- Email: [email protected]
Linear and non-linear analysis of flow instability in gas-lift wells
Journal of Petroleum Science and Engineering 108 (2013) 162–171
Contents lists available at ScienceDirect
Journal of Petroleum Science and Engineering journal homepage: www.elsevier.com/locate/petrol
Linear and non-linear analysis of flow instability in gas-lift wells I. Guerrero-Sarabia, Y.V. Fairuzov n Institute of Engineering, National Autonomous University of Mexico, Cd. Universitaria, 04510 Mexico City, Mexico
a r t i c l e i n f o
a b s t r a c t
Article history: Received 14 April 2012 Accepted 31 January 2013 Available online 13 February 2013
Linear and non-linear analyses of flow instability in continuous gas-lift wells were performed in this study. The linear analysis is based on a modified gas-lift stability criterion that takes into account compressibility of the mixture below the injection point and is applicable to saturated reservoirs. The analysis of non-linear dynamics and stability of the well was performed using direct numerical integration in the time domain of the governing equations describing the gas-lift system. The transient gas-lift well model developed comprises of a model of transient three-phase gas–oil–water flow in the wellbore, a transient model of gas flow in the casing annulus, and a pseudo-steady flow model in the reservoir. The multiphase flow model used is based on the drift-flux theory. Stability boundaries predicted by both linear and non-linear analysis were compared with field data published in a previous study; both types of analysis reproduced the data. The effects of the main well design and flow parameters on the frequency and amplitude of the oscillations during heading in a typical gas-lift well were studied. It was found that flow instability results in the oil production loss, which depends on severity of heading. The largest reduction in oil production takes place in case of the most severe heading in the well (flow instability with the largest amplitude of production rate oscillations). An increase in the lift gas consumption is required to compensate for the production losses caused by heading. An increase in the depth of the injection point may result in heading and an increase in the operating costs caused by the increase in the lift gas consumption. An increase in the separator pressure has a destabilizing effect. At high separator pressures the well can experience two modes of instabilities: casing heading and density-wave oscillations. & 2013 Published by Elsevier B.V.
Keywords: gas-lift well flow instability heading stability analysis
1. Introduction Flow instability (heading) in continuous flow gas-lift wells has been the subject of many studies over the last three decades (Alhanati et al., 1993; Asheim, 1988; Blick et al., 1988; Fairuzov et al., 2004; Grupping et al., 1984a, 1984b; Hu, 2004; Hu and Golan, 2003; Poblano et al., 2002). Heading is the reason of many problems in the operation of oil production facilities (Alhanati et al., 1993) and finally leads to an increase in the operating costs. Two types of instabilities in gas-lift systems have been identified: casing heading and density-wave instability. The former is associated to variations of the injected gas flow rate caused by variations of the density of the multiphase fluid in the tubing downstream the injection point (Asheim, 1988). The flow in a gaslift well can be also unstable even the downhole gas injection rate is constant due to density-wave oscillations (Hu, 2004; Hu and Golan, 2003). Self-excited pressure and flow rate oscillations in the tubing may either diverge (result in the complete loss of
n
Corresponding author. E-mail address: [email protected] (Y.V. Fairuzov).
0920-4105/$ - see front matter & 2013 Published by Elsevier B.V. http://dx.doi.org/10.1016/j.petrol.2013.01.012
liquid production and gas recirculation) or reach a self-sustained periodic mode (heading). Two different approaches have been proposed in the literature to analyze gas-lift instability. The linear analysis has been used to develop flow stability criteria in terms of flow and well design parameters by different authors (Alhanati et al., 1993; Asheim 1988; Blick et al., 1988). In this type of analysis, the response of the system, which is initially at equilibrium, to an infinitesimal perturbation of tubing pressure at the injection point is predicted. To obtain practical analytical criteria, several strong simplifications in the description of the system are made. The stability criteria can be used to develop gas-lift stability maps (Fairuzov et al., 2004; Poblano et al., 2002), which significantly reduce the time required for the analysis. The disadvantage of the linear stability analysis is that it only predicts the onset of instability and cannot be used to model the operation of well in the unstable region. The second approach to studying flow instability in gas liftwells, the non-linear analysis, is usually based on numerical modeling of multiphase flow in the tubing. This technique has been used to develop active control systems to eliminate heading (Dalsmo et al., 2002; Eikrem et al., 2002, 2004; Hu and Golan, 2003; Jansen et al., 1999; Scibilia et al., 2008; Sine gre et al., 2005).
I. Guerrero-Sarabia, Y.V. Fairuzov / Journal of Petroleum Science and Engineering 108 (2013) 162–171
Subscripts
Nomenclature CD CS d DH f g J K k M m p q R T t V v x y z
163
discharge coefficient flow coefficient orifice size, L, m (ft) vertical distance between the injection point and the perforated interval, L, m (ft) friction factor acceleration of gravity, L/t2, m/s2 (ft/s2) productivity index, L4t/m, std. m3/s Pa (scf/s psi) production choke’s loss coefficient ratio of specific heats molecular weight, M, kg/kg mol (lbm/lbm mol) mass flow rate, m/t, kg/s (lbm/s) pressure, m/Lt2, Pa (psia) flow rate, L3/t, m3/s (ft3/s) universal gas constant, L2m/Tt2M, N m/kmol K (ft lbf/ lbm mol R) temperature, T, K (R) time, t, s volume, L3, m3 (ft3) velocity, L/t, m/s (ft/s) axial coordinate, L, m (ft) ratio of downstream and upstream pressure of the downhole gas-lift orifice valve gas compressibility factor
avg c ch f g i L m o R r sc sv t w
average casing downhole gas-lift orifice valve reservoir fluids gas injection point liquid mixture oil reservoir relative standard conditions surface gas-lift valve tubing water
Greeks
a g r
holdup specific gravity of gas density, m/L3, kg/m3 (lbm/ft3)
The non-linear analysis can be used to study the system behavior when the operating parameters of a gas-lift well exceed the stability limits. The non-linear well models predict the amplitude and frequency of oscillations of flow parameters (tubing and casing pressure, liquid and gas flow rates, liquid holdup, etc.) The main disadvantage of this method is that is time-consuming. Also, it takes a lot of efforts to obtain an agreement between the model predictions and field data for all operating conditions of a well. In this paper, the results, obtained using the linear gas-lift stability theory for an offshore gas-lift well with unstable flow due to casing heading, are compared to those of a non-linear analysis. The analysis of non-linear dynamics and stability of the well was performed using direct numerical integration in the time domain of the governing equations of multiphase flow in the tubing. Stability boundaries predicted by both linear and nonlinear analysis are compared with field data using gas-lift stability maps. The effect of the main well design and flow parameters on the frequency and amplitude of the oscillations during heading is studied. Density-wave instability in a well with constant gas injection rate is also analyzed.
modeling of gas-lift wells producing below the bubblepoint pressure. To take into account the presence of gas below the injection point, the derivative of the wellbore fluid flow rate with respect to the tubing pressure at the injection point can be calculated as follows:
2. Linear stability analysis
3. Non-linear analysis
In the present study, the linear analysis was carried out using two gas-lift stability criteria proposed by Asheim (1988). The first Asheim’s stability criterion can expressed in the following form:
A transient gas-lift well model was developed to perform the non-linear stability analysis. It comprises of a model of transient three-phase gas–oil–water flow in the wellbore, a transient model of gas flow in the casing annulus, and a pseudo-steady flow model in the reservoir. The drift flux model (Kim and Doster, 1991; Liles and Reed, 1978; Xiao et al., 1994) was used to simulate multiphase flow in the well. The proposed model is based on three mass conservation equations (for mixture, oil, and water) and the mixture momentum equation (Appendix A). The flow patterns considered are bubble flow and slug flow. The thermodynamic properties of
F1 ¼
ð1=qf i ÞUðdqf i =dpti Þ 41 ð1=qgi ÞUðdqgi =dpti Þ
ð1Þ
The flow is stable when the inequality given by Eq. (1) is satisfied. To obtain this criterion, Asheim assumed that flow is incompressible below the injection point and the production rate at the perforated interval is a linear function of the bottomhole flowing pressure. These assumptions cannot be used for the
dqf i dqf i dpwf ¼ dpti dpwf dpti
ð2Þ
The first derivate from the right hand side of Eq. (2) can be easily calculated from the Vogel’s equation (Vogel, 1968). The second derivative can be calculated either numerically using a correlation of two-phase flow or analytically assuming that the flow is homogeneous and the pressure gradient due to friction is negligible, i.e. pwf ¼pti þ rm,avggDH. In the homogeneous flow, the mixture density is only a function of pressure and the calculation of its derivative with respect to pressure is straightforward. In this work, the gas solubility required for the calculation of the no-slip holdup was calculated using the Standing correlation (Standing, 1957). To take into account the design of gas injection system the original second criterion proposed by Asheim (1988) was used. Both criteria were used to carry out the linear stability analysis.
164
I. Guerrero-Sarabia, Y.V. Fairuzov / Journal of Petroleum Science and Engineering 108 (2013) 162–171
phases are calculated using empirical black oil correlations (Brill and Mukherjee, 1999). The system of non-linear partial differential equations was solved numerically using a semi-implicit method (Liles and Reed, 1978; Xiao et al., 1994). This method is based on replacing the system of differential equations with a system of finite-difference equations, which are partially implicit in time. The scalar quantities (pressure and density) are obtained at the cell centers, and vector quantities are defined at the cell boundaries. All implicit terms are linear in the new time variables. A detailed description of the gas-lift model including the initial boundary and boundary conditions is presented in Appendix A. To perform the non-linear analysis, the response of the system, which is initially at equilibrium, to a perturbation of finite amplitude of a system parameter (orifice size, gas injection rate, injection point depth, surface choke size, etc.) was predicted. Operation of the well during at least four hours was simulated to determine whether the system is stable or not. Then, this procedure was repeated for different initial equilibrium conditions to determine the stability boundary. Also, a series of transient simulations were performed to investigate the influence of well design parameters on the amplitude and frequency of oscillations of main flow parameters when the well operates in the unstable region.
