SPE-140935-PA (Gas Lift Optimization Using Proxy Functions in Reservoir Simulation)

February 2012 SPE Reservoir Evaluation & Engineering 109

Gas Lift Optimization Using Proxy Functions in Reservoir Simulation

Q. Lu and G.C. Fleming, Halliburton, SPE

SummaryArtificial lift by means of gas injection into production wells or risers is frequently used to increase hydrocarbon production, espe-cially when reservoir pressure declines. We propose an efficient optimization scheme that finds the optimal distribution of the available gas lift gas to maximize an objective function subject to surface-pipeline-network rate and pressure constraints. This pro-cedure is a nonlinearly constrained optimization problem solved by the generalized reduced-gradient (GRG) method. The values of objective function, constraint functions, and derivatives needed for optimization can be evaluated through two methods. The first method repeatedly solves the full-network equations using Newton iteration, which takes into account the flow interactions among wells; however, this method can be computationally expensive. The second and more efficient method is a new approach proposed in this paper. It constructs a set of proxy functions that approximates the objective function and constraints as functions of gas lift rates. The proxy functions are obtained by solving part of the network that consists of a gas lifted well or riser, assuming a stable pres-sure at the terminal node where the partial network is decoupled from the rest of the network, and are used to inexpensively evalu-ate the objective function, constraints, and necessary derivatives for the optimizer. A procedure to predict the proxy functions on the basis of previous values can be used to reduce the number of partial-network solves, and the partial-network solution has been parallelized for faster simulation. These two methods can be applied at different timesteps during the course of the simulation. The proposed methods are implemented within a general-purpose black-oil and compositional reservoir simulator and have been applied to real-field cases.

IntroductionContinuous gas lift injection to production wells or risers is an important method to maintain and improve hydrocarbon production. The availability of lift gas is limited because it is typically provided by produced gas; the gas lift operation is also constrained by the resources of surface facilities, such as the compression and separation capacities. Therefore, one goal of well management is to optimally allocate available lift gas to targeted wells or risers to maximize hydrocarbon production under various facility constraints. Furthermore, as the market price of gas continues to increase and produced-water treatment becomes more expensive as a result of stricter environment regulations, it is desirable to maximize the overall profit rather than just production.

The gas lift optimization schemes described in the literature can be categorized into two types. The first type is based on single-well analysis; all wells on gas lift are isolated from each other, assuming a fixed wellhead or gathering-center pressure. In many cases, this assumption is valid because wells are normally choked at the wellhead to maintain a stable wellhead pressure, or multiple wells are gathered at a separator under a pressure control, or the pressure drop across the well tubing is dominant in the pipeline network. A performance curve (oil- or liquid-production rate vs. gas lift rate or total gas rate) for each well can be constructed, and

the optimal gas lift rate allocation can be found using some form of optimization. Kanu et al. (1981) proposed an equal-slope method. The gas lift was allocated to wells through a manual procedure at an average slope of the performance curves. This procedure did not take into account the surface-facility constraints except for the limit of total gas lift rate. Fang and Lo (1996) proposed a linear programming method to optimally allocate gas lift under various flow-rate constraints; the weighting factors used to interpolate the objective function and constraint functions were taken as deci-sion variables. Stoisits et al. (1999) developed neural-network models to replace the well and surface pipeline hydraulics simu-lation to reduce computation time, and genetic algorithms were implemented to solve the optimization problems on the basis of individual-well performances.

The second type is based on well solutions coupled through a surface pipeline network. This method is more rigorous when flow interactions among wells are strong. Dutta-Roy and Kattapuram (1997) pointed out that flow interactions among wells could be significant when the backpressure in the flowline shared by the wells was relatively large. Therefore, a constrained nonlinear opti-mization problem based on full-network solutions was formulated and was solved using successive-quadratic-programming (SQP) method. Hepguler et al. (1997) coupled a reservoir simulator and a surface network simulator by iteratively solving the two simula-tors until the boundary values between them were converged. An SQP optimizer was applied for production optimization in the surface network simulator. Wang and Litvak (2008) presented an optimization strategy that iteratively adjusted gas lift allocation and solved the full network until a minimum lift efficiency was achieved at all wells. They developed a linear programming model to scale the gas lift and production rates to satisfy network flow-rate constraints. In general, the computational cost of the second type is significantly higher than that of the first type because a large number of full-network solutions may be needed for opti-mization calculations.

We propose an efficient gas lift optimization scheme that com-bines the two methods under the same optimization algorithm. A constrained optimization problem is defined, and an objective function is maximized using a highly efficient derivative-based GRG optimizer (Lasdon and Waren 1997, Fletcher 1987). The gas lift rates are specified as the decision variables. The opti-mizer requires the values of the objective function, the constraint functions, and the first-order derivatives of the functions with respect to the decision variables. The values of the functions and derivatives can be evaluated by two methods. The first method is based on the full-network solution; the network is solved using Newton iteration every time the function or derivative evaluations are needed by the optimizer. This method can be very expensive, especially if the derivatives are numerically evaluated because the optimizer may need tens (or more) of function and derivative evaluations before the optimal point is found. The second method constructs a set of proxy functions that are similar to performance curves; the proxy functions estimate the objective and constraint functions as functions of gas lift rates. In this method, each gas lifted well or riser (which includes all wells flowing to the riser) is decoupled from the rest of the network at the first gathering node so that a partial network is formed. A stable pressure at the decoupling node (terminal node of a partial network) is assumed; in other words, the interaction between wells is ignored. Proxy functions of each gas lifted well or riser are obtained by solving the partial network over a specific range of gas lift rates. They are

Copyright © 2012 Society of Petroleum Engineers

This paper (SPE 140935) was accepted for presentation at the SPE Reservoir Simulation Symposium, The Woodlands, Texas, USA, 21–23 February 2011, and revised for publication. Original manuscript received 24 March 2011. Revised manuscript received 9 August 2011. Paper peer approved 17 August 2011.

