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A novel method for evaluation of asphaltene precipitation titration data
Chemical Engineering Science 78 (2012) 181–185
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A novel method for evaluation of asphaltene precipitation titration data Amir H. Mohammadi a,b,n, Ali Eslamimanesh a, Farhad Gharagheizi c, Dominique Richon a,b ´ nerge´tique et Proce´de´s, 35 Rue Saint Honore´, 77305 Fontainebleau, France MINES ParisTech, CEP/TEP - Centre E Thermodynamics Research Unit, School of Chemical Engineering, University of KwaZulu-Natal, Howard College Campus, King George V Avenue, Durban 4041, South Africa c Department of Chemical Engineering, Buinzahra Branch, Islamic Azad University, Buinzahra, Iran a
b
a r t i c l e i n f o
abstract
Article history: Received 16 February 2012 Received in revised form 23 April 2012 Accepted 7 May 2012 Available online 18 May 2012
In this work, we propose a mathematical method for detection of the probable doubtful asphaltene precipitation titration data. The algorithm is performed on the basis of the Leverage approach, in which the statistical Hat matrix, Williams Plot, and the residuals of the model results lead to identify the probable outliers. This method not only contributes to outliers diagnostics but also defines the range of applicability of the applied models and quality of the existing experimental data. Two available scaling equations from the literature are used to pursue the calculation steps. It is found from the obtained results that: I. The applied models to represent/predict the weight percent of asphaltene precipitation are statistically valid and correct. II. All the treated experimental titration data seem to be reliable except one. III. The whole data points present in the dataset are within the domain of applicability of the employed models. & 2012 Elsevier Ltd. All rights reserved.
Keywords: Evaluation of data Leverage approach Asphaltene precipitation Outlier diagnostics Scaling equation Titration data
1. Introduction Saturates, aromatics, resins, and asphaltenes (SARA) are generally assigned as the portions of crude oils (Leontaritis and Mansoori, 1988; Leontaritis et al., 1992; Mohammadi and Richon, 2007, 2008a,b; Thawer et al., 1990; Pan and Firoozabadi, 1996; Andersen and Speight, 2001; Mohammadi et al., 2012). The latter fractions (asphaltenes) are normally toluene/benzene soluble but n-heptane/n-pentane insoluble. It can be stated that asphaltenes are the most aromatic with highest molecular weight fraction of petroleum fluids that generally contain simple heteroatoms (C, H, N, O, and S) or even particular metal constituents like Fe, Ni, and V (Mohammadi et al., 2012). Asphaltenes have potential to be precipitated from the bulk of crude oils mainly due to changes in pressure, temperature, and fluid composition. For instance, there is a possibility of asphaltene precipitation as the oil pressure drops during production of some crude oils. It is currently well accepted that asphaltene precipitations/ depositions in petroleum reservoirs, production and/or processes facilities cause remarkable problems (Leontaritis and Mansoori, 1988; Leontaritis et al., 1992; Mohammadi and Richon, 2007; Thawer et al., 1990; Mohammadi and Richon, 2008a,b; Pan and Firoozabadi, 1996; Andersen and Speight, 2001; Mohammadi et al., 2012) Nevertheless, the real mechanism of asphaltene
n Corresponding author at: MINES ParisTech, CEP/TEP - Centre E´nerge´tique et Proce´de´s, 35 Rue Saint Honore´, 77305 Fontainebleau, France. Tel.: þ33 1 64 69 49 70; fax: þ 33 1 64 69 49 68. E-mail address: [email protected] (A.H. Mohammadi).
0009-2509/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ces.2012.05.009
agglomeration, flocculation, and precipitation has been not completely understood so far mainly due to its complexity (Mohammadi and Richon, 2007; Mohammadi et al., 2012) There are still many debates on this subject (Mohammadi and Richon, 2007; Mohammadi et al., 2012). To deal with clarification of this issue, many experimental techniques have been proposed in the literature. Perhaps, the most widely used ones are the titration tests. They are generally performed using core flood investigations accompanied by addition of different asphaltene precipitants such as n-alkanes (mainly n-C5, n-C6, and n-C7). Different dilution ratios (the ratios of solvents/precipitants to the weight of the live/dead oil samples) are normally used to precipitate or dissolve asphaltene fraction in the bulk of crude at different temperatures (Andersen and Speight, 2001; Ashoori et al., 2003; Rassamdana et al., 1996; Rassamdana and Sahimi, 1996). The common experimental (filtration) procedure (which is generally performed on the basis of IP-143 procedure) is briefly as follows: (Rassamdana et al., 1999). The crude oil sample accompanied by the precipitants are mixed with each other (according to specified dilution ratios) in a suitable vessel using agitation system. Later, an asphaltene solvent (like toluene) is added to the system to dissolve the precipitated asphalts. The solvent would then evaporate and the remaining asphaltenes (precipitated asphaltenes) can be measured. Although the available titration data seem to be adequate to investigate the amounts and onsets of asphaltene precipitations in various oil samples, no method has been proposed to check their reliability up to now. Furthermore, comparison between the various titration data as a method for checking the quality of the corresponding datasets may be conservative due to non-similar
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phase behaviors and/or structures of asphaltene fractions present in different oil samples. Apart from that, the scaling equation developed by Rassamdana and co-workers (Rassamdana et al., 1996; Rassamdana and Sahimi, 1996) has generated number of attractions in the past decade to represent/predict the asphaltene precipitation titration data. Therefore, it is of much interest to propose a statisticallycorrect method for simultaneous detection of the doubtful titration data and their quality along with checking the validity and domain of applicability of the scaling equation. In this communication, our aim is to use the Leverage approach (Rousseeuw and Leroy, 1987; Goodall, 1993; Gramatica, 2007) for this purpose. To the best of our knowledge, this is the first time that this statistical method is used for evaluation of such data which are of interest in petroleum industry.
