585

Khan et al.

5 ]

Source: N/A

Datasets: 1500

ANN

Architecture:

5-6-1

P_wh, S, T,

AARE = 2.50

R² = 0.9940

R = 0.9970

Models’ structures have a single hidden layer that is simple to apply.

Al-Kadem et al.

[35]

Source: N/A

Datasets: 1854

ANN

Architecture:

3-10-1

P_wh, S, GOR

AARE = 3.70

R² = 0.80

R = 0.8944

Ghorbani et al.

6 ]

Source: Iran Oilfields

Datasets: 182

Architecture:

N/A

P_wh, S, GLR, BS&W

AARE = 7.33

ARE = -2.890

R² = 0.9970

R = 0.9985

Models’ structures are not provided to assess their simplicity and applicability.

Khan et al.

[36]

Source: Asian Oilfields

Datasets: 1400

ANN

Architecture:

N/A

S, P_wh, T,

AARE = 2.5618

R² = 0.9934

R = 0.9967

Al-Rumah et al.

7 ]

Source: Existing works

Datasets: 1111

ANN

Architecture:

3-39-23-1

P_wh, S, GLR

AARE = 0.2206

R² = 0.9292

R = 0.9640

Model architecture is too complex to be applicable

Marfo and Kporxah

8 ]

Source: Ghana Oilfields

Datasets: 1600

ANN

Architecture:

4-2-1

, THP, FBHP, t

MAPE = 3.18

R² = 0.9966

R = 0.9983

Model architecture is simple for application.

Ibrahim et al.

[37]

Source: Middle East Oilfields

Datasets: 548 wells

SVM

Architecture:

N/A

P_wh, S, GOR

AAPE = 1.40

R² = 0.930

R = 0.9644

The AI models’ structures are not visible for assessment.

Architecture:

N/A

AAPE = 0.75

R² = 0.940

R = 0.9695

Okorugbo et al.

[38]

Source: Niger Delta Oilfields

Datasets: 1595 from 7 fields

ANN

Architecture:

N/A

P_wh, S, GLR, GOR, , T, T/T_sc, BS&W

AARE = 28.44

APE = 7.64

R² = 0.8774

R = 0.9367

Models’ structures are not provided to assess their simplicity and applicability.

ANN-PSO

Architecture:

N/A

AARE = 35.83

APE = 12.20

R² = 0.8318

R = 0.9120

Alarifi

[27]

Source: N/A

Datasets: 1595 from 7 fields

ANN

Architecture:

N/A

S, P_wh, T, GLR, GOR, WCT

MAPE = 19.33

R² = 0.8649

R = 0.930

Azim

9 ]

Source: Egypt Oilfields

Datasets: 350 from 12 fields

ANN

Architecture:

6-10-1

WHT, GLR, WCT, BHT, H, A_t

MSE = 0.020

RMSE = 0.1414

R² = 0.9630

R = 0.9813

Model architecture is a single hidden layer with less complexity for application.

*N/A = not available

2. Methods

2.1. Data Collection and Preparation

This study collected 283 datasets from 21 producing wells across different oilfields in the Niger Delta and used them to develop neural network-based models for oil flow rate prediction. The datasets include wellhead pressure (P_wh), bean (choke) size (S), gas-liquid ratio (GLR), oil gravity (

), basic sediments and water (BS&W), well-flowing temperature (T), gas gravity (γ_g), as the input data, and oil flow rate (

) as the output datasets. The gathered datasets were as required for Gilbert

[14]

, Khorzoughi et al.

[22]

and Choubineh et al.

[12]

models’ parameters for the estimation of oil well flow rate. These oil flow rate models are in Equations 1 to 3;

Gilbert

[14]

model:

(1)

Khorzoughi et al.

[22]

model:

(2)

Choubineh et al.

[12]

model:

(3)

where A denotes the proportionality constant and the variables B to F are exponential constants of the indicated variable in the models. These datasets' statistical parameters (i.e., maximum, minimum, range, average and standard deviation values) before and after data preprocessing are in Tables 2 and 3, respectively. Two scaling approaches, namely the maximum-minimum method (in Equation 4) and clip method (in Equation 5), were used to minimize the differences (i.e., range) between the datasets. Maximum-minimum approach normalized the datasets to values between 0 and 1, while the clip method scaled the datasets to values between -1 and 1.

(4)

(5)

where

represents the scaled values for input or output parameters,

is the values of the not-scaled parameters,

and

denote the minimum and maximum values of the not-scaled parameters, respectively. According to Okon and Ansa

[46]

and Okon et al.

[60]

, scaling the datasets for the neural network training is necessary for the following reasons: adequate adjustment of the network connecting weights for optimum prediction and reducing the sensitivity of the sigmoidal (i.e., transfer or activation) function to large datasets values.

Table 2. Statistical description of the datasets for the development of the neural-based models before data preprocessing.

Parameters	Maximum	Minimum	Range	Average	Std. Dev.
Wellhead Pressure,	3600.0	53.65	3546.35	977.136	831.215
Bean (Choke) Size,	114.0	14.0	100.0	48.532	26.285
Gas-Liquid Ratio,	32851.7	20.175	32831.5	2851.31	5205.32
Oil Gravity,	0.9433	0.643	0.3003	0.8612	0.0564
Basic Sediments and Water,	0.9750	0.0005	0.9745	0.7626	0.2104
Well Flowing Temperature,	192.0	48.0	144.0	97.627	31.606
Gas Gravity,	0.9399	0.5227	0.4172	0.7410	0.0974
Oil Flow Rate,	14417.0	183.0	14234.0	2616.11	2933.33

Table 3. Statistical description of the datasets for developing the neural-based models after data preprocessing.

Parameters	Maximum	Minimum	Range	Average	Std. Dev.
Wellhead Pressure,	3300.0	95.0	3205.0	956.318	737.64
Bean (Choke) Size,	74.0	14.0	60.0	33.753	14.815
Gas-Liquid Ratio,	28597.0	29.0	28568.0	2435.81	5025.79
Oil Gravity,	0.9433	0.7339	0.2094	0.8361	0.0443
Basic Sediments and Water,	0.9430	0.0005	0.9425	0.7442	0.2053
Well Flowing Temperature,	186.0	48.0	138.0	99.747	29.736
Gas Gravity,	0.9399	0.5227	0.4172	0.7410	0.0974
Oil Flow Rate,	14417.0	183.0	14234.0	2616.11	2933.33

2.2. Neural Network Models Development

The neural fitting tool (nftool) in MATLAB (Matrix Laboratory) software 2020a was used to develop the neural network for predicting oil well flow rate (

). The scaled datasets from Equations 4 and 5 were grouped based on Gilbert

[14]

, Khorzoughi et al.

[22]

and Choubineh et al.

[12]

models’ input requirements. That is, P_wh, S and GLR (3-input) for Gilbert’s model, P_wh, S, GLR, T/T_sc and BS&W (5-input) for the Khorzoughi et al.’s model, and P_wh, S, GLR, T/T_sc,

and γ_g (6-input) for the Choubineh et al.’s model. These scaled data (i.e., inputs and output) were imported from Microsoft Excel to the MATLAB workspace, named accordingly, and saved. Afterward, the nftool environment was active from the command window, and the data files in the workspace moved to the nftool environment for neural network development and training. The neural network training to determine the number of hidden layer neurons was a trial-and-error approach. The networks learned the input and output datasets using the Leverberg-Marquardt algorithm based on the feed-forward back-propagation (FFBP) approach. The network generates the initial weights and biases using its random number generator. Thus, the input datasets, output data, weights, biases, and neurons at the hidden layer are combined to create the network. The outcomes of the network training (70% of the datasets), validation (15% of the datasets) and testing (15% of the datasets), based on the mean square error (MSE) and correlation coefficient (R) values, determine the network performance. For more details on the networks’ stopping criteria and weight adjustment, readers can obtain from published works by Mahmoudi and Mahmoudi

[61]

, Okon et al.

