SPE 75721 Simplified Correlations for Hydrocarbon Gas Viscosity and Gas Density — Validation and Correlation of Behavior Using a Large-Scale Database F.E. Londono, R.A. Archer, and T. A. Blasingame, Texas A&M U. Copyright 2002, Society of Petroleum Engineers Inc. For this study, we created a large-scale database of gas pro- This paper was prepared for presentation at the SPE Gas Technology Symposium held in perties using existing sources available in the literature. Our Calgary, Alberta, Canada, 30 April–2 May 2002. data-base includes: composition, viscosity, density, tempera- This paper was selected for presentation by an SPE Program Committee following review of ture, pressure, pseudoreduced properties and the gas com- information contained in an abstract submitted by the author(s). Contents of the paper, as presented, have not been reviewed by the Society of Petroleum Engineers and are subject to pressibility factor. We use this database to evaluate the appli- correction by the author(s). The material, as presented, does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Papers presented at cability of the existing models used to estimate hydrocarbon SPE meetings are subject to publication review by Editorial Committees of the Society of gas viscosity and gas density (or more specifically, the z- Petroleum Engineers. Electronic reproduction, distribution, or storage of any part of this paper for commercial purposes without the written consent of the Society of Petroleum Engineers is factor). Finally, we provide new models and calculation pro- prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied. The abstract must contain conspicuous cedures for estimating hydrocarbon gas viscosity and we also acknowledgment of where and by whom the paper was presented. Write Librarian, SPE, P.O. provide new optimizations of the existing equations-of-state Box 833836, Richardson, TX 75083-3836, U.S.A., fax 01-972-952-9435. (EOS) typically used for the calculation of the gas z-factor. Abstract Introduction The focus of this work is the behavior of gas viscosity and gas Hydrocarbon Gas Viscosity density for hydrocarbon gas mixtures. The viscosity of hydro- carbon gases is a function of pressure, temperature, density, NIST — SUPERTRAP Algorithm: The state-of-the-art mechan- and molecular weight, while the gas density is a function of ism for the estimation of gas viscosity is most likely the com- pressure, temperature, and molecular weight. This work pre- puter program SUPERTRAP developed at the U.S. National sents new approaches for the prediction of gas viscosity and Institute of Standards and Technology1 (NIST). SUPERTRAP gas density for hydrocarbon gases over practical ranges of was developed from pure component and mixture data, and is pressure, temperature and composition. These correlations stated to provide estimates within engineering accuracy from can be used for any hydrocarbon gas production or transport- the triple point of a given substance to temperatures up to ation operations. 1,340.33 deg F and pressures up to 44,100 psia. As the SUPERTRAP algorithm requires the composition for a parti- In this work we created an extensive database of measured gas cular sample, this method would not be generally suitable for viscosity and gas density (>5000 points for gas viscosity and applications where only the mixture gas gravity and composi- >8000 points for gas density). This database was used to tions of any contaminants are known. evaluate existing models for gas viscosity and gas density. In this work we provide new models for gas density and gas Carr, et al. Correlation: Carr, et al.2 developed a two-step viscosity, as well as optimization of existing models using this procedure to estimate hydrocarbon gas viscosity. The first database. step is to determine the gas viscosity at atmospheric condi- tions (i.e., a reference condition). Once estimated, the vis- The objectives of this research are: cosity at atmospheric pressure is then adjusted to conditions at (cid:122) To create a large-scale database of measured gas vis- the desired temperature and pressure using a second correla- cosity and gas density data which contains all of the in- tion. The gas viscosity can be estimated using graphical cor- formation required to establish the applicability of var- relations or using equations derived from these figures. ious models for gas density and gas viscosity over a wide range of pressures and temperatures. Jossi, Stiel, and Thodos Correlation: Jossi, et al.3 developed a relationship for the viscosity of pure gases and gas mixtures (cid:122) To evaluate a number of existing models for gas vis- which includes pure components such as argon, nitrogen, cosity and gas density. oxygen, carbon dioxide, sulfur dioxide, methane, ethane, pro- (cid:122) To develop new models for gas viscosity and gas den- pane, butane, and pentane. This "residual viscosity" relation- sity using our research database — these models are ship can be used to predict gas viscosity using the "reduced" proposed, validated, and presented graphically. density at a specific temperature and pressure, as well as the molecular weight. The critical properties of the gas (specifi- 2 SPE 75721 cally the critical temperature, critical pressure, and critical Y = 2.447−0.2224X ..........................................................