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Abstract This paper considers the use of extended Kalman Filtering as a soft-sensing technique for gas-lift wells. This technique is deployed for the estimation of dynamic variables that are not directly measured. Possible applications are the estimation of flow rates from pressure measurements or the estimation of parameters of a drift-flux model. By means of simulation examples different configurations of sensor systems are analyzed. The estimation of drift-flux model parameters is demonstrated on real data from a laboratory set-up.

Introduction The smart well paradigm involves the instrumentation of wells with sensors and actuators, which can be used for monitoring and control purposes. From a monitoring point of view, the use of sensors that measure different properties at several locations is preferred. However, because of practical and economical reasons such demands are unrealistic. Some measurements, like pressure measurements, are more readily available than others (e.g. oil flow rate). To have access to the unmeasured variables, the concept of soft sensing is used in this paper: the unmeasured dynamic variables are estimated from the measured ones by fitting a model to the measurements using extended Kalman Filtering [6].

In this paper a gas-lift well is considered. Possible applications of soft sensing for gas-lift wells are the estimation of gas and oil flow rates from pressure measurements and the parameter estimation for models that describe the multiphase flow phenomena. The use of soft sensing for well operations has been described in e.g. [9], [10]. In [9], [10] ensemble Kalman Filtering is used as the soft sensing algorithm, whereas in this paper extended Kalman Filtering is used. The main difference between these two algorithms is the prediction of the state-covariance matrix: ensemble Kalman filtering uses an ensemble of nonlinear state predictions to construct the predicted state-covariance matrix, whereas extended Kalman

Filtering uses a locally linearized model to predict the state-covariance matrix. For models that are moderately nonlinear, in the sense that the change of the dynamics is small within two subsequent sampling times, extended Kalman filtering works well since in such cases the linear approximation between two sampling times is accurate. For highly nonlinear models this approximation is no longer accurate, and the use of an ensemble of nonlinear predictions may improve the predicted state covariance matrix [4]. However, the nonlinearity of the gas-lift model considered in this paper proved to be modest in the investigated operating region, which justifies the use of local linearizations.

According to [9] a disadvantage of the extended Kalman Filter is the large computational demand of the numerical linearizations, which require a number of model evaluations that is of the same order as the number of state variables. However, in the ensemble Kalman Filter in [9] and [10] the number of nonlinear model predictions is set to 100, which is also of the same order as the number of state variables (159 in [10]: a discretization of 20 meters for a 1000 meter well results in 50 sections and each section consists of 3 states, additionally 9 states are used for the estimation of model parameters). Drawing the ensemble members randomly from a distribution introduces a stochastical component in the prediction of the state-covariance matrix of an ensemble Kalman Filter. This stochastic dependency on the random realization of the ensemble can be circumvented by choosing the realizations as suggested in [4] and [5], but this results in a number of ensemble members that is twice the number of state variables. Thus with respect to the computational load the extended Kalman Filter may be preferred over the ensemble Kalman Filter in the case of the application for gas-lift wells.

The organization of the paper is as follows: first brief descriptions are given of both the model for the gas-lift well and of the extended Kalman Filter. Next, different measurement configurations are analyzed by means of simulations: the use of pressure measurements along the tubing, and the use of topside measurements from the annulus and the tubing. These configurations can be used for the on-line estimation of the gas and oil flow rates in the tubing, acting as a multiphase-flow soft-sensor. Besides, unknown model parameters can be estimated on-line in order to keep the model on track. For the estimation of unknown parameters of a drift-flux model, the soft sensor is tested on experimental data from a laboratory set-up.

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Soft Sensing for Gas-Lift Wells H.H.J. Bloemen SPE, S.P.C. Belfroid SPE, W.L. Sturm SPE, F.J.P.C.M.G Verhelst SPE, TNO TPD

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Gas-lift model A schematic overview of a gas-lift well is presented in Fig. 1. The gas-lift well consists of a tubing which transports the oil and gas from a reservoir to the surface, and of an annulus which transports the lift gas to the bottom of the tubing. The purpose of the lift gas is to enhance the oil production from the reservoir by lowering the hydrostatic pressure in the tubing. The annulus and the tubing are connected through a gas-lift valve.