[62]

and Okon and Ansa

[46]

. Figures 3 to 5 depict the different neural network architectures obtained for the various oil flow rate models. Thus, the basic settings of the trained neural networks’ parameters are in Table 4. The summary of the neural network learning processes is as follows

[63]

1) read the input datasets and the expected output;

2) calculate the network output by performing weighted sums and transfer functions;

3) compare the network predictions with the expected output (target);

4) compute and update fitness (MSE) value based on the comparison;

5) repeat steps (ii) and (iii) until all training points are exhausted;

6) adjust weights appropriately to maximize fitness; and

7) repeat steps (i) to (vi) until an acceptable fitness value is established.

Thus, the stages involved in executing this study are represented in the flowchart in Figure 6.

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Figure 3. Neural network architecture for a 3-input-based model.

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Figure 4. Neural network architecture for a 5-input-based model.

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Figure 5. Neural network architecture for a 6-input-based model.

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Figure 6. Flowchart of the processes involved in performing the study.

Table 4. Basic settings of the neural networks training hyperparameters.

Hyperparameters	Values
Training Datasets	199 (70% of the datasets)
Validation Datasets	42 (15% of the datasets)
Testing Datasets	42 (15% of the datasets)
Hidden Layer	1
Hidden Layer Neurons	5, 6, 8
Hidden Layer Activation Function	tansig
Output Layer Activation Function	purelin
Learning Algorithm	Levenberg-Marquardt
Number of Epochs	1000
Rate of Learning	0.7
Architecture Selection	Trial-and-error
Target Goal MSE	10^-7
Minimum Performance Gradient	10^-7

3. Results and Discussion

3.1. The Neural Networks Performance

The neural networks developed for oil flow rate prediction are feed-forward back-propagation networks with input, hidden, and output layers. Their architectures indicated that the best predictions were with 3-5-1 for the 3-input-based model, 5-6-1 for the 5-input-based model and 6-8-1 for the 6-input-based model. This outcome implies that the 3-input-based network has three neurons at the input layer, five at the hidden layer, and one at the output layer. The 5-input-based network has five input neurons, six hidden neurons and a neuron at the input layer. Again, the 6-input-based network has six input neurons, eight hidden neurons, and one at the output layer. Therefore, the developed networks presented in Figures 3 to 5 are multiple-inputs single-output (MISO) neural networks. Table 5 depicts the performance indices: mean square error (MSE) and coefficient of determination (R²) values of the various networks during the training, validation and testing stages of the network development. As shown in Table 4, the 3-input-based network had overall MSE and R² values of 9.6185

\times

10^-4 and 0.9921 for the maximum-minimum normalization approach and 5.7986×10^-3 and 0.9915 for the clip normalization method. Also, the 5-input-based network resulted in overall MSE and R² values of 5.7790

\times

10^-4 and 0.9966 for the maximum-minimum scaling approach and 3.7243×10^-3 and 0.9929 for the clip scaling method. The 6-input-based network had overall MSE and R² values of 8.7523×10^-4 and 0.9952 for the maximum-minimum data normalization method and 3.8859

\times

10^-3 and 0.9947 for the clip normalization approach. The network performance indices showed that the maximum-minimum normalization approach performed slightly better than the clip counterparts. This observation is because of the sigmoidal (i.e., transfer) function, which ranged between 0 and 1, as with the maximum-minimum scaling method.

Furthermore, the overall performance of the various networks showed that their predictions were close to the actual oil flow rate (

) data. The observation is because the MSE and R² values are within acceptable limits for any model/network performance. Therefore, the networks can predict the fields’

with a 99.0% certainty based on the R² values obtained. Again, the closeness of the network predicted

with the actual

datasets is visible on the diagonal trend of the output (i.e., network predictions) and target (field datasets) for the overall performance, as in Figures 7 to 9. According to Al-Bulushi et al.

[64]

and Okon et al.

[62]

, when the experimental (or field) and predicted data points aligned along a unit slope, it implies a good agreement between them, as observed in Figures 7 to 9.

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Figure 7. Comparison of the normalization approaches overall performance for the 3-input-based network.

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Figure 8. Comparison of the normalization approaches overall performance for the 5-input-based network.

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Figure 9. Comparison of the normalization approaches overall performance for the 6-input-based network.

Table 5. Performance indices of the neural networks during training, validation and testing.

	Models	Indices	Maximum-minimum Method				Clip Method
	Models	Indices	Training	Validation	Testing	Overall	Training	Validation	Testing	Overall
i.	3-input-based model	MSE	1.3421x10^-3	9.6157x10^-4	1.5935x10^-3	9.6158x10^-4	5.5103x10^-3	5.7986x10^-3	6.5508x10^-4	5.7986x10^-3
i.	3-input-based model	R²	0.9918	0.9950	0.9905	0.9921	0.9929	0.9926	0.9884	0.9915
ii.	5-input-based model	MSE	5.8034x10^-4	5.7790x10^-4	4.7800x10^-4	5.779x10^-4	5.1033x10^-3	3.7243x10^-3	4.2300x10^-3	3.7243x10^-3
ii.	5-input-based model	R²	0.9971	0.9942	0.9932	0.9966	0.9929	0.9922	0.9945	0.9929
iii.	6-input-based model	MSE	8.2605x10^-4	8.7523x10^-4	7.2486 x10^-4	8.7523x10^-4	3.0894 x10^-3	6.9058 x10^-3	2.4769 x10^-4	3.8859x10^-3
iii.	6-input-based model	R²	0.9949	0.9962	0.9955	0.9952	0.9953	0.9923	0.9955	0.9947

Generally, the computations of the neural network variables (i.e., inputs, weights, biases and output) in vector form are related, as represented in Equation 6

[65]

;

(6)

where

is the network predicted output (in normalized form),

denotes the output neuron activation function (i.e., purelin),

represents the hidden layer neurons’ weights from the jth neuron to the kth output layer neuron,

is the transfer function (tansig) at the hidden neuron,

is the input layer weights from the ith neuron to the jth hidden layer neuron, and

represents an input variable. Then,

and

represent the hidden and output layers nodes’ biases, respectively. Therefore, based on the established architectures for the neural network-based models for

prediction, their computation notations are presented in Equations 7 to 9; 3-input-based model:

(7)

5-input-based model:

(8)

6-input-based model:

(9)

where

is the neural network predicted oil flow rate in normalized form. The variables

and

are the weights of the network inputs:

and

to the hidden layer neuron;

represents the hidden layer weights that connect the output layer neuron;

and

are biases at the hidden and output neurons, respectively. Then, purelin and tansig are activation functions at the output and hidden layers’ neurons. The weights and biases of the various neural network-based models for oil flow rate prediction are in Tables 6 to 11.

Table 6. Weights and biases of the 3-input-based model using max.-min. scaling method.

	Input layer weights			Hidden biases and weights		Output bias
	()	()	()
1	-0.5385402	-1.0111643	-0.3697213	1.974864	-1.1109503	-0.1585362
2	-1.2660117	5.7652163	2.0999111	-3.1918448	0.8373804
3	0.29895761	1.3578018	-1.9698251	-2.8393038	5.1375123
4	0.24131335	2.1912495	-0.5929832	-2.0673442	-2.6497395
5	1.01553212	-1.3477269	1.15601166	3.3939060	3.6110957

Table 7. Weights and biases of the 3-input-based model using the clip scaling method.

