Accurate Calculations of Compressibility Factor for Pure Gases and Gas Mixtures OBEIDA, Tawfic A, AGOCO, Libya and Mining University Leoben, Austria HEINEMANN, Zoltán E., Mining University Leoben , Austria KRIEBERNEGG, Michael, Mining University Leoben, Austria Paper presented at the Sth European Conference on the Mathematics of Oil Recovery, Leoben, Austria, 3-6 Sept. 1996 ABSTRACT in non-compositional simulators. An accurate method to prediet volumetrie behavior of gas INTRODUCTION mixtures, such as in the case of underground gas storage The compressibility factor is an important property for where the in-situ gas is mixed with the injected gas, is gases to calculate volume (formation volume factor) of presented in this paper. This method accurately calculates gases and the coefficient of isothermal compressibility the compressibility (Z) factor of pure hydrocarbon, non- under given conditions (pressure, temperature). hydrocarbon gases and gas mixtures. To account for the non-additive behavior of volumes of hydrocarbon and non- It is important to calculate the Z-factor more accurately, hydrocarbon gases, correction functions were developed specially for gas mixtures, in order to predict the volumetrie from correlation of data (Z-factors) generated by the Peng- gas behavior more reasonable. In compositional simulators Robinson equation of state. The correction functions are the calculations of the Z-factor are accurate, but for every function of gas composition, pressure and temperature, so condition the cubic equation of state is solved for Z-factor. the Z-factor can be calculated explicitly from gas The solution procedure involves iterations such as in composition under different reservoir conditions. Newton Raphson method. These iterations and convergence checking procedure consumes, some times, a considerable Several comparative examples are presented to compare part of CPU time for just calculating gas properties (Z- the Z-factors calculated by the correction functions with factor). The CPU time should be used more efficiently those calculated by the Peng-Robinsion equation of state and wisely in the simulator. On the other side in the (PR-EOS) and with measured data published in the non-compositional simulators, the Z-factoi values are literature. The comparison results indicated that the average tabulated for certain gas composition and pressures and a relative deviation (ARD) is 3% for gas mixtures, 2% for linear interpolation procedure is used to calculated those pure hydrocarbon, non-hydrocarbon gases and less than Z-factor values which are not listed in the table. This 1% for pure components (methane, nitrogen, carbon procedure leads to erroneous calculations of Z-factor dioxide). A stable method of calculating Z-factor for gases specially for gas mixtures where the linear interpolations from their composition is presented. This method is iteration are no longer accurate. The calculation procedure of Z- free so the CPU time is minimized. More accurate values factor using the correlation functions presented in this paper of Z-factor can be calculated which are much better than bas two advantages: Obtaining an accurate value of Z-factor those obtained by linear interpolations. and saving CPU time for other more important calculations The correction functions can be incorporated in any non- in the simulator. compositional simulator to calculate the Z-factor directly BACKGROUND without any iterative procedures, which occur in compositional simulators during the calculations of Z-factor Some impurities such as nitrogen and carbon dioxide are by the equation of state. These functions also eliminate often existed in appreciable amounts in natural gases. The the inaccurate linear interpolations of tabulated Z-values , Z-factor for non-hydrocarbon components of natural gas specially during calculations of Z-factor for gas mixtures, in certain corresponding states differ markedly from those of hydrocarbons. This makes the non-hydrocarbon and 517 2 Accurate CalcuJations of Compressibility ECMOR V, 1996 Factor for Pure Gases and Gas Mixtures hydrocarbon components not quit additive. Eilerts, Muller T = Temperature, and Carlson I studied the compressibility of natural gas p = Pressure. and nitrogen mixtures. They proposed a method to calcuJate the Z-factor for the gas mixture by introducing a correction The additive volume correction function (Fa) is a function factor into the additive foon in Eq. l. of temperature, pressure, mole fraction of the non- hydrocarbon gas in the mixture and the mole fraction of z, =c{nZn+(l-n)Zh} (1) carbon dioxide in the non-hydrocarbon gas. The function Where: actual Z-factor for gas mixture, z". = Fa. is symmetrical with respect to xm and equals to one at X Zn = Z-factor of the nitrogen in the mixture, m