Optimization of Advanced Well Type and Performance

Louis J. Durlofsky

(from www.halliburton.com) Department of Petroleum Engineering, Stanford University ChevronTexaco ETC, San Ramon, CA

1

Acknowledgments

•

B. Yeten, I. Aitokhuehi, V. Artus

•

K. Aziz, P. Sarma

2

Multilateral Well Types TAML, 1999

3

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 4

Encoding of Unknowns for GA 10101 10110 1011 010111 1101 1000 101 100110 1001 1010 ...

I1 J1 K1 l xy

q

h z Jn l xy

q

h z heel toe heel toe main trunk lateral multilateral well

• Representation allows well type to evolve (

Jn

 0 generates a lateral) 5

Nonconventional Well Optimization p

                 

l h x h y t h z

q

xy z

        

Unknowns

     

l J

q

t

1

xy

1

z

1 1

            

t l J

q

k z k k xy k

     

q d well

       

Objective Function

f



n Y

 

1

   

1

 

n

 

Q

 

Q Q w o g

   

T n

  

C

 

C C w o g

        

C well

• Objective function can be any simulation output (NPV, cumulative oil) 6

Flowchart for Single Geological Model 1 compose population 0101011101010111 1101001001111100 0010110111100010 1101011100111101 2 evaluate fitness y1 y2

Objective function

f

(or fitness): NPV, cumulative oil

reservoir simulator x1 x2 x3 x4 x5 x6 ANN skin transformer 3 perform a local search hill climber 4 rank based selection 5 reproduction 6 form children

7

Single Well Optimization Example

• Objective: optimum well and production rate that maximizes NPV, subject to GOR, WOR constraints

Optimum well (quad-lateral) 200.0

180.0

160.0

140.0

120.0

100.0

80.0

60.0

0 Best 10 20 Generation # 30 Average 40

8 (from Yeten et al., 2003)

Evolution of Well Types 100% 80% 60% 40% 20% 0% 1 3 5 7 9 11 invalid 13 15 monobore 17 19 1 lateral 21 23 2 laterals 25 27 29 3 laterals 31 33 4 laterals 35 37 39

9 (from Yeten et al., 2003)

Nonconventional Well Optimization with Geological Uncertainty ?

10

Optimization over Multiple Realizations

• Find well that maximizes

F

= <

f

> +

r

s 

< f >

is average fitness of well over

N

risk attitude, s 2 is variance in

f

realizations, over realizations)

r

is • Evaluate each individual (well) for each realization (well

i

, realization

j

)

E in iz (G

F f

i

s

i

11

Risk Neutral (r =0) Optimization (Primary Production, Maximize NPV)

Realization # 12

Risk Averse (r = -0.5) Optimization (Primary Production, Maximize NPV)

Realization # 13

Comparison of Optimization Results Risk neutral attitude (r = 0)

well cost = $ 759,158 expected NPV = $ 3,506,390 std = $ 935,720

Risk averse attitude (r = -0.5)

well cost = $ 1,058,704 expected NPV = $ 3,401,210 std = $ 404,920 Realization # 14

Proxy - Unsupervised Cluster Analysis

• Attributes can be combined into principal components

cluster #

15

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

actual fitness

16

Estimated Mean for All Realization (Primary Production, Monobore Wells) r = 0.97

actual mean fitness

17

www.halliburton.com

18

Smart Well Control: “Reactive” versus “Defensive”

• 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 19

Numerical Gradients

• Define cost function

J

(NPV, cumulative oil)

J

  1

N n

  0

L n



x n

 1 ,

u n



x

- dynamical states,

u

- controls • Numerically compute 

J

/ 

u



J



u

  

u

) 

u

 • Apply conjugate gradient technique to drive 

J

/ 

u

to 0 20

Adjoint Procedure

• Define

augmented

cost function

J A

J A

  1

N n

  0

L n



x n

 1 ,

u n

    1)

g n



x n

 1 ,

x n

,

u n

 

u

- Lagrange multipliers, - controls,

g x

- dynamical states, - reservoir simulation equations • Optimality requires first variation of