	Input layer weights			Hidden biases and weights		Output bias
	()	()	()
1	-1.9102399	0.1173669	-0.9406521	2.8339813	0.16789125	1.188847
2	2.1891470	-0.4798915	-3.5388877	-1.4183450	0.38368504
3	0.7112502	2.47403781	-3.539615	-1.3528274	-0.28206409
4	-1.0860498	-0.7007909	4.3089994	4.66603469	-2.55714526
5	1.6228026	-2.7474310	-2.0315176	-2.3284541	-0.37376004

Table 8. Weights and biases of the 5-input-based model using max.-min. scaling method.

	Input weights					Hidden biases	Hidden weights	Output bias
	()	()	()	()	()
1	3.4900924	-7.731092	-6.4360719	2.09776475	6.6313152	-11.863917	-2.3205470	0.3472364
2	3.4423928	-7.372197	-6.3975091	2.06356456	6.4071733	-11.511381	2.3664439
3	0.0299627	1.7319449	-0.2676342	-0.56557378	-0.4638709	1.4411783	0.17676961
4	0.8098289	1.0948423	-3.4110332	0.727664266	0.0829497	-4.0252434	1.6790642
5	-7.188589	5.6356843	6.7863179	7.263317569	-0.5029194	-4.1082139	4.9673305
6	-5.888010	4.8817139	0.50369342	6.2700012	-0.30080753	-8.6339629	-5.3101384

Table 9. Weights and biases of the 5-input-based model using the clip scaling method.

	Input weights					Hidden biases	Hidden weights	Output bias
	()	()	()	()	()
1	0.9010532	0.6868277	-3.3368676	0.3989323	-0.0023792	-3.695982	1.7599079	0.662883
2	-0.994761	2.1941044	0.4255441	-3.6930451	1.69289456	3.0435556	0.1249966
3	2.7616773	-2.832146	-2.1586329	-0.6128486	3.78528638	-2.1373435	0.0529069
4	0.4204269	0.0523971	-1.1852713	-0.3920006	0.44395427	-0.655956	-0.2171184
5	1.1663272	0.6718416	-2.0633890	-0.1329151	-0.1273617	0.2522338	0.133396
6	-0.269904	-2.297810	1.6909864	0.3070311	-1.4469666	1.7310819	-0.0644734

Table 10. Weights and biases of the 6-input-based model using max.-min. scaling method.

	Input weights						Hidden biases	Hidden weights	Output bias
	()	()	()	()	()	()
1	0.047515	0.130103	0.44245	-2.030657	0.581165	-0.089446	1.970767	0.4357025	0.235916
2	-1.011681	0.891035	0.020582	-1.195120	-1.019611	-0.161630	1.139384	0.0776338
3	0.316439	0.309080	-0.488147	1.140003	-0.367717	0.6344811	-0.450963	0.3284348
4	-0.316973	-0.908201	0.045713	0.607017	-1.051208	-1.110216	-0.172092	-0.1304821
5	0.632287	0.294746	-0.235050	-0.862207	-0.594977	-0.928184	0.637973	0.2560455
6	0.594650	0.439590	-0.827425	-1.193106	0.6061402	0.1739179	1.484508	-0.260832
7	-0.626795	0.869463	-0.980515	0.176416	-0.468085	-0.577010	-1.364878	-0.036701
8	-0.560974	-0.483019	1.727690	-0.821655	-0.648451	-0.043607	2.329126	-1.3554218

Table 11. Weights and biases of the 6-input-based model using the clip scaling method.

	Input weights						Hidden biases	Hidden weights	Output bias
	()	()	()	()	()	()
1	-0.802896	1.032175	0.523643	1.267974	0.494186	-0.555345	2.146365	0.071464	-0.65368
2	-0.5326472	0.391553	0.265645	-1.635991	0.504839	-0.119175	-0.794149	-0.523327
3	-0.000600	-0.864577	0.055065	0.576637	-0.537443	-0.721054	0.014171	-0.769143
4	0.688203	2.37251	-1.50568	2.36764	0.877263	0.617937	0.603996	-0.068565
5	0.523447	0.539761	-0.702578	2.091369	1.525157	0.449091	-1.461013	0.493293
6	-0.832304	-0.934466	0.977342	-0.880203	-0.200528	0.809025	-1.422903	0.385712
7	-1.877361	-0.722014	-0.02678	-0.280917	-0.139388	0.215438	-1.82908	-0.136098
8	0.233416	0.240730	0.911708	-0.095287	-0.241667	-1.41470	2.185845	0.587712

Aside from the network architectures, as visible in Figures 3 to 5, the weights and biases (Tables 5 to 10), and other details of the networks, it is imperative to establish the average contribution of the input variables on the network output

[60, 66]

. This average contribution of input parameters on the network output is the contribution factor (CF) or relative importance (RI). Thus, the Garson method expanded in Equation 10 to determine the RI of the input variables on the neural network's output

[60]

(10)

where

is the input layer weights,

denotes the hidden layer weights to the output neuron,

and

represent the numbers of inputs and hidden layer’s neurons. The outcomes of the RI assessment on the networks’ input variables are in Table 12.

The results from the RI assessment in Table 12 revealed that the bean (choke) size and GLR are the most significant parameters in the developed neural networks for oil flow rate prediction. The observation aligns with the position of Joshua et al.

[26]

. Also, it revealed that the dataset scaling (normalization) approach influenced the input variables’ RI on the networks’ output. This observation is visible in the RI ranking of the network input variables in Table 12. The 3-input-based network had an RI ranking of S>GLR>P_wh for the maximum-minimum scaling method and an RI ranking of GLR>S>P_wh for the clip scaling approach. The 5-input-based network had an RI ranking of S>GLR>T/T_sc>P_wh>BS&W for the maximum-minimum normalization approach and GLR>S>P_wh>BS&W>T/T_sc for the clip method. Likewise, the 6-input-based network resulted in T/T_sc>γ_o>GLR>S>P_wh>γ_g and T/T_sc>S>P_wh>γ_g>GLR>γ_o for maximum-minimum and clip methods, respectively. Thus, the overall RI ranking of the input variables on the developed networks’ output is S>GLR>P_wh>T/Tsc> γ_o>BS&W>γ_g.

Table 12. Relative importance ranking of the neural-based models’ input variables.

	Model	Scaling method	Input Variables Relative Importance (%)
	Model	Scaling method	P_wh	S	GLR	T/T_sc	BS&W	γ_o	γ_g
i.	3-input-based	Max.-min.	17.40	52.80	28.80	NA	NA	NA	NA
		RI ranking	3rd	1st	2nd	NA	NA	NA	NA
		Clip	26.01	26.64	47.35	NA	NA	NA	NA
		RI ranking	3rd	2nd	1st	NA	NA	NA	NA
ii.	5-input-based	Max.-min.	16.68	30.07	23.56	18.00	11.69	NA	NA
		RI ranking	4th	1st	2nd	3rd	5th	NA	NA
		Clip	16.68	19.51	35.06	12.93	15.82	NA	NA
		RI ranking	3rd	2nd	1st	5th	4th	NA	NA
iii.	6-input-based	Max.-min.	13.24	13.90	15.62	27.29	NA	17.36	12.59
		RI ranking	5th	4th	3rd	1st	NA	2nd	6th
		Clip	16.59	19.02	12.72	23.76	NA	12.20	15.71
		RI ranking	3rd	2nd	5th	1st	NA	6th	4th

*NA = not applicable

3.2. Simplified Neural Network-Based Models for Oil Flow Rate Prediction

According to Okon et al.

