Prediction of reservoir brine properties using radial basis function - Core

Petroleum 1 (2015) 349e357

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Original article

Prediction of reservoir brine properties using radial basis function (RBF) neural network Afshin Tatar a, **, Saeid Naseri b, Nick Sirach c, Moonyong Lee d, Alireza Bahadori c, * a

Young Researchers and Elite Club, North Tehran Branch, Islamic Azad University, Tehran, Iran Department of Petroleum Engineering, Ahwaz Faculty of Petroleum Engineering Petroleum University of Technology, Ahwaz, Iran c Southern Cross University, School of Environment, Science and Engineering, PO Box 157, Lismore, NSW, Australia d School of Chemical Engineering, Yeungnam University, Gyeungsan, Republic of Korea b

a r t i c l e i n f o

a b s t r a c t

Article history: Received 17 August 2015 Accepted 28 October 2015

Aquifers, which play a prominent role as an effective tool to recover hydrocarbon from reservoirs, assist the production of hydrocarbon in various ways. In so-called water flooding methods, the pressure of the reservoir is intensified by the injection of water into the formation, increasing the capacity of the reservoir to allow for more hydrocarbon extraction. Some studies have indicated that oil recovery can be increased by modifying the salinity of the injected brine in water flooding methods. Furthermore, various characteristics of brines are required for different calculations used within the petroleum industry. Consequently, it is of great significance to acquire the exact information about PVT properties of brine extracted from reservoirs. The properties of brine that are of great importance are density, enthalpy, and vapor pressure. In this study, radial basis function neural networks assisted with genetic algorithm were utilized to predict the mentioned properties. The root mean squared error of 0.270810, 0.455726, and 1.264687 were obtained for reservoir brine density, enthalpy, and vapor pressure, respectively. The predicted values obtained by the proposed models were in great agreement with experimental values. In addition, a comparison between the proposed model in this study and a previously proposed model revealed the superiority of the proposed GA-RBF model. Copyright © 2015, Southwest Petroleum University. Production and hosting by Elsevier B.V. on behalf of KeAi Communications Co., Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Keywords: Reservoir brine Intelligent method Density Enthalpy Vapor pressure Radial basis function neural network

1. Introduction Aquifers, which are rocks containing water, surround the majority of hydrocarbon reservoirs. The effect of the aquifer on reservoirs depends on the extent of the aquifer and the permeability of the rock. If these parameters are high enough, the

* Corresponding author. ** Corresponding author. Tel.: þ98 9398106818. E-mail addresses: [email protected] (A. Tatar), alireza.bahadori@scu. edu.au (A. Bahadori). Peer review under responsibility of Southwest Petroleum University.

Production and Hosting by Elsevier on behalf of KeAi

aquifer has a greater impact on the reservoir [1]. Aquifers, which play a prominent role as an effective tool to recover hydrocarbon from reservoirs, assist the hydrocarbon production in various ways such as: peripheral water drive, edge water drive, and bottom water drive [2]. In so-called water flooding methods, the pressure of the reservoir is intensified by the injection of water into the formation, increasing the capacity of the reservoir to allow for more hydrocarbon extraction [1,3]. In the aforementioned methods to recover hydrocarbon, brine would also be produced in addition to hydrocarbon [4]. Brine production increases when the reservoir pressure drops [5]. In some cases even if the most modern field management techniques are employed, produced fluid from the reservoir may comprise of 90% brine in volume [6]. Salty wet crude, recovered from the reservoir, lacks a good quality and causes some problems impeding hydrocarbon production. The brine production can have some adverse effects on the efficiency of hydrocarbon

http://dx.doi.org/10.1016/j.petlm.2015.10.011 2405-6561/Copyright © 2015, Southwest Petroleum University. Production and hosting by Elsevier B.V. on behalf of KeAi Communications Co., Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

