Prediction Of Heavy-Oil Viscosities With A Simple .
Prediction of Heavy-Oil ViscositiesWith a Simple Correlation ApproachA. Bahadori, Southern Cross University, and M. Mahmoudi and A. Nouri, University of AlbertaSummaryHeavy-oil development is becoming increasingly important because of the continuous decline in conventional-oil production. Forheavy-oil reservoirs, the oil viscosity usually varies dramaticallyduring production processes such as in thermal processes. Whenproducing heavy oil, the high viscosity is a major impediment torecovery. Oil viscosity is often correlated directly to the reservesestimate in heavy-oil formations and can determine the successor failure of a given enhanced-oil-recovery scheme. As a result,viscosity is an important parameter for performing numericalsimulation and determining the economics of a project.In this work, a simple-to-use correlation has been developed tocorrelate the viscosity of heavy oil to temperature and to a simplecorrelating parameter that can be used for heavy-oil characterization. The reported results are the product of the analysis of heavyoil data collected from the open literature for various heavy-oilfields around the world. The tool developed in this study can beof immense practical value for petroleum engineers, providing amethod for quick assessment of the viscosity of heavy oils. In particular, petroleum and production engineers would find the proposed correlation to be user-friendly, with transparent calculationsinvolving no complex expressions.The new proposed correlation shows consistently accurateresults. This consistency could not be matched by any of the widelyaccepted existing correlations within the investigated range. For allconditions, the new correlation provided better results than existingcorrelations in the literature.IntroductionThe continuing decline in conventional crude-oil reserves combined with the continuing high worldwide demand for oil has ledto the increased role of unconventional resources, especially heavycrude oil, in the world (Pedersen and Fredenslund 1984; Butler1991; Al-Maamari et al. 2006). There are many challenges to thesuccessful exploitation of these resources (Willman et al. 1961).One of the major issues is the high viscosity of the heavy crudeoils, which makes production and processing difficult (Monneryet al. 1995; Mehrotra et al. 1996). Heavy crude oil and tar-sandoil are petroleum or petroleum-like liquids or semisolids occurringnaturally in porous media. These oils are characterized by viscosityand density (Puttagunta et al. 1993).The viscosity of heavy oils is a critical property in predictingoil recovery (Mehrotra 1990) and selecting a production approach.Several viscosity correlations are available in the literature (Mehrotra 1991a, b; Miadonye et al. 1992; Puttagunta et al. 1992; Miadonye et al. 1993; Puttagunta et al. 1993; Singh et al. 1993a, b; DeCopyright 2015 Society of Petroleum EngineersThis paper (SPE 157360) was accepted for presentation at the SPE Heavy Oil ConferenceCanada, Calgary, 12–14 June 2012, and revised for publication. Original manuscriptreceived for review 25 June 2014. Revised manuscript received for review 23 September2014. Paper peer approved 5 November 2014.66Oil and Gas Facilities February 2015Ghetto et al. 1995; Petrosky and Farshad 1995; Wakabayashi 1997;Bennison 1998; Miadonye and Puttagunta 1998), and are mainly afunction of density (or API) and temperature. These correlationsare used when viscosity measurements are not available. Considerable errors may be introduced when these correlations are usedfor assessing heavy-oil viscosity. To increase accuracy, compositional terms, such as the percent of saturates, aromatics, resins, andasphaltenes, should be used in the correlation (Al-Maamari et al.2006). Therefore, we use a new correlating parameter called corrected API (CAPI), as proposed by Al-Maamari et al. (2006), whichcan be used for heavy-oil characterization.The results in this paper are the product of analysis of heavy-oildata collected from the open literature for various heavy-oil fieldsaround the world. Distinctive parameters that have been consideredare crude-oil gravity (API) and compound class distributions [i.e.,saturated hydrocarbons (Sa), aromatic hydrocarbons (Ar), resins(Re), and asphaltenes (As)] (Al-Maamari et al. 2006):Sa . .