Petroleum Science and Technology, 26:1993–2008, 2008

Copyright © Taylor & Francis Group, LLC

ISSN: 1091-6466 print/1532-2459 online

DOI: 10.1080/10916460701399493

Predicting Bubble-point Pressure and

Formation-volume Factor of Nigerian Crude Oil

System for Environmental Sustainability

E. O. Obanijesu1 and D. O. Araromi1

1Chemical Engineering Department, Ladoke Akintola University of Technology,

Ogbomoso, Nigeria

Abstract: This paper presents a model for predicting the bubble–point pressure (Pb )

and oil formation-volume-factor at bubble-point (Bob) for crude samples collected

from some producing wells in the Niger-Delta region of Nigeria. The model was

developed using artificial neural networks with 542 experimentally obtained Pressure-

Volume-Temperature (PVT) data sets. The model accurately predicts the Pb and Bob

as functions of the solution gas-oil ratio, the gas relative density, the oil specific

gravity, and the reservoir temperature. In order to obtain a generalized accurate model,

backpropagation with momentum for error minimization was used. The accuracy of

the developed model in this study was compared with some published correlations.

Apart from its accuracy, this model takes a shorter time to predict the PVT properties

when compared with empirical correlations. The immediate reason for this may have

to do with the non-linear nature of the empirical correlations.

Keywords: environmental sustainability, neural network, Niger-Delta, PVT

INTRODUCTION

The bubble point pressure (Pb), which is an important Pressure-Volume-

Temperature (PVT) property, determines the oil-water flow ratio during hy-

drocarbon production. If too high, the quantity of produced water obtained at

the surface may be higher than that of oil, production will be reduced, and

well efficiency will be low. The produced water, also known as brine (SPE,

2006), is the water associated with oil and gas reservoirs that is produced

along with the oil and gas. Produced water contains aromatic heterocycles

Address correspondence to E. O. Obanijesu, Chemical Engineering Department,

Ladoke Akintola University of Technology, Ogbomoso, Nigeria. E-mail: emmanuel

257@yahoo.com

1993

1994 E. O. Obanijesu and D. O. Araromi

and pyrogenics (CMI, 2002); trace (or heavy) metal such as cadmium (Cd),

chromium (Cr), lead (Pb), zinc (Zn), copper (Cu), and nickel (Ni) (Obanijesu

et al., 2004); and inorganic solutes like sulfur compounds; all of which could

be transported to air through evaporation. The volatile organic compounds

(VOCs), which are major components of the aromatic compounds, are pri-

mary contributors to the formation of photochemical oxidants (Mohammed

et al., 2002) and increase cancer risk in humans at certain levels of exposure

(Guo, 2004); hence, the need for its treatment before disposal. The disposal

of the concentrate at the end of treatment equally constitutes a pressing

technological problem. Dumping of the concentrate, which is the common

method of disposal in Nigeria, may result in the accumulation of heavy metals

in soils (Howari, 2004); and these toxic metals may pose an enormous threat

to soil, water, and air (Baba and Deniz, 2004). At certain concentrations,

exposure to heavy metals results to human death (Obanijesu et al., 2004).

The conventional way of obtaining the required PVT data is either

through experimental setup early in the production life of the reservoir or

by empirical correlation developed by some researchers. Oftentimes, exper-

imental data are very difficult and expensive to obtain while most of the

empirical correlations are complex in composition and highly non-linear.

Before the advent of ANN, various empirical correlations have been

in use for prediction (Gharbi and Elsharkawy, 2003; Petrosky and Farshad,

1993). They are essentially based on assumption that the bubble point pressure

normally increases with an increase in solution gas-oil ratio and reservoir tem-

perature and a decrease in oil API gravity and gas gravity. Other assumptions

include Bob increases with an increase in Rs , T , gas gravity, and oil gravity.

A major shortcoming with most of these correlations is that different crude

oil and gas systems used to develop them exhibit regional trends in chemical

and physical properties. They are either classified as paraffinic, naphthenic,

or aromatic. This has made it impossible to use correlations developed from

samples of certain regions to give satisfactory prediction fluid properties for

other regions.

Neural network-base modeling has been extensively used in process

engineering in the last decade (Gharbi and Elshakawy, 1997; Elsharkawy,

1998; Mujtaba and Hussain, 2001). ANN is a computer-based algorithm

which learns the behavior of a data population by self-turning its parameter in

such a way that the trained ANN matches the employed data accurately. For

a given set of input, ANN is able to produce a corresponding set of outputs

according to some mapping relationships that are encoded into some network

structure during a period of training (also called learning), and is dependent

upon the parameters of the network, that is, weight and biases.

