A condensation model for calculating pressure gradients in condensate wells

Journal of Petroleum Science and Engineering, 1 (1988) 315-321

315

Elsevier Science Publishers B.V., Amsterdam - - Printed in The Netherlands

A CONDENSATION MODEL FOR CALCULATING PRESSURE GRADIENTS IN CONDENSATE WELLS DAVID SUTTON and J. LANGLINAIS Petroleum Engineering Department, Louisiana State University, Baton Rouge, LA 70803-6417 (U.S.A.) (Accepted for publication May 6, 1988)

Abstract Sutton, D. and Langlinais, J., 1988. A condensation model for calculating pressure gradients in condensate wells. J. Pet. Sci. Eng., 1: 315-321. A method is presented to calculate pressure traverses in a condensate well utilizing existing techniques for twophase flow of fluids in pipes. Existing correlations have been developed for black oil flow where gas evolves from the oil as pressure and temperature are reduced: In the case of condensate flow, a liquid evolves as pressure and temperature are reduced, thereby making the existing correlations inapplicable. The Peng-Robinsonequation of state is used to model the gas/condensate fluid and predict the gas/liquid ratio at any pressure and temperature as a traverse is calculated for the depth of the flowing well. The existing two-phase flow correlations for black oil flow are used in this work to predict gas/liquid properties such as liquid holdup, friction factors, and flow regimes. This model is then applied to several wells for which extensive data was taken, including P V T properties and pressure measurements.

Introduction The area of two-phase flow in pipes as applied to the oil and gas industry has been extensively investigated both experimentally and theoretically. Of particular interest has been the flow of combinations of oil, water, and gas in vertical pipes for application in the production phase of naturally flowing oil wells and artificially lifted oil wells such as gas lift. All these applications involve oil with gas evolving from solution as the fluid mixture moves up the vertical pipe. However, the phenomena of gas flow in vertical pipes where a liquid phase condenses from the gas as temperature and pressure are reduced has not received sufficient attention. Recent developments in analysis methods applied

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to flowing wells such as the "systems analysis" technique requires a reliable predictive method for calculating the pressure traverse in a vertical pipe. This paper presents a calculation method for predicting pressure versus depth in flowing gas condensate wells utilizing a finite element technique. In each element, the fluid properties are predicted by an equation of state model, i.e. the Peng-Robinson equation (Peng and Robinson, 1976). The energy equation for pressure losses associated with vertical flow of multiple phases was solved numerically with techniques already available from the literature. Finally, this method was used to compare prediction with actual data taken in several wells representing different flowing conditions, resulting in excellent agreement.

© 1988 Elsevier Science Publishers B.V.

316

Notation D g G f f~ p R T V V x, y~ 0 p

pipe diameter (L) gravitational constant (L t ~) velocity (L t ~) fugacity (M L " ~t :~) friction factor (dimensionless) pressure (M L - ~t 2) university gas constant (M L ~ t 2 T ~) temperature (absolute) molar volume (L a) component partial molar volume (L :~) mole fraction (dimensionless) mole fraction (dimensionless) fugacity coefficient (dimensionless) density (M L -:~) gas volume ratio (dimensionless)

Subscripts a i f g L m

actual initial, ideal final gas liquid mixture

Existing correlations Several correlations for vertical flow in pipes have been published (Brill and Beggs, 1978), describing the flow of oil and water with evolving gas. All of these numerically solve the energy balance equation, written as:

(~Z)total = (~)

fric "~- ( ~-~ ) . . . . 1

+(

Lti ....

or:

gc(~_~__) \ O.Z ,/total

- ftG2flm i_pmGd__G_ G 2D

dz

+ [~pg + (1 --~)pa]g(sinO) where: dP/dz = pressure loss in an interval, dz, due to friction, acceleration or elevation; /t = friction factor; G = velocity; D = diameter of the pipe; Pm = mixture density; pg = gas density;