110 February 2012 SPE Reservoir Evaluation & Engineering

used to inexpensively evaluate the objective function, constraints, and necessary derivatives for the optimization calculations. The number of sample points (approximately 10) of gas lift rates for proxy-function construction is normally much smaller than the number of function or derivative evaluations needed by the opti-mizer, and the partial-network solve is much less costly than the full network solve. Therefore, the method using proxy functions is much faster than the method using full-network solutions. We solve the partial networks in the same way that we solve the full network, using Newton iterations, as explained in the Network Computations section.

We may change the solution methods during the course of the simulation. For instance, when gas lift first starts, optimization based on the full-network solution can be used because the interac-tions between wells may be strong. Once the wellhead pressures become stable, the method based on proxy functions can be applied for faster simulation.

Notably, Rashid (2010) and Rashid et al. (2011) proposed a procedure for gas lift optimization using gas lift performance curves at given wellhead pressures for each gas lifted well and performed optimization as a separable problem. The wellhead pressure is then updated by a call to the underlying network simulator to account for well interaction, and the optimization problem is solved again [a related paper is Gutierrez et al. (2007)], where the network simulator is coupled with the reservoir model described by look-up tables. This approach is similar to the proxy-function method proposed in this paper. However, there are some key differences between the two methods. First, in addition to constructing a proxy function of the objective function (the per-formance curve), we also construct proxy functions of some of the network constraints, which will be used to estimate the opti-mization constraints. Second, the performance curve in Rashid’s method is constructed by solving the gas lifted well using single-well nodal analysis, while in our approach we solve the partial-network equations using Newton iteration, as explained in the Network Computations section. Our approach allows solution for a gas lifted riser that includes multiple wells flowing to the riser. Third, the optimization problem is solved with a Newton reduc-tion method in Rashid (2010), a genetic algorithm in Gutierrez et al. (2007), and a mixed-integer nonlinear programming solverin Rashid et al. (2011), while in our approach it is solved using the GRG method. Fourth, in Rashid’s method the network con-straints are generally managed using penalty imposition in the optimization procedure. They did not give details on handling constraints at individual network connections and nodes and the manifold constraints. We provide a detailed procedure to strategi-cally handle various network constraints in the sections Treatment of Constraints and the Gas Lift Optimization Procedure section. Fifth, in Rashid’s method, the wellhead pressure is updated by a call to the network simulator to account for the effects of well interaction. No coupling is made with the reservoir other than the boundary conditions imposed (assuming fixed reservoir con-ditions at a give timestep). In our implementation, the network domain and the reservoir domain are fully coupled.

The optimization calculations proposed herein are implemented in a commercial general-purpose black-oil and compositional reservoir simulator featuring full implicit coupling of reservoir-grid domain and surface-pipeline-network domain (Coats et al. 2003). Multiple reservoirs can be connected with the same surface network. The governing equations of the two domains (reservoir-grid and surface network) are linearized and are put in a single Jacobian-matrix equation system, which is solved by a Newton iteration—namely, a reservoir/network global Newton iteration. However, the optimization is performed in a “standalone” network solve (also using Newton iteration), assuming fixed reservoir grid conditions (using the grid-cell pressures and phase compositions from the preceding global Newton step) at the beginning of the global Newton step. The determined optimal gas lift allocation will be applied in the current global Newton iteration. Although the optimization is explicit with regard to the global solve, the reservoir and network conditions used by optimization will get updated at the next global Newton step or timestep.

MethodologyMathematically, the optimal gas lift allocation can be formulated as a constrained nonlinear optimization problem in the following form:

Maximize ( )F x x Rn, ∈ , . . . . . . . . . . . . . . . . . . . . . . . . . . (1)

subject to constraints

L g x U i mi i i≤ ≤ =( ) , , ,1 . . . . . . . . . . . . . . . . . . . . . . . . (2)

S x H j nj j j≤ ≤ =, , ,1 , . . . . . . . . . . . . . . . . . . . . . . . . . (3)

where F(x) is the objective function; x is the vector of n decision variables; gi is the ith inequality constraint function; Li and Ui are the lower and upper bounds of constraint gi, respectively; m is the number of inequality constraints; and Sj and Hj are the lower and upper bounds of the decision variable xj, respectively. The gas lift rates are chosen as the decision variables.

We define an objective function as

F x Q R Q R Q R Q R( ) oil oil gas gas water wat glift gl= + − − iift , . . . . . . . . . . (4)

where, Qoil, Qgas, Qwater, and Qglift are the total volumetric flow rate of oil-phase production, gas-phase production, water-phase produc-tion, and gas lift injection at surface condition, respectively; Roil is the unit value of the oil phase; Rgas is the unit value of the gas phase; Rwat is the unit processing cost of water phase; and Rglift is the unit operating cost of gas lift.

The optimization problem is solved by a derivative-based opti-mizer using the GRG algorithm, ranked as one of the most efficient algorithms for the nonlinearly constrained optimization problem, especially for cases with a large number of decision variables. The function evaluations of the optimization are provided by network solutions.

Network ComputationsA surface pipeline network includes all wells and the connected pipelines and surface facilities. It consists of connections and nodes. Types of connections include well tubing strings, pipes, chokes, and pumps. Types of nodes include perforation inlets/out-lets, gathering centers, and separators. Produced- or injected-fluid streams flow through the connections and join at the nodes. The network flow computations are formulated as steady state, because the connections and nodes are treated as having no volume. Bound-ary conditions are set by network sink and source pressures, source fluid compositions and perforated reservoir grid-cell pressures, and fluid mobilities. The network equations consist of

1. Connection equations imposed at connections2. Perforation equations at perforations3. Mass-balance equations at nodes

These equations are linearized and solved at each Newton step. Further details can be found in Coats et al. (2003) and Shiralkar and Watts (2005).