(7) density) are also required. and, Our presumption is that the Jossi, et al. correlation (or at least ρ = Density at temperature and pressure, g/cc a similar type of formulation) can be used for the prediction of M = Molecular weight of gas mixture, lb/lb-mole w viscosity for pure hydrocarbon gases and hydrocarbon gas T = Temperature, deg R mixtures. We will note that this correlation is rarely used for µ = Gas viscosity at temperature and pressure, cp g hydrocarbon gases (because an estimate of the critical density Lee, et al.4 reported 2 percent average absolute error (low is required) — however; we will consider the formulation pressures) and 4 percent average absolute error (high pres- given by Jossi, et al. as a possible model for the correlation of sures) for hydrocarbon gases where the specific gravity is be- hydrocarbon gas viscosity behavior. low 1.0. For gases of specific gravity above 1.0 this relation is The "original" Jossi, et al. correlation proposed for gas vis- purported to be "less accurate." cosity is given by: The range of pressures used by Lee, et al.5 in the development 1 of this correlation is between 100 and 8,000 psia and the tem- ⎡(µ −µ∗)ξ+10−4⎤4 = f(ρ ) ..............................(1) perature range is between 100 and 340 deg F. This correlation ⎢⎣ g ⎥⎦ r,JST can also be used for samples which contain carbon dioxide — where: (in particular for carbon dioxide concentrations up to 3.2 mole percent). Fig. 3 shows the behavior of the Gonzalez, et al.5 f(ρr,JST) = 0.1023 + 0.023364 ρr,JST + 0.058533 ρr2,JST data (natural gas sample 3) compared to the Lee, et al.4 hydro- − 0.040758 ρ3 + 0.0093324 ρ4 carbon gas viscosity correlation. r,JST r,JST Hydrocarbon Gas Density ..............................................................................................(2) A practical issue pertinent to all density-based gas viscosity 1 T models is that an estimate of gas density must be known. Al- ξ= c6 ..................................................................(3) though there are many equation of state (EOS) correlations for 1 2 M p gas density (or more specifically, the gas z-factor) we found w2 c3 that these EOS models do not reproduce the measured gas and, densities in our database to a satisfactory accuracy. This ob- ρr,JST = ρ/ρc, JST Reduced density, dimensionless servation led us towards an effort to "tune" the existing models ρ = Density at temperature and pressure, g/cc (refs. 6-8) for the z-factor using the data in our database. ρ = Density at the critical point, g/cc c For reference, the definition of gas density for real gases is T = Critical temperature, deg K c given by: p = Critical pressure, atm c Mw = Molecular weight, lb/lb-mole ρ= 1 p Mw (ρ in g/cc)...........................................(8) µ = Gas viscosity, cp 62.37 z RT g µ* = Gas viscosity at "low" pressure, cp where: The Jossi, et al. correlation is shown in Figs. 1 and 2. Jossi, et ρ = Density at temperature and pressure, g/cc al.3 reported approximately 4 percent average absolute error p = Pressure, psia and also stated that this correlation should only be applied for M = Molecular weight, lb/lb-mole values of reduced density (ρr) below 2.0. The behavior of the z w = z-factor, dimensionless "residual" gas viscosity function is shown in Figs. 1 and 2. T = Temperature, deg R Lee, Gonzalez, and Eakin, Correlation: The Lee, et al.5 cor- R = Universal gas constant, 10.732 (psia cu ft)/ relation evolved from existing work in the estimation of (lb-mole deg R) hydrocarbon gas viscosity using temperature, gas density at a 62.37 = Conversion constant: 1 g/cc = 62.37 lbm/ft3 specific temperature and pressure, and the molecular weight of the gas. This correlation is given by: The real gas z-factor is presented as an explicit function of the µg =10−4Kexp(XρY) .....................................................(4) p"Lseauwd oorfe dCuocrerdes pporensdsiunrge Satnadte ste"m9 (pseerea tFuirges .a s4 -p6r,e wdihcetered wbye uthsee where: the data of Poettmann and Carpenter10). It is important to note that EOS models are implicit in terms of the z-factor, which K = (9.379 + 0.01607 Mw) T1.5 ..................................... (5) means that the z-factor is solved as a root of the EOS. This 209.2 + 19.26 M +T must be considered in the regression process — the regression w formulation must include the solution of the "model" z-factor X =3.448+⎡986.4⎤+0.01009 M ............................... (6) as a root of the EOS. ⎢ ⎥ w ⎣ T ⎦ SPE 75721 3 Dranchuk-Abou-Kassem,6 Nishiumi-Saito,7 and Nishiumi8 ⎡ ⎤ provide EOS representations of the real gas z-factor. In parti- z=1+⎢A − A2 − A3 − A4 − A5 ⎥ρ2 1 r cular, the Dranchuk-Abou-Kassem result is based on a Han- ⎢⎣ Tr Tr3 Tr4 Tr5⎥⎦ Starling form of the Benedict-Webb-Rubin equation of state (pEreOdSic)t iaonnd o fi sg acso dnesindseitrye.d to be the current standard for the +⎢⎡A6 − A7 − A8 − A9 − A10 ⎥⎤ρr2 ⎢⎣ Tr Tr2 Tr5 Tr24⎥⎦ z-Factor Model: Dranchuk-Abou-Kassem (ref. 6) The DAK-EOS6 is given by: ⎡ ⎤ A A A A z=1+⎢⎡A1+ A2 + A3 + A4 + A5 ⎥⎤ρr +A11⎢⎢⎣Tr7 +Tr82 +Tr95 +Tr1204⎥⎥⎦ρr5 ⎢⎣ Tr Tr3 Tr4 Tr5⎥⎦ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ +⎢A12 + A13 + A14 ⎥ρ2(1+A ρ2)exp(−A ρ2) + ⎢A6+ A7 + A8 ⎥ρr2− A9⎢A7 + A8 ⎥ρr5 ⎢⎣Tr3 Tr9 Tr18⎥⎦ r 15 r 15 r ⎢⎣ Tr Tr2⎥⎦ ⎢⎣Tr Tr2⎥⎦ ........................................................................................