Reviewing the Ashford and Pierce Relationship for Determining Multiphase Pressure Drops and Flow Capacities in Down-Hole Safety Valves

Gerardo Lobato-Barradas Pemex Exploracin y Produccin Exitep 2001, Mexico City, February 4 7, 2001

Abstract.

A reexamination of the Ashford and Pierce1 relationship is made to properly account for the water/oil ratio of the stream. The changes include a different form for the estimation of the total specific fluid volume, the liquid specific volume, the in situ gas/oil ratio and the total fluid flow rate. The Ashford and Pierce1 data are used on the new formulation for the estimation of oil flow rates and the results are compared to the original calculations. The data of Test 2 for a 14/64th in. choke are used to evaluate the influence of the water fraction on the oil flow rate estimation. Then, both relationships are evaluated using the data of nine well tests, resulting in a better performance for the modified correlation. Finally, the original criteria for estimating the critical pressure ratio is discussed.

Introduction.

Since the publication of the Ashford and Pierce1 paper in 1975, few correlations have proved to perform better than this relationship. Although this correlation was focused on the down-hole safety valves problem, it is widely used in the oil industry to estimate rates and pressure drops through superficial chokes. When analyzing an oil well during its different stages of production life, it can be seen that in a certain time, water appears in the flow stream. The amount of water raises gradually, increasing the value of Fwo, until either the well stops flowing or the well is no longer commercially productive. When a large value of Fwo is used in the Ashford and Pierce1 relationship, the oil flow rate calculated doesnt show a significant change, as compared if a value of Fwo=0 were used. Moreover, if a value of Fwo=1 is introduced, positive value for the oil flow rate will be calculated. This situation is because of the description of the fluid specific volume and the total rate calculation, made by Ashford and Pierce. Now, since 1975, additional concepts have been developed and used in the study of multiphase flow. This additional

concepts can be applied now to improve the Ashford and Pierce1 relationship.

Discussion.

In the derivation of the Ashford and Pierce1 relationship, the following are the original equations presented to estimate the liquid specific volume, the total specific volume, the total rate and the in situ gas/oil ratio, respectively:

wow

gso

woo

L

FR

FBv

++

+=

615.5

( 1 )

wow

go

wo

sc

scso

Lt

FR

FzTT

PPRR

Bv

+

+

+

+

=

615.5

615.5 11

1 (2)

+

+= wo

sc

scsootf FTP

zTPRRBqq

1

11

615.5 (3)

( )

=

615.5,

1

11 s

sc

sc RRTP

zTPTpR (4)

From the analysis of these equations, it can be seen that for a given conditions of pressure and temperature, an increase in the water/oil ratio doesnt follow a decrease in the oil flow rate. In equations 1 and 2, if the water fraction, Fwo, is changed, the variation is not reflected on the other components. In the case that Fwo tends to a value of 1.0, the specific volume of liquid should tend to the specific volume of water. For these equations this is not the case. In equation 3, the liquid rate is described by ql = qo( Bo + Fwo ). Again, in the case that Fwo tends to a value of 1.0, the liquid rate should be equal to the water rate and it is clear that this description does not follow this behavior. Finally, in equation 4, the presence of water

is ignored. In the derivation 1 of this equation it is stated that R(p, T) should replace the gas/liquid specific volume ratio ( vf vL )/vL. Instead, a formulation of gas/oil ratio is used. The importance of having a correct estimation of R(p, T) is that the critical pressure ratio, is a function this value.

In order to solve for these problems, an alternate set of equations is presented here. The detail of the derivation of equations 5 8 is shown in Appendix A.

The liquid specific volume is:

( ) ( )wow

scw

wo

o

scgdssco

L

FB

FB

Rv

,

,

,

1615.5

1

+

+

= (5)

the total fluid specific volume is:

LLggLt EE

v +

=

1 (6)

the total rate is:

+

=

1

11

615.51 PTzTPRR

BFBqq

sc

scs

wwo

ootf (7)

and the in situ gas/liquid ratio is:

( )

=

615.5,

1

11 sL

sc

sc RGLRTP

zTPTpR (8)

The use of these equations and the algorithm for the calculation of the oil flow rate is shown in Appendix B.

