Copyright 2004, Society of Petroleum Engineers Inc. This paper was prepared for presentation at the SPE Annual Technical Conference and Exhibition held in Houston, Texas, U.S.A., 26–29 September 2004. This paper was selected for presentation by an SPE Program Committee following review of information contained in a proposal submitted by the author(s). Contents of the paper, as presented, have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material, as presented, does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Papers presented at SPE meetings are subject to publication review by Editorial Committees of the Society of Petroleum Engineers. Electronic reproduction, distribution, or storage of any part of this paper for commercial purposes without the written consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to a proposal of not more than 300 words; illustrations may not be copied. The proposal must contain conspicuous acknowledgment of where and by whom the paper was presented. Write Librarian, SPE, P.O. Box 833836, Richardson, TX 75083-3836, U.S.A., fax 01-972-952-9435.

Abstract Although various gas-lift optimization algorithms have been proposed in literature, few of them is suitable for long-term reservoir development studies, which require the gas-lift optimizer to be highly efficient, flexible and powerful enough to handle complicated fluid flows and operational constraints, and have low impact on simulator convergences. This paper investigated methods to address these important issues.

The gas-lift optimization problem considered in this paper is to maximize the daily hydrocarbon production by selecting optimally the well production and lift gas rates subject to pressure and rate constraints in nodes of the surface pipeline network and to the amount of lift gas available. The problem is regarded as a well management problem in a commercial reservoir simulator capable of simulating multiphase compositional fluid flow in reservoirs, well tubing strings, surface pipeline network systems, and separation facilities. The problem is solved in selected iterations of a reservoir simulation time step.

This paper proposed a method for the described gas-lift optimization problem and investigated its performance against multiple existing methods. Case studies showed that the new method is capable of producing high quality results while requires less CPU time for optimization and has smaller impact on reservoir simulator convergence.

This paper also applied the concept of multiobjective optimization to smooth the rate oscillation between adjacent iterations by sacrificing a certain amount of oil production. In certain cases, this method reduces the simulation time significantly. Introduction When oil field matures, the hydrocarbon production is often assisted by continuous lift gas injection and constrained by the gas and/or liquid handling capacities of surface facilities. The optimal allocation of production rates and lift gas rates subject

to reservoir deliverability and surface facility capacities can have a big impact on facility design and other capital investment decisions and should be captured in long term reservoir studies. Compared to real time production optimization, the optimal rate allocation in long-term reservoir simulation studies poses unique problems: the rate allocation optimizer has to be highly efficient and have low impact on simulation convergence while is capable of generating quality results.

The stated problem has been addressed in different ways in existing literature. Fang and Lo1 proposed a linear programming technique to allocate lift gas rates and production streams subject to multiple flow rate constraints. The method was implemented in a reservoir simulator and proved to be efficient in several field studies. Based on Fang and Lo’s work, Wang et al.2 developed a procedure to optimally allocate the production rate, lift gas rate, and well connections to surface pipeline systems simultaneously. The optimization procedure is invoked at the Newton-iteration level of a commercial reservoir simulator. Hepgular et al.3 coupled a separate commercial surface pipeline network optimizer with a commercial reservoir simulator through an iterative procedure. The surface network optimizer employees a Sequential Quadratic Programming (SQP) optimization algorithm and has the ability to perform general operation and design optimizations. Davidson and Beckner4 presented an integrated facility and reservoir model in which the rate allocation problem is solved in the facility model using SQP methods. They also presented a detailed procedure on how to handle infeasible conditions.

Fang and Lo’s1 method is simple and efficient. However, it ignores the pressure interactions among wells through common flow lines and may result unsatisfying results. Hepgular et al.3 and Davidson and Beckner’s4 methods relied on powerful facility network optimizers that require significant effort to develop. This paper bridges this gap by presenting a simple yet robust and efficient rate allocation optimization procedure. In addition, this paper applied the concept of multiobjective optimization5 to minimize the impact of lift gas rate oscillation on simulator convergence. This method proved to be successful in certain cases. Problem Statement The optimization problem is to maximize the daily hydrocarbon production by selecting optimally the well production and lift gas rates subject to pressure and rate constraints of surface facilities. The optimization problem is considered a well management problem in VIP6, a commercial