(11) ρ2 +A10(1+A11ρr2) r exp(−A11ρr2) ..........(9) where: 3 Tr z = z-factor, dimensionless where: Tr = Reduced temperature, dimensionless ρ = ρρ, Reduced density, dimensionless z = z-factor, dimensionless r / c ρ = Density at temperature and pressure, g/cc T = Reduced temperature, dimensionless ρr = ρρ, Reduced density, dimensionless ρc = Critical density, g/cc (zc=0.27) r / c ρ = Density at temperature and pressure, g/cc Our perspective in utilizing the NS-EOS is that this relation is ρ = Critical density, g/cc (z=0.27) purported to provide better performance in the vicinity of the c c critical isotherm — which is traditionally a region where the We must note that the definition of critical density is a matter DAK-EOS has been shown to give a weak performance. We of some debate — in particular, how is critical density esti- will compare the performance of the DAK and NS-EOS mated when this is also a property of the fluid? As such, we relations in detail once these relations are regressed using our use the "definition" of ρ as given by Dranchuk and Abou- c gas density (z-factor) database. Kassem6: p Correlation of Hydrocarbon Gas Viscosity ρc =zc r (where zc = 0.27) zT In this section we provide comparisons and optimizations of r existing correlations for hydrocarbon gas viscosity (refs. 3 and and the "original" parameters given by Dranchuk and Abou- 4), as well as a new correlation for hydrocarbon gas viscosity Kassem (ref. 6) for hydrocarbon gases are: that is implicitly defined in terms of gas density and tempera- A1 = 0.3265 A7 =-0.7361 ture. Our approach is to use an extensive database of gas vis- A2 =-1.0700 A8 = 0.1844 cosity and gas density data, derived from a variety of literature A3 =-0.5339 A9 = 0.1056 sources (refs. 11-15). This database contains 2494 points ta- A4 = 0.01569 A10 = 0.6134 ken from pure component data and 3155 points taken from A5 =-0.05165 A11 = 0.7210 mixture data. More data were available in the literature — A6 = 0.5475..................................................................(10) however, the data points chosen for this study satisfy the fol- lowing criteria: The Dranchuk-Abou-Kassem (DAK-EOS) is compared to the (cid:122) Temperature is greater than 32 deg F. data of Poettmann and Carpenter10 in Figs. 7-9. We note that (cid:122) Density measurement is available for each measure- the "original" DAK-EOS agrees quite well with the data ment of gas viscosity. (This criterion was not ap- trends, and would, in the absence of data to the contrary, seem plied for selection of points for the correlation of to be adequate for most engineering applications. However, gas viscosity at one atmosphere.) we would like to extend the range of this relation as well as (cid:122) Gas composition must be representative of a natural provide a more statistically sound correlation of the EOS (i.e., gas (e.g., data for binary mixtures containing decane add more data to the regression process). were excluded (ref. 4)). z-Factor Model: Nishiumi-Saito (ref. 7) (cid:122) Liquid or liquid-like (i.e., unusually high) viscosi- The Nishiumi-Saito model (NS-EOS)7 adds a few more terms ties were excluded from consideration. to the original Dranchuk-Abou-Kassem expression, and is given by: 4 SPE 75721 Gas Viscosity: Jossi, et al. (ref. 3) compared to 5.26 percent for the original Jossi, et al. model). For reference, Jossi, et al. reported an average absolute error In this section we test the performance of the Jossi, et al. of 4 percent when they presented their model (fitted to a model for gas viscosity against viscosity values from our variety of fluids — including non-hydrocarbon samples). It is database. We then propose a "refitted" form of the Jossi, et al. relevant to note that Jossi, et al. used a relatively small model where the coefficients of the original model were ad- database of pure component data. justed using regression to better match the viscosity values provided in our database. Fig. 10 shows the results of the We must also note that our optimization of this model re- original Jossi, et al. model when applied to our database — we quires the gas viscosity at one atmosphere (µ∗) — in this case note that there are significant departures from the 45 degree we used an independent correlation for µ∗ based on the rele- straight line (conformance to this straight line would indicate perfect agreement between the measured and calculated gas vant data from our database. This correlation for µ∗ is an inde- viscosity values). There are 2494 pure component data points pendent development and is discussed in a later section. given on this plot, where this data match has an average Gas Viscosity: Lee, et al. (ref. 4) absolute error of 5.26 percent in the prediction of gas viscos- The Lee, et al. model for gas viscosity