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Comparison of scaling equation with neural network model for prediction of asphaltene precipitation
Journal of Petroleum Science and Engineering 72 (2010) 186–194
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Journal of Petroleum Science and Engineering j o u r n a l h o m e p a g e : w w w. e l s e v i e r. c o m / l o c a t e / p e t r o l
Comparison of scaling equation with neural network model for prediction of asphaltene precipitation S. Ashoori a, A. Abedini a,⁎, R. Abedini b, Kh. Qorbani Nasheghi a a b
Department of Petroleum Engineering, Petroleum University of Technology, Ahwaz, Iran Department of Chemical Engineering, Ferdowsi University of Mashhad, Mashhad, Iran
a r t i c l e
i n f o
Article history: Received 23 July 2009 Accepted 3 March 2010 Keywords: Asphaltene precipitation Scaling equation Artificial neural network Dilution ratio Molecular weight Temperature
a b s t r a c t The precipitation and deposition of crude oil polar fractions such as asphaltenes in petroleum reservoirs reduce considerably the rock permeability and the oil recovery. Therefore, it is of great importance to determine “how much” the asphaltenes precipitate as a function of pressure, temperature and liquid phase composition. Extensive new experimental data for the amount of asphaltene precipitated in an Iranian crude oil has been determined with various solvents at different temperatures and dilution ratios. All experiments were carried out at atmospheric pressure. The experimental data obtained in this study were used to examine the scaling equations proposed by Rassamdana et al. and Hu et al. We introduced a modified version of their proposed scaling equation. Our observation showed that the results obtained from the present scaling equation are more satisfactory. Furthermore, an Artificial Neural Network (ANN) model was also designed and applied to predict the amount of asphaltene precipitation at a given operating condition. The predicted results of asphaltene precipitation from ANN model was also compared with the results of Rassamdana et al., Hu et al. and our proposed scaling equations. It was observed that there is more acceptable quantitative and qualitative agreement between experimental data and predicted amount of asphaltene precipitation through using ANN model and this model can be a more accurate method than scaling equations to predict the asphaltene precipitation. © 2010 Elsevier B.V. All rights reserved.
1. Introduction Crude oils have complex composition; hence characterization by the individual molecular types is not possible. Instead, hydrocarbon group type analysis is commonly employed (Jewell et al., 1972; Leontaritis, 1997). The SARA separation is an example of such group type analysis, separating the crude oils in four main classes based on differences in solubility and polarity. Instead of molecules or atoms, certain structures are here considered as the components of the crude oil. The four SARA fractions are saturates, aromatics, resins and asphaltenes. Asphaltenes are high-molecular weight solids which are soluble in aromatic solvents such as benzene and toluene and insoluble in paraffinic solvents (Ali and Al-Ghannam, 1981; Speight et al., 1984). Asphaltene precipitation is one of the most common problems in both oil recovery and refinery processes. In oil recovery, especially in gas injection, formation of asphaltene aggregation, following their deposition causes blocking in the reservoir. This makes the remedial process costly and sometimes uneconomical. Unfortunately, there is no predictive model for asphaltene problem treatment. Hence it is necessary to predict the amount of asphaltene precipitation, as a pre-emptive measure. The major ques-
⁎ Corresponding author. Tel.: +98 911 354 1409. E-mail address: [email protected] (A. Abedini). 0920-4105/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.petrol.2010.03.016
tions in facing such problems are “How” and “How much” heavy organic compounds will precipitate in operational condition. Over the years, many researchers have tried to find the answer. They introduced experimental procedures or even analytical models, but a fully satisfactory interpretation is still lacking. The problem is very difficult mainly because of the fuzzy nature of asphaltene and the large number of parameters affecting precipitation. However the existing models fall into three categories: (I) Molecular thermodynamic models in which asphaltenes are dissolved in crude oil and crude oil forms a real solution (Hirschberg et al., 1988; Kawanaka et al., 1991; Nghiem et al., 1998). The validity of such models depends on the reversibility of asphaltene precipitation. Reversibility experiments strongly support this type of models (Kawanaka et al., 1991; Ramos et al., 1997; Hammami, 1999; Peramanu et al., 2001). (II) Colloidal models in which, asphaltene is suspended in crude oil and peptized by resins. The asphaltene precipitation is irreversible in such models (Pfeiffer and Saal, 1940; Leontaritis and Mansoori, 1987; Mansoori, 1997). Reversibility experiments are strongly against this type of models. (III) Models based on scaling equation, in which the properties of complex asphaltenes are not involved (Rassamdana et al., 1996; Rassamdana and Sahimi, 1996; Hu et al., 2000; Hu and Guo, 2001). Analytically, an EOS used for calculating thermodynamic parameters assuming asphaltene precipitation is completely reversible. The calculation process is often found to be a difficult task because of the complexity of asphaltene. Nevertheless,
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Table 1 SARA analysis of the oil under study. Fractions
Saturates
Aromatics
Resins
Asphaltenes (n-C7)
Wt.%
29.3
35.2
27.2
8.3
neither using EOS nor assuming asphaltene reversibility brings enough accuracy and trustable results. 2. Scaling equation Rassamdana et al. (1996) gathered extensive experimental data for the amount of precipitated asphaltene formed with crude oil and various solvents. All experiments were performed at atmospheric pressure and room temperature. They employed a thermodynamic model that uses Flory–Huggins theory of polymer solutions and an equation of state was also used for predicting the experimental data, and its predictions were found in disagreement with the data. As an alternative, they proposed a simple scaling equation that appears to be capable of providing accurate prediction for the data. They assumed that formation of asphaltene structure is to some extent similar to aggregation and gelation phenomena (Park and Mansoori, 1988). These phenomena are associated with universal properties independent of many microscopic properties of their structure. To develop such scaling equation, they manipulated three main variables: Wt Rv Mw
Amount of asphaltene precipitated (wt.%), Solvent to oil dilution ratio (mL/g), Molecular weight of solvent, and combined them into two new variable X and Y in which:
X=
Z Rv = Mw
Y = Wt = Rv
Z
ð1Þ
0
ð2Þ
Z and Z´ are two adjustable parameters and must be carefully tuned to obtain the best scaling fit of the experimental data. They suggested that Z´ is a universal constant of −2 and Z = 0.25 regardless of oil and precipitant used. The proposed scaling equation is expressed in terms of X and Y through a third-order polynomial function
Fig. 2. Variation of asphaltene precipitation with dilution ratio at T = 50 °C.
weight of the solvent with setting Y = 0. The result was Rc = 0.275M1/4 w which shows that at onset, Rc only depends on solvent molecular weight. A general form of above equation is Rc = CM1/4 w in which C is a temperature dependent constant. Despite the simplicity and accuracy of the scaling equation mentioned above, it is restricted to use at a constant temperature and since temperature is not involved in the scaling equation as a variable, it is not adequate for correlating and predicting the asphaltene precipitation data measured at different temperatures. Due to this issue, Rassamdana and Sahimi (1996) modified their scaling equation by implanting temperature parameter in the scaling equation. Based on the previous equation, they defined two new variables x and y: x = X=T
C1
ð4Þ
y = Y=T
C2
ð5Þ
in which X and Y are variables defined as in Eqs. (1) and (2) and constant C1 and C2 are adjustable parameters. They reported that the good fit of their experimental data can be achieved by setting C1 = 0.25 and C2 = 1.6. Again the new scaling equation is a 3rd order polynomial in general form of: 2
2
y = b1 + b2 x + b3 x + b4 x ðx N xc Þ
ð6Þ
where Xc is the value of X at the onset of asphaltene precipitation. They also determined the critical dilution ratio Rc, at which onset of asphaltene precipitation takes place as a function of molecular
Hu et al. (2000) performed a detailed study on the application of scaling equation proposed by Rassamdana et al. (1996) for asphaltene precipitation. They checked the predictive capability of the scaling equation in comparison with literature precipitation data and reported that the scaling equation is an attractive tool for modeling asphaltene precipitation. They examined the universality of
Fig. 1. Variation of asphaltene precipitation with dilution ratio at T = 30 °C.