[60]

, numerous researchers have presented neural networks in “black box” form. The developed neural models are not in a simplified mathematical form. This drawback limits the ability of ardent readers to understand the application of any developed neural network model

[46]

. Okon et al.

[62]

presented detailed workings of the neural network to achieve its prediction (output) from the input variables. Considering the 3-input-based network, the basic neural network computations steps are as follows

[67]

1) input variables (

and

) from the input neurons multiply with input weights (

and

), respectively, and are linked to hidden layer neurons;

2) at the first hidden layer neuron (i.e., i = 1), the input (i.e.,

) from the input layer combined with the neuron’s bias (

) and the sum (i.e.,

) is transformed by the sigmoid function (Equation 11), to the output neuron;

(11)

where

3) the transformed output from the hidden neuron (i.e.,

) multiplied by the hidden neuron weight (

) and linked to the output neuron;

4) at the output neuron, the hidden layer output combined with its bias (

), thus,

;

5) steps (i) to (iv) are repeated for values of i = 2,...5 for the neurons and at the output neuron, the sum

is transformed using the purelin function as the network’s output. Thus, the predicted values are

The values for the variables

and

are in Tables 5 and 6, and they could be applied to other networks (i.e., 5-input-based and 6-input-based) with appropriate adjustments to the network’s variables. The output from the neural network is presented in the normalized form and would require de-normalization to transform the network predictions to a required format (values). Thus, the simplified neural network-based models for oil flow rate prediction are in Equations 12 and 13, based on maximum-minimum and clip scaling approaches;

(12)

(13)

where

is the de-normalized oil flow rate,

and

are the predicted oil flow rates (in normalized form) based on the maximum-minimum and the clip scaling methods, respectively, from the neural network. Thus,

and

are expressed in Equations 14 and 15;

(14)

(15)

Then,

in Equations 14 and 15 are expressed as

and

, where

are the computations at the hidden layer neurons. For the 3-input-based network with the maximum-minimum scaling method,

are expanded as Equations 16 to 20;

(16)

(17)

(18)

(19)

(20)

Also, the variables

for the 3-input-based network with the clip scaling method are expressed as Equations 21 to 25;

(21)

(22)

(23)

(24)

(25)

where

, and

are the normalized input variables (i.e.,

, and

), presented as

, and

, for the maximum-minimum scaling method and

, and

, for the clip scaling method.

Similarly, the 5-input-based and 6-input-based models’

and

are adjusted to reflect the additional input parameters:

and

. Using the appropriate weights and biases presented in Tables 8 and 9 for 5-input-based models and Tables 10 and 11 for 6-input-based models,

and

would be established. The normalized additional input variables are expressed as

and

for the maximum-minimum normalization method and

and

for the clip scaling method.

3.3. Comparison of the Simplified Models with Some Empirical Correlations for Oil Flow Rate Estimation

The predictions of the simplified neural network-based models were compared with some empirical correlations for oil flow rate (

) estimations. The performance of (i.e., the closeness between) the models predicted

with the actual field data was related using some statistical indices: coefficient of determination (R²), root mean square error (RMSE), average relative error (ARE) and average absolute relative error (AARE), and regression plot. The statistical indices are estimated using Equations 26 to 31:

(26)

(27)

(28)

(29)

where

and

denote the field oil flow rate and average field flow rate, respectively,

and

represent the predicted oil flow rate and average predicted oil flow rate, respectively, from the neural-based models, and

denotes the number of datasets or data points.

The 3-input-based models’ (i.e., ANN-MS and ANN-CS) predictions and other empirical correlations: Gilbert

[14]

, Baxendell

[15]

, Ros

[16]

, Achong

[17]

, Owolabi et al.

[20]

, Beiranvand et al.

[4]

, Okon et al.

[23]

and Joshua et al.

[26]

, estimations were collated. The results are in Table 13 and Figures 10 and 11. In Table 13, the neural network-based models have the lowest statistical indices (i.e., RMSE, ARE and AARE) compared to the empirical correlations. The developed neural network model based on the maximum-minimum normalization approach (ANN-MS) resulted in an R² value of 0.9687, while the model based on the clip scaling method resulted in an R² value of 0.9663. The R² values for the ANN-MS and ANN-CS models were higher than the performance of the empirical correlations, even for the correlations (Owolabi et al.

[20]

; Okon et al.

[23]

; Joshua et al.

[26]

) based on the Niger Delta region. Statistically, these high R² values for the neural network-based models implied that their predictions would fit closer to the actual field data than the estimation of the empirical correlations

[60]

. Besides the statistical indicators in Table 13, the Taylor diagram of the 3-input-based models and empirical correlations is presented in Figure 10. This figure shows the statistical errors and agreement of the developed models and the empirical correlations with the field data

[68, 69]

. Thus, the predictions of the field data by the 3-input-based models (i.e., ANN-MS and ANN-CS) are more noticeable than the empirical correlations in Figure 10. Again, Figure 11 depicts the predictions of the neural models and empirical correlations with the field

. The figure indicates that the neural and empirical correlations estimated data points aligned along the diagonal trend in Figure 11. According to Al-Bulushi et al.

[64]

, the diagonal alignment of the neural-based model predictions and the estimation of the empirical correlations with the field

shows a good agreement between the predicted

and the actual field

datasets.

Table 13. Statistical performance of the developed Gilbert-based models with some empirical correlations for oil flow rate estimation.

	Models	Statistical Performance
	Models	R²	RMSE	ARE	AARE
i.	Gilbert (1954)	0.8717	1056.088	-0.1268	0.3177
ii.	Baxendell (1958)	0.9329	759.155	0.2513	0.3734
iii.	Ros (1960)	0.9402	716.155	0.1688	0.3354
iv.	Achong (1961)	0.8939	962.841	0.3927	0.4872
v.	Owolabi et al. (1991)	0.9073	970.250	0.3358	0.5766
vi.	Beiranvand et al. (2012)	0.9386	726.693	-0.0264	0.3002
vii.	Okon et al. (2014)	0.9627	877.850	0.4330	0.5187
viii.	Joshua et al. (2020)	0.9477	669.982	0.0547	0.2998
ix.	ANN-MS	0.9687	517.719	0.1012	0.2468
x.	ANN-CS	0.9663	537.677	0.0673	0.2582

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Figure 10. Taylor diagram of the 3-input-based models with some empirical correlations.

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Figure 11. Comparison of the 3-input-based neural models and empirical correlations predictions; the diagonal line represents 1:1 trend line.

On the other hand, Table 14 presents the statistical performance (i.e., R², RMSE, ARE and AARE) of the 5-input neural-based models and some empirical correlations. The results showed that the simplified neural-based models’ (i.e., ANN-MS and ANN-CS) statistical yardsticks were better than the empirical correlations by Beiranvand et al.

[4]

, Khorzoughi et al.

[22]

, Okon et al.

[23]

and Ghorbani et al.

[24]

. This assertion is revealed in the R² values obtained for the 5-input neural-based models. In Table 14, the ANN-MS model had an R² value of 0.9867, while the ANN-CS model resulted in an R² value of 0.9720. These R² values implied that the 5-input neural-based models would predict the field

with 98.67% and 97.20% for ANN-MS and ANN-CS models, respectively. Also, the RMSE, ARE, and AARE values obtained for the neural-based models were lesser than the empirical correlations’ estimations. Furthermore, the performance of the neural network-based models and the empirical correlations are in Figure 12. From the figure, the neural-based models and some correlations: Beiranvand et al.

[4]

, Okon et al.

[23]

, and Ghorbani et al.

[24]

predictions cluster along the diagonal trend. The figure reveals that the estimated correlation values between Khorzoughi et al.