was utilized in a similar ity. We wish to note that our database does include gas vis- manner to the Jossi, et al. model — i.e. the performance of the cosity values measured at reduced density values greater than original model was first assessed using our database, and then 2.0. We note that such points were not included in the original the coefficients of this relation were optimized using the data- study by Jossi, et al. base of gas viscosity. We note that in this work we have In an attempt to minimize the error between our data and the utilized data for both pure components and gas mixtures. Jossi, et al. model, we refitted the coefficients of the Jossi, et Fig. 12 shows the performance of the Lee, et al. model on al. model using non-linear regression techniques. The generic 4909 points from our database. This figure shows that gas vis- form of the Jossi, et al. model is written in the form: cosity is under predicted by the Lee, et al. model at the higher 1 end of the gas viscosity scale. The average absolute error ⎡(µ −µ∗)ξ+10−4⎤4 = f(ρ ) .............................(12) associated with the comparison of this model with our ⎢⎣ g ⎥⎦ r,JST viscosity database is 3.34 percent. where: The coefficients of the Lee, et al. model were then optimized using the gas viscosity database in order to improve the per- f(ρ ) = f + f ρ + f ρ2 + f ρ3 r,JST 1 2 r,JST 3 r,JST 4 r,JST formance of the model. These results are shown in Fig. 13. + f ρ4 For the optimization the Lee, et al. relation, the correlation 5 r,JST model was cast in the following form: ..........................................................................................(13) µ =10−4Kexp(XρY)..................................................(16) e g T 1 ξ= c ...............................................................(14) where: e e M 2p 3 w c The "optimized" coefficients obtained from the "refitting" the K = (k1 + k2 Mw) Tk3 ..................................................(17) Jossi, et al. model are: k4 + k5 Mw+T f1 = 1.03671E-01 f2 = 1.31243E-01 X =x +⎡x2⎤+x M ...............................................(18) f = 1.71893E-02 f = -3.12987E-02 1 ⎢ ⎥ 3 w 3 4 ⎣T ⎦ f = 8.84909E-03 5 Y = y1−y2X ................................................................(19) e = -1.21699E-01 e = 3.91956E-01 1 2 e = -1.50857E-01 The optimized coefficients for this model are: 3 ...................................................................................(15) k = 1.67175E+01 k = 4.19188E-02 1 2 k = 1.40256E+00 k = 2.12209E+02 3 4 and the variables are defined in the same fashion as the ori- k = 1.81349E+01 5 ginal correlation proposed by Jossi, et al. — where the most x = 2.12574E+00 x = 2.06371E+03 important issue is that this correlation is limited to pure com- 1 2 x = 1.19260E-02 ponent data. This means that the critical density is directly 3 tied to the component — no alternate definition is permitted. y1 = 1.09809E+00 y2 = -3.92851E-02 ....................................................................................(20) The performance of our "optimized" version of the Jossi, et al. model for gas viscosity is shown in Fig. 11. This plot shows The average absolute error for this "optimized" model is 2.29 better conformance to the 45 degree line than the original percent. Lee, et al. reported average absolute errors of 2 to 4 Jossi, et al. model shown in Fig. 10. The average absolute percent for their original model — where we recall that the error for the optimized Jossi, et al. model is 4.43 percent (as SPE 75721 5 original Lee, et al. correlations were generated using a less c = 1.01803E+00 c = 4.98986E+00 0 1 comprehensive database. c = 3.02737E-01 2 d = -9.90531E-01 d = 4.17585E+00 Gas Viscosity: Proposed "Implicit" Model 0 1 d = -6.36620E-01 2 Our correlation work based on a "non-parametric" regression e = 1.00000E+00 e = -3.19646E+00 algorithm16 shows that gas viscosity is primarily a function of 0 1 e = 3.90961E+00 the following variables: 2 f = -1.00364E+00 f = -1.81633E-01 (cid:122) Gas viscosity at 1 atm, 0 1 f = -7.79089E+00 (cid:122) Gas density, and 2 g = 9.98080E-01 g = -1.62108E+00 (cid:122) Temperature. 0 1 g = 6.34836E-04 2 We found that pressure and molecular weight could be dis- h = -1.00103E+00 h = 6.76875E-01 carded as explicit variables for this model (these are included 0 1 h = 4.62481E+00 implicitly in the gas density function). We then developed a 2 .....................................................................................