J A

= 0 ( d

J A

= 0): 

L n

 1 

x n

   1) 

g n



x n

 

Tn



g n

 1 

x n

adjoint equations

 0 

J



u A n

 

L n



u n

   1) 

g n



u n

 0

optimality criteria

21

Adjoint versus Numerical Gradient Approaches for Optimization

Numerical Gradients

Advantages

• Easily implemented • No simulator source code required Adjoint Gradients

Advantages

• Much faster for large number of wells & updates • Can also be used for HM

Main Drawback

• CPU requirements

Main Drawback

• Adjoint simulator required • Adjoint and numerical gradient procedures developed; implementation of smart well model into

GPRS

underway 22

Smart Well Model

• Numerical gradient approach (Yeten et al., 2002) allows use of existing (commercial) simulator • Applying

ECLIPSE

multi-segment wells option 23

Optimization Methodology - Fixed Geology

• Sequential restarts applied to determine optimal settings 24

Impact of Smart Well Control - Example

• Channelized reservoir, 4 controlled branches • Production at fixed liquid rate with GOR and WOR constraints (three-phase system) 25

Effect of Valve Control on Oil Production Oil rate - uncontrolled case Oil rate - controlled case

• Downhole control provides an increase in cumulative oil production of 47% (from Yeten et al., 2002) 26

Optimized Valve Settings

27

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 Optimization Step Pass # Restart Points New history-matched reservoir model 28

History Matching Procedure

• Facies-based probability perturbation algorithms (Caers, 2003) • Multiple-point geostatistics (training images) • Performs two levels of nonlinear optimization (facies and

k

f ) • History matching based on well pressure, cumulative oil and water cut (for each branch) • Initial models from same training image as “actual” models 29

History Matching Objective Functions

• Two levels of optimization – Single parameter facies optimization minimize

r D

 [ 0 , 1 ]

g

(

r D

)  

j

(

D j

(

r D

) 

D

 model data,

D obs D obs

,

j

) 2  observed data – Multivariate permeability-porosity optimization 0  

i

 1

α

 

j

(

D j



D

) 2

α

f

k

30

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 31

Optimization on Known Geology 3000 2500 2000 1500 1000 500 0 0 500

days

without valves with valves

1000 0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 0 500

days

without valves with valves

1000

• Valves provide ~40% gain in cumulative oil over no-valve base case 32

Dimensionless Increase in N

p

• Dimensionless cumulative oil difference, 

N



N



N p target model w/valves N p known geology, w/valves

 

N p k nown geology, no valves N p k nown geology, n o valves



N



N

= 0 (no valves result) = 1 (known geology result) 33

Illustration of Incremental Recovery 3000 2500 2000 1500 1000 500 0 0



N

=1 

N

=0.5



N

=0

500

days

HM with valves without valves with valves

1000

34

Optimization with History Matching 3000 2500 2000 1500 1000 500 0 0

Known geol. w/o valves HM w/valves Known geol. w/valves

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 0 500

days

1000 500

days

• Optimization with history matching gives 

N

=0.94

• Repeating for different initial models: 

N

=0.90

 0.18

1000

35

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 36

Optimization with History Matching - CM II

3000 2500 2000 1500 1000 500 0 0 

N

=0.41

Known geol. w/o valves HM w/valves Known geol. w/valves 200 400

days

600 800 1000 • Repeating for different initial models: 

N

=0.44

 0.27

• Inaccuracy may be due to nonuniqueness of HM 37

Optimization over Multiple HM Models Number of HM Models

1 3 5 

N (

s

)

0.44  0.27

0.85  0.16

0.84

• Use of multiple history-matched models provides significant gains 38

Effect of Conditioning (on Facies)

Model CM I CM II

Single HM Model



N

w/o HM, w/ cond 

N

w/ HM, w/o cond 0.58 ± 0.17

0.54 ± 0.27

0.90 ± 0.18

0.44 ± 0.27



N

w/HM, w/cond 0.88 ± 0.06

0.64 ± 0.17

• 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) 39

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 40

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 41