2. Theory
The points located in the range of R o 3 or 3oR (whether they are larger or smaller than the Hn value) are designated as outliers of the model or ‘‘Bad High Leverage’’ points. These erroneous representations/predictions may be attributed to the doubtful data. 2.2. Scaling equation In 1996, Rassamdana and co-workers (Rassamdana et al., 1996; Rassamdana and Sahimi, 1996) stated that the titration data of asphaltene precipitation including the dilution ratio, the molecular weight of solvent/precipitant, and the weight percent of precipitated asphaltene can be collapsed onto a single (and simple) equation (curve) (Rassamdana et al., 1996; Rassamdana and Sahimi, 1996). As a consequence, they proposed the following three-order polynomial for representation/prediction of the amounts/onsets of asphaltene precipitations against addition of n-alkanes: (Rassamdana et al., 1996). Y ¼ a þ bX þ cX 2 þ dX 3
2.1. Leverage approach
ð2Þ
where, Outlier diagnostics (or detection) are of much importance in developing the mathematical models. As a matter of fact, outlier detection is to identify individual datum (or groups of data) that may differ from the bulk of the data present in a dataset (Rousseeuw and Leroy, 1987). The corresponding methods generally consist of numerical and graphical algorithms (Rousseeuw and Leroy, 1987; Goodall, 1993; Gramatica, 2007; Gharagheizi et al., 2011). The Leverage approach (Rousseeuw and Leroy, 1987; Goodall, 1993; Gramatica, 2007) is considered to be a reliable algorithm for outlier detection that deals with the values of the residuals (i.e., the deviations of a model results from the experimental data) and a matrix known as Hat matrix composed of the experimental data and the represented/predicted values obtained from a model. The primary application criterion of this method is to use a model, which is capable of acceptable calculation/ estimation of the data of interest. The Leverage or Hat indices are calculated based on Hat matrix (H) with the following definition: (Rousseeuw and Leroy, 1987; Goodall, 1993; Gramatica, 2007; Gharagheizi et al., 2011) H ¼ XðX t XÞ1 X t
X ¼ Rd =M z
ð3Þ
and Y ¼ W=RZd
0
ð4Þ
In the preceding equations, W is the weight percent of the precipitated asphaltene, Rd stands for the dilution ratio, and M denotes the molecular weight of the solvent/precipitant. They recommended the values of 1/4 and 2 for z and z0 after adjusting these parameters against the obtained experimental titration data. Later, Rassamdana et al. (1999) and Ashoori et al. (2003) modified one of the parts of the original scaling equation to account for the effects of temperature on the amount of precipitations introducing a third parameter in Eq. (3) as follows: X ¼ Rd =T n M z
ð5Þ
where T is the temperature. The value of 0.15 for the n parameter in Eq. (5) shows acceptable results for representation of the experimental titration values reported by Ashoori et al. (2003).
ð1Þ
where X is a (n k) matrix, in which n stands for the data (rows) and k denotes the parameters of the model (columns), and t stands for the transpose matrix. The Hat values of the data in the feasible region of the problem are, as a matter of fact, the diagonal elements of the H value. Having evaluated the H values with Eq. (1), the Williams plot is sketched for graphical identification of the suspended data or outliers. This plot shows the correlation of Hat indices and standardized cross-validated residuals (R), which are defined as the difference between the represented/predicted values and the implemented data. A warning Leverage (H) is generally fixed at the value equal to 3n/p, where n is number of training points and p is the number of model parameters plus one. The Leverage of three is normally considered as a ‘‘cut-off’’ value to accept the points within 73 range (two horizontal red lines) standard deviations from the mean (to cover 99% normally distributed data). Existence of the majority of data points in the ranges 0 r H r Hn and 3 r R r3 reveals that both model development and its predictions are done in applicability domain, which result in a statistically valid model. ‘‘Good High Leverage’’ points are located in domain of Hn rH and 3r R r3. The Good High Leverage can be designated as the ones, which are outside of applicability domain of the applied model. In other words, the model is not able to represent/predict the following data at all.
3. Experimental data The experimental asphaltene precipitation titration data obtained and reported in the original publications of Rassamdana et al. (1996) and Ashoori et al. (2003) have been treated in this work.
4. Results and discussion Tables 1 and 2 show the absolute relative deviations of the results using the original (Rassamdana et al., 1996) and the modified scaling equation. (Ashoori et al., 2003). As can be seen, the deviations of the models representations/predictions from the corresponding experimental data reported in the original articles (Ashoori et al., 2003; Rassamdana et al., 1996) are generally acceptable to be used for the Leverage statistical approach. The evaluation steps have been followed on the basis of the aforementioned procedure (Section 2.1). The H values have been calculated through Eq. (1). The Williams plots have been sketched in Figs. 1 and 2 for the results using the first model (Rassamdana et al., 1996) and corresponding experimental dataset (Rassamdana et al., 1996) and the second ones, (Ashoori et al., 2003), respectively. All the calculated H and R values are presented in Tables 1 and 2. Two warning Leverages