SPE 90506

Gas Lift Optimization for Long-Term Reservoir Simulations Pengju Wang, SPE, BP, Michael Litvak, SPE, BP

2 SPE 90506

reservoir simulator capable of simulating multiphase compositional fluid flow in reservoirs, well tubing strings, surface pipeline network systems, and separation facilities. Wang et al.2 presented a procedure to integrate the optimization problem into the reservoir simulator. The procedure is presented below. 1. Start with pressure and fluid compositions in reservoir grid

blocks calculated in the previous Newton iteration. Use well lift gas rates from the previous Newton iteration as the initial guesses.

2. Solve the surface pipeline network problem, convert pressure constraints to flow rate constraints. This step was presented in detail by Litvak and Darlow7.

3. Perform production and lift gas rate allocation optimization.

4. Determine active constraints in wells and nodes. Linearize multiphase fluid flow equations for well tubing strings and the surface pipeline network system (numerical derivatives are used). Add these equations to the linearized fluid flow equations for reservoir grid blocks.

5. Solve the linearized system of equations established in Step 4.

6. Repeat Steps 1 to 5 until convergence. 7. March to the next time step.

Wang et al.2 adopted Fang and Lo’s1 separable programming (SP) method to solve the rate allocation problem in Step 3. This method works as follows. 1. Construct a gas-lift performance curve (oil rate versus lift

gas rate curve) and inflow performance curves (oil rate versus water rate curve and oil rate versus formation gas rate curve) for every well on automatic gas-lift allocation. In current implementation, a minimum gas-lift efficiency parameter (defined as the oil rate increase for a unit of lift gas injection) can be specified. A gas-lift performance curve is constructed in such a way that its slope at the end of the curve should be larger than or equal to the user-specified minimum gas-lift efficiency.

2. Approximate the gas-lift and inflow performance curves using piecewise linear curves. Formulate the constrained gas-lift optimization problem as a linear programming problem. Solve the linear programming problem and obtain the optimal lift gas rates.

A detailed description of the above method can be found in Fang and Lo1 and Wang8.

When Wang et al.’s2 procedure was applied to several field case studies; two major limitations of that procedure were exposed: 1. The SP method requires a gas-lift performance curve and

two inflow performance curves for each well on gas-lift optimization. Each curve has to be established after the corresponding well is isolated from the surface pipeline network (SPN) by ignoring the backpressure imposed by other wells. Consequently, the method may produce significantly suboptimal solutions when the flow interactions among wells are significant.

2. The gas lift optimization problem is solved in selected Newton iterations. Fluctuations of reservoir and operation conditions can cause significant oscillations of lift gas rate allocated in different iterations. These oscillations can make the simulator hard to converge.

This paper addresses the first and second problem by presenting a simple yet powerful gas-lift optimization algorithm. This paper also addresses the second problem by introducing a multiobjective optimization method to minimize the lift gas rate oscillations while maximizing the oil production.

Gas-lift Optimization Procedure Overall Method. The gas-lift optimization method developed in this study takes into account the flow interactions among wells through common surface pipelines. The method works as follows. 1. Start with the existing lift gas rates for all wells on

automatic lift gas allocation. Solve the multiphase flow problem in the surface pipeline network (SPN). Build a linear programming model (described below in section “Constraint Handling”) to scale production and lift gas rates to satisfy flow rate and/or velocity constraints. Denote the objective function value obtained in this step as 0f .

2. Select a well on automatic lift gas rate allocation, say well i . Denote its lift gas rate at this stage as 0

lg,iq . Increase its

lift gas rate by iqlg,δ . Solve the multiphase flow problem in the SPN with the updated lift gas rates and scale production and lift gas rates to satisfy the flow rate constraints. Denote the objective function value obtained in this step as 1f .

3. Compute the gas-lift efficiency for well i

iqff

elg,

01

δ−

= (1)

Compare the gas-lift efficiency e with a user-specified minimum gas-lift efficiency coefficient, mine . If minee ≥ ,

update 0f by setting 10 ff = and go to step 6 with the increased lift gas rate for well i . If min0 ee ≤≤ , reset the

lift gas rate of well i to 0lg,iq and go to step 6. If 0<e ,

reset the lift gas rate of well i to 0lg,iq and go to step 4.

