Model-based optimization of oil and gas production

Transcription

1 IPAM Long Program Computational Issues in Oil Field Applications - Tutorials UCLA, March 217 Model-based optimization of oil and gas production Jan Dir Jansen Delft University of Technology j.d.jansen@tudelft.nl IPAM Computational Issues in Oil Field Applications 1

2 Closed-loop reservoir management Noise Controllable input Input System (reservoir, wells & facilities) Output Noise Optimization algorithms Sensors System models Geology, seismics, well logs, well tests, fluid properties, etc. Predicted output Data assimilation algorithms Measured output IPAM Computational Issues in Oil Field Applications 2

3 Notation: time-discretized equations System eqs: States: Parameters: Inputs: Initial conditions: g u, x, x, m 1 T T x p s T T m φ T T u pwell T q x x T well T pressures, saturations permeabilities, porosities well pressures/rates Time interval: 1, 2,, K IPAM Computational Issues in Oil Field Applications 3

4 Production optimization: objective function Simple Net Present Value (NPV) N inj injectors, N prod producers N prod,,, r q r q r q K o o j wp wp j wi wi i j1 i1 t 1 1 b N inj t r = unit price or cost, b = discount factor, = 365 days Flow rates q functions of inputs u or outputs (states) x IPAM Computational Issues in Oil Field Applications 4

5 Production optimization: maximization problem Problem statement: max u1: K subject to u 1: K System equations: Initial conditions: Equality constraints: Inequality constraints: g u, x 1, x x x c u, x d u, x IPAM Computational Issues in Oil Field Applications 5

6 1) Open-loop flooding optimization Noise Controllable input Input System (reservoir, wells & facilities) Output Noise Optimization algorithms Sensors System model Geology, seismics, well logs, well tests, fluid properties, etc. Predicted output Data assimilation algorithms Measured output IPAM Computational Issues in Oil Field Applications 6

7 12-well example (1) 3D reservoir High-permeability channels 8 injectors, rate-controlled 4 producers, BHP-controlled Production period of 1 years 12 wells x 1 x 12 time steps Van Essen et al., 26 => 144 optimization parameters Bound constraints on controls Optimization of monetary value (oil revenues minus water costs) IPAM Computational Issues in Oil Field Applications 7

8 12-well example (2) IPAM Computational Issues in Oil Field Applications 8

9 12-well example (3) IPAM Computational Issues in Oil Field Applications 9

10 Why this wouldn t wor Real wells are sparse and far apart Real wells have more complicated constraints Field management is usually production-focused Long-term optimization may jeopardize short-term profit Production engineers don t trust reservoir models anyway We do not now the reservoir! IPAM Computational Issues in Oil Field Applications 1

11 2) Robust open-loop flooding optimization Noise Controllable input Input System (reservoir, wells & facilities) Output Noise Optimization algorithms Sensors System models Geology, seismics, well logs, well tests, fluid properties, etc. Predicted output Data assimilation algorithms Measured output IPAM Computational Issues in Oil Field Applications 11

12 Robust optimization example 1 realizations Optimize expectation of objective function N 1 r max, u N 1: K r i u1: K mi i1 Van Essen et al., 26 IPAM Computational Issues in Oil Field Applications 12

13 Robust optimization results 3 control strategies applied to set of 1 realizations: reactive control, nominal optimization, robust optimization Van Essen et al., 26 IPAM Computational Issues in Oil Field Applications 13

14 3) Closed-loop flooding optimization Noise Controllable input Input System (reservoir, wells & facilities) Output Noise Optimization algorithms Sensors System models Geology, seismics, well logs, well tests, fluid properties, etc. Predicted output Data assimilation algorithms Measured output IPAM Computational Issues in Oil Field Applications 14

15 Truth Noise Input System (reservoir, wells & facilities) Output Noise Controllable input Optimization algorithms Sensors System models Geology, seismics, well logs, well tests, fluid properties, etc. Predicted output Data assimilation algorithms Measured output IPAM Computational Issues in Oil Field Applications 15