Fig. 3. Variation of asphaltene precipitation with dilution ratio at T = 70 °C.
2
3
Y = A1 + A2 X + A3 X + A4 X ðX N Xc Þ
ð3Þ
188
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Fig. 4. Collapse of our experimental data onto a single curve using Rassamdana et al. (1996) scaling equation.
Fig. 5. Collapse of our experimental data onto a single curve using Hu et al. (2001) scaling equation.
exponents Z and Z´ and found that Z´ is a universal constant (Z´ = −2) while exponent Z depends on the oil composition and independent of specific precipitant (n-alkane) used. For the experimental data used, they found also that the optimum value of Z is generally within the range of 0.1 ≤Z ≤ 0.5. Hu and Guo (2001) studied the effect of temperature, molecular weight of n-alkane precipitants and dilution ratio on asphaltene precipitation in a Chinese crude oil experimentally. The amounts of asphaltene precipitation at four temperatures in the range of 293– 338 K were measured using seven n-alkanes as precipitants. They found that their experimental data could not be well correlated by setting C1 = 0.25 and C2 = 1.6 recommended by Rassamdana and
Sahimi (1996). They reported that their experimental data could be correlated successfully by choosing C1 = 0.5 and C2 = 1.6. They showed that the proposed new scaling equation is adequate for correlating and predicting the asphaltene precipitation data measured at different temperatures using various n-alkane precipitants. 3. Experimental work An asphaltenic crude oil with a specific gravity of 0.934 (ρo = 0.934 g/cc) was selected from an oil reservoir in southwest Iran. The n-heptane asphaltene content of this crude oil is 8.3% by weight. Table 1 shows the SARA analysis of the oil under study. The amount of
Table 2 Comparison of experimental data with predicted values for n-C5, n-C6 and n-C7 at three temperatures (30 °C, 50 °C and 70 °C). n-C5
30 °C
50 °C
70 °C
n-C6
n-C7
Rv
WExp
WRa
WFe
WAs
WANN
WExp
WRa
WFe
WAs
WANN
WExp
WRa
WFe
WAs
WANN
0.67 1 1.5 4 5 7 10 12 15 20 0.67 1 1.5 4 5 7 10 12 15 20 0.67 1 1.5 4 5 7 10 12 15 20
0.91 1.42 2.32 4.66 5.83 8.1 9.25 10 10.2 10.4 0.73 1.18 2.12 4.15 5.16 7.46 8.58 9.16 9.25 9.46 0.58 0.81 1.48 3.6 4.48 6.27 7.32 7.73 7.88 7.98
0.42 1.28 2.25 5.12 5.86 7.01 8.17 8.69 9.21 9.65 0.16 0.95 1.84 4.47 5.16 6.25 7.38 7.92 8.49 9.02 0.00 0.75 1.59 4.07 4.73 5.78 6.89 7.42 8.01 8.60
0.53 1.41 2.40 5.34 6.10 7.27 8.43 8.95 9.47 9.88 0.25 1.06 1.97 4.67 5.38 6.49 7.63 8.17 8.73 9.25 0.09 0.86 1.72 4.26 4.94 6.00 7.13 7.67 8.25 8.82
1.25 1.53 2.00 4.80 5.99 8.38 8.93 9.32 9.72 10.12 1.03 1.26 1.63 3.83 4.76 6.65 7.50 7.86 8.23 8.60 0.91 1.11 1.42 3.29 4.09 5.72 6.68 7.02 7.37 7.72
0.91 1.18 1.63 4.49 5.83 8.14 9.60 10.00 10.20 10.40 0.82 1.08 1.51 4.10 5.26 7.40 8.67 9.02 9.32 9.45 0.56 0.83 1.24 3.10 3.85 5.56 7.33 7.71 7.92 7.95
0.81 1.23 1.91 4.16 5.03 7.03 8.13 8.68 8.86 9.08 0.62 1.01 1.44 3.33 4.06 5.77 6.67 7.22 7.3 7.46 0.51 0.67 1.2 2.67 3.27 4.59 5.48 5.96 6.63 6.74
0.31 1.08 1.96 4.55 5.23 6.28 7.35 7.84 8.35 8.78 0.07 0.79 1.59 3.96 4.59 5.58 6.63 7.13 7.67 8.19 0.07 0.61 1.37 3.61 4.21 5.16 6.18 6.67 7.23 7.80
0.31 1.08 1.95 4.53 5.21 6.26 7.33 7.83 8.34 8.79 0.07 0.79 1.59 3.95 4.58 5.57 6.61 7.11 7.66 8.20 0.04 0.61 1.37 3.60 4.19 5.14 6.16 6.66 7.22 7.80
1.12 1.37 1.77 4.21 5.24 7.33 8.07 8.45 8.83 9.21 0.92 1.13 1.44 3.35 4.17 5.82 6.78 7.12 7.47 7.82 0.81 0.99 1.26 2.89 3.59 5.01 6.03 6.35 6.69 7.02
1.17 1.39 1.75 3.93 4.95 6.57 8.16 8.63 8.87 8.27 0.63 0.94 1.42 3.81 4.66 5.78 6.64 6.94 7.34 8.10 0.52 0.75 1.09 2.71 3.41 4.62 5.49 5.94 6.63 7.25
0.75 1.2 1.7 4 4.8 6.6 7.5 8 8.1 8.3 0.6 0.98 1.35 2.98 3.61 5.07 5.88 6.32 6.42 6.58 0.5 0.75 1.5 2.51 3.07 4.2 4.91 5.36 5.53 5.65
0.22 0.93 1.74 4.11 4.74 5.71 6.71 7.18 7.67 8.10 0.02 0.66 1.40 3.58 4.16 5.07 6.05 6.52 7.03 7.54 0.13 0.50 1.20 3.26 3.81 4.68 5.63 6.09 6.62 7.17
0.16 0.85 1.63 3.93 4.54 5.50 6.50 6.97 7.47 7.95 0.05 0.59 1.31 3.42 3.98 4.88 5.84 6.31 6.84 7.38 0.11 0.44 1.11 3.11 3.64 4.50 5.43 5.90 6.43 7.00
1.02 1.24 1.60 3.76 4.68 6.54 7.41 7.77 8.13 8.50 0.84 1.02 1.30 3.00 3.72 5.20 6.21 6.54 6.87 7.22 0.73 0.90 1.14 2.58 3.20 4.47 5.52 5.83 6.15 6.47
0.75 1.00 1.40 3.74 4.81 6.59 7.72 7.98 8.11 7.45 0.67 0.94 1.32 2.97 3.67 5.02 5.97 6.33 6.93 6.58 0.58 0.77 1.05 2.48 3.16 4.32 4.92 5.15 5.64 5.65
WExp: experimental data. WRa: predicted value using Rassamdana et al. (1996) equations, Eqs. (4) and (5). WFe: predicted value using Hu et al. equations, Eqs. (4) and (5). WAs: predicted value using our modified equations, Eqs. (7) and (2). WANN: predicted value using our proposed ANN model.
