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
101011011010110101111101100010110011010011010...
I1
J1
K1
lxy
heel
q
toe
hz Jn
lxy
heel
main trunk
q
hz
toe
lateral
multilateral well
•
Representation allows well type to evolve (Jn  0
generates a lateral)
5
Nonconventional Well Optimization
Unknowns
  hx 
 
  hy 
 
hz

p  
  l xy 
 
 q 
 t 
 z 
•


1
k
J 
J 

 1 
 k 

l
l
 xy    xy  q d 
well
q 1 
q k 

 1
 k

t
t
 z 
 z 



Objective Function

Y 
1

f 
n



1

i
n 1


 Qo   C o  
   
Qw   C w    C well
Q g  C g  
 n  
T
Objective function can be any simulation output
(NPV, cumulative oil)
6
compose
population
1
Flowchart for Single
Geological Model
0101011101010111
1101001001111100
0010110111100010
1101011100111101
2
evaluate
fitness
y1
x1
reservoir
simulator
x2
x3
y2
x4
x5
x6
ANN
6
Objective function f
(or fitness):
NPV, cumulative oil
form
children
skin
transformer
3
perform
a local search
hill
climber
4
rank based selection
7
5
reproduction
Single Well Optimization Example
• Objective: optimum well and production rate that
maximizes NPV, subject to GOR, WOR constraints
200.0
Fitness - NPV, MM$
180.0
160.0
140.0
120.0
100.0
80.0
Best
Average
60.0
0
Optimum well (quad-lateral)
10
20
Generation #
30
40
8
(from Yeten et al., 2003)
Evolution of Well Types
100%
80%
60%
40%
20%
invalid
(from Yeten et al., 2003)
monobore
1 lateral
2 laterals
3 laterals
4 laterals
39
37
35
33
31
29
27
25
23
21
19
17
15
13
11
9
7
5
3
1
0%
9
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 realizations, r is
risk attitude, s2 is variance in f over realizations)
{Individual}
i
Optimization Engine
(GA)
• Evaluate each individual (well) for each
realization (well i, realization j)
Fi = < f >i + rsi
11
Risk Neutral (r =0) Optimization
NPV ($)
(Primary Production, Maximize NPV)
Realization #
12
Risk Averse (r = -0.5) Optimization
NPV ($)
(Primary Production, Maximize NPV)
Realization #
13
Comparison of Optimization Results
Risk neutral attitude (r = 0)
well cost = $ 759,158
std = $ 935,720
NPV ($)
expected NPV = $ 3,506,390
Risk averse attitude (r = -0.5)
well cost = $ 1,058,704
expected NPV = $ 3,401,210
std = $ 404,920
Realization #
14
attribute
attribute11
Proxy - Unsupervised Cluster Analysis
fitness
attribute
attribute 22
• Attributes can be combined
into principal components
cluster #
15
Proxy Estimate for a Single Realization
(Primary Production, Monobore Wells)
estimated fitness
r = 0.93
actual fitness
16
Estimated Mean for All Realization
(Primary Production, Monobore Wells)
estimated mean fitness
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 
N 1
n
n 1
n
L
x
,
u

 
n 0
x - dynamical states, u - controls
• Numerically compute J/u
J
J (u  u )  J (u )

u
u
• Apply conjugate gradient technique to drive J/u to 0
20
Adjoint Procedure
• Define augmented cost function JA
JA 
N 1
n
n 1
n
T ( n 1)
n
n 1
n
n


L
x
,
u


g
x
,
x
,
u







n 0
 - Lagrange multipliers, x - dynamical states,
u - controls, g - reservoir simulation equations
• Optimality requires first variation of JA = 0 (dJA = 0):
n
n 1
Ln 1
T ( n 1) g
Tn g


0
n
n
n
x
x
x
adjoint equations
n
J A Ln

g
T ( n 1)



0
n
n
n
u
u
u
optimality criteria
21
Adjoint versus Numerical Gradient
Approaches for Optimization
Numerical Gradients
Adjoint Gradients
Advantages
• Easily implemented
• No simulator source
code required
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 g ( rD ) 
rD [ 0,1]
2
(
D
(
r
)

D
)
 j D
obs, j
j
D  model data, Dobs  observed data
– Multivariate permeability-porosity optimization
minimize f (α) 
0  i  1

( D j (α)  Dobs , j ) 2
j
α : statistics of f and log 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
2000
Water cut
Cum. oil, MSTB
2500
1500
1000
without valves
500
with valves
0
0
500
days
1000
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
without valves
with valves
0
500
1000
days
• Valves provide ~40% gain in cumulative oil over
no-valve base case
32
Dimensionless Increase in Np
• Dimensionless cumulative oil difference, N
N 
Np target model w/valves  Np known geology,no valves
Np known geology,w/valves  Np known geology,no valves
N = 0
N = 1
(no valves result)
(known geology result)
33
Illustration of Incremental Recovery
3000
N =1
Cum. oil, MSTB
2500
N =0.5
N =0
2000
1500
1000
HM with valves
without valves
500
with valves
0
0
500
1000
days
34
Optimization with History Matching
3000
2000
Water cut
Cum. oil, MSTB
2500
1500
1000
Known geol. w/o valves
HM w/valves
Known geol. w/valves
500
0
0
500
days
1000
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
500
1000
days
• Optimization with history matching gives N =0.94
• Repeating for different initial models: N =0.900.18
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
N =0.41
Cum. oil, MSTB
2500
2000
1500
1000
Known geol. w/o valves
HM w/valves
500
Known geol. w/valves
0
0
200
400
600
800
1000
days
• 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
N (s)
1
0.44  0.27
3
0.85  0.16
5
0.84
• Use of multiple history-matched models provides
significant gains
38
Effect of Conditioning (on Facies)
Single HM Model
Model
N
w/o HM, w/ cond
N
w/ HM, w/o cond
N
w/HM, w/cond
CM I
0.58 ± 0.17
0.90 ± 0.18
0.88 ± 0.06
CM II
0.54 ± 0.27
0.44 ± 0.27
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