pL=liquid density; ~=gas volume ratio; g = gravity constant (see also Notation ). These existing correlations fall into one of three categories. The first category, such as the P o e t t m a n n and Carpenter correlation (Brill and Beggs, 1978), ignores the concept of flow regime and assumes no slippage between the liquid and gas phases. Therefore, since the components are assumed to be traveling at the same velocity, the mixture density is calculated from the gas-liquid ratio. An empirical correlation is available tbr the two-phase friction loss factor. The second category, such as the Hagedorn and Brown correlation (Brill and Beggs, 1978) neglect flow regime but account for slippage; that is, the fact that the gas and liquid travel at different velocities in the tubing are accounted for. Now, two empirical correlations are needed, one for hold-up (slippage), and one for the twophase friction factor. The third category, such as the Duns and Ros, Orkiszewski, or Beggs and Brill correlations (Britl and Beggs, 1978 ), account for slippage as well as flow regime. Flow regimes are the flow patterns encountered in two-phase flow. The four commonly recognized regimes are bubble, slug, transition and mist. Bubble flow has liquid as the continuous phase with bubbles of gas. In slug flow, the gas bubbles coalesce to form plugs which may fill the pipe cross-sectional area. Transition is a slow change from a liquid continuous phase to a gas continuous phase with small droplets of liquid. Correlations are needed to predict the flow regimes, the slippage associated with each regime, and the two-phase friction factor under each of these conditions. Because of the simplistic approach and lack of accuracy of the first category correlations, only the second and third category correlations were used in this study. In addition to the empirical correlations for flow regime, slippage and two-phase friction factor, the fluid properties such as density, viscosity, and surface tension of each phase are needed by all category corre-

317

lations. Each correlation develops its own empirical correlations. For this study, the phase behavior and fluid properties correlations were changed to reflect a condensation phenomenon, but the slippage, two-phase friction factor, and flow regime correlations were retained. C o n d e n s a t i o n model

The first requirement in the development of a compositional model is to describe the phase behavior of the mixture. The basis of all phase behavior calculations is the vapor-liquid equilibria constant, or K-value, given as:

Ki= yi Xi

where Yi= mole fraction in the vapor phase of a component, and xi = mole fraction in the liquid phase of a component. Because of difficulties predicting the convergence pressure with Kvalues, the concept of fugacity (Bergman, 1977, 1977 ) is used to predict vapor-liquid equilibria. Fugacity relates the vaporization characteristic to the volumetric behavior of the phases. For a pure fluid, the fugacity is given by: P

In ~ - f ) = - ( R ~ ) f (Vi - Va) dp 0

where / - f u g a c i t y of a pure component, R=universal gas constant, T=absolute temperature, Va= actual molar volume of a pure component, Vi = ideal molar volume = - ( R T / p), p =pressure. For real mixtures, the component fugacity for a vapor is given by:

ln ( f V ~ = l n (~)v) \YiP /

The component K value, Ki, is given by:

K, -¢v If volumetric data were available for the mixture of interest, the data could be numerically differentiated to give the partial molar volume of each component. For complex systems like natural gases, other methods for estimating this volume are required. One method is to use an equation of state to describe how the volume of a phase is affected by changes in the composition of that phase, as well as changes in temperature and pressure. The equation of state is used to determine the partial molar volume of each component in the vapor and liquid phase. These volumes in turn predict the fugacity coefficients for each component in each phase and determines the equilibrium constants. In this work, the Peng-Robinson equation of state was used to predict the needed properties (Peng and Robinson, 1976), that is:

p_-- RT (v-b)

a(T) v(v+b)+b(v-b)

where P = absolute pressure, R = universal gas c6nstant, T = absolute temperature, v = molar volume, b = molecular repulsion parameter, and a (T) = molecular attraction parameter. For mixtures, a(T) and b can be calculated using the following mixing rules given by:

a= ~. ~. xixjAij t

3

b= ~ xib i t

where: Aij = ( 1 - Jij)

! P

where ¢v = fugacity coefficient of component i, and Vi = component partial molar volume.

xi = mole fraction of a given phase, and gii = Katz and Firoozabadi binary interaction coefficient characterizing the binary formed by components i a n d j (Katz and Firoozabadi, 1978). Now, the variables A and B can be defined as:

318

A=a(T)p/(R2T 2) B=(bp)/(RT) with the molar volume defined as:

V= (ZRT)/p we can write the Peng-Robinson equation as: Z :~_ ( I _ B ) Z ~ + ( A - 3 B 2 - 2 B ) Z