16 Closed-loop optimization NPV and contributions from water & oil production 1 month 1 year 2 years 4 years 1.5 x x NPV, $ Discounted water costs, $ Discounted oil revenues, $ open-loop reactive IPAM Computational Issues in Oil Field Applications 16

17 Optimization techniques Global versus local Gradient-based versus gradient-free Constrained versus non-constrained Classical versus non-classical (simulated annealing, particle swarms, etc.) We use optimal control theory or adjoint-based optimization Has been proposed for history matching (Chen et al. 1974, Chavent et al. 1975, Li, Reynolds and Oliver 23) and for flooding optimization (Ramirez 1987, Asheim 1988, Virnovsi 1991, Zairov et al. 1996, Sudaryanto and Yortsos, 2, Brouwer and Jansen 24, Sarma et al. 24) IPAM Computational Issues in Oil Field Applications 17

18 Optimal control theory, summary Gradient based optimization technique local optimum Gradients of objective function with respect to controls obtained from adjoint equation Gradients can be used with steepest ascent, quasi Newton, or trust-region methods Results in dynamic control strategy, i.e. controls change over time Computational effort independent of number of controls Output constraints not trivial; various techniques used Implementation is code-intrusive IPAM Computational Issues in Oil Field Applications 18

19 Adjoint-Based Optimization Part 1 - Theory IPAM Computational Issues in Oil Field Applications 19

20 Unconstrained optimization (1D) 2 u 2u u u 4u Objective function u 2 4 minimum 2 u IPAM Computational Issues in Oil Field Applications 2

21 Unconstrained optimization (2D) u2u u2 u u u 1 2 T u1 T 4u1 4u2 u2 u 2 4 minimum 4 2 u Objective function u 2 u 1 u 1 Contour lines IPAM Computational Issues in Oil Field Applications 21

22 Constrained optimization (elimination) 2 2 u u 1 u2 u 2 s.t. c u u u 2 u u 2 1 4u 2.4u u1.3 8u 2.4 u u1 8 minimum u 2 u 1 Contour lines IPAM Computational Issues in Oil Field Applications 22

23 Constrained optimization (Lagrange multipliers) 2 2 u u 1 u2 u 2 s.t. c u u u u c u u u1 u2 u1u2.6 u T u 4u1 4u2 u1u2.6 u second-order conditions more complex u u 1 2 T IPAM Computational Issues in Oil Field Applications 23

24 Lagrange multipliers interpretation (a) Recall elimination: 2 2 u u 1 u2 u 2 s.t. c u u u.6 u 2 1 u 4u 2.4u What if u 2 cannot be expressed in u 1 or v.v.? Consider the total differential: d u du u u u But how do we compute u u? IPAM Computational Issues in Oil Field Applications 24

25 Lagrange multipliers interpretation (b) Consider constraint Expressed in differential form: Can be rewritten as c u Implicit differentiation! c u, u 1 2 c u u u2 1 u 2 c c u u u IPAM Computational Issues in Oil Field Applications 25

26 Lagrange multipliers interpretation (c) d u du u u u 2 Given and u 2 c c u u u we can now write which, in an optimum, can also be written as 1 c c u u u u d c c du u u u u d du 1 IPAM Computational Issues in Oil Field Applications 26

27 Lagrange multipliers interpretation (d) c c If we have u u u u 1 we can also derive that IPAM Computational Issues in Oil Field Applications c c u u u u c c u1 u1 u2 u2 c u u Use of Lagrange multipliers = implicit differentiation T

28 Bac to the real thing: Production optimization Problem statement: max u1: K subject to u 1: K System equations: Initial conditions: Equality constraints: Inequality constraints: g u, x 1, x x x c u, x d u, x As a first step: disregard constraints c and d IPAM Computational Issues in Oil Field Applications 28

