Reviewing the Ashford and Pierce Relationship for Determining Multiphase Pressure Drops and Flow Capacities in Down-Hole Safety Valves Gerardo Lobato-Barradas Pemex Exploración y Producción Exitep 2001, Mexico City, February 4 – 7, 2001
Abstract. A reexamination of the Ashford and Pierce 1 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 14/64 th 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.
concepts can be applied now to improve the Ashford and Pierce1 relationship.
Discussion. In the derivation of the Ashford and Pierce 1 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: B o + F wo
v L =
ρ o
+
R s ρ g
+ ρ w F wo
R − Rs Psc T 1 z1 + F wo 5.615 P1 T sc R ρ g + ρ w F wo ρ o + 5 . 615
(2)
R − Rs Psc T 1 z1 + F wo 5.615 P1T sc
(3)
Bo + v Lt =
Introduction . Since the publication of the Ashford and Pierce 1 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 F wo wo, until either the well stops flowing or the well is no longer commercially productive. When a 1 large value of F wo wo is used in the Ashford and Pierce relationship, the oil flow rate calculated doesn’t show a significant change, as compared if a value of F wo wo=0 were used. Moreover, if a value of F wo wo=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
5.615
(1)
q tf = q o Bo +
PscT 1 z1 R − Rs 5 . 615 P T sc 1
R( p, T ) =
(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 doesn’t follow a decrease in the oil flow rate. In equations 1 and 2, if the water fraction, F wo wo, is changed, the variation is not reflected on the other components. In the case that F wo wo 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 + F wo wo ). Again, in the case that F wo wo 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 ( v f – v L )/v L. Instead, a formulation of gas/oil ratio is used. T he 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: v L =
1 Rs ρ gd ,sc ρ o,sc + 5.615 Bo
ρ w,sc (F wo ) (1 − F wo ) + Bw
(5)
the total fluid specific volume is: v Lt =
1 E g ρ g + E L ρ L
(6)
the total rate is:
Bo
qtf = qo
1 − F wo Bw
R − Rs PscT 1 z1 5.615 T sc P1
+
(7)
and the in situ gas/liquid ratio is:
PscT 1 z1 GLR − RsL 5 . 615 P T 1 sc
R( p, T ) =
(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 Pierce 1 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 v Lt . The new formulation estimates a smaller value for v Lt causing a reduction in the total rate calculated. This condition results in larger coefficients of discharge, than those reported by Ashford and Pierce 1. 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 F wo. After this point the slope smoothes until it reaches the value of F wo.= 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 F wo.= 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 F wo, 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 F wo 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.
1
Ashford and Pierce presented a relationship to determine the critical pressure ratio ε c, from which any reduction on the downstream pressure will not result in an oil flow rate increase. The modifications proposed here doesn’t substantially affect this equation. The only change needed is to replace the original R(p, T), using equation 8 instead.
1=
n −1 n R( p, T ) 1 − ε c n + (1 − ε c ) R( p, T ) n − 1 n
2
−1 nn+1 n 0.51 + R( p, T )ε c ε c
(9)
Conclusions 1.
2.
3.
4.
5.
6.
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. For F wo = 0, the application of both, the original and the modified relationship, resulted in very similar oil flow rates predictions. For F wo > 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 F wo axis, as described by Figure 1. The modified relationship was tested using field data, showing a better performance, when compared to the original formulation. For cases D to H, the modified relationship prediction was better that the one obtained by using the original formulation. The criteria for the calculation of εc is not changed, except for the use of equation 8, to estimate R(p, T).
Acknowledgements The author wants to thank Pemex Exploración y Producción 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., López, 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., FuentesNucamendi, 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 Bg B L Bo Bw C D E g E L F wo gc GLR n P1 P2 Psc qg q L
Orifice cross sectional area Gas volume factor Liquid volume factor Oil volume factor Water volume factor Orifice discharge coefficient Choke diameter Gas fraction Liquid fraction Water fraction Gravitational constant Gas liquid ratio Specific heat ratio Upstream pressure Downstream pressure Pressure at standard conditions = 14.7 Gas rate Liquid rate
ft2 ft3@p,T/
[email protected]. bl@p,T/
[email protected]. bl@p,T/
[email protected]. bl @p,T/
[email protected].
64th in.