_ (AB_B2_B3)=O The largest positive root of this equation is the compressibility factor of the vapor phase and the smallest positive root is the compressibility factor of the liquid phase. Once the compressibility factors of the respective phases have been determined, the fugacity coefficient of each phase can be calculated by: ln(Oj) = ~ ( Z - 1 ) - l n

2.82843 B

(Z-B)

~

In k Z + . 4 1 4 B ] which then defines the equilibrium constant, i.e. the K-value. By a trial-and-error successive substitution technique, the phase compositions are varied until ¢ ~ = ¢ v is found for each component. The components heavier than the heptane have been characterized using a lumping scheme proposed by Behrens and Sandler (1986) whereby a semicontinuous thermodynamic description is used. Satisfactory results were obtained by describing the heavy fraction using two pseudo-components.

Fluid properties The pressure gradient equation requires values of fluid density, surface tension, velocity, and viscosity. In this study, the vapor density was obtained from the Peng-Robinson equa-

tion of state with the molar density results. The liquid density was obtained from a correlation proposed by Hankison and Thomson (1979), using the mixing rules to determine the characteristic volume, critical temperature, and acentric factor. The vapor phase viscosity was computed using the correlation by Lee (Lee et al., 1966 ) and the liquid phase viscosity was obtained from the method proposed by Lohrenz (Lohrenz et al., 1964). The latter work requires the Stiel and Thodos (1961) correlation, the Herning and Zipperer (1936) equation, and the equation developed by Jossi et al. (1962). The surface tension between the liquid and vapor phase was calculated using the correlation of Weinaug and Katz ( Katz et al., 1959 ) using parachors for the pure constituents.

Results Field data furnished by an oil company on seven Gulf Coast condensate wells was used to evaluate this technique. Data furnished included the condensate, water and gas rates, condensate and gas gravity, well depth, tubing diameter, temperature profiles (see Table I) as well as the flowing pressure measurements at several points in the well. Also, the composition of the gas stream from each well was determined from samples (see Table II). Several two-phase flow correlations (Hagedorn and Brown, Orkiszewski, and Gray - - see Brill and Beggs, 1978) were modified to calculate a pressure traverse using the condensate phase behavior models described earlier along with the two phase friction factors, holdup, and flow regimes inherent to the various correlations (see Table III ). The phase behavior properties for the gas-condensate fluid were calculated using the equation of state computer program developed by the Gas Processors Association (GPA*SIM). The computer program actually used was the GPA*SIM program as modified by the Tulsa University Fluid Flow Project (TUFFP) to include the Peng-Robin-

319 TABLE I Well data Well No.

1 2 3 4 5 6 7

Oil (stb/d)

Oil (sg.)

149 206 114 351 302 455 260

Water (stb/d)

.791 .794 .784 .799 .799 .800 .802

35.0 0.0 0.0 0.0 12.6 0.0 8.0

Gas (mscf/d)

Gas (sg.)

6047 4600 2484 15715 14420 20394 13841

Gas-oil ratio (scf/stb)

.609 .607 .607 .607 .616 .606 .618

40583 22330 21788 44810 47745 44783 53234

Well depth (ft)

Tubing dia. (in.)

Temp. (°F) top

bot. 222 205 205 219 218 219 213

11857 11766 11766 11859 11834 11764 11816

2.764 2.441 2.441 2.992 2.992 2.992 2.992

81 102 102 82 91 80 84

T A B L E II Compositional data (mole percent of components in well effluent) No.

C02

H2S

N2

C1

C2

C3

I-C4

N-C4

I-C5

N-C5

C-C6

C7-b

1 2 3 4 5 6 7

0.51 0.92 0.92 0.58 0.49 0.50 0.52

0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.53 0.30 0.30 0.53 0.54 0.54 0.54