4. Comparison of linear and non-linear stability analysis results Both types of stability analysis were performed to study flow stability in a typical gas-lift well of an offshore oil field in the Gulf of Mexico (Fairuzov et al., 2004). Well design and operation data are presented in Tables 1–4. A mesh consisting of 66 numerical cells was used in the transient model of the well; the length of each cell was 150 ft. It was difficult to start up the well and maintain continuous flow at gas injection flow rates lower than 3 MMscf/D. In the range of injection rates from 3 to 4 MMscf/D, unstable flow was observed with the amplitude of wellhead pressure variations of 3–4 kg/cm2 and frequency of oscillations from 2 to 4 cycles per hour. Finally, stable flow was achieved by increasing the gas injection rate up to 4.5 MMscf/D. The well tests were carried out at wellhead pressures from 12 to 14 kg/cm2. So, according to the field data the onset of instability occurred at a gas injection rate between 4 and 4.5 MMscf/D. Stability and operability boundaries predicted by linear and non-linear analysis are presented in the gas-lift stability maps shown in Fig. 1. The region of stable operation is always above of the stability boundary. The operability boundary predicted by the linear analysis (Fig. 2) shows typical wellhead pressure and liquid flow rate variations predicted the well transient model when the well is operated in the unstable region. Field data (Fairuzov et al., 2004) corresponding to stable and unstable flow conditions are also presented in Fig. 1. It is important to note that initially the
non-linear analysis was performed using a transient well model, which did not include the surface choke model (fixed pressure was used as a boundary condition at the wellhead). As can be seen, this model significantly over predicts the unstable region. The model, which takes into account the choking at the wellhead, provides a better agreement with the data. Both types of analysis reproduce field data published by Fairuzov et al. (2004). The stability boundaries predicted by the two methods employed in this study practically coincide in the range gas injection rate 3.5–4.6 MMscf/D. At injection flow rates lower than 3.5 MMscf/D the discrepancy between the linear and non-linear analyses increases. The stability boundary obtained using the linear stability theory is above that determined with help of transient simulations. This may be attributed to the following. The linear stability analysis only predicts the onset of instability caused by an infinitesimal disturbance. Outside the unstable region, the disturbances can either decay or grow resulting in heading. The former condition cannot be predicted by the linear stability theory. Table 2 Injection/production data. Wellhead pressure Wellhead temperature Water cut Liquid rate Gauge depth Gauge pressure Formation gas–oil ratio Lift gas rate Injection depth Orifice diameter Injection pressure Choke size
13.1 140 0 7632 2695 93.7 360 5.5 1882 48 68 2.5
kg/cm2 1F Percent STB/D m kg/cm2 scf/STB MMscf/D m 64th in. kg/cm2 in.
Table 3 Deviation survey. Measured depth (m)
True vertical Tubing inside depth diameter (m) (in.)
Tubing outside diameter (in.)
Casing inside diameter (in.)
720.0 1290.0 1380.0 1582.9 1650.0 1800.0 1860.0 1878.5 1980.0 2100.0 2106.6 2112.0 2310.0 2610.0 2670.0 2740.0
720.0 1283.4 1368.6 1550.1 1610.1 1731.2 1775.9 1789.3 1862.9 1952.5 1957.6 1961.8 2115.2 2345.3 2389.8 2436.4
7.625 7.625 7.625 7.625 7.625 7.625 7.625 7.625 7.625 7.625 7.625 7.625 – – – –
10.685 10.685 10.685 10.685 10.685 10.685 10.685 10.685 10.685 10.685 10.685 8.53 8.53 8.53 8.53 8.53
6.765 6.765 6.765 6.765 6.765 6.765 6.765 6.765 6.765 6.765 6.765 6.765 – – – –
Table 1 Well data. Reservoir pressure Reservoir temperature Water cut Liquid rate Flowing bottomhole pressure Oil gravity Gas gravity Gas lift gravity Solution GOR
108.5 217 0 7632 95.9 21.4 0.922 0.675 360
kg/cm2 1F Percent STB/D kg/cm2 API Sp. gravity Sp. gravity scf/STB
Table 4 Field data. Lift gas rate (MMscf/D)
Wellhead pressure (kg/cm2)
Flow
4.0 4.5 5.0 5.5
12.0 13.0 12.5 13.5
Unstable Stable Stable Stable
I. Guerrero-Sarabia, Y.V. Fairuzov / Journal of Petroleum Science and Engineering 108 (2013) 162–171
Wellhead Pressure [kg/cm2]
20
165
Stability Boundaries Linear Analysis
NonOperational
18
Non-Linear Analysis (with surface choke)
Stable
16
Non-Linear Analysis (without surface choke)
Operability Boundaries 14
Steady State Model Transient Model
Unstable
12
Field Data Stable
10 1
2
3
4
5
6
Unstable
Gas Injection [MMscf/d]
14
4.5
12 4.4
10
4.3
8 6
4.2
4 4.1
2
Gas Injection, MMscf/D
4.6
16
Oil Rate, STB/D x 103
Wellhead Pressure, kg/cm2
Fig. 1. Comparison of gas-lift stability boundaries predicted by linear and non-linear analyses.
4.0
0 0
8
16
24
32
40
Time, hr Fig. 2. Predicted variations of wellhead pressure and oil flow rate caused by a reduction of the gas-lift flow rate.
The well operability boundaries predicted by linear and nonlinear models are also shown in Fig. 1. In the linear stability analysis, the region where the well does not produce was predicted by searching the conditions when the problem does not have a steady-state solution. Usually, these operating conditions occur at the upper left part of the stability map, in the region where the wellhead pressure is high and the injection rate is low. In the transient analysis, the operability boundary was determined by a condition when the pressure becomes so high or low that the numerical algorithm fails. The transient model predicts a larger region, where the well cannot produce, than the steadystate one. It is important to note that the transient model, which is based on the drift-flux theory, cannot simulate countercurrent flow in the well and back flow to the reservoir. For this reason, in the non-operational region predicted by the transient model under certain conditions the well can still produce, but the model used cannot find a numerical solution due to the above mentioned restrictions. When predicting the well operability boundary, a better agreement between the steady-state and transient models can be obtain using a more complex and robust model of transient multiphase flow, but this is out of the scope of the present study.
5. Effect of gas-lift valve size on oil production rate and flow stability Oil production losses caused by heading have been studied by several authors either by analyzing field data (Faustinelli et al., 1999; Gamaud et al., 1996; Tokar et al., 1996) or by comparing gaslift performance obtained using a steady-state well model to that predicted by a transient model (Avest and Oudeman, 1995; Eikrem
et al., 2004; Hu and Golan, 2003; Jansen et al., 1999; Scibilia et al., 2008). It was found that the optimum point of operation predicted by steady-state models in many cases lies in the unstable flow region and heading may cause a reduction up to 25% in oil production rate. In the present article, a more detailed study of effect of flow instability on the oil production rate was carried out. A series of numerical simulations were performed to predict oil flow rate variations for four sizes of gas-lift orifice valve. As all other parameters were kept constant in the simulations, a comparison of the oil flow rates predicted for different valve sizes enable us to quantify the influence of heading on oil production. The orifice size of 52/64 in. and smaller insures stable well operating conditions with oil production of 6761 STB/D (Fig. 3). The surface choke was simulated by Eq. (A-3). A normalized choke loss coefficient, K/K0, was used in the study to present simulation results for different choke apertures. The loss coefficient used for normalization was determined for a 2.5 in. choke size based on field data. The greater the normalized choke’s loss coefficient the smaller is the choke size. An increase in the orifice size up to 56/64 in. results in unstable well operation and oscillations of oil flow rate. The average daily oil production in this case is of 5721 STB/D; it was calculated by integrating the predicted instantaneous oil flow rate (Fig. 3) for a period of time of one day. A further increase in the orifice size results the increase in the amplitude of oil flow rate oscillations, the frequency of oscillations slightly increases too. Table 5 shows the average oil production rate calculated for the four cases analyzed in this section. The largest reduction in oil production (27%) takes place in case of the most severe heading in the well (flow instability with the largest amplitude of production rate oscillations, orifice size of 64/64 in.) The linear analysis performed in a previous paper (Fairuzov et al., 2004) demonstrated that the reduction of the orifice size of the downhole gas-lift valve has a strong stabilizing effect. In the present study, the effect of the orifice size on the flow behavior was investigated when the well is operated in the unstable region (Figs. 3 and 4); a detailed description of the results obtained was given in the preceding section. Here we only would like to note that in practice well designers tend to oversize the gas-lift valve to avoid its obstruction by solid particles that can enter the gas allocation pipeline network. The oil production losses caused by heading then are compensated by an increase in the gas injection rate. A prediction of the increase in the gas-lift efficiency obtained by decreasing the orifice size can be done by a well transient model to compare saving from the reduction in the lift-gas consumption to the cost of well intervention required to periodically clean up the gas-lift valve. Based on this comparison, an optimal decision can be taken regarding the orifice size and the well intervention program.
166
I. Guerrero-Sarabia, Y.V. Fairuzov / Journal of Petroleum Science and Engineering 108 (2013) 162–171
Oil Flow Rate, STB/D x 103
25
Orifice Size
20
64/64 in. (Qo avg = 4914 STB/D)
15 60/64 in. (Qo avg = 5126 STB/D)
10
56/64 in. (Qo avg = 5721 STB/D) 52/64 in.
5
(Qo avg = 6761 STB/D)
0 0
0.5
1
1.5
2
Time, hr Fig. 3. Effect of the size of the subsurface injection gas-lift valve on the instantaneous liquid flow rate variations (gas-lift flow injection rate ¼ 5 MMscf/D; separator pressure ¼10.5 kg/cm2; injection depth ¼1882 m; water cut ¼0%; GOR ¼345 scf/STB; normalized choke’s loss coefficient, K/K0 ¼ 1).
Table 5 Effect of instability on oil production rate. Orifice Stable size flow (64th in.) Yes/no
Amplitude of oil flow rate oscillation (STB/D)
Average oil production (STB/D)
Average oil production lost (%)
52 56 60 64
0 12,319 18,400 20,083
6761 5721 5126 4914
0 15 24 27
Yes No No No
6. Effect of gas-lift flow rate Operating conditions with several gas-lift injection rates, from 3.75 to 5.25 MMscf/D, were simulated with a practically fully open surface choke. The flow is unstable for gas injection rates lower than 5.25 MMscf/D (Figs. 5 and 6). In the most severe heading case analyzed (with a gas injection of 3.75 MMscf/D), the oil flow rate varies between 0 and 32,000 STB/D with a frequency of 2.8 cycles per hour (Fig. 5), while the amplitude of wellhead pressure variations is about 3.2 kg/cm2 (Fig. 6). The frequency of the oscillations increases and the amplitude decreases with the increase in the gas injection rate until the flow is stabilized.