[22]

did not agree with the actual field

. The observation is visible in the data points (trend) for Khorzoughi et al.

[22]

correlation in Figure 12 and the R² value obtained for the correlation in Table 14.

Table 14. Statistical performance of the developed 5-input-based models with some empirical correlations for oil flow rate estimation.

Models

Statistical Performance

R²

RMSE

ARE

AARE

Beiranvand et al.

[4]

0.9468

718.010

0.1737

0.3300

ii.

Khorzoughi et al.

[22]

0.6560

2229.569

0.4601

0.7499

iii.

Okon et al.

[23]

0.9023

1253.268

0.5515

0.6486

iv.

Ghorbani et al

[24]

0.9255

926.690

-0.1040

0.3742

ANN-MS

0.9867

383.276

0.1139

0.2367

vi.

ANN-CS

0.9720

491.487

0.1144

0.2642

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Figure 12. Comparison of the 5-input-based neural models and empirical correlations predictions; the diagonal line represents 1:1 trend line.

Table 15 presents the statistical performance of the 6-input-based neural network models and some empirical correlations: Choubineh et al.

[12]

and Joshua et al.

[26]

. The results (i.e., R² values) indicated that the prediction of the neural network-based models was closer to the actual field

than the empirical correlation estimations. Again, the error indicators (i.e., RMSE, ARE and AARE values) for the neural-based models were lesser than those obtained for the empirical correlations. Also, the neural models' predictions aligned diagonally more than the empirical correlations’ estimations in Figure 13. The observation implied that the neural-based models' predictions were in sync with the actual field datasets, with 98.08% and 97.91% certainty for the ANN-MS and ANN-CS models, respectively.

Table 15. Statistical performance of the 6-input-based models with some empirical correlations for oil flow rate estimation.

Models

Statistical Performance

R²

RMSE

ARE

AARE

Choubineh et al.

[12]

0.9443

721.958

0.0892

0.3316

ii.

Joshua et al.

[26]

0.9749

1089.123

0.3833

0.4196

iii.

ANN-MS

0.9808

407.186

0.0643

0.2192

iv.

ANN-CS

0.9791

424.931

0.0425

0.2594

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Figure 13. Comparison of the 6-input-based neural models and empirical correlations predictions; the diagonal line represents 1:1 trend line.

3.4. Generalization Performance of the Simplified Neural Network-Based Models for Oil Flow Rate Predictions

According to Alexander et al.

[70]

, applying datasets not involved in developing the model to assess its generalization robustness is the gold standard. In this regard, 63 and 113 datasets from the Niger Delta and Iran fields, respectively, in Okon et al.

[23]

and Choubineh et al.

[12]

, were used for the assessment. Tables 16 and 17 show the statistical description of the datasets. The generalization performances of the simplified 3-input-based and 6-input-based neural models were tested using these datasets. The models' performance is visible in statistical indices (i.e., R², RMSE, ARE and AARE) in Tables 18 and 19 and regression plots in Figures 14 to 19.

Table 16. Statistical description of Okon et al.

[23]

datasets for the models’ generalization performance.

Parameters	Data	Max.	Min.	Range	Mean	Std. Dev.	Kurt.	Coef. of Var. (%)
Wellhead Pressure,	63	2320.0	101.5	2218.6	592.93	445.31	2.957	75.10
Bean (Choke) Size,	63	72.0	16.0	56.0	35.46	13.96	0.268	39.38
Gas-Liquid Ratio,	63	4134.41	93.26	4041.15	889.64	925.15	4.802	103.99

Table 17. Statistical description of Choubineh et al.

[12]

datasets for the models’ generalization performance.

Parameters	Data	Max.	Min.	Range	Mean	Std. Dev.	Kurt.	Coef. of Var. (%)
Wellhead Pressure,	113	2940.0	50.0	2890.0	1280.1	348.69	1.275	77.13
Bean (Choke) Size,	113	80.0	24.0	56.0	54.51	16.88	0.122	43.89
Gas-Liquid Ratio,	113	3660.0	107.0	3553.0	858.58	440.25	8.518	206.33
Oil Gravity,	113	0.92	0.808	0.112	0.8583	0.0171	-0.047	5.30
Temperature,	113	135.0	90.0	45.0	124.12	12.61	0.384	29.81
Gas Gravity,	113	1.236	0.6886	0.5474	0.7313	0.0597	-0.805	3.14

Table 18 and Figures 14 to 17 present the 3-input-based neural models (i.e., ANN-MS and ANN-CS) generalization performance with the Okon et al.

[23]

and Choubineh et al.

[12]

datasets. For the Okon et al.

[23]

datasets, the statistical indicator showed that the neural-based models predicted

resulted in R² values of 0.9488 for ANN-MS, while the ANN-CS model had 0.9644. From a statistical standpoint, the R² values implied that the simplified neural-based models predicted

are 94.88% (for ANN-MS) and 96.44% (for ANN-CS) related to the Okon et al.

[23]

datasets. The results further depict RMSE, ARE, and AARE values of 251.926, 0.1158, and 0.1862 for the ANN-MS model and 205.871, 0.1158, and 0.1862 for the ANN-CS model. Again, the 3-input-based neural model predictions from the Choubineh et al.

[12]

datasets resulted in R² values of 0.8848 for the ANN-MS model and 0.9264 for the ANN-CS model. The errors (i.e., RMSE, ARE, and AARE) values obtained for the models were 2754.48, 0.0746 and 0.2573 for ANN-MS and 2089.93, 0.1656 and 0.2267 for ANN-CS. From the R² values obtained, the simplified 3-input-based neural models would predict Choubineh et al.

[12]

datasets with 88.48% and 92.64% certainty for ANN-MS and ANN-CS models, respectively. Aside from the R² values that depict the predicted

’s closeness with the test datasets, the generality robustness of these simplified neural-based models is visible on the cross-plots in Figures 14 to 17. As observed in these figures, the diagonal trend of the data points indicates a good agreement between the predicted

and the test datasets

[64]

Table 18. Statistical indices of the simplified 3-input-based models’ generalization robustness to predict the oil flow rate.

Datasets Source

Neural Model

Statistical Performance

R²

RMSE

ARE

AARE

Okon et al.

[23]

ANN-MS

0.9488

251.926

0.1158

0.1862

ANN-CS

0.9644

205.781

0.0248

0.1275

ii.

Choubineh et al.

[12]

ANN-MS

0.8848

2754.48

0.0746

0.2573

ANN-CS

0.9264

2089.93

0.1656

0.2267

Download: Download full-size image

Figure 14. Comparison of 3-input-based model (ANN-MS) predictions with oil flow rate datasets from Okon et al.

[23]

Download: Download full-size image

Figure 15. Comparison of 3-input-based model (ANN-CS) predictions with oil flow rate datasets from Okon et al.

[23]

Download: Download full-size image

Figure 16. Comparison of 3-input-based model (ANN-MS) predictions with oil flow rate datasets from Choubineh et al.

[12]

Download: Download full-size image

Figure 17. Comparison of 3-input-based model (ANN-CS) predictions with oil flow rate datasets from Choubineh et al.

[12]

On the other hand, the 6-input neural-based models’ generality performances are presented in Table 19 and Figures 18 and 19. The statistical indices obtained for these models were 0.8908, 2655.50, 0.1484, and 0.2219 for R², RMSE, ARE, and AARE for ANN-MS, while ANN-CS had 0.9264, 2089.93, 0.1656 and 0.2267. The R² values obtained indicated that the 6-input-based neural models would predict the Choubineh et al.

