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An Effective Reservoir Management by
Streamline-based Simulation, History
Matching and Optimization
Shusei Tanaka
May, 2014
• Development of a general purpose streamline-based reservoir simulator: Inclusion of diffusive flux via Orthogonal Projection Illustration by black oil model Extension to a multicomponent system
• Application to Brugge benchmark case: Streamline-based simulation Streamline-based BHP/WCT data integration Flow diagnostics for streamline-based NPV optimization
• Conclusion
Outline
2/50
Streamline Technology: Overview
3/50
• Key concept of Streamline: Fast IMPES-based reservoir simulation
History matching(HM) by calibration of travel time
Improves sweep efficiency by streamline information
Pressure field Streamlines Connection map
Problem Statement:
SL-based Reservoir Management
4/50
• Challenges for mature field, multiple well… Quick forecasting
HM for individual well
Improve NPV by reallocating well rate
• Streamline is efficient, but can we apply all the time? What if flow is not convective dominant?
How about prior to breakthrough for HM?
Can we improve NPV?
Mature field with multiple
wells
Development of a General Purpose Streamline-
based Simulator
Motivation
6/50
Solve 1D Convection EquationsCalculate Diffusive Flux on Grid
Compute Pressure & Velocity Field
• Streamline simulation is difficult to apply if…
System of equation is highly nonlinear (ex. Gas injection)
Capillary and gravity effects are dominant
Error by Operator-Split
Error by IMPES
; 0
w
w ut
S
Why Split the Equation?
• Water velocity does not follow total velocity with capillary (and gravity)
7/50
tuwu
Streamline
cowowtww pkFuFu
• Split equation by physical mechanisms
Convective
Transport
Capillary
Diffusion 0
cowow
w pkFt
S
0
w
w ut
S 0
wt
w Fut
S
cowowtww pkFuFu
Saturation Transport
Equation
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0 0.2 0.4 0.6 0.8 1
Wa
ter
Sa
tura
tio
n
Normalized Distance
Correct Solution
Convection Flow
Too much diffusion
with large time step
(SPE 163640)
Operator Splitting
Capillary after
convection
8/50
0~
wtw
w Fuut
S
cowowtwwwtw pkFuFFFuu
~
~
• Split equation by physical mechanisms
• Anti-diffusive corrections
Computationally expensive:Function of (P, T, composition,
Initial state) for each grid, time step
0
w
w ut
S 0
wt
w Fut
S
Splitting with anti-
diffusive flux
Convection Eq.
Corrected Operator Splitting
Anti-diffusive concave envelope
9/50
0
w
w ut
S
wtww uufu
Parallel component,
calculate along
streamline
Anti-diffusive correction
not needed
Orthogonal Projection
• Split equation into parallel and transverse flux terms
twf u
wu
tuwu
Streamline
10/50
twf u
wu
0
w
w ut
S 0
tw
w uft
S
0
w
w ut
S
tuwu
Orthogonal Projection
Parallel to Ut
(Solve along streamline)
Transverse to Ut
(Solve on grids)
Streamline
• Split equation into parallel and transverse flux terms
wtww uufu
Parallel component,
calculate along streamline
Anti-diffusive correction not
needed
11/50
1.Compute pressure & velocity field
Include capillary effects
2.Trace streamlines
Solve 1D convection equations
Include capillarity and gravity
3.Map back saturation to grid
Calculate corrector term
Predictor-Corrector Workflow
Iterative IMPES
Orthogonal Projection
12/50
• Pressure equation(IMPES)
• Transport equation (along SL)
Orthogonal Projection:
Application to Multicomponent System
0
owgj
j
owgj
j
owgj
jj
owgj
jjr Qupuct
pScc
i
sl
ii fmt
cfgDpFyu
kyFfSym
sl
ii
owgj owgj jogwmm
m
jmjmcmjjij
t
jijj
sl
ij
ogwj
jiji
1
,2 , ,
Δ
• Transport equation (on Grid, corrector)
ogwj jmogwm
m
jmcjmmjjijtti DgpFykuuI
t
m
,
ˆˆ
Pc,Gravity along streamline
Transverse Pc,Gravity on grid
0
owgj
