A structured approach to optimizing offshore oil and gas production with uncertain models

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Computers and Chemical Engineering 34 (2010) 163–176

Contents lists available at ScienceDirect

Computers and Chemical Engineering

journa l homepage: www.e lsev ier .com/ locate /compchemeng

structured approach to optimizing offshore oil and gas production withncertain models

teinar M. Elgsæter a,∗, Olav Slupphaug b, Tor Arne Johansen a

Norwegian University of Science and Technology, O.S Bragstads plass 2D, N-7491 Trondheim, NorwayABB, Ole Deviks vei 10, N-0666 Oslo, Norway

r t i c l e i n f o

rticle history:eceived 2 October 2008eceived in revised form 18 April 2009ccepted 6 July 2009vailable online 29 July 2009

a b s t r a c t

Optimizing offshore production of oil and gas has received comparatively little attention despite the largescale of revenues involved. The complexity of multiphase flow means that any model for use in productionoptimization must be fitted to production data for accuracy, but the low information content of productiondata means that the uncertainty in the fitted parameters of any such model will be significant. Due tocosts and risk the information content in production data cannot be increased through excitation unless

eywords:ncertaintyroduction optimizationarameter estimationil and gas productionxcitation planning

the benefits are documented.A structured approach is suggested which iteratively updates setpoints while documenting the benefits

of each proposed setpoint change through excitation planning and result analysis. In simulations on ananalog which mimics a real-world oil field and its typical low information content data the approach isable to realize a significant portion of the available profit potential while ensuring feasibility despite largeinitial model uncertainty.

esult analysis

. Introduction

The potential for optimizing offshore oil and gas productionay be significant as small increases in profits in relative termsay translate into large gains due to scales of revenues involved

Elgsaeter, Slupphaug, & Johansen, 2008a), and as this topic haseceived little attention compared to optimization and controlf downstream processing facilities. Both modeling and measur-

ng the offshore production of oil and gas present challenges forptimization. Exploiting the information in what measurementsre available to fit production models is complicated by the lownformation content of production data (Elgsaeter, Slupphaug, &ohansen, 2007). Practitioners in the oil and gas industry are riskverse as the scale of revenues mean cost and risks of implementinghanges in setpoints can be significant. An approach to optimiza-ion which can quantify these expected benefits may have increasedikelihood of industry acceptance. This paper proposes a struc-ured approach to optimizing production which takes the nature of

odels, measurements and data into account, while the expected
onetary benefits of each setpoint change are quantified.
Production in the context of offshore oil and gas fields, cane considered the total output of production wells, a mass flowith components including hydrocarbons, in addition to water,

∗ Corresponding author. Tel.: +47 918 06886.E-mail address: [email protected] (S.M. Elgsæter).

098-1354/$ – see front matter © 2009 Elsevier Ltd. All rights reserved.oi:10.1016/j.compchemeng.2009.07.011

© 2009 Elsevier Ltd. All rights reserved.

CO2, H2S, sand and possibly other components. Hydrocarbon pro-duction is for simplicity often lumped into oil and gas. Productiontravels as multiphase flow from wells through flow lines to a pro-cessing facility for separation. Water and gas injection is used foroptimizing hydrocarbon recovery of reservoirs. Gas-lift can increaseproduction to a certain extent by increasing the pressure differencebetween reservoir and well inlet.

Production is constrained by several factors, including: on thefield level, the capacity of the facilities to separate components ofproduction and the capacity of facilities to compress gas. The pro-duction of groups of wells may travel through shared flow linesor inlet separators which have a limited liquid handling capacity.The production of individual wells may be constrained due to slug-ging, other flow assurance issues or due to reservoir managementconstraints.

Multiphase flows are hard to measure and are usually not avail-able for individual flow lines in real-time, however measurementsof total single-phase produced oil and gas rates are usually available,and estimates of total water rates can often be found by adding dif-ferent measured water rates after separation. To determine the ratesof oil, gas and water produced from individual wells, the produc-tion of a single well is usually routed to a dedicated test separator
at intervals where the rate of each separated single-phase compo-nent is measured, a well test. Well tests allow biases in models ofindividual wells to be updated, and well tests which measure ratesfor different settings, multi-rate well tests, also allow responsespredicted by models to be validated.
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164 S.M. Elgsæter et al. / Computers and Chemical Engineering 34 (2010) 163–176

Nomenclature 1

u decision variablesx internal variablesd modeled and measured disturbances independent

of u� parameters to be determined through parameter

estimationM profit measurec production constraintsy measured variables exploited in parameter estima-

tiont timeN the number of time steps in the tuning data setZN a tuning data set spanning N time steps� residuals in parameter estimationDy normalization matrix for yw weighting of residuals in parameter estimationVs soft constraints on parameters �c� hard constraints on parameters �umin the lower end of the range of possible values for uumax the higher end of the range of possible values for uUprc a limit on the size of change in decision variablesU scaling of ux0, u0, d0 x, u, d at the time of optimizationf� probability function for parameters �Po the potential for production optimizationPo,r realizable potential for production optimizationLu lost potential of production optimizationuT a target value for an operational strategyU− the set of indices of u which are to be decreased as

an operational strategy approaches uT from u0U+ the set of indices of u which are to be increased as

an operational strategy approaches uT from u0JM a cost function against which measured profit

change during implementation of an operationalstrategy is to be compared to determine whether toterminate the operational strategy before the targetis reached

JTM a threshold value for JM

tos the time at which the operational strategy is initi-

atedtoe the time at which the operational strategy is termi-

natedf�Pr probability function for �Po,r

S the user-specified size of setpoint change at eachiteration of an operational strategy

fu probability function for uJr a cost function for the expected profit attainable

from moving setpoints toward a new targetJTr a threshold value for JM

BE the benefit of excitationCE the cost of excitation�n the nominal parameter estimate, in the context of

benefit of excitationE

s(

qgl gas-lift ratez relative production choke openingIa the set of indices of all wells considered in optimiza-

tionb bias calculated over interval [N − L, N]qtot,c a capacity for total produced ratesIMGS indices of wells with lower constraints on gas-lift

rate

J the cost–benefit tradeoff of an excitationJE,T a threshold value for JE

1 Subscript o, g and w indicate oil, gas and water, respectively. Numbered super-cripts indicate well indices, the superscript l indicates a local operating point. Bars.) indicate measured variables, while hats (.) indicate estimated variables.

ˇy bias estimated over interval [1, N]˛, � fitted parameters in gas-lift models

In the production of oil and gas it is common to divide the taskof optimization into subproblems on different time scales to limitcomplexity, and to consider separately reservoir management, theoptimization of reservoir injection and drainage on the time scalesof months and years, and production optimization, the maximizationof profit from the daily production of reservoir fluids (Saputelli etal., 2003). Reservoir management typically specifies constraints onproduction optimization to link these problems.

