Optimization of Gas Injection Allocation in Multi Gas Lift Wells System

Transcription

1 EngOpt International Conference on Engineering Optimization Rio de Janeiro, Brazil, June Optimization of Gas Injection Allocation in Multi Gas Lift Wells System Deni Saepudin 1,4, Pudjo Sukarno 2, Edy Soewono 1, Kuntjoro Adji Sidarto 1, Agus Yodi Gunawan 1, Septoratno Siregar 3 Yana Budicakrayana 5 1 Financial and Industrial Mathematics Research Group, Institut Teknologi Bandung, Indonesia 2 Drilling, Production Engineering and Oil & Gas Management Research Group, Institut Teknologi Bandung, Indonesia 3 Reservoir Engineering Research Group, Institut Teknologi Bandung, Indonesia 4 Departemen Sains, IT Telkom, Jl. Telekomunikasi No. 1, Bandung, Indonesia 5 PT PERTAMINA (Persero), Indonesia addresses: Deni Saepudin: denis301@students.itb.ac.id, Pudjo Sukarno: psukarno@pusat.itb.ac.id, Edy Soewono: esoewono@bdg.centrin.net.id, Kuntjoro Adji Sidarto: sidarto@dns.math.itb.ac.id, Agus Yodi Gunawan: ayodi@math.itb.ac.id, Septoratno Siregar: ss@tm.itb.ac.id, Yana Budicakrayana: cakra@pertamina.com 1. Abstract Optimization problem for oil production in a multi gas lift wells system is discussed. The main problem is to identify allocation of gas injection for each well to obtain maximum total oil production. The gas injection rate is constrained by a maximum limit. Oil production rate is a nonlinear function of gas injection rate, which is unknown explicitly. In existing approaches, the nonlinear function is estimated from empirical or numerical simulation data, by curve fitting using regression method, or estimated by piecewise linear function. We developed here, a mathematical model for gas lift well system, where the fluid flow in reservoir and pipes consists of liquid and gas, so the conditions represent two phase flow phenomena. Relationship between gas injection and oil production is given implicitly from the model. We have also developed a computation scheme to solve the optimization problem. Considering complexity of the problem, computation scheme is developed based on genetic algorithms. Our results show quite good estimation for optimum solution. The approach also gives better quality prediction over existing approach, since all computation results come from the model, not from regression or interpolation. 2. Keywords: Gas Lift, Gas Lift Performance Curve, Constrained Optimization, Genetic Algorithm. 3. Introduction Gas lift is one of the most common artificial lift methods which is used widely in oil production process. During the lift process, gas is injected into the tubing. Gas injection will lighten the fluid column along the tubing, so it will increase oil production. Normally oil production increases as gas injection increases. However, the gas injection has an optimum limit because too much gas injection will cause slippage, where gas phase moves faster than liquid, so that it reduces oil production. The main interest of gas lift optimization problem is to identify optimal gas injection allocation such that maximizing oil production. In real problem, oil is produced from an oil field consisting of a group of gas lift wells. The schematic of multi gas lift wells system is depicted in Figure 1. Oil and gas which are produced from each well are collected in a separator. After separation process, oil is distributed to the pipe line network. Some gas is used for injection and the remaining gas is for sale. Usually, gas available for injection is limited and should be allocated to each well. Here, the optimization problem to solve is : How to share the gas in optimal form such that yields maximum total oil production? The gas allocation optimization problem is a complicated long time problem of interest. Liquid production rate for each well is nonlinear function of gas injection rate, but unfortunately it is not known explicitly. In existing approaches, the optimization problem has been solved in three steps of procedure. In first step, a set of data relating gas injection to oil production from each well are collected. The data may be obtained from field data or numerical simulation data. In second step, a regression or interpolation method is applied to estimate the nonlinear function which relates gas injection to liquid production. Some functions usually used for regression are quadratic, see [1], combination quadratic with logarithmic, 1

