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A Semianalytical Approach tPseudopressure CalculationsM.A. Klins SPE, Pennsylvania State UM.B. Biterge SPE, Pennsylvania State U

Summary. Real-gas pseudopressures allow rigorous analytical solutions to the nonlinear mass/momentumequations for gas flow in porous media. These solutions, however, are in terms of pseudopressure rather thanreservoir pressure. To convert pseudopressures to their complementary reservoir pressures, one of threetechniques is traditionally applied: numerical integration, table look-up, or curve-fit analysis. All three interjectsome numerical error into the m p) calculations. This paper introduces a new, exact procedure for makingthe pseudopressure/pressure conversion. It is applicable to a wide range of reservoir properties including sourgases, temperatures up to 460F [238C], and pressures to 10,000 psia [70 MPa]. Sample calculations areshown and comparison with a number of other pseudopressure estimates is made.Semianalytical Approach tPseudopressuresAl-Hussainy et al 1 introduced the notion of pseudopressure, m p), to analyze real-gas flow phenomena in porousmedia. Since that time, at least three approaches have beenpresented that convert reservoir pressures to their complementary pseudopressures: numerical integration, tablelook-up and interpolation, 1 ~ 3 and curve-fit approximation. 4Of these three, the most commonly applied approachis numerical integration by use of known gas correlationsfor viscosity and supercompressibility. 19 Its majoradvantage is that it allows the user to select the p and zcorrelations of his/her choice before numerical integration. The drawbacks are that it requires a large number

of p and z calculations to create a table of p/ p z vs. pressure, that a numerical integration procedure and accompanying p and z programs must be developed, and thatthe magnitude of error inherent in numerical integrationis unknown and depends on the number of data points usedin the integration.The semianalytical approach, then, involves the actualsolution of the m p) integral defined in Eq. l f the viscosity and compressibility terms in Eq. 1 are replaced withselected correlations, the real-gas pseudopressure equation .can be solved analytically and an exact solution forthe correlations chosen) of m p) obtained.Model Development. Real gas pseudopressure with a zerobase pressure of integration is defined as

P rP p[m p)] 0 =2 J - dp. . 1)o p zSubstitutions for p/z, dp, and p with Eqs. 5, 9, and 10can be made with a change in the variable of integrationfrom p to P r introduced.

Copyright 1987 Society of Petroleum Engineers468

By using the imperfect-gas law, we can calculate gasdensities:pM

p 2)zRTand

PcMP c= - - 3)zcRTc

where Dranchuk et at 17 assume that the critical gas compressibility factor is 0.27 andPPr= 4)Pc

Then it follows that

~ = P r P c ) C : ; ) 5)and

With the Dranchuk et al 17 correlation for z factors,

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anddz= - 5 A p ~ +2Bp r +C+2Ep r(1 + p ~ ) e -Fp,)[ 2

Tr

+ 2 E F p ~ e - F p ~ ) - 2 E F p ~ 1 + F p 2 r ) e - F P ~ ) ] dPr.............................. (8)

Substitution of Eqs. 7 and 8 into Eq. 6 gives

1 2 - F p ~ ) 2EF2 6 (-FP;)]X + Pre - Pre dp r' .... (9)The Lee et ill. 8 viscosity correlation for natural gases canbe written as

J}-t=GeH(p,p,). . ......................... (10)

When Eqs. 5, 9, and 10 are substituted into the definition of m(p) (Eq. 1), gas pseudopressure becomesp ( 2 )- p cRT) 2 rP [[m(p)] 0 = GTr M J 6 A p ~ 3 B p ~

f he exponential terms are replaced with their series expansions and integrated, the analytical solution to Eq. 11 is

p 2 ) P cRT) 2[m(p)]o = GTr

MX [ 6 A P ~ [1+7 (-H)n(Pcpr) 'U]