Importantly, if the data used are sufficiently descriptive, the ANN pro-

vides a rapid and confident prediction as soon as a new case, which has not

been “seen” by the model during the training phase. Also, ANN has the ability

to discover patterns in data, which are so obscure as to be imperceptible to

normal observation and standard statistical methods.

Nigerian Crude Oil System for Environmental Sustainability 1995

Most of the PVT properties predicted have a very low mean relative error

of 0.5–2.5% with no data set having a relative error in excess of 5%.

Osman et al. (2001) used 803 data sets from the Middle East, Malaysia,

Colombia, and the Gulf of Mexico to develop an ANN model with 402 of

these data sets used to train it, 201 sets to cross validate the relationship

established during the training process, and the remaining 200 sets used to

test the model for accuracy evaluation.

Garbi and Elsharkawy (2003) developed a neural network to predict Pb

and Bob for crude oil samples collected from North and South America,

the North Sea, South East Asia, the Middle East, and Africa. In all, 5,200

experimentally obtained PVT data sets were used to train the model while

an additional 234 PVT data sets were used to investigate its effectiveness.

The petroleum industry in Nigeria is more interested in hydrocarbon

production and avoidance of the resultant problems emanating from brine

production, there is dire need to develop a predictive model on Nigerian crude

based on its peculiarity when compared to other regional oil. This will assist

in solving some of the emerging communal problems faced by the operating

companies in the Niger-Delta area of the country due to environmental

degradation.

METHODOLOGY

This work develops a predictive model for Nigeria crude oil PVT properties

using artificial neural network (ANN). This involves mapping of PVT data

of Nigeria crude oil systems using a generalized neutral network model; a

statistical tool was used to analyze and compare the accuracy of the neural

network model prediction with published PVT correlations data and using the

developed model to estimate the bubble point (Pb) and oil formation volume

factor (Bob) for Nigeria crude oil system.

The values of bubble point pressure (Pb) and oil formation volume factor

(Bob) were predicted. These two variables correspond to ANN output vector

(indicated by Y ). The inputs to ANN represent the amounts of information

that networks need to give output vector. The choice of inputs is made

using engineering judgment. Too much input would overload the network

and introduce unnecessary correlations among data, and can therefore disrupt

the network performance (Greaves et al., 2003). On the other hand, too few

inputs could lead to inaccurate predictions.

From the previous work done, it has been established that Pb and Bob are

strong functions of the solution gas-oil ratio Rs , the reservoir temperature T ,

gas specific gravity Yg, and oil specific gravity Yo. These variables constituted

our input vectors for the networks. A Marquardt-Levenberg feed forward

backpropagation ANN topology is used due to its robustness and efficient

performance (Hussain and Ho, 2004). The network structure adopted for this

work is multilayer perceptron (MLP) consisting of an input layer with four

1996 E. O. Obanijesu and D. O. Araromi

neurons (T , Rs , Yg, and Yo) and several hidden layers with 50 neurons each

(Figure 1). Input to the perceptron are individually weighted and summed.

The perceptron computes the output as a function of F of the sum. The

activation function F is needed to introduce non-linearities into the network.

This makes the multilayer network a powerful representation of the non-

linearity system. The output from the perceptron is given as

y.k/ D f .wT .k/:x.k//: (1)

The weights are dynamically updated using backpropagation algorithm. The

difference between the target output T and actual output (error e) is calculated

as

e.k/ D T .k/ � y.k/: : : : (2)

The errors are backpropagated through the layer z and weight changes are

made. The formula for adjusting the weight is given by

w.k C 1/ D w.k/ C �:e.k/: : : : (3)

Once the weights are adjusted, the feedforward process is repeated.

The training is intended to gradually update the connection weights to

minimize the mean square error E through

E D

mX

kD1

nX

iD1

.T.k/i � y

.k/i /2: (4)

The weights are adjusted according to the gradient decent rule, so that the

actual output of the MLP moves closer to the desired output.

The training algorithm and the network architecture were implemented

using the MATLAB program. The algorithm is designed to approach a second

order training speed without computing the Hessian matrix. The performance

function having the form of a sum of squares was approximated as

H D J T J; (5)

Figure 1. Diagram of the perceptron model.

