multi-objective optimisation techniques in reservoir ... christie.pdf · in reservoir simulation...
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![Page 1: Multi-Objective Optimisation Techniques in Reservoir ... Christie.pdf · in Reservoir Simulation Mike Christie Heriot-Watt University . Outline •Introduction ... 0 50 100 150 200](https://reader034.vdocuments.us/reader034/viewer/2022042110/5e8b4c9b95d35119f018ba26/html5/thumbnails/1.jpg)
Multi-Objective Optimisation Techniques in Reservoir Simulation
Mike Christie
Heriot-Watt University
![Page 2: Multi-Objective Optimisation Techniques in Reservoir ... Christie.pdf · in Reservoir Simulation Mike Christie Heriot-Watt University . Outline •Introduction ... 0 50 100 150 200](https://reader034.vdocuments.us/reader034/viewer/2022042110/5e8b4c9b95d35119f018ba26/html5/thumbnails/2.jpg)
Outline
• Introduction
• Stochastic Optimisation
• Model Calibration
• Forecasting
• Reservoir Optimisation
• Summary
![Page 3: Multi-Objective Optimisation Techniques in Reservoir ... Christie.pdf · in Reservoir Simulation Mike Christie Heriot-Watt University . Outline •Introduction ... 0 50 100 150 200](https://reader034.vdocuments.us/reader034/viewer/2022042110/5e8b4c9b95d35119f018ba26/html5/thumbnails/3.jpg)
Outline
• Introduction
• Stochastic Optimisation
• Model Calibration
• Forecasting
• Reservoir Optimisation
• Summary
![Page 4: Multi-Objective Optimisation Techniques in Reservoir ... Christie.pdf · in Reservoir Simulation Mike Christie Heriot-Watt University . Outline •Introduction ... 0 50 100 150 200](https://reader034.vdocuments.us/reader034/viewer/2022042110/5e8b4c9b95d35119f018ba26/html5/thumbnails/4.jpg)
Mathematics of Flow in Porous Media
• Conservation of Mass
• Conservation of Momentum
– replaced by Darcy’s law
• Conservation of Energy
– most processes isothermal
• Equation of State
kp
xv
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• Parabolic equation for pressure
• Hyperbolic equation for saturation
( ) ( ). ( ) ro rw
o w
k S k Spc k p
t
x
. 0o i o g i g
o i o g i g
x S y Sx y
t
x v v
Equations governing flow
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Data Collection
?
?k
x
x
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Model Calibration: Teal South
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0 200 400 600 800 1000 1200 1400
time (days)
simulated
observed
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time (days)
simulated
observed
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0 200 400 600 800 100012001400
time (days)
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observed
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Range of Possible Values for Unknown Parameters
log(kh1)
log(kh2)
boundary
porosity
log(kv/kh1)
log(kv/kh1)
log(cr)
aquifer strength
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Outline
• Introduction
• Stochastic Optimisation
• Model Calibration
• Forecasting
• Reservoir Optimisation
• Summary
![Page 10: Multi-Objective Optimisation Techniques in Reservoir ... Christie.pdf · in Reservoir Simulation Mike Christie Heriot-Watt University . Outline •Introduction ... 0 50 100 150 200](https://reader034.vdocuments.us/reader034/viewer/2022042110/5e8b4c9b95d35119f018ba26/html5/thumbnails/10.jpg)
Particle Swam Optimization (PSO)
• A swarm intellegence algorithm (Kennedy & Eberhart,1995).
• Particles are points in parameter space.
• Particles move based on their own experience and that of the swarm.
• PSO equations
− r1, r2 are random vectors
− w is the inertial weight
− c1, c2 are the cognition and social acceleration components
1
1 1 2 2
1 1
k k best k k k
i i i i best i
k k k
i i i
v v c r p x c r g x
x x v
w
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Dominance and Pareto Optimality
Solution 𝐱𝟏 dominates solution 𝐱𝟐, if :
1. 𝐱𝟏 is no worse than 𝐱𝟐 in all objectives, and
2. 𝐱𝟏 is strictly better than 𝐱𝟐 in at least one objective
Ob
j 2
Obj 1
Pareto front
Image of Pareto optimal set in objective space
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1
1 1 2 2
1 1
k k best k k k
i i i i best i
k k k
i i i
v v c r p x c r g x
x x v
w
• MOPSO equations
− r1, r2 are random vectors
− w is the inertial weight
− c1, c2 are the cognition and social acceleration components
• Pbest and Gbest now sampled from Pareto archive
MO Particle Swam Optimization (PSO)
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Why Use Multi-Objective?