[12]

data with 89.08% certainty for ANN-MS and 92.64% for ANN-CS. Also, the closeness of the model’s predictions with the test datasets is visible on the diagonal alignment of the predicted

and test datasets in the cross-plots (Figures 18 and 19). Thus, the generalization performance of the simplified neural-based models is about 90.0% certainty with the test datasets.

Download: Download full-size image

Figure 18. Comparison of 6-input-based model (ANN-MS) predictions with oil flow rate datasets from Choubineh et al.

[12]

Table 19. Statistical indices of the simplified 6-input-based models’ generalization robustness to predict the oil flow rate.

Datasets Source

Neural Model

Statistical Performance

R²

RMSE

ARE

AARE

Choubineh et al.

[12]

ANN-MS

0.8908

2655.50

0.1484

0.2219

ANN-CS

0.9264

2089.93

0.1656

0.2267

Download: Download full-size image

Figure 19. Comparison of 6-input-based model (ANN-CS) predictions with oil flow rate datasets from Choubineh et al.

[12]

The generalization performance of Okon et al.

[23]

and Choubineh et al.

[12]

datasets was extended to some existing empirical correlations and compared with the simplified neural-based models. The results of the assessment are in Tables 20 and 21. In Table 20, the statistical results (R², RMSE, ARE and AARE) for the Okon et al.

[23]

datasets showed that the simplified 3-input-based neural models predicted

values were closer to the actual test datasets than the estimated

from the empirical correlations. The neural-based models’ predictions for the Choubineh et al.

[12]

test data agreed with the estimated

from the empirical correlations as the R-values obtained for these models were close. On the other hand, the simplified 6-input-based neural models’ generalization performance was comparable with the correlations of Choubineh et al.

[12]

and Joshua et al.

[26]

. This observation is visible in the statistical indices in Table 21 obtained for the neural-based models and the empirical correlations.

Table 20. Generalization performances of the simplified 3-input-based models and some correlations with test datasets.

Models

Statistical Performance

Okon et al. (2015). Datasets

Choubineh et al. (2017) Datasets

R²

RMSE

ARE

AARE

R²

RMSE

ARE

AARE

Gilbert

[14]

0.5099

944.51

-0.4403

0.3349

0.8117

2451.04

0.1957

-0.1924

ii.

Baxendell

[15]

0.5062

1120.79

0.0504

0.3254

0.9596

1130.74

-0.0110

0.0904

iii.

Ros

[16]

0.5070

1111.48

-0.0093

0.3182

0.9548

1138.66

0.0261

0.0941

iv.

Achong

[17]

0.4805

1218.12

0.1445

0.3665

0.9555

1128.76

0.0161

0.0989

Pilehvari

[19]

0.4977

2971.07

0.8235

0.8828

0.9394

10379.76

0.9623

vi.

Beiranvand et al.

[4]

0.4574

1210.70

-0.1398

0.3833

0.9495

1205.77

0.0868

0.1339

vii.

Okon et al.

[23]

0.5319

1053.51

0.1658

0.3660

0.9345

1371.86

0.0157

0.1154

viii.

Owolabi et al.

[

20]

0.5210

1059.91

0.0323

0.3461

0.9302

141620

-0.0023

0.1184

ix.

Joshua et al.

[26]

0.4821

1246.46

-0.0756

0.3469

0.9334

1383.10

-0.0010

0.1234

ANN-MS

0.9488

251.926

0.1158

0.1862

0.8848

2754.48

0.0746

0.2573

xi.

ANN-CS

0.9644

205.781

0.0248

0.1275

0.9264

2089.93

0.1656

0.2267

Table 21. Generalization performances of the simplified 6-input-based models and some correlations with test datasets.

Models

Statistical Performance

R²

RMSE

ARE

AARE

Choubineh et al.

[12]

0.9695

1077.45

-0.0089

0.0941

ii.

Joshua et al.

[26]

0.8949

1786.34

0.2262

0.2450

iii.

ANN-MS

0.8908

2655.50

0.1484

0.2219

iv.

ANN-CS

0.9264

2089.93

0.1656

0.2267

3.5. Statistical Significance and Acceptability of the Developed Models Predictions

In addition to the earlier-mentioned statistical indicators, which are parametric statistical tests to evaluate the performance of the developed neural network-based models, the Kruskal-Wallis test (i.e., H-test or

), a non-parametric method alternative to the one-way analysis of variance (ANOVA), was evaluated using Equation 30 to test whether the models’ predicted and field test oil flow rate datasets have the same mean values. Therefore, the null hypothesis (*H₀) is whether a significant difference exists between predicted models and the field oil flow rate mean values.

(30)

where

is the H-test value, which is equivalent to the critical value of Chi-square (

represents the total number of data points in the test,

denotes the number of data groups in the test,

is the rank sum of the individual group in the test and

is the number of data points in each group.

Table 22 presents the Kruskal-Wallis test results at a 95% significance level for the field test, Okon et al.

[23]

and Choubineh et al.

[12]

datasets compared with the developed models predicted values. Based on the

values obtained, which are equivalent to the critical values of Chi-square (

). Based on the critical values table, at 2 degrees of freedom, the corresponding p-values visible in Table 22 for the

values are less than the p-value of 0.05. Therefore, the null hypothesis (*H₀) that no significant difference exists between the models predicted and field test oil flow rate mean values are accepted. Hence, the developed models (ANN-MS and ANN-CS) predictions of oil flow rate are statistically significant and acceptable.

Table 22. Kruskal-Wallis test of the models predicted and field test oil flow rate 95% significance level.

Datasets source

values

p-values

Null hypothesis (*H₀)

Field test

6.6721

0.0377

ii.

Okon et al.

[23]

6.5562

0.0398

iii.

Choubineh et al.

[12]

7.6463

0.0478

In summary, the study has put forward simplified 3-input, 5-input, and 6-input neural-based models based on Gilbert

[14]

, Khorzoughi et al.

[22]

, and Choubineh et al.

[12]

correlations for Niger Delta oilfield flow rate prediction. Extending the generalization performances of the neural-based models to Iran oilfield datasets resulted in an excellent performance that compares with the Choubineh et al.

[12]

correlation. However, it is expedient to state that the unavailability or accessibility of datasets from oilfields across the globe for training and development of neural-based models would limit their application in other oil-producing regions. Again, it was alluded that most available neural network models for oil flow rate prediction in Table 1 have complex structures (architectures), making their implementation time-consuming. Moreover, these models' reproducibilities are doubtful, as the authors did not report the details of their models (networks) to reproduce them for applications. Therefore, the gains of this study showed that:

1) the neural networks are reproducible as their basic details and network arrangement are presented in this study;

2) the neural network models are not reported as “black boxes” but rather as “white boxes,” meaning the mathematical expressions of the networks’ computations are presented,

3) clip normalized neural-based model generalizes unseen datasets more than the maximum-minimum neural-based model;

4) the parametric (relative) importance of the neural-based models’ input variables on the output variable - oil flow rate is normalization approach sensitive;

5) the bean (choke) size (S), gas-liquid ratio (GLR), wellhead pressure (P_wh), well-flowing temperature (T) and oil gravity (γ_o) are more substantial in predicting oil flow rate (

) than basic sediments and water (BS&W) and gas gravity (γ_g); and

6) This work should consider the simplified representation of the neural-based models and the models’ details for application.

