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. 2021 Aug 12;6(33):21499–21513. doi: 10.1021/acsomega.1c02376

Deep Learning Approach for Robust Prediction of Reservoir Bubble Point Pressure

Fahd Saeed Alakbari †,*, Mysara Eissa Mohyaldinn †,*, Mohammed Abdalla Ayoub †, Ali Samer Muhsan ‡
PMCID: PMC8388097  PMID: 34471753

Abstract

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The bubble point pressure (Pb) is a crucial pressure–volume–temperature (PVT) property and a primary input needed for performing many petroleum engineering calculations, such as reservoir simulation. The industrial practice of determining Pb is by direct measurement from PVT tests or prediction using empirical correlations. The main problems encountered with the published empirical correlations are their lack of accuracy and the noncomprehensive data set used to develop the model. In addition, most of the published correlations have not proven the relationships between the inputs and outputs as part of the validation process (i.e., no trend analysis was conducted). Nowadays, deep learning techniques such as long short-term memory (LSTM) networks have begun to replace the empirical correlations as they generate high accuracy. This study, therefore, presents a robust LSTM-based model for predicting Pb using a global data set of 760 collected data points from different fields worldwide to build the model. The developed model was then validated by applying trend analysis to ensure that the model follows the correct relationships between the inputs and outputs and performing statistical analysis after comparing the most published correlations. The robustness and accuracy of the model have been verified by performing various statistical analyses and using additional data that was not part of the data set used to develop the model. The trend analysis results have proven that the proposed LSTM-based model follows the correct relationships, indicating the model’s reliability. Furthermore, the statistical analysis results have shown that the lowest average absolute percent relative error (AAPRE) is 8.422% and the highest correlation coefficient is 0.99. These values are much better than those given by the most accurate models in the literature.

1. Introduction

Bubble point pressure (Pb) is known as the pressure at which the gas’s initial bubble emerges from the liquid oil solution.1 Obtaining accurate Pb is essential for performing proper reservoir simulations, production optimization, and oil recovery calculations.2−5 Thus, accurate calculation of Pb is hugely significant in the petroleum industry. Therefore, enhancing Pb correlations and models for robust and accurate determination of reservoir Pb has been one of the hot topics of petroleum engineering research.6

