Capolei, Andrea

Frydendall, Jan

Jørgensen, John Bagterp

Völcker, Carsten

Conventional recovery techniques enable recovery of 10-50% of the oil in an oil eld. Advances in smart well technology and enhanced oil recovery techniques enable signicant larger recovery. To realize this potential, feedback model-based optimal control technologies are needed to manipulate the injections and oil production such that ow is uniform in a given geological structure. Even in the case of conventional water ooding, feedback based optimal control technologies may enable higher oil recovery than with conventional operational strategies. The optimal control problems that must be solved are large-scale problems and require specialized numerical algorithms. In this paper, we combine a single shooting optimization algorithm based on sequential quadratic programming (SQP) with explicit singly diagonally implicit Runge-Kutta (ESDIRK) integration methods and a continuous adjoint method for sensitivity computation. We demonstrate the procedure on a water ooding example with conventional injectors and producers

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Single Shooting and ESDIRK Methods for adjoint-based optimization of an oil reservoir

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Downloaded from orbit.dtu.dk on: Dec 20, 2017Single Shooting and ESDIRK Methods for adjoint-based optimization of an oil reservoirCapolei, Andrea; Völcker, Carsten; Frydendall, Jan; Jørgensen, John BagterpPublication date:2012Document VersionPublisher's PDF, also known as Version of recordLink back to DTU OrbitCitation (APA):Capolei, A., Völcker, C., Frydendall, J., & Jørgensen, J. B. (2012). Single Shooting and ESDIRK Methods foradjoint-based optimization of an oil reservoir [Sound/Visual production (digital)]. 17th Nordic Process ControlWorkshop, Kongens Lyngby, Denmark, 25/01/2012, http://npcw17.imm.dtu.dk/Single Shooting and ESDIRK Methods for adjoint-basedoptimization of an oil reservoirAndrea Capolei, Carsten Vo¨lcker, Jan Frydendall, John BagterpJørgensenDepartment of Informatics and Mathematical ModelingTechnical University of Denmark17th Nordic Process Control Workshop26th of January, 20121 / 23Outline1 Introduction2 Optimal Control ProblemSingle Shooting Optimization3 ESDIRK integration methods4 Continuous Adjoint Method5 Production Optimization for a Conventional Oil Field6 Conclusions2 / 23IntroductionIncreasing Oil demand0100907080205040306010 Asia Pacific Africa Middle East Europe & Eurasia S. & Cent. America North America85 9590 0500 10Production by regionMillion barrels daily010090708020504030601085 9590 0500 10Consumption by regionMillion barrels daily(from BP statistical review 2011)3 / 23IntroductionAn Offshore Oil Reservoir4 / 23IntroductionClosed-loop reservoir management(after Jansen 2005)5 / 23IntroductionWater Flooding Modeled by a Two-Phase Flow ModelThe mass conservation of water (i ≡ w) andoil (i ≡ o)∂∂tCi(Pi, Si) = −∇ · Fi(Pi, Si) +QiThe mass concentrationsCi = φρi(Pi)SiFluxes through the porous mediumFi = ρi(Pi)ui(Pi, Si)Darcy’s lawui(Pi, Si) = −Kkri(Si)µi(∇Pi − ρi(Pi)g∇z)1 23 4kmin:0.024 Darcy,   kmax:11.312 Darcy  log 10(K)   (Darcy)−1.5−1−0.500.51A general formulation of thetwo-phase flow problemddtg(x(t)) = f(x(t), u(t))6 / 23Optimal Control ProblemContinuous formConsider the continuous-time constrained optimal control problem in theBolza formminx(t),u(t)J = Φˆ(x(tb)) +∫ tbtaΦ(x(t), u(t))dt (1a)subject tox(ta) = x0 (1b)ddtg(x(t))= f(x(t), u(t)), t ∈ [ta, tb], (1c)u(t) ∈ U(t) (1d)x(t) ∈ Rnx is the state vector and u(t) ∈ Rnu is the control vector. Thetime interval I = [ta, tb] as well as the initial state, x0, are assumed to befixed.7 / 23Optimal Control ProblemPath constraintsPath constraintsη(x(t), u(t)) ≥ 0 (2)are included as soft constraints using the following smooth approximationχi(x(t), u(t)) =12(√ηi(x(t), u(t))2 + βi2 − ηi(x(t), u(t))(3)to the exact penalty function max(0,−ηi(x(t))) for i ∈ {1, . . . , nη}.Withthis approximation of the path constraints, the resulting stage cost,Φ(x(t), u(t)), used in (11) consist of the inherent stage cost,Φ˜(x(t), u(t)), and terms penalizing violation of the path constraints (2)Φ(x, u) = Φ˜(x, u) + ‖χ(x, u)‖1,Q1 +12‖χ(x, u)‖22,Q2 (4)8 / 23Optimal Control ProblemSingle Shooting DiscretizationWe introduce the functionψ({uk}N−1k=0 , x0) ={J =∫ tbtaΦ(x(t), u(t))dt+ Φˆ(x(tb)) : x(t0) = x0,ddtg(x(t)) = f(x(t), u(t)), ta ≤ t ≤ tb,u(t) = uk, tk ≤ t < tk+1, k ∈ N = 0, .., N − 1}(5)such that (1) can be approximated with the finite dimensional constrainedoptimization problemminy:={uk}N−1k=0ψ = ψ(y, x0) (6a)s.t. umin ≤ uk ≤ umax k ∈ N (6b)∆umin ≤ ∆uk ≤ ∆umax k ∈ N (6c)ck(uk) ≥ 0 k ∈ N (6d)with N = {0, 1, . . . , N − 1}.9 / 23Optimal Control ProblemSequential Quadratic ProgrammingWe solve the NLP (6) iteratively improving a given estimate yi of thesolution byyi+1 = yi + αipi (7)where αi (0 < α ≤ 1) is determined by a linesearch (LS) strategy based onPowell’s exact l1-merit function. The search direction pi is given by solvingthe KKT solution (pi, λi, µi) of a quadratic approximation to (6)minp12p′H ip′ +∇ψ(yi)′ps.t ∇h(yi)′p = −h(yi)∇c¯(yi) ≥ −c¯(yi)(8)where H i ∈ Rn×n is an approximation for the Hessian ∇2yL of thelagrangian function L(y, λ, µ) = ψ(y)− λh(y)− µc¯(y) which starting froman initial estimate H0, is updated after every step using BFGS.10 / 23ESDIRK integration methodsRunge-Kutta ESDIRK MethodsWe use an embeddedESDIRK method0 0c2 a21 γc3 a31 a32 γ....... . .1 b1 b2 b3 · · · γxn+1 b1 b2 b3 · · · γxˆn+1 bˆ1 bˆ2 bˆ3 · · · bˆs1 Solve implicit systemTi = tn + hnci, i ∈ 2, . . . , sg(Xi) = g(xn) + hns∑j=1aijf(Tj , Xj , u)2 Compute xn+1 = Xs3 Compute the error/tolerance ratioen+1 = g(xn+1)− g(xˆn+1)= hns∑j=1(bj − bˆj)f(Tj , Xj , u)rn+1 =1√nx∥∥∥∥ en+1atol + |g(xn+1)|rtol∥∥∥∥211 / 23ESDIRK integration methodsTemporal step size controller performance(after Vo¨lcker 2010)12 / 23Continuous Adjoint MethodProposition (Gradients based on Continuous Adjoints)Consider the function ψ = ψ({uk}N−1k=0 ;x0) defined by (5).The gradients, ∂ψ/∂uk, may be computed as∂ψ∂uk=∫ tk+1tk(∂Φ∂u− λT ∂f∂u)dt k = 0, 1, . . . , N − 1 (9)in which x(t) is computed by solution of (1b)-(1c) and λ(t) is computedby solution of the adjoint equationsdλTdt∂g∂x+ λT∂f∂x− ∂Φ∂x= 0 (10a)∂Φˆ∂x(x(tb)) + λT (tb)∂g∂x(x(tb)) = 0 (10b)Remember that the dynamic model is given asddtg(x(t)) = f(x(t), u(t))13 / 23Production Optimization for a Conventional Oil FieldTest case for waterflooding optimizationsquared reservoir of size 450m × 450m × 10muniform cartesian grid of 25x25x1 grid blocksno flow boundaries, 4 injectors and 1 producer (rate controlled)We maximize an economic value for different discount factorsb ∈ 0, 0.06, 0.12minx(t),u(t)J(tb) = −NPV(tb) =∫ tbtaΦ˜(x(t), u(t))dtΦ˜ = − 1(1 + b)t/365∑j∈P(ro(1− fw,j)− fw,jrw) qj(t)(11)where fw = λw/(λw + λo), λi = ρikkri/µi, i ∈ {w, o}1 23 4kmin:0.024 Darcy,   kmax:11.312 Darcy  log 10(K)   (Darcy)−1.5−1−0.500.5114 / 23Production Optimization for a Conventional Oil FieldConstraintsThe manipulated variable at time period k ∈ N isuk = {{qw,i,k}i∈I , {qi,k}i∈P} (12)with I being the set of injectors and P being the set of producers.qw,i,k injection rate (m3/day) of water at injector i ∈ Iqi,k total flow rate (m3/day) at producer i ∈ PBound constraints0 ≤ qw,i,k ≤ qmax i ∈ I, k ∈ N (13a)0 ≤ qi,k ≤ qmax i ∈ P, k ∈ N (13b)Rate constraints|qi,k − qi,k−1| ≤ 5 i ∈ I ∪ P, k ∈ N (14a)|qw,i,k − qw,i,k−1| ≤ 5 i ∈ I, k ∈ N (14b)15 / 23Production Optimization for a Conventional Oil FieldConstraints (2)voidage replacement constraint∑i∈Iqi,k =∑i∈Iqw,i,k =∑i∈Pqi,k k ∈ N (15)total injection constraint∑i∈Iqw,i,k = qmax k ∈ N (16)We set qmax = 100 m3/day, tb = 4270 days, Ts = 35 days henceN = 122 periods.Injection of 1.05 pore volume during operation