4. Conclusion

The available intelligence-based models for oil flow rate in the Niger Delta region are not reproducible, as the models’ basic details for their reproducibility are not published or in the public domain. In other words, the available models are viewed or presented as a black box. Therefore, this study showcases the simplified neural-based models based on 3-input, 5-input and 6-input parameters for predicting oil flow rate using 283 datasets from 21 wells in the Niger Delta oilfields. The feedforward backpropagation (FFBP) neural networks were trained based on maximum-minimum and clip normalization methods using the Levenberg-Marquardt learning algorithm. With a trial-and-error approach, the architectures that give good performance for the 3-input, 5-input, and 6-input variables network were [3-5-1], [5-6-1], and [6-8-1], respectively. The simplified neural-based models' predictions and their generalization performance were compared with some existing correlations to assess the efficiency of the developed models. From the performances of the simplified neural-based models, the under-listed conclusions are visible:

1) the simplified 3-input-based neural models predicted that the oil flow rate was close to the field flow rate with overall MSE and R values of 9.6185

\times

10^-4and 0.9843 for maximum-minimum normalization approach and 5.7986

\times

10^-3and 0.9830 for the clip scaling method;

2) the 5-input-based neural models predicted oil flow rate had overall MSE and R values of 5.7790

\times

10^-4and 0.9932 for the maximum-minimum normalization approach and 3.7243×10^-3and 0.9859 for the clip scaling method;

3) also, the 6-input-based neural models had overall MSE and R values of 8.7520

\times

10^-4and 0.9904 for the maximum-minimum normalization approach and 3.8859

\times

10^-3and 0.9895 for the clip scaling method;

4) the relative importance (RI) of the neural-based models’ input variables on oil flow rate is normalization (i.e., scaling) approach dependent, and the overall RI ranking of the input parameters for oil flow rate prediction is S>GLR>P_wh>T/T_sc> γ_o>BS&W>γ_g;

5) when compared with some empirical correlations, the neural-based models predicted oil flow rate resulted in the highest R² and lowest RMSE, ARE and AARE values than the existing empirical correlations;

6) the generalization performance of the simplified neural-based models, 3-input-based and 6-input-based with the test datasets resulted in R², RMSE, ARE, and AAPRE of 0.9820, 205.78, 0.0248 and 0.1275, respectively, for 3-input-based neural model and R² of 0.9264, RMSE of 2089.93, ARE of 0.1656 and AARE of 0.2267 for the 6-input-based neural model; and

7) finally, the generalization performance of the simplified neural-based models was outstanding and comparable with the test datasets to some existing empirical correlations.

Abbreviations

AARE	Average Absolute Relative Error
AI	Artificial Intelligence
ANN	Artificial Neural Network
ANN-CS	Clip Scaling Trained ANN
ANN-MS	Maximum-Minimum Scaling trained ANN
ARE	Average Relative Error
A_t	Tubing Cross-Section Area
BHP	Bottomhole Pressure
BHT	Bottomhole Temperature
BS&W	Basic Sediment & Water
CF	Contribution Factor
CNN	Convolution Neural Network
DNN	Deep Neural Network
Coef. of Var.	Coefficient of Variance
	Open Hole Size
FBHP	Flowing Bottomhole Pressure
FFBP	Feed-Forward Back-Propagation
GLR	Gas-Liquid Ratio
GOR	Gas-Oil Ratio
H	Well Depth
H-test	Kruskal-Wallis Test
k	Permeability
Kurt.	Kurtosis
L_e	Effective Length
MAPE	Mean Absolute Percentage Error
MISO	Multiple-Inputs Single-Output
ML	Machine Learning
MLP	Multilayer Perceptron
MSE	Mean Square Error
	Number of Laterals
	Flowline Pressure
	Gas lift Pressure
	Reservoir Pressure
	Wellhead Pressure
	Critical Liquid flow Rate
	Gas Lift Rate
	Liquid Flow Rate
	Oil Flow Rate
	Critical Oil flow Rate
R	Correlation Coefficient
R²	Coefficient of Determination
RF	Random Forest
RBF	Radial Basis Function
RI	Relative Importance
RMSE	Root Mean Square Error
RNN	Recurrent Neural Network
S	Choke (Bean) Size
T	Temperature
THP	Tubing Head Pressure
T_sc	Temperature at Surface Condition
WHT	Wellhead Temperature
WCT	Water-Cut
	Chi-Square
	Gas Gravity
	Oil Gravity

Funding

The funding for this research was provided by Tertiary Education Trust Fund (TETFund) Nigeria under the National Research Fund. Grant Number: TETF/ES/DR&D-CE/NRF2021/SETI/GEO/00104/01/VOL.1.

Data Availability Statement

The Niger Delta datasets used in the study are available upon request and approval from the Nigerian Upstream Petroleum Regulatory Commission (NUPRC). The Iran Oilfields datasets used for the models’ generalization evaluation are available at https://doi.org/10.1016/j.fuel.2017.06.131.

Conflicts of Interest

The authors declare no conflicts of interest.

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Umana, U. K., Okon, A. N., Agwu, O. E. (2024). Simplified Neural Network-Based Models for Oil Flow Rate Prediction. Petroleum Science and Engineering, 8(2), 70-99. https://doi.org/10.11648/j.pse.20240802.12

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Umana, U. K.; Okon, A. N.; Agwu, O. E. Simplified Neural Network-Based Models for Oil Flow Rate Prediction. Pet. Sci. Eng. 2024, 8(2), 70-99. doi: 10.11648/j.pse.20240802.12

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Umana UK, Okon AN, Agwu OE. Simplified Neural Network-Based Models for Oil Flow Rate Prediction. Pet Sci Eng. 2024;8(2):70-99. doi: 10.11648/j.pse.20240802.12

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@article{10.11648/j.pse.20240802.12,
  author = {Uduak Koffi Umana and Anietie Ndarake Okon and Okorie Ekwe Agwu},
  title = {Simplified Neural Network-Based Models for Oil Flow Rate Prediction
},
  journal = {Petroleum Science and Engineering},
  volume = {8},
  number = {2},
  pages = {70-99},
  doi = {10.11648/j.pse.20240802.12},
  url = {https://doi.org/10.11648/j.pse.20240802.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pse.20240802.12},
  abstract = {Available neural network-based models for predicting the oil flow rate (qo) in the Niger Delta are not simplified and are developed from limited data sources. The reproducibility of these models is not feasible as the models’ details are not published. This study developed simplified and reproducible three, five, and six-input variables neural-based models for estimating qo using 283 datasets from 21 wells across fields in the Niger Delta. The neural-based models were developed using maximum-minimum (max.-min.) normalized and clip-normalized datasets. The performances and the generalizability of the developed models with published datasets were determined using some statistical indices: coefficient of determination (R2), mean square error (MSE), root mean square error (RMSE), average relative error (ARE) and average absolute relative error (AARE). The results indicate that the 3-input-based neural models had overall R2, MSE, and RMSE values of 0.9689, 9.6185x10-4 and 0.0310, respectively, for the max.-min. normalizing method and R2 of 0.9663, MSE of 5.7986x10-3 and RMSE of 0.0762 for the clip scaling approach. The 5-input-based models resulted in R2 of 0.9865, MSE of 5.7790×10-4 and RMSE of 0.0240 for the max.-min. scaling method and R2 of 0.9720, MSE of 3.7243x10-3 and RMSE of 0.0610 for the clip scaling approach. Also, the 6-input-based models had R2 of 0.9809, MSE of 8.7520x10-4 and RMSE of 0.0296 for the max.-min. normalizing approach and R2 of 0.9791, MSE of 3.8859 x 10-3 and RMSE of 0.0623 for the clip scaling method. Furthermore, the generality performance of the simplified neural-based models resulted in R2, RMSE, ARE, and AAPRE of 0.9644, 205.78, 0.0248, and 0.1275, respectively, for the 3-input-based neural model and R2 of 0.9264, RMSE of 2089.93, ARE of 0.1656 and AARE of 0.2267 for the 6-input-based neural model. The neural-based models predicted qo were more comparable to the test datasets than some existing correlations, as the predicted qo result was the lowest error indices. Besides, the overall relative importance of the neural-based models’ input variables on qo prediction is S>GLR>Pwh>T/Tsc>γo>BS&W>γg. The simplified neural-based models performed better than some empirical correlations from the assessment indicators. Therefore, the models should apply as tools for oil flow rate prediction in the Niger Delta fields, as the necessary details to implement the models are made visible.
},
 year = {2024}
}