29 Gradient with implicit differentiation? What we are looing for: d x du u x u K j j j j Effect of u on all subsequent time steps Contributions from time steps K x x x x x x u x x x x u j j j1 2 1 j1 j2 1 Requires a lot of implicit differentiation IPAM Computational Issues in Oil Field Applications 29

30 Gradient with Lagrange multipliers Adjoin constraints to objective function: Modified objective function, x K T 1: K, : K, : K 1 1 T λ g u, x 1, x u x λ λ x x u Proceed as before: tae first derivatives w.r.t. all independent variables and equate them to zero (i.e. force optimality conditions) g g 1 Note that we can write: (index shift) x x 1 IPAM Computational Issues in Oil Field Applications 3

31 IPAM Computational Issues in Oil Field Applications 31 Optimality conditions (1) 1, 2,, T T K g λ u u u 1 1 T T T g λ λ x x 1 1 1, 2,, 1 T T T K g g λ λ x x x x K T T K K K K K x g λ x x 1: : : 1 1 1,,,,, K T K K K T u y u x λ λ x x λ g u x x

32 Optimality conditions (2) λ T x x T T, y K 1: K, : K, : K 1 1 T λ g u, x 1, x u x λ λ x x u λ T 1 T g u, x, x 1,2,, K (Just recovers the initial conditions and system equations) The optimality conditions form a joint set of equations for the unnowns u1: K, x: K, λ: K Can in theory be solved simultaneously (Wathen et al.) but are usually treated sequentially. IPAM Computational Issues in Oil Field Applications 32

33 Solving the resulting equations (1) λ λ x x x T T g u, x, x x T T 1 1: K Running the simulator. (Requires g x ) Initial guess! IPAM Computational Issues in Oil Field Applications 33

34 Solving the resulting equations (1) λ λ x x x T T g u, x, x x T T 1 1: K Running the simulator. (Requires g x ) x K x K K g T K T λ K xk IPAM Computational Issues in Oil Field Applications 34

35 Solving the resulting equations (1) λ λ x x x T T g u, x, x x T T 1 1: K Running the simulator. (Requires g x ) x K T g K K λk xk xk T λ K Final condition x x T g g λ λ λ 1 1 K1:1 x x T T λ g λ λ 1 1 x Bacward integration (linear) IPAM Computational Issues in Oil Field Applications 35

36 Solving the resulting equations (2) u u λ T g u T??? Usually not! IPAM Computational Issues in Oil Field Applications 36

37 IPAM Computational Issues in Oil Field Applications 37 Solving the resulting equations (2) K j j j j d d x u y u u!!! d d u u Just what we need Can now be used, e.g., in steepest ascent: 1 T i i i d d u u u Recall T g λ u u u

38 Summary adjoint-based optimization Adjoint ~ implicit differentiation Computational effort independent of number of controls Gradient-based optimization local optimum Constraint handling: GRG, lumping, SQP, augmented Lagrangian, ; not trivial Beautiful, but code-intrusive and requires lots of programming => automatic differentiation Available in Eclipse (limited functionality), AD-GPRS, MRST, proprietary simulators Alternatives: ensemble methods (EnOpt, StoSAG), streamline-based methods, non classical methods (particle swarm, etc.; often in combination with proxies to reduce computational effort) IPAM Computational Issues in Oil Field Applications 38

39 Adjoint-Based Optimization Part 2 - Examples IPAM Computational Issues in Oil Field Applications 39

40 Classic example; smart horizontal wells 45 x 45 grid blocs 45 inj. & prod. segments 5 1 permeability field 1log() [m 2 ] p wf, q t at segments nown PV injected, q inj = q prod oil price r o = 8 $/m water costs r w = 2 $/m 3 discount rate b = % Brouwer and Jansen, 24, SPEJ IPAM Computational Issues in Oil Field Applications 4

41 Results; conventional production 8 water, oil and liquid production rates (m3/d) as function of time rates [m3/d] cum time [d] cumulative water, oil and liquid production (m3) as function of time cum time [d] cum. production [m3] 5 x 15 ref 5 wat 1 ref liq 15 ref oil 2 opt wat 25 opt wat 3 opt oil Equal pressures in all injector/producer segments IPAM Computational Issues in Oil Field Applications 41