4. Decrease the lift gas rate of well i by iqlg,δ ( 0lg, >iqδ ). Solve the multiphase flow problem in the SPN with the updated lift gas rates. Scale optimally the production rates and lift gas rates to satisfy flow rate constraints. Denote the objective function value obtained in this step as 2f .

5. Compute the gas-lift efficiency for well i ,

iqff

elg,

02

δ−−

= (2)

Compare this gas-lift efficiency with the user-specified minimum gas-lift efficiency coefficient mine . If

minee ≥ , update 0f by setting 20 ff = and go to step 6 with the decreased lift gas rate for well i . Otherwise,

reset the lift gas rate of well i to 0lg,iq .

6. Repeat step 2-5 for every well on automatic lift gas allocation.

SPE 90506 3

7. Repeat step 2-6 until no lift gas rate change can be made or the maximum number of iterations allowed is reached. Constraint Handling. Given a set of lift gas rates lgqv , the

corresponding production rates (oil rates oqv , water rates wqv , and formation gas rates gqv ) may exceed the flow rate and/or velocity constraints and be not feasible. The new gas-lift optimization procedure adopted a linear programming model developed by Lo and Holden9 to scale the infeasible lift gas and production rates to the feasible region. This linear programming model takes a set of flow streams (either from production wells or from satellite reservoirs) as the input and scales them to meet the flow rate and velocity constraints in a way that maximize the objective function. A flow stream is represented by the unconstrained oil, water, formation gas, and lift gas rates of a well or a satellite reservoir. For example, suppose we want to maximize the total oil rate of a field subject to a total gas rate constraint. The problem can be formulated as (Problem 3)

Maximize ∑=

wn

iioiqx

1,

(3a)

Subject to ( )∑=

≤+wn

igiigi Qqqx

1lg,,

(3b)

10 ≤≤ ix , wni ,...,1= (3c)

where wn is the number of flow streams, ix denotes the decision variable for Problem 3, gQ is the total gas flow rate

capacity of the field, and ioq , , igq , , and iqlg, are the oil, formation gas, and lift gas rate for well i , respectively. In the optimal solution, 0=ix indicates well i should be shut-in;

1=ix indicates well i should produce at rate ioq , , igq , , and

iqlg, ; ( )1,0∈ix indicates well i should be choked back. The optimal objective function value of Problem 3 is a

feasible value to the gas-lift optimization problem and is regarded as the function value for the set of lift gas rates lgqv in Step 1, 2, and 4 of the overall gas-lift optimization procedure.

Discussion. The results generated by the new gas-lift optimization procedure will be suboptimal because of the following two facts. First, the function evaluation procedure employed in the overall optimization algorithm uses another optimization procedure with simplified assumptions. Consequently the function value obtained from this procedure is only an approximation. Secondly, the new method is a local search method and can be stuck at a local sub-optimal point. Fortunately, it was demonstrated in several case studies that the new method produces quality results for long-term development studies.

Coefficient mine is a parameter used to control how easily a lift gas rate can escape from its current value. If mine is large, the lift gas rate is not sensitive to small changes in reservoir and operation conditions. Consequently the result will be less near the true optimum but there will be less simulator convergence problem resulted from lift gas rate oscillations. Conversely, if mine is small, the allocated lift gas rates will be more noisy but the solution will be closer to the true optimum.

To facilitate later discussions, the above overall gas-lift optimization method will be referred to as the GLINC method.

Lift Gas Rate Damping For constrained gas-lift optimization problem, there are cases that multiple vastly different lift gas distributions result similar oil rate increases. For such cases, although moving from current gas-lift injection scenario to another gas-lift injection scenario may increase the total oil production by a minuscule amount, the resulting production rates for individual wells can be significantly different, thus make the reservoir simulator hard to converge. When gas-lift injection scenarios oscillate frequently in different Newton iterations, the computational efficiency of the reservoir simulator deteriorates significantly. In the GLINC method, this problem can be mitigated by using a large value for coefficient mine . For the separable programming method, however, a different strategy was employed. This strategy was described below.