Results

The reviewed relationship was used to compute oil rates using the original Ashford and Pierce1 data. The results are shown in Table 1. In general, the rates obtained are smaller than those predicted using the original formulation. Because the water fraction is zero, the difference is explained by the calculation of the total specific volume of fluid vLt. The new formulation estimates a smaller value for vLt causing a reduction in the total rate calculated. This condition results in larger coefficients of discharge, than those reported by Ashford and Pierce1. It has to be pointed out that for these combinations of pressures, gas/oil ratios, fluid densities, etc., the oil rates calculated are smaller and that for another combination of properties and conditions, the comparisons will be different.

The rates calculated for the test number 2 for a choke size of 14/64 pg are very similar for both relationships. Using this criteria, this test was chosen to show the influence of the water fraction on the oil rate calculation. The gas/oil ratio, upstream pressure and downstream pressure are 478 scf/STB, 1,205 psia and 1,015 psia, respectively. The water fraction is increased from zero to 1. The results are shown in Figure 1. The A-P original curve, shows a sharp slope at the beginning, i.e. at low values of Fwo. After this point the slope smoothes until it reaches the value of Fwo.= 1.0 and Qo=223 bpd. On the other side, the A-P modified curve, shows a smooth slope at the beginning and after this point, the slope becomes sharper until it reaches the value of Fwo.= 1.0 and Qo=0 bpd. From Figure 1 it is clear that the prediction of the A-P original curve has not a correct logic, since a well stream with a large value of Fwo, must have a reduced oil rate. Instead, the A-P modified curve shows a more logic oil rate description. One final analysis of the curves shapes shows that the curves may cross at certain points. This implies that for certain values of Fwo the oil flow rate, estimated by either relationship, may be larger than the other one.

Table 2 contains the data of 9 tests of flowing wells with water fractions ranging from 0.1 to 0.82. The data are used in both formulations to estimate the oil rates. In order to calculate the discharge coefficient for both relationships, the work of Abdul-Majeed and Aswad2 is used. Bo, Rs are calculated using the Standing4 correlations. z1 factor is estimated using references 3 and 5. Reference 9 is used to calculate Bw. The results are plotted in Figure 2. It can be observed that the modified correlation gives a better estimation of the oil rates. In Table 3, the statistical information is shown. It has to be noticed that almost all of the rates estimated by the modified formulas, are greater than those estimated using the original formulation. This condition indicates that the estimated rates are in a position similar to the point where the A-P Modified curve in Figure 1, is greater than the A-P Original curve. The cases described as Wells D to H corresponds to several tests made on one well through different conditions and times. Cases D, E and F correspond to the first year of the production life of the well. At this time, the water fraction is zero. After fifteen years of production, the water fraction has increased up to 0.38, which correspond to the cases G and H. The estimation of oil rates for cases D F, is practically the same for both relationships. The difference is evident for cases G and H, where the modified correlation gives a good agreement, while the original formulation gives a poor approximation. The importance of the analysis of cases D to H, relies on the capability for a single relationship to determine good approximations for prediction purposes.

Ashford and Pierce1 presented a relationship to determine the critical pressure ratio c, from which any reduction on the downstream pressure will not result in an oil flow rate increase. The modifications proposed here doesnt substantially affect this equation. The only change needed is to replace the original R(p, T), using equation 8 instead.

( ) ( ) ( )

( ) nn

cn

c

cn

n

TpR

n

nTpRn

TpRc

121

1

,15.0

111

,

,

1+

+

+

=

(9)

Conclusions

1. A better description of the water fraction was used in the derivation of the relationship reviewed here. It was demonstrated that a more logic estimation of the oil rate was obtained by using the modified formulation.

2. For Fwo = 0, the application of both, the original and the modified relationship, resulted in very similar oil flow rates predictions.

3. For Fwo > 0, the results obtained for one relationship will be higher or lower than the other one, depending upon the relative position of the curves along the Fwo axis, as described by Figure 1.

4. The modified relationship was tested using field data, showing a better performance, when compared to the original formulation.

5. For cases D to H, the modified relationship prediction was better that the one obtained by using the original formulation.

6. The criteria for the calculation of c is not changed, except for the use of equation 8, to estimate R(p, T).

Acknowledgements

The author wants to thank Pemex Exploracin y Produccin for having supported this study.

References

1. Ashford, F. E. and Pierce P. E., Determining Multiphase Pressure Drops and Flow capacities in Down-Hole Safety Valves, JPT, September 1975, pages 1145-1152.