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Fig. 6. Collapse of our experimental data onto a single curve using our scaling equation at different values of dilution ratios, Rv.
Fig. 8. Comparison of experimental data with predicted values at three temperatures (30 °C, 50 °C and 70 °C) using Rassamdana et al. (1996) scaling equation.
asphaltene precipitated was determined at various temperatures ranging from 30 °C to 70 °C. At each temperature, three n-alkanes (pentane, hexane and heptane) were used as precipitants at various dilution ratios. All experiments were carried out at atmospheric pressure. To determine the amount of asphaltene precipitation in a mixture, the gravimetric method was selected. The procedure for this method is outlined as follows:
separate the adhered solid content from the bottom of the test tube and allowed to contact with the newly added n-heptane. The test tube is placed in the centrifuge again, and remains there for another 20 min under a rotational velocity of 10,000 rpm. Steps vi, vii and viii are repeated until the solution becomes colorless. At this time, the colorless solution is removed and the test tube with the solid content is transferred into the oven and remains there for 12 h at a temperature of 100°C. The dried test tube with its solid content is taken out of the oven, allowed to cool down and then weighed. The difference between recorded weights from step 1 and step 12 is the weight of the precipitated asphaltene content of the mixture which is denoted by w2. Knowing the weight of the added crude oil (w1) and precipitated asphaltene (w2), the weight percent of the precipitated asphaltene is determined as follows:
i. An empty test tube is weighed accurately with an electronic laboratory balance. ii. Specific volume of crude oil is injected into the test tube with a glass syringe. iii. The weight of the test tube plus the added crude oil is recorded. The difference between recorded weights from step 1 and step 2 is the exact weight of added crude oil which is denoted by w1. iv. A mixture with specific dilution ratio is prepared by adding adequate volume of solvent to the test tube. Then the test tube is closed by its cap. v. The tube and its content are shaken for approximately 20 min. vi. The tube is then placed in a centrifuge, where it is rotated for about 20 min at 10,000 rpm. vii. After remaining about 20 min in the centrifuge, the asphaltene particles aggregate and precipitate at the bottom of the test tube. After removing the test tube from the centrifuge, its cap is opened and the solution content, which has a dark color, is discarded. viii. 10 cm3 of n-hexane is added to the test tube. The test tube with its contents is shaken for approximately 20 min in order to
Fig. 7. Collpase of our experimental data onto a single curve using our scaling equation at low values of dilution ratios, Rv, up to 7.
ix. x. xi.
xii. xiii.
xiv.
Wt% =
w2 × 100 w1
ð7Þ
4. Results and discussion 4.1. Scaling equation Figs. 1 to 3 show our experimental results for three different solvents (n-alkanes) and various dilution ratios, performed at temperatures of 30 °C, 50 °C and 70 °C. This work was designed to
Fig. 9. Comparison of experimental data with predicted values at three temperatures (30 °C, 50 °C and 70 °C) using Hu et al. (2001) scaling equation.
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shown in this figure. The scaling function can be represented accurately by a 3rd order polynomial given by: 2
3
Y = 9:768−122:5X + 267:5X + 0:049X ðX≥Xc Þ
ð10Þ
We used this equation to predict the amount of asphaltene precipitated for dilution ratios greater than 7. Fig. 7 was also produced with the experimental data of low value of dilution ratios up to 7. 2
3
Y = −0:209 + 3:668X−5:377X + 138:9X ðX≥Xc Þ
Fig. 10. Comparison of experimental data with predicted values at three temperatures (30 °C, 50 °C and 70 °C) using our scaling equation.
find a scaling equation to be capable of providing accurate prediction of the amount of asphaltene precipitation. The experimental data were used to test the scaling equations proposed by Rassamdana and Sahimi (1996) and the one by Hu and Guo (2001). Figs. 4 and 5 show the collapse of our data onto scaling curves. These curves were used to back calculate the experimental data shown in Figs. 1 to 3. The results of calculation are shown in Table 2. As this table shows, the results predicted by these scaling equations are in a relatively good agreement with the experimental data at large values of Rv, but in disagreement or very poor agreement with the experimental data at Rv ≤ 4. Consequently, these scaling equations were found useless for our purpose. As an alternative, we modified Eq. (1), while Eq. (2) was held unchanged. Our proposed scaling model includes the following two equations: n Z X = Rv = T ⋅Mw
Y = Wt = Rv
Z
ð8Þ
0
ð9Þ
implementing temperature parameter but Y had no change. The exponent n is a constant chosen between 0.10 and 0.25. Two other constants, Z and Z', are the same as the first scaling equation, i.e. Z = 0.25 and Z' = −2. For the new modified scaling equation introduced in Eq. (8) and Eq. (2) the best match was achieved by setting n = 0.15. Our experimental data were used to test the capability of our model to predict the value of asphaltene precipitated. The results are shown in Figs. 6, 7. Fig. 6 was produced using all experimental data included in Figs. 1 to 3. All experimental data with various dilution ratios, Rv up to 20, collapsed onto a single curve as
ð11Þ
Eq. (11) represents the scaling function in this figure. This equation is used to predict the amount of asphaltene precipitated for dilution ratios, Rv, less or equal to 7. Table 1 illustrates the comparison 0f experimental data with the results predicted from different models. Figs. 8 through 10 also depict the comparison of experimental data of asphaltene precipitation with simulated values obtained using scaling equations. As it is shown in Table 3, the relative absolute deviation (σ) and average relative absolute deviation (σ̅) of our proposed equation is less than the deviation of two other equations and are defined as follows: n σ = ∑ j WtExp −WtCal = WtExp j i=1 n
σ= ∑
i=1
j WtExp −WtCal = WtExp j = n
ð12Þ
ð13Þ
4.2. Determination of onset point By equating Y to zero the equation gives the X at onset point, Xc. By arranging temperature and molecular weight of n-alkane used as solvent, one can find Rc as the ratio of the solvent at onset point which is a crucial parameter in many evaluations. In this work, we employed Eq. (11) to predict the onset point. If we set Y=0 in Eq. (11) and solve it for the Xc, then the critical dilution ratio Rc can be determined as follows: Rc = 0:06T
0:15
0:25
Mw
ð14Þ
Fig. 11 shows that there is an acceptable agreement between experimental data and predicted values of dilution ratios at the onset point of asphaltene precipitation for different n-alkanes and temperatures.
Table 3 Relative absolute deviation (σ) and average relative absolute deviation (σ̅) of simulated data for scaling equations. Method
Rassamdana et al. (1996) eq.
Hu et al. eq.
Our eq.
σa σ̅b
16.94 0.188
16.51 0.183
9.78 0.109
a
n σ = ∑ j WtExp −WtCal = WtExp j i=1 n
b
σ= ∑
i=1
j WtExp −WtCal = WtExp = n
Fig. 11. Comparison of experimental and predicted dilution ratios at the onset point of asphaltene precipitation for different n-alkanes and temperatures.
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Fig. 12. Schematic of network in an Artificial Neural Network Model.