ijijjjijjjij qyuySyt
• Governing equation
13/50
Illustrative Example
100 mD
5 mD
• Water injection 0.2PVI, then CO2 0.2PVI
• Single time step for each injection period
• Observe capillarity by parallel/transverse to Ut
tw uf
wu
14/50
Water Saturation and Capillary Flux
Distributions
• Capillarity traps water at center by J-Function
• Capillarity flows back water towards injector during gas injection period
Sw after water injection
Arrow: water capillary flux
Sw after gas injection
Arrow: water capillary flux
1)(
kSwJpcow
15/50
Water Capillary Flux:
Parallel and Transverse to Total Velocity
Total capillary fluxCapillary flux transverse
to total velocity
Capillary flux along
total velocity
• Most of the capillary effects can be included along the streamlines
cow
t
tt
t
ow
t
w pu
uuI
u
ku
2
Along streamline On grids
16/50
cow
t
t
t
ow pu
uk
2
Water Saturation Distribution
Commercial, FD Operator Splitting
(no correction)Orthogonal Projection
• OP can take large time step without anti-diffusive correction
17/50
Injection :: CO2
10 rb/D – 1000 [Days]
Production :: BHP
(1900 psi)
2D Cross-Section CO2 Flooding Model
Pc, Convection
Pc, Gravity
Simulation model:• 7 HC component + Water• Rel-Perm by Corey• Water-wet Capillarity
Initial & Boundary Condition• 2000+ psi , 212F˚• Constant production BHP, constant CO2 injection at 10 rb/D• 1000 days
18/50
CO2 Mole Fraction Distribution:
Along Streamline
Including Pc & GravityConvection only
19/50
Production Mole Fraction of CO2
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 200 400 600 800 1000
Pro
du
ctio
n M
ole
Fra
ctio
n(C
O2
)
Time [Days]
Streamline
Commercial Simulator
Number of time step:
Commercial FD = 56
Streamline = 21
20/50
CO2 Mole Fraction Distribution:
Final Distribution
Orthogonal Projection
(After corrector term)Commercial FD
(E300)
21/50
0
50
100
150
200
250
300
350
400
450
500
2D Areal 2D Cross-Section 2D Cross-SectionHetero
Goldsmith Field
E300 FIM
Streamline
Previous case
Comparisons of Number of
Time Step
Nu
mb
er
of
Tim
e S
tep
Tested simulation cases
in the paper
10×
2×4×
3×
22/50
0
50
100
150
200
250
300
350
400
450
500
2D Areal 2D Cross-Section 2D Cross-SectionHetero
Goldsmith Field
E300 FIM
Streamline
Previous case
Comparisons of Number of
Time Step
Nu
mb
er
of
Tim
e S
tep
Tested simulation cases
in the paper
10×
2×4×
3×
23/50
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 180 360 540 720 900 1080
Pro
du
ctio
n M
ole
Fra
ctio
n (
CO
2)
Time [Days]
Streamline
Commercial Simulator0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.0E+00 2.5E+04 5.0E+04 7.5E+04 1.0E+05
Pro
du
ctio
n M
ole
Fra
ctio
n (
CO
2)
Time [Days]
Streamline
Commercial Simulator
Conclusions
24/50
• Developed a new SL-based simulation method to incorporate capillarity and gravity and applied to CO2 injection cases
• Computational advantages:• Minimizes the saturation correction term
• Can take large time steps without anti-diffusive corrections
• Demonstrated by synthetic and field case:• Iterative IMPES approach handles nonlinearity
• Larger time stepping obtained compared with commercial FD simulator
Application to Brugge Benchmark:
- Streamline-Simulation
- History Matching
- NPV Optimization
Brugge Benchmark Example
26/50
• Benchmark model for HM, optimization problem• 20 producers, 10 injectors in complex geometry• Conduct 40 years of waterflood, 1000 stb/d per wel
Oil saturation and well location
Initial So Net gross ratio
PorosityRock table ID
ECLIPSE vs. Streamline Simulation:
Water-Cut (4 producers)
27/50
Circle : ECLIPSE
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 7500 15000
Pro
du
ctio
n W
ate
r C
ut
Time [Days]
BR-P-18 ECL
BR-P-8
BR-P-12
BR-P-1
BR-P-18 SL
BR-P-8