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production and can affect the extent of depletion. In some cases, wells should be closed in due to inadequate treatment facilities [4]. In addition to mentioned problems, some studies have indicated that oil recovery can be increased by modifying the salinity of the injected brine in water flooding methods [7]. Furthermore, various characteristics of brines are required for different calculations used within the petroleum industry [8,9]. Consequently, it is of great significance to acquire the exact information about PVT properties of brine extracted from reservoirs. In recent years, various properties of brines such as density, vapor pressure, and enthalpy have gained attention and several studies have been conducted regarding these characteristics. Two methods have been utilized by different studies to find accurate knowledge about different parameters relating to brine: (1) experimental studies (2) studies to present estimative models. However, laboratory approaches are costly and time consuming. Accordingly, if the experimental equipment is not available, the latter method is employed [10]. There have been several reports presented regarding brine density, which is considered as a crucial factor in many areas such as fluid inclusion studies, simulating fluid flow, and enhanced oil recovery. In experimental scope, Ghafri et al. [11] measured the density of NaCl (aq) at temperatures between 283 and 472 K and pressures up to 68.5 MPa and molality of 1.06, 3.16, and 6 mol/kg. Kumar [12] reported the density of SrCl2 (aq) for a temperature range of (50e200)  C at 20.27 bar pressure up to a concentration of 2.7 mol/kg. Moreover, there are other experimental studies about density in literature [13e17]. Concerning predictive models, Hass [18] used the empirical Masson's rule to develop a model to predict the density of vapor-saturated NaCl (aq). This model is capable of density prediction in the range of (75e325)  C and up to a saturation of 7.3 molal. Phillips et al. [18] presented another model for density of brine, which is applicable for temperature range of (10e350)  C, molality range of (0.25e5) mol/kg, and pressures up to 50 MPa. This model can predict the laboratory data with a maximum deviation of ±2%. Concerning enthalpy, Busey et al. [19] used a calorimeter to find the enthalpies of NaCl (aq) for dilute solutions. They carried out this experiment for concentrations of (0.1e5) mol/kg and at temperatures from (323e673) K. Comparing the results of the experiment with existing data in the literature, they indicated that the calorimeter can measure enthalpy accurately, which can be applied to find thermodynamic characteristics. Silvester and Pitzer [8] analyzed the thermodynamic parameters for NaCl (aq) and developed equations to predict them for the temperature range of (298.15e573.15) K and molality range of (0e6) mol/ kg. They also provided a table for values of thermodynamic parameters including enthalpy. Mayrath and Wood [20] measured the enthalpy of NaCl (aq) for molality in range of (0.1e6) mol/kg and temperature range of (348e476) K. They applied the result to calculate the other parameters. This study also showed that flow calorimeter has the ability to measure the thermodynamic characteristics at high temperatures in a quick and accurate way. Mayrath and Wood [21] also utilized flow calorimeter to measure the enthalpy of the different aqueous solutions. Concerning vapor pressure, Gibbard et al. [22] reported the vapor pressures of NaCl (aq) at a temperature range of (298e373) K and molality range of (1e6.1) mol/kg. Using the measured data, enthalpy, and freezing-point data, they computed parameters of the modified Debye-Huckelpower-series. They compared the results of this equation

