Post on 14-Mar-2020
© 2017 Chevron U.S.A. Inc. | All rights reserved.
A Physics-Based Data-Driven Model for History Matching,
Prediction and Characterization of Unconventional Reservoirs*
Yanbin Zhang
*This work has been submitted to SPEJ and under review for publication
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© 2017 Chevron U.S.A. Inc. | All rights reserved.
Motivation
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Reservoir Characterization with Play-Doh
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Reservoir Characterization with Play-Doh
Reservoir
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© 2017 Chevron U.S.A. Inc. | All rights reserved.
Reservoir Characterization with Play-Doh
Wellbore
Reservoir
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© 2017 Chevron U.S.A. Inc. | All rights reserved.
Reservoir Characterization with Play-Doh
Wellbore
Reservoir
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© 2017 Chevron U.S.A. Inc. | All rights reserved.
Reservoir Characterization with Play-Doh
Wellbore
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4 56
Reservoir
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Reservoir Characterization with Play-Doh
Wellbore
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2
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4 56
1
2
3
4
5
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Reservoir
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Reservoir Characterization with Play-Doh
Wellbore
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4 56
1
2
3
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5
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𝑥
𝑦
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Reservoir Characterization with Play-Doh – Fractures
fracture
1 12 2
3 3
3 3 2 21 1
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1 1
43 3 2 2
1 1
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𝑥
𝑦
4
3
3
2
2
1
1
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© 2017 Chevron U.S.A. Inc. | All rights reserved.
Diffusive Diagnostic Function (DDF)
𝑃𝑉1 𝑃𝑉2 𝑃𝑉3 𝑃𝑉4𝑤𝑒𝑙𝑙
𝑃𝑉5
𝑃𝑉1𝑃𝑉2
𝑃𝑉3𝑃𝑉4
𝑃𝑉5
𝑤𝑒𝑙𝑙
𝑇0 𝑇1 𝑇2 𝑇3 𝑇4
3D
1D
Set of contours of the
pressure solution in 3D
Each “ring” is
represented as a
single cell
d𝑠
d𝑙
ȁΣ 𝑃
ቚd𝑃𝑉Σ= ර
Σ
𝜙 d𝑙 ⋅ d𝑠ቚ𝑇Σ= ර
Σ
𝑘d𝑠
d𝑙
ቚd𝜉Σ≝
ȁd𝑃𝑉 Σ
ȁ𝑇 Σ=
Σׯ 𝜙 d𝑙 ⋅ d𝑠
Σׯ 𝑘d𝑠d𝑙
=𝜙d𝑙 Σ
𝑘d𝑙 Σ
ቚ𝜎Σ≝ ቚd𝑃𝑉
Σቚ⋅ 𝑇Σ= ර
Σ
𝜙 d𝑙 ⋅ d𝑠 රΣ
𝑘d𝑠
d𝑙= ቚ𝑆
Σ𝜙d𝑙 Σ
𝑘
d𝑙Σ
ȁ𝜉 Σ = Σ0Σd𝜉 where Σ0 is the completion sand face
A contour surface of the
pressure solution in 3D
ȁΣ 𝑃+d𝑃
ቚ𝑄Σ= ර
Σ
𝑣d𝑠 = රΣ
𝑘d𝑃
d𝑙d𝑠 = ቚ𝑇
Σd𝑃
From Darcy’s Law
ቐ𝑃𝑉 = 𝜎Δ𝜉
𝑇 =𝜎
Δ𝜉
𝜎(𝜉)
In physical space, we need two functions 𝑃𝑉 𝑥 and 𝑇(𝑥)
In dimensionless 𝜉 space, we only need one function 𝜎 𝜉which is called the DDF
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The 1D Simulation Model with DDF
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5
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𝑥
𝑦
𝜉1 𝜉2 𝜉3 𝜉𝑁
𝜎
𝜉𝑁−1
𝜎0
𝜎1
𝜎2
𝜎3𝜎𝑁−1
Well
𝑃𝑤𝑓
1-Dimension
N Grid blocks
𝜉
𝑤𝑒𝑙𝑙
𝜉4
𝜎4
PV𝑖 =𝜎𝑖−1 + 𝜎𝑖
2(𝜉𝑖 − 𝜉𝑖−1)
𝑇𝑖 =𝜎𝑖
(𝜉𝑖+1 − 𝜉𝑖−1)/2
𝐽 =𝜎0𝜉1/2
unit: ft/md1/2
unit: ft2md1/2
We are doing the same, old, regular reservoir simulation
except that we replace the 3D grid with DDF
✓ Complex fluid and rock model
✓ Changing well constraints
✓ Capillary pressure
✓ Adsorption
✓ Coupled with wellbore flow modeling and surface network
Caveat
Remember we reduce 3D reservoir into a 1D model and that is
an approximation
However, as we will show later, it is quite a good approximation
in many cases
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© 2017 Chevron U.S.A. Inc. | All rights reserved.