5. Neural network model Neural networks are composed of simple elements operating in parallel. These elements are inspired by biological nervous systems. As in nature, the network function is determined largely by the connections between elements. One can train a neural network to perform a particular function by adjusting the values of the connections (weights) between elements. Commonly neural networks are adjusted, or trained, so that a particular input leads to a specific target output. Such a situation is depicted in Fig. 12. There, the network is adjusted, based on a comparison of the output and the target, until the network output matches the target (Ripley, 1996). There are multitudes of different types of ANNs. Some of the popular include the multilayer perceptron (MLP), which is more popular and generally trained with the back-propagation of error algorithm, Radial Basis Function (RBF), Adaptive Linear Neuron (ADALINE) and ANFIS (Adaptive Network Based Fuzzy Inference System). Some ANNs are classified as feed-forward, while others are recurrent, depending on how data is processed through the network. Another way of classifying ANN types is by their method of learning, as some ANNs employ supervised training, while others are referred to as unsupervised or self organizing (Poggio and Girosi, 1990a,b). As illustrated earlier by Bakshi and Utojo (1998), all feed-forward neural networks (such as MLP and RBF) can be represented by the following equation: M yˆðx; α; βÞ = ∑ αj φj x; βj j=1
ð15Þ
where φj(·) can be chosen as any arbitrary non-linear function. The model is always linear with respect to αjs but may be non-linear with respect to the βjs. Fig. 13 represents a feed-forward neural network
with a single hidden layer for a Multiple Input Single Output (MISO) system. For the case of multiple independent variables, the non-linear parameters βjs are used to transform the input vector into a scalar argument for the basis function φj(·).Once the input transformation is specified, it remains to choose the functional form of each basis function with respect to its scalar argument. With a specified input transformation of zj(x, β) and a specified functional form for φj(·), the objective is to find a set of optimal (best-fit) linear α* and non-linear β⁎ parameters which minimize a suitably defined merit function for a given set of observations. Wang et al. (1996) showed that a feed-forward neural network with two hidden layers employing sigmoidal basis functions is capable of approximating many non-linear functions. The sigmoidal basis function is defined as below: P
Input Transformation : z = ∑ βj xj + βo j=1
Equation : φðzÞ =
1 1 + expð−azÞ
ð16Þ
ð17Þ
In MLP neural networks, the basic element is the artificial neuron shown in Fig. 14 which performs a simple mathematical operation on its inputs. The input of the neuron consists of the variables x1 … xP and a threshold (or bias) term. Each of the input values is multiplied by a weight, Wi, after which the results are added with the bias term. On the result, a known activation function, φ, performs a pre-specified (non-linear) mathematical operation. MLP networks may consist of many neurons ordered in layers. The neurons in the hidden layers do the actual processing, while the neurons in the input and output layer merely distribute and collect the signals. Although, many hidden layers can be used, however, one
Fig. 13. Architecture of the three-layered feed-forward neural network with a single hidden layer for a MISO system.
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Fig. 14. Schematic representation of a projection based neuron.
Fig. 15. Learning procedure for training MLP networks.
hidden layer networks are more popular for practical applications due to their simple structures. The MLP network is trained by adapting the synaptic weights using a back-propagation technique or any other optimization procedure. During training phase, the network output is compared with a desired output. The error between these two signals is used to adapt the weights. This rate of adaptation may be controlled by a learning rate (η). A high learning rate will make the network adapt its weights quickly, but will make it potentially unstable (Haykin, 1999). Setting the learning rate to zero will make the network keep its weights constant. Additional linear weights (αs, as shown in Fig. 13) were used in this work to accelerate the network convergence. The optimal values of these linear parameters were updated after each iteration of backpropagation method using the following equation: T T Φ Φ α=Φ y
Experimental data were used to train various single hidden layer, totally connected MLP networks trained with the learning algorithm shown in Fig. 15. Three sets of dilution ratio, molecular weight of n-alkane
ð18Þ
where Φi,j = φ(zi,j), i = 1, . . ., N and j = 1, . . .,M and y is the N × 1 vector of measured values. The parameters N and M represent number of training data and number of neurons, respectively. Fig. 15 illustrates the training flow chart of such MLP network.
Fig. 16. Comparison of training results of predicted values of asphaltene precipitation by proposed ANN model with experimental data.
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Fig. 17. Comparison of testing results of predicted values of asphaltene precipitation by proposed ANN model with experimental data. Table 4 Comparison of the deviation for predicted values obtained by ANN with the deviation of the results obtained by scaling equations. Method
Rassamdana et al. (1996) eq.
Hu et al. eq.
Our eq.
ANN model
σ σ̅
16.94 0.188
16.51 0.183
9.78 0.109
5.03 0.056
and temperature were used as an input data and the corresponding amount of asphaltene precipitation was used as a target data. The numbers of training data in each set, neurons of input layer, neurons of hidden layer was 58 (65% of all data), 10 and 5 respectively (N=58, P=10 and M=5). The sigmoidal activation functions with a=0.1 was used in all neurons. The back-propagation learning rate parameters were selected as η=10−12 for all data sets. The initial weights were chosen at random between zero and one. In order to evaluate the capabilities of the ANN model, 65% of experimental data was taken to train the network and the rest 35% of data was taken to test the network efficiency. The predicted amount of asphaltene precipitation by ANN model was compared with the actual values. Figs. 16 and 17 depict the comparison of experimental data with training and testing results of predicted amount of asphaltene precipitation by proposed ANN model. Table 2 also shows this comparison. Furthermore, as shown in Table 4, the deviation of predicted values using ANN model is lower than the other scaling equations. This fact confirms that models based on ANN can be a suitable method for prediction and simulation of heavy organics deposition such as asphaltenes.
6. Conclusion To test existing scaling equation of asphaltene precipitation prediction, some core flood tests were carried out in the laboratory. The results were tested against the models developed by Rassamdana et al. and Hu et al. These latter models were not found fully satisfactory. A scaling equation was developed, that had better fit than the above models and adequately simulated the asphaltene precipitation for the different percipient, dilution ratios and temperatures. As an introduction to a new method, an ANN model based on MLP networks was designed to predict the amount of asphaltene precipitation. The predicted results from ANN model was compared with the results of scaling equations. Finally it was concluded that ANN prediction models can be used to predict the amount of asphaltene precipitation with sufficient accuracy. Therefore, it is quite possible that the ANN prediction models may become an efficient tool, useful for prediction and simulation of asphaltene precipitation.
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Nomenclature Symbols Rv Solvent to oil dilution ratio, (mL/g) Mw Molecular weight of the solvent T Temperature (°C) Wt Amount of asphaltene precipitation, (wt.%) Z, Z´, c1, c2 and n Constants Ai Scaling equation coefficients of Eq. (3) X Function defined by Eq. (1) Y Function defined by Eq. (2) Xc X at the onset of asphaltene precipitation bi Scaling equation coefficients of Eq. (6) x Function defined by Eq. (4) y Function defined by Eq. (5) xc x at the onset of asphaltene precipitation Rc Dilution ratios at the onset point of asphaltene precipitation n Number of all data M Number of neuron in the hidden layer P Number of neuron in the input layer N Number of input training data z Input transformation of activation function y Network response from output layer neuron(s) a Coefficient of sigmoidal activation function W Weight
Greeks σ σ̅ φ α β η α* β* θ ρo
Relative absolute deviation defined by Eq. (13) Average relative absolute deviation defined by Eq. (14) Activation function (basis function) Linear parameter Non-linear parameter Learning rate Best-fit of linear parameter Best-fit of nonlinear parameter Threshold (bias) Oil density
Acknowledgment Special thanks of the authors go to the Research Center of Petroleum University of Technology (PUT) for providing equipment and data.