BR-P-12
BR-P-1
Line: Streamline
- ECLIPSE without NNC option
Comparisons of Oil Saturation
Distribution
28/50
Initial oil saturation
After 20 years
Streamline Commercial (ECL)
Presented at student paper contest 2013
Application to Brugge Benchmark:
- Streamline-Simulation
- History Matching
- NPV Optimization
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0 500 1000 1500 2000
Wat
er
Cu
t
Time [Days]
Streamline-based Inverse Modeling
30/50
min 𝛿𝐝𝑤𝑐𝑡 − 𝐒𝑤𝑐𝑡𝛿𝐤
𝛿𝐝
1. Run reservoir simulation by given model
2. Trace Streamlines and calculate parameter sensitivity
3. Update parameters to satisfy:
Observation
Prediction
Motivation and Objective
31/50
Streamlines0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0 500 1000 1500 2000
Wat
er C
ut
Time [Days]
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 500 1000 1500 2000
CO
2M
ole
Fra
ctio
n
Time [Days]
WCT
• What can we tell prior to breakthrough? Pressure data can be used while not considered previously
• Study objective New approach to calculate pressure sensitivity along SL
Simultaneous inversion of pressure and water-cut data
0
200
400
600
800
1000
1200
1400
1600
1800
2000
0 50 100 150 200
Bo
tto
m H
ole
Pre
ssu
re
Time [Days]
0
200
400
600
800
1000
1200
1400
1600
1800
2000
0 50 100 150 200
Bo
tto
m H
ole
Pre
ssu
re
Time [Days]
BHP
Observation
Initial
ik
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0 500 1000 1500 2000
Wat
er
Cu
t
Time [Days]
𝛿𝒕𝑤𝑐𝑡
Production WCT
Parameter Sensitivity Along Streamline
32/50
• TOF( ): Travel time of neutral tracer along streamlines
, ,
, ,x y z
Inlet
dsx y z
u
𝜕𝑡
𝜕𝑘𝑖= −
𝜕𝑆
𝜕𝜏
𝜕𝜏
𝜕𝑘𝑖∙𝜕𝑆
𝜕𝑡
−1
=1
𝑓′(𝑆)
∆𝜏𝑖𝑘𝑖
• Water-cut travel time sensitivity:
injectorProducer
[He et. al,2003]
𝜕𝑝𝑏ℎ𝑝𝜕𝑘𝑖
=𝜕∆𝑝𝑖𝜕𝑘𝑖
≈∆𝑝𝑖𝑘𝑖
• Bottom hole pressure sensitivity: [new]
0
200
400
600
800
1000
1200
1400
1600
1800
2000
0 50 100 150 200
Bo
tto
m H
ole
Pre
ssu
re
Time [Days]Production BHP
𝛿𝒑𝑏ℎ𝑝
𝜕𝑝𝑏ℎ𝑝𝜕𝑘𝑖
≈𝜏𝑖𝜏
𝜕∆𝑝𝑖𝜕𝑘𝑖
≈𝜏𝑖𝜏
∆𝑝𝑖𝑘𝑖
Rate-Rate constraint
Rate-BHP constraint
Sensitivity Results: 1D CPG
(3phase Gas Injection)
-20.0
-15.0
-10.0
-5.0
0.0
0.0 0.5 1.0
Pre
ssu
re S
en
siti
vity
, wrt
k
Normalized Distance
Analytical (Stremaline)
Adjoint Method
0.0
5.0
10.0
15.0
20.0
0.0 0.5 1.0
Pre
ssu
re S
en
siti
vity
, wrt
k
Normalized Distance
Analytical (Stremaline)
Adjoint Method
33/50
Inj: Gas Rate
Prd: Rate
Producer BHP sensitivity to k
Injector BHP sensitivity to k
Sensitivity Results: 2D Areal
34/50
Inj
P1
P2P3
P4
Injector BHP sensitivity by k
P1 BHP sensitivity of by k
Permeability field(Wells by rate constraint)
Adjoint Proposed
Inversion of Permeability by LSQR
35/50
• Run simulation and get following parameter
• Solve LSQR Matrix :
• Advantages:• Find pressure/WCT sensitivity during SL simulation• Localized (high resolution) changes in permeability
min 𝛿𝐝𝑤𝑐𝑡 − 𝐒𝑤𝑐𝑡𝛿𝐤 + 𝛿𝐝𝑏ℎ𝑝 − 𝐒𝑏ℎ𝑝𝛿𝐤 + 𝛽1 𝐈𝛿𝐤 + 𝛽2 𝐋𝛿𝐤
𝐒𝑤𝑐𝑡𝐒𝑏ℎ𝑝𝛽1𝐈𝛽2𝐋
∆𝐤 =
𝛿𝐝𝑤𝑐𝑡𝛿𝐝𝑏ℎ𝑝00
Water-Cut Pressure - Smoothness- Consistency with
static model
Scaled by stdev
History Matching of Brugge Field
• Use simulation result of Real.77 as observed data• Use Real.1 as initial model• Assume 3 years of data is available
Reference model Initial model
36/50
Available Observation Data
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
144.0 644.0 1144.0
Pro
du
ctio
n W
ate
r C
ut
[-]
Time [Days]
BR-P-11
BR-P-12
BR-P-15
BR-P-18
BR-P-11
BR-P-12
BR-P-15
BR-P-18
• Only 4 producers have water breakthrough• Pressure data is available for 30 wells
Water cut:
InitialObserved
37/50
Reference kx Initial kx
Change of kx, WCT Change of kx, WCT&BHP
High perm at middle layer
Change of Permeability
38/50
Reduction of Data Mismatch
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 20 40 60 80 100
No
rmal
ize
d A
bso
lute
Err
or.