2 Gas for Gas Lift Gas to sale Compressor Station Gas Separator SPU Oil Production Manifold Injction Gas Manifold Gas and Oil Out Gas In Figure 1: Schematic of a Cluster of Gas Lift Wells see [2], and exponential function, see [3]. Some researchers applied interpolation method to estimate the nonlinear function using piecewise linear function, see [4], [5] and [6]. In third step, a constrained optimization problem is built and solved numerically using non linear programming methods or another methods such as genetic algorithm. Liquid production as a non linear function of gas injection for each well can be obtained implicitly from gas lift performance model. The mathematical model for gas lift performance problem can be written as a two parameter family of a non linear differential equation (ODE) dp dz = F (z, P ; q g, q l ), (1) which representing the steady flow (gas and liquid) along the tubing, with the wellhead pressure and the bottomhole pressure P (0) = P wh, (2) P (1) = P wf, (3) as the boundary conditions. The real valued function F (z, P ; q g, q l ) is non negative, F : [0, 1] [P wh, P wf ] R +, (4) q g and q l, where 0 q g < and 0 q l 1 are gas injection rate and liquid production rate, respectively. The existence and uniqueness of liquid production rate as an implicit function of gas injection rate has been shown by Saepudin et. al. [7]. Also, a computation scheme using genetic algorithm to find optimal gas injection rate was proposed in [7]. The computation approach that proposed in [7] has reduced the collecting data and regression or interpolation procedure as required in previous approaches. In this paper, a computation scheme is proposed to solve the optimization problem in multi gas lift wells system. The computation scheme is an extension of the scheme proposed in [7]. 4. Mathematical Model In an oil producing-well, reservoir fluid consisting of oil and water and sometimes together with gas, flows from reservoir through a tubing toward surface facilities. In case where the reservoir pressure is high enough, the reservoir fluid can flows up to the surface naturally. However as time increases, the reservoir depletes and the pressure decreases. If this happened, oil production decreases so that an artificial lift method, such as gas lift method need to apply. 2

3 4.1 Single Gas Lift Well Model When gas lift is applied, gas is injected at selected point into the tubing. Assuming injection point near the well bottomhole and reservoir fluid consists of liquid only (oil and water), Saepudin et. al. [7] proposed a mathematical model for a single gas lift well. The mathematical model in normalized form can be written as a boundary value problem (1)-(3), where Eq.(1) is derived from the mechanical energy balance equation (see [8]) dp dz = g ρ m + 2 f ρ mū 2 m g c g c D + ρ m d 2g c dz ū2 m, (5) with initial condition P (0) = P wh, (6) where terms g g c ρ m, 2 f ρmū2 m ρm d g cd and 2g c dz ū2 m correspond to the pressure drop due to gravity, friction and acceleration respectively. Mixture density ρ m, velocity ū m = u sl + u sg, and friction factor f are function of pressure P, parameters gas injection q g and liquid production q l. Since the liquid and gas superficial velocities are given by u sl = q l A, u sg = ZP sct q g T sc P A, (7) with the cross section area of the tubing then Eq.(5) can be written as dp dz = A = π D2 4, (8) g g c ρ m + 2 f ρmū2 m g cd. (9) 1 + ρmūmusg g cp In the reservoir, one-phase fluid flow (liquid) in steady state can be expressed by Darcy s law So, by scaling P wf = P r q l J. (10) P = P P r, z = z L, q l = Eq. (5), (6) and (10) can be written as where q q l g J P r, q g = L D g D, (11) ρ = ρm L g g cp r, ũ = u m g D, d P ρ + 2f ρũ2 =, dz 1 + α ρũ qg P 2 P (0) = P wh, (12) P wf = 1 q l, α = 4ZT P sc πt sc P r. (13) The gas lift performance model for single well is given by the boundary value problem (1) - (3), where the right hand sides are given by the right hand side of (12). For simplicity, we drop tilde from (12). Liquid production q l as a function of gas injection q g q l = ϕ(q g ), (14) 3

4 can be obtained implicitly from the gas lift model (1)-(3), and the graph of (14) is called Gas Lift Performance Curve (GLPC). 4.2 Multi Gas Lift Wells Model In most cases, oil is produced using gas lift system from an oil field which consists of a group of gas lift wells as illustrated in Fig.1. Assuming the gas lift system consists of N gas lift wells, the separator position is close enough to the manifold (so the pressure difference between separator and manifold can be neglected), flow lines are horizontal and the separator capacity is large enough, the multi gas lift wells model can be written as dp 1k dz = F 1k (z, P 1k ; q gk, q lk ) (15) P 1k (0) = P sep (16) dp 2k dz = F 2k (z, P 2k ; q gk, q lk ), (17) P 2k (0) = P whk = P 1k (1), (18) P 2k (1) = P wfk = 1 q lk, (19) 0 z 1, 0 P 1k 1, 0 P 2k 1, 0 q gk <, 0 q lk 1, for k = 1, 2,..., N. The model is an extension of the single gas lift well model (1)-(3). The initial value problem (15)-(16) represents gas and liquid flow model along the flow line of k th well. For horizontal flow line, Eq. (15) can obtained from (5), by dropping the gravity terms, that is in normalized form, is given by F 1k (z, P 1k ; q gk, q lk ) = 2fρ k u 2 k 1 + α ρ ku k q gk. (20) P 2 1k While (17) represents gas and liquid flow model along the tubing for k th well, which is given by F 2k (z, P 2k ; q gk, q lk ) = ρ k + 2fρ k u 2 k 1 + α ρ ku k q gk. (21) P 2 1k For each k = 1, 2,..., N, the gas lift performance function of the k th well is given by where (22) satisfies (15)-(19). q lk = ϕ k (q gk ), (22) 5. Optimization Problem The most common optimization problem faced in multi gas lift wells system is maximization of total oil production. Let the total gas available for injection N gas lift wells be given by Q gav. How much gas should be injected to each well to maximize total oil production? Since then the problem can be written as a constrained maximization max Q o = q o = (1 W C)q l, (23) N (1 W C k )ϕ k (q gk ), (24) subject to N q gk Q gav. (25) 4

