Gas-Lift Optimization with Choke Control using a Mixed-Integer

ARTICLE pubs.acs.org/IECR

Gas-Lift Optimization with Choke Control using a Mixed-Integer Nonlinear Formulation Kashif Rashid,*,† S€uleyman Demirel,‡ and Benoît Cou€et† † ‡

Schlumberger-Doll Research, 1 Hampshire Street, Cambridge, Massachusetts 02139, United States Ross School of Business, University of Michigan, 701 Tappan Ave, Ann Arbor, Michigan 48109, United States ABSTRACT: This paper presents an extension to the iterative offline-online procedure proposed by Rashid4 to efficiently optimize gaslifted production networks. In particular, this work extends the original scheme to additionally include a single subsurface choke in each of the production wells. Hence, a revised offline formulation is presented that encompasses dual control in each well, comprising both liftgas injection and a choke setting. As the latter can be discrete (on/off), have fixed positions or be treated as continuous, a mixed-integer nonlinear program (MINLP) results, necessitating the use of a suitable MINLP solver. The open source Bonmin solver is utilized for this purpose. Moreover, as chokes enable better field pressure and constraint management, in addition to well activation and closure, a broader spectrum of production optimization scenarios can be tackled. Significantly, the paper demonstrates that not only does the offline-online methodology enable the application of a MINLP solver, but the results are obtained in an extremely efficient manner, as per the initial development. Results from a 26-well production network are presented, with and without the imposition of additional operating constraints.

’ INTRODUCTION An oilfield production network comprises a number of interconnected wells, branches, manifolds, separators, and storage facilities necessary to aid the transport of produced hydrocarbons from the source (the reservoir via the wells) to the sink (a downstream delivery point). More often than not, production engineers use computer models of the real network for production operation purposes.1 These models simulate multiphase flow behavior through the network and are used for investigation and prediction purposes, including but not limited to production monitoring, facility design and sizing, scenario analysis, multiphase flow assurance, and for field pressure management to ensure hydrocarbon flow to the delivery sink.2 From an operational point of view, the primary objective is often to maximize production (or profit) from the produced, and saleable, oil and gas components, while minimizing production costs and meeting all existing operating constraints. The constraints include, for example, limits on storage, fluid flow velocities through pipes to prevent erosion, temperature levels for hydrate formation prevention, and the availability of natural gas for artificial lift purposes. The latter is used to enhance production from a field by improving individual well productivity by injection of natural gas at high pressure directly into the wellbore containing fluid from the reservoir.3 The reduced density of the fluid column effectively lowers the flowing bottom-hole pressure, and the increased pressure differential induced across the sandface (the connection between the well and the reservoir) assists greater fluid production up to the surface (see Figure 1). However, too much gas injection will increase the frictional pressure drop and lower the possible production. Hence, each well has a desirable lift gas quantity, especially including the effects of interdependence because of common network connections.3,4 To establish these lift-gas injection rates, together with the settings of other components of r 2011 American Chemical Society

Figure 1. Well schematic.

concern in the network simulation, a nonlinear optimization problem must be solved.5,6 Furthermore, given that certain decision Received: June 1, 2010 Accepted: December 6, 2010 Revised: October 14, 2010 Published: January 27, 2011 2971

dx.doi.org/10.1021/ie101205x | Ind. Eng. Chem. Res. 2011, 50, 2971–2980

Industrial & Engineering Chemistry Research

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max s:t:

Fnw ¼ fonline ðX, Y; F Þ gj ðX, YÞ e 0

j ¼ ½1, :::, J1

hk ðX, YÞ ¼ 0

k ¼ ½1, :::, K1

X ∈R ,Y ∈N N1

M1

ð1Þ

where X is the vector of continuous decision variables (i.e., gaslift rates), Y is the vector of discrete decision variables (i.e., choke settings), and F describes the set of fixed model parameters (e.g., network topology, boundary conditions, flow correlations, fluid compositions, etc.). In contrast, offline refers to the solution of the decoupled problem based on the notion of well separability. While the details are presented in the next section, the offline

max s:t:

Fsep ¼ foffline ðX, Y; PÞ gj ðX, YÞ e 0 j ¼ ½1, :::, J2 hk ðX, YÞ ¼ 0 k ¼ ½1, :::, K2 X ∈ R N2 , Y ∈ NM2

ð2Þ

where, similarly, X and Y are the vectors of continuous and discrete decision variables, respectively. Note, however, that the number of variables and constraints defined (N2, M2, J2, K2) are not the same as in the online problem (N1, M1, J1, K1). The iterative offline-online procedure is then defined by the following steps. First, in a preprocessing step, production profiles are obtained for each well from the network model and validated for use.1,2 This step is necessary only once and the time cost is proportional to the number of wells in the model and, additionally, the range and number of pressure, gas-lift, and choke setting values selected in the sensitivity study. Hence, a judicious choice must be made to cover the anticipated operating range of interest with sufficient detail while minimizing the total expected preprocessing cost.4 Once completed, however, there can be a significant advantage in timecost reduction compared to other approaches if the optimization problem is rerun with a change in available lift gas, operating conditions or constraints, as the existing production profiles can be reused. Thus, the real cost is considered to be the number of simulation evaluations necessary to obtain a solution. Once the preprocessing step is completed, the vector of wellhead pressures (P) is initialized. This dictates the current operating conditions, that is, the production curve (PC) for each well. At subsequent iterations, the updated wellhead pressure estimate is used to set the current operating curve. If the desired wellhead pressure does not match the family of curves stored, it is generated by interpolation. When the operating curves have been set, the optimal decision variable set (X,Y) is obtained as the global solution to the offline problem (2) using the Bonmin MINLP solver (details of the offline formulation and solver are given later). Once the offline solution has been obtained, the optimal decision variable set (X,Y) is applied to the real network model (1). Although the change in control variables or the objective function could be compared for convergence purposes, either across iterations or between the offline and online solutions, the wellhead pressures (Pnew) are instead used. Notably, the assumption of well separability is based on the prevailing wellhead pressures, while in addition, the wellhead pressure solution obtained accounts for the effects of well interaction necessary for the offline solution to represent the real network problem. Further, as each offline problem is solved to global optimality and the online problem returns a specific wellhead pressure solution for a steadystate solution, the offline-online procedure will also converge (results from direct methods demonstrating this are given in Appendix A1). Hence, convergence of the offline-online procedure is assessed using the maximum absolute difference norm between the current and previous estimates of the wellhead pressure vector, given by: ε ¼ Pnew - P