Production optimization requires production models, equationswhich express the relationship between change in decision vari-ables and resulting change in production. Production models areusually nonlinear to describe significant nonlinear phenomena inproduction. Parameter estimation is adjustment of fitted param-eters so that predictions of the production model match a set ofrecent historical production data as closely as possible. Parametersof production models should be fitted to production data throughparameter estimation to compensate for un-modeled effects or dis-turbances and to set reasonable values for physical parameterswhich cannot be measured directly or determined in the labo-ratory. Erosion of production chokes is an example of a typicalsuch un-modeled disturbance. The term information content in thispaper refers the amount in variation in decision- and disturbancevariables observed in production data. Fitted parameters may beuncertain if fitted to data with low information content, as manyparameter estimates may fit data equally well for a given model.If there is sufficient information content in data a unique parame-ter estimate can be found robustly which matches data better thanall other estimates. A previous study concluded that uncertaintyin fitted parameters is likely to result when production models arefitted to production data describing normal operations, due to lowinformation content (Elgsaeter et al., 2007). Planned excitation isvariation in decision or disturbance variables introduced deliber-ately for the purpose of exposing some aspect of production andreducing parameter uncertainty. The concepts of information con-tent and excitation used in this paper are motivated by the fieldof system identification (Ljung, 1999). As planned excitation in theoffshore production of oil and gas requires a temporary reductionin production, planned excitation has a significant associated costand completely eliminating parameter uncertainty may thereforenot be cost-effective.

As a consequence of the complexity of the process considered,of measurement difficulties, and of the low information contentin data, production models may be subject to significant uncer-tainty that can be reduced against a cost or not at all. As a resultmore than one setpoint can be the plausible optimum, which raisesseveral practical challenges. One challenge is that the result of asetpoint change is uncertain, it may even be negative. Another chal-
lenge is that ensuring that an implemented setpoint is feasible, i.e.that it obeys production constraints, is non-trivial in the presenceof uncertainty. A third challenge is that when uncertainty can bereduced at a cost, some method of determining how and whenuncertainty reduction is cost-effective may be required.

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The approach to optimization with uncertain models is moti-ated by current practice in the optimization of offshore oil and gasroduction. In the current practice, production optimization beginsy implementing well tests, a form of planned excitation. Whichells and when to test are chosen based on the subjective insight

f practitioners. Production models are updated by fitting againsthe most recent well test. An target setpoint is then calculated by

athematical programming. If the target setpoint is expected toncrease profits significantly, a setpoint change is implemented. Thearget setpoint is not implemented instantaneously, instead pro-uction is moved toward the target setpoint by an operator in aeries of smaller steps to limit transients and to ensure feasibility.his type of gradual setpoint change to ensure feasibility post-ptimization will be referred to as an operational strategy in thisaper. Production optimization therefore involves excitation plan-ing, model updating, risk analysis and operational strategy, andurrent industry practice does not explicitly consider uncertainty inny of these steps. This is paradoxical as the state of production datand complexity of multiphase flow dictate that uncertainty will beignificant in each of these steps. The contribution of this paper iso suggest how uncertainty can be accounted for in a structured

anner in each of these steps.

.1. Prior work

.1.1. Modeling production and quantifying model uncertaintyQuantifying uncertainty in reservoir models has attracted much

nterest in recent years. Authors have described fitting multipleodels to data (Griess, Diab, & Schulze-Riegert, 2006), describ-

ng measurement uncertainties (Little, Fincham, & Jutila, 2006) andensitivity analysis using prior knowledge of parameter uncertaintyCosta, Schiozer, & Poletto, 2006). Very few references have beenound which attempt to quantify uncertainty in models for produc-ion optimization, which seems paradoxical as production modelsre fitted to much the same production data as reservoir models.

Monte Carlo methods are a class of computational algorithms,uitable for the study of physical or mathematical systems withandom or uncertain properties, which sample a probability dis-ribution using a pseudo-random number generator with uniformrobability and observe the fraction of the numbers obeying someroperty or properties (Metropolis & Ulam, 1949). A Monte Carloethod for planning single-rate well tests under uncertainty was

xplored in (Bieker, Slupphaug, & Johansen, 2006). Bootstrapping isMonte Carlo type method designed to estimate the uncertainty

n parameters fitted to a set of data through regression (Efron &ibshirani, 1993).

Elgsaeter, Slupphaug, and Johansen (2008b) considered aethod for modeling production for production optimization based

n local valid linear and nonlinear stationary models, motivated byhe concepts of system identification. Bootstrapping methods weresed to find a large number of multiple parameter estimates whichescribe the production data equally well. This set of multiplearameter estimates expresses the consequences of low informa-ion content on the fitted parameters for the model structurehosen. The magnitude of estimated parameter uncertainty of evenimple models was found to be significant. Although this paper con-idered black-box type system identification models, bootstrappingan be applied regardless of the choice of model, as bootstrapping iscomputational approach to quantifying uncertainty which makeso assumptions about the chosen model structure. Elgsaeter et al.2008a) estimated the lost potential of production optimization
aused by uncertainty in fitted parameters to be several percent ofevenue for the case of a North Sea oil and gas field, based on sce-ario simulations. These earlier findings motivate the focus of thisaper on handling quantified parameter uncertainty in productionptimization.
ical Engineering 34 (2010) 163–176 165

1.1.2. Optimization under uncertaintyProduction optimization can be seen as a form of real-time opti-

mization system, a closed-loop controller which attempts to locatethe optimal setpoint through a series of steps, each consisting of asmaller setpoint change. Real-time optimization systems, originallydeveloped for chemical processes, consider an economic objectivefunction maximized subject to a rigorous steady-state nonlinearprocess model and process constraints. Real-time optimization isa form of model-based control, and obtaining the process model isconsidered the single most difficult and time-consuming task in theapplication and maintenance of model-based control and optimiza-tion (Andersen, Rasmussen, & Jorgensen, 1991; Terwiesch, Agarwal,& Rippin, 1994; Zhu, 2006).

Some parameters in process models are typically fitted to data inreal-time optimization. Ogunnaike (1995) has suggested that mod-eling for control should be based on criteria related to the actualend use, and that “fitting equations to data” may be inadequate inthe context of controller design. A constraint on the maximum set-point change is often applied in real-time optimization, and smallerchanges in setpoints implemented iteratively with re-identificationand re-optimization between each step, a two-step approach.

Uncertainty in real-time optimization falls into four main cat-egories (Zhang, Monder, & Forbes, 2002): market uncertainty, theimprecise knowledge of process economics, process uncertainty, theimprecise knowledge of operation due to process disturbances oruncertain inputs, measurement uncertainty, the imprecise knowl-edge of measured process variables due to sensor or transmissionerrors, and model uncertainty, plant/model structural and paramet-ric mismatch.

Explicit analysis of uncertainty has gained some attention incontrol design, the methods of robust control address the prob-lem of designing linear multivariable controllers that adhere tosome robust stability and performance criterion (Skogestad andPostlethwaite, 1996)(Skogestad & Postlethwaite, 1996). Ensuringthat a process is designed with sufficient flexibility to operate underchanging and uncertain parameters represented by probabilitydistributions has been suggested, usually under the assump-tion of linear models and normal distributions, see for instancePistikopoulos and Mazzuchi (1990).