In this regard, substantial research efforts have been devoted by many researchers leading to the availability of valuable prediction correlation and reported procedures for determining Pb. Standing7 used 105 data points from California crude oil, USA, to obtain the Pb correlation. Lasater8 published a correlation to obtain Pb based on 158 experimental data points from the USA and Canada. Glaso9 applied linear and nonlinear regressions to obtain a Pb correlation utilizing 41 data points from the North Sea to estimate the Pb. Vasquez and Beggs10 implemented regression techniques to find correlations for the Pb prediction utilizing 6004 data points worldwide. Al-Marhoun11 used nonlinear multiple regression analysis and introduced a Pb correlation using 160 data points from the Middle East. Kartoatmodjo and Schmidt12 applied nonlinear regression analysis to develop a bubble point prediction correlation using 5392 data points from the USA and South-East Asia. They formed a correlation to determine Pb.12 Dokla and Osman13 utilized a regression method utilizing 51 data points from the UAE to build a Pb prediction correlation. Petrosky and Farshad14 proposed a bubble point pressure correlation utilizing 90 data points from the Gulf of Mexico. They applied regression methods using the statistical analysis system (SAS) software. Macary and El-Batanoney15 applied 90 data points from the Gulf of Suez to build a Pb correlation. Omar and Todd16 utilized a nonlinear regression method to obtain Pb utilizing 93 data points from Malaysia. De Ghetto et al.17 applied multiple, linear, and nonlinear regressions using the statistical analysis system (SAS) program to develop a bubble point pressure correlation utilizing 3700 data points from the Mediterranean Basin, Africa, Persian Gulf, and the North Sea. Frashad et al.18 applied multiple nonlinear regression techniques to develop a Pb correlation utilizing 43 data points from Colombia. Almehaideb19 proposed a Pb correlation utilizing 62 data points from the United Arab Emirates (UAE). He used regression analysis to obtain the correlation.19 Hanafy et al.20 applied regression analysis to develop a Pb prediction using 324 data points from Egypt. Velarde et al.2 applied nonlinear regression methods utilizing 2097 data points worldwide to create a Pb correlation. Gharbi et al.21 applied a neural network to find Pb. Al-Shammasi22 used linear and nonlinear regression to propose a Pb correlation utilizing 1661 data points worldwide. Dindoruk and Christman23 developed the equation to estimate Pb utilizing Microsoft Excel. The correlation was based on 100 Gulf of Mexico pressure–volume–temperature (PVT) laboratory. Mehran et al.24 and Bolondarzadeh et al.25 proposed a Pb correlation using 387 data points and 166 data points, respectively, from Iran fields. Malallah et al.26 used graphical alternating conditional expectation to calculate a Pb. Hemmati and Kharrat27 proposed a Pb correlation using 287 data points from Iranian crude oils. They used the nonlinear multiple regression techniques to build the Pb correlation.27 Mazandarani and Asghari28 applied multiple regression analysis to develop a Pb correlation using 55 data points from different locations in Iran. Khamehchi et al.29 applied a stepwise multiple linear regression analysis using a statistical software package Statistical Package for the Social Sciences (SPSS) for Windows to develop a Pb correlation, utilizing 94 crude oil data points.29 Dutta and Gupta30 used support vector regression to find a Pb calculation. Arabloo et al.6 introduced a Pb correlation by applying a multistart feature of LINGO software and using 750 data points worldwide. Patil et al.31 applied the multiple regression methods and utilized 23 data points from Niger Delta to calculate Pb. Gomaa32 used the multiple nonlinear regression analysis, utilizing 441 data points from the Middle East crude oil, to build a Pb correlation.32 Alakbari et al.33 utilized an artificial neural network and the fuzzy logic method to obtain Pb models. Sharrad and Abd-Alrahman34 presented a Pb correlation applying 35 data points from Libya crude oil, using linear regression analysis through EViews software. Seyyedattar et al.35 used 569 data points from different places applying extra trees, a least square support vector machine, and an adaptive network-based fuzzy inference system to predict Pb. They showed that the extra trees model is a robust tool to forecast Pb.35 Talebkeikhah et al.36 utilized a radial basis function neural network, multilayer perceptron, support vector regression, adaptive neuro-fuzzy inference system, decision trees, and random forest to obtain Pb. They indicated that the decision trees model is the best model with an average absolute relative error of 3.379%.36 Tariq et al.37 used a functional network coupled with particle swarm optimization to determine Pb. However, the previous models lack accuracy and have not applied trend analysis to identify the correct relationships between the inputs and outputs as part of the validation process. To our knowledge, this paper is the first study to apply the long short-term memory (LSTM) network and trend analysis for developing a bubble point pressure model.

In this work, we developed an accurate and reliable LSTM-based model by implementing trend analysis to predict Pb. LSTM was applied in this study because it has surpassed other deep learning techniques such as recurrent neural network (RNN) and can capture the nonlinear trends in data and remember previous information for over a long time.38 LSTM has self-connecting hidden layers and a gating structure to control long-sequence time-series data efficiently.39 Therefore, LSTM has been successfully used to solve many problems.38 Furthermore, the trend analysis is used to present the relationships between inputs (gas solubility (Rs), gas-specific gravity (γg), oil specific gravity (API), and reservoir temperature (Tf)) and output (bubble point pressure (Pb)) in the existing and LSTM models. The trend analysis defines errors in the existing and LSTM models to present unexpected relationships between inputs and output, highlighting the need to show the model’s reliability. Moreover, several statistical analyses such as average absolute percentage relative error (AAPRE), maximum absolute percent relative error (Emax), minimum absolute percent relative error (Emin), root mean square error (RMSE), standard deviation (SD), and correlation coefficient (R) have been performed to describe, validate, and compare the LSTM model with the previously published models.

2. Methodology

Figure 1 presents the workflow diagram of the previous and proposed LSTM models. The workflow mainly included six phases: data collection, testing of previous models, training the LSTM model, trend analysis, testing the LSTM model, and comparing the models.

Figure 1.

Figure 1

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Workflow diagram for the present study.