The annulus and the tubing are modelled by partitioning them in a number of sections. Each section of the annulus is modelled by:

( )

( )a

iaiaiaiaaqia

iaiaiaiaapia

Niqqppfq

dtd

qqppfpdtd

,,1,,,

,,,

,1,1,,,,

,1,1,,,,

=

=

=

+

+

.. (1)

pa,i and qa,i represent the pressure and the gas flow, respectively, in the annulus at section i, Na is the number of sections of the annulus. Each section of the tubing is modelled by:

( )

( )

( )t

iGiGiLiLititiLiLtqiL

iGiGiLiLititiLtpit

LiLtiL

Ni

qqqqppfqdtd

qqqqppfpdtd

qqfdtd

,,1

,,,,,,,

,,,,,,

,

,1,,1,1,,,1,,,

,1,,1,,1,,,,

1,,,

=

=

=

=

+

............................................................................................. (2)

Nt is the number of sections of the tubing, L is the liquid hold-up, pt is the pressure in the tubing, qL is the liquid (oil) flow rate, and the gas flow rate qG in the tubing follows directly from the above states:

( ) ( ) tbstiLiLiL

iG NiuAqCCq ,,1,

111

,0,0

,, =+

=

............................................................................................. (3)

At is the area of the tubing, and C0 and ubs are parameters of the drift-flux model, see also [7], [10], [12]:

( ) bsLsGsG uuuCu ++= 0 .............................................. (4) where uG is the absolute gas velocity, uGs and uLs are the superficial gas and liquid velocities, and ubs is the bubble rise velocity (Eqn. 3 follows from Eqn. 4 after substituting uG=qG/((1-L)At), uGs=qG/At, uLs=qL/At). The value of these drift flux parameters is flow-regime dependent. According to the Taitel criterion, the transition from bubble flow to slug flow occurs when [2]:

( ) bsGsLs uuu = 11 ........................................... (5)

for a critical void fraction . For the drift-flux parameters C0 and ubs different relations exist. For a fully developed turbulent bubbly flow [3] gives:

l

gC

2.02.10 = ........................................................(6)

g and l are the density of the gas phase and the density of the liquid phase respectively. The value of C0 may be corrected by a factor depending on the hold-up and/or a factor depending on the Sauter mean diameter, see [3]. If these additional corrections are ignored and if the density of the gas phase is small compared to the density of the liquid phase, then C0 is approximately equal to 1.2. This value is also valid in the slug-flow regime [7], [8]. However, (slightly) different values are encountered as well, for example 1.13 in [12]. The bubble rise velocity in the bubble flow regime is [2], [8]:

( ) 4141

2 53.153.1

=

ll

glbs

ggu

................(7)

g is the gravitational constant, is the surface tension. Instead of 1.53, also factors of 1.4 are encountered [3], [7], [12]. Moreover, different corrections to incorporate an effect of the hold-up may be incorporated [3], [7], [8]. In the slug flow regime the following relation holds [3], [7], [8], [12]:

( ) ( )2121

35.035.0 gDgD

ul

glbs

=

.................(8)

where D is the diameter of the tubing. Slightly other values for the factor 0.35 have been suggested as well [8].

From the above discussion on the drift-flux parameters C0 and ubs it follows that these parameter values are not well defined. In particular the value of ubs seems to be strongly dependent on the hold-up and on the flow regime [3]. Since these dependencies are not very well quantified, this motivates to estimate these parameters on-line from experimental data.

For the first and last sections of the annulus and the tubing boundary conditions need to be specified:

pcNt

mfcNa

inGG

inLL

tt

LL

inaa

pp

ppqqqqpp

qq

t

a

=

======

+

+

1,

1,

,0,

,0,

1,0,

1,0,

,0,

..................................................................(9)

ppc and pmfc are the pressure of the production choke and the pressure of the mass flow controller of the lift gas choke, respectively, and are measured inputs to the model. qa,in and qG,in form the connection, by means of the gas-lift valve, between the annulus and the tubing and can be modelled by:

( )( ) rGtavtinG

tavaina

qppfqppfq

,1,1,,,

1,1,,,

,,

+==

........................................(10)

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qG,r is the gas flow rate from the reservoir into the tubing. In this paper this flow is assumed to be zero. qL,in can be modelled by:

( )1,, tresinL ppJq = .................................................... (11) where pres is the pressure in the reservoir, and J is the productivity index. If the knowledge of pres and J is accurate, then qL,in can be estimated from the pressure in the bottom section of the tubing. If this knowledge is inaccurate, then qL,in needs to be estimated. Soft Sensing In order to do valuable predictions with the model in Eqn. 1 and Eqn. 2, this model has to be kept on track with the real process, which can be done by fitting the model to the available measurements. Kalman filtering is a technique that performs this task recursively: at each sampling time the states in Eqn. 1 and Eqn. 2 are updated using the observed error between the measurements and the model prediction. This state update depends on the state-covariance matrix, which can be predicted recursively using a linearized and discretized model of the process. At each sampling time the model is re-linearized and re-discretized about the most recent estimate of the state, in order to try to keep the linearization error small. This is the so-called extended Kalman filter [6]. A Kalman filter is a state estimator. The estimation of model-parameters (like the drift-flux parameters), or unknown inputs (e.g. if pres and J are unknown) can be incorporated into this paradigm by modelling these as state variables, for example by:

[ ] [ ] [ ]kwGkkxAkkx +=+ 11111 ||1 .................... (12) where x1 is a vector of model parameters and unknown inputs. Also estimates of the measured inputs ppc and pmfc can be incorporated into this vector in order to filter these inputs when they are affected by measurement noise, w[k] is a vector of unknown process noise, A11 and G1 are (user-defined) matrices. Often A11=I and G1=I are chosen. The notation x1[k+1|k] is used to specify a prediction at sampling time k+1, given measurements up to sampling time k. The combination of the linearized and discretized model of Eqn. 1 and Eqn. 2 with the model in Eqn. 12 gives:

[ ] [ ] [ ] [ ] [ ][ ] [ ] [ ]kvkkCzkky

kGwkKkkzkAkkz+=

++=+||

||1................ (13)

with:

[ ] [ ] [ ]

=

=

kAkAA

kA

xx

z

2212

11

2

1

0............................................. (14)

where x2 is a vector with states from Eqn. 1 and Eqn. 2, A12 and A22 are matrices that describe the linearized dynamics of Eqn. 1 and Eqn. 2, K is a vector that contains the off-sets due to linearization, G is a (user defined) matrix, C is the matrix

defining the outputs / measurements y from the states z, and v is a vector of unknown measurement noise.

Now, given the observation error e[k]:

[ ] [ ] [ ] [ ] [ ]1|1| == kkCzkykkykyke mm ..(15) where ym is a vector of measurements (the value of v[k] is set to zero in the above equation since v is an unknown zero mean white-noise signal), the state vector is updated according to:

[ ] [ ] [ ] [ ]kekLkkzkkz += 1|| ..................................(16) with:

[ ] [ ] [ ]( )[ ] [ ]( ) [ ][ ] [ ] [ ] [ ] TT

TT

GQGkAkkkAkk

kkCkLIkkRCkkCCkkkL

+=+

=+=

||1

1||1|1| 1

........(17)

where R and Q are the covariance matrices of the zero-mean uncorrelated white-noise signals v and w respectively. The linearization step of the extended Kalman filter is primarily required for the prediction of the state-covariance matrix in Eqn. 17. The nonlinear simulation model may replace the actual state prediction in Eqn. 13. For more information regarding the extended Kalman filter the reader is referred to [6] for example.

The main difference between the extended Kalman filter, explained above, and the ensemble Kalman filter used in [9] and [10] or unscented filter in [4] and [5], lies in the prediction of the state covariance matrix: the extended Kalman filter uses a linearized model to recursively predict the state-covariance matrix (Eqn. 17), whereas the ensemble Kalman filter uses an ensemble of predicted states from which the state-covariance matrix is estimated.

To summarize: the soft sensing algorithm, by means of the extended Kalman filter, uses the observation error in Eqn. 15 to update all the states z (Eqn. 16), including the model parameters and unknown inputs collected in x1, which is a part of the vector z, see Eqn. 14. In this way the model parameters and unknown inputs collected in x1 are estimated on-line. Results A multiphase-flow soft sensor using pressure measurements along the tubing.

