- Email: [email protected]
Improvement of multiphase flow rate model for chokes
Journal of Petroleum Science and Engineering 145 (2016) 321–327
Contents lists available at ScienceDirect
Journal of Petroleum Science and Engineering journal homepage: www.elsevier.com/locate/petrol
Improvement of multiphase flow rate model for chokes Juliana Mwalyepelo n, Milan Stanko Norwegian University of Science and Technology (NTNU), Norway
art ic l e i nf o
a b s t r a c t
Article history: Received 27 December 2015 Received in revised form 21 April 2016 Accepted 20 May 2016 Available online 24 May 2016
This paper evaluates three models for predicting the flow rate of a multiphase mixture through a choke that is commonly used in the oil and gas industry. The paper also presents the development and evaluation of a new and more accurate choke model based on one of the existing models. Evaluation of these models was done by using experimental data published by Schüller (2003, 2006). The three choke models selected for benchmarking were: Sachdeva et al. (1986), Perkins (1993), Al-Safran and Kelkar (2007). From the evaluation performed, the model by Sachdeva et al. (1986) was found to have a mean relative error of 1.76%, a mean absolute error of 10.52% and a standard deviation of 12.49%. The Perkins (1993) model gave a mean relative error of 24.17%, a mean absolute error of 30.74% and a standard deviation of 25.91%. The Al-Safran and Kelkar (2007) model exhibited a mean relative error of 10.10%, a mean absolute error of 17.48% and a standard deviation of 17.6%. All models both underpredict and overpredict the experimental data. Based on the evaluation results, the Sachdeva model was found to be the best model for predicting mass flow rate through restrictions. The Sachdeva model was therefore modified and improved by introducing a slippage factor. Error analysis of the modified Sachdeva model showed a mean relative error of 0.4%, a mean absolute error of 6.12% and a standard deviation of 7.66%. The modified Sachdeva model was calibrated by changing the discharge coefficient (CD) and the best value of CD for model calibration was found to be 0.65. The model improves the predictability considerably by reducing the deviation from the original Sachdeva model by half and all the predictions seem to be within the range of 10% accuracy. However, it is recommended to evaluate the new model using more experimental data. & 2016 Elsevier B.V. All rights reserved.
Keywords: Multiphase metering
1. Introduction The prediction of multiphase flow rate through restrictions using numerical models is of great importance in the oil and gas industry. Some typical applications are: multiphase metering, virtual metering, allocation to improve the accuracy of the results of the numerical models in the production system for choke design among others. At present, multiphase flow metering equipment frequently employs numerical models of restrictions as part of their workflow to complement the multiphase measurements taken. In virtual metering schemes, the choke numerical model is a fundamental component to estimate flow rates using several measured data points along the production system. Choke models for predicting mass flow rate through restrictions are categorized into two main groups, namely empirical and theoretical models. The empirical models are developed through curve fitting based on a specific range of experimental data and cannot be extended beyond their range, while theoretical models n
Corresponding author.
http://dx.doi.org/10.1016/j.petrol.2016.05.022 0920-4105/& 2016 Elsevier B.V. All rights reserved.
are developed from mass, momentum and energy balance laws of the fluid across the restriction. The resulting expressions of theoretical models will have more unknowns than equations. To solve these equations, analytical relationships, experimental correlations between variables and simplifications are introduced. Examples of theoretical models are Sachdeva et al. (1986), Perkins (1993), Al-Safran and Kelkar (2007) and the Hydro model developed by Schüller et al. (2003). Examples of empirical models are Gilbert (1954), Ashford (1974), Ros (1960), Pilehvari (1980). Even though analytical models are typically derived introducing many simplications and assumptions, they generally have a fair predictability and accuracy that are appropriate for many applications in the oil and gas industry. Additionally, some tuning constants are sometimes included in the analytical expressions to improve their predictability by calibration against measured data when available. The Sachdeva, Perkins, Al-Safran and Hydro models are all used in the oil and gas industry. The 1D Sachdeva and Perkins models poorly predict flow rates partly because they do not consider slippage between the liquid phase and the gaseous phase. Therefore, these models require frequent tuning of the discharge coefficient for better
322
J. Mwalyepelo, M. Stanko / Journal of Petroleum Science and Engineering 145 (2016) 321–327
Nomenclature Abbreviations GVF LVF MPFM MPFMs WC LHS RHS CD ANTHEI
Gas volume fraction Liquid volume fraction Multiphase flow meter Multiphase flow meters Water cut Left-hand side Right-hand side Discharge coefficient Angolan Norwegian Tanzanian Higher Education Initiative
ε2 v ρi
ε
G σ N
μ
R ai U C
Subscripts
Symbols
Ui d di g ṁ k xi n ε1 p
Pressure ratio(p2/p1) in modified Sachdeva model Density of the mixture, (kg/m3) Elevation(m) Total cross-section area, (m2) Error Mass flux, (kg/m2/s) Standard deviation Number of data points Viscosity (N s m 2). Slip ratio Constants defined in Table 4-1. Velocity of the mixture, (m/s) Constant
y ρm z A
Velocity of phase i, (m/sec) Diameter, (m) Pipe inner diameter, (m) Acceleration due to gravity(m/s2) Mass flow rate, (kg/sec) Ratio of specific heat (Cp/Cv) Phase fraction of component i Polytropic exponent of gas (-vdp/pdv) Mean relative error Pressure, (pa) Mean absolute error Specific volume, (m3/kg) Density of component i, (kg/m3)
predictions. The Al-Safran and Hydro models consider the slippage factor between the gaseous phase and the liquid phase. The Hydro model is considered to be the most accurate one so far with a mean relative error of 0.36%, a mean absolute error of 5.78% and a standard deviation of 7.76%. The only disadvantage with the Hydro model is its complexibility which makes it complicated in programming using simple analytical equations. The Schüller papers are employed in the present work because they report a large number of experimental conditions and they provide enough details to benchmark other choke models. In the present work the choke models of Sachdeva, Perkins and AlSafran were programmed and then evaluated for the experimental data of Schüller et al., (2003, 2006). The model with the best predictability was then modified to improve its performance even further. A new choke model is proposed based on the model developed by Sachdeva et al., 1986. The model was derived (as in the original work of Sachdeva) using integral conservation law balances of mass, momentum and energy in the choke. The new model includes the slippage factor when calculating the mixture density, which was not considered in the original work of Sachdeva. The detailed derivation of the model is presented in Appendix A.
1 l 2 m 2′ meas 3 o Pred p c w g
Upstream condition Liquid Choke (throat)position Mixture Position just downstream the choke Measured value Recovered condition of choke Oil Predicted value Pipe Throat/choke/critical Water Gas
suggested to be ( C = 2000*CD ) for SI units but from the model derivation the value of C should be ( C = 2*CD2 ). Another correction was done in the Perkins model where the formula for calculating pressure ratio was missing a square in one part of the equation. The model derivation from the measured value was estimated by calculating the mean percentage error ( ε1), the mean absolute percentage error ( ε2 ) and the standard deviation ( σ ) according to (Eqs. (1)–3).
⎛1 ε1 = ⎜ ⎝N
∑
⎛1 ε2 = ⎜⎜ ⎝N
∑
σ=
1 N
ṁ pre − ṁ meas ⎞ ⎟× 100 ṁ meas ⎠ ṁ pre − ṁ meas ṁ meas
⎞ ⎟⎟× 100 ⎠
⎛ ṁ pre − ṁ meas ε ⎞ − 1 ⎟ × 100 ṁ meas 100 ⎠ ⎝ N=1
(1)
(2)
N
∑⎜
(3)
2.1. Evaluation results 2. Model evaluation Three different choke models i.e. Sachdeva et al. (1986), Perkins (1993), Al-Safran and Kelkar (2007) have been evaluated in this paper. These models are evaluated using 85 data points from Schüller papers, 60 from Schüller et al. (2003) and 25 data points from Schüller et al. (2006). During evaluation of the choke model of Al-Safran and Kelkar (2007) some errors were detected and corrected, the value of a constant C in the original paper is
Evaluation of the models show that those from Sachdeva et al. (1986), Perkins (1993), Al-Safran and Kelkar (2007) underpredict and overpredict the mass flow rate with a large deviation, specifically for values greater than 0.5 kg/s.
