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(from www.halliburton.com)

Optimization of Advanced Well Type and Performance

Louis J. Durlofsky

Department of Petroleum Engineering, Stanford University

ChevronTexaco ETC, San Ramon, CA

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• B. Yeten, I. Aitokhuehi, V. Artus

• K. Aziz, P. Sarma

Acknowledgments

3TAML, 1999

Multilateral Well Types

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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

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101011011010110101111101100010110011010011010...

I1 J1 K1 lxy hz Jn lxy hz

heel toe

main trunk

heel toe

lateral

multilateral well

• Representation allows well type to evolve (Jn 0 generates a lateral)

Encoding of Unknowns for GA

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well

kz

k

kxy

k

z

xy

z

xy

z

y

x

dq

t

l

J

t

l

J

t

l

h

h

h

1

1

1

1

p well

Y

ng

w

o

T

ng

w

o

nC

C

C

C

Q

Q

Q

if

1 1

1

Unknowns Objective Function

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

Nonconventional Well Optimization

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Flowchart for Single Geological Model

evaluatefitness

reservoir sim ulator

0101011101010111110100100111110000101101111000101101011100111101

x1 x2 x3 x4 x5 x6

y 1 y 2

rank based selection

reproduction

ANN

hillclim ber

formchildren

skintransformer

4

1

2

com posepopulation

performa local search

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6

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Objective function f (or fitness):

NPV, cumulative oil

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60.0

80.0

100.0

120.0

140.0

160.0

180.0

200.0

0 10 20 30 40Generation #

Fitn

ess

- NP

V, M

M$

Best Average

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)

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Evolution of Well Types

0%

20%

40%

60%

80%

100%

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39

invalid monobore 1 lateral 2 laterals 3 laterals 4 laterals

(from Yeten et al., 2003)

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?

Nonconventional Well Optimization with Geological Uncertainty

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Optimization over Multiple Realizations

• Find well that maximizes F = < f > + r < f > 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)

Op

tim

i za t

ion

En

gi n

e ( G

A)

{In

div

idu

al} i

F f= < > + r i ii

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Realization #

NP

V (

$)

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

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Realization #

NP

V (

$)

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

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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 #

NP

V (

$)

Comparison of Optimization Results

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attr

ibu

te 1

attribute 2

attr

ibu

te 1

attribute 2

Proxy - Unsupervised Cluster Analysis

fitn

ess

cluster #

• Attributes can be combined into principal components

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

esti

mat

ed f

itn

ess

actual fitness

r = 0.93

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

esti

mat

ed m

ean

fit

nes

s

actual mean fitness

r = 0.97

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• 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”

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Numerical Gradients

• Define cost function J (NPV, cumulative oil)

1

1

0

,N

n n n

n

L x uJ

• Numerically compute J/u

x - dynamical states, u - controls

( ) ( )J J u u J u

u u

• Apply conjugate gradient technique to drive J/u to 0

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Adjoint Procedure

1

1 ( 1) 1

0

, , , N

n n n T n n n n nA

n

L x u g x x uJ

- Lagrange multipliers, x - dynamical states, u - controls, g - reservoir simulation equations

• Optimality requires first variation of JA = 0 (JA = 0):

1 1( 1) 0

n n nT n Tn

n n n

L g g

x x x

( 1) 0n n

T nAn n n

J L g

u u u

optimality criteriaadjoint equations

• Define augmented cost function JA

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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

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Smart Well Model

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

• Applying ECLIPSE multi-segment wells option

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• Sequential restarts applied to determine optimal settings

Optimization Methodology - Fixed Geology

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Impact of Smart Well Control - Example

• Channelized reservoir, 4 controlled branches

• Production at fixed liquid rate with GOR and WOR constraints (three-phase system)

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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)

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Optimized Valve Settings

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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

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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

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History Matching Objective Functions

• Two levels of optimization

– Single parameter facies optimization

– Multivariate permeability-porosity optimization

data observed data, model

))(( )( minimize 2,

]1,0[

obs

jjobsDjD

r

DD

DrDrgD

2,

0 1

( ( ) )minimize ( )

: statistics of and log

i

j obs j

j

D Df

k

αα

α

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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

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Optimization on Known Geology

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

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Dimensionless Increase in Np

• Dimensionless cumulative oil difference, N

N = 0 (no valves result)

N = 1 (known geology result)

valveson geology, nownkp w/valvesgeology, knownp

valvesno geology, nownkp w/valvesmodel targetp

NN

NNN

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Illustration of Incremental Recovery

N =0

N =1

N =0.5

HM with valves

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Optimization with History Matching

• Optimization with history matching gives N =0.94

• Repeating for different initial models: N =0.900.18

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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

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Optimization with History Matching - CM II

0

500

1000

1500

2000

2500

3000

0 200 400 600 800 1000

days

Cu

m.

oil,

MS

TB

Known geol. w/o valves

HM w/valves

Known geol. w/valves

N =0.41

• Repeating for different initial models: N =0.440.27

• Inaccuracy may be due to nonuniqueness of HM

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Optimization over Multiple HM Models

• Use of multiple history-matched models provides significant gains

Number of HM Models N ()

1

3

5

0.44 0.27

0.85 0.16

0.84

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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

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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

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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