7. Effect of injection depth The classical gas-lift design methods (Brown, 1967) recommend increasing the depth of the injection point as much as possible to achieve the maximum oil production for a given casing pressure (or compressor power). However, this may result in heading and an increase in the operating costs caused by an increase in the lift gas consumption. Fig. 7 shows a map with the stability boundaries predicted by linear analysis corresponding to three different depths of injection point (1682, 1882 and 2082 m, respectively). As can be seen the region of unstable flow grows as the depth of injection is increased; thus, a higher gas-lift flow rate is required to stabilize the flow for a given wellhead pressure. The transient simulations were carried out to determine the effect of injection depth on flow stability and on the reduction of liquid flow rate. Fig. 8 shows the predicted oil flow rate variations at the surface for the same injection depths considered in the linear analysis. As can be seen, the average liquid flow rate effectively increases from 5046 to 6230 STB/D when the injection
depth increases from 1682 to 1882 m, respectively; the flow instability does not occur. In case on the deepest injection point (2082 m), the average liquid production rate reduces to 5670 STB/ D due to severe heading; the oil flow rate oscillations have a frequency of 2.3 cycles per hour and amplitude of 17,000 STB/D.
8. Effect of choking at surface Choking at the surface has been widely used in practice to control unstable wells, though it may result in a decrease in oil production. In some cases, it was observed on the circular wellhead pressure charts that the amplitude of oscillations during heading is amplified by choking. So, to understand why this can happen, the effect of choking was investigated using the transient well analysis. It is interesting to observe that the amplitude of wellhead pressure oscillations during heading initially grows with choking, from 2.4 to 4.8 kg/cm2 with the increase of K/K0 from 0.1 up to 0.5 (Fig. 9); at greater values of K/K0, the amplitude of wellhead pressure variations diminishes until the flow is eventually stabilized at 12.1 kg/cm2 (K/K0 ¼ 1.3). On the other hand, the range of oil flow rate variations is always narrowed down to the stable value (5820 STB/D) as a result of choking. The most severe variation of oil flow rate (from 0 to 27,300 STB/D) occurs in the case of K/K0 ¼0.1 (a value corresponding to fully open choke). It should be mentioned that depending on the gas-lift flow rate, the oil flow rate oscillations can also be initially amplified (as well as the wellhead pressure), as the choke size is reduced from a fully open position; this behavior was predicted at low gas injection rates. Also, the transition to stable flow may occur rather abruptly and the pressure variation damping effect of choking cannot be seen in some cases. As mentioned on other sections, the amplitude of wellhead pressure oscillations increases when the lift-gas flow rate decreases, for the same choke size. As seen in Figs. 9 and 10, the oscillation frequency initially decreases when the choke size is reduced (at K/K0 from 0.1 and 0.5). Once the damping effect of choking on wellhead pressure oscillations is manifested (K/K0 greater than 0.5), the frequency slightly increases with subsequent choking. It is important to note that choking as a stabilizing method should be used with care: at low gas injection rates it can result in a complete loss of production.
I. Guerrero-Sarabia, Y.V. Fairuzov / Journal of Petroleum Science and Engineering 108 (2013) 162–171
167
Wellhead Pressure, kg/cm2
25
20
Orifice Size 64/64 in. 60/64 in.
15
56/64 in. 52/64 in.
10
5 0
0.5
1
1.5
2
Time, hr Fig. 4. Effect of the size of the subsurface injection gas-lift valve on the wellhead pressure variations (gas-lift flow injection rate¼ 5 MMscf/D; separator pressure¼ 10.5 kg/cm2; injection depth¼1882 m; water cut¼0%; GOR¼ 345 scf/STB; normalized choke’s loss coefficient, K/K0 ¼ 1).
Oil Flow Rate, STB/D x 103
40
32
Gas-Lift Flow Rate, MMscf/D 3.75
24
4.00 4.25
16
4.50 4.75 5.00
8
5.25
0 0
0.2
0.4
0.6
0.8
1
Time, hr Fig. 5. Effect of gas-lift flow rate on instantaneous oil flow rate variations (separator pressure¼ 10.5 kg/cm2; injection depth¼ 1882 m; orifice size ¼48/64 in.; water cut ¼0%; GOR ¼ 345 scf/STB; normalized choke’s loss coefficient, K/K0 ¼0.1).
Wellhead Pressure, kg/cm2
15
14 Gas-Lift Flow Rate, MMscf/D 13 3.75 4.00
12
4.25 4.50
11
4.75 5.00
10
5.25
9 0
0.2
0.4
0.6
0.8
1
Time, hr Fig. 6. Effect of gas-lift flow rate on wellhead pressure variations (separator pressure ¼ 10.5 kg/cm2; injection depth¼1882 m; orifice size¼ 48/64 in.; water cut ¼0%; GOR ¼345 scf/STB; normalized choke’s loss coefficient, K/K0 ¼ 0.1).
168
I. Guerrero-Sarabia, Y.V. Fairuzov / Journal of Petroleum Science and Engineering 108 (2013) 162–171
Wellhead Pressure, kg/cm2
20
Stability Boundaries Injection depth
Non-Operational
18
Stable
2082 m 1882 m
16
1682 m
Operational Boundaries
14
Lower limit 2082 m
12
Unstable
1882 m 1682 m
10 1
2
3
4
5
6
Gas Injection, MMscf/D Fig. 7. Effect of the injection depth on flow stability predicted by linear analysis.
Oil Flow Rate, STB/D x 103
20
16
Injection Depth 2082 m
12 (Qo avg = 5600 STB/D) 1882 m
8
(Qo avg = 6230 STB/D) 1682 m
4
(Qo avg = 5046 STB/D)
0 0
0.5
1
1.5
2
2.5
3
Time, hr Fig. 8. Effect of injection depth on instantaneous oil flow rate variations and average oil flow rate (gas-lift flow injection rate¼ 4.5 MMscf/D; separator pressure¼ 10.5 kg/cm2; orifice size¼48/64 in.; water cut¼0%; GOR¼ 345 scf/STB; normalized choke’s loss coefficient, K/K0 ¼ 1).
Wellhead Pressure, kg/cm2
17
Normalized Choke's Loss Coeficient, K/K0
15
0.1 0.5
13
1.0 1.15 1.20
11
1.25 1.30
9 0
0.5
1
1.5
2
Time, hr Fig. 9. Effect of choke size on instantaneous wellhead pressure variations (gas-lift flow injection rate¼ 4.25 MMscf/D; separator pressure¼ 10.5 kg/cm2; injection depth¼ 1882 m; orifice size¼ 48/64 in.; water cut ¼ 0%; GOR ¼ 345 scf/STB).
9. Effect of separator pressure The effect of separator pressure on flow stability was also analyzed. Wellhead pressure variations corresponding to separator pressures from 10 to 20 kg/cm2 are plotted in Fig. 11. In the
simulations, pressure downstream the choke was equal to the separator pressure and the choke size is 2.5 in. An increase in separator pressure has a destabilizing effect (Fig. 11). Stable flow is predicted for a separator pressure of 10 kg/cm2; in all other cases, flow is unstable. It is interesting to observe that the
I. Guerrero-Sarabia, Y.V. Fairuzov / Journal of Petroleum Science and Engineering 108 (2013) 162–171
169
Oil Flow Rate, STB/D x 103
30
Normalized Choke's Loss Coeficient, K/K0
25
20
0.10 0.50
15
1.00 1.15
10
1.20 1.25
5
1.30
0 0
0.5
1
1.5
2
Time, hr Fig. 10. Effect of choke size on instantaneous oil flow rate variations (gas-lift flow injection rate ¼4.25 MMscf/D; separator pressure ¼10.5 kg/cm2; injection depth¼ 1882 m; orifice size¼ 48/64 in.; water cut ¼0%; GOR ¼ 345 scf/STB).
Wellhead Pressure, kg/cm2
24
21
Separator Pressure, kg/cm 2
18
20.0 17.5
15
15.0 12.5
12
10.0
9 0
0.5
1
1.5
2
Time, hr Fig. 11. Effect of separator pressure on instantaneous wellhead pressure variations (gas-lift flow injection rate¼ 4.5 MMscf/D; injection depth¼ 1882 m; orifice size¼ 48/64 in.; water cut¼0%; GOR¼ 345 scf/STB; normalized choke’s loss coefficient, K/K0 ¼1).
amplitude of the wellhead pressure diminishes with the increase in the separator pressure; e.g., it decreases from 4 to 1 kg/cm2 when the separation pressure is increased from 12.5 to 20 kg/cm2. This behavior can be erroneously interpreted as a stabilizing effect, however actually this oscillation decay is the result of a decrease in the oil flow rate. In the two aforementioned examples, the oil flow rate oscillates from 900 to 12,000 STB/D and from non-production to 5000 STB/D, respectively. The frequency of the oscillations slightly grows as the separator pressure is increased. It was also found that, even if casing heading does not happen, density-wave instability can occur as a result of the reduction of the total pressure drop imposed on the system caused by the separator pressure increase. For the analyzed operational conditions, this kind of instability occurs at the separator pressures higher than 15 kg/cm2. So, in the range of high separator pressures the well can experience two modes of instabilities: casing heading and density-wave oscillations. The presented results coincide with the phenomenon that frequently occurs in offshore gas-lift wells: some wells turns unstable when the pressure in the gathering network increases. This increase can be caused by an excessive gas production from neighborhood wells with problems of gas coning or channeling. When such wells are shut-in, flow is stabilized in the wells experienced heading.
10. Conclusions Linear and non-linear analyses of flow instability in continuous gas-lift wells were performed in this study. The linear analysis is based on a modified gas-lift stability criterion that takes into account compressibility of the mixture below the injection point and is applicable to saturated reservoirs. The analysis of non-linear dynamics and stability of the well was performed using direct numerical integration in the time domain of the governing equations of multiphase flow in the tubing. Stability boundaries predicted by both linear and non-linear analysis were compared with field data published in a previous study; both types of analysis reproduced the data. The effects of the main well design and flow parameters on the frequency and amplitude of the oscillations during heading in a typical gas-lift well were studied. The following conclusions can be drawn from this work:
1. Flow instability results in the oil production loss, which depends on severity of heading. In the gas-lift well analyzed in this study, the largest reduction in oil production (27%) occurs in case of the most severe heading in the well (flow instability with the largest amplitude of production rate
170
2.