2.1. Data Collection and Preprocessing

Seven hundred and sixty data points, collected from different published sources11,13,16 relevant to this model, were utilized in this work. The preprocessing step involved data cleaning for outliers in the collected data. The collected data set was split into two subparts; 70% (532 data points) of the data was allocated for training. In contrast, 30% (228 data points) of the collected data was allocated for testing the model. Table 1 lists the statistical description of the collected training and testing data. The box and whisker plot was used to detect and remove outliers in the collected data to obtain normal data. The box and whisker plot was introduced by Tukey.40Figure S1 (Supporting Information) presents the box and whisker plot that displays lower quartile Q1, upper quartile Q3, interquartile range (IQR), lowest observation (Q1 – 1.5 × IQR), and the highest observation (Q3 + 1.5 IQR). Outliers are any value less than (Q1 – 1.5 × IQR) or more than (Q3 + 1.5 IQR), Figure S1. The box and whisker plots for the inputs and output parameters for the collected and the normal training data are presented in Figures S2 and S3, respectively. The testing data does not have any outlier to make the box and whisker plots of the inputs and output parameters for the collected and normal testing data the same (refer to Figure S4). The statistical description of the normal data is shown in Table S1 (Supporting Information). Figures S5 and S6 show histograms of the inputs and output parameters for the collected and the normal training data in the Supporting Information, respectively. The histograms of the inputs and output parameters for the collected and the normal testing data are shown in Figure S7.

Table 1. Statistical Description of the Collected Data (760 Data Points).

  training data
 testing data
parameters Pb (psi) Rs (SCF/STB) γg API (°API) Tf (°F) Pb (psi) Rs (SCF/STB) γg API (°API) Tf (°F)
minimum 126 9 0.589 15.3 74 130 26 0.589 19.4 74
maximum 7127 2637 1.367 59.5 294 4432 1850 1.367 51.7 271
standard deviation (SD) 1151 423 0.150 7.32 49.4 1135 424 0.160 6.4 45.3
mean 1790 536 0.905 36.05 171 1808 567 0.947 33.3 158
median 1697 433 0.876 36.7 169 1912 560 0.930 32.6 160
mode 500 585 0.802 33.3 100 4005 61 0.802 31.2 100
kurtosis 1.447 1.959 0.325 –0.260 –0.785 –1.134 0.188 –0.105 –0.470 –0.684
skewness 0.881 1.230 0.834 –0.011 0.097 0.107 0.766 0.734 0.182 0.224
range 7001 2628 0.778 44.2 220 4302 1824 0.778 32.3 197
interquartile range 1648 559 0.201 10.35 74 2086 731 0.230 9.9 67
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We considered what is reported in the literature regarding the dependency of Pb on other fluid properties. We found that Pb highly depends on Rs, γg, API, and Tf, as reported by most previous studies; Table 2. Randomization of data was applied using the shuffle method that changes the original list of the data, and it does not return a new list of the data. Randomization can be used to assure that each data set does not memorize the pattern to avoid generalization and model overfitting. The correlation coefficient (R) was calculated to assess each feature’s significance (Rs, γg, API, and Tf) to the output Pb. It is found that Pb is a strong function of Rs (the correlation coefficient of 0.876 and 0.851 for collected and normal data, respectively) but γg is moderate (the correlation coefficient of −0.513 and −0.544 for collected and normal data, respectively). Pb is a weak function of API (correlation coefficient of 0.383 and 0.410 for collected and normal data, respectively). Pb is also a weak function of Tf (correlation coefficient of 0.315 and 0.265 for collected and normal data, respectively); Figure 2.

Table 2. Comparison of the Used Data and Accuracy for the Previous and the Proposed LSTM Models.