Approaches to handling uncertainty in optimization can bedivided into stochastic optimization, sensitivity analysis, back-offand robust optimization methods. Stochastic optimization attemptsto directly solve a problem given uncertainty by formulating thestochastic optimization problem in deterministic form using prob-abilities (Kall & Wallace, 1994). The sensitivity analysis approachconsists of augmenting the objective function of the optimizationproblem with a penalty term intended to minimize parametric sen-sitivity (Becker, Hall, & Rustem, 1994). The back-off method addsa vector constraint on decision variables to ensure that operatingpoints are feasible, and the back-off vector is computed at intervals(Loeblein, Perkins, Srinivasan, & Bonvin, 1999). Robust optimizationoptimizes the expected value for a chosen performance index fora given level of risk, formulated in terms of worst-case, mean-variance and so forth (Darlington, Pantelides, Rustem, & Tanyi,1999; Mulvey & Vanderbei, 1995). All these approaches introduceconservativeness to ensure feasibility rather than exploiting mea-surements post-optimization.

Optimization methods which attempt exploit measurementsfall into the two main categories: modified two-step approachesand approaches which do not update process models. Roberts andWilliams (1981) suggests a modified two-step approach which
add a gradient modification term to the cost function of the opti-mization problem to ensure that iterates converge to a point atwhich the necessary conditions for optimality are satisfied. Meth-ods which do not require model updating can be classified intomodel-free and model-fixed methods. Some model-free meth-

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ds mimic various iterative numerical optimization algorithms,uch the Nelder-Mead simplex algorithm (Box & Draper, 1969).ther model-free methods recast the nonlinear programming prob-

em into a problem of choosing decision variables whose optimalalues are approximately invariant to uncertainty. Self-optimizingontrol considers determining which variables to keep at con-tant setpoints to keep the process acceptably close to optimumSkogestad, 2000). Extremum-seeking control are adaptive control

ethods related to self-optimizing control which attempt to moverocess setpoints toward values which result in an extreme valuemaximum or minimum) of a measured output without a process

odel. Extremum-seeking control usually introduces ample exci-ation in setpoints to achieve this goal, for instance in form of aow-frequency sinusoidal reference (Krstic, 2000). NCO-tracking is

model-free method which uses offline analysis of the processodel to determine functions for setpoint values at which the

ecessary conditions for optimum are enforced, parameterized inerms of measured variables (Francois, Srinivasan, & Bonvin, 2005).ixed-model methods utilize both the available measurements and arocess model for guiding the iterative scheme toward an optimalperating point, but update constraint and cost function rather thanodel at each iteration (Forbes & Marlin, 1994). Methods such as

xtremum-seeking or those which mimic iterative numerical opti-ization algorithms introduce many variations to setpoints. Theseethods may be unsuitable to the offshore production of oil and

as where the costs and risk of each change in setpoint may be sig-ificant. So-called model-free methods still require models in theesign of controllers, and obtaining accurate models of the offshoreroduction of oil and gas offline may be challenging.

There is some precedence for analyzing the risk of implementingsetpoint change. Miletic and Marlin (1998) have proposed result

nalysis using multivariable statistical hypothesis testing to deter-ine whether the predicted increase in profit from implementing

hanges in setpoints is statistically significant or a result of processoise.

A classic theoretic approach to uncertainty reduction is dual con-rol theory, which deals with the controller design for processes

hich are initially unknown (Fel’dbaum, 1961a, 1961b). The the-ry is called dual as the objectives of such a controller are twofold,rstly to control the system as well as possible based on cur-ent system knowledge, secondly to experiment with the systemo as to better learn how to control it in the future. The prob-em of determining uncertainty reduction is also related to theeld of reinforcement learning, an area of machine learning con-erned with how an agent ought to take actions in an environmento as to maximize some notion of long-term reward (Sutton &arto, 1998), focused on on-line implementation while making aradeoff between exploration of uncharted territory and exploita-ion of current knowledge. Reinforcement learning has seen somepplication in modeling of batch and semi-batch chemical reactorsartinez (2000). Optimal experiment design for control has focused

n deriving an input signal that minimizes some control-orientedeasure of plant/model mismatch under a constraint on total input

ower. Optimal experiment design is usually performed with theim of achieving control that is robust to disturbances (Gevers &jung, 1986). Yip and Marlin (2003) suggested including excitationlanning in real-time optimization and to weight the cost of thexcitation against benefit, under the assumption that the modelarameters are initially known and that the occurrence of a distur-ance which may necessitate excitation to re-fit parameters can be

dentified by measurements.

.2. Problem formulation

The aim of this paper is to suggest and study a structuredpproach to optimizing the offshore production of oil and gas with

ical Engineering 34 (2010) 163–176

uncertain production models. An iterative two-step approach tooptimization combined with post-optimization feasibility assur-ance through an operational strategy is suggested. Bootstrappingmethods are used to quantify parameter uncertainty, which isexploited for excitation planning and result analysis under uncer-tainty based on multivariate Monte Carlo-like methods.

The suggested approach is outlined in Section 2, a simulationcase study is described in Section 3 before conclusions are drawn.

2. Production optimization under uncertainty

2.1. Casting uncertainty in production optimization inmathematical terms

This section suggest how uncertainty resulting from low infor-mation content in production data can be cast in mathematicalterms to allow a structured treatment. Let x be a vector of inter-nal variables and let u be a vector of decision variables. This paperwill consider a production optimization problem on the form:

[ û(�) x(�) ] = argmaxu,xM(x, u, d) (1)

s.t 0 = f (x, u, d, �) (2)

0 ≤ c(x, u, d), (3)

where � is a vector of parameters to be determined. û(�) and x(�)are the optimal solution of (1)-(3) for a given �. d is a vector ofmodeled and measured disturbances independent of u. M(x, u, d)is a profit measure which is to be maximized subject to a processmodel (2) and process constraints (3) for a given parameter value �.For offshore oil and gas production, x may be the production ratesof each fluid from each well, u may be relative valve openings, c maydescribe constraints in total water and gas processing capacity andM may express total oil production.

In this paper we will consider “response surface” or “perfor-mance curve” type production models (2), stationary, nonlinear,locally valid equations which express production rates of each wellin terms of gas-lift rates, production choke openings, the mostrecent well test and fitted parameters.

As production data are often local in nature, i.e. setpoints areonly varied within a narrow range of values, a model fitted to datamay only be locally valid, only accurately able to describe productionfor a narrow range of setpoints. The local nature of data and of anymodel fitted to such data can be accounted for by enforcing

max{umin, u0 − Uprc · U} ≤ u ≤ min{umax, u0 + Uprc · U} (4)

when solving (1)–(3) and iteratively re-updating the productionmodel and re-optimizing, an two-step approach. U defines thescale elements of u, u0 is the initial value of the decision vari-able, Uprc < 1 is a design parameter which limits the magnitudeof setpoint change at each step, and umin, umax are minimum andmaximum values of u, respectively.