)

’ PROBLEM DEFINITION In recent work, an iterative offline-online procedure was demonstrated specifically for the gas-lift optimization problem.4 This concerns the distribution of a fixed amount of lift-gas over a number of wells to maximize oil production at the delivery sink. While the production optimization problem presented in this paper has been broadened, the methodology is unchanged and is described here. In the iterative offline-online approach, online refers to a simulation run of the actual network model, given by the following:

problem can be defined as:

)

variables can be discrete (e.g., block valves with on/off states or fixed position chokes) a mixed-integer nonlinear program (MINLP) results.6-8 Direct optimization concerns the application of an appropriate solver directly on the network simulation model. However, given that the inherent behavior of the numerical multiphase flow simulator can be nonsmooth because of the choice of flow correlations used, noncontinuous because of flow regime changes, and nondifferentiable as a consequence, this approach can be limited.9 Moreover, a single simulation can also be computationally costly to evaluate. The use of numerical derivatives or derivativefree methods only tends to exacerbate matters.10-12 Thus, direct optimization, if at all tractable, is limited when dealing with largescale production networks, comprising hundreds of wells. Alternatively, a proxy optimization approach entails the use of a surrogate model in place of the actual network simulation model to ease the computational cost expected.13 Neural networks, radialbasis functions or other simplified models can be used for such purposes. More often than not, an adaptive procedure is adopted in which a coarse model is iteratively refined in pursuit of an optimal solution that bears close fidelity to the real network simulation problem.14 While such approaches can help alleviate the computational burden expected, they are however restricted to a few dozen decision variables for practical reasons, namely, the computational explosion associated with sampling high dimensional search space. Thus, these approaches can also be limited when dealing with largescale production networks. In the following section, a scheme based on the notion of well separability is presented.4,15 Each of the wells in the network model is assumed decoupled and a descriptive set of production profiles is obtained (for varying wellhead pressure) using sensitivity analysis on a well-by-well basis.1,2 This information is used to construct an “offline” representation of the real problem for optimization purposes.

¥

ð3Þ

If the convergence test is within a specified tolerance, typically 0.5 ^, Y ^, P ^ , F^) are returned. Otherwise, the psia, the optimal results (X 2972

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procedure repeats with the updated pressure vector (Pnew) used to set the current operating production curves. The methodology is summarized in algorithmic form below, while step 3 (the definition and solution of the offline problem) is elaborated in subsequent sections.

step 3 :

X0 , Y0 W foffline ðX, Y; Pk Þ

step 4 :

Pkþ1 W fonline ðX0 , Y0 ; FÞ

step 5 :

ε ¼ Pkþ1 - Pk

step 6 :

kþ1 If : ε > εtols ; P W P ; k W k þ 1; Goto step 3

)

k ¼ 1 Pk W Pini

)

step 1 : step 2 :

¥

k

^, Y ^, P ^ , F^; Stop Else : ε e εtols ; return X

ð4Þ

’ MATHEMATICAL FORMULATION (OFFLINE PROBLEM) While many elements in a network model could be used to manage the production at the delivery sink in the online problem, for convenience, a restricted set of variables is considered in the offline formulation. In particular, each well is assumed to have a single gas-lift injection valve used for lift-gas injection, and a single subsurface choke used for flow-rate control. The former is an artificial lift method designed to stimulate production from low producing wells, while the latter is used to choke back production to meet operating constraints downstream (see Figure 1). As will be shown, with these controls alone, a broad spectrum of production optimization problems can be managed. We assume that the gathering network has n wells and denote by I = {1, 2, ..., n} the set of wells. In addition, it is assumed that the chokes in the wells can be set to a number of positions, including fully open, fully closed or some intermediate position thereof. We denote by C the set of the positions a choke may take, in which the feasible choke positions are given by integers, that is, C = {0, 1, 2, ..., k}. Hence, there are k þ 1 positions, with 0 and k indicating the fully closed and open position, respectively. Moreover, without loss of generality, this definition can accommodate block valves (on/off with k = 1) and, with large k, a continuous position valve. In particular, we define yi,cp as a binary variable indicating whether the choke belonging to well i is set to position cp, where cp ∈ C . Notably, as a choke for a given well can be set to only one position, Pk cp=0yi,cp = 1. However, for convenience, we exclude the closed position (yi,0) leading to the following condition instead: P k cp=1yi,cp e 1. If the latter is 1, the choke is at a particular open position, and if the value is 0, the choke is closed and the well makes no contribution to the objective function. Hence, as the closed position can be conveniently omitted, we define C þ = {1, 2, ..., k} as the set of valid choke positions. If the total lift gas available for injection is given by C and xi is the lift gasPallocated to well i, then the following inequality is imposed: ni=1xi e C. In the offline formulation, we seek to create a representation of the real network model using a descriptive set of curve data. Previously, with gas-lift injection alone, gas-lift performance curves (GLPC) were established.4 These curves show the relationship of production with increasing lift-gas injection at varying wellhead pressure values. The GLPCs are established on a well-by-well basis using a multiphase flow simulator based on nodal analysis