5 In case where the gas available for injection Q gav is large enough, then for each k = 1, 2,..., N, gas injection q gk is chosen such that maximizing liquid production ϕ k (q gk ). Gas available for injection Q gav is usually very limited and should be shared in optimal form for each well. The constrained optimization problem (24)-(25) is a complicated problem since functions ϕ k (q gk ), k = 1, 2,..., N are not known explicitly. For each k = 1, 2,..., N, let P 2k (z; q gk, q lk ) be solution of (15)-(18), and for a given gas injection, the liquid production (22) can be obtained implicitly by substituting P 2k (z; q gk, q lk ) to (19). Therefore, the constrained maximization problem (24)-(25) can be rewritten as subject to max Q o = N (1 W C k )q lk, (26) P 2k (1; q gk, q lk ) = 1 q lk, k = 1, 2,..., N, (27) and (25). Further, the solution of maximization problem (26) with constraints (27) and (25) is equivalent with solution of minimization problem in the domain D = { min Ω( q g, q l ) = ( q g, q l ) R 2N P 2k (1; q gk, q lk ) 1 q lk, 1 N (1 W C k)q lk, (28) } N q gk Q gav In the next section, we construct a numerical scheme to solve the minimization problem (28)-(29). 6. Numerical Scheme In the numerical scheme, for each k = 1, 2,..., N, for given q gk and q lk, ˆP2k (1; q gk, q lk ) is the value of pressure P 2k (1; q gk, q lk ) in (27) computed by Runge Kutta 4 th order method. Using penalty approach, the solution of in the domain min ˆΩ( q g, q l ) = (29) 1 N (1 W C k)q lk + λ max{0, ˆP 2k (1; q gk, q lk ) (1 q lk )}, (30) { ˆD = ( q g, q l ) R } 2N N q gk Q gav, 0 q lk 1, k = 1, 2,..., N, (31) converges to the solution of (28)-(29) for large enough λ. Using transformation q g1 = θ N cos 2 θ 1, q g2 = θ N sin 2 θ 1 cos 2 θ 2, q g3 = θ N sin 2 θ 1 sin 2 θ 2 cos 2 θ 3, (32), q gk = θ N sin 2 θ 1 sin 2 θ 2 cos 2 θ k 1, k = 2, 3,..., N, the domain (31) can written in terms of θ and q l D θ = { ( θ, q l ) 0 θk π/2, k = 1, 2,..., N 1, 0 θ N Q gav, 0 q lk 1, k = 1, 2,..., N }. (33) We construct here a computational procedure using Genetic Algorithm (GA). The domain D θ is chosen as the search space to keep the population always in D θ when genetic operators are applied. The computation procedure can be written as follows. 5

6 1. Initialize a population of chromosomes v 1, v 2,..., v r which correspond to pairs {( θ (k), q (k) l ), k = 1, 2,..., r} D θ. 2. Using transformation (32), we can obtain {( q (k) g 3. Compute ˆP (1; q g (k) i, q (k) l i ) for i = 1, 2,..., N, k = 1, 2,..., r. 4. Evaluate the fitness value (k) ˆΩ( q g, q (k) l ), for k = 1, 2,..., r., q (k) l ), k = 1, 2,..., r} ˆD. 5. Create new chromosomes by doing crossover and applying mutation. 6. Apply a selection procedure to get a new population. 7. Return to step 2 until stopping criteria is satisfied. 7. Computational Results In this section, some numerical simulations are conducted using the field data given in Table 1. The gas lift performance curves for each well are obtained using shooting method and are depicted in Figure 2. If gas available for injection is large enough, the total oil production can be obtained by gas lift is STBD with required gas injection MMSCFD. Here, the computation is conducted for varying maximum gas available. The computation result for population number N ind = 100, crossover probability P c = 0.9, mutation probability P m = 0.1 up to 500 generation are written in Table 2. Table 1: Field Data. Well # Well depth (ft) Res. Pres. (psia) GLRf (SCF/STB) SG Oil ( o API) Water Cut Res. Temp.( o F) Wellhead Temp.( o F) Tubing ID (inch) Casing ID (inch) PI (STBD/d/psi) Flowline ID (inch) F.line length (ft) SG Water SG Gas