42 Results; rate-constrained (1) Conventional (equal pressure in all segments, no control) Best possible (identical total rates, no pressure constraints) IPAM Computational Issues in Oil Field Applications 42

43 Results; rate-constrained (2) 8 water, oil and liquid production rates (m3/d) as function of time NPV +6% Production + 41% cum oil - 45% cum water rates [m3/d] cum. production [m3] cum time [d] cumulative water, oil and liquid production (m3) as function of time 5 x ref wat ref liq ref oil opt wat opt wat opt oil cum time [d] IPAM Computational Issues in Oil Field Applications 43

44 Pressure-constrained operation Limited energy available Total injection/production rate dependent on number of active wells IPAM Computational Issues in Oil Field Applications 44

45 Results: pressure-constrained 8 water, oil and liquid production rates (m3/d) as function of time Improvement in NPV +53% Production +16% cum oil -77% cum water Injection -32% cum water rates [m3] cum. production [m3] cum time [d] cumulative water, oil and liquid production (m3) as function of time 5 x ref wat ref liq ref oil opt wat opt liq opt oil cum time [d] IPAM Computational Issues in Oil Field Applications 45

46 Optimum valve-settings (1) optimum valve-position for injector segment 12 as function of time step 1 valve-setting optimum valve-position for injector segment 12 as function of time step inj. valve setting vs. time for all wells 1 prod. valve setting vs. time for all wells 1 inject segm time step (n) well number well number cum time [yr] cum time [yr] Bang-bang (on-off) solution Necessary condition: linear controls, linear constraints IPAM Computational Issues in Oil Field Applications 46

47 Optimum valve-settings (2) inj. valve setting vs. time for all wells 1 prod. valve setting vs. time for all wells 1 inj. valve setting vs. time for all wells 1 prod. valve setting vs. time for all wells well number well number well number well number cum time [yr] cum time [yr] cum time [yr] cum time [yr] All the action is around the heterogeneities IPAM Computational Issues in Oil Field Applications 47

48 Optimum valve settings (3) sw at 2 days sw at 12 days sw at 129 days sw at 199 days sw at 272 day s sw at 386 days sw at 63 days Streas act as well extensions Presence of heterogeneities essential for optimization IPAM Computational Issues in Oil Field Applications 48

49 Optimum valve-settings (4) inj. valve setting vs. time for all wells 1 prod. valve setting vs. time for all wells valves in injector well number well number valves in producer cum time [yr] cum time [yr] No need for 45 segments per well IPAM Computational Issues in Oil Field Applications 49

50 St. Joseph field re-development case Objective: to determine the value of down-hole control in planned water injectors, in terms of incremental cumulative oil production Maximum number of ICVs: 5 Water injection rate: 1, bbl/d per well Trajectory of water injector fixed Optimum number of ICVs? Optimum configuration of perforation zones? Optimum operation of the ICVs? Van Essen et al., 21, SPEREE IPAM Computational Issues in Oil Field Applications 5

51 Pilot study on sector model Strongly layered structure Very limited vertical communication Dips approximately 2º 21,99 active grid blocs Dimensions 16m x 5m x 45m No aquifer support 1 gas injection well 1 (planned) water injection well 7 production wells in sector IPAM Computational Issues in Oil Field Applications 51

52 Smart water injection well Properties Fixed flow rate of 1, bbl/d Fixed location and trajectory Horizontal section perforated Lift table captures pressure drop Variables Number of ICVs Length of the perforation zones Operation of ICVs Controls: dh multipliers IPAM Computational Issues in Oil Field Applications 52

53 Base case No control All dh multipliers in 12 layers equal to 1 Water injection into each layer result of permeability, pressure difference, etc. Performance quantified in terms of cumulative oil production Also water injection rate into each zone is determined Zones B, C, D and E No injection in A A B C D E IPAM Computational Issues in Oil Field Applications 53