For a connection without a rate or pressure constraint, the con-nection equation is a hydraulics equation that relates the pressure drop across the connection to fluid-component mass-flow rates using either a correlation or a hydraulics table. For a connection with constraints, the connection equation can be a hydraulics equa-tion, a rate-constraint equation, or a pressure-constraint equation, depending on which equation most constrains the flow rate. When a constraint determines the flow rate of a connection, the constraint is said to be “active.” Physically, this means that a choke or valve at the connection reduces the flow rate so that the constraint is exactly satisfied. Therefore, the pressure drop across the connection is larger than it would be if the hydraulics equation were active. It is critical to determine a correct set of active network constraints, otherwise the network solution could be incorrect because some constraints are violated, or if the network is overconstrained, the equations could be singular. Watts et al. (2009) proposed an


innovative systematic method using the slack variable for active network constraint determination, which has been implemented in our network simulation.

If needed, a part of the network can be constructed and solved using the same kind of equations and solution procedure as in the full-network solve, given the pressure constraints at its terminal nodes, or the rate constraints at the terminal connections (the terminal nodes and connections are where the partial network is decoupled from the rest of the full network) and constant reservoir grid conditions. In our simulator, we use the same program for partial- and full-network computations, which greatly simplified the implementation.

Treatment of ConstraintsVarious user-specified constraints can be set at different points of the network. Typically these are maximum flow-rate constraints at individual connections and minimum flowing-pressure constraints at individual nodes. There may also be a maximum flow-rate con-straint on the sum of flow rates at a list of controlled connections. This type of constraint is referred to as a “target” in the paper.

To allocate the target rate to controlled connections, a spe-cial network solution, referred to in the paper as the “potential” network solution, needs to be performed. That is, the network is solved with all targets removed, but all rate constraints at individual connections and all pressure constraints at nodes are retained (assuming fixed reservoir conditions). Therefore, all the constraints at individual connections and nodes are satisfied using the slack-variable method (Watts et al. 2009) in the potential network solve. Flow rates in this solution represent the potential productivity or injectivity of the connections. The target rate is then allocated to the connections according to their potential flow rate if the total potential rate exceeds the target rate. This way, a target is converted to rate constraints at individual connections. All constraints (including targets) will be satisfied in the following reservoir/network global Newton iteration.

There are two types of target: scale type and ranking type. With scale type, the target rate is allocated to the controlled connections according to some formula, such as in proportion to their potential flow rates or in proportion to a specified guide rate. With ranking type, the controlled connections are ranked by some criterion such as water cut. Then, in the ranked order, the connections are shut in until the next connection would result in the flow rate being less than the target rate. This connection is then produced at a reduced rate so that the target is exactly met, and the remaining open con-nections are produced at their maximum rate. Because each target has multiple controlled connections, it is possible to exactly satisfy multiple targets simultaneously.

In our implementation, the potential network solution described will be used as function evaluations in the optimization procedure, and the scale-type targets are converted to constraints of the opti-mization problem. For clarity, all the user-specified constraints set at the network are called “network constraints,” while the constraints defined for the optimization problem are referred to as “optimization constraints.”

Gas Lift Optimization ProcedureThe procedure works using the following steps:

1. At the beginning of a selected reservoir/network global Newton step for a timestep when the gas lift optimization is performed, solve the optimization problem with the GRG optimizer using the values of decision variables from the previous optimization solution as the initial guess. The function evaluations of the objective function and the constraint functions, as well as the necessary derivatives of the functions, are provided by the potential network solutions, assuming fixed reservoir grid conditions (pressure, phase mobilities, and phase compositions). All scale-type targets are treated as optimization constraints; the total available gas lift rate is also set as an optimiza-tion constraint. We propose two methods for the potential network solutions, as explained in the Function Evaluations section.

2. Allocate the gas lift rates according to the solution of the decision variables. All network rate constraints at individual con-nections and all network pressure constraints have been satisfied

through the function evaluations (network solves). All scale-type targets have been satisfied as optimization constraints.

3. After the optimization is performed, a target allocation procedure is performed on all ranking-type targets. Flow rates on the controlled connections are scaled back to meet these targets if necessary. The gas lift rates are scaled back accordingly to achieve the well flow rates, as described in the Gas Lift Rate Reduction section.

4. Repeat Steps 1 through 3 once if any gas lift rate has been scaled back in Step 3.

5. Solve the grid/network linearized system in the current global Newton step with determined gas lift rates.

6. Repeat Steps 1 through 5 until the global Newton iteration converges.

7. March to the next timestep.Fig. 1 shows the flow chart of the whole procedure. Table 1

summarizes how the various constraints are satisfied.

Function EvaluationsThe majority of computational time of the optimization procedure is spent on the function evaluations because each involves a poten-tial network solution. We propose two methods for these potential network solutions.

Full-Network Solution. The fi rst method is based on full-net-work solution (see the Network Computations section). During the iterative steps of the optimization, the values of the objective function and constraint functions are calculated using the connec-tion volumetric fl ow rates of the full-network potential solution. The derivatives of these functions with respect to the decision variables (gas lift rates) are combinations of the derivatives of the connection volumetric phase fl ow rates, Q, with respect to the decision variables, which can be calculated through the chain rule, as follows:

∂∂

= ∂∂

∂∂

==∑Q

x

Q

q

q

xkk

j

k

ll

nl

j

c

1

, oil, gas, water;; j n= 1, , , . . . . . (5)

where nc is the number of fluid components; ql is the mass flow rate of Component l, which is one of the primary unknowns in network equation system; xj is a decision variable; and n is the number of

decision variables. ∂∂Q

qk

l

is evaluated through flash calculations. To

obtain ∂∂

q

xl

j

, we perturb the right-hand side of the connection equa-

tion at the gas lift connection corresponding to xj in the linearized network equation system and solve the system. The matrix of the linearized equations has already been factored in the full-network solve, the solution of the perturbed system requires only a for-ward and backward substitution, which is relatively inexpensive. However, because a large number of function evaluations may be needed during optimization iterations and the computational cost of function evaluations can be very high for a relatively large net-work, this approach can be prohibitively expensive.

