Mathematical analysis of intermittent gas injection model in oil production

Transcription

1 Mathematical analysis of intermittent gas injection model in oil production Tasmi, D. R. Silvya, S. Pudjo, M. Leksono, and S. Edy Citation: AIP Conference Proceedings 1716, (2016); View online: View Table of Contents: Published by the American Institute of Physics Articles you may be interested in A combined kriging and stochastic method to map paraffin scale growth in oil pipeline AIP Conference Proceedings 1707, (2016); / Solving mixed integer nonlinear programming problems using spiral dynamics optimization algorithm AIP Conference Proceedings 1716, (2016); / Simulation of nonlinear surface waves generated by submarine landslides AIP Conference Proceedings 1716, (2016); /

2 Mathematical Analysis of Intermittent Gas Injection Model in Oil Production Tasmi 1,a), Silvya D.R. 2,b), Pudjo S. 2,c), Leksono M. 2,d) and Edy S. 1,e) 1 Department of Mathematic, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung 2 Department of Petroleum Engineering, Faculty of Mining and Petroleum Engineering, Institut Teknologi Bandung a) tasmi@students.itb.ac.id b) sdr@tm.itb.ac.id c) psukarno@tm.itb.ac.id d) lm@tm.itb.ac.id e) esoewono@math.itb.ac.id Abstract. Intermittent gas injection is a method to help oil production process. Gas is injected through choke in surface and then gas into tubing. Gas forms three areas in tubing: gas column area, film area and slug area. Gas column is used to propel slug area until surface. A mathematical model of intermittent gas injection is developed in gas column area, film area and slug area. Model is expanding based on mass and momentum conservation. Using assume film thickness constant in tubing, model has been developed by Tasmi et. al. [14]. Model consists of 10 ordinary differential equations. In this paper, assumption of pressure in gas column is uniform. Model consist of 9 ordinary differential equations. Connection of several variables can be obtained from this model. Therefore, dynamics of all variables that affect to intermittent gas lift process can be seen from four equations. To study the behavior of variables can be analyzed numerically and mathematically. In this paper, simple mathematically analysis approach is used to study behavior of the variables. Variables that affect to intermittent gas injection are pressure in upstream valve and in gas column. Pressure in upstream valve will decrease when gas mass in valve greater than gas mass in choke. Dynamic of the pressure in the gas column will decrease and increase depending on pressure in upstream valve. INTRODUCTION Intermittent gas injection is a method to help process of oil production in petroleum industry [4]. Brown discussed that the installation of the intermittent gas lift is considered for low bottom-hole pressure wells [8]. The advantages of the intermittent gas lift are to increase bottom-hole pressure and liquid can be propelling to the surface. In the low bottom-hole pressure wells, slug column will accumulate less than 40 % of the well depth [3]. Equipment of the intermittent gas lift consists of compressor, choke, and valve in mandrel. High pressure of gas is obtained from compressor and gas is injected in choke then casing pressure increasing to open the gas lift valve. Gas through valve enter to the tubing and propel the liquid slug to the surface. In intermittent gas lift, flow regime is vertical flow of liquid slugs with unsteady-state nature. This flow is more difficult than either multiphase flow in continuous vertical or horizontal flow [8]. There are some researchers who have studied the intermittent gas lift. Brown and Jessen [7], Brill et. al. [6], and Neely et. al.[1] presented the early works of the intermittent gas lift model based on measurements of field wells. During gas injection, the expanding of gas was created a droplet or a film on the tubing wall, gas area in tubing, and slug. A droplet or a film on the tubing wall will be fallback and will become to part of the next productions. Gas area advantages to propel liquid to surface. Slug is a liquid that produced to the surface. Intermittent gas lift model based on conservation laws of mechanics had been developed by Liao [11]. This model described flow pattern of intermittent gas lift. Model divided into 4 stages: gas enter to the tubing, slug production, liquid production by entrainment, and regeneration of slug. Liao model was simulated with numerical procedure. Result of numerical simulation was obtained good agreement with experimental works. Liquid fall-back has observed by Hernandez et. al. [2] in the intermittent gas lift with plunger. Liao model has been expanded to other variants of intermittent gas lift by Santos et al. [9]. Filho and Bordalo [4] developed the new intermittent gas lift model and simulating with coupled Application of Mathematics in Industry and Life AIP Conf. Proc. 1716, ; doi: / AIP Publishing LLC /$