This paper only considers instantaneous optimization, determin-ing (û(�), x(�)) at the current time. Let y be the vector of productionmeasurements, y may for instance be measured total rates of oil, gasand water. Let y(u, d, �) be an estimate of process measurementsbased on the model (2). Let the tuning data set be a set of histori-cal process data spanning N time steps spanning the time intervalt ∈ [1, N].

ZN = [ y(1) d(1) u(1) y(2) d(2) u(2) . . . y(N) d(N) u(N) ]

(5)

with residuals

�(t, �) = y(t) − y(u(t), d(t), �) ∀t ∈ {1, . . . , N}. (6)

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position of this boundary and the optimal setpoint to be uncertain.To ensure that the implemented setpoint remains in the feasibleregion, an operational strategy could alternately decrease u1 andincrease u2 in smaller steps as illustrated in Fig. 1.


arameter estimation determines � by minimizing the sum of thequared residuals for the tuning data set:

ˆ = argmin�

N∑

t=1

w(t)‖�(t, �)‖22 + Vs(�), s.t. c�(�) ≥ 0 (7)

here w(t) is a user-specified weighting function, Vs(�) is anptional soft constraint and c�(�) an optional constraint on �. Therocess model (2) should only be fitted to historical process datahat are recent, in the sense production during the time inter-al spanned by the tuning set should be consistent with currentroduction and the production model. It may be impossible toetermine an accurate estimate � with (7) when the informationontent in ZN is low. Extending the model structure to more long-erm effects may enable the use of a longer tuning set, which mayn turn reduce parameter uncertainty. Therefore parameter uncer-ainty is linked to other forms of uncertainty such as structuralncertainty.

Let the potential for production optimization Po be the increasen profit attainable if the production was moved from the initialperating point (x0, u0, d0) to the globally optimal operating pointx∗(d0), u∗(d0), d0):

o�=M(x∗(d0), u∗(d0), d0) − M(x0, u0, d0) ≥ 0. (8)

hen implementing û(�) some of the potential Po may be remainnrealized due to uncertainty, which motivates

potential of production optimization (Po)�=realized potential (Po,r)+lost potential due to uncertainty (Lu).

(9)

By repeatedly solving (7) with bootstrapping methods, uncer-ainty in � resulting from low information content in ZN can beuantified. In the influence of uncertainty in � on setpoints û(�) androfits M can be assessed by Monte Carlo simulations of (1)–(3). This

dea was explored for the estimation of Lu in Elgsaeter et al. (2008a).

.2. Motivation for the chosen structured approach toptimization with uncertain models

The strategy considered in this paper, motivated by the aboveiscussion, is outlined in Fig. 4.

This paper considers the operational strategy explicitly as a com-onent of a structured approach to handling uncertainty, as suchtrategies are expected to add robustness against some of the uncer-ainty considered. Operational strategies can be considered a formf uncertainty handling through feedback.

This paper considers planning of excitation in a single decisionariable, which will reduce only some of the present parame-er uncertainty. The cost of such planned excitation is associatedith its resulting temporary reduction in production and can be

stimated in a fairly straightforward manner. The benefit of alanned excitation is much harder to quantify in advance, as itill depend on the parameter value found, on the influence of

his updated parameter value on the target setpoint found throughroduction optimization, and on subsequent implementation ofhe operational strategy while uncertainty in other parameterstill persist. This paper considers estimating the benefit of plannedxcitation, the marginal increase in profits that can be expectedhen re-optimizing after the planned excitation, and this bene-t is estimated through stochastic simulations. Not all reduction

n parameter uncertainty will be profitable, for instance it shoulde intuitively clear that eliminating uncertainty against which theperational strategy is robust has zero benefit.

As the change in profits that result from implementing setpointhange toward a target is uncertain and can even be negative, this


paper considers basing the choice of whether to implement set-point change on the distribution of simulated profit changes foundthrough stochastic simulations of the operational strategy.

The choice of relating decisions to profits both in result analysisand excitation planning is motivated by a desire to make the costof uncertainty visible in business terms as this is what ultimatelydrives industry decision making.

Stochastic simulations were chosen as the considered methodfor their conceptual simplicity, as they can be applied to a widevariety of operational strategies without introducing simplifyingassumptions, and as the frequency of decision making in productionoptimization on the order of days or weeks allows computationallyintensive methods such as these to remain practical.

A formalization of an operational strategy is discussed in detailin Section 2.3, result analysis in Section 2.4, and excitation planningin Section 2.5.

2.3. Operational strategy

Example 1 illustrates how an operational strategy may operate.

Example 1. Consider a hypothetical offshore oil and gas field,producing from two wells with u = [ u1 u2 ]T , one decision vari-able affecting the production of each well. Each well produces oiland water at rates x = [ q1

o q1w q2

o q2w ]T which depend nonlin-

early on u, and the objective is to maximize total oil productionqtot

o = q1o + q2

o while the total rate of produced water qtotw = q1

w + q2w

must not exceed a constraint. Suppose that a target setpoint hasbeen determined by solving the production optimization problem.Suppose that at the current setpoint, feasible and infeasible regionsare as illustrated in Fig. 1. The boundary between the feasible andinfeasible regions is the set of setpoints at which all water process-ing capacity is utilized, and parameter uncertainty may cause the

Fig. 1. Example 1: Illustration of an operational strategy initiated at tos and terminat-

ing at toe . Solid line illustrates the unknown border between feasible and infeasible

regions, the dotted lines illustrate the uncertainty of this border, while the dashedline illustrates how an operational strategy might move setpoints in closed-loop. Thecircle illustrates the setpoint u0, and crosses illustrate uncertain calculated optimalsetpoints.

1 Chemical Engineering 34 (2010) 163–176

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In this section the concepts illustrated in Example 1 are gen-ralized and an operational strategy is described formally for theurposes of analysis. In practice a human operator would perhapsot be able to stringently reproduce the operational strategy astated in this paper, instead the algorithm could be used more as auideline.

The operational strategy attempts to implement changes in uoward a target uT by sequentially increasing and decreasing ele-

ents of u and monitoring responses in profits and constraints.T may be any desired value of u, for instance that found byolving the production optimization problem numerically. Theperational strategy is intended as a post-optimization strategy forhe implementation of the new desired setpoint, and no produc-ion optimization or parameter estimation is performed during theourse of the operational strategy. It is assumed that the sign ofhe response in profits and constraint utilization to a change is set-oints is known. This will usually be the case in offshore oil and gasroduction, where decision variables may for instance be chokepenings or gas-lift rates and the response to an increase in u issually positive for the range of setpoints considered. Let U− behe set of indices corresponding the components of uT − u0 whichre negative, and let U+ be the indices of the components of corre-ponding the components of uT − u0 which are positive. uT does notave to be feasible in practice, the operational strategy will ensure

easibility of implemented setpoints.The operational strategy should have a defined criterion for

hen to terminate. If the termination criterion considers only theign of profit change, the operational strategy may not be able toroceed if the initial setpoint is close to a local optimum. However,

f the termination criterion considers the difference between mon-tored and predicted profit, the operational strategy can proceed inhe face of profit decrease, as long as this decrease is in reasonablegreement with the model used in optimization. If profits decreaseelow the predicted values then the decision to terminate shouldeigh the magnitude of this decrease against the predicted profit

ncrease at the target. The first scenario illustrated in Fig. 2 showsow a profit decrease is predicted by the model and it is reason-ble for the operational strategy to continue, while in the secondcenario the profit decrease is not predicted by the model and its reasonable for the operational strategy to terminate and returno the initial setpoint. These considerations can be formulated inerms of estimated and measured profit change and evaluated byomparing a cost function JM against a threshold value JT