(31) new gas viscosity model, which is simply a generic expansion of the Jossi, et al. model using additional temperature and den- A total of 4909 points were used in the regression calculation sity dependent terms. of these parameters (2494 pure component data and 2415 gas The relationship between the residual viscosity function (i.e., mixture data). The performance of the model is shown in Fig. µ-µ ) and the gas density appears to be univariate, as 16. We note excellent agreement with the 45 degree straight- g 1atm shown in Fig. 14. A log-log plot of the residual viscosity data line trend. The average absolute error for this model as com- pared to our database is 3.05 percent. We also note there are from our database shows significant scatter at low densities — non-hydrocarbon components such as carbon dioxide (0.19 to where this behavior reveals a strong dependence of gas viscos- 3.20 percent), nitrogen (0.04 to 15.80 percent) and helium ity on temperature at low densities (see Fig. 15). (0.03 to 0.80 percent) present in some of the gas mixtures used By observation we found that the "uncorrelated" distribution to develop these correlations. of data formed in the low-density range is directly related to Gas Viscosity: Hydrocarbon Gas Viscosity at 1 Atmosphere temperature. We propose a rational polynomial model in terms of gas density with temperature-dependent coefficients In order to utilize both new and existing correlations for gas and used nonlinear regression to fit our proposed model to viscosity, it is imperative that we estimate the viscosity of a temperature and gas density data. This model is given as: hydrocarbon gas mixture at 1 atm. We propose a new correla- tion for this purpose — where this correlation is given only as µ = µ + f(ρ) .....................................................(21) g 1atm a function of the temperature (in deg R) and the gas specific a+bρ+cρ2+dρ3 gravity (as a surrogate for molecular weight of the mixture). f(ρ)= .........................................(22) The generic form of this relation is given by: e+ fρ+gρ2+hρ3 a =a0 +a1T +a2T2 ...................................................(23) ln(µ1atm)=⎢⎢⎣⎡a10++ba11 l nln(γ(γgg))++ba22 l nln(T(T))++b3a 3ln ln(γ(γgg) l)n ln(T(T))⎥⎥⎦⎤ ........(32) b=b +b T +b T2 ....................................................(24) In this correlation we used 261 data points for the gas viscos- 0 1 2 ity at 1 atm where 135 of these are pure component data and c=c +c T +c T2.....................................................(25) 126 are gas mixture data. This new correlation gives an aver- 0 1 2 age absolute error of 1.36 percent. Fig. 17 illustrates the com- d =d +d T +d T2...................................................(26) 0 1 2 parison of the calculated gas viscosity at 1 atm and the mea- e=e +e T +e T2.....................................................(27) sured gas viscosity at 1 atm. 0 1 2 The numerical values of the parameters obtained for the new f = f0 + f1T + f2T2...................................................(28) gas viscosity model for viscosity at 1 atm model (Eq. 32) are g = g +g T +g T2..................................................(29) given by: 0 1 2 a = -6.39821E+00 a = -6.045922E-01 0 1 h=h +h T +h T2....................................................(30) a = 7.49768E-01 a = 1.261051E-01 0 1 2 2 3 The numerical values for the parameters of our proposed "im- b1 = 6.97180E-02 b2 = -1.013889E-01 b = -2.15294E-02 plicit" model for gas viscosity (Eqs. 21 to 30) are given as fol- 3 ....................................................................................(33) lows: a0 = 9.53363E-01 a1 = -1.07384E+00 Correlation of Hydrocarbon Gas Density (z-factor) a = 1.31729E-03 2 Gas Density: Dranchuk-Abou-Kassem (DAK-EOS) b0 = -9.71028E-01 b1 = 1.12077E+01 b = 9.01300E-02 2 6 SPE 75721 In this section we present the results of our regression work velop relations for estimating the pseudocritical temperature where we fitted the DAK-EOS to our gas density database. and pressure. This was a multi-step process where we first perform regress- In Fig. 22 we present the calculated versus measured values of ion of the DAK-EOS onto the "standard" and pure component z-factor for the "mixtures/pure component" calibration. We databases. The "standard" database is a tabular rendering of note that 6032 data points were used (gas mixtures and pure the Standing and Katz z-factor chart. These data are presumed component samples), and that we achieved an average abso- to accurately represent an "average" trend according to the lute error of 3.06 percent for this case. In performing this "Law of Corresponding States". regression, we simultaneously defined the new mixture rules After the "calibration" of the EOS, we can then use the mix- for the DAK-EOS (i.e., correlations of pseudocritical tempera- ture (and pure component) data to establish correlations for ture and pressure as quadratic polynomials as a function of the pseudocritical properties (we must correlate pseudocritical gas gravity). temperature and pressure because we will not be able to esti- Figs. 23 and 24 present the results of the optimized DAK-EOS mate these parameters independently — recall that we pre- for the z-factor coupled with the optimized quadratic relations sume we have only the mixture gravity of the gas, not a full used to model the pseudocritical temperature and pseudocriti- compositional analysis). cal pressure (as a function of gas specific gravity). The opti- The "first step" regression (EOS to database) is shown on Fig. mized quadratic equations for the pseudocritical temperature 18, where we note a very strong correlation. The associated and pressure of a given sample are given in terms of the gas plots for comparing the models and data for this case are specific gravity as follows: (DAK-EOS case only) shown in Figs. 19-21 — we also observe a strong correlation p =725.89−70.27γ −9.05γ2 ..............................