3 FIGURE 1. Cycle of Intermittent Gas Lift scheme. In 2004, Bordalo and Filho also extended model to the inverted intermittent gas lift [10]. Simulation scheme to consider the order and concurrent of the new intermittent gas lift model has been developed by Filho et. al. [5]. Single simulator to consider all variants of intermittent gas lift has been developed by Pestana et. al. [13]. Liao model is modified by Tasmi et. al. [14] with assumed that film thickness is constant. This paper will analyze Tasmi model through approach mathematic. Dynamic of variables that affect to intermittent gas injection can know. Therefore, dynamic of variables can use to design installation of the intermittent gas lift more effective and efficient. MATHEMATICAL MODEL Liao [11], Santos[9], and Filho [4] were developed mathematical model of intermittent gas lift. Tasmi, et. al. [14] has been simplified Liao model with assumed film thickness is constant in tubing and analyzed use approach numerical simulation. Tasmi model has been develop model based on mass and momentum conservation in the tubing-casing annulus and tubing. In the tubing consists of gas-column, film area, and slug area. Model consists of 10 coupled ordinary differential equations. In this paper, pressure in gas column is assumed uniform. Therefore, model consist of 9 ordinary differential equations as follows dy i (t) = F i (1) with initial value y i (0) = y io, i = 1, 2,..., 9 and t [0, k] for k R +. Boundary condition of the model y 3 > y 9 and P i jc > y 4. F i are equation and y i (t) are unknown variable in Table 1 of Appendix. This model can be study from 4 ordinary differential equations. Equation 1 can be obtained connection of each variables as the following: y 3 (t) = β 1 y 2 (t), (2) y 9 (t) = β 2 y 3 (t), (3) y 4 (t) = β 3 y 5 (t), (4) y 9 (t) = β 4 y 8 (t). (5)

4 or y 2 (t) = y 3(t), γ 1 (6) y 3 (t) = y 9(t), γ 2 (7) y 5 (t) = y 4(t), γ 3 (8) y 8 (t) = y 9(t). γ 4 (9) Therefore simplify model of Equation 1 as the following or dy 3 (t) dy 4 (t) dy 6 (t) dy 7 (t) dy 2 (t) dy 5 (t) dy 6 (t) dy 7 (t) = c 31 k v y 3 (t)(y 3 (t) y 4 (t)) β 1 = c 41 v y 4 (t) g β 3 y 6 (t) k v A g y 6 (t) k ch ρ i jc (P i jc β 2 y 3 (t)) (10) y 3 (t)(y 3 (t) y 4 (t)) β 1 (11) = v g (12) = v lq (13) = c 21 k v y 2 ( y2 (t) γ 1 y 5 (t) = v g y 6 (t) + k v A g y 6 (t) y 5(t) γ 2 ( y2 (t) y 2 (t) ) ( k ch ρ i jc P i jc y ) 2(t) γ 2 γ 1 y ) 5(t) γ 2 = v g (16) = v lq (17) Behavior of the variables in Equation 1 can be analyzed from simplify model that is system equation that consists of equation 10, 11, 12, 13 or system equation that consists of equation 14, 15, 16, 17. ANALYSIS OF MODEL AND NUMERICAL SIMULATION Model intermittent gas lift with assumption film thickness constant is analyzed with a simple mathematic approach and numerical simulation. In this section, model will be analyzed from Equation10, 11, 12, and 13. A simple mathematic approach is used to study condition of variables that increasing or decreasing. This condition shown behavior of variables when gas is injected. To simulate this model, a runge kutta method was chosen. Numerical simulation using data in Guo book [3]. Behavior of variables from Equation 10, 11, 12, and 13 can be used to know behavior other variables. Equation 10 is model describing dynamic of pressure in upstream valve. Equation 11 is model describing dynamic of pressure in bottom of gas-column or downstream of valve. Equation 12 is model describing dynamic of height gas-column and Equation 13 is model describing dynamic of height liquid-column. Dynamic of Pressure in Upstream Valve Equation 10 describe dynamic of pressure in upstream valve. Noted that: (14) (15)

5 c 31 < 0 k v y3 (t)(y 3 (t) y 4 (t)) β 1 is mass of gas in valve, and k ch ρi jc (P i jc β 2 y 3 (t)) is mass of gas in choke. Furthermore, c 31 k v y3 (t)(y 3 (t) y 4 (t)) β 1 is dynamic of pressure of gas in valve c 31 k ch ρi jc (P i jc β 2 y 3 (t)) is dynamic of pressure gas in choke. Therefore, if mass in valve less then mass in choke, then pressure in upstream valve is increasing. It means there is no dynamic of pressure in upstream valve, because fluid can not flow when mass in valve less then mass in choke. If mass in valve more then mass in choke, then pressure in upstream valve is decreasing. It means gas injection is started and gas enter to the tubing. Figure 2 show combined results of numerical simulation from Tasmi [14] and analysis with approach mathematic for dynamic of pressure in upstream valve. Therefore, dynamic of gas density in upstream FIGURE 2. Dynamic of Pressure in Upstream Valve valve can be seen on Figure 3. Figure 4 shows dynamic of gas density and pressure in upstream valve is proportional