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Contents lists available at ScienceDirect
Journal of Petroleum Science and Engineering j o u r n a l h o m e p a g e : w w w. e l s e v i e r. c o m / l o c a t e / p e t r o l
Comparison of scaling equation with neural network model for prediction of asphaltene precipitation S. Ashoori a, A. Abedini a,⁎, R. Abedini b, Kh. Qorbani Nasheghi a a b
Department of Petroleum Engineering, Petroleum University of Technology, Ahwaz, Iran Department of Chemical Engineering, Ferdowsi University of Mashhad, Mashhad, Iran
a r t i c l e
i n f o
Article history: Received 23 July 2009 Accepted 3 March 2010 Keywords: Asphaltene precipitation Scaling equation Artificial neural network Dilution ratio Molecular weight Temperature
a b s t r a c t The precipitation and deposition of crude oil polar fractions such as asphaltenes in petroleum reservoirs reduce considerably the rock permeability and the oil recovery. Therefore, it is of great importance to determine “how much” the asphaltenes precipitate as a function of pressure, temperature and liquid phase composition. Extensive new experimental data for the amount of asphaltene precipitated in an Iranian crude oil has been determined with various solvents at different temperatures and dilution ratios. All experiments were carried out at atmospheric pressure. The experimental data obtained in this study were used to examine the scaling equations proposed by Rassamdana et al. and Hu et al. We introduced a modified version of their proposed scaling equation. Our observation showed that the results obtained from the present scaling equation are more satisfactory. Furthermore, an Artificial Neural Network (ANN) model was also designed and applied to predict the amount of asphaltene precipitation at a given operating condition. The predicted results of asphaltene precipitation from ANN model was also compared with the results of Rassamdana et al., Hu et al. and our proposed scaling equations. It was observed that there is more acceptable quantitative and qualitative agreement between experimental data and predicted amount of asphaltene precipitation through using ANN model and this model can be a more accurate method than scaling equations to predict the asphaltene precipitation. © 2010 Elsevier B.V. All rights reserved.
1. Introduction Crude oils have complex composition; hence characterization by the individual molecular types is not possible. Instead, hydrocarbon group type analysis is commonly employed (Jewell et al., 1972; Leontaritis, 1997). The SARA separation is an example of such group type analysis, separating the crude oils in four main classes based on differences in solubility and polarity. Instead of molecules or atoms, certain structures are here considered as the components of the crude oil. The four SARA fractions are saturates, aromatics, resins and asphaltenes. Asphaltenes are high-molecular weight solids which are soluble in aromatic solvents such as benzene and toluene and insoluble in paraffinic solvents (Ali and Al-Ghannam, 1981; Speight et al., 1984). Asphaltene precipitation is one of the most common problems in both oil recovery and refinery processes. In oil recovery, especially in gas injection, formation of asphaltene aggregation, following their deposition causes blocking in the reservoir. This makes the remedial process costly and sometimes uneconomical. Unfortunately, there is no predictive model for asphaltene problem treatment. Hence it is necessary to predict the amount of asphaltene precipitation, as a pre-emptive measure. The major ques-
⁎ Corresponding author. Tel.: +98 911 354 1409. E-mail address: [email protected] (A. Abedini). 0920-4105/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.petrol.2010.03.016
tions in facing such problems are “How” and “How much” heavy organic compounds will precipitate in operational condition. Over the years, many researchers have tried to find the answer. They introduced experimental procedures or even analytical models, but a fully satisfactory interpretation is still lacking. The problem is very difficult mainly because of the fuzzy nature of asphaltene and the large number of parameters affecting precipitation. However the existing models fall into three categories: (I) Molecular thermodynamic models in which asphaltenes are dissolved in crude oil and crude oil forms a real solution (Hirschberg et al., 1988; Kawanaka et al., 1991; Nghiem et al., 1998). The validity of such models depends on the reversibility of asphaltene precipitation. Reversibility experiments strongly support this type of models (Kawanaka et al., 1991; Ramos et al., 1997; Hammami, 1999; Peramanu et al., 2001). (II) Colloidal models in which, asphaltene is suspended in crude oil and peptized by resins. The asphaltene precipitation is irreversible in such models (Pfeiffer and Saal, 1940; Leontaritis and Mansoori, 1987; Mansoori, 1997). Reversibility experiments are strongly against this type of models. (III) Models based on scaling equation, in which the properties of complex asphaltenes are not involved (Rassamdana et al., 1996; Rassamdana and Sahimi, 1996; Hu et al., 2000; Hu and Guo, 2001). Analytically, an EOS used for calculating thermodynamic parameters assuming asphaltene precipitation is completely reversible. The calculation process is often found to be a difficult task because of the complexity of asphaltene. Nevertheless,
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Table 1 SARA analysis of the oil under study. Fractions
Saturates
Aromatics
Resins
Asphaltenes (n-C7)
Wt.%
29.3
35.2
27.2
8.3
neither using EOS nor assuming asphaltene reversibility brings enough accuracy and trustable results. 2. Scaling equation Rassamdana et al. (1996) gathered extensive experimental data for the amount of precipitated asphaltene formed with crude oil and various solvents. All experiments were performed at atmospheric pressure and room temperature. They employed a thermodynamic model that uses Flory–Huggins theory of polymer solutions and an equation of state was also used for predicting the experimental data, and its predictions were found in disagreement with the data. As an alternative, they proposed a simple scaling equation that appears to be capable of providing accurate prediction for the data. They assumed that formation of asphaltene structure is to some extent similar to aggregation and gelation phenomena (Park and Mansoori, 1988). These phenomena are associated with universal properties independent of many microscopic properties of their structure. To develop such scaling equation, they manipulated three main variables: Wt Rv Mw
Amount of asphaltene precipitated (wt.%), Solvent to oil dilution ratio (mL/g), Molecular weight of solvent, and combined them into two new variable X and Y in which:
X=
Z Rv = Mw
Y = Wt = Rv
Z
ð1Þ
0
ð2Þ
Z and Z´ are two adjustable parameters and must be carefully tuned to obtain the best scaling fit of the experimental data. They suggested that Z´ is a universal constant of −2 and Z = 0.25 regardless of oil and precipitant used. The proposed scaling equation is expressed in terms of X and Y through a third-order polynomial function
Fig. 2. Variation of asphaltene precipitation with dilution ratio at T = 50 °C.
weight of the solvent with setting Y = 0. The result was Rc = 0.275M1/4 w which shows that at onset, Rc only depends on solvent molecular weight. A general form of above equation is Rc = CM1/4 w in which C is a temperature dependent constant. Despite the simplicity and accuracy of the scaling equation mentioned above, it is restricted to use at a constant temperature and since temperature is not involved in the scaling equation as a variable, it is not adequate for correlating and predicting the asphaltene precipitation data measured at different temperatures. Due to this issue, Rassamdana and Sahimi (1996) modified their scaling equation by implanting temperature parameter in the scaling equation. Based on the previous equation, they defined two new variables x and y: x = X=T
C1
ð4Þ
y = Y=T
C2
ð5Þ
in which X and Y are variables defined as in Eqs. (1) and (2) and constant C1 and C2 are adjustable parameters. They reported that the good fit of their experimental data can be achieved by setting C1 = 0.25 and C2 = 1.6. Again the new scaling equation is a 3rd order polynomial in general form of: 2
2
y = b1 + b2 x + b3 x + b4 x ðx N xc Þ
ð6Þ
where Xc is the value of X at the onset of asphaltene precipitation. They also determined the critical dilution ratio Rc, at which onset of asphaltene precipitation takes place as a function of molecular
Hu et al. (2000) performed a detailed study on the application of scaling equation proposed by Rassamdana et al. (1996) for asphaltene precipitation. They checked the predictive capability of the scaling equation in comparison with literature precipitation data and reported that the scaling equation is an attractive tool for modeling asphaltene precipitation. They examined the universality of
Fig. 1. Variation of asphaltene precipitation with dilution ratio at T = 30 °C.