of the reservoirWe consider as a reference case a fixed water injection of 100/4 = 25m3/day from each injector.The prediction horizon tb is optimal in the reference case16 / 23Production Optimization for a Conventional Oil FieldOptimal solutionTIME: 1050 (Day)xy  200 4001002003004000.20.40.60.8TIME: 2135 (Day)xy  200 4001002003004000.20.40.60.8TIME: 3220 (Day)xy  200 4001002003004000.20.40.60.8TIME: 4270 (Day)xy  200 4001002003004000.20.40.60.8(a) Optimal solution (b = 0).TIME: 1050 (Day)xy  200 4001002003004000.20.40.60.8TIME: 2135 (Day)xy  200 4001002003004000.20.40.60.8TIME: 3220 (Day)xy  200 4001002003004000.20.40.60.8TIME: 4270 (Day)xy  200 4001002003004000.20.40.60.8(b) Reference solution.Oil saturations at different times for the optimal solution and the referencesolution.17 / 23Production Optimization for a Conventional Oil FieldCumulative oil and water production for different discountfactors0 500 1000 1500 2000 2500 3000 3500 400000.511.522.53x 105TIME (DAY)CUMULATIVE PRODUCTION (m3 )  opt oilref oilopt waterref water(a) Discount factor b = 0.0 500 1000 1500 2000 2500 3000 3500 400000.511.522.53x 105TIME (DAY)CUMULATIVE PRODUCTION (m3 )  opt oilref oilopt waterref water(b) Discount factor b =0.06.0 500 1000 1500 2000 2500 3000 3500 400000.511.522.53x 105TIME (DAY)CUMULATIVE PRODUCTION (m3 )  opt oilref oilopt waterref water(c) Discount factor b =0.12.Cumulative oil and water productions for different discount factors, b.18 / 23Production Optimization for a Conventional Oil FieldNPV, Water cut, Water fraction0 500 1000 1500 2000 2500 3000 3500 4000051015202530daysMillion dollars  opt. b=0ref.  b=0opt. b=0.06ref.  b=0.06opt. b=0.12ref.  b=0.12(a) NPV.0 500 1000 1500 2000 2500 3000 3500 400000.050.10.150.20.250.30.350.4days  opt. b=0opt. b=0.06opt. b=0.12reference(b) Water cut.0 500 1000 1500 2000 2500 3000 3500 400000.20.40.60.81days  opt. b=0opt. b=0.06opt. b=0.12reference(c) Water fraction (kg wa-ter/kg fluid).The net present value (NPV), water cut (accumulated water production perproduced fluid), and the water fraction as function of time for the scenariosconsidered.19 / 23Production Optimization for a Conventional Oil FieldOptimal input trajectories0 1000 2000 3000 4000050100injector 1 (m3 /day)time t (Days) 0 1000 2000 3000 4000050100injector 2 (m3 /day)time t (Days)0 1000 2000 3000 4000050100injector 3 (m3 /day)time t (Days)  0 1000 2000 3000 4000050100injector 4 (m3 /day)time t (Days)b=0b=0.06b=0.12referenceOptimal water injection rates for different discount factors, b.20 / 23Production Optimization for a Conventional Oil FieldImprovement tableTable: Key indicators for the optimized cases. Improvements are compared to thebase case.b NPV ∆NPV Cum. Oil ∆Oil Cum. water ∆Water106 USD % 105 m3 % 105 m3 %0 28.0 +8.7 3.05 +6.5 0.122 −13.20.06 22.1 +5.6 3.01 +5.2 0.126 −10.50.12 18.3 +4.8 2.98 +4.1 0.129 −8.2b Oil Rec. factor ∆Oil Rec. factor% %-point0 83.7 +5.20.06 82.6 +4.10.12 81.7 +3.221 / 23ConclusionsConclusionsA novel algorithm for large scale optimal control based ona novel formulation of the differential equationsan ESDIRK method for integration of differential equationsthe continuous adjoint method for gradient computationthe SQP method for optimizationthe single shooting principleThis algorithm is applied for production optimization of oil reservoirs. Forthis field, optimal control improves the NPV by 8.7%.Optimal control and nonlinear model predictive control can potentiallyhave a very big impact in oil reservoir management.22 / 23ConclusionsAcknowledgementThe Center for Energy Resources Engineering Consortiumwww.cere.dtu.dkThe Danish Research Council for Technology and Production Sciences, FTP Grant no 274-06-028423 / 23

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Single Shooting and ESDIRK Methods for adjoint-based optimization of an oil reservoir

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