Proxy Functions Based on Partial-Network Solution. The sec-ond method constructs a set of tabular proxy functions that approximate the objective function and constraints as functions of gas lift rates. This approach is based on the assumption that the gathering nodes where the gas lifted wells or risers join have stable pressures. Therefore, these wells/risers can be decoupled from each other at the gathering nodes, and the interaction between wells is ignored. This assumption is generally reasonable because wells are normally choked at the wellhead and the fl ow through the choke can be critical (at sonic velocity), or multiple wells are gathered at a separator under a pressure control, or the pressure drop across the well tubing is dominant in the pipeline network. Numerical experiments also showed that the gathering-node pressures tend to be stable after a short period of time (a few timesteps) once the gas lift injection starts. In addition, the gathering-node pressures get updated at every global Newton step, so the interactions between


wells have been taken into account in an explicit manner. Another assumption of the method is that each gas lifted well or riser has only one gas lift injection point, while the method based on full-network solution does not have this restriction.

For every gas lifted well/riser, a partial network consisting of this well/riser up to the first gathering node (or the sink node if the well/riser connects to the sink directly) is set up. The terminal node at the top of the partial network takes the pressure from the previous reservoir/network global Newton step as the minimum pressure constraint. Fig. 2 is an illustration of how a full network is decoupled into two partial networks at the gathering node (Node 2). Because the flow rates of the partial networks are determined by the boundary condition at Node 2 (node pressure is P2), Node 1 and Conn 1 become irrelevant and are excluded from the partial networks. For a gas lifted riser, the partial network includes all wells flowing to the riser. The gas lift rate of a partial network is a decision variable. The partial-network model is solved over a range of prespecified gas lift rates to give the tabular sets neces-sary to furnish the proxy models of the objective and constraint functions. The proxy value of the objective function of a partial network can be evaluated using Eq. 4 on the basis of solution of the partial network at a given gas lift rate. The same approach is used for the proxy values of constraint functions. Fig. 3 shows a sample proxy function of the objective function, and Fig. 4 shows a sample proxy function of an oil-phase rate corresponding to an oil-phase target, both as functions of a decision variable (gas lift rate of a gas lifted well). In this example, the well cannot flow until the gas lift rate is greater than approximately 600 Mscf/D.

The overall objective function (defined in Eq. 4) can be defined as the sum of the proxy functions of the objective function of all gas lifted wells:

F x F xpj jj

n

( ) ( )==

∑1

, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6)

where Fpj(xj) is the proxy function of the objective function of Well j, which is associated with Decision Variable j.

Optimization constraint function in Eq. 2 becomes

g x g x i mi pi jj

n

j( ) ( )= ==

∑ , , , ,1

1 , . . . . . . . . . . . . . . . . . . . . . (7)

where gpi,j is the proxy function of Well j, corresponding to Opti-mization Constraint i.

Once the tabular proxy functions for each gas lifted well have been constructed at selected reservoir/network global Newton steps, the gas lift optimization can be performed using the values of the overall objective function and constraint functions calculated from Eqs. 6 and 7. Either piecewise linear interpolation or piece-wise polynomial interpolation of the tabular proxy functions can be used at a given set of decision-variable values (gas lift rates). The derivatives of these functions with respect to a decision vari-able are simply the slopes of the proxy functions of this decision variable, or

∂∂

= =F x

x

F x

xj n

j

pj j

j

( ), , ,

d ( )

d1 . . . . . . . . . . . . . . . . . . . . . . (8)

Gas lift optimization with GRG

Rate allocation for ranking-type targets

Scale back gas lift rates

Reservoir/network global Newton

Next global Newton step

No

Next timestep

Yes

Converged?

Rate/pressure constraints at individual connections and nodes, and scale-type targets.

Satisfy:

Satisfy:

Ranking-type targets. Once

Start of timestep

Fig. 1—Flow chart of gas lift optimization procedure.

TABLE 1—TREATMENT OF VARIOUS CONSTRAINTS

Constraints Satisfied At

Rate constraints at individual connections and pressure constraints at nodes

“Potential” network solution for function evaluations

Scale-type targets Optimization constraints

Ranking-type targets Post-optimization scaling back procedure


and

∂∂

= = =g x

x

g x

xi m j ni

j

pi j j

j

( ) d ( )

d, , , , ; , ,1 1 . . . . . . . . . . (9)

The method using proxy functions is much faster than the method using the full-network solution for the following reasons: First, the number of gas lift rates used to construct proxy functions is relatively small (approximately 10) compared to the number of full-network solutions needed in the first method. Second, solving the partial network is much faster than solving the full network because the former is a much smaller linearized equation system. Third, the solving of partial networks can be easily parallelized because they are decoupled.

Proxy-Function ScalingWhen the network configuration and constraints do not change with time, and the pressure at the decoupling node of the partial network does not change much, the shapes of proxy functions may change

relatively little. Fig. 5 shows an example of the proxy functions of the objective function at two timesteps, t1 and t2 with t1 < t2, that are close in time. In this case, instead of calculating all actual points of t2, we could first evaluate the function values at three gas lift rates at time t2 using partial-network solutions and compare them with the values at time t1. If they change only slightly and the changes are in the same trend, we could safely estimate the values at other gas lift rates at time t2 by proportionally shifting the curve at time t1 to time t2, on the basis of the three points computed at time t2. In other words, the other points in curve t2 could be calculated by interpolating the three actual points. This way, we reduce the number of partial-network solutions required to construct the proxy functions at time t2. The three checking points can be computed at, respectively, a gas lift rate close to zero, the gas lift rate that gives the maximum objective-function value at time t1, and the gas lift rate at which the slope of objective function at t1 is largest when the gas lift rate is in logarithmic scale. If the third point is too close to the first or the second point, we instead compute the point at the average gas lift rate of the first and the second points.