        input parameters
    
no. model location no. of data points Bob (bbl/STB) Rs (SCF/STB) γg API (°API) Tf (°F) average absolute error % standard deviation
1 Standing7 California USA 105   √ √ √ √    
2 Lasater8 Canada West and Mid-cont U.S. and SA 158   √ √ √ √    
3 Glaso9 North Sea 41   √ √ √ √   6.98
4 Vazquez and Beggs10 Worldwide 6004   √ √ √ √    
5 Al-Marhoun11 Middle East 160   √ √ √ √ 3.66 4.536
6 Kartoatmodjo and Schmit12 Worldwide 5392   √ √ √ √ 20.17  
7 Dokla and Osman13 UAE 51   √ √ √ √ 7.61 10.378
8 Petrosky and Farshed14 Gulf of Mexico 90   √ √ √ √ 3.28 2.56
9 Macary and El-Batanoney15 Gulf of Suez 90   √ √ √ √ 7.04  
10 Omar and Todd16 Malaysia 93 √ √ √ √ √ 7.17 9.54
11 De Ghetto et al.17 Mediterranean Basin, Africa, Persian Gulf, and the North Sea 195   √ √ √ √ 12.8 10
12 Frashad et al.18 Colombia 43   √ √ √ √   37.02
13 Almehaideb19 UAE 62 √ √ √ √ √ 4.997 6.56
14 Hanafy et al.20 Egypt 90   √          
15 Velarde et al.2 USA 728   √ √ √ √ 11.7  
16 Al-Shammasi22 Worldwide 1709   √ √ √ √ 17.849 17.16
17 Dindoruk and Christman23 Gulf of Mexico 100   √ √ √ √ 5.7 7.51
18 Mehran et al.24 Iran 387   √ √ √ √    
19 Bolodazadeh et al.25 Iran 166   √ √ √ √    
20 Hemati and Kharrat27 Iran 287 √ √ √ √ √ 3.71 4.06
21 Mazandarani and Asghari28 Iran 55   √ √ √ √ 0.51  
22 Khamechi et al.29 Iran 106   √ √ √ √   0.061
23 Arabloo et al.6 Worldwide 756   √ √ √ √ 18.9  
24 Patil et al.31 Niger Delta 23   √ √ √ √    
25 Gomaa32 Middle East 441   √ √ √ √ 8.12 10.69
26 Sharrad and Abd-Alrahman34 Libya 35   √ √ √ √ 8.7  
27 Tariq et al.37 Worldwide 760   √ √ √ √ 0.9358  
28 proposed LSTM Worldwide 760   √ √ √ √ 8.422 0.088
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Figure 2.

Figure 2

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Relative importance of input parameters with the Pb output for the (a) collected and (b) normal data.

Before the proposed LSTM model was applied, all of the parameters (inputs and outputs) are normalized between −1 and 1 to scale the parameters within a specified range using the following equation

2.1. 1
2.1. 2

where Y is the parameter in the normalized form, Ymax is the maximum value of the normalized form (1), Ymin is the minimum value of the normalized form (−1), X is the parameter to be normalized, Xmin is the minimum value of the parameter, and Xmax is the maximum value of the parameter.

2.2. Proposed LSTM Model

LSTM is a special type of recurrent neural network (RNN) mostly applied to define the relationship between the current and previous input data.41 Its memory is utilized to save the internal information of the previously input data. The long short-term memory hidden state can contain some memory cells (recurrently connected cells). Each LSTM memory cell involves three gates, the forget gate, the input gate, and the output gate that maintains information movement, which moves through the given layer; Figure 3. A memory cell, identified as a cell state, aims to store all significant information that is to be passed in the next layer. Eq 3 is used to update the cell state to a new value relevant to the new cell state. The inputs are shown as x at the bottom, and the outputs are shown as h; Figure 4.42 The gates can learn and remember the input sequence data, which is significant.

2.2. 3

On the other hand, the method can discard unnecessary information. There are some multiple activation functions applied in each gate, namely, tanh and sigmoid activation. The activation functions can be used to determine which information passes through the layer.

Figure 3.

Figure 3

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Architecture of an LSTM memory cell.

Figure 4.

Figure 4

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Hidden state of LSTM for the time step 0 to t + 1.

2.2.1. tanh Activation

tanh activation can control the data passing through the layer and squish data to ensure that the range of data falls between −1 and 1. Moreover, it can regulate the output to ensure that the value is between −1 and 1 or otherwise make the other insignificant value.

2.2.1. 4

2.2.2. Sigmoid Activation

Sigmoid activation squishes values between one and zero and can update or forget the information. When it is zero, the information disappears or is forgotten, whereas the information is remembered when it is one. Therefore, the network can learn when the information can be significant and can be maintained. When the information is not significant, it can be forgotten.

2.2.2. 5

2.2.3. Forget Gate

Forget gate can be used to decide on retaining or discarding the information. It can accept both the present sequence and the previous state information and decide to discard irrelevant data. The insignificant data is extracted, applying the multiplication of a filter. Sigmoid activation can be required here to make the information be between one and zero. The operation in the forget gate can be obtained using

2.2.3. 6

The forget gate can accept data from the prior hidden state ht–1 and the sequence input xt. Then, the date can be added with a bias and multiplied by the weights. Next, the values are followed through the sigmoid activation to be kept in the cell state or removed.