3. New modified sachdeva model A new model was derived based on the model by Sachdeva. The
J. Mwalyepelo, M. Stanko / Journal of Petroleum Science and Engineering 145 (2016) 321–327
323
model derivations follow the same concept used by Sachdeva et al. (1986), the only difference is the introduction of a slippage factor which Sachdeva did not consider when developing their model. The model accounts for conservation of mass, momentum and energy for multiphase flow across restriction with the following assumptions. (1) (2) (3) (4) (5) (6)
The flow is one dimensional; The predominant pressure term is acceleration; The quality is constant for high speed process (frozen flow); The liquid phase is incompressible; The flow is adiabatic and frictionless; A slippage effect exists between the gas phase and the liquid phase The model derivation is explained in detail in Appendix A. Fig. 3-2. Result of the Perkins model.
3.1. Model expressions Before calculating the mass flow rate through restrictions it is necessary to determine if the flow in the restriction is critical or subcritical. During the early production phase of an oil field, when the pressure in the reservoir is high, the flow in chokes is usually critical. In these cases the pressure fluctuations downstream of the choke do not affect the pressure at the bottom of the well. However with time, the flow changes to subcritical due to the decline in reservoir pressure. Therefore, it is crucial that the boundary between the critical and subcritical flow is well defined in cases where the flow passes through restrictions. The expression for defining critical/ subcritical boundary for this model is as shown in Eq. (4).
⎧ ⎪ ⎪ y=⎨ ⎪ ⎪ ⎩
⎫0.5 ⎪ + ⎪ xg vg1 ⎬ 2 nR( 1 − xg )vl n ⎛ R( 1 − xg )vl ⎞ ⎪ + + 2⎜ x v ⎟ ⎪ xg vg 2 g g2 ⎝ ⎠ ⎭
(
k k −1
+
n 2
)
R 1 − xg vl( 1 − y)
k k −1
(4) Fig. 3-3. Results of the Al-Safran model.
3.1.1. Polytropic coefficient ( n) The polytropic coefficient of this model is the same as that used by Sachdeva et al., 1986 (Figs. 3-1, 3-2, and 3-3) as shown in Eq. (5)
n=1+
(
xg Cpg − Cvg
(
) )
xg Cvg + 1−xg Cl
(5)
3.1.2. Slip ratio (R) The slip model used is the general slip correlation suggested by Grolmes and Leung (1985) as shown in Eq. (6) and the applicable values of the constants are shown in Table 4-1. In this paper the constants proposed by Simpson have been used because the deviation with the experimental data was less than the deviation that occurred when using other constants (Table 4-2). a a +1 ⎛ 1−x ⎞( a1−1)⎛ ρ ⎞( 2 )⎛ μ ⎞ 3 g l⎟ l⎟ ⎜ ⎜ ⎜ ⎟ R = a 0⎜ ⎟ ⎜ρ ⎟ ⎜μ ⎟ ⎝ xg ⎠ ⎝ g⎠ ⎝ g⎠
(6)
Table 4-1 Values for use with the Grolmes and Leung equation (Grolmes and Leung, 1985).
Fig. 3-1. Results of the Sachdeva model.
Model
ao
a1
a2
a3
Homogenous (no slip) Constant slip Fauske Moddy Simpson Thom Baroczy Lockhart-Martenelli
1 K 1 1 1 1 1 0.28
1 1 1 1 1 1 0.74 0.64
1 1 1/2 2/3 5/6 0.89 0.65 0.36
0 0 0 0 0 0.18 0.13 0.07
324
J. Mwalyepelo, M. Stanko / Journal of Petroleum Science and Engineering 145 (2016) 321–327
Table 4-2 Table of results. Model
ε1 (%)
ε2 (%)
σ (%)
Modified Sachdeva model CD ¼0.64 Modified Sachdeva model CD ¼0.65 Modified Sachdeva model CD ¼0.66
1.94 0.40 1.13
6.31 6.12 6.17
7.55 7.66 7.78
3.1.3. Relation between upstream pressure and recovered pressure (Perry, 1950)
⎛ ⎞ p1 − p3 ⎜ ⎟ p2 = p1 − ⎜ ⎜ 1− d /d 1.85 ⎟⎟ ′ ⎝ c p ⎠
(
Fig. 4-2. Results of the modified Sachdeva model with CD ¼0.66.
)
(7) 4. Conclusions and recommendations
3.1.4. Critical/subcritical mass flow rate The final expression for predicting mass flow rate is as shown in Eq. (8).
ṁ = CDA2 0.5 ⎧ ⎛ ⎤⎫ ⎛ 1−xg ⎞⎞ ⎡ kxg ⎪ ⎪ 2 ⎜ ⎟ ⎬ ⎨ ⎢ ⎥ 2 P x R 1 x v 1 y v yv ⎜ ⎟ ρ ∙ + ∙ − − + − ( ) ⎟ g l g1 m2 1 ⎜ g g2 ⎪ ⎪ k−1 ⎝ R ⎠⎠ ⎣ ⎦⎭ ⎝ ⎩ (8)
(
)
(
)
3.2. Model calibration The discharge coefficient ( CD ) was employed as a calibration constant to improve the prediction of the new model. Three different values of CD have been tested and the value of 0.65 was found to provide the best match with experimental results. 3.2.1. Results from the new modified Sachdeva model The new modified Sachdeva model (Figs. 4-1 ,4-2) improves predictability considerably by reducing the deviation from the original Sachdeva model by half, all the predictions seem to be within the range of 10% accuracy.
4.1. Conclusions The main conclusions that can be pointed out from this work are: (1) From the three choke models analyzed, the model of Sachdeva provided the best match with the experimental data of Schüller et al. (2003, 2006); (2) A new choke model was developed based on the model of Sachdeva but included phase slip at the choke entrance and at the throat; (3) The new developed model overperforms the choke models of Sachdeva, Perkins and Al-Safran; (4) The new model is capable of predicting critical/subcritical mass flow rate with a mean error of 0.4%, a mean absolute error of 6.12% and a standard deviation of 7.66%; (5) The discharge coefficient can be used as a tuning parameter to improve the predictability of the model. The best value of discharge coefficient for the new modified Sachdeva model is 0.65 for the particular data set of Schüller et al. (2003, 2006). 4.2. Recommendations The work performed in this study used only 85 experimental data points from the Schüller papers, it is therefore recommended that more data should be added for model evaluation. It is also recommended that other experimental data from other researchers who have done experiments on multiphase flow should be used to evaluate the model in order to benchmark its accuracy further. An in-depth study of the discharge coefficient should be done to see if the same discharge coefficient can be used in the orifice and cage type choke geometry). Acknowledgements This work is supported by the Norwegian University of Science and Technology under ANTHEI programme which is sponsored by Statoil-Tanzania.
Appendix A Mathematical derivation of the new choke model
Fig. 4-1. Results of the modified Sachdeva model with CD ¼0.65.
Assumptions In deriving this model the following assumptions have been
J. Mwalyepelo, M. Stanko / Journal of Petroleum Science and Engineering 145 (2016) 321–327
used: (1) Flow is one dimensional; (2) The predominant pressure term is acceleration; (3) The quality is constant for high speed process; (4) The liquid phase is incompressible; (5) The flow is adiabatic and frictionless; (6) A slippage effect exists between the gas phase and the liquid phase. Concepts used in deriving this model are the same concepts used by Sachdeva et al., 1986 where the equations of the conservation of mass, momentum and energy were used to determine the relationship between critical and subcritical flow. The difference arose from equation A 16 when defining mixture density by introducing slippage factor. If the slip factor is not included the results would be the same as the results of the model developed by Sachdeva et al. in 1986.