with the experimental data and found good agreement between them. Gibbard and Scatchard [23] conducted a similar investigation to Gibbard et al. [22] to measure the vapor pressure of LiCl (aq). Using the data, they presented a 25parameter quantic equation and the results of the equation were consist with the experimental data. Liu and Lindsay [24] utilized laboratory approaches to find the vapor pressure of NaCl (aq) and water, and osmotic coefficients in the concentration range of 4 mol/kg to saturation and temperature range of (75e300)  C. Using the results of the experiments, they developed a group of equations expressing the free energies of NaCl (aq) over a wide range of temperatures and concentrations. Recently, Bahadori et al. [25] proposed an Arrhenius type function to prognosticate the characteristics of reservoir brine including density, vapor pressure, and enthalpy at a concentration range of (5e25)% salt content by mass and for temperatures above 30  C. This model has eliminated some of the complexities of mathematics and allows petroleum engineers to calculate brine characteristics with fewer calculations than the previous models. It is evident from preceding explanations that researchers have attempted to provide precise knowledge about the PVT properties of brine in order to apply them in computation with other important parameters. However, most of the studies use experimental or thermodynamic models that require a lot of time and calculations. In recent years, soft computing approaches such as Support Vector Machines (SVMs), Fuzzy Logic (FL), Genetic Algorithms (Gas), and Artificial Neural Networks (ANNs) have been adopted by different researchers in various parts of the petroleum industry to eliminate such difficulties because of their great capacity for analysis and modeling of complex subjects [26e31]. Regarding the use of these models in the prediction of brine PVT properties, Arabloo et al. [2] proposed a model employing the least squares support vector machine technique to estimate liquid saturation vapor pressure, density and enthalpy of formation water. They showed that the results of this model are in good agreement with experimental data. Although the presented models using different methods to study the brine characteristics are valuable, further studies are needed to provide a more straightforward and accurate estimation of brine properties. To achieve this goal, artificial neural network has been applied in this communication. This study aims to develop three different intelligent models to predict the reservoir brine properties including density, enthalpy, and vapor pressure at vapor saturation pressure. After the development of the models, their accuracy will be investigated. Moreover, the results will be compared with Bahadori et al. [25] and Arabloo et al. [2]. 2. Details of intelligent model Artificial neural networks (ANNs), considered as a branch of artificial intelligence, have the capacity to learn, store and recall information if a suitable database is provided [32]. ANNs, incorporating a set of interconnected nodes, are computational models. Complex relationships can be modeled employing ANNs [33]. Ability for processing a huge data bank and capability to generalize the relationships between various variables are the main merits of neural networks. In addition to the aforementioned advantages of ANNs, there are some prominent problems with such approaches such as: noticeable computation stress, and probability to over-fitting [34,35].

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A powerful strategy in ANN is the radial basis function (RBF) network. A crucial privilege of RBFN is that it possesses only three layers, making modeling efficient [36]. RBFNs can respond appropriately to a data set that is not applied in the training process [37]. Because of RBFNs own nonlinear approximation properties, they model complicated relationships, and in these processes, perceptron neural networks can conduct the modeling by only employing multiple intermediary layers [38,39]. When RBFNs are utilized to develop a model it is necessary to determine the number of processing units, the hidden unit activation function, a rule to model a specific task, and an algorithm for the training process to specify the variables of the network. In the training process, which is to find the RBFN weights, the network variables are optimized using a set of data in order to fit the outputs of the network to the determined inputs. A cost function, typically the mean square error, is used to assess the fit [40,41]. After the training process, the developed RBFN model can be utilized with test data that is not adopted in the training process. Online learning allows the network to estimate the new data set with good precision [39]. The common approximation theory creates a solid basis for RBFN [42]. There are three beneficial properties shared between RBFN architecture and classical regularization networks [43,44]: 1. It can approximate any continuous functions that have several statistical variables on a compact domain with an adjustable precision, if the number of units is enough. 2. The result is optimum in minimizing a function which computes its oscillation. 3. Because the unknown coefficients are linear, the approximation enjoys the best approximation characteristics. The RBFN structure is depicted in Fig. 1. It comprises three layers including the input layer, hidden layer, and output layer. Each node (neuron) acts as a nonlinear activation function and employs a RBF (f(r)) in hidden layer. An input vector which undergoes a nonlinear transform in the hidden layer constitutes the input layer, that is, the hidden layer is composed of RBF. The net input for the RBF activation function is the vector distance between its weight and the input vector multiplied by the relevant bias. The output layer, a linear combiner, maps the nonlinearity into a new space. Using an additional neuron in the hidden layer, it is feasible to model the biases of the output layer neurons. This additional neuron possesses a constant activation function f0(r) ¼ 1. Linear optimization method can be applied by RBF to gain a universal optimum