2.2.4. Input Gate

Input gate obtains information from ht–1 and xt to update information in the cell state. The data is passed into two functions: sigmoid and tanh. Sigmoid activation can be used to decide significant information and can be updated. The values of xt and ht–1 are added with a bias and multiplied by the weight to obtain a vector from zero or one. The value of the sigmoid is utilized to determine significant data applying the following equation

2.2.4. 7

Then, eq 8 can be used to obtain the tanh function used to convert the data between −1 and 1 using the following equation. This was to create a new vector that needed to be applied to the cell.

2.2.4. 8

Before being applied to the present cell state, the sigmoid and tanh function values were multiplied together. This step is significant to ensure that the data added to the cell is essential and not redundant.

2.2.5. Output Gate

It is the last gate that relies on current inputs to obtain the next hidden state. Two steps are included in the output gate to complete the process.

The first step can be to obtain the current input xt and the prior hidden state ht–1. It then provides values into the sigmoid function. The sigmoid activation can be similar to the other two gates, where they can be multiplied by the weight before being added with bias.

2.2.5. 9

The next step can be to obtain the cell state’s value and bring it into the tanh activation. Then, the tanh function’s value is multiplied with the sigmoid’s output to determine the information for the hidden state.43 Our study focuses on applying LSTM to predict Pb (more about LSTM is found in different references, such as (44−46)).

2.2.5. 10

where ft is the output of the forget gate, it is the output of the input gate, C̅t is the temporary memory cell in the memory block that is produced by a tanh layer, Ct is the temporary memory cell in the memory block for the current cell state, Ct–1 is the temporary memory cell in the memory block for the previous cell state, and ot is the output of the output gate. Wf, Wi, Wc, Wo are the recurrent weight matrices; bf, bi, bc are the related bias vectors; ht–1 represents the previous results in each LSTM cell; ht represents the results in each LSTM cell; xt is the current input in each LSTM cell; and σ is the sigmoid function that determines the propagation of the information.

MATLAB software was used to develop the proposed LSTM model. The pseudocode of the proposed LSTM model is presented in Algorithm 1 (see the Supporting Information). Table 3 shows the hyperparameters of the LSTM MATLAB code applied to find Pb.

Table 3. Hyperparameters of the LSTM Model.
parameter description/value
features (inputs) 4
responses (outputs) 1
hidden units 50
max epochs 256
mini-batch size 128
gradient threshold 0.2000
initial learn rate 0.3000
verbose (indicator to display training progress information) 1
training options sgdm (stochastic gradient descent with momentum)
momentum 0.9000
L2 Regularization (factor for L2 regularization) 0.0001
OutputMode (format of output) sequence
StateActivationFunction (activation function to update the cell and hidden state) tanh
GateActivationFunction (activation function to apply to the gates) sigmoid
shuffle once
InputWeightsInitializer (function to initialize input weights) Glorot (Glorot initializer)
LearnRateSchedule (option for dropping the learning rate during training) none (the learning rate remains constant throughout the training)
RecurrentWeightsInitializer (function to initialize recurrent weights) orthogonal
BiasInitializer (function to initialize bias) unit-forget-gate
GradientThresholdMethod (gradient threshold method) l2norm
SequenceLength (option to pad, truncate, or split input sequences) longest
SequencePaddingDirection (direction of padding or truncation) right
ExecutionEnvironment (hardware resource for the training network) auto
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3. Results and Discussion

3.1. Pb Prediction Model Using LSTM

The proposed LSTM model was validated using 228 unseen data points (30% testing data set) after being studied by applying trend analysis to ensure that the model was stable and had sound relationships between the inputs and the output (see Section 3.3). In addition, statistical error analysis was performed to evaluate the LSTM model, as seen in Table 4. As a result, the LSTM model has the lowest average absolute percentage relative error (AAPRE), maximum absolute percent relative error (Emax), minimum absolute percent relative error (Emin), root mean square error (RMSE), and standard deviation (SD) and the highest correlation coefficient (R). As per the tabulated results in Table 4, the accuracy of the LSTM model is outstanding.

Table 4. Statistical Error Analysis of the LSTM Model.

AAPRE (%) Emax (%) Emin (%) RMSE SD R
8.422 45.409 0.040 12.20 0.088 0.9900
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The cross-plotting analysis was applied to assist in visualizing the model’s accuracy. First, the measured Pb values are plotted vs the predicted Pb values. Then, a 45° straight line is plotted on the measured Pb values’ cross plot and the expected Pb values. The closer the plotted data points to the straight line, the higher the correlation or model’s accuracy.