7 n=1 (nJ+7)n

+ 3 B p ~ [ +4 (-H)n(Pcpr)rz.J]4 n=1 (nJ+4)n

+ 2 C p ~ [ +3 (-H)n(Pcpr)rz.J]3 n=1 (nJ+3)n

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00 00 - H)n( -F)m(pcp r rz.J P r 2m+4n=1 m=1 (nJ+2m+4)n m

00 -F)n p , 2n]+4n=1 (2n+4)n

EFp 6 [ 00 (-H)n(pcPr)rz.J+ - - 2 -r- 1+6 n ~ = (nJ+6)n

00 00 -H)n( - F)m(pcp r rz.J P r 2m+6 n=1 m=1 (nJ+2m+6)n m00 -F)n pr 2n]+6

n=1 (2n+6)n

EF2 pr 8 [1+8 (-H)n(pcPr)rz.J4 n=1 (nJ+8)n

00 00 -H)n( - F)m(pcp r)lU P r 2m+8n=1 m=1 (nJ+2m+8)n m

00 ( - F) np,2n ] ]+8 12)n=1 (2n +8)n

While Eq. 12 may seem overpowering at first, it is a simple, algebraic solution for real-gas pseudopressure thatuses the Dranchuk et al. 7 and Lee et at. 8 gas propertycorrelations.Applications The semianalytical approach to estimatingreal-gas pseudopressures is well suited for personal- orminicomputer applications and has the following features.

1 Eq. 12 yields an exact solution for real-gas pseudopressure using the above correlations. No interpolationor integration is needed.2. t is a simple solution with one equation. No multiple viscosity or supercompressibility calculations are needed because they are implicit in equation development.3. The approach is valid for sweet or sour gases. Theonly required change in the input data is that the gas critical pressure and temperature be adjusted for CO 2, H 2S,and N 2 by correlations such as Wichert and Aziz s.20If Tc and Pc have been corrected for sour gases, z and} t (which is a function of z) will be adjusted automatically.214. Values of m(p) can be converted to any base pressure of integration.

5. Applications of Eq. 12 are limited to reservoir temperatures between 60 and 460F [16 and 238C], reservoir pressures from 14.7 to 10,000 psia [101 to 69 000kPa], the range of the Lee et at. viscosity correlation.469

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TABLE 1-COMPARISON OF NUMERICALINTEGRATION AND SEMIANALYTiCAL APPROACHTO PSEUDOPRESSURESInput Datap = 4,000 psi aPc = 660.96 psia

Yg =0.85T= 660 0 RTc =435.90o R

Semi analytical SolutionNumber ofSummation Terms

24681015Numerical IntegrationIntegration Subinterval(psi)

2,0001,0005004002008040201041

Number ofSegments2481020501002004001,0004,000

m 4,OOO)(psi 2 /cp)1,019.610x 10 6936.441 x 10 6933.845 x 10 6933.800 x 106933.800 x 10 6933.800 x 10 6

m 4,OOO)(psi 2 /cp)878.440 x 10 6919.864x 106930.340 x 10 6931.588x10 6933.248 X 106933.712x 10 6933.778 x 10 6933.794 x 10 6933.798 x 10 6933.800 x 10 6933.800 x 10 6

Calculation Procedure. Real-gas pseudopressures can bedetermined by the semianalytical approach as follows.l. Enter data for p, Pc T To M, and z atp and T).Tc and Pc should be adjusted for sour gases.2. With Egs. 2 through 4, calculate the gas density, p,the reduced density, P r, and the critical density, Pc.3. Calculate the constants A through J for Eg. 12 fromthe following definitions given by Dranchuk et al. andLee et al.:

A = 0.06423B = 0.5353Tr -0.6123,C = 0.315ITr -l.0467-0.5783ITr2 ,D = TnE = 0.6816ITr 2 ,F = 0.6845,

9.4 +0.02M)T1.5G= (209+ 19M+T) X 1 4H = (3.5+986IT+0.OlM) (0.OJ6018)J, andJ = l.7 -J97.2IT-0.002M.