2. Abdul-Majeed, G. H., Aswad, z. A., A New Approach for Estimating the Orifice Discharge Coefficient required in the Ashford-Pierce Correlation, Journal of Petroleum Science and

Engineering, 5 (1990) pages 25-33. Elsevier Science Publishers B. V., Amsterdam.

3. Standing, M. B., Katz, D. L.: Density of Natural Gases, Trans. AIME, 1942, pages 140 149.

4. Standing, M. B.: A Pressure - Volume Temperature Correlation for Mixtures of California Oil and Gases, Drilling and Production Practices, API, 1947, pages 275 286.

5. Benedict, M. et. al,: An Empirical Equation for Thermodynamic Properties of Light Hydrocarbons and Their Mixtures. J. Chem. Phys. Vol. 8, 1940.

6. Garaicochea, F, Bernal, C., Lpez, O.: Transporte de Hidrocarburos por Ductos, CIPM, 1991, pages 97-103.

7. Aziz, K., Settari, A.: Petroleum Reservoir Simulation Elsevier Applied Science Publishers, 1979, pages 9 10.

8. Xiao, J., Alhanati, F. J., Reynolds, A. C., Fuentes-Nucamendi, F.: Modeling and Analyzing Pressure Buildup Data affected by Phase Redistribution in the Wellbore. SPE 26965. III LACPEC, Buenos Aires, Argentina, 1994.

9. Dodson, C. R., Standing, M. B.: Pressure Volume Temperature and Solubility Relations for Natural Gas Water Mixtures, Drilling and Production Practices., API, 1944 pages 173, 179.

Nomenclature

A Orifice cross sectional area

ft2

Bg Gas volume factor ft3@p,T/ ft3@s.c. BL Liquid volume factor bl@p,T/ bl@s.c. Bo Oil volume factor bl@p,T/ bl@s.c. Bw Water volume factor bl@p,T/ bl@s.c. C Orifice discharge

coefficient

D Choke diameter 64th in. Eg Gas fraction EL Liquid fraction Fwo Water fraction gc Gravitational constant (lbmft)/(sec2lbf) GLR Gas liquid ratio ft3/bl n Specific heat ratio P1 Upstream pressure psia P2 Downstream pressure psia Psc Pressure at standard

conditions = 14.7 psia

qg Gas rate bl/d qL Liquid rate bl/d

qo Oil rate bl/d qtf Total rate bl/d R In situ gas/oil ratio ft3/bl R In situ gas/liquid ratio

(eqs. 8, 9) ft3/bl

Rs Solution gas/oil ratio ft3/bl RsL Solution gas/liquid ratio ft3/bl T1 Upstream temperature R Tsc Temperature at standard

conditions = 520 R

vL Specific volume of liquid

ft3/lbm

vLt Total specific volume ft3/lbm w Mass rate lbm/s z1 Non ideal gas factor at

T1 and P1

Orifice downstream to upstream pressure ratio at critical conditions

c Orifice downstream to upstream pressure ratio P2/P1

g Gas density @ P1, T1 lbm/ft3 gd,sc Solution gas density at

standard conditions lbm/ft3

gs Gas density at standard conditions

lbm/ft3

L Liquid density lbm/ft3 M Mixture density lbm/ft3 o Oil density @ P1, T1 lbm/ft3 os Oil density at standard

conditions lbm/ft3

w Water density @ P1, T1 lbm/ft3

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0Water fraction

0

100

200

300

400

500

Oil

rate

(bp

d)

A-P Original

A-P Modified

Figure 1. Water Fraction Influence

0 500 1000 1500 2000Qo Measured (bpd)

0

500

1000

1500

2000

Qo ca

lcu

late

d (bp

d)

A-P Original

A-P Modified

Figure 2. Oil Rates Calculated vs. Oil rates Measured

Appendix A

Several references6, 7, 8 were used to develop the proposed set of equations. The derivation of the specific volume of liquid correlation is as follows:

LLv

1= (A1)

( ) wowwooL FF += 1 (A2)

o

scgdssco

o B

R615.5

,

,

+= (A3)

w

scw

w B,

= (A4)

by substituting equations A2 A4 in A1, the liquid specific volume is obtained:

( ) ( )wow

scw

wo

o

scgdssco

L

FB

FB

Rv

,

,

,

1615.5

1

+

+

= A5)

The derivation of the total specific volume is:

MLtv

1= A6

ggLLM EE += A7

where:

( )sgLL

L RRBBBE

+= A8

( ) ( )wowwooL FBFBB += 1 A9

sc

scg TP

zTPB1

11= A11

Lg EE = 1 A12

g

scgg B

, = A13

and from equation A5

LL

v

1= A14

Equations A8 A14 are used to solve for equation A7. By substituting equation A7 in A6, the solution for vLt, as presented in equation 6, is obtained.