Pre
ssu
re
Number of Iteration
0.0
0.3
0.6
0.9
1.2
1.5
0 20 40 60 80 100
No
rmal
ize
d A
bso
lute
Err
or,
Wat
er
Cu
t
Number of Iteration
Pressure RMSE error WCT RMSE error
Individual well
Mean
39/50
• Have developed a new SL-based method to integrate pressure data into prior geologic models
• Same advantages as prior streamline work:• Analytic calculation of streamline sensitivities• Requires only a single flow simulation per iteration
• Can be applied to field pressure/rate data prior to water breakthrough
• Can be integrate pressure with water-cut or GOR simultaneously, for black-oil and compositional simulation
Conclusion
40/50
Presented at student paper contest 2014
Application to Brugge Benchmark:
- Streamline-Simulation
- History Matching
- NPV Optimization
Overview
42/50
• Problem: Determining optimal injection/production rates to
maximize NPV• Solution:
Developed a new streamline and NPV-based rate allocation method
• Advantages: Visualize efficiency of injector and producer Extensible to any secondary recovery process with
commercial simulator
- Improve oil production rate
- Works only after breakthrough
SL-based Flow Rate Allocation
Optimization: Previous Study
43/50
• Use of Well Allocation Factors (WAFs): [Thiele et. al, 2003]
Well Allocation Factor map [SPE84080]
[SPE113628]
- WAFs: offset oil production of well-pair
• Equalize arrival time of injection fluid: [Al-Hutali et. al, 2009]
Norm Wt. - 0
Aft
er
2 y
ears
Aft
er
5 y
ears
Aft
er
10 y
ears
Base
Base Improved
Norm Wt. - 0
Aft
er
2 y
ears
Aft
er
5 y
ears
Aft
er
10 y
ears
Base
- Control well rate to have equivalent
‘breakthrough’ time
- Increase well rate of high WAFs
Decrease
Increase
Decrease
Decrease
Decrease
Increase
- Improves sweep efficiency- Works only before breakthrough
• Fast
• Not robust
• Does not optimize NPV
Proposed Optimization Method:
Overall Workflow
44/50
2. Trace Streamlines and Find connection map
3. Calculate NPV diagnostic plot
4. Reallocate well rate
via efficiency
1. Run simulation model
I1 I2 I3
I6
I5
I7 I8
NPV-based Efficiency of Streamline
P1 P2
P3 P4 P5
P6 P7
Hydrocarbon value, along SL
NPV along SL, integrate over reservoir life time
𝑣𝑠𝑙
= 𝑞𝑠𝑙
𝑛𝑜𝑑𝑒
𝑆𝑜𝑏𝑜𝑅𝑜 ∆𝜏
𝑟𝑠𝑙 = 𝑞𝑠𝑙
𝑛𝑜𝑑𝑒
𝑆𝑜𝑏𝑜𝑅𝑜 + 𝑆𝑤𝑏𝑤𝑅𝑤 ∆𝜏 ∙ 1 + 𝑑 −∆𝜏/365∉
𝑝𝑟𝑑
𝑛𝑜𝑑𝑒
∆𝜏 > 𝑡𝑟𝑠𝑚
• Hydrocarbon value and NPV along streamline
Pore volume × Saturation × FVF × Price
Discount rate Reservoir life
I4
45/50
NPV-based Flow Diagnostics
I1 I2 I3
I6
I5
I7 I8
P1 P2
P3 P4 P5
P6 P7
𝑒𝑝𝑎𝑖𝑟 =σ𝑠𝑙 𝑟𝑠𝑙σ𝑠𝑙 𝑣𝑠𝑙 Total value
NPV
5-connection from Inj-4
Total value (Normalized)