This section considers the possibility to estimate the gas flow rate and oil flow rate into the tubing from pressure measurements along the tubing. In this way the extended Kalman filter (EKF) acts as a multiphase-flow soft sensor. In this section it is assumed that the information about the annulus and the reservoir is not available (no measurements and no model), which implies that Eqn. 10 and Eqn. 11 cannot be used to estimate qG,in and qL,in.

Data were generated by simulating a model for the gas-lift well with a sampling time of 20 seconds. This simulation model uses a coupled model of the annulus, the tubing, the gas-lift valve, the reservoir, the production choke and the lift gas choke. This simulation model considers a 2.3 km deep well divided into 6 sections. Between 125 and 142 minutes the set point for the lift gas choke was gradually lowered to half of

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the set point during the first 125 minutes, after which it was kept constant till the end (250 minutes). This decreased gas flow causes casing-heading instability, see for example [1]. Measurement noise with a standard deviation of 1 (bar) was added to the pressure signals of the six sections of the tubing and the pressure signal from the production choke. These seven simulated and noisy pressure signals were used as measurements for the EKF to estimate the other states, including qG,in, qL,in and a filtered estimate for ppc. The performance of the EKF is demonstrated in Figure 2. In this figure both the estimates of the EKF and the signals of the coupled simulation model are plotted. From this figure it is clear that the estimated signals are close to the true signals. Even under the dynamics of casing-heading instability, at the end of the simulation, the EKF performs well. The largest deviations occur when the hold-up values are close to one. This is because in the linearization step of the EKF a maximum hold up of 0.95 is used (not for the estimated values though) in order to stay away from the singularity in the model that is caused by a hold-up equal to one.

In this section only the tubing is considered and the model for the annulus is not used. This implies that the results can be generalized to ordinary wells (i.e. without gas lift) that produce a mixture of gas and oil. The knowledge of the flow in the annulus in case of gas-lift wells enables the EKF to discriminate between the gas flow from the annulus and the gas flow from the reservoir.

Parameter estimation using experimental data.

This section considers the on-line estimation of the bubble rise velocity ubs in an airlift. The experimental setup of the airlift is described in [2]. The measured data are the inflow rates of air and liquid at the bottom of the airlift and the pressures at different heights (2, 4, 6, 8, 10, and 12 meters). The height of the pipe was 18 meters and the diameter was 72 mm. During the experiment the gas flow rate was increased stepwise. Most of the data are collected in the bubble flow regime, as indicated in Figure 3.

The model for the 18-meter high airlift consists of 8 sections of 2 meters and 2 boundary sections. The bottom boundary section is used to specify the inflow of gas and liquid and the top boundary section is used to specify the pressure at the top. Because of an unknown length of the down comer of the airlift (see [2]) the top pressure is estimated. The gas and liquid inflow rates are filtered within the EKF.

The initial values of the drift-flux parameters were set to:

41

0

53.1

2.1

=

=

lbs

gu

C

......................................................... (18)

Estimating C0 and ubs simultaneously resulted in unrealistic values for C0 and ubs (data not shown): ubs increasing and C0 decreasing to very low and even negative values. This counteracting effect can be understood from Eqn. 3 for the gas flow rate. This effect was also visible when comparing the results for the two cases in which one of the two parameters was fixed while the other was estimated: in both cases the

estimated parameters showed similar trends. From the results in [3] it follows that the differences between the experimentally determined values of C0 and its theoretical value are smaller than the differences between the experimentally determined values of ubs and its theoretical value. Therefore, since the value for C0 is better characterized and (almost) equal in both flow regimes, the value of C0 was fixed to 1.2.

The results for the estimation of Atubs are shown in Figure 4 and Figure 5 (instead of ubs the product Atubs was estimated because of scaling reasons). After an initial increase Atubs is slowly decreasing. This is probably caused by a decreasing liquid hold up, see Figure 4 and [3]. To account for a dependency on the hold-up the following relation from [3] is assumed (which varies per section):

75.1,, iLubsibst KuA = ................................................(19)

The EKF is then used to estimate the factor Kubs. The result for this implementation is displayed in Figure 6. Comparing Figure 5 and Figure 6 it can be concluded that the above dependency on the hold up removes a large part of the variation in the estimated variable. This validates Eqn. 19 proposed in [3]. In the slug flow regime the bubble rise velocity is larger and is no longer correlated with the hold up as in Eqn. 19, see [3]. It is expected that the transition from the bubble flow regime to the slug flow regime be accompanied by a gradual change in the bubble rise velocity. The slow increase of the estimated parameter value towards the end of the dataset probably indicates a gradual transition to the slug flow regime. This transition to the slug flow regime is also displayed in Figure 3.

