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History matching by joint perturbation of facies distribution and net-to-gross. Junrae Kim and Jef Caers. Objectives. Constraining geological models to production data, adjusting aspects of geological model. Assessing value of information of production data on N/G. - PowerPoint PPT Presentation
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History matching by jointperturbation of facies distribution
and net-to-gross
Junrae Kim and Jef Caers
Objectives
• Constraining geological models to production data, adjusting aspects of geological model.
• Assessing value of information of production data on N/G.
• Application to a realistic 3D reservoir and well test.
Motivation :N/G should not be fixed in history matching
P
I
50
50Reference
N/G=0.05
N/G=0.6
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Timesteps
fw
N/G=0.05
Ref
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N/G=0.6
Ref• Elliptical bodies of high permeability (1000 mD) in a lower permeability matrix (15 mD)• Constant injection rate of 700 STB/day• No boundary conditions assumed • N/G (Reference) = 0.32
Flow response of 20 realizations
Probability perturbation method
Define: P(A|D) = (1-rD) i(k)(u) + rD P(A)
Perturb the conditional probabilities P(A|B) using another conditional probability that depends on the production data
P(A|B)P(A|D)
Model for P(A|B,D) Generate i(k+1)(u)
rD = 1 Full perturbation
rD = 0 No perturbation
Jef Caers
rD is found by solving a simple 1D optimization problem.
• Problem
=>More difficult joint optimization of rD and N/G.
• Solution
=>Making proportional to rD : Back to 1D optimization
• Why?
=>High rD, : Big change in model
G/N
G/N
Couple perturbation of N/G with perturbation of rD.
Perturbing N/G
Make linearly proportional to GN / .Dr
GN /
GN /
max/GΔN
1Dr
max/GΔN
minDr
Couple perturbation of N/G with perturbation of rD.
Increase GN /
Decrease GN /
Probability perturbation method:basic algorithm
Generate an initial guess realization
Outer Loop
Change random seed
History match?
Done
Converged to best rD?
Inner Loop
Yes
No
Yes No
Choose value for rD
Define P(A|D)
Generate a new realization
Run flow simulation
The proposed method:basic algorithm
Choose value for rD
Calculate G/NG/N
Calculate O(+)
Choose best rD and Obest from O(-) and O(+)
Generate a newrealization & flow simulation
Calculate G/NG/N
Generate a newrealization & flow simulation
Calculate O(-)
P
I
50
50Reference
2-D reservoir model
Conditioning data (facies)
Training image
• Reference: 50 x 50• Elliptical bodies of high permeability (1000 mD) in a lower permeability matrix (15 mD)• Constant injection rate of 700 STB/day• No boundary conditions assumed • N/G (Reference) = 0.32• Training image: 150 x 150
GNGN // 0 GNGN // 0
Inner iteration
Optimal value GNGN // 0
GNGN // 0
Initial model
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
1 2 3 4 5 6 7Inner-iteration
Outer Iterations
12.0/,31.0
,11 GNr
Iteration
D 27.0/,26.0
,22 GNr
Iteration
D
31.0/,16.0
,33 GNr
Iteration
D 33.0/,12.0
,44 GNr
Iteration
D0
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0 5 10 15 20 25 30 35 40
Timesteps
fw
Ref
Init
Iter 1
Iter 2
Iter 3
Iter 4 (History matched)
32.0/ 0 GN
Reference Model
20 realizations
Ref=0.32Mean(H.M.)= 0.33IQR=[0.28, 0.38]
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0.9
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0 5 10 15 20 25 30 35 40
Timesteps
fw
Initial Pressure
Ref
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0 5 10 15 20 25 30 35 40
Timesteps
fw
History match
Ref
0.12 0.52Initial N/G :
Selected randomly from [0.12, 0.52]
0.32
20 initial realizations
20 history matched realizations
20 realizations
Ref=0.32Mean(H.M.)= 0.33IQR=[0.19, 0.45]
Initial N/G : Always 0.5
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Timesteps
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Initial Pressure
Ref
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Timesteps
fw
History match
Ref
Unbiased!!
20 initial realizations
20 history matched realizations
• Investigate the value of information provided by a well test in a realistic 3D reservoir (Stanford V).