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Table 1 The results of the applied model (Rassamdana et al., 1996) (the original scaling equation) and the Leverage approach (Rousseeuw and Leroy, 1987; Goodall, 1993; Gramatica, 2007). W
exp*
1.19 1.61 2.05 2.61 3.12 3.5 3.85 4.21 4.62 0.79 1.09 1.43 1.57 1.68 1.74 1.96 2.16 2.55 3.05 3.46 3.61 4.28 0.74 1.02 1.16 1.4 1.4 1.64 1.94 2.3 2.58 2.93 3.21 3.54 3.68 0.7 1.08 1.26 1.4 1.49 1.59 1.82 2.24 2.48 2.89 3.13 3.17 3.42 0.7 0.89 1.15 1.19 1.39 1.42 1.61 1.91 2.16 2.56 2.83 2.97 2.99
W
rep/pred **
0.6 1.13 1.72 2.38 2.83 3.14 3.55 4.01 4.48 0.54 1 1.37 1.5 1.67 1.8 1.92 2.13 2.54 2.82 3.2 3.63 4.07 0.41 0.84 1.18 1.33 1.46 1.7 1.91 2.29 2.55 2.92 3.33 3.57 3.74 0.29 0.7 1.03 1.15 1.3 1.51 1.7 2.06 2.33 2.69 3.09 3.32 3.48 0.17 0.52 0.81 0.96 1.1 1.28 1.46 1.79 2.03 2.36 2.72 2.93 3.08
Table 2 The results of the applied model (Ashoori et al., 2003) (the modified scaling equation) and the Leverage approach (Rousseeuw and Leroy, 1987; Goodall, 1993; Gramatica, 2007). exp*
Precipitant
Hat
Standardized residuals
W
n-C5 n-C5 n-C5 n-C5 n-C5 n-C5 n-C5 n-C5 n-C5 n-C6 n-C6 n-C6 n-C6 n-C6 n-C6 n-C6 n-C6 n-C6 n-C6 n-C6 n-C6 n-C6 n-C7 n-C7 n-C7 n-C7 n-C7 n-C7 n-C7 n-C7 n-C7 n-C7 n-C7 n-C7 n-C7 n-C8 n-C8 n-C8 n-C8 n-C8 n-C8 n-C8 n-C8 n-C8 n-C8 n-C8 n-C8 n-C8 n-C10 n-C10 n-C10 n-C10 n-C10 n-C10 n-C10 n-C10 n-C10 n-C10 n-C10 n-C10 n-C10
0.05 0.03 0.02 0.02 0.03 0.03 0.05 0.07 0.1 0.05 0.03 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.04 0.05 0.07 0.06 0.04 0.03 0.02 0.02 0.02 0.02 0.02 0.02 0.03 0.04 0.05 0.06 0.06 0.04 0.03 0.03 0.02 0.02 0.02 0.02 0.02 0.02 0.03 0.04 0.05 0.07 0.05 0.04 0.03 0.03 0.02 0.02 0.02 0.02 0.02 0.02 0.03 0.03
2.38 1.85 1.09 0.72 1.36 2.00 1.79 1.34 1.16 0.02 0.90 0.93 0.80 1.13 1.54 0.80 0.77 0.71 0.94 1.33 0.40 1.44 0.48 0.36 1.57 0.88 1.71 1.59 0.87 0.83 0.57 0.53 1.24 0.5 0.63 0.98 0.96 0.08 0.27 0.07 0.72 0.36 0.23 0.15 0.67 0.24 1.45 0.75 1.77 0.81 0.73 0.04 0.53 0.42 0.27 0.31 0.13 0.51 0.06 0.32 1.14
0 0.75 1.2 1.7 4 4.8 6.6 7.5 8 8.1 8.3 0.81 1.23 1.91 4.16 5.03 7.03 8.13 8.68 8.86 9.08 0.91 1.42 2.32 4.66 5.83 8.1 9.25 10 10.2 10.4 0.6 0.98 1.35 2.98 3.61 5.07 5.88 6.32 6.42 6.58 0.62 1.01 1.44 3.33 4.06 5.77 6.67 7.22 7.3 7.46 0.73 1.18 2.12 4.15 5.16 7.46 8.58 9.16 9.25 9.46 0.5 0.75 1.5 2.51 3.07 4.2 4.91 5.36
n Experimental weight percent of precipitated asphaltene (Rassamdana et al., 1996). The experiments have been done at approximately 299.2 72 K. These data have been taken from the figures in the original publication (Rassamdana et al., 1996). nn Represented/predicted weight percent of precipitated asphaltene by the original scaling equation (Rassamdana et al., 1996). The values of the parameters in Eq. 2 are as follows: a ¼1.18; b¼ 14.9; c¼ 39.16; d¼ 0.92.
183
W
rep/pred **
0 0.99 1.24 1.6 3.76 4.68 6.55 7.39 7.73 8.09 8.46 1.09 1.36 1.78 4.21 5.24 7.33 8.05 8.41 8.78 9.17 1.23 1.53 2.01 4.81 5.99 7.98 8.9 9.27 9.66 10.08 0.81 1.02 1.3 3 3.73 5.2 6.21 6.52 6.84 7.18 0.89 1.12 1.44 3.35 4.17 5.83 6.77 7.09 7.43 7.78 1 1.26 1.63 3.83 4.76 6.6 7.49 7.83 8.18 8.56 0.7 0.9 1.14 2.58 3.21 4.47 5.52 5.82
Precipitant Temperature Hat (K)
Standardized residuals
n-C7 n-C7 n-C7 n-C7 n-C7 n-C7 n-C7 n-C7 n-C7 n-C7 n-C7 n-C6 n-C6 n-C6 n-C6 n-C6 n-C6 n-C6 n-C6 n-C6 n-C6 n-C5 n-C5 n-C5 n-C5 n-C5 n-C5 n-C5 n-C5 n-C5 n-C5 n-C7 n-C7 n-C7 n-C7 n-C7 n-C7 n-C7 n-C7 n-C7 n-C7 n-C6 n-C6 n-C6 n-C6 n-C6 n-C6 n-C6 n-C6 n-C6 n-C6 n-C5 n-C5 n-C5 n-C5 n-C5 n-C5 n-C5 n-C5 n-C5 n-C5 n-C7 n-C7 n-C7 n-C7 n-C7 n-C7 n-C7 n-C7
0.49 0.27 0.24 0.57 0.68 0.26 0.15 0.09 0.30 0.44 0.94 0.39 0.02 0.63 0.14 0.69 1.18 0.25 0.22 0.33 0.84 0.52 0.01 1.08 0.48 0.64 0.13 0.38 1.36 0.80 0.16 0.17 0.26 0.47 0.08 0.27 0.47 1.12 0.81 1.44 1.96 0.34 0.06 0.32 0.04 0.3 0.36 0.58 0.00 0.73 1.29 0.35 0.13 1.61 0.89 0.99 2.01 2.52 3.13 2.40 1.90 0.13 0.02 1.32 0.00 0.26 0.76 1.79 1.42
303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15
0.04 0.03 0.03 0.02 0.01 0.01 0.02 0.02 0.02 0.03 0.03 0.03 0.03 0.02 0.01 0.01 0.02 0.03 0.03 0.03 0.04 0.03 0.02 0.02 0.01 0.01 0.03 0.04 0.04 0.05 0.05 0.03 0.03 0.03 0.02 0.01 0.01 0.01 0.02 0.02 0.02 0.03 0.03 0.03 0.01 0.01 0.01 0.02 0.02 0.02 0.02 0.03 0.03 0.02 0.01 0.01 0.02 0.02 0.02 0.03 0.03 0.03 0.03 0.03 0.02 0.01 0.01 0.01 0.01
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Table 2 (continued ) W
exp*
5.53 5.65 0.51 0.67 1.2 2.67 3.27 4.59 5.48 5.96 6.63 0.58 0.81 1.48 3.6 4.48 6.27 7.32 7.73 7.88 7.98
W
rep/pred **
6.12 6.44 0.78 0.99 1.26 2.89 3.59 5.01 6.03 6.33 6.65 0.88 1.1 1.42 3.29 4.1 5.72 6.67 7 7.33 7.68
Precipitant Temperature Hat (K)
Standardized residuals
n-C7 n-C7 n-C6 n-C6 n-C6 n-C6 n-C6 n-C6 n-C6 n-C6 n-C6 n-C5 n-C5 n-C5 n-C5 n-C5 n-C5 n-C5 n-C5 n-C5 n-C5
1.8 2.38 0.33 0.49 0.18 0.44 0.79 1.22 1.69 1.24 0.35 0.42 0.42 0.48 0.93 1.02 1.28 1.44 1.61 1.10 0.39
343.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15
0.01 0.02 0.03 0.03 0.03 0.02 0.01 0.01 0.01 0.01 0.02 0.03 0.03 0.03 0.01 0.01 0.01 0.02 0.02 0.02 0.02
n
Experimental weight percent of precipitated asphaltene (Ashoori et al., 2003). nn Represented/predicted weight percent of precipitated asphaltene by the modified scaling equation (Ashoori et al., 2003). The values of the parameters in Eq. 2 are as follows: a¼ 1.18; b ¼ 14.9; c ¼ 39.16; d ¼ 0.92.