7 data1 data2 data3 data4 data5 600 Oil Production Rate (STBD) q x 10 6 g Gas Injection Rate (SCFD) Figure 2: Gas Lift Performance Curves By Shooting Method for P sep = 150 Psi Table 2: Computational Results. Optimum Solution Exact By GA By GA By GA By GA Unit Gas Available unlimited unlimited MMSCFD q g MMSCFD q g MMSCFD q g MMSCFD q g MMSCFD q g MMSCFD q o STBD q o STBD q o STBD q o STBD q o STBD Total Oil Production STBD 8. Conclusions The following conclusions are obtained from the present study. 1. Gas lift optimization problem can be expressed in mathematical model as an optimization in a class of boundary value problems. 2. Computation scheme constructed in this paper has eliminated regression or interpolation procedure which is usually applied in previous approaches, and also it gives better quality prediction since all computation results come from the model, not from regression or interpolation. 3. Since the well data are considered here as the input parameters, then the computation scheme can accommodate the changes of the data with respect to time. 4. This approach is potential to develop for more complicated gas lift optimization problem regarding surface facilities. 9. Acknowledgment This research is partially funded by Hibah Bersaing Research Grant XV DP2M DIKTI The authors also thank the Research Consortium on Pipeline Network ITB (OPPINET) for providing relevant 7

8 data and field information. 10. Nomenclature D Pipe (tubing) diameter, in [m] f Friction Factor, dimensionless g c Gravitation force, ft/s [ 2 m/s 2] g Acceleration due to gravity, ft/s [ 2 m/s 2] J Productivity Index, stbd/psi [ m 3 /s P a ] L Tubing Length, ft [m] P Pressure a long production pipe, psi [P a] P r Reservoir Pressure, psi [P a] P wh Wellhead Pressure, psi [P a] q g Gas Injection Rate, scfd [ m 3 /s ] q l Liquid Production Rate, stbd [ m 3 /s ] q o Oil Production Rate, stbd [ m 3 /s ] q gav Available gas injection rate, scfd [ m 3 /s ] T Temperature a long production pipe, o F [ o K] u m W C W OR Z γ g Velocity of mixture, ft/s [m/s] Water Cut Water oil ratio Gas compressibility factor, dimensionless Gas specific gravity, dimensionless γ w Water specific gravity, dimensionless ρ l Density of liquid, lbm/ft [ 3 kg/m 3] ρ g Density of gas, lbm/ft [ 3 kg/m 3] ρ m Density of mixture, lbm/ft [ 3 kg/m 3] 11. References [1] Nishikori N, Redrer R A, Doty D R and Schmidt Z, An Improved Method for Gas Lift Allocation Optimization, SPE Paper 19711, [2] Alarcón G A, Torres C F, and Gómez L E, Global optimization of gas allocation to a group of wells in artificial lifts using nonlinear constrained programming, JERT - Journal of Energy Resources Technology, 2002, 124, [3] P. Sukarno, K. A. Sidarto, S. Dewi, et al., New Approach on Gas Lift Wells Optimization with Limited Available Gas Injected, Proc. of IATMI , 2006, Jakarta. 8

9 [4] T. Ray and R. Sarker, Genetic Algorithm for Solving a Gas Lift Optimization Problems, Journal of Petroleum Science and Engineering, 2007, 59, [5] T. Ray and R. Sarker, Optimum Oil Production Planning using an Evolutionary Approach, SCI - Studies in Computational Intelligence, 2007, 49, [6] E. Camponogara and P. Nakashima, Optimizing Gas-Lift Production of Oil Wells : Piecewise Linear Formulation and Computational Analysis, IIE Transactions, 2006, 38, [7] D. Saepudin, E. Soewono, K.A. Sidarto, A.Y. Gunawan, S. Siregar, and P. Sukarno, An Investigation on Gas Lift Performance Curve in an Oil Producing Well, IJMMS - International Journal of Mathematics and Mathematical Sciences, 2007, Article ID [8] M. J. Economides, A. D. Hill and C. E. Economides, Petroleum Production Systems, 1994, Prentice Hall Petroleum Engineering Series, NJ, USA. 9