Fig. 3. Variation of asphaltene precipitation with dilution ratio at T = 70 °C.
2
3
Y = A1 + A2 X + A3 X + A4 X ðX N Xc Þ
ð3Þ
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Fig. 4. Collapse of our experimental data onto a single curve using Rassamdana et al. (1996) scaling equation.
Fig. 5. Collapse of our experimental data onto a single curve using Hu et al. (2001) scaling equation.
exponents Z and Z´ and found that Z´ is a universal constant (Z´ = −2) while exponent Z depends on the oil composition and independent of specific precipitant (n-alkane) used. For the experimental data used, they found also that the optimum value of Z is generally within the range of 0.1 ≤Z ≤ 0.5. Hu and Guo (2001) studied the effect of temperature, molecular weight of n-alkane precipitants and dilution ratio on asphaltene precipitation in a Chinese crude oil experimentally. The amounts of asphaltene precipitation at four temperatures in the range of 293– 338 K were measured using seven n-alkanes as precipitants. They found that their experimental data could not be well correlated by setting C1 = 0.25 and C2 = 1.6 recommended by Rassamdana and
Sahimi (1996). They reported that their experimental data could be correlated successfully by choosing C1 = 0.5 and C2 = 1.6. They showed that the proposed new scaling equation is adequate for correlating and predicting the asphaltene precipitation data measured at different temperatures using various n-alkane precipitants. 3. Experimental work An asphaltenic crude oil with a specific gravity of 0.934 (ρo = 0.934 g/cc) was selected from an oil reservoir in southwest Iran. The n-heptane asphaltene content of this crude oil is 8.3% by weight. Table 1 shows the SARA analysis of the oil under study. The amount of
Table 2 Comparison of experimental data with predicted values for n-C5, n-C6 and n-C7 at three temperatures (30 °C, 50 °C and 70 °C). n-C5
30 °C
50 °C
70 °C
n-C6
n-C7
Rv
WExp
WRa
WFe
WAs
WANN
WExp
WRa
WFe
WAs
WANN
WExp
WRa
WFe
WAs
WANN
0.67 1 1.5 4 5 7 10 12 15 20 0.67 1 1.5 4 5 7 10 12 15 20 0.67 1 1.5 4 5 7 10 12 15 20
0.91 1.42 2.32 4.66 5.83 8.1 9.25 10 10.2 10.4 0.73 1.18 2.12 4.15 5.16 7.46 8.58 9.16 9.25 9.46 0.58 0.81 1.48 3.6 4.48 6.27 7.32 7.73 7.88 7.98
0.42 1.28 2.25 5.12 5.86 7.01 8.17 8.69 9.21 9.65 0.16 0.95 1.84 4.47 5.16 6.25 7.38 7.92 8.49 9.02 0.00 0.75 1.59 4.07 4.73 5.78 6.89 7.42 8.01 8.60
0.53 1.41 2.40 5.34 6.10 7.27 8.43 8.95 9.47 9.88 0.25 1.06 1.97 4.67 5.38 6.49 7.63 8.17 8.73 9.25 0.09 0.86 1.72 4.26 4.94 6.00 7.13 7.67 8.25 8.82
1.25 1.53 2.00 4.80 5.99 8.38 8.93 9.32 9.72 10.12 1.03 1.26 1.63 3.83 4.76 6.65 7.50 7.86 8.23 8.60 0.91 1.11 1.42 3.29 4.09 5.72 6.68 7.02 7.37 7.72
0.91 1.18 1.63 4.49 5.83 8.14 9.60 10.00 10.20 10.40 0.82 1.08 1.51 4.10 5.26 7.40 8.67 9.02 9.32 9.45 0.56 0.83 1.24 3.10 3.85 5.56 7.33 7.71 7.92 7.95
0.81 1.23 1.91 4.16 5.03 7.03 8.13 8.68 8.86 9.08 0.62 1.01 1.44 3.33 4.06 5.77 6.67 7.22 7.3 7.46 0.51 0.67 1.2 2.67 3.27 4.59 5.48 5.96 6.63 6.74
0.31 1.08 1.96 4.55 5.23 6.28 7.35 7.84 8.35 8.78 0.07 0.79 1.59 3.96 4.59 5.58 6.63 7.13 7.67 8.19 0.07 0.61 1.37 3.61 4.21 5.16 6.18 6.67 7.23 7.80
0.31 1.08 1.95 4.53 5.21 6.26 7.33 7.83 8.34 8.79 0.07 0.79 1.59 3.95 4.58 5.57 6.61 7.11 7.66 8.20 0.04 0.61 1.37 3.60 4.19 5.14 6.16 6.66 7.22 7.80
1.12 1.37 1.77 4.21 5.24 7.33 8.07 8.45 8.83 9.21 0.92 1.13 1.44 3.35 4.17 5.82 6.78 7.12 7.47 7.82 0.81 0.99 1.26 2.89 3.59 5.01 6.03 6.35 6.69 7.02
1.17 1.39 1.75 3.93 4.95 6.57 8.16 8.63 8.87 8.27 0.63 0.94 1.42 3.81 4.66 5.78 6.64 6.94 7.34 8.10 0.52 0.75 1.09 2.71 3.41 4.62 5.49 5.94 6.63 7.25
0.75 1.2 1.7 4 4.8 6.6 7.5 8 8.1 8.3 0.6 0.98 1.35 2.98 3.61 5.07 5.88 6.32 6.42 6.58 0.5 0.75 1.5 2.51 3.07 4.2 4.91 5.36 5.53 5.65
0.22 0.93 1.74 4.11 4.74 5.71 6.71 7.18 7.67 8.10 0.02 0.66 1.40 3.58 4.16 5.07 6.05 6.52 7.03 7.54 0.13 0.50 1.20 3.26 3.81 4.68 5.63 6.09 6.62 7.17
0.16 0.85 1.63 3.93 4.54 5.50 6.50 6.97 7.47 7.95 0.05 0.59 1.31 3.42 3.98 4.88 5.84 6.31 6.84 7.38 0.11 0.44 1.11 3.11 3.64 4.50 5.43 5.90 6.43 7.00
1.02 1.24 1.60 3.76 4.68 6.54 7.41 7.77 8.13 8.50 0.84 1.02 1.30 3.00 3.72 5.20 6.21 6.54 6.87 7.22 0.73 0.90 1.14 2.58 3.20 4.47 5.52 5.83 6.15 6.47
0.75 1.00 1.40 3.74 4.81 6.59 7.72 7.98 8.11 7.45 0.67 0.94 1.32 2.97 3.67 5.02 5.97 6.33 6.93 6.58 0.58 0.77 1.05 2.48 3.16 4.32 4.92 5.15 5.64 5.65
WExp: experimental data. WRa: predicted value using Rassamdana et al. (1996) equations, Eqs. (4) and (5). WFe: predicted value using Hu et al. equations, Eqs. (4) and (5). WAs: predicted value using our modified equations, Eqs. (7) and (2). WANN: predicted value using our proposed ANN model.
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189
Fig. 6. Collapse of our experimental data onto a single curve using our scaling equation at different values of dilution ratios, Rv.