6 FIGURE 3. Dynamic of Density in Upstream Valve FIGURE 4. Pressure vs Density in Upstream Valve

7 Dynamic of Pressure in Bottom of Gas-Column Equation 3 describe dynamic of pressure in bottom of gas-column. Noted that: c 41 < 0 v g y 4 (t) β 3 y 6 (t) k v A g y 6 (t) is mass of gas in bottom of gas-column per unit column, and y3 (t)(y 3 (t) y 4 (t)) β 1 is mass of gas in valve per unit column. Furthermore, c 41 v g y 4 (t) β 3 y 6 (t) c 41 k v A g y 6 (t) is dynamic of pressure in bottom of gas-column y3 (t)(y 3 (t) y 4 (t)) β 1 is dynamic of pressure in valve. Therefore, if mass in tubing less then mass valve, then pressure in tubing is increasing. It means gas injection is started and gas enter to the tubing. If mass in tubing more then mass in valve. Figure 5 show combined results of numerical simulation from Tasmi [14] and analysis with approach mathematic for dynamic of pressure in gas-column. FIGURE 5. Dynamic of Pressure in Bottom of Gas-Column

8 FIGURE 6. Dynamic of Density in Bottom of Gas-Column FIGURE 7. Pressure vs Density in Bottom of Gas-Column

9 Dynamic of The Gas-Column and The Liquid-Column Height Dynamics of the gas-column and the liquid-column height is influenced only by velocity of the gas-column and liquid-column. Because the velocity of the gas-column and liquid-column is assumed constant, then gas-column and liquid-column height will be linear. Figure 8 and 9 show combined results of numerical simulation from Tasmi [14] and analysis with approach mathematic for dynamic of the gas-column and liquid-column height. FIGURE 8. Dynamic of Height Gas-Column FIGURE 9. Height of Liquid-Column Dynamic of Other Variables From solution of system Equation 10, 11, 12, and 13, dynamic of other variables can be obtained. Figure 11, 12, 13, 14, and 9 respectively are dynamic of gas pressure in downstream choke, dynamic of gas density in downstream choke,

10 dynamic of gas pressure is proportional with density in downstream, dynamic of gas pressure in choke is proportional with gas pressure in valve, and dynamic of gas mass in tubing-casing. Figure 9 shows gas mass increasing. It is occurs because of gas injection from surface and gas mass in valve more then gas mass in choke. At later injection gas mass will be constant. FIGURE 10. Dynamic of Gas Pressure in Downstream Choke FIGURE 11. Dynamic of Gas Density in Downstream Choke

11 FIGURE 12. Pressure vs Density in Downstream Choke FIGURE 13. Pressure of Gas in Downstream Choke and in Upstream Valve

12 FIGURE 14. Mass of Gas in Tubing-Casing CONCLUSION This paper shows that approach simple-analysis mathematic and numerical simulations is consistent. If gas injected in surface, then mass in tubing less then mass valve and pressure in tubing is increasing and mass in valve more then mass in choke and pressure in upstream valve is decreasing. Dynamics height of the gas-column and liquid-column are influenced by gas-column velocity and liquid-column. In this model, velocity of gas and liquid assume constant. Therefore height of gas-column and liquid-column are linear with time. The research is founded by Desentralisasi DIKTI ACKNOWLEDGMENTS