To minimize the impact of lift gas oscillation on simulator convergence, the gas-lift optimization problem in VIP was reformulated as a multiobjective optimization problem5 with two competing objectives: 1. Maximize the total oil production rate subject to the flow

rate and velocity constraints. This objective can be expressed mathematically as

∑=

=wn

iioqf

1,

1 (4)

2. Minimize the absolute change of lift gas rates between two consecutive Newton iterations subject to the flow rate and velocity constraints. This objective can be expressed mathematically as

∑=

−=wn

iii qqf

1

0lg,lg,

2 (5)

where 0lg,iq is the lift gas rate of well i allocated in

previous iteration and iqlg, is the lift gas rate of well i to be allocated in current Newton iteration. The multiobjective optimization problem was solved by a

hierarchical optimization method10. This method allows the decision maker to rank and optimize the objectives in descending order of importance. For this particular gas-lift optimization problem, the first objective 1f is ranked the

most important; the second objective 2f is ranked the least important. Denote the decision variable of the optimization problem as xv . Denote the flow rate and velocity constraints as

( ) 0≤xciv , mi ,...,1= (6)

The solution procedure for the multiobjective optimization problem goes as follows: 1. Find the optimum point ,*1xv for the first objective, 1f ,

subject to the original set of constraints (Problem 7) Maximize ( )xf v1 (7a)

Subject to ( ) 0≤xciv

, mi ,...,1= (7b) As described by Fang and Lo1 and Wang8, Problem 7 can be reformulated to and solved as a linear programming problem. Let ( ),*,*1 1xf v denote the optimal objective function value for Problem 7 .

4 SPE 90506

2. Find the optimum point ,*2xv for the second objective function 2f subject to the original and an additional constraint (Problem 8)

Minimize ( )xf v2 (8a)

Subject to ( ) 0≤xciv , mi ,...,1= (8b)

( ) ( ),*,*11 )1( 1xfxf vv α−≥ (8c)

where [ ]1,0∈α is called the damping factor. If α equals

0, the solution of Problem 8 is ,*1xv , and there is no lift gas rate damping. If α equals 1, the solution of Problem 8 will

be the lift gas rates of previous Newton Iteration, 0lg,iq . By

adjusting the damping factor between 0 and 1, the competition between maximizing the total oil production,

1f , and minimizing the discrepancy of lift gas rates

between two consecutive Newton iterations, 2f , can be balanced. With appropriate reformulations11, Problem 8 can also be solved as a linear programming problem. Although two optimization problems need to be solved in

the hierarchical method, the CPU time is increased only slightly compared to the original single objective optimization problem provided that the separable programming method is used to solve Problem 7 and 8. The reason is that for the separable programming method, the majority of computational time is spent on constructing the gas-lift and inflow performance curves; once the performance curves are established, it takes relatively little extra time to formulate and solve Problem 7 and 8.

Field Examples The GLINC gas-lift optimization method and the lift gas damping method were successfully applied to the long-term development studies of two North Sea oil fields, respectively. These application examples are presented below to demonstrate the advantages and shortcomings of the developed methods.

Field Example 1. A full field model was developed to study the long-term development plan of a North Sea oil field. The reservoir model contains about 20 production wells. All but one production well in the model are on automatic lift gas allocation. All production wells are tied to a processing center through a surface pipeline network system. The production system is organized in such a way that the production wells can be classified into two groups, and the wells within each group share at least one common flow line. It is observed that the production rate of some wells interfere each other through the common flow lines.

One objective of this reservoir model is to investigate the appropriate surface facility capacities of the field. As a consequence, a total lift gas injection rate as well as total oil, gas, water, and liquid flow rate constraints are specified in the model, and the optimal allocation of lift gas rates and production rates during the simulation is crucial to identify the right facility capacities.

The full field model was first run with the separable programming rate allocation method. The minimum gas-lift efficiency was specified as 50 STB/MSCF. One concern with this run was whether the SP method is suitable for this model.