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way to solve the adaptable weights in the minimum MSE sense. The output of the network for an input pattern x is shown in the following form:

yi ðxÞ ¼

J2 X

wki fðkx  ck kÞ

(1)

k¼1

For i ¼ 1,...,J3, where yi(x) represents the i th output of the RBF, wki denotes the connection weight from the k th hidden unit to the i th output unit, ck is prototype of center of the k th hidden unit, and k : k indicates the Euclidean norm. The RBF f(.) is usually considered as the Gaussian function [42]. For a set of N pattern pairs of {(xp,yp)}, Equation (1) can be written in matrix form:

Y ¼ WT F

(2)

where W ¼ [w1,...,wJ3] shows a J2  J3 is weight matrix in which wi ¼ (u1i,...,uJ2i)T, F ¼ [f1,...,fN] denotes a J2  N matrix, fp ¼ (fp,1,...,fp,J2)T is the output of the hidden layer for the pwth   sample, fp;k ¼ fðxp  ck Þ, Y ¼ [y1 y2 yN] indicates a J3  N matrix, and yp ¼ (yp,1,...,yp,J3)T. The fluctuation of the value of the radial function is associated to the distance from a central point. Among several types of radial basis functions, the most common one is the Gaussian function, Equation (1) [37]. In addition to the Gaussian function, some other radial basis can be found in the literature [44e46]. In this study, Gaussian function is preferred due to its great flexibility. r2

fðrÞ ¼ e2s2

(3)

where, r > 0 indicates a distance between a data point x and a center c, s denotes the width parameter which the smoothness of the interpolating function is governed by this parameter, that is greater than zero. More information about RBFNs is provided in a previous study, namely, Tatar et al. [47].

3. Result and discussion 3.1. Data acquisition To develop a valid and reliable model it is necessary to incorporate authentic raw data, which covers a wide range of variables. A review of recent studies indicates that at the vapor saturation pressure, the thermodynamic properties of brine including density, enthalpy, and vapor pressure are functions of temperature and brine salt concentration. The same data set used by Bahadori et al. [25] and Arabloo et al. [2] is incorporated in this study. The details of the experimental data are listed in Table 1.

Table 1 Statistical parameters of the raw experimental data.

Fig. 1. Schematic representation of RBFN [42].

Parameter

Minimum Maximum Average

Standard deviation

Temperature Brine salt content, fraction Reported density, kg/m Reported enthalpy, kJ/kg Reported vapor pressure, kPa

38 0.05 750 151.849 111

88.66405 0.071362 107.9569 348.9729 3392.843

316 0.25 1199 1375.058 12,461

176.7273 0.15 1001.473 674.0685 3530.273

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3.2. Model development Matlab® 2014a implementation of RBF code was utilized to model the thermodynamic properties of the reservoir brine. This code has some adjusting parameters among which the most important ones are two parameters of Spread and Maximum Number of Neuron. The optimal values of the aforementioned parameters lead to the acquiring of the best performance of the networks. Although it is possible to determine the tuning parameters with trial and error, an optimization algorithm was utilized for this task. Owing to its great flexibility, Genetic Algorithm (GA) was utilized in this study. To develop the GA-RBF models, firstly the data set was divided by the ratio of 4:1 for train and test data sets for the trio density, enthalpy, and vapor pressure. The division was such that there is no local accumulation of train or test data. Then, 100 pairs of random values for Spread and MNN were

Table 2 The optimal adjusting parameters determined by GA. Model

Spread

MNN

Density Enthalpy Vapor pressure

1.97 0.83 0.57

22 34 73

generated. The convergence to the optimal values is depicted in Fig. 2. The optimum values were acquired after 40 generation, which are listed in Table 2. 3.3. Accuracy of the proposed model and validation 3.3.1. Models validation Both statistical and graphical methods are used in this study to validate the proposed GA-RBF models.

