1 (from Optimization of Advanced Well Type and Performance Louis J. Durlofsky Department of Petroleum Engineering, Stanford University ChevronTexaco ETC, San Ramon, CA

2 B. Yeten, I. Aitokhuehi, V. Artus K. Aziz, P. Sarma Acknowledgments

3 TAML, 1999 Multilateral Well Types

4 Optimization of NCW Type and Placement Applying a Genetic Algorithm that optimizes via analogy to Darwinian natural selection GA approach combines “survival of the fittest” with stochastic information exchange Algorithm includes populations with generations that reproduce with crossover and mutation General level of fitness as well as most fit individual tend to increase as algorithm proceeds

I1 J1 K1 l xy  h z Jn l xy  h z heeltoe main trunk heel toe lateral multilateral well Representation allows well type to evolve (Jn  0 generates a lateral) Encoding of Unknowns for GA

6 UnknownsObjective Function Objective function can be any simulation output (NPV, cumulative oil) Nonconventional Well Optimization

7 Flowchart for Single Geological Model Objective function f (or fitness): NPV, cumulative oil

8 Single Well Optimization Example Objective: optimum well and production rate that maximizes NPV, subject to GOR, WOR constraints (from Yeten et al., 2003) Optimum well (quad-lateral)

9 Evolution of Well Types (from Yeten et al., 2003)

10 ? Nonconventional Well Optimization with Geological Uncertainty

11 Optimization over Multiple Realizations Find well that maximizes F = + r   is average fitness of well over N realizations, r is risk attitude,   is variance in f over realizations) Evaluate each individual (well) for each realization (well i, realization j) i

12 Realization # NPV ($) Risk Neutral (r =0) Optimization (Primary Production, Maximize NPV)

13 Realization # NPV ($) Risk Averse (r = -0.5) Optimization (Primary Production, Maximize NPV)

14 Risk averse attitude (r = -0.5) well cost = $ 1,058,704 expected NPV = $ 3,401,210 std = $ 404,920 Risk neutral attitude (r = 0) well cost = $ 759,158 expected NPV = $ 3,506,390 std = $ 935,720 Realization # NPV ($) Comparison of Optimization Results

15 attribute 1 attribute 2 attribute 1 attribute 2 Proxy - Unsupervised Cluster Analysis fitness cluster # Attributes can be combined into principal components

16 Proxy Estimate for a Single Realization (Primary Production, Monobore Wells) estimated fitness actual fitness r = 0.93

17 Estimated Mean for All Realization (Primary Production, Monobore Wells) estimated mean fitness actual mean fitness r = 0.97

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19 Reactive control: adjust downhole settings to react to problems (e.g., water or gas production) as they occur Defensive control: optimize downhole settings to avoid or minimize problems. This requires: –Accurate reservoir description (HM models) –Optimization procedure Optimize using gradients computed numerically or via adjoint procedure Smart Well Control: “Reactive” versus “Defensive”

20 Numerical Gradients Define cost function J (NPV, cumulative oil) Numerically compute  J/  u x - dynamical states, u - controls Apply conjugate gradient technique to drive  J/  u to 0

21 Adjoint Procedure - Lagrange multipliers, x - dynamical states, u - controls, g - reservoir simulation equations Optimality requires first variation of J A = 0 (  J A = 0): optimality criteriaadjoint equations Define augmented cost function J A

22 Adjoint versus Numerical Gradient Approaches for Optimization Numerical Gradients Advantages Easily implemented No simulator source code required Main Drawback CPU requirements Adjoint Gradients Advantages Much faster for large number of wells & updates Can also be used for HM Main Drawback Adjoint simulator required Adjoint and numerical gradient procedures developed; implementation of smart well model into GPRS underway

23 Smart Well Model Numerical gradient approach (Yeten et al., 2002) allows use of existing (commercial) simulator Applying ECLIPSE multi-segment wells option

24 Sequential restarts applied to determine optimal settings Optimization Methodology - Fixed Geology

25 Impact of Smart Well Control - Example Channelized reservoir, 4 controlled branches Production at fixed liquid rate with GOR and WOR constraints (three-phase system)

26 Effect of Valve Control on Oil Production Oil rate - uncontrolled caseOil rate - controlled case Downhole control provides an increase in cumulative oil production of 47% (from Yeten et al., 2002)

27 Optimized Valve Settings

28 Optimization with History Matching Actual geology is unknown (one model selected randomly represents “actual” reservoir and provides “production” data) Update reservoir models based on synthetic history Optimize using current (history-matched) model

29 History Matching Procedure Facies-based probability perturbation algorithms (Caers, 2003) Multiple-point geostatistics (training images) Performs two levels of nonlinear optimization (facies and k-  ) History matching based on well pressure, cumulative oil and water cut (for each branch) Initial models from same training image as “actual” models

30 History Matching Objective Functions Two levels of optimization –Single parameter facies optimization –Multivariate permeability-porosity optimization

31 Channelized Model I Unconditioned 2 facies model, 20 x 20 x 6 grid Quad-lateral well with a valve on each branch –Constant total fluid rate (10 MSTB/D initial liquid rate) –Shut-in well if water cut > 80% OWG flow, M < 1; 4 optimization and HM steps

32 Optimization on Known Geology Valves provide ~40% gain in cumulative oil over no-valve base case

33 Dimensionless Increase in N p Dimensionless cumulative oil difference,  N  N = 0 (no valves result)  N = 1 (known geology result)

34 Illustration of Incremental Recovery  N =0  N =1  N =0.5 HM with valves

35 Optimization with History Matching Optimization with history matching gives  N =0.94 Repeating for different initial models:  N =0.90  0.18

36 Channelized Model II Unconditioned 2 facies model, 20 x 20 x 6 grid Different training image than Channelized Model I, same well and other system parameters

37 Optimization with History Matching - CM II  N =0.41 Repeating for different initial models:  N =0.44  0.27 Inaccuracy may be due to nonuniqueness of HM

38 Optimization over Multiple HM Models Use of multiple history-matched models provides significant gains Number of HM Models  N (  )  

39 Effect of Conditioning (on Facies) Partial redundancy of conditioning and production data reduces impact of conditioning in some cases For CM II, use of 3 conditioned and history matched models gives  N = 0.83  0.10 (~same as w/o cond) Single HM Model

40 Summary Presented genetic algorithm for optimization of nonconventional well type and placement Applied GA under geological uncertainty Developed combined valve optimization – history matching procedure for real-time smart well control Demonstrated that optimization over multiple history-matched models beneficial in some cases

41 Research Directions Developing efficient proxies for optimization of well type and placement under geological uncertainty Implementing adjoint approach (optimal control theory) and multisegment well model into GPRS for determination of valve settings Plan to incorporate additional data (4D seismic) and accelerate history matching procedure