sensitivity studies.1 However, as chokes are now also included in the model, the curves are additionally functions of the choke position. That is, there is a set of performance curves (PC) for varying wellhead pressure and choke position for each well. In practice, however, not all choke positions may be applicable for well stability purposes. The PC of a particular well can take several forms. On the basis of the well behavior observed we define the following four categories for modeling purposes (see Figure 2): SI = smooth-instantaneous SNI = smooth-noninstantaneous NSI = nonsmooth-instantaneous NSNI = nonsmooth-noninstantaneous A smooth-instantaneous (SI) profile is one comprising well production without any lift-gas assigned (naturally flowing), while a smooth-noninstantaneous (SNI) well requires some minimum level of lift-gas injection before it commences to produce. Nonsmooth-instantaneous (NSI) and nonsmoothnoninstantaneous (NSNI) wells are similarly described but with nonsmooth production profiles. These abrupt changes are mostly a result of flow regime changes along the well or a change in the flow correlation used to model the behavior of the threephase flow at given well inclinations. The nonsmooth and possibly noncontinuous profiles that result, while not strictly correct, must be managed nonetheless. Moreover, as the PC behavior will depend on the well-head pressure and choke setting, a general description of curve behavior is desirable. This is provided below. The most general curve type is NSNI (see Figure 3). We denote by qi,cp(xi) the production curve of well i as a function of the lift gas xi (at a fixed well-head pressure pi and choke position cp). We denote by ui,cp the upper bound on the lift gas that can be injected into well i. This is defined by the value of lift-gas at which the production is maximized and beyond which little or no gain results. In addition, we also identify the thresholds li,cp and mi,cp such that qi, cp(xi) = 0 for all xi < li,cp, and qi,cp(xi) is nonsmooth (and possibly discontinuous) at xi = mi,cp. For convenience, qi,cp(xi) is defined using two upper convex spline forms, g1i,cp(xi) and g2i,cp(xi), over the intervals [li,cp,mi,cp] and [mi,cp,ui,cp], respectively (see Figure 3). This ensures better modeling of the nonlinear physical system, in contrast to methods based on assumptions of linearity.6,16-18 Moreover, the spline-based approach reduces the number of binary variables necessary in the optimization problem and helps reduce the time required to obtain a solution. An algorithm is used to classify and fit performance curves by automatic detection of the thresholds li,cp, mi,cp, and ui,cp. While NSNI-wells have 0 < li,cp < mi,cp, other well types are special cases of the general curve type. For example, a SI-well has li,cp = mi,cp = 0, a SNI-well has li,cp = mi,cp > 0 and a NSI-well has li,cp = 0, mi,cp > 0. Notably, all wells are defined using the most general NSNI curve definition. We now address how these curves are defined and used in the offline formulation. While g1i,cp(xi) and g2i,cp(xi) are smooth and concave, the composite production function qi,cp(xi) is not. However, as the MINLP solver employed for the solution of the offline problem (Bonmin19) is able to return the globally optimal solution if the MINLP formulation admits a convex continuous relaxation (concave for maximization problems), it is desirable to have a formulation with this property. For this reason, we make the following definitions. 2973



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Figure 2. Performance curve types.

nonzero term. Lastly, we define l1i,cp = li,cp, l2i,cp = mi,cp, u1i,cp = mi,cp, u2i,cp = ui,cp. The introduction of the auxiliary variables can alternatively be interpreted as follows. We define well i as being composed of 2k subwells (indexed as {i, cp, t}), in which each has a smooth production curve defined by gti,cp(xti,cp). Moreover, only a given subwell is allowed to produce with the following bounds: lti,cp e xti,cp e uti,cp. Therefore, with n wells with k valid choke positions (∈C þ ), the model has 2nk continuous variables and 2nk binary variables at most. For example, with SI wells only, the model will have nk continuous variables and nk binary variables. With the foregoing definitions, the relationship between the production function (qi) and the decision variables is stated as follows: xi ¼

2 X k X t ¼ 1 cp ¼ 1

Figure 3. General performance curve model.