Fig. 2. Detection of the probable doubtful experimental data (Ashoori et al., 2003) and the applicability domain of the modified scaling equation (Ashoori et al., 2003) Squares, valid data; Circular, Probable suspended data; Horizontal (red) lines, Suspended data limit; Vertical (blue) line, Leverage limit. (For interpretation of the references to color in this figure legend, the reader is reffered to the web version of this article.)
Good High Leverage points are accumulated in the domains of 0:147 rH and 3 r R r3 for the first model þdata (Rassamdana et al., 1996) and 0:133 rH and 3 rR r 3 for the second ones (Ashoori et al., 2003). These points may be declared to be outside of applicability domains of the applied models. As can be seen, there are no such points in the investigated datasets. It should be noted that, in the case of existence of these kinds of points, it is recommended to use/develop other models on the basis of different theoretical concepts for their calculations/estimations, in order to avoid estimation through biased model calculations. The points located in the range of Ro3 or 3 oR (ignoring their H values) are designated as outliers of the model or ‘‘Bad High Leverage’’ points, as already mentioned. These erroneous representations/predictions can be attributed to the doubtful asphaltene precipitation data. There is only one point in the two treated datasets (Ashoori et al., 2003; Rassamdana et al., 1996), which is within this domain and consequently we can state it as probable doubtful datum. In the final analysis, we may conclude the following elements: Fig. 1. Detection of the probable doubtful experimental data (Rassamdana et al., 1996) and the applicability domain of the original scaling equation (Rassamdana et al., 1996). Squares, valid data; Horizontal (red) lines, Suspended data limit; Vertical (blue) line, Leverage limit. (For interpretation of the references to color in this figure legend, the reader is reffered to the web version of this article.)
(Hn ¼0.147) and (Hn ¼ 0.133) have been fixed at 3n/p for the first model and experimental data (Rassamdana et al., 1996) and the second ones (Ashoori et al., 2003), respectively. In addition, the recommended cut-off value of three has been applied (Gharagheizi et al., 2011). Accumulation of the whole dataset in the ranges 0 r H r 0:147 and 3 r Rr 3 for the data þmodel reported by Rassamdana et al. (1996) and 0 r H r 0:133 and 3 rR r 3 regarding the dataþ model of Ashoori et al. (2003) reveals that the applied models are statistically correct and valid. Furthermore, it shows that the whole data except one in the later dataset are located within the applicability domains of the applied models.
1. The applied scaling equations (Ashoori et al., 2003; Rassamdana et al., 1996) are statistically correct and valid. 2. All of the data points are within the applicability domains of the presented model. 3. There is only 1 point, which may be designated as outlier from the dataset of Ashoori et al. (2003) (see Table 2). 4. The quality of the treated data (even different data in the same dataset) is different. The data with lower absolute R values (near R ¼0 line) and lower H values may be declared as the more reliable experimental data. It is worth it to point out that the reliability of experimental phase equilibrium data can be checked normally by performing the thermodynamic consistency tests. However, these kinds of tests may be just applicable for homogeneous phases or the nonheterogeneous ones with small amounts of (solubility of) the solutes in the mixture. However, this is not the case for the studied systems. It may be the main reason why we cannot
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extend our previous works (Eslamimanesh et al., 2011a–d, 2012; Mohammadi et al., 2011a, b) on the data assessments tests for the treated experimental data in this work.
5. Conclusion A method for evaluation of asphaltene precipitation titration data was proposed on the basis of the Leverage statistical approach (Rousseeuw and Leroy, 1987; Goodall, 1993; Gramatica, 2007). Two asphaltene precipitation titration datasets from the literature (Ashoori et al., 2003; Rassamdana et al., 1996) were represented/ predicted by the original scaling equation (Rassamdana et al., 1996) and the modified one (Ashoori et al., 2003) to pursue the calculation steps of the evaluation method. The results show that the applied models are valid and statistically correct. Furthermore, only one of the data points was found to be outlier (doubtful experimental data) while all of the investigated precipitation data were interpreted to within the applicability of the employed models. The results can be further used to conclude about the quality of the data points, which are supposed to be applied in tuning the asphaltene precipitation models (thermodynamic or numerical ones).
Acknowledgement Ali Eslamimanesh is grateful to MINES ParisTech for providing him with a PhD scholarship. References Andersen, S.I., Speight, J.G., 2001. Petroleum resins: separation, character and role in petroleum. J. Pet. Sci. Technol. 19, 1–34. ¨ Ashoori, S., Jamialahmadi, M., Muller-Steinhagen, H., Ahmadi, K., 2003. A new scaling equation for modeling of asphaltene precipitation. SPE 85673, 1–15. Eslamimanesh, A., Mohammadi, A.H., Richon, D., 2011a. Thermodynamic consistency test for experimental solubility data in carbon dioxide/methaneþwater system inside and outside gas hydrate formation region. J. Chem. Eng. Data 56, 1573–1586. Eslamimanesh, A., Mohammadi, A.H., Richon, D., 2011b. Thermodynamic consistency test for experimental data of water content of methane. AIChE J. 57, 2566–2573. Eslamimanesh, A., Mohammadi, A.H., Richon, D., 2011c. Thermodynamic consistency test for experimental data of sulfur content of hydrogen sulfide. Ind. Eng. Chem. Res. 50, 3555–3563.