91.88 91.16 91.16 91.63 92.40 92.16 92.51

3.22 2.93 2.93 3.23 3.03 3.13 2.97

1.18 0.69 0.69 1.18 0.94 1.09 0.92

0.40 0.50 0.50 0.40 0.27 0.34 0.26

0.39 0.53 0.53 0.39 0.25 0.33 0.24

0.28 0.23 0.23 0.29 0.15 0.24 0.15

0.20 0.19 0.19 0.22 0.11 0.16 0.12

0.83 0.34 0.34 0.87 0.25 0.64 0.24

0.58 2.20 2.20 0.67 1.56 0.85 1.53

son equation of state and the transport properties of viscosity and surface tension. A modified correlation was constructed using the PVT properties predicted by GPA*SIM with the friction factor and holdup correlations published by Gray (1978). The Cullender and Smith (1956) equation for calculating the pressure traverse of a single-phase gas was included for comparison purposes. In this equation, the gas gravity was altered to reflect the condensate as suggested by Arnondin et al. (1980). However, this method is not suggested since the gravity correction does not account for the condensate being either a liquid or gas phase nor does it consider possible water production. Also, the Hagedorn and Brown correlation was run for a "black oil" situation, i.e., assuming the flow stream to be an oil evolving gas rather than a condensation phenomenon. This was done simply for comparison purposes.

First, Well No. 4 flowed 351 BCPD and 15,715 MCFPD. However, the equation of state model predicted a gas phase over the entire length of the tubing. All of the multiphase flow correlations using the equation of state for condensation gave the same results. This is expected since they all use the same method for approximating gas flow. Also, the Cullender and Smith equation gave good agreement in this case because of the single phase present corrected for gravity. However, the advantage of the equation of state model is its ability to predict the actual phase of the flowing well stream. The equation of state model predicted no condensation in Well No. 1, but did indicate most of the water produced to be in a liquid phase in the tubing string. The multiphase flow correlations indicated a slugging regime of flow. In this case, the Hagedorn and Brown correlation and the Gray correlation predicted the

320 T A B L E III Measured and calculated pressure traverses Well No.

Depth (ft)

Measured pressure (psi)

Compositional model Gray (psi)

Hag.Br. (psi)

Orkis. (psi)

Cullender and S m i t h (psi)

Black-oil Hag.-Br. (psi)