(35) (with only minor errors) near the critical isotherm. Coeffi- pc g g cients for the DAK-EOS obtained from regression (using the T =40.39+549.47γ −94.01γ2...............................(36) Poettmann-Carpenter10 "standard" database) are: pc g g A = 3.024696E-01 A =-1.118884E+00 where, 1 7 A = -1.046964E+00 A = 3.951957E-01 p = Pseudocritical pressure, psia 2 8 pc A = -1.078916E-01 A = 9.313593E-02 T = Pseudocritical temperature, deg R 3 9 pc A4 = -7.694186E-01 A10 = 8.483081E-01 γg = Gas specific gravity (air = 1.0) A = 1.965439E-01 A = 7.880011E-01 5 11 Eqs. 35 and 36 were calibrated using the DAK-EOS (and the A = 6.527819E-01 6 coefficients for the DAK-EOS were taken from Eq. 34b). For ...................................................................................(34a) the optimized DAK-EOS based on our research database, we note that only the combination of Eqs. 9, 34b, 35, and 36 can The average absolute error associated with this case is 0.412 be used be used to estimate the z-factor for gas mixtures. percent (5960 data points). For reference, the original work by Dranchuk and Abou-Kassem (ref. 6) was based on a data- In summary, we have recalibrated the DAK-EOS against three base of 1500 points and yielded an average absolute error of databases – the Poettmann-Carpenter10 data (5960 points), an 0.486 percent. extended database which includes the Poettmann-Carpenter10 data and additional pure component data (8256 points), and a Coefficients for the DAK-EOS regression using the "combin- database of pure component and mixture data (6032 points). ed" database (Poettmann-Carpenter10 data and pure component data) are: In the first two cases we provide new coefficients to replace the original DAK-EOS (which was similarly defined by the A = 2.965749E-01 A =-1.006653E+00 1 7 original authors using pure component data). The average ab- A = -1.032952E+00 A = 3.116857E-01 2 8 solute errors for these cases were 0.486 percent and 0.821 per- A = -5.394955E-02 A = 9.506539E-02 3 9 cent, respectively. A = -7.694000E-01 A = 7.544825E-01 4 10 A = 2.183666E-01 A = 7.880000E-01 Lastly, we applied the optimized DAK-EOS based on the 5 11 A = 6.226256E-01 "combined" database (Poettmann-Carpenter10 data and pure 6 ...................................................................................(34b) component data) (i.e., the combination of Eqs. 9 and 34b) for the case of gas mixtures and developed new models for the For this case we obtained an average absolute error of 0.821 pseudocritical pressure and pseudocritical temperature as percent (8256 points). We note that this error is higher than functions of gas gravity. This model resulted in an overall error we obtained using the Poettmann-Carpenter10 "stan- average absolute error of 3.06 percent for z-factors estimated dard" database — however, this error is (certainly) still accep- using the DAK-EOS (and the quadratic polynomials for T pc table. and p . pc We now pursue the "second step" of this development by Gas Density: Nishiumi-Saito (NS-EOS) applying the optimized DAK-EOS on our mixture and pure This section follows a procedure similar to the previous work component database (for the z-factor) as a mechanism to de- which provided new forms of the DAK-EOS. The first step SPE 75721 7 was to "refit" the coefficients of the NS-EOS model using the — those near the critical isotherm, high pressure/high tem- Poettmann-Carpenter10 database. perature data, and cases of very high molecular weight. We appreciate that this issue may cause concerns — however, The results from this regression are: based on our procedures and vigilance in the regression pro- cess, we remain confident that the T and p correlations for A = 2.669857E-01 A =-2.892824E-02 pc pc 1 9 this case (i.e., NS-EOS) are both accurate and robust. A = 1.048341E+00 A =-1.684037E-02 2 10 A3 = -1.516869E+00 A11 = 2.120655E+00 Conclusions A = 4.435926E+00 A =-5.046405E-01 4 12 The following conclusions are derived from this work: A = -2.407212E+00 A = 1.802678E-01 5 13 (cid:122) The new correlations presented in this work for gas A = 6.089671E-01 A = 8.563869E-02 6 14 viscosity, z-factor, and gas viscosity at 1 atm are ap- A = 5.174665E-01 A = 4.956134E -01 7 15 propriate for applications in petroleum engineering. A = 1.296739E+00 8 (cid:122) The original Jossi, et al.3 and Lee, et al.4 correlations ...................................................................................