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Eslamimanesh, A., Mohammadi, A.H., Richon, D., 2011d. Experimental data assessment test for diamondoids solubility in gaseous system. J. Chem. Eng. Data 56, 2655–2659. Eslamimanesh, A., Babaee, S., Mohammadi, A.H., Javanmardi, J., Richon, D., 2012. Experimental data assessment test for composition of vapor phase in equilibrium with gas hydrate and liquid water for carbon dioxide þ methane or nitrogenþwater system. Ind. Eng. Chem. Res. 2012 (51), 3819–3825. Goodall, C.R., 1993. Computation Using the QR Decomposition, Handbook in Statistics, 9. Elsevier/North-Holland, Amsterdam. Gramatica, P., 2007. Principles of QSAR models validation: internal and external. QSAR Comb. Sci. 26, 694–701. Gharagheizi, F., Eslamimanesh, A., Sattari, M., Mohammadi, A.H., Richon, D., 2011. Evaluation of thermal conductivity of gases at atmospheric pressure through a corresponding states method. Ind. Eng. Chem. Res. 2012 (51), 3844–3849. Leontaritis, K.J., Mansoori, G.A, 1988. Asphaltene deposition: a survey of field experience and research approaches. J. Petrol. Sci. Eng. 1, 229–239. Leontaritis, K.K., Amacfule, J.O., Charles, R.E., 1992. A Systematic Approach for the Prevention and Treatment of Formation Damage Caused by Asphaltene Deposition. SPE 23810, SPE Intl. Symposium on Formation Damage Control, Lafayette, Louisiana. Mohammadi, A.H., Richon, D., 2007. A monodispersethermodynamicmodel for estimating asphalteneprecipitation. AIChE J. 53, 2940–2947. Mohammadi, A.H., Richon, D., 2008a. The Scott-Magat polymer theory for determining onset of precipitation of dissolved asphaltene in the solventþ precipitant solution. Open Thermodyn. J. 2, 13–16. Mohammadi, A.H., Richon, D., 2008b. A Thermodynamic Micellar Model Based on the Scott-Magat Polymer Solution Theory for Predicting Asphaltene Precipitation. European Symposium on Applied Thermodynamics, ESAT, Cannes, France. Mohammadi, A.H., Eslamimanesh, A., Richon, D., 2012. Monodisperse thermo¨ dynamic model based on chemical þFlory–Huggins polymer solution theories for predicting asphaltene precipitation. Ind. Eng. Chem. Res. 2012 (51), 4041–4055. Mohammadi, A.H., Eslamimanesh, A., Richon, D., 2011a. Wax solubility in gaseous system: thermodynamic consistency test of experimental data. Ind. Eng. Chem. Res. 50, 4731–4740. Mohammadi, A.H., Eslamimanesh, A., Yazdizadeh, M., Richon, D., 2011b. Glycol loss in a gaseous system: thermodynamic assessment test of experimental solubility data. J. Chem. Eng. Data 56, 4012–4016. Pan, H., Firoozabadi, A., 1996. A Thermodynamic Micellization Model for Asphaltene Precipitation: Part I: Micellar Size and Growth. SPE 36741, ATCE, Denver. Rassamdana, H., Dabir, B., Nematy, M., Farhani, M., Sahimi, M., 1996. Asphalt flocculation and deposition: I. The onset of precipitation. AIChE J. 42, 10–22. Rassamdana, H., Sahimi, M., 1996. Asphalt flocculation and deposition: II. Formation and growth of fractal aggregates. AIChE J. 42, 3318–3332. Rassamdana, H., Farhani, M., Dabir, B., Mozaffarian, M., Sahimi, M., 1999. Asphalt flocculation and deposition. V. Phase behavior in miscible and immiscible injections. Energy Fuels 13, 176–187. Rousseeuw, P.J., Leroy, A.M., 1987. Robust Regression and Outlier Detection. John Wiley & Sons, New York. Thawer, R., Nicoll, D.C.A., Dick, G., 1990. Asphaltene deposition in production facilities. SPE Prod. Eng. 5, 475–480.
Contents lists available at SciVerse ScienceDirect
Chemical Engineering Science journal homepage: www.elsevier.com/locate/ces
Note
A novel method for evaluation of asphaltene precipitation titration data Amir H. Mohammadi a,b,n, Ali Eslamimanesh a, Farhad Gharagheizi c, Dominique Richon a,b ´ nerge´tique et Proce´de´s, 35 Rue Saint Honore´, 77305 Fontainebleau, France MINES ParisTech, CEP/TEP - Centre E Thermodynamics Research Unit, School of Chemical Engineering, University of KwaZulu-Natal, Howard College Campus, King George V Avenue, Durban 4041, South Africa c Department of Chemical Engineering, Buinzahra Branch, Islamic Azad University, Buinzahra, Iran a
b
a r t i c l e i n f o
abstract
Article history: Received 16 February 2012 Received in revised form 23 April 2012 Accepted 7 May 2012 Available online 18 May 2012
In this work, we propose a mathematical method for detection of the probable doubtful asphaltene precipitation titration data. The algorithm is performed on the basis of the Leverage approach, in which the statistical Hat matrix, Williams Plot, and the residuals of the model results lead to identify the probable outliers. This method not only contributes to outliers diagnostics but also defines the range of applicability of the applied models and quality of the existing experimental data. Two available scaling equations from the literature are used to pursue the calculation steps. It is found from the obtained results that: I. The applied models to represent/predict the weight percent of asphaltene precipitation are statistically valid and correct. II. All the treated experimental titration data seem to be reliable except one. III. The whole data points present in the dataset are within the domain of applicability of the employed models. & 2012 Elsevier Ltd. All rights reserved.
Keywords: Evaluation of data Leverage approach Asphaltene precipitation Outlier diagnostics Scaling equation Titration data
1. Introduction Saturates, aromatics, resins, and asphaltenes (SARA) are generally assigned as the portions of crude oils (Leontaritis and Mansoori, 1988; Leontaritis et al., 1992; Mohammadi and Richon, 2007, 2008a,b; Thawer et al., 1990; Pan and Firoozabadi, 1996; Andersen and Speight, 2001; Mohammadi et al., 2012). The latter fractions (asphaltenes) are normally toluene/benzene soluble but n-heptane/n-pentane insoluble. It can be stated that asphaltenes are the most aromatic with highest molecular weight fraction of petroleum fluids that generally contain simple heteroatoms (C, H, N, O, and S) or even particular metal constituents like Fe, Ni, and V (Mohammadi et al., 2012). Asphaltenes have potential to be precipitated from the bulk of crude oils mainly due to changes in pressure, temperature, and fluid composition. For instance, there is a possibility of asphaltene precipitation as the oil pressure drops during production of some crude oils. It is currently well accepted that asphaltene precipitations/ depositions in petroleum reservoirs, production and/or processes facilities cause remarkable problems (Leontaritis and Mansoori, 1988; Leontaritis et al., 1992; Mohammadi and Richon, 2007; Thawer et al., 1990; Mohammadi and Richon, 2008a,b; Pan and Firoozabadi, 1996; Andersen and Speight, 2001; Mohammadi et al., 2012) Nevertheless, the real mechanism of asphaltene
n Corresponding author at: MINES ParisTech, CEP/TEP - Centre E´nerge´tique et Proce´de´s, 35 Rue Saint Honore´, 77305 Fontainebleau, France. Tel.: þ33 1 64 69 49 70; fax: þ 33 1 64 69 49 68. E-mail address: [email protected] (A.H. Mohammadi).