qi ¼

yti,cp,

2 X k X

xti, cp

"i ∈ I

ðgit, cp ðxti, cp Þ - git, cp ð0Þð1 - yti, cp ÞÞ

ð5Þ "i ∈ I ð6Þ

t ¼ 1 cp ¼ 1

Define a binary variable, admitting a value of 1 if well i is set to choke position cp and the production is given by the smooth curve gti,cp(xi), where t = 1,2. The continuous variable, xti,cp, is similarly defined. Specifically, these auxiliary variables are related to the variables Poriginal model P P by the following conditions: yi,cp = 2t=1yti,cp and xi = 2t=1 kcp=1xti,cp. Here, whenever a binary variable yti,cp equals zero, the corresponding continuous variable xti,cp will also be equal to zero, giving at most, one

lti, cp yti, cp e xti, cp e uti, cp yti, cp 2 X k X t ¼ 1 cp ¼ 1

2974

"i ∈ I, cp ∈ C þ , t ¼ 1, 2 ð7Þ

yti, cp e 1

"i ∈ I

ð8Þ



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In the above formulation, eq 5 relates the auxiliary variables to the original variable for lift gas injection. Equation 8 implies that the choke can be set to at most one nonzero position, and only one curve region can be used to evaluate production. Whenever yti,cp = 0 for some cp and t, then eq 7 implies that xti,cp = 0. If yti,cp = 0 for all i ∈ C þ , then xti,cp = 0 for all cp and t, and consequently xi = 0. If yti,cp = 1 for some cp and t, then xi = xti,cp in eq 5. Also, eqs 6 and 8 imply that qi = gti,cp(xti,cp). In addition, eq 7 implies lti,cp e xti,cp e uti,cp, whenever yti,cp = 1, as desired. Now, although qi provides a measure of the total oil production possible from a particular well, it does not account for the other components produced, namely, gas and water. If the provision is available, the produced gas is a saleable quantity that can be appropriately processed and sold. Alternately, it can be treated as a by-product that may be used for gas-lift injection, reservoir pressure support through injector wells or simply disposed of by flaring. Thus, the gas has a process cost and a possible sale value. The produced water is an undesirable waste product that has a processing and disposal cost. A monetary value can therefore be assigned to each barrel of produced fluid, given by the following: vi ¼ po ð1 - ωi Þ - cw ωi þ pg γi ð1 - ωi Þ

constraints are defined using matrices U [nc, n], V [nc, n], and W [nc, 1], such that Uq þ Vx e W, where nc is the number of constraints applied, n is the number of wells, W is the right-hand side vector, and both U and V contain information about the subgroupings of concern. Hence, these constraint matrices contain the information necessary to specify the operating constraints imposed. Note, however, that only those constraints that can be represented offline can be treated, that is, simulation-dependent constraints, such as wellhead temperature or erosional velocity cannot be readily accommodated. The foregoing leads to the full MINLP formulation for the offline problem given by the following: n n X X vi q i - c g xi max ð15Þ i¼1

s:t:

vi q i - c g

i¼1

n X

xi ¼

ð9Þ

xi

ð12Þ

i∈M

ωi qi e qmax w

ð13Þ

i∈M

i∈M

qi ¼

X

xti, cp

2 X k X t ¼ 1 cp ¼ 1

xi e qmax g

ð14Þ

i∈M

where M denotes a subset of wells (ranging from 1 to n) connected to a particular node of interest. Thus, if q is defined as the vector of the liquid rates and x the vector of lift rates, the operating

"i ∈ I

ðgit, cp ðxti, cp Þ - git, cp ð0Þð1 - yti, cp ÞÞ

lti, cp yti, cp e xti, cp e uti, cp yti, cp

ð10Þ

i∈M

γi ð1 - ωi Þqi þ

2 X k X t ¼ 1 cp ¼ 1

t ¼ 1 cp ¼ 1

i¼1

X ð1 - ωi Þqi e qmax o

X

xi e C

2 X k X

where vi is the unit value of liquid flowing through well i, defined by eq 9. The second term in eq 10 accounts for the cost of lift gas injection in order to prevent excessive lift-gas use. Thus, a production engineer is able to optimize a particular financial objective, by suitably varying the revenue and cost factors, or simply chooose to maximize oil production with vi = 1 - ωi and cg = 0. The formulation cannot be considered complete without consideration of operating constraints. These include limits imposed at well, manifold (an internal node connecting a number of wells) or at sink (the terminating node comprising production from all wells) level, for capacity and storage limitations on the produced hydrocarbons. For example, maximum liquid, oil, water, and gas production constraints can be defined by the following: X qi e qmax ð11Þ l

X

i¼1

i¼1

where po is the profit per unit barrel of oil produced, pg is the profit per unit gas produced (MMscfd), cw is the water processing cost per unit barrel, γ is the gas-to-oil ratio (GOR), and ω is the watercut indicating the fraction of water present in the produced fluids. In addition, if cg is the cost per unit gas injection, we can state the composite objective function by the following: n X

n X

yti, cp e 1

"i ∈ I

"i ∈ I, cp ∈ C þ , t ¼ 1, 2

"i ∈ I

Uq þ Vx e W yti, cp ∈f0, 1g

"i∈I, cp ∈ C þ , t ¼ 1, 2

Note that in the above, variables xi and qi are shown for clarity, but can be replaced by substitution.

’ SOLUTION PROCEDURE The offline MINLP formulation, presented in the last section, is solved using the open source Bonmin (basic open-source nonlinear mixed-integer programming) solver to global optimality as the continuous relaxation of the problem is convex.19,20 Note, however, that the offline solution will differ slightly from the online solution at convergence due to the offline curve modeling process. This is due to the quality of the performance curves established (i.e., the number of samples and the quantity of curves with varying wellhead pressure provided for use) and to a lesser extent, the quality of the spline models developed. In addition, network effects imposed by interconnected wells may not be readily captured by the single well analysis used to elicit the performance curves. As a consequence of the aforementioned issues, and particularly when operating constraints are present, a solution that is feasible offline may actually be infeasible online. To counter this process, a bootstrapping procedure has been included that updates the properties of the constraint matrix U by a ratio of the mismatch observed between the offline and online (constrained) solution from the last iteration for each well. This helps ensure that the online solution matches as closely as possible the solution obtained offline and is also feasible. 2975



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Figure 4. 26-Well gathering network schematic. Key: O, wells; b, manifold nodes; |, branches; v, the transport line to the delivery sink O. The boundary conditions are all pressure specified.