3.
4. 5.
6. 7.
I. Guerrero-Sarabia, Y.V. Fairuzov / Journal of Petroleum Science and Engineering 108 (2013) 162–171
oscillations). An increase in the lift gas consumption is required to compensate for the production losses caused by heading. The frequency of the oil flow rate and wellhead pressure oscillations increases with the increase in the gas injection rate. The amplitude of oscillations decreases until the flow is stabilized. A reduction of the choke size at the surface to a certain value results in an increase in the amplitude of wellhead pressure oscillations. After that the wellhead pressure oscillations diminish by choking until the flow is eventually stabilized. The amplitude of oil flow rate oscillations always decreases when the choke size is reduced. An increase in the depth of the injection point may result in heading and an increase in the operating costs caused by the increase in the lift gas consumption. An increase in the separator pressure has a destabilizing effect. At high separator pressures the well can experience two modes of instabilities: casing heading and density-wave oscillations.
The volumetric gas injection rate at standard conditions is related to downstream and upstream pressures of a squared orifice through the following equation (Brill and Mukherjee, 1999): vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi u " 2 k þ 1 # 2 C s C D pc dch u k pti k pti k t qgisc ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðA 11Þ k1 pc pc g T c zc g
A.3. Inflow performance relationship A reservoir inflow performance relationship for pseudosteady-state flow in undersaturated reservoir: qLsc ¼ JUðpR pwf Þ
ðA 12Þ
A.4. State equations Black oil correlations are used to predict PVT properties of oil, gas and water, which are functions of pressure, temperature, gas– oil ratio, and oil, gas and water relative densities.
Appendix A. Governing equations A.5. Boundary conditions A.1. Wellbore model is based on the following equations Mass conservation equation for oil @ ao @ vo ao þ ¼0 @t Bo @x Bo Mass conservation equation for water @ aw @ vw aw þ ¼0 @t Bw @x Bw
ðA 1Þ
Dp ¼ K ðA 2Þ
Mixture mass conservation @rm @ n þ ðrm vm Þ ¼ mgi @x @t
ðA 3Þ
Mixture momentum conservation equation @ @p r vm 2 ðr vm Þ þ þ rm g þ f m @t m @x 2d @ 2 2 þ ðrL aL vL þ rg ag vg Þ ¼ 0 @x
vm ¼
ro ao vo þ rw aw vw þ rg ag vg , rm
ðA 4Þ
ðA 13Þ
2
The flowline was horizontal and short (less than 300), so the pressure downstream the choke was practically the separator pressure. In the reservoir, the specified boundary condition is the reservoir static pressure. In the casing annulus, a constant gas injection rate at surface is specified.
The distributions of flow parameters obtained by system nodal analysis for a specified operational condition are used as the initial condition.
ðA 5Þ A.7. Numerical method ðA 6Þ
rg ag vr , vL ¼ vm rm
ðA 7Þ
rL aL vr , rm
ðA 8Þ
vg ¼ vm þ
rm v2m
A.6. Initial condition
where
rm ¼ ro ao þ rw aw þ rg ag ,
At surface, a constant pressure boundary condition is specified: either the wellhead pressure (fully open choke) or the choke downstream pressure (the separator pressure). The pressure drop through the production choke for subcritical flow can be calculated as follows (Brill and Mukherjee, 1999):
The set of non-linear hyperbolic equations, Eqs. (A-1)–(A-4), is solved using a semi-implicit finite difference method (Kim and Doster, 1991). See Table A.1. Table A.1 SI metric conversion factors.
and
ao þ aw þ ag ¼ 1
ðA 9Þ
A.2. Casing model Casing annulus is modeled as a tank with a constant gas rate at surface and the orifice gas-lift valve at the injection point: VM g dpc þ mgich mgisv ¼ 0 ðA 10Þ zRT c dt
bbl bbl/(psi-D) cp ft ft3 in lbm psi kg/cm2 scf/bbl a
1.589873 2.305916 1.0a 3.048a 2.831685 2.54a 4.535924 6.894757 9.80665 1.801175
Conversion factor is exact.
E 01 E 02 E 03 E 01 E 02 E þ 00 E 01 E þ 00 E þ 01 E 01
¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼
m3 m3/(kPa d) Pa s m m3 cm kg kPa kPa std m3/m3
I. Guerrero-Sarabia, Y.V. Fairuzov / Journal of Petroleum Science and Engineering 108 (2013) 162–171
References Alhanati, F.J.S., Schmidt, Z., Doty, D.R., 1993. Continuous gas-lift instability: diagnosis, criteria, and solutions. Paper SPE 26554 Presented at the 1993 SPE Annual Technical and Exhibition. 3–6 October, Houston, Texas, USA. Asheim, H., 1988. Criteria for gas lift stability. JPT. November, pp. 1452–1456. Avest, D.T., Oudeman, P., 1995. A dynamic simulator to analyse and remedy gas lift problems. Paper SPE 30639, Presented at the 1995 SPE Annual Technical Conference and Exhibition. 22–25 October, Dallas, Texas, USA. Blick, E.F., Enga, P.N., Lin, P.C., 1988. Theoretical stability analysis of flowing oil wells and gas-lift wells. SPEPE. November, pp. 508–514. Brown, K.E., 1967. Gas Lift Theory and Practice. Prentice-Hall, Englewood Cliffs, New Jersey. Brill, J.P., Mukherjee, H., 1999. Multiphase Flow in Wells, first edition Henry L. Doherty Memorial Found of AIME. SPE, Richardson, Texas, pp. 70–71. Dalsmo, M., Halvorsen, E., Slupphaug, O., 2002. Active feedback control of unstable wells at the Brage field. Paper SPE 77650 Presented at the 2002 SPE Annual Technical Conference and Exhibition. 29 September–2 October, San Antonio, Texas, USA. Eikrem, G.O, Foss, B., Imsland, L., Hu, B., 2002. Stabilization of gas lifted wells. IFAC 2002. Eikrem, G.O., Imsland, L., Foss, B., 2004. Stabilization of gas lifted wells based on state estimation. IFAC 2004. Fairuzov, Y. V., Guerrero, I., Morales, C., Carmona, D., Cervantes, T., Herna´ndez, N., Rojas, A., 2004. Stability maps for continuous gas-lift wells: a new approach to solving an old problem. Paper 90644 Presented at the 2004 SPE Annual Technical Conference and Exhibition. 26–29 September, Houston, Texas, USA. Faustinelli, J., Bermu´dez, G., Cuauro, A., 1999. A solution to instability problems in continuous gas-lift wells offshore lake Maracaibo. Paper SPE 53959 Presented at the 1999 SPE Latin America and Caribbean Petroleum Engineering Conference. 21–23 April, Caracas, Venezuela. Gamaud, F., Casagrande, M., Fouillout, C., Lemetayer, P., 1996. New field methods for a maximum lift gas efficiency through stability. Paper SPE 35555 Presented at the 1996 SPE/NPF European Production Operations Conference. 16–17 April, Stavanger, Norway. Grupping, A.W., Luca, C.W.F., Vermulen, F.D., 1984a. Continuous flow gas lift: heading action analyzed for stabilization. Oil Gas J. (July 23), 47–51.
171
Grupping, A.W., Luca, C.W.F., Vermulen, F.D., 1984b. Continuous flow gas lift: these methods can eliminate or control annulus heading. OilGas J. (July 30), 186–192. Hu, B., 2004. Characterizing Gas-Lift Instabilities. Ph.D. Dissertation. Norwegian University of Science and Technology, Trondheim, Norway. Hu, B., Golan, M., 2003. Gas-lift instability resulted production loss and its remedy by feedback control: dynamical simulation results. Paper SPE 84917 Presented at the SPE International Improved Oil Recovery Conference in Asia Pacific. 20–21 October, Kuala Lumpur, Malaysia. Jansen, B., Dalsmo, M., Nøkleberg, L., Havre, K., Kristiansen, V., Lemetayer, P., 1999. Automatic control of unstable gas lifted wells. Paper SPE 56832 Presented at the 1999 SPE Annual Technical Conference and Exhibition. 6–9 October, Houston, Texas, USA. Kim, K., Doster, M., 1991. Appliaction of mixture drift flux equations to vertical separating flows. Nucl. Technol. 95 (July), 103–115. Liles, D.R., Reed, W.H., 1978. A semi-implicit method for two-phase fluid dynamics. J. Comput. Phys. 26, 390–407. Poblano, E., Camacho, R., Fairuzov, Y., 2002. Stability analysis of continuous-flow gas-lift wells. Paper SPE 77732 Presented at the 2002 SPE Annual Technical and Exhibition. 29 September–2 October, San Antonio, Texas, USA. Scibilia, F., Hovd, M., Bitmead, R., 2008. Stabilization of gas-lift oil wells using topside measurements. In: Proceedings of 17th World Congress of the International Federation of Automatic Control. 6–11 July, Seoul, Korea. Sinegre, L., Petit, N., Me´ne´gatti, P., 2005. Distributed delay model for density wave dynamics in gas lifted wells. In: Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference 2005. 12–15 December, Seville, Spain, pp. 7390–7397. Standing, M.B., 1957. A pressure–volume–temperature correlation for mixtures of California oils and gases. Drill. Prod. 275. Tokar, T., Schmidt, Z., Tuckness, C., 1996. New gas lift valve design stabilizes injection rates: case studies. Paper SPE 36597 Presented at the 1996 SPE Annual Technical Conference and Exhibition. 6–9 October, Denver, Colorado, USA. Vogel, J.V., 1968. Inflow Performance relationship for solution-gas drive wells. J. Pet. Technol.. Xiao, J., Alhanati, F.J., Reynolds, A.C., Fuentes-Nucamendi, F., 1994. Modeling and analyzing pressure buildup data affected by phase redistribution in the wellbore. Paper SPE 26965 Presented at the III Latin American and Caribbean Petroleum Engineering Conference. 27–29 April, Buenos Aires, Argentina.