0009-2509/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ces.2012.05.009
agglomeration, flocculation, and precipitation has been not completely understood so far mainly due to its complexity (Mohammadi and Richon, 2007; Mohammadi et al., 2012) There are still many debates on this subject (Mohammadi and Richon, 2007; Mohammadi et al., 2012). To deal with clarification of this issue, many experimental techniques have been proposed in the literature. Perhaps, the most widely used ones are the titration tests. They are generally performed using core flood investigations accompanied by addition of different asphaltene precipitants such as n-alkanes (mainly n-C5, n-C6, and n-C7). Different dilution ratios (the ratios of solvents/precipitants to the weight of the live/dead oil samples) are normally used to precipitate or dissolve asphaltene fraction in the bulk of crude at different temperatures (Andersen and Speight, 2001; Ashoori et al., 2003; Rassamdana et al., 1996; Rassamdana and Sahimi, 1996). The common experimental (filtration) procedure (which is generally performed on the basis of IP-143 procedure) is briefly as follows: (Rassamdana et al., 1999). The crude oil sample accompanied by the precipitants are mixed with each other (according to specified dilution ratios) in a suitable vessel using agitation system. Later, an asphaltene solvent (like toluene) is added to the system to dissolve the precipitated asphalts. The solvent would then evaporate and the remaining asphaltenes (precipitated asphaltenes) can be measured. Although the available titration data seem to be adequate to investigate the amounts and onsets of asphaltene precipitations in various oil samples, no method has been proposed to check their reliability up to now. Furthermore, comparison between the various titration data as a method for checking the quality of the corresponding datasets may be conservative due to non-similar
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phase behaviors and/or structures of asphaltene fractions present in different oil samples. Apart from that, the scaling equation developed by Rassamdana and co-workers (Rassamdana et al., 1996; Rassamdana and Sahimi, 1996) has generated number of attractions in the past decade to represent/predict the asphaltene precipitation titration data. Therefore, it is of much interest to propose a statisticallycorrect method for simultaneous detection of the doubtful titration data and their quality along with checking the validity and domain of applicability of the scaling equation. In this communication, our aim is to use the Leverage approach (Rousseeuw and Leroy, 1987; Goodall, 1993; Gramatica, 2007) for this purpose. To the best of our knowledge, this is the first time that this statistical method is used for evaluation of such data which are of interest in petroleum industry.
2. Theory
The points located in the range of R o 3 or 3oR (whether they are larger or smaller than the Hn value) are designated as outliers of the model or ‘‘Bad High Leverage’’ points. These erroneous representations/predictions may be attributed to the doubtful data. 2.2. Scaling equation In 1996, Rassamdana and co-workers (Rassamdana et al., 1996; Rassamdana and Sahimi, 1996) stated that the titration data of asphaltene precipitation including the dilution ratio, the molecular weight of solvent/precipitant, and the weight percent of precipitated asphaltene can be collapsed onto a single (and simple) equation (curve) (Rassamdana et al., 1996; Rassamdana and Sahimi, 1996). As a consequence, they proposed the following three-order polynomial for representation/prediction of the amounts/onsets of asphaltene precipitations against addition of n-alkanes: (Rassamdana et al., 1996). Y ¼ a þ bX þ cX 2 þ dX 3
2.1. Leverage approach
ð2Þ
where, Outlier diagnostics (or detection) are of much importance in developing the mathematical models. As a matter of fact, outlier detection is to identify individual datum (or groups of data) that may differ from the bulk of the data present in a dataset (Rousseeuw and Leroy, 1987). The corresponding methods generally consist of numerical and graphical algorithms (Rousseeuw and Leroy, 1987; Goodall, 1993; Gramatica, 2007; Gharagheizi et al., 2011). The Leverage approach (Rousseeuw and Leroy, 1987; Goodall, 1993; Gramatica, 2007) is considered to be a reliable algorithm for outlier detection that deals with the values of the residuals (i.e., the deviations of a model results from the experimental data) and a matrix known as Hat matrix composed of the experimental data and the represented/predicted values obtained from a model. The primary application criterion of this method is to use a model, which is capable of acceptable calculation/ estimation of the data of interest. The Leverage or Hat indices are calculated based on Hat matrix (H) with the following definition: (Rousseeuw and Leroy, 1987; Goodall, 1993; Gramatica, 2007; Gharagheizi et al., 2011) H ¼ XðX t XÞ1 X t
X ¼ Rd =M z
ð3Þ
and Y ¼ W=RZd
0
ð4Þ
In the preceding equations, W is the weight percent of the precipitated asphaltene, Rd stands for the dilution ratio, and M denotes the molecular weight of the solvent/precipitant. They recommended the values of 1/4 and 2 for z and z0 after adjusting these parameters against the obtained experimental titration data. Later, Rassamdana et al. (1999) and Ashoori et al. (2003) modified one of the parts of the original scaling equation to account for the effects of temperature on the amount of precipitations introducing a third parameter in Eq. (3) as follows: X ¼ Rd =T n M z
ð5Þ
where T is the temperature. The value of 0.15 for the n parameter in Eq. (5) shows acceptable results for representation of the experimental titration values reported by Ashoori et al. (2003).
ð1Þ
where X is a (n k) matrix, in which n stands for the data (rows) and k denotes the parameters of the model (columns), and t stands for the transpose matrix. The Hat values of the data in the feasible region of the problem are, as a matter of fact, the diagonal elements of the H value. Having evaluated the H values with Eq. (1), the Williams plot is sketched for graphical identification of the suspended data or outliers. This plot shows the correlation of Hat indices and standardized cross-validated residuals (R), which are defined as the difference between the represented/predicted values and the implemented data. A warning Leverage (H) is generally fixed at the value equal to 3n/p, where n is number of training points and p is the number of model parameters plus one. The Leverage of three is normally considered as a ‘‘cut-off’’ value to accept the points within 73 range (two horizontal red lines) standard deviations from the mean (to cover 99% normally distributed data). Existence of the majority of data points in the ranges 0 r H r Hn and 3 r R r3 reveals that both model development and its predictions are done in applicability domain, which result in a statistically valid model. ‘‘Good High Leverage’’ points are located in domain of Hn rH and 3r R r3. The Good High Leverage can be designated as the ones, which are outside of applicability domain of the applied model. In other words, the model is not able to represent/predict the following data at all.
3. Experimental data The experimental asphaltene precipitation titration data obtained and reported in the original publications of Rassamdana et al. (1996) and Ashoori et al. (2003) have been treated in this work.
