The use of the streamline-based method for reservoir management is receiving increased interest in recent years because of its computational advantages and intuitive appeal for reservoir simulation, history matching and rate allocation optimization. Streamline-based method uses snapshots of flow path of convective flow. Previous studies proved its applicability for convection dominated process such as waterflooding and tracer transport. However, for a case with gas injection with strong capillarity and gravity effects, the streamline-based method tends to lose its advantages for reservoir simulation and may result in loss of accuracy and applicability for history-matching and optimization problems. In this study, we first present the development of a 3D 3-phase black oil and compositional streamline simulator. Then, we introduce a novel approach to incorporate capillary and gravity effects via orthogonal projection method. The novel aspect of our approach is the ability to incorporate transverse effects into streamline simulation without adversely affecting its computational efficiency. We demonstrate our proposed method for various cases, including CO2 injection scenario. The streamline model is shown to be particularly effective to examine and visualize the interactions between heterogeneity which resulting impact on the vertical and areal sweep efficiencies. Next, we apply the streamline simulator to history matching and rate optimization problems. In the conventional approach of streamline-based history matching, the objective is to match flow rate history, assuming that reservoir energy was matched already, such as pressure distribution. The proposed approach incorporates pressure information as well as production flow rates, aiming that reservoir energy are also reproduced during production rate matching. Finally, we develop an NPV-based optimization method using streamline-based rate reallocation algorithm. The NPV is calculated along streamline and used to generate diagnostic plots of the effectiveness of wells. The rate is updated to maximize the field NPV. The proposed approach avoids the use of complex optimization tools. Instead, we emphasize the visual and the intuitive appeal of streamline methods and utilize flow diagnostic plots for optimal rate allocation. We concluded that our proposed approach of streamline-based simulation, inversion and optimization algorithm improves computational efficiency and accuracy of the solution, which leads to a highly effective reservoir management tool that satisfies industry demands.

1. An Effective Reservoir Management by
Streamline-based Simulation, History
Matching and Optimization
Shusei Tanaka
May, 2014

2. • 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

3. 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

4. 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

5. Development of a General Purpose Streamline-
based Simulator

6. 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

7. ;0


w
w
u
t
S

Why Split the Equation?
• Water velocity does not follow total velocity with capillary
(and gravity)
7/50
tuwu
Streamline
cowowtww pkFuFu  

8. • Split equation by physical mechanisms
Convective
Transport
Capillary
Diffusion
  0


cowow
w
pkF
t
S

0


w
w
u
t
S
   0


wt
w
Fu
t
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
WaterSaturation
Normalized Distance
Correct Solution
Convection Flow
Too much diffusion
with large time step
(SPE 163640)
Operator Splitting
Capillary after
convection
8/50

9.   0
~



wtw
w
Fuu
t
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
u
t
S
   0


wt
w
Fu
t
S

Splitting with anti-
diffusive flux
Convection Eq.
Corrected Operator Splitting
Anti-diffusive concave envelope
9/50

10. 0


w
w
u
t
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

11. twf u
wu
0


w
w
u
t
S
   0


tw
w
uf
t
S

0


w
w
u
t
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

12. 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

13. • Pressure equation(IMPES)
• Transport equation (along SL)
Orthogonal Projection:
Application to Multicomponent System
0


















   owgj
j
owgj
j
owgj
jj
owgj
jjr Qupuc
t
p
Scc
i
sl
ii fm
t








  cfgDpFy
u
k
yFfSym sl
ii
owgj owgj jogwmm
m
jmjmcmjjij
t
jijj
sl
ij
ogwj
jiji
1
,2
,, 
  



    

 Δ
• Transport equation (on Grid, corrector)
     



ogwj jmogwm
m
jmcjmmjjijtt
i
DgpFykuuI
t
m
,
ˆˆ 
Pc,Gravity along streamline
Transverse Pc,Gravity on grid
  0








owgj
ijijjjijjjij qyuySy
t

• Governing equation
13/50

14. 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

15. 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

16. Water Capillary Flux:
Parallel and Transverse to Total Velocity
Total capillary flux
Capillary 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 p
u
uu
I
u
k
u 





 2


Along streamline On grids
16/50
cow
t
t
t
ow
p
u
u
k 2



17. Water Saturation Distribution
Commercial, FD Operator Splitting
(no correction)
Orthogonal Projection
• OP can take large time step without anti-diffusive correction
17/50

18. 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

19. CO2 Mole Fraction Distribution:
Along Streamline
Including Pc & GravityConvection only
19/50

20. 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
ProductionMoleFraction(CO2)
Time [Days]
Streamline
Commercial Simulator
Number of time step:
Commercial FD = 56
Streamline = 21
20/50

