A Mean-Variance Objective For Robust Production .
A Mean-Variance Objective for Robust Production Optimization in Uncertain GeologicalScenariosAndrea Capoleia , Eka Suwartadib , Bjarne Fossb , John Bagterp Jørgensena, of Applied Mathematics and Computer Science & Center for Energy Resources EngineeringTechnical University of Denmark, DK-2800 Kgs. Lyngby, Denmark.b Department of Engineering Cybernetics, Norwegian University of Science and Technology (NTNU), 7491 Trondheim, Norway.a DepartmentAbstractIn this paper, we introduce a mean-variance criterion for production optimization of oil reservoirs and suggest the Sharpe ratio asa systematic procedure to optimally trade-off risk and return. We demonstrate by open-loop simulations of a two-phase syntheticoil field that the mean-variance criterion is able to mitigate the significant inherent geological uncertainties better than the alternative certainty equivalence and robust optimization strategies that have been suggested for production optimization. In productionoptimization, the optimal water injection profiles and the production borehole pressures are computed by solution of an optimalcontrol problem that maximizes a financial measure such as the Net Present Value (NPV). The NPV is a stochastic variable as thereservoir parameters, such as the permeability field, are stochastic. In certainty equivalence optimization, the mean value of thepermeability field is used in the maximization of the NPV of the reservoir over its lifetime. This approach neglects the significantuncertainty in the NPV. Robust optimization maximizes the expected NPV over an ensemble of permeability fields to overcomethis shortcoming of certainty equivalence optimization. Robust optimization reduces the risk compared to certainty equivalenceoptimization because it considers an ensemble of permeability fields instead of just the mean permeability field. This is an indirectmechanism for risk mitigation as the risk does not enter the objective function directly. In the mean-variance bi-criterion objectivefunction risk appears directly, it also considers an ensemble of reservoir models, and has robust optimization as a special extremecase. The mean-variance objective is common for portfolio optimization problems in finance. The Markowitz portfolio optimizationproblem is the original and simplest example of a mean-variance criterion for mitigating risk. Risk is mitigated in oil productionby including both the expected NPV (mean of NPV) and the risk (variance of NPV) for the ensemble of possible reservoir models.With the inclusion of the risk in the objective function, the Sharpe ratio can be used to compute the optimal water injection andproduction borehole pressure trajectories that give the optimal return-risk ratio. By simulation, we investigate and compare the performance of production optimization by mean-variance optimization, robust optimization, certainty equivalence optimization, andthe reactive strategy. The optimization strategies are simulated in open-loop without feedback while the reactive strategy is basedon feedback. The simulations demonstrate that certainty equivalence optimization and robust optimization are risky strategies. Atthe same computational effort as robust optimization, mean-variance optimization is able to reduce risk significantly at the cost ofslightly smaller return. In this way, mean-variance optimization is a powerful tool for risk management and uncertainty mitigationin production optimization.Keywords: Robust Optimization, Risk Management, Oil Production, Optimal Control, Mean-Variance Optimization, UncertaintyQuantification1. IntroductionIn conventional water flooding of an oil field, feedback basedoptimal control technologies may enable higher oil recoverythan with a conventional reactive strategy in which producers are closed based on water breakthrough (Chierici, 1992;Ramirez, 1987).Optimal control technology and Nonlinear Model PredictiveControl (NMPC) have been suggested for improving the oil recovery during the water flooding phase of an oil field (Jansen CorrespondingauthorEmail addresses: acap@dtu.dk (Andrea Capolei),eka.suwartadi@ieee.org (Eka Suwartadi), Bjarne.Foss@itk.ntnu.no(Bjarne Foss), jbjo@dtu.dk (John Bagterp Jørgensen)Preprint