Contents lists available at ScienceDirect
Journal of Petroleum Science and Engineering journal homepage: www.elsevier.com/locate/petrol
Linear and non-linear analysis of flow instability in gas-lift wells I. Guerrero-Sarabia, Y.V. Fairuzov n Institute of Engineering, National Autonomous University of Mexico, Cd. Universitaria, 04510 Mexico City, Mexico
a r t i c l e i n f o
a b s t r a c t
Article history: Received 14 April 2012 Accepted 31 January 2013 Available online 13 February 2013
Linear and non-linear analyses of flow instability in continuous gas-lift wells were performed in this study. The linear analysis is based on a modified gas-lift stability criterion that takes into account compressibility of the mixture below the injection point and is applicable to saturated reservoirs. The analysis of non-linear dynamics and stability of the well was performed using direct numerical integration in the time domain of the governing equations describing the gas-lift system. The transient gas-lift well model developed comprises of a model of transient three-phase gas–oil–water flow in the wellbore, a transient model of gas flow in the casing annulus, and a pseudo-steady flow model in the reservoir. The multiphase flow model used is based on the drift-flux theory. Stability boundaries predicted by both linear and non-linear analysis were compared with field data published in a previous study; both types of analysis reproduced the data. The effects of the main well design and flow parameters on the frequency and amplitude of the oscillations during heading in a typical gas-lift well were studied. It was found that flow instability results in the oil production loss, which depends on severity of heading. The largest reduction in oil production takes place in case of the most severe heading in the well (flow instability with the largest amplitude of production rate oscillations). An increase in the lift gas consumption is required to compensate for the production losses caused by heading. An increase in the depth of the injection point may result in heading and an increase in the operating costs caused by the increase in the lift gas consumption. An increase in the separator pressure has a destabilizing effect. At high separator pressures the well can experience two modes of instabilities: casing heading and density-wave oscillations. & 2013 Published by Elsevier B.V.
Keywords: gas-lift well flow instability heading stability analysis
1. Introduction Flow instability (heading) in continuous flow gas-lift wells has been the subject of many studies over the last three decades (Alhanati et al., 1993; Asheim, 1988; Blick et al., 1988; Fairuzov et al., 2004; Grupping et al., 1984a, 1984b; Hu, 2004; Hu and Golan, 2003; Poblano et al., 2002). Heading is the reason of many problems in the operation of oil production facilities (Alhanati et al., 1993) and finally leads to an increase in the operating costs. Two types of instabilities in gas-lift systems have been identified: casing heading and density-wave instability. The former is associated to variations of the injected gas flow rate caused by variations of the density of the multiphase fluid in the tubing downstream the injection point (Asheim, 1988). The flow in a gaslift well can be also unstable even the downhole gas injection rate is constant due to density-wave oscillations (Hu, 2004; Hu and Golan, 2003). Self-excited pressure and flow rate oscillations in the tubing may either diverge (result in the complete loss of
n
Corresponding author. E-mail address: [email protected] (Y.V. Fairuzov).
0920-4105/$ - see front matter & 2013 Published by Elsevier B.V. http://dx.doi.org/10.1016/j.petrol.2013.01.012
liquid production and gas recirculation) or reach a self-sustained periodic mode (heading). Two different approaches have been proposed in the literature to analyze gas-lift instability. The linear analysis has been used to develop flow stability criteria in terms of flow and well design parameters by different authors (Alhanati et al., 1993; Asheim 1988; Blick et al., 1988). In this type of analysis, the response of the system, which is initially at equilibrium, to an infinitesimal perturbation of tubing pressure at the injection point is predicted. To obtain practical analytical criteria, several strong simplifications in the description of the system are made. The stability criteria can be used to develop gas-lift stability maps (Fairuzov et al., 2004; Poblano et al., 2002), which significantly reduce the time required for the analysis. The disadvantage of the linear stability analysis is that it only predicts the onset of instability and cannot be used to model the operation of well in the unstable region. The second approach to studying flow instability in gas liftwells, the non-linear analysis, is usually based on numerical modeling of multiphase flow in the tubing. This technique has been used to develop active control systems to eliminate heading (Dalsmo et al., 2002; Eikrem et al., 2002, 2004; Hu and Golan, 2003; Jansen et al., 1999; Scibilia et al., 2008; Sine gre et al., 2005).
I. Guerrero-Sarabia, Y.V. Fairuzov / Journal of Petroleum Science and Engineering 108 (2013) 162–171
Subscripts
Nomenclature CD CS d DH f g J K k M m p q R T t V v x y z
163
discharge coefficient flow coefficient orifice size, L, m (ft) vertical distance between the injection point and the perforated interval, L, m (ft) friction factor acceleration of gravity, L/t2, m/s2 (ft/s2) productivity index, L4t/m, std. m3/s Pa (scf/s psi) production choke’s loss coefficient ratio of specific heats molecular weight, M, kg/kg mol (lbm/lbm mol) mass flow rate, m/t, kg/s (lbm/s) pressure, m/Lt2, Pa (psia) flow rate, L3/t, m3/s (ft3/s) universal gas constant, L2m/Tt2M, N m/kmol K (ft lbf/ lbm mol R) temperature, T, K (R) time, t, s volume, L3, m3 (ft3) velocity, L/t, m/s (ft/s) axial coordinate, L, m (ft) ratio of downstream and upstream pressure of the downhole gas-lift orifice valve gas compressibility factor
avg c ch f g i L m o R r sc sv t w
average casing downhole gas-lift orifice valve reservoir fluids gas injection point liquid mixture oil reservoir relative standard conditions surface gas-lift valve tubing water
Greeks
a g r
holdup specific gravity of gas density, m/L3, kg/m3 (lbm/ft3)
The non-linear analysis can be used to study the system behavior when the operating parameters of a gas-lift well exceed the stability limits. The non-linear well models predict the amplitude and frequency of oscillations of flow parameters (tubing and casing pressure, liquid and gas flow rates, liquid holdup, etc.) The main disadvantage of this method is that is time-consuming. Also, it takes a lot of efforts to obtain an agreement between the model predictions and field data for all operating conditions of a well. In this paper, the results, obtained using the linear gas-lift stability theory for an offshore gas-lift well with unstable flow due to casing heading, are compared to those of a non-linear analysis. The analysis of non-linear dynamics and stability of the well was performed using direct numerical integration in the time domain of the governing equations of multiphase flow in the tubing. Stability boundaries predicted by both linear and nonlinear analysis are compared with field data using gas-lift stability maps. The effect of the main well design and flow parameters on the frequency and amplitude of the oscillations during heading is studied. Density-wave instability in a well with constant gas injection rate is also analyzed.
modeling of gas-lift wells producing below the bubblepoint pressure. To take into account the presence of gas below the injection point, the derivative of the wellbore fluid flow rate with respect to the tubing pressure at the injection point can be calculated as follows:
2. Linear stability analysis
3. Non-linear analysis
In the present study, the linear analysis was carried out using two gas-lift stability criteria proposed by Asheim (1988). The first Asheim’s stability criterion can expressed in the following form:
A transient gas-lift well model was developed to perform the non-linear stability analysis. It comprises of a model of transient three-phase gas–oil–water flow in the wellbore, a transient model of gas flow in the casing annulus, and a pseudo-steady flow model in the reservoir. The drift flux model (Kim and Doster, 1991; Liles and Reed, 1978; Xiao et al., 1994) was used to simulate multiphase flow in the well. The proposed model is based on three mass conservation equations (for mixture, oil, and water) and the mixture momentum equation (Appendix A). The flow patterns considered are bubble flow and slug flow. The thermodynamic properties of
F1 ¼
ð1=qf i ÞUðdqf i =dpti Þ 41 ð1=qgi ÞUðdqgi =dpti Þ
ð1Þ
The flow is stable when the inequality given by Eq. (1) is satisfied. To obtain this criterion, Asheim assumed that flow is incompressible below the injection point and the production rate at the perforated interval is a linear function of the bottomhole flowing pressure. These assumptions cannot be used for the
dqf i dqf i dpwf ¼ dpti dpwf dpti
ð2Þ
The first derivate from the right hand side of Eq. (2) can be easily calculated from the Vogel’s equation (Vogel, 1968). The second derivative can be calculated either numerically using a correlation of two-phase flow or analytically assuming that the flow is homogeneous and the pressure gradient due to friction is negligible, i.e. pwf ¼pti þ rm,avggDH. In the homogeneous flow, the mixture density is only a function of pressure and the calculation of its derivative with respect to pressure is straightforward. In this work, the gas solubility required for the calculation of the no-slip holdup was calculated using the Standing correlation (Standing, 1957). To take into account the design of gas injection system the original second criterion proposed by Asheim (1988) was used. Both criteria were used to carry out the linear stability analysis.
164
I. Guerrero-Sarabia, Y.V. Fairuzov / Journal of Petroleum Science and Engineering 108 (2013) 162–171
phases are calculated using empirical black oil correlations (Brill and Mukherjee, 1999). The system of non-linear partial differential equations was solved numerically using a semi-implicit method (Liles and Reed, 1978; Xiao et al., 1994). This method is based on replacing the system of differential equations with a system of finite-difference equations, which are partially implicit in time. The scalar quantities (pressure and density) are obtained at the cell centers, and vector quantities are defined at the cell boundaries. All implicit terms are linear in the new time variables. A detailed description of the gas-lift model including the initial boundary and boundary conditions is presented in Appendix A. To perform the non-linear analysis, the response of the system, which is initially at equilibrium, to a perturbation of finite amplitude of a system parameter (orifice size, gas injection rate, injection point depth, surface choke size, etc.) was predicted. Operation of the well during at least four hours was simulated to determine whether the system is stable or not. Then, this procedure was repeated for different initial equilibrium conditions to determine the stability boundary. Also, a series of transient simulations were performed to investigate the influence of well design parameters on the amplitude and frequency of oscillations of main flow parameters when the well operates in the unstable region.