All this is good, but...
1
2
3
4 56
43 3 2 2
1 1
How do I know if I should “cut” my reservoir this way or that way?
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2
3
4
5
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𝑥
𝑦
𝑥
𝑦
4
3
3
2
2
1
1
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© 2017 Chevron U.S.A. Inc. | All rights reserved.
All this is good, but...
1
2
3
4 56
43 3 2 2
1 1
How do I know if I should “cut” my reservoir this way or that way?
1
2
3
4
5
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𝑥
𝑦
𝑥
𝑦
4
3
3
2
2
1
1
The bad news: we don’t know in
general
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© 2017 Chevron U.S.A. Inc. | All rights reserved.
All this is good, but...
1
2
3
4 56
43 3 2 2
1 1
How do I know if I should “cut” my reservoir this way or that way?
1
2
3
4
5
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𝑥
𝑦
𝑥
𝑦
4
3
3
2
2
1
1
𝜉
𝜎
DDFs
𝑡
𝑞
Production
Datahistory
matching
forward
modeling
The bad news: we don’t know in
general
The good news:
(1) We can guess and we’ll make a lot
of guesses
(2) We can adjust our guesses by
history matching
History Matching using ESMDA
Ensemble Smoother with Multiple Data Assimilation
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© 2017 Chevron U.S.A. Inc. | All rights reserved.
We don’t do wild guesses; we guess based on
DDF Characteristics
radial flow
𝑟𝑤𝑟
𝜉
𝜎
slope:2𝜋𝑘ℎinfinite acting
boundary
𝑟𝑤′ = 𝑟𝑤𝑒
−𝑠
𝜎0 = 2𝜋𝑟𝑤′ ℎ 𝑘𝜙
0𝜉 = 𝑟 𝜙/𝑘
linear flow
𝑥𝑓
𝑑
𝜉
infinite acting
boundary
𝜎0 = 4𝑥𝑓ℎ 𝑘𝜙
0𝜉 = 𝑑 𝜙/𝑘
𝜎
𝑥𝑓
𝐿𝑤
𝑁𝑓
𝑑𝑓
𝜉
boundary
𝜎0~4𝑥𝑓𝑁𝑓ℎ 𝑘𝜙
0
𝜉𝑆𝑅𝑉~𝑑𝑓2
𝜙/𝑘
𝜎
slope~𝛼𝑘ℎSRVp
𝜎~(4𝑥𝑓 + 2𝐿𝑤 + 4𝑑𝑓)ℎ 𝑘𝜙
Take out your Play-Doh and construct this DDF!
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© 2017 Chevron U.S.A. Inc. | All rights reserved.
DDF Characteristics - Summary
Characteristics of
the DDFDiagnostic Properties
Approximate
EquationsComments
𝜎 levelFlow area and
Reservoir quality𝜎~𝐴 𝑘𝜙
Fractures cause sharp increase in 𝜎 level.
Interferences or boundaries cause drop in 𝜎 level. 𝜎
level keeps constant for linear flow
Slope of linearly
increasing 𝜎 𝜉
Reservoir
permeability
Δ𝜎
Δ𝜉~𝛼𝑘ℎ
𝛼 depends on flow pattern. 𝛼 = 2𝜋 for radial flow.
Generally, 𝛼 > 2𝜋 for irregular flow pattern.
Area under the DDF
curvePore volume 𝐴𝑟𝑒𝑎 = 𝑉𝑝
A dramatic drop in 𝜎 level signifies boundary effect or
interferences. For unconventional reservoirs, SRV may
be identified in this way.
𝜉 at which 𝜎
behavior changesDistance 𝜉~𝑑
𝜙
𝑘
The estimation of distance is difficult to be precise
because the transition of 𝜎 behavior is usually not
clear-cut and may span a wide range.