54 Base case results 15 x 16 Cumulative production data oil production water production 1 8 Injection per group Volume [stb] Rate [stb/day] 6 4 Group B 2 Group C Group D Group E Van Essen et al., 21, SPEREE Cumulative oil production: MMstb IPAM Computational Issues in Oil Field Applications 54

55 Full 12 zone control ( technical limit ) 15 x 16 Cumulative production data 1 Injection per group 8 Volume [stb] 1 5 oil production, base case water production, base case oil production, full standard 12 control 4-group control water production, full standard 12 control 4-group control Rate [stb/day] 6 4 Group B 2 Group C Group D Group E Cumulative oil production: MMstb Increase of 11.7% (1.35 MMstb) IPAM Computational Issues in Oil Field Applications 55

56 Standard 4-group control (geological insight) 15 x 16 Cumulative production data 1 Injection per group 8 Volume [stb] 1 5 oil production, base case water production, base case oil production, standard 4-group control water production, standard 4-group control Rate [stb/day] 6 4 Group B 2 Group C Group D Group E Cumulative oil production: 12.4 MMstb Increase of 8.1% (.93 MMstb) IPAM Computational Issues in Oil Field Applications 56

57 Alternative 4-group control (optimal grouping) 15 x 16 Cumulative production data 1 Injection per group 8 Volume [stb] 1 5 oil production, base case water production, base case oil production, alternative 4-group control water production, alternative 4-group control Rate [stb/day] 6 4 Group B* 2 Group C* Group D* Group E* Cumulative oil production: MMstb Increase of 1.% (1.15 MMstb) IPAM Computational Issues in Oil Field Applications 57

58 Lin with short-term optimization Noise Controllable input Input System (reservoir, wells & facilities) Output Noise Optimization algorithms Sensors System models Geology, seismics, well logs, well tests, fluid properties, etc. Predicted output Data assimilation algorithms Measured output IPAM Computational Issues in Oil Field Applications 58

59 Life-cycle optimization vs. reactive control (1) IPAM Computational Issues in Oil Field Applications 59

60 Life-cycle optimization vs. reactive control (2) IPAM Computational Issues in Oil Field Applications 6

61 Life-cycle optimization vs. reactive control (3) Life-cycle optimization attractive for reservoir engineers Increased NPV due to improved sweep efficiency injector 1 injector 2 injector 3 injector 4 Net Present Value - No Discounting flow rate [bbl/d] flow rate [bbl/d] time [year] injector 5 time [year] producer 1 flow rate [bbl/d] flow rate [bbl/d] time [year] injector 6 time [year] producer 2 flow rate [bbl/d] flow rate [bbl/d] time [year] injector 7 time [year] producer 3 flow rate [bbl/d] flow rate [bbl/d] time [year] injector 8 time [year] producer 4 Revenues [M$] short term horizon -5% +1% flow rate [bbl/d] time [year] flow rate [bbl/d] time [year] flow rate [bbl/d] time [year] flow rate [bbl/d] time [year] time [year] Reactive Control Optimal Control Van Essen et al., 211, SPEJ Not so attractive from production engineering point of view Decreased short term production Erratic behavior of optimal operational strategy IPAM Computational Issues in Oil Field Applications 61

62 Hierarchical optimization Tae production objectives into account by incorporating them as additional optimization criteria: Formal solution: Order objectives according to importance Optimize objectives sequentially Optimality of upper objective constrains optimization of lower one Only possible if there are redundant degrees of freedom in input parameters after meeting primary objective IPAM Computational Issues in Oil Field Applications 62

63 Objective function with ridges IPAM Computational Issues in Oil Field Applications 63

64 Example: Hierarchical optimization using nullspace approach (1) 3D reservoir 8 injection / 4 production wells Period of 1 years Producers at constant BHP Rates in injectors optimized Primary objective: undiscounted NPV over the life of the field Secondary objective: NPV with very high discount factor (25%) to emphasize importance of short term production IPAM Computational Issues in Oil Field Applications 64