M .An operating strategy which is initiated at time to

s and termi-ated at time to

e , increases the realized potential by:

Po,r�=Po,r(to

s ) − Po,r(toe ), (10)

nd in the section below it is suggested how a probability den-ity function f�Pr(�Po,r) can be found prior to implementing theperational strategy. The operational strategy can continue alter-

ng setpoints beyond uT as long as each step of the operationaltrategy causes increased profit. As the operational strategy mayvershoot the most profitable setpoint, the final step should be tossess all steps and return to the most profitable one. The suggestedperational strategy is outlined in Algorithm 1.

lgorithm 1 (An operational strategy). Given a process that is ini-ially at a feasible setpoint (u0, d0) and given measured processonstraints c, measured profit M and decision variables u, a tar-et value uT and probability distributions f�Pr(�Po,r), let 0 < S ≤ 1enote step size.

1. Let the measured profit prior to implementing the operationalstrategy be M0. Let k be the index of the current step, and setk = 0 initially.

. Determine sets (U−, U+) to correspond with uT − u0.

Fig. 2. The two plots illustrate how measured profit (solid lines) may evolve com-pared to a calculated prediction interval (dotted lines) in two different cases. M0 isthe profit as the operational strategy is implemented, and M0 + �Po,r (dashed line)is the predicted increase in profit from implementing the operational strategy.

3. Repeat (3a), (3b), (3c)(a) k = k + 1(b) decrease ui, i ∈ U− by an amount proportional to ui

T −ui

0, i ∈ U−, so that uik

= uik−1 − S · (ui

T − ui0), i ∈ U−,

(c) increase ui, i ∈ U+ by an amount proportional to uiT −

ui0, i ∈ U+ while observing c and M, as long as all elements

of c obey c < 0 and (∂M/∂u) > 0, let the resulting measuredprofit be Mk, and the resulting setpoint uk,

while (‖uk − u0‖ > ‖uT − u0‖ and Mk ≥ Mk−1) or (‖uk − u0‖ <‖uT − u0‖ and JM(Mk, fP(�Po,r)) > JT

M).4. Implement the setpoint which resulted in the highest measured

profit in steps 1–3.

One of the operators’ tasks is to ensure that the operational con-straints are obeyed, therefore it is reasonable to assume that (u0, d0)is feasible in Algorithm 1. It is assumed that in practice cases ofinfeasible (u0, d0) are handled operationally, i.e. an operator willapply process knowledge instead of mathematical optimization tomove the process to a feasible setpoint.

The magnitude of S is a user preference and preferably S � 1.The magnitude of S will also depend on how often production opti-mization is implemented. Smaller S may result in an operationalstrategy which is more labor-intensive to implement, but may insome cases cause the operational strategy to terminate closer tooptimum.

2.4. Result analysis

This section aims to investigate whether an estimate of theparameter uncertainty can be exploited in result analysis. The
approach chosen is simulating the profit change �Po,r that resultsfrom implementing a setpoint change suggested by productionoptimization on the production model with different parameterestimates drawn from the estimated parameter distribution f�(�)in a Monte Carlo manner. The approach is outlined in Algorithm 2.

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lgorithm 2 (Estimating f�Pr(�Po,r)). Given an operational strat-gy, the current operating point (x0, u0, d0), Uprc and probabilityistributions f�(�) and fu(û),

repeat Nt times◦ draw a sample �t from f�(�),◦ obtain a point estimate of �Po,r by simulating the operational

strategy running on a process described by the process modelwith negligible structural model uncertainty and with param-eters equal to �t , while enforcing the constraint (4).

The distribution of Nt point estimates �Po,r is an estimate off�Pr(�Po,r).

As a copy of the production model cannot account for struc-ural uncertainty which will grow with the magnitude of u − u0, aonstraint of the form (4) should be enforced in simulations. Theagnitude of estimates of �Po,r will depend on the choice of Uprc ,

ut a conservative choice of Uprc should yield conservative esti-ates Po,r on which result analysis could be based.

The aim of the result analysis is to determine whether to imple-ent the setpoint change suggested by production optimization.

his decision could be based on the evaluation of a cost functionhich depends on f�Pr(�Po,r):

r(f�Pr(�Po,r)) ≥ JTr . (11)

r(f�Pr(�Po,r)) could for instance be the expected or worst-casealue. The threshold JT

r should ensure that setpoint changes are onlyerformed when the resulting increase in profits is expected to beignificant and to justify the risks and operational costs which aressociated with any setpoint change, such as the risk of triggeringn unplanned shutdown.

.5. A cost/benefit approach to excitation planning

The value of implementing a planned excitation prior to imple-enting production optimization is that the excitation may reduceodel uncertainty, allowing production optimization and the oper-

tional strategy to find a more profitable setpoint than otherwiseossible. This paper investigates exploiting estimates of parame-er uncertainty f�(�) found from recent historical production datao form the basis of a structured approach to rank the benefit of
xciting different decision variables. For simplicity this discussions restricted to wells which are decoupled, i.e. where implement-ng change in the production of one well will not influence theroduction of other wells, which is often a reasonable assumption
or so-called platform wells as production from such wells travel

Fig. 3. A schematic model of offsh


through separate flowlines and join first at the production man-ifold. The discussion is restricted to excitation of single decisionvariables, although simultaneously exciting several variables is alsoconceivable.

The benefit of excitation BE is proposed defined as the differ-ence between the profit which would be realized if no excitationis performed, �Po,r , and the profit which would be realized afterimplementing an excitation, �PE

o,r:

BE �=�PEo,r − �Po,r . (12)

The idea of the benefit of excitation is illustrated in Example 2.

Example 2. Consider a hypothetical field producing oil andwater from two decoupled wells. Well 1 produces (q1

o, q1w) at

rates which depend on u1, well 2 produces (q2o, q2

w) at rateswhich depend on u2. The relationships between (q1

o, q1w) and u1

and between (q2o, q2

w) and u2 are uncertain, but a model whichexpresses q1

o(u1, �1), q2o(u2, �2), q1

w(u1, �1), q2w(u2, �2) is available.

Models for well 1 depend on parameters �1 and models for well 2on �2. Two parameter estimates �a = [ �1

a �2a ] and �b = [ �1

b�2

b]

have been determined which result in estimates of qtoto = q1

o + q2o

and qtotw = q1

w + q2o which match measurements equally well for a

set of recent historical data. Optimization of the objective M =q1

o(u1, �1) + q2o(u2, �2) subject to the model and the constraint

q1w(u1, �1) + q2

w(u2, �2) < qmaxw is desired. If no excitation is imple-

mented the target uT = û(�a) is planned implemented using anoperational strategy. If an excitation could be performed whichwould reveal which of �a and �b better describe production, whatwould be the benefit of that excitation?