4. Comparison of linear and non-linear stability analysis results Both types of stability analysis were performed to study flow stability in a typical gas-lift well of an offshore oil field in the Gulf of Mexico (Fairuzov et al., 2004). Well design and operation data are presented in Tables 1–4. A mesh consisting of 66 numerical cells was used in the transient model of the well; the length of each cell was 150 ft. It was difficult to start up the well and maintain continuous flow at gas injection flow rates lower than 3 MMscf/D. In the range of injection rates from 3 to 4 MMscf/D, unstable flow was observed with the amplitude of wellhead pressure variations of 3–4 kg/cm2 and frequency of oscillations from 2 to 4 cycles per hour. Finally, stable flow was achieved by increasing the gas injection rate up to 4.5 MMscf/D. The well tests were carried out at wellhead pressures from 12 to 14 kg/cm2. So, according to the field data the onset of instability occurred at a gas injection rate between 4 and 4.5 MMscf/D. Stability and operability boundaries predicted by linear and non-linear analysis are presented in the gas-lift stability maps shown in Fig. 1. The region of stable operation is always above of the stability boundary. The operability boundary predicted by the linear analysis (Fig. 2) shows typical wellhead pressure and liquid flow rate variations predicted the well transient model when the well is operated in the unstable region. Field data (Fairuzov et al., 2004) corresponding to stable and unstable flow conditions are also presented in Fig. 1. It is important to note that initially the
non-linear analysis was performed using a transient well model, which did not include the surface choke model (fixed pressure was used as a boundary condition at the wellhead). As can be seen, this model significantly over predicts the unstable region. The model, which takes into account the choking at the wellhead, provides a better agreement with the data. Both types of analysis reproduce field data published by Fairuzov et al. (2004). The stability boundaries predicted by the two methods employed in this study practically coincide in the range gas injection rate 3.5–4.6 MMscf/D. At injection flow rates lower than 3.5 MMscf/D the discrepancy between the linear and non-linear analyses increases. The stability boundary obtained using the linear stability theory is above that determined with help of transient simulations. This may be attributed to the following. The linear stability analysis only predicts the onset of instability caused by an infinitesimal disturbance. Outside the unstable region, the disturbances can either decay or grow resulting in heading. The former condition cannot be predicted by the linear stability theory. Table 2 Injection/production data. Wellhead pressure Wellhead temperature Water cut Liquid rate Gauge depth Gauge pressure Formation gas–oil ratio Lift gas rate Injection depth Orifice diameter Injection pressure Choke size
13.1 140 0 7632 2695 93.7 360 5.5 1882 48 68 2.5
kg/cm2 1F Percent STB/D m kg/cm2 scf/STB MMscf/D m 64th in. kg/cm2 in.
Table 3 Deviation survey. Measured depth (m)
True vertical Tubing inside depth diameter (m) (in.)
Tubing outside diameter (in.)
Casing inside diameter (in.)
720.0 1290.0 1380.0 1582.9 1650.0 1800.0 1860.0 1878.5 1980.0 2100.0 2106.6 2112.0 2310.0 2610.0 2670.0 2740.0
720.0 1283.4 1368.6 1550.1 1610.1 1731.2 1775.9 1789.3 1862.9 1952.5 1957.6 1961.8 2115.2 2345.3 2389.8 2436.4
7.625 7.625 7.625 7.625 7.625 7.625 7.625 7.625 7.625 7.625 7.625 7.625 – – – –
10.685 10.685 10.685 10.685 10.685 10.685 10.685 10.685 10.685 10.685 10.685 8.53 8.53 8.53 8.53 8.53
6.765 6.765 6.765 6.765 6.765 6.765 6.765 6.765 6.765 6.765 6.765 6.765 – – – –
Table 1 Well data. Reservoir pressure Reservoir temperature Water cut Liquid rate Flowing bottomhole pressure Oil gravity Gas gravity Gas lift gravity Solution GOR
108.5 217 0 7632 95.9 21.4 0.922 0.675 360
kg/cm2 1F Percent STB/D kg/cm2 API Sp. gravity Sp. gravity scf/STB
Table 4 Field data. Lift gas rate (MMscf/D)
Wellhead pressure (kg/cm2)
Flow
4.0 4.5 5.0 5.5
12.0 13.0 12.5 13.5
Unstable Stable Stable Stable
I. Guerrero-Sarabia, Y.V. Fairuzov / Journal of Petroleum Science and Engineering 108 (2013) 162–171
Wellhead Pressure [kg/cm2]
20
165
Stability Boundaries Linear Analysis
NonOperational
18
Non-Linear Analysis (with surface choke)
Stable
16
Non-Linear Analysis (without surface choke)
Operability Boundaries 14
Steady State Model Transient Model
Unstable
12
Field Data Stable
10 1
2
3
4
5
6
Unstable
Gas Injection [MMscf/d]
14
4.5
12 4.4
10
4.3
8 6
4.2
4 4.1
2
Gas Injection, MMscf/D
4.6
16
Oil Rate, STB/D x 103
Wellhead Pressure, kg/cm2
Fig. 1. Comparison of gas-lift stability boundaries predicted by linear and non-linear analyses.
4.0
0 0
8
16
24
32
40
Time, hr Fig. 2. Predicted variations of wellhead pressure and oil flow rate caused by a reduction of the gas-lift flow rate.
The well operability boundaries predicted by linear and nonlinear models are also shown in Fig. 1. In the linear stability analysis, the region where the well does not produce was predicted by searching the conditions when the problem does not have a steady-state solution. Usually, these operating conditions occur at the upper left part of the stability map, in the region where the wellhead pressure is high and the injection rate is low. In the transient analysis, the operability boundary was determined by a condition when the pressure becomes so high or low that the numerical algorithm fails. The transient model predicts a larger region, where the well cannot produce, than the steadystate one. It is important to note that the transient model, which is based on the drift-flux theory, cannot simulate countercurrent flow in the well and back flow to the reservoir. For this reason, in the non-operational region predicted by the transient model under certain conditions the well can still produce, but the model used cannot find a numerical solution due to the above mentioned restrictions. When predicting the well operability boundary, a better agreement between the steady-state and transient models can be obtain using a more complex and robust model of transient multiphase flow, but this is out of the scope of the present study.
5. Effect of gas-lift valve size on oil production rate and flow stability Oil production losses caused by heading have been studied by several authors either by analyzing field data (Faustinelli et al., 1999; Gamaud et al., 1996; Tokar et al., 1996) or by comparing gaslift performance obtained using a steady-state well model to that predicted by a transient model (Avest and Oudeman, 1995; Eikrem
et al., 2004; Hu and Golan, 2003; Jansen et al., 1999; Scibilia et al., 2008). It was found that the optimum point of operation predicted by steady-state models in many cases lies in the unstable flow region and heading may cause a reduction up to 25% in oil production rate. In the present article, a more detailed study of effect of flow instability on the oil production rate was carried out. A series of numerical simulations were performed to predict oil flow rate variations for four sizes of gas-lift orifice valve. As all other parameters were kept constant in the simulations, a comparison of the oil flow rates predicted for different valve sizes enable us to quantify the influence of heading on oil production. The orifice size of 52/64 in. and smaller insures stable well operating conditions with oil production of 6761 STB/D (Fig. 3). The surface choke was simulated by Eq. (A-3). A normalized choke loss coefficient, K/K0, was used in the study to present simulation results for different choke apertures. The loss coefficient used for normalization was determined for a 2.5 in. choke size based on field data. The greater the normalized choke’s loss coefficient the smaller is the choke size. An increase in the orifice size up to 56/64 in. results in unstable well operation and oscillations of oil flow rate. The average daily oil production in this case is of 5721 STB/D; it was calculated by integrating the predicted instantaneous oil flow rate (Fig. 3) for a period of time of one day. A further increase in the orifice size results the increase in the amplitude of oil flow rate oscillations, the frequency of oscillations slightly increases too. Table 5 shows the average oil production rate calculated for the four cases analyzed in this section. The largest reduction in oil production (27%) takes place in case of the most severe heading in the well (flow instability with the largest amplitude of production rate oscillations, orifice size of 64/64 in.) The linear analysis performed in a previous paper (Fairuzov et al., 2004) demonstrated that the reduction of the orifice size of the downhole gas-lift valve has a strong stabilizing effect. In the present study, the effect of the orifice size on the flow behavior was investigated when the well is operated in the unstable region (Figs. 3 and 4); a detailed description of the results obtained was given in the preceding section. Here we only would like to note that in practice well designers tend to oversize the gas-lift valve to avoid its obstruction by solid particles that can enter the gas allocation pipeline network. The oil production losses caused by heading then are compensated by an increase in the gas injection rate. A prediction of the increase in the gas-lift efficiency obtained by decreasing the orifice size can be done by a well transient model to compare saving from the reduction in the lift-gas consumption to the cost of well intervention required to periodically clean up the gas-lift valve. Based on this comparison, an optimal decision can be taken regarding the orifice size and the well intervention program.
166
I. Guerrero-Sarabia, Y.V. Fairuzov / Journal of Petroleum Science and Engineering 108 (2013) 162–171
Oil Flow Rate, STB/D x 103
25
Orifice Size
20
64/64 in. (Qo avg = 4914 STB/D)
15 60/64 in. (Qo avg = 5126 STB/D)
10
56/64 in. (Qo avg = 5721 STB/D) 52/64 in.
5
(Qo avg = 6761 STB/D)
0 0
0.5
1
1.5
2
Time, hr Fig. 3. Effect of the size of the subsurface injection gas-lift valve on the instantaneous liquid flow rate variations (gas-lift flow injection rate ¼ 5 MMscf/D; separator pressure ¼10.5 kg/cm2; injection depth ¼1882 m; water cut ¼0%; GOR ¼345 scf/STB; normalized choke’s loss coefficient, K/K0 ¼ 1).
Table 5 Effect of instability on oil production rate. Orifice Stable size flow (64th in.) Yes/no
Amplitude of oil flow rate oscillation (STB/D)
Average oil production (STB/D)
Average oil production lost (%)
52 56 60 64
0 12,319 18,400 20,083
6761 5721 5126 4914
0 15 24 27
Yes No No No
6. Effect of gas-lift flow rate Operating conditions with several gas-lift injection rates, from 3.75 to 5.25 MMscf/D, were simulated with a practically fully open surface choke. The flow is unstable for gas injection rates lower than 5.25 MMscf/D (Figs. 5 and 6). In the most severe heading case analyzed (with a gas injection of 3.75 MMscf/D), the oil flow rate varies between 0 and 32,000 STB/D with a frequency of 2.8 cycles per hour (Fig. 5), while the amplitude of wellhead pressure variations is about 3.2 kg/cm2 (Fig. 6). The frequency of the oscillations increases and the amplitude decreases with the increase in the gas injection rate until the flow is stabilized.