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TY - JOUR
T1 - Simplified Neural Network-Based Models for Oil Flow Rate Prediction

AU - Uduak Koffi Umana
AU - Anietie Ndarake Okon
AU - Okorie Ekwe Agwu
Y1 - 2024/09/23
PY - 2024
N1 - https://doi.org/10.11648/j.pse.20240802.12
DO - 10.11648/j.pse.20240802.12
T2 - Petroleum Science and Engineering
JF - Petroleum Science and Engineering
JO - Petroleum Science and Engineering
SP - 70
EP - 99
PB - Science Publishing Group
SN - 2640-4516
UR - https://doi.org/10.11648/j.pse.20240802.12
AB - Available neural network-based models for predicting the oil flow rate (qo) in the Niger Delta are not simplified and are developed from limited data sources. The reproducibility of these models is not feasible as the models’ details are not published. This study developed simplified and reproducible three, five, and six-input variables neural-based models for estimating qo using 283 datasets from 21 wells across fields in the Niger Delta. The neural-based models were developed using maximum-minimum (max.-min.) normalized and clip-normalized datasets. The performances and the generalizability of the developed models with published datasets were determined using some statistical indices: coefficient of determination (R2), mean square error (MSE), root mean square error (RMSE), average relative error (ARE) and average absolute relative error (AARE). The results indicate that the 3-input-based neural models had overall R2, MSE, and RMSE values of 0.9689, 9.6185x10-4 and 0.0310, respectively, for the max.-min. normalizing method and R2 of 0.9663, MSE of 5.7986x10-3 and RMSE of 0.0762 for the clip scaling approach. The 5-input-based models resulted in R2 of 0.9865, MSE of 5.7790×10-4 and RMSE of 0.0240 for the max.-min. scaling method and R2 of 0.9720, MSE of 3.7243x10-3 and RMSE of 0.0610 for the clip scaling approach. Also, the 6-input-based models had R2 of 0.9809, MSE of 8.7520x10-4 and RMSE of 0.0296 for the max.-min. normalizing approach and R2 of 0.9791, MSE of 3.8859 x 10-3 and RMSE of 0.0623 for the clip scaling method. Furthermore, the generality performance of the simplified neural-based models resulted in R2, RMSE, ARE, and AAPRE of 0.9644, 205.78, 0.0248, and 0.1275, respectively, for the 3-input-based neural model and R2 of 0.9264, RMSE of 2089.93, ARE of 0.1656 and AARE of 0.2267 for the 6-input-based neural model. The neural-based models predicted qo were more comparable to the test datasets than some existing correlations, as the predicted qo result was the lowest error indices. Besides, the overall relative importance of the neural-based models’ input variables on qo prediction is S>GLR>Pwh>T/Tsc>γo>BS&W>γg. The simplified neural-based models performed better than some empirical correlations from the assessment indicators. Therefore, the models should apply as tools for oil flow rate prediction in the Niger Delta fields, as the necessary details to implement the models are made visible.

VL - 8
IS - 2
ER -

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Author Information

Uduak Koffi Umana

Department of Petroleum Engineering, University of Uyo, Uyo, Nigeria

http://orcid.org/0009-0009-5478-4975
Anietie Ndarake Okon

Department of Petroleum Engineering, University of Uyo, Uyo, Nigeria

http://orcid.org/0000-0003-1744-3310
Okorie Ekwe Agwu

Department of Petroleum Engineering, University of Uyo, Uyo, Nigeria

Contact Email

http://orcid.org/0000-0003-3855-8352

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Umana, U. K., Okon, A. N., Agwu, O. E. (2024). Simplified Neural Network-Based Models for Oil Flow Rate Prediction. Petroleum Science and Engineering, 8(2), 70-99. https://doi.org/10.11648/j.pse.20240802.12

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Umana, U. K.; Okon, A. N.; Agwu, O. E. Simplified Neural Network-Based Models for Oil Flow Rate Prediction. Pet. Sci. Eng. 2024, 8(2), 70-99. doi: 10.11648/j.pse.20240802.12

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AMA Style

Umana UK, Okon AN, Agwu OE. Simplified Neural Network-Based Models for Oil Flow Rate Prediction. Pet Sci Eng. 2024;8(2):70-99. doi: 10.11648/j.pse.20240802.12

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@article{10.11648/j.pse.20240802.12,
  author = {Uduak Koffi Umana and Anietie Ndarake Okon and Okorie Ekwe Agwu},
  title = {Simplified Neural Network-Based Models for Oil Flow Rate Prediction
},
  journal = {Petroleum Science and Engineering},
  volume = {8},
  number = {2},
  pages = {70-99},
  doi = {10.11648/j.pse.20240802.12},
  url = {https://doi.org/10.11648/j.pse.20240802.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pse.20240802.12},
  abstract = {Available neural network-based models for predicting the oil flow rate (qo) in the Niger Delta are not simplified and are developed from limited data sources. The reproducibility of these models is not feasible as the models’ details are not published. This study developed simplified and reproducible three, five, and six-input variables neural-based models for estimating qo using 283 datasets from 21 wells across fields in the Niger Delta. The neural-based models were developed using maximum-minimum (max.-min.) normalized and clip-normalized datasets. The performances and the generalizability of the developed models with published datasets were determined using some statistical indices: coefficient of determination (R2), mean square error (MSE), root mean square error (RMSE), average relative error (ARE) and average absolute relative error (AARE). The results indicate that the 3-input-based neural models had overall R2, MSE, and RMSE values of 0.9689, 9.6185x10-4 and 0.0310, respectively, for the max.-min. normalizing method and R2 of 0.9663, MSE of 5.7986x10-3 and RMSE of 0.0762 for the clip scaling approach. The 5-input-based models resulted in R2 of 0.9865, MSE of 5.7790×10-4 and RMSE of 0.0240 for the max.-min. scaling method and R2 of 0.9720, MSE of 3.7243x10-3 and RMSE of 0.0610 for the clip scaling approach. Also, the 6-input-based models had R2 of 0.9809, MSE of 8.7520x10-4 and RMSE of 0.0296 for the max.-min. normalizing approach and R2 of 0.9791, MSE of 3.8859 x 10-3 and RMSE of 0.0623 for the clip scaling method. Furthermore, the generality performance of the simplified neural-based models resulted in R2, RMSE, ARE, and AAPRE of 0.9644, 205.78, 0.0248, and 0.1275, respectively, for the 3-input-based neural model and R2 of 0.9264, RMSE of 2089.93, ARE of 0.1656 and AARE of 0.2267 for the 6-input-based neural model. The neural-based models predicted qo were more comparable to the test datasets than some existing correlations, as the predicted qo result was the lowest error indices. Besides, the overall relative importance of the neural-based models’ input variables on qo prediction is S>GLR>Pwh>T/Tsc>γo>BS&W>γg. The simplified neural-based models performed better than some empirical correlations from the assessment indicators. Therefore, the models should apply as tools for oil flow rate prediction in the Niger Delta fields, as the necessary details to implement the models are made visible.
},
 year = {2024}
}

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TY - JOUR
T1 - Simplified Neural Network-Based Models for Oil Flow Rate Prediction

VL - 8
IS - 2
ER -

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