4. Results and discussion Tables 1 and 2 show the absolute relative deviations of the results using the original (Rassamdana et al., 1996) and the modified scaling equation. (Ashoori et al., 2003). As can be seen, the deviations of the models representations/predictions from the corresponding experimental data reported in the original articles (Ashoori et al., 2003; Rassamdana et al., 1996) are generally acceptable to be used for the Leverage statistical approach. The evaluation steps have been followed on the basis of the aforementioned procedure (Section 2.1). The H values have been calculated through Eq. (1). The Williams plots have been sketched in Figs. 1 and 2 for the results using the first model (Rassamdana et al., 1996) and corresponding experimental dataset (Rassamdana et al., 1996) and the second ones, (Ashoori et al., 2003), respectively. All the calculated H and R values are presented in Tables 1 and 2. Two warning Leverages
A.H. Mohammadi et al. / Chemical Engineering Science 78 (2012) 181–185
Table 1 The results of the applied model (Rassamdana et al., 1996) (the original scaling equation) and the Leverage approach (Rousseeuw and Leroy, 1987; Goodall, 1993; Gramatica, 2007). W
exp*
1.19 1.61 2.05 2.61 3.12 3.5 3.85 4.21 4.62 0.79 1.09 1.43 1.57 1.68 1.74 1.96 2.16 2.55 3.05 3.46 3.61 4.28 0.74 1.02 1.16 1.4 1.4 1.64 1.94 2.3 2.58 2.93 3.21 3.54 3.68 0.7 1.08 1.26 1.4 1.49 1.59 1.82 2.24 2.48 2.89 3.13 3.17 3.42 0.7 0.89 1.15 1.19 1.39 1.42 1.61 1.91 2.16 2.56 2.83 2.97 2.99
W
rep/pred **
0.6 1.13 1.72 2.38 2.83 3.14 3.55 4.01 4.48 0.54 1 1.37 1.5 1.67 1.8 1.92 2.13 2.54 2.82 3.2 3.63 4.07 0.41 0.84 1.18 1.33 1.46 1.7 1.91 2.29 2.55 2.92 3.33 3.57 3.74 0.29 0.7 1.03 1.15 1.3 1.51 1.7 2.06 2.33 2.69 3.09 3.32 3.48 0.17 0.52 0.81 0.96 1.1 1.28 1.46 1.79 2.03 2.36 2.72 2.93 3.08
Table 2 The results of the applied model (Ashoori et al., 2003) (the modified scaling equation) and the Leverage approach (Rousseeuw and Leroy, 1987; Goodall, 1993; Gramatica, 2007). exp*
Precipitant
Hat
Standardized residuals
W
n-C5 n-C5 n-C5 n-C5 n-C5 n-C5 n-C5 n-C5 n-C5 n-C6 n-C6 n-C6 n-C6 n-C6 n-C6 n-C6 n-C6 n-C6 n-C6 n-C6 n-C6 n-C6 n-C7 n-C7 n-C7 n-C7 n-C7 n-C7 n-C7 n-C7 n-C7 n-C7 n-C7 n-C7 n-C7 n-C8 n-C8 n-C8 n-C8 n-C8 n-C8 n-C8 n-C8 n-C8 n-C8 n-C8 n-C8 n-C8 n-C10 n-C10 n-C10 n-C10 n-C10 n-C10 n-C10 n-C10 n-C10 n-C10 n-C10 n-C10 n-C10
0.05 0.03 0.02 0.02 0.03 0.03 0.05 0.07 0.1 0.05 0.03 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.04 0.05 0.07 0.06 0.04 0.03 0.02 0.02 0.02 0.02 0.02 0.02 0.03 0.04 0.05 0.06 0.06 0.04 0.03 0.03 0.02 0.02 0.02 0.02 0.02 0.02 0.03 0.04 0.05 0.07 0.05 0.04 0.03 0.03 0.02 0.02 0.02 0.02 0.02 0.02 0.03 0.03
2.38 1.85 1.09 0.72 1.36 2.00 1.79 1.34 1.16 0.02 0.90 0.93 0.80 1.13 1.54 0.80 0.77 0.71 0.94 1.33 0.40 1.44 0.48 0.36 1.57 0.88 1.71 1.59 0.87 0.83 0.57 0.53 1.24 0.5 0.63 0.98 0.96 0.08 0.27 0.07 0.72 0.36 0.23 0.15 0.67 0.24 1.45 0.75 1.77 0.81 0.73 0.04 0.53 0.42 0.27 0.31 0.13 0.51 0.06 0.32 1.14
0 0.75 1.2 1.7 4 4.8 6.6 7.5 8 8.1 8.3 0.81 1.23 1.91 4.16 5.03 7.03 8.13 8.68 8.86 9.08 0.91 1.42 2.32 4.66 5.83 8.1 9.25 10 10.2 10.4 0.6 0.98 1.35 2.98 3.61 5.07 5.88 6.32 6.42 6.58 0.62 1.01 1.44 3.33 4.06 5.77 6.67 7.22 7.3 7.46 0.73 1.18 2.12 4.15 5.16 7.46 8.58 9.16 9.25 9.46 0.5 0.75 1.5 2.51 3.07 4.2 4.91 5.36
n Experimental weight percent of precipitated asphaltene (Rassamdana et al., 1996). The experiments have been done at approximately 299.2 72 K. These data have been taken from the figures in the original publication (Rassamdana et al., 1996). nn Represented/predicted weight percent of precipitated asphaltene by the original scaling equation (Rassamdana et al., 1996). The values of the parameters in Eq. 2 are as follows: a ¼1.18; b¼ 14.9; c¼ 39.16; d¼ 0.92.