(1)CAPI API Ar Re As The units in Eq. 1 for Sa, Ar, Re, and As are mass fraction. Inview of the aforementioned issues and the importance of viscosityin heavy-oil production and processing, it is necessary to developan accurate and simple correlation for predicting the viscosity ofheavy crude oil as a function of temperature and for correlatingparameters that can be easily assessed. This paper discusses theformulation of such a predictive tool in a systematic manner. Theproposed method is an exponential function that leads to well-behaved (i.e., smooth and nonoscillatory) equations, enabling moreaccurate and nonoscillatory predictions.In the majority of cases, existing correlations indicated a goodprediction of crude-oil viscosity for the oils from which they werederived. However, when used with other crude oils from otherregions, these correlations are, in most cases, not accurate andcertain modifications are needed to obtain acceptable viscositypredictions.This correlation is only for dead-oil viscosity; therefore, the viscosity/API correlation is the focus of this paper. A new correlatingparameter, CAPI, is used for heavy-oil characterization. It is not afunction of pressure. The reported results are the product of analysis of data from many heavy oils collected from the open literaturefor various heavy-oil fields around the world.Methodology for the Development of a Novel CorrelationThe primary purpose of the present study is to accurately correlatethe viscosity of heavy crude oil as a function of temperature, andas a function of a simple correlating parameter that can be used forheavy-oil characterization.The Vandermonde matrix is a matrix with the terms of a geometric progression in each row (i.e., an m n matrix) (Bair et al.2006):February 2015 Oil and Gas Facilities1
Table 1—Tuned coefficients used in Eqs. 11 through 14. V 11112.222.23.131m2m 1n 1 2n 1 3n 1 .(2) mn 1 orVi , j ij 1 .(3)for all indices i and j. The determinant of a square Vandermondematrix (where m n) can be expressed as (Bair et al. 2006)det (V ) (1 i j nj) i . .(4)The Vandermonde matrix evaluates a polynomial at a set ofpoints. Formally, it transforms coefficients of a polynomial a0 a1x a2x2 . an–1xn–1 to the values that the polynomial takes atthe points αi. The nonvanishing of the Vandermonde determinantfor distinct points αi shows that for distinct points, the map from coefficients to values at those points is a one-to-one correspondence,and thus that the polynomial interpolation problem is solvable witha unique solution. This result is called the unisolvence theorem(Fulton and Harris 1991). They are thus useful in polynomial interpolation because solving the system of linear equations Vu yfor u, with V being an m n Vandermonde matrix, is equivalent tofinding the coefficients uj of the polynomial(s) (Bair et al. 2006;Fulton and Harris 1991; Horn and Johnson 1991).n 1P ( x ) u j x j . .(5)j 0For degree n 1, which has the propertyP ( i ) yi , for i 1, ., m, .(6)the Vandermonde matrix can easily be inverted in terms of Lagrange basis polynomials—each column is the coefficient of theLagrange basis polynomial, with terms in increasing order goingdown. The resulting solution to the interpolation problem is calledthe Lagrange polynomial.Suppose that the interpolation polynomial is in the form (Bairet al. 2006; Fulton and Harris 1991; Horn and Johnson 1991):P ( x ) an x n an 1 x n 1 . a2 x 2 a1 x a0 . .(7)The statement that P interpolates the data points means thatP ( xi ) yi , for all i {0,1,., n}. .(8)2Oil and Gas Facilities February 2015If we substitute Eq. 3 into Eq. 8, we obtain a system of linear equations in the coefficients ak. The system in matrix/vector form reads(Fulton and Harris 1991; Horn and Johnson 1991; Bair et al. 2006) x0nx0n 1x0n 2 x0n1n 11x1n 2 x1xxxnnxn 1nn 2nx xn1 an y0 1 an 1 y1 . .(9) 1 a0 yn We have to solve this system for ak to construct the interpolantP(x). The matrix on the left is commonly referred to as a Vandermonde matrix (Fulton and Harris 1991; Horn and Johnson 1991;Bair et al. 2006).Development of the Correlation. The data required to developthis correlation include the viscosity of heavy oil as a function oftemperature and the corrected API (CAPI) (Henshaw et al. 1998;Al-Maamari et al. 2006). The following methodology (Bahadoriand Vuthaluru 2009; Bahadori 2010, 2011) has been applied todevelop this correlation by use of Matlab technical computinglanguage (Matlab 2008):1. Correlate the viscosities of heavy oil as a function of theCAPI for a given temperature.2. Repeat Step 1 for other temperatures.3. Correlate the corresponding polynomial coefficients, whichwere obtained for different temperatures, vs. temperature: a f(T), b f(T), c f(T), d f(T) (see Eqs. 11 through 14].Eq. 10 represents the proposed governing equation in whichfour coefficients are used to correlate the viscosity of heavy oil as afunction of temperature and as a function of the CAPI:ln ( ) a bcd , .(10)CAPI ( CAPI )2 ( CAPI )3wherea A1 B1 T C1T 2 D1T 3, .(11)b A2 B2T C2T 2 D2T 3 , .(12)c A3 B3T C3T 2 D3T 3 , .(13)andd A4 B4T C4T 2 D4T 3 , .(14)with the relevant coefficients reported in Table 1.February 2015 Oil and Gas Facilities67