4. Calculate the summation terms in Eg. 12. Inclusionof 10 to 15 terms in each summation will yield errors onthe order of 10 12.5. Calculate m p) by Eg. 12.470

x10 ' - ' r---T--- '--- ' - ' -- ' -- ' -- ---Y-- --- ' -- ' - ' -- ---Y-- -- ' -- '3503253005 275250

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T BLE 2 COMPARISON OF RE L G S PSEUDOPRESSURES C LCUL TED YTR DITION L METHODS ND Y THE SEMIANALYTIC L PPRO CH

Average* Maximum*Absolute AbsoluteTemperature Difference DifferenceReference Gas Type Gas Gravity OF) ( ) ( )Gas Well Testing Sweet 0.61 120 4.21 5.63Theory andPractice 2Dake 22 Sweet 0.85 200 1.70 4.18Zana and Thomas 3 Sour 0.794 290 1.33 2.33Wattenbarger and Sweet 0.70 100 2.87 2.87Ramey 23Lee et a 8 Sweet 0.6324 340 0.30 0.84 Uses tabular values of m(p) presented in reference against those estimated with the semi analytical approach.

sure for a given reservoir; however, they are each legitimate answers for the PVT data given or the correlationsused. Five cases are presented in Table 2 for comparison. In all cases, a maximum difference of less than 6%between m p) estimates reported in the literature and thosecalculated with the semianalytical approach was observed.The data span a wide range of gas reservoir propertiesincluding sweet and sour gases, gas gravities from 0.61to 0.85, reservoir temperatures from 120 to 340F [49to 171C], and pressures up to 10,000 psia [70 MPa].Sample results from both a sweet-gas and a sour-gasreservoir are plotted in Figs. 1 and 2.ConclusionsA simple, analytical approach to calculating real-gas pseudopressures has been developed with the Dranchuk etal. 17 Z factor and Lee et al. 8 gas viscosity correlations.It eliminates the need for numerical integration anddelivers an exact solution for a wide range of reservoirconditions, including sweet and sour gases, temperaturesup to 460F [238C], and reservoir pressures of 10,000psia [70 MPa]. Values of pseudo pressure calculated withthis approach compare favorably with other published estimates for a wide variety of gas gravities, reservoir pressures, and temperatures.NomenclatureA,B,C,D,E,F = Dranchuk et al. 17 Z factor constants

G,B,] = Lee et al. 8 gas viscosity constantsm,n summation indices

m p) = real-gas pseudopressure, psi2/cp[Pa2/Pa' s]

M = gas apparent molecular weight,lbmllbm-mol [kg/kmol]

p = reservoir pressure, psi [kPa]Pc = pseudocritital pressure adjusted forsour gas, psi [kPa]p r = reduced pressure pip c

R = universal gas constant, 10.73 psift 3/lbm-mol-oR [8.314X 10-3kPam 3/g mol' K]T = reservoir temperature, OR [K]

Tc pseudocritical temperature adjustedfor sour gas, OR [K]Tr = reduced temperature T/Tc

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z = gas supercompressibility factorZc = critical gas supercompressibilityfactor (for mixtures of nonpolargases Zc =:: 0.27)l g = gas gravity (air= 1)

f t = gas viscosity at T and p, cp [Pa' s]p = gas density at T and p, Ibm/ft3

[kg/m3]Pc = critical gas density at Tc and Pc,Ibm/ft 3 [kg/m3]P r = reduced gas density p/Pc

ReferencesI. AI-Hussainy, R., Ramey, H.J. Jr., and Crawford, P.B.: The Flow

of Real Gases Through Porous Media, JPT(May 1966) 624-36;Trans., AIME, 237.2. Gas Well Tesling-lheory and Praclice, Energy ResourcesConservation Board, Calgary, Alta. (1978) 2-27.3. Zana, E.T. and Thomas, G.W.: Some Effects of Contaminantson Real Gas Flow, JPT Sept. 1970) 1157-68; Trans., AIME, 249.4. Schafer-Perini, A.L. and Miska, S.Z. : Cur ve Fitting of Real Gas