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Synthetic Example: Vertical Well
𝑟𝑒 = 500ft
ℎ = 100ft
𝑟𝑤 = 0.3ft
𝑘 = 0.001md 𝜙 = 0.05md
3D Cartesian grid 101 × 101 × 1DX = DY = 9.9 ftDZ = 100 ft
Black oil fluid model (𝑃𝑏 = 800 psia)
Initial reservoir pressure 𝑃𝑖 = 5000 psiaInitial water saturation 𝑆𝑤𝑖 = 𝑆𝑤𝑖𝑟 = 0.15Well producing at constant BHP = 1000 psia
(a) (b)
(c) (d)
𝝃 (ft/md1/2
) Time (days)
𝝃 (ft/md1/2
) Time (days)
Before HM
After HM
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Synthetic Example: Multiple Fractures
10 Infinite conductivity fractures
Other parameters are the same as previous slide
(a) (b)
(c) (d)
Time (days)
Time (days)
𝝃 (ft/md1/2
)
𝝃 (ft/md1/2
)
Before HM
After HM
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Applications
History matching and forecasting
for a Gas Well• Physics Based
✓ Complex fluid, multi-phase flow
• Data Driven
✓ Extremely fast history match
• Reservoir Characterization
✓ Total fracture area and SRV
• Integrated Workflow
✓ Coupled with surface network
✓ Optimization / uncertainty
analysis
Oil Well Examples
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© 2017 Chevron U.S.A. Inc. | All rights reserved.
Summary
1. Physics-Based:
DDF provides a general 1D
simulation framework to
approximate 3D reservoir
2. Data-Driven:
DDF is probabilistically
conditioned to production data
DDFMachine learning and big data
First-principle-based computation
Future Research
?
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© 2017 Chevron U.S.A. Inc. | All rights reserved.
Acknowledgement
• ETC/RPP
– Jincong He
– Jiang Xie
– Xian-huan Wen
• ETC/RPS
– Robert Fitzmorris
– Shusei Tanaka
• ETC/PEWP
– Jorge Acuna
• ETC/TRU
– Reza Banki
• MCBU
– Baosheng Liang
– Hannah Luk
• AMBU
– James Wing
– Richin Chhajlani
Please reach out to me for any
questions or to connect with me.
You may contact me at:
Yanbin.Zhang@chevron.com
© 2017 Chevron U.S.A. Inc. | All rights reserved.
Backup Slides
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History Matching using ESMDA with DDFEnsemble Smoother with Multiple Data Assimilation Diffusive Diagnostic Function
𝜉
𝜎
DDFs
𝑡
𝑞
Production
Datahistory
matching
forward
modeling
𝚫𝒎 = 𝐂𝑀𝐷 𝐂𝐷𝐷 + 𝛼𝑖𝐂𝐷−1(෩𝒅𝑜𝑏𝑠 − 𝒅𝑝𝑟𝑒𝑑)
෩𝒅𝑜𝑏𝑠 = 𝒅𝑜𝑏𝑠 + 𝛼𝑖𝐂𝐷1/2
𝑧𝑑 where 𝑧𝑑~𝑁(0, 𝐼𝑁𝑑)
𝒎𝑖+1 = 𝒎𝑖 + 𝚫𝒎
perturbed observations
model updatedata mismatch
Workflow
1. Come up with initial ensemble of DDFs
2. Perform forward modeling to obtain data
prediction
a) predictions way off, go back to 1
b) if predictions follow the trend and cover
the range of observed data, go to 3
3. Randomize the model to avoid ensemble
collapse
4. For the 𝑖th (out of 𝑛) iteration, use 𝛼𝑖 = 1/𝑛in the equation to update the ensemble of
models considering the mismatch of all
data points simultaneously
5. Model regularization by smoothing the DDF
curves and eliminate negative values (set
to 0)
6. Go to 4 for next iteration
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Synthetic Example: Single Fracture
Infinite Conductivity vs. Finite Conductivity
Time (days) 𝝃 (ft/md1/2
)
𝝃 (ft/md1/2
) Time (days)
Infinite conductivity fracture 𝑘𝑓 = 1 × 106 md
Finite conductivity fracture 𝑘𝑓 = 1 md
Infinite or finite conductivity single fracture
Other parameters are the same as previous slide
𝜎 starts at a much larger value
𝜎0 = 4𝑥𝑓ℎ 𝑘𝜙 = 353 ft2md1/2
𝜎 starts at a smaller value, but
increases rapidly near 𝜉 = 0
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© 2017 Chevron U.S.A. Inc. | All rights reserved.