The major assumption of the SP method is that for a given lift gas rate, the oil rate of a well in the entire production system follows the gas-lift performance curve built from the isolated single-well system (recall that a well is isolated from the surface pipeline network by fixing its well head pressure). Since some wells have noticeable interference through common flow lines, this assumption does not hold for this model. This was demonstrated in Fig. 1a, which shows that the oil rate of well A1 after the first optimization is far away from the gas-lift performance curve built for that optimization. It was observed that in subsequent Newton iterations, the post-optimization oil rate of a well converged to the gas-lift performance curve built for the SP gas-lift optimization in those iterations (Fig. 1b-1c). However, the concern that whether the lift gas rates allocated from the SP method are suitable for this long-term reservoir development study remained.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.00 0.20 0.40 0.60 0.80 1.00 1.20

Normalized Gas Lift Rate

No

rmal

ized

Oil

Rat

e

Gas lift performance curve

Allocated lift gas and oil rate

Fig. 1a - Gas-lift performance curve and the allocated lift gas and oil rates for well A1 at the first Newton iteration of the first time step. The oil rate and lift gas rate s are normalized.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.00 0.20 0.40 0.60 0.80 1.00 1.20Normalized Gas Lift Rate

Nor

mal

ized

Oil

Rat

e

Gas lift performance curve

Allocated lift gas and oil rate

Fig. 1b - Gas-lift performance curve and the allocated lift gas and oil rates for well A1 at the second Newton iteration of the first time step. The oil rate and lift gas rates are normalized.

SPE 90506 5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.00 0.20 0.40 0.60 0.80 1.00 1.20Normalized Gas Lift Rate

Nor

mal

ized

Oil

Rat

e

Gas lift performance curve

Allocated lift gas and oil rate

Fig. 1c - Gas-lift performance curve and the allocated lift gas and oil rates for well A1 at the third Newton iteration of the first time step. The oil rate and lift gas rate s are normalized.

To assess the performance of the SP method, another run was made with the GLINC method as the rate allocation method. In this run, parameter mine was set to a small value of 10 STB/MSCF at the beginning of the run so that the optimization will not stuck at a point far from the optimal lift gas rates. Parameter mine was set to a bigger value of 50 STB/MSCF after the first time step so that the lift gas rates change less frequently and the simulator runs faster.

Although the GLINC method handles explicitly the flow interactions among wells, it is a heuristic search method that does not guarantee local optimum. To further verify the performance of the SP and GLINC method, two conventional optimization methods were applied to the constrained gas-lift optimization problem.

The first method is a genetic algorithm12 (GA). In this method, the lift gas rates for wells on automatic lift gas allocation are selected as the decision variables and encoded as a binary string (or genes). The method encodes multiple sets of solutions (lift gas rate distributions) into populations, evaluate the fitness of each population (i.e. evaluate the total oil rate for a given set of lift gas rates subject to the flow rate constraints), and evolve the populations through the means of selection, crossover, and mutation. As in the GLINC method, the flow rate constraints of surface facilities are handled in the function evaluation procedure by using the linear programming model9 to scale infeasible flow rates to the feasible region.

The second method is a direct search algorithm that tries to maintain a regularly-shaped simplex throughout the iterations. The method used in this study was implemented by Powell13 and referred to in this paper as the COBYLA method. As in the GA method, the lift gas rate for each well on automatic lift gas allocation is selected as the decision variable and the flow rate constraints of surface facilities are handled in the function evaluation procedure using the linear programming model9.

The GA and COBYLA methods are time consuming. To reduce the number of calls to the GA and COBYLA methods, the runs with the GA method and the COBYLA method have a maximum time step of 10 days while the runs with the SP and GLINC methods have a maximum time step of 6 days.

Results from the four different rate allocation methods are shown in Fig. 2 through Fig. 5. Fig. 2 shows that the field cumulative oil productions from the four optimization methods are close; while Fig. 3 shows that the daily field lift gas injection volumes from the four methods are quite different. The reason that the SP and GLINC method allocated significantly less lift gas than the GA and COBYLA methods is that the SP and GLINC methods have parameters to control the balance of the injection cost and oil rate increase while the current implementations of the GA method and COBYLA method do not have. Fig. 4 and Fig. 5 compare the lift gas rate and oil rate allocated by the four optimization methods for well A2, respectively. It is observed that the lift gas rates allocated by the four methods follow roughly the same trend. Although the absolute lift gas rate differences between four methods are significant, the oil rates are similar. This is because that for a gas-lift well, the gas-lift efficiency decreases to a small value as the lift gas rate increases beyond certain value. In the actual field operations, the wells are operated with lift gas rates similar to those obtained from the GLINC method. Table 1 shows that the SP method and the GLINC methods require significantly less CPU time on both well management and overall simulation than the GA method and the COBYLA method do.