183
W
rep/pred **
0 0.99 1.24 1.6 3.76 4.68 6.55 7.39 7.73 8.09 8.46 1.09 1.36 1.78 4.21 5.24 7.33 8.05 8.41 8.78 9.17 1.23 1.53 2.01 4.81 5.99 7.98 8.9 9.27 9.66 10.08 0.81 1.02 1.3 3 3.73 5.2 6.21 6.52 6.84 7.18 0.89 1.12 1.44 3.35 4.17 5.83 6.77 7.09 7.43 7.78 1 1.26 1.63 3.83 4.76 6.6 7.49 7.83 8.18 8.56 0.7 0.9 1.14 2.58 3.21 4.47 5.52 5.82
Precipitant Temperature Hat (K)
Standardized residuals
n-C7 n-C7 n-C7 n-C7 n-C7 n-C7 n-C7 n-C7 n-C7 n-C7 n-C7 n-C6 n-C6 n-C6 n-C6 n-C6 n-C6 n-C6 n-C6 n-C6 n-C6 n-C5 n-C5 n-C5 n-C5 n-C5 n-C5 n-C5 n-C5 n-C5 n-C5 n-C7 n-C7 n-C7 n-C7 n-C7 n-C7 n-C7 n-C7 n-C7 n-C7 n-C6 n-C6 n-C6 n-C6 n-C6 n-C6 n-C6 n-C6 n-C6 n-C6 n-C5 n-C5 n-C5 n-C5 n-C5 n-C5 n-C5 n-C5 n-C5 n-C5 n-C7 n-C7 n-C7 n-C7 n-C7 n-C7 n-C7 n-C7
0.49 0.27 0.24 0.57 0.68 0.26 0.15 0.09 0.30 0.44 0.94 0.39 0.02 0.63 0.14 0.69 1.18 0.25 0.22 0.33 0.84 0.52 0.01 1.08 0.48 0.64 0.13 0.38 1.36 0.80 0.16 0.17 0.26 0.47 0.08 0.27 0.47 1.12 0.81 1.44 1.96 0.34 0.06 0.32 0.04 0.3 0.36 0.58 0.00 0.73 1.29 0.35 0.13 1.61 0.89 0.99 2.01 2.52 3.13 2.40 1.90 0.13 0.02 1.32 0.00 0.26 0.76 1.79 1.42
303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15
0.04 0.03 0.03 0.02 0.01 0.01 0.02 0.02 0.02 0.03 0.03 0.03 0.03 0.02 0.01 0.01 0.02 0.03 0.03 0.03 0.04 0.03 0.02 0.02 0.01 0.01 0.03 0.04 0.04 0.05 0.05 0.03 0.03 0.03 0.02 0.01 0.01 0.01 0.02 0.02 0.02 0.03 0.03 0.03 0.01 0.01 0.01 0.02 0.02 0.02 0.02 0.03 0.03 0.02 0.01 0.01 0.02 0.02 0.02 0.03 0.03 0.03 0.03 0.03 0.02 0.01 0.01 0.01 0.01
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Table 2 (continued ) W
exp*
5.53 5.65 0.51 0.67 1.2 2.67 3.27 4.59 5.48 5.96 6.63 0.58 0.81 1.48 3.6 4.48 6.27 7.32 7.73 7.88 7.98
W
rep/pred **
6.12 6.44 0.78 0.99 1.26 2.89 3.59 5.01 6.03 6.33 6.65 0.88 1.1 1.42 3.29 4.1 5.72 6.67 7 7.33 7.68
Precipitant Temperature Hat (K)
Standardized residuals
n-C7 n-C7 n-C6 n-C6 n-C6 n-C6 n-C6 n-C6 n-C6 n-C6 n-C6 n-C5 n-C5 n-C5 n-C5 n-C5 n-C5 n-C5 n-C5 n-C5 n-C5
1.8 2.38 0.33 0.49 0.18 0.44 0.79 1.22 1.69 1.24 0.35 0.42 0.42 0.48 0.93 1.02 1.28 1.44 1.61 1.10 0.39
343.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15
0.01 0.02 0.03 0.03 0.03 0.02 0.01 0.01 0.01 0.01 0.02 0.03 0.03 0.03 0.01 0.01 0.01 0.02 0.02 0.02 0.02
n
Experimental weight percent of precipitated asphaltene (Ashoori et al., 2003). nn Represented/predicted weight percent of precipitated asphaltene by the modified scaling equation (Ashoori et al., 2003). The values of the parameters in Eq. 2 are as follows: a¼ 1.18; b ¼ 14.9; c ¼ 39.16; d ¼ 0.92.
Fig. 2. Detection of the probable doubtful experimental data (Ashoori et al., 2003) and the applicability domain of the modified scaling equation (Ashoori et al., 2003) Squares, valid data; Circular, Probable suspended data; Horizontal (red) lines, Suspended data limit; Vertical (blue) line, Leverage limit. (For interpretation of the references to color in this figure legend, the reader is reffered to the web version of this article.)
Good High Leverage points are accumulated in the domains of 0:147 rH and 3 r R r3 for the first model þdata (Rassamdana et al., 1996) and 0:133 rH and 3 rR r 3 for the second ones (Ashoori et al., 2003). These points may be declared to be outside of applicability domains of the applied models. As can be seen, there are no such points in the investigated datasets. It should be noted that, in the case of existence of these kinds of points, it is recommended to use/develop other models on the basis of different theoretical concepts for their calculations/estimations, in order to avoid estimation through biased model calculations. The points located in the range of Ro3 or 3 oR (ignoring their H values) are designated as outliers of the model or ‘‘Bad High Leverage’’ points, as already mentioned. These erroneous representations/predictions can be attributed to the doubtful asphaltene precipitation data. There is only one point in the two treated datasets (Ashoori et al., 2003; Rassamdana et al., 1996), which is within this domain and consequently we can state it as probable doubtful datum. In the final analysis, we may conclude the following elements: Fig. 1. Detection of the probable doubtful experimental data (Rassamdana et al., 1996) and the applicability domain of the original scaling equation (Rassamdana et al., 1996). Squares, valid data; Horizontal (red) lines, Suspended data limit; Vertical (blue) line, Leverage limit. (For interpretation of the references to color in this figure legend, the reader is reffered to the web version of this article.)
(Hn ¼0.147) and (Hn ¼ 0.133) have been fixed at 3n/p for the first model and experimental data (Rassamdana et al., 1996) and the second ones (Ashoori et al., 2003), respectively. In addition, the recommended cut-off value of three has been applied (Gharagheizi et al., 2011). Accumulation of the whole dataset in the ranges 0 r H r 0:147 and 3 r Rr 3 for the data þmodel reported by Rassamdana et al. (1996) and 0 r H r 0:133 and 3 rR r 3 regarding the dataþ model of Ashoori et al. (2003) reveals that the applied models are statistically correct and valid. Furthermore, it shows that the whole data except one in the later dataset are located within the applicability domains of the applied models.
1. The applied scaling equations (Ashoori et al., 2003; Rassamdana et al., 1996) are statistically correct and valid. 2. All of the data points are within the applicability domains of the presented model. 3. There is only 1 point, which may be designated as outlier from the dataset of Ashoori et al. (2003) (see Table 2). 4. The quality of the treated data (even different data in the same dataset) is different. The data with lower absolute R values (near R ¼0 line) and lower H values may be declared as the more reliable experimental data. It is worth it to point out that the reliability of experimental phase equilibrium data can be checked normally by performing the thermodynamic consistency tests. However, these kinds of tests may be just applicable for homogeneous phases or the nonheterogeneous ones with small amounts of (solubility of) the solutes in the mixture. However, this is not the case for the studied systems. It may be the main reason why we cannot
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extend our previous works (Eslamimanesh et al., 2011a–d, 2012; Mohammadi et al., 2011a, b) on the data assessments tests for the treated experimental data in this work.
5. Conclusion A method for evaluation of asphaltene precipitation titration data was proposed on the basis of the Leverage statistical approach (Rousseeuw and Leroy, 1987; Goodall, 1993; Gramatica, 2007). Two asphaltene precipitation titration datasets from the literature (Ashoori et al., 2003; Rassamdana et al., 1996) were represented/ predicted by the original scaling equation (Rassamdana et al., 1996) and the modified one (Ashoori et al., 2003) to pursue the calculation steps of the evaluation method. The results show that the applied models are valid and statistically correct. Furthermore, only one of the data points was found to be outlier (doubtful experimental data) while all of the investigated precipitation data were interpreted to within the applicability of the employed models. The results can be further used to conclude about the quality of the data points, which are supposed to be applied in tuning the asphaltene precipitation models (thermodynamic or numerical ones).
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