(
)
(A1)
)
ṁ l2 = 1 − x g2 ṁ
( (
)
During critical flow condition equation A 13 holds and equation A 11 reduces to,
G2
dU2 =−1 dp2
G 22
dvm2 =−1 dp2
Defining momentum mixture density from Schüller, et al., 2003,
(
⎡ ⎤ −A2dp2 = d⎣ G 2A2 U2 1 − x g2 + U2x g2 ⎦
(A7)
−A2dp2 = d⎡⎣ G 2A2 U2⎤⎦
(A8)
)
(A9)
Simplifying equation A 8 we get,
(
(A10)
⎡ dx dx g2 ⎤ ⎢ g2 − ⎥ ⎢⎣ dp2 Rdp2 ⎥⎦
)
(
)
(A18)
Using the assumption of frozen flow, the gas quality remains constant therefore,
(A19)
Please note that the assumption of frozen flow might not hold for small throat areas. Equation A 18 is therefore reduced to,
⎤ ⎡ x gdvg2 dvm2 ⎡ dvl ⎤ 1 ⎥ = ⎢ xg + + R 1 − xg . 1 − x g ⎥∙⎢ ⎣ ⎦ ⎣⎢ dp dp2 R dp2 ⎥⎦ 2
(
)
(
)
(A20)
Considering the assumption of incompressible liquid phase dvl =0, dp2
equation A 20 is reduced to,
⎤ ⎡ x gdvg2 ⎤ dvm2 ⎡ 1 ⎥ = ⎢ xg + 1 − x g ⎥∙⎢ ⎣ ⎦ ⎣⎢ dp ⎥⎦ dp2 R 2
)
(A21)
Substituting equation A 21 into equation A 15.
⎡ ⎤ ⎡ x gdvg2 ⎤ 1 ⎥=−1 G 22. ⎢ x g + 1 − x g ⎥∙⎢ ⎣ ⎦ ⎢⎣ dp ⎥⎦ R 2
(
G 22 =
ṁ g2 + ṁ l2 A2
(A17)
)
(A22)
(A11)
And G2 at the throat can be defined as
G2 =
)
⎡ ⎤ 1 + ⎢ x g2 + 1 − x g2 ⎥ ⎣ ⎦ R ⎡ x dv v dx dx g2 ⎤ dvl ⎢ g2 g2 + g2 g2 + R 1 − x g2 . ⎥ − Rvl. ⎢⎣ dp2 dp2 dp2 dp2 ⎥⎦
(
Expanding the right-hand side of expression A 10 and then dividing both sides by the pressure differential dp2,
dU2 dG + U2 2 dp2 dp2
)
x g1 = x g2, dx g = 0
From equation A 5 velocity of the moving fluid at the choke can be expressed as,
−dp2 = d⎡⎣ G 2U2⎤⎦
(
(A3)
(A6)
−1 = G 2
)
dvm2 ⎡ = ⎣ x g2vg2 + R 1 − x g2 vl⎤⎦. dp2
⎤ ⎡ ⎛ U2 1 − x g2 ṁ U2x g2ṁ ⎞⎟⎥ −A2dp2 = d⎢ G 2⎜⎜ + ⎟ ⎢ G2 G 2 ⎠⎥⎦ ⎣ ⎝
U2 = G 2/ρm2 = G 2vm2
(A16)
⎡ ⎤ 1 vm2 = ⎡⎣ x g2vg2 + R 1 − x g2 vl⎤⎦⎢ x g2 + 1 − x g2 ⎥ ⎣ ⎦ R
(A4)
)
)
Differentiating equation A 17 with respect to downstream pressure p2
(A5)
)
(A15)
(
)
G 2 = ṁ /A2
( (
(A14)
Substituting equation A 9 into equation A 14
Dividing the right-hand side of equation A 4 by G2 and multiplying by G2 where,
(
(A13)
(A2)
And
ṁ g2 = x g2ṁ −A2dp2 = d U2 1 − x g2 ṁ + U2x g2ṁ
dG 2 =0 dp2
(
However we know,
(
to throat pressure p2 at the boundary can be defined as,
⎡x 1 − x g ⎤⎡ ⎤ 1 g ⎥⎢ x g + 1 1 − x g ⎥ =⎢ +R ⎣ ⎦ R ρm ⎣⎢ ρg ρl ⎥⎦
Momentum equation at the throat
−A2dp2 = d U2ṁ l2 + U2ṁ g2
325
(A12)
For the fixed set of upstream conditions, during critical flow the mass flux reaches a maximum value with respect to downstream (throat) pressure. Therefore, the change of mass flux with respect
⎡x + ⎣ g
−dp2
1 R
( 1 − x g )⎤⎦. x gdvg2
(A23)
Considering polytropic expansion of gas,
p2 vgn2 = C Differentiating equation A 24 with respect to vg2
(A24)
326
J. Mwalyepelo, M. Stanko / Journal of Petroleum Science and Engineering 145 (2016) 321–327
np2vgn2− 1 + vgn2
dp2
dp2
(A25)
+
vgn2
(A26)
vg2
(A27)
U22 =
Substituting equation A 27 into equation A 23,
np2 1 ⎡ vg2x g ⎣ x g + R 1 − x g ⎤⎦
G 22 =
(
(A28)
p U2 U2 + gz1 + 1 = 2 + gz2 + 2 ρm2 2 2
ρm1
tegrating from p1to p2 the term
∫
∫
∫
∫
dp ρm
can be deduced as,
⎛ p ⎞1/ k ⎜ ⎟ ⎝C⎠
dp 1/ k
( p/ C )
= C1/ k
(A31)
∫
+
(
+
) ⎤⎥⎡ R ⎢ ⎥⎦⎣
p2 p1
x gk
⎥⎦
( 1 − x g )vl( 1 − y) (A41)
+
x gk k−1
) ⎤⎥R ⎥⎦
( 1 − x g )vl( 1 − y)
⎫0.5 ⎪ ⎬ ⎪ ⎭
( vg1 − yvg2)
(A42)
⎡ 1 − xg p1⎢ x g + ⎢⎣ R
) ⎤⎥⎡ R ⎢ ⎥⎦⎣
( 1 − x g )vl( 1 − y)
⎤ np2 k x gvg1 − x gyvg2 ⎥ = 1 ⎡ ⎦ k−1 vg2x g ⎣ x g + R 1 − x g ⎤⎦. 2ρm2 2 (A43)
(
)
(
)
Substituting the mixture density into RHS equation A 43
(
(
)
(
2
))
(
)
(A44)
(A35)
⎤ U2 k x gvg1p1 − x gvg2p2 ⎥ = 2 ⎦ k−1 2
)
=
2 np2 ⎡⎣ x g2vg2 + R 1 − x g2 vl⎤⎦ . ⎡⎣ x g2 +
(
)
1 R
( 1 − x g2)⎤⎦
2. x gvg2
(A45)
Combining equation A 45 as the RHS of the equation and the LHS of equation A 43,
⎡ 1 − xg p1⎢ x g + ⎢⎣ R
(
(A36)
= , p2 = yp1
) ⎤⎥R
( vg1 − yvg2)
(
Defining pressure ratio (y ),
y=
(A40)
1 np2 ⎡⎣ x g2vg2 + R 1 − x g2 vl⎤⎦⎡⎣ x g2 + R 1 − x g2 ⎤⎦ = 1 2. vg2x g ⎡⎣ x g + R 1 − x g ⎤⎦
( 1 − x g )vl( p1 − p2)
(
)
⎧ ⎡ 1 − xg ⎪ G 2 = ⎨ 2ρm2 2 p ⎢ x g + 1⎢ ⎪ R ⎣ ⎩
(A34)
Following the assumption of adiabatic flow, incompressible liquid phase and defining the mixture density equation A 35 can be written as,
⎡ 1 − xg ⎢x + ⎢⎣ g R