Figures 5 and 6 demonstrate the cross-plotting of the training and testing data set of the proposed LSTM model with the coefficient of determination (R2) of 0.9837 and 9843 for the training and testing data sets, respectively.

Figure 5.

Figure 5

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Cross plot of the training data set using the LSTM model.

Figure 6.

Figure 6

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Cross plot of the testing data set using the LSTM model.

3.2. Proposed LSTM Model and Current Models’ Comparison

Some statistical analyses have been performed to describe, validate, and compare the proposed LSTM model with the previously published models that follow the correct relationships between the inputs and the output (see Section 3.3). The statistical error analysis used for the LSTM model and the previous models included cross plot, correlation coefficient, AAPRE, RMSE, SD, Emax, and Emin. In this study, AAPRE and R are used as key indicators.

3.2.1. Cross-Plotting Analysis

Figure 7 displays the cross-plotting and coefficient of determination (R2) of the LSTM model compared to the existing correlations. As presented in the figure, the LSTM model shows the highest accuracy and can predict Pb with the coefficient of determination (R2) of 0.9843.

Figure 7.

Figure 7

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Cross plot comparison of the LSTM model with the previously published models.

3.2.2. Statistical Error Analysis

Figure 8 shows the AAPRE and correlation coefficient (R) comparison of the LSTM model with the previously published models. As shown, the LSTM model indicates the lowest AAPRE of 8.422% and the highest correlation coefficient (R) of 0.99. On the other hand, the Velarde et al.2 and Mehran et al.24 models display AAPRE (%) of 9.0 and 9.75% and correlation coefficient (R) of 0.972 and 0.969, respectively. The Petrosky and Farshad14 model shows the highest AAPRE of 76.59%.

Figure 8.

Figure 8

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Correlation coefficient (R) and AAPRE (%) comparison of the LSTM model with previously published models.

The prediction performance of the published correlations was compared with the LSTM model. Table 5 shows the statistical parameters for the comparative evaluation of the existing correlations and the LSTM model. Statistical error analysis has been performed to describe the robustness of the LSTM model. AAPRE and correlation coefficient (R) were used as primary indicators of the model’s accuracy. As seen in the table, the LSTM model and the existing correlations are ranked based on AAPRE and the correlation coefficient (R). The LSTM model also has the lowest RMSE and SD compared to other models. This useful comparison of all correlations and the LSTM model forms an essential means of evaluating correlations and the LSTM performance. Investigation of these statistical error analyses proves that the LSTM model exceeds all of the current correlations.

Table 5. Statistical Error Analysis Comparison of the LSTM Model with the Previously Published Models.
rank model AAPRE (%) Emax (%) Emin (%) RMSE SD R
1 The proposed LSTM (2021) 8.422 45.409 0.040 12.20 0.088 0.9900
2 Velarde et al. (1997) 9.00 62.47 0.039 13.04 0.095 0.9724
3 Mehran et al. (2006) 9.75 63.86 0.035 13.60 0.095 0.9699
4 Lasater (1958) 11.07 66.08 0.016 15.31 0.106 0.9742
5 Standing (1947) 12.35 69.28 0.032 16.26 0.106 0.9753
6 Arabloo et al. (2014) 12.66 72.98 0.000 17.12 0.116 0.9589
7 Hemati and Kharrat (2007) 13.76 85.01 0.026 22.13 0.174 0.9741
8 Vazquez and Beggs (1980) 16.88 74.79 0.493 21.65 0.136 0.9767
9 Kartoatmodjo and Schmit (1991) 16.94 78.37 0.085 22.74 0.152 0.9722
10 Al-Shammasi (1999) 17.33 62.95 0.205 22.60 0.145 0.9663
11 Frashad et al. (1996) 18.23 74.23 0.042 24.30 0.161 0.9621
12 De Ghetto et al. (1994) 18.37 73.97 0.007 24.83 0.167 0.9720
13 Dindoruk and Christman (2001) 20.89 77.83 0.432 25.81 0.152 0.9369
14 Glaso (1980) 23.02 79.52 0.281 27.70 0.154 0.9701
15 Mazandarani and Asghari (2007) 23.91 120.93 0.127 34.19 0.245 0.9462
16 Almehaideb (1997) 26.15 234.92 0.037 44.18 0.357 0.9482
17 Macary and El-Batanoney (1993) 31.20 149.75 0.111 42.62 0.291 0.9499
18 Khamechi et al. (2009) 31.24 97.52 0.059 37.27 0.204 0.9652
19 Bolodazadeh et al. (2006) 40.42 434.20 0.175 84.69 0.746 0.9694
20 Sharrad and Abd-Alrahman (2019) 45.93 72.46 0.346 47.96 0.139 0.8929
21 Al-Marhoun (1988) 54.06 79.22 27.176 54.40 0.148 0.9538
22 Petrosky and Farshed (1993) 76.59 784.59 0.295 159.87 1.406 0.9703
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From Table 5, it can also be observed that the Velarde et al.2 correlation is the second-rank correlation with AAPRE of 9% and the correlation coefficient (R) of 0.972. The Petrosky and Farshad14 model shows AAPRE of 76.59% and the correlation coefficient (R) of 0.970 in predicting Pb.