Nigerian Crude Oil System for Environmental Sustainability 1997

and the gradient was computed as

g D J T e; (6)

where J is the Jacobian matrix that contains first derivatives of the network

errors with respect to the weights and biases, and e is a vector of network

errors.

The Jacobian matrix is then computed through a standard back-propaga-

tion technique and the generated Hessian matrix is then approximated using

quasi-Newton equation, thus,

xkC1 D xk � ŒJ T J C �I ��1J T e: (7)

During the training phase, the weights are adjusted according to the

generalized rule. To obtain the accurate models for predicting Pb and Bob as a

function of the other four variables, the number of neurons was systematically

varied to obtain a good fit to the data. Training was completed when the

network was able to predict the given output.

Of the 542 data sets obtained from some wells in Niger-Delta of Nigeria,

264 sets were used to train the model, 142 sets used to cross validate the

relationship established during the training process, and the remaining 136

sets were used to test the model for accuracy evaluation. Each data range

contains the reservoir temperature T , solution gas-oil ratio Rs , gas specific

gravity g, oil API gravity o, bubble point pressure Pb , and oil formation

volume factor Bob (Table 1).

For comparison purposes, Pb and Bob for some existing empirical models

such as Standing, Glasø, Labedi, and Elsharkawy were calculated through

their various equations.

For Standing model, Pb and Bob are respectively given as

Pb D

��

Rs

g

�

anti log10.0:00091T � 0:01425API/ � 1:4

�

; (8)

Bob D 0:972 C1:47 � 10�4

ŒRs. g= o/0:5 C 1:25T �1:175: (9)

Table 1. Range of PVT data used for training

Bubble point pressure (psia) 95–3,660

Bubble point oil formation volume factor (RB/STB) 1.0310–1.659

Solution gas oil ratio (SCF/STB) 22–1,234

Gas specific density (air D 1) 0.6690–1.1510

Stock tank oil gravity (ıAPI) 16.3–50.8

Reservoir temperature 108–220

1998 E. O. Obanijesu and D. O. Araromi

For Glasø model

Bob D 1:0 C 10Œ�6:58511C2:91329log.Bb�/�0:27683.log.Bb�//2� (10)

Bb� D Rs

�

g

o

�0:526

C 0:968T; (11)

while the Bob for Labedi and Elsharkawy were respectively calculated as

Bob D 0:9897 C 0:0001364

"

Rs

�

g

o

�0:5

C 1:25T

#1:175

; (12)

Bob D 1:0 C 40:428 � 10�5Rs C 63:802 � 10�5.T � 60/

C 0:0780 � 10�5

�

Rs.T � 60/

�

g

o

��

: (13)

For comparative performance evaluation, average relative percent error,

average absolute percent error, minimum and maximum absolute percent

error, standard deviation and correlation coefficient were used as statistical

tools for comparison between the developed Neural Network (NN) model

and these existing empirical models.

The average percent relative error, which is the relative deviation of the

estimated values from the experimental data, was calculated as

Er D1

n

NX

iD1

Ei ; (14)

where

Ei D.yi � Oyi /

yi

� 100 i D 1; : : : ; n: (15)

The average absolute percent relative error, which measures the relative

absolute deviation of the estimated values from the experimental values, was

calculated as

Ea D1

n

NX

iD1

jEi j: (16)

The minimum and maximum absolute percent relative error, which define the

ranges of error for each correlation, are respectively given by

Emin Dn

miniD1

jEi j (17)

Emax Dn

maxiD1

jEi j: (18)

Nigerian Crude Oil System for Environmental Sustainability 1999

The Standard deviation, which is a measure of the spread or dispersion of

the data distribution, was calculated as

� D

s

PniD1.xi � x/

n � 1: (19)

While the correlation coefficient that represents the degree of success in

reducing the standard deviation by regression analysis was calculated as

r D

v

u

u

t1 �

nX

iD1

.yi � Oyi /2=

nX

iD1

.yi � y/; (20)

where

y D1

n

nX

iD1

yi : (21)