Ideally, wells connected to the same manifold should have the same well-head pressure. However, the network simulator tolerates small differences due to the convergence conditions imposed, leading to slightly different values. As this variability can cause the offline-online procedure to take longer to converge, we smooth the wellhead pressures of wells connected to common manifolds. Thus, if M is the set of wells connected to a particular manifold (indexed 1, 2, ..., M) with wellhead pressures (P1, P2, ..., PM), we instead impose (P, P, ..., P), where P is the average well~ head pressure across the manifold. Furthermore, we denote by P the modified pressure profile (of length n) of all wells obtained by averaging wellhead pressures over common manifolds where ~ in the offline problem enhances the stabilnecessary. The use of P ity and convergence rate of the online-offline procedure (e.g., see results in Table 4 in Appendix A1). Inclusion of chokes in the model allows the offline procedure to shut down wells to meet operating constraints. However, the convergence of the offline-online procedure can be adversely affected if many wells are shut down or reactivated at each iteration leading to significant well-head pressure fluctuations in the resulting online case. To overcome this limitation, we deactivate wells one at a time based on the value metric vi (eq 9) until no further improvement is made to the objective value. More importantly, the lowest ranking well is permanently deactivated, imparting greater stability to the process. The revised offline-online procedure with a MINLP formulation is presented in Appendix A2. Results from some test cases are presented in the following section, while other comparative results are given in Appendix A1.

’ TEST RESULTS A 56-well curve-based test case was presented in Buitrago et al., comprising 10 noninstantaneous flow wells with 22,500 Mscf/d of lift gas available for allocation.21 This example is strictly used to evaluate the offline solution procedure.

The Buitrago model formulation involves 56 continuous variables for lift gas injection and 56 binary variables for well activation state. However, as all instantaneously flowing wells are assumed on, only 10 binary variables are required with the remaining fixed at 1. First, the global optimum is identified by enumerating all possible combinations of activation status (on/off) of the 10 noninstantaneous flow wells (210 = 1024 cases) and solving the corresponding nonlinear program for each case. The best objective value of 23,381 stb/d was identified after 527 s. Notably, the same result is obtained in 5 s using Bonmin and the proposed formulation. Hence, this validates the offline formulation and the efficacy of the solver to establish the global solution in the offline problem. It is worth noting that the same solution is obtained in 17 s when all 56 binary variables are included. Problem specific details, including CPU information, can be found in Appendix A3. Next, the iterative offline-online procedure is demonstrated on a gathering network of 26-wells, each comprising one subsurface choke, with 45 MMscf/d of lift gas available for allocation (see Figure 4). The wells use a black oil definition with varying water-cut (ω) and gas-to-oil ratio (γ) values embodied by the performance curves extracted at the preprocessing stage. As the focus is on the efficient evaluation of a number of production optimization scenarios using the formulation developed, for convenience, particular details of the network simulation are omitted. However, comparative results with other approaches are provided in Appendix A1. For demonstrative purposes, the 26-well model is optimized for oil production and alternately for profit maximization, with only a total lift-gas constraint applied. Each choke is set to the fully open position (2-in. diameter) and assumed invariant in the first instance. The results are given in Table 1. The offline-online procedure yields a solution with only 6 evaluations of the actual network simulator (the real cost involved) in both cases. For oil maximization (solution O), 38,784 STB are estimated with an equivalent fiscal value of $2,580,825. In contrast, for a profit objective (solution 2976



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Table 3. 26-Well Solution Choke Positionsa

Table 1. 26-Well Case with Lift Gas Constraint Only objective

max oil (solution O)

max profit (solution A)

oil (STB)

38,784

38,454

profit ($)

2,580,825

2,583,717

inactive wells

w3

w3, w22

simulator calls

6

6

solution A

B

C

well

lift gas constraint only

constrained with fixed open chokes

constrained with variable discrete chokes

w1

2

2

2

w2

2

0

0

w3

0

0

0

w4

2

2

2

solution

w5

2

2

2

A

B

C

w6 w7

2 2

0 2

0 2

Table 2. 26-Well Network Model Profit Maximization with Constraints a

constraint entity

type

value

mA mD

gas (MMscf) gas (MMscf)

20 12

25.63 16.49

19.99 12.01

19.99 12.01

w8

2

2

2

w9

2

2

2

mB

gas (MMscf)

18

22.09

17.99

17.99

w10

2

0

0

mC

gas (MMscf)

15

16.19

15.00

15.00

w11

2

2

1.75

mA

liquid (STB)

14,000

14,714

11,959

11,976

w12

2

2

2

mD

liquid (STB)

12,000

12,820

11,028

11,029

w13

2

2

1.75

mB

liquid (STB)

12,000

12,321

11,801

11,819

mC

liquid (STB)

15,000

14,583

13,800

13,800

w14 w15

2 2

2 2

2 2

sink sink

liquid (STB) oil (STB)

41,000 36,000

42,117 38,454

36,778 36,044

36,806 36,060

w16

2

2

2

w17

2

2

0

sink

water (STB)

8000

3663

743

746

w18

2

2

2

sink

gas (MMscf)

48

58

47

47

w19

2

0

0

network

lift gas (MMscf)

45

45

34.83

35

w20

2

2

2

profit ($)

2,583,717

2,444,715

2,445,716

w21

2

2

2

simulation evaluations

6

11

9

w22 w23

0 2

0 2

0 2

w24

2

0

0

w25

2

2

1.75

w26

2

2

2

a

Solution A = Lift gas constraint only with fixed open chokes. Solution B = Constrained case with fixed open chokes. Solution C = Constrained case with variable discrete chokes.