Fig. 8. Comparison of experimental data with predicted values at three temperatures (30 °C, 50 °C and 70 °C) using Rassamdana et al. (1996) scaling equation.
asphaltene precipitated was determined at various temperatures ranging from 30 °C to 70 °C. At each temperature, three n-alkanes (pentane, hexane and heptane) were used as precipitants at various dilution ratios. All experiments were carried out at atmospheric pressure. To determine the amount of asphaltene precipitation in a mixture, the gravimetric method was selected. The procedure for this method is outlined as follows:
separate the adhered solid content from the bottom of the test tube and allowed to contact with the newly added n-heptane. The test tube is placed in the centrifuge again, and remains there for another 20 min under a rotational velocity of 10,000 rpm. Steps vi, vii and viii are repeated until the solution becomes colorless. At this time, the colorless solution is removed and the test tube with the solid content is transferred into the oven and remains there for 12 h at a temperature of 100°C. The dried test tube with its solid content is taken out of the oven, allowed to cool down and then weighed. The difference between recorded weights from step 1 and step 12 is the weight of the precipitated asphaltene content of the mixture which is denoted by w2. Knowing the weight of the added crude oil (w1) and precipitated asphaltene (w2), the weight percent of the precipitated asphaltene is determined as follows:
i. An empty test tube is weighed accurately with an electronic laboratory balance. ii. Specific volume of crude oil is injected into the test tube with a glass syringe. iii. The weight of the test tube plus the added crude oil is recorded. The difference between recorded weights from step 1 and step 2 is the exact weight of added crude oil which is denoted by w1. iv. A mixture with specific dilution ratio is prepared by adding adequate volume of solvent to the test tube. Then the test tube is closed by its cap. v. The tube and its content are shaken for approximately 20 min. vi. The tube is then placed in a centrifuge, where it is rotated for about 20 min at 10,000 rpm. vii. After remaining about 20 min in the centrifuge, the asphaltene particles aggregate and precipitate at the bottom of the test tube. After removing the test tube from the centrifuge, its cap is opened and the solution content, which has a dark color, is discarded. viii. 10 cm3 of n-hexane is added to the test tube. The test tube with its contents is shaken for approximately 20 min in order to
Fig. 7. Collpase of our experimental data onto a single curve using our scaling equation at low values of dilution ratios, Rv, up to 7.
ix. x. xi.
xii. xiii.
xiv.
Wt% =
w2 × 100 w1
ð7Þ
4. Results and discussion 4.1. Scaling equation Figs. 1 to 3 show our experimental results for three different solvents (n-alkanes) and various dilution ratios, performed at temperatures of 30 °C, 50 °C and 70 °C. This work was designed to
Fig. 9. Comparison of experimental data with predicted values at three temperatures (30 °C, 50 °C and 70 °C) using Hu et al. (2001) scaling equation.
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S. Ashoori et al. / Journal of Petroleum Science and Engineering 72 (2010) 186–194
shown in this figure. The scaling function can be represented accurately by a 3rd order polynomial given by: 2
3
Y = 9:768−122:5X + 267:5X + 0:049X ðX≥Xc Þ
ð10Þ
We used this equation to predict the amount of asphaltene precipitated for dilution ratios greater than 7. Fig. 7 was also produced with the experimental data of low value of dilution ratios up to 7. 2
3
Y = −0:209 + 3:668X−5:377X + 138:9X ðX≥Xc Þ
Fig. 10. Comparison of experimental data with predicted values at three temperatures (30 °C, 50 °C and 70 °C) using our scaling equation.
find a scaling equation to be capable of providing accurate prediction of the amount of asphaltene precipitation. The experimental data were used to test the scaling equations proposed by Rassamdana and Sahimi (1996) and the one by Hu and Guo (2001). Figs. 4 and 5 show the collapse of our data onto scaling curves. These curves were used to back calculate the experimental data shown in Figs. 1 to 3. The results of calculation are shown in Table 2. As this table shows, the results predicted by these scaling equations are in a relatively good agreement with the experimental data at large values of Rv, but in disagreement or very poor agreement with the experimental data at Rv ≤ 4. Consequently, these scaling equations were found useless for our purpose. As an alternative, we modified Eq. (1), while Eq. (2) was held unchanged. Our proposed scaling model includes the following two equations: n Z X = Rv = T ⋅Mw
Y = Wt = Rv
Z
ð8Þ
0
ð9Þ
implementing temperature parameter but Y had no change. The exponent n is a constant chosen between 0.10 and 0.25. Two other constants, Z and Z', are the same as the first scaling equation, i.e. Z = 0.25 and Z' = −2. For the new modified scaling equation introduced in Eq. (8) and Eq. (2) the best match was achieved by setting n = 0.15. Our experimental data were used to test the capability of our model to predict the value of asphaltene precipitated. The results are shown in Figs. 6, 7. Fig. 6 was produced using all experimental data included in Figs. 1 to 3. All experimental data with various dilution ratios, Rv up to 20, collapsed onto a single curve as
ð11Þ
Eq. (11) represents the scaling function in this figure. This equation is used to predict the amount of asphaltene precipitated for dilution ratios, Rv, less or equal to 7. Table 1 illustrates the comparison 0f experimental data with the results predicted from different models. Figs. 8 through 10 also depict the comparison of experimental data of asphaltene precipitation with simulated values obtained using scaling equations. As it is shown in Table 3, the relative absolute deviation (σ) and average relative absolute deviation (σ̅) of our proposed equation is less than the deviation of two other equations and are defined as follows: n σ = ∑ j WtExp −WtCal = WtExp j i=1 n
σ= ∑
i=1
j WtExp −WtCal = WtExp j = n
ð12Þ
ð13Þ
4.2. Determination of onset point By equating Y to zero the equation gives the X at onset point, Xc. By arranging temperature and molecular weight of n-alkane used as solvent, one can find Rc as the ratio of the solvent at onset point which is a crucial parameter in many evaluations. In this work, we employed Eq. (11) to predict the onset point. If we set Y=0 in Eq. (11) and solve it for the Xc, then the critical dilution ratio Rc can be determined as follows: Rc = 0:06T
0:15
0:25
Mw
ð14Þ
Fig. 11 shows that there is an acceptable agreement between experimental data and predicted values of dilution ratios at the onset point of asphaltene precipitation for different n-alkanes and temperatures.
Table 3 Relative absolute deviation (σ) and average relative absolute deviation (σ̅) of simulated data for scaling equations. Method
Rassamdana et al. (1996) eq.
Hu et al. eq.
Our eq.
σa σ̅b
16.94 0.188
16.51 0.183
9.78 0.109
a
n σ = ∑ j WtExp −WtCal = WtExp j i=1 n
b
σ= ∑
i=1
j WtExp −WtCal = WtExp = n
Fig. 11. Comparison of experimental and predicted dilution ratios at the onset point of asphaltene precipitation for different n-alkanes and temperatures.
S. Ashoori et al. / Journal of Petroleum Science and Engineering 72 (2010) 186–194
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Fig. 12. Schematic of network in an Artificial Neural Network Model.