21. CO2 Mole Fraction Distribution:
Final Distribution
Orthogonal Projection
(After corrector term)
Commercial FD
(E300)
21/50

22. 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
NumberofTimeStep
Tested simulation cases
in the paper
10×
2×
4×
3×
22/50

23. 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
NumberofTimeStep
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
ProductionMoleFraction(CO2)
Time [Days]
Streamline
Commercial Simulator
0.0
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
ProductionMoleFraction(CO2)
Time [Days]
Streamline
Commercial Simulator

24. 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

25. Application to Brugge Benchmark:
- Streamline-Simulation
- History Matching
- NPV Optimization

26. 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
Porosity
Rock table ID

27. 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
ProductionWaterCut
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

28. Comparisons of Oil Saturation
Distribution
28/50
Initial oil saturation
After 20 years
Streamline Commercial (ECL)

29. Presented at student paper contest
2013
Application to Brugge Benchmark:
- Streamline-Simulation
- History Matching
- NPV Optimization

30. 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
WaterCut
Time [Days]
Streamline-based Inverse Modeling
30/50
min 𝛿𝐝 𝑤𝑐𝑡 − 𝐒 𝑤𝑐𝑡 𝛿𝐤
𝛿𝐝
1. Run reservoir simulation
by given model
2. Trace Streamlines and
calculate parameter sensitivity
3. Update parameters to satisfy:
Observation
Prediction

31. Motivation and Objective
31/50
Streamlines
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
WaterCut
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
CO2MoleFraction
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
BottomHolePressure
Time [Days]
0
200
400
600
800
1000
1200
1400
1600
1800
2000
0 50 100 150 200
BottomHolePressure
Time [Days]
BHP
Observation
Initial

32. 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
WaterCut
Time [Days]
𝛿𝒕 𝑤𝑐𝑡
Production WCT
Parameter Sensitivity Along Streamline
32/50
• TOF( ): Travel time of neutral tracer along streamlines
 
 , ,
, ,
x y z
Inlet
ds
x y z
u

  
𝜕𝑡
𝜕𝑘𝑖
= −
𝜕𝑆
𝜕𝜏
𝜕𝜏
𝜕𝑘𝑖
∙
𝜕𝑆
𝜕𝑡
−1
=
1
𝑓′(𝑆)
∆𝜏𝑖
𝑘𝑖
• Water-cut travel time sensitivity:
injector
Producer
[He et. al,2003]
𝜕𝑝 𝑏ℎ𝑝
𝜕𝑘𝑖
=
𝜕∆𝑝𝑖
𝜕𝑘𝑖
≈
∆𝑝𝑖
𝑘𝑖
• Bottom hole pressure sensitivity: [new]
0
200
400
600
800
1000
1200
1400
1600
1800
2000
0 50 100 150 200
BottomHolePressure
Time [Days]
Production BHP
𝛿𝒑 𝑏ℎ𝑝
𝜕𝑝 𝑏ℎ𝑝
𝜕𝑘𝑖
≈
𝜏𝑖
𝜏
𝜕∆𝑝𝑖
𝜕𝑘𝑖
≈
𝜏𝑖
𝜏
∆𝑝𝑖
𝑘𝑖
Rate-Rate constraint
Rate-BHP constraint

33. Sensitivity Results: 1D CPG
(3phase Gas Injection)
-20.0
-15.0
-10.0
-5.0
0.0
0.0 0.5 1.0
PressureSensitivity,wrtk
Normalized Distance
Analytical (Stremaline)
Adjoint Method
0.0
5.0
10.0
15.0
20.0
0.0 0.5 1.0
PressureSensitivity,wrtk
Normalized Distance
Analytical (Stremaline)
Adjoint Method
33/50
Inj: Gas Rate
Prd: Rate
Producer BHP
sensitivity to k
Injector BHP
sensitivity to k

34. 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

35. 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 𝛿𝐝 𝑤𝑐𝑡 − 𝐒 𝑤𝑐𝑡 𝛿𝐤 + 𝛿𝐝 𝑏ℎ𝑝 − 𝐒 𝑏ℎ𝑝 𝛿𝐤 + 𝛽1 𝐈𝛿𝐤 + 𝛽2 𝐋𝛿𝐤
෠𝐒 𝑤𝑐𝑡
෠𝐒 𝑏ℎ𝑝
𝛽1 𝐈
𝛽2 𝐋
∆𝐤 =
𝛿𝐝 𝑤𝑐𝑡
𝛿𝐝 𝑏ℎ𝑝
0
0
Water-Cut Pressure - Smoothness
- Consistency with
static model
Scaled by stdev