Table 2—Viscosity data used in developing the new correlation (Al-Maamari et al. 2006; Henshawet al. 1998). SG , .(15)where μ is the dynamic viscosity in cp, η is the kinematic viscositymeasured in cSt, and SG is specific gravity. These optimum tunedcoefficients help to cover temperatures up to 180 C and CAPI aslow as 6. The optimum tuned coefficients given in Table 1 can befurther refined according to the proposed approach if more data become available in the future.In this work, our efforts were directed toward formulatinga correlation that can be expected to assist engineers for rapidcalculation of the heavy-oil viscosity as a function of temperatureand the CAPI. The proposed tool is simple and novel. The selectedexponential function to develop the tool leads to well-behaved (i.e.,smooth and nonoscillatory) equations, enabling reliable and accurate predictions.ResultsTable 2 summarizes the data used to develop this correlation. Fig. 1shows the proposed correlation curves in comparison with the literature data (Henshaw et al. 1998; Al-Maamari et al. 2006) that wereused to calibrate the correlation. Note that the data that were usedin developing and calibrating the correlation had gravities less than15 API [corrected API (CAPI) less than 6] and temperatures in therange of 40 to 177 C. Figs. 2 and 3 show the results from the proposed method and its smooth performance in the prediction of theviscosity of heavy oil as a function of temperature and CAPI.The performance of the proposed correlation, as well as othercorrelations in the literature, was examined against the literaturedata. Table 3 compares the performance of various exsiting correlations with that of the proposed correlation in predicting the viscosity of crude oil. It should be noted that the same set of datathat was used for the correlation calibration was used for the performance assessment, which does not allow an unbiased performance comparison. It shows that the existing correlations can result68Oil and Gas Facilities February 2015in significant errors when it comes to predicting the viscosity ofheavy crude oils because the correlations do not involve the composition of heavy oil. The inclusion of the heavy-oil composition inthe proposed correlation ensures a more-accurate viscosity assessment because it recognizes the difference in the viscosity of oilsthat have the same gravity in API but have different compositions.Further viscosity and compositional data are required to make amore-objective performance comparison. Table 4 summarizes theerror assessments for various correlations at different temperatures.It can be seen from the table that most correlations result in highererrors at lower temperatures.It is expected that our efforts in formulating the tool will pavethe way for arriving at an accurate prediction of the viscosity ofheavy crude oil. The tool developed in this study can be of greatpractical value for experts and engineers, providing a method forobtaining a quick assessment of the viscosity of heavy crude oil.In particular, petroleum engineers would find the approach to beuser-friendly, with transparent calculations involving no complexexpressions.ConclusionsIn this work, simple-to-use equations are presented for the estimation of heavy-oil viscosity as a function of temperature, and of asimple correlating parameter that can be used for heavy-oil characterization (corrected API). The performance of the proposedcorrelation was examined against the performance of several correlations in the literature. It was found that the proposed correlationresulted in the most-accurate predictions for the data used in thatexercise. Note that the correlation was developed with viscositydata in the range of 8.7 to 14.5 API and in the temperature rangeof 40 to 177 C. Therefore, care must be exercised when the correlation is used in the assessment of viscosity (in API) and temperature ranges outside of those of the data used in calibrating thecorrelation.February 2015 Oil and Gas Facilities3
Kinematic Viscosity (mm2/s)105104103102101100T 40 CDataT 50 CDataT 60 CDataT 70 CDataT 100 CDataT 135 CDataT 177 CData11.522.533.544.555.5Correlating Parameter (CAPI)Kinematic Viscosity (mm2/s)Fig. 1—The calibration of the predictive tool for the estimation of viscosity of heavy oil (Al-Maamari et al. 2006; Henshaw et al. 1998).104103102T 40 CT 110 C1.522.533.544.555.5Correlating Parameter (CAPI)Fig. 2—The smooth results of the predictive tool in estimating the viscosity of heavy oils for temperatures less than 110 C.90Kinematic Viscosity (mm2/s)807060T 110 C504030201001.5T 180 C22.533.544.555.5Correlating Parameter (CAPI)Fig. 3—The smooth results of the predictive tool in estimating the viscosity of heavy oils for temperatures greater than 110 C.4Oil and Gas Facilities February 2015February 2015 Oil and Gas Facilities69