Synthetic Example: Multiple Fractures – DDF Diagnostics
(a) (b)
𝝃 (ft/md1/2
) 𝝃 (ft/md1/2
)
𝜉𝑖~140 ft/md1/2
𝜎2~1000 ft2⋅md
1/2
𝜎1~3500 ft2⋅md
1/2
Slope ~1 ft⋅md
𝑥𝑓
𝐿𝑤
𝑁𝑓
𝑑𝑓
𝜉
boundary
𝜎1~4𝑥𝑓𝑁𝑓ℎ 𝑘𝜙
0
𝜉𝑆𝑅𝑉~𝑑𝑓2
𝜙/𝑘
𝜎
slope~𝛼𝑘ℎSRVp
𝜎2~(4𝑥𝑓 + 2𝐿𝑤 + 4𝑑𝑓)ℎ 𝑘𝜙
Lw = 450 ft
Nf = 10df = 50 ft xf ~ 120 ft
𝜙 = 0.05𝜎1
𝜎2~ 3.5
𝜎1~ 3500 ft2md1/2
k ~ 0.001 md
SRVp ~ 6x105 ft3
2𝑥𝑓𝐿𝑤ℎ𝜙 = 6.25x105 ft3
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© 2017 Chevron U.S.A. Inc. | All rights reserved.
Field Example: Marcellus Gas Well
Well length (𝐿𝑤) 3202 ft
Number of hydraulic fractures (𝑁𝑓) 12
Reservoir thickness (ℎ) 150 ft
Reservoir porosity (𝜙) 0.065
Initial reservoir pressure (𝑃𝑖) 5008.8 psia
Reservoir temperature (𝑇) 160 F
Connate water saturation (𝑆𝑤𝑐) 0.39
Rock compressibility (𝑐𝑓) 3×10-6 psi-1
Gas specific gravity (𝛾) 0.570.0
1.0
2.0
3.0
4.0
5.0
6.0
0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
0 500 1000 1500 2000
Ga
s p
rod
uc
tio
n R
ate
(M
MS
CF
/D)
Bo
tto
mh
ole
Pre
ss
ure
(p
sia
)
Time (days)
1E+05
1E+06
1E+07
100 1000 10000 100000
No
rma
lize
d G
as
Po
ten
tia
l (p
si2
/cp
/MS
CF
D)
Material Balance Time (hr)
Integral of Normalized Gas Potential
Bourdet Derivative
predictionHM
Not half slope
Log-Log PlotSquare Root Time Plot
A straight line can be used to fit the data
even though it is not linear flow
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Field Example: HM and prediction with DDF
(a) (b)
(c) (d)
Time (days)
Time (days)
𝝃 (ft/md1/2
)
𝝃 (ft/md1/2
)
𝝃 (ft/md1/2
)
Before HM
After HM
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© 2017 Chevron U.S.A. Inc. | All rights reserved.
Field Example: DDF Diagnostics
𝝃 (ft/md1/2
)
𝑥𝑓
𝐿𝑤
𝑁𝑓
𝑑𝑓
𝜉
boundary
𝜎1~4𝑥𝑓𝑁𝑓ℎ 𝑘𝜙
0
𝜉𝑆𝑅𝑉~𝑑𝑓2
𝜙/𝑘
𝜎
slope~𝛼𝑘ℎSRVp
𝜎2~(4𝑥𝑓 + 2𝐿𝑤 + 4𝑑𝑓)ℎ 𝑘𝜙
Characteristic of finite
conductivity fractures
SRVp~2.6 × 107 ft3
𝐴𝑡𝑜𝑡𝑎𝑙 𝑘𝜙~1 × 104 ft2md1/2
𝜙 = 0.065𝐴𝑡𝑜𝑡𝑎𝑙 𝑘~4 × 104 ft2md1/2
Total fracture sandface area
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Field Example: Probabilistic Nature of the DDF Method
Time (days) 𝝃 (ft/md1/2
)
Time (days) 𝝃 (ft/md1/2
)
HM
HM
P50 SRVp decreases
SRVp uncertainty
range decreases