Fig. 2. The convergent to the optimal model for reservoir brine (a) density, (b) enthalpy, and (c) vapor pressure. The vertical values indicate the cost function (MSE value).

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1300 1400

Best Fit Line Predicted Enthalpy (kJ/kg)

Predicted Density (kg/m3)

1200 R = 0.99999 1100 1000 900 800

Train Data

Best Fit Line

1200

R =1

1000 800 600 400 Train Data

200

Test Data

Test Data 0

700 700

800

900 1000 1100 1200 Experimental Density (kg/m3)

0

1300

200

400 600 800 1000 1200 1400 1600 Experimental Enthalpy (kJ/kg)

(a)

(b) 14000

Predicted Vapor Pressure (kPa)

Best Fit Line 12000 R =1

10000 8000 6000 4000

Train Data

2000

Test Data 0 0

2000 4000 6000 8000 10000 12000 14000 Experimental Vapor Pressure (kPa)

(c) Fig. 3. The cross-plot of the performance of the proposed GA-RBF models for (a) density, (b) enthalpy, and (c) vapor pressure.

For the graphical method different plots are utilized. The first plot to discuss is the cross-plot. Fig. 3 shows the cross plot for train and test data sets for trio of the proposed models. In this plot, the vertical axis is the predicted values by the GA-RBF method and the horizontal axis shows the experimental values. Better prediction of the proposed models result in accumulation of data points in close vicinity of the 45 line. The error relative deviation for the proposed models is depicted in Fig. 4. As it is obvious the relative error deviation for the proposed models for the reservoir brine density, enthalpy, and vapor pressure is not more than 0.0015, 0.006, and 0.004, respectively. The error distribution is depicted in Fig. 5. As it is obvious, for trio models, the error distribution has a symmetric trend around the center of 0. Four different statistical parameters of correlation factor (R2), Average Absolute Relative Deviation (AARD), Standard Deviation (STD), and Root Mean Squared Error (RMSE) are utilized (Equations (2)e(5)) to investigate the accuracy of the proposed models. The formulation of these parameters is as follows.

2 lPred ðiÞ  lExp ðiÞ R ¼1 2 PN  i¼1 lPred ðiÞlExp PN  i¼1

2

%AARD ¼

  N lPred ðiÞ  lExp ðiÞ 100 X lExp ðiÞ N i¼1

0 10:5   P 2 N B lPred ðiÞ  lExp ðiÞ C B C RMSE ¼ B i¼1 C @ A N

STD ¼

N X i¼1



lPred ðiÞ  lExp ðiÞ N

(4)

(5)

(6)

2 !0:5 (7)

These parameters represent the accuracy and validity of the proposed models. The values of the mentioned parameters are presented in Table 3. The RMSE values of 0.270810, 0.455726, and

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0.002 Train Data Test Data

% Relative Deviation

0.0015 0.001 0.0005 0 -0.0005 -0.001 -0.0015 -0.002 700

800

900 1000 1100 Experimental Density (kg/m3)

1200

1300

(a) 0.006 Train Data Test Data

% Relative Deviation

0.004 0.002 0 -0.002 -0.004 -0.006 0

200

400

600 800 1000 Experimental Enthalpy (kJ/kg)

1200

1400

1600

(b) 0.004 Train Data Test Data

% Relative Deviation

0.003 0.002 0.001 0 -0.001 -0.002 -0.003 -0.004 0

2000

4000 6000 8000 10000 Experimental Vapor Pressure (kPa)

12000

14000

(c) Fig. 4. The relative error deviation of the proposed GA-RBF models for (a) density, (b) enthalpy, and (c) vapor pressure.

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Fig. 5. The error distribution plot of the proposed GA-RBF models for (a) density, (b) enthalpy, and (c) vapor pressure.