In summary, for this full field model, both the SP method and GLINC method produce good results and are much more efficient than the GA and the COBYLA methods.

Field Example 2. A full field model was developed to study the long-term development plan for another North Sea oil field. This model contains about 20 production wells. All wells are on automatic lift gas allocation when they are in production.

Two case studies of this model are presented here. In Case 1, total oil, water, and liquid flow rate constraints were specified. When the rate allocation problem in this case was solved using the SP method without lift gas rate damping, it was observed that some wells have significant lift gas rate oscillations. To overcome this problem, a second run was made with a lift gas rate damping factor of 0.02 specified. In the second run, the lift gas rate for individual wells was greatly smoothened (Fig. 6) while the field cumulative oil production has discernable differences with that of the first run only at the very end of the simulation (Fig. 7). Table 2 shows that the CPU time required by the second run is slightly less than the CPU time required by the first run.

Case 2 is a variation of case 1. The major difference is that Case 2 contains a total gas flow rate constraint as well as the total oil, water, and liquid flow rate constraints of Case 1. Again, two runs were made to demonstrate the effect of lift gas rate damping. The first run had no lift gas rate damping. The second run specified a lift gas rate damping factor of 0.02. It was observed that for this case, the run with lift gas rate damping consumed only 32% of the CPU time required by the run without lift gas rate damping (Table 3) while the lift rates for individual wells were smoothened with various degree (Fig. 8) and the cumulative oil production for the field was only slightly impacted (Fig. 9).

6 SPE 90506

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

0 2000 4000 6000 8000 10000

Time, Days

No

rmal

ized

Cu

mu

lativ

e O

il P

rod

uct

ion

SPGLINCCOBYLAGA

Fig. 2 - Field Example 1: normalized field cumulative oil production allocated by the four rate allocation methods.

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 2000 4000 6000 8000 10000

Time, Days

Nor

mal

ized

Lift

Gas

Rat

e

SP

GLINC

COBYLA

GA

Fig. 3 - Field Example 1: normalized field daily lift gas injection volume allocated by the four rate allocation methods.

0

0.2

0.4

0.6

0.8

1

1.2

0 2000 4000 6000 8000 10000

Time, Days

No

rmal

ized

Lift

Gas

Rat

e

SPGLINCCOBYLAGA

Fig. 4 - Field Example 1: normalized lift gas rate for well A2 allocated by the four rate allocation methods.

0

0.2

0.4

0.6

0.8

1

1.2

0 2000 4000 6000 8000 10000

Time, Days

Nor

mal

ized

Oil

Rat

e

SPGLINCCOBYLAGA

Fig. 5 - Field Example 1: normalized oil rate for well A2 allocated by the four rate allocation methods.

Table 1 - Field Example 1: performance statistics for simulation runs with different rate allocation methods.

SP GLINC GA COBYLA Number of Time Steps* 1361 1369 960 904 Number of Iterations* 4703 4522 5136 4562 Time on Well Management (min)

196 266 6793 712

Total CPU Time (min) 2401 2466 9750 3710 * The runs with SP and GLINC methods have a maximum time step of

6 days while the runs with the GA and COBYLA methods have a maximum time step of 10 days.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 2000 4000 6000 8000 10000

Time, Days

No

rmal

ized

Lift

Gas

Rat

e

Without Damping

With Damping

Fig. 6 - Field Example 2 Case 1: normalized daily lift gas rate for well A3 obtained from the SP method with damping and without damping.

SPE 90506 7

0

0.2

0.4

0.6

0.8

1

1.2

0 2000 4000 6000 8000 10000

Time, Days

No

rmal

ized

Cu

mu

lati

ve O

il P

rod

uct

ion Without Damping

With Damping

Fig. 7 - Field Example 2 Case 1: normalized field cumulative oil production obtained from the SP method with damping and without damping.