( 1 − x g )vl( 1 − y)
(
k−1
+
⎛ p p ⎞ U2 ⎜⎜ 1 − 2 ⎟⎟ = 2 1 ⎝ ρm1 2 ρm2 ⎠
⎢ ⎥⎦⎣
(A39)
⎤ G 22 k x gvg1 − x gyvg2 ⎥ = ⎦ 2ρ 2 k−1 m2
(
(A33)
k p k − 1 ρm
) ⎤⎥⎡ R
)
Calculating the critical/ subcritical boundary. Eliminating G2 by substituting equation A 28 into A - 40
Therefore, equation A 30 can be written as
k k−
(
Therefore, the critical/subcritical mass flow rate formula can be written as,
Then,
m
ρm2 2
(
(A32)
⎡ p ( k − 1) / k ⎤ dp ⎥ = k C1/ kp ( k − 1) / k = C1/ k⎢ ⎢⎣ ( k − 1)/k ⎥⎦ ρm k−1
= ∫ dp ρ
(A38)
2 ⎛⎡ x 1 − x g ⎤⎡ ⎤⎞ g ⎥⎢ x g + 1 1 − x g ⎥⎟ = G 22⎜⎜ ⎢ +R ⎦⎟ R ρl ⎥⎦⎣ ⎠ ⎝ ⎣⎢ ρg
⎡ 1 − xg G 22 = 2ρm2 2 p ⎢ x g + 1⎢ R ⎣
p−1/ k dp
⎛ ⎞1/ k k ⎜ p⎟ = p ( k − 1) / k k − 1 ⎜⎝ ρ k ⎟⎠ m
)
Simplifying the equation above:
(A30)
Assuming that the flow is adiabatic and U22> > >U12, then in-
dp = ρm
(
⎡ 1 − xg p1⎢ x g + ⎢⎣ R
(A29)
⎛ U2 ⎞ dp ⎟ = d⎜ ρm ⎝ 2 ⎠
( 1 − x g )vl( p1 − yp1)
⎤ U2 k x gvg1p1 − x gvg2yp1 ⎥ = 2 ⎦ k−1 2
G 22
+
With negligible elevation changes gz1 = gz2 equation A 29 reduces to,
p2 vgk1 = C , ρ =
⎢ ⎥⎦⎣
(
)
Now considering the general Bernoulli's equation,
p1
) ⎤⎥⎡ R
From equation A 9 velocity (U2) can be defined as:
−np2
=
dvg2
⎡ 1 − xg ⎢x + ⎢⎣ g R
(
=0
−np2vgn2− 1
=
dvg2
dp2 dvg2
(A37)
) ⎤⎥⎡⎢ R ⎥⎦⎣
kx
⎤
( 1 − x g )vl( 1 − y) + k −g1 ( vg1 − yvg2 )⎥⎦
2 nyp1⎡⎣ x g2vg2 + R 1 − x g2 vl⎤⎦ . ⎡⎣ x g2 +
(
)
2. x gvg2
Simplifying equation A 46
1 R
( 1 − x g2)⎤⎦ (A46)
J. Mwalyepelo, M. Stanko / Journal of Petroleum Science and Engineering 145 (2016) 321–327
(
)
R 1 − x g vl( 1 − y ) +
=
kx gy. y−1/ k vg1 k−1
+
Simplify equation A 52 then the critical/subcritical boundary formula can be written as
kx gvg1 k−1 2 ny⎡⎣ x g2vg2 + R 1 − x g2 vl⎤⎦
(
)
2. x gvg2
(A47)
Considering the LHS of equation A 47:
⎡ R 1 − x g vl( 1 − y ) ⎤ k ⎥ + x gvg1⎢ ⎢⎣ k − 1 ⎥⎦ x gvg1
(
327
)
⎧ ⎪ ⎪ y=⎨ ⎪ ⎪ ⎩
k
⎫ k−1 ⎪ + ⎪ x g vg1 ⎬ 2 ⎛ ⎞ ⎪ nR( 1 − x g )v l n R( 1 − x g )v l + + 2⎜ x v ⎟ ⎪ x g vg 2 g g 2 ⎝ ⎠ ⎭
k k−1 k k−1
+
n 2
(
)
R 1 − x g vl ( 1 − y)
(A53)
(A48)
Now considering the RHS of equation A 47:
kx gy ( k − 1) / k . vg1 k−1
kx gvg1y ( k − 1) / k k−1 +
+
+
2 ny⎡⎣ x g2vg2 + R 1 − x g2 vl⎤⎦
(
References
(
(A49)
2 −2/ k
)
n x gvg1 y 2 x gvg1y−1/ k
nR 1 − x g vlx gvg1. y ( k − 1) / k
(
)
2. x gvg2
)
x gvg2
+
ny R 1 − x g vl 2x gvg2
( (
2
) )
(A50)
⎡ nR 1 − x g vly ( k − 1) / k ⎢ ky ( k − 1) / k ny ( k − 1) / k x gvg1⎢ + + k−1 x gvg2 2 ⎢⎣
(
)
⎤ ⎛ ⎞2 n ⎜ R 1 − x g vl ⎟ ( k − 1) / k ⎥ + ⎜ ⎥ ⎟y 2⎝ x gvg2 ⎠ ⎥⎦
(
)
(A51)
Combining equation A 48 as the LHS of the equation and equation A 51 as the RHS of the equation:
⎡ ⎡ R 1 − x g vl( 1 − y ) ⎤ ⎢ k ⎢ ⎥ x gvg1 + = x vg1⎢ ⎢⎣ k − 1 ⎥⎦ x gvg1 ⎢⎣ g
(
)
nR 1 − x g vly ( k − 1) / k ky ( k − 1) / k ny ( k − 1) / k + + k−1 2 x gvg2
(
2⎤ ⎛ ⎞ n ⎜ R 1 − x g vl ⎟ ( k − 1) / k ⎥ + ⎜ ⎥ ⎟y 2⎝ x gvg2 ⎠ ⎥⎦
(
)
)
(A52)
Ashford, F.E., 1974. An Evaluation of critical Multiphase flow performance through wellhead chokes. Al-Safran, E.M., Kelkar, M., 2007. Prediction of two phase critical flow boundary and mass flow rate across chokes. Soc. Pet. Eng., Anaheim Gilbert, W., 1954. Flowing and Gas Gas Lift Well Performance Drilling Production Practise. Grolmes, A.M., Leung, C., 1985. J. Chem. Eng. Prog. 8 (p. 81). Perkins, T.K., 1993. Critical and subcritical flow of multiphase mixtures through chokes. Soc. Pet. Eng. Pilehvari, A.A., 1980. Eperimental study of critical two phase flow through well head chokes. s.l.: University of Tulsa. Perry John, H., 1950. Chemical Engineers Handbook. Mcgraw-Hill Book Co. Inc., NewYork, p. 404. Ros, J.N.C., 1960. An Analysis of Critical Simultaneous Gas/Liquid Flow Through a Restriction and its Application to Flow Metering, Netherlands, vol. 9. Sachdeva, R., et al., 1986. Two phase flow through chokes. Soc. Pet. Eng. Schüller, R.B., et al. 2006. Critical and Subcritical Oil/Gas/Water Mass Flow Rate Experiments and Predictions for Chokes. Schüller, R.B., Solbakken, T, Selmer-Oslen, S., 2003. Evaluation of multiphase flow rate models for chokes under subcritical oil/gas/flow conditions. Soc. Pet. Eng.