The derivation of the total flow rate equation is:

gLtf qqq += A15

where

=

1

11

615.5 PTzTPRRqq

sc

scsog A16

wwooL BqBqq += A17

but

woLw Fqq = A18

substituting A18 in A17 and solving for qL:

wwo

ooL BF

Bqq

=

1 A19

finally, by substituting A16 and A19 in A15 and rearranging the terms, the total flow rate is obtained:

+

=

1

11

615.51 PTzTPRR

BFBqq

sc

scs

wwo

ootf A20

The calculation of the in situ gas-liquid ratio is as follows. From reference 1, R(p,T) is defined as:

( )

=

615.5,

1

11 s

sc

sc RGORTP

zTPTpR A21

but the difference of GOR-Rs, only accounts for the oil, not for the total liquid. If GLR-Rsw is used instead, the total liquid is accounted for.

( )

=

615.5,

1

11 sL

sc

sc RGLRTP

zTPTpR A22

where:

( )woFGORGLR = 1 A23

( )wossL FRR = 1 A24

Appendix B

From reference 1, the following equations are presented:

86400615.5

tfLt qwv = B1

=

21

1 1442 cL

gPv

CAw

B2

where:

( ) ( )

( ) n

nn

PPTpR

PP

PP

n

TpnR

1

1

2

21

1

2

1

1

2

,1

111,

+

+

= B3

from equation A20:

otf qq = B4

where:

+

=

1

11

615.51 PTzTPRR

BFB

sc

scs

wwo

o B5

substituting equation B4 in B1 and solving for qo:

=

86400615.5Lt

o

wvq B6

The algorithm of calculation is as follows:

a) To start the calculations, estimate the terms o, g, w, Bo, Bg, Bw, z1, Rs, Rsw, GLR, with the desired correlations. To avoid instabilities, if Fwo = 0, then Bw=1.

b) Calculate R(p,T) using equation 8.

c) Using equation B3, calculate ,

d) Using equation 5, calculate vL.

e) Calculate the orifice area using the following equation:

=

5898244DA pi B7

f) Calculate w using equation B2. The orifice discharge coefficient, C, may be estimated using reference 2.

g) Calculate the terms described by equations A8 A14 and use equation 6 to estimate vLt.

h) Calculate , using equation B5.

i) Calculate the oil flow rate, qo, by using equation B6.

Table 1 Oil Rates Computed

Test D qom qo A-P original

qo A-P modified

C C

(64th in) (bpd) (bpd) (bpd) qom/qoAP qom/qoMod

1 16 559 615 573 0.9089 0.9756

2 16 484 402 374 1.2039 1.2941

1 14 261 224 208 1.1652 1.2548

2 14 427 432 411 0.9890 1.0389

3 14 409 358 332 1.1425 1.2319

4 14 382 308 286 1.2403 1.3357

5 14 596 489 461 1.2189 1.2928

1 20 232 270 251 0.8593 0.9243

2 20 345 363 336 0.9504 1.0268

3 20 551 493 456 1.1176 1.2083

Table 2 Well Test Data

Well Fwo GOR P1 P2 D ro rg Tth qom qoAP qoCorr

(scf/bl) (psi) (psi) (64th in.) ( F) (bpd) (bpd) (bpd)

A 0.79 7860 633 284 16 0.876 0.79 120 88 41 59

B 0.54 2543 1394 178 32 0.876 0.79 120 730 659 741

C 0.58 1218 875 149 16 0.876 0.79 120 258 150 201

D 0 1880 1086 995 48 0.838 0.782 171 2069 1753 1779

E 0 2021 1322 995 32 0.838 0.782 163 1598 1470 1461

F 0 1740 1749 995 14 0.838 0.782 129 465 522 540

G 0.38 595 332 66 32 0.838 0.782 81 868 474 879

H 0.38 909 420 72 24 0.838 0.782 79 570 254 460

I 0.82 14800 1450 412 32 0.810 0.704 150 258 238 245

Table 3 Statistical Data

Average percent error Absolute average percent error Standard deviation

Original formulation -16.7 20.8 24.3

Modified formulation -6.9 10.1 14.7