A), 38,454 STB are obtained with a value of $2,583,717. In the latter, the profit value is greater even though less oil is produced as the costs associated with water production and gas usage are minimized. However, these costs are ignored in the oil maximization problem therefore yielding a lower profit value. Note, that well-3 (shown in Figure 4) was shut down for the oil maximation case, while well-22 was additionally shut down to maximize the profit objective based on the following factors: po = $68/STB, cw = $8.5/ STB, pg = $1/MMscfd, and cg = $2.5 MMscfd. In Table 2, 12 additional operating constraints are introduced. These are gas, liquid, oil, and water constraints at manifold and sink level. Column 1 indicates the entity, column 2 states the constraint type, and column 3 the desired right-hand-side constraint value. Results for the profit maximization problem are shown in the last three columns. The unconstrained case (as given previously in Table 1) is repeated for convenience in column 4 (solution A). The constrained solution is given in column 5 (solution B) assuming the chokes are invariant, and, in the last column, the solution (C) when the chokes are permitted to take 6 possible positions (corresponding to 0, 1.0, 1.25, 1.5, 1.75, or 2.0 in. diameter opening). The highest objective function is achieved by the unconstrained case (solution A), where the desired network capacity constraints have been knowingly ignored and are hence in violation (for example, the produced gas flowing through manifold A is 25.63 MMscf, but the desired constraint limit is 20.0 MMscf). For the constrained case with fixed chokes (solution B), a lower solution is obtained but the constraints are now satisfied (e.g., the produced

a

Choke diameters given in inches.

gas in manifold A is 19.99 MMscf). Lastly, with the inclusion of choke control, a better result is returned that is still feasible with respect to the applied constraints. For solution A, well-3 and well-22 were shut down. For solution B, wells 2, 3, 6, 10, 19, 22 and 24 were shut down in order to meet the constraints as the choke position is otherwise unchangeable. For solution C, where the chokes can be varied, well 17 is additionally shut-down and the chokes in wells 11, 13, and 25 are choked back to ensure the constraints are managed more effectively. The respective choke sizes for the three solutions are given in Table 3. Hence, this shows the value of flow rate control with chokes to ensure constraint compliance, while still maximizing the possible profit. Note that similar results can be established for produced oil or liquid maximization objectives. Lastly, the cases presented required a remarkably low number of function evaluations (less than 11) of the actual network simulator as per the original development of the offline-online methodology.4 For the benefit of the reader, comparative results from the revised and original schemes are presented in Appendix A1 together with those from other optimization approaches for network problems of varying size. These confirm that comparable solutions are obtained with a minimal number of simulator calls compared to conventional approaches. Additional problem specific data is provided in Appendix A3. 2977



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Table 4. Network Model Test Resultsa method

polytope

proxy-polytope

GLO

GLO*

2-well Case Fopt (STB)

2837.2

2836.2

2837.2

2836.0

simulator calls

20

14

3

2

Fopt (STB)

5764.2

5750.0

5762.5

5759.0

simulator calls

59

18

4

2

Fopt (STB)

45,913

45,922

45,905

45,838

simulator calls

179

88

4

3

Fopt (STB)

27,438

27,365

27,336

simulator calls

369

8

3

4-well Case

26-well Case

100-well Case >101

a

GLO is the original method.4 GLO* is the revised offline-online procedure. Polytope is the downhill simplex method.22 Proxy-polytope uses a neural-network proxy model with the polytope solver.14 The available lift gas is 2, 4, 45, and 40 MMscf/d for the 2, 4, 26 and 100-well cases, respectively.

Table 5. Test Case Specific Informationa Buitrago21

network

case B1 case B2 case O

case A

case B

case C

actual problem objective

oil

oil

oil

profit

profit

profit

unit variables

STB 56

STB 56

STB 26

$ 26

$ 26

$ 26

linear consts

1

1

1

1

1

1

nonlinear consts 0

0

0

0

12

12

chokes

none

fixed

fixed

fixed

variable

none

particular choke setting. The latter includes the possibility of discrete or continuous position chokes, and the consideration of well activation state. The representative offline problem is solved to global optimality using the open source MINLP solver Bonmin. Successful test results were shown for a 26-well network model with and without additional operating constraints imposed. While the revised method no longer requires the explicit assumption of convexity of the well production curves and can handle operating constraints more readily than the original approach,4 it is still somewhat limited with respect to constraints that cannot easily be represented in the offline problem (e.g., mid-network erosional velocity constraints). Equally, due to its highly tailored nature, the method is not applicable to multiwell, multichoke or multicomponent problems. Nonetheless, the proposed method is significantly more efficient compared to standard optimization approaches (for the posed scenarios), achieving comparable results in only a fraction of the number of simulator evaluations. Moreover, the approach is suitable for solving large-scale production optimization scenarios efficiently, and can aid the development of real-time integrated asset solutions due to the speed of solution.