7. Effect of injection depth The classical gas-lift design methods (Brown, 1967) recommend increasing the depth of the injection point as much as possible to achieve the maximum oil production for a given casing pressure (or compressor power). However, this may result in heading and an increase in the operating costs caused by an increase in the lift gas consumption. Fig. 7 shows a map with the stability boundaries predicted by linear analysis corresponding to three different depths of injection point (1682, 1882 and 2082 m, respectively). As can be seen the region of unstable flow grows as the depth of injection is increased; thus, a higher gas-lift flow rate is required to stabilize the flow for a given wellhead pressure. The transient simulations were carried out to determine the effect of injection depth on flow stability and on the reduction of liquid flow rate. Fig. 8 shows the predicted oil flow rate variations at the surface for the same injection depths considered in the linear analysis. As can be seen, the average liquid flow rate effectively increases from 5046 to 6230 STB/D when the injection
depth increases from 1682 to 1882 m, respectively; the flow instability does not occur. In case on the deepest injection point (2082 m), the average liquid production rate reduces to 5670 STB/ D due to severe heading; the oil flow rate oscillations have a frequency of 2.3 cycles per hour and amplitude of 17,000 STB/D.
8. Effect of choking at surface Choking at the surface has been widely used in practice to control unstable wells, though it may result in a decrease in oil production. In some cases, it was observed on the circular wellhead pressure charts that the amplitude of oscillations during heading is amplified by choking. So, to understand why this can happen, the effect of choking was investigated using the transient well analysis. It is interesting to observe that the amplitude of wellhead pressure oscillations during heading initially grows with choking, from 2.4 to 4.8 kg/cm2 with the increase of K/K0 from 0.1 up to 0.5 (Fig. 9); at greater values of K/K0, the amplitude of wellhead pressure variations diminishes until the flow is eventually stabilized at 12.1 kg/cm2 (K/K0 ¼ 1.3). On the other hand, the range of oil flow rate variations is always narrowed down to the stable value (5820 STB/D) as a result of choking. The most severe variation of oil flow rate (from 0 to 27,300 STB/D) occurs in the case of K/K0 ¼0.1 (a value corresponding to fully open choke). It should be mentioned that depending on the gas-lift flow rate, the oil flow rate oscillations can also be initially amplified (as well as the wellhead pressure), as the choke size is reduced from a fully open position; this behavior was predicted at low gas injection rates. Also, the transition to stable flow may occur rather abruptly and the pressure variation damping effect of choking cannot be seen in some cases. As mentioned on other sections, the amplitude of wellhead pressure oscillations increases when the lift-gas flow rate decreases, for the same choke size. As seen in Figs. 9 and 10, the oscillation frequency initially decreases when the choke size is reduced (at K/K0 from 0.1 and 0.5). Once the damping effect of choking on wellhead pressure oscillations is manifested (K/K0 greater than 0.5), the frequency slightly increases with subsequent choking. It is important to note that choking as a stabilizing method should be used with care: at low gas injection rates it can result in a complete loss of production.
I. Guerrero-Sarabia, Y.V. Fairuzov / Journal of Petroleum Science and Engineering 108 (2013) 162–171
167
Wellhead Pressure, kg/cm2
25
20
Orifice Size 64/64 in. 60/64 in.
15
56/64 in. 52/64 in.
10
5 0
0.5
1
1.5
2
Time, hr Fig. 4. Effect of the size of the subsurface injection gas-lift valve on the wellhead pressure variations (gas-lift flow injection rate¼ 5 MMscf/D; separator pressure¼ 10.5 kg/cm2; injection depth¼1882 m; water cut¼0%; GOR¼ 345 scf/STB; normalized choke’s loss coefficient, K/K0 ¼ 1).
Oil Flow Rate, STB/D x 103
40
32
Gas-Lift Flow Rate, MMscf/D 3.75
24
4.00 4.25
16
4.50 4.75 5.00
8
5.25
0 0
0.2
0.4
0.6
0.8
1
Time, hr Fig. 5. Effect of gas-lift flow rate on instantaneous oil flow rate variations (separator pressure¼ 10.5 kg/cm2; injection depth¼ 1882 m; orifice size ¼48/64 in.; water cut ¼0%; GOR ¼ 345 scf/STB; normalized choke’s loss coefficient, K/K0 ¼0.1).
Wellhead Pressure, kg/cm2
15
14 Gas-Lift Flow Rate, MMscf/D 13 3.75 4.00
12
4.25 4.50
11
4.75 5.00
10
5.25
9 0
0.2
0.4
0.6
0.8
1
Time, hr Fig. 6. Effect of gas-lift flow rate on wellhead pressure variations (separator pressure ¼ 10.5 kg/cm2; injection depth¼1882 m; orifice size¼ 48/64 in.; water cut ¼0%; GOR ¼345 scf/STB; normalized choke’s loss coefficient, K/K0 ¼ 0.1).
168
I. Guerrero-Sarabia, Y.V. Fairuzov / Journal of Petroleum Science and Engineering 108 (2013) 162–171
Wellhead Pressure, kg/cm2
20
Stability Boundaries Injection depth
Non-Operational
18
Stable
2082 m 1882 m
16
1682 m
Operational Boundaries
14
Lower limit 2082 m
12
Unstable
1882 m 1682 m
10 1
2
3
4
5
6
Gas Injection, MMscf/D Fig. 7. Effect of the injection depth on flow stability predicted by linear analysis.
Oil Flow Rate, STB/D x 103
20
16
Injection Depth 2082 m
12 (Qo avg = 5600 STB/D) 1882 m
8
(Qo avg = 6230 STB/D) 1682 m
4
(Qo avg = 5046 STB/D)
0 0
0.5
1
1.5
2
2.5
3
Time, hr Fig. 8. Effect of injection depth on instantaneous oil flow rate variations and average oil flow rate (gas-lift flow injection rate¼ 4.5 MMscf/D; separator pressure¼ 10.5 kg/cm2; orifice size¼48/64 in.; water cut¼0%; GOR¼ 345 scf/STB; normalized choke’s loss coefficient, K/K0 ¼ 1).
Wellhead Pressure, kg/cm2
17
Normalized Choke's Loss Coeficient, K/K0
15
0.1 0.5
13
1.0 1.15 1.20
11
1.25 1.30
9 0
0.5
1
1.5
2
Time, hr Fig. 9. Effect of choke size on instantaneous wellhead pressure variations (gas-lift flow injection rate¼ 4.25 MMscf/D; separator pressure¼ 10.5 kg/cm2; injection depth¼ 1882 m; orifice size¼ 48/64 in.; water cut ¼ 0%; GOR ¼ 345 scf/STB).
9. Effect of separator pressure The effect of separator pressure on flow stability was also analyzed. Wellhead pressure variations corresponding to separator pressures from 10 to 20 kg/cm2 are plotted in Fig. 11. In the
simulations, pressure downstream the choke was equal to the separator pressure and the choke size is 2.5 in. An increase in separator pressure has a destabilizing effect (Fig. 11). Stable flow is predicted for a separator pressure of 10 kg/cm2; in all other cases, flow is unstable. It is interesting to observe that the
I. Guerrero-Sarabia, Y.V. Fairuzov / Journal of Petroleum Science and Engineering 108 (2013) 162–171
169
Oil Flow Rate, STB/D x 103
30
Normalized Choke's Loss Coeficient, K/K0
25
20
0.10 0.50
15
1.00 1.15
10
1.20 1.25
5
1.30
0 0
0.5
1
1.5
2
Time, hr Fig. 10. Effect of choke size on instantaneous oil flow rate variations (gas-lift flow injection rate ¼4.25 MMscf/D; separator pressure ¼10.5 kg/cm2; injection depth¼ 1882 m; orifice size¼ 48/64 in.; water cut ¼0%; GOR ¼ 345 scf/STB).
Wellhead Pressure, kg/cm2
24
21
Separator Pressure, kg/cm 2
18
20.0 17.5
15
15.0 12.5
12
10.0
9 0
0.5
1
1.5
2
Time, hr Fig. 11. Effect of separator pressure on instantaneous wellhead pressure variations (gas-lift flow injection rate¼ 4.5 MMscf/D; injection depth¼ 1882 m; orifice size¼ 48/64 in.; water cut¼0%; GOR¼ 345 scf/STB; normalized choke’s loss coefficient, K/K0 ¼1).
amplitude of the wellhead pressure diminishes with the increase in the separator pressure; e.g., it decreases from 4 to 1 kg/cm2 when the separation pressure is increased from 12.5 to 20 kg/cm2. This behavior can be erroneously interpreted as a stabilizing effect, however actually this oscillation decay is the result of a decrease in the oil flow rate. In the two aforementioned examples, the oil flow rate oscillates from 900 to 12,000 STB/D and from non-production to 5000 STB/D, respectively. The frequency of the oscillations slightly grows as the separator pressure is increased. It was also found that, even if casing heading does not happen, density-wave instability can occur as a result of the reduction of the total pressure drop imposed on the system caused by the separator pressure increase. For the analyzed operational conditions, this kind of instability occurs at the separator pressures higher than 15 kg/cm2. So, in the range of high separator pressures the well can experience two modes of instabilities: casing heading and density-wave oscillations. The presented results coincide with the phenomenon that frequently occurs in offshore gas-lift wells: some wells turns unstable when the pressure in the gathering network increases. This increase can be caused by an excessive gas production from neighborhood wells with problems of gas coning or channeling. When such wells are shut-in, flow is stabilized in the wells experienced heading.
10. Conclusions Linear and non-linear analyses of flow instability in continuous gas-lift wells were performed in this study. The linear analysis is based on a modified gas-lift stability criterion that takes into account compressibility of the mixture below the injection point and is applicable to saturated reservoirs. The analysis of non-linear dynamics and stability of the well was performed using direct numerical integration in the time domain of the governing equations of multiphase flow in the tubing. Stability boundaries predicted by both linear and non-linear analysis were compared with field data published in a previous study; both types of analysis reproduced the data. The effects of the main well design and flow parameters on the frequency and amplitude of the oscillations during heading in a typical gas-lift well were studied. The following conclusions can be drawn from this work:
1. Flow instability results in the oil production loss, which depends on severity of heading. In the gas-lift well analyzed in this study, the largest reduction in oil production (27%) occurs in case of the most severe heading in the well (flow instability with the largest amplitude of production rate
170
2.