Contents lists available at ScienceDirect
Journal of Petroleum Science and Engineering journal homepage: www.elsevier.com/locate/petrol
Improvement of multiphase flow rate model for chokes Juliana Mwalyepelo n, Milan Stanko Norwegian University of Science and Technology (NTNU), Norway
art ic l e i nf o
a b s t r a c t
Article history: Received 27 December 2015 Received in revised form 21 April 2016 Accepted 20 May 2016 Available online 24 May 2016
This paper evaluates three models for predicting the flow rate of a multiphase mixture through a choke that is commonly used in the oil and gas industry. The paper also presents the development and evaluation of a new and more accurate choke model based on one of the existing models. Evaluation of these models was done by using experimental data published by Schüller (2003, 2006). The three choke models selected for benchmarking were: Sachdeva et al. (1986), Perkins (1993), Al-Safran and Kelkar (2007). From the evaluation performed, the model by Sachdeva et al. (1986) was found to have a mean relative error of 1.76%, a mean absolute error of 10.52% and a standard deviation of 12.49%. The Perkins (1993) model gave a mean relative error of 24.17%, a mean absolute error of 30.74% and a standard deviation of 25.91%. The Al-Safran and Kelkar (2007) model exhibited a mean relative error of 10.10%, a mean absolute error of 17.48% and a standard deviation of 17.6%. All models both underpredict and overpredict the experimental data. Based on the evaluation results, the Sachdeva model was found to be the best model for predicting mass flow rate through restrictions. The Sachdeva model was therefore modified and improved by introducing a slippage factor. Error analysis of the modified Sachdeva model showed a mean relative error of 0.4%, a mean absolute error of 6.12% and a standard deviation of 7.66%. The modified Sachdeva model was calibrated by changing the discharge coefficient (CD) and the best value of CD for model calibration was found to be 0.65. The model improves the predictability considerably by reducing the deviation from the original Sachdeva model by half and all the predictions seem to be within the range of 10% accuracy. However, it is recommended to evaluate the new model using more experimental data. & 2016 Elsevier B.V. All rights reserved.
Keywords: Multiphase metering
1. Introduction The prediction of multiphase flow rate through restrictions using numerical models is of great importance in the oil and gas industry. Some typical applications are: multiphase metering, virtual metering, allocation to improve the accuracy of the results of the numerical models in the production system for choke design among others. At present, multiphase flow metering equipment frequently employs numerical models of restrictions as part of their workflow to complement the multiphase measurements taken. In virtual metering schemes, the choke numerical model is a fundamental component to estimate flow rates using several measured data points along the production system. Choke models for predicting mass flow rate through restrictions are categorized into two main groups, namely empirical and theoretical models. The empirical models are developed through curve fitting based on a specific range of experimental data and cannot be extended beyond their range, while theoretical models n
Corresponding author.
http://dx.doi.org/10.1016/j.petrol.2016.05.022 0920-4105/& 2016 Elsevier B.V. All rights reserved.
are developed from mass, momentum and energy balance laws of the fluid across the restriction. The resulting expressions of theoretical models will have more unknowns than equations. To solve these equations, analytical relationships, experimental correlations between variables and simplifications are introduced. Examples of theoretical models are Sachdeva et al. (1986), Perkins (1993), Al-Safran and Kelkar (2007) and the Hydro model developed by Schüller et al. (2003). Examples of empirical models are Gilbert (1954), Ashford (1974), Ros (1960), Pilehvari (1980). Even though analytical models are typically derived introducing many simplications and assumptions, they generally have a fair predictability and accuracy that are appropriate for many applications in the oil and gas industry. Additionally, some tuning constants are sometimes included in the analytical expressions to improve their predictability by calibration against measured data when available. The Sachdeva, Perkins, Al-Safran and Hydro models are all used in the oil and gas industry. The 1D Sachdeva and Perkins models poorly predict flow rates partly because they do not consider slippage between the liquid phase and the gaseous phase. Therefore, these models require frequent tuning of the discharge coefficient for better
322
J. Mwalyepelo, M. Stanko / Journal of Petroleum Science and Engineering 145 (2016) 321–327
Nomenclature Abbreviations GVF LVF MPFM MPFMs WC LHS RHS CD ANTHEI
Gas volume fraction Liquid volume fraction Multiphase flow meter Multiphase flow meters Water cut Left-hand side Right-hand side Discharge coefficient Angolan Norwegian Tanzanian Higher Education Initiative
ε2 v ρi
ε
G σ N
μ
R ai U C
Subscripts
Symbols
Ui d di g ṁ k xi n ε1 p
Pressure ratio(p2/p1) in modified Sachdeva model Density of the mixture, (kg/m3) Elevation(m) Total cross-section area, (m2) Error Mass flux, (kg/m2/s) Standard deviation Number of data points Viscosity (N s m 2). Slip ratio Constants defined in Table 4-1. Velocity of the mixture, (m/s) Constant
y ρm z A
Velocity of phase i, (m/sec) Diameter, (m) Pipe inner diameter, (m) Acceleration due to gravity(m/s2) Mass flow rate, (kg/sec) Ratio of specific heat (Cp/Cv) Phase fraction of component i Polytropic exponent of gas (-vdp/pdv) Mean relative error Pressure, (pa) Mean absolute error Specific volume, (m3/kg) Density of component i, (kg/m3)
predictions. The Al-Safran and Hydro models consider the slippage factor between the gaseous phase and the liquid phase. The Hydro model is considered to be the most accurate one so far with a mean relative error of 0.36%, a mean absolute error of 5.78% and a standard deviation of 7.76%. The only disadvantage with the Hydro model is its complexibility which makes it complicated in programming using simple analytical equations. The Schüller papers are employed in the present work because they report a large number of experimental conditions and they provide enough details to benchmark other choke models. In the present work the choke models of Sachdeva, Perkins and AlSafran were programmed and then evaluated for the experimental data of Schüller et al., (2003, 2006). The model with the best predictability was then modified to improve its performance even further. A new choke model is proposed based on the model developed by Sachdeva et al., 1986. The model was derived (as in the original work of Sachdeva) using integral conservation law balances of mass, momentum and energy in the choke. The new model includes the slippage factor when calculating the mixture density, which was not considered in the original work of Sachdeva. The detailed derivation of the model is presented in Appendix A.
1 l 2 m 2′ meas 3 o Pred p c w g
Upstream condition Liquid Choke (throat)position Mixture Position just downstream the choke Measured value Recovered condition of choke Oil Predicted value Pipe Throat/choke/critical Water Gas
suggested to be ( C = 2000*CD ) for SI units but from the model derivation the value of C should be ( C = 2*CD2 ). Another correction was done in the Perkins model where the formula for calculating pressure ratio was missing a square in one part of the equation. The model derivation from the measured value was estimated by calculating the mean percentage error ( ε1), the mean absolute percentage error ( ε2 ) and the standard deviation ( σ ) according to (Eqs. (1)–3).
⎛1 ε1 = ⎜ ⎝N
∑
⎛1 ε2 = ⎜⎜ ⎝N
∑
σ=
1 N
ṁ pre − ṁ meas ⎞ ⎟× 100 ṁ meas ⎠ ṁ pre − ṁ meas ṁ meas
⎞ ⎟⎟× 100 ⎠
⎛ ṁ pre − ṁ meas ε ⎞ − 1 ⎟ × 100 ṁ meas 100 ⎠ ⎝ N=1
(1)
(2)
N
∑⎜
(3)
2.1. Evaluation results 2. Model evaluation Three different choke models i.e. Sachdeva et al. (1986), Perkins (1993), Al-Safran and Kelkar (2007) have been evaluated in this paper. These models are evaluated using 85 data points from Schüller papers, 60 from Schüller et al. (2003) and 25 data points from Schüller et al. (2006). During evaluation of the choke model of Al-Safran and Kelkar (2007) some errors were detected and corrected, the value of a constant C in the original paper is
Evaluation of the models show that those from Sachdeva et al. (1986), Perkins (1993), Al-Safran and Kelkar (2007) underpredict and overpredict the mass flow rate with a large deviation, specifically for values greater than 0.5 kg/s.