36. 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

37. 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
ProductionWaterCut[-]
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:
Initial
Observed
37/50

38. Reference kx Initial kx
Change of kx, WCT Change of kx, WCT&BHP
High perm
at middle layer
Change of Permeability
38/50

39. Reduction of Data Mismatch
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 20 40 60 80 100
NormalizedAbsoluteError.Pressure
Number of Iteration
0.0
0.3
0.6
0.9
1.2
1.5
0 20 40 60 80 100
NormalizedAbsoluteError,WaterCut
Number of Iteration
Pressure RMSE error WCT RMSE error
Individual well
Mean
39/50

40. • 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

41. Presented at student paper contest
2014
Application to Brugge Benchmark:
- Streamline-Simulation
- History Matching
- NPV Optimization

42. 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

43. - 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
After2yearsAfter5yearsears
Base
Base Improved
Norm Wt. - 0
After2yearsAfter5yearsyears
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

44. 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

45. 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
𝑣 𝑠𝑙
= 𝑞 𝑠𝑙 ෍
𝑛𝑜𝑑𝑒
𝑆 𝑜 𝑏 𝑜 𝑅 𝑜 ∆𝜏
𝑟𝑠𝑙 = 𝑞 𝑠𝑙 ෍
𝑛𝑜𝑑𝑒
𝑆 𝑜 𝑏 𝑜 𝑅 𝑜 + 𝑆 𝑤 𝑏 𝑤 𝑅 𝑤 ∆𝜏 ∙ 1 + 𝑑 −∆𝜏/365
∉ ෍
𝑝𝑟𝑑
𝑛𝑜𝑑𝑒
∆𝜏 > 𝑡 𝑟𝑠𝑚
• Hydrocarbon value and NPV along streamline
Pore volume × Saturation × FVF × Price
Discount rate Reservoir life
I4
45/50

46. NPV-based Flow Diagnostics
I1 I2 I3
I6
I5
I7 I8
P1 P2
P3 P4 P5
P6 P7
𝑒 𝑝𝑎𝑖𝑟 =
σ 𝑠𝑙 𝑟 𝑠𝑙
σ 𝑠𝑙 𝑣 𝑠𝑙 Total value
NPV
5-connection from Inj-4
Total value (Normalized)
NPV(Normalized)
𝑰 𝟒
𝐆𝐨𝐨𝐝
𝑷 𝟒
𝑰 𝟒
𝐏𝐨𝐨𝐫
𝑷 𝟕
NPV-based diagnostic plot
I4
46/50

47. NPV(Normalized)
Streamline-based Rate Allocation:
A New Approach
47/50
𝑞 𝑛𝑒𝑤
= 𝑞 𝑜𝑙𝑑 𝑒 𝑝𝑎𝑖𝑟
ҧ𝑒𝑓𝑖𝑒𝑙𝑑
ത𝐞 𝐟𝐢𝐞𝐥𝐝
decrease rate
Increase rate
Before update After update
Total value (Normalized)

48. NPV(Normalized)
Total value (Normalized)
Streamline-based Rate Allocation:
A New Approach
48/50
𝑞 𝑛𝑒𝑤
= 𝑞 𝑜𝑙𝑑 𝑒 𝑝𝑎𝑖𝑟
ҧ𝑒𝑓𝑖𝑒𝑙𝑑
ത𝐞 𝐟𝐢𝐞𝐥𝐝
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

49. 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

50. 0.00
0.04
0.08
0.12
0.16
0.20
0 1200 2400 3600 4800 6000 7200
RecoveryFactor[-]
Time [Days]
SLNPV
EqArrive
WAFs
Uniform
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
NetPresentValue[$]
Time [Days]
NPV
EqArrive
WAFs
Uniform
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
ProductionRate[bbl/day]
Time [Days]
BR-P-1 BR-P-2
BR-P-3 BR-P-4
BR-P-5 BR-P-6
BR-P-7 BR-P-8
BR-P-9 BR-P-10
BR-P-11 BR-P-12
BR-P-13 BR-P-14
BR-P-15 BR-P-16
BR-P-17 BR-P-18
BR-P-19 BR-P-20
0
1000
2000
3000
4000
5000
6000
7000
0 1200 2400 3600 4800 6000 7200
InjectionRate[bbl/day]
Time [Days]
BR-I-1 BR-I-2
BR-I-3 BR-I-4
BR-I-5 BR-I-6
BR-I-7 BR-I-8
BR-I-9 BR-I-10
50/50

51. Streamlines by Sw
SLNPVUniformInjection
Streamlines by Injector
Example of SLs: After 10 Years
Not sweep aquifer region
Sweep aquifer region
Increased Inj-Prd
connection
51/50

52. 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