Combined flow rate and parameter estimation using topside annulus and tubing measurements.

In this section the EKF will be used for simultaneous flow rate and parameter estimation. Moreover, the possibility of using only the following topside measurements is investigated:

Pressure at the top section of the annulus Pressure of the mass flow controller of the lift

gas choke Pressure at the top section of the tubing Pressure of the production choke Gas flow into the annulus (at the top)

Data was generated as described before, about 1% measurement noise was added to the data. The flow rate of the lift gas through the gas-lift valve is estimated from the pressures in the bottom sections, Eqn. 10. Since the gas production form the reservoir is assumed to be zero, this flow determines the flow of gas into the tubing. The oil flow from the reservoir into the tubing is a remaining unknown input to be estimated by the EKF. Along with the estimation of this unknown input and the other state variables, the estimation of the drift-flux parameters is considered.

First C0 is considered. The EKF is initiated with the correct value. After 50 minutes (to remove initialization effects) the value of C0 in the model of the EKF is abruptly lowered, see Figure 7 and Figure 8. From Figure 7 it can be observed that the change in C0 affects the measured signals (in particular the gas flow rate in the annulus and the pressures at the top of the

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tubing) and that the EKF is able to recover from this error: the estimated value of C0 returns to its true value and is not affected by the casing-heading dynamics. Similar results have been obtained for an abruptly increased value of C0.

Secondly ubs is considered. The EKF is initiated with the correct model. After 50 minutes (to remove initialization effects) the value of ubs in the model of the EKF is abruptly lowered, see Figure 9 and Figure 10. From these figures it can be observed that the change in ubs hardly affects the measured signals, which makes it impossible for the EKF to recover from this error. Similar results have been obtained for an abruptly increased value of ubs. The states that are most affected by the change in ubs are the unmeasured liquid flow rates. The EKF compensates a too low value for ubs by a larger liquid flow rate, such that the estimated gas flow rates are hardly affected, see Eqn. 3. Moreover, analysis of the differential equations reveals that if all the estimated liquid flows get the same bias then this will not affect the estimated liquid hold up nor the estimated pressures. This indicates that a liquid flow rate needs to be measured in order to be able to estimate ubs (note that in the previous section the inflow of liquid was measured for example). In Figure 11 and Figure 12 it is demonstrated that the measurement of the liquid flow rate (5% measurement noise) out of the tubing helps to estimate ubs. Another possibility is to use an estimate of the liquid flow into the tubing from the estimated pressure at the bottom section and knowledge of the productivity index, Eqn. 11 (data not shown).

Discussion This paper demonstrates the possibility of using an extended Kalman filter as a soft sensor for estimating flow rates and drift-flux parameters. It was demonstrated that the EKF can be used as a multiphase-flow soft sensor using pressure measurements along the tubing. Next it was demonstrated that the EKF can be used as a soft sensor for model parameters. The combined estimation of model-parameters and flow rates is discussed in the previous section. It has been demonstrated that these requirements put demands on the measurement configuration. In the last case a liquid flow rate had to be either measured or estimated from the other variables in order to be able to estimate the bubble rise velocity ubs.

An alternative Kalman Filter is the ensemble Kalman Filter used in [9] and [10]. A discussion concerning the difference between an ensemble Kalman Filter and an extended Kalman Filter is given in the introduction of this paper. For the application of this paper the extended Kalman Filter proved to work well, which indicates that for this case the linearization error in the extended Kalman Filter is acceptable. The computational complexity of the linearization step in the extended Kalman Filter is comparable to the computational complexity of the nonlinear simulations of the ensemble in the ensemble Kalman Filter (see the introduction). The stochastic dependency on the random realization of the ensemble in the ensemble Kalman Filter in [9] and [10] is an unfavourable property that can be circumvented by choosing the realizations as suggested in [4] and [5], at the cost of a larger computational demand due to a larger number of ensemble members (equal to twice the number of state variables).