If �a describes production better BE = 0 as �PEo,r = �Po,r . If �b

describes production better BE will depend on how much profitincrease the operational strategy would be able to implement ifgiven the target uT = û(�b) as opposed to uT = û(�a). In the casethat �b describes production better while uT = û(�a), BE could befound through simulations on the production models, provided thatstructural model errors are small.

Example 2 illustrates that BE will depend on the interactionbetween uncertainty in �, uncertainty in û(�) and the ability of theoperational strategy to compensate for uncertainty in û(�). The ben-
efit of excitation will also depend on what target the operationalstrategy will pursue if no excitation is performed. In the offshoreproduction of oil and gas, exciting all components of u is usuallynot feasible, so some method for ranking components of u to beexcited is desirable, as is illustrated in Example 3.
ore oil and gas production.

1 Chem

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Fo


xample 3. Consider again the field described in Example 2. Con-ider that our aim is to choose whether to introduce excitation in1, in u2 or not at all. Assume that an excitation of u1 would allow uso distinguish between �1

a and �1b

, while �2 would still be uncertain,nd that an excitation of u2 would allow us to distinguish between

ˆ2a and �2

bwhile �1 would still be uncertain. With the assumptions

ade it can be expected that an excitation of well 1 will either resultn implementing uT = û([�1

a , �2a ]) if production is described by �a, or

mplementing uT = û([�1, �2a ]) if production is described by �b, as

b

t was planned to implement uT = û([�1a , �2

a ]) if no excitation waserformed, and �2 would still be uncertain after excitation of u1.he outcome on the production models can be simulated in bothases provided that structural model errors are small, which would

ig. 4. Flowchart of the proposed structured approach to optimization of offshoreil and gas production with uncertain models.


give two estimates of BE . A structured approach to planning exci-tation could then be to excite the well associated with the highestpositive average BE .

Examples 2 and 3 illustrate the principles of a Monte Carlo sim-ulation approach to estimating BE for the simplest possible case oftwo wells, two parameter estimates and one constraint. In our casestudy the same principles are applied to estimate BE for a largernumber of wells, constraints, and parameter estimates �.

Let �n be the nominal parameter estimate, the parameter estimatefound by solving (7) directly. The estimation of BE will considerthe case when the operational strategy attempts to implement thetarget uT = û(�n) if no excitation is performed. Note that althoughproduction of wells are decoupled, the components of û(�) as cal-culated by (1)–(3) are coupled, so updating models describing onewell may influence suggested settings for other wells, and thiswill need to be reflected when estimating BE . Simulations on theproduction model assume that structural model uncertainty is neg-ligible, and, as locally valid models are considered, this assumptionis reasonable as long as change in decision variables is small, whichmotivates enforcing the constraint (4) in simulations, and whichshould result in conservative estimates BE .

The approach considered is outlined in Algorithm 3.

Algorithm 3 (Benefit of excitation, ui). Given that it is planned
to implement û(�n) if no excitation is performed. Given Nt , thedistribution f�(�), an operational strategy, a process model whichdepends on parameters �, Uprc , and that the aim is to estimate thebenefit of exciting ui, a single decision variable in the vector u.
Fig. 5. Case study: Production data (dotted), the production analog (dashed), esti-mates with nominal nonlinear production model (solid).

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•

•

t

eci

lb

Fo(am

J (fB (B ), fC (C )) ≥ J , (13)

where JE,T is a user-specified threshold and JE(f EB (BE), f E

C (CE)) is auser-specified metric such as for instance the expected value of thedifference BE − CE .


repeat Nt times◦ draw a sample �t from f�(�).◦ determine �Po,r(�t) by simulating the operational strategy

using uT = û(�n) on the process model with parameters �t whilesetpoint change is limited by (4).

◦ let �it be parameters of �t excited by ui. Let �E(�n, �i

t) be a vector

of parameters that is equal to �it for parameters excited by ui

and equal to �n for all other parameters.◦ determine �PE,i

o,r(�t) by simulating implementing the opera-

tional strategy using uT = û(�E(�n, �it)) on the process model

with parameters �t while setpoint change is limited by (4).◦ let BE,i(�t) = �PE,i

o,r(�t) − �Po,r(�t).the distribution of simulated values BE,i is an estimate of theprobability density function for the benefit of exciting ui, f E

B (BE,i)

A structured approach to excitation planning could be to excitehose decision variables ui which have highest BE,i on average.

As components of û(�) as calculated by (1)–(3) are coupled,xcitation of ui can cause û(�E(�n, �i )) to differ from û(�n) in
tomponents other than ui. If uncertainty for non-excited wellss significant, implementing û(�E(�n, �i
t)) may inadvertently cause

ower profits than û(�n), and in such cases estimates BE,i maye negative. Negative BE,i is an indication that the profit increase

ig. 6. Case study: Iteration 1, formulation A. Each row shows modeled rates ofil, gas and water for different parameter estimates found through bootstrappingdashed) compared with modeled rates for the nominal parameter estimate (dotted)nd the production analog (solid), global optimum (crosses) and currently imple-ented setpoint (circles).


resulting from excitation of ui is variant to uncertainty in modelsdescribing other wells.

Except during startup it will in most cases be reasonable toassume that production is initially utilizing at least one process-ing capacity fully, so that any excitation will require production toback-off from full capacity for a period of time, incurring a cost.It is conceivable to simulate the cost of excitation CE in a MonteCarlo fashion for different samples of f�(�). To limit the scope ofthis paper, estimation of CE will not be considered. It is proposedthat candidate excitations can be ranked by the value of a function

E E E E E E,T

Fig. 7. Case study: Iteration 1, formulation B. Each row shows modeled rates of oil, gasand water for different parameter estimates found through bootstrapping(dashed)compared with modeled rates for the nominal parameter estimate (dotted) andthe production analog (solid), global optimum (crosses) and currently implementedsetpoint (circles).

1 Chem

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. Case study

This section describes the application of the suggested approacho a case study modeled on the North Sea oil and gas field and the setf real-world production data considered previously in Elgsaeter etl. (2007, 2008b, 2008a).

.1. Field description

The case study considered in this paper is motivated by a Northea oil and gas field with 20 gas-lifted platform wells producing pre-ominantly oil, gas and water. The field has a layout as depicted inig. 3, with one production separation train and one test separator.
easurements of the total rates of produced oil, gas and water are
vailable. The operator of the field requested that all data be keptnonymous, therefore all variables will be presented in normalizedorm. The production data are characterized by little variation inecision variables. The aim of production optimization on the field

ig. 8. Case study: Distribution fu(û) for formulations A and B shown for each wellfter iteration 1 (bars). Constraints on u (dotted), u0 (circle) and uT (stem), andptimal setpoint (crosses).


is to distribute available lift gas so as to maximize total oil pro-duction while keeping total produced water and gas rates belowcapacity constraints.