Sink

Sink Sink

1 krowteN laitraPkrowteN lluF 2 krowteN laitraP

Perf 1 Perf 2 Perf 1 Perf 2

conn 1

conn 2

conn 3 conn 6

conn 4 conn 7

node 1

node 2

node 3 node 6

node 4 node 7

Pmin = 14.7 psi

PPmin = P nim2 = P2

conn 2

conn 3

conn 4

conn 2

conn 6

conn 7

node 2 node 2

node 3

node 5

node 6

node 8

Decoupling

Gas lift 1 Gas lift 2 Gas lift 1 Gas lift 2

node 5 node 8

conn 5

node 4 node 7

conn 8 nnoc 5 nnoc 8

Fig. 2—Decoupling of a full network into partial networks.

-2.00E+04

0.00E+00

2.00E+04

4.00E+04

6.00E+04

8.00E+04

1.00E+05

1.20E+05

10 100 1000 10000 100000

Gas Lift Rate (Mscf/D)

Pro

xy F

un

ctio

n o

f O

bje

ctiv

e F

un

ctio

n

(Do

llars

/day

)

Fig. 3—Proxy function of the objective function associated with a decision variable.

0.00E+00

5.00E+03

1.00E+04

1.50E+04

2.00E+04

2.50E+04

3.00E+04

10 100 1000 10000 100000


Pro

xy F

un

ctio

n o

f O

il-P

has

e R

ate

Co

nst

rain

t (S

TB

/D)

Fig. 4—Proxy function of a constraint function associated with a decision variable.


Gas Lift Rate ReductionAs mentioned in Step 3 of the gas lift optimization procedure, certain connection rates may need to be reduced to satisfy rank-ing-type targets. A target allocation procedure determines if there are any ranking-type targets getting violated after the gas lift optimization. If any such target is violated, the flow rates of the controlled connections of the target are reduced on the basis of the ranking method (e.g., the flow rate of the connection with the highest water cut will be reduced to meet the target if the ranking method is water-cut based). The reduction factor is calculated and is propagated to all downstream and upstream connections. The gas lift rate can be reduced according to how much the flow rate of the associated well tubing connection has been reduced. When the proxy function method is used, we can interpolate a phase-flow rate (e.g., oil-phase-flow rate) in the corresponding proxy function to obtain the reduced gas lift rate.

When the full-network solution method is used, we may simply reduce the gas lift rate using the same reduction factor as is being applied to the flow rate at the gas lifted well tubing connection. This approach assumes a linear relationship between the well-tubing-connection flow rate and the gas lift flow rate, and this assumption can be inaccurate. To improve the solution, we construct a local proxy function (a phase-flow rate vs. gas lift rate, such as in Fig. 4) for the gas lifted well through a partial-network solution using the latest gathering-node pressure as the minimum pressure constraint

at the terminal node of the partial network. The proxy function can then be used to estimate the reduced gas lift rate. Note that in this approach, the proxy function is used just for gas lift rate reduction estimates. It is not involved in gas lift optimization calculations.

Examples and ComparisonsFor convenience, the simulator in which the proposed gas lift optimization methods are implemented is referred to as Simula-tor N. We compare our implementation with another commercial reservoir simulator, Simulator V (Litvak and Darlow 1995), in one of our examples. A scheme of automatic allocation of gas lift using a performance curve has been implemented in Simulator V. In this scheme, nodal analysis is used to construct the gas lift performance curve (produced-oil-phase rate vs. gas lift rate) at selected timesteps for each gas lifted well, assuming fixed minimum tubinghead pres-sures. The gas lift rate is calculated (it is basically a table-lookup procedure) on the basis of the minimum lift efficiencies of the wells specified by the user, where the lift efficiency is defined as the incremental oil produced per increment of gas lift gas, which is effectively the slope of the gas lift performance curve. In Simulator V, the reservoir-grid domain and the surface-network domain have partial implicit coupling because they are solved with an alternative Schwarz procedure; that is, at selected timesteps, the network is solved with fixed reservoir grid conditions, then the reservoir grid is solved with updated well bottomhole or well tubinghead condi-tions provided by the latest network solution. These network and reservoir grid solutions are repeated iteratively until the boundary values (well flow rates and bottomhole pressures) between them are converged. For comparison, we refer to the automatic gas lift allocation method in Simulator V as the “fixed-slope” scheme, the method using proxy functions in our implementation as the “proxy” scheme, and the method using full-network solution in our implementation as the “full-network” scheme.

Case 1. Case 1 is a black-oil model simulated for 900 days. It has four gas lifted production wells with fi xed minimum tubinghead pressures, one gas-injection well, and one water-injection well. All wells are connected directly to a sink or a source. To match the user-specifi ed minimum lift effi ciency (0.01 STB/Mscf) used by the “fi xed-slope” scheme, we set Roil = USD 30/STB, Rgas = 0, Rwat =0, and Rglift = USD 0.3/Mscf in the proxy and the full-network schemes so that Rglift/Roil = 0.01 STB/Mscf. There is an 8,000 STB/D maximum oil-phase rate constraint at every well but they are inac-tive during the simulation. The maximum gas lift rate allowed for each well is 10,000 Mscf/D. Fig. 6 compares the oil-production

40000

50000

60000

70000

80000

90000

100000

110000

0.1 10 1000 100000 10000000


Ben

efit

Fu

nct

ion

(D

olla

rs/d

ay)

t1

t2

Fig. 5—Proxy functions at two timesteps.