’ APPENDIX A1: NETWORK TEST RESULTS Comparative results for liquid rate optimization for network models of varying dimensionality are shown in Figure 4 based on several different approaches, including the revised offline-online (GLO*) procedure presented in this paper. ’ APPENDIX A2: REVISED OFFLINE-ONLINE PROCEDURE The revised offline-online procedure (GLO*) is given by the following algorithm: Revised Online-Offline Procedure

O Plug x = x0, a vector of initial lift gas allocation to wells in the network simulator and read P=P0. O Let P~ be the modified pressure profile based on averaging across all manifolds. O Let U = U. Offline constraints are initially the same as the online constraint targets. O Select εtols. O Let z = 0 (the current online objective value). O Continue = 1. while Continue=1 do

offline formulation continuous vars 56

56

52

52

52

260

binary vars

10

56

52

52

52

260

ave time (s)

5

17

1

1

1

510

objective value

23,381 23,381 38,784 2,583,717 2,444,715 2,445,716

time (s)

5

17

119

121

245

5832

simulations

0

0

6

6

11

9

solution

O Let zold = z. O Let ε = εtolsþ1. while ε > εtols do

a

Case B1 = original curve model with only 10 wells considered for activation/deactivation. Case B2 = Curve model with all 56 wells considered for activation/decactivation. Case O = Network model; lift gas constraint only with fixed open chokes. Case A = Network model; lift gas constraint only with fixed open chokes. Case B = Network model; nonlinear constrained problem with fixed open chokes. Case C = Network model; nonlinear constrained problem with variable discrete chokes, each with 5 positions given by the set {1.0, 1.25, 1.5, 1.75, 2.0 } in inches.

Let Pold = P~. ~. Fit curves to pregenerated lift data for P Solve the offline problem with constraint matrix U. Let qi be the liquid rate for well i returned by the offline procedure. O Plug the offline solution in the network simulator. O Let P be the pressure profile returned by the network simulator. O Let Qi be the liquid rate for well i returned by the online procedure. O Let P~ be the modified pressure profile based on averaging across all manifolds. if P~(i) = 0 for well i then

O O O O

’ CONCLUSION An iterative offline-online procedure for production optimization problems has been presented in which the actual online network model is replaced by an offline curve-based approximation under the assumption of well separability. Specifically, a convex continuous MINLP formulation is developed that encompasses dual control in each well, comprising gas-lift injection and a 2978



)

)

if Well i is permanently deactivated then O Set the production curve to zero. else O P~(i) = Manifold Pressure end if end if O Let ~uij = uijQi/qi for all i and j. If qi = 0, let ~uij = uij. Constraint bootstrapping procedure. O Let ε = P~-Pold . end while O Let z be the current online objective value. if z > zold then O Permanently deactivate the well with the lowest rank else O Continue = 0 end if end while

’ APPENDIX A3: TEST CASE SPECIFIC INFORMATION Problem specific data concerning the test cases explored in this paper are presented in Table 5. Note that all results were obtained on a PC (Intel(R) Core2 Duo CPU, T9500, 2.60 GHz, 3GB) running Windows XP.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected].

’ ACKNOWLEDGMENT The authors are grateful to the anonymous reviewers for their comments and suggestions on this manuscript. ’ NOMENCLATURE Acronyms

GL = gas-lift GLR = gas-liquid ratio GLV = gas-lift valve GLIR = gas-lift injection rate GLPC = gas-lift performance curve GLO = the original offline-online methodology for gas-lift optimization4 GLO* = the revised GLO methodology presented in this paper GOR = gas-to-oil ratio MINLP = mixed integer nonlinear programming MILP = mixed integer linear programming NLP = nonlinear programming NSI = nonsmooth-instantaneous flowing well NSNI = nonsmooth-noninstantaneous flowing well NN = neural network PC = production curve SI = smooth-instantaneous flowing well SNI = smooth-noninstantaneous flowing well Symbols

cp = choke position cg = cost per unit (MMscfd) gas used cw = cost per unit (STB) water processed

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C = available lift-gas constraint foffline = function representing offline model fonline = function representing network model Fnw = objective function value from the network model Fsep = objective function value from the offline model Fopt = optimal objective function value gj(X,Y) = j-th vector of inequality constraints t = ith well with choke position cp-production curve t gi,cp hk(X,Y) = kth vector of equality constraints i = counter j = counter J = number of inequality constraints k = counter or number of choke positions K = number of equality constraints li,cp = ith well with choke position cp-gas-lift lower bound t = ith well with choke position cp on region t-gas-lift lower li,cp bound mi,cp = i-th well with choke position cp-gas-lift intermediate bound M = number of integer control variables n = number of wells nc = number of operating constraints applied N = number of continuous control variables pg = profit per unit (MMscfd) gas sold po = profit per unit (STB) oil sold P = vector of wellhead pressures q = vector of liquid flow rates [n,1] qi,cp = ith well with choke position cp-flow rate qgmax = maximum produced gas constraint bound qlmax = maximum produced liquid constraint bound qomax = maximum produced oil constraint bound qwmax = maximum produced water constraint bound ui,cp = ith well with choke position cp-gas-lift upper bound t = ith well with choke position cp on region t-gas-lift upper ui,cp bound U = constraint definition matrix [nc,n] vi = ith well value metric V = constraint definition matrix [nc,n] W = constraint definition matrix [nc,1] xi = ith well continuous variable gas-lift rate t = auxiliary continuous variable of ith well with choke position xi,cp cp on region t x = vector of gas-lift rates [n,1] X = vector of continuous control variables Y = vector of integer control variables yi,cp = binary variable of ith well indicating choke position cp t = auxiliary binary variable of ith well with choke position cp yi,cp on region t C = set of choke positions {0, 1, ..., k} C þ = set of choke positions {1, ..., k} ε = error norm on wellhead pressure vectors εtols = error norm tolerance value γi = ith well gas-to-oil ratio ωi = ith well water-cut F = network model specific parameters