(37a) for gas viscosity appear to yield acceptable behavior The average absolute error achieved in this regression was compared to our database, the average absolute errors 0.426 percent (5960 points), which is slightly higher than the (AAE) for these correlations are as follows: DAK-EOS result for the same case (0.412 percent). — Jossi, et al. original:3 AAE = 5.26 percent The refitting procedure was also performed on the extended — Lee, et al. original:4 AAE = 3.34 percent database of the Poettmann-Carpenter10 data and pure com- However, the "refits" of these correlations (using our ponent data (8132 points). We note that we used fewer data research database) exhibit significantly better repre- for the regression as compared to the same case for the DAK- sentations of the data: EOS (8256 points) — we found it necessary to delete certain extreme points in this regression, particularly values near the — Jossi, et al. "refit:" AAE = 4.43 percent — Lee, et al. "refit:" AAE = 2.29 percent critical isotherm. The regression coefficients for this case are: For reference, the Jossi, et al. correlation was fit us- A = 4.645095E-01 A =-1.941089E-02 1 9 ing pure component data only (2494 points) — and A = 1.627089E+00 A =-4.314707E-03 2 10 can only be applied to pure component data (this is a A = -9.830729E-01 A = 2.789035E-01 3 11 requirement of the Jossi, et al. formulation). The A = 5.954591E-01 A = 7.277907E-01 4 12 Lee, et al. correlation was fit using both pure compo- A = 6.183499E-01 A =-3.207280E-01 5 13 nent and gas mixture data (4909 points), and should A = 4.109793E-01 A = 1.756311E -01 6 14 be considered appropriate for general applications. A = 8.148481E-02 A = 7.905733E -01 7 15 A = 3.541591E-01 (cid:122) Our new "implicit" viscosity correlation (given as a 8 function of density) works well for pure gases and for ...................................................................................(37b) gas mixtures over a wide range of temperatures, pres- The average absolute error for this model was 0.733 percent, sures, and molecular weights. The average absolute which is somewhat better than the DAK-EOS result for the error for the new "implicit" viscosity correlation is same case (0.821 percent). 3.05 percent for our combined database of pure com- In a similar manner to the DAK-EOS case, we also considered ponent and natural gas mixture data (4909 total gas mixtures by developing new relations for the pseudo- points). critical pressure and temperature for use with the NS-EOS. (cid:122) Our new correlation for gas viscosity at 1 atm gave The results for this case are: an average absolute error of 1.36 percent based on p =621.81+81.09γ −56.51γ2 ..........................(38) 261 data points (135 pure component data and 126 pc g g gas mixture data). Tpc =46.91+542.86γg −93.14γg2.............................(39) (cid:122) Although carbon dioxide, nitrogen, and helium were present in some of the gas mixtures, the new gas The performance for the NS-EOS (using the coefficients from viscosity correlations match our research database Eqs. 37a and 37b) is shown in Figs. 25 to 27 (NS-EOS case very well — and, by extension, these correlations only). The results for the "mixture" case are shown Fig. 28, should work well (without correction) for practical and we note that this version of the NS-EOS has an average applications where relatively small amounts of non- absolute error of 2.55 percent (with a database of 5118 points) hydrocarbon impurities are present. a nd is uniquely defined by Eqs. 11, 37b, 38, and 39. (cid:122) The original work by Dranchuk and Abou-Kassem We note that we again used fewer data in this regression than (DAK-EOS) for the implicit correlation of the real the corresponding case for the DAK-EOS (5118 points for the gas z-factor used 1500 data points and gave an aver- NS-EOS and 6032 points for the DAK-EOS). This was age absolute error of 0.486 percent.6 Refitting the necessary due to poor regression performance of the Tpc and DAK-EOS to our research database we considered ppc parameters — and, as before, we removed extreme values two cases — the "standard" database given by Poett- 8 SPE 75721 mann and Carpenter10 (5960 points) and the "standard µ = Gas viscosity at 1 atm, cp 1atm and pure component" database (the Poettmann and µ* = Gas viscosity at low pressures used by Jossi, Carpenter data combined with the pure component et al.4, cp data) (8256 points). µ = Gas viscosity, cp g The average absolute errors (AAE) for the DAK-EOS γ = Gas specific gravity (air=1.0), dimensionless g correlations are: y = Mole fraction of the non-hydrocarbon N2, CO2, H2S —DAK-EOS "standard" AAE = 0.412 percent component (fraction) —DAK-EOS "standard/pure" AAE = 0.821 percent Subscripts We performed a similar effort with the Nishiumi and c = critical value Saito EOS7 (NS-EOS) using the same databases as pc = pseudocritical value for the DAK-EOS and obtained the following results: r = reduced variable —NS-EOS "standard" AAE = 0.426 percent pr = pseudoreduced variable —NS-EOS "standard/pure" AAE = 0.733 percent Acknowledgements (cid:122) For the case of gas mixture densities, we developed The authors wish to acknowledge the Department of Petro- quadratic formulations to represent the pseudocritical leum Engineering at Texas A&M University for the use of temperature and pressure as functions of the gas computer and reference services. specific gravity. A combined database of pure com- ponent and gas mixture data was used in this optimi- References zation. 1. Huber, M. L: Physical and Chemical Properties Division, Using the "optimized" DAK-EOS as a basis, we ob- National Institute of Standards and Technology, Gaithers- tained an average absolute error of 3.06 percent burg, MD. (6032 data points) for the gas mixture correlation. 