1.264687 for density, enthalpy, and vapor pressure indicate the good performance of the proposed GA-RBF methods. 3.3.2. Comparison with other models At this point the model be will compared with the models developed by Bahadori et al. [25] and Arabloo et al. [2]. As it was

mentioned in the literature review, Bahadori et al. [25] developed an Arrhenius type function to predict the reservoir brine properties. Arabloo et al. [2] proposed two intelligent models based in Least Square Support Vector Machine (LSSVM) to predict the reservoir brine properties. Comparison of the statistical values is listed in Table 4e6. The RMSE is plotted in Fig. 6 in order to have a better understanding of the comparison.

Table 3 Statistical parameters of the proposed models.

Density (kg/m3)

Train Data Test Data All Data Enthalpy (kJ/kg) Train Data Test Data All Data Vapor pressure (kPa) Train Data Test Data All Data

R^2

AARD

STD

RMSE

N

0.999997 0.999981 0.999994 0.999999 0.999995 0.999998 1.000000 1.000000 1.000000

0.015685 0.032111 0.018971 0.041015 0.157145 0.064241 0.045491 0.088732 0.054139

0.000203 0.000492 0.000281 0.000544 0.002061 0.001047 0.000743 0.001377 0.000897

0.198078 0.457985 0.27081 0.342832 0.753854 0.455726 1.037936 1.920398 1.264687

44 11 55 44 11 55 88 22 110

Table 4 Comparison of the developed models with the predictors proposed by Bahadori et al. [25] and Arabloo et al. [2] to predict reservoir brine density. Parameter

Dataset

R2

Train Test All Train Test All

RMSE

Bahadori [25]

Arabloo [2]

GA-RBF

e 0.9999 e e 0.949

1.0000 0.9999 0.9999 0.341 0.481 0.454

0.999997 0.999981 0.999994 0.198078 0.457985 0.27081

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Table 5 Comparison of the developed models with the predictors proposed by Bahadori et al. [25] and Arabloo et al. [2] to predict reservoir brine enthalpy. Parameter

Dataset

Bahadori [25]

Arabloo [2]

GA-RBF

R2

Train Test All Train Test All

e e 0.9999 e e 2.199

0.9999 0.9999 0.9999 0.418 0.709 0.561

0.999999 0.999995 0.999998 0.342832 0.753854 0.455726

RMSE

Table 6 Comparison of the developed models with the predictors proposed by Bahadori et al. [25] and Arabloo et al. [2] to predict reservoir brine vapor pressure. Parameter

Dataset

Bahadori [25]

Arabloo [2]

GA-RBF

R2

Train Test All Train Test All

e e 0.9985 e e 139.972

1.0000 1.0000 1.0000 1.371 1.642 1.796

1.000000 1.000000 1.000000 1.037936 1.920398 1.264687

RMSE

The comparison between the previously developed models and the models developed in this study indicate the superiority of the GA-RBF models. 4. Conclusions Comprehensive and estimative models using radial basis function (RBF) neural network as a soft computing method were developed to predict some PVT characteristics of reservoir formation water such as: density, enthalpy, and liquid saturation pressure. A huge database incorporating a wide spectrum of experimental data points was collected from open literature in order to present and investigate the models. Coefficient of determination (R2) between the outcome of the proposed models and laboratory data for density, enthalpy, and liquid saturation pressure of formation brine are 0.999994, 0.999998, and 1.000000, respectively. To interpret the performance of the proposed models, the results of the models are compared with the laboratory data points as well as with other models presented in the literature, namely, Bahadori et al. [25] and Arabloo et al. [2]. The models of this study were superior to compared models, as evidenced by the results of the statistical quality measures. The developed GA-RBF model is capable of accurate prognostication of PVT characteristics of the formation brine without having the complexity of the thermodynamic models and without conducting some costly and lengthy experiments.

Fig. 6. Comparison Between the RMSE of the developed models in this study with the proposed model by Bahadori et al. [25] and Arabloo et al. [2] for reservoir brine (a) density, (b) enthalpy, and (c) vapor pressure.

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