3.3. Sensitivity Analysis

Sensitivity analysis or trend analysis is the study of the uncertainty in the output of a model.47 The trend analysis is utilized to determine the model’s reliability. Trend analysis may define the relationships in a model between output and input variables and look for model errors. Additionally, trend analysis can also identify and remove unnecessary parts of the model structure.48 An input parameter can be changed between the minimum value and the maximum value, while the others are fixed at their constant mean values.22,49−51 Graphs are plotted for the input parameter values as the x-axis against Pb as the y-axis for the previous models and the LSTM model.

Four input parameters (Rs, γg, API, and Tf) have been selected for the trend analysis. Oil formation volume factor at the bubble point pressure (Bob) (bbl/STB) has been discarded from the model’s inputs as Omar and Todd,16 Almehaideb,19 and Hemati and Kharrat27 only apply it. No apparent trend of Pb has been reported in the literature.

The trend of Rs is displayed in Figure 9. Petrosky and Farshad14 and Almehaideb19 developed correlations based on the Rs ranges of 217–1406 and 128–3871 SCF/STB, respectively. The Petrosky and Farshad14 and Almehaideb19 correlations show that Pb was negative, −812.6 and −207.5 psi, respectively, at Rs 26 SCF/STB (Figure 9). The LSTM model demonstrates that the trend obeys the trend of the correlations in the literature where Rs is directly proportional to Pb;Figure 10. Therefore, the LSTM model is successful in following the models’ trends.

Figure 9.

Figure 9

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Gas solubility trend analysis of the LSTM model and previously published models.

Figure 10.

Figure 10

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Gas solubility trend analysis of the LSTM model.

The trend of gas-specific gravity (γg) is demonstrated in Figure 11. As presented by the previous models, the gas-specific gravity (γg) is inversely proportional to Pb. Nevertheless, the Hanafy et al.20 correlation indicates that Pb is constant as they did not include the gas-specific gravity in their correlation. The Goma32 correlation has shown that a slight increase in gas-specific gravity can increase Pb, contradicting the previous correlation behavior. Omar and Todd’s16 correlation showed different trends in which the gas-specific gravity shows a decreasing trend and then increases. Consequently, Hanafy et al.,20 Omar and Todd,16 and Goma32 failed to represent the behavior accurately. The LSTM model trend followed the most current correlation trend; Figure 12 shows that the LSTM model follows the proper gas-specific gravity trend.

Figure 11.

Figure 11

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Gas-specific gravity trend analysis of the LSTM model and previously published models.

Figure 12.

Figure 12

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Gas-specific gravity trend analysis of the LSTM model.

The trend of oil API gravity (API) is displayed in Figure 13. The oil API gravity (API) is inversely proportional to Pb. Hanafy et al.20 showed a horizontal line behavior, which indicates that their correlation does not consider the oil API gravity. Dokla and Osman13 and Gomaa32 correlations followed the trend of existing correlations, but increasing the oil API gravity slightly leads to a decrease in Pb. The Tariq et al.37 model shows that increasing the oil API gravity leads to an increase of Pb.

Figure 13.

Figure 13

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Oil API gravity’s trend analysis of the LSTM model and previously published models.

Consequently, Hanafy et al.,20 Dokla and Osman,13 Gomaa,32 and Tariq et al.37 have not proven a proper trend for the Pb correlation. Petrosky and Farshad14 established their correlation based on the API oil gravity range of 16.3–45 °API. Thus, the Petrosky and Farshad14 correlation shows that Pb is negative, −37.35 and −145.91 psi at API oil gravity of 48.11 and 51.7 °API, respectively; Figure 13. Figure 14 displays that the oil API gravity trend of the LSTM model obeys the trend of the existing correlations.