submitted to Journal of Petroleum Science and Engineeringet al., 2008). In such applications, the controller adjusts the water injection rates and the bottom hole well pressures to maximize oil recovery or a financial measure such as the NPV. Inthe oil industry, this control concept is also known as closedloop reservoir management (CLRM) (Foss, 2012; Jansen et al.,2009). The controller in CLRM consists of a state estimator forhistory matching (state and parameter estimation) and an optimizer that solves a constrained optimal control problem for theproduction optimization. Each time new measurements fromthe real or simulated reservoir are available, the state estimatoruses these measurements to update the reservoir’s models andthe optimizer solves an open loop optimization problem withthe updated models (Capolei et al., 2013). Only the first part ofthe resulting optimal control trajectory is implemented. As newAugust 16, 2014
Expected Returnmeasurements become available, the procedure is repeated. Themain difference of the CLRM system from a traditional NMPCis the large state dimension (106 is not unusual) of an oil reservoir model (Binder et al., 2001). The size of the problem dictates that the ensemble Kalman filter is used for state and parameter estimation (history matching) and that single shootingoptimization is used for computing the solution of the optimalcontrol problem (Capolei et al., 2013; Jansen, 2011; Jørgensen,2007; Sarma et al., 2005a; Suwartadi et al., 2012; Völcker et al.,2011).In this paper, we focus on the formulation of the optimization problem in the NMPC for CLRM. In the study of differentoptimization formulations, we leave out data assimilation (history matching) as well as the effect of feedback from a movinghorizon implementation and consider only the predictions andcomputations of the manipulated variables in the open-loop optimization of NMPC. This can be regarded as an optimal controlstudy. The reason for this is twofold. First, in the initial development of a field, no production data would be available and theproduction optimization would be an open-loop optimal controlproblem, i.e without feedback from measurements. Secondly,the ability of different optimization strategies to mitigate the effect of the significant uncertainties present in reservoir modelsis better understood if investigated in isolation.In conventional production optimization, the nominal netpresent value (NPV) of the oil reservoir is maximized (Brouwerand Jansen, 2004; Capolei et al., 2013, 2012b; Foss, 2012; Fossand Jensen, 2011; Nævdal et al., 2006; Sarma et al., 2005b;Suwartadi et al., 2012). To compute the nominal NPV, nominalvalues for the model’s parameters are used. In certainty equivalence production optimization, the expected reservoir modelparameters are used in the maximization, while robust production optimization uses an ensemble of reservoir models tomaximize the expected NPV (Capolei et al., 2013; Van Essenet al., 2009). Certainty equivalence optimization is equivalentto robust optimization for the ideal case of unconstrained linear dynamics with Gaussian additive noise and quadratic costfunctions, i.e. for Linear Quadratic Gaussian (LQG) problems(Bertsekas, 2005; Stengel, 1994). For all other problems, thecertainty equivalence optimization and the robust optimizationare different. Both certainty equivalence optimization and robust optimization assume that the stochastic event is repeatedinfinitely many times such that only the expected value but notthe risk is of interest. The purpose of the robust production optimization is to (indirectly) mitigate the effect of the significantuncertainties in the parameters of the reservoir model. However, by the certainty equivalence and the robust productionoptimization methods, the trade-off between return (expectedNPV) and risk (variance of the NPV) is not addressed directly.Fig. 1 illustrates risk versus expected return (mean) for different optimization and operation strategies. This is a sketchthat shows the qualitative behavior of the results in this paper.As is evident in the sketch, a significant risk is typically associated with the certainty equivalence optimization and the ROstrategy. The implication is that the RO strategy may improvecurrent