0 100 200 300 400 500 600 700 800 900 1000

Time (days)

0.2

0.4

0.6

0.8

1.0

No

rmal

ized

QO

P —

Oil-

Pro

du

ctio

n R

ate

0

0.2

0.4

0.6

0.8

1.0

No

rmal

ized

QG

LG

— G

as L

ift

Inje

ctio

n R

ate

QOP : Fixed-slope QOP : Full-network QOP : Proxy

QGLG : Fixed-slope QGLG : Full-network QGLG : Proxy

Fig. 6—Oil-production rates and gas lift rates of Case 1.


rate and gas lift injection rate of the simulation results. Before 130 days, there is an active water-production target (8,000 STB/D) that prohibits any gas lift injection. Afterward, as the reservoir depletes, Simulators V and N manage to inject gas lift to improve oil produc-tion without violating the water-production target. The total lift gas available is 10,000 Mscf/D. The proxy scheme predicts a higher oil-production rate after approximately 200 days by injecting more lift gas. Fig. 7 shows that the proxy scheme also manages to reduce water production compared to the fi xed-slope scheme. A closer investigation of the results shows that the proxy scheme injects more lift gas in three production wells with high oil-production rates and low water cuts as well as less lift gas in a well with low oil-production rate and high water cut compared to the fi xed-slope scheme. This is probably because the proxy scheme performs an optimization, and handles constraints more strictly than the fi xed-slope method. Fig. 8 shows that the proxy scheme increases the cumulative benefi t by approximately 3% over the fi xed-slope scheme. The solution of the full-network scheme is virtually the same as that of the proxy scheme in Figs. 6 and 7, which provides a confi rmation of the proxy-function implementation.

The simulation CPU times (in a Windows XP machine with Intel Xeon 2.33 GHz CPU and 3 GB of RAM) using the fixed-

slope, proxy, and full-network schemes are 119, 124, and 316 seconds, respectively. As a result, the simulation speed of the proxy scheme in Simulator N is comparable with the fixed-slope scheme in Simulator V and is approximately 2.5 times as fast as the full-network scheme. This example demonstrates that the optimization method using proxy functions is much more efficient than the method using the full-network solution and can achieve a more optimal solution than the fixed lift efficiency scheme.

Case 2. Case 2 is used to investigate the effects of the interactions between wells. The properties of the reservoir domain are based on the SPE9 test case (Killough 1995), but the values of solution-gas/oil ratio (Rs) have been reduced by approximately three times (Table 2) to make the effect of gas lift more obvious. It is a black-oil model with 30 production wells, 20 of which are gas lifted starting at time of 50 days. There are no injection wells. All the wells are connected through a network that has three levels of gathering centers. The highest level connects to a sink (Fig. 9). All network constraints are listed in Table 3. We set Roil = USD 30/STB, Rgas = USD 3/Mscf, Rwat = USD 2/STB, and Rglift = USD 4.5/Mscf. Most of the oil-rate constraints at wells are inactive during the simulation, and there is no active pressure constraint except for a minimum pressure con-straint (100 psia) at the sink node. Therefore, the well fl ow rates are mostly determined by the hydraulics in the tubings and fl owlines, and the interactions between wells are strong, especially when the

0 100 200 300 400 500 600 700 800 900 1000

Time (days)

0.3

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1.0

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No

rmal

ized

QW

P —

Wat

er-P

rod

uct

ion

Rat

e

QWP : Fixed-slope QWP : Full-network QWP : Proxy

Fig. 7—Water-production rates of Case 1.

0

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1

1.2

0 200 400 600 800 1000

Time (days)

No

rmal

ized

Cu

mu

lati

ve B

enef

it Fixed-slope

Proxy

Fig. 8—Cumulative benefit of Case 1.

TABLE 2—SOLUTION-GAS/OIL RATIO IN CASE 2

Pressure (psia) Rs (Mscf/STB)

0005.0 0.0004

2364.0 0.0063

1324.0 0.0023

4673.0 0.0082

9723.0 0.0042

5572.0 0.0002

2122.0 0.0061

1661.0 0.0021

9011.0 0.008

9350.0 0.004

0.0 7.41


gas lift fi rst starts. The pressure drops after the well tubing heads are approximately at 400 psi. Fig. 10 shows that the oil-phase-produc-tion rates of the fi eld predicted by full-network and proxy methods are noticeably different at the early stage of the simulation after gas lift starts. We now apply these two methods in sequence: Before

time = 60 days, the full-network method is used to handle the strong coupling of wells shortly after the gas lift starts; after that time, the proxy method is used for faster simulation. We refer to this scheme as the “mixed” method. The switch time is chosen on the basis of the observation that the fl ow is about to be stable using the full-network

Sink

Pmin = 100 psia

Gas liftWell-17

Fig. 9—Surface-network structure of Case 2. The arrows at the bottom of the plot denote gas lift injection to the bottomhole nodes of the wells.

TABLE 3—NETWORK CONSTRAINTS IN CASE 2

Maximum oil-phase rate constraint at each well 15,000 STB/D, time < 50 days 5,000 STB/D, time 50 days

Maximum gas lift rate constraint at each well 0 Mscf/D, time < 50 days

20,000 Mscf/D, time 50 days

Mscf/D 000,02 tfil sag latot fo etar tegraT

aisp 001 edon knis ta tniartsnoc erusserp muminiM

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30000

40000

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QO

P —

Oil-

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ate

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B /

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LG

— G

as L

ift

Inje

ctio

n R

ate

(Msc

f / D

)

QOP : Full-network QOP : Proxy QOP : Mixed

QGLG : Full-network QGLG : Proxy QGLG : Mixed

Fig. 10—Case 2: Effects of interactions between wells.


method approximately 10 days after the gas lift injection starts. A general criterion to determine when the switch should happen is still under investigation. Fig. 10 shows that the mixed method achieved production rates very close to the full-network method. The total gas lift rates computed by the three methods are overlapped. Fig. 11 shows the oil-production rates and gas lift rates of Well-17. For a short period of time after the gas lift starts, the rates computed by the proxy method show some oscillations because unstable well-head pressures, as shown in Fig. 11. The mixed method achieves results very close to those of the full-network method. The CPU time of the network calculations of the three methods (full-network, proxy, and mixed) are 126, 50, and 71 seconds, respectively.