(lb mft)/(sec2lbf ) ft3 /bl psia psia psia bl/d bl/d
qo qtf R R
Oil rate Total rate In situ gas/oil ratio In situ gas/liquid ratio (eqs. 8, 9) Solution gas/oil ratio Solution gas/liquid ratio Upstream temperature Temperature at standard conditions = 520 Specific volume of liquid Total specific volume Mass rate Non ideal gas factor at T1 and P1
Rs RsL T 1 T sc v L v Lt w z1
Orifice downstream to upstream pressure ratio at critical conditions Orifice downstream to upstream pressure ratio P2 /P1 Gas density @ P1 , T 1 Solution gas density at standard conditions Gas density at standard conditions Liquid density Mixture density Oil density @ P1 , T 1 Oil density at standard conditions Water density @ P1 , T 1
ε
ε c
ρ g ρ gd,sc ρ gs ρ L ρ M ρ o ρ os ρ w
bl/d bl/d ft 3 /bl ft3 /bl
2000
A-P Original A-P Modified
1500 3
) d p b ( d e t a l u c l a c o Q
ft /bl ft 3 /bl °R °R ft3 /lbm
1000
500
ft 3 /lbm lbm /s 0 0
500
1000 1500 Qo Measured (bpd)
2000
Figure 2. Oil Rates Calculated vs. Oil rates Measured lbm /ft3 lbm /ft3
Appendix A
lbm /ft3
Several references 6, 7, 8 were used to develop the proposed set of equations. The derivation of the specific volume of liquid correlation is as follows:
lbm /ft3 lbm /ft3 lbm /ft3 lbm /ft3
v L =
lbm /ft3
1
ρ L
(A1)
ρ L = ρ o (1 − F wo ) + ρ w F wo
(A2)
500
ρ o ,sc
) d p b ( e t a r l i O
400
ρ o
=
300
ρ w
=
ρ w,sc Bw
+
Rs ρ gd ,sc
5.615 Bo
(A3)
(A4)
by substituting equations A2 – A4 in A1, the liquid specific volume is obtained:
200
100
v L =
A-P Original A-P Modified
0 0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Water fraction
Figure 1. Water Fraction Influence
0.8
0.9
1.0
1 Rs ρ gd ,sc ρ o,sc + 5.615 B o
ρ w,sc (F wo ) (1 − F wo ) + Bw
The derivation of the total specific volume is:
A5)
v Lt =
1
A6
ρ M
R − Rs PscT 1 z1 5.615 T sc P1
Bo
qtf = qo
+
1 − F wo Bw
ρ M = ρ L E L + ρ g E g
A7
A20
The calculation of the in situ gas-liquid ratio is as follows. From reference 1, R(p,T) is defined as:
where: E L =
B L B L + Bg ( R − Rs )
B L = Bo (1 − F wo ) + Bw (F wo ) PscT 1 z1
Bg =
P1T sc
A8
A9
PscT 1 z1 GOR − Rs P1T sc 5.615
R( p, T ) =
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.
A11
PscT 1 z1 GLR − RsL P1T sc 5.615
R( p, T ) =
E g = 1 − E L
A22
A12 where:
ρ g
=
ρ g ,sc Bg
A13
and from equation A5
ρ L
1
=
v L
A14
Equations A8 – A14 are used to solve for equation A7. By substituting equation A7 in A6, the solution for v Lt , as presented in equation 6, is obtained.
GLR = GOR(1 − F wo )
A23
RsL = Rs (1 − F wo )
A24
Appendix B From reference 1, the following equations are presented: wv Lt = qtf
5.615
B1
86400
The derivation of the total flow rate equation is: qtf = q L + q g
A15
where
CA 2 P1 g c 144 v L
1 2
= α
B2
where:
P T z R − Rs q g = qo sc 1 1 5.615 T sc P1
A16
q L = qo Bo + q w Bw
A17
but q w = q L F wo
A18
substituting A18 in A17 and solving for q L: q L =
w
qo Bo
1 − F wo Bw
( n−1) nR( p, T ) P2 n P2 + 1 − 1 − 1 n − P P 1 1 α = −1 P2 n 1 + R( p, T ) P1
finally, by substituting A16 and A19 in A15 and rearranging the terms, the total flow rate is obtained:
2
B3
from equation A20: qtf = qo β
A19
1
B4
where:
β =
Bo
1 − F wo Bw
P T z R − Rs sc 1 1 5.615 T sc P1
+
B5
π D 4 589824
substituting equation B4 in B1 and solving for qo: qo =
wv Lt
A =
B7
B6
5.615 β 86400
f) Calculate w using equation B2. The orifice discharge coefficient, C , may be estimated using reference 2.
The algorithm of calculation is as follows: a) To start the calculations, estimate the terms ρ o, ρ g, ρ w , Bo, Bg, Bw, z1, Rs, Rsw, GLR, with the desired correlations. To avoid instabilities, if F wo = 0, then Bw=1. b) Calculate R(p,T) using equation 8.
g) Calculate the terms described by equations A8 – A14 and use equation 6 to estimate v Lt . h) Calculate β , using equation B5. i) Calculate the oil flow rate, qo, by using equation B6.
c) Using equation B3, calculate α , d) Using equation 5, calculate v L. e) Calculate the orifice area using the following equation:
Table 1 Oil Rates Computed Test
D
qom
qo A-P original
qo A-P modified
C
C
(64th in)
(bpd)
(bpd)
(bpd)
q om /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
(scf/bl)
(psi)
(psi)
(64 th in.)
γ ro
γ rg
Tth
qom
qoAP
qoCorr
(° 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