operation, but you cannot be sure due to the significantrisk arising from the uncertain reservoir model. This is prob-Robust OptimizationTangency point:solution with the highestreturn vs risk ratioEfficient frontierMarket solutionReactive strategyCertainty equivalence optimizationRiskFigure 1: A sketch of the trade-off between risk and expectedreturn in different optimization methods implemented in the optimizer for model based production optimization.ably one of the reasons that NMPC for CLRM has not beenwidely adopted in the operation of oil reservoirs. The optimization problem in production optimization can be compared insome sense to Markowitz portfolio optimization problem in finance (Markowitz, 1952; Steinbach, 2001) or to robust designin topology optimization (Beyer and Sendhoff, 2007; Lazarovet al., 2012). The key to mitigate risk is to optimize a bicriterion objective function including both return and risk forthe ensemble of possible reservoir models. In this way, we canuse a single parameter to compute an efficient frontier (the bluePareto curve in Fig. 1) of risk and expected return. The robustoptimization is one limit of the efficient frontier and the otherlimit is the minimum risk minimum return solution. By properbalancing the risk and the return in the bi-criterion objectivefunction, we can tune the optimizer in the controller such thatan optimal ratio of return vs risk is obtained; such a solution iscalled the market solution and is illustrated in Fig. 1.The mean-variance optimization is based on a bi-criterionobjective function. Previously in the oil literature, multiobjective functions have been used in production optimizationto trade-off long- and short-term NPV (Van Essen et al., 2011),to robustify a non-economic objective function (Alhuthali et al.,2008), and to trade-off oil production, water production and water injection using a combination of mean value and standarddeviation for each term (Yasari et al., 2013). These approachespointed to the fact that a multi-objective function may be usedto trade-off risk for performance, but did not explicitly addressthe risk-return relationship studied in the present paper usinga mean-variance optimization strategy. Furthermore, these papers did not provide a systematic method for selection of therisk adverse parameter. The main contribution of the presentpaper is to demonstrate, that a return-risk bi-criterion objectivefunction is a valuable tool for the profit-risk trade-off and provide a systematic method for selection the risk-return trade-offparameter. We do this for the open loop optimization and donot consider the effect of feedback.2
piBHP is the wellbore pressure, and WIi is the Peaceman wellindex. The volumetric water flow rates at injection and production wells areThe paper is organized as follows. Section 2 defines thereservoir model. Section 3 states the constrained optimalcontrol problem and describes the mean-variance optimizationstrategy. The computation of economical and production keyperformance indicators is explained in Section 4 . Section 5 describes the numerical case study. Conclusions are presented inSection 6.qw,i qii I(5a)qw,i fw qii P(5b)The volumetric oil flow rates at production wells are2. Reservoir Modelqo,i (1 fw )qiWe assume that the reservoirs are in the secondary recovery phase where the pressures are above the bubble point pressure of the oil phase. Therefore, two-phase immiscible flow,i.e. flow without mass transfer between the two phases, is areasonable assumption. We focus on water-flooding cases fortwo-phase (oil and water) reservoirs. Further, we assume incompressible fluids and rocks, no gravity effects or capillarypressure, no-flow boundaries, and isothermal conditions. Thestate equations in an oil reservoir Ω, with boundary Ω andoutward facing normal vector n, can be represented by pressureand saturation equations. The pressure equation is described asv λt K p, ·v Xqi · δ(r ri )r ΩIn this section, we present the continuous-time constrainedoptimal control problem and its transcription by the singleshooting method to a finite dimensional constrained optimization problem. First we present the continuous-time optimalcontrol problem; then we parameterize the control function using piecewise