3. New modified sachdeva model A new model was derived based on the model by Sachdeva. The
J. Mwalyepelo, M. Stanko / Journal of Petroleum Science and Engineering 145 (2016) 321–327
323
model derivations follow the same concept used by Sachdeva et al. (1986), the only difference is the introduction of a slippage factor which Sachdeva did not consider when developing their model. The model accounts for conservation of mass, momentum and energy for multiphase flow across restriction with the following assumptions. (1) (2) (3) (4) (5) (6)
The flow is one dimensional; The predominant pressure term is acceleration; The quality is constant for high speed process (frozen flow); The liquid phase is incompressible; The flow is adiabatic and frictionless; A slippage effect exists between the gas phase and the liquid phase The model derivation is explained in detail in Appendix A. Fig. 3-2. Result of the Perkins model.
3.1. Model expressions Before calculating the mass flow rate through restrictions it is necessary to determine if the flow in the restriction is critical or subcritical. During the early production phase of an oil field, when the pressure in the reservoir is high, the flow in chokes is usually critical. In these cases the pressure fluctuations downstream of the choke do not affect the pressure at the bottom of the well. However with time, the flow changes to subcritical due to the decline in reservoir pressure. Therefore, it is crucial that the boundary between the critical and subcritical flow is well defined in cases where the flow passes through restrictions. The expression for defining critical/ subcritical boundary for this model is as shown in Eq. (4).
⎧ ⎪ ⎪ y=⎨ ⎪ ⎪ ⎩
⎫0.5 ⎪ + ⎪ xg vg1 ⎬ 2 nR( 1 − xg )vl n ⎛ R( 1 − xg )vl ⎞ ⎪ + + 2⎜ x v ⎟ ⎪ xg vg 2 g g2 ⎝ ⎠ ⎭
(
k k −1
+
n 2
)
R 1 − xg vl( 1 − y)
k k −1
(4) Fig. 3-3. Results of the Al-Safran model.
3.1.1. Polytropic coefficient ( n) The polytropic coefficient of this model is the same as that used by Sachdeva et al., 1986 (Figs. 3-1, 3-2, and 3-3) as shown in Eq. (5)
n=1+
(
xg Cpg − Cvg
(
) )
xg Cvg + 1−xg Cl
(5)
3.1.2. Slip ratio (R) The slip model used is the general slip correlation suggested by Grolmes and Leung (1985) as shown in Eq. (6) and the applicable values of the constants are shown in Table 4-1. In this paper the constants proposed by Simpson have been used because the deviation with the experimental data was less than the deviation that occurred when using other constants (Table 4-2). a a +1 ⎛ 1−x ⎞( a1−1)⎛ ρ ⎞( 2 )⎛ μ ⎞ 3 g l⎟ l⎟ ⎜ ⎜ ⎜ ⎟ R = a 0⎜ ⎟ ⎜ρ ⎟ ⎜μ ⎟ ⎝ xg ⎠ ⎝ g⎠ ⎝ g⎠
(6)
Table 4-1 Values for use with the Grolmes and Leung equation (Grolmes and Leung, 1985).
Fig. 3-1. Results of the Sachdeva model.
Model
ao
a1
a2
a3
Homogenous (no slip) Constant slip Fauske Moddy Simpson Thom Baroczy Lockhart-Martenelli
1 K 1 1 1 1 1 0.28
1 1 1 1 1 1 0.74 0.64
1 1 1/2 2/3 5/6 0.89 0.65 0.36
0 0 0 0 0 0.18 0.13 0.07
324
J. Mwalyepelo, M. Stanko / Journal of Petroleum Science and Engineering 145 (2016) 321–327
Table 4-2 Table of results. Model
ε1 (%)
ε2 (%)
σ (%)
Modified Sachdeva model CD ¼0.64 Modified Sachdeva model CD ¼0.65 Modified Sachdeva model CD ¼0.66
1.94 0.40 1.13
6.31 6.12 6.17
7.55 7.66 7.78
3.1.3. Relation between upstream pressure and recovered pressure (Perry, 1950)
⎛ ⎞ p1 − p3 ⎜ ⎟ p2 = p1 − ⎜ ⎜ 1− d /d 1.85 ⎟⎟ ′ ⎝ c p ⎠
(
Fig. 4-2. Results of the modified Sachdeva model with CD ¼0.66.
)
(7) 4. Conclusions and recommendations
3.1.4. Critical/subcritical mass flow rate The final expression for predicting mass flow rate is as shown in Eq. (8).
ṁ = CDA2 0.5 ⎧ ⎛ ⎤⎫ ⎛ 1−xg ⎞⎞ ⎡ kxg ⎪ ⎪ 2 ⎜ ⎟ ⎬ ⎨ ⎢ ⎥ 2 P x R 1 x v 1 y v yv ⎜ ⎟ ρ ∙ + ∙ − − + − ( ) ⎟ g l g1 m2 1 ⎜ g g2 ⎪ ⎪ k−1 ⎝ R ⎠⎠ ⎣ ⎦⎭ ⎝ ⎩ (8)
(
)
(
)
3.2. Model calibration The discharge coefficient ( CD ) was employed as a calibration constant to improve the prediction of the new model. Three different values of CD have been tested and the value of 0.65 was found to provide the best match with experimental results. 3.2.1. Results from the new modified Sachdeva model The new modified Sachdeva model (Figs. 4-1 ,4-2) improves predictability considerably by reducing the deviation from the original Sachdeva model by half, all the predictions seem to be within the range of 10% accuracy.
4.1. Conclusions The main conclusions that can be pointed out from this work are: (1) From the three choke models analyzed, the model of Sachdeva provided the best match with the experimental data of Schüller et al. (2003, 2006); (2) A new choke model was developed based on the model of Sachdeva but included phase slip at the choke entrance and at the throat; (3) The new developed model overperforms the choke models of Sachdeva, Perkins and Al-Safran; (4) The new model is capable of predicting critical/subcritical mass flow rate with a mean error of 0.4%, a mean absolute error of 6.12% and a standard deviation of 7.66%; (5) The discharge coefficient can be used as a tuning parameter to improve the predictability of the model. The best value of discharge coefficient for the new modified Sachdeva model is 0.65 for the particular data set of Schüller et al. (2003, 2006). 4.2. Recommendations The work performed in this study used only 85 experimental data points from the Schüller papers, it is therefore recommended that more data should be added for model evaluation. It is also recommended that other experimental data from other researchers who have done experiments on multiphase flow should be used to evaluate the model in order to benchmark its accuracy further. An in-depth study of the discharge coefficient should be done to see if the same discharge coefficient can be used in the orifice and cage type choke geometry). Acknowledgements This work is supported by the Norwegian University of Science and Technology under ANTHEI programme which is sponsored by Statoil-Tanzania.
Appendix A Mathematical derivation of the new choke model
Fig. 4-1. Results of the modified Sachdeva model with CD ¼0.65.
Assumptions In deriving this model the following assumptions have been
J. Mwalyepelo, M. Stanko / Journal of Petroleum Science and Engineering 145 (2016) 321–327
used: (1) Flow is one dimensional; (2) The predominant pressure term is acceleration; (3) The quality is constant for high speed process; (4) The liquid phase is incompressible; (5) The flow is adiabatic and frictionless; (6) A slippage effect exists between the gas phase and the liquid phase. Concepts used in deriving this model are the same concepts used by Sachdeva et al., 1986 where the equations of the conservation of mass, momentum and energy were used to determine the relationship between critical and subcritical flow. The difference arose from equation A 16 when defining mixture density by introducing slippage factor. If the slip factor is not included the results would be the same as the results of the model developed by Sachdeva et al. in 1986.