65 Example: Hierarchical optimization using nullspace approach (2) Optimization of secondary objective function - constrained to null-space of primary objective Optimization of secondary objective function - unconstrained 32 Secondary Objective Function 48 Primary Objective Function 32 Secondary Objective Function 48 Primary Objective Function Net Present Value - Discounted [M$] % Net Present Value - Undiscounted [M$] % Net Present Value - Discounted [M$] % Net Present Value - Undiscounted [M$] % Iterations Iterations Iterations Iterations Van Essen et al., 211, SPEJ IPAM Computational Issues in Oil Field Applications 65

66 Example: Hierarchical optimization using nullspace approach (3) NPV over Time - Undiscounted [1 6 $] * value of objective function J 1 resulting from u. 5 ~ * value of objective function J 1 resulting from u value of objective function J ~ 1 resulting from u time [days] IPAM Computational Issues in Oil Field Applications 66

67 Observability, controlability, identifiability input (p wf,q t ) System model state (p,s) parameters (,φ, ) output (p wf,q w,q o ) Controlability of a dynamic system is the ability to influence the states through manipulation of the inputs. Observability of a dynamic system is the ability to determine the states through observation of the outputs. Identifiability of a dynamic system is the ability to determine the parameters from the input-output behavior. All very limited for reservoir simulation models! Zandvliet, M. et al., 28: Computational Geosciences 12 (4) Van Doren, J.F.M., et al. 213: Computational Geosciences 17 (5) IPAM Computational Issues in Oil Field Applications 67

68 Model based optimization conclusions Well control optimization : Adjoint-based techniques wor well; constraints, regularization, storage, efficiency, still to be improved Alternatives: gradient-free, particle swarms, EnOpt, StoSAG Controllability very limited. Increased by heterogeneities Well location optimization (not discussed): Gradient-free seems to wor best Combination with well control optimization Field implementation: Well control optimization: none reported Acceptance will require combi with short-term optimization Computer-assisted history matching: thriving! Well location/trajectory optimization: up and coming! Advisory mode tools for discussion IPAM Computational Issues in Oil Field Applications 68

69 References adjoint-based optimization (1) Review paper (with additional references) Jansen, J.D., 211: Adjoint-based optimization of multiphase flow through porous media a review. Computers and Fluids 46 (1) DOI: 1.116/j.compfluid Early use in history matching Chavent, G., Dupuy, M. and Lemonnier, P., 1975: History matching by use of optimal theory. SPE Journal 15 (1) DOI: /4627-PA. Chen, W.H., Gavalas, G.R. and Wasserman, M.L., 1974: A new algorithm for automatic history matching. SPE Journal 14 (6) DOI: /4545-PA. Li, R., Reynolds, A.C., and Oliver, D.S., 23: History matching of three-phase flow production data. SPE Journal 8 (4): DOI: /87336-PA. Early use in flooding optimization Ramirez, W.F., 1987: Application of optimal control theory to enhanced oil recovery, Elsevier, Amsterdam. Asheim, H., 1988: Maximization of water sweep efficiency by controlling production and injection rates. Paper SPE presented at the SPE European Petroleum Conference, London, UK, October DOI: /18365-MS. Virnovsi, G.A., 1991: Water flooding strategy design using optimal control theory, Proc. 6th European Symposium on IOR, Stavanger, Norway, IPAM Computational Issues in Oil Field Applications 69