NP
V (N
orm
aliz
ed)
𝑰𝟒𝐆𝐨𝐨𝐝
𝑷𝟒
𝑰𝟒𝐏𝐨𝐨𝐫
𝑷𝟕
NPV-based diagnostic plot
I4
46/50
NP
V (N
orm
aliz
ed)
Streamline-based Rate Allocation:
A New Approach
47/50
𝑞𝑛𝑒𝑤 = 𝑞𝑜𝑙𝑑𝑒𝑝𝑎𝑖𝑟ҧ𝑒𝑓𝑖𝑒𝑙𝑑
ത𝐞𝐟𝐢𝐞𝐥𝐝
decrease rate
Increase rate
Before update After update
Total value (Normalized)
NP
V (N
orm
aliz
ed)
Total value (Normalized)
Streamline-based Rate Allocation:
A New Approach
48/50
𝑞𝑛𝑒𝑤 = 𝑞𝑜𝑙𝑑𝑒𝑝𝑎𝑖𝑟ҧ𝑒𝑓𝑖𝑒𝑙𝑑
ത𝐞𝐟𝐢𝐞𝐥𝐝
decrease rate
Increase rate
Before update After update
• Advantages:• Dynamically visualize efficiency of the injector and producer
• Able to propose ‘better’ well rate during SL-simulation
Oil Saturation and Well Location
• Constraints:- Field water injection qt <= 20,000 bbl/d- Well flow rate qti <= 6000 bbl/d- Producer BHP > 100 psi, Injector BHP < 6000 psi
• Simulation Model:- Synthetic water flooding - 20 producers, 10 injectors- 20 years of simulation- Relative oil, water price = 1, -0.2 $/bbl
Brugge Benchmark Application
• Compare developed model with 3 approaches: • Uniform injection (Uniform), Well allocation factors
(WAFs), Equalize Arrival Time (EqArrive), Developed model (SLNPV)
49/50
0.00
0.04
0.08
0.12
0.16
0.20
0 1200 2400 3600 4800 6000 7200
Re
cove
ry F
acto
r [-
]
Time [Days]
SLNPVEqArriveWAFsUniform
0.E+00
5.E+06
1.E+07
2.E+07
2.E+07
3.E+07
3.E+07
4.E+07
0 1200 2400 3600 4800 6000 7200
Net
Pre
sen
t V
alu
e [
$]
Time [Days]
NPVEqArriveWAFsUniform
Recovery Factor Net Present Value
Recovery Factor and NPV
Injection Rate Production Rate
Updated Well Rate by SLNPV
0
1000
2000
3000
4000
5000
6000
7000
0 1200 2400 3600 4800 6000 7200
Pro
du
ctio
n R
ate
[b
bl/
day
]
Time [Days]
BR-P-1 BR-P-2BR-P-3 BR-P-4BR-P-5 BR-P-6BR-P-7 BR-P-8BR-P-9 BR-P-10BR-P-11 BR-P-12BR-P-13 BR-P-14BR-P-15 BR-P-16BR-P-17 BR-P-18BR-P-19 BR-P-20
0
1000
2000
3000
4000
5000
6000
7000
0 1200 2400 3600 4800 6000 7200
Inje
ctio
n R
ate
[b
bl/
day
]
Time [Days]
BR-I-1 BR-I-2BR-I-3 BR-I-4BR-I-5 BR-I-6BR-I-7 BR-I-8BR-I-9 BR-I-10
50/50
Streamlines by Sw
SLN
PV
Un
ifo
rm In
ject
ion
Streamlines by Injector
Example of SLs: After 10 Years
Not sweep aquifer region
Sweep aquifer region
Increased Inj-Prd
connection
51/50
MCERI
• Have developed a new SL-based rate allocation method to improve recovery considering NPV
• Proposed a new diagnostic plot to visualize the relative value and efficiency of a well in the asset
• Results in greater NPV compared to prior streamline-based rate allocation methods
• Can be applied to IOR/EOR simulation study with any commercial simulator, with low computational cost
Conclusions
54
Recommended