3.
4. 5.
6. 7.
I. Guerrero-Sarabia, Y.V. Fairuzov / Journal of Petroleum Science and Engineering 108 (2013) 162–171
oscillations). An increase in the lift gas consumption is required to compensate for the production losses caused by heading. The frequency of the oil flow rate and wellhead pressure oscillations increases with the increase in the gas injection rate. The amplitude of oscillations decreases until the flow is stabilized. A reduction of the choke size at the surface to a certain value results in an increase in the amplitude of wellhead pressure oscillations. After that the wellhead pressure oscillations diminish by choking until the flow is eventually stabilized. The amplitude of oil flow rate oscillations always decreases when the choke size is reduced. An increase in the depth of the injection point may result in heading and an increase in the operating costs caused by the increase in the lift gas consumption. An increase in the separator pressure has a destabilizing effect. At high separator pressures the well can experience two modes of instabilities: casing heading and density-wave oscillations.
The volumetric gas injection rate at standard conditions is related to downstream and upstream pressures of a squared orifice through the following equation (Brill and Mukherjee, 1999): vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi u " 2 k þ 1 # 2 C s C D pc dch u k pti k pti k t qgisc ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðA 11Þ k1 pc pc g T c zc g
A.3. Inflow performance relationship A reservoir inflow performance relationship for pseudosteady-state flow in undersaturated reservoir: qLsc ¼ JUðpR pwf Þ
ðA 12Þ
A.4. State equations Black oil correlations are used to predict PVT properties of oil, gas and water, which are functions of pressure, temperature, gas– oil ratio, and oil, gas and water relative densities.
Appendix A. Governing equations A.5. Boundary conditions A.1. Wellbore model is based on the following equations Mass conservation equation for oil @ ao @ vo ao þ ¼0 @t Bo @x Bo Mass conservation equation for water @ aw @ vw aw þ ¼0 @t Bw @x Bw
ðA 1Þ
Dp ¼ K ðA 2Þ
Mixture mass conservation @rm @ n þ ðrm vm Þ ¼ mgi @x @t
ðA 3Þ
Mixture momentum conservation equation @ @p r vm 2 ðr vm Þ þ þ rm g þ f m @t m @x 2d @ 2 2 þ ðrL aL vL þ rg ag vg Þ ¼ 0 @x
vm ¼
ro ao vo þ rw aw vw þ rg ag vg , rm
ðA 4Þ
ðA 13Þ
2
The flowline was horizontal and short (less than 300), so the pressure downstream the choke was practically the separator pressure. In the reservoir, the specified boundary condition is the reservoir static pressure. In the casing annulus, a constant gas injection rate at surface is specified.
The distributions of flow parameters obtained by system nodal analysis for a specified operational condition are used as the initial condition.
ðA 5Þ A.7. Numerical method ðA 6Þ
rg ag vr , vL ¼ vm rm
ðA 7Þ
rL aL vr , rm
ðA 8Þ
vg ¼ vm þ
rm v2m
A.6. Initial condition
where
rm ¼ ro ao þ rw aw þ rg ag ,
At surface, a constant pressure boundary condition is specified: either the wellhead pressure (fully open choke) or the choke downstream pressure (the separator pressure). The pressure drop through the production choke for subcritical flow can be calculated as follows (Brill and Mukherjee, 1999):
The set of non-linear hyperbolic equations, Eqs. (A-1)–(A-4), is solved using a semi-implicit finite difference method (Kim and Doster, 1991). See Table A.1. Table A.1 SI metric conversion factors.
and
ao þ aw þ ag ¼ 1
ðA 9Þ
A.2. Casing model Casing annulus is modeled as a tank with a constant gas rate at surface and the orifice gas-lift valve at the injection point: VM g dpc þ mgich mgisv ¼ 0 ðA 10Þ zRT c dt
bbl bbl/(psi-D) cp ft ft3 in lbm psi kg/cm2 scf/bbl a
1.589873 2.305916 1.0a 3.048a 2.831685 2.54a 4.535924 6.894757 9.80665 1.801175
Conversion factor is exact.
E 01 E 02 E 03 E 01 E 02 E þ 00 E 01 E þ 00 E þ 01 E 01
¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼
m3 m3/(kPa d) Pa s m m3 cm kg kPa kPa std m3/m3
I. Guerrero-Sarabia, Y.V. Fairuzov / Journal of Petroleum Science and Engineering 108 (2013) 162–171
References Alhanati, F.J.S., Schmidt, Z., Doty, D.R., 1993. Continuous gas-lift instability: diagnosis, criteria, and solutions. Paper SPE 26554 Presented at the 1993 SPE Annual Technical and Exhibition. 3–6 October, Houston, Texas, USA. Asheim, H., 1988. Criteria for gas lift stability. JPT. November, pp. 1452–1456. Avest, D.T., Oudeman, P., 1995. A dynamic simulator to analyse and remedy gas lift problems. Paper SPE 30639, Presented at the 1995 SPE Annual Technical Conference and Exhibition. 22–25 October, Dallas, Texas, USA. Blick, E.F., Enga, P.N., Lin, P.C., 1988. Theoretical stability analysis of flowing oil wells and gas-lift wells. SPEPE. November, pp. 508–514. Brown, K.E., 1967. Gas Lift Theory and Practice. Prentice-Hall, Englewood Cliffs, New Jersey. Brill, J.P., Mukherjee, H., 1999. Multiphase Flow in Wells, first edition Henry L. Doherty Memorial Found of AIME. SPE, Richardson, Texas, pp. 70–71. Dalsmo, M., Halvorsen, E., Slupphaug, O., 2002. Active feedback control of unstable wells at the Brage field. Paper SPE 77650 Presented at the 2002 SPE Annual Technical Conference and Exhibition. 29 September–2 October, San Antonio, Texas, USA. Eikrem, G.O, Foss, B., Imsland, L., Hu, B., 2002. Stabilization of gas lifted wells. IFAC 2002. Eikrem, G.O., Imsland, L., Foss, B., 2004. Stabilization of gas lifted wells based on state estimation. IFAC 2004. Fairuzov, Y. V., Guerrero, I., Morales, C., Carmona, D., Cervantes, T., Herna´ndez, N., Rojas, A., 2004. Stability maps for continuous gas-lift wells: a new approach to solving an old problem. Paper 90644 Presented at the 2004 SPE Annual Technical Conference and Exhibition. 26–29 September, Houston, Texas, USA. Faustinelli, J., Bermu´dez, G., Cuauro, A., 1999. A solution to instability problems in continuous gas-lift wells offshore lake Maracaibo. Paper SPE 53959 Presented at the 1999 SPE Latin America and Caribbean Petroleum Engineering Conference. 21–23 April, Caracas, Venezuela. Gamaud, F., Casagrande, M., Fouillout, C., Lemetayer, P., 1996. New field methods for a maximum lift gas efficiency through stability. Paper SPE 35555 Presented at the 1996 SPE/NPF European Production Operations Conference. 16–17 April, Stavanger, Norway. Grupping, A.W., Luca, C.W.F., Vermulen, F.D., 1984a. Continuous flow gas lift: heading action analyzed for stabilization. Oil Gas J. (July 23), 47–51.
171
Grupping, A.W., Luca, C.W.F., Vermulen, F.D., 1984b. Continuous flow gas lift: these methods can eliminate or control annulus heading. OilGas J. (July 30), 186–192. Hu, B., 2004. Characterizing Gas-Lift Instabilities. Ph.D. Dissertation. Norwegian University of Science and Technology, Trondheim, Norway. Hu, B., Golan, M., 2003. Gas-lift instability resulted production loss and its remedy by feedback control: dynamical simulation results. Paper SPE 84917 Presented at the SPE International Improved Oil Recovery Conference in Asia Pacific. 20–21 October, Kuala Lumpur, Malaysia. Jansen, B., Dalsmo, M., Nøkleberg, L., Havre, K., Kristiansen, V., Lemetayer, P., 1999. Automatic control of unstable gas lifted wells. Paper SPE 56832 Presented at the 1999 SPE Annual Technical Conference and Exhibition. 6–9 October, Houston, Texas, USA. Kim, K., Doster, M., 1991. Appliaction of mixture drift flux equations to vertical separating flows. Nucl. Technol. 95 (July), 103–115. Liles, D.R., Reed, W.H., 1978. A semi-implicit method for two-phase fluid dynamics. J. Comput. Phys. 26, 390–407. Poblano, E., Camacho, R., Fairuzov, Y., 2002. Stability analysis of continuous-flow gas-lift wells. Paper SPE 77732 Presented at the 2002 SPE Annual Technical and Exhibition. 29 September–2 October, San Antonio, Texas, USA. Scibilia, F., Hovd, M., Bitmead, R., 2008. Stabilization of gas-lift oil wells using topside measurements. In: Proceedings of 17th World Congress of the International Federation of Automatic Control. 6–11 July, Seoul, Korea. Sinegre, L., Petit, N., Me´ne´gatti, P., 2005. Distributed delay model for density wave dynamics in gas lifted wells. In: Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference 2005. 12–15 December, Seville, Spain, pp. 7390–7397. Standing, M.B., 1957. A pressure–volume–temperature correlation for mixtures of California oils and gases. Drill. Prod. 275. Tokar, T., Schmidt, Z., Tuckness, C., 1996. New gas lift valve design stabilizes injection rates: case studies. Paper SPE 36597 Presented at the 1996 SPE Annual Technical Conference and Exhibition. 6–9 October, Denver, Colorado, USA. Vogel, J.V., 1968. Inflow Performance relationship for solution-gas drive wells. J. Pet. Technol.. Xiao, J., Alhanati, F.J., Reynolds, A.C., Fuentes-Nucamendi, F., 1994. Modeling and analyzing pressure buildup data affected by phase redistribution in the wellbore. Paper SPE 26965 Presented at the III Latin American and Caribbean Petroleum Engineering Conference. 27–29 April, Buenos Aires, Argentina.