70 References adjoint-based optimization (2) Zairov, I.S., Aanonsen, S.I., Zairov, E.S., and Palatni, B.M., 1996: Optimization of reservoir performance by automatic allocation of well rates. Proc. 5th European Conference on the Mathematics of Oil Recovery (ECMOR V), Leoben, Austria. Sudaryanto, B. and Yortsos, Y.C., 2: Optimization of fluid front dynamics in porous media using rate control. Physics of Fluids 12 (7) DOI: 1.163/ TU Delft series Brouwer, D.R. and Jansen, J.D., 24: Dynamic optimization of water flooding with smart wells using optimal control theory. SPE Journal 9 (4) DOI: /78278-PA. Van Doren, J.F.M., Marovinović, R. and Jansen, J.D., 26: Reduced-order optimal control of waterflooding using POD. Computational Geosciences 1 (1) DOI: 1.17/s Zandvliet, M.J., Bosgra, O.H., Van den Hof, P.M.J., Jansen, J.D. and Kraaijevanger, J.F.B.M., 27: Bang-bang control and singular arcs in reservoir flooding. Journal of Petroleum Science and Engineering 58, DOI: 1.116/j.petrol Lien, M., Brouwer, D.R., Manseth, T. and Jansen, J.D., 28: Multiscale regularization of flooding optimization for smart field management. SPE Journal 13 (2) DOI: /99728-PA. Zandvliet, M.J., Handels, M., Van Essen, G.M., Brouwer, D.R. and Jansen, J.D., 28: Adjoint-based well placement optimization under production constraints. SPE Journal 13 (4) DOI: /15797-PA. IPAM Computational Issues in Oil Field Applications 7

71 References adjoint-based optimization (3) Van Essen, G.M., Zandvliet, M.J., Van den Hof, P.M.J., Bosgra, O.H. and Jansen, J.D., 29: Robust waterflooding optimization of multiple geological scenarios. SPE Journal 14 (1) DOI: /12913-PA. Van Essen, G.M., Jansen, J.D., Brouwer, D.R. Douma, S.G., Zandvliet, M.J., Rollett, K.I. and Harris, D.P., 21: Optimization of smart wells in the St. Joseph field. SPE Reservoir Evaluation and Engineering 13 (4) DOI: / PA. Van Essen, G.M., Van den Hof, P.M.J. and Jansen, J.D., 211: Hierarchical long-term and short-term production optimization. SPE Journal 16 (1) DOI: / PA. Farshbaf Zinati, F., Jansen, J.D. and Luthi, S.M., 212: Estimating the specific productivity index in horizontal wells from distributed pressure measurements using an adjoint-based minimization algorithm. SPE Journal 17 (3) DOI: / PA. Namdar Zanganeh, M., Kraaijevanger, J.F.B.M., Buurman, H.W., Jansen, J.D., Rossen, W.R., 214: Challenges in adjoint-based optimization of a foam EOR process. Computational Geosciences 18 (3-4) DOI: 1.17/s de Moraes R.J., Rodrigues, J.R.P., Hajibeygi, H. and Jansen, J.D., 217: Multiscale gradient computation for subsurface flow models. Journal of Computational Physics. Published online. DOI: 1.116/j.jcp IPAM Computational Issues in Oil Field Applications 71

72 References adjoint-based optimization (4) Computational aspects Sarma, P., Aziz, K. and Durlofsy, L.J., 25: Implementation of adjoint solution for optimal control of smart wells. Paper SPE presented at the SPE Reservoir Simulation Symposium, Houston, USA, 31 January 2 February. DOI: / MS. Han, C., Wallis, J., Sarma, P. et al., 213: Adaptation of the CPR preconditioner for efficient solution of the adjoint equation. SPE Journal 18 (2) DOI: org/1.2118/1413-pa. Algebraic formulation Rodrigues, J.R.P., 26: Calculating derivatives for automatic history matching. Computational Geosciences 1 (1) DOI: 1.17/s Kraaijevanger, J.F.B.M., Egberts, P.J.P., Valstar, J.R. and Buurman, H.W., 27: Optimal waterflood design using the adjoint method. Paper SPE presented at the SPE Reservoir Simulation Symposium, Houston, USA, February. DOI: / MS. Constraint handling De Montleau, P., Cominelli, A., Neylon, K. and Rowan, D., Pallister, I., Tesaer, O. and Nygard, I., 26: Production optimization under constraints using adjoint gradients. Proc. 1th European Conference on the Mathematics of Oil Recovery (ECMOR X), Paper A41, Amsterdam, The Netherlands, September 4-7. IPAM Computational Issues in Oil Field Applications 72