(
)
(A1)
)
ṁ l2 = 1 − x g2 ṁ
( (
)
During critical flow condition equation A 13 holds and equation A 11 reduces to,
G2
dU2 =−1 dp2
G 22
dvm2 =−1 dp2
Defining momentum mixture density from Schüller, et al., 2003,
(
⎡ ⎤ −A2dp2 = d⎣ G 2A2 U2 1 − x g2 + U2x g2 ⎦
(A7)
−A2dp2 = d⎡⎣ G 2A2 U2⎤⎦
(A8)
)
(A9)
Simplifying equation A 8 we get,
(
(A10)
⎡ dx dx g2 ⎤ ⎢ g2 − ⎥ ⎢⎣ dp2 Rdp2 ⎥⎦
)
(
)
(A18)
Using the assumption of frozen flow, the gas quality remains constant therefore,
(A19)
Please note that the assumption of frozen flow might not hold for small throat areas. Equation A 18 is therefore reduced to,
⎤ ⎡ x gdvg2 dvm2 ⎡ dvl ⎤ 1 ⎥ = ⎢ xg + + R 1 − xg . 1 − x g ⎥∙⎢ ⎣ ⎦ ⎣⎢ dp dp2 R dp2 ⎥⎦ 2
(
)
(
)
(A20)
Considering the assumption of incompressible liquid phase dvl =0, dp2
equation A 20 is reduced to,
⎤ ⎡ x gdvg2 ⎤ dvm2 ⎡ 1 ⎥ = ⎢ xg + 1 − x g ⎥∙⎢ ⎣ ⎦ ⎣⎢ dp ⎥⎦ dp2 R 2
)
(A21)
Substituting equation A 21 into equation A 15.
⎡ ⎤ ⎡ x gdvg2 ⎤ 1 ⎥=−1 G 22. ⎢ x g + 1 − x g ⎥∙⎢ ⎣ ⎦ ⎢⎣ dp ⎥⎦ R 2
(
G 22 =
ṁ g2 + ṁ l2 A2
(A17)
)
(A22)
(A11)
And G2 at the throat can be defined as
G2 =
)
⎡ ⎤ 1 + ⎢ x g2 + 1 − x g2 ⎥ ⎣ ⎦ R ⎡ x dv v dx dx g2 ⎤ dvl ⎢ g2 g2 + g2 g2 + R 1 − x g2 . ⎥ − Rvl. ⎢⎣ dp2 dp2 dp2 dp2 ⎥⎦
(
Expanding the right-hand side of expression A 10 and then dividing both sides by the pressure differential dp2,
dU2 dG + U2 2 dp2 dp2
)
x g1 = x g2, dx g = 0
From equation A 5 velocity of the moving fluid at the choke can be expressed as,
−dp2 = d⎡⎣ G 2U2⎤⎦
(
(A3)
(A6)
−1 = G 2
)
dvm2 ⎡ = ⎣ x g2vg2 + R 1 − x g2 vl⎤⎦. dp2
⎤ ⎡ ⎛ U2 1 − x g2 ṁ U2x g2ṁ ⎞⎟⎥ −A2dp2 = d⎢ G 2⎜⎜ + ⎟ ⎢ G2 G 2 ⎠⎥⎦ ⎣ ⎝
U2 = G 2/ρm2 = G 2vm2
(A16)
⎡ ⎤ 1 vm2 = ⎡⎣ x g2vg2 + R 1 − x g2 vl⎤⎦⎢ x g2 + 1 − x g2 ⎥ ⎣ ⎦ R
(A4)
)
)
Differentiating equation A 17 with respect to downstream pressure p2
(A5)
)
(A15)
(
)
G 2 = ṁ /A2
( (
(A14)
Substituting equation A 9 into equation A 14
Dividing the right-hand side of equation A 4 by G2 and multiplying by G2 where,
(
(A13)
(A2)
And
ṁ g2 = x g2ṁ −A2dp2 = d U2 1 − x g2 ṁ + U2x g2ṁ
dG 2 =0 dp2
(
However we know,
(
to throat pressure p2 at the boundary can be defined as,
⎡x 1 − x g ⎤⎡ ⎤ 1 g ⎥⎢ x g + 1 1 − x g ⎥ =⎢ +R ⎣ ⎦ R ρm ⎣⎢ ρg ρl ⎥⎦
Momentum equation at the throat
−A2dp2 = d U2ṁ l2 + U2ṁ g2
325
(A12)
For the fixed set of upstream conditions, during critical flow the mass flux reaches a maximum value with respect to downstream (throat) pressure. Therefore, the change of mass flux with respect
⎡x + ⎣ g
−dp2
1 R
( 1 − x g )⎤⎦. x gdvg2
(A23)
Considering polytropic expansion of gas,
p2 vgn2 = C Differentiating equation A 24 with respect to vg2
(A24)
326
J. Mwalyepelo, M. Stanko / Journal of Petroleum Science and Engineering 145 (2016) 321–327
np2vgn2− 1 + vgn2
dp2
dp2
(A25)
+
vgn2
(A26)
vg2
(A27)
U22 =
Substituting equation A 27 into equation A 23,
np2 1 ⎡ vg2x g ⎣ x g + R 1 − x g ⎤⎦
G 22 =
(
(A28)
p U2 U2 + gz1 + 1 = 2 + gz2 + 2 ρm2 2 2
ρm1
tegrating from p1to p2 the term
∫
∫
∫
∫
dp ρm
can be deduced as,
⎛ p ⎞1/ k ⎜ ⎟ ⎝C⎠
dp 1/ k
( p/ C )
= C1/ k
(A31)
∫
+
(
+
) ⎤⎥⎡ R ⎢ ⎥⎦⎣
p2 p1
x gk
⎥⎦
( 1 − x g )vl( 1 − y) (A41)
+
x gk k−1
) ⎤⎥R ⎥⎦
( 1 − x g )vl( 1 − y)
⎫0.5 ⎪ ⎬ ⎪ ⎭
( vg1 − yvg2)
(A42)
⎡ 1 − xg p1⎢ x g + ⎢⎣ R
) ⎤⎥⎡ R ⎢ ⎥⎦⎣
( 1 − x g )vl( 1 − y)
⎤ np2 k x gvg1 − x gyvg2 ⎥ = 1 ⎡ ⎦ k−1 vg2x g ⎣ x g + R 1 − x g ⎤⎦. 2ρm2 2 (A43)
(
)
(
)
Substituting the mixture density into RHS equation A 43
(
(
)
(
2
))
(
)
(A44)
(A35)
⎤ U2 k x gvg1p1 − x gvg2p2 ⎥ = 2 ⎦ k−1 2
)
=
2 np2 ⎡⎣ x g2vg2 + R 1 − x g2 vl⎤⎦ . ⎡⎣ x g2 +
(
)
1 R
( 1 − x g2)⎤⎦
2. x gvg2
(A45)
Combining equation A 45 as the RHS of the equation and the LHS of equation A 43,
⎡ 1 − xg p1⎢ x g + ⎢⎣ R
(
(A36)
= , p2 = yp1
) ⎤⎥R
( vg1 − yvg2)
(
Defining pressure ratio (y ),
y=
(A40)
1 np2 ⎡⎣ x g2vg2 + R 1 − x g2 vl⎤⎦⎡⎣ x g2 + R 1 − x g2 ⎤⎦ = 1 2. vg2x g ⎡⎣ x g + R 1 − x g ⎤⎦
( 1 − x g )vl( p1 − p2)
(
)
⎧ ⎡ 1 − xg ⎪ G 2 = ⎨ 2ρm2 2 p ⎢ x g + 1⎢ ⎪ R ⎣ ⎩
(A34)
Following the assumption of adiabatic flow, incompressible liquid phase and defining the mixture density equation A 35 can be written as,
⎡ 1 − xg ⎢x + ⎢⎣ g R