Case 3. Case 3 is a multiple-reservoir compositional model. More than 10 reservoirs are connected through a common surface net-work. The fl uids of these reservoirs have up to 19 components.

There are 100 production wells and a number of gas- and water-injection wells. 60 of the production wells are assisted with gas lift. The network consists of approximately 500 connections, with a large number of rate and pressure constraints, and approximately 50 production and injection targets of either scale type or ranking type. There are multiple levels of gathering centers and separators in the network. The network eventually fl ows to a sink with a minimum pressure constraint of 14.7 psia. Complicated procedures are defi ned to dynamically change the constraints and network confi gurations during the simulation. We set Roil = USD 60/STB, Rgas = 0, Rwat = 0, and Rglift = USD 0.6 /Mscf in this case. The oil-phase-production rates computed by the full-network method agree well with those computed by the proxy method, which is shown in Fig. 12. The gas lift rates of the two methods follow approximately the same trend, but the former gives higher values than the later, as shown in Fig. 12. The case was run in parallel on four processors.

0 50 100 150 200 250

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QOP W17 : Full-network QOP W17 : Proxy

QOP W17 : Mixed QGLG W17 : Full-network

QGLG W17 : Proxy QGLG W17 : Mixed

QO

P —

Oil-

Pro

du

ctio

n R

ate

(ST

B /

D)

QG

LG

— G

as L

ift

Inje

ctio

n R

ate

(Msc

f / D

)

Fig. 11—Oil-production rates and gas lift rates of Well-17 in Case 2.

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Time (days)

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QG

LG

— G

as L

ift

Inje

ctio

n R

ate

QOP : Full-network QOP : Proxy

QGLG : Full-network QGLG : Proxy

Fig. 12—Oil-production rates and gas lift rates of Case 3.


The proxy method is much faster than the full-network method. The network solution CPU times of the two methods are 4.4 hours and 9.9 hours, respectively.

From our general observations of a number of test cases (not all shown in this paper), we have found that although the total produc-tion rates predicted by the different optimization methods normally match well, the gas lift rate allocations to individual wells can be quite different because the optimization can be trapped by different local optimal points. In addition, the network solution is typically sensitive by nature; a small difference in the solutions at a timestep may create very different flow patterns (different wells flow or are shut-in) at a later timestep. The sensitivity of solutions can be allevi-ated by adding rate or pressure constraints at individual wells.

Conclusions1. We have implemented an efficient gas lift optimization scheme

in a general-purpose black-oil and compositional reservoir simulator with fully coupled reservoir grid domain and sur-face-network domain. The nonlinearly constrained optimization problem is solved by the GRG method.

2. Network constraints are satisfied strategically in the optimiza-tion scheme: All surface-network constraints are satisfied at individual connections or nodes; all scale-type targets are satis-fied during optimization; and all ranking-type targets are satis-fied through a gas lift reduction procedure after optimization.

3. The function evaluations for optimization can be performed by two methods. The first method repeatedly solves the full-network equations using Newton iteration. The second method constructs a set of proxy functions using partial-network solu-tions.

4. The full-network method is more rigorous because it takes into account the flow interactions among wells; the proxy-function method neglects these interactions, assuming a fixed gather-ing-node pressure, so the solution can be suboptimal when the interactions are strong, especially during a short period of time after the gas lift first starts. However, the proxy-function method has the following advantages over the full-network method:

a. The computation speed of the proxy-function method is sig-nificantly faster because once the proxy functions have been constructed, they can be used to inexpensively approximate the objective function and constraints as functions of gas lift rates during optimization iterations. On the contrary, the full-network method has to resolve the whole network at every optimization iteration, which can be very expensive.

b. The number of partial-network solutions for proxy-function construction can be reduced through a simple interpolation scheme if the shapes of the proxy functions do not change much between two optimizations.

c. The calculations of partial networks in proxy-function construc-tion have been parallelized for faster simulation.

5. According to the test cases presented in the paper, the production values predicted by the two methods are in good agreement. We also observe that while the total gas lift rates given by the two methods generally follow the same trend, the gas lift allocations at individual wells can be different because optimization can be trapped by local optimal points using derivative-based optimizer, and solutions of the surface network are sensitive by nature. The two methods can be used in tandem during the simulation.

Nomenclature F = overall objective function, USD/D Fpj = proxy function of objective function of decision variable

j, USD/D gi = constraint function i gpi,j = proxy function of constraint function i of decision vari-

able j Hj = the upper bound of decision variable j i = subscript denoting constraint function j = subscript denoting decision variable k = subscript denoting fl uid phase

Li = the lower bound of constraint function i m = number of constraint functions n = number of decision variables nc = number of fl uid components p = node pressure, psi pmin = minimum allowed pressure, psi ql = mass fl ow rate of component l Qk = total volumetric fl ow rate of phase k Qoil = total volumetric fl ow rate of oil-phase production, STB/D QOP = volumetric fl ow rate of oil-phase production, STB/D Qgas = total volumetric fl ow rate of gas-phase production, Mscf/D Qglift = total volumetric fl ow rate of lift gas, Mscf/D Qwater = total volumetric fl ow rate of water-phase production, STB/D QWP = volumetric fl ow rate of water-phase production, STB/D QGLG = volumetric fl ow rate of lift gas, Mscf/D Rn = n-dimensional real coordinate space Roil = unit value of oil phase, USD/STB Rgas = unit value of gas phase, USD/Mscf Rglift = unit cost of gas lift operation, USD/Mscf Rwat = unit cost of water-phase treatment, USD/STB Sj = the lower bound of decision variable j Ui = the upper bound of constraint function i x = vector of decision variables xj = decision variable j

AcknowledgmentsBill Watts provided helpful comments during revision of this paper. The authors wish to thank Halliburton for permission to present this paper.

