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History matching under geological controlThe probability perturbation method
Jef Caers
Department of Petroleum Engineering
Stanford University, Stanford, California, USA
Stanford Center for Reservoir Forecasting
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Motivation
• The goal of history matching is not just to match history
Prediction power of models cannot be verified Geological realism enhances prediction
• Interaction between geological model and flow model
Not all geology matters for flow Iterative process between static/dynamic, not sequential
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HM itself is not difficult
P1
P2
P6
P5
P4
P3
I1I2
I3
Initial Model
ProposedMatched Model
P1
P2
P6
P5
P4
P3
I1I2
I3
P1
P2
P6
P5P4
P3
I1I2
I3
Eclipse (SimOpt)Matched Model
Regions (SimOpt) 500
300
0
100
Log Scale
md
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Geological scenario
A prior geological scenario defines what remainsconstant during history matching
prior geological scenario =
set of decisions about the styleof geological structures/features or aboutthe parameterizations of these structures/features
prior geological scenario =
set of decisions about the styleof geological structures/features or aboutthe parameterizations of these structures/features
• permeability/porosity variogram• Boolean model with shape distributions• Training image model and seismic derived facies probabilities• Training image with unknown Net-to-Gross
Example geological scenario
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Quantify geological scenario
Prior geological scenario defines conditional probabilities
P(A|B) A = “channel sand occurs”B = known “conditioning” data
Key idea: Perturb the probability P(A|B) such that* a history match is achieved* the geological scenario remains unchanged
Key idea: Perturb the probability P(A|B) such that* a history match is achieved* the geological scenario remains unchanged
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Probability perturbation
Perturb the conditional probabilities P(A|B) using another conditional probability that depends on the production data D
P(A|D) = (1-rD) i(o)(u)+ rD P(A)prior information on A
Binary case: A=“channel occurs”, then i(o)(u)=1
Combine P(A|D) and P(A|B) into P(A|B,D)
rD=0 P(A|B,D) = i(o)(u) : No perturbation
rD=1 P(A|B,D) = P(A|B) : equiprobable realization isgenerated if random seed is changed
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Example
Reference model
East
No
rth
0.0 50.0000.0
50.000
facies 0
facies 1
training image
East
No
rth
0.0 150.0000.0
150.000
facies 0
facies 1
Geological scenario 1. Two hard data 2. Training imageI
P
Assume permeability of each facies known (1500 vs 50mD)
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Creating perturbations
Seed= 76845
Seed= 36367
Perturbations preservegeology
Perturbation are betweentwo equiprobable models
Optimize on rD
i(o)(u)
P(A|D) = (1-rD) i(o)(u)+ rDP(A)
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Graphical representation
high
low
rD=0
rD=1
Initial model
East
No
rth
0.0 50.0000.0
50.000
facies 0
facies 1
Iteration 1, r_D=0.21
East
No
rth
0.0 50.0000.0
50.000
facies 0
facies 1
Iteration 3, r_D=0.52
East
No
rth
0.0 50.0000.0
50.000
facies 0
facies 1
Iteration 5, r_D=0.50
East
No
rth
0.0 50.0000.0
50.000
facies 0
facies 1
Iteration 7, r_D=0.31
East
No
rth
0.0 50.0000.0
50.000
facies 0
facies 1
Iteration 9, r_D=0.24
East
No
rth
0.0 50.0000.0
50.000
facies 0
facies 1
seed=76845
initial model
East
No
rth
0.0 50.0000.0
50.000
facies 0
facies 1
r_D = 0.05
East
No
rth
0.0 50.0000.0
50.000
facies 0
facies 1
r_D = 0.1
East
No
rth
0.0 50.0000.0
50.000
facies 0
facies 1
r_D = 0.2
East
No
rth
0.0 50.0000.0
50.000
facies 0
facies 1
r_D = 0.5
East
No
rth
0.0 50.0000.0
50.000
facies 0
facies 1
r_D = 1
East
No
rth
0.0 50.0000.0
50.000
facies 0
facies 1
seed=36367
Initial model
East
No
rth
0.0 50.0000.0
50.000
facies 0
facies 1
Iteration 1, r_D=0.21
East
No
rth
0.0 50.0000.0
50.000
facies 0
facies 1
Iteration 3, r_D=0.52
East
No
rth
0.0 50.0000.0
50.000
facies 0
facies 1
Iteration 5, r_D=0.50
East
No
rth
0.0 50.0000.0
50.000
facies 0
facies 1
Iteration 7, r_D=0.31
East
No
rth
0.0 50.0000.0
50.000
facies 0
facies 1
Iteration 9, r_D=0.24
East
No
rth
0.0 50.0000.0
50.000
facies 0
facies 1
Space ofAll realizations
Mismatch between simulated and field data
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Basic Algorithm
Generate initialguess realization
Outer Loop
Change random seed Choose value for rD
Define P(A|D)
Generate a new realization
and run flowsimulation
Convergedto best rD ?
Inner Loop
YES NO
History match ?
Done
YES
NO
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Result
Outer iteration 4
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General method
Training image
Reference Match
flow
pressure
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Multi-category
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Junrae KimHistory matching on N/G
History matching:
Most critical : Finding a good geological model
PP : searches within a fixed geological model
Junrae:
Critical parameters such as Net-to-Gross needto be part of the history matching process
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Todd HoffmanRegional probability perturbation
PP: one parameter creates same perturbation everywhere
Todd:
Create regionsAttach a parameter rD to each region
Regional perturbation method (RPP)
Challenges: no artifact discontinuities at region borderSolve a multiple-parameter problem
P(A|D) = (1-rD) i(o)(u)+ rDP(A)
P(A|D) = (1-rDk) i(o)(u)+ rDkP(A)
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Satomi SuzukiHierarchical history matching
1) Perturb Facies by Prob. Perturbation
Perm Fixed
Change Random #
History Matched ?
END
YES
NO
2) Perturb Perm within
Facies
Facies Fixed
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Inanc TureyenJoint fine and coarse scale HM
Static model : fine scaleFlow simulations : coarse scale
Traditional approach
Initial fine-scale realization
Downscaled realization
Non-uniformcoarsened realization
Initial coarsened realization
History matched coarsened realization
HistoryMatching
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Joint optimization
GridOptimization
Mismatch Between fine and coarse minimized
HistoryMatching
Mismatch on coarse scale usedfor fine scale perturbations
Result: History matched non-uniform gridded model Fine-scale model also History matched
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Joe VoelkerApplication to Ghawar Field
dep
th
BBL/day/ft
Super-K= extreme flowBut not caused by extreme K
Driving mechanism = combinedFacies and fracture model
Super-K determinedBy HM flow meter data