( 1 − x g )vl( 1 − y)
(
k−1
+
⎛ p p ⎞ U2 ⎜⎜ 1 − 2 ⎟⎟ = 2 1 ⎝ ρm1 2 ρm2 ⎠
⎢ ⎥⎦⎣
(A39)
⎤ G 22 k x gvg1 − x gyvg2 ⎥ = ⎦ 2ρ 2 k−1 m2
(
(A33)
k p k − 1 ρm
) ⎤⎥⎡ R
)
Calculating the critical/ subcritical boundary. Eliminating G2 by substituting equation A 28 into A - 40
Therefore, equation A 30 can be written as
k k−
(
Therefore, the critical/subcritical mass flow rate formula can be written as,
Then,
m
ρm2 2
(
(A32)
⎡ p ( k − 1) / k ⎤ dp ⎥ = k C1/ kp ( k − 1) / k = C1/ k⎢ ⎢⎣ ( k − 1)/k ⎥⎦ ρm k−1
= ∫ dp ρ
(A38)
2 ⎛⎡ x 1 − x g ⎤⎡ ⎤⎞ g ⎥⎢ x g + 1 1 − x g ⎥⎟ = G 22⎜⎜ ⎢ +R ⎦⎟ R ρl ⎥⎦⎣ ⎠ ⎝ ⎣⎢ ρg
⎡ 1 − xg G 22 = 2ρm2 2 p ⎢ x g + 1⎢ R ⎣
p−1/ k dp
⎛ ⎞1/ k k ⎜ p⎟ = p ( k − 1) / k k − 1 ⎜⎝ ρ k ⎟⎠ m
)
Simplifying the equation above:
(A30)
Assuming that the flow is adiabatic and U22> > >U12, then in-
dp = ρm
(
⎡ 1 − xg p1⎢ x g + ⎢⎣ R
(A29)
⎛ U2 ⎞ dp ⎟ = d⎜ ρm ⎝ 2 ⎠
( 1 − x g )vl( p1 − yp1)
⎤ U2 k x gvg1p1 − x gvg2yp1 ⎥ = 2 ⎦ k−1 2
G 22
+
With negligible elevation changes gz1 = gz2 equation A 29 reduces to,
p2 vgk1 = C , ρ =
⎢ ⎥⎦⎣
(
)
Now considering the general Bernoulli's equation,
p1
) ⎤⎥⎡ R
From equation A 9 velocity (U2) can be defined as:
−np2
=
dvg2
⎡ 1 − xg ⎢x + ⎢⎣ g R
(
=0
−np2vgn2− 1
=
dvg2
dp2 dvg2
(A37)
) ⎤⎥⎡⎢ R ⎥⎦⎣
kx
⎤
( 1 − x g )vl( 1 − y) + k −g1 ( vg1 − yvg2 )⎥⎦
2 nyp1⎡⎣ x g2vg2 + R 1 − x g2 vl⎤⎦ . ⎡⎣ x g2 +
(
)
2. x gvg2
Simplifying equation A 46
1 R
( 1 − x g2)⎤⎦ (A46)
J. Mwalyepelo, M. Stanko / Journal of Petroleum Science and Engineering 145 (2016) 321–327
(
)
R 1 − x g vl( 1 − y ) +
=
kx gy. y−1/ k vg1 k−1
+
Simplify equation A 52 then the critical/subcritical boundary formula can be written as
kx gvg1 k−1 2 ny⎡⎣ x g2vg2 + R 1 − x g2 vl⎤⎦
(
)
2. x gvg2
(A47)
Considering the LHS of equation A 47:
⎡ R 1 − x g vl( 1 − y ) ⎤ k ⎥ + x gvg1⎢ ⎢⎣ k − 1 ⎥⎦ x gvg1
(
327
)
⎧ ⎪ ⎪ y=⎨ ⎪ ⎪ ⎩
k
⎫ k−1 ⎪ + ⎪ x g vg1 ⎬ 2 ⎛ ⎞ ⎪ nR( 1 − x g )v l n R( 1 − x g )v l + + 2⎜ x v ⎟ ⎪ x g vg 2 g g 2 ⎝ ⎠ ⎭
k k−1 k k−1
+
n 2
(
)
R 1 − x g vl ( 1 − y)
(A53)
(A48)
Now considering the RHS of equation A 47:
kx gy ( k − 1) / k . vg1 k−1
kx gvg1y ( k − 1) / k k−1 +
+
+
2 ny⎡⎣ x g2vg2 + R 1 − x g2 vl⎤⎦
(
References
(
(A49)
2 −2/ k
)
n x gvg1 y 2 x gvg1y−1/ k
nR 1 − x g vlx gvg1. y ( k − 1) / k
(
)
2. x gvg2
)
x gvg2
+
ny R 1 − x g vl 2x gvg2
( (
2
) )
(A50)
⎡ nR 1 − x g vly ( k − 1) / k ⎢ ky ( k − 1) / k ny ( k − 1) / k x gvg1⎢ + + k−1 x gvg2 2 ⎢⎣
(
)
⎤ ⎛ ⎞2 n ⎜ R 1 − x g vl ⎟ ( k − 1) / k ⎥ + ⎜ ⎥ ⎟y 2⎝ x gvg2 ⎠ ⎥⎦
(
)
(A51)
Combining equation A 48 as the LHS of the equation and equation A 51 as the RHS of the equation:
⎡ ⎡ R 1 − x g vl( 1 − y ) ⎤ ⎢ k ⎢ ⎥ x gvg1 + = x vg1⎢ ⎢⎣ k − 1 ⎥⎦ x gvg1 ⎢⎣ g
(
)
nR 1 − x g vly ( k − 1) / k ky ( k − 1) / k ny ( k − 1) / k + + k−1 2 x gvg2
(
2⎤ ⎛ ⎞ n ⎜ R 1 − x g vl ⎟ ( k − 1) / k ⎥ + ⎜ ⎥ ⎟y 2⎝ x gvg2 ⎠ ⎥⎦
(
)
)
(A52)
Ashford, F.E., 1974. An Evaluation of critical Multiphase flow performance through wellhead chokes. Al-Safran, E.M., Kelkar, M., 2007. Prediction of two phase critical flow boundary and mass flow rate across chokes. Soc. Pet. Eng., Anaheim Gilbert, W., 1954. Flowing and Gas Gas Lift Well Performance Drilling Production Practise. Grolmes, A.M., Leung, C., 1985. J. Chem. Eng. Prog. 8 (p. 81). Perkins, T.K., 1993. Critical and subcritical flow of multiphase mixtures through chokes. Soc. Pet. Eng. Pilehvari, A.A., 1980. Eperimental study of critical two phase flow through well head chokes. s.l.: University of Tulsa. Perry John, H., 1950. Chemical Engineers Handbook. Mcgraw-Hill Book Co. Inc., NewYork, p. 404. Ros, J.N.C., 1960. An Analysis of Critical Simultaneous Gas/Liquid Flow Through a Restriction and its Application to Flow Metering, Netherlands, vol. 9. Sachdeva, R., et al., 1986. Two phase flow through chokes. Soc. Pet. Eng. Schüller, R.B., et al. 2006. Critical and Subcritical Oil/Gas/Water Mass Flow Rate Experiments and Predictions for Chokes. Schüller, R.B., Solbakken, T, Selmer-Oslen, S., 2003. Evaluation of multiphase flow rate models for chokes under subcritical oil/gas/flow conditions. Soc. Pet. Eng.