constant basis functions; and finally we convertthe problem into a constrained optimization problem using thesingle shooting method.Consider the continuous-time constrained optimal controlproblem in the Lagrange formZ tbmax J Φ(x(t), u(t))dt(7a)(1a)r Ω (1b)x(t),u(t)x(ta ) x0 , dg x(t) f (x(t), u(t), θ),dtu(t) U(t). X S w · fw (S w )v qw,i · δ(r ri ) ti I,P(3)φ is the porosity, s is the saturation, fw (s) is the water fractionalflow which is defined as λλwt , and qw,i is the volumetric waterrate at well i. We use the MRST reservoir simulator to solve thepressure and saturation equations, (1) and (3), sequentially (Lieet al., 2012). Specifically, MRST first computes the total mobility using the initial water saturation. Secondly, the pressureequation is solved explicitly using the initial water saturationand the computed total mobility value. Thirdly, with the obtained pressure solution, the velocity is computed and is usedin an implicit Euler method to solve the saturation equation.This procedure is repeated until the final time is reached. Wellsare implemented using the Peaceman well model (Peaceman,1983)qi λt WIi (pi pbhpi )(7b)t [ta , tb ],(7c)(7d)x(t) Rnx is the state vector, u(t) Rnu is the control vector,and θ is a parameter vector in an uncertain space Θ (in our casethe permeability field). The time interval I [ta , tb ] as wellas the initial state, x0 , are assumed to be fixed. (7c) representsthe dynamic model and includes systems described by index-1differential algebraic equations (DAE) (Capolei et al., 2012a,b;Völcker et al., 2009). (7d) represents linear bounds on the inputvalues, e.g. umin u(t) umax . In our formulations we donot allow nonlinear state or output constraints. Suwartadi et al.(2012) provide a discussion of output constraints.(2)The saturation equation is given byφtasubject tor is the position vector, ri is the well position, v is the Darcyvelocity (total velocity), K is the permeability, p is the pressure,qi is the volumetric well rate in barrels/day, δ is the Dirac’s deltafunction, I is the set of injectors, P is the set of producers, andλt is the total mobility. The total mobility, λt , is the sum of thewater and oil mobility functionsλt λw (s) λo (s) krw (s)/µw kro (s)/µo(6)3. Optimal Control Problemi I,Pv·n 0i P3.1. Production OptimizationProduction optimization aims at maximizing the net presentvalue (NPV) or the oil recovery for the life time of the oil reservoir. The stage cost, Φ, in the objective function for a NPVmaximization can be expressed as"X 1Φ(x(t), u(t)) rwi ql (u(t), x(t))d τ(t)(1 365)l I # (8)X ro qo,i (u(t), x(t)) rwp qw,i (u(t), x(t))(4)i P3
ro , rwp , and rwi represent the oil price, the water separation cost,and the water injection cost, respectively. qw,i and qo,i are thevolumetric water and oil flow rate at producer i; ql is the volumetric well injection rate at injector l; d is the annual interestrate and τ(t) is the integer number of days at time t. The disd τ(t))accounts for a daily compounded valuecount factor (1 365of the capital. Note that from the well model (4), it follows thatthe flow rates, q, are negative for the producer wells and positivefor the injector wells. Hence, the negative sign in front of thesquare bracket in the stage cost, Φ. Note that in the special casewhen the discount factor is zero (d 0) and the water injectionand separation costs are zero as well, the NPV is equivalent tothe quantity of produced oil.In the single shooting optimization algorithm, we define thefunctionN 1ψ({uk }k 0, x0 , θ) Z tb J Φ(x(t), u(t))dt :tax(t0 ) x0 ,dg(x(t)) f (x(t), u(t), θ), ta t tb ,dt u(t) uk , tk t tk 1 , k 0, 1, . . . , N 1such that (7) can be expressed as the optimization problem3.2. Control Vector ParametrizationmaxN 1{uk }k 0Let T s denote the sample time such that an equidistant meshcan be defined asta t0 . . . tS . . . t N tbs.t.N 1; x̄0 , θ)ψ ψ({uk }k 0(12a)N 1) 0c({uk }k 0(12b)Gradient based optimization algorithms for solution of (12) reN 1quire evaluation of ψ ψ({uk }k 0; x̄0 , θ), uk ψ for k N,N 1N 1c({uk }k 0 , and uk c({uk }k 0 ) for k N. For the cases studied inthis paper, the constraint function defines linear bounds. Consequently, the evaluation of these constraint functions and theirN 1gradients is trivial. Given an iterate, {uk }k 0, ψ is computed bysolving (7c) marching forwards. uk ψ for k N is computedby the adjoint method (Capolei et al., 2012a,b; Jansen, 2011;Jørgensen, 2007; Sarma et al., 2005a; Suwartadi et al., 2012;Völcker et al., 2011).To solve (12), we use Matlab’s fmincon function (MATLAB, 2011). fmincon provides an interior point and an activeset solver. We use the interior point method since we experienced the lowest computation times with this method. An optimal solution is reported if the KKT conditions are satisfied towithin a relative and absolute tolerance of 10 6 . The currentbest but non-optimal iterate is returned in cases when the optimization algorithm uses more than 200 iterations, the relativechange in the objective function is less than 10 8 , or the relative change in the step size is less than 10 8 . Furthermore, thecost function is normalized to improve convergence. We use 4different initial guesses when running the optimizations. Theseinitial guesses are constant bhp trajectories with the bhp close tothe maximal bhp for the injectors and the bhp close to the minimal bhp for the producers. About half of the simulations endedbecause they exceeded the maximum number of iterations butwithout satisfying the KKT conditions at the specified tolerancelevel. In these cases, the relative changes in the cost functionand step size were of the order of 10 6 . Even if these solutionsdo not reach our specified tolerances for the KKT conditions,the solutions are sufficiently close to optimality to demonstratequalitatively the behavior of the mean-variance (MV) optimization. This closeness to optimality is assessed by re-simulationof some of these scenarios with a tolerance limit of 10 8 . Inthese cases, the optimizer converged to a KKT point in about300 iterations; and we did not observe important differencesin these control trajectories compared to the already computedcontrol trajectories.(9)with t j ta jT s for j 0, 1, . . . , N. We use a piecewise constant representation of the control function in this equidistantmesh, i.e. we approximate the control vector in every subinterval [t j , t j 1 ] by the zero-order-hold parametrizationu(t) u j , u j Rnu , t j 6 t t j 1 , j 0, . . . , N 1(11)(10)The optimizer maximizes the net present value by manipulating the well bhps. A common alternative is to use the injection rates as manipulated variables (Capolei et al., 2012b).The manipulated variables at time period k N are uk bhp{{pbhpi,k }i I , {pi,k }i P } with I being the set of injectors and P being the set of producers. For i I, pbhpi,k is the bhp (bar) in timeperiod k N at injector i. For i P, pbhpi,k is the bhp (bar) atproducer i in time period k N.3.3. Single-Shooting OptimizationWe use a single shooting algorithm for solution of (7)(Ca
formance of production optimization by mean-variance optimization, robust optimization, certainty equivalence optimization, and the reactive strategy. The optimization strategies are simulated in open-loop without f
Bruksanvisning för bilstereo . Bruksanvisning for bilstereo . Instrukcja obsługi samochodowego odtwarzacza stereo . Operating Instructions for Car Stereo . 610-104 . SV . Bruksanvisning i original
Variance formula of a mean for surveys where households are sampling units and persons are elementary units. 5.3.4 Confidence Interval of Mean. For the confidence interval, we need the mean and variance of the mean. The variance of the sample mean is self-weighted, like the mean, as long as each household has the same probability of being selected.
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LÄS NOGGRANT FÖLJANDE VILLKOR FÖR APPLE DEVELOPER PROGRAM LICENCE . Apple Developer Program License Agreement Syfte Du vill använda Apple-mjukvara (enligt definitionen nedan) för att utveckla en eller flera Applikationer (enligt definitionen nedan) för Apple-märkta produkter. . Applikationer som utvecklas för iOS-produkter, Apple .
diction about the mean of yi, but in actuality we are really focused on the scale parameter i. Generally, . 3 Note the linkage between mean and variance The ratio of the mean to the variance is a constant{the same no matter how large or small the mean is. As a result, when the expected value is small{near zero{the variance is small as well. .