3.2. Method

The cost and risks of implementing a trial of the suggestedapproaches on an actual field are significant, and it may be diffi-cult to compare strategies by implementation on an actual field asproduction will vary with time due to disturbances. This motivatesthe choice of studying the suggested approach in simulations.

Due to uncertainty there may be many plausible descriptionsof production, and an approach to optimization under uncertaintyshould ideally perform well for all such plausible descriptions. Thispaper will consider simulating optimization on a model that isplausible in the sense that it conforms with production data andrefer to this model as the production analog. When optimizing pro-duction, the production analog is considered unknown, productionoptimization can only infer knowledge of the production analogthrough measurements.

When modeling the response to change in gas-lift rate, a first-order linear kernel with a single fitted parameter for each fluid and
phase with dim(�) = 63 (formulation A) will be compared with asecond-order nonlinear kernel with two fitted parameters for eachfluid and phase and dim(�) = 123 (formulation B). By comparingthese formulations the significance of model structure and param-
Fig. 9. Case study: The percentage of the initial potential �P∗o,r that was realized

in simulations of formulation A (circles) and B (squares) for Uprc = 0.2 (line) andUprc = 0.5 (dashed).

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a result uncertainty in the setpoint û(�) suggested by productionoptimization was highly uncertain, as illustrated in Fig. 8. Despitethis significant uncertainty, the suggested approach was able torealize a significant profit increase with either model formulationand with either choice of Uprc , as is illustrated in Fig. 9.


terization on the suggested methodology can be assessed. Physicalnowledge is included through constraints and regularization onarameter estimation.

Both for formulations A and B parameters were re-fitted and pro-uction re-optimized according to the workflow outlined in Fig. 4

n four iterations. Algorithms 2 and 3 were run for each iterationith Nt = 200, but no excitations were implemented and setpoint

hange was implemented regardless of risk-reward estimates. Foromparison this process was repeated for two choices of the designarameter Uprc = 0.2 and 0.5.

Further details on the simulation case study are given inppendix A.

.3. Results

A comparison of the predictions of the production analog againsteld data is shown in Fig. 5. The production analog is shown alongith models of formulation A and B at iteration 1 in Figs. 6 and 7.

he distribution of û found through Monte Carlo simulation prior toteration 1 are compared for the two models in Fig. 8. Fig. 9 compares
he realized profits Po,r of formulations A and B at each iteration,or Uprc = 0.2 and Uprc = 0.5, indicating that between 30% and 80%f the profit potential was realized.
Fig. 10 illustrates the changes in setpoint implemented by theperational strategy for formulation B. Figs. 11 and 12 compare

ig. 10. Case study: An example of how the operational strategy implements set-oint changes while obeying process constraints for formulation B. Top graphs showormalized profits (qtot

o ), gas capacity utilization (qtoto ) and water capacity utilization

qtotw ), lower graphs show normalized relative changes in gas-lift rates qgl for all wells.


the simulated benefit of excitation with estimates found withAlgorithm 3 for iterations 1 and 4. Fig. 13 compares f�Pr(�Po,r)found with Algorithm 2 with realized profit for each iteration offormulation A and B.

3.4. Discussion

The suggested optimization method can be classified as a two-step approach and the operational strategy as post–optimizationfeasibility assurance. In the simulation case considered, modeluncertainty was significant, as illustrated in Figs. 6 and 7, and as

Fig. 11. Case study: Distribution of BE of exciting well i in isolation at iteration 1 ofthe simulation, found with Algorithm 3 (bars), BE found by simulating excitation onthe production analog (stems). Numbers show average BE as predicted by Algorithm3. All values are expressed in percent of total unrealized potential.

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(1)–(3) is a nonlinear optimization problem and convergencef its solution toward the global optimum cannot be guaranteed

n general unless (1)–(3) can be shown to be a convex optimiza-ion problem and is solved with methods of convex optimizationBoyd & Vandenberghe, 2004). Analytical methods (Ljung, 1999)an only guarantee that the solution of (7) is the best descriptionf production if N approaches infinity, if the reservoirs, wells androcessing facilities can be considered stationary, and if ZN is suf-ciently informative to distinguish between all solutions of (7).ypically bootstrap estimates are not exact but have an inherentrror, while the bias in estimates is often small the variance can
ften be quite large due to the finite amount of data and the finiteumber of resamples (Efron & Tibshirani, 1993).
For the reasons listed above, no guarantees can be given thathe implemented setpoint will converge toward global optimumn practice when the suggested approach is applied to processes

ig. 12. Case study: Distribution of BE of exciting well i in isolation at iteration 4 ofhe simulation, found with Algorithm 3 (bars), BE found by simulating excitation onhe production analog (stems). Numbers show average BE as predicted by Algorithm. All values are expressed in percent of total unrealized potential.


such as these with low information content data and limits on theplanned excitation. We argue that lack of guaranteed convergenceis a result of the properties of the process considered and not of theproposed solution, and processes with these properties are never-theless interesting for academic study. It is because of the lack ofguaranteed convergence that this paper has focused on identify-ing setpoint change which increase profits with some measure ofconfidence.

The methods suggested would remain sensible in the specialcase that all the requirements stated above are satisfied and aunique parameter estimate � can be found from (7) and (1)–(3) isable to return the globally optimum setpoint. In this case, boot-strapping (7) would return the same parameter estimate for allresamples, all estimates of the benefit of excitation found withAlgorithm 3 would be zero, and and Algorithm 2 would predicta single profit change �Po,r with high confidence.

The operational strategy suggested will always ensure that pro-duction is feasible as long as production is initially at a feasiblesetpoint, constraints are measured, transients are negligible andas long as the sign of change in constraint utilization to change in
decision variables are known. Constraints were enforced at all timesduring the simulation case study, an example is shown in Fig. 10. Theassumption of negligible transients may be relaxed by expandingthe operational strategy method, for instance by model-predictivecontrol.
Fig. 13. Case study: Estimated f (�Po,r ) for formulations A and B, one row for eachiteration. The increase in profits that was implemented in practice is shown as stems.

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Excitation planning and result analysis suggested employ com-utational, Monte Carlo-like methods. These methods can beonsidered a form of non-parametric multivariate analysis, as aample of the entire parameter vector � is drawn at each iteration,nd co-variance between parameters is therefore accounted for inhe analysis. The fact that no assumptions about linearity of modelsr types of probabilities are made is a strength of the methods, andhis trait alone separates these methods from the majority of rivalpproaches found in the literature.

The method for excitation planning suggested in this paper dealsith identifying the decision variable which when excited will

ncrease the profit attained by production optimization the most.his is done by analyzing the influence of the modeled uncertaintyn profit function by stochastic simulations. The suggested methodiffer from optimum experiment design in that the object is not tochieve control that is robust to disturbance, but rather to increaserofits by as much as possible by targeting which decision variableo excite. In this case the actual benefit of excitation could be com-uted, as shown in Figs. 11 and 12, and a correlation between largeean BE and large actual BE is visible. This finding of the simula-

ion case study supports the assertion that testing wells with largeean BE is a sensible excitation planning approach, provided esti-ated benefits are significant compared to estimated costs. Furtherork could consider estimating the benefit of simultaneously excit-

ng a small number of decision variables simultaneously, as suchxcitation could be designed to cause less temporary reduction inroduction and hence incur lower costs.