’ REFERENCES (1) PipeSim Network Simulator; Schlumberger SIS: Abingdon, Oxon, U. K., 2007. (2) Beggs, H. D. Production Optimization using Nodal Analysis, 2nd ed.; OGCI Inc., PetroSkills, LLC: Tulsa, OK, 2006. 2979



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(3) Brown, K. E. Overview of Artificial Lift Systems. JPT, J. Pet. Technol. 1982, 34 (10), 2384–23969979A. (4) Rashid, K. Optimal Allocation Procedure for Gas-Lift Optimization. Ind. Eng. Chem. Res. 2010, 49, 2286–2294. (5) Dutta-Roy, K.; Kattapuram, J. A New Approach to Gas-Lift Allocation Optimization. Presented at the SPE Western Regional Meeting, California, June 1997, 38333-MS. (6) Barragan-Hernandez, V.; Vazquez-Roman, R.; Rosales-Marines, L.; Garcia-Sanchez, F. A Strategy for Simulation and Optimization of Gas and Oil Production. Comput. Chem. Eng. 2005, 30, 215–227. (7) Kosmidis, V. D.; Perkins, J. D.; Pistikopoulos, E. N. Optimization of Well Oil Rate Allocations in Petroleum Fields. Ind. Eng. Chem. Res. 2004, 43, 3513–3527. (8) Kosmidis, V. D.; Perkins, J. D.; Pistikopoulos, E. N. A Mixed Integer Optimization formulation for well scheduling problem on Petroluem Fields. Comput. Chem. Eng. 2005, 29, 1523–1541. (9) Fujii, H.; Horne, R. N. Multivariate Optimization of Production Systems.SPE Production & Facilities, Aug. 1995, 165-171, 27617-PA. (10) Martinez, E. R.; Moreno, W. J.; Moreno, J. A. Application of Genetic Algorithm on the Distribution of Gas Lift Injection. Presented at the SPE Latin American Petroleum Eng. Conference, April 1994, Argentina, 26993-MS. (11) Ray, T.; Sarker, R. Multi-objective Evolutionary Approach to the Solution of Gas Lift Optimization Problems. Presented at the IEEE Congress on Evolutionary Computation, July 2006, British Columia, Canada, 3182-3188. (12) Vazquez, M.; Suarez, A.; Aponte, H.; Ocanto, L.; Fernandez, J. Global Optimization of Oil Production Systems: A Unified Operational View. Presented at SPE ATCE, Oct. 2001, Louisana, 71561-MS. (13) Stoisits, R. F.; MacAllister, D. J.; McCormack, M. D.; Lawal, A. S.; Ogbe, D. O. Production Optimization at the Kuparak River Field Utilizing Neural Networks and Genetic Algorithms. Presented at SPE Mid-Continent Operations Symposium, March 1999, OK, 52177-MS. (14) Cou€et, B.; Djikpesse, H.; Tonkin, T.; Wilkinson, D. Production Enhancement through Integrated Asset Model Optimization. Presented at the SPE Production and Operations Conference, June 2010, Tunisia, 135901-MS. (15) Rashid, K. A Method for Optimal Lift-Gas Allocation and other Production Scenarios. Research Note; Schlumberger AbTC: Abindgon, U.K., 2006. (16) Fang, W. Y.; Lo, K. K. A Generalized Well-Management Scheme for Reservoir Simulation. SPE Reservoir Eng. 1996, 11 (2), 116–120 29124PA. (17) Nadar, M. S.; Schneider, T. S.; Jackson, K. L.; McKie, C. J. N.; Hamid, J. Implementation of a Total System Production Optimization Model in a Complex Gas Lifted Offshore Operation. SPE Prod. Oper. 2008, 5-13, 103670PA. (18) Misener, R.; Gounaris, C. E.; Floudas, C. A. Global Optimization of Gas Lifting Operations: A Comparative Study of Piecewise Linear Formulations. Ind. Eng. Chem. Res. 2009, 48, 6098–6104. (19) Bonami, P.; Biegler, L. T.; Conn, A. R.; Cornuejols, G.; Grossmann, I. E.; Laird, C. D.; Lee, J.; Lodi, A.; Margot, F.; Sawaya, N.; et al. An Algorithmic Framework for Convex Mixed Integer Nonlinear Programs. Discrete Optim. 2008, 5 (2), 186–204. (20) Fletcher, R. Practical Methods of Optimization; John Wiley and Sons: New York, 2000. (21) Buitrago, S.; Rodriguez, E.; Espin, D. Global Optimization Techniques in Gas Allocation for Continuous Flow Gas Lift Systems. Presented at the SPE Gas Technology Symposium, April 1996, Canada, 35616-MS. (22) Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. Numerical Recipes in Cþþ, 3rd ed.; Cambridge University Press: Cambridge, U.K., 2002.

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Gas-Lift Optimization with Choke Control using a Mixed-Integer

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