1

118 2000 4000 5999 7999 9985 11305 11857

4616 4863 5112 5383 5622 5868 6050 6119

4621 4870 5127 5379 5626 5864 6027 6097

4621 4872 5132 5385 5634 5869 6026 6094

4622 4912 5242 5588 5950 6312 6563 6672

4619 4848 5083 5312 5536 5755 5898 5957

4621 4874 5136 5396 5650 5893 6056 6126

2

1087 2146 3087 4087 5087 7010 8898 10803 11766

2932 3032 3114 3202 3293 3465 3631 3807 3894

2945 3036 3116 3201 3285 3446 3604 3764 3845

2946 3039 3122 3209 3296 3464 3630 3801 3889

2968 3083 3185 3294 3404 3584 3743 3905 3988

2959 3063 3155 3252 3348 3530 3708 3889 3981

2970 3086 3188 3297 3405 3611 3814 4021 4126

3

2146 4146 6105 8010 9811 11766

3755 4064 4261 4442 4634 4839

3852 4051 4239 4421 4591 4775

3855 4058 4255 4445 4625 4823

3874 4094 4303 4492 4657 2837

3882 4106 4320 4524 4712 4918

3883 4113 4333 4546 4746 4964

4

152 2152 4152 6152 8152 10006 11695 11859

4027 4243 4456 4671 4888 5098 5293 5311

4028 4248 4458 4667 4876 5070 5254 5272

4028 4248 4458 4667 4876 5070 5254 5272

4028 4248 4458 4667 4876 5070 5254 5272

4029 4266 4491 4715 4939 5148 5340 5358

4031 4294 4547 4801 5054 5288 5507 5529

5

152 8118 9980 11670 11834

3247 4008 4192 4365 4381

3249 4045 4220 4386 4402

3251 4175 4375 4558 4576

3248 3987 4144 4307 4323

3249 4083 4272 4444 4461

3252 4220 4439 4644 4664

6

157 2157 4157 6157 8157 9987 11426 11764

3837 4118 4341 4573 4805 5027 5208 5256

3880 4118 4348 4577 4806 5017 5197 5241

3881 4136 4384 4631 4880 5108 5303 5350

3879 4112 4339 4565 4792 5000 5179 5222

3881 4141 4389 4636 4885 5113 5294 5336

3883 4163 4434 4706 4977 5225 5434 5484

7

157 8118 10053 11473 11816

2828 3584 3764 3912 3950

2815 3531 3698 3831 3864

2818 3675 3860 4008 4044

2815 3520 3685 3816 3849

2815 3589 3770 3905 3938

2818 3717 3926 4092 4132

321

pressures very well. It is interesting to note that the Hagedorn and Brown correlation only handles slugging flow. Well No. 6 had a flow stream indicated to be entirely mist flow with condensation occurring over the entire length of the tubing. In this case the Orkiszewski correlation and the Gray correlation gave the best results. This is due to the fact that the Hagedorn and Brown correlation does not address mist flow, but the Orkiszewski correlation does. Well No. 2 exhibited condensation throughout the entire tubing length and the correlations indicated mostly a slugging regime, but with some regions of mist flow. In this case, the Hagedorn and Brown and the Orkiszewski correlations gave excellent results. Well No. 3 is the same well as No. 2, but represented a test taken at a significantly higher flowing tubing pressure. The results are the same as for Well No. 2. Well No. 7 showed all three flow regimes (slugging, transition, and mist) at some elevation in the tubing. Because of the mist flow being present and extensive, the Orkiszewski correlation gave the best agreement. This was also true for No. 5. It is interesting to note that the Hagedorn and Brown correlation overpredicted in both cases and this is attributable to the fact that only slugging is considered, giving a larger pressure gradient. Overall, the modified Gray correlation gave the best results for the variety of well data available. However, when water is present or other situations where the flow regime is dominated by a slugging phenomenon, the Hagedorn and Brown correlation best described the traverse data. Whenever a mist flow regime was dominant, the Orkiszewski correlation was the better correlation. In the case of a gas phase only, the multiphase correlations gave identical results and agreed with the Cullender and Smith calculation.

References Arnondin, M., van Pollen, M. and Farshad, F.F., 1980. Predicting bottomhole pressure for gas and gas condensate wells. Pet. Eng. Inter., Nov. Behrens, R.A. and Sandler, S.I., 1986. The use of semicontinuous description to model the C fraction in equation of state calculations. Paper SPE 14925, presented at the SPE/DOE Fifth Symposium on Enhanced Oil Recovery held in Tulsa, Okla., April, 1986. Bergrnan, D.F., 1977. Predicting the Phase Behavior of Natural Gas in Pipelines. Ph.D. dissertation, University of Michigan, Ann Arbor. Bergman, D.F., Tek, M.R. and Katz, D.L., 1977. Retrograde Condensation in Natural Gas Pipelines. Project PR26-69 of the Pipelines Research Committee of the American Gas Association, Jan., 1977. Brill, J.P. and Beggs, H.D., 1978. Two Phase Flow in Pipes. University of Tulsa, Okla. Cullender, M.H. and Smith, R.V., 1956. Practical solution of gas flow equations for wells and pipelines with large temperature gradient. Trans. AIME, 1956. Gray, H.E., 1978. SSCSV sizing computer program. API 14B Man 14BM, pp. 38-41. Hankison, R.W. and Thomson, G.H., 1979. A new correlation for saturated densities of liquids and their mixtures. Paper presented at the AIChE Meeting, Houston, Texas, April 1979. Herning, F. and Zipperer, L., 1936. Calculation of the viscosity of technical gas mixtures from the viscosity of individual gases. Gas Wasserfach, 79 (49): 69. Jossi, J.A., Stiel, L.I. and Thodos, G., 1962. The viscosity of pure substances in the dense gaseous and liquid phases. AIChE J., 8: 59. Katz, D.L. et al., 1959. Handbook of Natural Gas Engineering. McGraw-Hill, New York, N.Y. Katz, D.L. and Firoozabadi, A., 1978. Predicting phase behavior of condensate/crude-oil systems using methane interaction coefficients. J. Pet. Technol., Nov. 1978, pp. 1649-1655. Lee, A.L. et al., 1966. The viscosity of natural gases. Trans. AIME, 997. Lohrenz, J., Bray, B.G. and Clark, C.R., 1964. Calculating viscosities of reservoir fluids from their compositions. J. Pet. Technol., Oct. 1964, pp. 1171-1176. Peng, D.Y. and Robinson, D.B., 1976. A new two-constant equation of state. Ind. Eng. Chem. Fundam., 15 ( 1 ): 5964. Stiel, L.I. and Thodos, G., 1961. The viscosity of nonpolar gases at normal pressures. AIChE J., 7: 611.