Table 2 - Field Example 2 Case 1: performance statistics for the simulation run with damping and the simulation run without damping.

Without Damping With Damping

Number of Time Steps 2722 2716 Number of Iterations 10291 10388 Time on Well Management (min)

177 153

Total CPU Time (min) 1186 1119

0

0.2

0.4

0.6

0.8

1

1.2

0 2000 4000 6000 8000 10000

Time, Days

No

rmal

ized

Lift

Gas

Rat

e

Without Damping

With Damping

Fig. 8 - Field Example 2 Case 2: normalized daily lift gas rate for well A4 obtained from the SP method with damping and without damping.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 2000 4000 6000 8000 10000

Time, Days

No

rmal

ized

Cu

mu

lati

ve O

il P

rod

uct

ion Without Damping

With Damping

Fig. 9 - Field Example 2 Case 2: normalized field cumulative oil production obtained from the SP method with damping and without damping.

Table 3 - Field Example 2 Case 2: performance statistics for the simulation run with damping and the simulation run without damping.

Without Damping

With Damping

Number of Time Steps 3465 2223 Number of Iterations 48517 9270 Time on Well Management (min)

384 132

Total CPU Time (min) 3674 1180

Summary and Conclusion • The developed GLINC method is simple and easy to

implement. Though the method is a local search method and handles the flow rate constraints with approximations, it generates results of good quality for long-term simulation studies, as verified by the separable programming (SP) method, the GA method, and the COBYLA method.

• Although the SP method does not handle flow interactions through common flow lines, its execution in consecutive Newton iterations can mitigate this shortcoming.

• The GLINC and SP methods have distinctive characteristics. The GLINC method is more rigorous in function evaluation; however it does not guarantee local optima. The SP method uses significant simplifications in its function evaluation; but it guarantees the global optimum of the reformulated linear programming optimization problem. These two methods can be used as an alternative and verification method to each other.

• It was verified by the GA method and the COBYLA method that both the SP method and GLINC method are efficient and capable of generating quality results for some models with flow interactions among wells through common flow lines.

• For certain cases, the new lift gas rate damping method can significantly mitigate lift gas rate oscillations of individual wells and/or reduce the number of convergence failures of a simulation run, thus decrease the total CPU time requirement of the simulation studies.

8 SPE 90506

Nomenclature ( )xciv = i th constraint function of decision

variable xv e = gas-lift efficiency

mine = minimum gas-lift efficiency threshold used in method GLINC

f = objective function m = number of constraints

wn = number of wells

gQ = total gas flow rate capacity

igq , = formation gas rate of well i , MSCF/d

lgqv = well lift gas rates, MSCF/d

oqv

= well oil rate, STB/d

wqv

= well water rate, STB/d xv = decision variable of an optimization

problem

Symbol α = damping factor for the lift gas rate

damping method iqlg,δ = lift gas rate change for well i in the

GLINC gas-lift optimization method Acknowledgement The authors wish to acknowledge Peter Clifford and Chris Macdonald for their support and valuable opinions. The authors also thank the management of BP for granting permission to publish this paper.

References 1. Fang, W.Y. and Lo, K.K., “A Generalized Well-

Management Scheme for Reservoir Simulation”, SPE Reservoir Engineering, May 1996, 116-120.

2. Wang, P., Litvak, M.L. and Aziz, K., “Optimization of Production from Mature Fields”, paper presented at the 17th World Petroleum Congress, Rio de Janeiro, Brazil, Sep 1-5, 2002.

3. Hepguler, G., Barua, S. and Bard, W., “Integration of a Field Surface and Production Network with a Reservoir Simulator”, SPE Computer Applications, Jun 1997, 88.

4. Davidson, J.E. and Beckner, B.L., “Integrated Optimization for Rate Allocation in Reservoir Simulation”, paper SPE 79701 presented at the SPE Reservoir Simulation Symposium held in Houston, TX, U.S.A., Feb 3-5, 2003.

5. Miettinen, K., Nonlinear Multiobjective Optimization, Kluwer Academic Publishers, Boston, 1999.

6. Landmark, VIP-EXECUTIVE Technical Reference, 2003. 7. Litvak, M.L., and Darlow, B.L., “Surface Network and

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