13 APPENDIX y i TABLE 1. Appendix A.1 F(y) dy 1 (t) c 11 ( kv y2 (y 3 y 4 ) k ch ρi jc (P i jc y 9 ) ) dy 2 (t) c 21 ( kv y2 (y 3 y 4 ) k ch ρi jc (P i jc y 9 ) ) dy 3 (t) ( c 31 kv y2 (y 3 y 4 ) k ch ρi jc (P i jc y 9 ) ) ( y c 5 41 c51 y 6 + c 52 k v y2 (y 3 y 4 ) ) dy 4 (t) dy 5 (t) y c 5 51 dy 6 (t) dy 7 (t) dy 8 (t) y 6 + c 52 k v y2 (y 3 y 4 ) v g v lq ( c 81 kv y2 (y 3 y 4 ) k ch ρi jc (P i jc y 9 ) ) ( dy 9 (t) c 91 kv y2 (y 3 y 4 ) k ch ρi jc (P i jc y 9 )) ) NOMENCLATURE TABLE 2. Nomenclature Variables/Parameters y 1 : Mass of gas in tubing-casing annulus lbm lbm y 2 : Density of gas in upstream valve f t 3 y 3 : Pressure of gas in upstream valve psia y 4 : Pressure of gas in gas column psia y 5 : Density of gas in gas column lbm f t 3 y 6 : Height of gas column f t y 7 : Height of liquid column f t lbm y 8 : Density of gas in downstream valve f t 3 y 9 : Pressure of gas in downstream valve lbm y i0 : Initial value of variables y i k : Constant β 1 : Coefficient of variable β 2 : Coefficient of variable β 2 : Coefficient of variable β 3 : Coefficient of variable β 4 : Coefficient of variable γ 1 : Coefficient of variable γ 2 : Coefficient of variable γ 3 : Coefficient of variable γ 4 : Coefficient of variable

14 Variables/Parameters c i j : Coefficient of variable i, j N k v : Constant of valve k ch : Constant of choke lbm ρ i jc : Density of gas injection f t 3 P i jc : Pressure of gas injection psia f t v g : Velocity of gas s f t v lq : Velocity of liquid s A g : Cross-sectional area of gas column f t 2 REFERENCES [1] A.B. Neely, J.W. Montgomery, and J.V. Vogel, A Field Test and Analytical Study of Intermittent Gas Lift, Society of Petroleum Engineers Journal, 1974, 14(05): , SPE-4538-PA. [2] A. Hernandez, S. Caicado, and R. Cabunaru, Liquid Fall-Back Measurements in Intermittent Gas Lift With Plunger, Presented at the 68th Annual Technical Conference and Exhibition of the Society of Petroleum Engineers held, Houston, Texas, 3-6 October 1993, SPE [3] B. Guo, W.C. Lyons, and A. Ghalambor, Petroleum Production Engineering, A Computer-Assisted Approach, Lafayette: Elsevier Science and Technology Books, [4] C.O.C. Filho, and S.N. Bordalo, A simultaneous and coupled simulation scheme for the conventional intermittent gas lift, 2003, Proceeding of COBEM. [5] C.O.C. Filho and S.N. Bordalo, Assessment of Intermittent Gas Lift Performance Through Simultaneous and Coupled Dynamic Simulation, Presented at the Latin American and Carribean Petroleum Engineering Conference held, Rio de Janeiro, Brazil, June 2005, SPE [6] J.P. Brill, T.C. Doerr, and K.E. Brown, An Analytical Description of Liquid Slug Flow in Small-Diameter Vertical Conduits, J. Pet. Technol., 1967, 19 (3): , SPE-1526-PA. [7] K.E. Brown and F.W. Jessen, Evaluation of Valve Port Size, Surface Chokes and Fluid Fall-Back in Intermittent Gas Lift Installations, J. Pet. Technol., 1962, 14 (3): , SPE-179-PA. [8] K.E. Brown and H.D. Beggs, The Technology of Artificial Lift Methods, Tulsa: PennWell Books, volume 2a, pp 260, [9] O.G. Santos, S.N. Bordalo, and F.J.S Alhanati, Study of the dynamics, optimization and selection of intermittent gas-lift methodsa comprehensive model, Journal of Petroleum Science and Engineering, 2001, 32 (2001): [10] S.N. Bordalo, C.O.C. Filho, Modeling and Performance Assessement of Inverted Intermittent Gas Lift, Thermal Enginering, 2004, 6 (1): [11] T. Liao, Artificial Lift Project : Mechanistic Modeling of Intermittent Gas Lift. Disertation, University Of Tulsa, TX, EUA, [12] T. Liao, Z. Schmie, and D. R. Doty, Investigation of intermittent gas lift by using mechanistic modeling., Presented at the Production Opperations Symposium held, Olkahoma, OK, US, 2-4 April 1995, SPE MS. [13] T. Pestana, S.N. Bordalo, and M.A.B. Filho, Numerical Simulation in the Time Domain of the Intermittent Gas-Lift and its Variants for Petroleum Wells, Presented at the SPE Artificial Lift Conference America held, Cartagena, Colombia, May 2013, SPE MS. [14] T. Tasmi, S.D. Rahmawati, P. Sukarno, S. Siregar, and E. Soewono, A New-Simple-Effective Analytical Approach to Determine Intermittent Gas Lift Parameters, Presented at the SPE Asia Pacific Oil and Gas Conference bd Exhibition held in Bali, Indonesia, October 2015, SPE MS