A possible application of the soft sensor is in the design of

an Operator Support System. Such a system can be used to monitor the status of the well and to advice the operator in order to optimize the performance of the well [11].

Acknowledgements The authors would like to thank Sebastien Guet of Shell / Delft University of Technology for providing us with the experimental data.

Nomenclature pa,i = pressure in the annulus at section i qa,i = gas flow in the annulus at section i L,i = liquid hold-up in the tubing at section i pt,i = the pressure in the tubing at section i qL,i = liquid (oil) flow rate in the tubing at section i qG,i= gas flow rate in the tubing at section i Na = number of sections of the annulus Nt = number of sections of the tubing f..(..,..,..) = function At = area of the tubing, C0 = parameter of the drift-flux model ubs = parameter of the drift-flux model uG = absolute gas velocity, uGs = superficial gas velocity uLs = superficial liquid velocity ubs = bubble rise velocity = critical void fraction g = density of the gas phase l = density of the liquid phase g = gravitational constant = surface tension D = diameter of the tubing ppc = pressure of the production choke pmfc = pressure of mass flow controller of lift gas choke pres = pressure in the reservoir J = productivity index x1 = vector of model parameters and unknown inputs x2 = vector of states z = augmented state vector v = vector of measurement noise w = vector of process noise y = vector of outputs ym = vector of measurements e = observation error I = identity matrix k = sampling time A = matrix of the augmented model A11 = matrix A12 = matrix of linearized model A22 = matrix of linearized model G1 = matrix G = matrix C = output matrix K = vector that contains off-sets due to linearization, L = feedback matrix R = covariance matrix of v Q = covariance matrix of w = state-covariance matrix Kubs = gain factor

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References [1] G.O. Eikrem, B. Foss, L. Imsland, B. Hu, M. Golan. ''Stabilization

of gas lifted wells'', in Proc. 15th IFAC World Congress, Barcelona, Spain, 2002.

[2] S. Guet, G. Ooms, R.V.A. Oliemans. ''Influence of bubble size on the transition from low-Re bubbly flow to slug flow in a vertical pipe'', Experimental Thermal and Fluid Science 26, 635-641 (2002)

[3] T. Hibiki, M. Ishii. ''Distribution parameter and drift velocity of drift-flux model in bubbly flow'', International Journal of Heat and Mass Transfer 45, 707-721 (2002).

[4] S.J. Julier, J.K. Uhlmann. ''A new extension of the Kalman Filter to nonlinear systems'', in Proc. AeroSense: 11th Int. Symp. Aerosp./Defense Sensing, Simulat. Contr., Orlando, 1997.

[5] S. Julier, J. Uhlmann, H.F. Durrant-Whyte, 2000. ''A new method for the nonlinear transformation of means and covariances in filters and estimators.'' IEEE Transcations on Automatic Control 45(3), 477-482.

[6] T. Kailath, A.H. Sayed, B. Hassibi, Linear Estimation, Prentice Hall, NJ (2000)

[7] C. Kleinstreuer. Two-phase flow; theory and applications, Taylor and Francis, New York (2003)

[8] H.J.W.M. Legius. ''Propagation of pulsations and waves in two-phase pipe systems'', PhD thesis, Delft University of Technology, 1997.

[9] R.J. Lorentzen, K.K. Fjelde, J. Fryen, A.C.V.M. Lage, G. Naevdal, E.H. Vefring. ''Underbalanced and low-head drilling operations: real time interpretation of measured data and operational support.'' SPE Annual Technical Conference and Exhibition, New Orleans, Louisiana, 2001, SPE paper 71384.

[10] R.J. Lorentzen, G. Naevdal, A.C.V.M. Lage, 2003. ''Tuning of parameters in a two-phase flow model using an ensemble Kalman filter.'' International Journal of Multiphase Flow 29(8), 1283-1309.

[11] W.L. Sturm, S.P.C. Belfroid, O. van Wolfswinkel, M.C.A.M. Peters, F.J.P.C.M.G. Verhelst. ''Dynamic Reservoir Well Interaction.'' SPE Annual Technical Conference and Exhibition, Houston, 2004, SPE paper 90108.

[12] P.B. Whalley. Boiling, condensation, and gas-liquid flow, Clarendon Press, Oxford (1987)

Fig. 1-- Schematic overview of a gas-lift well

Oil reservoir

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