Table 3—Comparison of the prediction error for different correlations.Unlike complex mathematical approaches for estimating theviscosity of heavy oils, the proposed correlation is straightforwardand would be greatly helpful for engineers, especially those dealingwith petroleum engineering and heavy-oil production. Additionally, the level of mathematical formulations associated with theestimation viscosity of heavy oil can be easily handled by an engineer or practitioner without any in-depth mathematical abilities.The proposed method has a clear numerical background, whereinthe relevant coefficients can be retuned quickly if more data become available in the future.Nomenclaturea A API Ar As b B c C CAPI 70correlating parameter in the viscosity correlationfirst tuned coefficientoil gravity, (API 145/γo, 135)aromatic hydrocarbonsasphaltenescorrelating parameter in the viscosity correlationsecond tuned coefficientcorrelating parameter in the viscosity correlationthird tuned coefficientcorrelating parameterOil and Gas Facilities February 2015d D i j m n P Re Sa SG T u V x X y Y z α γo,R η µ correlating parameter in the viscosity correlationfourth tuned coefficientindexindexmatrix row index for m n matrixmatrix column index for m n matrixpolynomialresinssaturated hydrocarbonsspecific gravitytemperature, Kcoefficient of polynomialVandermonde matrixcorrelating parameter in viscosity correlationdata pointcorrelating parameter in viscosity correlationdata pointcorrelating parameter in viscosity correlationmatrix elementoil specific gravity at reservoir conditionskinematic viscosity, mm2/soil dynamic viscosity, cpFebruary 2015 Oil and Gas Facilities5
Table 4—Maximum prediction error for different correlations at several temperatures.µod dead-oil viscosity, cpρ density, lbm/ft3ReferencesAl-Maamari, R.S., Houache, O., and Abdul-Wahab, S.A. 2006. New Correlating Parameter for the Viscosity of Heavy Crude Oils. Energy &Fuels 20 (6): 2586–2592. http://dx.doi.org/10.1021/ef0603030.Bahadori, A. 2010. Determination of Well Placement and BreakthroughTime in Horizontal Wells for Homogeneous and Anisotropic Reservoirs. Journal of Petroleum Science and Engineering 75 (1–2): 196–202. adori, A. 2011. Estimation of Combustion Flue Gas Acid Dew PointDuring Heat Recovery and Efficiency Gain. Applied Thermal Engineering 31 (8–9): 1457–1462. .020.Bahadori, A. and Vuthaluru, H.B. 2009. A Novel Correlation for Estimationof Hydrate Forming Condition of Natural Gases. Journal of NaturalGas Chemistry 18 (4): 453–457. r, E., Hastie, T., Paul, D. et al. 2006. Prediction by Supervised PrincipalComponents. Journal of the American Statistical Association 101(473): 119–137. http://dx.doi.org/10.1198/016214505000000628.Beal, C. 1946. The Viscosity of Air, Water, Natural Gas, Crude Oils andIts Associated Gases at Oil Field Temperatures and Pressures. Transactions of the AIME 165 (01): 94–115. SPE-946094-G. http://dx.doi.org/10.2118/946094-G.Beggs, H.D. and Robinson, J.R. 1975. Estimating the Viscosity of CrudeOil Systems. J Pet Technol 27 (09): 1140–1141. SPE-5434-PA. http://dx.doi.org/10.2118/5434-PA.6Oil and Gas Facilities February 2015Bennison, T. 1998. Prediction of Heavy Oil Viscosity. Presented at the IBCHeavy Oil Field Development Conference, London, England, ic html/PETE310/Papers/Visco%20models.pdf.Butler, R.M. 1991. Thermal Recovery of Oil and Bitumen. EnglewoodCliffs, New Jersey: Prentice-Hall.De Ghetto, G., Paone, F., and Villa, M. 1995. Pressure-Volume-Temperature Correlations for Heavy and Extra Heavy Oils. Presented at theSPE International Heavy Oil Symposium, Calgary, Alberta, 19–21June. SPE-30316-MS. http://dx.doi.org/10.2118/30316-MS.Elsharkawy, A.M. and Alikhan, A.A. 1999. Models for Predicting the Viscosity of Middle East Crude Oils. Fuel 78: 891–903. lton, W. and Harris, J. 1991. Representation Theory: A First Course. NewYork City, New York: Graduate Texts in Mathematics: Readings inMathematics Series, No. 129, Springer-Verlag.Glaso, O. 1980. Generalized Pressure-Volume-Temperature Correlations. J Pet Technol 32 (05): 785–795. SPE-8016-PA. http://dx.doi.org/10.2118/8016-PA.Henshaw, P.C., Carlson, R.M.K., Pena, M.M. et al. 1998. Evaluation ofGeochemical Approaches to Heavy Oil Viscosity Mapping in SanJoaquin Valley, California. Presented at the SPE Western RegionalMeeting, Bak
PB Oil and Gas Facilities February 2015 66 Oil and Gas Facilities February 2015 February 2015 Oil and Gas Facilities 1 February 2015 Oil and Gas Facilities 67 Prediction of Heavy-Oil Viscosities With a Simple Correlation Approach A. Bahadori, Southern Cross University, and M. Mahmoudi and A. Nouri, University of Alberta Ghett
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