ReferencesCoats, B.K., Fleming, G.C., Watts, J.W., Rame, M., and Shiralkar, G.S.

2003. A Generalized Wellbore and Surface Facility Model, Fully Coupled to a Reservoir Simulator. Paper SPE 79704 presented at the SPE Reservoir Simulation Symposium, Houston, 3–5 February. http://dx.doi.org/10.2118/79704-MS.

Dutta-Roy, K. and Kattapuram, J. 1997. A New Approach to Gas-Lift Allocation Optimization. Paper SPE 38333 presented at the SPE Western Regional Meeting, Long Beach, California, USA, 25–27 June. http://dx.doi.org/10.2118/38333-MS.

Fang, W.Y. and Lo, K.K. 1996. A Generalized Well-Management Scheme for Reservoir Simulation. SPE Res Eng 11 (2): 116–120. SPE-29124-PA. http://dx.doi.org/10.2118/29124-PA.

Fletcher, R. 1987. Practical Methods of Optimization, second edition, 317–322. New York: John Wiley & Sons.

Gutierrez, F.A., Hallquist, A.E., Shippen, M.E., and Rashid, K. 2007. A New Approach to Gas Lift Optimization Using an Integrated Asset Model. Paper IPTC 11594 presented at the International Petro-leum Technology Conference, Dubai, 4–6 December. http://dx.doi.org/10.2523/11594-MS.

Hepguler, G., Barua, S., and Bard, W. 1997. Integration of a Field Surface and Production Network With a Reservoir Simulator. SPE Comp App 9 (3): 88–92. SPE-38937-PA. http://dx.doi.org/10.2118/38937-PA.

Kanu, E.P., Mach, J., and Brown, K.E. 1981. Economic Approach to Oil Production and Gas Allocation in Continuous Gas Lift. J Pet Technol 33 (10): 1887–1892. SPE-9084-PA. http://dx.doi.org/10.2118/9084-PA.

Killough, J.E. 1995. Ninth SPE Comparative Solution Project: A Reexami-nation of Black-Oil Simulation. Paper SPE 29110 presented at the SPE Reservoir Simulation Symposium, San Antonio, Texas, USA, 12–15 February. http://dx.doi.org/10.2118/29110-MS.

Lasdon, L.S. and Waren, A.D. 1997. GRG2 User’s Guide. Technical docu-ment, University of Texas at Austin, Austin, Texas (2 October 1997).

Litvak, M.L. and Darlow, B.L. 1995. Surface Network and Well Tubinghead Pressure Constraints in Compositional Simulation. Paper SPE 29125 presented at the SPE Reservoir Simulation Symposium, San Antonio, Texas, USA, 12–15 February. http://dx.doi.org/10.2118/29125-MS.

Rashid, K. 2010. Optimal Allocation Procedure for Gas-Lift Optimization. Ind. Eng. Chem. Res. 49 (5): 2286–2294. http://dx.doi.org/10.1021/ie900867r.


Rashid, K., Demirel, S., and Couet, B. 2011. Gas-Lift Optimization with Choke Control using a Mixed-Integer Nonlinear Formulation. Ind. Eng. Chem. Res. 50 (5): 2971–2980. http://dx.doi.org/10.1021/ie101205x.

Shiralkar, G.S. and Watts, J.W. 2005. An Efficient Formulation for Simultaneous Solution of the Surface Network Equations. Paper SPE 93073 presented at the SPE Reservoir Simulation Symposium, The Woodlands, Texas, USA, 31 January–2 Feburary. http://dx.doi.org/10.2118/93073-MS.

Stoisits, R.F., Crawford, K.D., MacAllister, D.J., McCormack, M.D., Lawal, A.S., and Ogbe, D.O. 1999. Production Optimization at the Kuparuk River Field Utilizing Neural Networks and Genetic Algo-rithms. Paper SPE 52177 presented at the SPE Mid-Continent Opera-tions Symposium, Oklahoma City, Oklahoma, USA, 28–31 March. http://dx.doi.org/10.2118/52177-MS.

Wang, P. and Litvak, M. 2008. Gas Lift Optimization for Long-Term Reservoir Simulations. SPE Res Eval & Eng 11 (1): 147–153. SPE-90506-PA. http://dx.doi.org/10.2118/90506-PA.

Watts, J.W., Fleming, G.C., and Lu, Q. 2009. Determination of Active Constraints in a Network. Paper SPE 118877 presented at the SPE Reservoir Simulation Symposium, The Woodlands, Texas, USA, 2–4 February. http://dx.doi.org/10.2118/118877-MS.

Qin Lu is a technical advisor for Landmark Software and Services at Halliburton where he works on the development of Nexus reservoir simulator. He holds a BS and a MS degrees, both in mechanical engineering, from the University of Science and Technology of China, respectively, and a PhD degree in petro-leum engineering from the University of Texas at Austin.

Graham Fleming is a chief technical advisor for Landmark Software and Services at Halliburton where he is the devel-opment lead for the Nexus reservoir simulator. He has a PhD degree from the California Institute of Technology and a BE degree from the University of Auckland.

SPE-140935-PA (Gas Lift Optimization Using Proxy Functions in Reservoir Simulation)

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Transcript of SPE-140935-PA (Gas Lift Optimization Using Proxy Functions in Reservoir Simulation)