The suggested method for result analysis uses stochastic sim-lations to determine a distribution for the change in profit thatill result from changing decision variables toward a target byeans of the operational strategy. While methods have been found

n the literature which assess the significance of estimated profithange to noise, no reference was found to result analysis in theresence of uncertainty in all fitted parameters, either with orithout post-optimization feasibility assurance. Fig. 13 illustrates

hat for Formulation B, the potentially more accurate model withhigher number of parameters, estimates of realized potential Po

re comparable to the actual realized potential Po and estimatesecline with each iteration just as do actual Po. For the first iterationstimates Po are lower than the actual Po due to the conserva-iveness in the design of Algorithm 2, but the algorithm correctlydentifies a significant above-zero potential, which would haveupported the decision to update setpoints toward the suggestedarget.

Much further work on design of alternative algorithms foruantifying uncertainty, excitation planning, result analysis andperational strategies which fit the framework suggested here isossible. An element of this paper has not considered in detail isow active decision variables can be chosen to manage uncertainty,his is left for further work.

. Conclusion

An approach for casting in mathematical terms the uncer-ainty in production optimization arising from low informationontent in data has been suggested. A structured approach toandling this uncertainty by combining an iterative two-steppproach to optimization and post-optimization feasibility assur-nce was suggested and by uncertainty estimation, result analysisnd excitation planning based on multivariate Monte Carlo-likeethods.

In the simulation case study the method suggested realizedetween 30% and 80% of the available profit potential while feasibil-

ty was ensured at all times, despite that models were fitted to dataith low information content similar to that found on real-world

il fields.


Acknowledgments

The authors wish to acknowledge the Norwegian ResearchCouncil, StatoilHydro and ABB for funding this research.

Appendix A. Details of simulation case study

To ensure that the production analog is a plausible descrip-tion of the field considered, parameters in (15) are estimated withbootstrapping methods. (15) combined with one of the plausibleparameters determined in this manner is chosen as the the pro-duction analog for setpoint values similar to those observed in thetuning set.

Let qmax,igl

be the maximum values observed in the tuning set. Tosimulate our lack of knowledge about process behavior for valuesof u outside those observed in the tuning set, the kernel function fgl

is replaced in (15) with

f iu(qi

gl) = 12

ci(qigl − qi,max

gl)2 + bi(qi

gl − qi,maxgl

) + ai (14)

when qigl

> qmax,igl

. ai and bi are chosen so that qi(u) is smooth and

continuous for u around umax, and the curvature coefficient ci ischosen as a random negative value. Operating points ql,i,a of the pro-duction analog are chosen so that the tuning set can be described bythe production analog with bias terms. Fig. 5 illustrates that the pro-duction analog is a plausible description of production, as it matchesproduction data well.

Production modeling of the field considered based on the con-cepts of system identification has been considered previously inElgsaeter et al. (2008b), and similar methods are applied here. Theoil, gas and water rates qi

o, qig , qi

w of each well i are modeled asthe product of two kernel functions, one describing the effects ofchanges in production valve opening zi, and one describing theeffects of changes in gas-lift rates qi

gl. Models are local around the

most recent well test.The production model

qip = max{0, ql,i

p · f iz (zi, zl,i) · f i

gl(qigl, ql,i

gl)}, (15)

is considered, which is intended to be valid locally around the mostrecent well test rates ql,i

p ∀i = 1, . . . , nw∀p ∈ {o, g, w}. zi ∈ [0, 1] is therelative valve opening of well i, qi

glis the gas-lift rate of well i and

�qigl

�=(qigl

/ql,igl

) − 1 is the normalized, relative gas-lift rate for well i.Kernels are chosen as

f iz (zi, zl,i) = 1 − (1 − zi)

k

1 − (1 − zl,i)k, k = 5 (16)

f igl(z

i, zl,i) = (1 + ˛ip�qi

gl + �ip(�qi

gl)2) (17)

for all wells i.The measurement vector y(t) = [ qtot

o (t) qtotg (t) qtot

w (t) ]T

isconsidered and an attempt is made to find � so that estimates

y(�, t)�=1x(�, t) + ˆ y, (18)

fit measurements as close as possible for the tuning set, where ˆ y

is the vector of measurement biases due to calibration inaccuraciesto be determined and 1 is a matrix of ones.

In addition an upper and lower constraint on u of the form (4)was enforced. Some wells are in danger of slugging if gas-lift is
decreased, and on these wells a constraint which prohibits gas-liftfrom being decreased was implemented.
Based on the knowledge that the watercut is rate-independent,soft constraints which penalize deviation from ˛o = ˛w and �o = �w

were added to the objective function. A decline in qtoto was visible

1 Chem

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f

û

s

q

m

∀

∀

IBeFbm

soNs(

idm

R

A

B

B

B

B

C

D

E


n the tuning set, and it is chosen to de-trend qtoto and weigh older

easurements less than newer ones using the weighting-term w(t)n (7). In formulation B, ˛ and � for all wells and for oil, gas and water,s well as ˇy are considered part of the vector of fitted parameters.nly the first-order term in (17) is considered in formulation A.

The production optimization problem considered was of theorm

(�) = argmaxu

∑

∀i ∈ Ia

qio(u, d, �) + bo(�) (19)

.t. (20)l,igl

≤ qigl ∀i ∈ IMGS (21)

ax{umin, u0 − Uprc · U} ≤ u ≤ min{umax, u0 + Uprc · U} (22)∑

i ∈ Ia

qig(u, d, �)

∑

∀iIB

qigl + bg(�) ≤ qtot,c

g (23)

∑

i ∈ Ia

qiw(u, d, �) + bw(�) ≤ qtot,c

w . (24)

a is the indices of all wells considered in production optimization.iases bo(�), bg(�), bw(�) were determined for each � so that mod-led and measured production match at the time of optimization.or wells with indices IMGS = {1, 4, 10, 13, 14, 16} gas-lift could note decreased without the risk of triggering slugging. qi

o, qig , qi

w areodeled rates.

The case study was implemented in MATLAB 2. Linear least-quares parameter estimation problems and linear real-timeptimization problems were solved using the TOMLAB 3 solver lssol.onlinear production optimization problems were solved using the

equential quadratic programming solvers based on Schittkowski1983).

The simulated benefit of excitation of ui was found by replac-ng fitted parameters excited by variation of ui with those that bestescribe the production analog, and simulating production opti-ization and implementation with the operational strategy.

eferences

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ecker, R., Hall, S., & Rustem, B. (1994). Robust optimal decisions with stochas-tic nonlinear economic systems. Journal of Economic Dynamics and Control, 18,125–147.

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A structured approach to optimizing offshore oil and gas production with uncertain models

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