73 References adjoint-based optimization (5) Sarma, P., Chen, W.H. Durlofsy, L.J. and Aziz, K., 28: Production optimization with adjoint models under nonlinear control-state path inequality constraints. SPE Reservoir Evaluation and Engineering 11 (2) DOI: /99959-PA. Suwartadi, E., Krogstad, S. & Foss, B., 212: Nonlinear output constraints handling for production optimization of oil reservoirs. Computational Geosciences 16 (2) DOI 1.17/s Kourounis, D., Durlofsy, L.J., Jansen, J.D. and Aziz, K., 214: Adjoint formulation and constraint handling for gradient-based optimization of compositional reservoir flow. Computational Geosciences 18 (2) DOI: 1.17/s Kourounis, D. and Schen, O., 215: Constraint handling for gradient-based optimization of compositional reservoir flow. Computational Geosciences DOI:1.17/s Closed-loop reservoir management Jansen, J.D., Brouwer, D.R., Nævdal, G. and van Kruijsdij, C.P.J.W., 25: Closed-loop reservoir management. First Brea, January, 23, Naevdal, G., Brouwer, D.R. and Jansen, J.D., 26: Waterflooding using closed-loop control. Computational Geosciences 1 (1) DOI: 1.17/s Sarma, P., Durlofsy, L.J., Aziz, K., Chen, W.H., 26: Efficient real-time reservoir management using adjoint-based optimal control and model updating. Computational Geosciences 1 (1) DOI: 1.17/s z. IPAM Computational Issues in Oil Field Applications 73

74 References adjoint-based optimization (6) Jansen, J.D., Bosgra, O.H. and van den Hof, P.M.J., 28: Model-based control of multiphase flow in subsurface oil reservoirs. Journal of Process Control 18, DOI: 1.116/j.jprocont Sarma, P., Durlofsy, L.J. and Aziz, K., 28: Computational techniques for closed-loop reservoir modeling with application to a realistic reservoir. Petroleum Science and Technology 26 (1 & 11) DOI: 1.18/ Jansen, J.D., Douma, S.G., Brouwer, D.R., Van den Hof, P.M.J., Bosgra, O.H. and Heemin, A.W., 29: Closed-loop reservoir management. Paper SPE presented at the SPE Reservoir Simulation Symposium, The Woodlands, USA, 2-4 February. DOI: /11998-MS. Wang, C., Li, G. and Reynolds, A.C., 29: Production optimization in closed-loop reservoir management. SPE Journal 14 (3) DOI: /1985-PA. Foss, B. and Jensen, J.P., 21: Performance analysis for closed-loop reservoir management. SPE Journal 16 (1) DOI: / PA. Chen, C., Li, G. and Reynolds, A.C., 212: Robust constrained optimization of short- and long-term net present value for closed-loop reservoir management. SPE Journal 17 (3) DOI: / PA. Bushtynov, V., Volov, O., Durlofsy, L.J. and Aziz, K., 215: Comprehensive framewor for gradient-based optimization in closed-loop reservoir management. Computational Geosciences 19 (4) DOI:1.17/s IPAM Computational Issues in Oil Field Applications 74

75 Acnowledgments Colleagues and students of TU Delft Department of Geoscience and Engineering TU Eindhoven (TUE) Department of Electrical Engineering TU Delft Delft Institute for Applied Mathematics TNO Built Environment and Geosciences Especially for the optimization results presented in this tutorial: Prof. Paul Van den Hof (TUE), Prof. Arnold Heemin (TUD), and (former) PhD students Roald Brouwer, Maarten Zandvliet and Gijs van Essen Sponsors: Shell (Recovery Factory program), ENI, Petrobras, Statoil (ISAPP program) IPAM Computational Issues in Oil Field Applications 75