Pseudopressure and Its Practical Applicat ion, paper SPE 15034presented at the 1986 SPE Permian Basin Conference, Midland,March 13-14.5. Bicher, L.B. Jr. ,'nd Katz, D.L.: Viscosity of Natural Gases,Trans., AIME (1944) 155, 246-52.6. Carr, N.L., Kobayashi, R., and Burrows, D.B.: Viscosity ofHydrocarbon Gases Under Pressure, JPT (Oct. 1954) 47-55;Trans., AIME, 201.7. Dempsey, J.R.: Com put er Routine Treats Gas Viscosity as aVariable, Oil & GasJ. (Aug. 16, 1965) 141-43.8. Lee, A.L., Gonzalez, M.H., and Eakin, B.E.: The Viscosity ofNatural Gases, JPT(Aug. 1966) 997-1000; Trans., AIME, 237.9. Standing, M.B. and Katz, D.L.: Density of Natural Gases,Trans., AIME (1942) 146, 140-49.

10. Gray, E.H. and Sims, H.L.: Z-Factor Determination in a DigitalComputer, Oil & Gas J. (July 20, 1959) 80-81.II. Sarem, A.M.: Z-factor Equation Developed for Use in Digital

Computers, Oil & Gas J. (Sept. 18, 1961) 118.12. Carlile, R.E. and Gillett, B.E.: Digi tal Solutions of an Integral,Oil & Gas J. (July 19, 1971) 68-72.13. Papay, J.: A Termelestechnol6giai Parameterek Valtozasa aGaztelepek Miivelese Soran, OGIL Musz, Tud Kilz/., Budapest(1968) 267-73.14. Hall, K.R. and Yarborough, L.: A New Equation of State forZ-Factor Calculations, Oil & Gas J. (June 18, 1973) 82-92.15. Yarborough, L. and Hall, K.R.: How to Solve Equation of Statefor Z-Factors, Oil & Gas J. (Feb. 18, 1974) 86-88.16. Two-Phase Flow in Pipes, Intercomp, Houston (1974).17. Dranchuk, P.M., Purvis, R.A., and Robinson, D.B.: ComputerCalculations of Natural Gas Compressibility Factors Using theStanding and Katz Correlation, Technical Series No. IP 74-008,Inst. of Petroleum, London (1974).

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18. Dranchuk, P.M. and Abou-Kassem, J.H.: Calculation ofZ-Factorsfor Natural Gases Using Equations of State, J Cdn. Pet. Tech.(July-Sept. 1975) 34-36.19. Robinson, D.B., Macrygeorgas, C.A., and Govier, G.W.: TheVolumetric Behavior of Natural Gases Containing Hydrogen Sulfide and Carbon Dioxide, Trans. AIME (1960) 219, 54-60.20. Wichert, E. and Aziz, K.: Calculate Z's for Sour Gases,Hydrocarbon Processing (May 1972) 119-22.21. Meehan, D.N. and Ramey, H.J. Jr.: HP-4IC Petroleum Fluids PacHewlett-Packard, Corvallis, OR (1981) 40-41.22. Dake, L.P.: Fundamentals ofReservoir Engineering. Elsevier Scientific Publishing Co., New York City (1978).23. Wattenbarger, R.A. and Ramey, H.J. Jr.: Gas Well Testing withTurbulence, Damage and Wellbore Storage, PT (Aug. 1968)877-87; Trans. AIME, 243.

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5 Metric Conversion Factorscp x 1.0* E-03O OF-32)/1.8psi x 6.894757 E OOOR R/1.8

Conversion factor is exact

Pa'sCkPaK

JPT

Original manuscript SPE 15914) received in the Society of Petroleum Engineers officeJune 23.1986 . Paper accepted for publication Jan. 27.1987. Revised manuscript r oceived Dec. 23, 1986.

Journal of Petroleum Technology, April 1987