• Sum of objectives – limited exploration of Pareto front
– Example minimising two objectives
Objective sum Multi-Objective
Figure from Hajizadeh, PhD Thesis (Chapter 6, Figure12), Heriot-Watt, 2011
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Outline
• Introduction
• Stochastic Optimisation
• Model Calibration
• Forecasting
• Reservoir Optimisation
• Summary
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Model Calibration
• Called ‘History Matching’ in oil business
• Synthetic example
– IC Fault Model
• Real example
– Zagadka Field
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IC Fault Model
*
* Data from Z. Tavassoli, Jonathan N. Carter, and Peter R. King,
Imperial College, London
• Synthetic 2D Model
• 2 Wells: 1 Inj, 1 Prod
• 6 Layers
• 1,3,5 Poor Sand (blue)
• 2,4,6 Good Sand (red)
• 1 Fault
• 3 Uncertain Inputs: 1. khigh = [100,200] mD
2. klow = [0,50] mD
3. throw = [0,60] ft
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IC Fault Model
p : oil/water rates
Truth Profile (Observed) Observed
khigh = 131.6 mD
klow = 1.3 mD
throw = 10.4 ft
Misfit Definition:
• Simulator controlled to match BHP at the wells
2
2
1
2
0.03
p t
sim obsM
n
obs
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IC Fault Model Misfit Surface
high low
misfit
klow
khigh
throw
Database (DB) 159,661 uniformly generated models
truth
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Convergence Speed – 20 Runs
MOPSO
SOPSO
Iteration 12
Iteration 28
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Zagadka Field
• Waterflood/aquifer support
• 95 wells in 10 groups
• 15 years history
• Compartmentalized
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Model 4 Convergence (PSO)
500
700
900
1100
1300
1500
1700
1900
0 50 100 150 200 250
Mis
fit
Iteration
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Model 4 Field Level History Match
• Match with PSO (single objective)
• 250 simulations (overnight, 12 core workstation)
Field oil rate Field water rate
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Match on Group Rates – Single Objective
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Multi-Objective vs Single Objective
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550
600
650
700
750
800
850
900
950
1000
0 50 100 150 200 250 300
Min
imum
Mis
fit
Iteration
PSO
MOPSO
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Multi-Objective Trade Off
• MO balances changes to fit one quantity vs another
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MO Not Always Faster than SO
• Study on EoN field
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Histogram of 10 Final Misfits
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Outline
• Introduction
• Stochastic Optimisation
• Model Calibration
• Forecasting
• Reservoir Optimisation
• Summary
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Forecasting with MO
• Generic Bayesian integral
• MC approximation
• Resample to calculate P(mk)dVk
( ) ( )
eg mean oil rate ( ) ( )o
J g P d
Q P d
m m m
m m m
1 1
1 1N Nk k
k k k
k kk
g PJ g P dV
N h N
m m
m mm
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Calculating Weights for Sample
• Gibbs sampler
– Assume misfit constant in Voronoi cell
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Single & Multi Objective HM for PUNQ-S3
Based on real field
An industry standard benchmark
Multiple wells and production variables
Algorithm used: PSO
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SO vs. MO – Forecasting
Time (days)
Time (days)
FO
PT (
sm3)
Time (days)
FO
PT (
sm3)
FW
PT (
sm3)
Time (days)
FW
PT (
sm3)
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Outline
• Introduction
• Stochastic Optimisation
• Model Calibration
• Forecasting
• Reservoir Optimisation
• Summary
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Field Development Optimisation
• Based on Scapa field
– Original HM done as part of MSc project
– HM to field rates (from DECC website)
– Wells are different from real field
• 4 injectors, 4 producers
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Field Level History Match
• Match to first 10 years of history
– Remaining data used to check forecast
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Optimisation
• Multi-Objective
– Maximise cumulative oil (FOPT)
– Minimise maximum water rate (FWPR)
• Optimisation variables
– Well locations (16 variables – i, j for each well)
– Injection well rates
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Field Development Optimisation (Scapa)
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Optimise Field Development Plan
Original MSc development plan (4 injectors, 4 producers)
10%
55%
77 models
Current Scapa production
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Scapa: Optimise Mid-Life Infill Wells
Trade-off : 1.5 bbls oil in year 10 for 1 bbl oil in year 2
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Optimisation Under Uncertainty
• PUNQ-S3
• Multiple HM models
• Optimise infill wells
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Optimisation Under Uncertainty
Uncertainty in Cumulative Oil
Mean C
um
ula
tive O
il
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Summary
• History Matching
– Multi-objective may increase convergence speed, but not always
• Forecasting
– Multi-objective provides greater coverage of pareto front; can lead to better forecasts
• Optimisation
– Explore the full range of trade-offs
– Can include uncertainty
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Acknowledgements
• EoN for funding the SO vs MO comparison
• BG, EoN, JOGMEC, RFD for funding the uncertainty group at HW
• Schlumberger for licences of ECLIPSE
• RFD for licences of tNavigator