RESULTS AND DISCUSSION

The training plots of the network, which show that the performance goal

is met, are displayed in Figures 2 and 3. The accuracy of the ANN de-

veloped in this study is evaluated against those developed by Standing,

Glasø, Elsharkawy, and Labedi due to their global acceptability. The statistical

results of the comparison for both Pb and Bob are given in Tables 2 and

3. As shown in the tables, the proposed model shows high accuracy in

predicting both the Pb and Bob values, and achieves the lowest minimum

error, lowest maximum error, lowest standard deviation, and correlation co-

efficient. The model achieves 99.2% and 98.9% correlation coefficient for Pb

and Bob, respectively, which are highest when compared with the existing

correlations (Figures 4–6; summarized as Figures 7 and 8). Absolute percent

error was used to test the accuracy of the models and the result compared

with those of other correlations. The ANN model has the lowest value of

4.36% for bubble point error (Figure 9) and lowest error of 1.73% for

oil formation volume factor (Figure 10). The scatter plots in Figures 11–

15 depict the predicted Bob versus experimental Bob values. These cross

plots indicate the degree of agreement between the experimental and the

predicted values. If the agreement is perfect, then all points should lie on

the 45ı line indicating the excellent agreement between the experimental

and the calculated data values. The best plot is obtained for ANN data as

shown in Figure 11 and the most scattered points are shown in Figure 13

representing Standing correlation, indicating their poor performance for the

set of data.

2000 E. O. Obanijesu and D. O. Araromi

Figure 2. ANN training plot for Bob .

Figure 3. ANN training plot for Pb .

Nigerian Crude Oil System for Environmental Sustainability 2001

Table 2. Statistical parameters for the Bob correlations

ANN Standing Glasø Elsharkawy Labedi

Average relative error (%) �0.67 25.28 18.35 1.26 15.57

Average absolute relative error (%) 1.73 25.28 18.35 4.65 15.66

Minimum absolute relative error (%) 0.29 5.72 1.45 1.89 0.39

Maximum absolute relative error (%) 5.90 41.41 34.06 12.30 31.24

Standard deviation (%) 2.49 12.12 10.84 6.36 10.55

Correlation coefficient 0.989 — 0.881 0.880 0.889

Table 3. Statistical parameters for the Pb correlations

ANN Standing Glasø

Average relative error (%) �2.68 �12.59 �23.08

Average absolute relative error (%) 4.36 24.71 23.17

Minimum absolute relative error (%) 0.68 0.91 0.36

Maximum absolute relative error (%) 15.18 80.00 86.56

Standard deviation (%) 6.14 15.64 27.46

Correlation coefficient 0.992 0.864 0.868

Figure 4. Cross plot of Pb for ANN model.

2002 E. O. Obanijesu and D. O. Araromi

Figure 5. Cross plot of Pb for Standing correlation.

Figure 6. Cross plot of Pb for Glasø correlation.

Nigerian Crude Oil System for Environmental Sustainability 2003

Figure 7. Comparison of Pb correlation coefficient for different correlations.

Figure 8. Comparison of Bob correlation coefficient for different correlations.

Figure 9. Comparison of Pb average absolute percent relative error (AAPRE) for

different correlations.

2004 E. O. Obanijesu and D. O. Araromi

Figure 10. Comparison of Bob average absolute percent relative error (AAPRE) for

different correlations.

Figure 11. Cross plot of Bob for artificial neural network model.

Nigerian Crude Oil System for Environmental Sustainability 2005

Figure 12. Cross plot of Bob for Standing correlation.

Figure 13. Cross plot of Bob for Glasø correlation.

2006 E. O. Obanijesu and D. O. Araromi

Figure 14. Cross plot of Bob for Elsharkawy correlation.

Figure 15. Cross plot of Bob for Labedi correlation.

Nigerian Crude Oil System for Environmental Sustainability 2007

CONCLUSION

The developed model provides better predictions and higher accuracy for

Pb and Bob values, and achieves the lowest minimum error, lowest standard

deviation, and correlation coefficient when compared to empirical correlation

developed by the earlier researchers. Since it successfully predicts the bubble

point pressure and oil formation volume factor for data falling within the

range of data used in this study which were specifically acquired from Niger-

Delta region of Nigeria, this model is recommended for prediction of the

reservoirs within the region.

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NOMENCLATURE

API oil API gravity

Rs solution gas oil ratio

Pb bubble point pressure (psia)

Ps separator pressure

T reservoir temperature (ıF)

Tr reservoir temperature (ıR)

Ts separator temperature (ıF)

Tk reservoir temperature (ıK)

o oil specific gravity (air D 1.0)

g gas specific gravity (air D 1.0)

gs separator gas specific gravity (air D 1.0)

Bob bubble point oil formation volume factor (RB/STB)

T.k/i the target value of the output neuron for the given kth data pattern

y.k/i the prediction for the i th output neuron given the kth data pattern

M number of training data pattern

N number of neurons in the output layer