5. Neural network model Neural networks are composed of simple elements operating in parallel. These elements are inspired by biological nervous systems. As in nature, the network function is determined largely by the connections between elements. One can train a neural network to perform a particular function by adjusting the values of the connections (weights) between elements. Commonly neural networks are adjusted, or trained, so that a particular input leads to a specific target output. Such a situation is depicted in Fig. 12. There, the network is adjusted, based on a comparison of the output and the target, until the network output matches the target (Ripley, 1996). There are multitudes of different types of ANNs. Some of the popular include the multilayer perceptron (MLP), which is more popular and generally trained with the back-propagation of error algorithm, Radial Basis Function (RBF), Adaptive Linear Neuron (ADALINE) and ANFIS (Adaptive Network Based Fuzzy Inference System). Some ANNs are classified as feed-forward, while others are recurrent, depending on how data is processed through the network. Another way of classifying ANN types is by their method of learning, as some ANNs employ supervised training, while others are referred to as unsupervised or self organizing (Poggio and Girosi, 1990a,b). As illustrated earlier by Bakshi and Utojo (1998), all feed-forward neural networks (such as MLP and RBF) can be represented by the following equation: M yˆðx; α; βÞ = ∑ αj φj x; βj j=1
ð15Þ
where φj(·) can be chosen as any arbitrary non-linear function. The model is always linear with respect to αjs but may be non-linear with respect to the βjs. Fig. 13 represents a feed-forward neural network
with a single hidden layer for a Multiple Input Single Output (MISO) system. For the case of multiple independent variables, the non-linear parameters βjs are used to transform the input vector into a scalar argument for the basis function φj(·).Once the input transformation is specified, it remains to choose the functional form of each basis function with respect to its scalar argument. With a specified input transformation of zj(x, β) and a specified functional form for φj(·), the objective is to find a set of optimal (best-fit) linear α* and non-linear β⁎ parameters which minimize a suitably defined merit function for a given set of observations. Wang et al. (1996) showed that a feed-forward neural network with two hidden layers employing sigmoidal basis functions is capable of approximating many non-linear functions. The sigmoidal basis function is defined as below: P
Input Transformation : z = ∑ βj xj + βo j=1
Equation : φðzÞ =
1 1 + expð−azÞ
ð16Þ
ð17Þ
In MLP neural networks, the basic element is the artificial neuron shown in Fig. 14 which performs a simple mathematical operation on its inputs. The input of the neuron consists of the variables x1 … xP and a threshold (or bias) term. Each of the input values is multiplied by a weight, Wi, after which the results are added with the bias term. On the result, a known activation function, φ, performs a pre-specified (non-linear) mathematical operation. MLP networks may consist of many neurons ordered in layers. The neurons in the hidden layers do the actual processing, while the neurons in the input and output layer merely distribute and collect the signals. Although, many hidden layers can be used, however, one
Fig. 13. Architecture of the three-layered feed-forward neural network with a single hidden layer for a MISO system.
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Fig. 14. Schematic representation of a projection based neuron.
Fig. 15. Learning procedure for training MLP networks.
hidden layer networks are more popular for practical applications due to their simple structures. The MLP network is trained by adapting the synaptic weights using a back-propagation technique or any other optimization procedure. During training phase, the network output is compared with a desired output. The error between these two signals is used to adapt the weights. This rate of adaptation may be controlled by a learning rate (η). A high learning rate will make the network adapt its weights quickly, but will make it potentially unstable (Haykin, 1999). Setting the learning rate to zero will make the network keep its weights constant. Additional linear weights (αs, as shown in Fig. 13) were used in this work to accelerate the network convergence. The optimal values of these linear parameters were updated after each iteration of backpropagation method using the following equation: T T Φ Φ α=Φ y
Experimental data were used to train various single hidden layer, totally connected MLP networks trained with the learning algorithm shown in Fig. 15. Three sets of dilution ratio, molecular weight of n-alkane
ð18Þ
where Φi,j = φ(zi,j), i = 1, . . ., N and j = 1, . . .,M and y is the N × 1 vector of measured values. The parameters N and M represent number of training data and number of neurons, respectively. Fig. 15 illustrates the training flow chart of such MLP network.
Fig. 16. Comparison of training results of predicted values of asphaltene precipitation by proposed ANN model with experimental data.
S. Ashoori et al. / Journal of Petroleum Science and Engineering 72 (2010) 186–194
Fig. 17. Comparison of testing results of predicted values of asphaltene precipitation by proposed ANN model with experimental data. Table 4 Comparison of the deviation for predicted values obtained by ANN with the deviation of the results obtained by scaling equations. Method
Rassamdana et al. (1996) eq.
Hu et al. eq.
Our eq.
ANN model
σ σ̅
16.94 0.188
16.51 0.183
9.78 0.109
5.03 0.056
and temperature were used as an input data and the corresponding amount of asphaltene precipitation was used as a target data. The numbers of training data in each set, neurons of input layer, neurons of hidden layer was 58 (65% of all data), 10 and 5 respectively (N=58, P=10 and M=5). The sigmoidal activation functions with a=0.1 was used in all neurons. The back-propagation learning rate parameters were selected as η=10−12 for all data sets. The initial weights were chosen at random between zero and one. In order to evaluate the capabilities of the ANN model, 65% of experimental data was taken to train the network and the rest 35% of data was taken to test the network efficiency. The predicted amount of asphaltene precipitation by ANN model was compared with the actual values. Figs. 16 and 17 depict the comparison of experimental data with training and testing results of predicted amount of asphaltene precipitation by proposed ANN model. Table 2 also shows this comparison. Furthermore, as shown in Table 4, the deviation of predicted values using ANN model is lower than the other scaling equations. This fact confirms that models based on ANN can be a suitable method for prediction and simulation of heavy organics deposition such as asphaltenes.
6. Conclusion To test existing scaling equation of asphaltene precipitation prediction, some core flood tests were carried out in the laboratory. The results were tested against the models developed by Rassamdana et al. and Hu et al. These latter models were not found fully satisfactory. A scaling equation was developed, that had better fit than the above models and adequately simulated the asphaltene precipitation for the different percipient, dilution ratios and temperatures. As an introduction to a new method, an ANN model based on MLP networks was designed to predict the amount of asphaltene precipitation. The predicted results from ANN model was compared with the results of scaling equations. Finally it was concluded that ANN prediction models can be used to predict the amount of asphaltene precipitation with sufficient accuracy. Therefore, it is quite possible that the ANN prediction models may become an efficient tool, useful for prediction and simulation of asphaltene precipitation.
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Nomenclature Symbols Rv Solvent to oil dilution ratio, (mL/g) Mw Molecular weight of the solvent T Temperature (°C) Wt Amount of asphaltene precipitation, (wt.%) Z, Z´, c1, c2 and n Constants Ai Scaling equation coefficients of Eq. (3) X Function defined by Eq. (1) Y Function defined by Eq. (2) Xc X at the onset of asphaltene precipitation bi Scaling equation coefficients of Eq. (6) x Function defined by Eq. (4) y Function defined by Eq. (5) xc x at the onset of asphaltene precipitation Rc Dilution ratios at the onset point of asphaltene precipitation n Number of all data M Number of neuron in the hidden layer P Number of neuron in the input layer N Number of input training data z Input transformation of activation function y Network response from output layer neuron(s) a Coefficient of sigmoidal activation function W Weight
Greeks σ σ̅ φ α β η α* β* θ ρo
Relative absolute deviation defined by Eq. (13) Average relative absolute deviation defined by Eq. (14) Activation function (basis function) Linear parameter Non-linear parameter Learning rate Best-fit of linear parameter Best-fit of nonlinear parameter Threshold (bias) Oil density
Acknowledgment Special thanks of the authors go to the Research Center of Petroleum University of Technology (PUT) for providing equipment and data.
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