2. Carr, N.L. Kobayashi, R., and Burrows, D.B.: "Viscosity Proceeding in a similar fashion using the "optimized" of Hydrocarbon Gases Under Pressure," Trans., AIME NS-EOS, we obtained an average absolute error of (1954) 201, 264-272. 2.55 percent (5118 data points). 3. Jossi, J.A., Stiel, L.I., and Thodos G.: "The Viscosity of Pure Substances in the Dense Gaseous and Liquid Recommendations and Future Work Phases," AIChE Journal (Mar. 1962) Vol. 8, No.1; 59-62. 1. Further work should include investigations of the ex- 4. Lee, A.L., Gonzalez, M.H., and Eakin, B.E.: "The Vis- plicit effects of non-hydrocarbon components such as cosity of Natural Gases," JPT (Aug. 1966) 997-1000; water, nitrogen, carbon dioxide, and hydrogen sulfide Trans., AIME (1966) 234. on both gas viscosity and gas density (i.e., the gas z- 5. Gonzalez, M.H., Eakin, B.E., and Lee, A.L.: "Viscosity of factor). Natural Gases," American Petroleum Institute, Mono- graph on API Research Project 65 (1970). 2. This work could be extended to consider density and 6. Dranchuk, P.M., and Abou-Kassem, J.H.: "Calculation of viscosity behavior of rich gas condensate and volatile z-Factors for Natural Gases Using Equations of State," oil fluids — however, we are skeptical that any sort Journal of Canadian Petroleum (Jul.-Sep. 1975) 14, 34- of "universal" viscosity relation can be developed. 36. Nomenclature 7. Nishiumi, H. and Saito, S.: "An Improved Generalized AAE = Absolute error, percent BWR Equation of State Applicable to Low Reduced Tem- p = Pressure, psia peratures," Journal of Chemical Engineering of Japan, p = Critical pressure, atm Vol. 8, No. 5 (1975) 356-360. c p = Pseudocritical pressure, psia 8. Nishiumi, H.: "An Improved Generalized BWR Equation pc p = Pseudoreduced pressure, dimensionless of State with Three Polar Parameters Applicable to Polar pr p = Reduced pressure, dimensionless Substances," Journal of Chemical Engineering of Japan, r M = Molecular weight, lbm/lb-mole Vol. 13, No. 3 (1980) 178-183. w T = Temperature, deg F 9. Standing, M.B., Katz, D.L.: "Density of Natural Gases," T = Critical temperature, deg K Trans., AIME (1942) 146, 140. c T = Pseudocritical temperature, deg R 10. Poettmann, H.F., and Carpenter, P.G.: "The Multiphase pc T = Pseudoreduced temperature, dimensionless Flow of Gas, Oil, and Water Through Vertical Flow pr T = Reduced temperature, dimensionless String with Application to the Design of Gas-lift Instal- r R = Universal gas constant, 10.732 (psia cu ft)/ lations," Drilling and Production Practice, (1952) 257- (lb-mole deg R) 317. z = z-factor, dimensionless 11. Lee, A.L.: "Viscosity of Light Hydrocarbons," American ρ = Density, g/cc Petroleum Institute, Monograph on API Research Project ρ = Reduced density, dimensionless 65 (1965). r SPE 75721 9 12. Diehl, J., Gondouin, M., Houpeurt, A., Neoschil, J., Thelliez, M., Verrien, J.P., and Zurawsky, R.: "Viscosity and Density of Light Paraffins, Nitrogen and Carbon Di- oxide," CREPS/Geopetrole (1970). 13. Golubev I.F., "Viscosity of Gases and Gas Mixtures, a Handbook," This paper is a translation from Russian by the NTIS (National Technical Information Service) (1959). 14. Stephan, K., and Lucas, K.: "Viscosity of Dense Fluids," The Purdue Research Foundation (1979). 15. Setzmann U., and Wagner, W.: "A New Equation of State and Tables of Thermodynamic Properties for Methane Covering the Range from the Melting Line to 625 K at Pressures up to 1000 MPa," J. Phys. Chem. Ref. Data (1991) Vol. 20, No 6; 1061-1155. 16. Xue, G., Datta-Gupta, A., Valko, P., and Blasingame, T.A.: "Optimal Transformations for Multiple Regression: Application to Permeability Estimation from Well Logs," SPEFE (June 1997), 85-93. 17. McCain, W. D., Jr.: "The Properties of Petroleum Fluids," Second Edition, Penn Well Publishing Co., Tulsa, OK (1990) 90-146. 18. Brill, J. P. and Beggs, H. D.: "Two-Phase Flow in Pipes," Figure 2 – The "residual viscosity" function versus reduced University of Tulsa. INTERCOMP Course, The Hague, density for different pure components — note the (1974). effect of temperature at low reduced densities (Jossi, et al.3). Figure 3 – Gas viscosity versus temperature for the Gonzalez, et al.5 data (natural gas sample 3) compared to the Lee, et al.4 hydrocarbon viscosity correlation. Figure 1 – The "residual" gas viscosity function versus reduc- ed density for different pure substances of similar molecular weights (Jossi, et al.3). 10 SPE 75721 Figure 4 – Real gas z-factor, as attributed to Standing and Figure 6 – Real gas z-factor, as attributed to Standing and Katz,9 plotted as a function of the pseudoreduced Katz,9 plotted as a function of the pseudoreduced pressure (data of Poettmann and Carpenter10). density function (data of Poettmann and Carpen- ter10). Figure 5 – Real gas z-factor, as attributed to Standing and Figure 7 – Real gas z-factor, as attributed to Standing and Katz,9 plotted as a function of the pseudoreduced Katz,9 plotted as a function of the pseudoreduced pressure divided by the pseudoreduced tempera- pressure (data of Poettmann and Carpenter10) ture (data of Poettmann and Carpenter10). compared to the original DAK-EOS (coefficients from Eq. 10).