Figure 14.

Figure 14

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Oil API gravity’s trend analysis of the LSTM model.

Figure 15 reveals the trend of the reservoir temperature. Pb is an increasing function regarding reservoir temperature. However, the Hanafy et al.20 correlation shows a horizontal line, which indicates that their correlation too did not include the reservoir temperature parameter. Dindoruk and Christman23 and Arabloo et al.6 correlations indicate that Pb is almost constant, including the reservoir temperature in their correlations. Dokla and Osman’s13 correlation suggests that Pb decreases with the increase of the reservoir temperature. Therefore, the Dokla and Osman’s13 correlation shows that the trend fails to show dependency adequately. The trend demonstrated by the LSTM model also follows the correct relationships between inputs and output, Figure 16, indicating that it represents the proper trend for the reservoir temperature.

Figure 15.

Figure 15

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Reservoir temperature trend analysis of the LSTM model and previously published models.

Figure 16.

Figure 16

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Reservoir temperature trend analysis of the LSTM model.

In conclusion, all of the input parameters (Rs, γg, API, and Tf) of the LSTM model follow the proper trends. Consequently, the trend analysis represented the reliability of the proposed LSTM model. In contrast, Hanafy et al.,20 Goma,32 Omar and Todd,16 Dokla and Osman,13 and Tariq et al.37 correlation trends fail to present the behavior correctly. Finally, Petrosky and Farshad14 and Almehaideb19 correlations show the predicted Pb’s negative values because the used values of Rs and oil API are out of their considered ranges.

4. Conclusions

The LSTM-based prediction model was developed, validated, and compared against the previous models available in the literature to predict Pb using 760 data points. As a result, the following conclusions can be drawn.

The model’s trend analysis results indicate that the model can accurately describe the profile Pb as a function of all of the considered independent variables following the actual trend expected from the physical relationship.

The LSTM model generates a substantially higher accuracy of Pb compared to all published models. This accuracy is proven by the highest correlation coefficient of 0.99, the lowest AAPRE of 8.422%, the smallest root mean squared error of 12.20, and the lowest standard error deviation (SD) of 0.088 compared to 27 published correlations. The model accuracy and reliability were further enhanced by data randomization to ensure that each data set does not memorize the pattern and avoid generalization and model overfitting.

5. Limitations and Recommendations

The LSTM model was built based on bulk properties (gas solubility (Rs), gas-specific gravity (γg), oil specific gravity (API), and reservoir temperature (Tf)) without considering the compositional data of the crude oil because of the data limitations. The proposed model provides high-accuracy results with the four input parameters considered (Rs, γg, API, and Tf), especially if the input values are within the data ranges used to develop the model. However, the proposed model’s benefits outweigh the limitations. The results presented here are self-explanatory. The proposed LSTM model was the most robust and accurate model compared to the published models. We recommend using more data to enhance the performance of LSTM for other applications with optimal hyperparameters. For example, deep learning (LSTM) can be applied for many petroleum industry calculations, such as production engineering and drilling engineering calculations.

Acknowledgments

The authors express their appreciation to the Universiti Teknologi PETRONAS for supporting this work under the YUTP-Grant cost center #015LC0-098. They also especially thank COREOR, Petroleum Engineering, Universiti Teknologi PETRONAS, for supporting this work. The first author, in particular, is grateful to Universiti Teknologi PETRONAS for supporting his Ph.D. study under the Graduate Assistance (GA) scheme.

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsomega.1c02376.

  • Appendix for statistical error analysis; pseudocode of the proposed LSTM model; statistical description of the normal data; box and whisker plot; box and whisker plots for the collected training data; box and whisker plots for the normal training data; box and whisker plots for the collected and normal testing data; histograms of the inputs and output parameters for the collected training data; histograms of the inputs and output parameters for the normal training data; and histograms of the inputs and output parameters for the collected and normal testing data (PDF)

Author Contributions

F.S.A. implemented the writing: original draft preparation, review and editing, methodology, software, and data collection. M.E.M. implemented the problem conceptualization, research supervision and funding, and drafting. M.A.A. implemented model development, result verification, and review and editing. A.S.M. implemented the review and editing.

The authors declare no competing financial interest.

Supplementary Material

ao1c02376_si_001.pdf (997.5KB, pdf)

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Supplementary Materials

ao1c02376_si_001.pdf (997.5KB, pdf)

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