equals to zero (pure hydrocarbon gas) also at xm equals ~ = Z-factor of hydrocarbon gas, to one (pure non-hydrocarbon gas) as shown in Fig. l. n = mole fraction of nitrogen in the mixture Since the function Fa. is symmetrical with respect to ~ ' it was calcuJated only for 0 < xm < 0.5 as following: Where c is an arbitrary factor to account for the fact that 2 volumes of hydrocarbon and non-hydrocarbon are not quit Fav =aO+alP+~p (3) additive. The values of the factor c were presented in Where ao' ~ and ~ are parameters function of xm which charts for different mixtures at certain pressures and can be calcuJated from Eq. 4: temperatures. ao(xm) = bo + qXm } Olds, Sage and Lacey 4 studied the effects of carbon dioxide al(xm)=CO+ClXm ; (4) on the compressibility of methane. They computed the additive-volume correction factors for few methane-carbon a2(xm) = do +d1xm dioxide mixtures at different pressures and temperatures. If~ is grater than 0.5, then ~ in Eq. 4 should be substituted To obtained the full range of mixture compositions , by (l-xJ. The parameters bo, bI' co' cl' <10 and dl are interpolations were used. function of the mole fraction of carbon dioxide in the non-hydrocarbon gas (XcOO>and they can be calculated The main disadvantages of the above method are that the from Eq. 5 to Eq. 7 as following: values of the correction factor (c) were computed for a few and limited gas mixtures using pure components such as nitrogen or carbon dioxide to be mixed withthe 1.0 1 5,......~~,....~~"T'""~~""T""~~....,.~~..., hydrocarbon gas and to obtained a fuIl range of mixtures, interpolations were the only option left to be used. In this paper a more comprehensive method is proposed to eliminate the above mentioned limitations . We propose ~ . a similar method to calculate the Z-factor for any gas ~ 1.01 ················r· ··T······......···· mixture, but instead of a constant c, a function was ~ Ol introduced to repJace the constant c as shown by Eq. 2. § ..Q METHODOLOGY ~Ol ~ 1.005 . In this section we like to present the calculation methods "Cl of the Z-factor for gas mixtures, pure hydrocarbon gases < and pure non-hydrocarbon gases. Z~factor of Gas Mixtures To calculate the Z-factor for a mixture of hydrocarbon 1~--~--~--------L-~----~ o o.z 0.4 0.6 0.8 and non-hydrocarbon gases, we propose the following equation: Mole Fraction Flue gas (Xm) Zm = Fav(T.p,Xm,xcCY).)X {ZHc(1-Xm)+ ZNHcXm} Figure 1: Additive volume correction function of methane (2) and fIue gas (87%N2) at 60T Where: Fa. = Additive volume correction function, ~c = Z- factor for the non-hydrocarbon gas bO(XC02) bOl + b02XCO + bfJ3X:o + b04X: } ~c = Z-factor for the hydrocarbon gas, = , a , Mole fraction of the non-hydrocarbon ~ = bI (XC02) = bil + bl2Xco ,+ b13X:o , gas in the mixture, (5) Xco2 = Mole fraction of CO in the non- 2 Co(Xco2) =cOl +c02Xco2 +C03X';02} hydrocarbon gas, (6) Cl (Xco2)= <u +C12Xco2 +C13X';o2 518 ECMOR V, 1996 OBEIDA, T. A.; HEINEMANN, Z.E.; KRIEBERNEGG, M. 3 Z-factor of hydrocarbon gases do(Xco2) = dOl + d02Xco2 + d03Xco2 The Z-factor of any hydrocarbon gas (dry gas or rich 2 } (7) gas) can be calculated in two steps. First, the Z-factor of dl (Xco2) = dll + d12Xco2 + d13X;o2 pure methane is calculated, then the Z-factor of the richer The parameters bOl' b02, b03' bw bil' b12, b13, cOl' c02' coo' gas can be calculated second. cII' C12' cl3' dOl' d02, <103' dil. d12, and dl3 are function of Z-Factor of pure Methane temperature and they can be calculated from Eq. 8 to Eq. The compressibility factor of pure methane can be 13 as following : 2 calculated directly by the correlation in Eq. 14 as bOl = aOl + /301 T +801T following. 2 b02 = a02 + /302 T + 802T Zet = llo + alP + ~p2 (14) (8) Where: ZCI = Z-factor of pure methane. b03 = a03 +/303T +803T2 30, al and ~ are parameters function of temperature which b04 = a04 +/304T +804T2 can be calculated from Eq. 15: 2 ao(T) = bo +b1T+b2T bll = al1 + /311T} T (9) 2 b12 = a12 +/312 ~(T)=co +c1T+c2T (15) b13 = a13 + /3l3T 2 ~(T)=do +d1T+d2T 2 bo, b., ....., d2 are constants obtained from regression. COl = <TOl + AolT+80lT } 2 Z-Factor of Natural Gas C02 =<T02 +Ào2T+802T (10) If the hydrocarbon gas is not pure methane, such as natural C03 = <T03 + Ào3T + 803T2 gases (dry and rich gases), the compressibility factor can be calculated in terms of Z-factor of methane as following: )X ZCl T T2 ZHC = Fhc(p,T,KCI (16) Cll =<Tll +All +811 } 2 Where: Fhc (p,T'~I) and ZHCare the correction function C12 <T12 + A12T + 8 T (11) = 12 and the Z-factor of the hydrocarbon gas. C13 = <T13 + A13T + 8l3T2 The correction function of hydrocarbon gases is a function of pressure, temperature and methane deviation factor (KCI)' T 2 a factor depends on the composition of the natural gas, dOl = EOI +f/JOl + POlT } which is defmed by Eq.