• First: Spatial re-sampling to model the uncertainty on N/G without well test data: Journel (1993).
• Second: Apply the proposed method. 1. Use re-sampled N/G as initial guess 2. History match 3. Check final N/G from history matched realization
Quantification of uncertainty on N/G from single well test
3-D reservoir model
•Dimensions 100x130x30 cells, three layers.
•Fluvial channels (1500mD) in (50mD) permeability matrix.
•Reference True N/G = 0.39.
•A single vertical well (N/G = 0.5) available at x=52, y=65.
Quantification of uncertainty on N/G
What is the uncertainty without well test? => Spatial bootstrapping
1. Generate a geostatistical model conditioned to the single vertical well log data only; N/G realization = 0.5
2. Randomly sample 20 vertical wells from the single geostatistical model.
3. Calculate N/G from 20 resampled wells.=> N/G histogram
Quantification of uncertainty on N/G (First: without well test)
Quantification of uncertainty on N/G (Bootstrap)
True N/G = 0.4Mean = 0.46IQR = [0.33, 0.58]
Wide confidence intervalUpward bias
A spatial resamplingof 20 wells
N/G realization= 0.5
Section z=1
50 100 150 200
50
100
150
200Section z=2
50 100 150 200
50
100
150
200Section z=3
50 100 150 200
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100
150
200
Section z=4
50 100 150 200
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200Section z=5
50 100 150 200
50
100
150
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True Reference Training image
Resampled N/G
Quantification of uncertainty on N/G (Conditioning to well test data)
1. Each of the 20 re-sampled N/G is used as an initial guess.
2. 20 final history match generated- perturbing facies and N/G.
3. Calculate N/G from final HM models.=> N/G Histogram
What is the uncertainty with well test?
Well test model
•A drawdown test for 100 days.
•The rate of production is fixed to 1000 STB/day.
•Boundary effect starts at 10 days.
•
0.01
0.1
1
10
100
0.001 0.01 0.1 1 10 100 1000
t
d(Dp)/d(lnt)Dp
Radial flow LateTransient
PSS/Closed boundary
][][][ STB/RB.B,cp,psi/. ootc 00011111028356
Quantification of uncertainty on N/G (Conditioning to well test data)
Quantification of uncertainty on N/G (Conditioning to well test data)
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
0.001 0.01 0.1 1 10 100
days
psia
(WBH
P)
ref
HistorymatchedPressure
987
992
997
0.001 0.01 0.1 1 10 100
days
psia (W
BHP)
ref
Initial Pressure
WBHP (initial guesses) WBHP (history matched)
Initial guesses (N/G) without well-testMean = 0.46IQR = [0.33, 0.58]
History matched (N/G) with well-testMean = 0.44IQR = [0.40, 0.46] Reduction of uncertainty on N/G is significant.
True N/G = 0.4Original well log N/G = 0.5
Quantification of uncertainty on N/G (Conditioning to well test data)
Pseudo steady state versus transient state
• Variance in all cases are smaller than the case, where no well test is available.
• The uncertainty is largest in the 3-day case.
• Assumption: Averaging volume known (PSS)
Quantification of uncertainty on N/G (Conditioning to well test data)
100 days 10 days 3 days
Conclusions
•N/G should not be fixed in a history matching process.
•The proposed method jointly parameterizes perturbation of
N/G and of facies => simple & robust.
•The method quantifies uncertainty of N/G based on
production data.
•Well test: N/G can be quantified if averaging volume is well known.
Facies anisotropy and facies distribution
History matching by joint perturbation of
Case 1: Always starts from -45°
P
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50
50Reference
-90° 90°
Case 3 : Select randomly from [-90°, 90°]
Mean: 47.6[36, 58]
Mean: 39.7[30, 52]
Mean: 44.0[35, 52]
Case 2: Always starts from 0°
IQR:
A history match was achieved in all cases…
45° 75° -45°-15°
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Ref
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-45
But still with large effort
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0.32
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VR
N/G
3 days
Reservoir N/G
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Days
Vo
lum
e R
atio
PSS
The global N/G corresponding to the volume ratio
volumereservoirTotal
welltestaroundvolumelCylindricaVR
Volume ratio corresponding to the number of days for which well test is run.